# Fragility of spectral flow for topological phases in non-Wigner-Dyson
classes

Alexander Altland  Institute for Theoretical Physics, University of Cologne,
50937 Cologne, Germany  Piet W. Brouwer  Dahlem Center for Complex Quantum
Systems and Physics Department, Freie Universität Berlin, Arnimallee 14, 14195
Berlin, Germany  Johannes Dieplinger  Institute for Theoretical Physics,
Universität Regensburg, 93040 Regensburg, Germany  Matthew S. Foster
Department of Physics and Astronomy, Rice University, Houston, Texas 77005,
USA  Rice Center for Quantum Materials, Rice University, Houston, Texas 77005,
USA  Mateo Moreno-Gonzalez  Institute for Theoretical Physics, University of
Cologne, 50937 Cologne, Germany  Luka Trifunovic  Laboratoire de Physique
Théorique, CNRS, Université Paul Sabatier, 31400 Toulouse, France

###### Abstract

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
principle becomes _fragile_ 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.

###### Contents

  1. I Introduction
  2. II Spectral flow: Bulk perspective
    1. II.1 Surface and bulk spectra
    2. II.2 Case study: AIII insulator in three dimensions
    3. II.3 Other tenfold-way classes
  3. III Spectral-flow: Boundary perspective
    1. III.1 General considerations
    2. III.2 Case study: Class-A Chern insulator in two dimensions
    3. III.3 Case study: Class AIII insulator in three dimensions
  4. IV AIII insulator in three dimensions
    1. IV.1 Lattice model
    2. IV.2 Surface Dirac approximation
    3. IV.3 Detaching bulk and surface bands
    4. IV.4 Chiral-symmetric chiral modes: surface Hall conductance
  5. V Surface localization properties
    1. V.1 Surface Berry curvature and disorder: Numerics
    2. V.2 Implications for experiment
  6. VI Field theory
    1. VI.1 Two-dimensional Chern insulator
    2. VI.2 Three-dimensional AIII insulator
    3. VI.3 Surface of the three-dimensional AIII insulator
  7. VII Discussion and conclusion
  8. A Field theory analysis
    1. A.1 Replica field theory
    2. A.2 Gradient expansion
  9. B Details of numerical calculation of the multifractal spectra
    1. B.1 Effective multifractal exponent
    2. B.2 Distribution functions of inverse participation ratios
    3. B.3 Convergence in transverse direction
  10. C Class AIII superconductors

##  I Introduction

Topological insulators are subject to a powerful bulk-boundary principle
according to which their insulating (yet topologically nontrivial) bulk
implies conducting boundaries Bernevig and Hughes (2013); Ando and Fu (2015);
Hasan and Kane (2010); Qi and Zhang (2011). 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 of a
three-dimensional topological insulator. In these three cases, the topological
boundary states are continuously connected to a band of likewise delocalized
bulk states at high energies, without interruption by a spectral gap at the
boundary or a window of localized states at intermediate energies. For the
quantum Hall effect, the connection of delocalized boundary states to a band
of delocalized bulk states follows from the celebrated Laughlin gauge argument
Laughlin (1981), which shows that a quantized Hall conductance necessitates
the existence of delocalized bulk states at energies below the Fermi energy,
continuously connected to the conducting boundary states. Similar arguments
have been made for the quantum spin-Hall effect and the three-dimensional
topological insulator Prodan (2011).

The principle that the anomalous boundary states must be continuously
connected to a delocalized band of bulk states — which we will refer to as the
“spectral-flow principle” — essentially relies on the topological equivalence
of all boundary states inside the bulk gap, irrespective of their energy. This
freedom to choose a reference energy inside the gap exists for the three
“Wigner-Dyson classes” A, AI, and AII from the tenfold-way classification of
topological insulators and superconductors Kitaev (2009); Schnyder  _et al._
(2009); Ryu  _et al._ (2010)111All insulators and superconductors considered
in this article are _topological_ , and so are all classifications and
equivalences; to simplify the language we omit the attribute ‘topological’ in
most cases.. (In the tenfold-way nomenclature, the quantum Hall effect is in
class A, whereas the quantum spin-Hall effect and the three-dimensional
topological insulator are in class AII.) In this article we question the
general validity of the spectral-flow principle for the non-Wigner-Dyson
classes. These have a defining symmetry that forces the spectrum to be
symmetric around a special point $E=0$ Altland and Zirnbauer (1997), so that
the topological equivalence of all boundary states inside the bulk gap is no
longer guaranteed. Considering the three-dimensional topological insulator in
symmetry class AIII (chiral symmetry, no other tenfold-way symmetries) as an
example, we show that, in contrast to the Wigner-Dyson classes, (i) the
surface spectrum may be detached from the bulk and that (ii) there is no
topological obstruction to the simultaneous localization of bulk states at all
energies. The two opposing scenarios — with and without spectral-flow
principle — are illustrated schematically in Fig. 1. The absence of a robust
connection of surface states to a band of delocalized bulk states in class
AIII has profound consequences for even very basic surface state signatures,
such as the spatial extension of states in the presence of static disorder and
the resulting transport coefficients.

Figure 1: (a) If the spectral flow principle holds, there is a robust
continuous attachment of anomalous boundary states (red) to a bulk band of
delocalized states (black), which may be embedded in a wider bulk band of
localizable states (grey). (b) Without spectral flow principle, anomalous
surface bands (red) may in principle be detached from the bulk spectrum by a
spectral gap or by band of Anderson localized states. In this case there is no
obstruction to localization of the bulk states (grey). The figure shows a
schematic one-dimensional boundary spectrum, assuming translation symmetry
parallel to the boundary.

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
“${\bm{k}}\cdot{\bm{p}}$” approximations, physically relevant parts of the
boundary spectrum are thus described by minimal models of manageable
complexity. The anomalous nature of the boundary states — i.e., the fact that
they do not exist as stand-alone lattice models — is reflected in the
impossibility to close the Dirac theories in the limit of high momenta and
exists for all tenfold-way classes. Nevertheless, as we show in detail for
class AIII in three dimensions, absence of an ultraviolet closure of the
surface Dirac theory does not automatically imply a continuous connection of
the low-energy surface theory to the bulk spectrum.

Our work is motivated in part by a series of studies, which predicted — in
contrast to the message of the present article — a spectrum-wide
delocalization at the surface of class-AIII insulators and other tenfold-way
classes with integer-valued invariants in three dimensions Ostrovsky  _et al._
(2007); Ghorashi  _et al._ (2018). That prediction was verified numerically as
a robust feature of the effective two-dimensional Dirac surface theories of
class CI, AIII, and DIII superconductors Ghorashi  _et al._ (2018); Sbierski
_et al._ (2020); Ghorashi  _et al._ (2020); Karcher and Foster (2021), and it
was backed up by numerical studies of a three-dimensional lattice model for
class AIII Sbierski  _et al._ (2020). On the other hand, robust topological
arguments Essin and Gurarie (2015), or even rigorous mathematical proof
Schulz-Baldes and Stoiber (2022) for surface state delocalization exist for
states at zero energy only. With our demonstration that for class AIII all
bulk states may be localized and that surface states can be energetically
detached from the bulk, the spectrum-wide (de)localization of Refs. Ostrovsky
_et al._ (2007); Ghorashi  _et al._ (2018); Sbierski  _et al._ (2020) is up
for renegotiation. We will present numerical and analytical evidence that for
a suitably chosen model realization all states away from zero energy are
localized, albeit with a localization length diverging at zero energy.

To understand how these findings relate to the numerical results obtained
previously, we have to anticipate some of the results derived in the remainder
of this article. We there show that for an AIII insulator with winding number
one — the “minimal” nontrivial topological insulator — perturbations
interrupting the connection of the low-energy surface band of the class-AIII
insulator to the bulk spectrum induce a surface Berry curvature in the surface
band without breaking the chiral symmetry. This Berry curvature of the surface
states is intimately tied to their localization behavior Altland  _et al._
(2002): The surface states localize away from zero energy if the (spatially
averaged) surface Berry curvature is nonzero, but not if it is zero. The
spectrum-wide delocalization observed in Ref. Sbierski  _et al._ (2020)
therefore reflects a statistical symmetry of a model with zero average surface
Berry curvature, not a topological obstruction to localization. On the other
hand, a minimal two-dimensional Dirac theory with chiral symmetry — the two-
dimensional surface theory that the numerical studies of Refs. Ghorashi  _et
al._ (2018); Sbierski  _et al._ (2020); Ghorashi  _et al._ (2020); Karcher and
Foster (2021) were based on — is strictly without Berry curvature and,
therefore, unable to capture the localizing geometric effects deriving from
surface states at high energies.

The remainder of this article is organized as follows: In Sec. II we review
the arguments relating delocalization of boundary states to the existence of
(possibly high-energy) bulk states that are protected from localization in the
Wigner-Dyson classes. Considering three-dimensional AIII as an example, we
then discuss the principles which may lift this obstruction, and state the
localizability status of all symmetry classes in dimensions up to three. In
Sec. III we derive the same conclusions from the boundary perspective: We show
that the effective Dirac-like surface theory of a class-AIII insulator is
protected from being gapped at the special energy $E=0$, but not for $E\neq
0$, and that it may have nonzero Berry curvature at any energy $E\neq 0$. Both
features may require the addition of trivial bands. (The minimal realization
of the surface Dirac theory, on the other hand, cannot be gapped at any energy
and has manifestly zero Berry curvature at all energies.) In Sec. IV we
consider a canonical four-orbital topological-insulator lattice model for the
class-AIII insulator in three dimensions Ryu  _et al._ (2010), analogous to
the model that was analyzed in Ref. Sbierski  _et al._ (2020) to demonstrate
spectrum-wide delocalization, and show that the surface bands can be detached
from the bulk bands at a high energy by a suitably chosen perturbation. In
Sec. V we present numerical evidence for surface localization if the average
surface Berry curvature is nonzero and explain why delocalization is possible
if surface regions with positive and negative Berry curvature occur with the
same probability. We also discuss ramifications of our results for the
experimental detection of 3D topological superconductors. Implications for the
field-theoretic perspective are discussed in Sec. VI. We conclude in Sec. VII.

##  II Spectral flow: Bulk perspective

In the following, we will address the spectral flow principle from two
different perspectives. The first, taken in this section, substantiates the
connection between surface and bulk states alluded to in the introduction. In
the next section we will then discuss spectral flow entirely on the basis of
surface state physics.

