Fragility of surface states in non-Wigner Dyson topological insulators

Topological insulators and superconductors support extended surface states protected against the otherwise localizing effects of static disorder. Specifically, in the Wigner-Dyson insulators belonging to the symmetry classes A, AI, and AII, a band of extended surface states is continuously connected to a likewise extended set of bulk states forming a ``bridge'' between different surfaces via the mechanism of spectral flow. In this work we show that this mechanism is absent in the majority of non-Wigner-Dyson topological superconductors and chiral topological insulators. In these systems, there is precisely one point with granted extended states, the center of the band, $E=0$. Away from it, states are spatially localized, or can be made so by the addition of spatially local potentials. Considering the three-dimensional insulator in class AIII and winding number $\nu=1$ as a paradigmatic case study, we discuss the physical principles behind this phenomenon, and its methodological and applied consequences. In particular, we show that low-energy Dirac approximations in the description of surface states can be treacherous in that they tend to conceal the localizability phenomenon. We also identify markers defined in terms of Berry curvature as measures for the degree of state localization in lattice models, and back our analytical predictions by extensive numerical simulations. A main conclusion of this work is that the surface phenomenology of non-Wigner-Dyson topological insulators is a lot richer than that of their Wigner-Dyson siblings, extreme limits being spectrum-wide quantum critical delocalization of all states vs. full localization except at the $E=0$ critical point. As part of our study we identify possible experimental signatures distinguishing between these different alternatives in transport or tunnel spectroscopy.

Topological insulators are subject to a powerful bulkboundary principle according to which their insulating (yet topologically nontrivial) bulk implies conducting boundaries [1][2][3][4].Examples include the chiral edge states of the quantum Hall effect, the helical edge states of the quantum spin-Hall effect, or the single Dirac cones in the surface spectrum arXiv:2308.12931v2[cond-mat.mes-hall]29 Mar 2024 of a three-dimensional topological insulator.In these three cases, the boundary states extended along surfaces are continuously connected to a band of likewise delocalized bulk states at high energies.The presence of such "bulk bridges" between surface states is behind numerous physical phenomenana that make topological insulators stand out against conventional ones.Examples include anomalous transport, i.e. forces applied along one surface driving currents directed towards another, as in the integer quantum Hall effect, or topological protection against Anderson localization, safeguarding intra-surface conduction at arbitrary values of the chemical potential.These phenomena relate to the principle of spectral flow, whereby adiabatic transport between disconnected surfaces is enabled by a bulk bridge.
In this work we show that spectral flow is not as closely tied to the physics of topological insulators as one might think.On the contrary, we will demonstrate that for the majority of three-dimensional topological insulators outside the three Wigner-Dyson classes A, AI, AII, the two prerequisites for the spectral flow principle -a robustly delocalized bulk state and an uninterruptible band of boundary states continuously connected to it -are absent.(In the tenfold-way nomenclature, the quantum Hall effect is class A, whereas the threedimensional topological insulator resides in class AII.In practice, the non-Wigner-Dyson classes refer to topological superconductors [5][6][7].)The two opposing scenarios -with and without spectral flow -are illustrated schematically in Fig. 1.The fragility of the surface bands in the absence of a spectral flow principle leaves room for a wider phenomenology of surface physics than in the Wigner-Dyson classes.It has profound consequences for the Anderson localization of surface states and, hence, for the observable transport characateristics of topological matter in the presence of disorder.
The absence of a spectral flow principle can be seen from both bulk and boundary perspectives.From the bulk perspective, it is ruled out if the bulk is "Wannier localizable," i.e., if the bulk admits a complete basis of exponentially localized quantum states.From the boundary perspective, its absence means that boundary states can be localized by disorder or that they may be gapped out by a local boundary perturbation.In the topological non-Wigner-Dyson classes, such "fragility" of the boundary states exists away from the distinguished energy E = 0 only.The existence of delocalized boundary states at E = 0 is robustly protected so long as the bulk remains topologically nontrivial [8,9].Conversely, a non-localizable phase possesses at least some Anderson delocalized bulk states above and below the Fermi energy, as well as an uninterruptable band of Anderson delocalized boundary states that are continuously connected (in energy) to these bulk states.The spectral flow principle exists in non-localizable phases only.
The bulk and boundary perspectives are closely intertwined in topological matter.Examples of Wannier localizable phases include topologically trivial band insulators, but also all topological bands in one dimension [10], including topological superconductors [11].(In one dimension, a continuous band of boundary states is trivially ruled out by phase-space considerations.)By contrast, two-dimensional topological insulators 1.(a) A topological free-fermion phase with spectral flow possesses a robust continuous attachment of anomalous boundary states (red) to delocalized bulk states (black) residing in a Wannier nonlocalizable bulk band (grey).(b) Without a spectral-flow principle, anomalous surface bands (red) may in principle be detached from the bulk spectrum by a spectral gap.In this case there is no obstruction to Wannier localization of all bulk states (grey).The figure shows a schematic one-dimensional boundary spectrum, assuming translation symmetry parallel to the boundary.
are non-localizable [1][2][3][4].A case in point is the integer quantum Hall insulator, where the celebrated Laughlin gauge argument [12] shows that a quantized Hall conductance necessitates the existence of delocalized bulk states at energies below the Fermi energy, continuously connected to the edge states.
The bulk bridge states in this case are critically Andersondelocalized in the presence of disorder, and associated to the topological Hall plateau transitions [13,14].Similar arguments have been made for the quantum spin-Hall effect and the three-dimensional topological insulator [15].In this article, we demonstrate that most topological phases of non-Wigner-Dyson type are Wannier localizable and, hence, do not possess a spectral flow principle.In three dimensions, the only exception is the topological class-DIII superconductor for odd values of its integer invariant.In all other cases -classes AIII, CII, CI, and DIII for even values -surface states in these phases are fragile in the sense defined above.
Boundary fragility implies that the surface band can be detached from the bulk band and that surface states at E ̸ = 0 can Anderson localize in the presence of weak disorder, but not that this must happen.In fact, numerical studies of surface states of three-dimensional topological superconductor classes AIII, CI, and DIII revealed surface states that remain delocalized in the presence of disorder at all energies [16][17][18][19].Delocalization was observed as a robust feature of the effective two-dimensional Dirac surface theories of class CI, AIII, and DIII superconductors [16][17][18][19] and it was backed up by numerical studies of a three-dimensional lattice model for class AIII [17].On the other hand, topological arguments [8], or even rigorous mathematical proof [9] for surface state delocalization exist for states at zero energy only -consistent with the notion that boundary states in these classes are fragile for E ̸ = 0.
To answer the question of when fragile boundary states localize in the presence of weak disorder and when they do not, we have considered the case of a "minimal" topological insulator in class AIII in detail, which has a single surface Dirac cone at zero energy.Using a field-theoretic analysis, we show that whether or not surface states localize in the presence of disorder is intimately tied to the presence of surface Berry curvature [20]: The surface states localize for E ̸ = 0 if the surface Berry curvature is nonzero, but not if it is zero.The surface Berry curvature must necessarily be nonzero if the surface band is detached (in energy) from the bulk, in which case the fragility of the surface states is manifest, but it can vanish if there is a continuous connection between surface and bulk bands.
Without exception, models of class-AIII insulators studied in the literature -not only Refs.[16][17][18][19], but also earlier work on class AIII [7] -have vanishing surface Berry curvature and a continuous connection between surface and bulk bands.For these models, it takes the addition of a particular "surface potential" to the Hamiltonian to impose surface Berry curvature and, hence, expose the fragility of the surface bands.We present numerical evidence that, once such a potential is included, all surface states away from zero energy are localized in the presence of disorder, albeit with a localization length diverging at zero energy.
Even if the surface band structure without disorder does not possess Berry curvature, a sufficiently random disorder potential itself contains terms that would induce it.A spatially inhomogeneous potential will then parcel the surface into domains of positive and negative Berry curvature.Whereas states inside each domain may localize, domain boundaries host onedimensional chiral edge states, similar to the chiral edge states between domains of different filling fraction in the quantum Hall effect.These edge states form a percolating network if the spatial average of the Berry curvature is zero, leading to critical delocalization of surface states at all energies.This is precisely the "spectrum-wide quantum criticality" of quantum-Hall plateau-transition type previously observed numerically in Refs.[16][17][18][19].The spectrum-wide delocalization of Refs.[16][17][18][19] therefore reflects a statistical symmetry of a model with zero average surface Berry curvature, not a topological obstruction to Anderson localization.Physical properties of boundary states -such as their spatial structure at a given energy and the resulting conduction properties -are commonly addressed in terms of effective low-energy approaches that zoom in on linear crossing points in the boundary spectra.Employing such Dirac, or "k • p" approximations, physically relevant parts of the boundary spectrum are thus described by minimal models of manageable complexity.Our findings show that such Dirac descriptions can be a dangerously oversimplification if they are used to capture localization properties of topological surface states in the non-Wigner-Dyson classes.Case in point is the minimal Dirac theory with chiral symmetry, which is the effective description of topological surface states of an insulator in class AIII used in the numerical studies of Refs.[16][17][18][19].This effective two-dimensional description is strictly without Berry curvature and, therefore, unable to capture the geometric effects responsible for surface-state localization of the full three-dimensional insulator.

