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The holographic interface of a fractional (2+1)D topological insulator at finite temperature Wong, Anson W.C. 2014

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The Holographic Interface of aFractional (2+1)D TopologicalInsulator at Finite TemperaturebyAnson W.C. WongB.Sc., The University of Waterloo, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2014c© Anson W.C. Wong 2014AbstractTopological insulators are materials that are insulating in the bulk but con-ductive on the boundary. Although standard condensed matter techniqueselucidate the dissipationless boundary physics of topological insulators wellat weak coupling, they fail to do the same at strong coupling where excit-ing phenomena such as emergence and fractionalization are likely to occur.Fortunately the AdS/CFT correspondence offers an alternative perspectiveof the strong coupling limit in the form of a classical supergravity dual.In this thesis we realize the interface of a strongly-interacting fractional(2+1)D time-reversal invariant topological insulator at finite temperatureby embedding a D5-brane with a U(1) chemical potential into (AdS5 blackhole)×S5 supergravity. The thermodynamics of our interface are found tobe considerably fermionic. Study of the interface has promising applica-tions ranging from the design of spin channels in quantum computing, tothe deeper understanding of highly-entangled systems.iiPrefaceThis thesis is original and unpublished work of mine. An existing templatefor the implementation of spectral methods was provided by M. Rozali, butotherwise, I was responsible for all other analytical, numerical, and writtenaspects of this thesis. M. Rozali was my primary research advisor and wasinvolved in the overall guidance and formulation of the project including therevisions of this thesis transcript. J. Karczmarek was the second reader ofthis thesis.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The AdS/CFT correspondence . . . . . . . . . . . . . . . . . 42.1 AdS/CFT from D3-branes . . . . . . . . . . . . . . . . . . . 52.2 The strong coupling limit . . . . . . . . . . . . . . . . . . . . 62.3 Finite temperature AdS/CFT . . . . . . . . . . . . . . . . . 83 Topological insulators . . . . . . . . . . . . . . . . . . . . . . . 93.1 The Z2 phases . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 The strong coupling limit and fractionalization . . . . . . . . 143.3 A topological interface . . . . . . . . . . . . . . . . . . . . . 144 Gravity side set-up . . . . . . . . . . . . . . . . . . . . . . . . . 164.1 The field theory to realize . . . . . . . . . . . . . . . . . . . . 164.2 The D5-brane embedding . . . . . . . . . . . . . . . . . . . . 194.3 DBI action and equations of motion . . . . . . . . . . . . . . 214.4 Time-reversal invariance . . . . . . . . . . . . . . . . . . . . 224.5 Classes of D-brane embeddings . . . . . . . . . . . . . . . . . 224.6 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 25ivTable of Contents4.7 Scaling symmetry . . . . . . . . . . . . . . . . . . . . . . . . 274.8 Realizing a holographic topological interface . . . . . . . . . 285 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 315.1 The phase diagram . . . . . . . . . . . . . . . . . . . . . . . 315.2 The holographic topological interface . . . . . . . . . . . . . 335.3 Thermal properties of the interface . . . . . . . . . . . . . . 366 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 Prospective research . . . . . . . . . . . . . . . . . . . . . . . . 417.1 Zero temperature solution . . . . . . . . . . . . . . . . . . . 417.2 Integrating nb(x) over the interface . . . . . . . . . . . . . . 417.3 Turning on an external magnetic field . . . . . . . . . . . . . 427.4 Searching for the Majorana fermion . . . . . . . . . . . . . . 43Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44AppendicesA Chern-Simons terms of the integer QSH state . . . . . . . 50B Calculation of the AdS black hole Hawking temperature . 51vList of Tables4.1 Directions occupied by the branes in our D3-D5 system. Thefilled circles • represent the directions occupied by the brane,and the unfilled circles ◦ represent the orthogonal directionsunoccupied by the brane in the world volume. The D5-braneprobes the D3-brane stack background. . . . . . . . . . . . . 18viList of Figures1.1 The quantum critical point gc and quantum critical phase(sandwiched between the ordered and disordered state) as afunction of temperature T and running coupling g. In thequantum critical phase, the theory is independent of lengthscale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 A stack of coincident D3-branes with open strings connectingthe brane with either itself or another identical brane to en-hance the gauge group to SU(Nc). Gauge fields are sourcedat the end-points of the connected strings. . . . . . . . . . . . 52.2 The D3-brane stack as a black 3-brane. The geometry isAdS5×S5 near the horizon, and R1,9 Minkowski near infinity.The compactified S5 manifold gives a minimum size to the‘throat’ of the near-horizon brane geometry. . . . . . . . . . . 72.3 We take the large Nc limit before we take the strong couplingλ → ∞ limit to achieve a classical gravity dual. The planarlimit frees the strings and the strong coupling limit allows forclassical treatment. . . . . . . . . . . . . . . . . . . . . . . . . 83.1 The band structure diagrams for different (2+1)D insulators.The bulk is gapped and non-conducting for each insulatorabove. The trivial insulator is completely gapped in boththe bulk and the boundary. The boundary for the quantumHall and QSH states are gapless and can thus support bound-ary transport. The Hall current travels in one direction withno spin dependence, but the QSH current has two counter-propagating channels of opposite spins. . . . . . . . . . . . . . 10viiList of Figures3.2 Schematic diagrams of the conduction (or non-conduction) ofedge states in different (2+1)D insulators. The trivial insula-tor in Fig 3.2a is fully gapped at the boundary and supportszero conductivity. The quantum Hall state in Fig 3.2b con-ducts Hall current on the edge given a strong external mag-netic field Bext. The spin current on the edge of a QSH stateas shown in Fig 3.2c comes as Kramers doublets of oppo-site spin electrons traveling in opposite directions. Spin-orbitcoupling drives spin transportation in the QSH effect. Unlike(3+1)D, the boundary states are restricted to only two direc-tions of travel along the edge of a (2+1)D insulator. Notethat QSH states have no net electric charge flowing along theedge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.1 The D3-D5 brane system for a single Nf = 1 D5 probe brane.The Aµ on the D5-brane is a global U(Nf ) .= U(1) gaugefield, and the A˜µ on the D3-brane stack is an internal SU(Nc)‘phonon’ gauge field. The string connecting the D3 and D5branes is the Dirac fermion ψ. Separating the D5-brane fromthe D3-branes gives mass to the fermion. . . . . . . . . . . . . 174.2 The supersymmetric D3-D5 brane intersection. By passingthe D5-brane through the D3-brane stack in the transverse x9direction, the mass of the fermion can switch signs to realizea gapless interface. Positive and negative mass embeddingscorrespond to which side of the D3-brane stack the D5-braneis being pulled away from. . . . . . . . . . . . . . . . . . . . . 184.3 Schematic diagrams of the Minkowski, critical, and black holeembeddings. The Minkowski embedding (Fig 4.3a) smoothlyterminates outside of the black hole and forms a mass gap.Minkowski embeddings necessarily correspond to zero densitystates. The black hole embedding (Fig 4.3b) closes the massgap by having the brane falling through the horizon. Blackhole embeddings on the other hand can induce dense states.The critical embedding (Fig 4.3c) is the embedding interme-diate to both the Minkowski and the black hole embeddingwhere the brane just pinches off from the black hole. Wefocus mainly on black hole embeddings in this thesis. . . . . . 23viiiList of Figures4.4 Boundary conditions on the horizon and boundary of theAdS5 black hole. Not included here are the homogenous Neu-mann boundary conditions at the spatial boundaries x = ±∞.The horizon requires regularity conditions, and the boundaryrequires the specification of ultraviolet sources from the fieldtheory. In our case, we feed in a mass profile m(x) and achemical potential µ into the bulk from the boundary. . . . . 274.5 The ideal and our constructed mass profile m˜(x) at finite tem-perature T . In the ideal profile (Fig 4.5a), we have a gaplessinterface at x = 0 sandwiched between two gapped insulatorsof differing topologies. It is realized with both Minkowski(M) and black hole (BH) embeddings. In our construction(Fig 4.5b), we consider a mass profile m˜(x) that has a fi-nite interface width and a chemical potential µ˜ & µ˜c(M˜) thatmakes the embedding a black hole embedding for all x. . . . . 294.6 A sketch of the quark density nb(x) as a function of the spatialdirection x for the ideal and our constructed topological in-terface situated at x = xint. A finite density d˜ is induced as aconsequence of our construction but its effect can be reducedby keeping µ˜ & µ˜(M˜). . . . . . . . . . . . . . . . . . . . . . . 305.1 The homogenous (T˜ ≡ 1/M˜, µ˜/M˜) phase diagram. Thereare two types of embeddings: the nb > 0 black hole embed-dings and (implicitly) the nb = 0 Minkowski embeddings.A chemical potential threshold µ˜c > 0 must be overcome attemperatures below T˜c ≈ 0.87 to favour the black hole embed-ding. However, above the critical temperature T˜c, black holeembeddings are favoured regardless of how much chemical po-tential is fed in. The blue star on the left of the phase diagramare the parameters we use for our representative topologicalinterface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2 The numerical solution for χ(z, x) at low temperature T˜ (µ˜ =2.000, M˜ = 2.954) = 0.339. The interface is at x = 0. Themass is positive for x > 0 (red curves in Fig 5.4), and negativefor x < 0 (blue curves in Fig 5.4). The asymptotic behaviourgoes as χ ≈ m˜(x)z + c˜(x)z2 at the boundary and achieves anabsolute value of less than one at the horizon. . . . . . . . . . 