# Topology of critical chiral phases: Multiband insulators and superconductors

Oleksandr Balabanov1,2 Daniel Erkensten3 Henrik Johannesson2 1Department of
Physics, Stockholm University, SE 106 91 Stockholm, Sweden  2Department of
Physics, University of Gothenburg, SE 412 96 Gothenburg, Sweden  3Department
of Physics, Chalmers University of Technology, SE 412 96 Gothenburg, Sweden

###### Abstract

Recent works have proved the existence of symmetry-protected edge states in
certain one-dimensional topological band insulators and superconductors at the
gap-closing points which define quantum phase transitions between two
topologically nontrivial phases. We show how this picture generalizes to
multiband critical models belonging to any of the chiral symmetry classes
AIII, BDI, or CII of noninteracting fermions in one dimension.

##  I Introduction

The presence of topological edge states, decoupled from the bulk, is a key
characteristic of symmetry-protected topological phases of quantum many-body
systems Chiu2016 . In one dimension (1D), these states are exponentially
localized at the physical boundaries of the system, making their energy vanish
identically in the thermodynamic limit (and sometimes also at certain fine-
tuned points in the phase diagram of a finite system). With no interactions
present, the possible 1D fermionic topological phases are those of the
topological band insulators and mean-field superconductors Hasan ; Qi ,
classified by the ”ten-fold way” Kitaev2009 ; Schnyder2008 ; Ryu2010 . For
this class of models the very existence of edge states is a consequence of the
topologically nontrivial phase structure of the single-particle bulk states
(”bulk-boundary correspondence” Teo2010 ), with their robustness against local
perturbations (or uncorrelated disorder) being ensured by the symmetries
enforced on the perturbations. Well-known examples include the fractionalized
soliton mode of the Su-Schrieffer-Heeger model SSH1979  and the Majorana zero-
energy mode of the Kitaev chain Kitaev2001 .

The existence of edge states has conventionally been thought to require that
perturbations do not close the insulating band (or quasiparticle) gap. This
assumption was proven wrong in 1D in a work by Verresen, Jones, and Pollmann
VJP . These authors showed that exponentially localized edge states may
survive at quantum criticality, at the gap-closing quantum phase transition
(QPT) between two topologically nontrivial gapped phases, i.e. phases which
both support topological edge states. Earlier works Kestner ; Cheng ;
Fidkowski ; Sau ; Kraus ; Keselman1 ; Iemini ; Lang ; Montorsi ; Ruhman ;
Jiang ; Zhang ; Parker ; Keselman2  exploring specific models in 1D had
anticipated that distinct topological phases $-$ supporting robust edge states
$-$ may in fact form also at quantum critical points (and possibly also in
higher dimensions in the presence of additional gapped degrees of freedom
Grover ; Scaffidi ). However, why and how this happens was first unveiled in
detail by Verresen et al. VJP , providing important intuition.

Their theory is underpinned by a study of QPTs within the BDI symmetry class
of the ten-fold way, where the single-particle Hamiltonians $H$ exhibit
spinless time-reversal symmetry, $THT^{-1}=H,$ particle-hole symmetry
$CHC^{-1}=-H$, and chiral symmetry $SHS^{-1}=-H$, where $T,\,C,$ and $S$ are
the corresponding first-quantized symmetry operators Ludwig2016  with
$T^{2}\\!=\\!C^{2}\\!=\\!1$. The critical point separating the two
topologically nontrivial phases is labeled by two numbers: a topological
invariant $\nu$, which, if positive, counts the edge states, and the central
charge $c$ of the conformal field theory (CFT) which describes the scaling
limit of the unperturbed Hamiltonian, with no gapped degrees of freedom
present foot . The key to the analysis is provided by a meromorphic function
which encodes the properties of BDI Hamiltonians, the zeros and poles of which
control the values of $\nu$ and $c$ VJP . Given this, Verresen et al. argue
that two critical Hamiltonians in the BDI class can be smoothly connected (by
tuning a control parameter) only if they share the same values of $\nu$ and
$c$. In a follow-up work it is shown that $\nu$ and $c$ can be encoded also by
correlation functions of certain nonlocal string order parameters Jones2019 .
The theory was subsequently put into a larger framework of “symmetry-enriched
quantum criticality” Verresen2019 ; Vasseur  where the presence of nonlocal
symmetry operators implies localized and topologically robust edge modes.
Also, a generalization to all symmetry classes where the topological
classification is larger than $\mathds{Z}_{2}$, independent of dimension, was
put forward by Verresen in Ref. Verresen2020, .