A. Outline
The remainder of this article is organized as follows: In Sec.II we present a general argument showing that Wannier localizability of the bulk implies the possibility to spectrally detach surface states from the bulk -thus underlining the equivalence the bulk and boundary perspectives on the prerequisites for the spectral flow principle.We also state the Wannier localizability status of all tenfold-way symmetry classes in dimensions up to three.In Sec.III we derive the same conclusion for the Wannier-localizable class AIII from the boundary perspective.In Sec.IV we consider a canonical fourorbital topological-insulator lattice model for the class-AIII insulator in three dimensions [7], analogous to the model that was analyzed in Ref. [17] to demonstrate spectrum-wide delocalization.For this model, we show that surface Berry curvature can be induced and the surface bands can be detached from the bulk bands at a high energy by the addition of a suitably chosen potential.
Sections II, III, and IV consider topological insulators without disorder.Yet, the implications of the findings of these Sections have profound consequences for the localization properties of surface states in the presence of disorder.Disorder is considered in Sec.V, where we present numerical evidence for Anderson localization of the surface states of a 3D class-AIII insulator, if (and only if) the average surface Berry curvature is nonzero.We also discuss ramifications of our results for the three-dimensional topological superconductors.The field-theoretic analysis, which relates surface-state localization in the presence of disorder to the presence of surface Berry curvature, follows in Sec.VI.We conclude in Sec.VII.

II. SPECTRAL FLOW: BULK PERSPECTIVE
In the following, we show that Wannier localizability of a topological phase necessarily implies absence of the spectral flow principle.Furthermore, for class AIII in three dimensions we demonstrate its Wannier localizabilty, whereas for the remaining symmetry classes we only state the final results.In the next section we will then discuss gapability of surface states.

A. Wannier localizability implies absence of spectral flow
A free-fermion insulator is described in terms of a set of Bloch bands.It is defined to be Wannier localizable if the subspaces defined by conduction and valence bands admit bases of states, |Ψ Rα ⟩, exponentially localized around centers R, where α is an additional index.Wannier localizability implies the absence of spectral flow, as the ground state defined by the occupied bands can be represented in terms of individually flux-insensitive states.Wannier localizability also is in contradiction to the presence of protected delocalized states.Finally, Wannier localizability is a weaker condition than retractability to an "atomic limit" [21][22][23][24][25][26][27][28].The possibly small but finite exponential overlap between neighboring unit cells remains essential to the definition of ground state topology [29], and to the stabilization of anomalously delocalized surface states.
While the Wannier states |Ψ Rα ⟩ are not, in general, eigenstates of the parent Hamiltonian, it is straightforward to show [30] that a symmetry and gap preserving deformation can be employed to bring the latter into the form with parameters ϵ Rα assuming the role of state energies.Now consider a system with surfaces, Wannier localizable in directions transverse to them.(The existence of states extended along the surfaces in topological insulators excludes unconditional localizabilty.)Referring to the above representation, we consider the decompostion [31] where the surface Hamiltonian, H ∂ , is the contribution of states to (1) with centers within a Wannier localization radius of the surface, and H bulk its complement.We now have the option to rescale H ∂ → λH ∂ to shrink the surface band defined by H ∂ to a width narrower than the band gap implied by H bulk , see Fig. 2(b).This construction demonstrates the topological equivalence of our system to one with surface bands detached from the bulk bands by gaps of adjustable width.Referring to section IV for a microscopic realization of this construction, we anticipate that detachable surface bands in the above sense are best suited to address the surface phenomenology of insulators without protected spectral flow.

B. Case study: AIII insulator in three dimensions
The arguments of Sec.II A suggest that a continuous attachment of boundary states to the bulk spectrum and the existence of bulk delocalized states are flipsides of the same coin.The two-dimensional class A insulator is case in point for a situation where both exist.We now turn to the opposite situation, as realized in the three-dimensional AIII insulator, where spectral flow and delocalized bulk states are generically absent.
A Hamiltonian in class AIII can be written as where A is a complex square matrix acting on the subspaces defined by the condition Γ = ±1, and Γ = τ z defines the chiral symmetry S. For example, in a lattice system the subspaces corresponding to Γ = ±1 may define two bipartite sublattices.We assume that H and, hence, A are local matrices: The matrix elements ⟨Rα|A|R ′ α ′ ⟩ between atomic orbitals α and α ′ at lattice sites R and R ′ decay exponentially with the distance |R − R ′ |.
For an insulator subject to periodic boundary conditions the spectrum of H has a finite gap around zero energy.Hence, following Ref.[7], we may deform the Hamiltonian (3) by sending the positive (negative) eigenvalues of H to 1 (−1).Such a flattening deformation does not change the bulk topology and preserves the locality of the Hamiltonian matrix.It defines the Hamiltonian Locality of H f implies that U is a local unitary operator.(U is unitary because H 2 f = 1 1.)We may then easily construct a basis of localized eigenstates at energy ±1 [29], where α is an additional index.(Eigenstates |Ψ ± Rα ⟩ are localized near lattice site R because the unitary operator U is local.)This simple construction proves the existence of a basis of localized eigenstates of H f , regardless of the underlying topology [32].Since the (many-body) ground states of H and H f are the same, Eq. ( 5) proves Wannier localizability of this topological phase.The discussion of the previous subsection then implies the corollary that surface and bulk spectra can be separated from each other.
For later reference we mention that the Wannier functions {|Ψ η Rα ⟩} [η ∈ ±, Eq. ( 5)] and, hence, the decomposition of H f into surface and bulk contributions as in Eq. ( 2) is not unique: Each local unitary matrix V generates another set of Wannier functions {|Ψ η,V Rα ⟩} by multiplying both elements of the two-component spinor in Eq. ( 5) by V .In Sec.IV C, we demonstrate that the surface states described by H ∂ in Eq. ( 2) may have a nonzero Chern number Ch.This number is constrained to have the same parity as the winding number of the bulk Hamiltonians H and H f .However, for a given parity its (integer) value depends on the choice of the localized basis for the bulk states: basis changes |Ψ ± Rα ⟩ → |Ψ ±,V Rα ⟩ lead to a change [33] where, for a periodically extended definition of the local transformation, ν[V ] ∈ Z is the third winding number of V over momentum space.(An analogous relation was found for the surface response theory of a three-dimensional topological insulator [34].)

C. Other tenfold-way classes
The discussion above shows that there are two different classes of topological insulators: those with and without a spectral flow principle.Given that this dichotomy presides over the spectrum-wide robustness of boundary states, it seems necessary to tag each entry in the periodic table of topological insulators and superconductors [5][6][7] according to its Wannier localizability status.Referring for the full classification program to the upcoming publication [35], Table I summarizes the result for dimensions up to three.It adds to the topological status of a given symmetry class and dimensionality (Z, Z 2 , or 0) information on the localizability of its states: The absence of topology is denoted by the entry "0," topological classes with symmetry-compatible exponentially Wannier localizable bulk states are labeled by "×," and those with non-localizable bulk by "✓." The information provided by Table I confirms that the three Wigner-Dyson classes A, AI, and AII are always nonlocalizable, provided they are topological in the first place.Indeed, since for the Wigner-Dyson classes one is free to choose a reference energy inside the bulk gap, the existence of a topologically protected boundary state at one energy implies that such states must exist for all energies inside the gap.On the other hand, in the non-Wigner-Dyson classes, the presence of charge-conjugation symmetry C and/or chiral symmetry S forces the spectrum to be mirror-symmetric around the distinguished energy E = 0.The equivalence of all energies inside the gap therefore no longer holds, and the presence of a protected uninterruptable band of boundary states must be reconsidered.
For the non-Wigner-Dyson classes, what determines Wannier localizability is whether C or S are essential for the topology, or whether they are "spectator symmetries" and the topol-ogy of the bulk Hamiltonian remains nontrivial if all constraints imposed by C and/or S are lifted.In three dimensions, class AIII is an example of the former category, which we refer to as "genuine" non-Wigner-Dyson classes.Other examples of genuine non-Wigner-Dyson classes are classes BDI and D in one dimension and classes CI and CII in three dimensions.Examples of non-genuine non-Wigner-Dyson classes are classes C and D in two dimensions, which have chiral edge states [36,37], and which remain topological if the particle-hole symmetry is lifted.Class DIII in three dimensions, which describes time-reversal-invariant superconductors with broken spin-rotation symmetry is a special case, as it is a localizable genuine non-Wigner-Dyson class only if the topological invariant is even.
The Wannier localizability of one-dimensional topological insulators (second column in Table I) is well known in the literature [10].Examples are the Su-Schrieffer-Heeger model (class AIII) [38][39][40] and the topological superconductor Kitaev chain (class D) [11], which in their nontrivial phases have localized bases stretching across adjacent unit cells.The possibility of Wannier localizable topological superconductors in dimensions larger than one was mentioned by Ono, Po, and Watanabe [41] in the context of topological superconductors with additional crystalline symmetries (see also Ref. [42]).Furthermore, the existence of exponentially localized Wannier basis for topological phases with (possibly higher-order) winding number invariant was discussed in Ref. [43].
In Sec.II A, we have shown that Wannier localizability of the insulating bulk implies that a gap in the surface spectrum may be opened by a perturbation at the surface.Similarly, away from the energy E = 0, surface states of a Wannier localizable insulator may undergo Anderson localization in the presence of disorder.We refer to both properties as the fragility of surface states.It is important to point out that such fragility only signals a possibility of a gap opening or of Anderson localization; it does not imply that a gap in the surface state spectrum must open or that surface states must localize in the presence of disorder.The question, whether or not surface states of a specific band structure will Anderson localize in the presence of disorder remains to be answered.Either way, fragility leaves room for a wider spectrum of surface phenomenologies than in insulators with protected surfaces.We will consider such phenomena in concrete detail for the case of a "minimal" topological insulator in class AIII in Secs.V and VI.

