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Dirac materials and the response to elastic lattice deformation Liu, Tianyu 2019

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Dirac materials and the response toelastic lattice deformationbyTianyu LiuM.Sc., The University of British Columbia, 2015B.Sc., University of Science and Technology of China, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2019c Tianyu Liu 2019The following individuals certify that they have read, and recommend to theFaculty of Graduate and Postdoctoral Studies for acceptance, a dissertationentitled:Dirac materials and the response to elastic lattice deformationsubmitted by Tianyu Liu in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin PhysicsExamining Committee:Marcel FranzResearch SupervisorFei ZhouSupervisory Committee MemberJoshua FolkSupervisory Committee MemberJoanna KarczmarekUniversity ExaminerRoman KremsUniversity ExaminerAdditional Supervisory Committee Members:Ariel ZhitnitskySupervisory Committee MemberiiAbstractDirac materials have formed a thriving and prosperous direction in moderncondensed matter physics. Their bulk bands can linearly attach at discretepoints or along curves, leading to arc or drumhead surface states. The candi-date Dirac materials are exemplified by Dirac/Weyl semimetals, Dirac/Weylsuperconductors, and Dirac/Weyl magnets. Owing to the relativistic bandstructure, these materials have unique responses to the applied elastic crys-talline lattice deformation, which can induce pseudo-magnetic and pseudo-electric fields near the band crossings and produce transport distinguishedfrom that caused by ordinary magnetic and electric fields. In this disserta-tion, I will demonstrate the exotic transport due to the strain-induced gaugefield in Weyl semimetals, Weyl superconductors, and Weyl ferromagnets.I will first elucidate that a simple bend deformation can induce a pseudo-magnetic field that can give rise to the Shubnikov-de Haas oscillation in Weylsemimetals. Then I will elaborate that strain can Landau quantize chargeneutral Bogoliubov quasiparticles as well and result in thermal conductiv-ity quantum oscillation in Weyl superconductors. Lastly, I will consider thestrain-induced gauge field beyond the fermionic paradigm and explain vari-ous quantum anomalies of magnons in Weyl ferromagnets.iiiLay SummaryThe first decades of 20th century have witnessed the birth of quantum me-chanics, while the first decades of 21st century see its great impacts to ourlife through a variety of quantum materials. A deep understanding on thephysics of quantum materials will undoubtedly help us make better use ofthem to benefit our life. Motivated by this goal, this dissertation systemat-ically investigates a special type of quantum materials – Dirac materials –with particular attention paid to their response to elastic deformation. Byanalytical derivation and numerical simulation, I elucidate that the responseof Dirac materials to static and dynamic elastic deformation highly mimicsthat to electromagnetic fields. The currents driven by electromagnetic fieldscan then be alternatively created by properly deforming Dirac materials.This unique feature makes Dirac materials stand out from other quantummaterials and renders Dirac materials useful for the future application inquantum technologies.ivPrefaceThis dissertation summarizes several projects I did during my doctorate,focusing on the transport properties of Dirac materials in the presence ofstrain-induced gauge field. I am responsible for most of the analytical deriva-tions and numerical simulations under supervision of Prof. Marcel Franz andin consultation with Prof. Satoshi Fujimoto – my supervisor during my visitto Osaka University. Dr. Dmitry Pikulin and Dr. Zheng Shi also share someworkload with me. The contribution of each researcher is detailed as below.• Chapter 2 and Appendix A are published in the paper Physical ReviewB 95, 041201(R) (2017), coauthored by Tianyu Liu, Dmitry Pikulin,and Marcel Franz. The initial idea of the project is brought up by Prof.Franz. I am in charge of all numerical simulation presented in Figs. 2.3-2.6 and Fig. A.1 and a portion of analytical calculation. Prof. Franzplotted Figs. 2.1-2.3, 2.5. Dr. Pikulin plotted Fig. 2.6(b). The otherfigures are plotted by me and further edited by Dr. Pikulin. Sometext in Chapter 2 and Appendix A is directly adapted from the paper,where Prof. Franz and Dr. Pikulin share the credit for preparing themanuscript.• Chapter 3 and Appendix B are published in the paper Physical Re-view B 96, 224518 (2017), coauthored by Tianyu Liu, Marcel Franz,and Satoshi Fujimoto. The initial idea is formulated by myself in con-sultation with Prof. Franz and Prof. Fujimoto. I did all the analyticaland numerical calculations presented in Chapter 3 and Appendix B.Some text in Chapter 3 and Appendix B is directly adapted from thepaper, where Prof. Franz, Prof. Fujimoto, and I contributed equallyto the preparation of the manuscript.• Chapter 4 and Appendices C – E are published Physical Review B 99,214413 (2019), coauthored by Tianyu Liu and Zheng Shi. The initialidea of the project is formulated by myself in consultation with Prof.Franz. I did all of the analytical and numerical calculations presentedin Chapter 4 and Appendices C – E except for Figs. 4.8 and D.2, whichvPrefaceare produced by Dr. Shi. Some text in Chapter 4 and Appendices C– E is directly adapted from the manuscript, which is prepared by Dr.Shi and myself. Prof. Franz has read the paper and made some minorchanges.The reuse of the three published papers is approved by the AmericanPhysical Society (APS) under Licenses RNP/19/JUN/015910-015912.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xxiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Dirac materials . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Dirac and Weyl semimetals . . . . . . . . . . . . . . . 31.1.3 Dirac and Weyl superconductors . . . . . . . . . . . . 61.1.4 Weyl magnets . . . . . . . . . . . . . . . . . . . . . . 71.1.5 Other Dirac materials . . . . . . . . . . . . . . . . . . 81.2 Dirac-Landau levels . . . . . . . . . . . . . . . . . . . . . . . 81.3 Strain-induced Landau levels in Graphene . . . . . . . . . . . 111.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Zero-field quantum oscillations in Weyl semimetals . . . . 162.1 Model of Weyl semimetals . . . . . . . . . . . . . . . . . . . 172.2 Strain-induced pseudo-magnetic field . . . . . . . . . . . . . 182.3 Band structure of Weyl semimetals . . . . . . . . . . . . . . 21viiTable of Contents2.4 Longitudinal electric conductivity . . . . . . . . . . . . . . . 242.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Dirac-Landau levels in Weyl superconductors . . . . . . . . 313.1 Model of Weyl superconductors . . . . . . . . . . . . . . . . . 323.2 Strain-induced pseudo-magnetic field . . . . . . . . . . . . . 363.3 Weyl superconductors with chemical potential . . . . . . . . 413.4 Longitudinal thermal conductivity . . . . . . . . . . . . . . . 463.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Magnon quantum anomalies in Weyl ferromagnets . . . . . 534.1 Model of Weyl ferromagnets . . . . . . . . . . . . . . . . . . 544.2 Weyl ferromagnets under electromagnetic fields and strain . . 604.2.1 Landau quantization in the presence of electric field . 604.2.2 Landau quantization in the presence of pseudo-electricfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2.3 Magnon motion in the presence of magnetic field . . . 654.2.4 Magnon motion in the presence of pseudo-magneticfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3 Magnon quantum anomalies and the anomalous transport . . 694.3.1 Magnon chiral anomaly due to electric and magneticfields . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3.2 Magnon chiral anomaly due to pseudo-electric and pseudo-magnetic fields . . . . . . . . . . . . . . . . . . . . . . 744.3.3 Magnon heat anomaly due to electric and pseudo-magneticfields . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3.4 Magnon heat anomaly due to pseudo-electric and mag-netic fields . . . . . . . . . . . . . . . . . . . . . . . . 804.4 Field dependence of anomalous spin and heat currents . . . . 824.5 Experimental measurement of magnon quantum anomalies . 844.5.1 Experimental signature of magnon chiral anomaly dueto electric and magnetic fields . . . . . . . . . . . . . 854.5.2 Experimental signature of magnon chiral anomaly dueto pseudo-electric and pseudo-magnetic fields . . . . . 864.5.3 Experimental signature of magnon heat anomaly dueto electric and pseudo-magnetic fields . . . . . . . . . 874.5.4 Experimental signature of magnon heat anomaly dueto pseudo-electric and magnetic fields . . . . . . . . . 884.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89viiiTable of Contents5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A Electronic structure of Dirac semimetal Cd3As2 . . . . . . 105B Weyl superconductor with a vortex lattice . . . . . . . . . . 107C Weyl ferromagnets under electric field and strain . . . . . 112D Magnon bands of multilayer Weyl ferromagnets . . . . . . 114E Circular bend induced pseudo-electric field . . . . . . . . . 120F The tetrahedron method . . . . . . . . . . . . . . . . . . . . . 123ixList of Tables4.1 Summary of field (gradient) dependence of anomalous spinand heat currents in magnon quantum anomalies. . . . . . . . 84A.1 Parameters of Dirac semimetal Cd3As2. All quantities aremeasured in terms of electron volt (eV). . . . . . . . . . . . . 105xList of Figures1.1 Graphene lattice and chemical bonds. (a) Graphene is a singlelayer of carbon atoms arranged on a honeycomb lattice. (b)Each carbon atom is connected with 3 adjacent carbon atomsby the  bonds due to the head-to-head overlap of the sp2hybridized orbitals. (c) The electrons on each lattice site canhop to neighboring sites along the ⇡ bonds due to the side-by-side overlap of the unhybridized 2pz orbitals. . . . . . . . . 41.2 Schematic band crossings of Dirac and Weyl semimetals. (a)In Dirac semimetals, the doubly degenerate conduction andvalence bands touch linearly forming cone-like band structure.The vertex of a cone represents a four-fold Dirac point. (b) InWeyl semimetals, the non-degenerate conduction and valencebands also touch linearly forming cone-like band structure.The vertex of a cone represents a two-fold Weyl point. . . . . 51.3 Schematic band structure plot for Weyl semimetals. (a) Apair of Weyl points in momentum space, one being the sourceof the Berry flux while the other acting as the drain. Theyare connected by Fermi arc states on the open boundaries.(b) Weyl cones and surface states in momentum space. Thesurface states cutting through two Weyl cones can be under-stood as a combination of chiral edge states of 2D subsystemsbetween two Weyl points. When tuning Fermi energy to theWeyl points, the Fermi surface becomes an arc connecting thetwo Weyl points. . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Schematic depiction of the low-energy electron excitation spec-trum in Weyl semimetals. a) A pair of Weyl cones appear onkz axis. b) Contours of constant energy for ky = 0. . . . . . 19xiList of Figures2.2 Proposed setup for strain-induced quantum oscillation obser-vation in Weyl semimetals. a) Bent film is analogous, in termsof its low-energy properties, to an unstrained film subjectedto magnetic field B. b) Detail of the atomic displacements inthe bent film. Displacements have been exaggerated for clarity. 212.3 Band structure and DOS for lattice Hamiltonian Eq. 2.2. Inall panels, films of thickness 500 lattice points are studiedwith parameters t0 = 2.522eV, t1 = 1.042eV, t2 = 0.75eV,and ⇤ = 0.148eV. (a) Band structure and DOS for zero fieldand zero strain. The inset shows the first Brillouin zone. Thedashed parabolic curve is the expected DOS (Eq. 2.16) forideal Weyl dispersion (Eq. 2.15). (b) Band structure and nor-malized DOS for B = 1.5T. The solid black curve comprisedof spikes at Landau levels is the expected DOS (Eq. 2.18) cal-culated from Dirac-Landau levels (Eq. 2.17). Red crosses in-dicate the peak positions expected on the basis of the Lifshitz-Onsager quantization condition. (c) Band structure and DOSfor b = 1.5T. The solid black curve comprised of spikes atpseudo-Landau levels is the expected DOS (Eq. 2.20) calcu-lated from pseudo-Landau levels (Eq. 2.19). . . . . . . . . . . 232.4 Normalized density of states for both fields present, B = 1Tand b = 0.0184T. Each of the DOS peaks due to ordinarymagnetic field splits due to torsion thus proving the equiva-lence of the external and gauge fields. Inset gives closer viewof the first two peaks. . . . . . . . . . . . . . . . . . . . . . . 252.5 Strain-induced quantum oscillations. Top panel shows oscil-lations in DOS at µ = 10meV as a function of inverse strainstrength expressed as 1/b. For comparison ordinary magneticoscillations are displayed, as well as the result of the bulkcontinuum theory Eq. 2.18. Crosses indicate peak positionsexpected based on the Lifshitz-Onsager theory. Bottom panelshows SdH oscillations in conductivity yy assuming chemi-cal potential µ = 10meV. To simulate the effect of disorder alldata are broadened by convolving in energy with a Lorentzianwith width  = 0.25meV. The same geometry and parametersare used as in Fig. 2.3. . . . . . . . . . . . . . . . . . . . . . . 28xiiList of Figures2.6 Quantum oscillations above Lifshitz transition. (a) QOs abovethe Lifshitz transition due to ordinary magnetic field and dueto strain-induced pseudo-magnetic field. Period difference byapproximately a factor of 2 is seen. The low-energy analytictheory does not apply anymore, as expected. (b) Correspond-ing hypothesized quasi-classical trajectories of electrons in theBrillouin zone. Green for By field and red for by field. . . . . 293.1 Schematic plot for (a) undeformed and (b) bent TI-SC multi-layer Weyl superconductor. The alternating TI and SC layersare omitted in the bulk but explicitly drawn at ends to illus-trate that there are integer number of unit cells comprised ofone TI layer and one SC layer. . . . . . . . . . . . . . . . . . 333.2 Band structure of a Weyl superconductor plotted (a) along kzaxis with kx = 0 and (b) along kx axis with kz = 0. Periodicboundary conditions are applied in x, z directions while thesystem is chosen to have l¯y = 500 layers in y direction. Theparameters are listed below Eq. 3.4. . . . . . . . . . . . . . . 353.3 Band structure of a Weyl superconductor with open bound-ary conditions and l¯y = 150 layers along the y-direction. Allpanels are plotted along kz-axis with kx = 0 and with pa-rameters as in Fig. 3.2. (a) Weyl superconductor phase for(m,) = (10.26, 1). A Fermi arc connecting two Weyl pointsappears due to the chiral Majorana edge states of the effec-tive px + ipy superconductors that emerge for fixed kz be-tween the Weyl nodes. (b) Topological superconductor phasefor (m,) = (19.82, 1). The increase of m will separate twoWeyl points and extend Fermi arc. When two Weyl pointsmeet at Brillouin zone boundary, they annihilate and openup a SC gap but leave behind the Fermi arc extended overthe whole BZ. (c) Trivial superconductor phase for (m,) =(9.98, 1). The decrease of m makes two Weyl points meet atBrillouin zone center and annihilation and leads to the disap-pearance of the Fermi arc. (d) Trivial superconductor phasewith (m,) = (10.26, 2.56). The increase of  is equivalentto decrease of m and Weyl points again annihilate at the BZcenter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37xiiiList of Figures3.4 Phase diagram of theWeyl superconductor described by Hamil-tonian Eq. 3.3 in terms of (m, ||) with labels (a)-(d) cor-respond to spectra shown in Fig. 3.3(a)-(d). The two blackcurves mark the phase boundaries given in Eq. 3.10 and Eq. 3.11.The dotted line indicates the asymptote for the two phaseboundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5 Energy spectra and DOS for our Weyl superconductor withopen boundaries and l¯y = 150 along the y direction and peri-odic along x and z. (a) The spectrum of undeformed system;the flat band at zero energy is the Fermi arc. (b) The spectrumof a bent Weyl superconductor as shown in Fig. 3.1(b) with" = 8% corresponding to a pseudo-magnetic field b = 10.45T.For both (a) and (b) the spectrum is plotted along X--Z asshown in the inset. For comparison, energy levels of Eq. 3.34are overlain as black dots. (c) DOS of the unstrained sam-ple (blue curve) compared to the ideal ⇠ E2 DOS expectedfor a massless Dirac fermion in continuum (black parabola).(d) DOS of the strained system (red curve) compared to DOScalculated for ideal Dirac-Landau levels with b = 10.45T. . . 423.6 Energy spectrum of the Weyl superconductor with the chem-ical potential of the TI layers tuned away from the surfaceDirac points to µ = 0.19. (a) Quasiparticle spectrum cal-culated from the lattice model Eq. 3.13. It is worth notingthat only the left moving chiral mode is due to the Landauquantization while the other is a surface mode. (b) Quasipar-ticle spectrum predicted by Eq. 3.63. To compare with thefirst panel, the chiral modes (orange lines) due to the surfacestates have been added manually. . . . . . . . . . . . . . . . . 463.7 Strain-induced quantum oscillation in a Weyl superconductor.The upper panel shows oscillations in DOS as a function ofinverse strain strength expressed as 1/b at zero-energy. Thelower panel shows oscillations in the longitudinal quasiparti-cle thermal conductivity xx. To simulate the effect of dis-order, all data are broadened by convolving in energy with aLorentzian with width ✏ = 1.67⇥ 103. . . . . . . . . . . . . 49xivList of Figures4.1 Schematic plot for the Weyl ferromagnet multilayer. (a) 2Dhoneycomb ferromagnet sheet. The Weyl ferromagnet multi-layer is constructed by stacking many sheets in the z direction.(b) Conventional crystal cells of the Weyl ferromagnet within-plane nearest (second nearest) neighbors connected by ↵i(i), i = 1, 2, 3. . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Magnon dispersion and spectral functions for the Weyl ferro-magnet multilayer. For all panels, we setDS = 1 and measureenergies in terms of DS such that J1S = 4.56, J2S = 1.14,JS = 7.22, K+S = 2.77 and KS = 1.12. (a) Magnonband structure for the nanowire with a pair of zigzag edgesand a pair of armchair edges in the cross section. The crosssection of the nanowire is illustrated in Fig. 4.3(b). Themagnon bands exhibit two Weyl points on the kz axis andare connected by a set of almost flat states analogous tothe arc states in Weyl semimetals. (b) Magnon bands forthe nanowire with periodic boundary conditions for the crosssection. The flat bands disappear, indicating their surfaceorigin. The red curves are the analytical dispersion ✏k =[K+ + 3J1 + 6J2]S ± [K + J(1  cos kza)  3p3D]S forthe Bloch Hamiltonian Hk at the honeycomb lattice Brillouinzone corner k? = (4⇡/3p3a, 0). (c) Surface spectral func-tion of the Bloch Hamiltonian which confirms that the almostflat states reside on surfaces. (d) Bulk spectral function whichindicates the positions of Weyl cones. . . . . . . . . . . . . . 594.3 Schematic plot for theWeyl ferromagnet nanowire. (a) Nanowireunder an inhomogeneous electric field and nanowire under atwist deformation. Landau quantization takes place in bothcases. (b) Cross section of the Weyl ferromagnet nanowirewith a pair of zigzag edges (x-direction) and a pair of arm-chair edges (y-direction). We use m = n = 30 unless other-wise specified so that all numerical simulations could be im-plemented with available computational resources. . . . . . . 60xvList of Figures4.4 Magnon dispersion of the Weyl ferromagnet nanowire underan inhomogeneous electric field. For all panels, we take gµBEa2ec2 =0.01240 where E represents the gradient of the externalelectric field and 0 = h/2e is the magnetic flux quantum. (a)Magnon bands are Landau-quantized by the external inhomo-geneous electric field due to the Aharonov-Casher effect. Thetwo resulting zeroth Landau levels at different Weyl pointshave opposite velocities ±|v⌘z | and are connected by a set ofalmost flat states. (b) Surface spectral function, which revealsthat these flat bands are localized at the surface of the Weylferromagnet nanowire. (c) Bulk spectral function highlightingthe Dirac-Landau levels at each Weyl cone. . . . . . . . . . . 624.5 Schematic plot for the exchange integrals J2. The most im-portant impact of the twist deformation is to modulate J2spatially. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.6 Magnon dispersion of a twisted Weyl ferromagnet nanowire.For all panels, we take gµB"a2ec2 = 0.01240 where " representsthe gradient of the strain-induced pseudo-electric field. (a)Magnon bands are Landau-quantized by the strain-inducedpseudo-electric field. The resulting zeroth Landau levels atthe two Weyl points are both right-moving and are connectedby a set of left-moving states. (b) Surface spectral function,which reveals that these left-moving states are localized at thesurface of the Weyl ferromagnet nanowire. (c) Bulk spectralfunction highlighting the Dirac-Landau levels at each Weylcone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65xviList of Figures4.7 Schematic plot of the magnon band structures and distribu-tions in various quantum anomalies of a Weyl ferromagnet,which is in contact with two magnon reservoirs in a uniformmagnetic field B0. (a)-(c) Magnon Dirac-Landau levels dueto an inhomogeneous electric field. (d)-(f) Magnon Dirac-Landau levels due to a strain-induced pseudo-electric field.(a, d) Magnon distributions in the absence of pumping. (b,e) Magnon chiral anomaly with chirality imbalance createdby ordinary magnetic field pumping in (b) and by pseudo-magnetic field pumping in (e). (c, f) Magnon heat anomalywith magnon concentration variation created by pseudo-magneticfield pumping in (c) and by ordinary magnetic field pumpingin (f). For all panels, only the distributions (green dots) on thezeroth Landau levels (red) are plotted. In principle, magnonscan occupy all bands above the population edges providedthat the relaxation time is sufficiently long. . . . . . . . . . . 704.8 Bulk-surface separation for the twistedWeyl ferromagnet nanowire.(a) Schematic plot of a Weyl ferromagnet nanowire with arectangular cross section. The spin current propagates alongthe z direction in the bulk but along the +z direction onthe surface, while the heat current propagates along the +zdirection in the bulk but along the z direction on the sur-face. (b) Spatially resolved spin current on the cross section ofthe cuboid Weyl ferromagnet nanowire. (c) Spatially resolvedheat current on the cross section of the cuboid Weyl ferromag-net nanowire. The directions of currents are color coded withblue (orange) representing z (+z). (d)-(f) Same as (a)-(c)but for a Weyl ferromagnet nanowire with an (almost) circu-lar cross section. The total spin current on the rectangular(circular) cross section is 0.002DS (0.0001DS) while the to-tal heat current on the rectangular (circular) cross section is0.0473D2S2/~ (0.0018D2S2/~). . . . . . . . . . . . . . . 77xviiList of FiguresA.1 Numerically calculated band structure and density of statesfor Dirac semimetal Cd3As2 with both spin sectors and particle-hole asymmetric term ✏k considered. Top row is for the pseudo-magnetic field b = 4.25T and the bottom row is for the or-dinary magnetic field B = 4.25T. From left to right – bandstructure of spin up sector, band structure of spin down sec-tor, and normalized total DOS. The appearance of Landaulevels is obviously showed in all panels. . . . . . . . . . . . . 106B.1 Schematic plot of square vortex lattice. The red and blue dotscorrespond to two vortex sublattices. The orange square is themagnetic unit cell with vortices placed on the diagonal. Thedimension of the magnetic unit cell is chosen to be L = 30ain the simulation. . . . . . . . . . . . . . . . . . . . . . . . . . 108B.2 Spectra of Weyl superconductor with vortex lattice. The sizeof magnetic unit cell is L ⇥ L = 30a ⇥ 30a. The spac-ings between two vortices in the magnetic unit cell are (a)d = (15a, 15a) (b) d = (10a, 10a) (c) d = (5a, 5a) (d) d =(15a, 15a) (e) d = (10a, 10a) (f) d = (5a, 5a) The orangecurves in panel (d)-(f) are analytical Dirac-Landau levels withn = 1 band matched to the numerics. . . . . . . . . . . . . . 111C.1 Magnon dispersion for a twisted Weyl ferromagnet nanowirein the presence of an inhomogeneous electric field. For allpanels, we take gµBEa2ec2 =gµB"a2ec2 = 0.01240. (a) Magnonband structure. Due to the chiral nature of the strain-inducedpseudo-electric field, the effective electric field at the left Weylcone vanishes while the effective field at the right Weyl coneis doubled. Therefore, the left Weyl cone is not Landau-quantized but the right Weyl cone exhibits Dirac-Landau lev-els. (b) Surface spectral function, which shows a set of left-moving surface states connecting the left Weyl cone and theright zeroth Landau level. (c) Bulk spectral function, whichclearly unveils the linear band touching at the left Weyl cone,and Dirac-Landau levels at the right Weyl cone. Compared toFig. 4.4(c) and Fig. 4.6(c), the Landau level spacing is doubleddue to the doubling of the effective electric field. . . . . . . . 113xviiiList of FiguresD.1 Magnon dispersion for the Weyl ferromagnet nanowire. Forall panels, the parameters are same as those of Fig. 4.2 inChapter 4 except that we reintroduce a nonzero J+S = 4.08.(a) Magnon band structure of a nanowire without externalfields. Due to the nonzero J+ the Weyl cones and arc statesare tilted. (b) Magnon band structure of a nanowire underan inhomogeneous external electric field whose gradient Esatisfies gµBEa2ec2 = 0.01240. The Dirac-Landau levels aretilted by J+ such that the velocity of the right (left) zerothLandau level is |v⌘z | + |v⌘0 | (|v⌘z |  |v⌘0 |). (c) Magnon bandstructure of a twisted nanowire. The gradient of the strain-induced pseudo-electric field " satisfies gµB"a2ec2 = 0.01240.The Dirac-Landau levels are tilted by J+ such that the ve-locity of the right (left) zeroth Landau level is |v⌘z | + |v⌘0 |(|v⌘z | |v⌘0 |). (d) Magnon band structure of a twisted nanowireunder an inhomogeneous external electric field, with gµBEa2ec2 =gµB"a2ec2 = 0.01240. Due to the chiral nature of the strain-induced pseudo-electric field, the effective electric field at theleft Weyl cone vanishes while the effective electric field at theright Weyl cone is doubled. Therefore, the left tilted Weylcone is not Landau-quantized but the right tilted Weyl coneexhibits tilted Landau levels. . . . . . . . . . . . . . . . . . . 115D.2 Reproduction of bulk-surface separation for the twisted Weylferromagnet nanowire (Fig. 4.8) with a nonzero J+S = 4.08.Though the Weyl cones are displaced and tilted, the bulk-surface separation of spin and heat currents is preserved forboth the rectangular cross section (b, c) and the circular crosssection (e, f). The total spin current on the rectangular (cir-cular) cross section is 0.0017DS (0.0016DS) while the to-tal heat current on the rectangular (circular) cross section is0.046D2S2/~ (0.0423D2S2/~). . . . . . . . . . . . . . . . . . 119E.1 Schematic plot for theWeyl ferromagnet nanowire. (a) Nanowireunder a circular bend deformation. (b) Lattice site positionswithout deformation (left) and with a circular bend (right). . 121xixList of FiguresF.1 Discretization of energy band in the tetrahedron method. TheBrillouin zone spanned by (k, k⌫) is first discretized into rect-angular grid. Then each rectangular plaquette is cut into apair of right triangular plaquettes colored grey and white. Then-th energy band ✏n(k, k⌫) is then discretized on the bothtypes of triangular plaquettes. On each triangular plaquetteSk, the discretized piece of energy band can be approximatedas having zero curvature and each piece has vertex energies✏n1 (Sk), ✏n2 (Sk), and ✏n3 (Sk). . . . . . . . . . . . . . . . . . . . 