The case when a lattice system is not strictly translational invariant (with
respect to the underlying lattice), but instead has a repeating enlarged unit
cell, is briefly touched upon in Ref. VJPSM, , but then only for the BDI
symmetry class, and with no proof of the existence of the critical edge
states. In this paper we wish to add to the picture by addressing the unit-
cell problem for all three chiral symmetry classes of noninteracting fermions
in 1D, i.e. the symmetry classes in 1D with a topological classification
beyond $\mathds{Z}_{2}$: BDI, CII, and AIII, containing models subject only to
perturbations which respect chiral invariance. Note that other symmetry
classes in 1D exhibit at most a single topologically nontrivial gapped phase,
and therefore, in keeping with Refs. VJP, ; Verresen2020, , do not support
critical edge states. One should here recall that an enlarged unit cell
implies that the spectrum of the model $-$ be it a band insulator or a mean-
field superconductor $-$ displays a multiband structure in the Brillouin zone.
Aside from the possible relevance for experiments $-$ including studies of
multiband topological nanowires Lutchyn ; Stanescu ; Setiawan ; Samokhin ,
quasi-1D fermionic gases in synthetic gauge fields Mizushima , and 1D
topological quantum phase transitions out of equilibrium Mendl ; Maslowski
$-$ an analysis of the multiband problem introduces several new facets which
may advance our general understanding of topology at quantum criticality. One
may here mention that the extended scenario presented in Ref. Verresen2020,
builds on a low-energy representation of an appropriate Hamiltonian and is
therefore not directly applicable to multiband systems where higher-energy
bands may impact the topological classification. This provides yet another
motivation for a more thorough study of the multiband problem, here narrowed
to critical chiral phases in 1D.

The paper is organized as follows. In the next section we address the problem
of 1D gapless chiral phases in symmetry class AIII. This is the most general
case supporting such phases in 1D since only chiral invariance is enforced on
the allowed perturbations; time-reversal and particle-hole symmetries, if at
all present, are considered as accidental symmetries. As a warm-up, in Sec.
II.A we construct the topological invariant for the simple two-band case,
borrowing some of the machinery from Ref. VJP, , but slightly adapted so as to
be immediately extendable to the multiband case, as shown in Sec. II.B. In
Secs. II.C and II.D the critical edge states are constructed explicitly for
the two-band and multiband case respectively. While our approach for the two-
band case in Sec. II.C again closely follows Ref. VJP, , the approach in Sec.
II.D is new. Section III contains a test of our approach in Sec. II when
applied to symmetry class BDI, here treated as a special case of AIII, where,
in addition to chiral symmetry, time-reversal and particle-hole symmetries are
also enforced. This section aims to thoroughly compare our results to those in
Ref. VJP,  where class BDI is treated explicitly. In Sec. IV we turn to the
CII symmetry class, with the same symmetries enforced as for BDI but with
$T^{2}=C^{2}=-1$, thus comprising spinful free fermion models protected by all
three symmetries; chiral, time-reversal, and particle-hole symmetries.  For
transparency $-$ and also to chisel out the similarities and differences
between the treatment of critical BDI models in Ref. VJP,  $-$ we here focus
on the case of a CII spinful Majorana chain with four bands. Section V
contains a numerical check of the robustness of critical edge states against
uncorrelated disorder,  employing the spinful Majorana chain from the previous
section as benchmark model. Section VI, finally, briefly summarizes our work.

##  II symmetry class AIII

###  II.1 Topological invariant for two-band gapless AIII systems in 1D

A noninteracting fermion system with first-quantized single-particle
Hamiltonian $H$ is said to possess chiral symmetry if there exists a unitary
operator $S$ such that $SHS^{-1}=-H$. For a lattice system one can then define
two effective sublattices associated with this symmetry by using projectors
$P_{A}=(1+S)/2$ and $P_{B}=(1-S)/2$ where $P_{A/B}$ projects states onto the
sublattices labelled by $A$ and $B$, respectively. Note that by using this
definition, $A$ and $B$ act indeed as sublattices: the terms present in $H$
cannot couple states from the same sublattice due to the restriction imposed
by $SHS^{-1}=-H$.

Now, let us consider a two-band fermionic lattice system from the AIII
symmetry class, defined by requiring that all perturbations, and also the
Hamiltonian, respect chiral symmetry. By this, any AIII Hamiltonian $H$ can
connect only sites from different sublattices, implying the generic expression

$\displaystyle\begin{split}H&=\sum_{j,n}t_{n}|A,j\rangle\langle
B,j+n|+\mbox{H.c.},\\\ \end{split}$ (1)

where $j$ ($n$) runs from 1 to $N$ (from $-N$ to $N$), with $N$ the number of
lattice sites. The two sublattices are denoted by $A$ and $B$, and $t_{n}$ are
the corresponding hopping amplitudes, here allowed to be complex but
restricted to a finite range, i.e. $t_{n}=0$ for large enough $|n|$. Note that
$A$ and $B$ denote states which correspond to a pair of internal degrees of
freedom. This can be spin, the two sites of a unit cell, or Nambu degrees of
freedom.