III. SPECTRAL-FLOW: BOUNDARY PERSPECTIVE
In the previous section, we considered the status of surface states on the basis of a connection to bulk states via the spectral flow correspondence.We here approach the question from a complementary perspective, which focuses entirely on the surfaces themselves.

A. General considerations
The vicinity of the Fermi crossing points at topological insulator surfaces is often described in terms of effective Dirac Hamiltonians, which in the two-dimensional case assume the form where k i are two momenta along the surface measured relative to the crossing point k = 0, and the two Gamma matrices satisfy {Γ 1 , Γ 2 } = δ ij .Additional Gamma matrices may appear at higher order in k or as prefactors of a random surface potential.At large momenta, the Hamiltonian ( 7) is ultraviolet divergent.These divergences cannot be cured by embedding H b into the periodic structure of a two-dimensional Brillouin zone, reflecting that the surface theory does not define the lowenergy limit of a two-dimensional stand-alone lattice model.
In the context defined by Eq. ( 7), the spectral flow principle translates to the statement of an anomaly: Coupled to an external vector potential, H b , supplemented with an ultraviolet regularization, lacks gauge invariance.The absence of gauge invariance indicates (quasi-)particle number nonconservation.Specifically, under adiabatic insertion of a flux quantum through the bulk, high-lying boundary states get pushed up in energy, leading to a drain out of the window of momentum states below a fixed cutoff.If the spectral flow principle applies, overall particle number conservation is restored by conversion of boundary states into bulk states and eventually to states at the opposite surface.The observable consequence is adiabatic transport from one boundary to the other, i.e., the quantized transverse conductance characteristic of topological insulators.
In the previous Section, we have argued that the spectral flow principle does not apply to all symmetry classes.How can this be reconciled with the intrinsic absence of an ultraviolet closure of the Dirac surface theory?To resolve the anomaly of the latter there must exist a "sink" of highlying states absorbing spectral weight being pushed up by an anomalous gauge operation.If the spectral flow principle applies, these are extended bulk states.If it does not apply, implying that the boundary states can be detached from the bulk, these states must be supported by the boundaries themselves.
In the following we illustrate these complementary scenarios on two case studies, class A in two, and class AIII in three dimensions.In either case, the focus will be entirely on the boundaries, no explicit reference to bulk states is made.

B. Case study: Class-A Chern insulator in two dimensions
Two-dimensional class A is the paradigmatic example of an insulator with spectral flow, as realized, e.g., in the physics of the integer quantum Hall effect.In this case, the boundary theory (linearized around any Fermi energy) is governed by the effective Hamiltonian describing a single branch of chiral fermions.Assuming zero temperature, and the Fermi energy at E = 0 for convenience, states with k < 0 are occupied.Coupling the system to a gauge potential representing adiabatic magnetic flux insertion through the bulk, k → k + A, causes an upward shift of all levels.After the insertion of one flux quantum, the full quantized single particle spectrum is restored, but the occupations of the states have changed, as shown schematically in Fig. 3(a).Specifically, one occupied state previously at k < 0 now occupies the lowest state at k > 0. By repeated insertion of flux quanta, the range of occupied states will extend up to arbitrarily high energies and, eventually, a state previously sitting at the upper cutoff of the low-energy theory gets pushed beyond it [see the arrows in Fig. 3(a)].
Topological features must be stable with respect to arbitrary perturbations at the boundary.To see how this robustness manifests itself in the anomaly of the boundary Hamiltonian (8), consider adding a band of trivial localized boundary states at energy ε c > 0. Assuming weak coupling γ to the chiral band, the boundary Hamiltonian generalizes to The coupling matrix element γ now generates an avoided crossing between the chiral and the flat band, see Fig. how a construction similar to the one above leads to very different conclusions.

C. Case study: Class AIII insulator in three dimensions
The minimal surface theory of a class AIII insulator with winding number one is described by a two-dimensional generalization of Eq. ( 8), where the chiral symmetry is realized as Again, this Hamiltonian has no ultraviolet closure.(To see why, note that the off-diagonal element k x − ik y defines a winding number around the origin in two-dimensional k-space.This is incompatible with the k-space periodicity required by a genuine two-dimensional lattice Hamiltonian.)A continuous deformation of this two-band Hamiltonian cannot open a gap at zero energy.The required perturbation would have to be proportional to τ z , in violation of the chiral symmetry.The conduction and valence band then connect the single touching point k = 0 to the ultraviolet divergences at large momentum; Within this two-band representation, the dispersion is continuous without gap openings at finite momenta.It is for this Hamiltonian, augmented with a chiral symmetry respecting random vector potential, that Refs.[16,17,19] established a spectrum-wide resilience to Anderson localization.
As in our previous discussion of class A, we now introduce a band of localized surface states at energies ±ε c .As a two-dimensional surface Hamiltonian analogous to Eq. ( 9) we consider where τ ± = τ x ± iτ y .As in class A, the band-coupling γτ − defines an avoided crossing between the localized and the linearly dispersive bands, see Fig. 4(b).However, unlike in class A, the continuous interpolation of the bands at the boundaries of the Brillouin zone no longer presents an obstruction to the opening of a global gap, see Fig. 4(b).This option to disrupt the surface spectrum was to be expected from the discussion of bulk Wannier localizability in Sec.II, but here follows from inspection of the surface alone.We note that the Hamiltonian (11) still has, and needs to have, an ultraviolet divergence.However, unlike in class A the repeated insertion of bulk flux quanta no longer leads to a spectral flow anomaly: the occupied states get shifted as indicated by the arrows in Fig. 4(b), and eventually fill the large-k part of band of localized boundary states.However no particles reach the upper energy cut-off.This observation indicates that the ultraviolet divergence of effective surface Hamiltonians -which reflects their non-existence without a supporting bulk -does not necessarily imply spectral flow from the surface into the bulk.
In the literature, topological properties that are robust to the addition of trivial bands are called "stable".The sensitivity of the minimal model to the addition of bands, i.e., the opening of gaps and the disruption of spectral flow, should therefore be considered a manifestation of "unstable" or "fragile" topology: the conclusion is that minimal models are insufficient to fathom the full spectrum of phenomenologies displayed by the surfaces of three-dimensional AIII insulators.
Although the concrete worked-out example here is for an AIII insulator with winding number ν = 1, the general conclusion about the fragility of the spectrum-wide protection of the surface bands in a minimal effective 2 × 2 theory equally applies to higher winding numbers.A 2 × 2 surface theory with a winding number larger than one may be realized as a nodal point involving higher powers of the momentum k [16] or by having multiple nodal points at different locations in reciprocal space.In either case, the combination of chiral symmetry and the restriction to two-component spinors precludes the interruption of the surface states by a spectral or mobility gap.
Below Eq. ( 10), we linked the absence of intrinsic UV regularization of effective surface theories to the presence of a winding number.It is interesting to observe that the isolation of a detached surface band introduces another invariant, namely a two-dimensional surface Chern number [33,44].Describing the momentum space topology of the band through the map k → |α k ⟩, where k runs through the two-dimensional Brillouin zone, and |α k ⟩ are the positive energy states of the finite band indicted in Fig. 4b, we define the Berry curvature and from it the Chern number, This is the surface Chern number mentioned previously in section II B; the prefactor of 2 accounts for the identical contributions of the positive-and negative-energy surface bands, which are related by chiral symmetry.While the numerical value of Ch may vary depending on the realization of the coupling between the trivial and the chiral bands, its parity is determined by that of the bulk winding number.For further discussion of this point, we refer to section IV C.

IV. AIII INSULATOR IN THREE DIMENSIONS
In the previous two sections we discussed generic features of topological insulator bulk states and of their asymptotically linearizable surface spectra.We will now turn to a more concrete analysis and discuss the band structure of a microscopic model in class AIII.In the next section we then generalize to the presence of static disorder and discuss localization properties of the model's surface states.
We start our discussion with the definition of the model in subsection IV A. Following standard protocol, we then project down to its low-energy Dirac approximation in subsection IV B. [Impatient readers may just take notice of the two principal definitions ( 16) of the lattice model and ( 24) of its Dirac approximation, and directly proceed to Sec.IV C.] On the basis of these model definitions, we then discuss the consequences of the absence of a spectral flow principle in Sections IV C-IV D.

A. Lattice model
We consider a cubic lattice model with four orbitals per site defined by the Bloch Hamiltonian [7] where we set the hopping strength, which is parametrically of the same order as the total band width, to unity for convenience.The Pauli matrices σ a and τ a act on two independent degrees of freedom of the 4 = 2×2 orbitals in the unit cell.An application of the standard mapping [7] between chiral lattice Hamiltonians and three-dimensional winding number invariants, ν, shows that The Hamiltonian ( 14) is invariant under the symmetry operations S and C, with Here T denotes the matrix transpose, M S = τ z , and M C = τ y σ y .We note that the combination CS = T of chargeconjugation and chiral symmetry satisfies T 2 = −1, putting our system into class DIII [45].
To define a class AIII model, we break C, while preserving S. For our purposes, it will be convenient to realize this symmetry breaking by adding C-breaking disorder.To this end, we turn to a real space representation of the model (14), which reads with where the R are lattice vectors on the cubic lattice, and the e a , a = x, y, z unit vectors in the lattice directions.The amplitudes t a R , constant in the clean case, are now chosen as, where a ∈ {x, y, z} specifies the direction of the nearestneighbor bond, and the {ϕ a x } are static random phase variables with variance [46] The effects of this disorder on the (de)localization properties of the surface states will be discussed in Sec.V.

