Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Can, Oguzhan
Doctor of Philosophy - PhD
University of British Columbia
In this thesis we explore the physics of moiré heterostructures of few-layer d-wave superconductor thin films. We show that such a twisted bilayer stack exhibits spontaneous time reversal symmetry breaking near 𝜃 = 45º twist, leading to a chiral superconductor state with topological edge modes. We then discuss the signatures of this prediction in polar Kerr effect experiments and find that expected Kerr rotation is three orders of magnitude larger compared to strontium ruthenate, one of the most studied chiral topological superconductor candidates. Another experimental signature of this novel phase we predict turns out to be the superconducting diode effect, which has recently been observed in twisted cuprate bilayers. Inspired by this experimental observation and the Josephson current-phase behaviour of this system, we introduce a novel superconducting circuit element made out of arrays of conventional Josephson junctions which can be printed on a Al or Nb superconducting chip relying on available fabrication technology for practical applications. Finally, we propose a novel superconducting qubit design and discuss its advantages over existing superconducting qubits.
Topological superconductivity intwisted cupratesand device applications inspired by their JosephsonphysicsbyOguzhan CanB.Sc., The University of Toronto, 2016M.Sc., The University of British Columbia, 2018A 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)February 2024c© Oguzhan Can, 2024The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the thesis entitled:Topological superconductivity in twisted cuprates and device applicationsinspired by their Josephson physicssubmitted by Oguzhan Can in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Physics.Examining Committee:Marcel Franz, Professor, Physics and Astronomy, UBCSupervisorAndrea Damascelli, Professor, Physics and Astronomy, UBCSupervisory Committee MemberJoshua Folk, Professor, Physics and Astronomy, UBCUniversity ExaminerLukas Chrostowski, Professor, Electrical and Computer Engineering, UBCUniversity ExaminerJesko Sirker, Professor, Physics and Astronomy, University of ManitobaExternal ExaminerAdditional Supervisory Committee Members:Mona Berciu, Professor, Physics and Astronomy, UBCSupervisory Committee MemberJeremy Heyl, Professor, Physics and Astronomy, UBCSupervisory Committee MemberiiAbstractIn this thesis we explore the physics of moire´ heterostructures of few-layerd-wave superconductor thin films. We show that such a twisted bilayer stackexhibits spontaneous time reversal symmetry breaking near θ = 45o twist,leading to a chiral superconductor state with topological edge modes. Wethen discuss the signatures of this prediction in polar Kerr effect experimentsand find that expected Kerr rotation is three orders of magnitude largercompared to strontium ruthenate, one of the most studied chiral topologicalsuperconductor candidates. Another experimental signature of this novelphase we predict turns out to be the superconducting diode effect, whichhas recently been observed in twisted cuprate bilayers. Inspired by thisexperimental observation and the Josephson current-phase behaviour of thissystem, we introduce a novel superconducting circuit element made out ofarrays of conventional Josephson junctions which can be printed on a Alor Nb superconducting chip relying on available fabrication technology forpractical applications. Finally, we propose a novel superconducting qubitdesign and discuss its advantages over existing superconducting qubits.iiiLay SummaryA superconductor is a quantum state of matter that emerges at low tem-peratures, characterized by its zero resistance and the complete expulsionof magnetic fields. In this thesis, we demonstrate that when two supercon-ducting thin films are stacked with a twist, the resulting system becomes anovel quantum state of matter known as a topological superconductor. Thissystem is known as “twisted cuprates” with “cuprate” referring to a specificclass of copper and oxygen based superconductors we studied in this work.The novelty of this state is marked by the presence of exotic particlesconfined to its boundaries, including defects like holes. These boundaryparticles are unique in that they can only exist at the superconductor’sedges and cannot exist independently in free space, unlike more familiarparticles like electrons. We also discuss experimental methods, particularlyoptical measurements, which can validate our findings by examining howlight reflects off the sample surface.Finally, inspired by the physics of twisted cuprates, we propose two in-novative superconducting device designs. The first one is a superconductingdiode, a two-terminal device permitting supercurrent flow in one directiononly. The second is a qubit, or a quantum bit which is memory element fora quantum computer.ivPrefaceContent of this thesis has been adapted from published work except forChapter 5.Chapter 2: Can, O. et al. “High-temperature topological superconduc-tivity in twisted double-layer copper oxides”. Nature Physics 17, 519–524.issn: 1745-2481 (2021): Microscopic lattice model for twisted cuprates in[1] have been developed by myself and Tarun Tummuru. I have derived thegeneralized gap equation using path integral methods, generated the latticemodel phase diagram and produced Fig. 3 (except for panels g,h). Mar-cel Franz who came up with the original idea and project design and alsocarried out the GL theory and continuum model calculations. Ryan Daycarried out the ab-initio calculations under supervision of Ilya Elfimov andAndrea Damascelli. The complete code used to obtain results shown in Fig.2.3 is publicly available at GitHub. The manuscript was drafted by MarcelFranz with input from all authors.Chapter 3: Can, O., Zhang, X.-X., Kallin, C. and Franz, M. “ProbingTime Reversal Symmetry Breaking Topological Superconductivity in TwistedDouble Layer Copper Oxides with Polar Kerr Effect”. Phys. Rev. Lett.127, 157001 (2021). Hall conductance and Kerr angle calculations are doneby myself and Xiao-Xiao Zhang. I have derived an expression for Hall con-ductance suitable for numerical calculations utilizing the lattice model fromthe original proposal and generated all the figures in the manuscript. Theproject has been conceived by Marcel Franz and we consulted CatherineKallin throughout the project. Manuscript has been drafted by myself withinput from all authors.Chapter 4: Haenel, R. and Can, O. “Superconducting diode from fluxbiased Josephson junction arrays.” arXiv:2212.02657 A reproduction of thepublished pre-print (currently under peer-review) [2] based on an originalidea of mine to use Josephson junctions to harness the superconductingdiode effect with quantum circuits which are easily scalable rather thanrelying on quantum materials/heterostructures with fabrication challenges.Rafael Haenel suggested the SNAIL geometry, a maximal efficieny ansatzand carried out the numerics. I have worked out the generalization of thevList of publications related to the thesismodel to N-arms, including the RSJ equations of motion. The main texthas been drafted by Rafael Haenel with input from myself.Chapter 5 :Can, O. and Franz, M. “NMon, a novel superconducting qubitproposal” This is a reproduction of a manuscript based on an original ideaof mine. The manuscript has been written by myself with input from MarcelFranz.List of publications related to the thesis1. “d-Mon: A Transmon with Strong Anharmonicity Based on Planar c-axis Tunneling Junction between d-wave and s-wave Superconductors”H Patel, V Pathak, OC, AC Potter, M Franz. Phys. Rev. Lett. 132,017002 (2024)2. “Superconducting diode from flux biased Josephson junction arrays”R Haenel, OC arXiv preprint arXiv:2212.02657 (2022)3. “Probing time reversal symmetry breaking topological superconduc-tivity in twisted double layer copper oxides with polar Kerr effect”OC, XX Zhang, C Kallin, M Franz. Physical Review Letters 127(15), 157001 (2021)4. “Chiral p-wave superconductivity in a twisted array of proximitizedquantum wires” T Tummuru, OC, M Franz. Physical Review B 103(10), L100501 (2021)5. “High-temperature topological superconductivity in twisted double-layer copper oxides” OC, T Tummuru, RP Day, I Elfimov, A Dama-scelli, M Franz. Nature Physics 17, 519–524 (2021)viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 A brief introduction to cuprates . . . . . . . . . . . . 21.1.2 Novel degree of freedom: twisting 2D materials . . . . 41.2 Twisted bilayer cuprate stack: a topological superconductorcandidate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Polar Kerr effect as a probe of time reversal breaking intwisted cuprates . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Superconducting diode effect in twisted cuprates and Joseph-son junction arrays . . . . . . . . . . . . . . . . . . . . . . . 81.5 A brief introduction to mesoscopic circuits . . . . . . . . . . 102 High temperature topological superconductor . . . . . . . . 132.1 Ginzburg-Landau theory . . . . . . . . . . . . . . . . . . . . 152.2 Microscopic models . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Temperature dependence . . . . . . . . . . . . . . . . 212.2.2 Josephson tunneling . . . . . . . . . . . . . . . . . . . 232.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24viiTable of Contents3 Polar Kerr effect as a probe of time reversal symmetrybreaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1 PKE and optical Hall conductivity. . . . . . . . . . . . . . . 283.2 10-band model of coupled twisted copper oxide layers. . . . . 293.3 Effective 2-band model. . . . . . . . . . . . . . . . . . . . . . 323.4 Kerr angle estimate. . . . . . . . . . . . . . . . . . . . . . . . 343.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . 364 Superconducting diode from JJ arrays . . . . . . . . . . . . 374.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 The diode effect in a superconducting Josephson circuit . . . 384.2.1 Minimal circuit . . . . . . . . . . . . . . . . . . . . . 384.2.2 N -arm interferometers and the ideal diode limit . . . 404.3 Parasitic components . . . . . . . . . . . . . . . . . . . . . . 424.3.1 IV-characteristic . . . . . . . . . . . . . . . . . . . . . 424.3.2 Finite inductance . . . . . . . . . . . . . . . . . . . . 454.4 Discussion and Summary . . . . . . . . . . . . . . . . . . . . 455 NMon: a novel superconducting qubit . . . . . . . . . . . . 475.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 A variant on split transmon with enhanced anharmonicity . 495.3 Large odd N, M=2 . . . . . . . . . . . . . . . . . . . . . . . 545.4 Population dynamics with flux drive . . . . . . . . . . . . . . 605.5 Discussion, Conclusion and Summary . . . . . . . . . . . . . 606 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . 62Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65AppendicesA Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . 82A.0.1 Microscopic derivation of GL coefficients . . . . . . . 82A.0.2 Self-consistent mean field theory on the lattice . . . . 84A.0.3 Topological phases . . . . . . . . . . . . . . . . . . . . 86A.1 Derivation of the gap equation from path integrals . . . . . 88A.1.1 s-wave pairing . . . . . . . . . . . . . . . . . . . . . . 88A.1.2 Extended pairing terms . . . . . . . . . . . . . . . . . 89viiiTable of ContentsB Appendix for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . 94B.1 Details on the 2-band model . . . . . . . . . . . . . . . . . . 94B.1.1 d+ id pairing . . . . . . . . . . . . . . . . . . . . . . 94B.1.2 d+ is pairing . . . . . . . . . . . . . . . . . . . . . . 98B.2 Hall Conductivity . . . . . . . . . . . . . . . . . . . . . . . . 98B.2.1 Explicit formula for the 2-band model . . . . . . . . . 101B.2.2 Comparison of the 2-band and the 10-band model re-sults . . . . . . . . . . . . . . . . . . . . . . . . . . . 102B.3 Estimating the Kerr angle . . . . . . . . . . . . . . . . . . . 103B.3.1 Material properties . . . . . . . . . . . . . . . . . . . 103B.3.2 Scaling properties of the Kerr angle . . . . . . . . . . 104B.3.3 Results for other commensurate angles . . . . . . . . 106C Appendix for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . 107C.1 Quantizing the NMon for time dependent flux - using irrota-tional constraint . . . . . . . . . . . . . . . . . . . . . . . . . 107C.2 Numerical approach . . . . . . . . . . . . . . . . . . . . . . . 109C.3 Noise Calculations . . . . . . . . . . . . . . . . . . . . . . . 110C.4 Transition rate matrix elements . . . . . . . . . . . . . . . . 112C.4.1 Split transmon . . . . . . . . . . . . . . . . . . . . . . 112C.4.2 Fluxonium . . . . . . . . . . . . . . . . . . . . . . . . 114ixList of Figures1.1 Typical lattice structure of a layered cuprate compound. Cop-per and oxygen atoms are shown by red and green sites re-spectively. Purple sites represent the spacing layers betweenthe CuO2 planes. Figure taken from ref. [3] Reproduced withpermission from Springer Nature . . . . . . . . . . . . . . . . 21.2 Generic phase diagram of cuprates. Figure taken from ref.[3] Reproduced with permission from Springer Nature. . . . . 31.3 Moire´ pattern formed by twisted bilayer graphene . . . . . . . 41.4 d-wave order parameters ∆k plotted over the Brillouin zone.In this picture d and d’ refer to dx2−y2 and dxy represen-tations, respectively, which are identical but rotated by 45degrees with respect to one another. Right panel shows thetopological state of d + id′ which is a linear combination ofthe two states with a relative complex phase. Note that thecomplex phase of this state winds around the origin twice,corresponding to the Chern number C = 2. . . . . . . . . . . 61.5 a) Characteristic I-V curve of a conventional Josephson junc-tion. b) Non-reciprocal I-V current I+c 6= I−c means that themaximum supercurrent the device can support before reach-ing the dissipative regime depends on the direction in whichwe drive the supercurrent. . . . . . . . . . . . . . . . . . . . . 91.6 LC circuit (left) Transmon (right) . . . . . . . . . . . . . . . 101.7 Energy levels of the Hamiltonian 1.9 for EJ = 1 and EC = 0.03 12xList of Figures2.1 Schematic view of two copper-oxygen square lattices withtwist angle close to 45o. Superconducting order parameterwith a dx2−y2 symmetry in the twisted layer (red) resemblesa dxy order parameter in the original untwisted coordinateframe (blue). Although each Bi2212 monolayer contains twoCuO2 planes, for simplicity and concreteness we primarily fo-cus on models with a single CuO2 plane per monolayer, aswould be the case in Bi2Sr2CuO6+δ and other high-Tc com-pounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Twisted double layer d-wave superconductor. Panel (a) illus-trates the lattice geometry at small twist angle θ < θ−c wherethe free energy is minimized for interlayer phase differenceϕ = 0. This results in a gapless spectrum with weakly splitDirac points. For twist angles θ > θ−c in panel (b) the pre-ferred state has ϕ 6= 0. It breaks time reversal and is fullygapped as indicated by the massive Dirac spectrum. Thespectra are obtained by diagonalizing the BdG Hamiltonian(2.10); k⊥ and k‖ denote momentum components perpendicu-lar and parallel to the Fermi surface near a nodal point. Panel(c) shows the GL free energy F(ϕ) for twist angles θ rangingfrom 0o to 45o in 3o increments for K = 0.125. When θ > θ−cminima occur away from ϕ = 0. Panel (d): predicted phasediagram based on the the GL theory (2.2) and T˜c(θ) given byEq. (2.6). TSC denotes the gapped topological phase, dSCstands for gapless d-wave superconductor. Panel (e): Fermisurface and d-wave gap in the continuum formulation of themicroscopic model. Panel (f) displays the relative phase be-tween the layers ϕmin that minimizes the GL and the BdGfree energy as a function of twist angle θ. Orange symbolsrepresent the spectral gap. . . . . . . . . . . . . . . . . . . . . 17xiList of Figures2.3 Lattice model results. Panels (a,b) show the temperaturedependence of the minimum gap ∆min, the maximum gap∆max and phase ϕ, based on a fully self-consistent lattice cal-culation for coupled layers with commensurate twist anglesθ1,2 ' 53.13o, and θ2,5 ' 43.60o corresponding to a unit cellwith 10 and 58 sites, respectively. Panels (c-e) show zero tem-perature phase diagrams of the system for three twist anglesas a function of chemical potential µ and interlayer couplingg0. The range of chemical potentials µ ∈ (−1.6t,−1.0t) cor-respond to range of fillings n ∈ (0.033, 0.04) near optimaldoping. Each data point in the phase diagram correspondsto an independent self-consistent solution and the radius ofmarkers is proportional to the size of the minimum gap ∆min.The color indicates the Chern number as shown in the legend.Panel (f) displays tunneling conductance σ(E)/σN calculatedfor θ ' 53.13o, µ/t = −1.3, g0 = 20meV and temperaturesranging from 0 to 1.2Tc as indicated by color scale. Curves fordifferent temperatures have been offset for clarity. Panels (g)and (h) illustrate the edge modes for C = 4 and C = 2 topo-logical phases, respectively, in θ2,5 configuration for parame-ters µ = −1.3t and g0 = 20, 52 meV. The energy spectrumis shown for a bilayer system in the infinite strip geometrywith width of 90 unit cells. The color scale represents thenormalized position expectation value of the eigenstate alongthe direction of finite length. . . . . . . . . . . . . . . . . . . 222.4 Temperature dependence of the phase diagram is shown forθ2,5 ' 43.6◦ at T = 0 and T = Tc/2 in panels (a) and (b) re-spectively. Panel (c) shows the minimum gap as a function oftemperature for g0 = 32meV and µ = −1.35t (the parametersare indicated in panels (a) and (b) by a dashed circle) . . . . 232.5 (a) The current-phase relation in units of the critical currentat twist θ = 41.4◦ when K = 0.125. (b) Twist angle de-pendence of the critical Josephson current normalized by thecurrent in the untwisted case. While the two curves showsimilar behaviour close to θ = 0◦ and 90◦, a non-negative Kresults in a finite value of the current at θ = 45◦. . . . . . . . 25xiiList of Figures3.1 a) Optical Hall conductivity for the 10-band lattice modelcalculated from Eq. (3.4) for various values of the interlayercoupling strength g0. Solid and dashed lines show the realand the imaginary parts of σH(ω), respectively. The tightbinding model parameters are chosen to be t = 0.153eV, t′ =−0.45t, µ = −1.3t. The SC order parameter is calculatedself consistently with Vij chosen such that the maximum gapis 40meV. Grey dotted curve shows the ω−2 scaling whichis the expected behavior of ReσH(ω) in the high frequencylimit. b) The 10-band model unit cell with two stacked layersdepicted in blue and red. The signs characteristic of d-waveorder parameter along with the relative complex phases areindicated in the right panel. Note that the complex phases inthe order parameters break the mirror reflection symmetriesalong the x, y and x± y directions. . . . . . . . . . . . . . . . 313.2 a) Zero frequency limit of the real part of the optical Hallconductivity σH(ω) for the 10-band lattice model for the sameparameters as Fig. 3.1. The data shown in the main panelcorresponds to the highlighted slice in the inset (b) whichdisplays the phase diagram of this configuration as a functionof g0 and µ/t. The color corresponds to the Chern number Cwhile the radius of the dots is proportional to the minimumexcitation gap. . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 Kerr angle θK as a function of photon energy ~ω for variousvalues of interlayer coupling g0. a) Thick sample limit Eq.(3.1). b) Thin sample limit Eq. (3.8). c) Semi-log plot withthe expected ω−γ scaling at high frequencies shown by thedotted grey line. Vertical dashed lines indicate the frequencyωc above which σxx shows the power-law form ∝ (−iω)γ−2 . . 354.1 Superconducting circuit consisting of two inductive arms withone and two Josephson junctions in series, respectively. Anexternal flux Φext may thread the superconducting loop. Here,the inductors L1, L2 represent the geometric inductance of thesuperconducting wires. . . . . . . . . . . . . . . . . . . . . . . 39xiiiList of Figures4.2 Generalized Josephson interferometer consisting of N armswith current Ik and nk identical Josephson junctions withcritical current I(k)c that each experience a superconductingphase drop ϕk. Geometrical inductances have been lumpedinto the elements Lk. A magnetic flux Φk,l may be threadedbetween arms k and l. . . . . . . . . . . . . . . . . . . . . . . 414.3 (a) Current phase relation for number of interferometer armsN = 1, 2, 5, 20 and (b) diode efficiency as a function of N thatis given by η = (N − 1)/(N + 1). . . . . . . . . . . . . . . . . 434.4 I-V curves computed within RSJ model for β = 0.6√n1 andN = 1, 2, 5, 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.5 Current phase relation (a) for the exemplary case N = 5 and(b) diode efficiency as a function of N for varied strength ofthe inductance parameter λ/n1. The geometrical inductancedistorts the current-phase relation and diminishes the diodeefficiency η. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.1 NMon geometry. Two parallel arms with N and M Joseph-son junctions shunted by large capacitances CNS and CMS , re-spectively. The external flux Φext through the circuit loopintroduces a relative phase difference between the harmonicscontributing to the Josephson potential given in Eq. 5.1 . . . 495.2 Top panel: Spectrum for N = 2, M = 3 NMon for β = 75,η = 15 with ϕext = 0. Code space is defined as n = 0 → |0〉and n = 3→ |1〉. Model parameters can be chosen (by over-all rescaling of all energies) such that ω = E01 = 6.08GHzwhich is a typical transmon frequency. In terms of frequency,closest transition from the excited state |1〉 is to n = 6. Forthese parameters the relative anharmonicity is found to beαr = −0.105. Note that this is larger than typical rela-tive anharmonicity of the transmon where −αr ≈ 0.03− 0.05corresponding to 200 − 300MHz detuning from a harmonicoscillator with energy spacing of 6GHz. Bottom panel: Tran-sition frequencies near ω01 including all transitions betweenlevels shown in the top panel (up to n = 6) . . . . . . . . . . 51xivList of Figures5.3 Charge fluctuation matrix elements for N = 2, M = 3 NMonfor β = 75, η = 15 with ϕext = 0. Code space is defined asn = 0→ |0〉 and n = 3→ |1〉 and this choice is dictated by thenature of the matrix elements: Note that the most significantmatrix element that the ground state has with the excitedstates is 〈3|∂ngH|0〉 which is larger than any other matrixelement (in the energetic vicinity of |0〉) by at least threeorders of magnitude. Similarly, once the system is excited ton = 3, the two most likely transitions are 3→ 0 (back to theexcited state) and 3→ 6 (information leaking out of the codespace) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.4 Spectrum for N = 2, M = 3 NMon for β = 75, η = 15 a) asa function of charge offset ng ∈ [−2, 2) and b) as a functionof the external flux ϕext ∈ [0, 2pi) through the circuit loop.Model parameters chosen to be the same as Fig. 5.2 whereϕext = 0 is shown. The spectrum is virtually independentof ng as seen in panel a). Inset panel b) shows that relativefluctuations in the qubit frequency are of the order ∆ω/ω ≈10−9 as a function of offset charge. Note that the ”sweet spot”external flux value is found to be ϕext = 0 in panel c) wherethe band dispersion for the energy levels in the computationalspace have zero dispersion. . . . . . . . . . . . . . . . . . . . . 