The Hamiltonian in Eq. (1) is easily diagonalized by a Fourier transformation:

$\displaystyle\begin{split}H=\sum_{k,n}t_{n}\exp{(ikn)}|A,k\rangle\langle
B,k|+\mbox{H.c.},\end{split}$ (2)

with $k=0,2\pi/N,...,2\pi(N-1)/N$. Or simply

$\displaystyle\begin{split}H(k)=\begin{pmatrix}0&f(k)\\\
f^{\dagger}(k)&0\end{pmatrix},\end{split}$ (3)

with $f(k)=\sum_{n}t_{n}\exp{(ikn)}$ in the basis spanned by
$|A,k\rangle,|B,k\rangle$. Importantly, the function $f(k)$ is seen to be in
one-to-one correspondence with $H(k)$. It follows from Eq. (3) that the
eigenenergies are given by $\epsilon_{k}=\pm|f(k)|$ (which means that
$f(k)=\epsilon_{k}e^{i\varphi_{k}}$ for some $\varphi_{k}$) and therefore the
zeros of $\epsilon_{k}$ coincide with the zeros of $f(k)$. If nondegenerate,
such a zero, $k=k_{0}$, implies that $\epsilon_{k}\sim k-k_{0}$, bringing
about a massless relativistic excitation of the critical theory (with the bulk
energy gap being closed). When the internal states $A$ and $B$ in Eq. (1) are
Nambu degrees of freedom, allowing for a nontrivial Majorana representation of
the Hamiltonian in Eq. (1), the excitations are those of Majoranas (as in Ref.
VJP, ), else they are ordinary fermions. Provided that all zeros of $f(k)$ are
nondegenerate, precluding the appearance of dispersions
$\epsilon_{k}\sim(k-k_{0})^{m}$ with $m\neq 1$ a dynamical exponent, one
infers that the effective field theory which describes the critical phase is
that of a conformal field theory VJP .

For gapped systems, with $\epsilon_{k}\neq 0$,
$f(k)=\epsilon_{k}\mathrm{e}^{i\varphi_{k}}$ is a well-defined function on the
unit circle with the phase factor $\mathrm{e}^{i\varphi_{k}}$ prescribing a
mapping $S^{1}\rightarrow S^{1}$. By this, the winding number $\nu$ which
defines the topological invariant for the 1D AIII symmetry class can be
identified with the number of times that $f(k)$ winds around the origin in the
complex plane as $k$ is swept through the Brillouin zone (BZ), $k\in[0,2\pi]$,

$\nu\equiv\frac{1}{2\pi
i}\int_{\mathrm{BZ}}dk\,{\color[rgb]{0,0,0}\partial_{k}\ln(f(k)).}$ (4)

This standard formula for a winding number topological invariant breaks down
for gapless AIII phases, since now $\epsilon_{k}$ vanishes for at least one
value of $k$. However, as realized by Verresen et al. for the BDI symmetry
class VJP , one may circumvent this difficulty by performing an analytic
continuation of $f(k)$ to the entire complex plane, i.e., taking
$f(k)\rightarrow f(z)$ in such a way that

$f(z)=\sum_{n}t_{n}z^{n}.$ (5)

The winding number $\nu$ can now be calculated using Cauchy’s argument
principle as

$\nu=N_{z}-N_{p},$ (6)

with $N_{z}$ the number of zeros of $f(z)$ inside the unit disk, $|z|<1$, and
where $N_{p}$ is the number of poles (counting multiplicities) also inside the
unit disk. Importantly, the right-hand side of (6) remains well defined also
in the gapless case. The only difference to the BDI case is that $t_{n}$ are
now allowed to be complex-valued constants since time-reversal invariance is
not enforced for AIII. The quantity on the right-hand side of Eq. (6) is well
defined even for gapless systems and hence can be used for topological
distinction of such systems in symmetry class AIII. To readers familiar with
the work by Verresen et al., let us point out that for the BDI symmetry class
(with real-valued hopping amplitudes), a unitary transformation connects the
function $f(z)$ in Eq. (5) to its namesake in Ref. VJP, , constructed using a
Majorana representation. For details, see Sec. IV.

###  II.2 Multiband case

We next consider the extension to the case with $2n$ distinct states per unit
cell, with $n>1$. The corresponding single-particle Hamiltonian $H(k)$ in
$k$-space will now be represented by a $2n\times 2n$ matrix, implying $2n$
bands in the Brillouin zone.

To set the stage, let us choose a basis for the unit cell that diagonalizes
the chiral operator $S$, and call the corresponding basis states $|X,i\rangle$
(with eigenvalue $+1$) and $|Y,i\rangle$ (with eigenvalue $-1$) where the
labels $X$, $Y$ run over the $2n$ different states within a cell and $i$ runs
over the cells. We can always go to this basis by a unitary transformation. To
satisfy the chiral symmetry condition $SHS^{-1}=-H$ we can have couplings only
between states of opposite eigenvalues, in other words, $|X,i\rangle$ can
couple only to $|Y,j\rangle$. Therefore, the most general single-particle
multiband Hamiltonian can be written as $H=\sum_{X,Y}H_{XY}$, where

$\displaystyle\begin{split}H_{XY}&=\sum_{i,j}t^{XY}_{j}|X,i\rangle\langle
Y,i+j|+{\mbox{H}.c.},\\\ \end{split}$ (7)

with $i,j$ running over all cell indices. The labels $X$ and $Y$ run over the
internal degrees of freedom $A,B,C,D,...$, partitioned according to the
evenness and oddness of the corresponding states under chirality,
$X\in\\{A,B,...\\}$ and $Y\in\\{C,D,...\\}$. $t^{XY}_{j}$, finally, are
hopping amplitudes, which, as for the two-band case, are allowed to be
complex.