B. Surface Dirac approximation
A bulk winding number ν generically implies the appearance of |ν| species of gapless Dirac fermions at the surface [7].Specifically, we consider a ν = 1 realization of the model (14) with 1 < M < 3, a vacuum interface at x = 0 in the xdirection, and infinite extension in yand z-directions.In this case, a Dirac surface state appears in the surface Brillouin zone at (k y , k z ) = (0, 0).
Considering µ ≡ 3 − M , with 0 < µ ≪ 1, a continuum approximation near the bottom of the band leads to the effective Hamiltonian with The zero modes of H 0 are where m ∈ {↑, ↓} denotes the polarization of the surface state, |m z ⟩ τ and |m x ⟩ σ are eigenspinors of τ z and σ x , respectively, and |ψ⟩ is an envelope function decaying exponentially into the bulk in the x-direction.The projection of the transverse part of the Hamiltonian H x into the space of zero modes gives where k ≡ {k z , k y } and Γ = {Γ x , Γ y } are the standard Pauli matrices acting in the space of zero modes.
Equation (24) defines the minimal two-component massless Dirac approximation to the surface states of the bulk model in Eq. ( 14) with winding number ν = 1.As discussed in Sec.III, the minimal Dirac surface theory has a fragile obstruction to localization, which is lifted if additional degrees of freedom are added to the surface theory.In Sec.III the additional degrees of freedom were added in the form of a trivial band of localized states at the surface.In the next subsections, we will show that the inclusion of the quantum geometric structure of the high-lying states in the full three-dimensional theory has the same effect.

C. Detaching and characterizing surface bands
The model ( 14) has a surface band that is continuously connected to the bulk, see Fig. 5(a).We add a term where the R-summation runs over the outermost n surface layers.(We set n = 1 or n = 3 in our numerical calculations.)The perturbation U f breaks C and T = CS symmetries, but preserves the chiral symmetry S. We refer to this perturbation as the "fragmenting surface potential."For values |u f,x | > u c f , we observe the opening of an indirect global gap between the x-surface band and bulk bands, see Fig. 5(b).(Detaching surface and bulk bands for surfaces perpendicular to the yand z-directions would require nonzero u f,y and u f,z , respectively.)The threshold parameter u c f depends on how many surface layers are perturbed by the fragmenting surface potential, see Fig. 5(c).However, regardless of its particular value, the present construction demonstrates the absence of spectral flow.
The band geometry of the surface band is characterized by its Berry curvature Ω k , see Eq. ( 12).If the surface band is fully detached from the bulk, the integral of Ω k over the Brillouin zone is well-defined and gives the integer Chern number Ch in Eq. ( 13).The parity of the Chern number must be the same as that of the winding number ν [47], but its (integer) numerical value depends on the sign of the potential (25) used to separate the surface band from the bulk, In the language of Sec.II B, the sign of u f,x thus corresponds to two different bulk gauge choices for the detachment of surface and bulk, see Eq. ( 6).Alternatively, in the language of the phenomenological surface theory of Sec.III C, different signs of u f,x represent different perturbations coupling the dispersing surface band and the degrees of freedom of the external band, see Eq. (11).The parity constraint on the Chern number of the surface band implies that for the minimal class-AIII insulator, detaching the surface band is not possible without inducing surface Berry curvature.
In Sec.VI we show that the question whether or not states in a surface band with winding number one (as is the case for the model we investigate here) are localized at energy E in the presence of weak disorder can be answered by considering the integrated Berry curvature carried by the states with energy ε k between 0 and E. Hereto we define The field theoretical analysis of such a system in the presence of disorder, see Sec.VI, then shows that states are delocalized at energy E if Since θ(0) = π, this condition is consistent with the topological surface states at E = 0 being delocalized.f , surface and bulk bands are detached, so that θ(E) can be determined for the full surface band; for u f,x = 0, θ(E) can be calculated only for energy E inside the bulk gap.(b) The same as panel (a) but for the minimal surface theory (10) (red) and the non-minimal model (11) (black) with |γ|/εc = 0.05.According to the criterion of Ref. [20], surface states at energy E are delocalized if θ(E) = π mod 2π.
Fig. 6(a) shows the angle θ(E) as a function of E for u f,x = −0.3 and u f,x = −1.0 and with the fragmenting perturbation supported on the three outermost layers (n = 3).We note that the presence of U f has little effect on the surface Berry curvature for energies close to E = 0 [48].This is consistent with the absence of Berry curvature in the twocomponent Dirac approximation, which in turn is a consequence of the chiral symmetry (i.e., the absence of terms on the diagonal of the 2 × 2 matrix operator.)However, we observe significant deviations from θ(E) = π for energies approaching the edges of the bulk bands.In the light of our discussion of section III C, these reflect the onset of effective hybdridization with extraneous bands, whose role in the present context is assumed by the bulk.
For comparison, in Fig. 6(b) we also show θ(E) for the two-dimensional band that was obtained by coupling a surface Dirac cone to a localized surface band, see Eq. (11).Analogous to the detached surface band of the full 3d model shown in Fig. 6(a), the Berry curvature is concentrated mainly near the band edges.

D. Chiral-symmetric chiral modes: surface Hall conductance
We have seen that addition of the perturbation (25) detaches the surface bands from the bulk and that the now isolated surface band has a nonzero Chern number Ch, which for the present ν = 1 model is given by Eq. (26).For a surface perpendicular to the x direction, it is interesting to ask what happens if u f,x changes sign, for example along an intra-surface domain wall.As we will see, the ensuing phenomenology is key to the understanding of the disordered system discussed in Secs.V and VI.
To explore this situation in the simplest possible setting, we consider a flattened version of the model (14).The latter is obtained from the Hamiltonian ( 14) by keeping its Bloch states unchanged, while sending the energy eigenvalues to ±1.To describe an insulator with a surface, we then switch to the position representation and impose open (or vacuum) boundary conditions at two coordinates in the x-direction.More specifically, we consider an annular cylinder geometry with two surfaces in the radial x-direction, periodic boundary conditions in circumferential y-direction, and the cylinder axis in z-direction.
Figure 7 shows that for a constant fragmenting potential u f,x , see Eq. ( 25), the flattened model shows a global gap between surface (red) and flat bulk (black) bands.However, the spectrum becomes more interesting, once we introduce two surface domain walls parallel to the z-direction where u f,x switches sign.(Periodicity in y-direction requires the presence of two of these.)In this case, we observe the formation of two counterpropagating chiral modes bound to the respective domain walls.The spectrum of these modes, indicated in green in Figure 7(a), connects the surface and the bulk spectrum.
In the clean model, these chiral modes are supported only by states inside the high-lying band gap.Below it, they hybridize with the extended surface states.In the presence of disorder, the surface states at E ̸ = 0 will localize, but the chiral edge modes do not, so that, for sufficiently strong disorder, the chiral modes will eventually extend their support over the entire surface spectrum, as shown in Fig. 7(b).
At energies E ̸ = 0 the chiral symmetry is effectively broken, so that we may think of each surface as a twodimensional system in class A. In this reading, the counterpropagating chiral modes at the domain walls surrounding a surface region with a different sign of the fragmenting potential acquire a status equivalent to the edge modes of a quantum Hall insulator, and Laughlin's gauge argument applies.It requires that the branches of chiral modes must eventually hybridize with extended states, at E = 0 as well as at high energies.The extended states at E = 0 are the topologically protected delocalized surface states of the class-AIII insulator.At high energies, the chiral modes must hybridize with delocalized bulk states or with energetically high-lying delocalized surface states.Either way, the presence of the domain wall modes prevents a full localizability of all states at large energies.
In anticipation of our later in-depth discussion of disorder, it will be rewarding to link the presence of chiral edge modes to transport coefficients.To this end, consider a surface geometry where a puddle of given value of u f,x is surrounded by an outer region with u f,x of opposite sign.The presence of a chiral edge mode at the puddle boundary implies that a fictitious four-terminal measurement of its Hall conductance would yield the result σ(E) = sgn(E) sgn(u f,x )/2 for all energies inside the mobility gap where the chiral mode exists.(The factor 1/2 reflects the fact that the surface is governed by a single Dirac fermion species, with its characteristic halfintegral transverse conductance.Chiral symmetry requires σ(E) to be an odd function of E [49].At the band center, σ(0) = 0, again by chiral symmetry.).Now imagine a profile u f,x (y, z) (in the n outermost layers) smoothly varying in such a way that the spatial average  14) with open boundary conditions along x and two symmetrically placed domain walls and periodic boundary boundary conditions along y.The fragmenting potential is of the form u f,x (x, y), where u f,x (x, y) switches from 0.2 to −0.2 and back at the two domain walls in the n = 1 outermost surface layers, and is zero otherwise.The bulk, surface, and domain-wall part of the spectrum are indicated in black, red, and green, respectively.Solid and dashed green curves are for the domain wall for which u f,x goes from negative to positive and from positive to negative upon increasing y, respectively.Note that the addition of the fragmenting potential to the surface raises part of the surface spectrum above the flattened bulk band.(b) In the presence of disorder, only surface states at E = 0 are delocalized.In this case, the chiral domain-wall modes are expected to extend from the bulk spectrum all the way down to zero energy.
of u f,x vanishes and puddles with fragmenting potentials of opposite sign form with equal probability.Since each puddle is surrounded by its own chiral mode, we expect the formation of a network in which co-and counterpropagating loops occur with equal likelihood.This system is topologically equivalent to the Chalker-Coddington network [50] of the integer quantum Hall effect at criticality.At this point, the network model predicts the percolation of quantum states evading Anderson localization in the presence of even strong disorder.This simple picture -which is the mechanism behind the "statistical topological insulator" [51] -is compatible with the observation of model realizations with a spectrum-wide existence of delocalized surface states [17].(The same statistical mechanism underlies delocalization of a topological-insulator surface in a random magnetic field [52] and of the surface states of a weak topological insulator [53][54][55].)On the other hand, we expect localization of finite-energy surface states if a nonvanishing average of the fragmenting potential causes an imbalance between puddles with opposite signs of u f,x .Note that these predictions are consistent with the expectation that it is the presence or absence of Berry curvature, corresponding to the presence or absence of an average surface fragmenting potential, that decides over localization.In the next section, we will back these hypotheses by a quantitative analysis of disorder.