125xxList of AbbreviationsAFM Atomic force microscopeDSC Dirac superconductorDSM Dirac semimetalNI Normal insulatorQO Quantum oscillationSC SuperconductorSdH Shubnikov - de HaasSTM Scanning tunneling microsocpeTI Topological insulatorWSC Weyl superconductorWSM Weyl semimetalxxiAcknowledgementsDuring my program in UBC, many people offered me help on my academicand daily life. It may not be possible for me to thank them one by one here.But their help will be appreciated and cherished forever.My heartfelt gratefulness first goes to my supervisor Prof. Marcel Franz,who is an exceptional and considerate supervisor, supporting me both aca-demically and financially throughout my graduate education and researchcareer. Prof. Franz offers me maximal flexibility and always encourages meto pursue projects I am interested in and to conduct research independentlybut always provides timely help. He puts my personal development on highpriority and always recommends me useful workshops and conferences butalways allows me to attend those I find interesting myself. His deep insights,extensive knowledge, and outstanding professionalism greatly influence meas a physicist more than anyone else. I feel honorable and fortunate to workwith him during my early research career.Secondly, I would like to thank my supervisory committee – Prof. FeiZhou, Prof. Ariel Zhitnitsky, and Prof. Joshua Folk. They help me to un-derstand the physics in my dissertation from various points of view. Prof.Zhou helps me clarify the physics in magnon systems. Prof. Zhitnisky facil-itates my understanding in the strain-induced gauge field. Prof. Folk guidesme in experimental implementation of some of my proposals on observingpseudo-electromagnetic transport. Their challenging questions push me tothink harder and deeper. Without them, the dissertation would never be asconsolidate as it is now.A part of my research was done during my visit to Osaka University whereI held a funding awarded by “Junior Research Program (JREP)” under Topo-Q network offered by the “Topological Materials Science (TMS)” project. Iowe particular thanks to Prof. Yoshiteru Maeno (Kyoto University) whobrings about this opportunity and guides me through the application. Iam also very grateful to Prof. Norio Kawakami (Kyoto University) – theTMS project leader for the financial supports during my time in Japan.My particular gratefulness goes to Prof. Satoshi Fujimoto, my supervisorin Osaka University. His rigorous scholarship and extraordinary patience inxxiiAcknowledgementsconducting research impressed and benefited me a lot.I offer my enduring gratitude to my collaborators – Dr. Dmitry Pikulin(Microsoft Station Q) and Dr. Zheng Shi (Freie Universität Berlin) for theirprominent contribution to the work presented in Chapter. 2 and Chapter. 4,respectively. I am indebted to Dr. Ying Su (Hong Kong University of Sci-ence and Technology), Prof. Tobias Meng (Technische Universität Dresden),Prof. Atsushi Tsuruta (Osaka University), Prof. Takeshi Mizushima (Os-aka University), Taiki Matsushita (Osaka University), Prof. Masatoshi Sato(Kyoto University), Prof. Danshita Ippei (Kyoto University), Prof. YongBaek Kim (University of Toronto), Prof. Akira Furusaki (RIKEN) and myfellow graduate students in UBC and Osaka University for their insightfuland illuminating discussions.My special gratefulness and appreciation goes to Prof. Philip C. E. Stampwho is an awesome mentor and friend to me. I have greatly benefited fromhis extensive knowledge in quantum field theory and perhaps learned evenmore from his attitude towards life, which influences me profoundly andhelps shape my character.Lastly, I would like to thank my family. My parents offered me uncon-ditional love and support throughout my doctorate. Their encouragementhelp me come through the most difficult and depressed times during the pro-gram. I am also thankful to my wife Margaret for her consistent care andher optimistic attitude that always cheers me up.xxiiiTo My FamilyxxivChapter 1IntroductionIn 1928, Paul Dirac invented the celebrated 4-component spinor (bispinor)wave equation named after him [1], making an important contribution tothe reconciliation of special relativity and quantum mechanics1, 23 yearsafter Albert Einstein’s Annus mirabilis paper [3] and 2 years after ErwinSchrödinger’s milestone equation [4]. The most important consequence ofDirac equation is the prediction of the existence of anti-matter, which islater justified by the discovery of positron [5] – the antiparticle of electron. Inthe relativistic limit, the elementary fermionic particles (quarks and leptons)and their antiparticles obey Dirac equation, constituting the so-called “Diracfermions.”Shortly after Dirac’s groundbreaking work, German mathematician Her-mann Weyl proposed a way that decouples Dirac equation into a pair ofwave equations for 2-component spinors2, provided that the mass term inDirac equation is vanishing [7]. The 2-component spinor wave function rep-resents a massless relativistic fermion with definite chirality (handedness).This fermion is referred to as the “Weyl fermion.” Since Wolfgang Pauli’sfamous letter addressed to “Dear radioactive ladies and gentlemen” [8], neu-trinos are once thought of as Weyl fermions [9–11]. However, the discovery ofneutrinos in 1956 [12] prompts the examination of the consequence of smallbut nonzero neutrino mass. Now, through the oscillation experiment [13, 14],we know that a neutrino does have mass and cannot be a Weyl fermion.The complexity of generating relativistic Dirac fermions in acceleratorsand the hopeless search of elementary particles as Weyl fermions cast shadowon experimental studies of the interplay between Dirac/Weyl fermions andgauge fields (e.g., the electromagnetic field) in the context of particle physics.1Another contribution worthy to be mentioned is the Klein-Gordon equation, in whichthe probability density unfortunately cannot be well defined due to the lack of positivedefiniteness [2]. But this drawback motivates the search for new relativistic quantumtheories and eventually paves the way to Dirac equation.2The other way of decoupling Dirac equation is proposed by Ettore Majorana in 1937[6]. In Majorana’s representation, the 2-component spinor wave function characterizes achargeless fermion that is its own antiparticle. This fermion is referred to as the “Majoranafermion.”1Chapter 1. IntroductionFortunately, such difficulty can be overcome in the context of condensedmatter physics, where physics is encoded by the collective behavior of all theparticles comprising a condensed matter system. The collective behavioritself is characterized by a quasiparticle, which is the quantum mechanicalsuperposition of all the component particles. Although the component parti-cles are generally massive, the resulting quasiparticles established from themcan have zero effective mass, thus mimic the massless relativistic Dirac andWeyl fermions.In the context of condensed matter physics, the materials that host quasi-particles obeying Dirac and Weyl equations constitute the so-called “Diracmatter,” characterized by linear energy band crossings in the bulk. It isexemplified by graphene [15, 16], Dirac/Weyl semimetals [17] and super-conductors [18, 19], Weyl ferromagnets and anti-ferromagnets [20–27], andvarious photonic [28–31]/acoustic [32]/mechanical [33] meta-materials3. Theeasiness of being obtained and the convenience of being manipulated makeDirac matter ideal experimental venue to investigate the relativistic theo-ries of Dirac and Weyl fermions, especially their interplay with gauge fields.Remarkably, the advantage of studying Dirac and Weyl physics with Diracmatter over elementary particles is that the Dirac/Weyl quasiparticles canexhibit either fermionic or bosonic statistics, which extends the scope of theoriginal Dirac/Weyl theory.My dissertation is devoted to study Dirac and Weyl quasiparticles inDirac matter from a theoretical point of view, paying close attention totheir interplay with a specific type of gauge field that is induced by elasticstrain. And the present chapter aims at briefly reviewing the research ac-complishments and progress on gauge fields in Dirac matter. In Section 1.1,we briefly introduce several Dirac materials. In Section 1.2, we derive thecommon spectra of Dirac-Landau levels of Dirac materials in the presence ofgauge fields incorporated through the minimal substitution. In Section 1.3,we demonstrate that elastic strain can induce a gauge field incorporated tographene through the minimal substitution, thus resulting in Landau quanti-zation. In Section 1.4, we post our central motivation – extending the idea ofstrain-induced elastic gauge field beyond graphene to other Dirac materials.3Though the quasiparticles constituting Dirac matter are quite diversified (electrons,Bogoliubov quasiparticles, magnons, photons, phonons, etc.), the Majorana particle maynot be a legitimate candidate for Dirac matter. This is because the Majorana particlesin topological superconductors are either confined to boundaries or vortices [34]. Theresulting in-gap Majorana bands characterize bound states rather than bulk states. In fact,the bulk physics of topological superconductors are characterized by gapped Bogoliubovquasiparticle bands, which do not encode Dirac physics.21.1. Dirac materialsThis will be elaborated gradually in Chapter 2-4 for Weyl semimetals, Weylsuperconductors, and Weyl ferromagnets, respectively.1.1 Dirac materials1.1.1 GrapheneThe simplest and best understood Dirac material is the single layer graphite– graphene [15, 16]. Though graphene has be theorized for decades, it isfirst successfully exfoliated from graphite in 2004 [35], 440 years after theinvention of the graphite pencil. Graphene has many unusual properties,such as large thermal conductivity, low electric resistivity, and ultra highelectron mobility, making it valuable for the future application as electronicdevices.The unusual properties of graphene reflect its Dirac nature. Graphene iscomposed of carbon atoms connected with 3 nearest neighbors (Fig. 1.1(a))through  bonds formed by the “head to head” overlap of the sp2 hybridizedorbitals (Fig. 1.1(b)). The 3  bonds of a carbon atom are identical; thus aremutually 120 apart, resulting in a honeycomb lattice spreading in the x-yplane. The 2pz orbitals perpendicular to the honeycomb sheet (Fig. 1.1(c))do not engage in hybridization. Their “side by side” overlap leads to ⇡ bondsconnecting a carbon atom with its neighbors (nearest, second nearest, etc.).Since a carbon atom has 4 valence electrons and 3 of them are  electronsoccupying the sp2 hybridized orbitals, the 2pz orbital is thus half filled. It isthis ⇡ electron on the 2pz orbital that can hop to the neighboring sites, givingrise to energy bands. The conduction band and valence touches linearly atcorners of the hexagonal Brillouin zone, producing massless quasiparticles.The linearity is measured by Fermi velocity vF ⇡ 106m/s [36]. Consequently,the fast drifting massless quasiparticles highly emulate Dirac fermions.The discovery of graphene has profound impacts on the subsequent re-search on Dirac materials. The Dirac/Weyl semimetals can be viewed as 3Dgraphene. The superconducting and bosonic Dirac materials are the gen-eralized Dirac/Weyl semimetals whose component particles are replaced byBogoliubov quasiparticles and various bosonic excitations.1.1.2 Dirac and Weyl semimetalsDirac and Weyl semimetals [17, 37–39] can be understood as 3 dimensionalgeneralization of graphene, because they both exhibit linearly dispersingcone structure formed by energy band crossing as illustrated schematically31.1. Dirac materials↵1↵2↵3sp2 orbitals pz orbitals(a) (b) (c)Figure 1.1: Graphene lattice and chemical bonds. (a) Graphene is a singlelayer of carbon atoms arranged on a honeycomb lattice. (b) Each carbonatom is connected with 3 adjacent carbon atoms by the  bonds due to thehead-to-head overlap of the sp2 hybridized orbitals. (c) The electrons oneach lattice site can hop to neighboring sites along the ⇡ bonds due to theside-by-side overlap of the unhybridized 2pz orbitals.in Fig. 1.2. One major difference between the two types of semimetals isthe degeneracy of band crossings. A Dirac semimetal exhibits four-fold bandcrossing referred to as the Dirac point (Fig. 1.2(a)), while a Weyl semimetalpossesses two-fold band crossing known as the Weyl point (Fig. 1.2(b)). Inthe view of band crossing degeneracy, a Weyl semimetal is like half a Diracsemimetal.The difference in band crossing degeneracy may be further clarified bysymmetry considerations. In the presence of inversion symmetry P , a quan-tum state |k, si of momentum k and spin s is mapped to |k, si, whilein the presence of time reversal symmetry T , the same state is mapped to|k,si. Since a Dirac semimetal has both P and T , the Dirac cone atmomentum k has its P partner (of same spin) and T partner (of oppositespin) both located at momentum k, giving rise to a doubly degenerateband structure, which accounts for the four-fold degeneracy of Dirac points.On the other hand, a Weyl semimetal does not simultaneously have P andT . The lacking of aforementioned symmetry protection lifts the two-folddegeneracy of energy bands. Therefore, the Weyl points formed from theresulting non-degenerate bands are only two-fold degenerate.It is worth noting that P and T only determine the fold of degeneracy ofthe Dirac point. The robostness of Dirac points must rely on other crystallinesymmetries. For example, Dirac semimetal Cd3As2 [40–45] is protected byfour-fold rotational symmetry and Dirac semimetal NaBi3 [46–48] is pro-41.1. Dirac materialsDirac point Weyl point(a) (b)Figure 1.2: Schematic band crossings of Dirac and Weyl semimetals. (a)In Dirac semimetals, the doubly degenerate conduction and valence bandstouch linearly forming cone-like band structure. The vertex of a cone rep-resents a four-fold Dirac point. (b) In Weyl semimetals, the non-degenerateconduction and valence bands also touch linearly forming cone-like bandstructure. The vertex of a cone represents a two-fold Weyl point.tected by three-fold rotational symmetry. Besides these two intrinsic Diracsemimetals, more candidate materials can be obtained by fine tuning theparameters of topological insulators to induce phase transition towards triv-ial insulators or vice versa. This physics has been examined in Bi2xInxSe3[49, 50], which is a Dirac semimetal existing intermediately between topo-logical insulator Bi2Se3 and trivial insulator In2Se3.Weyl semimetals have been proposed and observed in P breaking systemssuch as the TaAs family of materials [51–53] and T breaking systems (all-in-all-out pyrochlore iridates [37], HgCr2Se4 [54], TI-NI multilayer [38]). Thetwo types of Weyl semimetals may be differentiated by examining the numberof Weyl points. For a Weyl semimetal with inversion symmetry P broken,the time reversal symmetry T requires a Weyl cone at k with chirality to be paired with another Weyl cone at k with same chirality . But theno-go theorem [55] requires the total chirality to be vanishing, indicatingthat there must be two more Weyl cones of chirality  located at ±k0.Therefore, P breaking Weyl semimetals must have at least four Weyl cones.On the other hand, for a Weyl semimetal with time reversal symmetry Tbroken, the inversion symmetry P requires a Weyl cone at k with chirality to be paired with another Weyl cone at k with chirality , satisfyingthe no-go theorem automatically. Therefore, T breaking Weyl semimetalscan have just two Weyl cones.51.1. Dirac materialsThe band crossings in Weyl semimetals rely on the translational sym-metry under which the Weyl cone chirality can be well defined. The Weylpoints are protected topologically by their chirality [17], which works as ei-ther the source or the drain of Berry flux (Fig. 1.3(a)). Any 2D subsystembetween the Weyl points encloses a nontrivial Berry flux thus resembles aChern insulator. The edge states of all the 2D subsystems constitute thesurface state of the Weyl semimetal (Fig. 1.3(b)). When tuning the Fermienergy EF to the Weyl points, the surface state traverses the EF iso-energysurface, leaving an arc state connecting the Weyl points, as illustrated inFig. 1.3(b). The discrete “Fermi arc” surface state is a fingerprint of Weylsemimetals.Dirac and Weyl semimetals exhibit many exotic properties due to theirunusual electronic structure. These include the chiral anomaly [56–58], thechiral magnetic effect4 [62, 63, 65–67], the anomaly-induced negative magne-toresistance [51, 63, 65, 66, 68–70], Majorana flat bands [71], and Fermi arcquantum oscillations [72, 73]. These properties may make Dirac and Weylsemimetals valuable in the future application to novel electronic devices.1.1.3 Dirac and Weyl superconductorsDirac and Weyl superconductors [18, 19] are the superconducting analogs ofDirac and Weyl semimetals. The only difference is that the Dirac and Weylcones are formed by the touching of 3D Bogoliubov quasiparticle energybands. Due to the similarity to Dirac and Weyl semimetals, the Dirac andWeyl superconductors inherit unusual properties from their semi-metalliccounterparts, such as the arc-like Fermi surface [19].Currently, there are about 20 nodal superconductors [74] having the po-tential to be Dirac and Weyl superconductors. Among those, the most4A more generalized chiral magnetic effect is first derived in the context of high energyphysics [59, 60], where both chiral fermions and their antiparticles are considered. Inthe presence of an applied magnetic field B, a parallel axial current emerges J5 ⇠ µB,provided that there is a concentration imbalance between particles and antiparticles (i.e.,chemical potential µ 6= 0). The interpretation of µ is subtle. In neutrino systems, par-ticles are always left-handed (LH) while antiparticles are always right-handed (RH) [61].Therefore µ is analogous to the chiral chemical potential µ5 in Weyl semimetals [62, 63].In quark systems, the particles/antiparticles can be either left-handed or right-handed[64]. Then axial current can occur even within a single chiral species (RH or LH), if anonzero chemical potential (µR 6= 0 or µL 6= 0) exists. And the total axial current iscontributed by both quarks and antiquarks, making it slightly different from the chiralmagnetic current [62, 63] in Weyl semimetals, where only electrons are responsible for thetransport. For this reason, the axial current generated by the applied magnetic field andquark-antiquark imbalance is sometimes referred to as the “chiral separation effect” [64].61.1. Dirac materialsbulkFermi arcsurfaceFermi arc(a) (b)kxkykzkxkykzFigure 1.3: Schematic band structure plot for Weyl semimetals. (a) A pairof Weyl points in momentum space, one being the source of the Berry fluxwhile the other acting as the drain. They are connected by Fermi arc stateson the open boundaries. (b) Weyl cones and surface states in momentumspace. The surface states cutting through two Weyl cones can be understoodas a combination of chiral edge states of 2D subsystems between two Weylpoints. When tuning Fermi energy to the Weyl points, the Fermi surfacebecomes an arc connecting the two Weyl points.promising candidate materials may be CuxBi2Se3 [19, 75] and NbxBi2Se3[76]. Recent nuclear magnetic resonance experiments [77] and specific heatexperiments [78] suggest CuxBi2Se3 to be a Dirac superconductor. However,symmetry and energetic considerations [79, 80] suggest CuxBi2Se3 to havea small but non-vanishing gap. According to Ref. [76], the low temperaturepenetration depth of NbxBi2Se3 exhibits quadratic temperature dependence,which is a characteristic feature of linearly dispersing point nodes in threedimensions. This is consistent with NbxBi2Se3 being a Weyl superconductor.1.1.4 Weyl magnetsWeyl magnets are a special type of spin crystals existing in various ferromag-netic and antiferromagnetic ordered systems such as pyrochlore magnets [20–23], double perovskites [24], and multilayer magnets [25–27]. They exhibitenergy bands touching linearly at discrete points in the momentum-energyspace, akin to Weyl semimetals. However, unlike the Weyl semimetals whoseenergy bands result from Bloch waves, the energy bands of Weyl magnetsassociate with spin waves [81], whose quantum is a magnon [82].71.2. Dirac-Landau levelsA magnon is a bosonic quasiparticle carrying spin, dipole moment, andheat. Though charge neutral, it can be controlled by either an electric fieldor a magnetic field through the Aharonov-Casher effect [83] or the Zeemaneffect. Therefore, magnons can mimic electrons in many ways, enablingWeyl magnets to reproduce many properties of Weyl semimetals, such asthe chiral anomaly [20, 25], the spin Hall effect [26], and the thermal Halleffect [84]. Superior over Weyl semimetals, Weyl magnets carry magnoncurrents, which are inherently free of energy dissipation due to Ohmic losses.This advantage may make Weyl magnets potentially useful to process andtransport information.1.1.5 Other Dirac materialsIn Section 1.1.4, we have seen that substituting the spin wave for the Blochwave produces a new Dirac material – Weyl magnets. Such generalizationfrom Weyl semimetals to Weyl magnets reveals a standard way of search-ing for new Dirac materials – finding other substitutes for the Bloch wave.The first example is the electromagnetic wave carried by photonic meta-materials. By fine tuning the permittivity and the permeability of photonicmeta-materials, the energy bands can touch linearly, resulting in “photonicDirac semimetals” [28] and “photonic Weyl semimetals” [29–31]. Another ex-ample is the sound wave harbored by acoustic meta-materials. According toRef. [32], a multilayer acoustic crystal composed of precisely designed hollowhexagonal unit cells can exhibit Weyl type energy band crossing for audi-ble sound waves. Moreover, even the classical mechanical wave in coupledmechanical oscillators can produce linearly crossed energy bands, giving aso-called “mechanical graphene” [33].1.2 Dirac-Landau levelsIn Section 1.1, we have seen various Dirac materials comprised of fermionicand bosonic particles. In this section, we will quantitatively understand theirphysics in the framework of band theory. We will pay particular attentionto the band structure of such materials in the presence of gauge fields.The physics of Dirac materials is encoded in the Dirac Hamiltonians.The simplest Dirac Hamiltonian isH = vxpxx + vypyy + vzpzz, (1.1)which is a 2⇥ 2 matrix defined on a Hilbert space spanned by some degrees81.2. Dirac-Landau levelsof freedom. vi and i are the i-th component of velocity vector5 and Paulimatrices. For example, graphene has vz = 0 and is defined on the Hilbertspace spanned by sublattice degrees of freedom. More generically, DiracHamiltonian is 4⇥ 4 due to other degrees of freedom. Explicitly, it readsH = vxpx↵x + vypy↵y + vzpz↵z +m, (1.2)where ↵ and  are 4 ⇥ 4 Dirac matrices and m is the mass of the particle.Hamiltonian Eq. 1.2 applies to Dirac semimetal Cd3As2 defined on spin-orbitbasis [85]. For simplicity, we will only consider Dirac materials characterizedby Eq. 1.1, whose eigenvalueE(p) = ±qv2xp2x + v2yp2y + v2zp2z, (1.3)exhibits linearly dispersing Dirac cone structure in the momenta-energyspace. When a gauge field A is applied to Dirac materials, the transla-tional invariance in certain direction will generally be broken, which can gapout the Dirac cone Eq. 1.3. In the rest of this section, we will discuss theband structure of Dirac Hamiltonian Eq. 1.1 in the presence of gauge fields.We consider a gauge fieldA that can be incorporated into Eq. 1.1 throughthe standard Peierls substitution p ! p  qA. In electronic systems, Ais the magnetic vector potential due to the Aharonov-Bohm effect [86]. Inmagnonic systems, A ⇠ E⇥µ withE being the external electric field appliedto magnons of moment µ due to the Aharonov-Casher effect [83]. Withoutloss of generality, we choose A = (⌦y, 0, 0), whose curl r ⇥ A = ⌦zˆ is auniform field. Under the gauge field, the Dirac Hamiltonian can be writtenasH = vx⇣ i~ @@x+ q⌦y⌘x + vy⇣ i~ @@y⌘y + vz⇣ i~ @@z⌘z. (1.4)5Generically, the momentum pi can couple to Pauli matrices j 6=i as well, making veloc-ity a tensor, under which the generalized Dirac Hamiltonian reads H =Pij pivijj . Thehermicity of the generalized Dirac Hamiltonian requires the entries of the velocity tensorto be real. This implies that the singular value decomposition of velocity tensor existsand reads vij =Pmn Limv˜mnRnj , where Lim and Rnj are 3⇥ 3 orthogonal matrices andv˜mn = v˜nmn is a diagonal matrix with Kronecker delta mn being the 3⇥3 unity matrix.The singular value decomposition allows us to rewrite the generalized Dirac Hamiltonianas H =Pijmn piLimv˜mnRnjj =Pmn p˜mv˜mn˜n =Pn v˜np˜n˜n, where p˜m =Pi piLimand ˜n =Pj Rnjj are orthogonally transformed momenta and Pauli matrices. Usingthe properties of orthogonal matrices, it is straightforward to prove that the orthogonal-ity of momenta and the commutation/anti-commutation relations of Pauli matrices areall preserved. Therefore, the generalized Dirac Hamiltonian can always be simplified tothe standard Dirac Hamiltonian Eq. 1.1 where the velocity is a vector. Without loss ofgenerality, Dirac materials with vector velocity will be our central concern in the rest ofthis dissertation.91.2. Dirac-Landau levelsTo solve for the eigenvalues of Eq. 1.4, we first find the eigenvalues ofH2 =⇣ ~2v2x@2@x2 ~2v2y@2@y2 ~2v2z@2@z2⌘ 2i~q⌦yv2x@@x+ q2⌦2y2v2x  ~q⌦vxvyz. (1.5)Because [H2,z] = 0, the eigenstate wave function of H2 can be written ass(x, y, z) = ei~ (pxx+pzz)fs(y)s, (1.6)where + = (1, 0)T and  = (0, 1)T are the up/down spinor and functionfs(y) satisfies~2v2y@2fs@y2 v2x(px + q⌦y)2fs + (v2zp2z + s~q⌦vxvy)fs = E2fs, (1.7)where E is the eigenvalue of the Dirac Hamiltonian Eq. 1.4. We define anew variable ⇠ satisfying⇠2 =vxq⌦~vy⇣y +pxq⌦⌘sgn(q⌦vxvy). (1.8)Therefore, Eq. 1.7 can be written as a differential equation with respect to⇠ asd2fsd⇠2 ⇠2fs + asfs = 0, (1.9)whereas =E2  v2zp2z~|vxvyq⌦| + s · sgn(q⌦vxvy).By assuming fs(⇠) = e12 ⇠2us(⇠), Eq. 1.9 is reduced tod2usd⇠2 2⇠us + (as  1)us = 0. (1.10)The solution to Eq. 1.10 is Hermite polynomial us = Hn(⇠) when the follow-ing condition is satisfiedas =E2  v2zp2z~|vxvyq⌦| + s · sgn(q⌦vxvy) = 2n+ 1 n = 0, 1, 2, · · · . (1.11)Since s = ±1, we define integer ⌫ such that2⌫ = 2n+ 1 s · sgn(q⌦vxvy). (1.12)101.3. Strain-induced Landau levels in GrapheneTherefore, the eigenvalue E of the Dirac Hamiltonian Eq. 1.4 satisfiesE2 = v2zp2z + 2⌫~|q⌦vxvy|. (1.13)According to Eq. 1.12, when s · sgn(q⌦vxvy) = 1, ⌫ = 1, 2, · · · . But whens · sgn(q⌦vxvy) = 1, ⌫ = 0, 1, 2, · · · . Since there are two occasions that ⌫ canbe positive integers, the associated eigenvalues can then be determined asE⌫ = ±qv2zp2z + 2⌫~|q⌦vxvy| ⌫ = 1, 2, · · · . (1.14)These are the famously known Dirac-Landau levels. It is worth to note thatthe dispersion of the zeroth Landau level is not ±vzpz, because there is onlyone occasion (s = sgn(q⌦vxvy)) making ⌫ = 0. When s = sgn(q⌦vxvy) =±1, according to Eq. 1.6, the eigenstate wave function is± ⇠ e i~ (pxx+pzz)e12vxq⌦~vy (y+pxq⌦ )2sgn(q⌦vxvy)±. (1.15)The zeroth Landau level dispersion can be determined by using H± =E0±. This leads to a chiral stateE0 = sgn(q⌦vxvy)vzpz. (1.16)In summary, Eqs. 1.14 and 1.16 comprise the complete spectrum ofEq. 1.4. The higher (⌫  1) Dirac-Landau levels always come in pairs withopposite energies. On the other hand, the zeroth Landau levels is chiral. InChapters 2-4, we will see Dirac-Landau levels occur in various Dirac mate-rials and profoundly influence the charge, heat, and spin transports in thosematerials.1.3 Strain-induced Landau levels in GrapheneIn Section 1.2, we have seen that if a gauge field is incorporated to Diracmatter by Peierls substitution, the resulting energy bands exhibit Dirac-Landau levels. We already know that the magnetic field leads to Landauquantization. In this section, following Refs. [87, 88], by using graphene asan example, we will review that properly designed elastic strain behaves likea pseudo-magnetic field that also gives rise to Landau quantization.Our starting point is the nearest neighbor tight binding Hamiltonian ona honeycomb latticeH =Xr,itb†r+↵iar + h.c., (1.17)111.3. Strain-induced Landau levels in Graphenewhere the i-th nearest neighbor is connected by ↵i illustrated in Fig. 1.1(a)and t is the overlap integral of the ⇡ bond connecting to the i-th nearestneighbor. Apply Fourier transform✓arbr◆=1pNXkeik·r✓akbk◆, (1.18)where N is the number of unit cells. In the sublattice basis k = (ak, bk)T ,the tight binding Hamiltonian can be written asH =Xk†kHkk, (1.19)whereHk =Xit cos(k ·↵i)x Xit sin(k ·↵i)y. (1.20)The first quantized Hamiltonian (Eq. 1.20) is gapless at the corners of Bril-louin zone K⌘ = ⌘(4⇡/3p3a, 0). In the vicinity of K⌘, Hk can be expandedasHK⌘+q ⇡ HK⌘ + q ·rkHk = 32 ta⌘qxx +32taqyy = h⌘q, (1.21)which is a 2D Dirac Hamiltonian same as Eq. 1.1 if the velocity parametersin Eq. 1.1 are set as(vx, vy, vz) =1~⇣ 32ta⌘,32ta, 0⌘.When the honeycomb lattice of graphene is deformed by external strain,the 2pz orbital on each lattice site R will be translated to a new positionR+u(R), where u(R) is the displacement field. Then the overlap t betweenthe 2pz orbital originally located at R and the adjacent 2pz orbital originallylocated at R0 = R+↵i will be changed tot(R0 + u(R0)R u(R)) ⇡ t(R0 R) +rt · (u(R0) u(R))⇡ t(R0 R) +rt · (R0 R) ·ru. (1.22)We may estimate rt = t↵ia2 ⇡ t↵ia2 , where we have taken the Grüneisenparameter  ⇡ 1. Then we conclude that the overall effect of the elasticstrain can be considered by doing the following overlap integral substitutiont! t⇣1 1a2↵i ·↵i ·ru⌘. (1.23)121.3. Strain-induced Landau levels in GrapheneIt is worth noting that ↵i · ↵i · ru = ↵µi ↵⌫i @µu⌫ = ↵µi ↵⌫i uµ⌫ where uµ⌫ =12(@µu⌫ + @⌫uµ) is the symmetrized strain tensor. Plug Eq. 1.23 to Eq. 1.17and assume constant strain tensor, following the Fourier transform, we findthat the first quantized Hamiltonian Hk (Eq. 1.20) needs to be modified byan extra termHk = t⇣34uxx +14uyy +p32uxy⌘cos(k ·↵1)x+ t⇣34uxx +14uyy +p32uxy⌘sin(k ·↵1)y t⇣34uxx +14uyy