How spectrum-wide quantum criticality protects surface states of topological superconductors from Anderson localization: Quantum Hall plateau transitions (almost) all the way down
HHow spectrum-wide quantum criticality protects surface states oftopological superconductors from Anderson localization:Quantum Hall plateau transitions (almost) all the way down
Jonas F. Karcher
Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, GermanyInstitut f¨ur Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
Matthew S. Foster
Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USARice Center for Quantum Materials, Rice University, Houston, Texas 77005, USA
Abstract
We review recent numerical studies of two-dimensional (2D) Dirac fermion theories that exhibitan unusual mechanism of topological protection against Anderson localization. These describesurface-state quasiparticles of time-reversal invariant, three-dimensional (3D) topological super-conductors (TSCs), subject to the e ff ects of quenched disorder. Numerics reveal a surprisingconnection between 3D TSCs in classes AIII, CI, and DIII, and 2D quantum Hall e ff ects inclasses A, C, and D. Conventional arguments derived from the non-linear σ -model picture implythat most TSC surface states should Anderson localize for arbitrarily weak disorder (CI, AIII),or exhibit weak antilocalizing behavior (DIII). The numerical studies reviewed here instead indi-cate spectrum-wide surface quantum criticality , characterized by robust eigenstate multifractalitythroughout the surface-state energy spectrum. In other words, there is an “energy stack” of criti-cal wave functions. For class AIII, multifractal eigenstate and conductance analysis reveals iden-tical statistics for states throughout the stack, consistent with the class A integer quantum-Hallplateau transition (QHPT). Class CI TSCs exhibit surface stacks of class C spin QHPT states.Critical stacking of a third kind, possibly associated to the class D thermal QHPT, is identified for nematic velocity disorder of a single Majorana cone in class DIII. The Dirac theories studied herecan be represented as perturbed 2D Wess-Zumino-Novikov-Witten sigma models; the numericalresults link these to Pruisken models with the topological angle ϑ = π . Beyond applications toTSCs, all three stacked Dirac theories (CI, AIII, DIII) naturally arise in the e ff ective descriptionof dirty d -wave quasiparticles, relevant to the high- T c cuprates. Keywords:
Topological Superconductor, Quantum Hall, Multifractality, Disordered systems,Kubo conductivity
Contents1 Introduction 2
Preprint submitted to Annals of Physics January 25, 2021 a r X i v : . [ c ond - m a t . d i s - nn ] J a n Fundamentals 5 n ), Sp(4 n ), and O(2 n ) . . . . . 92.4 Class A, C, and D quantum Hall criticality . . . . . . . . . . . . . . . . . . . . . 11 T c cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Multifractal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Topological phases of non-interacting fermions are classified according to the “10-fold way”[1, 2, 3, 4]. The same scheme (also referred to as the Altland-Zirnbauer or Cartan classification)applies to a seemingly unrelated problem, that of the Anderson (de)localization in the presence ofquenched disorder [5, 6, 7]. In fact, topology and disorder are closely intertwined in condensedmatter physics. In both cases, one seeks to characterize not the details of a particular band struc-ture or disorder configuration, but the physics that robustly persists under smooth deformationsof the Hamiltonian that preserve defining symmetries.In each spatial dimension, five of the ten classes admit topologically nontrivial phases. Threeof the five topological classes are characterized by an integer-valued winding number ν ∈ Z [2, 3]; the other two classes in each spatial dimension have Z invariants. In two dimensions(2D), the three classes correspond to three di ff erent versions of the integer (non-interacting)quantum Hall e ff ect. These are the charge, spin, and thermal quantum Hall e ff ects in classes A,C, and D; the latter two arise in theories of 2D d + id and p + ip topological superconductors(TSCs). In three dimensions (3D), the topological classes with winding numbers ν ∈ Z candescribe time-reversal invariant TSCs [1]. The three TSC classes are distinguished by the degreeof spin symmetry preserved in every quasiparticle band structure or disorder realization; these areU(1), SU(2), and no spin symmetry for classes AIII, CI, and DIII, respectively. Although TSCshave yet to be conclusively identified in nature, fermionic topological superfluids in classes Aand DIII are believed to be realized in thin-film He- A and bulk He- B , respectively [1, 8, 9].2opologically nontrivial phases host gapless edge or surface states at the sample boundary[10, 11, 12] that are robust to local perturbations. In particular they should be protected fromAnderson localization [1, 13, 14]. The topological protection for 1D and 2D boundary modesis in conflict with the natural tendency of low-dimensional states to localize in the presence ofarbitrarily weak disorder [7, 15]. For 1D chiral or helical edge modes, the route of escape isthat elastic backscattering is strictly prohibited [10, 11, 16, 17]. Surfaces o ff er a richer varietyof possibilities, where topological bands often feature massless 2D Dirac or Majorana fermions.The suppression of pure backscattering for the single 2D Dirac fermion cone is insu ffi cient toprevent quantum interference. Without the restriction to 1D (only forward and backward) in awire, elastic impurity scattering can still occur at all other angles. In order to resolve the puzzle inthis case, it is necessary to use more technical tools like the nonlinear sigma model to gain furtherinsight. For the simplest 3D topological insulator (TI), one finds protection of the 2D surfacestates from localization throughout the entire bulk energy gap. This is understood as due to weakantilocalization enabled by strong spin-orbit coupling, and the presence of a Z topological termthat nullifies the metal-insulator transition in the symplectic class [7, 18, 19, 20, 21, 22].The 2D surface states of bulk 3D TSCs in classes AIII, CI, and DIII are typically predictedto appear as massless Dirac or Majorana fermions. Di ff erent from graphene or TI surface states,time-reversal invariant quenched disorder enters into these surface theories in a peculiar way.Due to the “fractionalization” of the Hilbert space associated to confinement at the sample bound-ary and the natural particle-hole symmetry present in a superconductor, 2D Dirac TSC surfacetheories admit only quenched gauge-field disorder [1, 6, 23]. In classes AIII and CI, minimalrealizations involve U(1) and SU(2) vector potentials. The minimal realization of a class DIIIsurface consists of a single Majorana cone; in this case, disorder can only modulate the velocitycomponents of the cone. Since it couples to the stress tensor, we call this “quenched gravitationaldisorder” (QGD) [24]. Although class CI and AIII 2D Dirac models with gauge disorder couldbe robustly realized as TSC surface states, these were originally studied two decades ago in thecontext of the high- T c cuprate superconductors [25]. Indeed, by suppressing interpair, internode,and / or intranode elastic impurity scattering in a 2D d -wave superconductor, one can realize allthree minimal surface models in classes CI [26, 27, 28, 29, 30], AIII [31, 32], and DIII [24].In this paper, we review the recent numerical evidence of Refs. [24, 33, 34], indicating thatthe class AIII, CI, and DIII Dirac surface theories evade Anderson localization via a highly un-usual mechanism. These 2D Dirac models exhibit a “stack” of critical states at finite energies,see Fig. 1. The statistics of these states at di ff erent energies (away from zero) are identical. Inparticular, the multifractal spectrum of wave function fluctuations and the distribution of the Lan-dauer conductance for finite-energy class AIII Dirac surface states appear to match the universalvalues associated to the quantum Hall plateau transition (QHPT) in class A [33]. This is sur-prising for a number of reasons. First, the critical state associated to the QHPT typically obtainsonly with fine-tuning of the magnetic field or particle density. This is because the QHPT is aquantum phase transition separating topologically distinct plateaux. Instead, at the surface of aTSC, every finite-energy state appears to feature its own plateau transition. Second, the quantumHall e ff ect lacks time-reversal symmetry (TRS), yet the findings in Ref. [33] show an energy-stacking of QHPT states without TRS breaking. For the TSC with full spin SU(2) symmetry(class CI), the finite-energy surface states [34] mimic the class C spin QHPT phenomenologyprecisely [7, 35, 36, 37, 38, 39, 40, 41]. Finally, for the minimal realization of a class DIII sur-face with QGD, stacking occurs for a new class of wave function quantum criticality. This ishypothesized to be related to the thermal QHPT in class D [42, 43, 