We now diagonalize $H$ by performing a Fourier transformation, writing

$\displaystyle\begin{split}{\color[rgb]{0,0,0}H=\sum_{X,Y}\sum_{k,j}t^{XY}_{j}}\exp{(ikj)}|X,k\rangle\langle
Y,k|+\mbox{H.c.},\end{split}$ (8)

with $k=0,2\pi/N,...,2\pi(N-1)/N$. Reading off the Hamiltonian in $k$-space:

$\displaystyle\begin{split}{\color[rgb]{0,0,0}H(k)}=\begin{pmatrix}0&F(k)\\\
F^{\dagger}(k)&0\end{pmatrix},\end{split}$ (9)

with the $n\times n$ matrix $F(k)$ composed of elements
$f_{XY}(k)=\sum_{j}t^{XY}_{j}\exp{(ikj)}$. For example, with four bands we
have

$\displaystyle\begin{split}F(k)=\begin{pmatrix}f_{AC}&f_{AD}\\\
f_{BC}&f_{BD}\end{pmatrix}.\end{split}$ (10)

Clearly, the eigenenergies are zero if and only if $\det F(k)$ is zero.
Therefore, the zeros of $H(k)$ coincide with the zeros of $d(k)=\det F(k)$,
with $F(k)$ in one-to-one correspondence with $H(k)$.

The winding number $\nu$ for a multiband gapped system is now calculated as

$\displaystyle\begin{split}\nu=\frac{1}{2\pi i}\int
dk\,{\color[rgb]{0,0,0}\partial_{k}\ln({\color[rgb]{0,0,0}d(k)}).}\end{split}$
(11)

For the same reason as for the two-band case, this formula becomes
inapplicable for gapless phases. To get around this we again use analytic
continuation, $d(k)\rightarrow d(z)$, with

$\displaystyle d(z)$
$\displaystyle=\det\sum_{XY}\sum_{j}t^{XY}_{j}z^{j}|X\rangle\langle Y|$ (12)
$\displaystyle=\det\sum_{j}T_{j}z^{j},$

where $T_{j}$ is the corresponding hopping matrix constructed out of the
$t^{XY}_{j}$ coefficients. By employing Cauchy’s argument principle, we obtain
the same expression for the winding number $\nu$ as in the two-band case,
$\nu=N_{z}-N_{p}$, now with $N_{z}$ the number of zeros and $N_{p}$ the number
of poles (including multiplicity) of $d(z)$ inside the unit disk. By
construction, this expression for the winding number is valid also for a
gapless multiband system in the AIII symmetry class and hence can be used to
label its distinct critical phases.

####  II.2.1 Extending the unit cell

Any lattice system invariant under a translation by a unit cell has an
equivalent description in which the unit cell has been enlarged, but now with
more internal degrees of freedom. In particular, chiral symmetry is preserved
under an extension of the unit cell since the new Hamiltonian, written in a
basis with the enlarged unit cell, still only couples sites belonging to
distinct sublattices. We can enlarge the unit cell until we end up with
hopping only between nearest-neighbor cells, i.e. only the matrices $T_{-1}$,
$T_{0}$, and $T_{1}$ in Eq. (12) are nonzero (with the row- and column-indices
of these hopping matrices now running over all internal states in the extended
unit cell). This drastically simplifies the analysis since we are left with
only three terms in the summation over $j$ in Eq. (12). As a result, the
winding number $\nu=N_{z}-N_{p}$ can now be calculated using

$d(z)=\det F(z)=z^{-n}\det(T_{-1}+T_{0}z+T_{1}z^{2}).$ (13)

The poles in this expression appear only in the prefactor $z^{-n}$ and it
follows that we can write $\nu={\bar{N}}_{z}-n$ where ${\bar{N}}_{z}$ is the
number of zeros inside the unit disk of $\det{\bar{F}}(z)$, where

$\bar{F}(z)=T_{-1}+T_{0}z+T_{1}z^{2}.$ (14)

Note that the counting of zeros include possible zeros at the origin that will
cancel one or several of the poles of $z^{-n}$ in Eq. (13). This is a useful
result that we shall exploit when proving the existence of critical edge
states for a multiband system. Also important to note here that the enlarging
of the unit cell does not change the generalized winding number
$\nu=N_{z}-N_{p}$VJPSM .  In passing, let us stress that the overbars in the
formulas above and in the rest of the paper are used to denote different
variables, not complex conjugation.

####  II.2.2 Short comment

Any chiral critical noninteracting fermion system in 1D $-$ i.e. any system
exhibiting a critical phase and belonging to symmetry class AIII, BDI, or CII
$-$ can be classified using the approach above. While we have here exploited
only the presence of chiral invariance, assuming symmetry class AIII, the
additional symmetries enforced by CII and BDI will only add restrictions on
$F(z)$, and hence on $d(z)$ in Eq.(13). It may also be worth emphasizing that
we have not assumed any specific form for the chiral symmetry operator $S$,
but only used that $S$ is unitary and that hence there exists a basis which
diagonalizes $S$.