p32uxy⌘cos(k ·↵2)x+ t⇣34uxx +14uyy p32uxy⌘sin(k ·↵2)y tuyy cos(k ·↵3)x + tuyy sin(k ·↵3)y. (1.24)The physics in the vicinity of K⌘ is then characterized byhq + HK⌘ = 32 ta⌘⇣qx + ⌘uyy  uxx2a⌘x +32ta⇣qy + ⌘uxya⌘y. (1.25)It is obvious to see that the strain-induced extra Hamiltonian Hk is in-corporated into the 2D Dirac Hamiltonian hq through Peierls substitutionq ! q + e~A, where the emergent vector potential isA = ⌘~2ea(uyy  uxx, 2uxy). (1.26)Though we assumed constant strain tensor uµ⌫ when deriving A, we arguethat even when uµ⌫ is spatially varying, the strain effect can be treated as anemergent vector potential expressed in Eq. 1.26, as long as it varies slowly onthe lattice scale. By fine tuning the displacement field as in Ref. [87], we canget a uniform strain-induced emergent gauge field B = r ⇥ A. Therefore,according to our analysis in Section 1.2, the resulting band structure will beflat Landau levelsEn =q2n~e|B||vxvy| =3ta2r2ne|B|~ n = 0, 1, 2, · · · . (1.27)From the point of view of band structure, the strain-induced gauge field Bhighly mimics the ordinary magnetic field. Therefore, many magnetic trans-port properties associated with Landau levels can be in principle reproduced131.4. Motivationby proper strains. For this reason B is often referred to as a “pseudo-magneticfield.” However, it is worth noting that the emergent vector potential A andthe pseudo-magnetic field B only exist near Dirac points and couple to dif-ferent Dirac points oppositely. Therefore, the strain-induced Landau levelsonly exist in the vicinity of Dirac points and are fundamentally different fromthe ordinary Landau levels due to an ordinary magnetic field.1.4 MotivationIn Section 1.3, we have seen that strain couples to low-energy 2D Diracfermions in graphene as an emergent U(1) gauge field. And according tothe analysis in Section 1.2, this strain-induced elastic gauge field leads toLandau quantization in graphene. This discovery has profound influence onthe study of graphene, because it is the first instance that the presence ofLandau levels does not depend on applying magnetic fields.Since the proposal [87] and experimental implementation [89] of strain-induced Landau levels in graphene, the idea of strain-induced gauge fieldsand Landau quantization has be broadened to other systems such as “pho-tonic graphene” [90] and “magnonic graphene” [91]. Since there are lots ofother members in the family of Dirac matter, we hypothesize that strain willalso produce Landau quantization to these materials and can affect theirindividual transport drastically. Motivated to justify this hypothesis, thepresent dissertation is organized as follows.In Chapter 2, we show that a circular bend lattice deformation gives riseto Dirac-Landau levels in Weyl semimetal thin films. Though there havebeen several proposals of generating Landau levels in Weyl semimetals withstrain [92–97], their experimental implementation and transport measure-ment are not as feasible as ours. Due to the simplicity of the strain design,our induced elastic gauge field can be tuned continuously, allowing us toperform dynamic measurement of quantum oscillations, which are difficultto measure in other existing proposals. In the chapter, we demonstrate thatthe scanning strain-induced pseudo-magnetic field results in Shubnikov-deHaas (SdH) oscillation. This is the first instance that quantum oscillationsoccur without applying magnetic field.In Chapter 3, we show that the same bend deformation can also Landauquantize linearly dispersing Bogoliubov quasiparticles in Weyl superconduc-tors. This sheds new light on observing Landau quantization based transportin the presence of superconductivity. Particularly, the strain-induced Lan-dau levels allow superconducting regime quantum oscillations. On the other141.4. Motivationhand, QOs have only been previously observed in normal [98–103] and vor-tex state [104–106] superconductors. We elucidate that the strain-inducedLandau levels give rise to quasiparticle Wiedemann-Franz law and thermalconductivity quantum oscillations when the strain-induced gauge field is con-tinuously tuned.In Chapter 4, we show that a twist deformation can Landau quantize rel-ativistic magnons in Weyl ferromagnets. It thus resembles an inhomogeneouselectric field. We further show that a time-dependent uniaxial deformationcan drive magnons along the energy bands. Therefore, such deformationis analogous to an inhomogeneous magnetic field. The combination of thestrain-induced elastic gauge field and the electromagnetic field produces vari-ous magnon quantum anomalies in Weyl ferromagnets. These anomalies arecharacterized by the non-conservation of chirality or bulk thermal energy.The anomalous spin and heat transport due to these anomalies is also de-rived. Its unique field dependence may play a key role in detecting magnonquantum anomalies experimentally.Chapter 5 concludes the whole dissertation, envisages a few worthwhiledirections on the thermal transport in the presence of strain-induced gaugefields, and discusses the possibility of reproducing Landau quantization inother Dirac materials.15Chapter 2Zero-field quantum oscillationsin Weyl semimetalsDirac and Weyl semimetals [17, 37–39] are known to exhibit a variety ofexotic behaviors owing to their unusual electronic structure comprised oflinearly dispersing electron bands at low energies. These include the pro-nounced negative magnetoresistance [51, 62, 63, 65, 66, 68–70] attributed tothe phenomenon of the chiral anomaly [56–58], theoretically predicted non-local transport [107, 108], Majorana flat bands [71], as well as an unusualtype of quantum oscillations (QOs) that involve both bulk and topologicallyprotected surface states [72, 73].Materials with linearly dispersing electrons respond in peculiar ways tothe externally imposed elastic strain. In graphene, for instance, the effect ofcurvature is famously analogous to a pseudo-magnetic field [87, 88] that canbe quite large and is known to generate pronounced Landau levels observedin the tunneling spectroscopy [89]. Recent theoretical work [92–95, 109]showed that similar effects can be anticipated in three-dimensional Diracand Weyl semimetals, although the estimated field strengths in the geome-tries that have been considered are rather small (below 1 T in Ref. [94]).Ordinary quantum oscillations, periodic in 1/B, have already been observedin Dirac semimetals Cd3As2 and Na3Bi [40, 73, 110, 111] but the magneticfield required is B & 2T. This, then, would seem to rule out the observationof strain-induced QOs in the geometries considered previously.In this chapter, by considering a new geometry – circular bend, we estab-lish a completely new mechanism for QOs in Weyl semimetals with strain-induced pseudo-magnetic fields but in the complete absence of ordinary mag-netic fields. The chapter is organized as follows. In Section 2.1, we discuss a2-band toy model of Weyl semimetals. Such model is referred to as 12 -Cd3As2because it characterizes one of the spin sectors of Dirac semimetal Cd3As2.In Section 2.2, we derive the pseudo-magnetic field induced by a circularbend deformation and show that it can be as large as 15T, which shouldbe sufficient for the observation of QOs without ordinary magnetic fields.In Section 2.3, we show that the pseudo-magnetic field gives rise to Landau162.1. Model of Weyl semimetalslevels, the necessary QO ingredient. In Section 2.4, we derive the longitu-dinal electric conductivity and demonstrate that it exhibits Shubnikov-deHaas oscillations [112] both below and above Lifshitz transition in zero or-dinary magnetic field as long as the external strain is continuously tuned.Section 2.5 concludes the chapter and discusses the promising venue for re-alizing the strain-induced QOs.2.1 Model of Weyl semimetalsIn order to implement quantum oscillations with purely strain-induced pseudo-magnetic fields, the minimal model we require is a 2-band Dirac model ofWeyl semimetals. In this section, we will introduce a toy model that cancharacterize Weyl semimetals.Our starting point is Dirac semimetal Cd3As2 [40–45] which may bethe best characterized representative of this class of materials. Our resultsare directly applicable also to Na3Bi [46–48] whose low-energy descriptionis identical, and are easily extended to other Dirac and Weyl semimetals[52, 53, 85, 113–115]. We start from the tight-binding model formulated inRefs. [41, 46] which describes the low-energy physics of Cd3As2 by includingthe band inversion of its atomic Cd-5s and As-4p levels near the  point.In the basis of the spin-orbit coupled states |P 32, 32i, |S 12 ,12i, |S 12 ,12i, and|P 32,32i the model is defined by a 4⇥ 4 matrix HamiltonianHk = ✏k +✓Hk 00 Hk◆, (2.1)on a simple rectangular lattice with lattice constants ax,y,z, whereHk = mk⌧ z + ⇤(⌧x sin kxax + ⌧y sin kyay). (2.2)⌧x,y,z are Pauli matrices in the orbital space and mk = t0 + t1 cos kzaz +t2(cos kxax+cos kyay). The upper diagonal blockHk realizes a toy model of aWeyl semimetal that we will use for our analytical and numerical calculationsthroughout the present chapter. It exhibits a pair of Weyl points, shown inFig. 2.1(a), which are located at crystal momenta K⌘ = (0, 0, ⌘Q) with Qgiven by cos(aQ) = (t0 + 2t2)/t1, where we have assumed cubic latticegeometry ax = ay = az = a. In the vicinity of such Weyl points, HK⌘+qrealizes the standard Dirac HamiltonianHK⌘+q = HK⌘ + q ·rkHk|K⌘ =Xi~v⌘i qi⌧i = h⌘q, (2.3)172.2. Strain-induced pseudo-magnetic fieldwhere the velocity is given by(v⌘x, v⌘y , v⌘z ) =a~(⇤,⇤,⌘t1 sinQa). (2.4)The chirality associated with each Weyl point is then determined by⌘ = sgn(v⌘xv⌘yv⌘z ) = ⌘sgn(t1), (2.5)indicating that the two Weyl points work as source and drain of Berry flux,respectively. The non-trivial topology encoded by Hk can be unveiled byexamining the electronic structure of Hk regularized on a thin film as il-lustrated in Fig. 2.2(a). The energy bands of the Weyl semimetal thin filmexhibit Weyl cones and arc states terminated at the Weyl points as shown inFig. 2.3(a). To understand the appearance of such arc states, we calculatethe Chern number of Hk with momentum kz fixed and it readsCkz =12[sgn(t0 + t1 cos kza+ 2t2) + sgn(t2)]. (2.6)For the parameters used in Fig. 2.3(a), we obtainCkz =(1 |kz| < Q0 Q < |kz| < ⇡. (2.7)Therefore, each of the 2D slices of Hk with topological kz realizes a Cherninsulator with chiral edge states. The combination of these edge states leadsto the surface states of the Weyl semimetal thin film.The lower diagonal block Hk in Eq. 2.1 describes the spin-down sectorof Cd3As2 and has identical spectrum. For our purpose, this term will notbe considered in this chapter. But we will add it back in Appendix A toshow that our theory for zero-field quantum oscillations in Weyl semimetalscan be easily transplanted to Dirac semimetals. The diagonal term ✏k =r0 + r1 cos kzaz + r2(cos kxax + cos kyay) in Eq. 2.1 encodes the particle-hole asymmetry in Cd3As2. It shifts the energy bands but does not affectthe band topology. Therefore, we will neglect this term and add it back inAppendix A as well.2.2 Strain-induced pseudo-magnetic fieldIn this section, we first derive the theory of the respondence of Weyl semimet-als to a generic lattice deformation due to external strain. Then we apply182.2. Strain-induced pseudo-magnetic fieldkzkx,kyEa kzkxbK+K+Figure 2.1: Schematic depiction of the low-energy electron excitation spec-trum in Weyl semimetals. a) A pair of Weyl cones appear on kz axis. b)Contours of constant energy for ky = 0.the theory to a specific type of lattice deformation – circular bend and givethe expression for the induced pseudo-magnetic field.Following Refs. [92–94, 109], the most important effect of elastic straincan be included in the lattice model Eq. 2.2 by modifying the electron tun-neling amplitude along the z direction according tot1⌧z ! t1(1 u33)⌧ z + i⇤Xj 6=3u3j⌧j , (2.8)where uij = 12(@iuj+@jui) is the strain tensor and u = (u1, u2, u3) representsthe displacement of the atoms. The hopping integral substitution Eq. 2.8produces a correction to the unstrained lattice Hamiltonian (Eq. 2.2), makingthe full HamiltonianH˜k = Hk + Hk, (2.9)where the correction isHk = ⇤u31 sin kza⌧x + ⇤u32 sin kza⌧y  t1u33 cos kza⌧ z. (2.10)In the vicinity of Weyl points, the low-energy theory for the strained Hamil-tonian isHK⌘+q + HK⌘+q ⇡ h⌘q + HK⌘ =Xi~v⌘i⇣qi +e~Ai⌘⌧ i, (2.11)192.2. Strain-induced pseudo-magnetic fieldwhere the strain-induced gauge potential isA = ⌘~ea(u31 sinQa, u32 sinQa, u33 cotQa). (2.12)We see that elements uj3 of the strain tensor act on the low-energy Weylfermions as components of a chiral gauge field, i.e., the strain-induced gaugepotential A, which oppositely couples to Weyl cones with different chirality⌘. On the other hand, ordinary electromagnetic vector potential couplesthrough the same Peierls substitution q ! q+ e~A, but the vector potentialAis independent of ⌘. Ref. [94] noted that application of a torsional strain to ananowire made of Cd3As2 (grown along the [001] crystallographic direction)results in a uniform pseudo-magnetic field b = r⇥A pointed along the axisof the wire. The strength of this pseudo-magnetic field was estimated as b .0.3T which would be insufficient to observe QOs. Our key observation hereis that a different type of distortion – circular bend, illustrated in Fig. 2.2(a),can produce a much larger field b.One reason why the torsion-induced b-field is relatively small lies in thefact that it originates from the Ax and Ay components of the vector po-tential. According to Eq. 2.12, these are suppressed relative to the straincomponents by a factor of sin aQ. This is a small number in most Dirac andWeyl semimetals because the distance 2Q between the Weyl points is typi-cally a small fraction of the Brillouin zone size 2⇡/a. Specifically, we haveaQ ' 0.132 in Cd3As2 [41]. Note on the other hand that the Az compo-nent of the chiral gauge potential comes with a factor cot aQ ' 1/aQ and istherefore enhanced. A lattice distortion that produces nonzero strain tensorelement u33 will therefore be much more efficient in generating large b thanu13 or u23. Specifically, for the same amount of strain the field strength isenhanced by a factor of cot aQ/ sin aQ ' 1/(aQ)2 ' 57 for Cd3As2.To implement this type of strain we consider a thin film (or a nanowire)grown such that vectorK⌘ lies along the z direction as defined in Fig. 2.2(a).More generally we require that K⌘ has a nonzero projection onto the sur-face of the film or on the long direction for the nanowire. Cd3As2 films[110], microribbons [116] and nanowires [117, 118] satisfy this requirement.Bending the film as shown in Fig. 2.2(b) creates a displacement field u =(0, 0, 2↵xz/d), where d is the film thickness and ↵ controls the magnitudeof the bend. (If R is the radius of the circular section formed by the bentfilm then ↵ = 2d/R. ↵ can also be interpreted as the maximum fractionaldisplacement ↵ = umax/a that occurs at the surface of the film.) This distor-tion gives u33 = 2↵x/d which, through Eq. 2.12, yields a pseudo-magnetic202.3. Band structure of Weyl semimetalsfieldb = r⇥A = yˆ2↵d⌘~eacotQa. (2.13)Noting that 0 = h/e = 4.12⇥ 105TA˚2, we may estimate the magnitude ofthe pseudo-magnetic field for a d = 100nm film asb ' ↵⇥ 246T. (2.14)The maximum pseudo-magnetic field that can be achieved will depend onthe maximum strain that the material can sustain. Ref. [117] characterizedthe Cd3As2 nanowires as “greatly flexible” and their Fig. 1(a) shows somewires bent with a radius R as small as several microns. This implies that ↵of several percent can likely be achieved. From Eq. 2.13 we thus estimatethat field magnitude b ' 10  15T can be reached, providing a substantialwindow for the observation of the strain-induced QOs.zxyumaxaxzbbBFigure 2.2: Proposed setup for strain-induced quantum oscillation observa-tion in Weyl semimetals. a) Bent film is analogous, in terms of its low-energyproperties, to an unstrained film subjected to magnetic field B. b) Detail ofthe atomic displacements in the bent film. Displacements have been exag-gerated for clarity.2.3 Band structure of Weyl semimetalsIn Section 2.2, we argued that the strain-induced pseudo-magnetic field in abent Weyl semimetal thin film can be sufficiently large for the observation ofquantum oscillations. Since the necessary ingredient of quantum oscillationis Landau levels, in the present section, we will numerically examine the212.3. Band structure of Weyl semimetalspresence of strain-induced pseudo-Landau levels, paying close attention toits similarity to the Dirac-Landau levels due to the ordinary magnetic field.We first consider an unstrained Weyl semimetal thin film whose geometryis outlined in Fig. 2.2 with periodic boundary conditions along y and z,open boundary condition along x. Fig. 2.3(a) summarizes the numericallycalculated electronic structure and density of states (DOS) of such thin filmin the absence of external magnetic field. The band structure shows bulkWeyl nodes close to kza = ±0.2 and a pair of linearly dispersing surfacestates corresponding to Fermi arcs. The DOS exhibits the quadratic behaviorD(E) ⇠ E2 at low energies with some deviations apparent for |E| & 12meV.The numerical results can be understood by analytically deriving the effectivelow-energy theory band structureEaq = ±~q(v⌘xqx)2 + (v⌘yqy)2 + (v⌘z qz)2, (2.15)and the associated DOSDa(E) = Vˆd3q(2⇡)3(E  Eaq) =LxLyLzE22⇡2~3|v⌘xv⌘yv⌘z | . (2.16)And the deviation of D(E) of numerical results from the analytical results athigher energies is due to the departure of the lattice model from the perfectlylinear Weyl dispersion. At ELif ' 20meV, Lifshitz transition occurs wheretwo small Fermi surfaces associated with each Weyl point merge into a singlelarge Fermi surface as illustrated in Fig. 2.1(b).In Fig. 2.3(b), magnetic field B = yˆB is seen to reorganize the linearlydispersing bulk bands into flat Landau levels. In the continuum approxi-mation given by Eq. 2.11, according to Eq. 1.14, the bulk spectrum of suchDirac-Landau levels readsEbq = ±~r(v⌘yqy)2 + 2neB~ v⌘xv⌘z  n = 1, 2, · · · , (2.17)whose DOS is given byDb(E) = Vˆd3q(2⇡)3(E  Ebq) =LxLz2⇡l2BLy⇡~|v⌘y |XnsE2E2  2n| eB~ v⌘xv⌘z |,(2.18)where the magnetic length is lB =p~/e|B|. Such DOS shows a series ofspikes at the onset of each new Landau level and is in a good agreement withthe DOS calculated from the lattice model by using the tetrahedron method(see Appendix F for details). Deviations occur above ⇠ 12meV because the222.3. Band structure of Weyl semimetals0 5 10 15 20 25 30E[meV]0.00.1DOSlattice modelcontinuumLifshitz-0.4 -0.2 0.0 0.2 0.4k-30-20-100102030E[meV]0 5 10 15 20 25 30E[meV]0.00.1DOSlattice modelcontinuum-0.4 -0.2 0.0 0.2 0.4k-30-20-100102030E[meV]0 5 10 15 20 25 30E[meV]0.00.1DOSlattice modelcontinuum-0.4 -0.2 0.0 0.2 0.4k-30-20-100102030E[meV]kykzZYΓΓ ZY kzkya) B=b=0 Γ ZY kzkyb) B=1.5T Γ ZY kzkyc) b=1.5T~E2Figure 2.3: Band structure and DOS for lattice Hamiltonian Eq. 2.2. Inall panels, films of thickness 500 lattice points are studied with parameterst0 = 2.522eV, t1 = 1.042eV, t2 = 0.75eV, and ⇤ = 0.148eV. (a) Bandstructure and DOS for zero field and zero strain. The inset shows the firstBrillouin zone. The dashed parabolic curve is the expected DOS (Eq. 2.16)for ideal Weyl dispersion (Eq. 2.15). (b) Band structure and normalizedDOS for B = 1.5T. The solid black curve comprised of spikes at Landaulevels is the expected DOS (Eq. 2.18) calculated from Dirac-Landau levels(Eq. 2.17). Red crosses indicate the peak positions expected on the basis ofthe Lifshitz-Onsager quantization condition. (c) Band structure and DOSfor b = 1.5T. The solid black curve comprised of spikes at pseudo-Landaulevels is the expected DOS (Eq. 2.20) calculated from pseudo-Landau levels(Eq. 2.19).energy dispersion of the lattice model is no longer perfectly linear at higherenergies. The peak positions En agree perfectly with the Lifshitz-Onsagerquantization condition [112], which takes into account these deviations. It re-quires that S(En) = 2⇡n(eB/~), where S(E) is the extremal cross-sectionalarea of a surface of constant energy E in the plane perpendicular to B andn = 1, 2, · · · .In Fig. 2.3(c), we find that the pseudo-magnetic field b induced by strainusing Eq. 2.8 with u33 = 2↵x/d, also generates flat Landau levels and DOScomprised of spikes at each Landau level. Analytically, the low-energy spec-trum and DOS can be directly written down by comparing to Eq. 2.17 and232.4. Longitudinal electric conductivityEq. 2.18 asEcq = ±~r(v⌘yqy)2 + 2neb~ v⌘xv⌘z  n = 1, 2, · · · , (2.19)andDc(E) = Vˆd3q(2⇡)3(E  Ecq) =LxLz2⇡l2bLy⇡~|v⌘y |XnsE2E2  2n| eb~ v⌘xv⌘z |,(2.20)where lb =p|~/eb| is the magnetic length of the pseudo-magnetic field. Itis worth noting that Eq. 2.20 agrees with the numerically calculated DOSperfectly for all energies up to ELif , unlike Eq. 2.18 which begins to deviatefrom the numerics at ⇠ 12meV. We attribute this interesting result to thefact that strain couples as the chiral vector potential only to Weyl fermions.If we write the full Hamiltonian as h(p) = hW (p) + h(p) where hW isstrictly linear in momentum p and h is the correction resulting from thelattice effects, then strain causes p ! p + eA only in hW but does not tothe leading order affect h. On the other hand, the vector potential A ofmagnetic field B affects hW and h in the same way.Before we leave this section, we further test the equivalence of the strain-induced pseudo-magnetic field b and the ordinary magnetic field B. Wepropose to apply external magnetic field of fixed strength and then slowlyturn on strain (or vice versa, whichever is more convenient in a particularexperimental design). We find this will result in splitting of the first peakin DOS as illustrated in Fig. 2.4. This happens because the two Weyl coneswill feel different effective magnetic fields. Due to the chiral nature of thegauge field Eq. 2.12, the strain-induced pseudo-magnetic field takes oppositevalues at different Weyl cones (Eq. 2.13), while the ordinary magnetic fieldis uniform in the crystal. Consequently, the two Weyl cones feel effectivemagnetic field B+b and Bb, respectively, which results in two independentsequences of peaks in DOS. Observation of the splitting would prove theidentical nature of the gauge and external magnetic fields in each of theWeyl cones.2.4 Longitudinal electric conductivityIn Section 2.3, we have confirmed that the necessary QO ingredient – Lan-dau levels occur in the presence of strain. In Section 2.2, we have demon-strated that the strain-induced pseudo-magnetic field has a substantial win-242.4. Longitudinal electric conductivity0 5 10 15 20 25 300.000.020.040.060.080.105 6 7 8 90.0000.015  DOSE[meV] DOS E[meV]Figure 2.4: Normalized density of states for both fields present, B = 1T andb = 0.0184T. Each of the DOS peaks due to ordinary magnetic field splitsdue to torsion thus proving the equivalence of the external and gauge fields.Inset gives closer view of the first two peaks.dow. These findings make the observation of QOs possible with strain-induced pseudo-magnetic field only. Specifically, when continuously tuningthe strain-induced pseudo-magnetic field b, nearly all observables (e.g., elec-tric conductivity, thermal conductivity, magnetization) exhibit oscillatingbehaviors periodic in 1/b because these observables are dominated by theDOS at Fermi surface, which is periodic in 1/b as elucidated in Eq. 2.20.From the experimental point of view, the longitudinal electric conductivityQO may be the easiest to measure. Therefore, in the present section, wewill derive the longitudinal electric conductivity yy as a function of pseudo-magnetic field and show it indeed exhibits the Shubnikov-de Haas (SdH)oscillation.We use Boltzmann equation approach [81] to calculate the longitudinalelectric conductivity. Without loss of generality, we will assume positivechemical potential µ > 0. The conductivity of the n-th band readsn(µ) = e2LxLz2⇡l2bLyˆdky2⇡⌧n(E2n(ky))(vn(ky))2✓@f(E  µ)@E◆En(ky),(2.21)where f(✏) is the Fermi function and ⌧n(E2n(ky)) is the relaxation time. The252.4. Longitudinal electric conductivityvelocity can be derived from Eq. 2.19 asvn(ky) =1~@En@ky= v⌘ykyqk2y + 2n eb~v⌘xv⌘z(v⌘y )2 . (2.22)We assume zero temperature, angle-independent relaxation time, and sub-stitute the dispersion relation Eq. 2.19 to obtainn(µ) =e2⌧nv⌘y⇡~LxLz2⇡l2bLysµ2  2n| eb~ ~2v⌘xv⌘z |µ2n = 1, 2, · · · , nmax, (2.23)where nmax = [µ2/2| eb~ ~2v⌘xv⌘z |]6. Physically, this indicates only those Dirac-Landau levels that traverse chemical potential contribute to the conductivity.The total longitudinal electric conductivity is thenyy(µ) =e2v⌘y⇡~LxLz2⇡l2bLynmaxXn=0⌧n(µ)sµ2  2n| eb~ ~2v⌘xv⌘z |µ2. (2.24)Finally, we estimate the relaxation time in the lowest order Born approxi-mation [119]1⌧=2⇡~ D(µ)nimpC, (2.25)where D(µ) is the density of states at the Fermi level and nimp is the im-purity concentration. Constant C depends on the details of scattering fromimpurities. Thus the final formula we use for the conductivity computationin Fig. 2.4 isyy(µ) =e2v⌘y⇡~LxLz2⇡l2bLy12⇡~ nimpC1D(µ)nmaxXn=0sµ2  2n| eb~ ~2v⌘xv⌘z |µ2= yy(0)nmaxXn=0sµ2  2n| eb~ ~2v⌘xv⌘z |µ2,nmaxXn=0sµ2µ2  2n| eb~ ~2v⌘xv⌘z |, (2.26)where we have used Eq. 2.20 and we have defined zero strain electric con-ductivity asyy(0) =e2(v⌘y)22⇡~ nimpC. (2.27)6The notation [x] means an integer that is no greater than x.262.4. Longitudinal electric conductivityIt is easy to see that when continuously tuning the external strain, the in-duced pseudo-magnetic field scans, and the Landau levels successively falloff the chemical potential µ. Every time a new Landau level hits the chem-ical potential, we have condition µ2 = 2n| eb~ ~2v⌘xv⌘z |, yy(µ) hits a valleywhile the corresponding DOS Dc(µ) hits a peak, according to Eq. 2.26 andEq. 2.20, respectively. Both of them exhibit quantum oscillation periodic in1/b. The same is true for the ordinary magnetic field as long as b is replacedby B.Numerically, we can use Eq. 2.26 to calculate the longitudinal electricconductivity but input the actual numerically calculated velocities and ener-gies into it. In order to obtain satisfying resolution, we apply the tetrahedronmethod (see Appendix F for details) when calculating DOS and longitudinalelectric conductivity. As illustrated in Fig. 2.5, both DOS and longitudinalelectric conductivity show oscillations periodic in 1/B and 1/b, at chemicalpotential µ = 10meV. The latter realizes zero-field QOs, which is the keyresult of the present chapter. The period of the strain-induced QO 0.329T1in Fig. 2.5 is in a good agreement with the period 0.324T1 expected on thebasis of the Lifshitz-Onsager theory and the period 0.336T1 obtained fromEq. 2.19. Small irregularities that appear at low fields can be attributed tothe finite size effects as the Landau level spacing becomes comparable to thesub-band spacing, e.g., Fig. 2.3(a). We further verify that similar oscillationsoccur at other energies below the Lifshitz transition ELif ⇠ 20meV.Above the Lifshitz transition, at chemical potential µ = 28meV, wesee the QOs still happen for the strain-induced pseudo-magnetic field inFig. 