44, 45, 46, 47, 48, 49].The 2D Dirac surface models studied here are equivalent to Wess-Zumino-Novikov-Witten3 igure 1: “Stacked” quantum criticality at the surface of a bulk topological superconductor (TSC). Panel (a) depicts clas-sical geometric critical phenomena in 2D, as can occur when fluid floods a landscape. Criticality arises at the percolationthreshold (middle), where fine-tuning of the fluid level makes travel across the landscape equally di ffi cult by land or bysea. By contrast, at the surface of a bulk TSC with quenched disorder that preserves time-reversal symmetry, the numer-ical studies [24, 33, 34] reviewed in this paper demonstrate a “stacking” of critical eigenstates throughout the surfaceenergy spectrum, schematically indicated in (b). Panel (c) depicts position-space probability density maps for dirty TSC2D surface eigenstates, as could be measured in the local density of states probed by scanning tunneling microscopy(STM). Eigenstates at di ff erent energies ε are shown for a single class DIII surface Majorana cone, subject to a particularrealization of nematic quenched disorder in the components of its velocity (nematic “quenched gravitational disorder”[24]). Eigenenergies ε are measured in units of the momentum cuto ff Λ (with average Fermi velocity set equal to one),and lengths are measured in units of 2 π/ Λ . While low-energy states are plane-wave like in this case, states with energies0 . < ε < . ε and of the disorder strength, forming a “stack” of quantum-critical states. Importantly,evidence for Anderson localization is observed only at high energies, well above the ultraviolet cuto ff for TSC surfacestates in all three classes CI, AIII, and DIII; stacking statistics improve for increasing system sizes and disorder strengths[24, 33, 34]. Results are obtained by exact diagonalization of the continuum Dirac theory with periodic boundary con-ditions, defined in momentum space (so as to avoid fermion doubling) [24]. The “stacked” critical states for class AIIIand CI TSC surface states match the known critical statistics of the class A charge and class C spin quantum Hall plateautransitions, respectively [33, 34]. [Since the class C transition shares a few exactly known critical exponents with 2Dclassical percolation [36], one can say that the stacking in class CI realizes critical percolation without fine-tuning [34],as sketched in panel (b).] The finite-energy critical fluctuations observed in class DIII, shown in (c), may correspond tothe thermal quantum Hall plateau transition in class D. (WZNW) nonlinear sigma models [6], modified by the addition of the nonzero quasiparticle en-ergy. The latter couples to the trace of the principal chiral field, a strongly relevant perturbation.At zero energy (the surface Dirac point), these models are also quantum critical, and have beenlong understood thanks to the exact solution via conformal field theory [23, 25, 26, 28, 29, 30, 31,50]. By contrast, there is very little known analytically of the finite-energy behaviour in the per-turbed WZNW models. Ludwig et al . [31] investigated the minimal single-node class AIII Diracmodel (corresponding to the surface of a class AIII TSC with winding number ν = et al . [21]. They showed that agradient expansion yields the Pruisken model that describes the integer quantum Hall e ff ect. Forodd winding numbers ν , the Pruisken model has a theta term with topological angle ϑ = π , corre-sponding to the class A QHPT. This result was confirmed for ν = ν , it predicts Anderson localization for even ν (despite the Z classification for class AIII) [13, 21, 22]. There is no indication of this even / odde ff ect numerically [33], and both ν = , nematic QGD matches the phenomenologyobserved in STM studies of the high- T c cuprate superconductor BSCCO [59, 60, 61, 62, 63, 64],see Sec. 5.1. The field of experimental studies of disordered superconductors is very rich byitself. Many theoretical scenarios for the disorder-driven superconductor-to-insulator transitioninvolve enhanced Cooper pairing, due to multifractal rarification [65, 66, 67, 68, 69]. Reportingan increase in T c with increasing disorder, Ref. [70] recently added experimental support for this.Furthermore there are indications that multifractal superconductor physics provides an adequatedescription of transition metal dichalcogenides [71, 72, 73]. It is also interesting to note thatthe chiral model for twisted bilayer graphene is e ff ectively described by a class CI surface DiracHamiltonian [74, 75, 76]. However, the most prominent and well-studied experiments revolvearound the mystery of the spatial inhomogeneity in the high- T c cuprate superconductors [64];the results reviewed here call for a re-evaluation of the role of disorder in these materials. In this paper, we first give a brief overview about topological surface theories. We startwith the topological classification and the corresponding σ models (see Table 1), including theconventional expectations for the finite-energy behaviour. Next we review the key results ofWZNW theory relevant for zero-energy states of time-reversal invariant superconducting classes.We explain the role of the energy perturbation and how the modified WZNW model can bedeformed “by hand” into the Pruisken model.The main part is organized as a review of the most important numerical results for the classAIII, CI, and DIII surface theories. Each section about these theories covers the multifractalspectra, and for class AIII the Landauer conductance distribution. New results in this reviewinclude Kubo conductivity computations for classes AIII and CI, as well as larger system sizesand a finite-size analysis for class CI, winding number ν =
2. Fundamentals
We consider 2D surface theories of 3D bulk, time reversal ( T )-invariant topological super-conductors (TSCs) with di ff erent degrees of spin symmetry. These reside in classes AIII, CI,or DIII, as indicated in Table 1. For any superconducting realization of a class, physical time-reversal symmetry T corresponds to the e ff ective chiral symmetry S in this table. This symmetrytransmutation is due to the “automatic” particle-hole invariance associated to the self-conjugate(Balian-Werthammer) spinor formulation of any Bogoliubov-de Gennes Hamiltonian [1, 77].These bulk phases can be topologically non-trivial, and are indexed with integer-valued windingnumbers.The form of the 2D surface band structure for a clean topological phase in general dependsupon some details of the bulk and of the surface orientation. A large class of TSC surface statesin classes AIII, CI, and DIII take the form of massless Dirac or Majorana fermions. This hasbeen demonstrated using bulk lattice models in (e.g.) Refs. [78, 79, 80, 81]. Generic T -invariant5lass T P S
Spin sym. d = d = σ MC 0 -1 0 SU(2) 2 Z - SQHE Sp(4 n )U(2 n )A 0 0 0 U(1) Z - IQHE U(2 n )U( n ) ⊗ U( n )D 0 + Z - TQHE O(2 n )U( n )CI + Z
3D TSC Sp(4 n ) ⊗ Sp(4 n )Sp(4 n )AIII 0 0 1 U(1) - Z
3D TSC, chiral TI U(2 n ) ⊗ U(2 n )U(2 n )DIII -1 + Z Z
3D TSC ( He- B ) O(2 n ) ⊗ O(2 n )O(2 n )AI + n )Sp(2 n ) ⊗ Sp(2 n )AII -1 0 0 - Z Z n )O( n ) ⊗ O( n )BDI + + n )Sp(2 n )CII -1 -1 1 - - Z
3D chiral TI U(2 n )O(2 n ) Table 1: The 10-fold way classification for strong (fully gapped), d -dimensional symmetry-protected topological phasesof fermions, i.e. topological insulators (TIs) and topological superconductors (TSCs) [1, 2, 3, 4]. The 10 classes aredefined by di ff erent combinations of the three e ff ective discrete symmetries T (time-reversal), P (particle-hole), and S (chiral or sublattice). For a d -dimensional bulk, any deformation of the clean band structure that preserves T , P , and S and does not close a gap preserves the topological winding number. For a ( d − T , P , and S also preserves the “topological protection” against Anderson localization. Of particular interest here are classes C, A,D on one hand, and classes CI, AIII, DIII on the other. Classes C, A, and D are topological in d =
2, and describe thespin (SQHE), integer or charge (IQHE), and thermal (TQHE) quantum Hall e ff ects; all three can be realized as TSCswith broken T . Classes CI, AIII, and DIII are topological in d =