54xvList of Figures5.5 Phase diagram for a range of ENJ and EMJ in dimensionlessparameters β = ENJ /EC and η = EMJ /EC at ϕext = 0 andng = 0 for κ = 0.5 a) absolute value of the relative anhar-monicity |αr|. b) renormalized transition matrix element forcharge fluctuations c) matrix element for flux fluctuations.Note that this is the only plot that depends on the value ofκ for ϕext = 0. Decreasing 0 < κ < 1 rotates the “line ofzeros” clockwise. (See Fig. C.1 in the Appendix) This meansthe flux matrix elements for a point of interest in the param-eter space can be made arbitrarily small by tuning the ratioof shunt capacitances. d,e) lowest two energy levels abovethe ground state relevant for computational space |0〉, |1〉 andinformation loss |1〉 → |2〉 due to leakage outside the compu-tational space. State |1〉 (n = n1) is chosen such that it isa low energy excited state with the highest matrix elementwith the ground state |0〉 which is always n = 0. State |2〉(n = n2) is chosen such that |En2−En1 | is closest to the qubitfrequency, therefore it is the state that is most relevant forinformation loss when the qubit is driven externally. . . . . . 555.6 Top panel: Spectrum for N = 13, M = 2 NMon for β = 140,η = 55 with ϕext = 0 and κ = 1 Code space is defined as n =0 → |0〉 and n = 3 → |1〉. Model parameters can be chosen(by overall rescaling of all energies) such that ω = E03 =5.39GHz which is a typical transmon frequency. Note thatn = 5 as the next excited state accessible from n = 3. In thiscase the relative anharmonicity is found to be αr = −0.359. 565.7 Phase diagram for NMon for N = 13 and M = 2 for a rangeof ENJ and EMJ in dimensionless parameters β = ENJ /EC andη = EMJ /EC at zero flux ϕ = 0 and ng = 0 for κ = 1 Samequantities calculated as in Fig. 5.5. Note that this configu-ration is able to achieve higher anharmonicity compared totransmon or the N = 2,M = 3 configuration we have dis-cussed in the previous section. . . . . . . . . . . . . . . . . . 575.8 Flux fluctuation matrix elements for N = 13, M = 2 NMonfor β = 140, η = 55 with ϕext = 0. Code space is defined asn = 0→ |0〉 and n = 3→ |1〉 . . . . . . . . . . . . . . . . . . 58xviList of Figures5.9 a) Flux drive in dimensionless flux ϕext(t) as a function oftime with amplitude≈ 0.3 and frequency ω = ω03 = 5.39GHz.b) Rabi oscillations as a result of the drive between |0〉 and|1〉 of the code space, demonstrating that state preparation ispossible with minimal information leakage into other energylevels. c) Rabi oscillations shown in the context of populationof the low-lying energy levels where the dot size for each timepoint is proportional to |cn(t)|2 for energy level n. . . . . . . 59A.1 Analysis of the energy integrands. (a) Integrand fB(ξ) enter-ing the expression Eq. (A.3) for the GL theory coefficient B.(b) Integrand fC(ξ) entering the expression for the GL coeffi-cient C compared to fB(ξ)2. Note that the amplitude of thelatter is scaled by a constant factor of 1/30 for easier compar-ison. The temperature is chosen such that kBT = 0.2g butthe qualitative properties of the integrands remain the samein a wide range of temperatures. . . . . . . . . . . . . . . . . 83A.2 Geometry of the commensurate unit cells for bilayers withrelative twists of (a) θ1,2 ' 53.13o(b) θ2,5 ' 43.60o and (c)θ5,12 ' 45.24o. Twisting procedure is illustrated in panel (a)where orange and blue arrows indicate integer valued vectorsv = (m,n) and v¯ = (−m,n) introduced in the text. Themoire´ unit cell is shown by the shaded area. . . . . . . . . . 85A.3 Schematic picture for the normal state band structure of thecontinuum model defined in Eq. (2.7). Both electron andhole bands that enter the BdG Hamiltonian are shown whilethe Fermi level is indicated by a dashed red line. Panel (a)shows decoupled layers with doubly degenerate bands yieldinga circular Fermi surface. In the presence of a d-wave orderparameter we have eight Dirac cones. (b) Interlayer couplingg splits the bands and Fermi surface now consists of a pairof concentric circles. (c) The inner Fermi circle shrinks to apoint and half of the Dirac cones disappear. (d) Four Diraccones survive in the strong interlayer coupling (g > µ) case. . 91A.4 Evolution of the density of states with temperature in theC = 2 phase with parameters g0 = 40 meV and µ = −1.35t.Temperature T/Tc shown as color scale. The superconductinggap persists up until T = Tc. . . . . . . . . . . . . . . . . . . 92xviiList of FiguresA.5 The topological phase transition between C = 0 (µ < −1.34t)and C = 4 (µ > −1.34t) phases is marked by a closing ofthe gap as the chemical potential is tuned while keeping theinterlayer coupling fixed at g0 = 20 meV and θ2,5 ' 43.6◦ . . 93B.1 Normal parts of energy dispersions (top) and the order pa-rameter (bottom) for decoupled layers that are naively ro-tated by ±θ/2 with θ = 2 arctan(1/2). We observe that thethe patterns are no longer periodic when plotted in the orig-inal Brillouin zone. . . . . . . . . . . . . . . . . . . . . . . . . 95B.2 Normal state dispersion (top) and order parameter (middle)of the effective 2-band model constructed for θ = 2 arctan(1/2).The black square encloses the moire´ unit cell with length 2pi/band we observe that the patterns are now properly periodic.The last term in Eq. (B.3) is plotted separately (bottom row)as it is very small compared to the other terms. However,inclusion of this term is very important since without it wewould have ξ1 = ξ2 and the Hall conductivity would identi-cally vanish. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96B.3 Comparison of optical Hall conductivities σxy(ω) between the2-band (shown in orange) and the 10-band (green) models.Solid and dashed lines show real and imaginary parts respec-tively. For the 10-band model, tight binding parameters arechosen to be t = 0.153eV, t′ = 0, µ = −3.9t, ∆max = 0.4eVand g(10band)0 = 0.008eV. The phase between monolayers ischosen to be φ = 0.4pi, breaking the time reversal symmetry.Note that in this plot we calculate the Hall conductance in adifferent regime and using different parameters (larger orderparameter and lower chemical potential compared to the cal-culations in Fig. 3.1) to show agreement between 2-band and10-band models. . . . . . . . . . . . . . . . . . . . . . . . . . 102B.4 Scaling collapse of θK/g20 indicating leading g20 dependenceof the Kerr angle on the interlayer coupling. Top and thebottom panels show thick and thin sample limits, respectively. 104B.5 Kerr angle estimates for other commensurate twist anglesθm,n at ω = 0.5eV in the thin sample limit (h λ), anno-tated by the twist vector [m,n] (blue dots). Each data pointis mirrored (red dots) with respect to θm,n = 45◦ to show theexpected behaviour for the entire range of twist angles. . . . 105xviiiList of FiguresC.1 Flux fluctuation matrix elements for theN = 2, M = 3 NMonparameter space shown in Fig. 5.5. Note that by pickingdifferent values of κ we observe that the line where the matrixelements vanish can be moved. β = ENJ /EC and η = EMJ /EC 112C.2 Model parameter space plots tuning EΣ and d showing a)relative anharmonicity b) charge fluctuation matrix elementsc) flux fluctuation matrix elements for split transmon at ng =0 and ϕext = 0. Panels d,e,f) show same quantities as a,b,c)except for ng = 0 and ϕext = pi. . . . . . . . . . . . . . . . . 113C.3 Fluxonium spectrum for N = 10 and M = 1. At half fluxquantum ϕext = pi the potential well develops two minimawhich leads to giant anharmonicity. . . . . . . . . . . . . . . 115C.4 Flux fluctuation matrix elements for fluxonium with N = 10and M = 1. Spectrum shown in Fig. C.3 . . . . . . . . . . . . 116C.5 Charge fluctuation matrix elements for fluxonium with N =10 and M = 1. Spectrum shown in Fig. C.3 . . . . . . . . . 117C.6 N = 13, M = 2 NMon charge fluctuation matrix elementsrenormalized by qubit frequency ω03 as described in sectionC.3 computed at β = 140, η = 55 with ϕext = 0. Code spaceis defined as n = 0 → |0〉 and n = 3 → |1〉. The table showsall transitions between levels n = 0− 7 . . . . . . . . . . . . 118C.7 N = 2, M = 3 NMon flux fluctuation matrix elements renor-malized by qubit frequency ω03 as described in section C.3computed at β = 75, η = 15 with ϕext = 0. Code space isdefined as n = 0→ |0〉 and n = 3→ |1〉. The table shows alltransitions between levels n = 0− 6 . . . . . . . . . . . . . . 119xixAcknowledgementsI am very fortunate to have Marcel Franz as my advisor throughout my grad-uate studies. Thank you Marcel, for your support and guidance through-out the years and the hours we spent standing in front of the whiteboard,scratching our heads. I have always appreciated the academic freedom youafforded me to explore any idea that I was interested in and for always beingavailable for discussions that kept me grounded.I would like to acknowledge Tarun Tummuru and Ryan Day for twistingthe cuprates with me, Rafael Haenel for exploring the physics of Josephsonjunctions together and our forays into superconducting electronics, Xiao-Xiao Zhang for our collaboration on the Kerr effect project and VedangiPathak for chasing together the supercurrents around the edges of the topo-logical superconductor.Furthermore, my research and personal growth have been significantlyenriched by conversations with a remarkable community of scholars, includ-ing Amirreza Ataei, Mona Berciu, Sayak Dasgupta, Se´amus Davis, ColbyDelisle, Alannah Hallas, Niclas Heinsdorf, Daniel Huerga, Catherine Kallin,Ali Lavasani, Chengshu Li, Pedro Lopes, Anne Matthies, Alberto Nocera,Dima Pikulin, Drew Potter, Stephan Plugge, Aline Ramires, SharmisthaSahoo, Urban Seifert, David Stephen, Gabrielle Tournaire, Rui Wen, Xiao-Chuan Wu, Cagin Yunus and Benjamin Zhou. Each of you has contributedto my perspective in unique and valuable ways, deepening my understandingof our field.This journey has received very significant support from my friends, fam-ily and guitars. Special thanks go to my parents Seyhan Can and AbdullahCan for their unconditional support in every possible way.xxChapter 1IntroductionThis thesis can be organized into two parts. The first part outlines a recipefor the realization of a novel quantum phase of matter called a topologi-cal superconductor and discusses predictions for optical measurements inthe characterization of this novel phase in an experimental setting. Theproposal is presented in Chapter 2, where we apply the novel paradigm of“twisting and stacking of 2D materials” to unconventional superconductorsand show that the heterostructure of two twisted cuprate monolayers formsa topological superconductor with d + id′ order parameter [1]. In Chapter3, we consider the optical response of the time reversal breaking d + id′order of the topological superconductor by calculating the polar Kerr effect[4], which rotates the polarization of the reflected light off the surface withrespect to the polarization of the incident light. We predict large intrinsiccontribution to the optical Hall conductivity resulting in Kerr rotation atleast two orders of magnitude larger than the one measured in the leadingchiral superconductor candidate Sr2RuO4.The second part of this thesis discusses superconducting device applica-tions, inspired by the physics of twisted cuprate bilayer system introducedin Chapter 2. Experiments looking for signatures of the chiral d+ id′ phaseof twisted cuprate bilayers have revealed that the system also exhibits thesuperconducting diode effect. [5] During our attempts to understand themechanism behind the effect in twisted cuprate bilayers, in a more recentwork [2] which is discussed in Chapter 4, we have demonstrated that thiseffect can be controlled and optimized using arrays of Josephson junctionswhich are widely used in scalable superconducting electronics applications.We have shown that the current-phase relation of the Josephson potentialcan be engineered to optimize the efficiency of a superconducting diode.Finally, this perspective on the Josephson potential engineering by combin-ing arrays of junctions leads to a novel qubit design which is presented inChapter 5 of this thesis.11.1. Overview1.1 Overview1.1.1 A brief introduction to cupratesSince the discovery of LaBaCuO [6] in 1986, high temperature superconduc-tors (HTS) captured the imagination of physicists, perhaps most strikinglywith the question of whether achieving room temperature superconductivityis possible. Critical temperatures of HTS can not be explained by conven-tional BCS theory of superconductivity [7] which has been extremely suc-cessful in understanding conventional (BCS) superconductors where Cooperpairs are formed by electron-phonon coupling.Figure 1.1: Typical lattice structure of a layered cuprate compound. Copperand oxygen atoms are shown by red and green sites respectively. Purple sitesrepresent the spacing layers between the CuO2 planes. Figure taken fromref. [3] Reproduced with permission from Springer NatureCuprates are a specific class of HTS formed by weakly coupled CuO2planes (Fig.1.1) separated by insulating spacer layers. In contrast to BCSsuperconductors, pairing mechanism in cuprates originate from electron-electron interactions. Although there are many cuprate compounds, a genericphase diagram as a function of temperature and hole doping will help demon-strate the variety of many strongly correlated orders. At zero doping strongCoulomb repulsion pins down electrons and the system becomes a Mott in-sulator while developing antiferromagnetic (AF) order. Upon hole dopingthis state, electrons start to move around in a very correlated fashion lead-ing to d-wave superconductivity, with the highest critical temperature atoptimal doping. There has been a plethora of work confirming the d-wave21.1. Overviewnature of cuprates. These include the measurement of predicted pi-shift intunnel junction SQUID loop experiments [8], observation of vortices withhalf magnetic flux quantum in tri-crystal junctions [9], superconducting gapanisotropy measured in the Cu-O planes in angle resolved photo emissionspectroscopy (ARPES) [10] and microwave measurements of temperaturedependence of the magnetic penetration depth [11] and specific heat mea-surements consistent with the low lying quasiparticle excitations due to thenodes of a d-wave superconductor. Above the superconducting dome, thepseudogap phase appears where charge density wave and spin density waveorders develop, and finally the strange metal phase, a non-Fermi liquid,where quasiparticle descriptions fail and unusual transport signatures areobserved such as T-linear resistivity. The strange metal phase being theparent state of superconductivity near optimal doping is poorly understoodand better understanding of it might have more clues about the pairingmechanism. We refer the reader to Ref. [3] for a recent and accessible re-view of the rich physics of cuprates.Figure 1.2: Generic phase diagram of cuprates. Figure taken from ref. [3]Reproduced with permission from Springer Nature.The consensus in the community has been that the essential ingredientfor superconductivity in cuprates is the CuO2 planes. Recently, single mono-layers of Bi2212 (short for Bi2Sr2CaCu2O8+x) containing only two CuO231.1. Overviewplanes have been isolated [12] by mechanical exfoliation from the bulk com-pound. Remarkably, the critical temperature Tc measured in the monolayersis within the error margin of the known bulk values. Obtaining supercon-ducting monolayers is difficult since they must be exfoliated in a cold glovebox filled with an inert gas with very low oxygen content. Otherwise theybecome insulating, as oxygen in ambient conditions can enter the monolayerfreely and change the doping level. This effect can be controlled, however, sothat oxygen can be allowed to slowly enter the sample and tune the dopinglevel as measurements are done on a single sample. In this way, authorsof Ref. [12] was able to generate a phase diagram and they found that themonolayer phase diagram is virtually the same as the bulk phase diagram:In addition to superconductivity, they observe the pseudogap in tunnelingspectroscopy and transport measurements, in addition to the linear T resis-tivity of the strange metal phase. This work then conclusively proves thatthe superconductivity among with other strongly correlated phenomena aretruly two dimensional in cuprates.1.1.2 Novel degree of freedom: twisting 2D materialsWhen 2D materials such as graphene or transition metal dichalcogenides arestacked on top of another with a twist, a repeating moire´ pattern emergesas shown in Fig 1.3. Due to interlayer coupling between the two layers,the moire´ potential can drastically modify the low energy physics where thetwist angle becomes another tunable parameter of the system in addition tothe more conventional degrees of freedom such as external electric or mag-netic fields, pressure, strain and doping. Enhanced tunability can help usexplore phase diagrams, approaching critical behaviour and realizing topo-logical phases.Figure 1.3: Moire´ pattern formed by twisted bilayer grapheneMost studied moire´ system so far is the twisted bilayer graphene (TBLG)where flat bands can be achieved by tuning the twist angle [13] to its ”magic41.2. Twisted bilayer cuprate stack: a topological superconductor candidatevalue” ≈ 1.1o, enhancing electronic interactions (with reduced screeningdue to low dimension) which lead to superconductivity, correlated insulatorstates and quantum anomalous Hall effects (see references in [14]). Introduc-ing additional layers can lead to further emergent phenomena such as reen-trant and potentially unconventional triplet superconductivity with largePauli limit violation, as observed in twisted trilayer graphene [15, 16]. How-ever, a recent experiment on rhombohedral trilayer (untwisted) graphenesuggests that twist may not be necessary to observe superconductivity. [17]More complex 2D materials can also be twisted and stacked, such as tran-sition metal dichalcogenides (TMD) with the chemical formula MX2 whereM is a transition metal (Zr,Mo,W,..) and X is a chalcogen (S,Se,Te,..).Unlike graphene, a semimetal where spin-orbit coupling is negligible, manyTMDs are semiconductors (they can also be metals or semimetals) withstrong spin-orbit coupling which results in a wide range of desirable elec-tronic and optical properties [18] with promising applications in digital elec-tronics, energy harvesting and quantum communications. [19] Twisted bi-layers of TMDs (t-TMDs) were shown to feature flat bands [20] which cansupport correlated insulator states [21]. In this case t-TMDs are more ad-vantageous over TBLG as they do not require high angular precision in thetwist angle to realize such correlated states, although the sample quality ofTBLG has been superior. t-TMDs can also host topological insulator phases[22] and their moire´ flat bands can also be used to simulate the triangularlattice Hubbard model [23] where the interaction strength can be tuned bythe twist angle.Van der Waals (vdW) materials are characterized by their structureformed by weakly coupled (by vdW interaction) stacks of monolayers withstrong in-plane covalent bonding. Heterostructures of different 2D vdW ma-terials, have been known to reveal unusual properties and novel phenomena[24] and the number of different 2D materials is thought to be around 500[19]. There could be many different combinations of 2D heterostructureswith emergent novel physics where we could now also explore effects of thetwist degree of freedom between the layers which could result in furthernovel effects and device applications.1.2 Twisted bilayer cuprate stack: a topologicalsuperconductor candidateAn example of this paradigm could be stacking and twisting of recentlyexfoliated [12] 2D cuprate monolayers, which is what we have explored in51.2. Twisted bilayer cuprate stack: a topological superconductor candidatereference [1] and reproduced as Chapter 2 in this thesis. In contrast tomost of the previous studies, we focus on the large twist angle regime whereθ ≈ 45o. We found that for a finite range of twist angles near θ ≈ 45o thesystem becomes a topological superconductor.A topological superconductor is a type of superconductor characterizedby a gap function ∆k that exhibits non-trivial topological winding through-out the Brillouin zone. This winding manifests as an integer known as theChern number, a topological invariant. This means the Chern number re-mains independent of the specific details of the Hamiltonian, provided thatthe quasiparticle spectrum remains gapped across the entire Brillouin zone.Topological superconductors support in-gap excitations at their boundaries,which are known as edge modes. Notably, the quantity of these edge modescorresponds directly to the Chern number. These edge modes are topolog-ically protected; they cannot be eliminated unless the bulk gap closes as aresult of tuning in the Hamiltonian parameters. This protection is a keyfeature of topological superconductors, lending them stability and potentialutility in various applications. We refer the reader to a review article [25]for more details on topological superconductivity.0kxa0k yadx2 y20kxa0dxy0kxa0Arg(dx2 y2+ idxy)101OP 0 ArgFigure 1.4: d-wave order parameters ∆k plotted over the Brillouin zone. Inthis picture d and d’ refer to dx2−y2 and dxy representations, respectively,which are identical but rotated by 45 degrees with respect to one another.Right panel shows the topological state of d + id′ which is a linear com-bination of the two states with a relative complex phase. Note that thecomplex phase of this state winds around the origin twice, corresponding tothe Chern number C = 2.While each monolayer is a d-wave superconductor, characterized by anodal gap function ∆k resembling a d-orbital over the Brillouin zone, weshow in reference [1] that relative spatial twist near 45 degrees between thetwo monolayers in the weak interlayer coupling regime leads to a mixed d+id′order parameter where d and d’ refer to dx2−y2 and dxy representations,61.2. Twisted bilayer cuprate stack: a topological superconductor candidaterespectively, which are identical but rotated by 45 degrees with respect toone another, as shown in Fig. 1.4. Note that θ = 450 is not a ”magic” angleas in the case of TBLG. There is a finite interval of twist angles withinθ = 450 where the topological superconductivity should be observed.Such mixing of the order parameters has nontrivial consequences. First,the new order parameter of the bilayer system spontaneously breaks the timereversal symmetry since the d+id′ order parameter is complex as opposed tothe real d-wave order parameter of the isolated cuprate monolayer (or bulk)samples. Experimental consequences of this symmetry breaking will be dis-cussed in Chapters 3. Second, the onset of the d+ id′ order parameter gapsout the quasiparticle spectrum and leads to a topologically nontrivial phasewhich can be distinguished by a topological invariant called the Chern num-ber. This number is an integer and equal to the number of the topologicaledge states which appear at the boundaries of the superconductor.In addition to d + id′ topological superconductor phase, we also finda time reversal symmetry breaking but topologically trivial d + is phase.