V. SURFACE LOCALIZATION PROPERTIES
Previous sections demonstrated that the surface states of a class AIII topological insulator can be detached from the bulk.
Concomitant with the opening of the spectral gap, the 2D surface acquires a nonzero Chern number due to induced surface Berry curvature, as explicated above in Secs.III C and IV C.
Here and in the next section, we consider the implications of surface Berry curvature for the Anderson localization properties of the surface states in the presence of symmetrypreserving disorder.Since the minimal 2-component surface Dirac theory is void of curvature, we work with a slab of the 3D lattice model defined in Eq. ( 16).Surface Berry curvature is induced along the slab boundary via the fragmenting potential introduced in Eq. ( 25), above.
We demonstrate below (see Fig. 8) that in the presence of a nonzero uniform fragmenting potential all surface states are localized by weak disorder, except the zero-energy one, which remains topologically protected from Anderson localization.By contrast, spectrum-wide criticality -i.e., critical delocalization linked to the plateau transition of the quantum Hall effect [17] -survives when either (a) the surface fragmenting potential is set to zero, or (b) the surface fragmenting potential is randomly distributed with zero mean.Scenario (b) explains the origin of spectrum-wide criticality as the percolation of chiral edge modes discussed in the end of the previous section.
We discuss detectable ramifications of our results for experiment in Sec.V B.

A. Surface Berry curvature and disorder: Numerics
We perform a numerical study of localization in a disordered version of the model ( 14), with and without a uniform fragmenting potential U f , see Eq. (25).Disorder is implemented by random Peierls phases as introduced in Eq. (18).We apply multifractal analysis to decide whether the surface wave functions are localized or delocalized [14,56].
It turns out that the spectral weight of surface wave functions is dominantly (> 75%) concentrated on the outermost surface layer.We define the surface inverse participation ratios (IPR) of these wave functions via the moments, where ψ E σ (y, z) ≡ ψ E σ (x = 0, y, z) are 3d wave functions of energy E evaluated at x = 0.The IPRs are normalized such that P E 1 = 1.To improve statistics, we consider P E q averaged over disorder and a small window of energy.(For system sizes from N y = N z = L = 16 to L = 128, the number of wave functions over which the moments are averaged ranges from 3 × 10 5 to 10 3 .)Details concerning the convergence of our data with the slab thickness and the distribution functions of the IPR are provided in Appendices B 2 and B 3.
For surfaces of large linear extension L [57] the IPRs are expected to asymptotically scale as 8.The scaled exponent, ∆q/q(1 − q) for q = −0.5, 0.5, 0.75 (left to right) and system sizes Nx = 8 and Ny = Nz = L = 24 to L = 128 of the model (14).Black data: no fragmenting potential u f = 0 [Eq.( 25)] and disorder strength W = 0.15; Blue data: a constant potential with u f = 0.3 applied to the surface layers, and disorder strength W = 0.2; Green data: random fragmenting potential with zero average and standard deviation u f = 0.3 [Eq.( 34)], and disorder strength W = 0.2.The horizontal dashed lines marks the value 0.25 of quantum Hall criticality, and the localization limit, τq = 0 corresponds to ∆q/q(1 − q) → ∞, 4, 8/3 for q = −0.5, 0.5, 0.75 .(The value ∞ reflects the formal divergence of the IPR in the localized limit for negative q.) with an effective dimension τ E q [14].Multifractality manifests itself in the appearance of a non-trivial anomalous dimension measuring deviations from the naive dimension d(q − 1) expected for uniformly distributed states.The opposite extreme is that of localized states, for which τ q = 0, reflecting the absence of scaling in system size.Presently, we are discussing a system with two distinct realizations of critical points.The first is the mirror symmetric point, E = 0, marking the position of a topologically protected critical state.In the vicinity of this point, the system is expected to show multifractality with the anomalous dimensions [14,58], where Θ AIII is a non-universal coefficient depending on the disorder strength.The second is the quantum criticality otherwise realized by states sitting at the center of Landau levels in quantum Hall systems.For these states, the spectrum of scaling dimensions is approximately parabolic [13] with Θ QH ≃ 0.25 [14,[59][60][61].In either case, the spectrum is expected to be approximately parabolic up to a threshold |q| ≃ q c = 2/Θ [14,58,62].
Figure 8 shows the anomalous multifractal exponent ∆ q /q(1 − q) for q = −0.5, 0.5, 0.75 (left to right) and vanishing (dashed), constant (blue) and random (green) fragmenting potential.The different curves show data obtained for increasing system sizes N y = N z = L = 24 to 128 as a function of energy, E .Numerically, we calculate an effective L-dependent multifractal exponent, with a finite logarithmic difference between IPRs of increasing system size L. The details of this procedure are delegated to Appendix B. At E = 0 we obtain Θ AIII ≈ 0.85 ± 0.2 (u f = 0), 0.32 ± 0.05 (constant u f ), and 0.18 ± 0.03 (random u f ).Here u f ≡ u f,x determines the fragmenting potential U f (25).The approximate independence of these values on the value of q and L indicates that we are observing the anomalous dimension (32) of the E = 0 quantum critical point.Away from E = 0, our results sensitively depend on the chosen model for u f : Zero fragmenting surface potential: For u f = 0 the data quickly drops to the value Eq. ( 33) expected at quantum Hall criticality.This value is maintained, including for large energies inside the bulk gap.In this way we confirm the observation of spectrum-wide criticality of Ref. [17].
Constant fragmenting surface potential: Upon application of a constant u f = 0.3, we observe a clear tendency away from criticality and towards localized behavior upon increasing the system size.We note that the perturbation of strength u f = 0.3, presently applied to only one surface layer, is by a factor of two below the threshold u c f ≃ 0.6 required to induce an indirect gap below surface and bulk band, see Fig. 5(c).(We restrict our attention to |u f | < u c f , because larger values require stronger disorder to observe effects in finite size.)This finding is consistent with the expectation that fragility of the surface-bulk connection -and consequently eigenstate localization at large length scales -will be induced by any constant non-vanishing u f .
Random fragmenting surface potential: Motivated by the scenario laid out at the end of Sec.IV D, we consider a spatially varying surface deformation u f (x, y, z) of unit layer depth, vanishing average, and variance for x = 0 and x = N x .The green data shows that this perturbation leads to delocalized and quantum Hall critical behavior at finite energies, much as that of the unperturbed model.However, the convergence towards the quantum-Hall exponent is slower than in the absence of a fragmenting potential.
For further discussion of this point we refer to Appendix B 2. We conclude from the three sets of data (black, blue, green) presented in Fig. 8 that an arbitrary perturbation of the surfaces is not necessarily sufficient to localize the finite-energy states.Instead, a deformation that induces a nonzero average surface Berry curvature [e.g., ⟨u f (x, y, z)⟩ y,z ̸ = 0] is needed.This conclusion is consistent with the considerations of Sec.IV C, where it was argued that a uniform perturbation of the form in Eq. ( 25) induces surface Berry curvature.
As discussed in Sec.IV D, spatial fluctuations of the fragmenting potential with zero average, on the other hand, lead to a percolating network of chiral domain-wall modes.This percolating network appears at all nonzero surface-state energies as quantum-Hall criticality.
Localization was not observed in previous continuum studies [16,17,19] that employed a 2D minimal Dirac description, as this carries exactly zero Berry curvature as long as chiral symmetry is preserved.The fragmenting potential projects to zero in the minimal Dirac approximation.Although we consider a phase with only a single surface Dirac cone (2 surface bands), the Berry curvature necessary to localize the surface states appears in the full 4-component description of the surface-state wave functions, when the fragmenting potential is applied to the lattice Hamiltonian in Eq. ( 16).Alternatively, localization should occur in the continuum Dirac description when the latter is wedded to a trivial band in such a way so as to induce surface Berry curvature, see Sec.III C. In both cases, it is essential to retain additional degrees of freedom beyond the minimal Dirac description in order to decide the fate of the surface states in the presence of disorder.