###  II.3 Edge states in two-band gapless
AIII systems in 1D

We now turn to the demonstration of AIII critical edge states. Our plan of
attack for the case of two bands is very similar to that of Verresen et al.
for the BDI symmetry class VJP,  (with substantial changes when we later turn
to multiband systems). The goal is to construct $\nu$ linearly independent
states (per edge) “by hand”, with the properties that (i) their energies
vanish identically for a semi-infinite chain, and (ii) their wave functions
decay exponentially as one moves away from the edge, by this establishing a
bulk-boundary correspondence Hasan  for 1D critical two-band systems in the
AIII symmetry class.

####  II.3.1 $\nu=N_{z}-N_{p}>0$

For clarity, let us again write the Hamiltonian in Eq. (1), but now explicitly
for a semi-infinite chain:

$\displaystyle\begin{split}H&=\sum^{\Lambda}_{n=-\Lambda}\sum_{i\geq
1}\,t_{n}|A,i\rangle\langle B,i+n|+\mbox{H.c.}\\\ \end{split}$ (15)

Here $\pm\Lambda$ are cutoffs beyond which the range of hopping vanishes. The
state $|B,i+n\rangle$ is a null vector for $i+n\leq 0$. Introducing a state

$\displaystyle\begin{split}|\psi_{\alpha}\rangle=\sum_{i\geq
1}\left(a^{(\alpha)}_{i}|A,i\rangle+b^{(\alpha)}_{i}|B,i\rangle\right),\end{split}$
(16)

this state will represent a zero mode of $H$ if

$\displaystyle\begin{split}&H|\psi_{\alpha}\rangle=\\\
&=\sum_{n=-\Lambda}^{\Lambda}\sum_{i,j\geq 1}\\!\left(t_{n}|A,i\rangle\langle
B,i\\!+\\!n|+\mbox{H.c.}\right)\\!(a^{(\alpha)}_{j}|A,j\rangle+b^{(\alpha)}_{j}|B,j\rangle)\\\
&=\sum_{n=-\Lambda}^{\Lambda}\sum_{i\geq
1}\left(t_{n}b^{(\alpha)}_{i+n}|A,i\rangle+t^{*}_{n}a^{(\alpha)}_{i}|B,i\\!+\\!n\rangle\right)=0,\\\
\end{split}$ (17)

with the sum over unit cells constrained by $i\\!+\\!n\\!\geq\\!1$. This gives
us the following conditions on the coefficients which multiply the
$|A,i\rangle$ and $|B,i\rangle$ states in Eq. (17), call them $C_{A,i}$ and
$D_{B,i}$ respectively:

$\displaystyle C_{A,i}$
$\displaystyle\equiv\sum_{{\color[rgb]{0,0,0}m=i-\Lambda}}^{i+\Lambda}t_{m-i}b^{(\alpha)}_{m}=0$
(18a) $\displaystyle D_{B,i}$
$\displaystyle\equiv\sum_{{\color[rgb]{0,0,0}m=i-\Lambda}}^{i+\Lambda}t^{*}_{i-m}a^{(\alpha)}_{m}=0,$
(18b) with $i\geq 1$. The idea is now to use the zeros of $f(z)$, Eq. (5), to
prove that the coefficients in Eqs. (18a) and (18b) can be chosen so as to
yield precisely $\nu$ edge states $|\psi_{\alpha}\rangle$ at edge edge of the
chain, $\alpha=1,...,\nu$, with the properties (i) and (ii) above. We denote
by $z_{\alpha}$ the largest $\nu$ zeros within the unit disk, with the rest of
the zeros denoted $\tilde{z}_{s}$: $f(z_{\alpha})=0$ with $\alpha=1,...,\nu$,
and $f(\tilde{z}_{s})=0$ with $s=1,...,N_{p}$.

Case $N_{p}=0\,$: Let us first consider the case when there are no poles in
$f(z)$. This means that $t_{n<0}=0$ in Eq. (5) since the corresponding terms
are the ones that create poles in $f(z)$. In this case Eqs. (18a) and (18b)
contain all hopping amplitudes $t_{n}$ present in $f(z)$. It is therefore easy
to construct the zero modes by taking $b^{(\alpha)}_{m}=z_{\alpha}^{m-1}$ and
$a^{(\alpha)}_{m}=0$. It follows that the coefficients reduce to
$C_{A,i}=z_{\alpha}^{i-1}f(z_{\alpha})=0$ (using that
$t_{n<0}=t_{n>\Lambda}=0$ in the expression for $f(z_{\alpha}$)) and
$D_{B,i}=0$. The choice $b^{(\alpha)}_{m}=z_{\alpha}^{m-1}$, rather than the
more intuitive $b^{(\alpha)}_{m}=z_{\alpha}^{m}$, is mandated by the
normalizability of a zero mode: With $b^{(\alpha)}_{m}=z_{\alpha}^{m-1}$, the
inner product $\langle\psi_{\alpha}|\psi_{\alpha}\rangle$ is guaranteed to be
nonzero,

$\displaystyle\langle\psi_{\alpha}|\psi_{\alpha}\rangle$ $\displaystyle=$
$\displaystyle\sum_{i,j\geq 1}b_{i}^{(\alpha)\ast}b_{j}^{(\alpha)}\langle
B,i|B,j\rangle=\sum_{i\geq 1}|z_{\alpha}|^{2(i-1)}$ (19) $\displaystyle=$
$\displaystyle\frac{1}{1-|z_{\alpha}|^{2}},$

with the second line obtained by summing the geometric series, given that
$|z_{\alpha}|<1$. Note that we here use a notation where $(z_{\alpha})^{0}=1$
also for $z_{\alpha}=0$.