2.6(a). However, the period of such QOs is different from that of theQOs due to the ordinary magnetic field by a factor of 2 approximately.The physics behind this can be clarified by considering the extremal cross-sectional area of Fermi surface. For the ordinary magnetic field B, abovethe Lifshitz transition, the two Weyl cones merge into a larger Fermi sur-face so that the effective area of Fermi surface is approximately doubled asillustrated by the green contour in Fig. 2.6(b). On the other hand, straincouples only to the linear part of the Hamiltonian as a gauge field, there-fore only the oscillations around each of the Weyl points are possible asillustrated by the red contours in Fig. 2.6(b), on which the electron in thepseudo-magnetic field travels clockwise around one of the Weyl points andcounterclockwise around the other. The precise nature of the correspond-ing quasi-classical trajectories above the Lifshitz transition is therefore aninteresting open question which we leave for further study. We speculatethat it includes tunneling between the opposite points of the Fermi surfaceas depicted in Fig. 2.6(b). Such trajectories would define an extremal area272.4. Longitudinal electric conductivity0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.01/B, 1/b [T-1]0.000.010.02DOS(10meV)0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.01/B, 1/b [T-1]0.00.20.40.60.8σyy(b)/σyy(0)strainmagnetic fieldcontinuumFigure 2.5: Strain-induced quantum oscillations. Top panel shows oscilla-tions in DOS at µ = 10meV as a function of inverse strain strength expressedas 1/b. For comparison ordinary magnetic oscillations are displayed, as wellas the result of the bulk continuum theory Eq. 2.18. Crosses indicate peak po-sitions expected based on the Lifshitz-Onsager theory. Bottom panel showsSdH oscillations in conductivity yy assuming chemical potential µ = 10meV.To simulate the effect of disorder all data are broadened by convolving inenergy with a Lorentzian with width  = 0.25meV. The same geometry andparameters are used as in Fig. 2.3.consistent with our numerical results.To better simulate QOs for realistic Weyl semimetals, the curves (Eq. 2.20and Eq. 2.26) shown in Fig. 2.5 and Fig. 2.6(a) have been convolved in energywith a Lorentzian of width  = 0.25meV, which corresponds to Landau levelbroadening due to scattering from phonons and impurities at finite temper-ature. Such broadening may be derived by the Born approximation [120]and exhibitspB dependence for Weyl semimetals [121]. Experimentally,for Dirac semimetal Cd3As2 to which our simulation is closely relevant, thebroadening can be obtained by phenomenologically fitting to experimentaldata such as the zero magnetic field optical conductivity [122] and SdH oscil-lations [123]. Specifically, the former suggests a 15meV broadening at T=10Kwhile the later exhibits a broadening within 0.02⇠20meV at T=2.5K. Fromthe width of differential conductance peaks in STM measurements [44, 124],the Landau level broadening of several milli-electronvolts can be approxi-mated. For the sake of transparency, we have chosen a B-independent value282.5. Summary = 0.25meV for the Lorentzian width, which should be experimentally avail-able in sufficiently clean sample of Cd3As2 at low temperature.0.10 0.15 0.20 0.250.20.40.60.040.080.12 σyy(B)/σyy(0)1/B,1/b [T-1] strain magnetic field continuum  DOS(28meV)a)b)Figure 2.6: Quantum oscillations above Lifshitz transition. (a) QOs abovethe Lifshitz transition due to ordinary magnetic field and due to strain-induced pseudo-magnetic field. Period difference by approximately a fac-tor of 2 is seen. The low-energy analytic theory does not apply anymore,as expected. (b) Corresponding hypothesized quasi-classical trajectories ofelectrons in the Brillouin zone. Green for By field and red for by field.2.5 SummaryIn this chapter, we have seen that quantum oscillations occur in the absenceof magnetic fields as long as a circular bend deformation is applied and tunedcontinuously. Our argument is numerically supported by studying a 2-bandtoy model of Weyl semimetal – 12 -Cd3As2, which can be understood as one ofthe spin sectors of real Dirac semimetal Cd3As2. As discussed in Section 2.1,the other spin sector contributes to the band structure identically and ✏konly shifts bands trivially; we thus argue that the full Cd3As2 HamiltonianEq. 2.1 also exhibits strain-induced Dirac-Landau levels (Appendix A). Sinceall the numerical simulation presented in this chapter is conducted usingthe parameters of Cd3As2, we argue that Dirac semimetal Cd3As2 may bean ideal candidate material to implement strain-induced QOs. Moreover,experimental studies [42–45] indicate that the linear dispersion in Cd3As2extends over a much wider range of energies than theoretically anticipated[41] with Lifshitz transition occurring near 200 meV. All these findings render292.5. SummaryCd3As2 even more promising for the observation of zero-field QOs.30Chapter 3Dirac-Landau levels in WeylsuperconductorsQuantum oscillations [112] furnish an essential experimental tool for mea-suring the Fermi surface of metals. They also help to understand electronicstructures of the recently discovered topological insulators [125–129] andtopological Dirac and Weyl semimetals [40, 73, 111, 130]. However, prob-ing superconductors by the quantum oscillation technique has been thoughtimpossible because such measurements require strong magnetic fields whichare either expelled from the SC due to the Meissner effect or render the ma-terial normal. Type-II superconductors allow the field to penetrate but formthe Abrikosov vortex state, whose quasiparticle eigenstates are known to beBloch waves rather than Landau levels [131–133].Quantum oscillations in resistivity [98–100], Hall coefficient [101], ther-mal conductivity [102], and torque [103] have already been observed in under-doped cuprates when magnetic field suppresses superconductivity. Quantumoscillations with 1/pB periodicity have also been predicted to appear in vor-tex lattice [105] and vortex liquid states [104] in cuprates and are observedin 2H-NbSe2 [106]. However, reports on conventional quantum oscillationsperiodic in 1/B in the superconducting state are lacking presumably due tothe reasons listed above.Inspired by the work presented in Chapter 2, we hypothesize that suchdifficulty can be overcome by using the strain-induced pseudo-magnetic fieldwhose quantum oscillations rely on continuously tuned lattice deformationrather than scanning magnetic field [134]. It has been proved that the strain-induced pseudo-magnetic field is not subject to Meissner effect or vortexstates [135]. Such unique feature makes the pseudo-magnetic field com-patible with superconductivity. Therefore, pseudo-magnetic field quantumoscillations periodic in 1/b are in principle available in the superconductingregime. Since pseudo-magnetic field is a common feature of Dirac materi-als such as graphene [87–89] and Dirac/Weyl semimetals [92–97, 134], thebest candidate materials to exhibit pseudo-magnetic field may be the 2D d-wave superconductors and the 3D Dirac/Weyl superconductors [18, 19]. The313.1. Model of Weyl superconductorsd-wave superconductors have been verified to host strain-induced pseudo-magnetic fields [136, 137]. Whether similar effect could be obtained in theDirac/Weyl superconductors remains as an open question and is the centralconcern of the present chapter.In this chapter, through a combination of analytical calculations andnumerical simulations, we demonstrate that quantum oscillations indeed oc-cur in Weyl superconductors under certain types of elastic deformations atzero magnetic field. Remarkably, these quantum oscillations arise due to theformation of Landau levels comprised of charge neutral Bogoliubov quasipar-ticles deep in the superconducting state. The chapter is organized as follows.In Section 3.1, we formulate a model of a Weyl superconductor and discussits spectrum and phase diagram. In Section 3.2, we incorporate strain toour Hamiltonian and show that to the leading order it produces a pseudo-magnetic field in the low-energy sector. In Section 3.3, we derive the bandstructure of a Weyl superconductor away from the neutrality point and showthat this is necessary to obtain a non-zero Fermi surface. In Section 3.4, weshow that the strain-induced pseudo-magnetic field can give rise to quantumoscillations in density of states (DOS) and longitudinal thermal conductivity.Section 3.5 concludes the chapter by discussing the experimental feasibilityin candidate materials and outlines various potentially interesting directions.3.1 Model of Weyl superconductorsTo implement strain-induced quantum oscillation, we first require a Weylsuperconductor. We thus employ the multilayer model invented by Mengand Balents [18] as illustrated in Fig. 3.1. The model comprises alternatingtopological insulator (TI) and s-wave superconductor (SC) layers stackedalong the z direction. For the TI layers, for simplicity, only the surfacestates are considered. In the following we modify the Meng-Balents modelslightly by adding anisotropy to the Zeeman mass term, which will allow usregularize the Hamiltonian on the tight binding lattice without adding extraWeyl points near the corners of the Brillouin zone.The Hamiltonian of such a TI-SC multilayer system readsH = HTI +HSC +Htd +Hts , (3.1)whereHTI =Xk?,z †k?z[~vFz(zˆ ⇥ s) · k? + (mm0a2k2?)sz] k?z,323.1. Model of Weyl superconductorsxzy(a)(b)FFigure 3.1: Schematic plot for (a) undeformed and (b) bent TI-SC multilayerWeyl superconductor. The alternating TI and SC layers are omitted in thebulk but explicitly drawn at ends to illustrate that there are integer numberof unit cells comprised of one TI layer and one SC layer.HSC =Xk?,z(c†k?z,1"c†k?z,1# +c†k?z,2"c†k?z,2#) + h.c.,Htd =Xk?,z⇣12td †k?z+1+ k?z +12td †k?z1 k?z⌘,Hts =Xk?,z †k?ztsx k?z.The basis  k?z = (ck?z,1", ck?z,1#, ck?z,2", ck?z,2#)T is written in terms ofannihilation operators ck?z,s for electrons located in the z-th unit cell withan in-plane momentum k? = (kx, ky) and spin projection sz =", #. “Sublat-tice” labels z = 1, 2 specify the TI-SC interface in the single unit cell. Paulimatrices s and  act in spin and sublattice space, respectively. Physically,HTI, HSC, Htd , and Hts can be interpreted as describing the Zeeman gappedtopological insulator surface states, proximity-induced pairing, hopping be-tween adjacent unit cells, and hopping within a single unit cell, respectively.The m0 term in HTI represents the above mentioned modification of theMeng-Balents model (it is easy to check that it has no significant effect atsmall k as long as m0 is chosen appropriately small).As written Hamiltonian Eq. 3.1 is k · p in x-y plane and tight-bindingin z direction. It will be useful to apply lattice regularization. We use a333.1. Model of Weyl superconductorssimple cubic lattice with lattice constant a and replace kx,y ! 1a sin akx,y andk2x,y ! 2a2 (1 cos akx,y). After partial Fourier transform in the z direction,the Hamiltonian can be written asH =12Xk †kHk k, (3.2)with  k = (ck,1", ck,1#, ck,2", ck,2#, c†k,1", c†k,1#, c†k,2", c†k,2#)T andHk = (m 4m0 + 2m0 cos kxa+ 2m0 cos kya)sz⌧ z+td sin kzay⌧ z + (ts + td cos kza)x⌧ z +~vFasin kyasxz ~vFasin kxasyz⌧ z  Imsy⌧x  Resy⌧y. (3.3)The spectrum of Hk reads✏2k,± =~2v2Fa2(sin2 kxa+ sin2 kya) +⇣m 4m0 + 2m0 cos kxa+ 2m0 cos kya±qt2s + t2d + 2tstd cos kza+ ||2⌘2. (3.4)We plot the spectrum in Fig. 3.2 for a system with l¯y = 500 layers and openboundary conditions along the y-direction and periodic boundary conditionsin the other two dimensions. We set  = 1 and measure all other parametersin terms of . We take m = 10.26, m0 = 2.53, ~vF /a = 1, td = 4.79,ts = 14.86, and the lattice constant is set to be a = 6A˚. These values willalso be used in our numerical simulations unless other values are specified.Without loss of generality, we have assumed m,m0 > 0 in the followingdiscussion. Thus, the sector ✏k,+ is fully gapped while ✏k, can be gaplesswhen p(|ts| |td|)2 + ||2 < m <p(|ts|+ |td|)2 + ||2. (3.5)If Eq. 3.5 holds, non-degenerate quasiparticle bands exhibit a pair of nodesat kW = (0, 0, ⌘Q) withQa = cos1✓m2  t2s  t2d  ||22tstd◆, (3.6)and ⌘ = ±1. As expected, the system is a Weyl superconductor.343.1. Model of Weyl superconductors-0.4 -0.2 0.0 0.2 0.4-0.30-0.150.000.150.30-0.4 -0.2 0.0 0.2 0.4-0.30-0.150.000.150.30  Ekz(b)  Ekz(a)Figure 3.2: Band structure of a Weyl superconductor plotted (a) along kzaxis with kx = 0 and (b) along kx axis with kz = 0. Periodic boundaryconditions are applied in x, z directions while the system is chosen to havel¯y = 500 layers in y direction. The parameters are listed below Eq. 3.4.To understand the low-energy physics better, we introduce 2⇥2 auxiliarymatricesDk,± =~vFasin kyax  ~vFasin kxay + (Mkz ,±  4m0+ 2m0 cos kxa+ 2m0 cos kya)z, (3.7)where  are Pauli matrices in transformed Nambu space andMkz ,± = m±qt2s + t2d + 2tstd cos kza+ ||2.As ✏k,± is also the dispersion for Dk,±, the low-energy physics of Eq. 3.3may be understood by studying Dk, because there always exists a unitarytransformation U that can block diagonalize HkU1HkU = diag(Dk,, Dk,, Dk,+, Dk,+).For fixed kz value, we rewrite Dk, in a k · p fashion,Dk·p =✓m0a2(k2x + k2y) + µeff i~vF (kx  iky)i~vF (kx + iky) m0a2(k2x + k2y) µeff◆. (3.8)We notice that Dk·p can be regarded as describing a px+ ipy superconductorwith an effective chemical potential µeff = Mkz ,. It is characterized by353.2. Strain-induced pseudo-magnetic fieldChern numberC =12[sgn(µeff) + sgn(m0)]. (3.9)If Eq. 3.5 holds, for those kz’s that satisfy µeff = Mkz , > 0, this SC is inthe weak pairing phase with Chern number C = 1. As a result, for each ofsuch kz’s, there exist counter propagating chiral Majorana states on a pairof boundaries open in the y-direction. Therefore, the edge states of Eq. (3.1)are Majorana-Fermi arcs as illustrated in Fig. 3.3(a).To understand the phases of this model, consider a value of m that satis-fies Eq. 3.5 with fixed. Now increase it such thatm >p(|ts|+ |td|)2 + ||2;according to Eq. 3.4, our Weyl superconductor will be gapped into a topo-logically superconducting phase, whose spectrum is shown in Fig. 3.3(b). Itexhibits a surface mode because the Chern number is still C = 1 for all kz.On the other hand, if m is decreased to m <p(|ts| |td|)2 + ||2, the sys-tem enters a trivially superconducting phase with no edge modes, as shownin Fig. 3.3(c). If m is fixed to a value obeying Eq. 3.5, but || is graduallyincreased, eventually, m is overwhelmed byp(|ts| |td|)2 + ||2 and thesystem becomes a trivial superconductor, as indicated by Fig. 3.3(d).Based on the above considerations, we plot the global phase diagram ofour Weyl superconductor in ||m plane in Fig. 3.4. The phase boundariesare given by two hyperbolas,m2  ||2 = (|ts|+ |td|)2, (3.10)m2  ||2 = (|ts| |td|)2. (3.11)Above the upper bound Eq. 3.10, the multilayer is a topological superconduc-tor, which can be viewed as a stack of 2D px+ipy superconductors. These areknown to possess counter propagating chiral Majorana edge modes on a pairof parallel boundaries. Since switching off the superconductivity will give a3D quantum anomalous Hall (QAH) insulator [38], the multilayer topologicalsuperconductor structure may be referred to as “3D QAH superconductor.”Below the lower bound Eq. 3.11, the multilayer is a trivial superconductorwhile between them it is a Weyl superconductor.3.2 Strain-induced pseudo-magnetic fieldIn Section 3.1, we studied electronic structure and phase diagram of multi-layer model of Weyl superconductor. In this section, we will understand howthe electronic structure is changed under generic strain.363.2. Strain-induced pseudo-magnetic field-0.4 -0.2 0.0 0.2 0.4-0.30-0.150.000.150.30-3 -2 -1 0 1 2 3-0.30-0.150.000.150.30-0.4 -0.2 0.0 0.2 0.4-0.30-0.150.000.150.30-0.4 -0.2 0.0 0.2 0.4-0.30-0.150.000.150.30(b)(d)(c) Ekz(a)  Ekz  Ekz  EkzFigure 3.3: Band structure of a Weyl superconductor with open boundaryconditions and l¯y = 150 layers along the y-direction. All panels are plottedalong kz-axis with kx = 0 and with parameters as in Fig. 3.2. (a) Weylsuperconductor phase for (m,) = (10.26, 1). A Fermi arc connecting twoWeyl points appears due to the chiral Majorana edge states of the effectivepx + ipy superconductors that emerge for fixed kz between the Weyl nodes.(b) Topological superconductor phase for (m,) = (19.82, 1). The increaseof m will separate two Weyl points and extend Fermi arc. When two Weylpoints meet at Brillouin zone boundary, they annihilate and open up a SCgap but leave behind the Fermi arc extended over the whole BZ. (c) Trivialsuperconductor phase for (m,) = (9.98, 1). The decrease of m makes twoWeyl points meet at Brillouin zone center and annihilation and leads tothe disappearance of the Fermi arc. (d) Trivial superconductor phase with(m,) = (10.26, 2.56). The increase of  is equivalent to decrease of m andWeyl points again annihilate at the BZ center.When elastic strain distorts the lattice, the chemical bonds are stretchedand compressed. Orbital orientations are also rotated, making the symmetry-373.2. Strain-induced pseudo-magnetic field0 500 1000 1500 2000 2500 3000 3500 400005001000150020002500300035004000(a)(b)(c)(d)Weyl SCTSCtrivial SCm||Figure 3.4: Phase diagram of the Weyl superconductor described by Hamil-tonian Eq. 3.3 in terms of (m, ||) with labels (a)-(d) correspond to spectrashown in Fig. 3.3(a)-(d). The two black curves mark the phase boundariesgiven in Eq. 3.10 and Eq. 3.11. The dotted line indicates the asymptote forthe two phase boundaries.prohibited hoppings now non-zero. For our purposes, the most importantmodification comes from the replacement of hopping amplitudes along z-direction [94, 109, 134]td± ! td(1 u33)±  i~vFa u31syz + i~vFau32sxzts ! ts(1u33), (3.12)with the strain tensor uij = 12(@iuj + @jui), where uj is the j-th componentof the displacement vector u. Under such hopping parameter substitution,the Hamiltonian in Eq. 3.3 is changed toH˜k = Hk + Hk, (3.13)where the correction due to strain isHk = (tsu33 + tdu33 cos kza)x⌧ z  tdu33y⌧ z sin kza ~vFau31syz⌧ z sin kza+~vFau32sxz sin kza. (3.14)To understand the effect of strain on the low-energy physics we consider H˜kin the vicinity of Weyl points kW = (0, 0, ⌘Q)H˜kW+q = HkW + hq + HkW +O(q2) +O(q)Q(uij), (3.15)383.2. Strain-induced pseudo-magnetic fieldwhere the linearized Hamiltonian hq readshq = q ·rkHk|k=kW = ~vF qxsyz⌧ z + ~vF qysxz ⌘tdqza sinQax⌧ z + tdqza cosQay⌧ z. (3.16)And HkW represents the Bloch Hamiltonian Eq. 3.3 at Weyl points. Accord-ing to Eq. 3.4, HkW has eigenvalues {2m, 2m, 0, 0, 0, 0,2m,2m} and thusencodes both low-energy Weyl points and high-energy gapped sector. Toextract the low-energy physics only, we assume real order parameter  2 Rand solve for the eigenstates of HkW associated with Weyl points as|1i = 1p2⇣z1  iz2m, 0,1, 0, 0, 0, 0, m⌘T, (3.17)|2i = 1p2⇣ m, 0, 0, 0, 0, 1,z1 + iz2m⌘T, (3.18)|3i = 1p2⇣0,z1 + iz2m, 0,1, 0, 0, m, 0⌘T, (3.19)|4i = 1p2⇣0,m, 0, 0, 1, 0,z1  iz2m, 0⌘T. (3.20)where we have used following quantitiesz1 = ts + td cos kza, (3.21)z2 = td sin kza. (3.22)We then project hq+HkW onto the four-dimensional Hilbert space spannedby |i=1,2,3,4i. We get(hq + HkW ) =0BB@ z1z˜1+z2z˜2m 0 ix˜ y˜ 00  z1z˜1+z2z˜2m 0 ix˜+ y˜ix˜ y˜ 0 z1z˜1+z2z˜2m 00 ix˜+ y˜ 0 z1z˜1+z2z˜2m1CCA ,(3.23)where, to keep our derivation transparent, we further defined the followingquantitiesx˜ =~vFa(qxa+ ⌘u31 sinQa), (3.24)y˜ =~vFa(qya+ ⌘u32 sinQa), (3.25)z˜1 = ⌘td sinQa⇣qza+ ⌘u33ts + td cosQatd sinQa⌘, (3.26)z˜2 = td cosQa(qza ⌘u33 tanQa). (3.27)393.2. Strain-induced pseudo-magnetic fieldThe projected 4⇥4 matrix Hamiltonian can be written in terms of standardDirac matrices which we express as a tensor product of Pauli matrices ↵ and as(hq + HkW ) =~vFa(qxa+ ⌘u31 sinQa)↵zy ~vFa(qya+ ⌘u32 sinQa)↵zx+⌘tstd sin aQm✓qza+ ⌘u33m2 2tstd sin aQ◆z. (3.28)From here, one can read off the strain-induced gauge fieldA = ⌘~ea⇣u31 sinQa, u32 sinQa, u33m2 2tstd sinQa⌘. (3.29)Clearly, the vector A can be understood as the gauge potential of a strain-induced chiral magnetic field. In most cases we expect Qa⌧ 1. In this limitthe z component of A given in Eq. 3.29 scales as 1/aQ but x(y) componentsscale as aQ. Thus only Az / u33 will be considered in the following.According to Fig. 3.1, we may characterize the bend deformation by ansmall angle ✓ = a/⇢ where ⇢ is the radius associated with the circular bend.The lattice constant of the outermost y direction layer is then a + a witha = 12 l¯ya✓. Here l¯y is the number of layers in y direction. Thus⇢ =l¯ya2a/a=l¯ya", (3.30)with " = 2a/a being the bending parameter used in the numerics. Now ifwe consider a generic y direction layer, its lattice constant will change bya(y) = (y l¯ya/2)✓. Then for a point with coordinate z on this layer, its zdirection displacement is u3 = za(y  l¯ya/2)2al¯ya . Thus,u33 =@u3@z= (y  l¯ya/2) 2aa2 l¯y= (y  l¯ya/2) "l¯ya. (3.31)Therefore, we expect a pseudo-magnetic field7b = @yAzxˆ = ⌘ ~ea2m2  ||2tstd sinQa"l¯yxˆ. (3.32)7Unlike the applied external magnetic field B, which can either be expelled from thesuperconductor bulk by the formation of surface currents (i.e., Meissner effect) or broughtinto the superconductor bulk by the formation of supercurrent vortices, the pseudo-magnetic field b couples to Weyl cones deep in the superconductor bulk and vanishesoutside the superconductor. It may be interpreted as a modular field [138] that can nei-ther propagate nor be screened.403.3. Weyl superconductors with chemical potentialAccording to Eq. 1.14, such pseudo-magnetic field will give rise to Dirac-Landau levels at energies✏˜n(k) = ±r~2v2xk2x + 2neb~ ~vy~vz, (3.33)for all integers n 6= 0 and ✏˜0(kx) = ~vFkx as the zeroth Landau levels forboth valleys. In view of Eq. 3.32, we get✏˜n(k) = ±s~2v2xk2x + 2n"l¯ym2  ||2m~vya. (3.34)We have numerically checked Eq. 3.34 by applying hopping substitutionsEq. 3.12 in the multilayer Hamiltonian Eq. 3.1 with l¯y = 150, as summarizedin Fig. 3.5(b). Indeed we observe that the Dirac-Landau levels in Eq. 3.33capture the features of the low-energy spectrum of the Weyl superconductormultilayer. For comparison we also plot the spectrum and DOS for theunstrained system and show the results in Fig. 3.5(a,c).For the sake of completeness we in addition calculate the spectrum ofour model Weyl superconductor in the presence of the magnetic field B k zˆand the Abrikosov vortex lattice. This is summarized in Appendix B. Wefind that all bands become completely flat Landau levels in the x-y plane.The zeroth Landau level, which is associated with Weyl nodes before B isswitched on, is still linearly dispersive along the z-direction in the vicinity ofnodes. In contrast, it is well known that the magnetic field does not lead toflat Landau levels in dx2y2 superconductors. This is because the spatiallyvarying supercurrent in the vortex lattice strongly scatters the Bogoliubovquasiparticles [131]. The difference between the dx2y2 and 3D Weyl super-conductors has been recently elucidated in Ref. [139] with which our resultsare in accord. In short, the zeroth Landau level cannot be scattered by vor-tices due to the protection of Weyl node chirality, which is a topologicallynontrivial and unique feature of the Weyl superconductor.3.3 Weyl superconductors with chemical potentialIn Section 3.2, we found that a bend deformation results in Dirac-Landaulevels of Bogoliubov quasiparticles, which unlike those in Weyl semimetals,are charge neutral on average. Therefore, the Shubnikov-de Haas quantumoscillation discussed in Section 2.4 cannot be observed in Weyl superconduc-tors. But Bogoliubov quasiparticles do carry heat, making thermal transportmeasurements, such as the thermal Hall effect, possible [140–142].413.3. Weyl superconductors with chemical potential-0.4 -0.2 0.0 0.2 0.4-0.30-0.150.000.150.30-0.4 -0.2 0.0 0.2 0.4-0.30-0.150.000.150.300.00 0.06 0.12 0.18 0.24 0.300.000.020.040.060.080.00 0.06 0.12 0.18 0.24 0.300.000.020.040.060.08  Ek  Ek lattice continuum(d)(c)(b)  DOSE lattice continuum(a) lattice continuum  DOSEkzXZkx X XZ ZFigure 3.5: Energy spectra and DOS for our Weyl superconductor with openboundaries and l¯y = 150 along the y direction and periodic along x and z. (a)The spectrum of undeformed system; the flat band at zero energy is the Fermiarc. (b) The spectrum of a bent Weyl superconductor as shown in Fig. 3.1(b)with " = 8% corresponding to a pseudo-magnetic field b = 10.45T. For both(a) and (b) the spectrum is plotted along X--Z as shown in the inset. Forcomparison, energy levels of Eq. 3.34 are overlain as black dots. (c) DOSof the unstrained sample (blue curve) compared to the ideal ⇠ E2 DOSexpected for a massless Dirac fermion in continuum (black parabola). (d)DOS of the strained system (red curve) compared to DOS calculated forideal Dirac-Landau levels with b = 10.45T.Our concern will be calculating the thermal conductivity quantum oscil-lation of Weyl superconductors. However, in our analysis above, the chemicalpotential µ of TI layers was assumed to lie at the Weyl points (µ = 0) andthe Fermi surface is then comprised of two separate points carrying vanishingquasiparticle DOS. According to Lifshitz-Onsager relation [112], the associ-ated quantum oscillation has infinite period thus cannot be observed. In thissection, we will attack this difficulty by tuning µ away from Weyl points. Wewill show that this results in a finite size Fermi surface compatible to the423.3. Weyl superconductors with chemical potentialthermal conductivity quantum oscillation.Away from Weyl points, we have nonzero chemical potential µ 6= 0 andEq. 3.3 will be modified by an extra term Hk = µ⌧ z. The modifiedHamiltonian is thenHk  µ⌧ z = H0 + V, (3.35)whereH0 = m0sz⌧ z + z1x⌧ z + z2y⌧ z sy⌧y, (3.36)V = µ⌧ z + ysxz  xsyz⌧ z. (3.37)Again, for simplicity we assume real  and define the following quantitiesm0 = m 4m0 + 2m0 cos kxa+ 2m0 cos kya, (3.38)x =~vFasin kx, (3.39)y =~vFasin ky. (3.40)Based on the parameter values we have chosen,  is of the same order asz2, but is one order of magnitude smaller than z1 and m0. We set chemicalpotential µ to be an order of magnitude smaller than . Also note that inthe vicinity of Weyl points, both x and y are sufficiently small. Therefore,we may treat V as a perturbation to H0, whose low-energy eigenvectors canbe easily resolved as|D1,1i =1p2⇣ z1  iz2pz21 + z22 +2, 0,1, 0, 0, 0, 0, pz21 + z22 +2⌘T, (3.41)|D1,2i =1p2⇣ pz21 + z22 +2, 0, 0, 0, 0, 1, 0,z1 + iz2pz21 + z22 +2⌘T, (3.42)|D2,1i =1p2⇣0,z1 + iz2pz21 + z22 +2, 0,1, 0, 0, pz21 + z22 +2, 0⌘T, (3.43)|D2,2i =1p2⇣0,pz21 + z22 +2, 0, 0, 1, 0,z1  iz2pz21 + z22 +2, 0⌘T. (3.44)These correspond to the degenerate subspace D1 with eigenvalueE(0)D1,1(2) = m0 qz21 + z22 +2, (3.45)and D2 with eigenvalueE(0)D2,1(2) =qz21 + z22 +2 m0. (3.46)433.3. Weyl superconductors with chemical potentialWe then project the perturbation V to the degenerate subspaces D1 and D2,respectively. In a compact form, it readsVDi =✓hDi,1|V |Di,1i hDi,1|V |Di,2ihDi,2|V |Di,1i hDi,2|V |Di,2i◆, (3.47)where i = 1, 2. Individually, we can write VDi in terms of the Pauli matrix⌫VD1 = dx⌫x + dy⌫y + dz⌫z,VD2 = dx⌫x  dy⌫y + dz⌫z,(3.48)wheredx =z1z21 + z22 +2µ, (3.49)dy =  z2z21 + z22 +2µ, (3.50)dz =  z21 + z22z21 + z22 +2µ. (3.51)Matrices VD1 and VD2 can be diagonalized through unitary transformationsU1Di VDiUDi = diag(d,d) i = 1, 2, (3.52)whered =sz21 + z22z21 + z22 +2µ ⇡ µ, (3.53)because for our purpose 2 ⌧ z21 + z22 . The transformation matrices areUD1 =0B@qd+dz2ddxidypd2x+d2yqddz2dqddz2dqd+dz2ddx+idypd2x+d2y1CA , (3.54)UD2 =0B@qd+dz2ddx+idypd2x+d2yqddz2dqddz2dqd+dz2ddxidypd2x+d2y1CA . (3.55)We can immediately write down the first order correction to energyE(1)D1,1(2) = E(1)D2,1(2)= ±d ⇡ ±µ. (3.56)443.3. Weyl superconductors with chemical potentialBecause the 2-fold degeneracy is lifted, the zeroth order eigenvectors are nowuniquely determined(|˜D1,1i , |˜D1,2i) = (|D1,1i , |D1,2i)UD1 , (3.57)(|˜D2,1i , |˜D2,2i) = (|D2,1i , |D2,2i)UD2 . (3.58)We may now calculate the second order correction to the energyE(2)D1,1(2) =X↵2D2| h˜↵|V |˜D1,1(2)i |2ED1  E↵=x2 + y22(m0 pz21 + z22 +2), (3.59)E(2)D2,1(2) =X↵2D1| h˜↵|V |˜D2,1(2)i |2ED2  E↵=  x2 + y22(m0 pz21 + z22 +2), (3.60)where we ignore the contribution from high-energy sector, if any. This isbecause for high energies E↵ = ±(m0+pz21 + z22 +2) the denominator inthe second order correction is either ±2m0 or ±2pz21 + z22 +2, whose mag-nitude is much larger than that of ±2(m0pz21 + z22 +2) and thus are lessimportant. Combining all the corrections, we can estimate the quasiparticleenergy at nozero µ asED1,1(2) ⇡rx2 + y2 + (m0 qz21 + z22 +2)2 ± µ, (3.61)ED2,1(2) ⇡ rx2 + y2 + (m0 qz21 + z22 +2)2 ± µ. (3.62)We observe that to the leading order the spectrum of Weyl superconductormultilayer is now biased. The original 2-fold degeneracy in Eq. 3.4 has beenlifted. One copy of the spectrum moves up while the other copy moves down.As a result, the strain-induced pseudo-Landau levels will also be biased asEn(k) = ±r~2v2xk2x + 2neb~ ~vy~vz± µ, (3.63)as illustrated in Fig. 3.6(b). The corresponding DOS at the chemical poten-tial isD(0) =LyLz2⇡l2bXnLxX±ˆdkx2⇡(En(k))=V2⇡l2b1⇡~vxXnsµ2µ2  2n| eb~ ~vy~vz|, (3.64)453.4. Longitudinal thermal conductivitywhere lb =p~/eb is the magnetic length. It can be easily seen fromEq. 3.64 that when µ 6= 0, the D(0) exhibits oscillating behavior whenpseudo-magnetic field b scans. Therefore, the thermal conductivity will alsooscillate because the number of heat carriers varies periodically with respectto 1/b.-0.30 -0.15 0.00 0.15 0.30-0.30-0.150.000.150.30-0.30 -0.15 0.00 0.15 0.30-0.30-0.150.000.150.30  Ekx  Ekx(a) (b)Figure 3.6: Energy spectrum of the Weyl superconductor with the chemicalpotential of the TI layers tuned away from the surface Dirac points to µ =0.19. (a) Quasiparticle spectrum calculated from the lattice model Eq. 3.13.It is worth noting that only the left moving chiral mode is due to the Landauquantization while the other is a surface mode. (b) Quasiparticle spectrumpredicted by Eq. 3.63. To compare with the first panel, the chiral modes(orange lines) due to the surface states have been added manually.3.4 Longitudinal thermal conductivityIn Section 3.3, we have justified that the needed nonzero Fermi surface forthe observation of QOs can be obtained by tuning the chemical potential µaway from the Weyl points. In this section, we derive the quantum oscillationof longitudinal thermal conductivity xx.We compute the thermal conductivity using the Boltzmann equation ap-proach [143–145] and it reads =1TXnXkE2n(k)⌧n(En(k))vn(k)vn(k)✓ @f@En◆, (3.65)whereEn(k) is the quasiparticle energy given by Eq. 3.63, vn(k) = 1~rkEn(k)is the associated velocity, ⌧n(k) is the corresponding scattering time, and463.4. Longitudinal thermal conductivityf(En) = (eEn/kBT + 1)1 is Fermi-Dirac distribution function. For our pur-poses, it is useful to rewrite the thermal conductivity (Eq. 3.65) in the low-Tlimit. To achieve this, we first introduce the auxiliary tensor(✏) =XnXk⌧n(En(k))(✏ En(k))vn(k)vn(k), (3.66)which may be understood as a thermal analogue of the usual conductivitytensor. It is easy to see that =1Tˆ +11d✏✏2(✏)✓ @f@✏◆. (3.67)We further define an auxiliary functionK(✏) = ✏2(✏), (3.68)through which the thermal conductivity can be written as =1Tˆ +11d✏1Xs=11s!dsKd✏s0✏s✓ @f@✏◆. (3.69)Note that @f@✏ is an even function of ✏. Therefore, we only need to considereven s. The thermal conductivity is further simplified as =1T1Xs=1ˆ +11d✏(kBT )2s(2s)!