34ixList of Figures5.3 The numerical low temperature solution for A˜t(z, x) withT˜ (µ˜ = 2.000, M˜ = 2.954) = 0.339. The interface is atx = 0. The dip in the gauge field near x = 0 is a detec-tion of higher density at the interface, which is most likely tobe fermionic from our thermodynamic analyses. The asymp-totic behaviour goes as A˜t ≈ µ˜− n˜b(x)z at the boundary andvanishes at the horizon. . . . . . . . . . . . . . . . . . . . . . 345.4 The probe D5-brane embeddings on the ρx-ρy plane whereρx = ρ cos θ = ρ√1− χ2, ρy = ρ sin θ = ρχ, and ρ2 = r2 +√r4 − 1. Each curve represents a spatial grid point x withred, black, and blue curves corresponding to positive, zero,and negative mass black hole embeddings, respectively. . . . . 355.5 The condensate c˜(x) across the x = 0 interface as a function ofthe spatial direction x. The red points are the actual spatialgrid points, and the black curve interpolates between them. . 365.6 The quark density n˜b(x) across the x = 0 interface as a func-tion of the spatial direction x. The red points are the actualspatial grid points, and the black curve interpolates betweenthem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.7 The scaling behaviour of the interface density, n˜b(x = 0), asa function of the chemical potential. The inset plot describeshow the scaling exponent α from n˜b ∼ µ˜α varies with thechemical potential. At high temperatures (low µ˜) the scalinggoes as n˜b ∼ µ˜, and at low temperatures (high µ˜) the scalinggoes as n˜b ∼ µ˜2. . . . . . . . . . . . . . . . . . . . . . . . . . . 397.1 Our candidate interface width definition (cyan-filled area) onan nb(x) plot. The red dotted line represents the backgrounddensity, and the blue dashed lines represent the density rangeof the interface width. . . . . . . . . . . . . . . . . . . . . . . 42xAcknowledgementsI thank first and foremost my parents for putting up with me throughoutmy whole life. Next up would be my caring aunts, uncles, and cousins thathave taken care of me unconditionally – I am forever grateful. I thank all myclosest friends from high school because I do not know where I would be inthis world you guys were not around to help me transform from an immaturelittle boy to a fully grown immature adult. I thank my former housemateJames for all the funny, intellectual and emotional late-night conversationswe have had during our ups and downs. I thank my storage room officematesAlex and Joel for making it much less lonely in our prison cell of an office,and the rest of the String Theory group for the fruitful physics discussionsthat I believe cannot possibly exist outside of the Hennings building. Ithank Joanna for saying yes when I asked her to be my second reader forthis thesis. And lastly, but most definitely not least, I thank my advisorMoshe for his constant guidance and patience throughout this project.xiDedicationTo my lovely grandfather who sadly passed away during the writing of thisthesis.xiiChapter 1IntroductionHolography is the study of how much information inside of a region of spaceis projected onto its own boundary. Often it is possible for the ‘shadow’of an object to describe a lot, if not all, about the object itself. In stringtheory, the notion of holography is rooted in the physics of gravity andquantum mechanics. Take for example the Bekenstein-Hawking entropy ofa black hole which is proportional to its event horizon area and not to itsvolume as classically expected [1]. The horizon here plays the role of theblack hole’s ‘shadow’ and tells a quantum story about the interior volume.First signs of a holographic principle coming to fruition in string theoryoriginated from the work of ’t Hooft and Susskind in the early 1990s [2, 3],but it was Maldecena in 1997 who provided an explicit hands-on example ofhow quantum mechanics could manifest itself as a hologram of gravity nowinfamously known as the AdS/CFT correspondence [4–8] (or gauge/gravityduality) which conjectures the equivalence between Type IIB string theoryon AdS5×S5 and N = 4 super Yang-Mills theory in four dimensions. Whatis the practical use of the AdS/CFT duality though? The correspondencestates the existence of a bijective mapping between the boundary fields of agravity theory and the physical observables of its field theory dual. Althoughthe specific mapping is not given freely to us a priori, with some work one canconstruct a holographic dictionary to infer the physics of a quantum fieldtheory from an often simpler gravity calculation. Holography to this dayremains as one of the forefronts of modern high energy physics research withapplications in modelling high-temperature quark-gluon plasmas, computingholographic entanglement entropies, realizing confinement/deconfinement innon-abelian gauge theories, etc. Our intention in this thesis is to study theinterface of a strongly-correlated topological insulator from a gravity pointof view.In condensed matter, majority of weakly-interacting systems follow theprescriptions of Fermi Liquid theory and/or Landau theory. These tech-niques however tend to fail when the interactions are more complicated:Fermi Liquid theory loses validity when the non-perturbative effects of strongcoupling come into play, and Landau theory fails to track down novel phase1Chapter 1. Introductiontransitions in materials where the ordering mechanism is no longer an ex-plicit breaking of symmetry i.e spin liquids. High-Tc superconductors andone-dimensional Luttinger liquids are examples of materials that are stillpoorly understood because of the lack of available techniques in field theory.Despite the current downfall of condensed matter at strong coupling, it isexactly at strong coupling where holography shines the brightest – a highlyquantum system can be described by a gravity dual that is completely classi-cal. At zero temperature, these are the scale-invariant systems that sit at thequantum critical point gc for some physical running coupling g as illustratedin Fig 1.1. As the temperature increases, the quantum critical point extendsinto the quantum critical phase. A plethora of strongly-correlated versionsof high-Tc superconductivity [9], the Josephson effect [10], the Kondo effect[11], non-Fermi liquids [12, 13], the integer/fractional quantum Hall effect[14–16], and striped phases [17] have been holographically realized in thisphase already to name a few. Although holographic techniques in condensedmatter are somewhat contrived because they demand the strict conditionsof supersymmetry and a large number of colours, they do on the very up sideserve as promising indicators for universal behaviour in the strong couplingregime – currently we have no simpler alternative.We are interested in applying holography to topological insulators [18–22]. Topological insulators are insulators that are gapped in the bulk butgapless on the boundary, as opposed to conventional insulators where it isgapped in both places. Moreover the admitted gapless boundary states ofa topological insulator are protected by bulk symmetries (typically time-reversal invariance) from dissipative processes like electron-impurity scat-tering. A convenient mental picture of the topological insulator is a con-ventional insulator with a thin coating of metal, but the reality is that thetopological insulator is uniform throughout.The peculiar behaviour of the boundary of a topological insulator comesas a consequence of topology. Insulators of differing topological classes com-ing in contact with one another necessarily forms a gapless interface forfermionic transport. In this thesis we study in particular the interface of a(2+1)D time-reversal invariant topological insulator at finite temperature.The topological invariant is well known to be Z2-valued at weak coupling.At strong coupling, it has been proposed that the fermions in the bulk ofthe topological insulator can fractionalize into partons [23, 24]. We holo-graphically realize this fractional topological insulator by using a top-downapproach of embedding a probe D5-brane with a U(1) chemical potentialinto an (AdS5 black hole) × S5 background similar to what was done inRefs [25–27]. The interface is localized by placing two different topologi-2Chapter 1. IntroductionFigure 1.1: The quantum critical point gc and quantum critical phase (sand-wiched between the ordered and disordered state) as a function of temper-ature T and running coupling g. In the quantum critical phase, the theoryis independent of length scale.cal phases in contact with eachother via flipping the fermionic mass signalong some spatial direction. We then construct a phase diagram and studythe thermodynamics of a representative interface solution. The (3+1) di-mensional version of our problem was investigated in Refs [28, 29] but itremained to be done in (2+1) dimensions until now. Sanity checks of ournumerical calculations were made and found to agree with the relevant re-sults of Refs [25, 28–32].The thesis will be structured as follows. We introduce the AdS/CFTcorrespondence in Chapter 2. Afterwards, we stray from string theory for awhile and introduce the topological insulators in Chapter 3. In Chapter 4,we construct step-by-step the holographic interface of a fractional (2+1)Dtime-reversal invariant topological insulator. The numerical results are pre-sented and discussed in Chapter 5. We conclude in Chapter 6 and provideprospective research in Chapter 7.3Chapter 2The AdS/CFTcorrespondenceWe briefly introduce the AdS/CFT correspondence conjecture with D-branesand its use towards studying strongly coupled systems. The AdS/CFT cor-respondence states the duality:Field theory side Gravity sideN = 4 super Yang-Mills theorySU(Nc) in (3+1) dimensions⇔ Type IIB superstring theoryon AdS5 × S5 spacetimeGauge-invariant operators ↔ Supergravity fieldson the boundaryOn the field theory side of the correspondence, N = 4 super Yang-Millstheory is a conformal non-abelian gauge theory with a vanishing β function(at all perturbative orders) and is maximally supersymmetric. The quarksof the theory transform in the fundamental representation of the specialunitary SU(Nc) gauge group with Nc being the number of quark colours.The field content of the theory includes a gauge field, four Weyl spinors, andsix scalars. There are 4N = 16 supercharges due to each of the four Weylspinors holding two complex degrees of freedom. The combination of theSO(4, 2) Lorentzian conformal group and the SO(6) R-symmetry betweenthe six scalars gives for an SO(4, 2)× SO(6) symmetry group.On the gravity side of the correspondence, Type