(See Fig. 2.3) Since the topological phases we find in this system have evenChern numbers, we do not expect to find Majorana zero modes (MZM)in the vortex cores which ara topologicala defects in the superconductingorder parameter, which appear in the presence of external perpendicularmagnetic field. However, recent work [26] showed that when the twistedbilayer cuprate film is placed on top of a topological insulator, MZMs canbe found in the vortex cores.In small twist angle regime, as independently discussed in reference [27],there is a magic angle near which low energy degrees of freedom result infinite density of states and correlations can then drive the system into atopological superconducting state. Authors also show that the system canbe driven into a topological superconductor state by driving currents acrossthe bilayer junction.Inspired by the recipe for the d + id′ topological superconductor, weshowed in reference [28] that one can engineer a p+ip′ topological supercon-ductor by stacking two 2D p-wave superconductors with a θ = 90o twist. Inthis case, we obtain a gapped topological superconductor induced by spon-taneous time reversal symmetry breaking similar to the d+id′ scenario, withthe difference that p+ ip′ superconductor has odd-integer Chern number asopposed to the even-integer Chern numbers we observed in d + id′ super-conductor. Odd-integer Chern number means that p + ip′ superconductorcan host Majorana zero modes in its vortex cores. [29, 30]71.3. Polar Kerr effect as a probe of time reversal breaking in twisted cuprates1.3 Polar Kerr effect as a probe of time reversalbreaking in twisted cupratesOptical measurements can be made for an experimental observation of thespontaneous time reversal symmetry breaking in the topological supercon-ductor phase of the twisted cuprates, as introduced in section 1.2. In thepolar Kerr effect, linearly polarized light is normally incident on the surfaceof a material sample, and the polarization axis of the reflected light will berotated by the Kerr angle θK . The Kerr angle depends on the anomalousHall conductivity σH = [σxy−σyx]/2 and the index of refraction n where σxyis the Hall conductivity [25]. Finite Kerr rotation unambiguously originatesfrom time reversal symmetry breaking.In Chapter 3 we compute the intrinsic multiband contribution for anoma-lous Hall conductivity for the twisted cuprate system [4] and compare ourresults to predicted values [31] for one of the leading chiral superconductorcandidates Sr2RuO4 which also shows time reversal symmetry breaking inmuon spin relaxation [32] and polar Kerr effect measurements [33]. We findthat the anomalous Hall conductivity for the twisted cuprate model is 3orders of magnitude larger than the corresponding values for Sr2RuO4. Wealso estimate the Kerr angle θK using material specific data and find thatthe Kerr angle for the twisted cuprate system is in the range of 10−100µradsif we assume if we assume g0 = 4 − 16 meV while the predicted [33] andexperimentally measured values [31] for Sr2RuO4 are 50nrads and 65nradsrespectively. We therefore expect that such a large Kerr angle should be ob-served in the twisted cuprate systems. At the time of writing this proposal,Ye group at QMI is working on measuring this effect. As of now their Kerrangle measurement resolution appears to be about 50nrads, which shouldbe enough to observe an effect measured in µrads.1.4 Superconducting diode effect in twistedcuprates and Josephson junction arraysNonreciprocal cooper pair tunneling (directional asymmetry in critical cur-rent I+c 6= I−c , see Fig. 1.5), also known as the superconducting diode effect,has been a focus of recent experimental observations, which include Joseph-son junctions fabricated on InAs quantum wells [34], of Dirac semimetals[35], Nb/V/Ta heterostructures [36], and twisted bilayer [37, 38] and tri-layer graphene [39] as well as recent experiments in twisted cuprate bilayers[5, 40, 41] which are discussed in a recent theory work [42]. On symmetry81.4. Superconducting diode effect in twisted cuprates and Josephson junction arraysgrounds, finite superconducting diode effect requires both inversion and timereversal symmetry breaking [43]. Therefore, experimental observation of thesuperconducting diode effect is a strong evidence of time reversal symmetrybreaking which confirms our predictions in Section 1.2 and [1]. In addition,we note that the fractional Fraunhofer patterns and Shapiro steps observedin [5] further confirm the suppressed first harmonic in the Josephson poten-tial, which is the mechanism leading to the time reversal symmetry brokenstate, discussed in Chapter 2In addition to the recent academic interest it has generated due to itsconnections with unconventional superconductivity [44], motivation to studythe superconducting diode effect also comes from its potential applications insuperconducting electronics, given its traditional analogue in semiconduct-ing diode which has found a wide range of practical applications such asrectifiers, AC/DC current converters or photo-detectors. For more details,we refer the reader to the recent comprehensive review of the superconduct-ing diode effect. [44]Figure 1.5: a) Characteristic I-V curve of a conventional Josephson junction.b) Non-reciprocal I-V current I+c 6= I−c means that the maximum supercur-rent the device can support before reaching the dissipative regime dependson the direction in which we drive the supercurrent.One can ask whether we can reproduce this effect using conventionalaluminum or niobium Josephson junctions which are easily scalable and inwide use in superconducting electronics, rather than relying on materialheterostructures such as twisted cuprates with complex physics such as un-conventional superconductivity. With a circuit design involving arrays ofJosephson junctions, we have shown in reference [2] which is reproduced inChapter 4 that the answer is affirmative. In conjunction with theory, we91.5. A brief introduction to mesoscopic circuitshave designed a blueprint for a device to be printed on a microchip, work-ing with CMC Microsystems in Canada, on which we plan to make mea-surements with the help of our experimentalist colleagues at the QuantumMatter Institute at UBC.1.5 A brief introduction to mesoscopic circuitsIn this subsection we will present a brief discussion of the theory of super-conducting circuits as a primer to a novel qubit design we call NMon whichwe describe in detail in Chapter 5 .C EJL CFigure 1.6: LC circuit (left) Transmon (right)We start with the discussion of the canonical quantization of the LC circuitwhere the Lagrangian is given by L = T − U . We defineT =CΦ˙22, U =LΦ22where C and L are the capacitance and the inductance, respectively. Weidentify the generalized flux coordinate Φ and compute the conjugate mo-mentum QΦ = ∂L/∂Φ˙ = CΦ˙ which we need for the Legendre transform theobtain the Hamiltonian H = QΦΦ˙− T + U which is given asH =QΦ22C+LΦ22(1.1)We will rewrite the charge variable QΦ = 2en in terms of the number ofCooper pairs n and define the charging energy EC = e2/2C. It is convenientto work with the dimensionless flux variable ϕ = 2piΦ/Φ0 which motivatesus to define the inductive energy EL = LΦ20/(2pi)2 which in turn allows usto write the quantum mechanical Hamiltonian 1.1 asH = 4EC nˆ2 +ELϕˆ22(1.2)101.5. A brief introduction to mesoscopic circuitsupon canonical quantization with the commutation relation[ϕˆ, nˆ] = i (1.3)Note that this Hamiltonian is identical to the quantum harmonic oscillatorand we can find its energy spectrum algebraically by definingnˆ = i(EL32EC)1/4(aˆ− aˆ†) (1.4)ϕˆ =(2ECEL)1/4(aˆ+ aˆ†) (1.5)where one can check[aˆ, aˆ†]= 1 for bosonic ladder operators aˆ and aˆ† andthat the Hamiltonian 1.3 becomes (we set ~ = 1)H =√8ELEC aˆ†aˆ (1.6)with energy spacing ωLC =√8ELEC . Note that we have arrived at amesoscopic quantum system with discrete energy levels. The problem withusing this system as a qubit is that all transition energies between adjacentlevels are the same. Suppose we initialize our system in its ground staten = 0 or |0〉 (from now on we will use notation |n〉 for the nth level.)Once we try to excite the system to energy level |1〉 with a microwave drivefrequency ω ≈ ωLC the system can get excited further to higher energylevels |2〉 and above since the transition frequency for 1 → 2 is exactly thesame. To avoid this so-called information leaking out of the computationalspace, we would like to modify our energy levels such that ω2 − ω1 = ω12must be different than ω1 − ω0 = ω01 such that the drive near ω01 will onlyinduce transitions within the computational space {|0〉, |1〉}. This detuningis defined as the anharmonicityα = ω12 − ω01 (1.7)as it is zero for a “harmonic” potential such as the one for the LC circuit.We also define the relative anharmonicity which is the dimensionless versionof ααr =ω12 − ω01ω01(1.8)To achieve an energy spectrum with finite anharmonicity, we can replaceinductor with a Josephson junction, which can be thought of as a nonlinearinductor. We would then modify Hamiltonian 1.1 asH = 4EC nˆ2 − EJ cos ϕˆ (1.9)111.5. A brief introduction to mesoscopic circuitsNote that if we expand the cosine, we can interpret the quadratic term as theinductor term in Eq. 1.1 and all higher order contributions can be thoughtof as perturbations to the harmonic potential, leading to anharmonicity inthe energy levels as shown in Fig. 1.7 In the limit where Josephson energy3 2 1 0 1 2 30.000.250.500.751.001.251.501.752.00EFigure 1.7: Energy levels of the Hamiltonian 1.9 for EJ = 1 and EC = 0.03EJ is much larger compared to the charging energy EC , Hamiltonian 1.1describes the transmon [45] which is a popular qubit architecture due toits reduced susceptibility to charge noise stemming from fluctuations in thenumber of Cooper pairs in the superconducting condensate.While the transmon limit EJ EC suppresses information loss to chargefluctuations, it also reduces the anharmonicity, which is required so that theinformation does not leak out the computational space. [45] Therefore theEJ/EC ratio can not be made arbitrarily large and there is a typical rangeof EJ/EC where these two competing factors can be optimized for the mostefficient transmon designs where EJ/EC ≈ 50 − 100. In Chapter 5 weoutline a novel proposal we call NMon, which is a combination of parallelarrays of Josephson junctions. We show that NMon can exhibit enhancedanharmonicity while having comparable susceptibility to charge noise to thatof transmon.12Chapter 2High temperaturetopological superconductorIn a remarkable recent development Yu and co-workers [46] succeeded inisolating and probing single monolayer samples of high-temperature super-conductor Bi2Sr2CaCu2O8+δ (Bi2212) with critical temperatures Tc essen-tially indistinguishable from the bulk Tc ' 90 K of optimally doped Bi2212crystals. This result delivers the long-awaited proof that high-Tc supercon-ductivity in Bi2212 (and possibly all the cuprates) is an intrinsic property ofa single two-dimensional monolayer. It also opens exciting new possibilitiesfor engineering structures with novel behaviour and functionality, from indi-vidual high-Tc cuprate monolayers. Motivated by the results of Ref. [46] andrecent developments in van der Waals-bonded materials [47–52], we considerhere structures composed of two cuprate monolayers with a twist angle θbetween them.Twisted double layer graphene famously exhibits a sequence of gate-voltage controlled correlated insulating and superconducting phases [47–51]when θ is close to the magic angle 1.1o. Similar physics is reported [52]in twisted bilayer WSe2 in the range of angles between 4.0o and 5.1o. Inboth cases, the observed behaviour is thought to reflect a subtle interplaybetween the band structure energetics and topology brought about by themoire´ patterns, electron-electron interactions and disorder [53–62]. As such,many aspects of the reported experimental observations in twisted grapheneand WSe2 still await full theoretical understanding [63]. As we argue below,the problem of twisted Bi2212 bilayers is potentially simpler to analyzetheoretically, at least at the level where each Bi2212 layer is treated asa BCS superconductor with a dx2−y2 order parameter. Results of such atheoretical analysis are, nevertheless, non-trivial: over a continuous range oftwist angles θ in the vicinity of 45o, we find that below a critical temperatureT˜c(θ) the bilayer system spontaneously breaks time reversal symmetry Tand forms a chiral topological phase that can be thought of as an emergentdx2−y2 ± idxy superconductor (d± id′ for short). Such a superconductor hasa fully gapped bulk and is characterized by Chern number C = ±2 per layer13Chapter 2. High temperature topological superconductorFigure 2.1: Schematic view of two copper-oxygen square lattices with twistangle close to 45o. Superconducting order parameter with a dx2−y2 symme-try in the twisted layer (red) resembles a dxy order parameter in the originaluntwisted coordinate frame (blue). Although each Bi2212 monolayer con-tains two CuO2 planes, for simplicity and concreteness we primarily focuson models with a single CuO2 plane per monolayer, as would be the case inBi2Sr2CuO6+δ and other high-Tc compounds.with a gap of dx2−y2 ± idxy symmetry, respectively. It exhibits two gapless,topologically protected chiral edge modes per monolayer. These modes arecoherent superpositions of electron and hole states and can be thereforeclassified as Majorana particles [64]. When θ ≈ 45o, our results indicatethat T˜c ' Tc, suggesting that, remarkably, in carefully assembled Bi2212samples, the topological phase could set in well above the liquid nitrogentemperature.The physics underlying the formation of the d ± id′ topological phaseis illustrated in Fig. 2.1. In momentum space, the dx2−y2 order parameterthat characterizes each monolayer can be represented as ∆k = ∆0(kˆ2x− kˆ2y),where ∆0 denotes the amplitude and (kˆx, kˆy) = (kx, ky)/k are momentumcomponents defined relative to the principal crystal axes of the underlyingcopper-oxygen lattice. When one of the layers is rotated by angle θ, its orderparameter is transformed in the original coordinate frame to∆(θ)k = ∆0[cos(2θ)(kˆ2x − kˆ2y) + sin(2θ)2kˆxkˆy]. (2.1)From the vantage point of the unrotated monolayer, the second term inEq. (2.1) has the functional form of a dxy order parameter. For coupledlayers, we then expect the Cooper pair tunneling to induce a subdominantdxy order parameter in the unrotated layer. By symmetry, the same should142.1. Ginzburg-Landau theoryhappen in the other layer. Furthermore, if the two order parameters combinewith a complex phase, the result is a fully gapped topological superconductorwith broken time reversal symmetry. The gapped nature of the phase followsfrom the fact that the excitation gap is given by |∆k| which, for a d± id′ gapfunction, is non-vanishing everywhere at the Fermi level, Fig. 2.2(a,b). Timereversal symmetry breaking is evident because time reversal maps d+ id′ tod− id′. We demonstrate below that for a range of twist angles θ between θ−cand θ+c , such a time reversal symmetry broken phase becomes energeticallyfavorable below a critical temperature T˜c(θ).We note that time reversal symmetry broken d ± id′ and d ± is phaseshave been discussed theoretically in various contexts, including bulk high-Tc cuprates in an applied magnetic field [65–67], c-axis twist Josephsonjunctions [68–71], and highly doped single-layer graphene [72, 73]. However,conclusive experimental evidence for these states has never been reported.A time reversal symmetry broken topological phase with px± ipy symmetryhas long been a leading candidate for superconducting order in Sr2RuO4[74–76]. Recent Knight shift measurements [77], however, appear to ruleout this possibility. By placing our emphasis on monolayer-thin samplesof an established dx2−y2 superconductor, we argue that the present workpromises to finally break the logjam in the quest for a chiral topologicalphase. As we show below, for ' 45o twist, the time reversal symmetrybreaking d± id′ phase is nucleated robustly and reliably through the simpleand well understood mechanism of interlayer Cooper pair tunneling that islargely insensitive to microscopic details.2.1 Ginzburg-Landau theoryThe phenomenon outlined above is most easily quantified by examining thephenomenological Ginzburg-Landau (GL) theory. For the sake of clarityand simplicity, we begin by considering models with a single CuO2 planeper monolayer. This is directly relevant to Bi2Sr2CuO6+δ (Bi2201), but,as we show in the subsequent sections, our results are broadly applicableto cuprates with multiple CuO2 plane structure, such as Bi2212, which hasalready been exfoliated [46] to the monolayer limit.The GL free energy density for two coupled monolayers can be writtenasF [ψ1, ψ2] = f0[ψ1] + f0[ψ2] +A|ψ1|2|ψ2|2 (2.2)+ B(ψ1ψ∗2 + c.c.) + C(ψ21ψ∗22 + c.c.),152.1. Ginzburg-Landau theorywhere ψa=1,2 are complex order parameters of the two layers and f0[ψ] =α|ψ|2 + 12β|ψ|4 is the free energy of a monolayer. We are interested inspatially uniform solutions, so the gradient terms have been omitted. Ifthe two layers are physically identical, then the order parameter amplitudesmust be the same and the most general solution (up to an overall phase) isψ1 = ψ, ψ2 = ψeiϕ, (2.3)where we take ψ to be real and positive.Because ψa describe order parameters with d-wave symmetry, they trans-form as ψa → −ψa under a 90o rotation. Suppose we increase the twist angleθ continuously from 0 to pi/2. Since ψ2 changes sign under this operation,we conclude that for the free energy (2.2) to remain invariant, the parame-ter B must also change sign. The simplest dependence consistent with thisconstraint is B = −B0 cos(2θ), where we take B0 > 0. The negative sign ischosen to allow the phase difference between the layers to vanish as the twistangle approaches zero. This also implies that B = 0 for θ = pi/4, consistentwith the intuition that a dx2−y2 Cooper pair cannot tunnel between the lay-ers in this configuration. Parameters A and C are not required by symmetryto depend on θ, so we treat them as constants henceforth. Incorporatingthese ingredients into Eq. (2.2), we findF(ϕ) = F0 + 2B0ψ2 [− cos(2θ) cosϕ+K cos(2ϕ)] , (2.4)where K = Cψ2/B0 and F0 contains all terms independent of ϕ. It is nowa simple matter to minimize F(ϕ) for a given twist angle θ. An interestingscenario arises when K > 0, as indicated by Fig. 2.2(c). Although symmetryalone does not constrain the sign of K, a simple physical argument suggeststhat K should, in general, be positive. One can interpret the ψ21ψ∗22 term inEq. (2.2) as representing the coherent tunneling of two Cooper pairs betweenthe layers. Up to an overall scale factor, its coefficient should, therefore, beproportional to the square of the coefficient B, which itself corresponds tosingle pair tunneling. Coefficient C (and, hence, alsoK) will then be positive,regardless of the sign of B. Below, we demonstrate that microscopic modelsfully support this argument, yielding K > 0.At θ = 45o and for K > 0, the free energy minimum clearly occurs atϕmin = ±pi/2 which indicates a d± id′ superconducting state. For θ 6= 45o,the free energy is minimized byϕmin = arccos(cos 2θ4K). (2.5)162.1. Ginzburg-Landau theoryπ/40 π/21.00.50.0π0-π1.00.50.0-0.5θφθ = 9.53o < θ−c θ = 36.87o > θ−cθ θ+cθ−cT/TcTSCϵk ϵkk⊥ k⊥k∥ k∥a) b) c)θ = 0θ = π /4θ = θ−cdx2−y2 + idxydx2−y2 + dxynormaldSC dSCℱ(φ)d)φ minf)π/40 π/20π/2π1020θgap[meV]TSCBdGGLe)θkxkyθ−c θ+cFigure 2.2: Twisted double layer d-wave superconductor. Panel (a) illus-trates the lattice geometry at small twist angle θ < θ−c where the free energyis minimized for interlayer phase difference ϕ = 0. This results in a gaplessspectrum with weakly split Dirac points. For twist angles θ > θ−c in panel(b) the preferred state has ϕ 6= 0. It breaks time reversal and is fully gappedas indicated by the massive Dirac spectrum. The spectra are obtained bydiagonalizing the BdG Hamiltonian (2.10); k⊥ and k‖ denote momentumcomponents perpendicular and parallel to the Fermi surface near a nodalpoint. Panel (c) shows the GL free energy F(ϕ) for twist angles θ rangingfrom 0o to 45o in 3o increments for K = 0.125. When θ > θ−c minima occuraway from ϕ = 0. Panel (d): predicted phase diagram based on the the GLtheory (2.2) and T˜c(θ) given by Eq. (2.6). TSC denotes the gapped topolog-ical phase, dSC stands for gapless d-wave superconductor. Panel (e): Fermisurface and d-wave gap in the continuum formulation of the microscopicmodel. Panel (f) displays the relative phase between the layers ϕmin thatminimizes the GL and the BdG free energy as a function of twist angle θ.Orange symbols represent the spectral gap.As illustrated in Fig. 2.2(f), a nontrivial phase difference occurs for a rangeof angles θ−c < θ < θ+c with θ±c =12 arccos (∓4K). This indicates a timereversal symmetry broken phase with order parameter dx2−y2 + e±iϕmindxywhich is fully gapped and topologically non-trivial as long as ϕmin 6= 0, pi, seealso Fig. 2.2(a,b). As K depends on temperature through its dependence onψ, the critical temperature T˜c of the topological phase will be a function ofthe twist angle θ. If we adopt the standard GL temperature dependence for172.2. Microscopic modelsthe order parameter ψ(T ) = ψ0√1− T/Tc, it is straightforward to deduceT˜c(θ) = Tc(1− | cos 2θ|4K0), θ−c < θ < θ+c , (2.6)where K0 = Cψ20/B0. This defines the phase diagram in Fig. 2.2(d). Weobserve that, remarkably, T˜c(45o) coincides with the Tc of a single monolayerwhich can be as high as 90 K in carefully prepared Bi2212 flakes. Awayfrom the ideal 45o twist, the critical temperature falls approximately asT˜c(θ) ' Tc(1 − |θ − pi/4|/2K0). Microscopic models with realistic Bi2212parameters discussed in the following give typical values K0 ' 0.1 − 0.2,implying a significant extent for the topological phase, as indicated in Fig.2.2(d).2.2 Microscopic modelsWe now turn to the microscopic theory. Although the pairing mechanismin high-Tc cuprates remains a subject of debate, it is widely accepted thatmost physical properties of the superconducting state in the optimally dopedand overdoped regime are accurately described within the framework of thestandard BCS theory with a dx2−y2 order parameter. Therefore, we begin bymodeling the twisted bilayer using a simple continuum model of coupled d-wave superconductors, and complement this with a calculation based on anattractive Hubbard model on the square lattice. The continuum formulationhas the advantage of being applicable to an arbitrary twist angle θ, similarto the Bistritzer-MacDonald model for graphene [53]. In contrast with thecontinuum model above, the lattice calculation captures various microscopicdetails of the CuO2 planes more effectively, such as the shape of the Fermisurface. It is, however, limited to commensurate twist angles that producerelatively small moire´ unit cells due to practical constraints.In keeping with the strategy outlined above, we begin with a single CuO2plane per monolayer, as for Bi2201. The continuum theory for such a bilayeris defined by the HamiltonianH =∑kσaξkac†kσackσa + g∑kσ(c†kσ1ckσ2 + h.c.)(2.7)+∑ka(∆kac†k↑ac†−k↓a + h.c.)