✓✏kBT◆2s✓ @f@✏◆d2sKd✏2s0. (3.70)Define x = ✏kBT andas =ˆ +11dxx2s(2s)!✓ ddx1ex + 1◆=2(2s)ˆ +10dxx2s1ex + 1= 2⌘(2s) = 2(1 212s)⇣(2s), (3.71)where (s) =´ +10 dxxs1ex , ⌘(s) =´ +10 dxxs1ex+1 , and ⇣(s) =´ +10 dxxs1ex1are Gamma function, Dirichlet eta function, and Riemann zeta function,respectively. The thermal conductivity now reads =1T1Xs=12(1 212s)⇣(2s)d2sKd✏2s0(kBT )2s. (3.72)473.4. Longitudinal thermal conductivityFor low temperatures kBT ⌧ µ, we keep only the s = 1 term and use⇣(2) = ⇡26 to get =1T⇡2k2BT23(0) =⇡2k2BT3XnXk⌧n(En(k))(En(k))vn(k)vn(k).(3.73)This relation can be regarded as the Wiedemann-Franz law for Bogoliubovquasiparticles. By comparing to Eq. 2.26, we obtain the longitudinal thermalconductivityxx =⇡2k2BT3vx⇡~LyLz2⇡l2bLxXn⌧n(0)sµ2  2n| eb~ ~vy~vz|µ2. (3.74)The scattering time can be determined by Fermi’s golden rule1⌧n(En(k))=2⇡~Xn0Xk0| hn0k0|V (r)imp⌧ z|nki |2(En(k) En0(k0)), (3.75)where |nki is the eigenvector of chemical potential biased Weyl superconduc-tor under strain, characterized by the Hamiltonian Hµ⌧ z. As discussed inSection 3.3, when µ2 ⌧ 2 ⌧ t2s+ t2d+2tstd cos kza, we can use perturbativecalculation to write down the Schrödinger equations for Weyl superconductorwith TI layer chemical potential µ 6= 0 and µ = 0, respectively, as(H  µ⌧ z) |nk0i ⇡ ✏˜n(k) |nk0i± µ |nk0i , (3.76)H |nk0i = ✏˜n(k) |nk0i , (3.77)where ✏˜n(k) is determined by Eq. 3.34 and |nk0i is the exact eigenvector ofH and the zeroth order eigenvector of H  µ⌧ z. If apply hn0k00| to Eq. 3.76and Eq. 3.77 and subtract, we gethn0k00|⌧ z|nk0i ⇡ ± hn0k00|nk0i , (3.78)then we can approximate ⌧n(En(k)) by1⌧n(En(k))⇡ 2⇡~Xn0Xk0| hn0k00|V (r)imp|nk0i |2 ⇥ (En(k) En0(k0)).(3.79)The righthand side is the same as the scattering rate of a Weyl semimetal[134] with electronic structure characterized by En(k). Therefore, the scat-tering rate in a Weyl superconductor should also be the same, which leadsto⌧1n (0) =2⇡~ nimpCimpD(0), (3.80)483.4. Longitudinal thermal conductivity0.03 0.06 0.09 0.12 0.15 0.180.250.500.750.040.080.120.160.20 1/b [T-1] κ xx(b)/κxx(0) lattice continuum DOSFigure 3.7: Strain-induced quantum oscillation in a Weyl superconductor.The upper panel shows oscillations in DOS as a function of inverse strainstrength expressed as 1/b at zero-energy. The lower panel shows oscillationsin the longitudinal quasiparticle thermal conductivity xx. To simulate theeffect of disorder, all data are broadened by convolving in energy with aLorentzian with width ✏ = 1.67⇥ 103.where nimp and Cimp denote the impurity concentration and the scatteringpotential strength, respectively. The longitudinal thermal conductivity thenbecomesxx(b) = xx(0)Pnrµ22n| eb~ ~vy~vz |µ2Pnrµ2µ22n| eb~ ~vy~vz |, (3.81)with the zero-field thermal conductivityxx(0) =⇡2k2BT3v2x2⇡~ nimpCimp. (3.82)Fig. 3.7 shows our results for DOS and xx(b) calculated from the approx-imate analytical formulas Eqs. 3.64 and 3.81, and based on the full latticecalculation using Eqs. 3.64 and 3.73 with the assistance of the tetrahedronmethod (Appendix F). They agree well and exhibit pronounced quantumoscillations periodic in 1/b.We note that due to Landau quantization, quantum oscillations in thex direction are most pronounced, while in the other directions, quantum os-cillations are expected to be weaker. Based on our results for the electronicstructure in Fig. 3.5(b), the z direction drift velocity of Bogoliubov quasi-particles is nonzero only at the edges of bands (kza ⇠ 0.3). In contrast,493.5. Summarythe x direction drift velocity is nonzero for almost all momenta. There-fore, zz should be small and its quantum oscillation is weaker than thatin xx. In y-direction, quasi-particle wave functions are Gaussian-localizedwith the characteristic decay lengthp~vy/ebvz ⇠ lB and localization cen-ters 2⇡⌫l2B/Lz with ⌫ = 1, 2, · · · , Lz/a. The localization makes transportdifficult unless the localization center is pumped across the system when bvaries. Therefore, we do not expect pronounced quantum oscillations alongthe y-direction.It is also worth noting that quasiparticle thermal conductivity can beobscured in real materials by phonons because phonons also carry heat. Fortemperature T ⌧ Tc, the thermal conductivity of acoustic phonons followsDebye T 3 law Aph ⇠ T 3. The less dominant optical phonon thermal conduc-tivity is Oph ⇠ 1T 2 exp( 1T ). At low temperatures both will be overwhelmedby quasiparticle contribution. At higher temperatures, the phonon contri-bution ph = Aph+Oph can dominate over the quasiparticle thermal conduc-tivity but quantum oscillations should remain visible, because ph does notshow quantum oscillations due to the bosonic nature of phonons.3.5 SummaryIn this chapter, we studied a minimal model for a Weyl superconductorwith a single pair of Weyl points based on the Meng-Balents multilayerconstruction. A Majorana-Fermi arc appears and connects the two Weylpoints if a pair of boundaries are open. This arc can be understood as beingformed of two counter propagating chiral Majorana modes at the edge ofan effective topological px + ipy SC that results from fixing one componentof the momentum in the 3D Hamiltonian describing the original Weyl SC.The phase diagram shows that the Weyl SC phase appears intermediatelybetween a fully gapped trivially superconducting phase and a topologicallysuperconducting phase. These features elucidate similarities between Weylsuperconductors and Weyl semimetals.In the low-energy sector of the theory, we showed that elastic strain actsas a chiral gauge potential incorporated in the Weyl Hamiltonian throughthe standard minimal substitution. Therefore, strain can mimic the effectof real physical magnetic field in the Weyl superconductor. One importantdifference is that the strain-induced pseudo-magnetic field is not subject tothe Meisner effect. Remarkably, this fact allows the pseudo-magnetic field toLandau quantize the spectrum of Bogoliubov quasiparticles instead of beingexpelled from the sample or creating Abrikosov vortices as would be the case503.5. Summaryfor physical magnetic field B.Landau quantization generates pronounced quantum oscillations that canbe observed by quasiparticle spectroscopy and by longitudinal thermal con-ductivity. These quantum oscillations occur deep in the superconductingstate and are thus fundamentally different from various theoretical propos-als and experimental results that pertain to mixed and normal states ofsuperconductors.To experimentally test our proposal, we require a Weyl superconductor.Such can be in principle artificially engineered through the Meng-Balentsconstruction or can occur naturally in a suitable crystalline solid. Currently,there are roughly twenty different nodal superconductors known to science[74]. One of the most promising candidates may be CuxBi2Se3 [19, 75]. Nu-clear magnetic resonance experiments [77] revealed broken spin rotationalsymmetry in CuxBi2Se3, suggesting the superconducting gap structure to beeither 4x where the nodes appear due to the protection of mirror symme-try or 4y where small gaps (or nodes) are expected. Recent experimentalresults on the specific heat [78] of CuxBi2Se3 are consistent with nematicsuperconductivity and favor 4y pairing structure. Unfortunately, the gapminima and nodes cannot be straightforwardly differentiated based on thereported specific heat data alone. However, symmetry and energetic consid-erations [79, 80] suggest gap minima in nematic superconductivity. Anotherpromising candidate is NbxBi2Se3, whose low temperature penetration depthexhibits quadratic temperature dependence characteristic of linearly dispers-ing point nodes in three dimensions [76]. This is consistent with NbxBi2Se3being a Weyl superconductor. Although it is too early to draw a firm con-clusion regarding the pairing state of the candidate materials, the existingexperimental data give hope that CuxBi2Se3 and NbxBi2Se3 could eventuallybe identified as 3D Dirac or Weyl superconductors.The second requirement is that the candidate material should be suffi-ciently flexible to allow a few percent elastic deformation in order to generatea sufficiently strong pseudo-magnetic field. The candidate material should beprepared in a nanoscale thin film geometry in order to maximize its flexibility.To the best of our knowledge, detailed data on the mechanical properties ofCuxBi2Se3 is lacking and further experimental work is needed to determinewhether or not this could be a suitable material.There are several future directions that might be interesting to pursuebased on our current work. The first is to test other properties associatedwith pseudo-Landau levels. Recent work on the fractional Josephson effect instrained 2D graphene superconductor [146] motivates the interest in studyinga similar effect in one dimension higher using strained Weyl superconductor.513.5. SummaryThe second lies in the study of the chiral anomaly, chiral magnetic effect,and gravitational anomaly with strain-induced gauge field.52Chapter 4Magnon quantum anomalies inWeyl ferromagnetsA magnon is a bosonic collective excitation of electron spins on a crystallattice, carrying spin, magnetic dipole moment, and heat [82]. It can bequantum-mechanically characterized by a quantized spin wave [147] and ex-perimentally detected by neutron scattering [148]. Though charge neutral,a magnon can be manipulated by electric fields and magnetic fields throughthe Aharonov-Casher effect [83] and the Zeeman effect, respectively. Dueto their unique reaction to electromagnetic (EM) fields, magnons stand outfrom other bosons (photons, phonons, polarons, etc.) but are akin to elec-trons. Consequently, magnons are proposed to mimic some electronic trans-port properties such as the chiral anomaly [20, 25], the magnon Josephsoneffect [149], the spin Hall effect [26], the Wiedemann-Franz law [150, 151],the thermal Hall effect [152–157], the spin Seebeck effect [158–162], the spinPeltier effect [163, 164], and the spin Nernst effect [165–168].On the other hand, electronic transport does not necessarily require theparticipation of EM fields or temperature gradients. One example is thatrelativistic electrons can be manipulated by elastic strains. As we have seenin Chapter 2, a properly designed strain can induce Landau levels in the ab-sence of magnetic field and gives rise to quantum oscillations. In Chapter 3,we have showed that strain can manipulate charge neutral relativistic Bo-goliubov quasiparticles, resulting in the Wiedemann-Franz law and quantumoscillations. These discoveries shed new light on novel magnon transport withonly electric (magnetic) fields or even in the complete absence of EM fields aslong as a suitable strain is present. It has been reported that strain is able toLandau-quantize 2D Weyl magnons hosted by “magnon graphene” and leadsto a non-quantized Hall viscosity [91]. Since 3D Weyl magnons have beenpredicted to exist in pyrochlore magnets [20–23], double perovskite magnets[24] and multilayer magnets [25–27], it is natural to ask how strain engagesin the transport of 3D Weyl magnons harbored by these Weyl magnets.In this chapter, we answer this question by a combination of analyti-cal calculations and numerical simulations. We show that a static torsional534.1. Model of Weyl ferromagnetsstrain can induce a pseudo-electric field to Landau-quantize the 3D Weylmagnons, and a dynamic uniaxial strain can induce a pseudo-magnetic fieldto pump the magnons. We also demonstrate that these elastic strain-inducedpseudo-EM fields result in novel magnon transport in the form of quantumanomalies. To support these findings, we organize the chapter as follows. InSection 4.1, we formulate the model of a multilayer Weyl ferromagnet anddiscuss its magnon band structure and topology. In Section 4.2, we study themagnon band structures of the Weyl ferromagnet under an electric field andunder a static torsional strain-induced pseudo-electric field respectively. Wealso study the magnon dynamics due to either a magnetic field or a dynamicuniaxial strain-induced pseudo-magnetic field. In Section 4.3, we derive themagnon quantum anomalies and the associated anomalous spin and heatcurrents under different combinations of EM fields and pseudo-EM fields. InSection 4.4, we derive the field (gradient) dependence of the anomalous spinand heat currents in various magnon quantum anomalies, and establish a du-ality to the anomalous electric current in the chiral magnetic effect [62, 63]and the chiral torsional effect [93, 94] of Weyl semimetals. In Section 4.5, wediscuss the proposals for experimentally measuring magnon quantum anoma-lies by atomic force microscopy (AFM) force sensing. Section 4.6 concludesthe chapter, proposes several promising materials for the implementationof magnon quantum anomalies, and envisages a few worthwhile directionsbased on the current work.4.1 Model of Weyl ferromagnetsWe consider a multilayer Weyl ferromagnet model proposed in Ref. [25] butamend the model with an inter-layer next nearest neighbor interaction. Itwill be shown in Section 4.2.2 that such an interaction is necessary for thestrain engineering for Landau quantization. The building block of this modelis a honeycomb ferromagnet layer of spins of size S (Fig. 4.1(a)), which hasbeen experimentally realized in CrI3 [169] and proposed in other compounds[170]. Multiple layers are then stacked in the z direction (Fig. 4.1(b)). Thein-plane nearest neighbors are labelled by ↵1 =p32 axˆ+12ayˆ, ↵2 = p32 axˆ+12ayˆ, and ↵3 = ayˆ, where a is the lattice constant of the honeycomb lattice;the in-plane next nearest neighbors are labelled by 1 = ↵3  ↵2, 2 =↵1 ↵3, and 3 = ↵2 ↵1. The inter-layer spacing (out-of-plane directionlattice constant) is denoted as z. We choose the unit cells of the honeycomblayer such that each contains an A site and a B site connected by vector ↵1.544.1. Model of Weyl ferromagnetsThe Heisenberg Hamiltonian of the Weyl ferromagnet readsH = H0 +H1 +H2 +Hz +HD + Z, (4.1)with its components listed explicitly belowH0 = XR?,zXµ=A,BKµSzµ(R?, z)Szµ(R?, z),H1 = XR?,zXiJ1(↵i)SA(R?, z) · SB(R? +↵i ↵1, z),H2 = XR?,zXiXs=±1J2(↵i + sz zˆ)SA(R?, z) · SB(R? +↵i ↵1, z + sz)Hz = XR?,zXµ=A,BJµ(z zˆ)Sµ(R?, z) · Sµ(R?, z + z),HD =XR?,zXiXµ=A,BDµ · [Sµ(R?, z)⇥ Sµ(R? + i, z)],Z = gµBBzXR?,zXµ=A,BSzµ(R?, z),where R? labels the unit cells of a honeycomb layer, and z is the layer index.H0 is an on-site interaction term with strengthKA(B) on A(B) sites; H1 is theintra-layer nearest neighbor interaction with strength J1(↵i) between A andB sites connected by vector ↵i; H2 is the inter-layer next nearest neighborinteraction with strength J2(↵i ± z) between A and B sites connected byvector ↵i±z; Hz is the inter-layer nearest neighbor interaction with strengthJA(B) between same-sublattice sites spaced by z in the stacking directionz; HD is the intra-layer next nearest neighbor Dzyaloshinskii-Moriya (DM)interaction [171, 172] between same-sublattice sites connected by i; and thelast term Z is the Zeeman energy. For simplicity, we choose the interactionstrength as DA = DB = D > 0, J1(↵i) = J1 > 0, and J2(↵i± z zˆ) = J2 >0. Moreover, since the Zeeman term only determines the spin polarizationof the ground state, we henceforth set Bz ! 0+. We further assume theinter-layer lattice constant z = a.554.1. Model of Weyl ferromagnetsThe Heisenberg Hamiltonian (Eq. 4.1) can be rewritten in terms of magnonsthrough the Holstein-Primakoff transformation [147],S+µ (R) =p2Ss1 µ†RµR2SµR,Sµ (R) =p2Sµ†Rs1 µ†RµR2S,Szµ(R) = S  µ†RµR,whereR = (R?, z) indicates the position of the unit cell, µ = A,B is the sub-lattice index, and µ†R/µR is the corresponding magnon creation/annihilationoperator. In the single-particle limit, the Hamiltonian becomesH = EFM +HM, (4.2)where the ferromagnetic ground state energy EFM = NS2(KA+KB+JA+JB + 3J1 + 6J2) and the tight binding magnon Hamiltonian isHM = 2KASXR?,za†R?,zaR?,z + 2KBSXR?,zb†R?,zbR?,z+ (3J1 + 6J2)SXR?,z(a†R?,zaR?,z + b†R?,zbR?,z)+ 2JASXR?,za†R?,zaR?,z + 2JBSXR?,zb†R?,zbR?,z+⇢iDSXR?,zXi(a†R?,zaR?+i,z + b†R?,zbR?+i,z) JASXR?,za†R?,zaR?,z+a  JBSXR?,zb†R?,zbR?,z+a J1SXR?,zXia†R?,zbR?+↵i↵1,z J2SXR?,zXiXs=±1a†R?,zbR?+↵i↵1,z+sa +H.c.. (4.3)We then perform Fourier transformaR?,z =1pNXkeik?·R?eikzzak, (4.4)bR?,z =1pNXkeik?·(R?+↵1)eikzzbk, (4.5)564.1. Model of Weyl ferromagnetsAB↵2↵1↵3123(b)(a)Figure 4.1: Schematic plot for the Weyl ferromagnet multilayer. (a) 2D hon-eycomb ferromagnet sheet. The Weyl ferromagnet multilayer is constructedby stacking many sheets in the z direction. (b) Conventional crystal cellsof the Weyl ferromagnet with in-plane nearest (second nearest) neighborsconnected by ↵i (i), i = 1, 2, 3.where k = (k?, kz) andN is the number of unit cells in the Weyl ferromagnetmultilayer. Then the tight-binding Hamiltonian can be written asHM =Xk †kHk k, (4.6)where the sublattice basis is  k = (ak, bk)T and the first-quantized BlochHamiltonian isHk = (J1  2J2 cos kza)SXicos(k ·↵i)x+ (J1 + 2J2 cos kza)SXisin(k ·↵i)y+ [K + J(1 cos kza) + 2DXisin(k · i)]Sz+ [3J1 + 6J2 +K+ + J+(1 cos kza)]S0, (4.7)where  and 0 are Pauli matrices and the identity matrix, respectively,and we have used the notations K± = KA ±KB and J± = JA ± JB. It iseasy to find that Hk can only be gapless at the corners of the 2D hexagonalBrillouin zone of the honeycomb ferromagnet. For simplicity, we requireK + 3p3D > 0 but 2J1 < K  3p3D < 0. In this case, there is onlyone pair of nodal points k⌘W = ( 4⇡3p3a , 0, ⌘Q) with ⌘ = ±1 andQ =1acos1✓K + J  3p3DJ◆. (4.8)574.1. Model of Weyl ferromagnetsIn the vicinity of these Weyl points, the Bloch Hamiltonian can be expandedto the lowest order asHk⌘W+q = Hk⌘W + ~v⌘0qz0 +Xi=x,y,z~v⌘i qii, (4.9)with the corresponding velocity parameters defined asv⌘x = 32~(J1 + 2J2 cosQa)Sa,v⌘y = 32~(J1 + 2J2 cosQa)Sa,v⌘z =⌘~JSa sinQa,v⌘0 =⌘~J+Sa sinQa.Without loss of generality, we take J > 0. Then the Berry flux (in units of⇡) that flows into/out of the Weyl points is⌘ = sgn(vxvyvz) = sgn(vz) = ⌘, (4.10)which is locked to the momentum space position of Weyl points. The topo-logical nature of Hk can be verified by evaluating the Chern number of a 2Dslice with fixed kz, which isCkz =12sgn⇥K + J(1 cos kza) 3p3D⇤ 12. (4.11)Based on the restrictions we impose on the parameters, we obtainCkz =(1 |kz| < Q0 Q < |kz| < ⇡. (4.12)Therefore, we expect arc surface states, which are akin to Fermi arcs in anelectronic Weyl semimetal, connecting magnon Weyl points at the topologi-cal crystal momenta |kz| < Q. To confirm this, we numerically calculate theband structure (Fig. 4.2(a)-(d)) of a Weyl ferromagnet nanowire whose crosssection containing 1800 lattice sites is schematically plotted in Fig. 4.3(b).The open boundary of the cross section consists of one pair of zigzag edges(x-direction) and one pair of armchair edges (y-direction). For simplicityand visual clarity, we set J+ = 0 in Eq. 4.7 while taking care to preservethe positive-definiteness of the magnon energy/spin wave frequency; the J+term will be reinstated in Appendix D. As illustrated in Fig. 4.2(a), the Weyl584.1. Model of Weyl ferromagnetsferromagnet nanowire exhibits a pair of Weyl cones connected by a set of al-most flat bands. When the boundaries are closed, these flat bands disappearin Fig. 4.2(b), indicating that the flat bands only reside on the surface of thenanowire. This can be further confirmed by evaluating the spectral functions:these flat states have a large spectral density at the surface (Fig. 4.2(c)) butdisappear deep in the bulk (Fig. 4.2(d)).(a) (b)(c) (d)Figure 4.2: Magnon dispersion and spectral functions for the Weyl ferro-magnet multilayer. For all panels, we set DS = 1 and measure energies interms of DS such that J1S = 4.56, J2S = 1.14, JS = 7.22, K+S = 2.77and KS = 1.12. (a) Magnon band structure for the nanowire with a pairof zigzag edges and a pair of armchair edges in the cross section. The crosssection of the nanowire is illustrated in Fig. 4.3(b). The magnon bands ex-hibit two Weyl points on the kz axis and are connected by a set of almost flatstates analogous to the arc states in Weyl semimetals. (b) Magnon bands forthe nanowire with periodic boundary conditions for the cross section. Theflat bands disappear, indicating their surface origin. The red curves are theanalytical dispersion ✏k = [K++3J1+6J2]S±[K+J(1cos kza)3p3D]Sfor the Bloch HamiltonianHk at the honeycomb lattice Brillouin zone cornerk? = (4⇡/3p3a, 0). (c) Surface spectral function of the Bloch Hamilto-nian which confirms that the almost flat states reside on surfaces. (d) Bulkspectral function which indicates the positions of Weyl cones.594.2. Weyl ferromagnets under electromagnetic fields and strainbˆ2mnaˆ2mnaˆ(2m1)nbˆ(2m1)nbˆ(2m1)n+1aˆ(2m1)n+1aˆ(2m2)n+1bˆ(2m2)n+1bˆn+1aˆn+1 aˆ2nbˆ2nbˆnaˆnaˆ1 bˆ1(a) (b)gµBEec2gµB"ec2Figure 4.3: Schematic plot for the Weyl ferromagnet nanowire. (a) Nanowireunder an inhomogeneous electric field and nanowire under a twist deforma-tion. Landau quantization takes place in both cases. (b) Cross section ofthe Weyl ferromagnet nanowire with a pair of zigzag edges (x-direction) anda pair of armchair edges (y-direction). We use m = n = 30 unless other-wise specified so that all numerical simulations could be implemented withavailable computational resources.4.2 Weyl ferromagnets under electromagneticfields and strainIn Section 4.1, we introduced a multilayer model for the Weyl ferromag-net and discussed the magnon band topology. In order to obtain magnonquantum anomalies, gauge fields are needed to manipulate magnons. In thepresent section, we will first discuss the Landau quantization of magnonbands in the presence of an inhomogeneous electric field E (Section 4.2.1) oran inhomogeneous chiral pseudo-electric field e⌘ (Section 4.2.2) induced bya static twist deformation of the Weyl ferromagnet nanowire. Then we de-rive the equation of motion for magnons under an inhomogeneous magneticfield B (Section 4.2.3) or an inhomogeneous chiral pseudo-magnetic field b⌘(Section 4.2.4) induced by a dynamic uniaxial strain.4.2.1 Landau quantization in the presence of electric fieldTo begin with, we study the magnon band structure under an electric fieldE. Dual to the Aharonov-Bohm phase AB =  e~´ r+r A · dl acquired by604.2. Weyl ferromagnets under electromagnetic fields and strainelectrons moving in a magnetic field B = r ⇥ A, magnons moving in anelectric field can acquire an Aharonov-Casher phase [83]AC =  e~ˆ r+r1ec2(E ⇥ µ) · dl, (4.13)where the curl of the integrand 1ec2r⇥(E⇥µ) is dual to the magnetic field inelectronics and will Landau-quantize the magnon bands. When transformedto the reciprocal space, the Aharonov-Casher phase results in the Peierlssubstitution k! k+ e~ 1ec2E⇥µ. Explicitly, if we take the magnon magneticmoment µ = gµB zˆ and the electric field E = (12Ex, 12Ey, 0), which maybe experimentally realized by periodically arranging scanning tunneling mi-croscope (STM) tips [150]8. The resulting Dirac-Landau levels are given by✏⌘n(qz) = ±~rv⌘z qz2+ 2ngµBE~c2 v⌘xv⌘y  n = 1, 2, · · · , (4.14)and✏⌘0(qz) = sgn✓gµBE~c2 v⌘xv⌘y◆~v⌘z qz = ⌘sgn(gE)~|v⌘z |qz. (4.15)The higher (n > 1) Landau levels at both Weyl points are identical and theyalways come in pairs with opposite energies at each momentum qz. However,the zeroth Landau levels at the two Weyl points are not identical but counter-propagating. Without loss of generality, we choose sgn(gE) = 1 such thatthe right (left) Weyl cone hosts a right (left) moving zeroth Landau level~|v⌘z |qz (~|v⌘z |qz). Unlike higher Landau levels, at each Weyl point, thezeroth Landau levels are unpaired.To numerically verify the Landau quantization due to the Aharonov-Casher effect, we consider the nanowire geometry under an electric field (leftpanel, Fig. 4.3(a)). The calculated band structure of the Weyl ferromagnetnanowire under an inhomogeneous electric field is shown in Fig. 4.4(a). Thezeroth Landau levels can be easily identified, and are connected by a set ofalmost flat states whose surface origin can be confirmed by calculating thesurface spectral function (Fig. 4.4(b)). The obscured higher Landau levels,on the other hand, are revealed by the bulk spectral function as illustratedin Fig. 4.4(c).8At least in principle, there is another proposal to experimentally realize such electricfield. In particular, a uniformly charged cylindrical Weyl ferromagnet nanowire producesan electric field E = 12Errˆ, where the field gradient is E = ⇢✏0✏r with ⇢ being the chargedensity and ✏0✏r being the permittivity of the nanowire. For the field gradient used inFig. 4.4, a charge density ⇠ 1010C/m3 is needed.614.2. Weyl ferromagnets under electromagnetic fields and strain(a) (b) (c)Figure 4.4: Magnon dispersion of the Weyl ferromagnet nanowire under aninhomogeneous electric field. For all panels, we take gµBEa2ec2 = 0.01240where E represents the gradient of the external electric field and 0 = h/2eis the magnetic flux quantum. (a) Magnon bands are Landau-quantizedby the external inhomogeneous electric field due to the Aharonov-Cashereffect. The two resulting zeroth Landau levels at different Weyl points haveopposite velocities ±|v⌘z | and are connected by a set of almost flat states. (b)Surface spectral function, which reveals that these flat bands are localizedat the surface of the Weyl ferromagnet nanowire. (c) Bulk spectral functionhighlighting the Dirac-Landau levels at each Weyl cone.4.2.2 Landau quantization in the presence ofpseudo-electric fieldIn the context of Weyl semimetals, besides applying a magnetic field B, theelectronic bands are also Landau-quantized by a suitable lattice deformation[92–95, 134], which spatially modulates the overlap integrals. The overalleffect of such a deformation on the low-energy Hamiltonian is a minimalsubstitution analogous to that for a magnetic field. In other words, thelattice deformation due to external strain induces a pseudo-magnetic field.It is worth noting that such a strain-induced pseudo-magnetic field onlycouples to the Weyl points and becomes negligible at energies far away fromthe Weyl points. In this section, we will show that a similar strain-inducedpseudo-electric field can be generated by spin lattice deformation and leadsto Landau quantization of Weyl magnons.For concreteness, we consider the spin lattice deformation due to a twist(Fig. 4.3(a), right panel). The resulting displacement field isu =zL(⌦⇥R?), (4.16)624.2. Weyl ferromagnets under electromagnetic fields and strainwhere ⌦ is the angle of rotation of the uppermost layer with respect tothe lowermost layer, and L is the spacing between these two layers, i.e., thelength of the nanowire. For this displacement field, the non-zero componentsof the symmetric strain tensor uij = 12(@iuj+@jui) are u13 = u31 = ⌦y/2Land u23 = u32 = ⌦x/2L. Based on the experience of the overlap integralsubstitution (Eq. 1.23) in Section 1.3, we only need to consider the exchangeintegrals whose arguments simultaneously have out-of-plane and in-planecomponents. In our model, this refers to the six J2’s as illustrated in Fig. 4.5.J2(z + ↵1)J2(z + ↵1)J2(z + ↵2)J2(z + ↵3)J2(z + ↵3)J2(z + ↵2)Figure 4.5: Schematic plot for the exchange integrals J2. The most importantimpact of the twist deformation is to modulate J2 spatially.The substitutions for these J2’s areJ2(↵1 ± azˆ)! J2(1⌥p32 u31 ⌥ 12u32),J2(↵2 ± azˆ)! J2(1±p32 u31 ⌥ 12u32),J2(↵3 ± azˆ)! J2(1± u32),under which an additional term appears in HkHek = 2J2S sin kzaXidi(sink ·↵ix + cosk ·↵iy), (4.17)where(d1, d2, d3) =⇣p32u31 +12u32,p32u31 +12u32,u32⌘.634.2. Weyl ferromagnets under electromagnetic fields and strainIn the vicinity of the Weyl points k⌘W , the Bloch Hamiltonian of the twistedWeyl ferromagnet nanowire readsHk⌘W+q + Hek⌘W+q ⇡ Hk⌘W + ~v⌘0qz0 +Xi~v⌘i qii + Hek⌘W= Hk⌘W + ~v⌘0⇣qz +e~a⌘S,z⌘0 +Xi~v⌘i⇣qi +e~a⌘S,i⌘i. (4.18)It is worth noting that the strain-induced term can be incorporated into thelinearized Hamiltonian through a minimal substitution q ! q+ e~a⌘S , wherethe strain-induced vector potential isa⌘S = ⌘2~eaJ2 sinQaJ1 + 2J2 cosQa(u31, u32, 0), (4.19)which is a chiral gauge field taking opposite values at different Weyl points.We note that the multilayer model in Ref. [25] does not have such a strain-induced vector potential because the inter-layer next nearest neighbor inter-action is not considered, i.e., J2 = 0. As r ⇥ a⌘S 6= 0, this strain-inducedvector potential will result in Landau quantization of magnon bands. InSection 4.2.1, we demonstrate that electric field E = (12Ex, 12Ey, 0) pro-duces Landau quantization through the Aharonov-Casher effect. Therefore,we may interpret a⌘S as the vector potential of a chiral pseudo-electric fielde⌘ = ⌘e = ⌘(12"x,12"y, 0), which only differs from E by a chiral charge⌘. The field gradient of this pseudo-electric field can be determined byr⇥ a⌘S = 1ec2r⇥ (e⌘ ⇥ µ) = ⌘ gµB"ec2 zˆ. Explicitly, the gradient reads" = 2~eaJ2 sinQaJ1 + 2J2 cosQa⌦Lec2gµB. (4.20)It is critically important to note that the strain-induced vector potential isa unique feature of relativistic particles. We thus expect that the strain-induced pseudo-electric field and magnon Dirac-Landau levels only reside inthe vicinity of magnon Weyl points. Unlike the electric field E which pro-duces counter-propagating zeroth Landau levels, the chiral pseudo-electricfield e⌘ results in co-propagating zeroth Landau levels on the Weyl points asillustrated in Fig. 4.6(a,c). At the first glance, this causes an imbalance in thenumbers of left-moving and right-moving channels, which must be the samein a lattice model. Considering that we have only discussed bulk Landaulevels so far, we argue that there should be a set of surface states connectingthe bulk Landau levels at the two Weyl cones, providing the needed counter-propagating channels to compensate the imbalance. Our argument has been644.2. Weyl ferromagnets under electromagnetic fields and strainconfirmed by the numerical simulation of the surface spectral function asillustrated in Fig. 4.6(b). Remarkably, we note that inside the gap spannedby the two first Landau levels, the left-moving surface channels and right-moving bulk channels are spatially well separated and cannot be scatteredinto each other. Therefore, if the reservoirs are fine-tuned to populate thisgap with magnons [25, 173], the bulk magnon particle current and surfacemagnon particle current are counter-propagating in the ballistic regime. Thisleads to exotic spin and heat transport known as the bulk-surface separationwhich will be elaborated in Section 4.3.