B. Implications for experiment
Interpreted as a topological insulator with sublattice symmetry, the model in Eqs. ( 14) and ( 16) is rather artificial.Although clean systems with approximate sublattice chiral symmetry appear in nature (e.g., graphene), simple onsite impurity potentials destroy the symmetry and revert the system to a Wigner-Dyson class.For this reason, topological phases in classes CI, AIII, and DIII have received far less attention than the quantum-Hall and quantum-spin Hall insulators.
However, the non-Wigner-Dyson classes admit natural interpretations as 3D topological superconductors [5].Then a lattice model as in Eq. ( 16) can be viewed as the Bogoliubovde Gennes quasiparticle Hamiltonian in static mean field theory [7,62]; indeed Eq. ( 16) can be interpreted as a lattice regularization of the topological superfluid 3 He-B [45].Although quantum fluctuations are inevitable in a non-s-wave topological superconductor, the notion of topology is expected to carry through to fully interacting phases of matter [63].
Three-dimensional topological superconductors are protected by physical time-reversal symmetry (which transmutes into the chiral S condition in Eq. ( 15) within the Bogoliubovde Gennes language [5]), and varying degrees of spin SU(2) symmetry.Class CI, AIII, and DI superconductors respectively possess SU(2), U(1), and no spin rotational symmetry.Beyond time-reversal and spin symmetries, no other restrictions are placed upon lattice structure or disorder realizations; see Appendix C for the explicit mapping in class AIII.This means that generic non-magnetic impurities do not alter the symmetry class for these superconductors.
The surface fluid of a bulk topological superconductor consists of unpaired fermion quasiparticles ("Dirac fermions" for classes CI and AIII, "Majorana fermions" for class DIII).This fluid can dominate certain observables at low temperature T .In particular, a clean surface-Dirac cone gives a power-lawin-T contribution to the Meissner effect, due to the paramagnetic current of the surface [64].The surface quasiparticles also contribute to the longitudinal thermal and (for classes CI and AIII) spin conductivities.By contrast, the contribution of the fully gapped bulk is exponentially suppressed for these quantities in the T ≪ ∆ 0 limit, where ∆ 0 is the bulk superconducting gap.
Disorder is inevitable in real materials, and particularly at crystal boundaries.Then, the alternative scenarios of spectrum-wide criticality versus surface Anderson localization produce very different phenomenologies.Localization suppresses surface conduction, which can then be mediated at finite temperature only through inelastic processes.This should suppress the surface contribution to the Meissner effect.Without inelastic scattering, the finite-T surface thermal conductivity vanishes with surface localization in the thermodynamic limit; this is because the single delocalized state at zero energy is a set of measure zero in the surface spectrum.In reality, dephasing stabilizes a surface contribution at finite T , analogous to the longitudinal conductivity measured at the plateau transition of the quantum Hall effect [65].By contrast, spectrum-wide quantum criticality should yield a universal surface thermal conductivity determined (via the Wiedemann-Franz relation) by the average zero-temperature electrical conductivity of the quantum Hall plateau transition [17].
Which scenario is expected to be realized experimentally?A main message of this paper is that microscopics are necessary to determine the presence or absence of average surface Berry curvature; the latter is responsible for surface localization with disorder.We note that for the cubic lattice model in Eq. ( 14), the fragmenting surface potential in Eq. ( 25) used to induce the localization in Fig. 8 breaks the average cubic rotational symmetry.Equivalently, different surface perturbations are needed to induce average surface Berry curvature on different surfaces.This suggests that point-group symmetrybreaking perturbations tailored to particular crystal terminations may be necessary to induce surface localization.Alternatively, as described in Sec.III C, surface localization for E ̸ = 0 can be induced by coupling the surface of the AIII superconductor to a non-magnetic atomic-limit insulator.Hybridization between the insulator states and the superconductor, which is most effective if the band edge of the insulator is close to the Fermi energy of the superconductor, then yields the required Berry curvature.On the other hand, magnetic impurities or a weak external field should be sufficient to localize even the zero-energy surface state because they break the (physical) time-reversal symmetry protecting the class-AIII superconductor.This would exponentially suppress the longitudinal surface thermal conductivity at low temperatures.

VI. FIELD THEORY
In this section, we discuss how the physics discussed above presents itself from the perspective of effective field theory.The presentation is self-contained, and included to provide an analytical justification for the criterion (28) of state delocalization in the presence of weak disorder.Readers willing to accept the empirical application of the criterion as convincing enough may consider the section as optional reading.
By "field theory," we here mean theoretical frameworks in which averaging over static disorder is performed at an early stage to describe d-dimensional systems in a given symmetry class in terms of (d + 0) dimensional [66] nonlinear σmodels.Such theories have been in use for a long time in the physics of conventional disordered metals (see the textbook [67] for review) and were extended to the description of various topological insulators [68][69][70] even before the momentum space topologies of clean insulators became understood.In parallel to that developement, the approach was upgraded to a full classification of disordered topological insulators [7] alternative to, say, the mathematical framework of non-commutative geometry [71].From an applied perspective, its strength is that it can predict, e.g, the flow of transport coefficients [55,72,73] as a function of disorder strength or system size.It is this latter aspect that will be important in our discussion below.
We begin with a short review of the physics of the twodimensional class A, and the three-dimensional AIII insulator, extending the discussion of Sec.II B to the presence of disorder.While these are known structures, included here to provide perspective, our discussion of the AIII surface in section VI C, and specifically that of a connection between field theoretical topological θ-angles and momentum space Berry curvature in Appendix A is new material.

A. Two-dimensional Chern insulator
The starting point of field theoretical representations of topological insulators is an intermediate action (see Appendix A for a brief review of its derivation) of the form where k and x are momentum and position, respectively, Ĥ(k) is the clean Hamiltonian (throughout, we will omit carets on operators in their eigenbasis representation), κ a parameter measuring the effective disorder scattering rate, and X = {X rr ′ ss ′ } a matrix valued slowly fluctuating field carrying a replica index r = 1, . . ., R (sometimes traded for the mathematically more rigorous internal supersymmetry structure [67]), and a second index s = ± = ±1 distinguishing between propagators of retarded and advanced causality.Further details of the internal structure of X depend on the symmetry class under consideration.For example, in class A, X ≡ Q = T τz T −1 , where τz will be a Pauli matrix in s-space throughout this section (not to be confused with the earlier sublattice/chiral τ z ) or in AIII just Q = T .
The further processing of the action reflects a notion of real and momentum space duality, according to which the momentum space symmetries and topology encoded in H(k) determine the real space symmetries and topology of Q(x).To demonstrate the principle, consider the first step towards a gradient expansion in class A and use the unitary invariance of the trace to transform the action to Here, we encounter the impurity broadened Green function of the system Ĝ−1 (k) ≡ E − Ĥ(k) + iκτ 3 in conjunction with a term in which the real-and momentum-space dependent terms of the action talk to each other.Assuming smooth variation of its two constituents, a first order Wigner-Moyal expansion leads to [T −1 (x), Ĥ(k)]T (x) ≈ F i (k)Φ i (x), with F i = i∂ i Ĥ and Φ i ≡ (∂ i T −1 )T , where the derivatives are with respect to k i and x i , respectively.The effective action describing the system at large distance scales then is obtained by expansion of the tr ln up to second order in the derivative terms Φ i .Notice that the real space Φ i always appear in conjunction with momentum space F i .Also note that to leading order in a derivative expansion, tr Â(x) B(k) = dxdk tr Â(x) B(k) (37) where dx = d d x and dk = d d k/(2π) d , and the trace on the right hand side is over the internal matrix structure of the operators in question.In this way, terms appearing in the action naturally assume the form of (momentum space integrals) × (real space integrals), where in the case of topological terms, the two partners encode "dual" aspects of the topology of the system.Specifically, for the case of the two-dimensional Chern insulator, the result of the expansion to second order in gradients is Pruisken's nonlinear σ-model, which first appeared in the context of the integer quantum Hall effect [72], Here, the first term describes the diffusion and eventually Anderson localization of a two-dimensional electron gas in the presence of disorder, where the bare value of the coupling constant g = σ xx /8 is set by the system's longitudinal conductance.The second term is of topological nature and counts the number of times the Q-matrix field winds around its target manifold.In the classical reference its weighing topological angle θ = 2πσ xy was identified with the Hall conductance.
Of more relevance to our present discussion is its interpretation as a momentum space dual of the real space term, namely θ = θ(E) as given in Eq. (27).(This expression is derived for general two-dimensional systems in class A in Appendix A.) As derived, the action Eq.(38) describes the system at "bare length scales," with a minimal distance cutoff set by the scattering mean free path.Upon integrating out short distance fluctuations, and for generic values of E, the coupling constant g renormalizes to zero (Anderson localization), while the effective angle θ renormalizes to a multiple of 2π (Hall quantization).For these fixed point values, the topological θ-action reduces to a boundary action 1  8 where and x now is a one-dimensional boundary coordinate.This single derivative action describes the dissipationless chiral circulation of boundary currents against the protecting background of a localized bulk.As with the chiral Hamiltonian of the clean system it lacks gauge invariance, signalling spectral flow through the delocalized states at the energies E + or E − .We next compare this physics to that in our reference system without protected spectral flow.