###  II.1 Surface and bulk spectra

The spectral flow principle essentially relies on the topological equivalence
of all surface states inside the bulk gap, irrespective of their energy. That
the freedom to choose a reference energy inside the bulk gap implies the
existence of a band of extended bulk states can be seen from a simple
argument: Suppose, on the contrary, that for the bulk there exists a basis of
(exponentially) localized eigenstates $|\Psi_{\mathbf{R}\alpha}\rangle$ at
energies $\varepsilon_{{\mathbf{R}}\alpha}$ and with localization centers
${\mathbf{R}}$. (The index $\alpha$ is an additional index to label the
eigenstates.) Then, up to an exponentially small correction, the Hamiltonian
$H$ in the presence of a (possibly conducting) boundary can be written as the
direct sum of boundary and bulk contributions Trifunovic (2020),

$H=H_{\partial}\oplus H_{\rm bulk},$ (1)

where $H_{\rm bulk}$ is the projection of $H$ onto eigenstates
$|\Psi_{\mathbf{R}\alpha}\rangle$ with ${\mathbf{R}}$ farther than a cut-off
distance from the boundary,

$H_{\rm
bulk}=\sum_{{\mathbf{R}}\in\textrm{bulk},\alpha}\varepsilon_{{\mathbf{R}}\alpha}|\Psi_{{\mathbf{R}}\alpha}\rangle\langle\Psi_{{\mathbf{R}}\alpha}|.$
(2)

Because the eigenstates $|\Psi_{\mathbf{R}\alpha}\rangle$ are localized,
$H_{\rm bulk}$ and, hence, $H_{\partial}$ are local operators. Choosing
(without loss of generality) the Fermi energy at $E=0$ and defining
$E^{\partial}$ and $E_{\rm gap}$ such that eigenvalues of $H_{\partial}$ and
$H_{\rm bulk}$ are $\leq E^{\partial}$ and $\geq E_{\rm gap}$, respectively,
see Fig. 2(a), $H$ may be continuously deformed to

$H^{\prime}=\left(\frac{E_{\rm gap}}{2E^{\partial}}H_{\partial}\right)\oplus
H_{\rm bulk},$ (3)

which has a spectral gap for boundary and bulk for energies between $E_{\rm
gap}/2$ and $E_{\rm gap}$, see Fig. 2(b), contradicting the requirement that a
topological insulator must have conducting boundary states at all energies in
the bulk gap.

Figure 2: Schematic picture of surface (red) and bulk (black) spectra of a
topological-insulator Hamiltonian $H$ (a) and the deformed Hamiltonian
$H^{\prime}$ (b). If there exists a basis of localized eigenstates in the
bulk, $H^{\prime}$ and $H$ are related via a continuous transformation.

The freedom to choose a reference energy inside the bulk gap exists for the
three Wigner-Dyson classes A, AI, and AII, but not of the other tenfold-way
symmetry classes. In these, the presence of charge-conjugation symmetry
$\mathcal{C}$ and/or chiral symmetry $\mathcal{S}$ forces the spectrum to be
mirror-symmetric around $E=0$. The equivalence of all energies inside the gap
therefore no longer holds, and the spectral flow scenario must be
reconsidered. Indeed, one may now argue in different directions, illustrated
schematically in Fig. 1. One option is to say that deviations away from zero
energy explicitly break the $\mathcal{C}$ or $\mathcal{S}$ symmetry in
question, and so we are limited to statements of surface state delocalization
at $E=0$. Indeed, this is as far as the rigorous mathematical proofs go
Schulz-Baldes and Stoiber (2022). Alternatively, one may give the anomalous
nature of the boundary states — which also holds for the non-Wigner-Dyson
classes — priority and argue that these remain protected against acquiring a
gap or being Anderson localized. However, from the simple argument presented
after Eq. (1), a spectrum-wide protection of boundary states would necessarily
imply the presence of a band of delocalized bulk states. We will consider the
three-dimensional class AIII insulator as an example to demonstrate where this
reading of a bulk-boundary correspondence can fail.

###  II.2 Case study: AIII insulator in three dimensions

The discussion above suggests 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

$\displaystyle H$ $\displaystyle=\begin{pmatrix}0&A\\\
A^{\dagger}&0\end{pmatrix},$ (4)

where $A$ is a complex square matrix acting on the subspaces defined by the
condition $\Gamma=\pm 1$, and $\Gamma=\tau_{3}$ defines the chiral symmetry.
For example, in a lattice system the subspaces corresponding to $\Gamma=\pm 1$
may define two bipartite sublattices. We assume that $H$ and, hence, $A$ are
local matrices: The matrix elements
$\langle{\mathbf{R}}\alpha|A|{\mathbf{R}}^{\prime}\alpha^{\prime}\rangle$
between atomic orbitals $\alpha$ and $\alpha^{\prime}$ at lattice sites
${\mathbf{R}}$ and ${\mathbf{R}}^{\prime}$ decay exponentially with the
distance $|{\mathbf{R}}-{\mathbf{R}}^{\prime}|$.

For an insulator subject to periodic boundary conditions the spectrum of $H$
has a finite gap around zero energy. Hence, following Ref. Ryu  _et al._
(2010), we may deform the Hamiltonian (4) 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

$\displaystyle H_{\textrm{f}}$ $\displaystyle=\begin{pmatrix}0&U\\\
U^{\dagger}&0\end{pmatrix}.$ (5)

Locality of $H_{\rm f}$ implies that $U$ is a local unitary operator. ($U$ is
unitary because $H_{\rm f}^{2}=\openone$.) We may then easily construct a
basis of localized eigenstates at energy $\pm 1$ Read (2017),

$|\Psi_{\mathbf{R}\alpha}^{\pm}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}|\mathbf{R}\alpha\rangle\\\
\pm U^{\dagger}|\mathbf{R}\alpha\rangle\end{pmatrix},$ (6)

where $\alpha$ is an additional index. (Eigenstates
$|\Psi_{{\mathbf{R}}\alpha}\rangle$ are localized near lattice site
${\mathbf{R}}$ because the unitary operator $U$ is local.) This simple
construction proves the existence of a basis of localized eigenstates of
$H_{\rm f}$, regardless of the underlyling topology 222This result is in
conflict with the conclusions of Ref. Song  _et al._ , 2014.. As shown by the
argument presented around Eq. (1), localizability of the bulk Hamiltonian
implies that surface and bulk spectra can be detached in principle.

For later reference we mention that the localized basis
$\\{|\Psi^{\eta}_{\mathbf{R}\alpha}\rangle\\}$ and, hence, the decomposition
of $H_{\rm f}$ into surface and bulk contributions as in Eq. (1) is not
unique: Each local unitary matrix $V$ generates another localized basis
$\\{|\Psi^{\eta,V}_{\mathbf{R}\alpha}\rangle\\}$ by multiplying both elements
of the two-component spinor in Eq. (6) by $V$. In Sec. IV.3, we demonstrate
that the surface states described by $H_{\partial}$ in Eq. (1) 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 $H_{\rm f}$. However,
for a given parity its value depends on the choice of the localized basis for
the bulk states: basis changes
$|\Psi^{\eta}_{\mathbf{R}\alpha}\rangle\to|\Psi^{\eta,V}_{\mathbf{R}\alpha}\rangle$
lead to a change Lapierre  _et al._ (2021)

$\displaystyle\delta\text{Ch}$ $\displaystyle=2W[V],$ (7)

where, for a periodically extended definition of the local transformation,
$W[V]\in\mathbb{Z}$ 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 Qi  _et al._ (2008).)

###  II.3 Other tenfold-way classes

class | $\,d=1$ | $d=2$ | $d=3$
---|---|---|---
A | $0$ | $\mathbb{Z}^{\checkmark}$ | $0$
AIII | $\mathbb{Z}^{\times}$ | $0$ | $\mathbb{Z}^{\times}$
AI | $0$ | $0$ | $0$
BDI | $\mathbb{Z}^{\times}$ | $0$ | $0$
D | $\mathbb{Z}_{2}^{\times}$ | $\mathbb{Z}^{\checkmark}$ | $0$
DIII | $\mathbb{Z}_{2}^{\times}$ | $\mathbb{Z}_{2}^{\checkmark}$ | $\mathbb{Z}^{\checkmark/\times}$
AII | $0$ | $\mathbb{Z}_{2}^{\checkmark}$ | $\mathbb{Z}_{2}^{\checkmark}$
CII | $2\mathbb{Z}^{\times}$ | $0$ | $\mathbb{Z}_{2}^{\times}$
C | $0$ | $2\mathbb{Z}^{\checkmark}$ | $0$
CI | $0$ | $0$ | $2\mathbb{Z}^{\times}$
Table 1: Localizability of topological insulators and superconductors. The
absence of topological phase is denoted by $0$, whereas entries where
topological phases exist are labeled by “$\times$” for localizable and
“$\checkmark$” for non-localizable phase. For class DIII in three dimensions,
superconductors with even bulk invariant are localizable, whereas
superconductors with odd bulk invariant are non-localizable. The spectral flow
principle only applies to non-localizable classes (denoted with $\checkmark$).

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 Kitaev (2009); Schnyder  _et al._ (2008); Ryu
_et al._ (2010) according to its localizability status. Referring for the full
classification program to the upcoming publication 333P. W. Brouwer, B.
Lapierre, T. Neupert, L. Trifunovic, in preparation, Table 1 summarizes the
result for dimensions up to three. It adds to the topological status of a
given symmetry class and dimensionality ($\mathbb{Z},\mathbb{Z}_{2}$, 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 localizable bulk states are labeled by “$\times$,” and those
with not fully localizable bulk by “$\checkmark$.”

Localizability or, as is is often called in the literature, Wannierizability
Thouless (1984); Brouder  _et al._ (2007); Thonhauser and Vanderbilt (2006);
Soluyanov and Vanderbilt (2011); Winkler  _et al._ (2016); Cornean  _et al._
(2017), is not the same as being deformable to an “atomic limit”. The
requirement that bulk states can be exponentially localized is essential here:
A stronger requirement, e.g., that there exists a basis of bulk states with
compact support leads to a trivial classification for dimensions $d>1$ Read
(2017). An “atomic-limit” insulator has eigenstates that can be localized
inside unit cell. For the topologically nontrivial non-Wigner-Dyson classes,
there still is an obstruction to localizing the eigenstates inside a unit
cell, which is consistent with the presence of anomalous delocalized boundary
states, in spite of the bulk bands being localizable.