−∑ka∆ka〈c†k↑ac†−k↓a〉.Here, c†kσa creates an electron with crystal momentum k and spin σ in layera = 1, 2 while ξka and g represent the in-plane kinetic energy and inter-plane182.2. Microscopic modelstunneling amplitude, respectively. The superconducting order parameter inlayer a can then be expressed as∆ka =∑p′V(a)kp 〈c−p↓acp↑a〉. (2.8)The prime on the summation indicates a restriction to momentum stateswith energy within c of the Fermi level. The Hamiltonian (2.7) should beregarded as a mean-field approximation to the BCS pairing Hamiltonianwith an interaction term∑kp V(a)kp c†k↑ac†−k↓ac−p↓acp↑a, where V(a)kp denotesthe interaction matrix element in layer a. We shall use a simple separableformV(a)kp = −2V cos (2αk) cos (2αp), (2.9)where αk represents the polar angle of the vector k. This is known to yielda robust solution with dx2−y2 symmetry for a single CuO2 layer, namely∆ka = ∆d cos (2αk).In order to incorporate the twist, we take the interaction in layer 1 asin Eq. (2.9), but we rotate the interaction in layer 2 by angle θ such thatV(2)kp = −2V cos (2αk − 2θ) cos (2αp − 2θ). For the sake of simplicity, weconsider a circular Fermi surface generated by ξka = ~2k2/2m − µ thatremains invariant under rotation, see Fig. 2.2(e). The problem posed byHamiltonian (2.7) is then solved by defining a four-component Nambu spinorΨk = (ck↑1, c†−k↓1, ck↑2, c†−k↓2)T in terms of which H = ∑k Ψ†khkΨk + E0.Here, the Bogoliubov-de Gennes (BdG) Hamiltonian readshk =ξk ∆k1 g 0∆∗k1 −ξk 0 −gg 0 ξk ∆k20 −g ∆∗k2 −ξk (2.10)and E0 =∑k 2ξk −∑ka ∆ka〈c†k↑ac†−k↓a〉. Diagonalizing hk gives two pairsof energy eigenvalues ±Ekα (α = 1, 2) for each momentum k. With theHamiltonian (2.7) expressed in diagonal form, the free energy of the systemcan be calculated from the standard expressionFBdG = E0 − 2kBT∑kαln [2 cosh (Ekα/2kBT )] . (2.11)By performing a systematic expansion of FBdG in powers of the order param-eter amplitudes ∆d, it is possible to ascertain various coefficients enteringthe GL free energy (2.2). This confirms the form of the GL coefficients192.2. Microscopic modelsB and C deduced previously on the basis of general symmetry arguments,and that K in Eq. (2.4) is positive. This calculation is summarized in theAppendix A.0.1We proceed with a fully self-consistent solution which follows from min-imizing FBdG with respect to ∆ka and can be performed for any given V,twist angle θ and temperature T . To facilitate this calculation we assumethat the order parameters in the two layers have the same amplitude ∆d,but can differ in phase as in Eq. (2.3). Free energy computed from Eq.(2.11) then shows the same qualitative behaviour as the GL theory (2.4)with K > 0: for small twist angles, FBdG(ϕ) has a single minimum at ϕ = 0,while for twist angles close to 45o there are always two degenerate minimaat ϕ = ±ϕmin, indicating formation of a time reversal symmetry brokentopological phase. Fig. 2.2(f) shows the calculated ϕmin as a function of thetwist angle θ for realistic Bi2212 parameters [78–80] ∆d = 40 meV, g = 30meV, c = 60 meV at T = 0. We observe an excellent agreement with theprediction of the GL theory when K = 0.125. The BdG theory also allowsus to predict the minimum excitation gap which, in this simple model, canbe as large as 20 meV when θ is close to 45o.The BdG theory can be formulated directly on the lattice by startingfrom the Hubbard model with on-site repulsion and nearest-neighbour at-traction. This is known to produce a dx2−y2 superconductor when appliedto a single CuO2 monolayer. The Hamiltonian isH = −∑ij,σatijc†iσacjσa − µ∑iσaniσa+∑ij,aVijnianja −∑ijσgijc†iσ1cjσ2, (2.12)where tij encodes the normal-state band structure of the single layer, gijdescribes the interlayer tunneling and Vij denotes density-density inter-actions. The mean-field calculations are performed in bilayer geometriescharacterized by a twist vector v = (m,n) and commensurate twist angleθm,n = 2 arctan (m/n) where m,n are integers, as explained in AppendixA.0.2. Fig. 2.3 summarizes our main results. These lend further supportto our conclusions drawn on the basis of the GL and continuum BdG ap-proaches and show additional interesting features. The lattice model con-firms the onset of the time reversal symmetry breaking phase below criticaltemperature T˜c(θ) whose dependence on θ is consistent with the GL pre-diction, see Fig. 2.3(a,b). The results are in reasonable quantitative agree-ment with the continuum BdG theory but the lattice model gives a smaller202.2. Microscopic modelsspectral gap. This may be attributed to more complicated Fermi surfacegeometry resulting from Brillouin zone folding that accompanies the largereal-space moire´ unit cell.The lattice model allows for a direct evaluation of the Chern numberC, as discussed in Appendix A.0.3. In addition to the C = 4 topologicalphase anticipated on the basis of the continuum BdG theory, the T = 0phase diagrams in Fig. 2.3(c-e) reveal the existence of C = 2 and C = 0gapped phases that can be reached by varying the chemical potential µ andthe interlayer coupling strength g0. In an experiment, the former can betuned over a wide range by oxygen annealing [46], while the latter maydepend on the twist angle θ and could be further controlled by applyinghydrostatic pressure as demonstrated in twisted graphene [49]. As explainedin Appendix A.0.3, the C = 0 phase corresponds to a d+ is superconductor,while the C = 2 phase can be effectively thought of as a single-layer d+ id′superconductor. For a strip geometry, the lattice model predicts protectedchiral edge modes traversing the bulk gap, Fig. 2.3(g,h), confirming thetopological nature of the d+ id′ phase.Additional interesting behaviour is observed at nonzero temperature.Fig. 2.4 shows a phase transition from d + is to d + id phase driven byincreasing T . As we discuss in more detail below, all these phenomena areexperimentally accessible using standard spectroscopic, thermodynamic andtransport techniques.2.2.1 Temperature dependenceThe self-consistent gap equations can be also solved at non-zero temperatureT . Although the Chern number is well defined only at T = 0 we may stillcompute C from Eq. (A.10) using the Bloch eigenstates obtained at T > 0.The resulting integer then describes the underlying mean-field band struc-ture of the system at non-zero temperature. In this case, there will be nostrictly quantized physical response (such as the thermal Hall conductance),but a nonzero C indicates the existence of protected chiral edge states thatcan be observed experimentally. In this sense the T > 0 phases with non-zero C can be regarded as topological and their properties are related to theT = 0 counterparts. In Fig. 2.4 we compare phase diagrams at T = Tc/2and T = 0. We observe that, remarkably, increasing temperature tends tosuppress the topologically trivial d+ is phase in favor of topological d+ id′phases. Specifically at some points in the phase diagram increasing temper-ature can trigger a transition from the d + is phase to the d + id′. This isillustrated in Fig. 2.4(c). As a result, we find that at elevated temperatures212.2. Microscopic modelsk kσ(E)/σNmina)b) c) d) e)f ) g) h)Figure 2.3: Lattice model results. Panels (a,b) show the temperature de-pendence of the minimum gap ∆min, the maximum gap ∆max and phaseϕ, based on a fully self-consistent lattice calculation for coupled layers withcommensurate twist angles θ1,2 ' 53.13o, and θ2,5 ' 43.60o correspondingto a unit cell with 10 and 58 sites, respectively. Panels (c-e) show zero tem-perature phase diagrams of the system for three twist angles as a functionof chemical potential µ and interlayer coupling g0. The range of chemicalpotentials µ ∈ (−1.6t,−1.0t) correspond to range of fillings n ∈ (0.033, 0.04)near optimal doping. Each data point in the phase diagram corresponds toan independent self-consistent solution and the radius of markers is propor-tional to the size of the minimum gap ∆min. The color indicates the Chernnumber as shown in the legend. Panel (f) displays tunneling conductanceσ(E)/σN calculated for θ ' 53.13o, µ/t = −1.3, g0 = 20meV and tempera-tures ranging from 0 to 1.2Tc as indicated by color scale. Curves for differenttemperatures have been offset for clarity. Panels (g) and (h) illustrate theedge modes for C = 4 and C = 2 topological phases, respectively, in θ2,5configuration for parameters µ = −1.3t and g0 = 20, 52 meV. The energyspectrum is shown for a bilayer system in the infinite strip geometry withwidth of 90 unit cells. The color scale represents the normalized positionexpectation value of the eigenstate along the direction of finite length.222.2. Microscopic modelsFigure 2.4: Temperature dependence of the phase diagram is shown forθ2,5 ' 43.6◦ at T = 0 and T = Tc/2 in panels (a) and (b) respectively. Panel(c) shows the minimum gap as a function of temperature for g0 = 32meVand µ = −1.35t (the parameters are indicated in panels (a) and (b) by adashed circle)(but still well below Tc) essentially all of the phase diagram is taken over bythe C = 4 and C = 2 topological phases.2.2.2 Josephson tunnelingIn the past, several works have studied the problem of Cooper pair tunnelingacross c-axis twist junctions in order to understand the order parametersymmetry of cuprates [69, 81]. Assuming a purely dx2−y2 order parameter,it can be shown that the critical Josephson current must vanish when thetwist angle is 45◦. Recent experiments in twisted ultra-thin layers of Bi2212,however, observed that the critical current is essentially independent of thetwist [82]. Here, we show the current is indeed non-vanishing when timereversal symmetry is broken.For the Josephson junction formed by the bilayer, the phase dependent232.3. Outlooksupercurrent for a given c axis twist can be determined within the frameworkof the phenomenological GL theory via the relationI(ϕ, θ) =2e~∂F(ϕ, θ)∂ϕ, (2.13)where the free energy has the form given in Eq. (2.4). A typical current-phase relation thus obtained is plotted in Fig. 2.5(a). The critical Josephsoncurrent across the two layers for a given twist configuration, Ic(θ), may nowbe obtained by maximizing (2.13) with respect to ϕ.As discussed in the main text, for twists in the vicinity of 45◦, a K > 0ushers in a nonzero phase difference between the order parameters in thestatic case. This non-trivial character is manifest also in the behaviour ofIc(θ), as shown in Fig. 2.5(b). To elucidate this, we note that setting K = 0reproduces the behaviour of a junction formed by superconductors with pured-wave character [69, 81]. Note that Ic(45◦) = 0. For a positive K, however,the critical current at 45◦ is non-vanishing. Given that rotations by θ andpi/2 − θ are equivalent from the perspective of the bilayer, the functionalform of the current respects this symmetry and so does its global maximum.The cusps at θ = 45◦ can be seen as a consequence of this.After the publication of this work, recent transport experiments reportedfractional Shapiro states and Fraunhofer patterns consistent with strong sec-ond harmonic. [5]. A detailed theoretical analysis of the Josephson physicsfor twisted cuprates appeared in [83]2.3 OutlookOur results establish a new avenue for engineering high-temperature topolog-ical superconductors by combining monolayers of van der Waals-bonded d-wave superconducting cuprates, such as Bi2Sr2CuO6+δ or Bi2Sr2CaCu2O8+δ[46]. The topological phase is fully gapped, breaks time reversal symmetryand supports protected chiral Majorana modes at the sample boundary.The chiral d ± id′ phase arises robustly, driven by Cooper pair tunnelingbetween the layers when they are twisted by an angle close to 45o. Unlike ingraphene, where the twist angle must be tuned very accurately, we predictthat a wide range of angles (45o ± 10o) will produce nontrivial physics inthese systems.The resulting fully gapped topological phase can be probed by con-ventional spectroscopic or transport techniques. For example, on loweringthe sample temperature, angle resolved photoemission or tunneling spec-troscopies [79, 80] will detect the transition at T = T˜c(θ) from the gapless242.3. Outlookb) a) Figure 2.5: (a) The current-phase relation in units of the critical currentat twist θ = 41.4◦ when K = 0.125. (b) Twist angle dependence of thecritical Josephson current normalized by the current in the untwisted case.While the two curves show similar behaviour close to θ = 0◦ and 90◦, anon-negative K results in a finite value of the current at θ = 45◦.dx2−y2 superconductor with Dirac nodes to a fully gapped excitation spec-trum. This change should also be observable via thermodynamic probes suchas those measuring the specific heat [84] and superfluid density [85, 86]. Sig-natures of the broken time reversal symmetry characteristic of the d±id′ andd+is phases may be identified by circular birefringence and dichroism exper-iments [87]. In transport, the non-zero Chern number manifests as quantizedelectronic contribution to the thermal Hall conductance κxy = C(pi2k2BT/3h)[67]. In mesoscopic samples, longitudinal thermal conductance κxx will also252.3. Outlookbe quantized due to the ballistic contribution of the edge states. The d± id′phase exhibits anomalous interlayer Josephson effect due to the two-minimastructure of the the free energy depicted in Fig. 2.2(c); the Josephson cur-rent IJ = (2e/~)∂F/∂ϕ will develop pronounced deviations from the usualsinusoidal dependence on the phase bias ∆ϕ = ϕ− ϕmin.It is worth noting that existing experimental results on c-axis twistJosephson tunneling in cuprates already offer some support for the ideasadvanced here. According to symmetry analysis, which assumes unbrokentime reversal and pure dx2−y2 order parameter, the Josephson current IJmust vanish when θ = 45o [68–70]. We have shown that for a d + id′ stateIJ , however, remains nonzero at θ = 45o. Non-vanishing IJ has often beeninterpreted as evidence for conventional s-wave order parameter in experi-ments [82, 88]. Given what we know about the order parameter symmetryin cuprates today, one may argue that a more plausible explanation fornon-vanishing IJ is the spontaneously time reversal symmetry broken phaseadvocated in this work.In next chapter we will study polar Kerr effect (PKE) in twisted cupratebilayers. If observed in an experiment, PKE is an unambiguous signature oftime reversal symmetry breaking which drives the topological phase transi-tion in our system.26Chapter 3Polar Kerr effect as a probeof time reversal symmetrybreakingWhile topological insulators and semimetals appear to be abundant in na-ture [89] materials exhibiting topological superconductivity have been no-toriously hard to find [90]. The leading candidate for the spin triplet px +ipy chiral superconductor, Sr2RuO4, which showed time reversal symme-try breaking in muon spin relaxation [32] and polar Kerr effect measure-ments [33], has been recently shown to be inconsistent with triplet pairing byKnight shift experiment [77]. Other suspected topological superconductors(TSCs) include FeTexSe1−x [91] and copper-doped Bi2Se3 [92] but at presentthe balance of evidence remains inconclusive. Perhaps the most convincingcase for TSC can be made in artificially engineered proximitized semiconduc-tor quantum wires where strong evidence for Majorana end modes has beenreported by multiple groups and in a variety of experimental configurations[93–97].In this chapter, we consider a platform recently proposed to artificiallyengineer a chiral topological superconductor based on a pair of monolayerthick cuprate superconductors assembled with a twist [1, 27]. For a range oftwist angles close to 45o such a bilayer is predicted to form a time reversalsymmetry breaking chiral phase that can be thought of as a dx2−y2 ± idxysuperconductor (which we abbreviate as d ± id′ henceforth). We addresshere the pressing question of how such a topological time reversal symmetrybroken phase can be reliably identified in experiment. It is to be noted thata direct detection of the topological phase through its chiral edge modes isexpected to be extremely challenging. In topological superconductors thesemodes are neutral and must therefore be probed via the thermal Hall effect,which is expected to attain a quantized contribution at low temperatures.Here we show that owing to its multiband nature and broken symme-tries the twisted bilayer exhibits strong anomalous electrical Hall conductiv-273.1. PKE and optical Hall conductivity.ity which can be probed optically through a conventional polar Kerr effect(PKE) measurement.Finite magneto-optic Kerr effect is a manifestation of time reversal sym-metry breaking [98] in a material where linearly polarized incident light isreflected with elliptical polarization whose major axis is rotated with respectto the incident polarization axis. Kerr angle θK is directly related to theanomalous Hall conductivity σH(ω) defined below. Time reversal symmetrybreaking is a necessary but not sufficient condition for finite Kerr rotation.In addition, any mirror reflection symmetry perpendicular to the incidentwave vector must also be broken for σH(ω) to be nonzero [99]. In this work,we calculate the anomalous Hall conductivity of two stacked CuO2 mono-layers twisted with respect to one another by an angle close to 45o. Such aconfiguration develops a complex phase between the d-wave order parame-ters in the two layers, spontaneously breaking the time reversal symmetryas well as all mirror symmetries. The resulting system can be in a d + id′or d + is phase, depending on parameters such as doping and interlayercoupling strength [1]. Both phases break time reversal symmetry and arefully gapped but only the former exhibits topologically protected chiral edgemodes. In this phase we find a large intrinsic contribution to σH(ω) arisingfrom interband transitions that involve chiral Cooper pairs – a mechanismoriginally identified in Ref. [100].3.1 PKE and optical Hall conductivity.In the presence of time reversal symmetry breaking, the polarization axis ofa linearly polarized incident light will be rotated by the Kerr angle θK givenby [101]θK(ω) =4pi2dωIm(σH(ω)n(n2 − 1))(3.1)where n is the complex index of refraction, d denotes the separation ofmonolayer pairs and σH(ω) is the antisymmetric part of the optical Hallconductivity per monolayerσH(ω) =12limq→0[σxy(q, ω)− σyx(q, ω)] . (3.2)Eq. (3.1) is valid when the sample thickness h λ, the wavelength ofthe incident light. As such it would apply to a stack composed of manybilayers. In the opposite limit h λ, relevant to a single bilayer, theKerr angle is given by a different expression [102] which in addition involves283.2. 10-band model of coupled twisted copper oxide layers.the longitudinal optical conductivity σxx(ω) and is specified in Eq. (3.8)below. Crucially, in both limits θK can be non-zero only when the Hallconductivity is finite. The latter can be computed from the Kubo formulaas σxy(q, ω) =1ω (pixy(q, ω) − pixy(q, 0)). Here ω is the frequency of theincident light andpixy(q, ω) =∫ ∞0dteiωt〈[Jˆ†x(q, t), Jˆy(q, 0)]〉 (3.3)is the current-current correlator. The total current operatorJˆi = e∑ktrΨ†kvˆiΨkuses vˆi = (1⊗σz)∂kih0k where h0k and h∆k are the normal and superconductingparts of the system Bloch Hamiltonian hk = h0k+h∆k , respectively, expressedin the orbital basis and σz acts in the particle-hole (Nambu) space.Under time reversal the current correlator transforms as pixy → piyx whileunder mirror reflections along x or y axes pixy → −pixy. Therefore, in orderto obtain a nonzero Kerr effect, both time reversal and all mirror symmetriesmust be broken [99].Equations (3.2,3.3) can be rewritten in terms of the eigenspectrum of hkasσH(ω) =ie2ω∑k,ab(ω + i)Qab(Eak − Ebk)2 − (ω + i)2nF (Eak) (3.4)where a, b are band indices and Qab = 2i Im{vxabvyba}k with the matrix ele-ment vjab = 〈ak|vˆj |bk〉 computed from the Bloch eigenstates. This formula-tion is convenient for numerical calculations based on the lattice models oftwisted bilayers discussed below. Details of the derivation are given in theAppendix B.2.3.2 10-band model of coupled twisted copperoxide layers.Following Ref. [1] (presented in Chapter 2) we consider two stacked cupratemonolayers twisted with respect to each other by a commensurate twist angleθm,n = 2 arctan (m/n). Each CuO2 plane is modeled by a minimal Hubbardmodel with nearest neighbor attractive potential, known to produce a dx2−y2293.2. 10-band model of coupled twisted copper oxide layers.superconductor within the standard Bogoliubov-de Gennes (BdG) theory.The Hamiltonian has the formH = −∑ij,σαtijc†iσαcjσα − µ∑iσaniσa+∑ij,αVijniαnjα −∑ijσgijc†iσ1cjσ2, (3.5)where c†iσα is the electron creation operator at site i with spin σ on top(bottom) layer with index α = 1(2), and niσα = c†iσαciσα. The nearest andnext nearest neighbor intralayer tunneling amplitudes between sites i andj are denoted by tij . Chemical potential µ controls the doping, Vij are theattractive interactions. The interlayer coupling strengths gij are assumed todecay exponentially with distance rij =√d2 + ρ2ij asgij = g0e−(rij−d)/ζ , (3.6)where ρij is the in-plane separation between sites i and j, d is the distancebetween the monolayers and ζ denotes the characteristic lengthscale forinterlayer tunneling.For simplicity and concretness we will focus here on a specific commen-surate twist angle θ1,2 = 2 arctan (1/2) ' 53.13o which leads to the smallestnontrivial moire´ unit cell with 10 sites, as sketched in Fig. 3.1 and is suffi-ciently close to 45o to illustrate all the interesting physics. We will refer tothis lattice model as the 10-band model in the following.The phase diagram of the 10-band model, obtained through the stan-dard mean-field decoupling of the interaction term in the pairing channeland then solving the resulting gap equations [1], is displayed in Fig. 3.2(b).At weak interlayer coupling g0 we find that d-wave superconducting orderparameters in twisted monolayers acquire a complex relative phase, spon-taneously breaking the time reversal symmetry and forming a d+ id′ TSC.This phase is characterized by Chern number C = 4 which can be under-stood as two layers of d + id′ TSC, each contributing C = 2. As g0 getsstronger the system transitions into the time reversal symmetry broken buttopologically trivial d+ is state with C = 0. For even larger g0 the complexphase between layers disappears resulting in a gapless phase that can bethought of as two independent dx2−y2 superconductors in the band basis.We note that for twist angles closer to 45o the gapless phase is replaced byanother TSC phase characterized by C = 2 [1] – this can be regarded as asingle-layer d+ id′ superconductor in the band basis.303.2. 