B. Three-dimensional AIII insulator
As with its lower-dimensional cousin, the gradient expansion of the prototypical action (35) leads to a nonlinear σmodel [74,75] enriched by a topological term [7], Compared to Eq. ( 38), the field manifold has changed to group-valued matrix fields, T ∈ U(2R).Otherwise, we again have a job division between a gradient term describing bulk conduction properties, and a topological term now measuring three-dimensional windings over the unitary group.The second term describes the symmetry breaking induced by departures away from E = 0, where ν is proportional to the three-dimensional density of states.There are different physical limits that may be investigated on the basis of this representation: at the particle-hole symmetric point E = 0 we are sitting inside the bulk spectral gap.The bare conduction parameter g may nevertheless be finite, due to impurity states smearing the band gap of the clean system.At large length scales, we expect renormalization to an Anderson insulator, g = 0, where a value ϑ = 2πn with n a non-vanishing integer will signal topological non-triviality.In this limit, and in analogy to the Pruisken action, the topological term becomes a boundary term, Γ[T ]/12π, with the physical interpretation of a Wess-Zumino term [76] of an emerging surface action.In the immediate vicinity of the surface, the gradient term remains finite and now describes intra-surface conduction.The net effect is the stabilization of a surface Wess-Zumino action through the localization of the bulk.This action is the AIII analog of Eq. ( 39) for the A system.At large length scales, this theory renormalizes [76,77] to the conformally invariant action with g = 1/8π representing a single two-dimensional Dirac point at zero energy; this is the field theoretical interpretation of zero energy surface delocalization in the AIII insulator.However, we may also investigate what happens at finite deviations E ̸ = 0 away from chiral symmetry.In this case, the (strongly RG relevant) "mass term" in Eq. ( 40) only admits configurations T → Q = T τz T −1 for which tr Q + Q −1 is a constant vanishing in the replica limit.These are the Qmatrices of the model of lower symmetry AIII → A. Substitution into the bulk action annihilates the second and third term, while the gradient term becomes the conventional action of a disordered three-dimensional metal below the Anderson transition point: away from zero energy, the AIII insulator behaves like a conventional Anderson insulator.A more interesting limit is the case of small but finite E ̸ = 0 in the vicinity of the surface.The symmetry breaking now collapses the Wess-Zumino term Γ[Q] to the Pruisken term of a two-dimensional class A action, at topological angle θ = (2Z + 1)π [76].We conclude that the naive extension of the zero-energy WZW action to finite energies equals the action Eq. ( 38) of a twodimensional Chern insulator fine tuned into criticality.This is a field theoretical indication of a tendency to extended surface quantum criticality.The question is what happens for larger deviations E away from zero.To answer it, we need to go beyond the present level of high level reasoning and turn to a first principle approach.

C. Surface of the three-dimensional AIII insulator
In order to understand the physics of the disordered surface at arbitrary E, we again start from the prototypical representation (35).For the slow field, we take X(x) = (T τz T −1 )(x), where x is a two-dimensional surface coordinate.(The justification behind this surface projection is that states of finite extension into the bulk have eigenenergies much larger than E, which we assume to be way below the bulk gap.)For the surface Hamiltonian Ĥ(k), we assume a spectral decomposition where {|α k ⟩} are the system eigenstates at a given transverse momentum.This formal spectral decomposition actually is less innocent than it looks: Naively, it should include all eigenstates at a given k.However, this is not the case.Going back to the tr ln (35), only eigenstates of Ĥ(k) with a finite spatial overlap with the surface Hubbard-Stratonovich field X(x) contribute to the expansion.The obvious candidates here are the two eigenstates forming the chiral partners of the surface band.However, the internal spinor representation space at a given k of the lattice model is four-dimensional, implying that two states are insufficient to span it.We must, therefore, assume a contribution of bulk states (with finite surface amplitude), and an associated state-dependent weight κ = κ α .As we do not have full access to this information, we sidestep the problem by considering Eq. ( 42) as a formal complete sum.We also consider the surface band for the flattened model, i.e. we assume a finite spectral gap to higher-lying bands.
In Appendix A we show that under these conditions, the surface action assumes the form of a two-dimensional class A action Eq. ( 38), with the topological angle given by Eq. ( 27), or Eq.(A2) in a more explicit representation.As discussed above, the added curvature integrals of the upper and lower surface band computed in this way need not add to zero.In view of the above discussion, this phenomenon relates back to the embedding of the surface band into a larger Hilbert space of bulk states.Unlike with intrinsic two-dimensional lattice bands, whose Chern numbers would have to add to zero, we are here considering a single two-dimensional shadow of a three-dimensional bulk (the other lives at the opposite surface) and the cancellation principle does not apply.
To summarize, away from E = 0 the surface of the AIII insulator is described by the action otherwise describing the physics of the integer quantum Hall effect at distance scales exceeding the mean free scattering path.(Equivalently, we may think of it as the continuum version of the network structure discussed in the end of Section IV D.) Its two coupling constants specify the localization properties of the system in terms of the bare longitudinal conductance, g, and the topological angle θ-angle, respectively.The latter is remarkable in that it links the real-space long range localization properties of states at a fixed energy E to momentum-space short-distance structures at all other band energies, via the integrated Berry curvature.Such extreme forms of infrared-ultraviolet mixing are rare (and strictly absent in generic free fermions systems), but here enabled by topology.

VII. DISCUSSION AND CONCLUSION
For topological insulators in the Wigner-Dyson classes A, AI, and AII -the most prominent realizations being the twodimensional class A integer quantum Hall insulator and class AII quantum spin Hall insulator, and the three-dimensional class AII topological insulator -boundary states are continuously attached to delocalized bulk states, without interruption by a spectral or mobility gap.In this paper, we showed that for the complementary class of genuinely non-Wigner-Dyson class topologlogical insulators this key principle is broken.(The attribute "genuine" indicates that the constraints imposed by charge conjugation symmetry C or chiral symmetry S that define the non-Wigner-Dysnon classes and force the spectrum to be symmetric around E = 0 are essential for the protection of the bulk topology.By contrast, non-genuine classes remain topological after lifting constraints due to C and S and behave effectively as Wigner-Dyson insulators.) In the literature, nontrivial topology is often associated with an obstruction to the construction of a localized basis of conduction and valence bands, referred to as "Wannierizability".Our general results -see Tab.I -show that, by contrast, all genuine non-Wigner-Dyson class insulators enjoy this property, and can be topologically non-trivial nonetheless.We arrived at these conclusions both from a bulk perspective, showing that Wannier localizability of the bulk implies that the connection between surface and bulk bands becomes fragile, and from an intrinsic boundary perspective, showing that the effective surface theory admits a gap-opening perturbation.A key conclusion following from this observation is that the surface states of genuine non-Wigner-Dyson topological insulators themselves are localizable, except at the center E = 0 where state delocalization is topologically protected.
The existence of gapless or conducting surfaces is the key signature distinguishing topological from conventional insulators.Our analysis shows that, in this regard, the physics of genuine non-Wigner-Dyson topological insulators is different from that of their Wigner-Dyson siblings: Their surface states can, but need not be delocalized away from one isolated energy, E = 0.As a concrete case study, we considered the three-dimensional AIII insulator and showcased its rich boundary phenomenology: the surface states can be detached from the bulk and electrons residing in them can form different phases of matter distinguished by their Chern number.Furthermore, at the transition point between any of these surface phases one observes spectrum wide quantum critical delocalization of states.
Indeed, our study was motivated by recent work, which reported a spectrum-wide delocalization of the surface states of a class-AIII insulator and other non-Wigner-Dyson classes [16][17][18][19].These observations were based both on a numerical analysis of an effective Dirac surface theory and a numerical study of a three-dimensional lattice model of a class-AIII insulator.In this work we identified the principles that led to this seemingly robust prediction.We showed that the minimal cally nontrivial context.In the two models mentioned above, that criterion signals global delocalization, if for different reasons: the minimal surface Dirac theory is Berry-flat; in the lattice model, the disorder model considered in Ref. [17] leads to a statistical cancellation of curvature in the integral.However, both the embedding of the minimal 2 × 2 theory into a four-component spinor theory (the minimal framework to describe topological non-triviality in three-dimensional AIII), or the lifting of the statistical symmetry in the lattice model by addition of a "fragmenting surface potential" of non-zero average, lead to state localization in a manner discussed in detail in section V.
At the same time, our analysis indicates that for winding numbers |ν| > 1, the precise meaning of the term "minimal model", and the identification of quantitative localization measures must be reconsidered.In conclusion, the principle that a localizable bulk implies a gapable surface spectrum (and vice versa) applies to all genuine non-Wigner-Dyson classes.Other observations, such as the precise ways in which the minimal Dirac description is fragile, and the relation of localizability and Berry curvature, need not straightforwardly generalize beyond the ν = 1 AIII context and invite future work.
Most of the previous work on surface state localization in the AIII insulator considered a low-energy, continuum two-dimensional Dirac description with two-component Dirac spinors [62,78].Our results imply that these theories are fun-damentally incomplete, because the surface Berry curvature responsible for Anderson localization is strictly ruled out.To readers who trust in the predictive power of minimal models, it is a surprising and possibly disturbing notion that such Lorentz-covariant and renormalizable field theories cannot encode the most basic characteristics (localized versus extended) of surface-state wave functions.At the same time, it may be reassuring that the origin of the problem does not lie in the notorious and difficult-to-handle lack of ultraviolet closure of the Dirac theories, but that the problem can be cured by the simple addition of trivial degrees of freedom.Allowing for the addition of trivial bands is common practice in topological classifications based on stable equivalence, and our results show that it is equally important when determining the existence of a topological obstruction to Anderson localization.
where in the first equality we used Ĝ(E) = − ∞ E dϵ Ĝ2 (ϵ) to increase the number of Green functions, thereby mitigating the UV issues.To keep the notation slim, we omit the energy arguments throughout.In the second equality we applied another Moyal expansion (with ∂ i Ĝ = ∂ ki Ĝ(k)), and used that only the imaginary part F ( Ĝ) → Turning to second order term and using the first of the above auxiliary relations, it is straightforward to derive an analogous expression, S (2) = I 2 dx L top (Q), where and we again retain only contributions which combine to a non-vanishing trace.It remains to make sense of the momentum integrals, I 1 and I 2 .To this end, we engage the eigenfunction representation, where we anticipate that momentum derivatives of energies will not contribute to an expression of topological significance (this can be checked by explicit computation).Substituting the first of these identities into I 1 , the energy dependent denominators can all be pulled out and integrated over.As a result, we obtain where we assumed the disorder to be weak enough to justify the approximation δ(E − ϵ α ) = − 1 π Im(E + iκ − ϵ α ) −1 .In I 2 no energy integral needs to be done, and the substitution of the spectral decomposition leads to We observe that in the combination S top = (I 1 + I 2 ) dx L top (Q) the on-Fermi-shell term I 2 cancels against the on-shell contributions of I 1 , a phenomenon which in the context of the quantum Hall effect is known as the cancellation of the Streda I Fermi surface conductance against a contribution to the Streda II conductance [72].In a final step, we substitute the second of the above spectral decompositions to compute the matrix elements as ⟨α|∂ i Ĥ|β⟩ = ϵ β ⟨α|∂ i β⟩ + ϵ α ⟨∂ i α|β⟩ = (ϵ α − ϵ β )⟨∂ i α|β⟩.Substitution into I 1 + I 2 leads to the final result where Ω k = i⟨dα| ∧ dα⟩ = iϵ ij ⟨∂ i α|∂ j α⟩.(Here, the topological control parameter is M = 2.0 and the disorder strength is W = 0.2.)The horizontal axis has been rescaled as discusseed in the text.Inset: the convergence of the mean of the IPRs, scaled by the trivial scaling exponent of fully extended states for constant (blue), zero (yellow) and random (green) u f .
Appendix B: Details of numerical calculation of the multifractal spectra