3, and can describe 3D time-reversal-invariant TSCs.(In this case, the physical time-reversal symmetry appears as the e ff ective chiral symmetry S in the table [1, 77].) Thecolumn “spin sym.” denotes the amount of spin SU(2) symmetry preserved for TSC realizations of these 6 classes.The 3D TSCs can host 2D massless Dirac (CI, AIII) or Majorana (DIII) surface theories. In the presence of disorderthat preserves physical time-reversal symmetry, these are equivalent to Wess-Zumino-Novikov-Witten (WZNW) sigmamodels, modified by a relevant perturbation for nonzero surface eigenenergy ε . While the ε = ε (cid:44) S symmetry, producing a standard Wigner-Dyson class, so that [7] CI → AI(Anderson localized), AIII → A (Anderson localized), and DIII → AII (Anderson localized or weak antilocalization).The numerical results of [33, 34] instead establish that the 2D Dirac surface theories in classes CI and AIII exhibit“critical stacking” (see text) of quantum Hall plateau transition states in classes C and A, respectively. Critical stackingfor class DIII is conjectured to correspond to the class D TQHE [24]. The last column gives the symmetry structure ofthe non-linear sigma model (NL σ M) description for each class, in terms of fermionic replicas [7]. / or nonabelianvector potentials in the low-energy surface Dirac theory. A generic Hamiltionian is [1, 23] H = ˆ σ · ( − i ∇ ) + (cid:88) i A i (r) ˆ τ i . (1)Although this is a single-particle Hamiltonian for (2 + H as the Lagrangian density for an imaginary time (2 + σ , then act separately on the spacesof left- and right-movers [23]. The matrices { ˆ τ i } act upon an N -dimensional color space, andcouple to the nonabelian vector potential A i . The color generators have to be compatible with thesymmetry of the class. In class AIII, for a bulk winding number ν = N , there can be generic U( N )disorder that encodes elastic scattering between the colors. Thus all hermitian N × N generators { ˆ τ i } are allowed, including the identity matrix [U(1) abelian vector potential disorder]. For classDIII, these are restricted to antisymmetric generators of SO( N ). In class CI, the winding number ν = N ≡ M is always even, and the matrices { ˆ τ i } generate the Lie algebra Sp(2 M ).The key defining characteristic of a topological surface is the anomalous representation of adefining bulk symmetry. For surface states of 3D TSCs, this is the chiral / physical time-reversalsymmetry. For the Hamiltonian in Eq. (1), it is encoded by the conditionˆ σ H + H ˆ σ = . (2)This version of chiral symmetry is anomalous, i.e. cannot arise without fine-tuning from the con-tinuum Dirac description of a 2D lattice model [1, 23]. It can be shown that Eq. (2) implies thatthe class CI, AIII, or DIII nonlinear sigma model (NL σ M) encoding Anderson (de)localizationphysics [7] is augmented by a Wess-Zumino-Novikov-Witten (WZNW) term [1, 6]. Without theWZNW terms, the NL σ Ms in these classes are termed “principal chiral models” or principal chi-ral NL σ Ms. The minimal realizations of Eq. (1) for topological class CI, AIII, and DIII surfaceshave winding numbers ν = N = { , , } , respectively.By contrast, the minimal “non-topological” version of class CI possesses four colors of 2DDirac fermions [25]. Incorporating disorder, the generic continuum Dirac model correspondingto a dirty 2D d -wave superconductor is perturbed by random mass, vector, and scalar potentialterms. This model is believed to Anderson localize for arbitrarily weak disorder at all energies;it corresponds to the class CI principal chiral nonlinear sigma model without the WZNW term[25, 82]. At the same time, the d -wave model can be fine-tuned to realize any of the threetopological models as exemplified by Eq. (1) [24, 25]. Suppressing elastic scattering between pairs of nodes gives two copies of the ν = T ). Further suppressing scattering between nodes within a pair breaks each ν = ν = ff erentDirac colors. These restrictions cannot, however, typically be realized exactly in a microscopic2D model with lattice-scale disorder.The averages over ensembles of disordered H in Eq. (1) can be described by the NL σ Mtheory [7]. Using fermionic replicas, the topological surface-state WZNW Dirac models areassociated to the group manifolds G (2 n ) ∈ { U(2 n ) , Sp(4 n ) , O(2 n ) } for classes AIII, CI, and DIII,7espectively, as shown in Table 1. Here n → ε (cid:44)
0. This is because ε (cid:44) G ⊗ G symmetry at zero energy, but this is reduced to the diagonal subgroup G for ε (cid:44) σ M from the group manifolds to the corresponding Grassmannians, G (2 n ) ⊗ G (2 n ) G (2 n ) (cid:39) G (2 n ) → G (2 n ) G ( n ) ⊗ G ( n ) , (3)see Table 1. Class AI always localizes in 2D, as does class A unless fine-tuned to the QHPT;class AII can exhibit weak antilocalization for su ffi ciently weak disorder [7, 15].Although we limit our focus in this review to Dirac surface theories, there are other possibil-ities. Bulk TSCs or fermionic topological superfluids that arise by pairing higher-spin fermions(e.g., S = /
2) can give rise to surface Hamiltonians that also exhibit the anomalous chiral sym-metry in Eq. (2). These have larger minimal winding numbers | ν | >
1, and the bulk windingnumber can be reflected through nonlinearity of the surface band structure, instead of N = | ν | colors of linearly-dispersing Dirac fermions [34, 83, 84, 85, 86]. Numerical studies suggest thatthe disorder-induced physics of these surfaces are the same as the Dirac models studied here[34, 85, 86]. Critical disordered systems can be characterized by the scaling of the distribution of powersof the wave function | ψ | , the so called inverse-participation ratios P q : P q ≡ (cid:90) d r | ψ (r) | q ∼ L − τ q . (4)In the limit that the system-size L → ∞ , the multifractal exponents τ q are self-averaging [7].Numerically it is favorable to subdivide a L × L system into N boxes of size b , with N ≡ L / b .The box probability µ i ≡ (cid:82) b i d r | ψ (r) | shows scaling behavior suitable to extract τ q : N (cid:88) i = ( µ i ) q ∼ (cid:32) bL (cid:33) τ q . (5)This way one can handle correlated disorder more easily, by excluding boxes smaller than thedisorder correlation length.The anomalous dimensions ∆ q ≡ τ q − q −
1) (6)can be interpreted as the deviation from the scaling of a fully delocalized metallic wave function.Localized states instead have τ q = q > .3. Class AIII, CI, and DIII WZNW models over U (2 n ) , Sp (4 n ) , and O (2 n )The statistics of the spatial fluctuations for eigenstates of 2D disordered systems is describedby the non-linear sigma model (NL σ M) framework [7]. Specifically, for the class AIII, CI,or DIII topological surface-state Hamiltonian in Eq. (1), this sigma model becomes a Wess-Zumino-Novikov-Witten (WZNW) model, familiar from conformal field theory (CFT). Usingnon-abelian bosonization [25, 26, 28, 29, 30, 31, 50] and conformal embedding theory [23],one can derive exact results for the scaling of generic operators in the energy ε → ρ ( ε ) as a function of the surface quasiparticle energy ε , one has lim ε → ρ ( ε ) (cid:39) | ε | x / z . (7)Here x = − z is the scaling dimension of the operator encoding the first moment of the localdensity of states at ε =
0, and z denotes the dynamic critical exponent. For surface states ofa bulk TSC with winding number ν , the scaling exponent x / z is summarized for the di ff erentWZNW models in Table 2. The multifractal spectrum for the WZNW models at ε = ∆ q = θ q (1 − q ) . (8)Table 2 summarizes how θ depends upon the class and winding number.The WZNW action for 2D dirty Dirac or Majorana TSC surface states with winding number ν reads S = ν π l φ (cid:90) d r Tr (cid:104) ∇ ˆ Q † · ∇ ˆ Q (cid:105) − i ν π l φ (cid:90) d r dR (cid:15) abc Tr (cid:104)(cid:16) ˆ Q † ∂ a ˆ Q (cid:17) (cid:16) ˆ Q † ∂ b ˆ Q (cid:17) (cid:16) ˆ Q † ∂ c ˆ Q (cid:17)(cid:105) − λ A ν π (cid:90) d r Tr (cid:104) ˆ Q † ∇ ˆ Q (cid:105) · Tr (cid:104) ˆ Q † ∇ ˆ Q (cid:105) + i ω (cid:90) d r Tr (cid:104) ˆ Λ (cid:16) ˆ Q + ˆ Q † (cid:17)(cid:105) . (9)See e.g. Ref. [23] for a derivation of this action from the disordered Dirac theory defined byEq. (1). The zero-energy surface theory for classes CI and AIII is described by the top lineEq. (9). The WZNW term is the second one on this top line, and requires extending the fieldconfigurations from the 2D surface into the 3D bulk [23, 87]; the parameter l φ is the Dynkinindex of the corresponding group. For class AIII only, an additional parameter appears evenat zero energy, which is the marginal disorder strength λ A that encodes the strength of abelianvector potential disorder.The parameter ω on the second line of Eq. (9) is the ac frequency at which the conductivity ofthe NL σ M is to be evaluated. With ω (cid:44)
0, states at finite energy can be accessed. This parametercouples to the imaginary (“tachyonic”) mass term ( i /
2) Tr (cid:104) ˆ Λ ( ˆ Q + ˆ Q † ) (cid:105) , where ˆ Λ = diag { ˆ1 n , − ˆ1 n } grades in the retarded / advanced space [7]. Since it is a strongly relevant perturbation, nonzero ω drives the theory away from quantum critical point solved by the WZNW conformal field theory.The field ˆ Q (r) is a (2 n ) × (2 n ) [(4 n ) × (4 n )] element of the matrix group U(2 n ), O(2 n ), [Sp(4 n )]for classes AIII, DIII [CI]. It satisfies the nonlinear constraint ˆ Q † (r) ˆ Q (r) = ˆ1, where ˆ1 denotesthe identity matrix. In the end, the replica limit n → ω =
0, the WZNW model in Eq. (9) is exactly solvable via CFT [23, 26, 27, 28, 29,30, 31, 50, 88]. Exact results for the DOS scaling [Eq. (7)], multifractal spectrum [Eq. (8)], andconductivity are summarized in Table 2. 9III CI DIII ν Z Z Z x / z π − ν λ A π (2 ν − + ν λ A [31] | ν | + [26] − | ν |− ( | ν | ≥
3) [23, 89] θ ( ε = | ν |− ν + λ A π [28, 29, 31] | ν | + [28, 29, 67] | ν |− ( | ν | ≥
3) [23] σ xx ( ε = νπ νπ νπ θ ( ε (cid:44) (cid:39) / (cid:39) / (cid:39) /
13 (TQHPT?) [24] σ xx ( ε (cid:44) (cid:39) . ± .