IIB string theory is on amaximally symmetric spacetime of 5-dimensional Anti-de Sitter (AdS) spaceproducted with a 5-dimensional sphere S5. AdS space naturally exhibits theconformal invariance of the field theory with an ‘extra’ radial variable z thatadjusts the lattice renormalization length so that the physics are indepen-dent of length-scale. The isometries of AdS5 and S5 producted togethergive for the same SO(4, 2) × SO(6) symmetry as it is on the field theoryside. Performing a dimensional reduction of the compact S5 sphere makes42.1. AdS/CFT from D3-branesFigure 2.1: A stack of coincident D3-branes with open strings connecting thebrane with either itself or another identical brane to enhance the gauge groupto SU(Nc). Gauge fields are sourced at the end-points of the connectedstrings.the gravity side effectively a (4+1)-dimensional AdS gravity theory with aninfinite number of fields – one dimension higher than the (3+1)-dimensionalfield theory ‘hologram’.2.1 AdS/CFT from D3-branesDp-branes are solitonic hyperplanes in string theory that extend in one tem-poral direction and p spatial directions [33–35]. They provide world vol-ume surfaces for open strings to attach their ends onto to source Ramond-Ramond gauge fields at the string ends so that scalars, vectors, and fermionscan live on them as dynamical degrees of freedom. Neumann boundary con-ditions are satisfied in (p+ 1) of the transverse directions of the brane, andDirichlet conditions are satisfied in the remaining 10 − (p + 1) orthogonaldirections. The gauge/gravity conjecture can come about by consideringextremal D3-branes from two different perspectives: the open string per-spective as a stack of coincident D3-branes connected by open strings, andthe closed string perspective as a black 3-brane gravity solution.The open-string persectiveThe coincident Nc D3-brane stack in the open-string perspective, as depictedin Figure 2.1, has an action of the form52.2. The strong coupling limitSFT = SFTD3-brane + SFTbulk + SFTint , (2.1)which is comprised of the action of the D3-branes, the bulk they sit in,and the interactions between them. In the Maldecena decoupling limit ofvanishing string tension α′ → 0, the interaction part SFTint → 0 vanishesand the D3-brane stack becomes N = 4 super Yang-Mills theory with anSU(Nc) colour symmetry embedded in flat spaceSFT =α′→0SFTN=4 Yang-Mills + SFTflat-space. (2.2)The closed-string persectiveNow consider the same D3-brane stack but in the closed-string perspectiveof an extremal black 3-brane solution. The black 3-brane metric is given byds2D3 = H(r)−12(−dt2 + d~x2)+H(r) 12(dr2 + r2dΩ25), (2.3)where H(r) = 1 + L4/r4 is the warp factor, r is the radial parameter,L4 = 4pigsNcα′2 is the AdS length, and dΩ25 is the 5-sphere metric. The near-horizon limit H(r  L) ≈ L4/r4 corresponds to AdS5 × S5 geometry, andthe boundary limit H(r  L) ≈ 1 corresponds to flat R1,9 Minkowski spaceas illustrated in Fig 2.2. Taking the Maldecena decoupling limit decouplesthe near-horizon and the boundary geometries of the curved backgroundand leaves the action asSG =α′→0SGAdS5×S5 + SGflat-space. (2.4)Equating the actions from the open and closed string perspectives andsubtracting the shared flat space component leaves us with the perturbativeintuition that N = 4 SU(Nc) super Yang-Mills theory is dual to Type IIBstring theory on AdS5 × S5 in the large Nc ∼ 1/α′ →∞ limit for arbitrarycoupling strength λ ≡ g2YMNc = 2pigsNc (gYM and gs being the Yang-Millsand string couplings respectively) .2.2 The strong coupling limitThe large Nc ∼ 1/α′ → ∞ limit with λ fixed, also known as the planarlimit, provides integrability by suppressing the 1/N2gc contributions fromnon-planar genus g > 0 Feynman diagrams to give precedence to only theplanar g = 0 diagrams. Although the strings are free in the α′ → 0 limit,62.2. The strong coupling limitrzR1,9AdS5 × S5Figure 2.2: The D3-brane stack as a black 3-brane. The geometry is AdS5×S5 near the horizon, and R1,9 Minkowski near infinity. The compactifiedS5 manifold gives a minimum size to the ‘throat’ of the near-horizon branegeometry.they are still quantum-like and can form massive towers of string modes thatcorrespond to higher-order curvature terms in the gravity action. However,if we take an additional strong coupling λ → ∞ limit as well, such stringyeffects get suppressed and the strings become not only free but classical asillustrated in Fig 2.3. This is the beauty of the strong coupling limit inAdS/CFT. Validity of perturbation theory on the field theory side holdsonly at weak-couplingλ = g2YMNc  1, (2.5)whereas the perturbative validity on the gravity side (for string lengthls) holds only at strong couplingλ = gsNc ∼ L4/l4s  1, (2.6)making the perturbative physics on one side of the duality resemble thenon-perturbative physics on the other side. We will make use of both thelarge Nc →∞ and strong coupling λ→∞ limits in our holographic set-up.72.3. Finite temperature AdS/CFTStrong coupling limitfreeclassical stringsPlanar limitinteractingclassical stringsinteractingquantum stringsfreequantum stringsFigure 2.3: We take the large Nc limit before we take the strong couplingλ → ∞ limit to achieve a classical gravity dual. The planar limit frees thestrings and the strong coupling limit allows for classical treatment.2.3 Finite temperature AdS/CFTThe use of extremal D3-branes in the AdS/CFT duality keeps the physicspertinent to only zero temperature. Holography at finite temperature in-volves adding thermal excitations to the AdS space to get ‘hot’ AdS –this corresponds to a thermal bath in the dual field theory. By replac-ing the extremal D3-branes with non-extremal ones we move out to finitetemperature. The near-horizon geometry of the D3-branes changes fromAdS5 × S5 to (AdS5 black hole)× S5 with a radial horizon proportional tothe field theory temperature. Overall the promotion to finite temperaturechanges the gravity side of the correspondence to Type IIB string theory on(AdS5 black hole)×S5. Note that the boundary geometry is unchanged fromreplacing the D3-branes; the horizon of the black hole effectively cuts off allinfrared physics below temperature T but leaves the ultraviolet physics atthe AdS boundary unscathed. We work with finite temperature exclusivelyin this thesis.8Chapter 3Topological insulatorsQuantum mechanics is responsible for the inert behaviour of conventionalinsulators. At sufficiently low temperatures, the valence electrons cannotovercome the energy threshold of ∆ > 0 to climb up into the conductionband. The band structure diagram for electrons in a conventional insulator isillustrated in Fig 3.1a with both the bulk and boundary being gapped. Suchinsulators are rather dull and luckily represent only a subset of materialswith gapped bulks. We focus here on another class of insulators called thetopological insulators.Topological insulators fit somewhere between an insulator and a con-ductor in the sense that it has a gapped bulk but a gapless boundary. Theboundary can conduct fermions even in the presence of an insulating bulk.In non-interacting models, the (2+1)D topological insulator has been ex-haustively classified in a periodic table of discrete symmetry classes in thebulk that includes time-reversal symmetry, charge conjugation symmetry,and sublattice symmetry [36, 37]. The most studied example is the (2+1)Dtime-reversal invariant quantum spin Hall (QSH) state and unlike most dis-coveries in condensed matter, the prediction of their existence well precededits experimental realization by almost 20 years when it was first observed intwo-dimensional HgTe quantum wells in 2007 [38, 39]. QSH states are time-reversal invariant in the bulk and can exhibit the QSH effect of spin-lockedelectronic transport along the insulator edge analogous to Hall current inthe quantum Hall effect. We illustrate both the QSH effect and quantumHall effect in Fig 3.2b and Fig 3.2c, and their corresponding band struc-ture diagrams in Fig 3.1b and Fig 3.1c. The spin-orbit coupling from latticeelectrons induce a magnetic-like field to drive Kramers pairs of opposite-spinfermions to counter-propagate on the edge of the QSH state [40, 41], whereasthe external magnetic field in the quantum Hall effect breaks time-reversalsymmetry and forces mixed-spins to propagate along the edge in the samedirection.The edge of the QSH state, topologically speaking, is simply the sharedinterface between a trivial conventional insulator (like vacuum) and a non-trivial topological insulator. We want to holographically construct such an9Chapter 3. Topological insulatorsMomentumValence BandConduction BandEnergyEF ∆(a) trivial insulatorMomentumValence Bandspin up/downConduction BandEnergyEF ∆(b) quantum Hall stateMomentumValence Bandopposite spinsConduction BandEnergyEF ∆(c) QSH stateFigure 3.1: The band structure diagrams for different (2+1)D insulators.The bulk is gapped and non-conducting for each insulator above. The triv-ial insulator is completely gapped in both the bulk and the boundary. Theboundary for the quantum Hall and QSH states are gapless and can thussupport boundary transport. The Hall current travels in one direction withno spin dependence, but the QSH current has two counter-propagating chan-nels of opposite spins.103.1. The Z2 phasesinterface for the (2+1)D time-reversal invariant topological insulator. To doso, we begin by discussing the possible topological phases that can exist inour insulator.3.1 The Z2 phasesSymmetry brings about the concept of topology to insulators. In mathemat-ics, the topology of a geometrical object is characterized by its topologicalinvariant – an intrinsic quantity that is insensitive to smooth deformationsof the object. A sphere is topologically equivalent to an ellipsoid but not toa torus without the use of scissors. Analogously in non-interacting topolog-ical insulators, the dispersion spectra of topologically distinct insulators arenot smoothly connected from one to another without closing the bulk gap(take for example the band structure diagrams of the trivial and non-trivialinsulators in Fig 3.1a and Fig 3.1c). Topology in general is not restricted toinsulators – superconductors can be topological as well because of a gappedbulk from Cooper pairing, but gapless materials such as metals and dopedsemi-conductors cannot be given such a classification.The topological invariant of a topological insulator is encoded