10-band model of coupled twisted copper oxide layers.0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 [eV]0.0020.0000.0020.0040.0060.0080.010H [e2 /]g0 = 4.0meVg0 = 8.0meVg0 = 32.0meVb)a)Figure 3.1: a) Optical Hall conductivity for the 10-band lattice model cal-culated from Eq. (3.4) for various values of the interlayer coupling strengthg0. Solid and dashed lines show the real and the imaginary parts ofσH(ω), respectively. The tight binding model parameters are chosen tobe t = 0.153eV, t′ = −0.45t, µ = −1.3t. The SC order parameter is calcu-lated self consistently with Vij chosen such that the maximum gap is 40meV.Grey dotted curve shows the ω−2 scaling which is the expected behavior ofReσH(ω) in the high frequency limit. b) The 10-band model unit cell withtwo stacked layers depicted in blue and red. The signs characteristic of d-wave order parameter along with the relative complex phases are indicatedin the right panel. Note that the complex phases in the order parametersbreak the mirror reflection symmetries along the x, y and x± y directions.Fig. 3.1 shows our results for the Hall conductivity σH(ω) in the 10-band model computed using Eq. (3.4). The parameters are chosen to cap-ture Bi2Sr2CaCu2O8+δ near optimal doping where we expect the mean-fieldBCS theory to provide an accurate description of the superconducting state.Because the strength of the interlayer coupling g0 is difficult to accuratelyestimate for the twisted configuration we display σH(ω) for several repre-sentative values spanning its likely range. When g0 < 20meV the groundstate has Chern number C = 4. The Hall conductivity is non-zero with anamplitude that is three to four orders of magnitude larger than the predic-313.3. Effective 2-band model.tion for Sr2RuO4 given in Ref. [100]. We show below that this leads to agiant Kerr angle that should be straightforward to detect experimentally.By contrast in the C = 0 phase, obtained for g0 = 32meV, σH(ω) is muchsmaller (≈ 10−7e2/~) and will lead to very weak Kerr signal.3.3 Effective 2-band model.In order to gain some insight into the Hall response of twisted bilayers wenow consider an effective 2-band model designed to capture the essentialfeatures of the system. The 2-band BdG Hamiltonian heffk and its derivationare given in the Appendix B.1 The Hall conductivity Eq. (3.2) for the 2-bandmodel is found to beσH(ω) = 2e2∑kIm[∆∗1∆2] [∂k(ξ2 − ξ1)× ∂k|g|2]z×1E1E2[1− nF (E1)− nF (E2)E+(E2+ − (ω + i)2)− nF (E1)− nF (E2)E−(E2− − (ω + i)2)],(3.7)where ξ1(2) and ∆1(2) are the effective dispersions and gap functions definedin Eqs. (B.8) and (B.9) of Appendix B.1. E1,2 denote the positive energyeigenvalues of heffk and E± = E1±E2. The derivation of Eq. (3.7) is given inB.2.1. Comparison with σH(ω) obtained directly from the 10-band model ispresented in Appendix B.2.2Eq. (3.7) is useful because it clarifies conditions under which a systemcan exhibit non-vanishing anomalous Hall effect. Specifically, three physicalingredients are required, in addition to broken time reversal symmetry andmirror symmetries: (i) the two normal-state dispersions must be different(ξ1 6= ξ2), (ii) a non-vanishing SC phase difference must exist between thelayers such that Im[∆∗1∆2] is non-zero, and (iii) the interlayer coupling gmust be k-dependent. In the two-band model defined by Hamiltonian (S7)the first two conditions are satisfied for a d + id′ state and when second-neighbor in-plane tunneling t˜′ is non-zero. The third condition will be metif we allow interlayer hopping between sites that are not directly aboveone another. For instance we may assume the two square lattices to beoffset by (b/2, b/2) and allow nearest neighbor hopping; this gives g(k) =4g0 cos (kxb/2) cos (kyb/2). Importantly, we expect the above conditions tobe met in a physical sample of a twisted Bi2Sr2CaCu2O8+δ bilayer.To produce finite and significant Hall signal, the integrand in Eq. (3.7)must be even under the reflection x↔ y. This is most easily ensured whenall factors are even: the time reversal symmetry breaking Im[∆∗1∆2], the323.3. Effective 2-band model.0 10 20 30 40 50 60 70 80g0 [meV]0.0000.0050.0100.0150.0200.0250.0300.0350.040ReH(0) [e 2/]g 20a)1.6 1.4 1.2 1.0/t 10 20 30 40 50g [meV]0C=0C=2C=4d+id d+isb)Figure 3.2: a) Zero frequency limit of the real part of the optical Hall con-ductivity σH(ω) for the 10-band lattice model for the same parameters asFig. 3.1. The data shown in the main panel corresponds to the highlightedslice in the inset (b) which displays the phase diagram of this configurationas a function of g0 and µ/t. The color corresponds to the Chern number Cwhile the radius of the dots is proportional to the minimum excitation gap.vertex, and the energy factors are all even in the d+ id′ case. On the otherhand Im[∆∗1∆2] becomes odd in the d + is case which renders the signalvanishing. This exemplifies a situation that breaks both mirror and time-reversal symmetries but still exhibits no Kerr signal. This distinguishingfeature between the two time reversal symmetry breaking cases, trivial d+isand topological d + id′, is indeed confirmed in the 10-band calculation inFig. 3.2, where the former exhibits negligibly small signal compared to thelatter.At zero temperature, only the first term on the second line of Eq. (3.7)contributes. This term describes a process of breaking a Cooper pair intotwo quasiparticles on different branches of the spectrum and indicates theonset of the absorptive part Im[σH(ω)] at ω∗ = mink[E1(k) + E2(k)]. Note333.4. Kerr angle estimate.that ω∗ is roughly set by the energy scale of the interlayer coupling g whenthe pairing is small compared to the chemical potential and is in generalmuch larger than the quasiparticle gap minimum. We expect this physicalpicture and the form of ω∗ to hold approximately in the 10-band modelwith E1,2 being the two lowest energy levels. This is indeed confirmed inour numerical calculations shown in Fig. 3.1 and Fig. B.3 in the Appendix.Lastly, Eq. (3.7) shows that the Hall signal amplitude exhibits quadraticdependence on g and ∆ to leading order. This is numerically confirmed tohold in the 10-band model as indicated in Fig. 3.2 for the g-dependence.This observation suggests a plausible explanation for why Fig. 3.1 showssignal four orders of magnitude larger than that predicted for Sr2RuO4 [90]:in Bi2Sr2CaCu2O8+δ the gap ∆ is about two orders of magnitude largerwhile g is roughly the same in the two materials.3.4 Kerr angle estimate.We use Eq. (3.1) and its thin sample (λ h) counterpart [102] given byθK = Re arctan( −σxyσxx + 4pi(σ2xx + σ2xy))(3.8)to estimate the expected Kerr angle. Here the optical conductivities aremade dimensionless by attaching the fine structure constant α, i.e., settinge2/~ = α in the natural units. The C4 rotation symmetry of the twistedbilayer implies σxy = −σyx which, in combination with Eq. (3.2), givesσxy = σH . The diagonal part of the conductivity tensor and the complexindex of refraction required in Eq. (3.1) are dominated by various relaxationmechanisms whose microscopic origin is poorly understood in the cuprates.To estimate σxx we thus rely on an empirical power-law formulaσxx ' 2dC(−iω)γ−2, γ = 1.447, (3.9)that was shown in Ref. [103] to accurately describe the ab-plane reflectancedata on Bi2Sr2CaCu2O8+δ for frequencies above ωc ' 0.12eV at all temper-atures. System thickness h = 2d is inserted to convert between the bulk andthin film conductivity.Fig. 3.3 shows our results for PKE with further details provided inAppendix B.3. For simplicity we assume the power law Eq. (3.9) to bevalid at all frequencies and therefore expect our predictions for θK to be lessaccurate for ω . ωc. We note, however, that PKE experiments are typically343.4. Kerr angle estimate.performed at frequencies above ∼ 0.5 eV [33, 98], well within the range ofapplicability of Eq. (3.9).In this experimentally relevant regime it is possible to obtain a simpleapproximate expression for θK in the thin-sample limit that can be used asguidance in experimental studies. As shown in Appendix B.3.2, for ω & ωcone may approximate Eq. (3.8) asθK ' Λg20ω−γ , (3.10)where θK is in radians and Λ = 0.3623[eV ]γ−2 is a constant we extractby fitting the curves in Fig. 3.3, which show excellent agreement with thescaling form (3.10). For the typical photon frequency ~ω = 0.5eV Eq. (3.10)predicts θK ' 10− 100 µrad if we assume g0 = 4− 16 meV.0.0 0.1 0.2 0.3 0.4 0.50.00060.00040.00020.00000.0002K [rad] (h)g0 = 4.0meVg0 = 8.0meVg0 = 16.0meVg0 = 32.0meV0.0 0.1 0.2 0.3 0.4 0.5 [eV]0.0000.0020.0040.0060.0080.0100.0120.014K [rad] (h)10 1 2 × 10 1 3 × 10 1 4 × 10 110 510 3a)b)c)Figure 3.3: Kerr angle θK as a function of photon energy ~ω for variousvalues of interlayer coupling g0. a) Thick sample limit Eq. (3.1). b) Thinsample limit Eq. (3.8). c) Semi-log plot with the expected ω−γ scaling athigh frequencies shown by the dotted grey line. Vertical dashed lines indicatethe frequency ωc above which σxx shows the power-law form ∝ (−iω)γ−2353.5. Summary and conclusions3.5 Summary and conclusionsOur modelling predicts a large intrinsic contribution to the anomalous Hallconductivity of twisted Bi2Sr2CaCu2O8+δ bilayer in the spontaneously timereversal symmetry broken topological phase, which is expected to occurfor a range of twist angles close to 45o. There could be other ‘extrin-sic’ contributions to σH(ω) from various scattering mechanisms discussedin the literature [87, 104–106] which we have not considered here. Typically,such contributions arise from higher-order diagrams in the expansion of thecurrent-current correlator Eq. (3.3) and are thus subdominant with respectto the intrinsic component identified here. To the extent that various con-tributions are additive our estimate for the Kerr angle should therefore beviewed as a lower bound. Its large magnitude, measured in µrad ratherthan nrad, typical of Sr2RuO4 and other chiral SC candidates such as UPt3[107, 108], gives hope that spontaneous time reversal symmetry breaking intwisted cuprate bilayers can be reliably and unambiguously detected withexisting laboratory instruments.36Chapter 4Superconducting diode fromJJ arrays4.1 IntroductionThe superconducting diode effect is present in a superconducting two-terminaldevice when it supports a larger critical current in forward direction than inbackward direction. In the ideal limit of maximal imbalance, current appliedin one direction is dissipationless (zero resistance), while it is always dissi-pative (resistive) in the opposite direction. This is a natural generalizationof the semiconductor diode that is only weakly resistive in one direction andhighly resistive in the opposite direction.Recent interest in the superconducting diode effect has lead to a surgein experimental [34, 36, 39, 109–116] and theoretical studies [43, 117–128]including device proposals [129]. The common factor among them is relianceon a rather complicated material platform, precluding near-term commercialapplications. The diode effect requires both inversion- and time-reversalsymmetry breaking. Inversion symmetry can be absent in systems with spin-orbit coupling [34, 127], or it can be explicitly broken by twisting bilayers[39, 123, 130] or applying a current bias [131].Two recent theoretical studies proposed a Josephson junction circuitrealization of the superconducting diode effect, independent of an inversionsymmetry breaking material platform [132, 133]. However, these proposalsrely on high-harmonic content in the current phase relation of the junctions,i.e., a non-sinusoidal current phase relation. Inversion symmetry is brokenby asymmetry of Josephson junctions in the circuit, which differ in theirharmonic content.In the present work, we propose a realization of the diode effect thatremoves all of the above requirements. It is based on a classical circuit ofJosephson junctions with standard sinusoidal current phase relation. Junc-tions can in principle even be identical, and inversion symmetry is brokenby the connectivity of the circuit. Time-reversal symmetry is broken by374.2. The diode effect in a superconducting Josephson circuitmagnetic fluxes that can be applied using flux bias lines. The present diodeimplementation is agnostic to its underlying material platform and can berealized in industry standard Nb-Al processes based on scalable semicon-ductor technology.This chapter is organized as follows. In Sec. 4.2.1 we introduce the sim-plest circuit that gives rise to a diode effect. The approach is based on asimilar scheme of leveraging high-harmonics in the current phase relation asin [132, 133], with the key difference that effective high-harmonics are engi-neered by chaining identical junctions in series. We generalize this picturein Sec. 4.2.2, where we give an analytical form for junction and flux param-eters that achieve the most optimal diode for this geometry. In Sec. 4.3 wediscuss effects of parasitic resistors, capacitors, and inductors. We concludewith a summary in Sec. 4.4.4.2 The diode effect in a superconductingJosephson circuit4.2.1 Minimal circuitWe consider the superconducting circuit depicted in Fig. 4.1 consisting oftwo arms of Josephson junctions (JJ). The first arm consists of two junctionsacross which the superconducting phase jumps by ϕ1a, ϕ1b, respectively. Thesecond arm consists of single Josephson junction with phase difference ϕ2.The parasitic inductance of the wires is modeled by the lumped inductorsL1, L2. This circuit has been previously introduced in the context of secondharmonic generation, where it is known as the Superconducting NonlinearAsymmetric Inductive Element (SNAIL) [134].The supercurrents in both arms of the SNAIL are determined by theJosephson relationsI1 = I(1a)c sinϕ1a = I(1b)c sinϕ1b (4.1)I2 = I(2)c sinϕ1 . (4.2)We will be interested in the case of identical junctions in the first arm, i.e.,I(1a) = I(1b). Here, the phases across the junctions are equal, ϕ1a = ϕ1b ≡ϕ1. Continuity of the phase along the superconducting loop then yields thephase quantization condition2ϕ1 − ϕ2 + 2pin = 2piΦ/Φ0 (4.3)384.2. The diode effect in a superconducting Josephson circuitFigure 4.1: Superconducting circuit consisting of two inductive arms withone and two Josephson junctions in series, respectively. An external flux Φextmay thread the superconducting loop. Here, the inductors L1, L2 representthe geometric inductance of the superconducting wires.where Φ is the total flux threading the junction and Φ0 = h/2e is thesuperconducting flux quantum.For small circuits the geometric inductance can be neglected, La, Lb → 0.Then, flux through the loop is solely determined by the external flux, i.e.,Φ = Φext. The total current isI(ϕ) = I1 + I2 = I(1)c sinϕ1 + I(2)c sin (2ϕ1 − ϕext) (4.4)where we have defined ϕext = 2piΦext/Φ0. The critical current in positiveand negative directions are defined asI+c ≡ maxϕ I(ϕ)I−c ≡∣∣∣∣minϕ I(ϕ)∣∣∣∣ (4.5)394.2. The diode effect in a superconducting Josephson circuitwhere we note that I+c and I−c are positive quantities. Importantly, forϕext/2pi /∈ Z, i.e., when ϕext is a nontrivial phase, one generally hasI+c 6= I−c , (4.6)i.e., the forward and backward applied critical currents are imbalanced. Thisconstitutes the superconducting diode effect. The degree of imbalance be-tween these two critical current values is quantified by the superconductingdiode efficiencyη =|I+c − I−c |I+c + I−c, (4.7)which is a positive number and η < 1. We find the maximum efficiencyof η = 1/3 for the two-arm geometry in Fig. 4.1 when I(2)c = I(1)c /2 andΦext = Φ0/4.4.2.2 N-arm interferometers and the ideal diode limitNext we consider a N -ladder of JJ arrays which we enumerate by k = 1, .., Nwith nk consecutive junctions forming each arm k and flux Φk,l enclosedbetween the arms k and l. By phase continuity, the Josephson phase for asingle junction in arm k in terms of the phase variable ϕ1 is given byϕk =n1ϕ1 − 2piΦ0 Φext1,k + 2pinnk(4.8)in the limit of zero inductance.We assume the zero-vortex state n = 0 and define φextk =1nk2piΦ0Φext1,k .Then, the total current isI(ϕ1) =∑kIk =N∑k=1I(k)c sin(n1nkϕ1 − φextk). (4.9)If the integer coefficients nk are chosen in such a way thatn1/nk = k , (4.10)Eq. (4.9) represents a Fourier series that allows for engineering of arbi-trary current phase relations. One such set of integers is given by nk =N !/ [floor(N/2)! k]. For an efficient diode layout, one should however reducethis set by its greatest common denominator, GCD({nk}).404.2. The diode effect in a superconducting Josephson circuitFigure 4.2: Generalized Josephson interferometer consisting of N arms withcurrent Ik and nk identical Josephson junctions with critical current I(k)cthat each experience a superconducting phase drop ϕk. Geometrical induc-tances have been lumped into the elements Lk. A magnetic flux Φk,l maybe threaded between arms k and l.The Fourier series is fully specified by the individual magnetic flux pa-rameters φextk and critical currents I(k)c . The fluxes are easily tuned via fluxbias lines and critical currents are determined by the junction area in thefabrication design.We are now left to discuss the problem of finding the set of parametersthat yield the greatest diode efficiency η. Let us propose the particularchoiceI(k)c = I0N + 1− kN(4.11)φextk = (k − 1)pi/2 (4.12)This yields the following analytical expression for the total current of the414.3. Parasitic componentsinterferometer:I(ϕ1)/I0 =N∑k=1N + 1− kNsin (kϕ1 + (1− k)pi/2)=cos ((N + 1)(ϕ1 − pi/2))− 12N (sinϕ1 − 1) −N + 12N(4.13)A plot of Eq. (4.13) for various N is shown in Fig. 4.3(a). As N is increased,the function develops a narrow peak of height I/I0 = (N+1)/2 at ϕ1 = pi/2over a seemingly flat background at I/I0 = (N+1)/2N , yielding a significantimbalance of critical currents with diode efficiencyη =N − 1N + 1(4.14)that approaches unity in the large-N limit. In fact, for N →∞, the current-phase relation approaches a series of δ-functions centered at 2pi(n+1/4) andshifted by a constant current −I0/2, according tolimN→∞I/I0 = pi∑kδ(ϕ1 − pi/2 + 2pik)− 12. (4.15)While the flux and critical current parameters defined in Eqs. (4.11-4.12)yield an ideal diode in the large-N limit, they likely also constitute thefastest converging series. For small N ≤ 5, where the parameter space isstill amenable to numerical optimization, we have numerically confirmedthat it represents the optimal solution.4.3 Parasitic components4.3.1 IV-characteristicWe now consider a resistively and capacitively shunted junction (RCSJ)model where each junction on arm k in Fig. 4.2 is shunted by an internalcapacitor Ck and resistor Rk. For an external dc current bias I(k)dc in arm kthe equation of motion is given byI(k)dc =Φ02piϕ˙kRk+Φ02piCkϕ¨k + I(k)c sinϕk (4.16)We assume that every junction is identical for a given arm and currentconservation implies that ϕk is the same for every junction. Using the phase424.3. Parasitic components0.0 0.5ϕ1/2pi−1.0−0.50.00.51.02I/[I 0(N+1)]25 50 75 100N0.00.20.40.60.81.0Diodeefficiencyη(N − 1)/(N + 1)0.0 0.5ϕ1/2pi0246810I/I025 50 75 100N0.00.20.40.60.81.0Diodeefficiencyη(N − 1)/(N + 1)Figure 4.3: (a) Current phase relation for number of interferometer armsN = 1, 2, 5, 20 and (b) diode efficiency as a function of N that is given byη = (N − 1)/(N + 1).continuity condition (4.8) and adding all currents one can show that thesystem is governed by a single differential equationIdc/I0 =d2ϕ1dτ2+ βdϕ1dτ+ I(ϕ1)/I0 , (4.17)where Idc =∑k I(k)dc is the total, experimentally applied current bias. Wehave also defined the dimensionless parameters τ = ωJ t, β = (ωJReffCeff)−1,ωJ =√2e~I0Ceffand the effective resistance and capacitance parametersReff =(N∑k=1n1Rknk+n1Rext)−1(4.18)Ceff =N∑k=1n1Cknk+ n1Cext . (4.19)Rk and Ck will depend on the geometries of individual junctions. For bettertunability, one may shunt the entire device with an additional resistor Rextand a capacitor Cext. The voltage across each arm is given by V = n1Φ02pi ϕ˙1.434.3. Parasitic components−0.5 0.0 0.5V/(n1RI0(N + 1))−0.50.00.5I dc/[I 0(N+1)]N = 8−0.5 0.0 0.5V/(n1RI0(N + 1))−0.50.00.5I dc/[I 0(N+1)]N = 2−0.5 0.0 0.5V/(n1RI0(N + 1))−0.50.00.5I dc/[I 0(N+1)]N = 1−0.5 0.0 0.5V/(n1RI0(N + 1))−0.50.00.5I dc/[I 0(N+1)]N = 5Figure 4.4: I-V curves computed within RSJ model for β = 0.6√n1 andN = 1, 2, 5, 8In the limit where Rext Rk and Cext Ck, dynamics is independent ofinternal resistances and capacitances of the junctions, where we find Reff ≈Rext/n1 and Ceff ≈ n1Cext. In this limit, β defined just below 4.17 scalesas β ≈ √N ! where ω˜ =√2e~I0Cextand n1 is the number of junctions on armk = 1 which depends on N as nk ≈ N ! in the large-N limitWe numerically solve the differential Eq. (4.17) for voltage as a functionof external current Idc. The resulting I-V -curves are shown in Fig. 4.4for various N and β = 0.6√n1. For N > 1, the critical currents becomeimbalanced. The I-V -curves display a well-known hysteretic behavior which,however, can be suppressed for large N , i.e., when β is sufficiently large.444.4. Discussion and SummaryN η NJ nk Φk,k+1/Φ0 I(k)c N/I02 1/3 3 2 1 1/4 2 13 1/2 11 6 3 2 3/4 1/4 3 2 14 3/5 25 12 6 4 3 3/2 1/2 1/4 4 3 2 15 2/3 137 60 30 20 15 12 15/2 5/2 5/4 3/4 5 4 3 2 1Table 4.1: Optimal parameters for superconducting diode circuits with Narms. NJ =∑k nk is the total number of Josephson junctions required andΦk,k+1 is the magnetic flux threaded between arms k and k + 1.4.3.2 Finite inductanceIn the case of finite geometric inductance, Lk > 0, the flux through thesuperconducting loops is no longer defined just by the external flux. Instead,Eq. (4.8) must be modified to also include the contribution of the inducedflux,ϕk =1nk2piΦ0(Φext1,k + L1I1 − LkIk). (4.20)Now, Eq. (4.9) must be solved self-consistently, since a change in currentimplies a change in ϕk, which again induces a change in the Ik. For suffi-ciently large inductances, the induced fluxes will develop hysteretic behaviorin the externally applied flux.Since the geometrical inductances are expected to be roughly equalfor each arm, we parametrize them by a single dimensionless parameter2piLi/Φ0 ≡ λ. In Fig. 4.5, we show the results of the self-consistent calcula-tion for various λ, small enough that the diode is still in the non-hystereticregime. The inductance can be seen to distort the current phase relation andyields an overall decrease in diode efficiency. Assuming that the λ roughlyscales with the maximum number of junctions in a branch n1, this efficiencydecrease is more pronounced at large N .4.4 Discussion and SummaryWe have proposed a superconducting diode circuit element that requiresonly conventional Josephson junctions and flux bias loops which can befabricated utilizing existing integrated circuit technology.A Josephson interferometer will exhibit the superconducting diode effectif current arms carry different harmonics sinnϕ of the current phase relationand if these harmonics are phase-shifted with respect to each other.454.4. Discussion and Summary0.0 0.5 1.0ϕ1/2pi−10123I/I05 10 15 20N0.00.20.40.60.81.0DiodeefficiencyηFigure 4.5: Current phase relation (a) for the exemplary case N = 5 and(b) diode efficiency as a function of N for varied strength of the inductanceparameter λ/n1. The geometrical inductance distorts the current-phase re-lation and diminishes the diode efficiency η.We have shown that higher harmonics can effectively be generated in gen-eralized SNAIL geometries when multiple conventional Josephson junctionsare connected in series. Flux biases in the loops are crucial for generatingthe phase differences, which we have optimized for diode efficiency. Notethat polarity of the superconducting diode can be switched by reversal of allfluxes, i.e., by setting ϕk,ext → −ϕk,ext in Eq. (4.12). We have also consid-ered the effects of geometric inductance, which lead to an overall decreasein diode efficiency.We work in the limit EJ EC where charging energy is Ec = e2/2CJ .Here, fluctuations in the phase variable ϕ are suppressed. Therefore ϕ canbe treated as a classical variable, justifying the use of RSJ model in Eq. (4.4).Table 4.1 summarizes the various optimal circuit parameters for differentN , along with the resulting diode efficiency in the non-inductive case. Cir-cuits with more than a few arms are likely mostly of academic interest, as thetotal number of required junctions scales exponentially in N , even thoughmore than 106 Josephson junctions and on the order of 105 flux biases havebeen integrated on a single chip [135, 136]. Nevertheless, for reasonablysmall N , our proposed diode device could be promising for practical use.46Chapter 5NMon: a novelsuperconducting qubit5.1 IntroductionTo effectively process quantum information, we must first learn how to storeand access it reliably. We aim to create a memory element that is not onlysufficiently isolated to prevent data loss from environmental interference,but also constructed in a way that it is easy to access as needed. One popu-lar approach to this problem has been to store quantum information in theenergy levels of an artificial anharmonic oscillator which can be engineeredusing superconducting circuit elements [137], most notably the Josephsonjunctions. Of course, there are many other approaches to quantum informa-tion processing in different physical platforms which include quantum dots[138], Rydberg atoms [139], NV centers [140] and photonic circuits [141]Transmon [45] is one of the most extensively studied superconductingqubits, where the Josephson energy EJ and the charging energy EC arechosen such that EJ/EC 1. This limit suppresses the noise due to