The information provided by Tab. 1 confirms the argument formulated at the
beginning of this section: topological classes without $\mathcal{C}$ or
$\mathcal{S}$ symmetries, i.e., A, AI, and AII, are not localizable. For the
non-Wigner-Dyson classes, what determines localizability is whether
$\mathcal{C}$ or $\mathcal{S}$ are essential for the topology, or whether they
are “spectator symmetries” and the topology of the bulk Hamiltonian remains
nontrivial if all constraints imposed by $\mathcal{C}$ and/or $\mathcal{S}$
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 Senthil and Fisher (2000); Read and Green (2000), 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 existence of a localized basis for the one-dimensional topological
tenfold-way classes (second column in Table 1) is well known in the literature
Kohn (1959). Example are the Su-Schrieffer-Heeger model (class AIII) Su  _et
al._ (1979); Kivelson (1982); Bradlyn  _et al._ (2017) and the topological
superconductor (class D, the “Kitaev chain” Kitaev (2001)), which in their
nontrivial phase have localized bases that stretch across adjacent unit cells.
The possibility of localizable topological superconductors in dimensions
larger than one was mentioned by Ono, Po, and Watanabe Ono  _et al._ (2020) in
the context of topological superconductors with additional crystalline
symmetries (see also Ref. Zhang  _et al._ (2020)).

Since bulk localizability implies the breakdown of the spectral flow argument,
we expect that _boundary states_ away from $E=0$ will be localizable, too.
However, “localizable” does not automatically mean “localized.” The
considerations of this Section therefore raise the follow-up question how the
localizability of surface states relates to the recent observations of
delocalized surface spectra in classes AIII or CI Karcher and Foster (2021),
and to the remarkable resilience of two-dimensional Dirac surface theories to
localization. In the rest of the paper we provide answers to these questions
for our principal case study, three-dimensional AIII.

##  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.

###  III.1 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

$\displaystyle H_{\textrm{b}}=k_{1}\Gamma_{1}+k_{2}\Gamma_{2},$ (8)

where $k_{i}$ are two momenta along the surface measured relative to the
crossing point ${\bm{k}}=0$, and the two Gamma matrices satisfy
$\\{\Gamma_{1},\Gamma_{2}\\}=\delta_{ij}$. Additional Gamma matrices may
appear at higher order in $k$ or as prefactors of a random surface potential.
At large momenta, the Hamiltonian (8) is ultraviolet divergent. These
divergences cannot be cured by embedding $H_{\textrm{b}}$ into the periodic
structure of a two-dimensional Brillouin zone, reflecting that the surface
theory does not define the low-energy limit of a two-dimensional stand-alone
lattice model.

In the context defined by Eq. (8), the spectral flow principle translates to
the statement of an anomaly: Coupled to an external vector potential,
$H_{\textrm{b}}$, supplemented with an ultraviolet regularization, lacks gauge
invariance. The absence of gauge invariance indicates (quasi-)particle number
non-conservation. 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 high-lying
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.

###  III.2 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

$H_{\textrm{b}}=k,$ (9)

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, $k\to k+A$,
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)].

Figure 3: Schematic dispersions of the minimal edge Hamiltonian (9) (a) and of
the non-minimal model (10), 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
$k\to\infty$ and $k\to-\infty$, to reflect the periodicity of the edge
Brillouin zone. Unbounded bands are continuously connected to the bulk
spectrum for $k\to\infty$ and $k\to-\infty$. The arrows indicate how the
occupation of states is changed after insertion of a flux quantum through the
bulk.

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 (9), consider adding a band of trivial localized
boundary states at energy $\varepsilon_{\mathrm{c}}>0$. Assuming weak coupling
$\gamma$ to the chiral band, the boundary Hamiltonian generalizes to

$H_{\textrm{b}}^{\prime}=\begin{pmatrix}k&\gamma\\\
\gamma^{*}&\varepsilon_{\rm c}\end{pmatrix}.$ (10)

The coupling matrix element $\gamma$ now generates an avoided crossing between
the chiral and the flat band, see Fig. 3(b), and a _local_ gap close to the
momentum $k\sim\varepsilon_{\mathrm{c}}$ . However, the _global_ spectrum of
the boundary Hamiltonian remains gapless, the reason being that the band
structure of the localized states must be continuous throughout the boundary
Brillouin zone. For the same reason, the addition of the localized band does
not resolve the ultraviolet anomaly of the boundary theory. Indeed, if flux
quanta are inserted repeatedly through the bulk, the occupied states will
eventually completely fill the flat band of localized boundary states and
continue to reach the upper cutoff [see arrows in Fig. 3(b)].

We now turn to three-dimensional class AIII and discuss how a construction
similar to the one above leads to very different conclusions.

###  III.3 Case study: Class AIII insulator in three dimensions

The minimal surface theory of a class AIII insulator with winding number one
described by a two-dimensional generalization of Eq. (9),

$H_{\textrm{b}}=k_{x}\tau_{1}+k_{y}\tau_{2}.$ (11)

where the chiral symmetry is realized as $H_{\textrm{b}}=-\Gamma
H_{\textrm{b}}\Gamma$ with $\Gamma=\tau_{3}$. 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
$\tau_{3}$, in violation of the chiral symmetry. The conduction and valence
band then connect the single touching point ${\bm{k}}=\mathbf{0}$ 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. Ghorashi  _et al._ (2018);
Sbierski  _et al._ (2020); Karcher and Foster (2021) established a spectrum-
wide resilience to Anderson localization.

Figure 4: Schematic dispersions of the minimal edge Hamiltonian (11) (a) and
the non-minimal model (12), 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 $k\to\infty$ and $k\to-\infty$, 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
$k\to\infty$ and $k\to-\infty$. 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.

As in our previous discussion of class A, we now introduce a band of localized
surface states at energies $\pm\varepsilon_{\rm c}$. As a two-dimensional
surface Hamiltonian analogous to Eq. (10) we consider

$H_{\textrm{b}}^{\prime}=\begin{pmatrix}k_{x}\tau_{1}+k_{y}\tau_{2}&\gamma\tau_{-}\\\
\gamma^{*}\tau_{+}&\varepsilon_{\rm c}\tau_{1}\end{pmatrix},$ (12)

where $\tau_{\pm}=\tau_{1}\pm i\tau_{2}$. As in class A, the band-coupling
$\gamma\tau_{-}$ 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 state localizability in Sec. II, but here follows from
inspection of the surface alone.

We note that the Hamiltonian (12) 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 $\nu=1$, the general conclusion about the fragility of the
spectrum-wide protection of the surface bands in a minimal effective $2\times
2$ theory equally applies to higher winding numbers. A $2\times 2$ surface
theory with a winding number larger than one may be realized as a nodal point
involving higher powers of the momentum ${\bm{k}}$ or by having multiple nodal
points at different locations in reciprocal space. In either case, the
restriction to two-component spinors precludes the interruption of the surface
states by a spectral or mobility gap.

Below Eq. (11), 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 detached surface band introduces
another invariant, namely a two-dimensional surface Chern number. Describing
the momentum space topology of the band through the map
${\bm{k}}\to|\alpha_{\bm{k}}\rangle$, where ${\bm{k}}$ runs through the two-
dimensional Brillouin zone, and $|\alpha_{\bm{k}}\rangle$ are the positive
energy states of the finite band indicted in Fig. 4b, we define the Berry
curvature

$\Omega_{\bm{k}}=i\langle d\alpha_{\bm{k}}|\wedge d\alpha_{\bm{k}}\rangle,$
(13)

and from it the Chern number,

$\displaystyle\textrm{Ch}\equiv\frac{1}{\pi}\int_{\mathrm{BZ}}\Omega_{\bm{k}}.$
(14)

This is the surface Chern number mentioned previously in section II.2. While
its numerical value 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.3.

##  IV AIII insulator in three dimensions

In the previous section we discussed generic features of topological insulator
bulk states and of their asymptotically linearizable surface spectra. We will
now turn to a more concrete level 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.1.
Following standard protocol, we then project down to its low-energy Dirac
approximation in subsection IV.2. [Impatient readers may just take notice of
the two principal definitions (17) of the lattice model and (25) of its Dirac
approximation, and directly proceed to Sec. IV.3.] On the basis of these model
definitions we then discuss spectral flow in sections IV.3-IV.4.

###  IV.1 Lattice model

We consider a cubic lattice model with four orbitals per site defined by the
Bloch Hamiltonian Ryu  _et al._ (2010)

$\displaystyle H({\bm{k}})=$ $\displaystyle\,\left(M-\sum_{a=1}^{3}\cos
k_{a}\right)\tau_{2}\sigma_{0}+\sum_{a=1}^{3}\tau_{1}\sigma_{a}\sin k_{a},$
(15)

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
${\sigma}_{\mu}$ and ${\tau}_{\mu}$ act on two independent degrees of freedom
of the $4=2\times 2$ orbitals in the unit cell. An application of the standard
mapping Ryu  _et al._ (2010) between chiral lattice Hamiltonians and three-
dimensional winding number invariants, $W$, shows that

$\displaystyle\left.\begin{array}[]{l}W=1\cr W=-2\cr
W=0\end{array}\right\\}\qquad\textrm{for}\qquad\left.\begin{array}[]{l}1<|M|<3\cr|M|<1\cr\textrm{else}\end{array}\right\\}.$

The Hamiltonian (15) is invariant under the symmetry operations $\mathcal{S}$
and $\mathcal{C}$, with $\mathcal{C}^{2}=-1$,

$\displaystyle H({\bm{k}})=$
$\displaystyle\,-M_{\mathcal{S}}\,H({\bm{k}})\,M_{\mathcal{S}}$
$\displaystyle=$
$\displaystyle\,-M_{\mathcal{C}}\,H^{\mathsf{T}}({\bm{k}})\,M_{\mathcal{C}}.$
(16)

Here $\mathsf{T}$ denotes the matrix transpose, $M_{\mathcal{S}}=\tau_{3}$,
and $M_{\mathcal{C}}=\tau_{2}\sigma_{2}$. We note that the combination
$\mathcal{C}\mathcal{S}=\mathcal{T}$ of charge-conjugation and chiral symmetry
satisfies $\mathcal{T}^{2}=-1$, putting our system into class DIII 444 The
model in Eq. (15) can be viewed as the mean-field description of a bulk
topological superconductor, see also Sec. V.2. In particular, the Hamiltonian
(15) defines a lattice representation of the DIII topological superfluid
3He-$B$, treated at the level of static mean-field theory Volovik (2003). In
this interpretation, the Pauli matrices ${\tau}_{\nu}$ and ${\sigma}_{\mu}$
act on particle-hole and spin-1/2 space, respectively, and the $p$-wave
pairing is encoded by the sine terms. The pairing amplitude has been set equal
to unity, and the topology is controlled by the “chemical potential” $M$. The
winding numbers are consistent with those of the continuum representation,
where $W=0$ corresponds to a topologically trivial BEC phase. In the 3He-$B$
interpretation, the $\mathcal{C}$-symmetry follows from the reality of the
Balian-Werthammer spinor used to define the Bogoliubov-de Gennes Hamiltonian
Coleman (2015); Foster  _et al._ (2014)..