(a) (b) (c)Figure 4.6: Magnon dispersion of a twisted Weyl ferromagnet nanowire. Forall panels, we take gµB"a2ec2 = 0.01240 where " represents the gradientof the strain-induced pseudo-electric field. (a) Magnon bands are Landau-quantized by the strain-induced pseudo-electric field. The resulting zerothLandau levels at the two Weyl points are both right-moving and are con-nected by a set of left-moving states. (b) Surface spectral function, whichreveals that these left-moving states are localized at the surface of the Weylferromagnet nanowire. (c) Bulk spectral function highlighting the Dirac-Landau levels at each Weyl cone.4.2.3 Magnon motion in the presence of magnetic fieldIn Section 4.2.1, we have Landau-quantized the magnon bands by applyingan inhomogeneous electric field E. To implement quantum anomalies withmagnons, we need to pump them through the zeroth Landau levels. In thissection, we will elucidate how magnons are pumped by an inhomogeneousmagnetic field B. The equation of motion of magnons will be derived.Due to the magnetic moment it carries, a magnon has a Zeeman energyU = µ · B in the presence of a magnetic field B. The force exerted on654.2. Weyl ferromagnets under electromagnetic fields and strainthe magnon is then F = rU = r(µ ·B) = ~dkdt . Practically, we are onlyconcerned about the transport in the z direction, which is governed by~dkzdt= @z(µ ·B). (4.21)Therefore, an inhomogeneous magnetic field in the z-direction will be capa-ble of driving magnons from one magnon Weyl point to the other. Moregenerally, we consider the case where there is also an electric field E in addi-tion to B. As discussed in Section 4.2.1, due to the Aharonov-Casher phase,the canonical momentum will be shifted through the Peierls substitutionp! p+ 1c2E ⇥ µ in the presence of E. Then the magnon Hamiltonian canbe written down asHM = Kp+ 1c2E ⇥ µ µ ·B, (4.22)whereK(p) is the magnon kinetic energy whose specific form does not matterfor our purpose. The Hamilton’s equations of motion givev =@HM@p=@K@p, (4.23)anddpdt= @HM@r=1c2r[v · (µ⇥E)] +r(µ ·B). (4.24)Therefore, the equation of motion for the crystal momentum k = 1~p+1~c2E⇥µ is~dkdt= r(µ ·B) @@t1c2µ⇥E + 1c2v ⇥ [r⇥ (µ⇥E)], (4.25)which clearly shows the duality to electronics with the magnon Zeeman en-ergy µ ·B (the electric momentum µ⇥E) playing the role of the electricpotential energy e (the magnetic momentum eA). For the inhomogeneouselectric field E = (12Ex, 12Ey, 0) and the magnetic moment µ = gµB zˆ usedin Section 4.2.1, the z component of Eq. 4.25 is exactly Eq. 4.21.In summary, an inhomogeneous magnetic field can be used to pumpmagnons along the z direction and is capable of producing magnon quantumanomalies. This will be discussed in detail in Section 4.3.1 and Section 4.3.4.In the meanwhile, a static inhomogeneous electric field only provides Landauquantization and does not affect the magnon transport in the z direction.664.2. Weyl ferromagnets under electromagnetic fields and strain4.2.4 Magnon motion in the presence of pseudo-magneticfieldIn Section 4.2.2, we have shown that magnon bands can be Landau-quantizedby a chiral pseudo-electric field e⌘ induced by a static twist, under whichthe magnon Hamiltonian is modified by the Peierls substitution q ! q +e~1ec2e⌘ ⇥ µ. Therefore, by comparing to Eq. 4.25, the magnon equation ofmotion in the presence of e⌘ and B can be immediately written down as~dkdt= r(µ ·B) @@t1c2µ⇥ e⌘ + 1c2v ⇥ [r⇥ (µ⇥ e⌘)]. (4.26)Because e⌘ field has exactly the same spacetime dependence as the inhomo-geneous electric field E except for the chiral nature, our analysis for E inSection 4.2.3 can be transplanted to e⌘. Therefore we argue that magnonscannot be pumped by the pseudo-electric field induced by a static twist defor-mation. For this reason, in order to pump magnons, we resort to a dynamicdeformation.We consider a dynamic uniaxial strain whose only nonzero strain tensorcomponent is u33. Such a strain can be generated by the displacement fieldu = uz(z, t)zˆ. The knowledge of the explicit form of uz is not necessary forour purpose. Experimentally, a legitimate uz may be obtained by applyingultrasonic sound waves along the z direction. Based on our experience of theoverlap integral substitution (Eq. 1.23) in Section 1.3, under this uniaxialstrain u33, we need to modify the exchange integrals whose arguments hasnonzero z components. Such exchange integrals are JA, JB, and the six J2’sillustrated in Fig. 4.5. The substitutions areJA ! JA(1 u33),JB ! JB(1 u33),J2 ! J2(1 12u33),under which an extra term enters the Bloch Hamiltonian (Eq. 4.7),Hbk = u33J2S cos kzaXicos(k ·↵i)x  u33J2S cos kzaXisin(k ·↵i)y u33JS(1 cos kza)z  u33[3J2 + J+(1 cos kza)]S0. (4.27)In the vicinity of the Weyl points k⌘W , the Bloch Hamiltonian under the674.2. Weyl ferromagnets under electromagnetic fields and strainuniaxial deformation becomesHk⌘W+q + Hbk⌘W+q ⇡ Hk⌘W + ~v⌘0⇣qz +e~a⌘D,z⌘0+Xi~v⌘i⇣qi +e~a⌘D,i⌘i  3J2Su330. (4.28)Similar to the static twist (Eq. 4.18), the dynamic uniaxial strain also shiftsthe momentum through the minimal substitution q ! q + e~a⌘D, where thestrain-induced vector potential readsa⌘D = ⌘~ea1 cosQasinQa(0, 0, u33). (4.29)Similar to the its static counterpart a⌘S , this dynamic strain-induced vec-tor potential a⌘D is also chiral. However, it is critically important to notethat this vector potential cannot be interpreted as the vector potential ofa pseudo-electric field because r ⇥ a⌘D = 0. In fact, a⌘D is associated witha chiral pseudo-magnetic field. To see this, by replicating the derivation inSection 4.2.3, we first write down the equation of motion for magnons underthe dynamic uniaxial strain~dkdt= e@a⌘D@t ev ⇥ (r⇥ a⌘D) +r(3J2Su33). (4.30)The extra non-chiral gradient termr(3J2Su33) appears because the dynamicstrain induces in the Hamiltonian an on-site term 3J2Su330, which cannotbe characterized by the minimal substitution. However, in systems whereJ2 is reasonably small, namely J2S ⌧ ~cs/a (cs is the speed of sound inthe nanowire), this non-chiral term becomes negligible relative to the chiraltime-derivative term9. We concentrate on such systems in the remainder ofthis paper. Therefore, considering the fact thatr⇥a⌘D = 0, the z componentof Eq. 4.30 is reduced to~dkzdt= e@a⌘D,z@t= @z(µ · b⌘), (4.31)whereb⌘ =µµ2ˆdze@a⌘D,z@t=⌘gµB~a1 cosQasinQaˆdz@u33@tzˆ (4.32)9In systems where the non-chiral gradient term is no longer negligible, its effects aresimilar to those of a inhomogeneous magnetic field B. Therefore, the resulting magnonquantum anomalies will be a combination of purely magnetic field induced anomalies andpurely chiral pseudo-magnetic field induced anomalies.684.3. Magnon quantum anomalies and the anomalous transportcan be understood as an inhomogeneous pseudo-magnetic field. Due to itschiral nature, b⌘ pumps magnons oppositely in the vicinity of the two Weylpoints.To sum up, a dynamic uniaxial strain can induce a chiral pseudo-magneticfield, accompanied by a non-chiral field whose effects can be neglected forreasonably small J2. The pseudo-magnetic field pumps magnons in oppositedirections±z at the twoWeyl points. Therefore, it may also result in magnonquantum anomalies as will be detailed in Section 4.3.2 and Section 4.3.3. Inthe meanwhile, the static torsional strain only provides the needed Landauquantization and does not affect the magnon transport in the z direction.4.3 Magnon quantum anomalies and theanomalous transportIn Section 4.2, we have found that magnon bands can be Landau-quantizedby either an inhomogeneous transverse electric field E or a chiral pseudo-electric field e⌘ induced by a static twist. To drive the magnons along thezeroth Landau levels, we may use either an inhomogeneous longitudinal mag-netic field B, or a chiral pseudo-magnetic field b⌘ induced by a dynamicuniaxial strain. In this section, we will show that each of the four possi-ble combinations of an electric/pseudo-electric field and a magnetic/pseudo-magnetic field gives rise to a magnon quantum anomaly. We will derivethe anomaly equations and discuss the associated anomalous spin and heattransport.4.3.1 Magnon chiral anomaly due to electric and magneticfieldsIn the present section, for completeness, we will derive the magnon chiralanomaly and the associated anomalous spin and heat currents in the presenceof E and B, though these have been briefly mentioned in Ref. [25].We consider a Weyl ferromagnet nanowire aligned in the z direction un-der the electric field E = (12Ex, 12Ey, 0), whose magnon Landau levels areillustrated in Fig. 4.4. The left and right ends are attached to magnon reser-voirs subjected to a uniform magnetic field B0 = B0zˆ. Magnons will thenattain a Zeeman energy µ ·B0 = gµBB0 and the magnon population edge,which is originally located in the band minima, can be lifted up to the gapof the first Landau levels (Fig. 4.7(a)) by a suitable B0. This is an ana-log to putting the chemical potential in the gap of Landau levels in Weyl694.3. Magnon quantum anomalies and the anomalous transportsemimetals [25, 173]. While electrons contributing to ballistic transport canhave energies above or below the chemical potential, magnons must resideabove the population edge gµBB0 in the magnon reservoirs to participatein ballistic transport. Then a spatially varying magnetic field B = Bz zˆ isoverlaid, where Bz has a nonzero gradient @zBz = B. Based on Eq. 4.21,the magnons begin to propagate along the Landau levels in the z directionaccording to semiclassical equation of motion qz(t) = qz(0) gµB´ t0 Bdt0/~.Thus the magnons originally on the left zeroth Landau level will be pumpedto the right zeroth Landau level across the Brillouin zone boundary, whichresults in a chirality imbalance, a key feature of the magnon chiral anomaly.(a) (b) (c)(d) (e) (f)-3 -2 -1 0 1 2 3-20-15-10-505101520gµBB0 gµBB0kzE-3 -2 -1 0 1 2 3-20-15-10-505101520gµBBLgµBBRkzE-3 -2 -1 0 1 2 3-20-15-10-505101520gµBBL gµBBRkzE-3 -2 -1 0 1 2 3-20-15-10-505101520gµBBL gµBBRkzE-3 -2 -1 0 1 2 3-20-15-10-505101520gµBB0gµBB0kzE-3 -2 -1 0 1 2 3-20-15-10-505101520gµBBLgµBBRkzEFigure 4.7: Schematic plot of the magnon band structures and distributionsin various quantum anomalies of a Weyl ferromagnet, which is in contact withtwo magnon reservoirs in a uniform magnetic fieldB0. (a)-(c) Magnon Dirac-Landau levels due to an inhomogeneous electric field. (d)-(f) Magnon Dirac-Landau levels due to a strain-induced pseudo-electric field. (a, d) Magnondistributions in the absence of pumping. (b, e) Magnon chiral anomaly withchirality imbalance created by ordinary magnetic field pumping in (b) andby pseudo-magnetic field pumping in (e). (c, f) Magnon heat anomaly withmagnon concentration variation created by pseudo-magnetic field pumpingin (c) and by ordinary magnetic field pumping in (f). For all panels, only thedistributions (green dots) on the zeroth Landau levels (red) are plotted. Inprinciple, magnons can occupy all bands above the population edges providedthat the relaxation time is sufficiently long.704.3. Magnon quantum anomalies and the anomalous transportDuring this pumping process, the magnon population edge is graduallyelevated (lowered) on the left (right) zeroth Landau level (Fig. 4.7(b)). Thedifference between the left and right magnon population edges is analogousto the chiral chemical potential in Weyl semimetals. Here we will see thedifference in magnon population edges as a magnetic field bias B5 ⌧ B0 suchthat the left (right) Weyl cone experiences a magnetic field BL = B0 + B5(BR = B0 B5). From the semiclassical equation of motion, we obtainB5 = ~|v⌘z |´dqzgµB=ˆ t0B|v⌘z |dt0. (4.33)We now derive the chiral anomaly equation for magnons. The magnon con-centration variation on the right zeroth Landau level can be written downas a Taylor seriesnE,BR =ˆ gµBB0gµBBRgE(✏)nB(✏)d✏=nB(gµBB0)4⇡2l2EgµBB5~|v⌘z | Xn=1n(n)B (gµBB0)(n+ 1)!(gµBB5)n+14⇡2l2E~|v⌘z |, (4.34)where gE(✏) = 12⇡l2E12⇡~|v⌘z | is the density of states with the electric length lE =(~c2/gµBE)1/2 10, and nB(✏) = (e✏/kBT  1)1 is the magnon distributionfunction. Similarly, the magnon concentration variation on the left zerothLandau level isnE,BL =ˆ gµBB0gµBBLgE(✏)nB(✏)d✏= nB(gµBB0)4⇡2l2EgµBB5~|v⌘z | Xn=1n(n)B (gµBB0)(n+ 1)!(gµBB5)n+14⇡2l2E~|v⌘z |. (4.35)In the low bias limit gµBB5 ⌧ kBT , the net chirality pumping rate can bewell approximated asd⇢E,B5dt= RdnE,BRdt+ LdnE,BLdt⇡  g2µ2B2⇡2~2c2nB(gµBB0)EB, (4.36)where, as in Eq. 4.10, R = +1 and L = 1. More generally, for arbitrarilyoriented E and B, the magnon chiral anomaly equation readsd⇢E,B5dt+r · jE,B5 ⇡nB(gµBB0)2⇡2~2c2 r(µ ·B) · [r⇥ (E ⇥ µ)]. (4.37)10The minus sign comes from the assumption that sgn(gE) = 1 in Section 4.2.1714.3. Magnon quantum anomalies and the anomalous transportSuch a magnon chiral anomaly is dual to the electron chiral anomaly [56–58]. It is worth noting that Eq. 4.37 is true to the first order in B5; at thisorder the number of magnons pumped out of the left zeroth Landau level isapproximately equal to the number of magnons pumped into the right zerothLandau level nL ⇡ nR. However, at the second order in B5, we havenE,BR + nE,BL = Xn=1n(n)B (gµBB0)(n+ 1)!(gµBB5)n+1[1 + (1)n+1]4⇡2l2E~|v⌘z |⇡ egµBB0/kBT(egµBB0/kBT  1)21kBT(gµBB5)24⇡2l2E~|v⌘z |> 0, (4.38)indicating that the magnon number is not conserved, because magnons arebosonic collective excitations rather than fundamental particles. From an-other point of view, if the magnon number were conserved, higher-energymagnons pumped out of the left zeroth Landau level would result in thesame number of lower-energy magnons populating the right zeroth Landaulevel, and the magnon distribution in Fig. 4.7(b) would have a lower totalenergy than that in Fig. 4.7(a), making the pumping process spontaneous.However, magnon pumping actually requires energy injection by EM fields.Explicitly, the energy injection into the right zeroth Landau level isUE,BR =ˆ +1gµBBR✏gE(✏)nB(✏)d✏ˆ +1gµBB0✏gE(✏)nB(✏)d✏=14⇡2l2E~|v⌘z |⇢12nB(gµBB0)g2µ2B(2B0B5 B25)Xn=1n(n)B (gµBB0)n!gµBB0(gµBB5)n+1n+ 1Xn=1n(n)B (gµBB0)n!(gµBB5)n+2n+ 2⇡ 14⇡2l2E~|v⌘z |⇢12nB(gµBB0)g2µ2B(2B0B5 B25) n0B(gµBB0)gµBB0(gµBB5)22, (4.39)where the approximation is still taken in the low bias limit gµBB5 ⌧ kBT .Similarly, the energy depletion on the left zeroth Landau level can be easily724.3. Magnon quantum anomalies and the anomalous transportwritten down by making the substitution B5 ! +B5,UE,BL =ˆ +1gµBBL✏gE(✏)nB(✏)d✏ˆ +1gµBB0✏gE(✏)nB(✏)d✏⇡ 14⇡2l2E~|v⌘z |⇢12nB(gµBB0)g2µ2B(2B0B5 B25) n0B(gµBB0)gµBB0(gµBB5)22. (4.40)And the total energy variationUE,BR + UE,BL = 14⇡2l2E~|v⌘z |nB(gµBB0)(gµBB5)2 14⇡2l2E~|v⌘z |n0B(gµBB0)gµBB0(gµBB5)2=  (gµBB5)24⇡2l2E~|v⌘z |dd✏✏nB(✏)gµBB0> 0 (4.41)is indeed positive as expected, because ✏nB(✏) is a decreasing function.Therefore, in the magnon pumping process, higher-energy magnons pumpedout of the left zeroth Landau level should result in more lower-energy magnonson the right zeroth Landau level with the assistance of electromagnetic en-ergy injection. The total energy after pumping will then increase.Before we leave this section, we calculate the anomalous spin and heatcurrents due to the magnon chiral anomaly. For the Weyl ferromagnet weconsidered, the spin and heat currents are given by the Landauer-Büttikerformalism [173],JE,Bspin = ˆ +1gµBBR~gE(✏)nB(✏)vER (✏)d✏ˆ +1gµBBL~gE(✏)nB(✏)vEL (✏)d✏,(4.42)JE,Bheat =ˆ +1gµBBR✏gE(✏)nB(✏)vER (✏)d✏+ˆ +1gµBBL✏gE(✏)nB(✏)vEL (✏)d✏, (4.43)where the magnon drifting velocity in the zeroth Landau levels are vER (✏) =1~d✏dqz|R and vEL (✏) = 1~ d✏dqz |L. Further simplification gives the spin and heatcurrents asJE,Bspin = ~|v⌘z |(nE,BR  nE,BL ) ⇡ ~g2µ2B2⇡2~2c2nB(gµBB0)B5E , (4.44)734.3. Magnon quantum anomalies and the anomalous transportJE,Bheat = |v⌘z |(UE,BR  UE,BL ) ⇡ g3µ3B2⇡2~2c2nB(gµBB0)B0B5E , (4.45)where vER (✏) = vEL (✏) = |v⌘z | is used. As discussed in Section 4.2.3, theelectric field E for magnons is dual to the vector potential A for electrons;the electric field gradient E thus plays the role of the magnetic field for elec-trons. gµBB5 measures the difference of the magnon population edges, andis therefore dual to the electron chiral chemical potential µ5. Consequently,the anomalous spin and heat currents of the magnon chiral anomaly is akinto the chiral magnetic current [62, 63] of the electron chiral anomaly. Wewill refer to Eqs. 4.44, 4.45 as the “chiral electric effect.”4.3.2 Magnon chiral anomaly due to pseudo-electric andpseudo-magnetic fieldsWe now consider another possibility of implementing the magnon chiralanomaly. We use a chiral electric field e⌘ = ⌘e = (12⌘"x,12⌘"y, 0) induced bya static torsional strain, whose Landau levels are illustrated in Fig. 4.6 withboth zeroth Landau levels being right-moving. The twisted Weyl ferromag-net is aligned in the z direction and in contact with magnon reservoirs, whichwill lift the magnon population edge to µ ·B0 = gµBB0. As with the caseof an electric field E, in principle, the magnon population edge can be tunedinto the gap of the first Landau levels (Fig. 4.7(d)) by a proper choice ofB0. Then a chiral magnetic field b⌘ = ⌘b = ⌘bz zˆ, which is induced by a dy-namic uniaxial strain, is overlaid, where bz has a nonzero gradient @zbz = .Based on Eq. 4.31, the magnons are pumped through Landau levels accord-ing to the semiclassical equation of motion qz(t) = qz(0)  ⌘gµB´ t0 t0/~.Thus magnons originally on the left zeroth Landau level are pumped intothe right zeroth Landau level, giving rise to a magnon chiral anomaly.Unlike the magnon pumping due to B, magnons on different Weyl conesare pumped oppositely by the chiral magnetic field b⌘ such that the magnonpopulation edge on left (right) zeroth Landau level is elevated (lowered) asillustrated in Fig. 4.7(e). The difference between magnon population edgescan be characterized by a magnetic field bias b5 ⌧ B0 under which the left(right) Weyl cone experiences a magnetic field bL = B0 + b5 (bR = B0  b5).According to the semiclassical equation of motion, we haveb5 = ~|v⌘z |⌘´dqzgµB=ˆ t0|v⌘z |dt0. (4.46)To derive the chiral anomaly due to e⌘ and b⌘, we need to know the magnon744.3. Magnon quantum anomalies and the anomalous transportconcentration variations on both zeroth Landau levels, which can be directlywritten down by referring to Eqs. 4.34, 4.35 asne,bR =nB(gµBB0)4⇡2l2egµBb5~|v⌘z | Xn=1n(n)B (gµBB0)(n+ 1)!(gµBb5)n+14⇡2l2e~|v⌘z |, (4.47)ne,bL = nB(gµBB0)4⇡2l2egµBb5~|v⌘z | Xn=1n(n)B (gµBB0)(n+ 1)!(gµBb5)n+14⇡2l2e~|v⌘z |, (4.48)where the pseudo-electric length le = (~c2/gµB")1/2. We assume sgn(g") =1 in order to be parallel to the assumption that sgn(gE) = 1 (see Sec-tion 4.2.1). In the low bias limit gµBb5 ⌧ kBT , the chirality pumping rateisd⇢e,b5dt= Rdne,bRdt+ Ldne,bLdt⇡  g2µ2B2⇡2~2c2nB(gµBB0)". (4.49)In the more general case, the magnon chiral anomaly equation can be writtenasd⇢e,b5dt+r · je,b5 ⇡nB(gµBB0)2⇡2~2c2 r(µ · b) · [r⇥ (e⇥ µ)], (4.50)which is parallel to Eq. 4.37. Similarly, this anomaly equation only containsthe leading order terms in Eqs. 4.47, 4.48. More rigorously, ne,bR + ne,bL > 0;otherwise the magnon pumping will become spontaneous. This can also beconfirmed by checking the energy variations on both zeroth Landau levels.By referring to Eqs. 4.39, 4.40, the estimated energy variations on the zerothLandau levels areUe,bR ⇡14⇡2l2e~|v⌘z |⇢12nB(gµBB0)g2µ2B[2B0b5  b25] n0B(gµBB0)gµBB0(gµBb5)22, (4.51)Ue,bL ⇡14⇡2l2e~|v⌘z |⇢12nB(gµBB0)g2µ2B[2B0b5  b25] n0B(gµBB0)gµBB0(gµBb5)22. (4.52)Ue,bR + Ue,bL > 0, indicating energy injection due to the pseudo-EM fieldsduring the pumping process.754.3. Magnon quantum anomalies and the anomalous transportWe now derive the anomalous spin and heat currents resulting from thechiral anomaly due to e⌘ and b⌘. First, it is important to note that be-fore the chiral magnetic field b⌘ is switched on, there must be an equalnumber of right-moving magnons and left-moving magnons; otherwise thenet spin/heat current will be nonzero in the absence of a driving force. Asdemonstrated in Section 4.2.2, under the chiral electric field e⌘, the bulk onlyhosts right-moving magnons while the left-moving magnons are localized atthe surface. Therefore, the bulk spin/heat current must be balanced by thesurface spin/heat current when b⌘ = 0. Explicitly, we haveJ bulkspin = ˆ +1gµBB0~ge(✏)nB(✏)veR(✏)d✏ˆ +1gµBB0~ge(✏)nB(✏)veL(✏)d✏ = J surfacespin , (4.53)J bulkheat =ˆ +1gµBB0✏ge(✏)nB(✏)veR(✏)d✏+ˆ +1gµBB0✏ge(✏)nB(✏)veL(✏)d✏ = J surfaceheat , (4.54)where the density of states is ge(✏) = 12⇡l2e12⇡~|v⌘z | and the velocities are veR =veL = |v⌘z |. To verify our argument, we have numerically calculated thespatial distribution of magnon spin and heat currents on the cross section ofthe Weyl ferromagnet nanowire illustrated in the right panel of Fig. 4.3(a).As shown in Fig. 4.8(b), the spin current in the bulk of the rectangular crosssection (Fig. 4.3(b)) propagates along the z direction while the spin currenton the edges of the cross section propagates along the +z direction. On theother hand, the bulk heat current of the rectangular cross section propagatesalong the +z direction while the edge heat current propagates along the zdirection, as illustrated in Fig. 4.8(c). We have carefully evaluated the totalspin and heat currents through the rectangular cross section under variousnumerical settings, and find that both currents are vanishingly small whenthe gradient of bz vanishes,  = 0. We further examine a Weyl ferromagnetnanowire with an almost circular cross section, whose bulk-surface separationfor spin/heat transport is exhibited again in Fig. 4.8 (e) and (f).764.3. Magnon quantum anomalies and the anomalous transportx0 20 40 60y0102030x0 20 40 60y0102030x0 20 40 60y0122436x0 20 40 60y0122436JspinsurfJspinbulkJheatbulk JheatsurfJspinsurfJspinbulk(f)(c)(b)(e)(a)(d)Jheatbulk JheatsurfFigure 4.8: Bulk-surface separation for the twisted Weyl ferromagnetnanowire. (a) Schematic plot of a Weyl ferromagnet nanowire with a rect-angular cross section. The spin current propagates along the z directionin the bulk but along the +z direction on the surface, while the heat currentpropagates along the +z direction in the bulk but along the z direction onthe surface. (b) Spatially resolved spin current on the cross section of thecuboid Weyl ferromagnet nanowire. (c) Spatially resolved heat current onthe cross section of the cuboid Weyl ferromagnet nanowire. The directionsof currents are color coded with blue (orange) representing z (+z). (d)-(f) Same as (a)-(c) but for a Weyl ferromagnet nanowire with an (almost)circular cross section. The total spin current on the rectangular (circular)cross section is 0.002DS (0.0001DS) while the total heat current on therectangular (circular) cross section is 0.0473D2S2/~ (0.0018D2S2/~).Then, we switch on the chiral pseudo-magnetic field b⌘, and the magnonsbegin to propagate along the Landau levels, giving rise to net anomalous spinand heat currentsJe,bspin = ˆ +1gµBbR~ge(✏)nB(✏)veR(✏)d✏ˆ +1gµBbL~ge(✏)nB(✏)veL(✏)d✏ J bulkspin ,(4.55)774.3. Magnon quantum anomalies and the anomalous transportJe,bheat =ˆ +1gµBbR✏ge(✏)nB(✏)veR(✏)d✏+ˆ +1gµBbL✏ge(✏)nB(✏)veL(✏)d✏ J bulkheat .(4.56)Again, in the low bias limit gµBb5 ⌧ kBT , we obtain the spin and heatcurrents to the lowest non-vanishing orderJe,bspin = ~|v⌘z |(ne,bR + ne,bL ) ⇡ ~g3µ3B4⇡2~2c2n0B(gµBB0)b25", (4.57)Je,bheat = |v⌘z |(Ue,bR + Ue,bL ) ⇡g3µ3B4⇡2~2c2hnB(gµBB0) + n0B(gµBB0)gµBB0ib25".(4.58)Unlike the magnon chiral electric spin and heat currents Eqs. 4.44, 4.45,which are proportional to the magnetic field bias B5, the anomalous spin andheat currents due to (e⌘, b⌘) are quadratic in the pseudo-magnetic field biasb5, because magnons on both zeroth Landau levels have the identical velocity.These small but nonzero spin and heat currents reflect the non-conservationof magnon number, in contrast to the chiral anomaly of electrons with purestrain-induced pseudo-EM fields, where the anomalous current is exactly zerodue to charge conservation.4.3.3 Magnon heat anomaly due to electric andpseudo-magnetic fieldsBesides the chiral anomalies discussed in Section 4.3.1 and Section 4.3.2,Weyl magnons can exhibit another type of quantum anomaly in which thethermal energy in the bulk is not conserved. In this section, we will quan-titatively characterize such a magnon “heat anomaly” in the presence of aninhomogeneous electric field E and an inhomogeneous pseudo-magnetic fieldb⌘.We again consider a Weyl ferromagnet nanowire subjected to an inho-mogeneous electric field E = (12Ex, 12Ey, 0). The two ends of the wire areattached to magnon reservoirs with a uniform magnetic field B0 = B0zˆ suchthat the magnon population edge gµBB0 is located in the gap of the firstLandau levels (Fig. 4.7(a)). Rather than using an inhomogeneous magneticfield B as is the case in Section 4.3.1, we drive magnons with an inhomo-geneous pseudo-magnetic field b⌘ = ⌘b = ⌘bz zˆ, where bz has a nonzerogradient @zbz = . As analyzed in Section 4.3.2, the magnon motion is gov-erned by the semiclassical equation of motion q(t) = q(0) ⌘gµB´ t0 dt0/~.784.3. Magnon quantum anomalies and the anomalous transportThus magnons are pumped into both zeroth Landau levels, indicating ther-mal energy non-conservation in the bulk, which is a clue of the magnon heatanomaly.Due to the chiral nature of the pseudo-magnetic field b⌘, magnons ondifferent zeroth Landau levels are oppositely pumped such that the magnonpopulation edge on both zeroth Landau levels are lowered as illustrated inFig. 