in theanomalous Chern-Simons terms that come from integrating out massivefermions from the action. For the (2+1)D time-reversal invariant topologi-cal insulator, the bulk topological invariant is Z2-valued meaning only twodistinct topological classes exist and they happen to be the trivial (conven-tional) insulator and the non-trivial (topological) insulator. The even/oddparity of the number of edge pair modes Nf crossing through the Fermi en-ergy EF identifies the Z2 phases – an odd Nf being of the same class as thetrivial insulator, and an even Nf being of the same class as the non-trivialinsulator.Another way of identifying the Z2 invariant of the QSH state is throughthe mass sign of the bulk fermions. Consider the microscopic (2+1)D time-reversal invariant model for a single Nf = 1 flavour pair of free complexDirac fermions ψ1 and ψ2 in spinor notationL = ψ¯1 (iγµ∂µ +m)ψ1 + ψ¯2 (iγµ∂µ −m)ψ2, (3.1)with oppositely signed masses ±m, a Dirac conjugate ψ¯ ≡ ψ†σz, andgamma matrices γµ made of Pauli matrices σi. Let us inspect the massterms ±m. They are necessarily real by time-reversal invariance. Al-though the ±mψ¯ψ terms are not time-reversal invariant individually because113.1. The Z2 phasesbulkboundary(a) trivial insulatorBext bulkboundary(b) quantum Hall statebulkboundary(c) QSH stateFigure 3.2: Schematic diagrams of the conduction (or non-conduction) ofedge states in different (2+1)D insulators. The trivial insulator in Fig 3.2ais fully gapped at the boundary and supports zero conductivity. The quan-tum Hall state in Fig 3.2b conducts Hall current on the edge given a strongexternal magnetic field Bext. The spin current on the edge of a QSH stateas shown in Fig 3.2c comes as Kramers doublets of opposite spin electronstraveling in opposite directions. Spin-orbit coupling drives spin transporta-tion in the QSH effect. Unlike (3+1)D, the boundary states are restrictedto only two directions of travel along the edge of a (2+1)D insulator. Notethat QSH states have no net electric charge flowing along the edge.123.1. The Z2 phasesT ψ¯ψT−1 = −ψ¯ψ is parity odd, the combination of them together is. Time-reversal symmetry forces the fermions to exist as degenerate Kramers pairswith oppositely-signed masses and spin. At the edge where the fermionicmodes are massless m = 0, the theory exhibits a global U(1)×SU(2Nf ) sym-metry. This SU(2Nf ) .= SU(2) symmetry allows for the massless fermionsto transform non-trivially from one another under the non-abelian flavourgroup. In the bulk where the mass is finite m 6= 0 (this could be nega-tive or positive), the global symmetry breaks down to U(Nf ) × U(Nf ) .=U(1) × U(1). These two U(1) groups represent Maxwell electromagnetismand R-symmetry in supersymmetric gauge theories. Moreover, under thelexicon of condensed matter, the R-symmetry represents the conservationof the z-component of spin. All fermions in this model carry the same U(1)Maxwell charge of qMaxwell = +1, but as for the U(1) R-charges, only halfof them carry qR = +1 with the remaining half carrying qR = −1.The classical description of our massive (2+1)D fermionic model is chi-rally invariant to axial rotations φ. At the quantum level though, chiralityis no longer a symmetry and the massive theory develops an anomalousquadratic AR ∧ dAMaxwell Chern-Simons term [19] of levelk = 12∑iqMaxwelli qRi sgn(mi) = sgn(m)Nf.= sgn(m) (3.2)from integrating out the Nf = 1 massive fermion pair, where AR andAMaxwell are the U(1) R and Maxwell gauge fields respectively, and the indexi runs through all fermionic fields. A quick summary of this Chern-Simonsterm can be found in Appendix A. What is important is that the topologi-cal invariant k = sgn(m) = ±1 is exactly the discrete Z2-valued topologicalinvariant we were expecting. The result of having the mass sign fully de-termine the topological phase is in agreement with what is allowed of thechiral rotation parameter φ: only φ = 0 and φ = pi rotations are permit-ted by time-reversal symmetry and these anomalously rotate the mass byexp(−iφ)|φ=0,pi to give the same m = ±M classification.1 For the remainderof this thesis, we will label the m > 0 phase as the trivial insulator, and them < 0 phase as the non-trivial insulator without any loss of generality.1Time-reversal symmetry demands φ = −φ (mod 2pi) to be satisfied.133.2. The strong coupling limit and fractionalization3.2 The strong coupling limit andfractionalizationWhat is less understood, but much more interesting, is the case whenfermionic interactions are turned on. One naturally relates back to the non-interacting case and asks questions like: Do the same topological phaseshold? If not, what do they become? How do strong correlations affect theedge physics? For the (2+1)D time-reversal invariant topological insulator,it has been proposed that the electrons inside of the bulk of the non-trivialinsulator fractionalize into partons at large Nc and strong coupling [23, 24].Implementing fractionalization involves breaking up each fermion into Ncpartonic fragments with each piece holding a 1/Nc fraction of the R andMaxwell charge of the fermion which for Nf = 1 gives an AR ∧ dAMaxwellChern-Simons term of levelkfrac =12∑iqMaxwellfrac,i qRfrac,i sgn(mi) =sgn(m)NfNc.= sgn(m)Nc. (3.3)This is the fractionalized version of our integral Chern-Simons term from(3.2); it takes the integral topological invariant k = ±1 and divides it byNc. Note that the fractional topological invariant is still Z2-valued anddependent on the mass sign – an interface will still be localized from a dis-continuous flip of the fermionic mass sign. The partons outside of the bulkof the non-trivial insulator are confined, but inside the bulk, the large Nclimit of the SU(Nc) gauge group adds in light matter that can drive thepartons to deconfine [28, 42]. Our holographic description of the topologicalinsulator incorporates fractionalization naturally because AdS/CFT exten-sively utilizes both the large Nc and large λ limits. In the next chapter wewill discuss how the corresponding Wess-Zumino terms on the gravity sidematches exactly with this fractionalized Chern-Simons term.3.3 A topological interfaceTo deform from one topological phase to the other, the sign of the mass mustflip regardless of whether the insulator is fractional or not. This necessarilylocalizes gapless modes simply by interpolation. If we put a trivial insulatorin contact with a non-trivial insulator at some spatial slice x = xint, theshared interface will be gapless and localized at x = xint; for if the interfacegap did not close, it would suggest that the two insulators in contact were143.3. A topological interfacenot of different topological classes to start off with. Spatially varying the realmass profile of a fermion m(x) so that it discontinuously switches signs atx = xint will do what we wish for. On the gravity side, this will correspondto the passing of a D5-brane through a D3-brane stack.15Chapter 4Gravity side set-upWe construct the gravity dual of the static interface of a (2+1)D holographictopological insulator infixed in a (3+1)D N = 4 super Yang-Mills theory byembedding a D5-brane into (AdS5 black hole) × S5 using the ideologies ofChapters 2 and 3. Our D5-brane embedding follows closely the embeddingsexplored in Refs [25, 26], and it was shown in Refs [26, 43] that our set-updoes in fact fractionalize the bulk fermions at large Nc and large λ. FromRef [26], the relevant Wess-Zumino termSWZ =NcNf4pi∫d2x∫dt(AR ∧ F +A ∧ FR)(4.1)has a coefficient that is equal to the level of the fractionalized Chern-Simons term kfrac = NcNf divided by 4pi (in their notation2). The exactmatching of the anomalous terms on both sides of the duality suggests thenaturalness of fermions to fractionalize in holographic topological insulators.As the anomalous terms are independent of the details of the D5-braneembedding, what remains is the construction of an interface that can supportthe QSH effect by spatially varying a fermionic mass profile m(x) so thatit has a root at some spatial slice x = xint. Additionally, we feed in ahomogenous chemical potential µ to the system via a U(1) Maxwell gaugefield on the D5-brane.4.1 The field theory to realizeN = 4 super Yang-Mills theory with a non-abelian SU(Nc) gauge groupwill serve as an appropriate (3+1)D background field for us to add our de-fect fermions into. As discussed in Chapter 2, an Nc stack of coincidentD3-branes will actualize this background. Although the statistical SU(Nc)gauge fields are gapless, they do not contribute to electronic transport be-cause they are electrically neutral – they act as the thermodynamic gauge2The notation of Ref [26] has kfrac = NcNf from defining the charge of the fermion tobe Nc so that each parton holds a charge of magnitude one (instead of 1/Nc).164.1. The field theory to realizeFigure 4.1: The D3-D5 brane system for a single Nf = 1 D5 probe brane.The Aµ on the D5-brane is a global U(Nf ) .= U(1) gauge field, and theA˜µ on the D3-brane stack is an internal SU(Nc) ‘phonon’ gauge field. Thestring connecting the D3 and D5 branes is the Dirac fermion ψ. Separatingthe D5-brane from the D3-branes gives mass to the fermion.