To define a class AIII model, we break $\mathcal{C}$, while preserving
$\mathcal{S}$. For our purposes, it will be convenient to realize this
symmetry breaking by adding $\mathcal{C}$-breaking disorder. To this end, we
turn to a real space representation of the model (15), which reads

$\displaystyle H=H_{0}+\sum_{a=1}^{3}H_{a},$ (17)

with

$\displaystyle H_{0}=$
$\displaystyle\,M\sum_{{\mathbf{R}}}|{\mathbf{R}}\rangle\tau_{2}\sigma_{0}\langle{\mathbf{R}}|,$
(18) $\displaystyle H_{a}=$
$\displaystyle\,\frac{1}{2}\sum_{{\mathbf{R}}}\Big{[}t_{{\mathbf{R}}}^{a}|{\mathbf{R}}+{\bm{e}}_{a}\rangle(\tau_{2}\sigma_{0}-i\tau_{1}\sigma_{a})\langle{\mathbf{R}}+{\bm{e}}_{a}|+\mbox{h.c.}\Big{]},$

where the ${\mathbf{R}}$ are lattice vectors on the cubic lattice, and the
$\mathbf{e}_{a}$, $a=1,2,3$ unit vectors in the lattice directions. The
amplitudes $t_{{\mathbf{R}}}^{a}$, constant in the clean case, are now chosen
as,

$\displaystyle t_{{\mathbf{R}}}^{a}\rightarrow
t\,e^{-i\phi^{a}_{\bm{\mathrm{R}}}},$ (19)

where $a\in\\{1,2,3\\}$ specifies the direction of the nearest-neighbor bond,
and the $\\{\phi^{a}_{\bm{\mathrm{x}}}\\}$ are static random phase variables
with variance 555In the numerical simulations disorder is introduced both in
the bulk and surface layers, however, on the surface layers it is stronger by
a factor of $\times 5$. This is to enhance surface multifractality of the
$E=0$ state.

$\langle\phi_{\mathbf{R}}^{a}\phi_{\mathbf{R^{\prime}}}^{a^{\prime}}\rangle=W^{2}\delta_{\mathbf{R}\mathbf{R}^{\prime}}\delta_{aa^{\prime}}.$
(20)

The effects of this disorder on the (de)localization properties of the surface
states will be discussed in Sec. V.

###  IV.2 Surface Dirac approximation

A bulk winding number $W$ generically implies the appearance of $|W|$ species
of gapless Dirac fermions at the surface Ryu  _et al._ (2010). Specifically,
we consider a $W=1$ realization of the model (15) with $1<M<3$, a vacuum
interface at $x=0$ in the 1-direction, and infinite extension in $2$\- and
$3$-directions. In this case, a Dirac surface state appears in the surface
Brillouin zone at $(k_{2},k_{3})=(0,0)$.

Considering $\mu\equiv 3-M$, with $0<\mu\ll 1$, a continuum approximation near
the bottom of the band leads to the effective Hamiltonian

$\displaystyle{H}\simeq$ $\displaystyle\,{H}_{0}+{H}_{1},$ (21)

with

$\displaystyle{H}_{0}=$
$\displaystyle\,\left(-\frac{1}{2}\frac{d^{2}}{dx^{2}}-\mu\right){\tau}_{2}{\sigma}_{0}+{\tau}_{1}{\sigma}_{1}\left(-i\frac{d}{dx}\right),$
(22) $\displaystyle{H}_{1}=$
$\displaystyle\,\left(\frac{k_{2}^{2}+k_{3}^{2}}{2}\right){\tau}_{2}{\sigma}_{0}+k_{2}\,{\tau}_{1}{\sigma}_{2}+k_{3}\,{\tau}_{1}{\sigma}_{3}.$
(23)

The zero modes of ${H}_{0}$ are

$\displaystyle|0,m\rangle\equiv|m_{z}\rangle_{\tau}|m_{x}\rangle_{\sigma}|\psi\rangle,$
(24)

where $m\in\\{\uparrow,\downarrow\\}$ denotes the polarization of the surface
state, and $|\psi\rangle$ is an envelope function decaying exponentially into
the bulk in the $x$-direction. The projection of the transverse part of the
Hamiltonian ${H}_{1}$ into the space of zero modes gives

$\displaystyle{H}_{1}\rightarrow\begin{pmatrix}0&k_{3}-ik_{2}\\\
k_{3}+ik_{2}&0\end{pmatrix}=\bm{\mathrm{k}}\cdot\bm{\mathrm{\Gamma}},$ (25)

where $\bm{\mathrm{k}}\equiv\\{k_{3},k_{2}\\}$ and
${\bm{\mathrm{\Gamma}}}=\\{{\Gamma}_{1},{\Gamma}_{2}\\}$ are the standard
Pauli matrices acting in the space of zero modes.

Equation (25) defines the minimal two-component massless Dirac approximation
to the surface states of the bulk model in Eq. (15) with winding number $W=1$.
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.

###  IV.3 Detaching bulk and surface bands

Figure 5: (a) Spectrum of the topological insulator (15) with $M=2$ for open
boundary conditions the $x$-direction. The bulk (surface) spectrum is shown in
black (red). There is a single low-energy Dirac cone at
$(k_{y},k_{z})=\Gamma=(0,0)$. Surface and bulk bands merge at high energy. (b)
The same as panel (a) but with the additional perturbation $U_{\rm f}$ of Eq.
(26) with $u_{\rm f}=0.5$ and $n=3$. (c) Minimal value of the strength $u_{\rm
f}$ of the fragmenting potential (26) 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 $\Delta_{\rm s}$ between surface and bulk bands.

The model (15) has a surface band that is continuously connected to the bulk,
see Fig. 5(a). We add a term

$\displaystyle U_{\rm f}$
$\displaystyle=\sum_{{\mathbf{R}}\in\textrm{surface}}u_{\rm
f}({\mathbf{R}})|{\mathbf{R}}\rangle{\tau}_{2}\,{\sigma}_{1}\langle{\mathbf{R}}|,$
(26)

where we set $u_{\rm f}({\mathbf{R}})=u_{\rm f}$ nonzero for ${\mathbf{R}}$ in
the $n$ outermost surface layers, and $u_{\rm f}({\mathbf{R}})=0$ otherwise.
(We set $n=1$ or $n=3$ in our numerical calculations.) The perturbation
$U_{\rm f}$ breaks $\mathcal{C}$ and $\mathcal{T}=\mathcal{C}\mathcal{S}$
symmetries, but preserves $\mathcal{S}$. We refer to this perturbation as
“fragmenting surface potential”. The definition of the fragmenting potential
(26) depends on the surface orientation and transforms the same way as the
low-energy surface Dirac cone (25). For values $u_{\rm f}>u_{\rm f}^{\rm c}$,
we observe the opening of an indirect global gap between the surface band and
bulk bands, see Fig. 5(b). The threshold parameter $u_{\rm f}^{\rm c}$ depends
on how many surface layers are perturbed by the fragmenting surface potential,
see Fig. 5(c). However, regardless of its concrete value, the present
construction demonstrates the interruptibility of spectral flow.

For later reference, we mention 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 $E$ can be answered by considering
the integrated Berry curvature carried by the states with energy
$\varepsilon_{k}$ between $0$ and $E$. Hereto we define

$\theta(E)=\pi+\int_{0\leq\varepsilon_{{\bm{k}}}\leq E}\Omega_{{\bm{k}}}.$
(27)

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

$\theta(E)=\pi\mod 2\pi.$ (28)

Since $\theta(0)=\pi$, this condition is consistent with the topological
surface states at $E=0$ being delocalized.

Figure 6: (a) Integrated Berry curvature $\theta(E)$ of the surface band of
the model (15) vs. energy $E$ for the case of zero fragmenting potential
$u_{\rm f}$ (red), $u_{\rm f}=-0.3$ (black, solid), and $u_{\rm f}=-1$ (black,
dotted). The fragmenting potential is added on the three outermost layers
adjacent to the surface, $n=3$. The vertical red lines denote the position of
the bulk gap. For the two nonzero values of $u_{\rm f}$, surface and bulk band
are detached, so that $\theta(E)$ can be determined for the full surface band;
for $u_{\rm f}=0$, $\theta(E)$ can be calculated only for energy $E$ inside
the bulk gap. (b) The same as panel (a) but for the minimal surface theory
(11) (red) and the non-minimal model (12) (black) with
$|\gamma|/\varepsilon_{\rm c}=0.05$. According to the criterion of Ref.
Moreno-Gonzalez  _et al._ (2023), surface states at energy $E$ are delocalized
if $\theta(E)=\pi\mod 2\pi$.

Fig. 6(a) shows the angle $\theta(E)$ as a function of $E$ for $u_{\rm f}=0.3$
and $u_{\rm f}=1.0$ and with the fragmenting perturbation supported on the
three outermost layers ($n=3$). We note that the presence of $u_{\rm f}$ has
little effect for energies close to $E=0$ 666Close inspection of Fig. 6 (solid
curve) shows that $\theta(E)=\pi$ for a finite value $E\approx 0.8$ for
$u_{\rm f}=-0.3$, corresponding to delocalization of surface states at that
energy according to the criterion (28). This delocalization does not
contradict our general conclusion that states are localizable in principle at
$E\neq 0$, as one finds, e.g., that all states with $E\neq 0$ can be localized
upon increasing the value of $u_{\rm f}$ [dashed curve in Fig. 6(a)].. This is
consistent with the absence of Berry curvature in the two-component Dirac
approximation, which in turn is a consequence of the chiral symmetry (i.e. the
absence of terms on the diagonal of the $2\times 2$ matrix operator.) However,
we observe significant deviations from $\theta(E)=\pi$ for energies
approaching the edges of the bulk bands. In the light of our discussion of
section III.3, these reflect the onset of effective hybdridization with
extraneous bands, whose role in the present context is assumed by the bulk
bands.

For comparison, in Fig. 6(b) we also show $\theta(E)$ for the two-dimensional
band that was obtained by coupling a surface Dirac cone to a localized surface
band, see Eq. (12). 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.

We finally note that the integral of the Berry curvature over the entire
surface band gives the Chern number Ch of the surface band. Its numerical
value depends on the sign of the fragmenting potential,

$\displaystyle\text{Ch}=-\text{sign}(u_{\rm f}).$ (29)

Comparing the above result with Eq. (7), we conclude that the sign of $u_{\rm
f}$ corresponds to two different bulk gauge choices for the detachment of
surface and bulk. Alternatively, comparing with the phenomenological surface
theory (12), different signs of $u_{\rm f}$ represent different perturbations
coupling the dispersing surface band and the degrees of freedom of the
external band.

###  IV.4 Chiral-symmetric chiral modes: surface Hall conductance

We have seen that addition of the perturbation (26) detaches the surface bands
from the bulk and that the now isolated surface band has a nonzero Chern
number Ch, which for the present $\nu=1$ model is given by Eq. (29). It is
interesting to ask what happens if $u_{\rm f}$ 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 below.