4.7(c). The variation of the magnon population edge is denoted byb ⌧ B0, and correspondingly both Weyl cones experience a magnetic fieldb0L = b0R = B0  b. According to the semiclassical equation of motion, wehaveb = ~|v⌘z |⌘´dqzgµB=ˆ t0|v⌘z |dt0. (4.59)By comparing to Eqs. 4.34, 4.35, we can directly write down the magnonconcentration variation on both zeroth Landau levels asnE,bR = nE,bL ⇡nB(gµBB0)4⇡2l2EgµBb~|v⌘z | , (4.60)where we make the approximation gµBb ⌧ kBT . Because the magnonpopulation edge is shifted identically on both chiral Landau levels, there is nonet chirality transport between two Weyl cones. Nevertheless, the fact nE,bR +nE,bL > 0 indicates that there are more magnons in the bulk11. Therefore,the thermal energy in the bulk increases. By comparing to Eqs. 4.39, 4.40,we immediately obtain the heat injection into the two zeroth Landau levelsin the limit gµBb ⌧ kBT asUE,bR = UE,bL ⇡nB(gµBB0)4⇡2l2EgµBB0gµBb~|v⌘z | . (4.61)Thus the bulk heat injection rate can be written down asd⇢E,bheatdt=dUE,bRdt+dUE,bLdt⇡ g3µ3BnB(gµBB0)2⇡2~2c2 B0E. (4.62)More generally, the bulk heat injection rate readsd⇢E,bheatdt+r · jE,bheat ⇡gµBB0nB(gµBB0)2⇡2~2c2 r(µ · b) · [r⇥ (E ⇥ µ)]. (4.63)11In the limit gµBb ⌧ kBT , the total magnon concentration variation on zeroth Landaulevels nE,bR +nE,bL ⇠ gµBb is linear in magnetic field variation b. And it is fundamentallydifferent from the total magnon concentration variation nE,BR + nE,BL ⇠ g2µ2BB25 (ne,bR +ne,bL ⇠ g2µ2Bb25) associated with magnon chiral anomaly in the low bias limit, which is ofquadratic order of magnetic field bias B5 (b5).794.3. Magnon quantum anomalies and the anomalous transportThis magnon heat anomaly equation is analogous to the magnon chiralanomaly equation (Eqs. 4.37, 4.50). It is a heat continuity equation witha source indicating that the bulk thermal energy is not conserved. Unfor-tunately, this heat anomaly does not have measurable anomalous currentsJE,bspin = 0, (4.64)JE,bheat = 0. (4.65)Because left-moving and right-moving magnons are always created/annihilatedin pairs, as illustrated in Fig. 4.7(c).We note that though the bulk thermal energy is not conserved, the totalenergy of a closed system must be conserved. Since Eq. 4.63 only charac-terizes the variation rate of the bulk thermal energy, we need to considerhow the surface thermal energy is altered by the pseudo-magnetic field. Asillustrated in Fig. 4.4(a), the bulk zeroth Landau levels are connected bya set of surface states. The magnons residing in these states can enter thebulk such that the thermal energy is transferred from the surface to the bulk.During this process, the external pseudo-magnetic field also does work onmagnons; otherwise the heat pumping from surface to bulk would be spon-taneous. The thermal energy from the surface and the mechanical energyfrom the pseudo-magnetic field constitute the heat injection into the bulkof the Weyl ferromagnet nanowire. In particular, if the pseudo-magneticfield is induced by the dynamic lattice deformation resulting from applyingan ultrasonic sound wave, the energy loss during sound propagation in theWeyl ferromagnet will lead to sound attenuation, which, in principle, shouldbe experimentally measurable.4.3.4 Magnon heat anomaly due to pseudo-electric andmagnetic fieldsWe consider another possibility of implementing the magnon heat anomaly.We apply a strain-induced chiral pseudo-electric field e⌘ = ⌘e = ⌘(12"x,12"y, 0)to a Weyl ferromagnet nanowire aligned in z direction. The two ends ofthe wire are attached to magnon reservoirs subjected to a magnetic fieldB0 = B0zˆ with the magnon population edges lying in the gap of the firstLandau levels (Fig. 4.7(d)). We drive magnons with an inhomogeneous mag-netic field B = Bz zˆ, where Bz has a nonzero gradient @zBz = B. Accordingto Section 4.3.1, the magnon motion is governed by the semiclassical equa-tion of motion qz(t) = qz(0)  gµB´ t0 Bdt0/~. Consequently, magnons are804.3. Magnon quantum anomalies and the anomalous transportpumped into both zeroth Landau levels and the thermal energy in the bulkincreases, indicating a magnon heat anomaly.The magnetic fieldB pumps magnons on different Weyl cones identically.However, due to the chiral nature of pseudo-electric field e⌘, the magnonpopulation edge on both zeroth Landau levels are lowered as illustrated inFig. 4.7(f). The variation of the magnon population edge is denoted as B ⌧B0 and both Weyl cones experience a magnetic field B0L = B0R = B0  B.According to the semiclassical equation of motion, we haveB = ~|v⌘z |´dqzgµB=ˆ t0B|v⌘z |dt0. (4.66)By comparing to Eqs. 4.34, 4.35, the magnon concentration variation onboth zeroth Landau levels can be calculated asne,BR = ne,BL ⇡nB(gµBB0)4⇡2l2egµBB~|v⌘z | , (4.67)in the limit gµBB ⌧ kBT . Because magnons on both zeroth Landau levelsare always created in pairs, there is no net chirality transport between thetwo Weyl points, but the larger total number of magnons on the zerothLandau levels indicates a thermal energy injection into the bulk of the Weylferromagnet. By comparing to Eqs. 4.39, 4.40, we can directly write down theheat injection into the two zeroth Landau levels in the limit gµBB ⌧ kBTasUe,BR = Ue,BL ⇡nB(gµBB0)4⇡2l2egµBB0gµBB~|v⌘z | . (4.68)Thus the rate of heat injection into the bulk of theWeyl ferromagnet nanowirereadsd⇢e,Bheatdt=dUe,BRdt+dUe,BLdt⇡ g3µ3BnB(gµBB0)2⇡2~2c2 B0"B. (4.69)More generally, the bulk heat injection rate readsd⇢e,Bheatdt+r · je,Bheat ⇡gµBB0nB(gµBB0)2⇡2~2c2 r(µ ·B) · [r⇥ (e⇥ µ)], (4.70)which is similar to the heat anomaly equation (Eq. 4.63), showing non-conservation of the bulk thermal energy. Again, when we take into accountthe contribution of the surface states and the applied pseudo-electric field,the heat anomaly will be removed and the total energy is conserved. In814.4. Field dependence of anomalous spin and heat currentscontrast to the heat anomaly due to E and b⌘ in Section 4.3.3, the heatanomaly due to e⌘ and B does result in anomalous spin and heat currentsJe,Bspin = ~|v⌘z |(ne,BR + ne,BL ) ⇡ ~g2µ2B2⇡2~2c2nB(gµBB0)B", (4.71)Je,Bheat = |v⌘z |(Ue,BR + Ue,BL ) ⇡ g3µ3B2⇡2~2c2nB(gµBB0)B0B", (4.72)both of which should be experimentally measurable. Unlike the magnon “chi-ral electric effect” (Eqs. 4.44, 4.45, 4.57, 4.58) whose spin and heat currentsresult from either EM fields or pseudo EM fields, the anomalous currentsEqs. 4.71, 4.72 result from their combination. Since the required pseudo-electric field e⌘ is induced by torsional strain, we refer to Eqs. 4.71, 4.72 asthe magnon “chiral torsional effect”.4.4 Field dependence of anomalous spin and heatcurrentsIn Section 4.3, we listed the magnon quantum anomaly equations (Eqs. 4.37,4.50, 4.63, 4.70) and the associated anomalous spin (Eqs. 4.44, 4.57, 4.64,4.71) and heat (Eqs. 4.45, 4.58, 4.65, 4.72) currents. The anomaly equationshave explicit EM/pesudo-EM field dependence while the anomalous currentsdo not, because the explicit field dependence of magnetic field bias (B5/b5)and magnetic field variation (b/B) is unknown. In this section, we willderive how these quantities depend on EM/pseudo-EM fields, and eventuallygive the full field dependence for the anomalous spin and heat currents inboth the semiclassical limit and the quantum limit.We take the magnon chiral anomaly due to E and B as an example anddetermine how B5 depends on these fields. In the semiclassical limit, themagnetic field bias between the two zeroth Landau levels dominates overthe magnon Landau spacing gµBB5  ~p2|gµBEv⌘xv⌘y/~c2|. Therefore, thechirality transported between the Weyl cones is⇢E,B5 =⇢Rˆ +1gµBBR+Lˆ +1gµBBLDs(✏)nB(✏)d✏⇡ g2µ2BB202⇡2~3|v⌘xv⌘yv⌘z |nB(gµBB0) · 2gµBB5, (4.73)824.4. Field dependence of anomalous spin and heat currentswhere the magnon density of states can be estimated using the dispersion"k = ~p(v⌘xkx)2 + (v⌘yky)2 + (v⌘zkz)2 asDs(✏) =1VXk(✏ "k) = ✏22⇡2~3|v⌘xv⌘yv⌘z | .In Section 4.2.1, we have obtained the chirality pumping rate (Eq. 4.36) inthe absence of scattering of magnons between the two zeroth Landau levels.However, in a realistic magnet, the chirality mixing scattering mechanismalways exists; otherwise the chirality transported between Weyl cones goesto infinity. Due to such scattering, the chirality pumping rate is changed tod⇢E,B5dt=  g2µ2B2⇡2~2c2nB(gµBB0)EB ⇢E,B5⌧E,B, (4.74)where ⌧E,B is the mean free time of magnons due to chirality mixing scat-tering. The solution for ⇢E,B5 at sufficiently long times t  ⌧E,B is then⇢E,B5 = g2µ2B2⇡2~2c2nB(gµBB0)EB⌧E,B. (4.75)By referring to Eq. 4.73, we obtain the field dependence of the effectivemagnetic field bias asB5 =  ~|v⌘xv⌘yv⌘z |2gµBc2B20EB⌧E,B / EB. (4.76)where we have assumed that the chirality mixing scattering is characterizedby a constant magnon mean free time ⌧E,B. In that case, the field depen-dence of the anomalous spin/heat current can be obtained as JE,Bspin/heat /E2B, which is dual to the electron chiral magnetic current since magnon EMfields are dual to electron EM potentials.In the quantum limit, the Landau level spacing becomes comparableto the magnetic field bias gµBB5 . ~p2|gµBEv⌘xv⌘y/~c2|, and the magnondensity of states is proportional to the electric field gradient (Dq(✏) ⇠ 1/l2E /E). The chirality transported between Weyl cones is then ⇢E,B5 / EB5,distinct from the semiclassical result Eq. 4.73. By comparing to Eq. 4.75, weobtain the field dependence of the magnetic field bias B5 / B, provided thatthe magnon mean free time is constant. As a result, the field dependence ofthe anomalous spin/heat current in the quantum limit can be immediatelyobtained as JE,Bspin/heat / EB.By replicating the analysis above, we can determine the field dependenceof other magnetic field bias/variations as b5 / ", b / E, and B / "B834.5. Experimental measurement of magnon quantum anomaliesin the semiclassical limit and b5 / , b / , and B / B in the quan-tum limit. The resulting field dependence of the corresponding spin/heatcurrent is summarized in Table. 4.1. The magnon chiral electric spin/heatcurrent (Eqs. 4.44, 4.45) is dual to the chiral magnetic current [62, 63], andthe magnon chiral torsional spin/heat current (Eqs. 4.71, 4.72) is dual tothe chiral torsional current given in Ref. [94]. However, it is worth notingthat the magnon anomalous spin/heat current (Eqs. 4.57, 4.58) due to eand b shows unprecedented field dependence without any anomalous electriccurrent counterpart.Table 4.1: Summary of field (gradient) dependence of anomalous spin andheat currents in magnon quantum anomalies.Field Semiclassicallimit Quantum limitE,B JE,Bspin/heat / E2B JE,Bspin/heat / EBe, b Je,bspin/heat / "32 Je,bspin/heat / "2E, b JE,bspin/heat = 0 JE,bspin/heat = 0e,B Je,Bspin/heat / "2B Je,Bspin/heat / "B4.5 Experimental measurement of magnonquantum anomaliesIn Section 4.4, we have obtained the field (gradient) dependence of anomalousspin and heat currents in magnon quantum anomalies as summarized inTab. 4.1. Unfortunately, the direct measurement of such anomalous spinand heat transport is experimentally non-trivial due to the lack of inducedelectric field from spin currents and the dissipation of thermal energy fromheat currents. For this reason, an easily measurable signature quantity isrequired for the experimental detection of magnon quantum anomalies.We propose that the force on the magnon current carrying Weyl ferro-magnet nanowire exerted by an external inhomogeneous electric or pseudo-electric field could be such a signature quantity. Intuitively, this force ontoa magnon current can be understood as an analog of Ampère force, which844.5. Experimental measurement of magnon quantum anomaliesemerges when applying an external magnetic field to an electric current.In the following, we will derive this force quantitatively for each magnonquantum anomaly.4.5.1 Experimental signature of magnon chiral anomalydue to electric and magnetic fieldsIn Section 4.2.1, we have demonstrated that magnon bands are Landau-quantized by an external inhomogeneous electric field E = (12Ex, 12Ey, 0).We now apply an additional electric field E0 = (0, E 0z, 0) with E 0 ⌧ E tothe Weyl ferromagnet nanowire. This weak additional electric field will notlead to further Landau quantization of magnon bands. But, according to themagnon equation of motion (Eq. 4.25), it gives rise to a “magnon Lorentzforce”~dkdt=1c2v ⇥ [r⇥ (µ⇥E0)], (4.77)which results from the Aharonov-Casher effect [83]. For the magnon popula-tion illustrated in Fig. 4.7(a), the net force contributed by magnons from twozeroth Landau levels is zero, because for each magnon drifting at velocity|v⌘z |, there is always another magnon drifting at velocity |v⌘z |.However, when magnons are pumped by an inhomogeneous magneticfield in the z direction B = Bz zˆ, a magnon imbalance develops on thezeroth Landau levels as illustrated in Fig. 4.7(b) such that there are moreright-moving magnons than left-moving magnons. This imbalance is thecause of the magnon chiral anomaly (Eq. 4.37) and the associated anomaloustransport (Eqs. 4.44, 4.45). It also produces a force acting on the wholenanowire, which may serve as a signature quantity for the experimentalconfirmation of the magnon chiral anomaly due to E and B. Explicitly,the force readsFE,B =XLL01c2v ⇥ [r⇥ (µ⇥E0)] = V (nE,BR  nE,BL )|v⌘z |gµBE 0c2xˆ, (4.78)where the summation goes over the zeroth Landau levels (LL0) and V refersto the volume of the nanowire. To estimate the magnitude of this force, wefirst assume the additional electric field gradient E 0 = 0.1E such that E 0 haslittle effects on the magnon band structure in Fig. 4.4. We then take themagnon drifting velocity |v⌘z | ⇠ 102m/s of the same order as that of yttriumiron garnet (YIG) [174]. For a typical nanowire with cross section radius ⇠100nm and length ⇠ 100µm, the volume can be estimated as V ⇠ 1018m3.854.5. Experimental measurement of magnon quantum anomaliesTo estimate the magnon concentration imbalance, we make use of Eqs. 4.34,4.35 and getnE,BR  nE,BL ⇡nB(gµBB0)2⇡2l2EgµBB~ ⌧E,B. (4.79)For the electric field gradient used in Fig. 4.4, we obtain 1/2⇡2l2E ⇠ 1014m2.We further estimate the magnon mean free time ⌧E,B ⇠ 106s, which is ofthe same order as that of YIG [175]. Lastly, an inhomogeneous magneticfield with gradient B ⇠ 10T/m should be experimentally available. Theselead to a magnon concentration imbalance nE,BR  nE,BL ⇠ 1020m3 and aforce FE,B ⇠ 1015N. By means of atomic force microscopy (AFM), thissmall force can be sensed as a clue of magnon chiral anomaly due to E andB.Before we leave this section, we briefly analyze the mechanical effectsof magnons on the surface and the higher Landau levels. According toFig. 4.4(a, b), the almost flat surface states indicate a vanishing magnondrifting velocity, thus a negligible force contribution is expected from thesurface magnons. For the higher Landau levels, the magnon population al-ters little during the whole pumping process because all states in these bandsare occupied. The force contribution is thus ideally zero because the numbersof left-moving magnons and right-moving magnons are equal. In conclusion,the net force on the nanowire is mostly contributed by the magnon imbalanceon the zeroth Landau levels.4.5.2 Experimental signature of magnon chiral anomalydue to pseudo-electric and pseudo-magnetic fieldsIn Section 4.2.2, we have demonstrated that magnon bands are Landau-quantized by an inhomogeneous pseudo-electric field e⌘ = ⌘(12"x,12"y, 0)induced by a static twist. Again, by applying an additional electric fieldE0 = (0, E 0z, 0), a magnon will experience a Lorentz force given by Eq. 4.77.However, for the magnon population illustrated in Fig. 4.7(d), the net forcecontributed by magnons on two zeroth Landau levels is nonzero becausethese two bands are now co-propagating.Nevertheless, when magnons are pumped by a pseudo-magnetic field inthe z direction b⌘ = ⌘bz zˆ, a magnon imbalance develops as illustrated inFig. 4.7(e), with slightly more right-moving magnons on the zeroth Landaulevels. The force acting on the nanowire increases byF e,b = V (ne,bR + ne,bL )|v⌘z |gµBE 0c2xˆ, (4.80)864.5. Experimental measurement of magnon quantum anomalieswhere the magnon concentration variation can be estimated by Eqs. 4.47,4.48 asne,bR + ne,bL ⇡egµBB0/kBTegµBB0/kBT  1gµB|v⌘z |⌧e,b2kBTnB(gµBB0)2⇡2l2egµB~ ⌧e,b. (4.81)If we assume similar parameters, i.e., le ⇠ lE ,  ⇠ B, and ⌧e,b ⇠ ⌧E,B,we can estimate the force increase F e,b ⇠ 1021N at room temperature.Though this force increase reflects the magnon number non-conservation aswell as the magnon anomalous transport (Eqs. 4.57, 4.58), the difficulty offorce sensing is greatly increased.To avoid this difficulty, we propose the force sensing experiment using anadditional pseudo-electric field e0⌘ = ⌘(0, "0z, 0), which can be generated by acircular bend deformation (Appendix E). For the magnon population shownin Fig. 4.7(d), the total Lorentz force on magnons on the zeroth Landau levelsis restored to zero due to the chiral nature of the additional pseudo-electricfield. However, as the pseudo-magnetic field is switched on, the magnonimbalance illustrated in Fig. 4.7(e) will produce a nonzero force F e,b actingon the whole nanowire. Explicitly, the force readsF e,b = V (ne,bR  ne,bL )|v⌘z |gµB"0c2xˆ, (4.82)where the magnon concentration imbalance is given by Eqs. 4.47, 4.48 asne,bR  ne,bL ⇡nB(gµBB0)2⇡2l2egµB~ ⌧e,b. (4.83)Using the parameters above, we obtain the magnon imbalance ne,bR  ne,bL ⇠1020m3 and the force exerted on the nanowire is then F e,b ⇠ 1015N, whichcan be measured by AFM. It is worth noting that the surface magnons,despite possessing a finite drifting velocity, do not have appreciable forcecontribution, because the additional pseudo-electric field e0⌘ only lives in thevicinity of Weyl cones deep in the bulk and thus has no effect on the surface.Unlike F e,b which reflects the anomalous magnon transport (Eqs. 4.57,4.58), the force F e,b indicates the magnon population imbalance on the ze-roth Landau levels and is thus a signature of the magnon chiral anomaly dueto e⌘ and b⌘.4.5.3 Experimental signature of magnon heat anomaly dueto electric and pseudo-magnetic fieldsFor the magnon heat anomaly due to E and b⌘, the Landau quantizationis still provided by the inhomogeneous electric field, but the pumping field874.5. Experimental measurement of magnon quantum anomaliesis a chiral pseudo-magnetic field due to a dynamic uniaxial strain, leadingto an equal number of left-moving and right-moving magnons injected intothe bulk as illustrated in Fig. 4.7(c). This renders E0 = (0, E 0z, 0) in Sec-tion 4.5.1 useless because the force contributed by the magnons on the zeroLandau levels is always zero. However, an additional pseudo-electric fielde0⌘ = ⌘(0, "0z, 0) produces a measurable mechanical effect.Due to the chiral nature of e0⌘, the magnons on the zeroth Landau levels(Fig. 4.7(a)) contribute constructively to the force exerted on the nanowire.When the magnons are pumped by the pseudo-magnetic field, more magnonsare driven into the zeroth Landau levels, leading to an increase in this forceFE,b = V (nE,bR + nE,bL )|v⌘z |gµB"0c2xˆ, (4.84)where the magnon concentration variation is given by Eq. 4.60 asnE,bR + nE,bL ⇡nB(gµBB0)2⇡2l2EgµB~ ⌧E,b. (4.85)Assuming ⌧E,b ⇠ 106s leads to an AFM measurable force increase FE,b ⇠1015N. As analyzed in Section 4.5.1, this force is the actual force expe-rienced by the nanowire because there is no force contribution from themagnons on the surface or the higher Landau levels. The force increase FE,bis of great experimental importance, because there is no magnon anomaloustransport (Eqs. 4.64, 4.65) in the magnon quantum anomaly due to E andb⌘.4.5.4 Experimental signature of magnon heat anomaly dueto pseudo-electric and magnetic fieldsFor the magnon heat anomaly due to e⌘ and B, the Landau quantization isprovided by the inhomogeneous pseudo-electric field resulting from a statictwist, but the pumping field is an ordinary magnetic field, leading to anequal number of magnons injected into each zeroth Landau level as illus-trated in Fig. 4.7(f). Consequently, the magnons on the zeroth Landaulevels contribute zero total force in the presence of the chiral pseudo-electricfield e0⌘ = ⌘(0, "0z, 0). We thus resort to an additional ordinary electric fieldE0 = (0, E 0z, 0) as is used in Section 4.5.1.Because the two zeroth Landau levels are co-propagating under the pseudo-electric field e⌘, which results from a static twist, the magnons on these twozeroth Landau levels contribute constructively to the force exerted on thenanowire in the presence of the additional electric field E0. When more884.6. Summarymagnons are pumped by B into the zeroth Landau levels, the force increasesbyF e,B = V (ne,BR + ne,BL )|v⌘z |gµBE 0c2xˆ, (4.86)where the magnon concentration variation is given by Eq. 4.67 asne,BR + ne,BL ⇡nB(gµBB0)2⇡2l2egµBB~ ⌧e,B. (4.87)Assuming ⌧e,B ⇠ 106s leads to an AFM measurable force increase F e,B ⇠1015N as a signature of both the magnon heat anomaly (Eq. 4.70) and theanomalous transport (Eqs. 4.71, 4.72). Unlike the case analyzed in Sec-tion 4.5.1 and Section 4.5.3 where surface magnons have a vanishing driftingvelocity or the case analyzed in Section 4.5.2 where the effective surfacepseudo-electric field is zero, the surface magnons in heat anomaly due to e⌘and B give rise to non-trivial mechanical effect due to non-vanishing drift-ing velocity and effective surface electric field E0. However, because sur-face states and the zeroth Landau levels are counter-propagating and theirmagnon numbers always change in opposite directions, the force due to sur-face magnon concentration variation always adds constructively to the bulkforce variation F e,B, leading to an even larger AFM measurable mechanicaleffect.4.6 SummaryIn this chapter, we have derived the magnon quantum anomalies and theanomalous spin and heat currents in a Weyl ferromagnet under electromag-netic fields and strain-induced chiral pseudo-electromagnetic fields. We firstanalyze a multilayer model of a Weyl ferromagnet whose spin wave structurepossesses two linearly dispersive Weyl cones located on the kz axis at thecorners of the honeycomb lattice Brillouin zone, akin to the electronic struc-ture of Weyl semimetals. We show that the two Weyl cones are connectedby a set of surface states analogous to the “Fermi arcs” in Weyl semimetals.These surface states can be understood as the combination of chiral edgestates of each kz-fixed 2D slice of the Weyl ferromagnet, which realizes amagnon Chern insulator whose Chern number is nontrivial for the momentabetween the two Weyl points.We then analyze how the Weyl ferromagnet reacts to EM fields andstrain-induced chiral pseudo-EM fields. Under an inhomogeneous electric894.6. Summaryfield E, due to the Aharonov-Casher effect [83], the magnons will be Landau-quantized. Similar Landau quantization can be obtained by a static twist(around z axis) of the Weyl ferromagnet nanowire because an inhomoge-neous pseudo-electric field e⌘ is induced. Such a chiral pseudo-electric fieldonly lives in the vicinity of the magnon Weyl points and couples to themoppositely, leading to a pair of co-propagating zeroth Landau levels. TheLandau-quantized magnons can be manipulated by applying an inhomoge-neous magnetic field B, which contributes a Zeeman energy whose gradientacts as a driving force. A similar pumping process is realized by applyinga dynamic uniaxial strain to the Weyl ferromagnet, so that an inhomoge-neous chiral pseudo-magnetic field b⌘ is induced. Again, this field stronglydepends on the “Diracness” of the spin wave structure and only couples toWeyl magnons. Due to its chiral nature, magnons are pumped oppositely ondifferent Weyl cones.Furthermore, we show that the four possible combinations of electric field(E/e⌘) and magnetic field (B/b⌘) give rise to magnon quantum anomaliesand anomalous spin and heat currents. For (E,B), magnons are pumpedfrom one zeroth Landau level to the other, resulting in a chirality imbalancebetween the two Landau levels. Anomalous spin and heat currents ariseand are proportional to this imbalance, resembling the chiral magnetic ef-fect in Weyl semimetals. We thus call this anomaly-related transport themagnon “chiral electric effect.” For (e⌘, b⌘), magnons are also injected intoone zeroth Landau level and extracted out of the other, leading to a chi-rality imbalance as well. Remarkably, this magnon chiral anomaly due topure pseudo-electromagnetic fields has weak but non-zero anomalous spinand heat currents, unlike the electron chiral magnetic current which mustbe zero in the presence of pseudo-electromagnetic fields. This is becausemagnons are quasiparticles immune to the particle conservation law. For(E, b⌘), magnons are pumped between the surface and the bulk, giving riseto a heat imbalance between the two. For this reason, we refer to such aphenomenon as the magnon “heat anomaly” because the energy in the bulkitself is not conserved. Unfortunately, there are always an equal number ofright-moving magnons and left-moving magnons; thus no net spin and heatcurrents exist. For (e⌘,B), magnons are also pumped between the surfaceand the bulk. Due to the chiral nature of e⌘, the bulk and the surface al-ways have opposite velocities. Such a bulk-surface separation thus causesa magnon “chiral torsional effect” that gives spin and heat currents propor-tional to the bulk-surface magnon imbalance.Lastly, considering the difficulty of the direct measurement of magnonanomalous spin and heat currents, we propose AFM based force sensing904.6. Summaryexperiments. For (E,B), an additional electric field on the nanowire exertsa force FE,B caused by the magnon imbalance on zeroth Landau levels,reflecting the magnon chiral anomaly and the magnon chiral electric effect.For (e⌘, b⌘), an additional pseudo-electric field on the nanowire gives riseto a force F e,b, also caused by the magnon imbalance on zeroth Landaulevels, indicating the magnon chiral anomaly due to pseudo-EM fields. For(E, b⌘), we demonstrate that the additional pseudo-electric field causes aforce increase FE,b on the nanowire, which is a signature of the magnonheat anomaly. This allows us to detect such an anomaly experimentallythough the anomalous transport is lacking. For (e⌘,B), we elucidate thatthe additional electric field produces a force increase F e,B on the nanowire,which is a clue of the magnon heat anomaly and the magnon chiral torsionaleffect.To experimentally test the magnon quantum anomalies, we first requirea Weyl ferromagnet, which may be artificially engineered by layering thehoneycomb ferromagnet CrX3 (X = F, Cl, Br, I) [169, 170]. Weyl ferromag-nets are also proposed to occur intrinsically in the pyrochlore oxide Lu2V2O7[20]. However, our theory constructed for multilayer ferromagnets cannot bedirectly transplanted to the pyrochlore lattice, where a twist around [111]rather than z axis may be the natural choice. The second requirement is thatthe candidate materials should be flexible to allow for sufficient twist in or-der to generate a strong pseudo-electric field to Landau-quantize the magnonbands. Unfortunately, the mechanical properties regarding the flexibility ofcandidate materials are lacking, and further experimental work is neededto verify whether or not layered honeycomb ferromagnets and Lu2V2O7 aresuitable materials.There are several future directions that might be interesting to pursuebased on the present work. The first is to test whether other types of spin lat-tice deformation can induce chiral pseudo-EM fields. In the context of Weylsemimetals, Ref. [93] shows that a screw dislocation can lead to the chiraltorsional effect. It will be interesting to examine whether such deformationdesign will induce a pseudo-electric field in a Weyl magnet. Another direc-tion is to study how other Weyl bosons react to strain-induced chiral gaugefields. To date, strain-induced Landau levels have been observed in photonicgraphene [90]. Since “photonic Weyl semimetals” [29–31] have been proposedand realized, checking whether or not strain can induce chiral anomalies inthese photonic systems will also be rewarding.91Chapter 5ConclusionsIn the present dissertation, we have studied three types of Dirac materials– Weyl semimetals, Weyl superconductors, and Weyl ferromagnets, payingclose attention to their band structure under elastic lattice deformation dueto strain. The associated electronic/thermal/spin transport is carefully in-vestigated.In Chapter 2, we study a Weyl semimetal thin film under a circular bendlattice deformation. The resulting spatial tuning of the electronic orbitalsgives rise to a pseudo-magnetic field and the Landau quantization of energybands. When the curvature of the thin film is continuously adjusted, thestrain-induced pseudo-magnetic field begins to scan and the Landau levelsbegin to fall of the Fermi surface successively, leading to a periodic popula-tion on the Fermi surface. Consequently, the Shubnikov-de Haas quantumoscillation occurs in the complete absence of magnetic fields.In Chapter 3, we apply a circular bend lattice deformation to a Weylsuperconductor multilayer composed of alternately stacked topological in-sulator layers and s-wave superconductor layers. The spatially tuning ofthe electronic orbitals due to the lattice deformation is shown to Landau-quantize the charge neutral Bogoliubov quasiparticles, while the ordinarymagnetic field is not capable of bringing Landau quantization due to theMeissner effect. When the lattice deformation varies, the thermal conduc-tivity begins to oscillate. Remarkably, such quantum oscillation occurs inthe superconducting regime, which is not accessible by the ordinary magneticfield, because strain cannot be screened by superconductivity.In Chapter 4, we investigate the Weyl ferromagnet nanowire under astatic torsional strain and a dynamic uniaxial strain and discover that theformer lattice deformation provides magnon Landau quantization while thelatter is responsible for magnon pumping. Considering the magnon Landaulevels under an inhomogeneous electric field and magnon pumping under aninhomogeneous magnetic field, we propose that the torsional strain induces apseudo-electric field while the uniaxial strain induces a pseudo-magnetic field.The combination of electromagnetic/pseudo-electromagnetic fields gives riseto magnon chiral anomaly and magnon heat anomaly. And the associated92Chapter 5. Conclusionsspin and heat transport known as magnon chiral electric effect and magnonchiral torsional effect are derived to possess unique electromagnetic field(gradient) dependence.The work presented in this dissertation provides a novel way of manipu-lating fermionic and bosonic quasiparticles and may have potential applica-tion to experiments which require large electromagnetic fields. For example,the observation of magnon Dirac-Landau levels may require an extremelyinhomogeneous electric field with gradient ⇠ 1022V/m2, which may not beeasily accessible by an ordinary electric field but can be fairly easy to berealized by deforming the lattice constant of the spin lattice by just a fewpercent.Based on the present dissertation, there are two further directions worthyto be investigated. 