02 [58] = √ [38] ? Table 2: Summary of known properties for the 2D disordered Dirac models [Eq. (1)] in classes AIII, CI, DIII thatcan exhibit Wess-Zumino-Novikov-Witten (“stacked”) criticality at zero (nonzero) energy. Here these are the allowedbulk TSC winding numbers ν , the scaling of the surface density of states ρ ( ε ) ∝ | ε | x / z , the curvature of the parabola θ controlling the surface multifractal spectrum via ∆ q = − θ q (1 − q ), and the longitudinal surface conductivity σ xx (for spinor heat transport at the boundary of the TSC, in units of the appropriate conductance quantum [1, 81]). The top four rowsdescribe the zero or near-zero energy critical features of dirty 2D TSC surface states, which are known analytically fromconformal field theory. In class AIII, these results depend on the winding number ν and the abelian disorder strength λ A , which is defined in Eqs. (13) and (14). The additional parameter λ A is RG-marginal and addresses a continuum ofdistinct zero-energy fixed points [28, 29, 31]. The last two rows detail the recent numerical findings of Refs. [33], [24],and [34]. These characterize the “stacked” criticality of TSC surface states at finite energy, where each state in the stackexhibits identical statistical properties. The results for the finite-energy multifractal spectra and conductance statisticsare consistent with a stacking of the integer class A and spin class C quantum Hall plateau transition (QHPT) statesfor class AIII and CI Dirac models, respectively. The stacked criticality observed for finite-energy class DIII states isconjectured to describe the thermal QHPT in class D [24]. We note that the numerical results for finite-energy class DIIIstates have only been obtained for quenched gravitational disorder, i.e. modulation of the velocity components for thesingle Majorana cone associated to winding number ν =
1. The multicolor DIII model with | ν | = N > Finite energy behavior.
The ac frequency parameter ω in Eq. (9) reduces the G ⊗ G group sym-metry of the WZNW model down to the diagonal subgroup G . Real nonzero ω gives oscillatorycontributions to the functional integral over ˆ Q unless a further constraint is imposed,ˆ Q = ˆ Q † , Tr (cid:104) ˆ Q (cid:105) = Λ by a left-group translation ˆ Λ ˆ Q → ˆ Q ). Then, the ω term and the λ A term (class AIII) on the second line of Eq. (9) are projected to zero.In this constrained case, Bocquet, Serban, and Zirnbauer [43] (see also [22, 25]) derived adeformation of the WZNW term to the topological term in the Pruisken model: S → σ x , x (cid:90) d r Tr (cid:104) ∇ ˆ Q · ∇ ˆ Q (cid:105) − σ x , y (cid:90) d r (cid:15) i j Tr (cid:104) ˆ Q ∂ i ˆ Q ∂ j ˆ Q (cid:105) , (11)where σ x , x = ν/π, σ x , y = ν/ . (12)For classes CI and DIII, the Pruisken model only applies if the target manifold for the con-strained ˆ Q is taken to be that of classes C and D, respectively. Although ω (cid:44) G subgroup, this information is insuf-ficient to determine the target manifold G / H of the e ff ective NL σ M governing the Anderson(de)localization physics of the finite-energy states. In the case of class CI with G = Sp(4 n ) (us-ing fermionic replicas, Table 1), there are two possible scenarios for the finite-energy NL σ M.Either H = Sp(2 n ) ⊗ Sp(2 n ) (the orthogonal Wigner-Dyson class AI), or H = U(2 n ) (classC). The former choice is the conventional one that guarantees Anderson localization at all finiteenergies [7]; the latter is realized in the stacking scenario, wherein Eq. (11) describes the spin quantum Hall plateau transition [7, 35, 36, 37, 38, 39, 40, 41].Although there is no ambiguity in the target manifold for finite-energy class AIII states(which reside in class A), the “derivation” of Eq. (11) from Eq. (9) via the imposition of the con-straint in Eq. (10) poses another problem. Eq. (12) implies that the topological angle ϑ = πσ x , y is an odd (even) multiple of 2 π for odd (even) winding numbers ν . This even-odd e ff ect is not observed in the numerics here and in Refs. [33, 34]. In other words, imposing Eq. (10) by handdirectly to the fields gives coe ffi cients of the Pruisken model that are not compatible with numer-ics. This does not rule out this analytical ansatz as a description of the problem, as the followingexplanation clarifies. The actual physical RG flow of the full WZNW theory is that ω runs to thestrong coupling regime. The other coe ffi cients are likely to receive renormalization well beforethe ˆ Q field is reduced to the target manifold associated to Eq. (10). Consequently, the physicalPruisken model parameters can deviate from the values stated in Eq. (12). We claim that time-reversal invariant 3D TSCs show stacks of quantum Hall plateau tran-sition (QHPT) states at finite energy. Therefore, a brief recapitulation of the features in QHsystems that we compare with our numerics is appropriate.
Integer Quantum Hall Plateau Transition – class A.
Studies of the network model [57] indicateparabolic multifractality with θ (cid:39) / σ xx IQHPT (cid:39) . ± .
02 [58] from numerical Kubo computations. Excitingly, the IQHPT transitionis conjectured to itself be described by a class AIII n = reverse RG flow from the Pruiskenmodel [Eq. (11)] to the class AIII WZNW action [Eq. (9) with ν = λ A = π/
16, and ω = ν = same conformal field theory (albeit with a renormalized value of λ A ) at both zero and finiteenergies [33]. Spin Quantum Hall Plateau Transition – class C.
The SQHPT transition is well-studied an-alytically by a mapping to 2D classical percolation, and numerically with the network model[35, 36, 37, 38, 39, 40, 41]. The conductance distribution and exact average longitudinal con-ductivity σ xx SQHPT = √ / θ = / Thermal Quantum Hall Plateau Transition – class D.
Class D permits a variety of distinct net-work models and Hamiltonians [42, 43, 44, 45, 46, 46, 47, 48], and there is non-universality. Theconjectured thermal QHPT transition is di ffi cult to observe, since it may be shaded by a weaklyantilocalizing thermal metal phase. 11 . Axial U(1) spin symmetry: class AIII In this Section we review the numerical results from Ref. [33] for the 2D Dirac TSC sur-face theory in class AIII, and the AIII WZNW → A IQHPT stacking conjecture. As pointedout already in Sec. 2.3, it is important to distinguish between even and odd winding numbers.Computationally it is easiest to look at ν = , One Dirac node, U(1) vector potential dirt (AIII, ν = ). The winding number ν = σ , ). H (1) AIII = ˆ σ · [ − i v ∇ + A(r)] , A a (r) A b (r (cid:48) ) = λ A δ ab δ (2) ξ (r − r (cid:48) ) . (13)Here v is the Fermi velocity (which will be set equal to one). Disorder enters as a random abelian U (1) vector potential ˆ σ · A = σ A + σ A with disorder strength λ A ; the overline · · · denotes anaverage over disorder configurations. The delta function δ (2) ξ (r − r (cid:48) ) is smeared out by a correlationlength ξ in the numerics described below. Two Dirac nodes, U(1) ⊕ SU(2) vector potential dirt (AIII, ν = ). The winding number ν = τ = ± with strength λ A , but also a non-abelianSU(2) vector potential A i ( i ∈ { , , } ) with strength λ : H (2) AIII ≡ ˆ σ · [ − i v ∇ + A (r) + A i (r) ˆ τ i ] , A a (r) A b (r (cid:48) ) = λ A δ ab δ (2) ξ (r − r (cid:48) ) , A ai (r) A bj (r (cid:48) ) = λ δ ab δ i j δ (2) ξ (r − r (cid:48) ) . (14)In the limiting case of λ A =
0, the full SU(2) spin symmetry as well as particle hole symmetryare restored. This puts the model into class CI, identical to the ν = For the class AIII dirty Dirac theories described above, at zero energy (the surface quasi-particle Dirac point) the WZNW theory [Eq. (9) with ω =
0] predicts exact parabolicity withcurvature [Eqs. (6), (8), and Table 2] θ ν = | ν | − ν + λ A π , (15)depending on winding number ν and abelian disorder strength λ A . With momentum-space exactdiagonalization in Ref. [51], the multifractal statistics of the low-energy states for the ν = ν = L = ξ isanalyzed. Here ξ denotes the common correlation length of the disorder potentials, which aretaken to be Gaussian distributed and correlated [33]. At finite energies, the anomalous mul-tifractal spectrum ∆ q is compared to the parabolic approximation for the class A IQHPT with θ QHPT = .