To explore this situation in the simplest possible setting, we consider a
flattened version of the model (15). The latter is obtained from the
Hamiltonian (15) by keeping its Bloch states unchanged, while sending the
energy eigenvalues to $\pm 1$. 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 $x$-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 mass (26), 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 $u_{\rm f}$ 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 $E\neq 0$ 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, cf. Fig. 7 (b).

At energies $E\neq 0$ the chiral symmetry is effectively broken, so that we
may think of each surface as a two-dimensional 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.

Figure 7: (a) Band structure for a slab geometry of the flattened version of
the model (15) with open boundary conditions along $x$ and two symmetrically
placed domain walls and periodic boundary boundary conditions along $y$. The
domain-wall height $\Delta u_{\rm f}$ takes the values $\pm 0.2$ and has
support in $n=1$ layer closest to the surface. 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
$u_{\rm f}$ 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.

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
$u_{\rm f}$ is surrounded by an outer region with $u_{\rm f}$ 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 $\sigma(E)=\mathrm{sgn}(E)\,\mathrm{sgn}(u_{\rm f})/2$ 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 half-integral transverse conductance. Chiral
symmetry requires $\sigma(E)$ to be an odd function of $E$ Trifunovic and
Brouwer (2019). At the band center, $\sigma(0)=0$, again by chiral symmetry.).

Now imagine a mass profile $u_{\rm f}(y,z)$ smoothly varying in such a way
that the spatial average $\langle u_{\rm f}(y,z)\rangle=0$ and puddles with
masses 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 Chalker
and Coddington (1988) 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” Fulga  _et al._ (2014) — is compatible with the observation of
model realizations with a spectrum-wide existence of delocalized surface
states Sbierski  _et al._ (2020). (The same statistical mechanism underlies
delocalization of a topological-insulator surface in a random magnetic field
Nomura  _et al._ (2008) and of the surface states of a weak topological
insulator Ringel  _et al._ (2012); Mong  _et al._ (2012); Fu and Kane (2012).)
On the other hand, we expect localization of finite-energy surface states if a
non-vanishing average $\langle u_{\rm f}(y,z)\rangle\neq 0$ causes an
imbalance between puddles with opposite signs of $u_{\rm f}$. 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.3 and
IV.3.

In this section, we consider the implications of surface Berry curvature for
the Anderson localization properties of the surface states in the presence of
symmetry-preserving 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. (17). Surface Berry curvature is induced along the slab
boundary via the fragmenting potential introduced in Eq. (26), above.

We demonstrate below (see Fig. 8) that a nonzero uniform fragmenting potential
localizes all surface states except the zero-energy one, which remains
topologically protected. By contrast, spectrum-wide criticality (critical
delocalization linked to the plateau transition of the quantum Hall effect
Sbierski  _et al._ (2020)) survives when either (a) the surface fragmenting
potential is set to zero, or (b) 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.2.

###  V.1 Surface Berry curvature and disorder: Numerics

We perform a numerical study of localization in a disordered version of the
model (15), with and without $U_{\rm f}[u_{\rm f}]$ (26). Disorder is
implemented by random Peierls phases as introduced in Eq. (19). We apply
multifractal analysis to decide whether the surface wave functions are
localized or delocalized Evers and Mirlin (2008); Rodriguez  _et al._ (2011).

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,

$P^{E}_{q}=\frac{\sum_{y,z}\left(\sum_{\sigma}\mid\psi^{E}_{\sigma}(y,z)\mid^{2}\right)^{q}}{[\sum_{\sigma,y,z}\mid\psi^{E}_{\sigma}(y,z)\mid^{2}]^{q}},$
(30)

where $\psi_{\sigma}^{E}(y,z)\equiv\psi_{\sigma}^{E}(x=0,y,z)$ 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 $3\times 10^{5}$ 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.

Figure 8: The scaled exponent, $\Delta_{q}/q(1-q)$ for $q=-0.5,0.5,0.75$ (left
to right) and system sizes $N_{x}=8$ and $N_{y}=N_{z}=L=24$ to $L=128$ of the
model (15). Black data: no fragmenting potential $u_{\rm f}=0$ [Eq. (26)] and
disorder strength $W=0.15$; Blue data: a constant potential with $u_{\rm
f}=0.3$ applied to the outermost surface layers, and disorder strength
$W=0.2$; Green data: random fragmenting potential with zero average and
standard deviation $u_{\rm f}=0.3$ [Eq. (35)], and disorder strength $W=0.2$.
The horizontal dashed lines marks the value $0.25$ of quantum Hall
criticality, and the localization limit, $\tau_{q}=0$ corresponds to
$\Delta_{q}/q(1-q)\to\infty,4,8/3$ for $q=-0.5,0.5,0.75$ . (The value $\infty$
reflects the formal divergence of the IPR in the localized limit for negative
$q$.)

For surfaces of large linear extension $L$ 777The computational cost of the
sparse matrix diagonalization, carried out using the ARPACK library scales
with the Hilbert space dimension of the 3d system, i.e., with $4\times
N_{x}\times L^{2}\leq 5\times 10^{5}$. the IPRs are expected to asymptotically
scale as

$\displaystyle P^{E}_{q}\propto L^{-\tau^{E}_{q}},$ (31)

with an effective dimension $\tau^{E}_{q}$ Evers and Mirlin (2008).
Multifractality manifests itself in the appearance of a non-trivial anomalous
dimension

$\Delta_{q}^{E}=\tau_{q}^{E}-d(q-1),$ (32)

measuring deviations from the naive dimension $d(q-1)$ expected for uniformly
distributed states. The opposite extreme is that of localized states, for
which $\tau_{q}=0$, 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 Ludwig  _et al._ (1994); Evers and Mirlin (2008),

$\displaystyle\Delta_{q}^{\text{AIII}}=\Theta^{\mathrm{AIII}}q(1-q),$ (33)

where $\Theta^{\mathrm{AIII}}$ 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
Huckestein (1995)

$\Delta_{q}^{\text{QH}}\simeq\Theta^{\text{QH}}q(1-q),$ (34)

with $\Theta^{\text{QH}}\simeq 0.25$ Evers  _et al._ (2008); Evers and Mirlin
(2008); Zirnbauer (2019). In either case, the spectrum is expected to be
approximately parabolic up to a threshold $|q|\simeq q_{c}=\sqrt{2/\Theta}$
Evers and Mirlin (2008); Ludwig  _et al._ (1994); Foster  _et al._ (2014).

Figure 8 shows the anomalous multifractal exponent $\Delta_{q}/q(1-q)$ 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 $\Theta^{\text{AIII}}\approx 0.85\pm 0.2$
($u_{\rm f}=0$), $0.32\pm 0.05$ (constant $u_{\rm f}$), and $0.18\pm 0.03$
(random $u_{\rm f}$). Here $u_{\rm f}$ determines the fragmenting potential
$U_{\rm{f}}$ (26). The approximate independence of these values on the value
of $q$ and $L$ indicates that we are observing the anomalous dimension (33) of
the $E=0$ quantum critical point. Away from $E=0$, our results sensitively
depend on the chosen model for $u_{\rm f}$:

#### Zero fragmenting surface potential:

For $u_{\rm f}=0$ the data quickly drops to the value Eq. (34) 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. Sbierski  _et al._ (2020).

#### Constant fragmenting surface potential:

Upon application of a constant $u_{\rm f}=0.3$, we observe a clear tendency
away from criticality and towards localized behavior upon increasing the
system size. We note that the perturbation of strength $u_{\rm f}=0.3$,
presently applied to only one surface layer, is by a factor of two below the
threshold $u_{\rm f}^{\rm c}\simeq 0.6$ required to induce an indirect gap
below surface and bulk band, see Fig. 5(c). (We restrict our attention to
$|u_{\rm f}|<u_{\rm f}^{\rm c}$, 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 $u_{\rm f}$.

#### Random fragmenting surface potential:

Motivated by the scenario laid out at the end of Sec. IV.4, we consider a
spatially varying surface deformation $u_{\rm f}(x,y,z)$ of unit layer depth,
vanishing average, and variance

$\langle u_{\rm f}(x,y,z)\,u_{\rm
f}(x,y^{\prime},z^{\prime})\rangle_{y,z}=u_{\rm
f}^{2}\,\delta_{y,y^{\prime}}\,\delta_{z,z^{\prime}}$ (35)

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., $\langle u_{\rm
f}(x,y,z)\rangle_{y,z}\neq 0$] is needed. This conclusion is consistent with
the considerations of Sec. IV.3, where it was argued that a uniform
perturbation of the form in Eq. (26) induces Berry curvature.

As discussed in Sec. IV.4, 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 Ghorashi  _et al._
(2018); Sbierski  _et al._ (2020); Karcher and Foster (2021) 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. (17). 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.3. 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.

###  V.2 Implications for experiment

Interpreted as a topological insulator with sublattice symmetry, the model in
Eqs. (15) and (17) 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 Schnyder  _et al._ (2008). Then a lattice model as
in Eq. (17) can be viewed as the Bogoliubov-de Gennes quasiparticle
Hamiltonian in static mean field theory Ryu  _et al._ (2010); Foster  _et al._
(2014); indeed Eq. (17) can be interpreted as a lattice regularization of the
topological superfluid 3He-$B$ Note (4). 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
Chiu  _et al._ (2016).

Three-dimensional topological superconductors are protected by physical time-
reversal symmetry (which transmutes into the chiral $\mathcal{S}$ condition in
Eq. (IV.1) within the Bogoliubov-de Gennes language Schnyder  _et al._
(2008)), 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-law-
in-$T$ contribution to the Meissner effect, due to the paramagnetic current of
the surface Wu  _et al._ (2020). 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 $T\ll\Delta_{0}$ limit, where
$\Delta_{0}$ 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
Wang  _et al._ (2000). 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 Sbierski  _et al._ (2020).

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. (15), the fragmenting surface potential in Eq. (26) 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
symmetry-breaking perturbations tailored to particular crystal terminations
may be necessary to induce surface localization. Alternatively, as described
in Sec. III.3, surface localization for $E\neq 0$ 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. It is self contained, and
included to provide an analytical justification for the criterion of state
delocalization used empirically in previous sections. Readers willing to
accept the evidence provided so far as convincing enough may consider this
section as optional or skip it.