‘phonons’ that make up the insulator. Conformal symmetry at large Ncpushes the phonons to deconfine our partons [44].Adding Nf D5 flavour branes to the D3-brane stack yields Nf pairsof fermions that are restricted to propagate on a (2+1)D hypersurface ofthe (3+1)D field theory in addition to the original N = 4 field content[45–47]. The (2+1)D defect comes from the D5-branes omitting one of theflat directions occupied by the D3-brane stack. Adding in Nf D5-branescouples Nf N = 2 fundamental hypermultiplets to N = 4 super Yang-Mills theory which halves the original number of supersymmetries. Extrascalars are inevitably added to the field content as part of introducing theN = 2 hypermultiplet but they can be safely ignored because of their non-contribution to the anomalous physics [26]. We illustrate our D3-D5 branepicture for Nf = 1 in Fig 4.1. The defect Dirac fermion is the string betweenthe D3 and D5 branes and it has a generated mass proportional to the spatialseparation of the branes.We localize the zero modes by considering the supersymmetric D3-D5brane intersection. This is illustrated in Fig 4.2 as a D5-brane passingthrough the D3-brane stack. Prior to the decoupling limit, both branesoccupy directions in the world volume as tabulated in Table 4.1. There is174.1. The field theory to realizeFigure 4.2: The supersymmetric D3-D5 brane intersection. By passing theD5-brane through the D3-brane stack in the transverse x9 direction, themass of the fermion can switch signs to realize a gapless interface. Positiveand negative mass embeddings correspond to which side of the D3-branestack the D5-brane is being pulled away from.Poincare´ invariance in the 012 directions and SO(3) rotational symmetryin the 456 directions of the D5-brane. Pulling the D5-brane away from theD3-brane stack in a common transverse direction, say x9, gives mass to thehypermultiplet by reducing the SO(3) symmetry to SO(2) ≈ U(1) in thetransverse 789 directions of both branes. After taking both the decouplingand strong coupling limits at finite temperature, the holographic picturegreatly simplifies to Nf D5-branes being embedded into (AdS5 black hole)×S5 geometry. Furthermore the gravitational backreaction of the D5-braneon the metric gµν can be neglected because of an Nf/Nc quenching of quarkloop corrections in the stress energy tensor Tµν [48, 49]. Therefore our D5-branes can be treated as probe branes in a fixed supergravity background.x0 x1 x2 x3 x4 x5 x6 x7 x8 x9D3 (background): • • • • ◦ ◦ ◦ ◦ ◦ ◦D5 (probe): • • • ◦ • • • ◦ ◦ ◦Table 4.1: Directions occupied by the branes in our D3-D5 system. The filledcircles • represent the directions occupied by the brane, and the unfilledcircles ◦ represent the orthogonal directions unoccupied by the brane in theworld volume. The D5-brane probes the D3-brane stack background.184.2. The D5-brane embedding4.2 The D5-brane embeddingOur D5-brane embedding occupies 6 directions of the possible 10 directionsof (AdS5 black hole) × S5. The embedding extends along a 4-dimensionalslice of the 5-dimensional AdS5 black hole by neglecting one of the threespatial directions, and wraps an S2 around the S5.The metricWe write the (AdS5 black hole)× S5 metric gµν asds2 = r2L2[−f(r)dt2 + d~x23]+ L2r2[ dr2f(r) + r2dΩ25], (4.2)d~x23 = dx2 + dy2 + dz2, (4.3)dΩ25 = dθ2 + cos2 θdΩˆ22 + sin2 θdΩ22, (4.4)f(r) = 1− r4hr4 , (4.5)where rh is the radial horizon of the AdS5 black hole, d~x23 is the metricfor 3-dimensional Euclidean space, and dΩ2m is the metric for the m-sphereSm. The first half of the metric is the AdS5 black hole written in globalcoordinates (t, x, y, z, r) with an AdS length of L, and the remaining half isa 5-sphere of radius L. One can convert from these coordinates to isotropiccoordinates with a ρ2 = r2 +√r4 − r4h transformation; this convention isused extensively in Refs [50, 51] for the D3-D7 system and it gives a metricof the formds2 = 12ρ2L2[−f(ρ)2h(ρ) dt2 + h(ρ)d~x23]+ L2ρ2[dρ2 + ρ2dΩ25], (4.6)f(ρ) = 1− r4hρ4 , (4.7)h(ρ) = 1 + r4hρ4 . (4.8)Assuming thermal equilibrium, the gauge theory temperature T can beidentified as the Hawking temperature through the elimination of the con-ical singularity at the horizon. Euclideanizing the metric with a τ = it194.2. The D5-brane embeddingWick rotation sets the inverse temperature β to be τ -periodic, which inturn gives a Hawking temperature of T = β−1 = rh/(piL2). The details ofthis calculation can be found in Appendix B.The embeddingWe choose our 5-sphere metric parameterization to bedΩ25 = dθ2 + cos2 θdΩˆ22 + sin2 θdΩ22, (4.9)for the convenience of wrapping an S2 around the S5 where dΩ22 =dψ2 + sin2 ψdφ2 is the S2 metric with the hat indicating which of two 2-spheres is the one being embedded. The parameter θ ∈[−pi2 , pi2]here controlsthe S2 ‘slipping’ mode around the S5.With the full metric (4.2) in hand, the D5-brane embedding in terms ofthe 10 world sheet coordinates Xµ can be catalogued asXµ = (t, r, x, y, z, θ(r, x), ψ1, ψ2, φ1, φ2) , (4.10)where θ ≡ θ(r, x) is our embedding parameter from metric (4.9). Forour purposes it is only dependent on the the radial coordinate r and thespatial coordinate x because of Poincare´ invariance and rotational symmetry.Additionally we set up a Maxwell U(1) gauge field Aµ to live on the worldvolume of the D5 brane asAµ = (At(r, x),~0), (4.11)with the only non-vanishing component being the temporal one. Includ-ing non-vanishing spatial components Ai into the gauge potential breakstime-reversal invariance in the action [52]. The gauge field At ≡ At(r, x)too depends only on r and x by symmetry. Our D5-brane has 6 embeddingcoordinatesξa = (t, r, x, y, ψ1, φ1) , (4.12)which give for an induced metric on the D5-brane (with L = 1 set tomake use of conformal symmetry) ofds2D5 = r2[−f(r)dt2 +(1 + (∂xθ)2r2)dx2 + dy2]+ 2(∂rθ)(∂xθ)drdx+( 1r2f(r) + (∂rθ)2)dr2 + cos2 θ(dψ21 + sin2 ψ1dφ21). (4.13)204.3. DBI action and equations of motionMoving the D5-brane through the D3-brane stack in a common trans-verse direction to switch mass signs is implicitly controlled by the embeddingparameter θ. Later on we will see that the mass profile is actually related tothe asymptotic expansion of the embedding function by θ(r →∞, x) ≈ m(x)r .4.3 DBI action and equations of motionGiven the embedding in terms of the {θ,At} bulk fields, we now proceedon with finding the brane action and its associated equations of motion.An appropriate action to use for the brane embedding in the decouplinglimit is the Dirac-Born-Infeld (DBI) action. It generalizes the Nambu-Gotoaction and recovers the Yang-Mills action in the weak α′ expansion. TheDBI action contains two types of D-brane excitations: the external rigidisometries and deformations of the brane geometry, and the internal worldvolume gauge fields sourced by open string ends. In general, the DBI actionfor Nf decoupled D5-branes takes the formSD5 = −NfTD5gs∫d6ξ√−det (γab + 2piα′Fab), (4.14)where γab is the induced (or pullback) metric (4.13) on the D5-branedefined by γab ≡ ∂Xµ∂ξa∂Xν∂ξb gµν with ξa being the embedding coordinates from(4.12), Fab is the pullback of the field tensor living on the world volume ofthe D5-brane, and TD5 is the tension of the D5 brane. The latin indicesa, b run over the 6 embedding coordinates. The determinant inside of thesquare root is responsible for mixing the tensor terms together to make fornon-trivial interaction dynamics.With a re-parameterization of the embedding field χ ≡ χ(r, x) = sin θ(r, x)and a convention of 2piα′ = 1, the DBI action (4.14) for our D5-brane canbe explicitly written asSD5 ∼∫drdx√r2(1− χ2) (Iχ + IAt + Imixed), (4.15)where we omit the factors of the integral that do not contribute to thefield equations. The terms inside of the square-root of (4.15) areIχ = r2(1− χ2) + r4f(r)(∂rχ)2 + (∂xχ)2, (4.16)IAt = −(1− χ2)(r2(∂rAt)2 +(∂xAt)2r2f(r)), (4.17)Imixed = − (∂rχ∂xAt − ∂xχ∂rAt)2 . (4.18)214.4. Time-reversal invarianceUsing Mathematica to minimize the embedding surface action (4.15)with respect to the fields χ(r, x) and At(r, x), we obtain two coupled second-order equations of motion. There are no Einstein equations to solve becauseof the fixed background metric. The exact form of our field equations areomitted here because of their length and cumbersomeness. We do notethough that our action and field equation solutions agree with the findingsof Refs [25, 26, 30, 31] and is of similar form to its D3-D7 analogue [29]. Itis the inclusion of spatial dependence that complicates the field equationsenough that numerical methods are needed to solve them.4.4 Time-reversal invarianceThe DBI action for our Nf D5 flavour probe branes in (4.14) is the DBIaction of a single probe brane times by Nf . This set-up holographicallyrealizes a Chern-Simons term of a level proportional to Nf , but this shouldnot be mistaken as a Z classification because of the freedom of Nf as aninteger. Rather, as mentioned in Ref [26], the even/odd parity of Nf isthe Z2-valued quantity protected by time-reversal symmetry. The reasoningbehind this is that the Nf pairs of massless fermions have a non-abelianSU(2Nf ) flavour symmetry group that transforms them non-trivially. It isalways possible to turn on non-trivial components of the SU(2Nf ) group tocreate time-reversal invariant mass terms that add or subtract multiples oftwo from Nf such that the even/odd parity of the resulting N ′f is preserved.This shows how our holographic description is at least representing a time-reversal invariant system.3 We set Nf = 1 for the remainder of our braneconstruction without any loss of generality.4.5 Classes of D-brane embeddingsWe take a moment now and classify what type of brane embeddings arepossible in our set-up. At finite temperature, there are three distinct classesof D-brane embeddings that have been well studied and they are namely theMinkowski, critical, and black hole embeddings [30, 31, 50, 51, 55]. Eachembedding class is defined by its behaviour near the black hole horizon whichdepends on the mass (controlled by θ), the chemical potential (controlled byAt), and the temperature. Sketches of each embedding class are in Fig 4.3.3Moreover in Refs [53, 54], the non-time-reversal invariant fields were projected outfrom the brane set-up and were found to give a Z2-valued D-brane charge.224.5. Classes of D-brane embeddings(a) Minkowski (b) black hole(c) criticalFigure 4.3: Schematic diagrams of the Minkowski, critical, and black holeembeddings. The Minkowski embedding (Fig 4.3a) smoothly terminatesoutside of the black hole and forms a mass gap. Minkowski embeddingsnecessarily correspond to zero density states. The black hole embedding(Fig 4.3b) closes the mass gap by having the brane falling through thehorizon. Black hole embeddings on the other hand can induce dense states.The critical embedding (Fig 4.3c) is the embedding intermediate to boththe Minkowski and the black hole embedding where the brane just pinchesoff from the black hole. We focus mainly on black hole embeddings in thisthesis.234.5. Classes of D-brane embeddingsMinkowski embeddingsMinkowski embeddings are the brane configurations where the probe D-brane does not reach the black hole horizon because either its surface ten-sion is too strong or the gravitational pull of the black hole is too weak. Theembedding ends smoothly at some finite distance rD5 > rh outside of theevent horizon leading to χ(r = rD5) = ±1, or equivalently θ(r = rD5) = ±pi2 ,as seen in Fig 4.3a. Minkowski embeddings are gapped, exhibit a discretemesonic spectrum, and necessarily correspond to states with zero density.If it was dense then it would imply electric field lines terminating abruptlybetween the D-brane and the black hole