25 [Eqs. (6), (8), and Table 2]. Great agreement is found over a wide energy range. Inthe high-energy tail ε ∼ (cid:126) v /ξ , there are larger deviations. This can be explained since close tothe energy cuto ff the system seems untouched by the disorder and matches the clean DOS [see12 E / v v (a) EDclean E / v N ( E ) (b) ED E z L x /050100150200 L y / G ( L x ) h / e (c) = e / hE / v =0.00.0031.2 E / v / (d) E z E / v h / e (e) QHPT= e / h (AIII) q ( q ) (f) QHPT =0.25
Figure 2: Numerical Landauer conductance and multifractal analysis from Ref. [33] for the winding number ν = √ λ A ≡ W = .
3. (a) The DOS ρ ( E )versus energy E , as calculated from momentum-space exact diagonalization (ED), is most strongly a ff ected by disorderaround the Dirac point ( E = N ( E ) = (cid:82) E d ε ρ ( ε ) is plotted versus energy. The predictedscaling form implied by Eq. (7) is governed by the disorder-dependent dynamical critical exponent z = + W /π .(c) Quantum transport results for the resistance normalized to system width. The energies are from top to bottom E ξ/ (cid:126) v = , . , . , . , . , . , . , .
2. (d) The crossover correlation scale from the transport calculation scales as ζ ( E ) ∼ E − / z . This scale (not to be confused with the fixed disorder correlation length ξ ) governs the crossover at energy E between WZNW and class A IQHPT criticalities at smaller and larger length scales, respectively. (e) Conductivitiesextracted from the slope of the curves in panel (c), compared to the established value of the class A IQHPT criticalconductivity (see Table 2). (f) Anomalous part of the multifractal spectrum ∆ ( q ) extracted from box-size scaling ofED eigenstates for box sizes beyond the crossover correlation length ζ ( E ), as extracted in (d). The data correspond to E ξ/ (cid:126) v = . , . , . , . , . , . Fig. 2(a)]. Panel (b) in Fig. 2 moreover confirms that the low-energy integrated DOS N ( ε ) scalesas expected from WZNW theory [Eq. (7) and Table 2].The results for the ν = L = ξ are shownin Fig. 3. In Panel (f) there is a comparison of finite-energy multifractal spectra to the class AIQHPT parabola with θ QHPT = .
25. In the high-energy tail ε ∼ (cid:126) v /ξ there are deviations forthe same reason as in the ν = The transport calculations are performed by slicing the system in the x direction and sub-sequently recasting the time-independent Schr¨odinger equation H (1) AIII ψ = ε ψ in terms of thetransfer matrix, using the method of Ref. [18]. Clean, highly doped leads are attached to thesystem at x = x = L x . The conductance G is then computed from the transmission block t of the scattering matrix S between the leads. 13 E / v v (a) EDclean E / v N ( E ) (b) ED E z L x /0100200300400 L y / G ( L x ) h / e (c) =2 e /( h ) E / v =0.00.0031.2 E / v / (d) E z E / v h / e (e) QHPT=2 e /( h ) q ( q ) (f) QHPT =0.25
Figure 3: Numerical results from Ref. [33] for the topological class AIII surface model with a two Dirac nodes( ν = √ λ A ≡ W A = . √ λ ≡ W N = .
5, respectively. (a) The DOS as a function of energy, as calculated from ED. (b) The in-tegrated DOS N ( E ) = (cid:82) E d ε ρ ( ε ) plotted versus energy. The predicted scaling form implied by Eq. (7) is gov-erned by the dynamical critical exponent z = / + W A /π , which depends only on the abelian disorder strength.(c) Quantum transport results for the resistance normalized to system width. The energies are from top to bottom E ξ/ (cid:126) v = , . , . , . , . , . , . , .
2. (d) The crossover correlation length from the transport calculation scalesas ζ ( E ) ∼ E − / z . (e) Conductivities extracted from the L x ≥ ξ slopes of the curves in panel (c), compared to the es-tablished value of the class A IQHPT critical conductivity (see Table 2). (f) Anomalous part of the multifractal spectrum ∆ ( q ) extracted from box-size scaling of ED eigenstates for box sizes beyond the correlation length ζ ( E ) as extracted in(d). The data correspond to E ξ/ (cid:126) v = . , . , . , . , . , . , . The finite-size resistance normalized to the sample width L y / G ( L x ) is expected depend lin-early on L x L y / G ( L x ) = L y R + σ L x . (16)Gauge invariance and chiral symmetry force the contact resistance R to zero for each configu-ration [88, 92]. The data in Fig. 2(c) for zero energy E = σ xx AIII ,ν = = e / h π , see Table 2. (Here the quoted conductance quantum e / h is appropriate to charged electrons at the surface of a chiral topological insulator in class AIII[79]. At the surface of a class AIII TSC, this should instead be replaced by the spin conductancequantum (cid:126) / π [1, 81].)The finite-energy crossover scale ζ ( E ) is defined as the length L x where L y / G deviates by5% from the E = ζ ( E ) ∼ E − / z ,consistent with the z determined from the DOS scaling. Physically, ζ ( E ) separates class AIIIWZNW critical scaling for shorter length scales from class A IQHPT scaling at larger ones;14 igure 4: Numerical transport results from Ref. [33] for the topological two-node class AIII Dirac model [ ν = W A = . W N = , , , E = . (cid:126) v /ξ . The left panel shows the bare resistance data, while the right panel depicts the bulk conductivitiesobtained from linear fits to the bare resistance data above L x = ξ . These plots establish the crossover of the two-nodemodel from the finite-energy conductivity plateau equal to 2 × σ xx IQHPT in the absence of internode scattering, to a plateauwith value 1 × σ xx IQHPT in its presence. ζ ( E ) → ∞ as E → < E (cid:46) (cid:126) v /ξ , thereis a plateau at σ (cid:39) . e / h ) in fair agreement with the value σ xx IQHPT = . ± . e h ob-tained by Schweitzer and Markoˇs [58] via the Kubo formula for a lattice model tuned to theclass A IQHPT (Table 2). At larger energies E , the conductivity at the accessible length scalesincreases with energy. This is expected for the semiclassical Drude conductivity, which goes as σ xx ∼ ( e / h )(1 / W ), where W is the disorder strength [93]. For these large energies, the avail-able length scales are insu ffi cient to decide which scenario, Anderson localization or IQHPTcriticality, is realized at the largest length scales.Results for the ν = ν = L x = ξ ) × ( L y = ξ ). The consistency check of the crossover length ζ in Fig. 3(c) works just as in the ν = z matches withthe expected DOS scaling in Fig. 3(b). The conductivity as function of energy shown Fig. 3(e)matches σ xx AIII ,ν = for very low energies. At finite energies it slightly drops to σ xx IQHPT . This dropcannot be resolved in the Kubo computations in Fig. 6 that we performed, discussed in the nextsubsection.A complementary perspective on the results in Fig. 3 for the ν = W N . InFig. 4, results for the conductivity are shown for W N = , , , W A = . E = . (cid:126) v /ξ . For W N =
0, the two nodes are decoupled and theconductivity is close to 2 × σ xx IQHPT , as expected for two replicas of the single node case. For W N = , , × σ xx IQHPT .Finally, we mention additional numerical evidence for the IQHPT-stacking scenario obtainedin Ref. [33]. First, the full Landauer conductance distribution was computed for both ν = , ν = , In this subsection, we present new results testing the AIII WZNW → A IQHPT stackingconjecture, this time computing the dc surface conductivity via the Kubo formula. In naturalunits e = , (cid:126) =
1, the Kubo formula relates the dc conductivity σ ab dc to the current-currentresponse function K ab ( ε ): σ ab dc = π (cid:90) ∞−∞ (cid:34) − d f ( ε ) d ε (cid:35) K ab ( ε ) , (17) K ab ( ε ) ≡ L (cid:90) r , r (cid:48) Tr (cid:104) ˆ σ a ˆ ρ ( ε ; r , r (cid:48) ) ˆ σ b ˆ ρ ( ε ; r , r (cid:48) ) (cid:105) . (18)The finite size conductivity σ ab dc must be evaluated with spectral densities ˆ ρ ( ε ; r , r (cid:48) ) broadened bya finite η of the order of the level spacing around energy ε :ˆ ρ ( ε ; r , r (cid:48) ) = i (cid:104) ˆ G R ( ε ; r , r (cid:48) ) − ˆ G A ( ε ; r , r (cid:48) ) (cid:105) = π (cid:88) l (cid:34) η/π ( ε − ε l ) + η (cid:35) ψ l ( r ) ψ † l ( r (cid:48) ) , (19) A AA AAA
Figure 5: Numerical Kubo conductivity σ xx dc computed with Eq. (20) for the strongly disordered AIII ν = η . The disorder strength is λ A =
5. A logarithmic scale for η in units of the locallevel spacing ∆ ε is chosen. There is convergence to a plateau of σ xx dc as function η as the linear system size N increases.At small energies ε (cid:28) Λ , σ xx dc tends to the WZNW value associated to the zero-energy state of the Dirac theory (reddashed). For finite energies ε (cid:46) Λ , we find a value of σ xx dc compatible with the universal IQHPT result σ xx IQHPT (greendashed). States at ε ≈ Λ are not a ff ected much by the disorder and therefore do not show universal conductance values.This confirms the Landauer computation in Fig. 2(e). ψ l (r) is an exact eigenstate. We employ K ab ( ε ) = (cid:32) π L (cid:33) (cid:88) l , m (cid:34) η/π ( ε − ε l ) + η (cid:35) (cid:34) η/π ( ε − ε m ) + η (cid:35) (cid:104) l | ˆ σ a | m (cid:105)(cid:104) m | ˆ σ b | l (cid:105) (20)to compute the Kubo conductivity with eigenenergies ε l and states | l (cid:105) from exact diagonaliza-tion. The result should be virtually independent of the broadening η chosen around the locallevel spacing ∆ ε . Calculations are performed for the momentum-space version of the continuumHamiltonians in Eqs. (13) and (14), with quantized momenta corresponding to a finite-size torusand an ultraviolet energy cuto ff Λ .We computed the Kubo conductivity σ xx dc via Eq. (20) for the strongly disordered AIII ν = η . A logarithmic scale for η in units of the local levelspacing ∆ ε is chosen (the critical DOS ν ( ε ) ∝ ε α is responsible for the dependence ∆ ε ∝ ε − α ). Forlarge enough systems, σ xx dc as function η should depend only weakly on η . We use the tendencyof σ xx dc to converge towards a plateau value as a measure of finite size e ff ects. The results areshown in Fig. 5. At small energies ε (cid:28) Λ , σ xx dc tends towards the WZNW value σ xx AIII ,ν = = /π (Table 2). For finite energies ε (cid:46) Λ , we find a value of σ xx dc compatible with the universal IQHPTresult σ xx IQHPT ≈ .