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 888The
‘(d+0)’ means that we are considering non-dynamical and non-interacting
frameworks and thus can avoid the introduction of a time-like coordinate; we
will always be considering a fixed reference energy. nonlinear
$\sigma$-models. Such theories have been in use for a long time in the physics
of conventional disordered metals (see the textbook Efetov (1997) for review)
and were extended to the description of various topological insulators
Gruzberg  _et al._ (1999, 2001); Altland and Merkt (2001) 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 Ryu  _et al._ (2010) alternative to, say,
the mathematical framework of non-commutative geometry Schulz-Baldes (2016).
From an applied perspective, its strength is that it can predict, e.g, the
flow of transport coefficients Pruisken (1984); Altland  _et al._ (2015); Fu
and Kane (2012) 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 two-dimensional class A,
and the three-dimensional AIII insulator, extending the discussion of Sec.
II.2 to the presence of disorder. While these are known structures, included
here to provide perspective, our discussion of the AIII surface in section
VI.3, and specifically that of a connection between field theoretical
topological $\theta$-angles and momentum space Berry curvature in Appendix A
is new material.

###  VI.1 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

$S[X]\equiv-\tr\ln{(E-\hat{H}(\textbf{k})+i\kappa\hat{X}(\textbf{x}))},$ (36)

where k and x are momentum and position, respectively, $\hat{H}(\textbf{k})$
is the clean Hamiltonian (throughout, we will omit carets on operators in
their eigenbasis representation), $\kappa$ a parameter measuring the effective
disorder scattering rate, and $\hat{X}=\\{X^{rr^{\prime}}_{ss^{\prime}}\\}$ a
matrix valued slowly fluctuating field carrying a replica index $r=1,\dots,R$
(sometimes traded for the mathematically more rigorous internal supersymmetry
structure Efetov (1997)), and a second index $s=\pm=\pm 1$ distinguishing
between propagators of retarded and advanced causality. Further details of the
internal structure of $\hat{X}$ depend on the symmetry class under
consideration. For example, in class A, $\hat{X}\equiv
Q=T\hat{\tau}_{3}T^{-1}$, where $\hat{\tau}_{3}$ will be a Pauli matrix in
$s$-space throughout this section (not to be confused with the earlier
sublattice/chiral ${\tau}_{3}$) 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 $H(\textbf{k})$ determine the real space symmetries and topology of
$Q(\textbf{x})$. 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

$S[Q]\equiv-\tr\ln\left(\hat{G}^{-1}(E,\textbf{k})-[T^{-1}(\textbf{x}),\hat{H}(\textbf{k})]T(\textbf{x})\right).$
(37)

Here, we encounter the impurity broadened Green function of the system
$\hat{G}^{-1}(\textbf{k})\equiv E-\hat{H}(\textbf{k})+i\kappa\hat{\tau}_{3}$
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
$[T^{-1}(\textbf{x}),\hat{H}(\textbf{k})]T(\textbf{x})\approx
F_{i}(\textbf{k})\Phi_{i}(\textbf{x})$, with $F_{i}=i\partial_{i}\hat{H}$ and
$\Phi_{i}\equiv(\partial_{i}T^{-1})T$, 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 $\Phi_{i}$. Notice that the real space
$\Phi_{i}$ always appear in conjunction with momentum space $F_{i}$. Also note
that to leading order in a derivative expansion,

$\displaystyle\tr(\hat{A}(\mathbf{x})\hat{B}(\mathbf{k}))=\int
dxdk\,\tr(\hat{A}(\mathbf{x})\hat{B}(\mathbf{k}))$ (38)

where $dx=d^{d}x$ and $dk=d^{d}k/(2\pi)^{d}$, 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) $\times$ (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
$\sigma$-model, which first appeared in the context of the integer quantum
Hall effect Pruisken (1984),

$\displaystyle S[Q]=g\int
d^{2}x\,\tr(\partial_{i}Q\partial_{i}Q)+\frac{\theta\epsilon_{ij}}{16\pi}\int
d^{2}x\,\tr(Q\partial_{i}Q\partial_{j}Q).$ (39)

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 $g=\sigma_{xx}/8$ 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
$\theta=2\pi\sigma_{xy}$ 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 $\theta=\theta(E)$ 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. (39) 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 $\theta$ renormalizes to a multiple of $2\pi$ (Hall
quantization). For these fixed point values, the topological $\theta$-action
reduces to a boundary action $\frac{1}{8}\epsilon_{ij}\int
d^{2}x\,\tr(Q\partial_{i}Q\partial_{j}Q)=S_{\textrm{1d}}[T]$, where

$\displaystyle S_{\mathrm{1d}}[T]\equiv\frac{1}{2}\oint
dx\,\tr(T^{-1}\hat{\tau}_{3}\partial_{x}T),$ (40)

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.

###  VI.2 Three-dimensional AIII insulator

As with its lower-dimensional cousin, the gradient expansion of the
prototypical action (36) leads to a nonlinear $\sigma$-model Gade and Wegner
(1991); Altland and Simons (1999) enriched by a topological term Ryu  _et al._
(2010),

$\displaystyle S_{\textrm{3d}}[T]$ $\displaystyle=\int
d^{3}x\,\bigg{[}g\,\tr(\partial_{i}T\partial_{i}T^{-1})+E\nu\tr(T+T^{-1})\bigg{]}+$
(41) $\displaystyle+\frac{\vartheta\epsilon^{ijk}}{24\pi^{2}}\int
d^{3}x\,\tr(T^{-1}\partial_{i}TT^{-1}\partial_{j}TT^{-1}\partial_{k}T).$ (42)

Compared to Eq. (39), the field manifold has changed to group-valued matrix
fields, $T\in\mathrm{U}(2R)$. 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 $\nu$ 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 $\vartheta=2\pi n$ 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,
$\Gamma[T]/12\pi$, with the physical interpretation of a Wess-Zumino term
Altland  _et al._ (2002) 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

$\displaystyle S_{\textrm{2d}}[T]$ $\displaystyle=g\int
d^{2}x\,\tr(\partial_{i}T\partial_{i}T^{-1})+\frac{1}{12\pi}\Gamma[T],$ (43)

through the localization of the bulk. This action is the AIII analog of Eq.
(40) for the A system. At large length scales, this theory renormalizes
Nersesyan  _et al._ (1994); Altland  _et al._ (2002) to the conformally
invariant action with $g=1/8\pi$ 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 $E\not=0$
away from chiral symmetry. In this case, the (strongly RG relevant) “mass
term” in Eq. (41) only admits configurations $T\to Q=T\hat{\tau}_{3}T^{-1}$
for which $\tr(Q+Q^{-1})$ is a constant vanishing in the replica limit. These
are the $Q$-matrices of the model of lower symmetry AIII $\to$ 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 $E\not=0$ in the vicinity of the
surface. The symmetry breaking now collapses the Wess-Zumino term $\Gamma[Q]$
to the Pruisken term of a two-dimensional class A action, at topological angle
$\theta=(2\mathbb{Z}+1)\pi$ Altland  _et al._ (2002). We conclude that the
naive extension of the zero-energy WZW action to finite energies equals the
action Eq. (39) of a two-dimensional 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.

###  VI.3 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 (36). For the slow field,
we take $\hat{X}(\mathbf{x})=(T\hat{\tau}_{3}T^{-1})(\mathbf{x})$, where
$\mathbf{x}$ 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 $\hat{H}(\mathbf{k})$, we assume a
spectral decomposition

$\displaystyle\hat{H}(\mathbf{k})\equiv\sum_{\alpha}|\alpha_{\textbf{k}}\rangle\,\epsilon_{\alpha_{\mathbf{k}}}\langle\alpha_{\mathbf{k}}|,$
(44)

where $\\{|\alpha_{\textbf{k}}\rangle\\}$ 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 $\mathbf{k}$. However,
this is not the case. Going back to the tr ln (36), only eigenstates of
$\hat{H}(\textbf{k})$ with a finite spatial overlap with the surface Hubbard-
Stratonovich field $\hat{X}(\textbf{x})$ 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 $\kappa=\kappa_{\alpha}$. As we do not have full access to this
information, we sidestep the problem by considering Eq. (44) 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. (39), with the topological
angle given by Eq. (27), or Eq. (46) 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.

##  VII Discussion and conclusion

For topological insulators in the Wigner-Dyson classes A, AI, and AII — the
most prominent realizations being the two-dimensional 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 ${\cal C}$ or chiral symmetry ${\cal
S}$ 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 ${\cal C}$ and ${\cal S}$ 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. 1 — 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 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 showed that the spectrum of different scenarios ranges from full
_de_ localization of all states in a surface band connected to a bulk band, to
localization of all states away from zero energy. The situation is “fragile”
in that the tuning of local lattice potentials may continuously interpolate
between these extremes. Depending on which option is realized, one expects
different behavior of surface thermodynamic and transport properties, as
discussed in section V.2.

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 Ghorashi  _et al._ (2018); Sbierski  _et al._ (2020);
Ghorashi  _et al._ (2020); Karcher and Foster (2021). 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 $2\times 2$ Dirac
theory considered in these references is intrinsically protected against
localization. However this protecition is _topologically fragile_ in that it
is lifted if additional trivial surface bands are added (which can be
achieved, e.g., by coupling an extraneous surface layer). The absence of
localization in the full three-dimensional lattice model considered in Ref.
Sbierski  _et al._ (2020) results from a statistical symmetry, similar to that
realized in the “statistical topological insulator” Fulga  _et al._ (2014).

For the case of minimal topological winding number $\nu=1$, we identified a
powerful indicator for the localization properties of states at energy $E$,
namely the integrated Berry curvature of all states energetically below (or
equivalently above) that energy: an integer-quantized integral implies
delocalization, departures from these values localization. While the energetic
non-locality of this criterion may be unexpected for a model of non-
interacting particles, it reflects the importance of global momentum space
quantum geometry in a topologically 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. Sbierski  _et al._ (2020)
leads to a statistical cancellation of curvature in the integral. However,
both the embedding of the minimal $2\times 2$ 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 $|\nu|>1$,
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 $\nu=1$ 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 Bernard and LeClair (2002); Foster  _et al._ (2014).
Our results imply that these theories are fundamentally 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.

#### Acknowledgements:

We thank Jennifer Cano, Sayed Ali Akbar Ghorashi, Alex Kamenev, Bastien
Lapierre, Herrmann Schulz-Baldes, Haruki Watanabe, and Justin Wilson for
helpful discussions. This work was supported by the Deutsche
Forschungsgemeinschaft (DFG) project grant 277101999 within the CRC network TR
183 (subproject A03) (A.A., P.W.B., M.M.-G.), the German Academic Scholarship
Foundation and the German Research and the Collaborative Research Center SFB
1277 (J.D.), the Welch Foundation Grant No. C-1809 (M.S.F.), and by the
FNS/SNF Ambizione Grant No. PZ00P2_179962 (L.T.).