chargefluctuations, which is a major challenge with the Cooper pair box [142],operated in the regime where EJ ≈ EC . On the other hand, approaching thislimit also reduces anharmonicity, a crucial property of the energy spectrumthat prevents information from leaking out the computational space. [45]Therefore the EJ can not be made arbitrarily larger than EC and typicallychosen such that EJ/EC ≈ 100. Larger anharmonicity is desirable not onlyfor separating the computational space from other energy levels, but alsofor decreasing the lower bound on the pulse durations τp ≈ |ω01αr|−1 for themanipulation of the two level system, leading to faster gate operations [45].As an alternative to transmon, the flux qubit [143] solves the anhar-monicity problem by introducing three junctions in a ring geometry, cre-ating a double-well potential where the two lowest lying energy levels areenergetically well separated from further excited states. However, such ge-ometries with loops are susceptible to flux fluctuations [144], reducing coher-475.1. Introductionence. Fluxonium qubit [145] alleviates this problem by shunting a Josephsonjunction by an array of N Josephson junctions where the flux fluctuationsare supressed by a factor of ∼ 1/N .In this work we introduce a novel qubit design we call NMon, consist-ing of two parallel arrays of identical Josephson junctions with N and Marrays with Josephson energies ENJ and EMJ , respectively. The geometry isillustrated in Fig. 5.1. This setup can be thought of as a variant on splittransmon for a certain choice of parameters. Notably, NMon exhibits en-hanced anharmonicity compared to transmon while maintaining transitionmatrix elements comparable in magnitude to those of the split transmon.We also show that if we increase the number of junctions in NMon, ithas the following advantage over the fluxonium qubit: for the same numberof Josephson junctions, while the matrix elements for charge fluctuationsare comparable for the two systems, NMon transition matrix elements forflux fluctuations can be made smaller by an order of magnitude comparedto the fluxonium. We expect NMon to work best near zero external fluxthrough the loop, which happens to be the sweet spot where the energylevels show minimum dispersion as a function of the external flux as seenin Fig. 5.4. This is in contrast to fluxonium (described in Appendix C.4.2)which is operated at half magnetic flux quantum bias, requiring currentsdriven through nearby flux bias loops, complicating the design and makingscalability more difficult.Our perspective in this work is engineering the free energy potentialusing arrays of identical Josephson junctions for enhanced tunability andhigher fidelity in superconducting qubits. NMon is described by the follow-ing Hamiltonian:H = 4EC nˆ2 −NENJ cos(Mφ− κϕextN)−MEMJ cos(Nφ+ (1− κ)ϕextM)(5.1)where we defined 0 < κ < 1 as κ = CMS /(CNS + CMS ) which is a parametercontrolled by the ratio of the shunt capacitances for each arm. The effectivecharging energy becomesEC =2e24(NM)2(CMS + CNS )(5.2)This Hamiltonian (Eq. 5.1) is derived in Appendix C.1 using canonical quan-tization with irrotational constraints [146, 147] in mind since we will consider485.2. A variant on split transmon with enhanced anharmonicityCSMCSNEJNEJMFigure 5.1: NMon geometry. Two parallel arms with N and M Joseph-son junctions shunted by large capacitances CNS and CMS , respectively. Theexternal flux Φext through the circuit loop introduces a relative phase differ-ence between the harmonics contributing to the Josephson potential givenin Eq. 5.1the time dependent flux as well, as a noise source and drive. If the externalflux Φext = ϕextΦ02pi through the loop of the circuit is time independent, weare free to allocate it in either cosine term when we impose the continuitycondition for the superconducting order parameter. This can be consid-ered as a gauge choice. However, time dependent case is more subtle andshould be treated by imposing the irrotational constraints [146]. Differentallocations controlled by the parameter κ corresponds to different values ofjunction capacitances or shunt capacitances in a specific limit which is de-scribed in detail in Fig.C.1 of the Appendix. Note that we will also use thedimensionless quantities β = ENJ /EC and η = EMJ /EC (see Fig. 5.5) whichwill simplify our analysis. In addition to the external parameters such asflux or charge offset, the energy spectrum is determined by these quantities,which are invariant under rescaling of all energy scales which we will use tomatch the qubit frequency to a desired value.5.2 A variant on split transmon with enhancedanharmonicityThe energy spectrum of the transmon is controlled by the ratio EJ/EC . Aswe increase this ratio the system becomes exponentially less susceptible to495.2. A variant on split transmon with enhanced anharmonicitycharge noise but at the same time anharmonicity is reduced algebraically[45]. This fact makes it possible to find a regime where charge noise issuppressed while sufficient anharmonicity is retained. A typical parameterchoice for the transmon is EJ/EC ≈ 113 which results in anharmonicityαr = −0.04 which is shown in Fig. C.2 for a split transmon. Increasing theanharmonicity is desirable for higher fidelity qubits and faster gate opera-tions.The idea is to modify the Josephson potential to increase nonlinearity,which in turn can increase anharmonicity. We start with the more generalexpression Eq. C.4 and pick α = M . This results inH = 4ECn2 −NENJ cos(φ− κϕextN)−MEMJ cos(NMφ+ (1− κ)ϕextM)(5.3)For simplicity, we focus on regime where ϕext = 0 though we will still con-sider flux fluctuations. If we ignore the second term (set EMJ = 0) theexpression above is identical to transmon or Cooper pair box Hamiltonian.Note that the split transmon is a special case of NMon for N,M = 1. SeeAppendix C.4.1 for more discussion on the split transmon.Introducing another harmonic to the free energy in Eq. 5.3 comes at theprice of spurious energy levels localized in additional potential wells whichare outside the computational space. (Fig. 5.2) We argue that they can beignored because the transition matrix elements between the energy levelslocalized in different potential wells are negligible compared to the oneswithin the computational space (at least three orders of magnitude smallerfor our specific example below, in Fig. 5.3) See Appendix C.4 for tables ofmatrix elements for flux and charge fluctuations.Note that for parameters chosen in Fig. 5.2 we have ENJ = 4.5GHz, andEMJ = 0.9GHz and EC = 60MHz. Note that the Hamiltonian in this caseis H = 4ECn2 − 2ENJ cos(3φ)− 3EMJ cos(2φ) where even if we only considerthe first term we see that 2ENJ /EC ≈ 150 which is well into the transmonregime where for a typical transmon with EJ = 22.5GHz and EC = 200MHzwe find ENJ /EC ≈ 113. In this sense, while reducing charge fluctuations wecompromise less on anharmonicity.In our analysis, we will consider transition matrix elements that arerenormalized by the qubit frequency. This is a universal metric for compar-ing different qubit architectures since we don’t have to match the gap tocompare transition rates for the same kind of noise channel. This can bethought about as a unit system where the qubit frequency ω01 = 1. Rescaling505.2. A variant on split transmon with enhanced anharmonicityFigure 5.2: Top panel: Spectrum for N = 2, M = 3 NMon for β = 75,η = 15 with ϕext = 0. Code space is defined as n = 0→ |0〉 and n = 3→ |1〉.Model parameters can be chosen (by overall rescaling of all energies) suchthat ω = E01 = 6.08GHz which is a typical transmon frequency. In termsof frequency, closest transition from the excited state |1〉 is to n = 6. Forthese parameters the relative anharmonicity is found to be αr = −0.105.Note that this is larger than typical relative anharmonicity of the transmonwhere −αr ≈ 0.03 − 0.05 corresponding to 200 − 300MHz detuning from aharmonic oscillator with energy spacing of 6GHz. Bottom panel: Transitionfrequencies near ω01 including all transitions between levels shown in thetop panel (up to n = 6)515.2. A variant on split transmon with enhanced anharmonicityall the energies in the system allows us to tune the qubit frequency, howeverthe renormalized matrix elements remain invariant under such rescaling.Figure 5.3: Charge fluctuation matrix elements for N = 2, M = 3 NMonfor β = 75, η = 15 with ϕext = 0. Code space is defined as n = 0 → |0〉and n = 3 → |1〉 and this choice is dictated by the nature of the matrixelements: Note that the most significant matrix element that the groundstate has with the excited states is 〈3|∂ngH|0〉 which is larger than any othermatrix element (in the energetic vicinity of |0〉) by at least three ordersof magnitude. Similarly, once the system is excited to n = 3, the twomost likely transitions are 3 → 0 (back to the excited state) and 3 → 6(information leaking out of the code space)We have calculated the transition matrix elements due to flux and chargefluctuations. These amplitudes are used for estimating the relaxation times.525.2. A variant on split transmon with enhanced anharmonicityThe details of the calculation are given in Appendix C.3. The charge fluc-tuation matrix elements |〈0|∂ngH|1〉|/ω = 0.2 and |〈1|∂ngH|2〉|/ω = 0.27(Fig. 5.3) which are smaller (same order of magnitude) than any value in thevicinity of the typical transmon parameter regime (See Fig. C.2). Flux ma-trix elements are found to be |〈0|∂ϕH|1〉|/ω = 0.16 and |〈1|∂ϕH|2〉|/ω = 0.18(taken from Fig. C.7 in the Appendix) which are comparable to what we findfor split transmon (Fig. C.2) depending on the junction asymmetry param-eter d defined in section C.4.1. Note that in the limit of perfect symmetry(d → 0) between two junctions of the split transmon, the matrix elementsvanish. We will also see this vanishing of matrix elements for NMon in thenext section. (See Fig. 5.5.c) This line can be identified with the d = 0 linein Figs. C.2.c and C.2.f We note that this line can be ‘rotated’ by tuningκ such that flux matrix elements can be made arbitrarily small by movingthis line closer to the chosen (β, η) values in the phase diagram. We there-fore conclude that matrix elements of NMon can be tuned to be comparableto those of typical transmon values while having enhanced relative anhar-monicity, with at least a factor of two in this specific case we studied inFig. 5.2 compared to the typical optimum value αr = −0.04 for transmon.Next, we compute the dispersion of the energy levels as a function ofthe external flux ϕext and charge offset ng in Fig. 5.4. We confirm that theenergy levels are virtually independent (Fig. 5.4.a) of the charge offset (thismeans we are well into the transmon regime) and that ϕext = 0 is the “sweetspot” in terms of minimizing noise due to the external flux fluctuations as thedispersion of the energy levels making up the computational space (blue-|0〉and red-|1〉 curves) vanishes, as shown in Fig. 5.4.cIn order to show that this behaviour is observed for a range of modelparameters, we now explore the parameter space for a fixed external fluxϕext = 0 and charge offset ng = 0 and tune the Josephson energies ENJ (β)and EMJ (η) well into the transmon regime and compute anharmonicitiesand matrix elements while using an algorithm to determine the most suitableenergy levels |0〉 (n = 0), |1〉 (n = n1) for a qubit. |2〉 (n = n2) is determinedas the energy level that the wavefunction in the computational space is mostlikely to leak out to. The algorithm and the phase space is described inFig. 5.5. Note that we find an extended region (red regions) in the phasespace where the relative anharmonicity is enhanced to αr ≈ −0.1 which ismore than twice the typical transmon value.535.3. Large odd N, M=2a) b)c)Figure 5.4: Spectrum for N = 2, M = 3 NMon for β = 75, η = 15 a) as afunction of charge offset ng ∈ [−2, 2) and b) as a function of the externalflux ϕext ∈ [0, 2pi) through the circuit loop. Model parameters chosen to bethe same as Fig. 5.2 where ϕext = 0 is shown. The spectrum is virtuallyindependent of ng as seen in panel a). Inset panel b) shows that relativefluctuations in the qubit frequency are of the order ∆ω/ω ≈ 10−9 as afunction of offset charge. Note that the ”sweet spot” external flux value isfound to be ϕext = 0 in panel c) where the band dispersion for the energylevels in the computational space have zero dispersion.5.3 Large odd N, M=2The ideas presented in the previous section can be developed further bypicking a large integer N while keeping M = 2 fixed. We will show that this545.3. Large odd N, M=2a) b) c)d) e)n=3n=1n=2n=6n=5n=4Figure 5.5: Phase diagram for a range of ENJ and EMJ in dimensionlessparameters β = ENJ /EC and η = EMJ /EC at ϕext = 0 and ng = 0 forκ = 0.5 a) absolute value of the relative anharmonicity |αr|. b) renormalizedtransition matrix element for charge fluctuations c) matrix element for fluxfluctuations. Note that this is the only plot that depends on the value ofκ for ϕext = 0. Decreasing 0 < κ < 1 rotates the “line of zeros” clockwise.(See Fig. C.1 in the Appendix) This means the flux matrix elements for apoint of interest in the parameter space can be made arbitrarily small bytuning the ratio of shunt capacitances. d,e) lowest two energy levels abovethe ground state relevant for computational space |0〉, |1〉 and informationloss |1〉 → |2〉 due to leakage outside the computational space. State |1〉(n = n1) is chosen such that it is a low energy excited state with the highestmatrix element with the ground state |0〉 which is always n = 0. State |2〉(n = n2) is chosen such that |En2 − En1 | is closest to the qubit frequency,therefore it is the state that is most relevant for information loss when thequbit is driven externally.configuration leads to smaller transition matrix elements (this would meanlonger relaxation/dephasing times) as well as even larger anharmonicity val-ues compared to what we found in the previous section.For a concrete example, we consider the configuration with N = 13555.3. Large odd N, M=2Figure 5.6: Top panel: Spectrum for N = 13, M = 2 NMon for β = 140,η = 55 with ϕext = 0 and κ = 1 Code space is defined as n = 0 → |0〉and n = 3 → |1〉. Model parameters can be chosen (by overall rescalingof all energies) such that ω = E03 = 5.39GHz which is a typical transmonfrequency. Note that n = 5 as the next excited state accessible from n = 3.In this case the relative anharmonicity is found to be αr = −0.359.and M = 2 depicted in Fig. 5.6. This choice leads to a deep double wellpotential created by the cos2φ term which is modulated by a faster oscil-lating term cosNφ which breaks the degeneracy between the energy lev-els of the double well pair. It is important that N is an odd integer asthe degeneracy is due to Cooper pair parity, which is broken when even565.3. Large odd N, M=2a) b) c)d) e)n=3 n=5n=6Figure 5.7: Phase diagram for NMon for N = 13 and M = 2 for a rangeof ENJ and EMJ in dimensionless parameters β = ENJ /EC and η = EMJ /ECat zero flux ϕ = 0 and ng = 0 for κ = 1 Same quantities calculated as inFig. 5.5. Note that this configuration is able to achieve higher anharmonicitycompared to transmon or the N = 2,M = 3 configuration we have discussedin the previous section.and odd sectors are mixed since a term coskφ corresponds to a k Cooperpair tunneling process. Despite introducing more junctions, this approachhas the following advantages: the overlap between states localized in dif-ferent wells is suppressed further by large-N , which decouples the compu-tational space further than the spurious energy levels which also appearedfor the N = 3,M = 2 case earlier, in addition to even higher nonlinear-ity which is preferable for faster gate operations. In addition, similar tofluxonium, large-N limit suppresses the flux fluctuation matrix elements,therefore this is a setup where one can control both charge (by increasingEJ) and flux matrix elements (by increasing N). We compute the matrixelements for the parameters in Fig. 5.6. Flux matrix elements are found tobe |〈0|∂ϕH|1〉|/ω = 0.01 and |〈1|∂ϕH|2〉|/ω = 0.10 taken from Fig. 5.8 and|〈0|∂ngH|1〉|/ω = 0.11 and |〈1|∂ngH|2〉|/ω = 0.12 taken from Fig. C.6. Com-paring these values to those of fluxonium (|〈0|∂ϕH|1〉|/ω = 6.9 (Fig. C.4)and |〈0|∂ngH|1〉|/ω = 0.24 (Fig. C.5)) which, we note that while the charge575.3. Large odd N, M=2matrix elements are comparable to those of fluxonium (see Appendix C.4.2for a discussion of fluxonium), the effect of flux fluctuations are suppressedby an order of magnitude, potentially leading to longer relaxation and de-phasing times.Figure 5.8: Flux fluctuation matrix elements for N = 13, M = 2 NMon forβ = 140, η = 55 with ϕext = 0. Code space is defined as n = 0 → |0〉 andn = 3→ |1〉In terms of frequencies, the closest transition from the excited state |1〉 is ton = 6. However, the matrix elements for the transition 3 → 6 is extremelysmall (≈ 10−16). Therefore we consider n = 5 as the next excited stateaccessible from n = 3. In this case the relative anharmonicity is found to beαr = −0.359. (For n = 6 it would be αr =0.23) Bottom panel: Transitionfrequencies near ω01 including all transitions between levels shown in the585.3. Large odd N, M=2top panel (up to n = 7)Finally, we present the anharmonicity and transition matrix elements fora range of parameters in Fig. 5.7. Note that it is possible to achieve evenlarger anharmonicity in this configuration compared to the N = 2,M = 3setup we studied above.a)b)c)Dynamics of system described in Fig. 5.6 for sinusoidal external flux drive.Figure 5.9: a) Flux drive in dimensionless flux ϕext(t) as a function of timewith amplitude ≈ 0.3 and frequency ω = ω03 = 5.39GHz. b) Rabi os-cillations as a result of the drive between |0〉 and |1〉 of the code space,demonstrating that state preparation is possible with minimal informationleakage into other energy levels. c) Rabi oscillations shown in the contextof population of the low-lying energy levels where the dot size for each timepoint is proportional to |cn(t)|2 for energy level n.595.4. Population dynamics with flux drive5.4 Population dynamics with flux driveThe two-level system we are considering can be manipulated by driving anoscillating flux through the circuit loop with the frequency of the desiredtransition, which is the qubit frequency ω03 in the specific example we havebeen considering for the parameters in Fig. 5.6 One can solve the timedependent Schrodinger equation numerically to study the response of thequbit to flux drive. Starting with the initial state |Ψ0〉 =∑m cm(t)|m〉 (inthe absence of perturbation) where H0|m〉 = Em|m〉 the time dependentperturbation V (t) leads toi~∂tcn(t) =∑n,m〈n|V (t)|m〉ei(En−Em)t/~cm(t) (5.4)which we solve for cn(t) determining the time evolution of the initial state inthe presence of the perturbation |Ψ(t)〉 = ∑n cn(t)e−iEnt|n〉. For flux drivethe perturbation we consider is given byV (t) =∂H∂ΦδΦ(t) (5.5)In Fig. 5.9 we plot the solution with the initial condition that at t = 0 thequbit is in the ground state |0〉. In order to observe complete populationinversion, flux drive amplitude must be above a certain threshold which isroughly around |ϕext| ≈ 0.18. We pick 0.3 for this value and observe thatincreasing this value increases the Rabi oscillation frequency.5.5 Discussion, Conclusion and SummaryWe have shown that the NMon circuit geometry is capable of exhibitingtwice as much anharmonicity (for N = 2, M = 3) comparable to existingtransmon architectures while having comparable (same order of magnitude)transition matrix elements. We have also demonstrated this architecture canbe improved by considering arrays with large number of junctions, makingNMon in this regime an alternative to fluxonium due to its even lower sus-ceptibility to flux noise. Our results suggest that large-N configuration wehave studied above might outperform fluxonium in terms of relaxation timedue to an improvement by an order of magnitude in terms of flux transitionmatrix elements.We emphasize that we are assuming junctions within the same array areall identical. However, fabrication imperfections will inevitably lead to a605.5. Discussion, Conclusion and Summarydistribution of Josephson energies and capacitances. Such imperfections areunavoidable and have been treated perturbatively for the case of fluxonium[148] (M = 1 and N 1), justifying the superinductance mode approxi-mation. Similar calculations can be carried out for the more general case ofNMon, which we leave for future work.Having explored only a small part of the configuration space, we havedemonstrated that further improvements are possible for the existing qubitarchitectures. One can also consider a flux-qubit type scenario in this sys-tem, choosing the lowest lying two levels (see Fig. 5.6) as the code spacewhere the two wavefunctions are localized in separate wells. Note that thematrix elements between the two are extremely suppressed (see 0-1 ma-trix elements in Figs. 5.8,C.6) making direct transitions between these twostates very difficult. Nevertheless, it may be possible to construct schemeswhere transitions between the code space energy levels through higher lyingintermediary states.61Chapter 6Conclusion and OutlookIn this thesis I presented a thread of research starting from the twisted bi-layer cuprates (Chapter 2) where my main contribution was to develop amicroscopic lattice model for the moire superlattice structure and to derivea generalized gap equation for this system where we were able to confirmthe initial predictions from Ginzburg-Landau arguments and calculationsperformed on the continuum model proposed by Marcel. We have demon-strated that depending on the details of the lattice model, including the theinterlayer couplings between monolayers being one of the difficult aspectsof it, a twist in a finite window around 45 degrees result in a chiral super-conducting state which is driven by spontaneous breaking of time reversalsymmetry. This work provides a recipe for creating a topological supercon-ductor, relying solely on readily accessible material ingredients and availableexperimental techniques. As a consequence, our work has motivated multi-ple experimental efforts: Recent observation of dominant second harmonicdriving the time reversal symmetry breaking in Fraunhofer patterns andShapiro steps as well as the appearance of the superconducting diode effectwithin a finite range of 45-degree twist [5] support our predictions. Sim-ilarly, the Kerr effect as a signature of time reversal symmetry breaking,which we discuss in Chapter 3, is another direction actively being pursuedby Ye group at UBC. Our work has also inspired other theoretical proposalsinvolving twisted stacks of superconductors which predict p + ip′ [28] andf + if ′ [149] order parameters which are also topologically nontrivial andare expected to host Majorana fermions in their defects.In Chapter 3 we turned to optical measurements (magneto optic Kerreffect) for the experimental observation of the spontaneous time reversalsymmetry breaking of the topological superconductor phase of the twistedcuprates, which is our prediction from Chapter 2. Finite Kerr rotationunambiguously originates from time reversal symmetry breaking. We havecomputed the expected order of magnitude of the Kerr rotation for thetwisted cuprate system [4] and compared our results to predicted values [31]for strontium ruthenate (SRO) which also shows time reversal symmetrybreaking in muon spin relaxation [32] and polar Kerr effect measurements62Chapter 6. Conclusion and Outlook[33]. We found that the anomalous Hall conductivity for the twisted cupratemodel is 3 orders of magnitude larger than the corresponding values for SROand