58 (green dashed). States at high energies ε ≈ Λ are only weakly a ff ectedby the disorder and therefore do not show universal conductance values. We are able to furtherconfirm the Landauer computation in Ref. [33], see Fig. 2(e).For the AIII ν = σ xx AIII ,ν = = /π (red dashed) and the IQHPT result σ xx IQHPT ≈ .
58 (green dashed) are numerically close toeach other. The convergence of σ xx dc is not as clear as for the ν = N ≤
80 are
Figure 6: Same as Fig. 5 for the AIII ν = λ =
5, while theabelian strength is λ A = .
2. The WZNW value (red dashed) and the universal IQHPT result (green dashed) are veryclose to each other. The convergence of σ xx dc is not as clear as in Fig. 5. Since the model incorporates the 2D color space[Eq. (14)], only systems with N ≤
80 are numerically accessible. The data does not show signs of the conventionallyexpected Anderson localization σ xx dc → ε ≈ Λ are not a ff ected much by the disorder and therefore do not show universal conductance values. σ xx dc → N → ∞ at finite energies for ν = ff ects estimated by the fluctuation of σ xx dc ( η ) are larger than the numerical di ff erence of theexpected conductivities σ xx IQHPT ∼ σ xx AIII ,ν = . States at ε ≈ Λ are not a ff ected much by the disorderand therefore do not show universal conductance values.
4. Full SU(2) spin symmetry: class CI
A class CI topological surface can be described by the following Dirac Hamiltonian: H CI = ˆ σ · (cid:104) ( − i ∇ ) + A i (r) ˆ τ i (cid:105) , A ai (r) A bj (r (cid:48) ) = λ δ ab δ i j δ (2) ξ (r − r (cid:48) ) , (21)where ˆ σ ≡ ˆ σ ˆ x + ˆ σ ˆ y . The winding number ν = k is always even for class CI; disorder thatpreserves physical time-reversal and spin SU(2) symmetry appears as the color gauge potentialA i , where the 2 k × k color-space matrices { ˆ τ i } generate the color group Sp(2 k ) [23, 67, 78]. Forthe minimal case k =
1, Eq. (21) is identical to the ν = λ A = ν lead to the class CI WZNW → C spinquantum Hall plateau transition (SQHPT) stacking conjecture. Instead of employing Sp(2 k )generators, for k > N max =
72 vs. N max = ν =
2. In this case we use Eq. (21), where the color generators { ˆ τ i } are Paulimatrices. Figure 7: Multifractality in the CI ν = N =
72, at weak λ = .
5, intermediate λ = .
5, and strong disorder λ = .
0. The system is a (2 N + × (2 N +
1) grid in momentum space; Λ is the ultravioletenergy cuto ff for the clean Dirac spectrum. Near-zero energy states and finite-energy states are compared to the classCI-WZNW (red dashed) and class C-SQHPT parabolic spectra (green dashed). With increasing disorder, there are fewerand fewer states that match the CI-WZNW prediction, and the crossover scale moves towards zero energy. Fig. 8 exhibitsa finite-size analysis of ∆ q for q = , = =
330 40 50 60 70 - - - - - Δ q CI, W = ν = q = =
330 40 50 60 70 - - - - - Δ q CI, W = ν = Figure 8: Finite energy ∆ q for q = , ν = N = , . . .
72. The greenlines are exact analytical predictions ∆ = − / ∆ = − / ∆ q in the energy range 0 . (cid:46) ε/ Λ (cid:46) Λ is the ultraviolet cuto ff for the clean Dirac spectrum. Errorbars indicate the variance. When increasing N , the ∆ q converge and fluctuations diminish. We numerically compute the multifractal spectrum ∆ q [Eqs. (5) and (6)] using exact diag-onalization of the continuum Dirac Hamiltonian in Eq. (21) in momentum space. Results areshown for ∆ q , computed for the largest available system size, at various energies in Fig. 7. Withincreasing disorder strength λ , fewer and fewer states match the zero-energy class CI WZNWprediction [Eq. (8) with θ = / θ SQHPT = .
125 (see Table 2). In Fig. 8 we show ∆ q for q = , ε/ Λ (cid:38) .
2. The red dots mark the average values over that part of the spectrumwith the standard deviation given by the error bars. With increasing system size, the error barsshrink and there is convergence towards the θ SQHPT = / density of criticalstates (DOCS). The latter is defined as follows. The DOCS is determined by the proportionof critical states at finite energy matching the class C SQHPT θ SQHPT = / τ q with q ∈ [0 , q c ] have to match the parabolic τ θ q up to 4% accuracy. Here q c ( θ ) ≡ √ /θ is the termination threshold [7, 32, 34]. In addition tothe DOS and SQHPT DOCS, in Fig. 9 we also exhibit the DOCS for matching the zero-energyclass CI WZNW prediction [Eq. (8) with θ = /
4, Table 2]. As indicated by the results in Fig. 9,more (less) of the spectrum matches the class C SQHPT (class CI WZNW) prediction as thesystem size N or disorder strength λ is increased. Since the class C SQHPT states exhibit weaker multifractality than the CI WZNW states, this is strong evidence against Anderson localization. We compute the Kubo conductivity the same way as for the AIII surfaces in Sec. 3.3. InFig. 10, we show the numerical Kubo results for selected energies across the spectrum. Nearzero energy, we expect the exact WZNW result σ xx CI ,ν = = /π (Table 2). This result holds for19 igure 9: The density of critical states (DOCS) versus the total density of states (DOS) for the class CI model. A stateis termed critical when it matches the expected multifractal spectrum τ q for the class C-SQHPT within 4% for at least75% of the 0 < q < q c , where q c =
4. At zero energy we expect class CI WZNW-criticality and at finite energiesSQHPT-criticality, see Fig. 8. With increasing system size N or disorder strength λ , the amount of WZNW-critical statesdecreases in favor of class C-SQHPT critical finite-energy states. (The green curve labeled “WZW” denotes the densityof critical class CI-WZNW states). Superimposed in light gray is the inverse-participation ratio P , which shows thatstates away from zero energy are less rarified, as predicted by the stacking conjecture. igure 10: Numerical Kubo conductivity σ xx dc for the class CI model, computed with Eq. (20), for moderate disorder asfunction of the level broadening η . A logarithmic scale for η in units of the local level spacing ∆ ε is chosen. Numericalresults are compared to the the exact zero-energy WZNW result σ xx CI ,ν = = /π (red dashed) and to the exact averagevalue for the class C SQHPT σ xx SQHPT = √ / η as N increases, and σ xx dc does not depend on η significantly, i.e. shows a plateau. Forour purposes only the finite energies are crucial, but at strong disorder λ (cid:38) ff ects near zero energy disable us from going to the strongly disordered regime, where more of thespectrum is SQHPT-critical, according to the multifractal analysis presented in Figs. 7–9. Crucially, though, we onlyobserve evidence for Anderson localization deep in the high-energy Lifshitz tail. This again contradicts the conventionalpicture that all finite-energy states localize in the orthogonal class AI (Secs. 2.1 and 2.3). every disorder configuration, in the infinite system-size limit. For weak disorder λ (cid:46) ff ects seem to grow quickly with increasing disorder,which makes the most interesting regime λ (cid:38)
3, where SQHPT multifractality has spread overa wide range of the energy spectrum, di ffi cult to reach numerically. This is the reason for thederivation from the exact WZNW result for λ = , . b c d e Figure 11: Visualization of quenched gravitational disorder (QGD): the spatial components of the random metric tensorprojected on a flat 2D space. A 2 × v ij can be visualized by the quadratic form { x : v ij x i x j = r } for somefixed r >
0. We consider di ff erent classes of disorder (a)–(e) (see text): (a) flattening / steepening + nematic disorder v = v =
0, (b) rotations + nematic disorder δ v = δ v =
0, (c) flattening / steepening + rotations δ v = δ v and δ v = − δ v (conformal spin s = δ v = − δ v and δ v = δ v (conformal spin s = − , be σ xx SQHPT = √ / ff ects remain relevant as near zero energy, the results indicate conver-gence towards a finite conductivity, coincident with the Cardy value, at finite energies. This isagain in contrast with the conventional expectation of Anderson localization in the orthogonalclass AI (Secs. 2.1 and 2.3). On the other hand, for very high energies ε (cid:38) Λ (in the Lifshitz tail),we observe localization in ∆ q . This is consistent with the σ xx dc → ε = Λ in Fig. 10.