due to the physical separation.Minkowski embeddings are thermodynamically favoured at low tempera-tures with a low µ/M ratio. In the D3-D7 system, Minkowski embeddingsare stable whenever the chemical potential satisfies µ < µc = M at zerotemperature.Black hole embeddingsBlack hole embeddings on the other hand are the embeddings that doreach the black hole horizon. These states are gapless, exhibit a con-tinuous mesonic spectrum, and are typically dense. The probe D-braneacquires a value of χ(r = rh) ∈ (−1, 1) at the horizon, or equivalentlyθ(r = rh) ∈(−pi2 , pi2), as depicted in Fig 4.3b. For the types of black holeembeddings that we find, negative mass embeddings take on χ < 0, posi-tive mass embeddings take on χ > 0, and zero mass embeddings take onχ = 0 at the black hole horizon. Black hole embeddings are thermodynam-ically favoured at high temperatures or at least whenever the ratio µ/M issufficiently high enough. A high chemical potential favours the black holeembedding because it fills the vacuum up with a Fermi sea of baryons to givedensity to the state. For the D3-D7 system, black hole embeddings can beachieved at zero temperature if the chemical potential satisfies µ > µc = M .The chemical potential threshold µc lowers with temperature for both theD3-D5 and D3-D7 systems as a result of thermal fluctuations assisting withthe melting of mesons.Critical embeddingsCritical embeddings are the configurations where the probe D-brane ‘just’reaches the black hole horizon with χ(r = rh) = ±1, or equivalently θ(r =rh) = ±pi2 . They have zero density and are considered as the transitional244.6. Boundary conditionsstate between Minkowski and black hole embeddings. We do not considerthem in this thesis.4.6 Boundary conditionsIn AdS/CFT, the physical observables of the strongly-coupled field theoryare sourced in from the AdS boundary as boundary conditions, but thesealone do not make up for all the boundary conditions – we also need regular-ity conditions. We require conditions for both χ(r, x) and At(r, x) on each ofthe 4 boundaries of the two-dimensional [r, x] grid and these one-dimensionalboundaries are:(B1) r = [rh,∞) at negative spatial infinity x = −∞(B2) r = [rh,∞) at positive spatial infinity x = +∞(B3) x = (−∞,+∞) at the black hole horizon r = rh(B4) x = (−∞,+∞) at radial infinity r =∞The simplest boundary conditions are on (B1) and (B2) – the spatialboundaries x = ±∞ that extend along the radial direction. We apply ho-mogenous Neumann conditions to both fields, ∂xχ = 0 and ∂xAt = 0, onthese boundaries. This boundary condition is not a strict choice but it isconsistent with having a constant gauge field and mass profile spatially faraway from the interface.The boundary conditions on (B3) ensure the regularity of the field equa-tions at the horizon r = rh. At the horizon where dt → ∞, we require Atto vanish in order to keep Atdt as a well-defined 1-form gauge field witha finite norm [56, 57]. This is purely a physical condition; a near-horizonexpansion of the field equations can only constrain the derivatives of At andnot At itself. The horizon regularity condition on χ on the other hand canbe derived by expanding the field equations at r = rh and doing so gives anembedding condition of ∂rχ = χ.Our final boundary conditions, which are on (B4), feeds in the ultravi-olet field theory from the AdS boundary. Generally one needs to cancel outthe ultraviolet divergences of scalar quantities with the pertinent counter-terms to ensure a finite partition function [58], but these results are alreadywell known for many D3-Dq systems including ours so we simply state howour asymptotic fields match with the physical observables of the field theorydual [25, 50, 51, 59]. Expanding our field equations around r = ∞, theasymptotic behaviour of our two fields follow an ansatz of254.6. Boundary conditionsχ(r, x) = m(x)r +c(x)r2 + . . . , (4.19)At(r, x) = µ−nb(x)r + . . . . (4.20)The holographic dictionary states that the leading term in our asymp-totic expansion corresponds to a non-normalizable mode with an amplitudeproportional to the coefficient of a gauge theory operator O, and the sub-leading term corresponds to a normalizable mode with an amplitude pro-portional to the vacuum expectation of the operator 〈O〉.4For our embedding field χ, the dual operator is Om, the mass variation∂∂m of the super Yang-Mills Lagrangian. The leading term m(x) as a resultis proportional to the bare quark mass (or baryon mass). This quantityeffectively describes the minimum length of the D3-D5 string. The sub-leading term c(x) is assigned as the vacuum expectation 〈Om〉 which isproportional to the quark condensate.For our gauge field At, the dual operator is Oq, the quark charge density.The homogenous leading term µ is proportional to the chemical potentialin the grand canonical ensemble. The sub-leading term nb(x) is matchedwith 〈Oq〉 which is proportional to the quark number density (or baryonnumber density); this is a quantity related to the density of the D3-D5strings pulling the D5-brane towards the black hole. We will use ‘density’and ‘number density’ synonymously throughout this thesis.The set of parameters {m(x), c(x)} for χ are conjugate pairs, as with{µ, nb(x)} for At. Solving the field equations with only one parameter ofthe conjugate pair specified at the boundary will fully determine the otherconjugate variable as a ‘response’ from the bulk. We are free to choosewhich parameter of each conjugate pair to keep fixed, and therefore fix thechemical potential µ and the mass profile m(x) so that c(x) and nb(x) canbe read off from the behaviour of χ and At at the boundary.5 By inspectionof ansatz (4.19) and (4.20), the total dimensionality of each conjugate pairis 2 + 1 = 3 as expected from a (2+1)-dimensional Lagrangian [25].6 The4See the appendix of Ref [50] for more details.5In Refs [25, 50, 51], a Legendre transformation was performed on the action to find thenumber density nb to be a conserved quantity. This can reduce the spatially homogenoussecond-order field equation to first-order. Our spatially non-homogenous second-orderfield equations as far as we know do not allow for such a simplification unfortunately.Moreover, we find it more relevant to fix the chemical potential µ instead of nb.6The sub-leading terms c(x) and nb(x) are of one less dimension in (2+1)D than theyare in (3+1)D. The total dimensionality of the conjugate pairs adds up to 4 in (3+1)D.264.7. Scaling symmetryFigure 4.4: Boundary conditions on the horizon and boundary of the AdS5black hole. Not included here are the homogenous Neumann boundary con-ditions at the spatial boundaries x = ±∞. The horizon requires regularityconditions, and the boundary requires the specification of ultraviolet sourcesfrom the field theory. In our case, we feed in a mass profile m(x) and a chem-ical potential µ into the bulk from the boundary.final schematic of how our boundary conditions are imposed at r = rh andr =∞ is shown in Fig 4.4.4.7 Scaling symmetryOur DBI action (4.15) enjoys scaling symmetry. It is crucial to identify allscaling symmetries from our problem to not only remove any redundancies,but to work in dimensionless parameters. We find the following scalingtransformationr → ar, x→ x/a, χ→ χ, At → At/a, (4.21)to be a scaling symmetry, and we make use of it by setting the radialhorizon to unity so that our remaining physical degrees of freedom {m(x), µ}are in units of L, rh, and α′. Our theory is now fully characterized by twodimensionless parametersm˜(x) ≡ m(x)piT , µ˜ ≡µpiT , (4.22)274.8. Realizing a holographic topological interfacewhich are namely the dimensionless quark mass profile m˜(x), and thedimensionless chemical potential µ˜. The dimensionless conjugate pairs tothese variables arec˜(x) ≡ c(x)(piT )2 , n˜b(x) ≡nb(x)(piT )2 , (4.23)which are namely the dimensionless quark condensate c˜(x), and the di-mensionless quark density number n˜b(x). We will use tilde symbols to denotedimensionless variables in this thesis.4.8 Realizing a holographic topological interfaceVarying the mass m(x) along the spatial direction x so that it discontin-uously flips signs at x = xint will localize a domain wall at x = xint. Inpractice, we have to interpolate m(x) at x = xint with a steep but contin-uous function for the sake of numerical stability. We choose a mass profilem(x) of the formm(x) = 2M1 + e−ax −M, (4.24)with a mass parameter M > 0, and a steepness parameter a 1 chosensufficiently high to approximate the sudden mass sign flip at x = 0. Themass profile takes on values m(x) ≈ M > 0 (trivial insulator) when x & 0,and m(x) ≈ −M < 0 (non-trivial insulator) when x . 0. We find that ourresults are rather insensitive to the steepness parameter a, as it should befor sensible choices that are not too low nor too high.A gapped system is naturally described by a Minkowski embedding, anda gapless one by a black hole embedding. Ideally one would like to sandwich agapless system in between two distinct gapped phases by setting µ˜ < µ˜c(M˜)as shown in Fig 4.5a, but it is numerically difficult to incorporate bothMinkowski and black hole embeddings together as one solution; it is mucheasier to implement a black hole embedding that extends through all valuesof x by increasing the chemical potential so that µ˜ > µ˜c(M˜) as shown inFig 4.5b. The slight downside of this simplification is that it will unavoidablyinduce edge effects n˜b(x = ±∞) = d˜ > 0 outside of the interface as shownin Fig 4.6b, but the overall effect can be minimized by tuning µ˜ to be closerbut above µ˜c(M˜).284.8. Realizing a holographic topological interfacexint0xM˜M BH−M˜M(a) An ideal T > 0 interface.xint0xM˜BH−M˜d˜d˜(b) Our T > 0 interface.Figure 4.5: The ideal and our constructed mass profile m˜(x) at finite tem-perature T . In the ideal profile (Fig 4.5a), we have a gapless interface atx = 0 sandwiched between two gapped insulators of differing topologies.It is realized with both Minkowski (M) and black hole (BH) embeddings.In our construction (Fig 4.5b), we consider a mass profile m˜(x) that has afinite interface width and a chemical potential µ˜ & µ˜c(M˜) that makes theembedding a black hole embedding for all x.294.8. Realizing a holographic topological interfacexint0 xM BH Mdensityd˜(a) An ideal T ≥ 0 interfacexint0 xdensityBHd˜edgeeffects(b) Our T > 0 interfaceFigure 4.6: A sketch of the quark density nb(x) as a function of the spatialdirection