that the Kerr angle for the twisted cuprate system is in the range of10− 100µrads while the predicted [33] and experimentally measured values[31] for SRO are 50nrads and 65nrads respectively. We therefore expect thatsuch a large Kerr angle should be observed in the twisted cuprate systems.At the time of writing this proposal, Ye group at UBC SBQMI is workingon measuring this effect. As of now their Kerr angle measurement resolutionappears to be about 50nrads, which should be enough to observe an effectmeasured in µrads.The dominant second harmonic in the current phase relation (CPR),which can be interpreted as Cooper pair co-tunneling, leading to the timereversal symmetry breaking in twisted cuprates has been observed in Joseph-son tunneling experiments in the Fraunhofer patterns and Shapiro steps [5],which are consistent with the dominant second harmonic in CPR [83] whichwe discussed Chapter 2. Recent tunneling experiments show that the su-perconducting diode effect in twisted cuprates appears only within a finitewindow of 45±5 degree twist [150] which is another evidence for time rever-sal breaking as it is a necessary condition [42] for a finite superconductingdiode effect. In addition to the Kerr rotation and the Josephson tunnelingexperiments, yet another signature of the chiral superconducting phase inthe twisted cuprates is the magnetic field due to chiral edge currents whichcan be measured by SQUID microscopy. Discussion of these edge states isanother direction for future work.Inspired by the observation of the diode effect in twisted cuprates, Chap-ter 4 affirmatively answers the question of whether superconducting diodeeffect can be observed in a superconducting circuit that is made out of con-ventional Josephson junctions. We show that parallel arrays of Josephsonjunctions can exhibit the diode effect and discuss how this effect can beoptimized by tuning the external fluxes through the loops of the circuit aswell as the optimal numbers of Josephson junctions that need to be includedin each arm for maximal superconducting diode efficiency. More recently,we have implemented this design on a superconducting chip which has re-cently been fabricated at a foundry, facilitated by CMC Microsystems. Thedevice is currently being characterized by our experimentalist colleagues atthe Stewart Blusson Quantum Matter Institute, UBC. Preliminary measure-ments so far exhibit superconducting diode effect in quantitative agreementwith our theoretical predictions.In Chapter 4 we learned that parallel arrays of Josephson junctions canbe used to generate higher harmonics in the CPR, giving rise to enhanced63Chapter 6. Conclusion and Outlooktunability by controlling the number of Josephson junctions as well as theirsizes. In Chapter 5 we show that this tunability gives rise to a qubit designwhich has larger anharmonicity compared to that of transmon while havingcomparable transition matrix elements in the charge and flux noise channels.This work serves as a proof of concept, demonstrating that engineering aCPR with arrays of Josephson junctions can lead to higher fidelity qubits.Time reversal symmetry breaking due to first harmonic cancellation inthe CPR can be observed in other systems than twisted cuprates as well.One example would a bilayer stack of an s-wave and a d-wave superconduc-tor. As long as s-wave order parameter is completely isotropic or is extendedwhile respecting C4 rotational symmetry, GL arguments as well as micro-scopic calculations confirm that the first harmonic vanishes, leading to atime reversal broken state although it is topologically trivial with a d + isorder parameter. Our recent work [151] has shown that such a system canbe utilized as a qubit. 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Qubits as spectrometers of quantum noise, 2002.81Appendix AAppendix for Chapter 2A.0.1 Microscopic derivation of GL coefficientsFrom the continuum BdG model, defined by the matrix Hamiltonian (2.10),it is possible to derive the corresponding GL theory by directly expandingthe free energy FBdG given in Eq. (2.11) in the powers of the superconductingorder parameters. This procedure is made feasible by the fact that explicitexpressions for the two positive-energy eigenvalues of hk can be obtainedand are given byEkα =√(∆2k1 + ∆2k2)/2 + ξ2k + g2 + (−)αDk (A.1)where D2k = (∆2k1 −∆2k2)2/4 + g2(∆2k1 + ∆2k2 + 4ξ2k)− 2g2∆k1∆k2 cosϕ andα = 1, 2. We now express the two d-wave gap functions as∆k1 = ψ cos(2αk), ∆k2 = ψ cos(2αk − 2θ), (A.2)substitute into (A.1) and expand the free energy (2.11) in the powers of theorder parameter amplitude ψ to fourth order. This expansions results inmany terms and is best carried out with the help of Mathematica or similarsoftware capable of symbolic manipulations.We are primarily interested in terms governing the phase difference be-tween the two layers, that is, coefficients B and C in the GL free energy(2.2). These can be isolated by focusing on terms containing powers of cosϕ.Coefficient B is thus found as a prefactor of ψ2 cosϕ in the above expansionand readsB = cos (2θ)NF∑s=±∫ c0dξ−pisgξ|ξ − sg|tanh |ξ − sg|2kBT(A.3)In deriving this expression we replaced the crystal momentum sum usingthe usual prescription∑k → NF∫ c−c dξ∫ 2pi0 dφ where NF = L2m/~2 is thenormal-state density of states at the Fermi level, L is the system size andthe angular integral has been performed. The remaining integral over theenergy variable cannot be completed in the closed form but one can show82Appendix A. Appendix for Chapter 2Out[!]= fB(ξ)0 1 2 3 4 5-5-4-3-2-10Out[!]=fC(ξ)fB2(ξ)0 1 2 3 4 50.00.20.40.60.81.0Out[!]= fB(ξ)0 1 2 3 4 5-5-4-3-2-10Out[!]=fC(ξ)fB2(ξ)0 1 2 3 4 50.00.20.40.60.81.0a) b) ξ/g ξ/gFigure A.1: Analysis of the energy integrands. (a) Integrand fB(ξ) enteringthe expression Eq. (A.3) for the GL theory coefficient B. (b) IntegrandfC(ξ) entering the expression for the GL coefficient C compared to fB(ξ)2.Note that the amplitude of the latter is scaled by a constant factor of 1/30for easier comparison. The temperature is chosen such that kBT = 0.2g butthe qualitative properties of the integrands remain the same in a wide rangeof temperatures.that the integrand fB(ξ) is negative for all values of ξ and thus the integralitself is also negative, as expected on physical grounds. This is illustratedin Fig. A.1(a).Coefficient C can likewise be identified as a prefactor of the ψ4 cos 2ϕterm in the expansion and has the form C = (2 + cos 4θ)IC . Here IC =NF∫ c0 fC(ξ)dξ is the corresponding energy integral similar to Eq. (A.3).Its integrand fC(ξ) has a more complicated structure and for the sake ofbrevity we do not show it explicitly. Importantly fC(ξ) is positive for all ξand, as anticipated in the main text, is approximately equal to the integrandof Eq. (A.3) squared, as shown in Fig. A.1(b). We thus conclude that thecontinuum BdG model indeed supports the GL phenomenology deduced inthe main text on the basis of general physical arguments. Specifically thecalculation explicitly shows that prefactor K of the cos 2ϕ term in Eq. (2.4)is positive which we argued was the key condition underlying the emergenceof the time reversal symmetry broken topological phase in the twisted dSCbilayers.83Appendix A. Appendix for Chapter 2A.0.2 Self-consistent mean field theory on the latticeTo describe the two-layer system microscopically, we employ the HubbardHamiltonian (2.12) and treat the interaction term at the mean field level toaccount for d-wave superconductivity in each layer. The MF HamiltonianbecomesH = −t∑〈ij〉σac†iσacjσa − t′∑〈〈ij〉〉σac†iσacjσa − µ∑iσaniσa+∑〈ij〉a(∆ij,ac†i↑ac†j↓a + h.c.)−∑ijσgijc†iσ1cjσ2, (A.4)where t and t′ are the hopping amplitudes between first and second neigh-bour sites on the square lattice, µ is the chemical potential that controlson-site particle density niσa and ∆ij,a = V 〈ci↑acj↓a〉 is the complex orderparameter. It is believed that the interlayer tunneling processes in cupratesare mediated, to leading order, by the spherically symmetric Cu 4s orbitals[78], although other orbitals such as those belonging to Bi and Sr atoms,probably also play an important role. With this in mind, we adopt a phe-nomenological form for the interlayer tunneling amplitude that decays ex-ponentially with distance:gij = g0e−(rij−c)/ρ, (A.5)where r2ij = c2 + d2ij , with dij denoting the in-plane separation betweentwo sites i and j on different layers and c being the interlayer distance.In practice, we limit these terms to a few unit cells, beyond which theamplitudes are negligible. The characteristic radial extent of the Cu 4sorbital has been denoted by ρ. We have also considered interlayer couplingsof the form gij = g0c2/r2ij [153]. Self consistent calculations still result intopological phases, but this form of coupling, when compared to (A.5) forsame g0, effectively corresponds to stronger interlayer coupling as the decayof 1/r2 is slower compared to the exponential.While the above real space Hamiltonian is valid for any twist angle,the bilayer forms a crystal only for a discrete set of commensurate rotationangles. To carry out the lattice mean-field calculations we focus on com-mensurate twist geometries that are amenable to numerical study using thestandard Bloch representation. We note that any commensurate twist canbe described by an integer-valued ‘twist’ vector v = (m,n) as shown in Fig.A.2. The corresponding moire´ unit cell comprises of 2q = 2(m2 + n2) sitesand the twist angle is determined by the relationθm,n = 2 arctan (m/n). (A.6)84Appendix A. Appendix for Chapter 2a) b) c)θ/2Figure A.2: Geometry of the commensurate unit cells for bilayers with rel-ative twists of (a) θ1,2 ' 53.13o(b) θ2,5 ' 43.60o and (c) θ5,12 ' 45.24o.Twisting procedure is illustrated in panel (a) where orange and blue arrowsindicate integer valued vectors v = (m,n) and v¯ = (−m,n) introduced inthe text. The moire´ unit cell is shown by the shaded area.This can be understood as a rotation of two perfectly aligned square latticesin opposite directions by θm,n/2, which leads to the sites originally at thelocations v and v¯ = (−m,n) to lie atop each other. The unit cells for aselection of commensurate twist angles close to 45o are illustrated in Fig.A.2.To model the Fermi surface of Bi2212 we follow Ref. [69] and chooset = 153 meV, t′ = −0.45t with µ = −1.35t near optimal doping. Otherparameters that determine the interlayer tunneling amplitudes are as follows:the average spatial extent of electrons in the 4s orbitals is ρ ≈ 4a.u. = 2.11A˚,the typical inter-layer distance is c ≈ 12A˚[154] and the in-plane separationbetween the Cu atoms is a ≈ 5.4A˚[155].In the remainder of this section we outline the procedure for obtaininggap equations for the order parameters ∆ij,a using the imaginary time pathintegral formalism. After decoupling the attractive pairing potential in theCooper channel with the Hubbard-Stratonovich transformation, we obtainthe action corresponding to the Hamiltonian (A.4) (cf. Ref. [156] for details)S =∑knΨ†kn[−iωn + hk]Ψkn +βNV∑i,j∈u.c.a∆ij,a∆∗ij,a. (A.7)Here hk is the BdG Hamiltonian that follows from Fourier transforming Eq.(A.4), β = 1/kBT is the inverse temperature, ωn = (2n + 1)pi/β are theMatsubara frequencies, V is the strength of attractive nearest neighbourinteractions, N is the number of unit cells and Ψk = (Φk1,Φk2)T is the85Appendix A. Appendix for Chapter 2Nambu spinor such that Φka = (c(1)k↑a, . . . , c(q)k↑a, c(1)†−k↓a, . . . , c(q)†−k↓a)T , with thesuperscripts 1, . . . , q labelling the sublattice degrees of freedom in layer a.The effective action for ∆ij,a can be determined by integrating out fermionicdegrees of freedomSeff = −∑knTr log[−G(k, iωn)−1] + βNV∑i,j∈u.c.a∆ij,a∆∗ij,a, (A.8)where we defined the Matsubara Green’s function G(k, iωn) = −(−iωn +hk)−1. To evaluate the saddle point condition for this effective action,∂Seff/∂∆∗ij,a = 0, we employ the identity ∂∆(Tr logA) = Tr[∂∆AA−1]. Afterperforming the Matsubara sum, this simplifies to∆ij,a = − VN∑kTr[∂hk∂∆∗ij,aUknF (Ek)U†k]. (A.9)In the above, Uk is the unitary matrix that diagonalizes the Bloch Hamilto-nian as U †khkUk = Ek and nF (Ek) is a diagonal matrix where Fermi functionis applied to the eigenvalues Ek. For decoupled layers (g0 = 0), with thetight binding parameters given above, a self-consistent solution converges toa d-wave form with maximal gap ∆max = 40 meV, when V = 146 meV. Weuse this value for calculations where two layers are coupled (g0 6= 0). Fig. 2.3in the main text summarizes the results obtained from these calculations.A.0.3 Topological phasesHere, we discuss in greater detail the various phases found in the self-consistent treatment of the lattice model.To reliably determine the topology of the band structure, we turn tothe Chern number, which is defined as an integral of the Berry curvatureover the full Brillouin zone: C = 12pi∫BZ Ω(k) d2k. Computation of theBerry curvature directly from the eigenstates is complicated by the arbitraryphases that accompany numerical diagonalization. To counter this, we relyon a gauge independent formulation where [157]Ω(k) =∑m6=n〈n|∇khk|m〉 × 〈m|∇khk|n〉(Em − En)2 . (A.10)Here |n〉 and |m〉 are eigenstates of the Bloch Hamiltonian with energies Enand Em respectively.86Appendix A. Appendix for Chapter 2The C = ±4 and ±2 phases (indicated in Fig. 2.3(e) in red and greenrespectively) can be understood by considering the continuum model witha circular Fermi surface in the normal state, which we defined in Eq. (2.7).Similar arguments hold for the more realistic Fermi surface with hole pock-ets. When two layers are decoupled (g = 0), Fermi surfaces of each layeroverlap, Fig. A.3(a), and each layer hosts four Dirac cones in the presenceof a dx2−y2 order parameter. Upon breaking of time reversal symmetry, eachDirac point is gapped out and can contribute a Berry flux of ±pi, thereby al-lowing for the maximal value C = ±4. As we turn on the interlayer coupling,bands from each monolayer split, Fig. A.3(b). If the interlayer coupling gis strong enough to push one of the bands above the Fermi level, half of theDirac cones disappear, leaving four Dirac cones behind. In this case, themaximum Chern number is ±2, a situation realized in the phase diagramFig. 2.3(e) for large g0.It is interesting to note that the spectral gap tends to be larger in the C =2 phase. We attribute this to the fact that the C = 2 phase is stabilized atstronger values of the interlayer tunneling g0 which enhances the Cooper pairtunneling amplitude. Moreover, in second order perturbation theory, the gapis expected to scale as g20/∆0. Temperature evolution of the correspondingtunneling density of states is shown in Fig. A.4.The gapped, topologically trivial C = 0 phase in the phase diagram canbe understood as follows. As a representative of this phase, we consider thepoint g0 = 20 meV and µ/t = −1.3 for v = (1, 3) in the phase diagramFig. 2.3(c). A self-consistent solution of the gap equation (A.9) yields thefollowing forms for order parameters on the two layers∆k1 =∆0(cos kx − eiφ cos ky) (A.11)∆k2 =∆0(cos ky − eiφ cos kx) (A.12)where ∆1 and ∆2 correspond to top and bottom layers respectively. Thenumerically obtained phase φ tends to be close to −pi/4. If we define ∆d =∆0eiφ/2 cosφ/2 and ∆s = ∆0eiφ/2 sinφ/2, order parameters can be cast as∆k1 =∆d(cos kx − cos ky)− i∆s(cos kx + cos ky)∆k2 =−∆d(cos kx − cos ky)− i∆s(cos kx + cos ky),which shows that both layers have d + is type superconducting order pa-rameter. The transition between the d + is and the d + id phases can beseen by tuning the chemical potential µ for a fixed interlayer coupling g0, asshown in Fig. A.5.87A.1. Derivation of the gap equation from path integralsA.1 Derivation of the gap equation from pathintegralsA.1.1 s-wave pairingAs a warm up, let us work out the s-wave on-site pairing. We neglect thekinetic hopping terms for now,S[c, c] =∫ β0dτ[∑rcrσ(∂τ − µ)crσ − gcr↑cr↓cr↓cr↑](A.13)by introducing complex fields ∆,∆, we can decouple the interaction as−gcr↑cr↓cr↓cr↑ → ∆rcr↓cr↑+∆rcr↑cr↓+g−1∆r∆r Defining the Nambu spinorsΨr =(cr↑ cr↓)Ψr =(cr↑cr↓)(A.14)the action can be rewritten as:S[Ψ,∆] =∫ β0dτ[∑rΨr(∂τ − µ ∆r∆r ∂τ + µ)Ψr +∆r∆rg](A.15)for the mean field analysis, we assume that ∆ is temporally homogeneous,i.e. is not a function of τ . Notice that the fact that we omitted hoppingssimplifies our notation. This can then be easily Fourier transformed toimaginary frequencies:S[Ψ,∆] =∑r∑ωΨr(iω − µ ∆r∆r iω + µ)︸ ︷︷ ︸G−1Ψr + β∆r∆rg(A.16)note the Gorkov Green’s function G above. We can now integrate out theNambu ”fermions” and obtain an effective action for which we want to treatat the saddle point level (aka mean field). After evaluating the Gaussianintegral,Seff =∑r(−∑ωtr log G−1 + β∆r∆rg)(A.17)In mean field approximation, ∂S/∂∆ = 0 where we use the identity ∂tr log G−1∂∆=tr[∂G−1∂∆G]then, at saddle point:β∆rg=∑ωtr[(0 01 0)G](A.18)88A.1. Derivation of the gap equation from path integralsin this particular case, G−1 can be analytically inverted, and we obtain thegap equationG = 1(iω)2 − µ2 −∆∆(iω + µ −∆−∆ iω − µ)β∆rg=∑ωn∆rω2n + µ2 + ∆∆(A.19)we want to generalize these steps to extended (beyond on-site) pairings andlarger moire´ unit cells.A.1.2 Extended pairing termsAn extended attractive interaction that looks likeV ↑↓δ =∑r−gc†r↑c†r+δ↓cr+δ↓cr↑can be expressed in momentum spaceV ↑↓δ = −g1N∑k,qei(k−q).δc†k↑c†−k↓c−q↓cq↑which is separableV ↑↓δ = −gN∑keik.δc†k↑c†−k↓∑qe−iq.δc−q↓cq↑defining the pair creation operatorPδ =∑qe−iq.δc−q↓cq↑ (A.20)this interaction can be decoupled in the Cooper channel by introducing thebackground term which appears in the effective actionSeff =∑k,nΨ†k [−iωn + hk] Ψk −gNP †P (A.21)asNg∆∆→ Ng∆∆ + ∆P + ∆P † +gNP †P89A.1. Derivation of the gap equation from path integralswith spinors defined as Ψk = (c†−k↑, ck↓) more explicitly, each interactionterm V ↑↓δ can be replaced withV ↑↓δ = ∆↑↓δ∑keik.δc†k↑c†−k↓ + ∆↑↓δ∑ke−ik.δc−k↓ck↑ +Ng∆↑↓δ ∆↑↓δ (A.22)For a generic bond interaction, we must also include the ↓↑ part of theinteraction which takes the following formVδ = V↑↓δ + V↓↑δ =∑k(∆↑↓δ eik.δ −∆↓↑δ e−ik.δ)c†k↑c†−k↓+ (A.23)∑k(∆↑↓δ e−ik.δ −∆↓↑δ eik.δ)c−k↓ck↑ (A.24)+Ng(∆↑↓δ ∆↑↓δ + ∆↓↑δ ∆↓↑δ)(A.25)We restrict our analysis to singlet pairing ∆δ over bond δ which means∆δ := ∆↑↓δ = −∆↓↑δ (A.26)such that the interaction term simplifies toVδ = 2∑k∆δ cos(k.δ)c†k↑c†−k↓ + 2∑k∆δ cos(k.δ)c−k↓ck↑ (A.27)+2Ng(∆δ∆δ)(A.28)The spinors can be generalized to the case of multiple orbitals α, β, ... suchthat we can define Ψαk = (c†−kα↑, ckα↓) where pairing between orbitals canbe described by generalizing Eq. A.29 such thatV αβδ = 2∑k∆αβδ cos(k.δ)c†kα↑c†−kβ↓ + 2∑k∆αβδ cos(k.δ)c−kβ↓ckα↑ (A.29)+2Ng(∆αβδ ∆αβδ)(A.30)where the bond variable δ denotes the reciprocal lattice vector connecting thesites that participate in the pairing. Such extended pairings are necessaryfor description of unconventional pairing states (beyond s-wave which wedescribed above) Such pairing states can be described in the same way wediscuss in section A.1.1 and leads to Eq. A.890A.1. Derivation of the gap equation from path integralsa) b)d)c)Figure A.3: Schematic picture for the normal state band structure of thecontinuum model defined in Eq. (2.7). Both electron and hole bands thatenter the BdG Hamiltonian are shown while the Fermi level is indicated bya dashed red line. Panel (a) shows decoupled layers with doubly degeneratebands yielding a circular Fermi surface. In the presence of a d-wave orderparameter we have eight Dirac cones. (b) Interlayer coupling g splits thebands and Fermi surface now consists of a pair of concentric circles. (c) Theinner Fermi circle shrinks to a point and half of the Dirac cones disappear.(d) Four Dirac cones survive in the strong interlayer coupling (g > µ) case.91A.1. Derivation of the gap equation from path integrals40 20 0 20 40E[meV]0246810(E)/N0.00.20.40.60.81.01.2Figure A.4: Evolution of the density of states with temperature in the C = 2phase with parameters g0 = 40 meV and µ = −1.35t. Temperature T/Tcshown as color scale. The superconducting gap persists up until T = Tc.92A.1. Derivation of the gap equation from path integrals1.40 1.38 1.36 1.34 1.32 1.30 1.28/t0246810min [meV]min vs for g0=0.02C=4C=0Figure A.5: The topological phase transition between C = 0 (µ < −1.34t)and C = 4 (µ > −1.34t) phases is marked by a closing of the gap as thechemical potential is tuned while keeping the interlayer coupling fixed atg0 = 20 meV and θ2,5 ' 43.6◦93Appendix BAppendix for Chapter 3B.1 Details on the 2-band modelIn this section we give a derivation of the 2-band model discussed in themain text. We assume the regime of strong hole doping such that the Fermisurface is small and confined to the center of the Brillouin zone Γ. Whilethis may be relevant for strongly overdoped Bi2Sr2CaCu2O8+δ the primarygoal of this derivation is to obtain a simple, analytically tractable model,that will provide insights into the physics of the Hall conductivity in thegeneral class of twisted bilayer d-wave superconductors.To construct the 2-band model we employ the following strategy. Webegin with two initially decoupled CuO2 layers described as BCS d-wavesuperconductors. We expand the electron dispersions and gap functionsto leading order in small momentum k around the Γ point and perform arotation by angle ±θ/2 in the two layers. Next we regularize these expres-sions on two perfectly aligned (i.e. unrotated) square lattices with the latticeconstant b given by the moire´ unit cell and write down the correspondingtight-binding model. Finally we couple the two layers by a generic single-electron tunneling g(k) consistent with the system symmetries. The final2-band model is thus defined on a pair of aligned square lattices with thetwist encoded in the effective dispersion relations and gap functions.B.1.1 d+ id pairingStarting with two decoupled CuO2 planes we expand the dispersion of eachmonolayer near the BZ center, keeping terms up to O(k4)k = −2t(cos kxa+ cos kya) (B.1)' −2t[2− 12k2a2 +18k4a4 − 116(2kxky)2a4].The fourth order terms are required to capture the 4-fold anisotropy of thedispersion which, as we shall see, turns out to be important for the Hall94B.1. Details on the 2-band modelFigure B.1: Normal parts of energy dispersions (top) and the order param-eter (bottom) for decoupled layers that are naively rotated by ±θ/2 withθ = 2 arctan(1/2). We observe that the the patterns are no longer periodicwhen plotted in the original Brillouin zone.response. The pair functions are expanded to second order∆k =∆02(cos kxa− cos kya) ' −∆04(k2x − k2y)a2. (B.2)Now we account for the twist by counter-rotating each layer by angle ±θ/2as shown in Fig. B.1. Under a rotation by θ/2 various terms transform ask2x − k2y → (k2x − k2y) cos θ + (2kxky) sin θ,2kxky → (2kxky) cos θ − (k2x − k2y) sin θ.Next we construct a tight binding model whose low-energy expansionnear the Γ point gives the above rotated dispersion relation and gap func-tions. This is achieved by replacingkxa→ absin(kxb), (kxa)2 → 2a2b2[1− cos (kxb)],95B.1. Details on the 2-band modelFigure B.2: Normal state dispersion (top) and order parameter (middle) ofthe effective 2-band model constructed for θ = 2 arctan(1/2). The blacksquare encloses the moire´ unit cell with length 2pi/b and we observe that thepatterns are now properly periodic. The last term in Eq. (B.3) is plottedseparately (bottom row) as it is very small compared to the other terms.However, inclusion of this term is very important since without it we wouldhave ξ1 = ξ2 and the Hall conductivity would identically vanish.and similar for ky. Here b is the lattice constant of the moire´ lattice thatresults from the twisted geometry (we assume commensurate twist angle