5. Broken spin rotation symmetry: class DIII c cuprates The simplest bulk topological superconductor resides in class DIII, with the minimal windingnumber ν =
1; this would be a solid-state analog of He- B [1, 8, 9], as has been proposed e.g.in Cu x Bi Se [97, 98, 99, 100, 101, 102, 103] Nb x Bi Se [104, 105, 106, 107], and β -PdBi [108, 109]. In this case, superconductivity and strong spin-orbit coupling imply that neithercharge nor spin transport are well-defined at the surface.The surface theory consists of a single massless Majorana cone. In contrast to the surfaceHamiltonian in Eq. (1) (which applies for class DIII with winding numbers | ν | ≥ ff ects of “quenched gravitational dis-order” (QGD) for a single 2D cone were studied. Time-reversal invariant perturbations, such asa charged impurity, couple only to the spatial-spatial components of the stress-energy tensor T ab ,with a , b ∈ { , } [24]. The most generic surface Hamiltonian takes the form H = − (cid:88) a , b = , (cid:90) d r v ab (r) (cid:18) ¯ ψ i ˆ σ a ↔ ∂ b ψ (cid:19) , (22)where the bidirectional derivative A ↔ ∂ B = A ∂ B − ( ∂ A ) B . For the Majorana surface theory,¯ ψ = ψ T ˆ σ . The four velocity components are the isotropic Fermi velocity of the clean Majoranacone, perturbed by quenched random fluctuations: (cid:8) v (r) ≡ + δ v (r), v (r) ≡ + δ v (r),22 (r), v (r) (cid:9) . In Ref. [24], five di ff erent variants of that model were considered. The variantsare visualized in Fig. 11,(a) Independent { δ v , δ v } , v = v =
0. Local isotropic flattening or steepening of theDirac cone and nematic squishing of the cone.(b) Independent { v , v } , δ v = δ v =
0. Local pseudospin rotations (antisymmetric part v a = − v a ) and nematic squishing of the Dirac cone (symmetric part v s = + v s ).(c) Independent { δ v = δ v , v = − v } . Local isotropic flattening or steepening of theDirac cone and pseudospin rotations.(d) Independent { δ v = − δ v , v = v } . Local nematic squishing of the Dirac cone.(e) Independent { δ v , δ v , v , v } . The generic model.Without further restrictions, the generic theory (e) is realized. For a fully isotropic bulksuperfluid, it can be shown that electric potentials couple only through the isotropic flatteningor steepening of the surface Majorana cone, model (c) [24]. Crystal field e ff ects will howevergenerically enable o ff -diagonal QGD (nonzero { v , v } ). Model (d) is another interesting specialcase, since pure nematic QGD couples only to the holomorphic T ( z ) and antiholomorphic ¯ T (¯ z )stress tensor components (using the language of 2D conformal field theory) [24].We also emphasize that QGD as in Eq. (22) will generically be present in any
2D mass-less Dirac material. At zero energy (the Dirac point), short-range correlated QGD is stronglyirrelevant. This is why it is typically ignored, compared to mass, scalar, or vector potential per-turbations [as in Eq. (1)]; short-ranged correlated disorder in the latter is marginal at tree level.The very surprising finding in [24], reviewed below, is that while weak QGD is indeed irrelevantnear zero energy, it appears to induce quantum-critical stacking of states with weak, but univer-sal multifractality, similar to the class AIII and CI systems studied above. This occurs becausenonzero energy is a strongly relevant perturbation to the (2 + ν = λ A =
0; again ω (cid:44) T c cuprate superconductors [24]. As reviewed inSec. 2.1, the generic Bogoliubov-de Gennes Hamiltonian for a 2D d -wave superconductor withnon-magnetic disorder resides in the non-topological version of class CI; in contrast to Eq. (1),the low-energy Dirac theory for the 4 independent quasiparticle colors features mass, scalar, andvector potentials. This model is known to Anderson localize at all energies [25]. If however weassume that su ffi ciently long-wavelength disorder dominates, which does not scatter between thecolors, then one obtains four independent copies of the class AIII ν = → class A IQHPT stacking conjecture, this wouldimply relatively strong multifractality (wave function rarification) at finite energy, along with astrongly renormalized low-energy density of states. This is not seen in experiment. However, ne-matic QGD [as in models (b) and (d), described above] in fact produces a phenomenology similarto that observed in STM maps of the local density of states in BSCCO [24, 59, 60, 61, 62, 63, 64].This includes plane-wave like states at low energy, with a linear-in-energy density of states, butenergy-independent, nanometer-scale critical inhomogeneity at finite energies [24]. See alsoFig. 1 and Table 2.Models (b) and (d) with nematic
QGD show the most robust stacking of critical eigenstateswith universal statistics, as reviewed below. In the context of the cuprates, this is interesting23ecause of the potential role of nematicity in these materials [110, 111]. In particular, evidencefor quenched random nematicity has emerged in recent studies of the pseudogap phase [112].The natural generalization of the stacking conjectures in classes AIII and CI is that the classDIII WZNW → class D thermal quantum Hall plateau transition (TQHPT). As reviewed inSec. 2.4, very little is known about the TQHPT. The multifractal spectra presented below mayconstitute the first (indirect) results substantiating a universal description of this transition. Figure 12: This figure shows the total density of states (DOS) and the density of critical states (DOCS) for the QGD,models (a)–(e), defined in and below Eq. (22). Results obtain by diagonalizing Hamiltonian from Eq. (22) over a (2 N + × (2 N +
1) grid in momentum space, with N =
96 here [24]. Data is plotted for the five di ff erent models at fixeddimensionless disorder strength λ = .
2; strong disorder corresponds to λ (cid:38) . P (grey dots), defined by Eq. (4). For models (a,c,e), a large swathof the spectrum appears critical for weak disorder. However, as the disorder strength is increased, the swath shrinks [24].The IPR P shows that states outside of the swath are more rarified or localized than the critical ones. The linear-in-energy DOS of the clean limit is strongly distorted and filled-in at low energies, which happens in models (a,c,e) for all λ (cid:38) .