x for the ideal and our constructed topological interface situated atx = xint. A finite density d˜ is induced as a consequence of our constructionbut its effect can be reduced by keeping µ˜ & µ˜(M˜).30Chapter 5Results and discussionWe solve our two field equations numerically on a two-dimensional Chebshevgrid using spectral methods in MATLAB for dimensionless parameters µ˜ andM˜ .7 The Chebyshev grid is prepared by compactifying the infinitely large[rmin, rmax] × [xmin, xmax] = [1,∞) × (−∞,+∞) grid with z = 1/r andy = tanh(x) coordinate transformations. Chebyshev differentiation matricesare used to evolve the initial guess solution until convergence is met via theNewton-Raphson method. Our criteria for convergence requires both fieldsχ and A˜t to have a normalized residual error of less than 10−8, and for thiserror to not change by more than 10−8 between the final two iterations.We present first the results of the phase diagram for a spatially homo-geneous mass profile m˜(x) = M˜ . Afterwards, we implement the spatiallynon-homogenous mass profile m˜(x) = 2M˜1+e−ax −M˜ from (4.24) and study thethermodynamics of a representative interface solution.5.1 The phase diagramThe spatially homogenous topological insulator with mass M˜ can be studiedby setting a flat mass profile m˜(x) = M . This is essentially the same as theone-dimensional case with no spatial dependence.8 We present the (T˜ , µ˜/M˜)phase diagram in Fig 5.1 by maneuvering through the parameter space ofblack hole embeddings with dimensionless temperature T˜ ≡ M˜−1 and ratioµ˜/M˜ . We find the low T˜ solutions to be fairly sensitive to the initial guesssolution so we obtain them by lowering the temperature adiabatically fromhigh T˜ solutions.The black hole embeddings with positive density nb > 0 occupy thephase region above the black curve in Fig 5.1, this implicitly makes thephase region below the curve the nb = 0 Minkowski embeddings. A chem-ical potential threshold µ˜c = µ˜c(T˜ ) must be overcome at low temperatures7See Ref [60] for more details on implementing spectral methods in MATLAB.8This is not exactly true because the boundary conditions at the horizon differ albeitslightly.315.1. The phase diagramFigure 5.1: The homogenous (T˜ ≡ 1/M˜, µ˜/M˜) phase diagram. There aretwo types of embeddings: the nb > 0 black hole embeddings and (implicitly)the nb = 0 Minkowski embeddings. A chemical potential threshold µ˜c > 0must be overcome at temperatures below T˜c ≈ 0.87 to favour the blackhole embedding. However, above the critical temperature T˜c, black holeembeddings are favoured regardless of how much chemical potential is fedin. The blue star on the left of the phase diagram are the parameters weuse for our representative topological interface.325.2. The holographic topological interfacein order for black hole embeddings to be thermodynamically favoured – en-tropic excitations alone are not enough to pull the probe brane into the blackhole. The chemical potential threshold ratio µ˜c/M˜ (the black curve) lowersas we raise the temperature and vanishes at a critical temperature T˜c ≈ 0.87;finite density black hole embeddings are stable past this critical tempera-ture for any µ˜ > 0. The (T˜ > T˜c, µ˜/M˜ = 0) line is where the black holeembeddings have zero density nb = 0 and are in fact the high-temperaturelimits of the Minkowski embeddings.There are strong qualitative similarities between the D3-D5 phase di-agram and its D3-D7 counterpart [50, 61]. The high temperature phasetransition between the Minkowski embedding and the black hole embeddingis of first-order with discontinuous jumps in thermodynamical quantitiesi.e. the charge condensate and the entropic density. At low temperatureshowever, the system develops a critical point and the Minkowski/black holephase transition disappears to allow for both phases to co-exist [25].5.2 The holographic topological interfaceNow we move on with a spatially non-homogenous mass profile m˜(x) =2M˜1+e−ax − M˜ to localize an interface at x = 0. The representative low tem-perature numerical solution we use takes on the dimensionless parameters(µ˜ = 2.000, M˜ = 2.954) as indicated by the blue star in Fig 5.1. The holo-graphic interface was solved on a 40 × 40 Chebyshev grid and the χ(z, x)and A˜t(z, x) solutions are plotted in Fig 5.2 and Fig 5.3, respectively. Theboundary behaviour is consistent with our previous analyses: Neumannboundary conditions ∂xχ = ∂xA˜t = 0 at the spatial infinities x = ±∞,χ ≈ m˜(x)z + c˜(x)z2 and A˜t ≈ µ˜ − n˜b(x)z near the radial boundary z = 0,and |χ| < 1 and A˜t ≈ 0 at the radial horizon z = 1.The embedding field χ(z, x) in Fig 5.2 is an x-odd function, χ(z, x) =−χ(z,−x), as with m˜(x) about the interface. At each spatial point x, χmonotonically decreases in magnitude from the boundary to the horizonand has the same sign as m˜(x). At m˜(x) = 0, the D5-brane splits the twotopologies and is completely flat. The geometry of the brane embeddingscan be visualized in isotropic coordinates as plotted in Fig 5.4. The D5-brane falls into the black hole horizon at all spatial points, with oppositemass sign embeddings doing so on opposite sides of the black hole about theχ = 0 equator.Our gauge field A˜t(z, x) solution in Fig 5.3 is rather simple to understand:chemical potential gets fed into the system from the boundary and the gauge335.2. The holographic topological interfaceFigure 5.2: The numerical solution for χ(z, x) at low temperature T˜ (µ˜ =2.000, M˜ = 2.954) = 0.339. The interface is at x = 0. The mass is positivefor x > 0 (red curves in Fig 5.4), and negative for x < 0 (blue curves inFig 5.4). The asymptotic behaviour goes as χ ≈ m˜(x)z + c˜(x)z2 at theboundary and achieves an absolute value of less than one at the horizon.Figure 5.3: The numerical low temperature solution for A˜t(z, x) with T˜ (µ˜ =2.000, M˜ = 2.954) = 0.339. The interface is at x = 0. The dip in the gaugefield near x = 0 is a detection of higher density at the interface, which is mostlikely to be fermionic from our thermodynamic analyses. The asymptoticbehaviour goes as A˜t ≈ µ˜ − n˜b(x)z at the boundary and vanishes at thehorizon.345.2. The holographic topological interfaceFigure 5.4: The probe D5-brane embeddings on the ρx-ρy plane where ρx =ρ cos θ = ρ√1− χ2, ρy = ρ sin θ = ρχ, and ρ2 = r2 +√r4 − 1. Eachcurve represents a spatial grid point x with red, black, and blue curvescorresponding to positive, zero, and negative mass black hole embeddings,respectively.355.3. Thermal properties of the interfacefield monotonically decreases from µ˜ to zero upon arrival at the horizon.We find also that the gauge field is x-even, A˜t(z, x) = A˜t(z,−x), about theinterface. Most interestingly there is a significant dip of A˜t at the interfaceprovoked by the sub-leading term n˜b(x). The deeper the dipping of A˜taround x = 0, the more pronounced the interface density n˜b(x = 0) isrelative to the background density n˜b(x→∞).5.3 Thermal properties of the interfaceThe quark condensate c˜(x) and density n˜b(x) can be read off as the sub-leading terms of χ and A˜t at the boundary – they are the bulk responsesof fixing m˜(x) and µ˜ as sources. We fit third-order polynomials p(x)fit =p0(x) + p1(x)z + p2(x)z2 + p3(x)z3 near the z = 0 boundary at each spatialgrid point x, and collect the series coefficients c˜(x) = p2(x)χ for χ, andn˜b(x) = −p1(x)A˜t for A˜t.9 The plots for our fitted c˜(x) and n˜b(x) across theinterface are given in Fig 5.5 and Fig 5.6, respectively.Figure 5.5: The condensate c˜(x) across the x = 0 interface as a function ofthe spatial direction x. The red points are the actual spatial grid points,and the black curve interpolates between them.9Sanity checks of m˜(x) = p1(x)χ and µ˜ = p0(x)A˜t were also made.365.3. Thermal properties of the interfaceFigure 5.6: The quark density n˜b(x) across the x = 0 interface as a functionof the spatial direction x. The red points are the actual spatial grid points,and the black curve interpolates between them.Let us analyze the condensate c˜(x) from Fig 5.5 first. Our x-odd massprofile makes c˜(x) vanish at x = 0 by parity symmetry. Therefore chiralsymmetry is preserved at the interface implying the existence of non-trivialfluid behaviour. Moreover we have chiral symmetry broken c˜(x) 6= 0 every-where outside of x = 0. As with Ref [25], we find that c˜(x) asymptotes to asmall but finite constant at large masses. The large condensate peaks foundnear the vicinity of x = 0 are numerical artifacts of having an interfacewith finite width. What is important from the plot is that c˜(x) = 0 at theinterface and c˜(x) 6= 0 everywhere else.Just as A˜t is an x-even function, the quark density n˜b(x) from Fig 5.6is also an x-even function with a density peak at x = 0. The density risesmonotonically from the the spatial boundaries to the interface. The ratioof the interface density n˜b(x = 0) to the background density n˜b(x → ±∞)can be increased by approaching the Minkowski/black hole transition atlow temperatures; doing so lessens the significance of edge effects, but hasthe trade-off of being more computationally demanding because of the lowtemperature limit for a finite temperature problem.More can be said about the interface by considering the interface den-sity n˜b(x = 0) as a function of chemical potential. Due to our theory beingfully characterized by the parameters µ˜ and m˜(x), the physics at the m˜ = 0375.3. Thermal properties of the interfaceinterface will depend only on µ˜. Therefore the leading-behaviour of the di-mensionless free energy at the interface is restricted by dimensional analysisto be of the form F˜ = F˜(µ˜). For the fractional (3+1)D topological insulatorstudied in Ref [29], the leading behaviour of F˜ at the interface was found tomatch with the free energy of non-interacting massless relativistic fermionsF˜3+1(µ˜) ∼ µ˜4 at low temperatures [62]. We can expect our (2+1)D topo-logical insulator by analogy to have an interface free energy scaling of oneless dimension F˜2+1(µ˜) ∼ µ˜3, and thus a density scaling of n˜b,2+1(µ˜) ∼ µ˜2at low temperatures (high µ˜). We plot in Fig 5.7 how our interface den-sity scales as a function of chemical potential with M˜ set to some constant.The ansatz n˜b ∼ µ˜α is fitted at each µ˜ by computing the scaling exponentα(µ˜) = ∂∂ log(µ˜) log(n˜b(µ˜)). We find an α ≈ 1.96 plateau at high µ˜, whichis close to the theoretical α = 2 scaling we would expect for free