The $\nu$ states obtained by inserting
$b^{(\alpha)}_{i}\\!=\\!z_{\alpha}^{i-1}$ and $a^{(\alpha)}_{i}\\!=\\!0$,
$\alpha=1,...,\nu$, into Eq. (16) have zero energy by construction, and
moreover, they decay with the unit cell index $i$ since $z_{\alpha}$ is inside
the unit disk, implying an exponential decay $\sim\exp(i/\xi_{\alpha})$ with
localization length $\xi_{\alpha}=-1/\ln|z_{\alpha}|$. Thus, both conditions
(i) and (ii) above are satisfied. Let us note in passing that when a zero
$z_{\alpha}$ approaches the unit circle, the localization length is seen to
diverge, signaling criticality, with the corresponding edge mode hybridizing
with the bulk spectrum, leaving room for a massless bulk excitation at
$|z_{\alpha}|=1$, in accord with the discussion in Sec. II.A.

All cases with distinct $z_{\alpha},\alpha=1,...,N_{z}$, yield $N_{z}$ linear
independent edge states. But what if there are $m$ degenerate zeros of $f(z)$
(meaning that for $\alpha_{1}$, …, $\alpha_{m}$ the zeros are
$z_{\alpha_{1}}=...=z_{\alpha_{m}}=z_{\alpha}$)? Following Ref. VJP,  we can
here take

$b^{(\alpha_{\ell})}_{m}=\frac{d^{\ell-1}z^{m-1}}{dz^{\ell-1}}|_{z=z_{\alpha}}=\frac{(m-1)!}{(m-\ell)!}z^{m-\ell}_{\alpha},\
\ \ell=1,2,...,m,$ (20)

and one verifies that this implies

$C_{A,i}=z_{\alpha}^{i-1}\frac{d^{\ell-1}f(z)}{dz^{\ell-1}}|_{z=z_{\alpha}}=0,\
\ \ell=1,2,...,m,$ (21)

thus producing $m$ linear independent edge states as required.

Summarizing, when $N_{p}=0$ we can readily construct $N_{z}$ edge states with
the required properties (i) and (ii), valid also when the system is critical.
Note that the edge states thus obtained are nonzero only on sublattice $B$
since we have chosen $a^{(\alpha)}_{m}=0$. One should here note that given a
basis in which the chiral symmetry operator $S$ is diagonal, zero-energy edge
states of any chiral-symmetric model necessarily have support on only one
sublattice of a bipartite lattice AsbothBook .

Case $N_{p}\neq 0\,$: In this case we look for
$b^{(\alpha)}_{m}=z_{\alpha}^{m}+\sum_{s=1}^{N_{p}}\lambda^{(\alpha)}_{s}\tilde{z}_{s}^{m}$,
with $\\{\lambda^{(\alpha)}_{s}\\}$ complex numbers to be determined. As
before we take all $a^{(\alpha)}_{m}=0$. We know that $t_{n<-N_{p}}=0$ since
otherwise the multiplicity of the pole at the origin would be larger than
$N_{p}$. It follows from Eq. (18a) that the condition $C_{A,i>N_{p}}=0$ is
trivially satisfied. It remains to find a set of constants
$\lambda^{(\alpha)}_{s}$ that will make the rest of the expansion coefficients
$C_{A,1\leq i\leq N_{p}}$ vanish identically. Again invoking Eq. (18a), these
constants must satisfy the equations

$\displaystyle\begin{split}A_{is}\lambda_{s}^{(\alpha)}=-\sum_{{\color[rgb]{0,0,0}m=i-\Lambda}}^{i+\Lambda}t_{m-i}z_{\alpha}^{m},\
\ i=1,2,...,N_{p},\end{split}$ (22)

where
$A_{is}=\sum_{{\color[rgb]{0,0,0}m=i-\Lambda}}^{i+\Lambda}t_{m-i}\tilde{z}_{s}^{m}$
and summation over $s$ is implied. These equations are equivalent to a matrix
equation of the type $A\lambda=b$, with $A$ an $N_{p}\times N_{p}$ matrix with
elements $A_{is}$. As follows from the discussion in Ref. VJPSM, , provided
that $z_{\alpha}$ is nondegenerate, a row reduction turns $A$ into an
invertible Vandermonde matrix, implying the existence of unique solutions for
the $\lambda^{(\alpha)}_{s}$.

If there are degenerate zeros of $f(z)$ one can employ the same device as in
the case with degeneracies when $N_{p}=0$, working with derivatives instead;
cf. Eq. (20). However, we shall not go into details here.

####  II.3.2 $\nu=N_{z}-N_{p}<0$

Turning to the case of negative winding numbers, we cannot directly employ the
elegant inversion argument used in Ref. VJP,  since in our case, with complex
hopping amplitudes allowed in the AIII symmetry class, inversion symmetry is
in general broken. However, the necessary modification of the inversion
argument is minor, and essentially comes down to doing some “relabeling” in
the preceding equations for positive winding numbers.