Effective multifractal exponent
In practice to analyze the convergence of the numerical calculation in the linear surface dimension L we define an effective L-dependent multifractal dimension τ E q (L) = − ln P E q (L) − ln P E q (L/2) ln L − ln L/2 .(B1) For a critical point with multifractal scaling, this quantity will converge to the true multifractal exponent lim L→∞ τ E q (L) → τ E q according to Eq. (30).[14] However, away from a critical point, where wave functions localize for sufficiently large system sizes, L, its value will not converge until L is larger than the localization length and τ E q → 0. In this way the effective exponent allows us to distinguish between a localizing and critical behavior.In the latter case, it also quantifies the convergence of the multifractal spectrum.

Distribution functions of inverse participation ratios
In a numerical experiment, the moments P q are randomly distributed quantities, whose mean values are shown in Fig. 8.The probability distribution of the moments P 0.5 is shown in Fig. 9, again for the three cases constant, zero and random u f .For ideal quantum Hall criticality, we expect P 0.5 = c L −τ0.5 , with the exponent [cf.Eqs. ( 31) and ( 33)] τ 0.5 ≈ −1 + 1  16 .This implies that the variable ln[P 0.5 + τ 0.5 ln(L)] should be distributed around the non-universal constant ln c [14,84].The figure shows for u f = 0 (center) this variable is indeed narrowly distributed around a maximum, with data collapse for all values of L. Qualitatively similar behavior is found for random u f (right).While, for the numerically accessible system sizes the scaling limit is harder to reach in this case, the collapse becomes more pronounced for our largest values of L, shown in blue.However, for non-vanishing constant u f (left) the data cannot be scaled to collapse, including for different values of τ 0.5 .In this way, the absence of criticality reveals itself.
In the inset we show the scaling of the mean of the moments with system size, rescaled by the trivial extended scaling behavior.Blue (yellow, green) data represents the finite (vanishing, random) u f , where the latter two asymptote towards the QH scaling indicated as a black dotted line.

Convergence in transverse direction
We here demonstrate that already small slab widths L x are sufficient to make quantitative statements about the localization properties of the surface states.As an example, Fig. 10 shows the distrubtion function of the moments P 0.5 for u f = 0 and a surface extension L = 64 for different values of L x .The full distribution function, including mean and tails, coincide, indicating that for all shown slab widths the distribution function are already converged.
The reason for the observed width-independence is the exponential transverse localization of surface states, with a decay length of order one layer or less.Here, and in the numerical calculations shown above, only surface states with a surface weight of more than 75% are taken into account to avoid artefacts due to low lying bulk states.(These are rare but they exist due to disorder inside the clean bulk gap.)

7 IV 15 VII
localizability implies absence of spectral flow 3 B. Case study: AIII insulator in three dimensions 4 C. Other tenfold-way classes 5 III.Spectral-flow: Boundary perspective 5 A. General considerations 6 B. Case study: Class-A Chern insulator in two dimensions 6 C. Case study: Class AIII insulator in three dimensions -symmetric chiral modes: surface Hall conductance 10 V. Surface localization properties 11 A. Surface Berry curvature and disorder: Numerics 11 B. Implications for experiment 13 VI.Field theory 14 A. Two-dimensional Chern insulator 14 B. Three-dimensional AIII insulator 15 C. Surface of the three-dimensional AIII insulator FIG. 1. (a)A topological free-fermion phase with spectral flow possesses a robust continuous attachment of anomalous boundary states (red) to delocalized bulk states (black) residing in a Wannier nonlocalizable bulk band (grey).(b) Without a spectral-flow principle, anomalous surface bands (red) may in principle be detached from the bulk spectrum by a spectral gap.In this case there is no obstruction to Wannier localization of all bulk states (grey).The figure shows a schematic one-dimensional boundary spectrum, assuming translation symmetry parallel to the boundary.

FIG. 2 .
FIG. 2. Schematic picture of surface (red) and bulk (black) spectra of the topological insulator Hamiltonian H ′ = H ∂ ⊕ H bulk , see Eq.(2), before (a) and after (b) a rescaling H ∂ → λH ∂ to shrink the surface band to a width narrower than the bulk gap.

FIG. 3 .
FIG.3.Schematic dispersions of the minimal edge Hamiltonian (8) (a) and of the non-minimal model(9), which has an additional band of localized states at the edge (b).Since it derives from a lattice model, the asymptotically flat band from the localized states must be continuously connected for k → ∞ and k → −∞, to reflect the periodicity of the edge Brillouin zone.Unbounded bands are continuously connected to the bulk spectrum for k → ∞ and k → −∞.The arrows indicate how the occupation of states is changed after insertion of a flux quantum through the bulk.

FIG. 4 .
FIG.4.Schematic dispersions of the minimal edge Hamiltonian (10) (a) and the non-minimal model(11), which has two additional bands of localized states at the edge (b).The asymptotically flats band from the localized states must be continuously connected for k → ∞ and k → −∞, to reflect the periodicity of the edge Brillouin zone.The ultraviolet divergent bands from the chiral edge states are continuously attached to the bulk spectrum for k → ∞ and k → −∞.In the non-minimal model, the high-energy band is detached from the low-energy band containing the linear crossing at zero energy.The arrows indicate how the occupation of states is changed after insertion of a flux quantum through the bulk.

FIG. 5 .
FIG. 5. (a) Spectrum of the topological insulator (14) with M = 2for open boundary conditions the x-direction.The bulk (surface) spectrum is shown in black (red).There is a single low-energy Dirac cone at (ky, kz) = Γ = (0, 0).Surface and bulk bands merge at high energy.(b) The same as panel (a) but with the additional perturbation U f of Eq. (25) with u f,x = 0.5 and n = 3. (c) A minimal value of the strength u f,x of the fragmenting potential(25) required to detach surface and bulk bands, for a perturbation with support on the outermost surface layer only (n = 1, solid) and for a perturbation with support on the three outermost surface layers (n = 3, dashed).The vertical axis shows the minimal value of the indirect gap ∆s between surface and bulk bands.

FIG. 6 .
FIG. 6.(a) Integrated Berry curvature θ(E) of the surface band of the model (14) vs. energy E for the case of zero fragmenting potential u f,x = 0 (red), u f,x = −0.3(black, solid), and u f,x = −1 (black, dotted).The fragmenting potential is added on the three outermost layers adjacent to the x-surface, n = 3 and the vertical red line marks the position of the bulk gap.For |u f,x | > u c f , surface and bulk bands are detached, so that θ(E) can be determined for the full surface band; for u f,x = 0, θ(E) can be calculated only for energy E inside the bulk gap.(b) The same as panel (a) but for the minimal surface theory (10) (red) and the non-minimal model (11) (black) with |γ|/εc = 0.05.According to the criterion of Ref.[20], surface states at energy E are delocalized if θ(E) = π mod 2π.

FIG. 7 .
FIG. 7. (a) Band structure for a slab geometry of the flattened version of the model(14) with open boundary conditions along x and two symmetrically placed domain walls and periodic boundary boundary conditions along y.The fragmenting potential is of the form u f,x (x, y), where u f,x (x, y) switches from 0.2 to −0.2 and back at the two domain walls in the n = 1 outermost surface layers, and is zero otherwise.The bulk, surface, and domain-wall part of the spectrum are indicated in black, red, and green, respectively.Solid and dashed green curves are for the domain wall for which u f,x goes from negative to positive and from positive to negative upon increasing y, respectively.Note that the addition of the fragmenting potential to the surface raises part of the surface spectrum above the flattened bulk band.(b) In the presence of disorder, only surface states at E = 0 are delocalized.In this case, the chiral domain-wall modes are expected to extend from the bulk spectrum all the way down to zero energy.

FIG. 9 .
FIG. 9. Distribution functions of Pq at q = 0.5 for u f = 0.3 (left), u f = 0 (middle), and u f random (right) with rms u f = 0.5.(Here, the topological control parameter is M = 2.0 and the disorder strength is W = 0.2.)The horizontal axis has been rescaled as discusseed in the text.Inset: the convergence of the mean of the IPRs, scaled by the trivial scaling exponent of fully extended states for constant (blue), zero (yellow) and random (green) u f .

FIG. 10 .
FIG. 10.Distribution functions of Pq at E = u f = 0, W = 0.15 and linear surface extension L = 64 for different slab widths Nx on a linear (main) and semi-logarithmic scale (inset).