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##  Appendix A Field theory analysis

In this section we discuss the field theoretical analysis of the disordered
surface. To make the paper self-contained, we start with a quick review of the
derivation of the intermediate representation Eq. (36), here formulated for
class A for concreteness. In a second step we then show that the expansion of
that action establishes a connection between the $\theta$-term describing the
real space surface topology in the presence of disorder and the integrated
momentum space Berry curvature.

###  A.1 Replica field theory

We begin by adding disorder a potential $\hat{V}(\textbf{x})$ with variance
$\langle\hat{V}(\textbf{x})\hat{V}(\textbf{x}^{\prime})\rangle=\frac{\gamma_{0}}{2}\delta(\textbf{x}-\textbf{x}^{\prime})$
to the clean Hamiltonian $\hat{H}$ . Transport observables such as the
longitudinal or transverse conductance at characteristic energy $E$ may be
computed from the $R$-fold replicated partition sum Wegner (1979)

$\displaystyle Z^{R}$ $\displaystyle=\int D\psi\exp(-S[\psi]),$ $\displaystyle
S[\psi]$ $\displaystyle=-i\int
dV\,\bar{\psi}(E+i\delta\hat{\tau}_{3}-\hat{H}-\hat{V})\psi,$

where $\psi=\\{\psi_{s,\mu}^{r}(\mathbf{x})\\}$ is a Grassmann field, the
index $\mu=1,\dots,4$ labels the components of the lattice spinor,
$r=1,\dots,R$ is a replica index, $s=1,2$ distinguishes between advanced and
retarded components, and $\hat{\tau}_{3}$ is a Pauli matrix in
advanced/retarded space. We average the partition sum over disorder to obtain
a quartic interaction potential between replicas,

$\displaystyle S[\psi]$ $\displaystyle=S_{0}[\psi]+\frac{\gamma_{0}}{2}\int
dV(\bar{\psi}\psi)^{2},$ (45)

where $S_{0}$ is the clean action. To decouple the quartic term, we introduce
a Hubbard-Stratonovich matrix field
$B(\textbf{x})=\\{B_{ss^{\prime},ii^{\prime}}^{rr^{\prime}}(\textbf{x})\\}$.
Integrating out the $\psi$ fields yields,

$\displaystyle\left<Z^{R}\right>$ $\displaystyle=\int
DB\exp\left(-\frac{1}{2\gamma_{0}}\int dV\tr B^{2}+\tr\ln\hat{G}[B]\right),$

with $\hat{G}[B]=(E+i\delta\hat{\tau}_{3}-\hat{H}-B)^{-1}$. A variation of the
action in $B$ leads to
$\bar{B}(\textbf{x})=\gamma_{0}\tr\hat{G}[\bar{B}](\textbf{x},\textbf{x})$,
which has the structure of a self-consistent Born equation. According to it,
the mean field $\bar{B}$ pays the role of an impurity scattering “self energy”
whose strength is determined by the impurity-broadened local spectral density,
$\hat{G}[\bar{B}](\textbf{x},\textbf{x})$. The equation is solved by the
diagonal ansatz $\bar{B}=-i\kappa\hat{\tau}_{3}$, where $\kappa$ an effective
scattering rate determined by the bare strength $\gamma_{0}$. For our
purposes, we need not discuss the self-consistent dependence
$\kappa(\gamma_{0})$ in detail. However, what does matter is that the
stationarity equation affords a whole manifold of solutions besides the
matrix-diagonal one, $B=-i\kappa T\hat{\tau}_{3}T^{-1}=-i\kappa Q$, where
$T\in U(2R)/(U(R)\times U(R))$, and $Q=T\hat{\tau}_{3}T^{-1}$. Physically,
these are the Goldstone modes associated to the “spontaneous symmetry
breaking” $i\delta\to i\kappa$ reflected in the upgrade of the infinitesimal
causal parameter $i\delta$ to the finite damping $i\kappa$.

Substituting these modes into the action, and noting that
$\tr(Q^{2})=\textrm{const}.$ is a constant (vanishing in the replica limit),
we arrive at the soft mode action (36) which will be our starting point for
all further considerations.

###  A.2 Gradient expansion

We now discuss the steps required to advance from Eq. (36) and its equivalent
representation (37) to a local action containing of lowest non-vanishing order
in gradient operators. There are only two of these consistent with the
symmetries of the model, namely $\tr(\partial_{i}Q\partial_{i}Q)$ and
$\epsilon_{ij}\,\tr(Q\partial_{i}Q\partial_{j}Q)$. The derivation of an action
containing the first via expansion of the tr ln is textbook material Efetov
(1997) (see also Ref. Moreno-Gonzalez  _et al._ (2023) for the specific case
of the two-dimensional topological class A insulator) and is not of primary
relevance to our present discussion. However, the construction of a
topological action containing the second terms is concerned, we need to start
afresh; previous derivations of this action where specific to the quantum Hall
effect Pruisken (1984), or other genuinely two-dimensional materials Ostrovsky
_et al._ (2007). By contrast, we here want to allow for situations where the
effective $H(\mathbf{k})$ is given by a general spectral decomposition as in
Eq. (44). In this way, we will address all class A situations relevant to our
discussion (bulk two-dimensional insulators for the sake of comparison, and
the AIII surface at finite energies) in one go.

The second order expansion of the tr ln in the combinations $F_{i}\Phi_{i}$
defined after Eq. (37) leads to two terms, $S\to
S_{\textrm{top}}=S^{(1)}+S^{(2)}$, where
$S^{(1)}=-\,\tr(\hat{G}F_{i}\Phi_{i})$, and
$S^{(2)}=\frac{1}{2}\,\tr(\hat{G}F_{i}\Phi_{i})^{2}$, and the subscript “top”
indicates that we wish to isolate the topological contribution to the action.
Naively, one might think that the first order term drops out by symmetry.
However, this is not so because the trace over a single Green function leads
to ultraviolet divergent expressions, i.e. we are facing a $0\times\infty$
situation. The way out, originally suggested by Pruisken, is to process the
first order term as

$\displaystyle S^{(1)}$
$\displaystyle=-\int_{E}^{\infty}d\epsilon\tr(\hat{G}F_{i}\Phi_{i}\hat{G})\approx-\frac{i}{2}\int
dxdk\int_{E}^{\infty}d\epsilon\,\tr((\partial_{j}\hat{G})F_{i}(\partial_{j}\Phi_{i})\hat{G}-\hat{G}F_{i}(\partial_{j}\Phi_{i})\partial_{j}\hat{G})=$
$\displaystyle=-\frac{i}{4}\int_{E}^{\infty}d\epsilon\int
dk\sum_{s}s\,\tr([\hat{G}^{s},\partial_{j}\hat{G}^{s}]F_{i})\int
dx\,\tr(\hat{\tau}_{3}\partial_{j}\Phi_{i}),$

where in the first equality we used
$\hat{G}(E)=-\int_{E}^{\infty}d\epsilon\,\hat{G}^{2}(\epsilon)$ 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
$\partial_{i}\hat{G}=\partial_{k_{i}}\hat{G}(\mathbf{k})$), and used that only
the imaginary part
$F(\hat{G})\to\frac{1}{2}(F(\hat{G}^{+})-F(\hat{G}^{-}))\hat{\tau}_{3}=\frac{1}{2}\sum_{s}sF(\hat{G}^{s})\hat{\tau}_{3}$
will contribute to a non-vanishing trace. With the second of the two auxiliary
identities

$\displaystyle-4\epsilon_{lm}\sum_{s}\,\mathrm{tr}(sP^{s}\Phi_{l}P^{-s}\Phi_{m})=4\epsilon_{lm}\mathrm{tr}(\hat{\tau}_{3}\partial_{l}\Phi_{m})=\epsilon_{ij}\tr(Q\partial_{i}Q\partial_{j}Q)\equiv\mathcal{L}_{\mathrm{top}}(Q),$

we reduce this expression to $S^{(1)}=I_{1}\int
dx\,\mathcal{L}_{\mathrm{top}}(Q)$, with the energy-momentum integral

$\displaystyle I_{1}=\frac{i}{32}\int_{E}^{\infty}d\epsilon\int
dk\sum_{s}s\epsilon_{ij}\,\tr([\hat{G}^{s},\partial_{j}\hat{G}^{s}]F_{i}).$

Turning to second order term and using the first of the above auxiliary
relations, it is straightforward to derive an analogous expression,
$S^{(2)}=I_{2}\int dx\,\mathcal{L}_{\mathrm{top}}(Q)$, where

$\displaystyle I_{2}=-\frac{1}{32}\epsilon_{ij}\int
dk\sum_{s}s\,\tr(\hat{G}^{s}F_{i}\hat{G}^{-s}F_{j}),$

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,

$\displaystyle\hat{G}(E)=\sum_{\alpha}|\alpha\rangle\frac{1}{E+is\kappa-\epsilon_{\alpha}}\langle\alpha|,\qquad
F_{i}=-i\partial_{i}\hat{H}=-i\sum_{\alpha}\epsilon_{\alpha}\partial_{i}|\alpha\rangle\langle\alpha|,$

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

$\displaystyle I_{1}=\frac{i\pi}{16}\int
dk\epsilon_{ij}\sum_{\alpha\beta}\frac{1}{\epsilon_{\alpha}-\epsilon_{\beta}}\left(\delta(E-\epsilon_{\alpha})+\delta(E-E_{\beta})-\frac{2}{\epsilon_{\alpha}-\epsilon_{\beta}}\left(\Theta(\epsilon_{\alpha}-E)-\Theta(\epsilon_{\beta}-E)\right)\right)\langle\alpha|\partial_{j}\hat{H}|\beta\rangle\langle\beta|\partial_{i}\hat{H}|\alpha\rangle,$

where we assumed the disorder to be weak enough to justify the approximation
$\delta(E-\epsilon_{\alpha})=-\frac{1}{\pi}\mathrm{Im}(E+i\kappa-\epsilon_{\alpha})^{-1}$.
In $I_{2}$ no energy integral needs to be done, and the substitution of the
spectral decomposition leads to

$\displaystyle
I_{2}=-\frac{i\pi}{16}\epsilon_{ij}\sum_{\alpha\beta}\int(dk)\frac{1}{\epsilon_{\alpha}-\epsilon_{\beta}}\left(\delta(E-\epsilon_{\alpha})+\delta(E-\epsilon_{\beta})\right)\langle\alpha|\partial_{i}\hat{H}|\beta\rangle\langle\beta|\partial_{j}\hat{H}|\alpha\rangle.$

We observe that in the combination $S_{\textrm{top}}=(I_{1}+I_{2})\int
dx\,\mathcal{L}_{\textrm{top}}(Q)$ 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 Pruisken (1984). In a final step, we substitute the second of the
above spectral decompositions to compute the matrix elements as
$\langle\alpha|\partial_{i}\hat{H}|\beta\rangle=\epsilon_{\beta}\langle\alpha|\partial_{i}\beta\rangle+\epsilon_{\alpha}\langle\partial_{i}\alpha|\beta\rangle=(\epsilon_{\alpha}-\epsilon_{\beta})\langle\partial_{i}\alpha|\beta\rangle$.
Substitution into $I_{1}+I_{2}$ leads to the final result

$\displaystyle S_{\textrm{top}}[Q]=\frac{1}{16\pi}\int
d^{2}k\,\Omega_{k}\Theta(\epsilon_{\alpha}-E)\,\int
dx\,\epsilon_{ij}\tr(Q\partial_{i}Q\partial_{j}Q),$ (46)

where $\Omega_{k}=i\langle d\alpha|\wedge
d\alpha\rangle=i\epsilon_{ij}\langle\partial_{i}\alpha|\partial_{j}\alpha\rangle$.