2. These strong disorder e ff ects are likely induced by the velocity component responsible for isotropic flattening orsteepening of the cone. By contrast, models (b,d) show plane-wave states near zero energy for weak disorder. Rarificationnear zero energy sets in for strong disorder; for model (d) the crossover is already visible at λ = . .2. Multifractal analysis In Ref. [24], models (a)–(e) were studied for a large range of N = , . . . ,
96. In all fivemodels, there are critical states at finite energy matching a θ = /
13 parabolic ansatz for ∆ q [Eqs. (6) and (8), Table 2]. As for class CI in Sec. 4 and Fig. 9, we define the density of criticalstates (DOCS) as follows. This is the proportion of states at finite energy matching the tolerancecriterion that 85% of the τ q within the range q ∈ [0 , q c ] match the parabolic τ θ q ansatz with θ = /
13 up to 4% accuracy; here q c = . P (gray) in Fig. 12. At high energies near the cuto ff there is a crossover to localized Lifshitztail states.In the absence of the isotropic flattening and steepening, models (b), (d) instead exhibit an in- Figure 13: Anomalous multifractal spectrum ∆ q [Eq. (6)] for an energy bin of states selected from the DOS with thehighest percentage of critical states from Ref. [24]. Here the spectrum is shown for model (b), evaluated for the sixdi ff erent disorder strengths. The solid red curve denotes an average over the 15 states in the bin; the shaded red regionindicates the standard deviation. The green curve is the parabolic ansatz for ∆ q = − θ q (1 − q ), with θ = /
13. Statescontributing to the critical count (DOCS) in Fig. 12(b) match the parabolic ansatz within a certain threshold (see text)over the range 0 < q ≤ q c = . reasing number of critical states (improved “stacking”) with increasing disorder. Models (b) and(d) are similar, except that for the latter, rarified zero-energy states set in at intermediate disorder,see Fig. 12. Indeed models (b) and (d) are related by local di ff eomorphisms in the gravitationalformulation of the problem [113]. Since there are more independent disorder terms in (d), it ismapped to a (b) with e ff ectively stronger disorder. We note however that the interpretation ofdi ff eomorphisms is di ff erent than in general relativity or 2D quantum gravity [114, 115, 116]: atthe surface of a dirty class DIII TSC, there is a preferred coordinate system ( x , y ) that measuresphysical distances across the surface, given the flatness of physical spacetime. Disorder modu-lates the Majorana surface fluid in a way that is mathematically identical to gravity, but geodesicdistances are not directly measurable.Model (e) containing generic disorder exhibits a wide swath of critical states for weak disor-der. However, this shrinks with increasing disorder strength, similar to models (a) and (c).The robustness of the anomalous multifractal spectra selected from an energy bin where theratio of the DOCS to DOS is maximized with respect to disorder is shown in Fig. 13 for theexemplary model (b). Tuning over a whole order of magnitude in the disorder strength, theresults are robust for q not to close to the termination threshold q c . The deviations are easilyunderstood by the absence of ensemble averaging in the numerics.Excitingly the behavior of models (b), (d) closely parallels observations from STM studies ofBSCCO [63, 64]. The cuprate superconductor shows quasiparticle interference at low energies,suggestive of plane wave states modified by rare internode scattering, but strongly inhomoge-neous, energy-independent spectra at higher energies. The inhomogeneous, energy-independentspectra could potentially be associated to the “stacked” multifractality, robustly exhibited herefor models with nematic QGD [24].
6. Discussion
The quantum-critical stacking conjecture for protecting class CI, AIII, and DIII surface statesfrom Anderson localization was outlined in detail in Sec. 2. The main predictions tested numeri-cally are summarized in Table 2. Below we quickly recap the key results presented in Secs. 3–5.It is worth emphasizing that at the surface of a 3D topological phase, at energies of order thegap, the 2D surface states deconfine and merge with the bulk continuum. This is not captured bythe pure continuum 2D Dirac theories studied here, which are instead implemented with a hardcuto ff in momentum space. Boundary states of a bulk TSC lattice model in the slab geometrywere studied in [33], with results consistent with the stacking scenarios articulated in this review. AIII.
The numerical analysis in Ref. [33] confirms the class AIII → A IQHPT stacking predic-tion for ν = ff ect implied by the NL σ M-based derivation in Ref. [21] isabsent [see also Eqs. (9) and (11), above]. Remarkably, both ν = , σ xx dc conductivity add support to these results. CI.
In Ref. [34], time-reversal invariant TSCs with full spin-rotation symmetry in class CI whereinvestigated for both small and large winding numbers ν . From the multifractal spectra, the stack-ing conjecture was posited for finite-energy surface states, CI WZNW → class C spin QHPT.26ere we reproduced some of the results for the minimal winding number ν = ff ects.We were able to additionally show that the finite-energy Kubo conductivity is reasonablyclose to the class-C SQHPT prediction (see Table 2). Most important, it trends towards a finitevalue, in contrast to the conventional expectation of Anderson localization. DIII.
Finally, the generic time-reversal topological superconductor with no spin symmetry at allresiding in class DIII also shows clear signs of stacking at finite energy. In Ref. [24], reviewedin Sec. 5, only the minimal ν = λ (since it determines the bare value of the conductance in the conventionalscenario). For QGD, the finite-energy multifractality is found to be approximately parabolic, andthe spectra change little despite the variation of the disorder strength λ by an order of magnitude.In analogy with classes CI and AIII, the natural stacking conjecture in this case relates DIIIWZNW → the class D thermal quantum Hall plateau transition (TQHPT). Although it is naturalto expect a TQHPT with universal multifractal and conductance statistics, no such observationor calculation has yet succeeded indicating this kind of transition in class D, see Sec. 2.4.The coexistence of low-energy plane-waves and finite-energy critical states (with fixed, uni-versal multifractal fluctuations in the latter) over a wide range of disorder strengths means thatmodel (b) is similar to the phenomenology of STM observations in the high- T c cuprate BSCCO[24, 59, 60, 61, 62, 63, 64]. Although not a topological system, the low-energy Dirac quasiparti-cle description of a 2D d -wave superconductor reduces to independent “topological” componentswhen interpair and / or internode scattering is suppressed by hand [24, 25]. The critical stackingfor a single cone is found to be most robust for nematic QGD.A remarkable property of the class CI, AIII, and DIII WZNW topological surface theories isthe precisely quantized longitudinal surface spin or thermal conductivity at zero energy, which isrobust to the presence of both disorder and interactions [7, 27, 31, 81, 88]. We reviewed studies[24, 33, 34] augmented by unpublished numerical evidence indicating an even more surprisingconnection between 2D class C, A, and D quantum Hall e ff ects and 3D class CI, AIII, and DIIItopological superconductors.
7. Outlook and further directions
There are several numerical open tasks:1. Computing the finite-energy Landauer conductance in the other classes CI, DIII. In class CIone can compare against the Cardy result [38] for the class C network model conductance.2. Analyzing higher winding numbers in class DIII systems. The winding ν = ff erent from the WZNW models at higher ν . An important questionis whether the observed θ ε (cid:44) = .
077 (Table 2) is also robust to changes in the windingnumber. 27. Increase system sizes using Arnoldi techniques instead of full diagonalization. The net-work models for QH e ff ects can be studied at more than an order of magnitude larger linearsizes L = O (2000). Network models can be described by sparse matrices. What actuallyenters the algorithm is the (usually not sparse) resolvent operator of the Hamiltonian, thatcan be computed e ffi ciently for sparse Hamiltonians. For generic dense matrices com-puting the resolvent is as hard as full diagonalization. Using the banded structure of theHamiltonian matrix in k -space, it is possible to construct the resolvent operator faster thanin the generic case.4. We need to find evidence for the existence of the conjectured thermal quantum Hall plateautransition in class D in the presence of disorder, and compute the thermal conductivity andmultifractal spectrum in order to compare with the stacked criticality in our DIII results.The current analytical understanding of the stacking phenomena observed numerically isprimitive. Goals include the following:1. Ref. [14] derives an interesting form of bulk-boundary correspondence for all symmetryclasses. A winding number of the critical surface states is constructed and shown to beequal to the winding number of the bulk. A fruitful direction of research would be toinvestigate the applicability of these concepts to the critical stacks at finite energies.2. Another goal is to improve the NL σ M expansion that gives the class AIII WZNW → AIQHPT prediction for odd winding numbers.3. Gain better understanding of the class DIII gravitational theory [113].4. Investigate the e ff ects of interparticle interactions on stacked critical states.Finally, a key problem is to understand the depth of the relationship implied by the stacking ofclass A, C, D topological quantum phase transitions (the Hall plateaux transitions) at the surfaceof class AIII, CI, and DIII TSCs. Although this connection has been revealed indirectly here,through studies of the e ff ects of disorder on TSC surface theories, it suggests a more intrinsic,topological relationship between these classes (which govern topological phases with integer-valued invariants in two and three dimensions). One idea is the following. Is it possible toreproduce the critical statistics studied here by studying an entanglement cut [117] for a clean bulk TSC Hamiltonian? Instead of averaging over disorder configurations at a physical surface,perhaps one can average over bulk quasiparticle band structures .
8. Acknowledgments
We thank our coauthors on the work reviewed here [24, 33, 34]: S. A. A. Ghorashi, Y.Liao, B. Sbierski, and S. M. Davis. JFK acknowledges funding by Graduate Funding fromthe German States awarded by KHYS. MSF acknowledges support by NSF CAREER GrantNo. DMR-1552327.
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