masslessrelativistic fermions.Having α not being exactly equal to 2 at high µ˜ suggests possible sub-leading contributions. These sub-leading contributions can reveal the natureof the interactions i.e. are they bosonic or fermionic? This is the essenceof the hyperscaling violation exponent γ, it controls the scaling of the sub-leading terms in the number density by n˜b ∼ µ˜2 + cγµ˜2−γ for constant cγ .Degenerate fermions are assigned with γ = 2 and bosons with γ = 0. If theinterface interactions are indeed Fermi surface like, then we would expectour density to scale by n˜b ∼ µ˜2 + cγ=2 for non-zero cγ=2 at high µ˜. We fitn˜b ∼ µ˜2 + cγ=2 to our data at high µ˜ and find that cγ=2 is in fact non-zerowith more than 99% confidence to suggest that the interface interactions arefermionic.10The last piece of evidence we present here is the fact that the interfaceis almost compressible [63]: it has translational invariance, a global U(1)symmetry, and a density n˜b that is smooth with µ˜. What is left to confirmcompressibility is that these conditions remain true in the zero temperatureground state. In order for this to happen the global U(1) gauge field mustpersist, and the system cannot break translational invariance by crystalliz-ing. If this can be shown, we are guaranteed that our interface is a Fermisurface as a corollary of being compressible.10Note that the bosonic hyperscaling violation exponent γ = 0 adds towards the overallleading behaviour of n˜b ∼ µ˜2 + cγ=0µ˜2 ∼ µ˜2, so the case of purely bosonic interactions isignored in our analysis.385.3. Thermal properties of the interface−2 −1.5 −1 −0.5 0 0.5 1 1.5−2−1.5−1−0.500.511.522.53log(µ˜)log(n˜ b) −2 −1 0 111.21.41.61.82αlog(µ˜)α ≈ 1α ≈ 2Figure 5.7: The scaling behaviour of the interface density, n˜b(x = 0), as afunction of the chemical potential. The inset plot describes how the scalingexponent α from n˜b ∼ µ˜α varies with the chemical potential. At high tem-peratures (low µ˜) the scaling goes as n˜b ∼ µ˜, and at low temperatures (highµ˜) the scaling goes as n˜b ∼ µ˜2.39Chapter 6ConclusionIn this thesis, we realized the static holographic interface of a fractional(2+1)D time-reversal invariant topological insulator by embedding a D5probe brane into (AdS5 black hole) × S5 with a spatially non-homogenousmass profile m(x) that switches signs at x = 0. The field equations wereobtained by minimizing the DBI action of the embedding probe D5 branewith respect to the embedding parameter χ(r, x) and the U(1) Maxwellgauge field At(r, x), and these in turn were numerically solved using spec-tral methods on a two-dimensional Chebyshev grid. Our theory is fullycharacterized by two parameters: the dimensionless chemical potential µ˜,and the dimensionless fermionic bulk mass M˜ .We considered first the spatially homogenous mass profile m˜(x) = M˜ andconstructed its (T˜ ≡ 1/M˜, µ˜/M˜) phase diagram. Our D3-D5 phase diagramis qualitatively similar to its D3-D7 counterpart. Minkowski embeddings arethermodynamically favoured when the chemical potential µ˜ is below a finitecritical threshold µ˜c, but black hole embeddings are favoured once µ˜ exceedsthis threshold. At higher temperatures where T˜ > T˜c ≈ 0.87, µ˜c vanishesand finite density black hole embeddings are favoured for any given µ˜ > 0.Next we studied the holographic topological interface of a representa-tive low temperature phase point on our D3-D5 phase diagram close tothe Minkowski/black hole phase transition. By applying the spatially non-homogenous mass profile m˜(x) ≡ 2M˜1+e−ax − M˜ as a boundary condition andsolving the corresponding field equations, we successfully realized a holo-graphic topological interface. Furthermore we studied the thermodynamicsof said interface by computing its n˜b(x = 0) scaling behaviour and foundn˜b ∼ µ˜2 at high µ˜ as expected of free massless relativistic electrons in (2+1)dimensions. The interface also exhibits a fermionic hyperscaling violationexponent. We can be guaranteed a Fermi surface by compressibility if theground state preserves translational invariance, a global U(1) gauge field,and a smooth density n˜b(µ˜).40Chapter 7Prospective researchHolography can provide a qualitative phenomenological guide on what tolook out for in strongly-interacting materials. We provide here some di-rect and indirect avenues of research to further our study on holographictopological insulators.7.1 Zero temperature solutionPhysically, the discreteness of the protected symmetries in topological insu-lators strongly suppresses small angle scattering at the boundary. Unfortu-nately the strength of this topological protection can get washed away bythermal fluctuations. The features of a degenerate Fermi gas are expectedbe sharper at zero temperature. Moving to zero temperature requires oneto embed the D5-brane into AdS5×S5 instead of (AdS5 black hole)×S5 aswe did so in this thesis. The difficulty with solving the corresponding zerotemperature field equations comes from keeping the equations regular atthe degenerate double pole horizon r = 0; it is simpler to deal with at finitetemperature because the pole at the horizon r = rh > 0 is non-degenerate.A zero temperature solution can also be used to confirm compressibility asdiscussed before. Computing the entanglement entropy and retarded Greensfunction of a zero temperature interface would also be interesting [64].7.2 Integrating nb(x) over the interfaceThe density nb(x) represents a number density per unit area. Integratingnb(x) over the spatial x direction as∫nb(x)dx gives the density per unitlength in the spatial y direction. Furthering our thermodynamical studyby considering also the scaling behaviour of∫nb(x)dx as a function of µis potentially significant experimentally. The difficulty in performing sucha calculation lies in defining the spatial width of the interface [xmin, xmax]because it is dynamically controlled by the interface width parameter a 1which we do not yet know the explicit dependence of. It would be useful417.3. Turning on an external magnetic fieldFigure 7.1: Our candidate interface width definition (cyan-filled area) on annb(x) plot. The red dotted line represents the background density, and theblue dashed lines represent the density range of the interface width.to know this dependence. Regardless, an example of a candidate interfacewidth definition could be the maximum interval that satisfiesnb(x) > nb,background + 0.05 (nb,max − nb,background) (7.1)everywhere in the interval. Here nb,max = nb(x = 0) is the interfacedensity and nb,background = nb(x = ±∞) is the background density. Thisparticular example defines the interface width as the maximum interval thathas nb above the background density plus 5% of the background-subtracteddensity peak (nb,max − nb,background). We illustrate this interface width defi-nition in Fig 7.1.7.3 Turning on an external magnetic fieldOur D3-D5 system has a phase diagram that has been well studied in thepresence of a constant external magnetic field [32, 65]. When the mass iszero, it is possible for a quantum phase transition to exist in the presenceof an external magnetic field with the order parameter being the chiralcondensate c. Turning up the magnetic field smoothly and studying how427.4. Searching for the Majorana fermionit breaks away the topological protection of the interface states could beinteresting.7.4 Searching for the Majorana fermionThe Majorana fermion is a fermion which is its own antiparticle. 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Rev. B, vol. 85,p. 035121, Jan. 2012.[64] B. Swingle, L. Huijse, and S. Sachdev, “Entanglement entropy of com-pressible holographic matter: loop corrections from bulk fermions,”ArXiv e-prints, Aug. 2013.[65] K. Jensen, A. Karch, D. T. Son, and E. G. Thompson, “HolographicBerezinskii-Kosterlitz-Thouless Transitions,” Physical Review Letters,vol. 105, p. 041601, July 2010.49Appendix AChern-Simons terms of theinteger QSH stateIt is only in (2+1)-dimensions that Chern-Simons theory is quadratic inthe gauge field. The Aa ∧ dAb Chern-Simons terms in our theory has aChern-Simons term of levelk = 12∑iqai qbi sgn(mi), (A.1)where the index i runs through all the fermionic fields, the latin indicesa, b run through our two U(1) Maxwell and R symmetries, q is the fermioniccharge, and sgn(mi) is the sign of the fermionic mass. The Maxwell and Rcharges can take values of either ±1. There are in general qMaxwelli qMaxwelliand qRi qRi contributions to the summation of (A.1) associated with theAMaxwell ∧ dAMaxwell and AR ∧ dAR Chern-Simons terms, but these posi-tive charge-squared (qa)2 = 1 terms get cancelled out from the calculationbecause the fermion pairs have oppositely-signed masses. Be that as it maythe AR∧dAMaxwell Chern-Simons term is the mixed term that remains non-zero because even though the Maxwell charges of a fermion pair are thesame, both the R-charges and masses are of opposite signs. This leads tothe integer QSH effect with an anomalous Chern-Simons term of levelk = 12∑iqMaxwelli qRi sgn(mi) (A.2)= sgn(m)Nf . (A.3)These arguments are directly transferrable to the fractional case withthe only difference being the magnitude of the charges.50Appendix BCalculation of the AdS blackhole Hawking temperatureThe simplest and most typical way of calculating the Hawking temperatureof the AdS black hole is by Euclideanizing it and removing the conical sin-gularity at the horizon by identifying the period of τ = it with its inversetemperature β = T−1. Physically, what this does is it implicitly matchesthe trace of the Euclideanized quantum mechanical evolution operator e−τHwith the thermodynamic partition function Z = tr[e−βH]. Starting withthe AdS5 black hole metricds2 = − r2L2 f(r)dt2 + L2r2dr2f(r) +r2L2d~x23, (B.1)f(r) = 1− r4hr4 , (B.2)we Wick rotate the metric with τ = it and expand it just outside of thehorizon r = rh +  to getds2 ≈ 4rhL2 dτ2 + L24rhd2 + r2L2d~x23 (B.3)from r2f(r) ≈ 4rh+O(2) near the horizon. To realize the conical sin-gularity explicitly, we move into spherical coordinates using transformationsρ2 = L2rh and χ =2rhτL2 to get the first two components of our metric (radialand temporal) to beds22 = ρ2dχ2 + dρ2. (B.4)Eliminating the conical singularity requires χ to have a period of 2piand thus requires τ to have a period of piL2rh . Matching this with β gives aHawking temperature ofT ≡ β−1 = rhpiL2 . (B.5)51

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