##  Appendix B Details of numerical calculation of the multifractal spectra

###  B.1 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

$\tau_{q}^{E}(L)=-\frac{\ln P_{q}^{E}(L)-\ln P_{q}^{E}(L/2)}{\ln L-\ln L/2}.$
(47)

For a critical point with multifractal scaling, this quantity will converge to
the true multifractal exponent
$\lim_{L\to\infty}\tau_{q}^{E}(L)\to\tau_{q}^{E}$ according to Eq. (31). Evers
and Mirlin (2008) 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 $\tau_{q}^{E}\to 0$. 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.

###  B.2 Distribution functions of inverse participation ratios

Figure 9: Distribution functions of $P_{q}$ at $q=0.5$ for
$u_{\mathrm{f}}=0.3$ (left), $u_{\mathrm{f}}=0$ (middle), and $u_{\mathrm{f}}$
random (right) with $\mathrm{rms}\,u_{\mathrm{f}}=0.5$. (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)
$u_{\mathrm{f}}$.

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 $u_{\rm f}$. For ideal quantum Hall
criticality, we expect $P_{0.5}=c\,L^{-\tau_{0.5}}$, with the exponent [cf.
Eqs. (32) and (34)] $\tau_{0.5}\approx-1+\frac{1}{16}$. This implies that the
variable $\ln[P_{0.5}+\tau_{0.5}\ln(L)]$ should be distributed around the non-
universal constant $\ln c$ Evers and Mirlin (2008); Puschmann  _et al._
(2021). The figure shows for $u_{\rm f}=0$ (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 $u_{\mathrm{f}}$
(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
$u_{\mathrm{f}}$ (left) the data cannot be scaled to collapse, including for
different values of $\tau_{0.5}$. 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) $u_{\mathrm{f}}$, where the latter
two asymptote towards the QH scaling indicated as a black dotted line.

###  B.3 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 $u_{\mathrm{f}}=0$ 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.)

Figure 10: Distribution functions of $P_{q}$ at $E=0.1$, $u_{\mathrm{f}}=0$,
$W=0.15$ and linear surface extension $L=64$ for different slab widths $N_{x}$
on a linear (main) and semi-logarithmic scale (inset).

##  Appendix C Class AIII superconductors

In a superconductor with a spin rotational invariance around a fixed axis, the
Bogoliubov-de Gennes (BdG) Hamiltonian
$H=\mbox{diag}\,(h_{\uparrow},h_{\downarrow})$ splits into two blocks,
corresponding to “spin up” and “spin down” sectors. (The BdG Hamiltonian acts
on four-component spinors with spin and particle-hole degrees of freedom.)
Particle-hole conjugation $\mathcal{C}$ and time-reversal symmetry
$\mathcal{T}$ map these two blocks onto each other,

$h_{\uparrow}=-\mathcal{C}^{-1}h_{\downarrow}\mathcal{C}=\mathcal{T}^{-1}h_{\downarrow}\mathcal{T},$
(48)

so that it is sufficient to consider the “spin-up” block $h\equiv
h_{\uparrow}$ only. The product $\mathcal{S}=\mathcal{CT}$ acts as an
antisymmetry constraint on the Hamiltonian,

$h=-\mathcal{S}^{-1}h\mathcal{S}.$ (49)

It follows that a superconductor with time-reversal symmetry and a remnant
U(1) of spin SU(2) rotational invariance resides in class AIII Schnyder  _et
al._ (2008); Foster and Ludwig (2008). We now make these arguments more
explicit, using a formulation in terms of fermion creation and annihilation
operators, so that particle-hole symmetry is automatically encoded in the
fermion anticommutation relations and need not be implemented explicitly.

For a system of spin-1/2 electrons, we can form the spin-triplet Cooper-pair
annihilation operator in position space

$\displaystyle b_{a}(\bm{\mathrm{r}},\bm{\mathrm{r^{\prime}}})\equiv$
$\displaystyle\,c_{\sigma}(\bm{\mathrm{r}})\,c_{\sigma^{\prime}}(\bm{\mathrm{r^{\prime}}})\,\left(\sigma_{2}\,\sigma_{a}\right)_{\sigma,\sigma^{\prime}}$
$\displaystyle=$
$\displaystyle\,c^{\mathsf{T}}(\bm{\mathrm{r}})\,\sigma_{2}\,\sigma_{a}\,c(\bm{\mathrm{r^{\prime}}}).$
(50)

Here the electron annihilation operator $c_{\sigma}(\bm{\mathrm{r}})$ carries
spin indices $\sigma\in\\{\uparrow,\downarrow\\}$, and the Pauli matrices
$\sigma_{a}$ act on this space; repeated indices are summed. On the second
line of Eq. (C), we suppress indices and $\mathsf{T}$ denotes the transpose,
viewing $c$ ($c^{\mathsf{T}}$) as a column (row) spinor. The pair operator is
antisymmetric (“$p$-wave”) under the exchange of
$\bm{\mathrm{r}}\leftrightarrow\bm{\mathrm{r^{\prime}}}$, and transforms like
a vector under spin SU(2) rotations of the fermions. Under the physical
$\mathcal{T}^{2}=-1$ antiunitary time-reversal transformation,

$\displaystyle c(\bm{\mathrm{r}})\rightarrow
i\sigma_{2}\,c(\bm{\mathrm{r}}),\quad i\rightarrow-i,$ (51)

the pair operator $b_{a}(\bm{\mathrm{r}},\bm{\mathrm{r}}^{\prime})$ is
_invariant_. By contrast the ordinary magnetization density inverts under
$\mathcal{T}$.

In Bogoliubov-de Gennes (BdG) static mean field theory, the Hamiltonian for a
spin-triplet superconductor can be expressed as

$\displaystyle H=$
$\displaystyle\,\int\frac{d^{d}\bm{\mathrm{k}}}{(2\pi)^{d}}\,\tilde{\varepsilon}(\bm{\mathrm{k}})\,c^{\dagger}(\bm{\mathrm{k}})\,c(\bm{\mathrm{k}})$
$\displaystyle\,+\frac{1}{2}\sum_{\bm{\mathrm{r}},\bm{\mathrm{r^{\prime}}}}\,\left[\Delta_{a}(\bm{\mathrm{r}}-\bm{\mathrm{r^{\prime}}})\,b_{a}^{\dagger}(\bm{\mathrm{r}},\bm{\mathrm{r^{\prime}}})+\mathrm{H.c.}\right],$
(52)

where $\mathrm{H.c.}$ denotes the Hermitian conjugate. Here
$c(\bm{\mathrm{k}})$ is the Fourier transform of the position-space
annihilation spinor, and $\tilde{\varepsilon}(\bm{\mathrm{k}})$ denotes the
normal-state band structure, incorporating the chemical potential. The vector-
valued function $\Delta_{a}(\bm{\mathrm{r}})=-\Delta_{a}(-\bm{\mathrm{r}})$ is
the mean-field BCS order parameter. With a particular gauge choice, Eq. (C) is
invariant under the time-reversal transformation in Eq. (51) if the band
structure is invariant and
$\Delta_{a}(\bm{\mathrm{r}})=\Delta_{a}^{*}(\bm{\mathrm{r}})$.

If we restrict to a real-valued
$\Delta_{a}(\bm{\mathrm{r}})=\delta_{a,3}\,\Delta(\bm{\mathrm{r}})$, then Eq.
(C) describes a time-reversal invariant superconductor with a remnant U(1) of
spin SU(2) invariance, corresponding to rotations about the $z$-axis in spin
space. To see that this is class AIII, we reformulate in terms of the Nambu
spinor,

$\displaystyle\eta(\bm{\mathrm{r}})\equiv\begin{bmatrix}c_{\uparrow}(\bm{\mathrm{r}})\\\
c_{\downarrow}^{\dagger}(\bm{\mathrm{r}})\end{bmatrix},\quad\eta^{\dagger}(\bm{\mathrm{r}})=\begin{bmatrix}c_{\uparrow}^{\dagger}(\bm{\mathrm{r}})&c_{\downarrow}(\bm{\mathrm{r}})\end{bmatrix}.$
(53)

In the Nambu language, a $z$-axis spin rotation becomes the U(1)
transformation $\eta\rightarrow e^{i\phi/2}\,\eta$,
$\eta^{\dagger}\rightarrow\eta^{\dagger}\,e^{-i\phi/2}$. Time-reversal [Eq.
(51)] becomes

$\displaystyle\eta(\bm{\mathrm{r}})\rightarrow\begin{bmatrix}c_{\downarrow}(\bm{\mathrm{r}})\\\
-c_{\uparrow}^{\dagger}(\bm{\mathrm{r}})\end{bmatrix}=i\tau_{2}\left[\eta^{\dagger}(\bm{\mathrm{r}})\right]^{\mathsf{T}}.$
(54)

Here the Pauli matrix $\tau_{2}$ acts on the components of the Nambu spinor,
and $\left[\eta^{\dagger}\right]^{\mathsf{T}}$ is the column spinor
corresponding to the row $\eta^{\dagger}$. Eq. (54) implies that time-reversal
acts like an _antiunitary particle-hole transformation_ in the Nambu language,
because spin (unlike electric charge) inverts under $\mathcal{T}$. This is in
fact chiral symmetry in second quantization. To see this, we recast Eq. (C)
compactly as

$\displaystyle H=\frac{1}{2}\eta^{\dagger}h\eta,$ (55)

where $h$ is the Hermitian BdG Hamiltonian that acts on position and Nambu
components. This form is manifestly invariant under spin U(1) rotations.
Imposing invariance under $\mathcal{T}$ in Eq. (54) leads to the chiral
condition on $h$,

$\displaystyle-\tau_{2}\,h\,\tau_{2}=h.$ (56)

Physical time-reversal symmetry is thus transmuted into a chiral condition on
the BdG Hamiltonian. Since there are no other constraints on $h$, the
superconductor resides in class AIII.

The other topological superconductor classes in three dimensions are CI and