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In this section, we argue thatsimilar zero-field quantum oscillations can be realized in Dirac semimetalCd3As2. And we will verify our argument by examining the band structureof Cd3As2.We model Cd3As2 using Hamiltonian Eq. 2.1 with parameters taken fromfirst principles band structure calculation [85]. We use rectangular lattice,which captures the actual geometry of the crystal lattice of Cd3As2, ratherthan assuming cubic lattice as we did in Section 2.1. The lattice constantsare ax = ay = 3A˚ and az = 5A˚. The other parameters to be used in Eq. 2.1are listed in Table. A.1.Table A.1: Parameters of Dirac semimetal Cd3As2. All quantities are mea-sured in terms of electron volt (eV).t0 t1 t2 ⇤ r0 r1 r2-7.8411 1.5016 3.0 0.296 5.9439 -0.8472 -2.5556The results for the energy dispersion and DOS for the realistic particle-hole asymmetric case are shown in Fig. A.1. We note the similarity tothe results displayed in Fig. 2.3 in Section 2.3. Specifically, both ordinarymagnetic field B and strain-induced pseudo-magnetic field b give rise topronounced Landau levels. We thus conclude that all our predictions remainvalid for the Dirac semimetal Cd3As2.105Appendix A. Electronic structure of Dirac semimetal Cd3As2-30-20-100102030-0.4 -0.2 0.0 0.2 0.4-30-20-100102030-0.4 -0.2 0.0 0.2 0.4-0.4 -0.2 0.0 0.2 0.4-30-20-100102030-0.4 -0.2 0.0 0.2 0.4-30-20-1001020300.000.020.040.060.080.10-20 -10 0 10 20-20 -10 0 10 200.000.050.100.150.200.25  E[meV]  E[meV]  E[meV]  E[meV]E[meV]  DOS  DOSE[meV]Γ ZY kzky Γ ZY kzky↑ ↓ total DOSb=4.25TB=4.25TFigure A.1: Numerically calculated band structure and density of states forDirac semimetal Cd3As2 with both spin sectors and particle-hole asymmetricterm ✏k considered. Top row is for the pseudo-magnetic field b = 4.25T andthe bottom row is for the ordinary magnetic field B = 4.25T. From left toright – band structure of spin up sector, band structure of spin down sector,and normalized total DOS. The appearance of Landau levels is obviouslyshowed in all panels.106Appendix BWeyl superconductor with avortex latticeIn this section we study Weyl superconductors under ordinary magnetic fieldB and compare the results to Section 3.2. Due to the Meissner effect, B fieldis known to generate quasiparticle Bloch waves rather than Dirac-Landaulevels in 2D nodal superconductors, such as those with a d-wave symmetryof the gap function [131, 132]. It is however not known how this resulttranslates to three-dimensional Weyl SC.We consider magnetic field along z-direction, so that kz remains a goodquantum number. Thus, the system can, in principle, stay gapless. To studythe vortex lattice, we write Eq. 3.1 asH =12Xk †rHr r = 12Xk †r✓H11r H12rH21r H22r◆ r, (B.1)with the real space basis to be written as r = (cr,",1, cr,#,1, cr,",2, cr,#,2, c†r,",1,c†r,#,1, c†r,",2, c†r,#,2)T and the blocks are defined asH11r =0BBBBB@m4b+bPsˆ i ~vF2aP⌘ˆ⇤ ts+tdeikza 0i ~vF2aP⌘ˆ m+4bbPsˆ 0 ts+tdeikzats+tdeikza 0 m4b+bPsˆ i~vF2aP⌘ˆ⇤0 ts+tdeikza i~vF2aP⌘ˆ m+4bbPsˆ1CCCCCA , (B.2)H22r =0BBBBB@m+4bbPsˆ i ~vF2aP⌘ˆ tstdeikza 0i ~vF2aP⌘ˆ⇤ m4b+bPsˆ 0 tstdeikzatstdeikza 0 m+4bbPsˆ i~vF2aP⌘ˆ0 tstdeikza i ~vF2aP⌘ˆ⇤ m4b+bPsˆ1CCCCCA , (B.3)H12r =✓ 0  0 0 0 0 00 0 0 0 0  0◆, H21r =✓ 0 ⇤ 0 0⇤ 0 0 00 0 0 ⇤0 0 ⇤ 0◆. (B.4)107Appendix B. Weyl superconductor with a vortex latticeHere the shift operator is defined assˆf(r) = f(r + )  = ±axˆ,±ayˆ, (B.5)and⌘ˆ =(⌥isˆ if  = ±axˆ±sˆ if  = ±ayˆ. (B.6)To model vortex lattice, the phase of (r) = 0ei(r) is taken to wind by 2⇡around each vortex center. We solve the problem by performing a unitarytransformation [131] in the Nambu space defined byU =✓eiA(r) 00 eiB(r)◆, (B.7)where we have partitioned vortices into two sublattices A and B such thatA(r)+B(r) = (r). This removes the phase winding from the off-diagonalpart of the Hamiltonian and makes it periodic in real space with a unit celldepicted in Fig. B.1.Figure B.1: Schematic plot of square vortex lattice. The red and blue dotscorrespond to two vortex sublattices. The orange square is the magnetic unitcell with vortices placed on the diagonal. The dimension of the magnetic unitcell is chosen to be L = 30a in the simulation.The eigenstates of the transformed Hamiltonian are Bloch waves [131–133] that read nK(r) = eiK·r[UnK(r), VnK(r)]T with crystal momentumK associated with the vortex lattice (Fig. B.1). The BdG type Bloch Hamil-tonian isHK = eiK·rU1HrUeiK·r with its 4 blocksH ijK = eiK·rU1H ijr UeiK·r108Appendix B. Weyl superconductor with a vortex latticedefined asH11K =0BBBBB@m 4b+ bPeiK·eiVA sˆ i~vF2aPeiK·eiVA ⌘ˆ⇤i~vF2aPeiK·eiVA ⌘ˆ m+ 4b bPeiK·eiVA sˆts + tdeikza 00 ts + tdeikzats + tdeikza 00 ts + tdeikzam 4b+ bPeiK·eiVA sˆ i~vF2aPeiK·eiVA ⌘ˆ⇤i~vF2aPeiK·eiVA ⌘ˆ m+ 4b bPeiK·eiVA sˆ1CCCCCA, (B.8)H22K =0BBBBB@m+ 4b bPeiK·eiVB sˆ i~vF2aPeiK·eiVB ⌘ˆi~vF2aPeiK·eiVB ⌘ˆ⇤ m 4b+ bPeiK·eiVB sˆts  tdeikza 00 ts  tdeikzats  tdeikza 00 ts  tdeikzam+ 4b bPeiK·eiVB sˆ i~vF2aPeiK·eiVB ⌘ˆi~vF2aPeiK·eiVB ⌘ˆ⇤ m 4b+ bPeiK·eiVB sˆ1CCCCCA,(B.9)H12K =0BB@0  0 0 0 0 00 0 0 0 0  01CCA , (B.10)H21K =0BB@0 ⇤ 0 0⇤ 0 0 00 0 0 ⇤0 0 ⇤ 01CCA , (B.11)where the phase factors associated with two types of vortices areVµ =m~ˆ r+rvµs (r) · dl µ = A,B. (B.12)109Appendix B. Weyl superconductor with a vortex latticeThe integral is along the bond connecting lattice point r to its nearest neigh-bor r + . The superfluid velocity isvµs (r) =~mrµ  e~A(r) µ = A,B. (B.13)Following the standard derivation [132] an expression for Vµ can be derivedin terms of summation over the reciprocal lattice vectors G of the vortexlattice,Vµ (r) =2⇡L2XGˆ r+reiG·(rµ) iG⇥ zˆG2· dl. (B.14)We apply Eq. B.14 to the real space Hamiltonian Hr and exactly diagonal-ize Hr for various vortex lattice configurations. The dispersions along thekz-axis are summarized in Fig. B.2(a-c). We observe that the Weyl pointssurvive as we change the A-B vortex distance within each unit cell. Sur-prisingly, the variation of the vortex positions barely changes the dispersion.Therefore, we conclude that the kz component of the Weyl dispersion isstable under magnetic field B as long as vortices form a periodic lattice.Dispersion in the Kx  Ky plane however changes dramatically. InFig. B.2(d-f), we plot dispersions along Kx for the vortex configurationsused in panels (a-c). We see that the energy bands are reorganized into al-most completely flat Dirac-Landau levels which are qualitatively similar tothose reported by Ref. [139]. For comparison we also indicate the expectedenergies ⇠ pn of Dirac-Landau levels (orange curves) by matching to then = 0, 1 bands. It is worth noting that the deviation of numerically calcu-lated bands (green curves) from the idealpn sequence is due to the factthat Dirac-Landau levels exist only in the low-energy regime in the vicinityof the Weyl points. For our model, Lifshitz transition occurs at ELif = 0.138.Therefore, we do not expect a perfect match to thepn behavior beyond thelowest few energy levels.110Appendix B. Weyl superconductor with a vortex lattice0.000.060.120.180.240.300.0 0.1 0.2 0.3 0.40.000.060.120.180.240.300.0 0.1 0.2 0.3 0.40.000.060.120.180.240.300.0 0.1 0.2 0.3 0.40.000 0.025 0.050 0.075 0.1000.000.060.120.180.240.300.000 0.025 0.050 0.075 0.1000.000.060.120.180.240.300.000 0.025 0.050 0.075 0.1000.000.060.120.180.240.30(f)(e)(d)(c)(b) kzE  kzE   kzE  EKx(a)  EKx  EKxFigure B.2: Spectra of Weyl superconductor with vortex lattice. The sizeof magnetic unit cell is L ⇥ L = 30a ⇥ 30a. The spacings between twovortices in the magnetic unit cell are (a) d = (15a, 15a) (b) d = (10a, 10a)(c) d = (5a, 5a) (d) d = (15a, 15a) (e) d = (10a, 10a) (f) d = (5a, 5a) Theorange curves in panel (d)-(f) are analytical Dirac-Landau levels with n = 1band matched to the numerics.111Appendix CWeyl ferromagnets underelectric field and strainIn Chapter 4, we have seen that the magnon bands of the Weyl ferromag-net can be Landau-quantized by either applying an inhomogeneous electricfield E (Section 4.2.1) or applying a twist which induces an inhomogeneouspseudo-electric field e⌘ (Section 4.2.2). The former acts on the whole Weylferromagnet, while the latter only couples to Weyl magnons and is greatlysuppressed at higher energies where the “Diracness” of magnon bands is nolonger preserved. Moreover, the pseudo-electric field is chiral and oppositelycouples to different Weyl cones, resulting in two identically dispersing zerothLandau levels, as illustrated in Fig. 4.6.We now further test the chiral nature of the strain-induced pseudo-electric field e⌘ = ⌘e. When an inhomogeneous electric field E is applied inaddition to the twist, the effective electric field for the right (left) magnonWeyl cone is ER = E + e (EL = E  e). For the special case that E = e,the left magnon Weyl cone feels no electric field but the electric field atthe right magnon Weyl cone is doubled. Therefore, the left Weyl cone isunchanged but the right Weyl cone is Landau-quantized as illustrated inFig. C.1(c). Compared to Fig. 4.4(c) and Fig. 4.6(c), the number of magnonDirac-Landau levels in Fig. C.1(c) is halved due to the doubling of the effec-tive electric field. These Dirac-Landau levels correspond to the even orderLandau levels (n = 2, 4, · · · ) in Fig. 4.4(c) and Fig. 4.6(c).112Appendix C. Weyl ferromagnets under electric field and strain(a) (b) (c)Figure C.1: Magnon dispersion for a twisted Weyl ferromagnet nanowirein the presence of an inhomogeneous electric field. For all panels, we takegµBEa2ec2 =gµB"a2ec2 = 0.01240. (a) Magnon band structure. Due to thechiral nature of the strain-induced pseudo-electric field, the effective electricfield at the left Weyl cone vanishes while the effective field at the right Weylcone is doubled. Therefore, the left Weyl cone is not Landau-quantizedbut the right Weyl cone exhibits Dirac-Landau levels. (b) Surface spectralfunction, which shows a set of left-moving surface states connecting the leftWeyl cone and the right zeroth Landau level. (c) Bulk spectral function,which clearly unveils the linear band touching at the left Weyl cone, andDirac-Landau levels at the right Weyl cone. Compared to Fig. 4.4(c) andFig. 4.6(c), the Landau level spacing is doubled due to the doubling of theeffective electric field.113Appendix DMagnon bands of multilayerWeyl ferromagnetsIn Chapter 4, we have neglected the J+(1  cos kza)S0 term in the first-quantized Bloch HamiltonianHk (Eq. 4.7) for ease of presentation. (We havecarefully ensured that the magnon energy remains positive-definite relativeto the ferromagnetic ground state.) In general, however, J+ > 0 becausethe couplings JA and JB are both ferromagnetic. For this reason, we addthe J+ term back in this section and discuss its effects on the magnon bandstructure, pumping, and transport.First, we examine the magnon band structure with the advent of theJ+ term. In the absence of an inhomogeneous electric field and strain, thisterm only shifts the magnon bands in Fig. 4.2 by J+(1 cos kza)S withoutaltering the band topology. Thus the Chern number (Eq. 4.11) is still validand guarantees that there are surface states akin to Fermi arcs connectingthe magnon Weyl cones. When a transverse inhomogeneous electric fieldE = (12Ex, 12Ey, 0) is switched on, Aharonov-Casher phases must be addedto the magnon “hopping” terms (the first three terms of Eq. 4.7). However,the diagonal termmk0 = [K++J+(1cos kza)+3J1+6J2]S0 correspondsto an “on-site” magnon energy whose Aharonov-Casher phase is zero; thusthe J+ term is invariant under the electric field. On the other hand, whena static twist is applied, an extra term Hek must be added to the first-quantized Bloch Hamiltonian Hk, but as shown in Eq. 4.17, J+ does notcontribute to Hek. Therefore, even in the presence of (pseudo-)electric fields,the effect of the J+ term is simply shifting the magnon bands in Fig. 4.4,Fig. 4.6 and Fig. C.1 by J+(1  cos kza)S. The magnon band structure inthe presence of the J+ term is summarized in Fig. D.1.114Appendix D. Magnon bands of multilayer Weyl ferromagnetskz-π π/2 0 π/2 πE2124273033kz-π π/2 0 π/2 πE2124273033kz-π π/2 0 π/2 πE2124273033kz-π π/2 0 π/2 πE2124273033(a) (b)(c) (d)Figure D.1: Magnon dispersion for the Weyl ferromagnet nanowire. For allpanels, the parameters are same as those of Fig. 4.2 in Chapter 4 exceptthat we reintroduce a nonzero J+S = 4.08. (a) Magnon band structureof a nanowire without external fields. Due to the nonzero J+ the Weylcones and arc states are tilted. (b) Magnon band structure of a nanowireunder an inhomogeneous external electric field whose gradient E satisfiesgµBEa2ec2 = 0.01240. The Dirac-Landau levels are tilted by J+ such thatthe velocity of the right (left) zeroth Landau level is |v⌘z |+ |v⌘0 | (|v⌘z | |v⌘0 |).(c) Magnon band structure of a twisted nanowire. The gradient of the strain-induced pseudo-electric field " satisfies gµB"a2ec2 = 0.01240. The Dirac-Landau levels are tilted by J+ such that the velocity of the right (left) zerothLandau level is |v⌘z |+|v⌘0 | (|v⌘z ||v⌘0 |). (d) Magnon band structure of a twistednanowire under an inhomogeneous external electric field, with gµBEa2ec2 =gµB"a2ec2 = 0.01240. Due to the chiral nature of the strain-induced pseudo-electric field, the effective electric field at the left Weyl cone vanishes whilethe effective electric field at the right Weyl cone is doubled. Therefore, theleft tilted Weyl cone is not Landau-quantized but the right tilted Weyl coneexhibits tilted Landau levels.115Appendix D. Magnon bands of multilayer Weyl ferromagnetsThen, we discuss the effect of the J+ term on the magnon equations ofmotion (Eqs. 4.25, 4.30). In the presence of a magnetic field B, a Zeemanenergy U = µ · B exists for both sublattices. It is thus diagonal in thesublattice basis k = (ak, bk)T , in which the Bloch Hamiltonian Hk is de-fined. For this reason, the total potential energy of magnons is U 0 = U+mk.When mk has no spatial dependence, the gradient of the potential energy isunchanged by mk: rU 0 = rU irrespective of the value of J+. Therefore,the magnon equation of motion Eq. 4.25 is not affected when consideringJ+. On the other hand, when a dynamic uniaxial strain is applied, an ex-tra term Hbk must be added to the first-quantized Bloch Hamiltonian Hk.It is worth noting that the contribution from J+ to Hbk has already beenconsidered in Eq. 4.27. Therefore, the J+ term is still a constant correctionto the potential energy and will not affect the magnon equation of motionEq. 4.30.Lastly, we consider how the magnon quantum anomalies and the anoma-lous spin and heat currents are affected by the J+ term. It is straightforwardto see such a diagonal term has two effects. First, it shifts the magnon Weylpoints in the energy dimension by an amount of J+(1 cosQa)S. Second, italters the velocities of the two zeroth Landau levels by v⌘0 = ⌘J+Sa sinQa/~.To derive a generic theory of magnon transport in the presence of thesetwo effects, we consider a Weyl ferromagnet nanowire aligned in the z di-rection and subjected to a generalized electric field gradient E , which canbe generated by either an inhomogeneous electric field or a twist. Fol-lowing the set-up in Section 4.3, the Weyl ferromagnet nanowire is at-tached to magnon reservoirs in a uniform magnetic field B˜0 = B˜0zˆ whereB˜0 = B0 + J+(1  cosQa)S/gµB, so that the magnon population edges onboth zeroth Landau levels remains tuned into the gap spanned by the firstLandau levels. Then a generalized magnetic field gradient B generated byeither an inhomogeneous magnetic field or a dynamic uniaxial strain is ap-plied to the nanowire, under which magnons are pumped along Landau levelsaccording to the generalized semiclassical equation of motionqR/Lz (t) = qR/Lz (0) sR/LgµBˆ t0Bdt0/~, (D.1)where the index sR/L indicates the nature of the magnetic field such that(sR, sL) =((+1,+1) for magnetic field(+1,1) for pseudo-magnetic field . (D.2)During the pumping process, the magnon population edge of the right/leftzeroth Landau level is slightly shifted by R/L ⌧ B˜0, so that the right/left116Appendix D. Magnon bands of multilayer Weyl ferromagnetsWeyl cone experiences a magnetic field BR/L = B˜0  R/L. From the gener-alized semiclassical equation of motion Eq. D.1, we obtain the magnetic fieldvariation for the right/left Weyl cone asR/L = sR/LvR/Lˆ t0Bdt0. (D.3)By comparing to Eqs. 4.34, 4.35, the magnon concentration variation on theright/left zeroth Landau level readsnR/L =ˆ gµBB˜0gµBBR/LgR/L(✏)nB(✏)d✏⇡ nB(gµBB˜0)4⇡2l2EgµBR/L~|vR/L| n0B(gµBB˜0)2(gµBR/L)24⇡2l2E ~|vR/L|, (D.4)where we take the limit gµBR/L ⌧ kBT . The magnon density of stateson the right/left zeroth Landau level is gR/L(✏) = 12⇡l2E12⇡~|vR/L| with thegeneralized electric length lE = (|~c2/gµBE |)1/2. Due to the variation ofmagnon population, thermal energy will be injected into or depleted fromthe right/left zeroth Landau level. Explicitly, the thermal energy variationcan be obtained by referring to Eqs. 4.39, 4.40 asUR/L =ˆ gµBB˜0gµBBR/L✏gR/L(✏)nB(✏)d✏⇡ 14⇡2l2E ~|vR/L|⇢nB(gµBB˜0)gµBB˜0gµBR/L 12[nB(gµBB˜0) + gµBB˜0n0B(gµBB˜0)](gµBR/L)2, (D.5)where we again approximate gµBR/L ⌧ kBT . The magnon chiral anomalyand heat anomaly to the linear order in gµBR/L are given byd⇢5dt= RdnRdt+ LdnLdt⇡ nB(gµBB˜0)4⇡2l2E ~gµBB(sRsgn(vR) sLsgn(vL)),(D.6)d⇢heatdt=dURdt+dULdt⇡ nB(gµBB˜0)4⇡2l2E ~g2µ2BB˜0B(sRsgn(vR)+ sLsgn(vL)).(D.7)117Appendix D. Magnon bands of multilayer Weyl ferromagnetsThese anomalies only depend on the signs of velocities of the zeroth Lan-dau levels. In the presence of an electric field, the inclusion of J+ onlychanges the magnitude of vR/L without altering the directions of magnonpropagation. Consequently, Eq. D.6 is reduced to Eq. 4.36 and Eq. D.7 isreduced to Eq. 4.62, and we still have the chiral anomaly for (E,B) andthe heat anomaly for (E, b). In the presence of a pseudo-electric field, J+changes the velocity of the right (left) zeroth Landau level to vR = |v⌘z |+ |v⌘0 |(vL = |v⌘z |  |v⌘0 |). For a type-I Weyl ferromagnet, |v⌘z | > |v⌘0 |, thus theadvent of the J+ term does not flip the sign of vL. Consequently, Eq. D.6 isreduced to Eq. 4.49 and Eq. D.7 is reduced to Eq. 4.69, preserving the chiralanomaly for (e, b) and the heat anomaly for (e,B). On the other hand, fora type-II Weyl ferromagnet, |v⌘z | < |v⌘0 |, and the sign of vL is flipped. In thiscase, (e, b) will have a heat anomaly but (e,B) will have a chiral anomaly.The anomalous spin and heat currents can be derived asJspin = ~(nRvR + nLvL) ⇡ nB(gµBB˜0)4⇡2l2EgµB[Rsgn(vR) + Lsgn(vL)]+n0B(gµBB˜0)8⇡2l2Eg2µ2B[2Rsgn(vR) + 2Lsgn(vL)], (D.8)Jheat = vRUR + vLUL ⇡ nB(gµBB˜0)4⇡2l2E ~g2µ2BB˜0[Rsgn(vR) + Lsgn(vL)] n0B(gµBB˜0)8⇡2l2E ~g3µ3BB˜0[2Rsgn(vR) + 2Lsgn(vL)] nB(gµBB˜0)8⇡2l2E ~g2µ2B[2Rsgn(vR) + 2Lsgn(vL)]. (D.9)Unlike the anomalies (Eqs. D.6, D.7), the anomalous spin and heat currentsdepend on the magnitude of vR/L as well because R/L is proportional tovR/L according to Eq. D.3. For (E,B), both zeroth Landau levels are steeperwhen the diagonal term is considered (vR = vL = |v⌘z | + |v⌘0 |). Therefore,the anomalous spin and heat currents are enhanced and the chiral electriceffect becomes more pronounced. For (E, b), both the spin current and theheat current are zero because there are an equal number of right-movingmagnons at the speed of vR = |v⌘z | + |v⌘0 | and left-moving magnons at thespeed of vL = |v⌘z |  |v⌘0 |. For (e, b), importantly, the terms linear inR/L in Eqs. D.8, D.9 are non-vanishing regardless of the sign of vR/L, be-cause |R| 6= |L| when the J+ term is considered. Therefore, the anomalous118Appendix D. Magnon bands of multilayer Weyl ferromagnetsspin/heat current Je,bspin/heat ⇠ b25 (Eqs. 4.57, 4.58) will be obscured by thedominant linear terms. Nevertheless, these less dominant chiral electric cur-rents can in principle be extracted because they have unique pseudo-EM fielddependence. For (e,B), the chiral torsional spin/heat current Je,Bspin/heat ⇠ B(Eqs. 4.71, 4.72) is linear in the magnon population edge variation B, whichis now replaced by 12 [Rsgn(vR)+ Lsgn(vL)] in Eqs. D.8, D.9. Such a varia-tion, however, affects no qualitative changes to anomalous currents. To provethis, we reproduce Fig. 4.8 with the J+ term considered. As illustrated inFig. D.2, for a Weyl ferromagnet nanowire with a either rectangular or circu-lar cross section, the bulk-surface separation for anomalous currents persists.JspinsurfJspinbulkJheatbulk JheatsurfJspinsurfJspinbulkJheatbulk Jheatsurf(f)(c)(b)(e)(a)(d)Figure D.2: Reproduction of bulk-surface separation for the twisted Weylferromagnet nanowire (Fig. 4.8) with a nonzero J+S = 4.08. Though theWeyl cones are displaced and tilted, the bulk-surface separation of spin andheat currents is preserved for both the rectangular cross section (b, c) and thecircular cross section (e, f). The total spin current on the rectangular (cir-cular) cross section is 0.0017DS (0.0016DS) while the total heat currenton the rectangular (circular) cross section is 0.046D2S2/~ (0.0423D2S2/~).119Appendix ECircular bend inducedpseudo-electric fieldIn Chapter 4, we propose that the magnon chiral anomaly (Eq. 4.50) andthe magnon heat anomaly (Eq. 4.63) may have experimentally measurablemechanical effects if an additional chiral pseudo-electric field e0⌘ = ⌘(0, "0z, 0)is applied. In this section, we will elaborate the implementation of such anadditional pseudo-electric field.We consider a simple circular bend lattice deformation as illustrated inFig. E.1. As explained in Refs. [134, 176], to the lowest order of approxima-tion, the displacement field of such a circular bend is u = U xzzˆ, resultingin nonzero strain tensor components u13 = u31 = 12U z and u33 = U x. Thestrain effect can then be incorporated by the following exchange integralsubstitutionsJ2(↵1 ± azˆ)! J2(1⌥p32 u31  12u33),J2(↵2 ± azˆ)! J2(1±p32 u31  12u33),J2(↵3 ± azˆ)! J2(1 12u33),JA ! JA(1 u33),JB ! JB(1 u33),which result in an effective HamiltonianHk⌘W+q + Hk⌘W+q ⇡ Hk⌘W + ~v⌘0⇣qz +e~a⌘z⌘0+Xi~v⌘i⇣qi +e~a⌘i⌘i  3J2Su330. (E.1)Here the strain-induced vector potential isa⌘ = ⌘ ~ea✓2J2 sinQaJ1 + 2J2 cosQau31, 0,1 cosQasinQau33◆, (E.2)which is incorporated by a minimal substitution in the same way as a⌘Sand a⌘D. Again, a non-chiral on-site term 3J2Su330 appears, but such a120Appendix E. Circular bend induced pseudo-electric fieldyxzxzy(a)(b)Figure E.1: Schematic plot for the Weyl ferromagnet nanowire. (a) Nanowireunder a circular bend deformation. (b) Lattice site positions without defor-mation (left) and with a circular bend (right).term is negligible when ~|v⌘z |/a  J2S, and it can be otherwise cancelledby an additional magnetic field B0 = 3J2Su33µ2 µ. Therefore, its effect onmagnon mechanics can be safely neglected. On the other hand, by referringto Eq. 4.30, the chiral gauge vector potential a⌘ gives rise to a magnonLorentz force~dkdt= ev ⇥ [r⇥ a⌘]. (E.3)Comparing to the magnon Lorentz force (Eq. 4.77) due to an additionalelectric field E0 = (0, E 0z, 0), we may interpret a⌘ as the vector potential ofthe additional chiral pseudo-electric field e0⌘ = ⌘(0, "0z, 0), which only differsfrom E0 by a chiral charge ⌘. The gradient of this additional pseudo-electricfield can be determined by r⇥a⌘ = 1ec2r⇥(e0⌘⇥µ) = ⌘ gµB"0ec2 yˆ. Explicitly,the field gradient reads"0 = ⌘ ec2gµByˆ · (r⇥ a⌘) = ~c2gµBaJ2 sinQaJ1 + 2J2 cosQa 1 cosQasinQaU . (E.4)As discussed in Section 4.5.2 (Section 4.5.3), when "0 is sufficiently small,i.e., "0 ⌧ " ("0 ⌧ E), the bend-induced pseudo-electric field e0⌘ generates atransverse force that may be measured by AFM, while the magnon Landaulevels are not strongly affected. On the other hand, if "0 is sufficiently large,the bend-induced pseudo-electric field e0⌘ also results in Landau quantiza-121Appendix E. Circular bend induced pseudo-electric fieldtion. In contrast to the Landau quantization due to a twist, the Landauquantization due to a bend takes place in the transverse direction.122Appendix FThe tetrahedron methodIn Section 2.4, we have analytically calculated DOS and longitudinal electricconductivity yy of WSM with semiclassical methods. We then applied sametechnique to calculate DOS and longitudinal thermal conductivity xx ofWSC in Section 3.4. In order to obtain high quality numerical data shownin Fig. 2.5 and Fig. 3.7, we have numerically calculated yy and xx usingthe tetrahedron method, which is detailed as below.We consider a multi-band lattice model with good quantum numbers kand k⌫ . The energy bands satisfy ✏n(k, k⌫) = ✏n(k, k⌫) = ✏n(k,k⌫) =✏n(k,k⌫). Therefore, the full electronic structure of this lattice modelcan be obtained by diagonalizing in the first quadrant of Brillouin zone.Numerically, the diagonalization only happens at discrete k points, whichconstitute the grid shown in Fig. F.1. We further cut each rectangular pla-quette into a pair of right triangular plaquettes, colored grey and whiterespectively, as illustrated in Fig. F.1. For a generic triangular plaquetteSk, the momentum at the right angle vertex is denoted as [k(Sk), k⌫(Sk)].We then discretize the n-th energy band on Sk. When Sk is sufficientlysmall, the n-th energy band can be approximated as a combination of manytriangular pieces in the energy-momentum space.For the triangular piece defined on Sk, the energies of vertices are denotedas ✏n1 (Sk), ✏n2 (Sk), and ✏n3 (Sk). Particularly, for the grey triangular plaquetteswe have ✏n1 (Sk) = ✏n[k(Sk), k⌫(Sk) + 2⇡L⌫ ], ✏n2 (Sk) = ✏n[k(Sk), k⌫(Sk)], and✏n3 (Sk) = ✏n[k(Sk) +2⇡L, k⌫(Sk)], while for the white triangular plaquetteswe have ✏n1 (Sk) = ✏n[k(Sk), k⌫(Sk) 2⇡L⌫ ], ✏n2 (Sk) = ✏n[k(Sk), k⌫(Sk)], and✏n3 (Sk) = ✏n[k(Sk) 2⇡L , k⌫(Sk)] with L and L⌫ measuring the dimensionof the lattice model. Thus, the dispersion of a triangular piece of energyband can be written down as✏nSk(k, k⌫) = ✏n2 (Sk)±✏n3 (Sk) ✏n2 (Sk)2⇡/L(k  k(Sk))± ✏n1 (Sk) ✏n2 (Sk)2⇡/L⌫(k⌫  k⌫(Sk)), (F.1)123Appendix F. The tetrahedron methodwhere the plus (minus) sign is for the grey (white) triangular plaquettes.And the corresponding DOS at energy µ of the lattice model isg(µ) =XnXk,k⌫(µ ✏n(k, k⌫))⇡ LL⌫⇡2XnXSk‹Skdkdk⌫(µ ✏n(k, k⌫)) = 4XnXSkgnSk(µ), (F.2)wheregnSk(µ) =[µ ✏n1 (Sk)]✓[µ ✏n1 (Sk)][(✏n2 (Sk)) ✏n1 (Sk)][(✏n3 (Sk)) ✏n1 (Sk)]+[µ ✏n2 (Sk)]✓[µ ✏n2 (Sk)][(✏n1 (Sk)) ✏n2 (Sk)][(✏n3 (Sk)) ✏n2 (Sk)]+[µ ✏n3 (Sk)]✓[µ ✏n3 (Sk)][(✏n1 (Sk)) ✏n3 (Sk)][(✏n2 (Sk)) ✏n3 (Sk)], (F.3)where ✓(✏) is the Heaviside step function. According to the Boltzmann equa-tion approach, at low temperature, the electric conductivity reads = e2XnXk,k⌫⌧n(✏n(k, k⌫))vn(k, k⌫)vn(k, k⌫)✓ @f(✏)@✏◆✏n(k,k⌫)⇡ e2LL⌫⇡2XnXSk‹Skdkdk⌫⌧n(✏nSk(kk⌫))✓1~@✏nSk@k◆2(µ ✏nSk(k, k⌫))=e2~2L2⇡2XnXSk⌧n(µ)(✏n3 (Sk) ✏n2 (Sk))2gSk(µ), (F.4)where we have used group velocity vn(k, k⌫) =1~@✏n(k,k⌫)@kand f(✏) =1e(✏µ)/kBT+1 is the Fermi function with chemical potential µ and temperatureT . On the other hand, Boltzmann equation gives thermal conductivity as =1TXnXk,k⌫(✏n(k, k⌫) µ)2⌧n(✏n(k, k⌫))⇥ vn(k, k⌫)vn(k, k⌫)✓ @f(✏)@✏◆✏n(k,k⌫), (F.5)124Appendix F. The tetrahedron methodSk✏n1 (Sk)✏n2 (Sk)✏n3 (Sk)kk✏n(k, k)Sk✏n2 (Sk)✏n1 (Sk)✏n3 (Sk)Figure F.1: Discretization of energy band in the tetrahedron method. TheBrillouin zone spanned by (k, k⌫) is first discretized into rectangular grid.Then each rectangular plaquette is cut into a pair of right triangular pla-quettes colored grey and white. The n-th energy band ✏n(k, k⌫) is thendiscretized on the both types of triangular plaquettes. On each triangularplaquette Sk, the discretized piece of energy band can be approximated ashaving zero curvature and each piece has vertex energies ✏n1 (Sk), ✏n2 (Sk), and✏n3 (Sk).which, by using the Sommerfeld expansion, can be further simplified as =⇡2k2BT3XnXk,k⌫⌧n(✏n(k, k⌫))vn(k, k⌫)vn(k, k⌫)✓ @f(✏)@✏◆✏n(k,k⌫),(F.6)which reflects the Wiedemann-Franz law. According to Eq. F.4, we candirectly write down the thermal conductivity as ⇡ ⇡2k2BT31~2L2⇡2XnXSk⌧n(µ)(✏n3 (Sk) ✏n2 (Sk))2gSk(µ). (F.7)We now consider the longitudinal electric conductivity yy in Weyl semimet-125Appendix F. The tetrahedron methodals. According to Eq. 2.25, Eq. F.4 can be rewritten asyy(b) =12⇡~ nimpCimpg(µ)e2~2L2y⇡2XnXSk(✏n3 (Sk) ✏n2 (Sk))2gSk(µ)=e2(v⌘y)22⇡~ nimpCimp1~2(v⌘y)2L2y4⇡2PnPSk(✏n3 (Sk) ✏n2 (Sk))2gSk(µ)PnPSkgSk(µ)= yy(0)PnPSk(✏n3 (Sk) ✏n2 (Sk))2gSk(µ)2⇡~v⌘yLy2PnPSkgSk(µ), (F.8)where ✏n3 (Sk) and ✏n2 (Sk) can be extracted by numerically diagonalizingEq. 2.9. And yy(b)/yy(0) is plotted in Fig. 2.5 for µ = 10meV and inFig. 2.6 for µ = 28meV, both of which exhibit Shubnikov-de Haas oscilla-tion.For the thermal conductivity of Weyl superconductor, according to Eq. 3.80,we rewrite Eq. F.7 asxx(b) =⇡2k2BT31~2L2x⇡2XnXSk⌧n(0)(✏n3 (Sk) ✏n2 (Sk))2gSk(0)=12⇡~ nimpCimp⇡2k2BT31~2L2x4⇡2PnPSk(✏n3 (Sk) ✏n2 (Sk))2gSk(0)PnPSkgSk(0)= xx(0)PnPSk(✏n3 (Sk) ✏n2 (Sk))2gSk(0)2⇡~vxLx2PnPSkgSk(0). (F.9)Again, ✏n3 (Sk) and ✏n2 (Sk) can be extracted by numerically diagonalizingEq. 3.13. And xx(b)/xx(0) is plotted in Fig. 3.7, exhibiting quantum os-cillation.126

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