θ).96B.1. Details on the 2-band modelWe thus obtainξk(θ) = − µ− 4t(1− a2/b2) (B.3)− 2t[a2b2(cos kxb+ cos kyb)− 14a4b4(ηxy cos θ + ηx2−y2 sin θ)2],and the pairing∆k(θ) =∆02a2b2[ηx2−y2 cos θ − ηxy sin θ], (B.4)whereηxy = sin kxb sin kyb, (B.5)ηx2−y2 = cos kxb− cos kyb. (B.6)The resulting dispersions can be seen in Fig. B.2.If we define µ˜ = µ + 4t(1− a2/b2), t˜ = ta2/b2, t˜′ = t2a4/b4 and ∆˜ =∆0(a2/b2) as the effective parameters, we obtain the 2-band model definedby the BdG Hamiltonian H = ∑k Ψ†kheffk Ψk withheffk =ξ1(k) ∆1(k) g(k) 0∆1(k)∗ −ξ1(k) 0 −g(−k)∗g(k)∗ 0 ξ2(k) ∆2(k)0 −g(−k) ∆2(k)∗ −ξ2(k) (B.7)and the Nambu spinor Ψk = (ck↑1, c†−k↓1, ck↑2, c†−k↓2)T . The effective dis-persions and gap functions are given byξ1(2)(k) = −µ˜− 2t˜(cos kxb+ cos kyb) (B.8)+ t˜′(ηxy cos θ ± ηx2−y2 sin θ)2∆1(2)(k) =∆˜2(ηx2−y2 cos θ ∓ ηxy sin θ]e±iφ/2 (B.9)The requisite time reversal symmetry breaking is encoded through the com-plex phase difference eiφ between superconducting order parameters in twolayers.In relation to the 10-band model the lattice constant is b =√12 + 22a =√5a. The corresponding BZ will be smaller, with the dimension 2pi/b. The97B.2. Hall Conductivity10-band model can be recovered from the 2-band model by folding the dis-persions in Fig. B.1 back into the smaller moire´ BZ. If we are interested infeatures that are near the center of the BZ and small enough energies suchthat they do not backfold, the 2-band model with the above parametersshould provide an accurate description.B.1.2 d+ is pairingFor the d+is case, the pairing form obtained from self-consistent calculationis∆1(2)(k) = ±∆dηx2−y2 − i∆sχ (B.10)in the local frame of each layer with ∆d = ∆˜eiφ/2 cosφ/2, ∆s = ∆˜eiφ/2 sinφ/2and χ = cos kxb+ cos kyb. We can follow the transformation procedure de-scribed above to obtain∆1(2)(k) = ± ∆d(ηx2−y2 cos θ ∓ ηxy sin θ) (B.11)− i∆s[χ− 14(ηxy cos θ ± ηx2−y2 sin θ)2].B.2 Hall ConductivityFor a BdG two band Hamiltonian B.7 we compute a Hall conductivity ex-pression in the following. This is a 4× 4 matrix but our calculation general-izes to a larger matrix with more orbital degrees of freedom we obtain fromthe larger moire´ unit cell, all steps below will still follow through. We willcall this matrix hk We will compute [100]pixy(q, νm) = e2 1β∑k,ωntr [vˆxG0(k, ωn)vˆyG0(k + q, ωn + νm)] (B.12)where G0 is the Green’s function in the absence of Hint. This quantityappears in the expression for Hall conductivityσH(ω) =i2ωlimq→0[pixy(q, ω)− piyx(q, ω)] (B.13)We start with the spectral representationG0(k, ωn) =∫dωρk(ω)−iωn + ω (B.14)98B.2. Hall Conductivitywhere ρ is a matrix containing all orbital degrees of freedom. Plugging thisinto Eq. (B.12) and rearranging:pixy(q, νm) = e2∫dωdν∑ktr [vˆxρk(ω)vˆyρk+q(ν)]1β∑ωn1iωn − ω1iωn + iνm − vthe Matsubara sum can be evaluated as1β∑ωn1iωn − ω1iωn + iνm − ν =nF (ω)− nF (v −iνm )iνm + ω − νwhere the bosonic Matsubara frequency drops out because they are even andthey vanish in the exponential in the definition of nF (ω) =1β∑neiωn0+iωn−ω Wenow see that we ended up with an expression that can be easily analyticallycontinued, which we should do at the end.pixy(q, νm) = e2∫dωdν∑ktr [vˆxρk(ω)vˆyρk+q(ν)]nF (ω)− nF (ν)iνm + ω − νA useful result is the following. We define unitary matrices Uk that di-agonalize the Hamiltonian hk such that U†khkUk = Ek Now by definitionG−1 = −iω + hk. After unitary rotation U †kG−1Uk = −iω + Ek. Invertingthis expression (can do it elementwise as it is a diagonal matrix on RHS) wegetU †kGUk =1−iω + Ekwhich is consistent with (B.14) only if U †kρk(ω)Uk = δˆ(ω−Ek). We can usethis identity iIf we insert unitaries U †kUk = 1 into the trace abovepixy(q, νm) = e2∫dωdν∑ktr[U †k+qvˆxUkδˆ(ω − Ek)U †kvˆyUk+qδˆ(ν − Ek+q)]× nF (ω)− nF (ν)iνm + ω − νrecall that the δˆ functions in the above expression represent diagonal matri-ces, which have the explicit form[δˆ(ω − Ek)]ij= δ(ω − Eik)δij (B.15)99B.2. Hall Conductivitywhere δij on the RHS is the Kronecker delta. Writing the trace explicitlyand evaluating the integrals with the delta functions we see thatpixy(q, ωn) = e2∑k,ab[U †k+qvˆxUk]ab[U †kvˆyUk+q]banF (Ebk)− nF (Eak+q)iωm + Ebk − Eak+qwe now write down the Hall conductance (3.2) (analytically continued)σH(iωn) = − 12iωlimq→0e2∑k,ab( [U †k+qvˆxUk]ab[U †kvˆyUk+q]ba−[U †k+qvˆyUk]ab[U †kvˆxUk+q]ba)× nF (Ebk)− nF (Eak+q)iωn + Ebk − Eak+qin the limit q → 0, if we define the shorthand uˆx = U †kvˆxUkσH(ω) = − 12iωe2∑k,ab([uˆx]ab [uˆy]ba − [uˆy]ab [uˆx]ba) nF (Ebk)− nF (Eak)ω + Ebk − Eak + i(B.16)Since vˆx is Hermitian since it is the derivative of a Hermitian operator (recallit is the normal part of the Hamiltonian), note that[uˆx]∗ab = [uˆ†x]ba = [U†vˆ†xU ]ba = [U †vˆxU ]ba = [uˆx]bawhich tells us that Qab = [uˆx]ab [uˆy]ba − [uˆy]ab [uˆx]ba = 2iIm{[uˆx]ab [uˆy]ba}in parenthesis above is imaginary. We can now rewrite (B.16) as followingσH(ω) =i2ωe2∑k,abQabnF (Ebk)ω + Ebk − Eak + i−Qba nF (Ebk)ω + Eak − Ebk + iwhere we relabeled indices a, b for the second term. Note that Qab = −Qbais antisymmetric. This simplifies our expression toσH(ω) =iωe2∑k,ab(ω + i)Qab(Eak − Ebk)2 − (ω + i)2nF (Eak) (B.17)We also compute the off-diagonal part of the optical conductivity:σxy(ω) =iωlimq→0[pixy(q, ω)− pixy(q, 0)] (B.18)using the expression for pixy above, we arrive at:σxy(ω) = ie2∑k,ab[uˆx]ab [uˆy]banF (Eak)− nF (Ebk)(ω + i+ Ebk − Eak)(Ebk − Eak + i)(B.19)100B.2. Hall ConductivityB.2.1 Explicit formula for the 2-band modelWe start from the general two-band BdG Hamiltonian hk in the followingformhk =ξ1(k) ∆1(k) g(k) 0∆∗1(k) −ξ1(−k) 0 −g∗(−k)g∗(k) 0 ξ2(k) ∆2(k)0 −g(−k) ∆∗2(k) −ξ2(−k) . (B.20)Once we assume intralayer inversion symmetry ξi(−k) = ξi(k), it becomesEq. (B.7). Now we have to further specify the relation between g(k) andg(−k). In general, if we work with interlayer coupling that does not explic-itly break time reversal symmetry, the complex phase of g(k) only entersvia the ei(r1i−r2j)·k factors (where raj are positions of atoms in the unit cell)and thus g∗(−k) = g(k). Then we havehk =ξ1(k) ∆1(k) g(k) 0∆∗1(k) −ξ1(k) 0 −g(k)g∗(k) 0 ξ2(k) ∆2(k)0 −g∗(k) ∆∗2(k) −ξ2(k) . (B.21)For this 2-band model the trace indicated in Eq. (B.12) can be explicitlyperformed. This leads to the following expression for the Hall conductivityσH(ω) =ie22ωβ∑k,ωn2νm(νm + 2ωn)(E2− + ω2n) (E2+ + ω2n)× (A+B)[E2− + (νm + ωn)2] [E2+ + (νm + ωn)2]∣∣∣∣νm→ω+i(B.22)whereA = E+(W − E−X)[|g|2 + ωn(νm + ωn)− E1E2]−W (E1|∆2|2 + E2|∆1|2) + Im[∆∗1∆2]U(νm + 2ωn)B = X{Re [E+E−∆1∆∗2 + ∆∗1∆∗2(∆21 −∆22)]+ (|∆1|2 − |∆2|2)[|g|2 − ωn(νm + ωn) + E1E2]}withX = (∂kg × ∂kg∗)z = 2i Im[∂kxg∂kyg∗]W = 2i (δv× Im[g∗∂kg])zU =(δv× ∂k|g|2])z101B.2. Hall ConductivityWe defined the velocity anisotropy of the single-particle spectrum δv =∂k(ξ2− ξ1) and E± = E1±E2 with E1,2 the positive eigenvalues of hk. Herethe energy denominators come from detG−10 (k, ωn) =(E2− + ω2n) (E2+ + ω2n).Performing the requisite fermionic frequency summation and the analyticcontinuation leads to Eq. (3.7).0.0 0.1 0.2 0.3 0.4 0.5 [eV]0.000050.000000.000050.000100.000150.000200.00025H [e2 /]xy (10-band)xy (2-band)Figure B.3: Comparison of optical Hall conductivities σxy(ω) between the2-band (shown in orange) and the 10-band (green) models. Solid and dashedlines show real and imaginary parts respectively. For the 10-band model,tight binding parameters are chosen to be t = 0.153eV, t′ = 0, µ = −3.9t,∆max = 0.4eV and g(10band)0 = 0.008eV. The phase between monolayers ischosen to be φ = 0.4pi, breaking the time reversal symmetry. Note that inthis plot we calculate the Hall conductance in a different regime and usingdifferent parameters (larger order parameter and lower chemical potentialcompared to the calculations in Fig. 3.1) to show agreement between 2-bandand 10-band models.B.2.2 Comparison of the 2-band and the 10-band modelresultsIn Fig. B.3 we contrast our results for σH(ω) computed for the two mod-els. We use parameters relevant to the highly overdoped (low filling) regimewhere we expect some degree of agreement in the optical conductivity be-102B.3. Estimating the Kerr angletween the two models. The tight binding parameters are related as indicatedabove Eq. (B.7). The main difference between the two Hamiltonians is theassumed form of the interlayer couplings. In the 10-band model we assumegij exponentially decaying with distance as given in Eq. (3.6) of the maintext, while in the 2-band model we use g(k) = 4g0 cos (kxb/2) cos (kyb/2)corresponding to a nearest neighbor hopping between two monolayers offsetby (b/2, b/2). We fix the amplitude of the interlayer coupling in the 10-bandmodel and adjust g0 in the 2-band model such that the onset frequenciesof the ImσH(ω) for two models match. For g(10band)0 = 8meV this happenswhen g(2band)0 ≈ 1.67g(10band)0 .Results in Fig. B.3 show that over the entire frequency range the behaviorof σH(ω) is qualitatively similar in the two models, with the amplitudesdiffering by about a factor of 2. This level of agreement confirms thatthe two-band model captures the essential physics of the bilayer system.The discrepancy in the amplitude can be attributed to the difference in theform of the interlayer coupling and to the mismatch between the two bandstructures away from the BZ center.B.3 Estimating the Kerr angleB.3.1 Material propertiesWe use data in the Fig. 6 of Ref. [103] to estimate the diagonal part of theoptical conductivity σ(ω) of the bulk Bi2Sr2CaCu2O8+δ near optimal dop-ing. As remarked by the authors the data above ωc = 0.12 eV is consistentwith the power law behaviour σ(ω) = C(−iω)γ−2. The measured quantity is|σ(ω)| = Cωγ−2 and we extract the coefficient C and the exponent γ by con-sidering two points on the graph p1 = (103, 2×103) and p2 = (104, 5.6×102)in units of ([cm−1], [Ω−1cm−1]). We find that C = 9.122×104Ω−1(cm−1)3−γand γ = 1.447.For the index of refraction n appearing in Eq. (1) we use [103]n =√H +iσ(ω)0ω(B.23)where H = 4.77 is the background dielectric tensor. We estimate the 2Dconductivity σxx (required in Eq. 8) from the bulk conductivity throughσ = σxx/2d where d = 12.6A˚is the interlayer spacing [1] between Cu-Oplanes. Note that we consider two Cu-O monolayers in our 2D sample.103B.3. Estimating the Kerr angleB.3.2 Scaling properties of the Kerr angleHere we discuss the dependence of the Kerr angle on the interlayer couplingstrength g0 and frequency ω focusing specifically on the the experimentallyrelevant high frequency region. We observe a scaling collapse of the curves inFig. B.4, valid at small g0, when we plot θK/g20 in both thin and thick samplelimits. This is consistent with the expected leading quadratic dependenceon the interlayer coupling found in the 2-band model.0.0 0.1 0.2 0.3 0.4 0.52.52.01.51.00.50.00.51.0K/g2 0 [rad/(eV)2 ] (h) g0=4.0meVg0=8.0meVg0=16.0meVg0=32.0meV0.0 0.1 0.2 0.3 0.4 0.5 [eV]0102030405060K/g2 0 [rad/(eV)2 ] (h)10 1 2×10 1 3×10 1 4×10 1100101Figure B.4: Scaling collapse of θK/g20 indicating leading g20 dependence ofthe Kerr angle on the interlayer coupling. Top and the bottom panels showthick and thin sample limits, respectively.At high frequencies comparison of Fig. 3.1 and Eq. (3.9) indicates that104B.3. Estimating the Kerr angleσxy is smaller than σxx by at least three orders of magnitude (comparingboth real and imaginary parts). It is also true that, at high frequencies|Reσxy| |Imσxy| and Reσxy ∝ ω−2. Therefore, in Eq. (3.8), it is permis-sible to neglect σxy compared to σxx, which leaves us withθK ' Re arctan( −σxyσxx + 4piσ2xx). (B.24)Because σxx shows∝ ωγ−2 scaling with γ < 2, at large enough frequencies wecan drop the σ2xx term as well. Finally, combining the frequency dependencesof σxy and σxx and noting that σxy/σxx 1, we expand arctanx ≈ x for|x| 1 and obtain Eq. (3.10) quoted in the main text. Therein Λ is aprefactor dependent on microscopic details and we included the g20 factor tocapture the leading dependence on the interlayer coupling strength observedin Fig. B.4.30 40 50 60m, n10 410 310 2K [rad] (h)[1, 2] [1, 3] [1, 4] [2, 5] [3, 8] = 0.5eVFigure B.5: Kerr angle estimates for other commensurate twist angles θm,nat ω = 0.5eV in the thin sample limit (h λ), annotated by the twist vector[m,n] (blue dots). Each data point is mirrored (red dots) with respect toθm,n = 45◦ to show the expected behaviour for the entire range of twistangles.105B.3. Estimating the Kerr angleB.3.3 Results for other commensurate anglesOur lattice model calculations so far focused on a single commensurate twistangle θ1,2. We have also estimated the Kerr angle in the thin sample limitfor other commensurate angles which are computationally accessible andFig. B.5 shows these results. We find that the Kerr angle is increased as thetwist angle θ approaches 45◦. Compared to the θ1,2 case we studied above,θ2,5 ≈ 43◦ configuration results in a θK at least two orders of magnitudelarger.106Appendix CAppendix for Chapter 5C.1 Quantizing the NMon for time dependentflux - using irrotational constraintThe Lagrangian of the circuit is given by (assuming the junctions in eacharm are identical - and therefore constrained in a way that there is only onephase-flux variable describing all of them) L = T − U whereT =12MCM Φ˙2M +12NCN Φ˙2NandU = −NENJ cos(2piΦN/Φ0)−MEMJ cos(2piΦM/Φ0)where we identify a generalized flux ”coordinate” Φα for each junction suchthat 2piΦα/Φ0 = φα where α = M,N . Continuity of order parameter impliesthe constraintΦext = MΦM −NΦNWe define a new effective coordinate as a linear combination of the originalflux variablesΦ = αΦM + βΦNthese two equations can be invertedΦM =NΦ + βΦextαN + βM, ΦN =MΦ− αΦextαN + βMWith these expressions, we can write the Lagrangian L in the new variableswe have defined above and compute the conjugate momenta QΦ = ∂L/∂Φ˙and QΦext = ∂L/∂Φ˙ext which we need for the Legendre transform the obtainthe HamiltonianH = QΦΦ˙ +QΦextΦ˙ext − T + U107C.1. Quantizing the NMon for time dependent flux - using irrotational constraintwhich becomesH =NM(NCM +MCN )2(αN + βM)2Φ˙2 +NM(βCM − αCN )(αN + βM)2Φ˙Φ˙ext+β2MCM + α2NCN2(αN + βM)2Φ˙2ext + U(Φ,Φext) (C.1)whereU(Φ,Φext) = −NENJ cos(2piΦ0[MΦ− αΦextαN + βM])−MEMJ cos(2piΦ0[NΦ + βΦextαN + βM])(C.2)We pick the new coordinate transformation coefficients α, β obeying theirrotational constraint [146] such that the second term proportional to Φ˙extvanishes.β =CNCMαWe also drop the term proportional to Φ˙2ext as it will be a constant offset inthe quantum mechanical problem when we quantize Φ only. Rewriting theHamiltonian in these new variables as well as the conjugate momentumQΦ =NM(NCM +MCN )(αN + βM)2Φ˙→ 2enˆand rescalingα→ CMCMN + CNMα (which implies αN + βM → α)we find thatH =2e2nˆ2α2NM(CMN + CNM)−NENJ cos(Mφα− CMϕextCMN + CNM)−MEMJ cos(Nφα+CNϕextCMN + CNM)(C.3)where α can be any finite value which we can pick α = 1 for convenience.Note that we have also converted flux variables to phase variables usingφ =2piΦΦ0ϕext =2piΦextΦ0108C.2. Numerical approachWe can also consider the case where each junction array is shunted by acapacitor with capacitances CNS and CMS , respectively. We can show that inour calculation above, we simply need to make the replacements (comes fromadding the capacitance energies with voltage matching constraint acrossparallel N(M) junctions) CN → CN + NCNS and CM → CM + MCMS .In the limit where shunt capacitances are much larger than the junctioncapacitances, which is conducive for reaching the transmon limit EC EJwhere charge noise is suppressed, we simply need to replace CN → NCNS andCM →MCMS this large shunt capacitance limit simplifies the HamiltonianH =2e2nˆ2α2(NM)2(CMS + CNS )−NENJ cos(Mφα− CMSCMS + CNSϕextN)−MEMJ cos(Nφα+CNSCMS + CNSϕextM)(C.4)let us pick α = 1 and define 0 < κ < 1κ =CMSCNS + CMSwhich is a parameter controlled by the ratio of the shunt capacitances.Hamiltonian becomesH = 4EC nˆ2−NENJ cos(Mφ− κϕextN)−MEMJ cos(Nφ+ (1− κ)ϕextM)(C.5)whereEC =2e24(NM)2(CMS + CNS )C.2 Numerical approachStarting with the Hamiltonian in the phase representation, we switch tothe charge representation which is more tractable for exact diagonalization.Given the Hamiltonian of the formH = 4EC(−i∂φ + ng)2 − EJ cos(kφ+ ϕ)109C.3. Noise Calculationswhere k is an integer and ϕ = ϕext represents dimensionless flux or a phaseshift. NMon Hamiltonian above consists of these two types of terms only.We define the conjugate variable, number of Cooper pairs n, which we canswitch to by defining|n〉 = 1√N0∑φe−inφ|φ〉where N0 is the number of points in our discretization scheme. We thencompute the matrix elements 〈n|H|m〉 in the charge representation〈n|H|m〉 = 4EC(n+ ng)2δn,m − EJ2eiϕδn,m−k − EJ2e−iϕδn,m+k (C.6)We also need the matrix elements of the derivative of the Hamiltonian (in theeigenbasis of H0, which we will call |u〉) with respect to external parameters,which we can compute in the following way. Consider the derivative of thematrix element which we have just computed (note the particle eigenstatesdo not depend on the external parameter λ)∂〈n|H|m〉∂λ= 〈n|∂H∂λ|m〉 (C.7)= 〈n|u〉〈u|∂H∂λ|v〉〈v|m〉 (C.8)= Mnu〈u|∂H∂λ|v〉M †vm (C.9)where we defined Mmv = 〈m|v〉 which diagonalizes M †H0M = E0 Invertingthe last line, we obtain〈u|∂λH|v〉 = M † ∂∂λ(〈n|H|m〉)Mwhere we know the expression 〈n|H|m〉 from above. We need to computethe derivatives given the fluctuating parameter and evaluate the expressionin the box above.C.3 Noise CalculationsExternal parameters (which we will call λ for general discussion) of theHamiltonian such as charge offset ng and external flux Φext fluctuate fromthe values we intend to set for optimal performance of a qubit. These clas-sical variables therefore contribute to the decoherence in addition to other110C.3. Noise Calculationschannels such as quasiparticle excitations or radiation. For small fluctua-tions of the classical parameter δλ(t) around λ0, we can expandH(λ) = H(λ0) +∂H∂λδλ(t)We use time dependent perturbation theory to first order to find thetransition amplitude cni(t) = 〈n|U(t, 0)|i〉 that we will find the system instate |n〉 provided that the system is initially in state |i〉 before we turnon the perturbation V (t) = ∂H∂λ δλ(t) where we treat H0 = H(λ0) as theunperturbed Hamiltonian satisfying H0|m〉 = Em|m〉 We use the textbookresult to first orderc(1)ni (t) = −i~∫ t0eiωnit′Vni(t′)where we defined Vni(t) = 〈n|V (t)|i〉 and ~ωni = En − Ei. The transitionprobability pni(t) = |cni(t)|2 is then given bypni(t) =|〈n|∂λH|i〉|2~2∫ t0dt1∫ t0dt2eiωni(t1−t2)δλ(t1)δλ(t2)After a change of variables by defining τ = t1 − t2 and T = (t1 + t2)/2and time averaging (bounds of τ integral can be sent to infinity for smallautocorrelation times [158])pni(t) =|〈n|∂λH|i〉|2~2∫ t0dT∫ ∞−∞dτeiωniτ 〈δλ(T + τ/2)δλ(T − τ/2)〉If we further assume time translation invariance (stationary noise correlationfunction), then the T integral becomes trivial.pni(t) =|〈n|∂λH|i〉|2~2tSλ(ωni)where we introduce the noise spectral densitySλ(ω) =∫ ∞−∞dτeiωτ 〈δλ(τ)δλ(0)〉which characterizes the time dependence of the fluctuations of the exter-nal parameter λ. The transition rate Γni is found by computing the timederivative of pni:Γni =|〈n|∂λH|i〉|2~2Sλ(ωni)111C.4. Transition rate matrix elementsAs we have just seen, the transition rates are directly related to transitionmatrix elements due to external parameter perturbations. In this work wewill not be estimating relaxation/dephasing times. Instead we will studythe transition matrix elements to compare performances of different qubitarchitectures.C.4 Transition rate matrix elementsIn this work we present the transition rates renormalized by the qubit fre-quency since this number is independent of overall energy scales, (only de-pends on EJ/EC for transmon or EN(M)J /EC for NMon and universal inthe sense that this allows us to compare different qubits for the same tran-sition frequency. This section is intended as a reference for supplementaryinformation.Figure C.1: Flux fluctuation matrix elements for the N = 2, M = 3 NMonparameter space shown in Fig. 5.5. Note that by picking different values ofκ we observe that the line where the matrix elements vanish can be moved.β = ENJ /EC and η = EMJ /ECC.4.1 Split transmonNMon reduces to split transmon for parameters N = 1 and M = 1. DefiningEΣ = ENJ + EMJ and asymmetry parameter d = ENJ − EMJ (note thatjunction asymmetries typically lead to d = ±10% [45] in fabrication process).112C.4. Transition rate matrix elementsEquation 5.1 reduces toH = 4EC nˆ2 − EΣ cos ϕext2cosφ− d sin ϕext2sinφfor κ = 1/2 which we can obtain by choosing equivalent shunt capacitancesfor both junctions. We compute the transition matrix elements and relativeanharmonicity for a range of model parameters in Fig. C.2 for ϕext = 0. Atypical transmon with EJ = 22.5GHz and EC = 200MHz we find ENJ /EC ≈113 where anharmonicity is αr = −0.04a) b) c)d) e) f )Figure C.2: Model parameter space plots tuning EΣ and d showing a) rela-tive anharmonicity b) charge fluctuation matrix elements c) flux fluctuationmatrix elements for split transmon at ng = 0 and ϕext = 0. Panels d,e,f)show same quantities as a,b,c) except for ng = 0 and ϕext = pi.113C.4. Transition rate matrix elementsC.4.2 FluxoniumNMon reduces to the fluxonium qubit if we pick M = 1 and N = 10, alarge number which leads to suppression of the flux fluctuations in additionto having giant anharmonicity. Starting with the master Hamiltonian Eq.C.4, for M = 1 we find thatH =2e2nˆ2(CMS + CNS )−NENJ cos(φN− CMSCMS + CNSϕextN)− EMJ cos(φ+CNSCMS + CNSϕext)(C.10)To connect with the existing literature, we picked α = N . In the large-Nlimit, the second term generates the quadratic potential which is called theinductance term. Experimentally relevant case of inductor allocation [147]corresponds to CMS CNS :H = 4EC nˆ2 −NENJ cos(φ− ϕextN)− EMJ cos (φ) (C.11)where EC = e2/2CMS . This expression reduces to the familiar fluxoniumHamiltonian:H = 4EC nˆ2 +EL2(φ− ϕext)2 − EMJ cosφ (C.12)where we have dropped the constant term and defined EL = ENJ /N . Wecompute the matrix elements at ϕext = pi flux and ng = 0 offset chargein Fig. C.4 and Fig. C.5. For parameters given in C.3 , we find that|〈0|∂ϕH|1〉|/ω = 6.9 and |〈0|∂ngH|1〉|/ω = 0.24114C.4. Transition rate matrix elements1.0 0.5 0.0 0.5 1.0 1.5051015202530E [GHz]N = 10, M = 1, EJN = 20.0, EJM = 10.0 EC = 0.02, = 1 ext/ 0 = 1.0 , ng = 0, V= 10ENJ cos( ext/10) EMJ cos(10 + (1 ) ext/1)2 4 6 8 10 12nm0.00.501 0212 03132304142434Transition frequenciesFigure C.3: Fluxonium spectrum for N = 10 and M = 1. At half fluxquantum ϕext = pi the potential well develops two minima which leads togiant anharmonicity.115C.4. Transition rate matrix elementsFigure C.4: Flux fluctuation matrix elements for fluxonium with N = 10and M = 1. Spectrum shown in Fig. C.3116C.4. Transition rate matrix elementsFigure C.5: Charge fluctuation matrix elements for fluxonium with N = 10and M = 1. Spectrum shown in Fig. C.3117C.4. Transition rate matrix elementsFigure C.6: N = 13, M = 2 NMon charge fluctuation matrix elementsrenormalized by qubit frequency ω03 as described in section C.3 computedat β = 140, η = 55 with ϕext = 0. Code space is defined as n = 0→ |0〉 andn = 3→ |1〉. The table shows all transitions between levels n = 0− 7118C.4. Transition rate matrix elementsFigure C.7: N = 2, M = 3 NMon flux fluctuation matrix elements renor-malized by qubit frequency ω03 as described in section C.3 computed atβ = 75, η = 15 with ϕext = 0. Code space is defined as n = 0 → |0〉 andn = 3→ |1〉. The table shows all transitions between levels n = 0− 6119
Thesis/Dissertation
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Physics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Franz, Marcel
Topological superconductivity in twisted cuprates and device applications inspired by their Josephson physics
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