We start off with the same Hamiltonian $H$ as before, Eq. (15), and
interchange $A$ with $B$, and $n$ with $-n$. We thus obtain

$\displaystyle\begin{split}H^{\prime}&=\sum_{n=-\Lambda}^{\Lambda}\sum_{i\geq
1}\,t^{\prime}_{n}|A,i-n\rangle\langle B,i|+\mbox{H.c.},\\\ \end{split}$ (23)

with $t^{\prime}_{n}=(t_{-n})^{*}$ and with $|A,i-n\rangle$ a null vector for
$i-n\leq 0$. Note that this is just a rewriting of the Hamiltonian and hence
the edge states of $H$ and $H^{\prime}$ must be the same. Let us focus on
$H^{\prime}$, in one-to-one correspondence with
$f^{\prime}(z)=\sum_{n}(t_{-n})^{*}z^{n}=f^{*}(1/z^{*})$ (where the summation
is over $n\in[-\Lambda,\Lambda]$). Now assume that $f(z)$ has
$N=N_{\text{Z}}+N_{\text{C}}+N_{\text{O}}$ total number of zeros, with
$N_{\text{C}}\,(N_{\text{O}})$ the number of zeros on (outside) the unit
circle. We can then write

$f^{\prime}(z)=\frac{1}{z^{-N_{\text{P}}}}\prod_{i=1}^{N}(\frac{1}{z}-z_{i}^{*})=\frac{(-1)^{N}}{z^{N-N_{\text{P}}}}\prod_{i=1}^{N}(z-\frac{1}{z_{i}^{*}})z_{i}^{*},$
(24)

with, as before, $N_{\text{P}}$ the multiplicity of the pole of $f(z)$ at the
origin. It follows that the multiplicity of the pole of $f^{\prime}(z)$ is
$N_{P}^{\prime}=N-N_{\text{P}}$. Moreover, by the inversion $z\rightarrow
1/z$, $f^{\prime}(z)$ must have the same number of zeros inside the unit
circle as $f(z)$ has outside the unit circle, that is,
$N_{\text{Z}}^{\prime}=N_{\text{O}}$. The winding number $\nu^{\prime}$ for
the inverted system is now easily calculated as

$\displaystyle\nu^{\prime}$ $\displaystyle=$ $\displaystyle
N_{\text{Z}}^{\prime}-N_{\text{P}}^{\prime}=N_{\text{O}}-(N-N_{\text{P}})$
(25) $\displaystyle=$ $\displaystyle
N_{\text{O}}-(N_{\text{Z}}+N_{\text{O}}+N_{\text{C}}-N_{\text{P}})=-\nu-
N_{\text{C}}.$

Given this result we can construct the edge states for $\nu<0$ in exact
analogy to the case $\nu>0$, using the zeros of $f^{\prime}(z)$. Acting with
$H^{\prime}$, Eq. (23), on a state $|\psi_{\alpha}\rangle=\sum_{i\geq
1}(a^{(\alpha)}_{i}|A,i\rangle+b^{(\alpha)}_{i}|B,i\rangle)$, the condition
that this state is a zero mode of $H^{\prime}$ takes the form

$\displaystyle\begin{split}&H^{\prime}|\psi_{\alpha}\rangle=\\\
&=\sum_{n=-\Lambda}^{\Lambda}\sum_{i\geq
1}\left(t^{\prime}_{n}b^{(\alpha)}_{i}|A,i-n\rangle+(t^{\prime}_{n})^{*}a^{(\alpha)}_{i-n}|B,i\rangle\right)=0,\\\
\end{split}$ (26)

with the sum over unit cells constrained by $i-n\geq 1$. It follows that the
coefficients $C^{\prime}_{A,i}$ and $C^{\prime}_{B,i}$ which multiply the
$|A,i\rangle$ and $|B,i\rangle$ states respectively must satisfy

$\displaystyle C^{\prime}_{A,i}$
$\displaystyle=\sum_{{\color[rgb]{0,0,0}m=i-\Lambda}}^{i+\Lambda}t^{\prime}_{m-i}b^{(\alpha)}_{m}=0$
(27a) $\displaystyle D^{\prime}_{B,i}$
$\displaystyle=\sum_{{\color[rgb]{0,0,0}m=i-\Lambda}}^{i+\Lambda}(t^{\prime}_{i-m})^{*}a^{(\alpha)}_{m}=0$
(27b)

with $i\geq 1$. These are the same equations as for the case with $\nu>0$,
Eqs. (18a) and (18b), and hence we can construct the edge states in exactly
the same manner, but now using the zeros of $f^{\prime}(z)$ that will give us
$\nu^{\prime}=-\nu-N_{\text{C}}$ edge states. It may be worth pointing out
that these states live on sublattice $A$, reflecting the fact that we
interchanged $A$ and $B$ when rewriting the Hamiltonian, replacing $H$ by
$H^{\prime}$.

###  II.4 Edge states in $\boldsymbol{2n}$-band gapless
AIII systems in 1D

We now proceed to generalize our finding from the previous section and
establish a bulk-boundary correspondence for 1D critical multiband systems in
the AIII symmetry class, i.e. systems with $2n$ bands (with $n\geq 1$,
treating two bands as a special case). As for the simple two-band case, our
objective is to construct $\nu$ linearly independent states per edge with the
properties that (i) their energies vanish identically for a semi-infinite
chain, and (ii) their wave functions decay exponentially as one moves away
from the edge. Here we shall take the unit cell large enough so that only the
hopping matrices $T_{0},T_{\pm 1}$ in Eq. (12) are nonzero. This is a
convenient choice which simplifies the analysis, and which does not impact the
result for the number of edge states.

####  II.4.1 $\nu=N_{z}-N_{p}>0$

For transparency, let us begin by explicitly writing down the Hamiltonian in
Eq. (7) on a semi-infinite lattice, having chosen a sufficiently large unit
cell so that there are only three sets of nonzero hopping amplitudes
$t_{j}^{XY}$, with $j=0,\pm 1$:

$\displaystyle\begin{split}H&=\\!\sum_{-1\leq j\leq 1}\sum_{i\geq
1}t^{XY}_{j}|X,i\rangle\langle Y,i+j|+\mbox{H.c.},\\\ \end{split}$ (28)

where $|Y,0\rangle$ is a null vector. As spelled out in Sec. II.B, the labels
$X$ and $Y$ run over the $2n$ different states within a unit cell,
$X\\!\in\\!\\{A,B,...\\}$ on one sublattice and $Y\\!\in\\!\\{C,D,...\\}$ on
the other, with $X$ and $Y$ implicitly summed over in Eq. (28). From now on,
all occurrences of repeated indices $X$ and $Y$ are summed over. Next, we
write a general expression for a multiband state,

$\displaystyle\begin{split}|\psi_{\alpha}\rangle=\sum_{i\geq
1}\left(a^{(\alpha)}_{i,X}|X,i\rangle+b^{(\alpha)}_{i,Y}|Y,i\rangle\right).\end{split}$
(29)

This state is a zero mode of $H$ if

$\displaystyle H|\psi_{\alpha}\rangle$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\sum_{-1\leq j\leq 1}\,\sum_{i\geq
1}(t^{XY}_{j}b^{(\alpha)}_{i\\!+\\!j,Y}|X,i\rangle\\!+\\!(t^{XY}_{j})^{*}a^{(\alpha)}_{i,X}|Y,i\\!+\\!j\rangle)$
(30) $\displaystyle=$ $\displaystyle 0.$

Similarly to the two-band case in Sec. II.C, Eqs. (18a) and (18b), this gives
us the following constraints on the coefficients which multiply the
$|X,i\rangle$ and $|Y,i\rangle$ states:

$\displaystyle C_{X,i}$
$\displaystyle=\sum_{{\color[rgb]{0,0,0}m=i-1}}^{i+1}t^{XY}_{m-i}b^{(\alpha)}_{m,Y}=0,$
(31a) $\displaystyle D_{Y,i}$
$\displaystyle=\sum_{{\color[rgb]{0,0,0}m=i-1}}^{i+1}(t^{XY}_{i-m})^{*}a^{(\alpha)}_{m,X}=0,$
(31b)

with $i\geq 1$. Using matrix notation,

$\displaystyle\boldsymbol{C}_{i}$
$\displaystyle=\sum_{{\color[rgb]{0,0,0}m=i-1}}^{i+1}T_{m-i}\boldsymbol{b}^{(\alpha)}_{m}=0$
(32a) $\displaystyle\boldsymbol{D}_{i}$
$\displaystyle=\sum_{{\color[rgb]{0,0,0}m=i-1}}^{i+1}T^{*}_{i-m}\boldsymbol{a}^{(\alpha)}_{m}=0$
(32b)

where $T_{m-i}$ is the hopping matrix constructed out of the amplitudes
$t^{XY}_{i}$ (and similarly for $T^{*}_{i-m}$), and where
$\boldsymbol{a}^{(\alpha)}_{m}$ and $\boldsymbol{b}^{(\alpha)}_{m}$ are
vectors with elements $a^{(\alpha)}_{m,X}$ and $b^{(\alpha)}_{m,Y}$
respectively.

Case $T_{-1}\\!=\\!0$: For this case, with only $T_{0}$ and $T_{1}$ being
nonzero, it is straightforward to construct the zero modes. Recall from Sec.
II.B that the winding number $\nu$ can be calculated from
$F(z)=T_{-1}z^{-1}+T_{0}+T_{1}z$ as $\nu=N_{z}-N_{p}$ where $N_{z}$ ($N_{p}$)
is the number of zeros (poles) of $\det F(z)$. When $T_{-1}=0$ there are no
poles, and the expression for the winding number simplifies to $\nu=N_{z}$.
Let us take a more pragmatic route here and simplify the computations by
assuming that every zero of $\det F(z)$ is nondegenerate. This is motivated by
the fact that since a degeneracy will split unless the hopping amplitudes are
fine-tuned, any generic experimental uncertainty will wash away the
degeneracies, making the degenerate case irrelevant for applications.

Any zero of $\det F(z)$, denoted by $z_{\alpha}$ with $\alpha=1,...,N_{z}$,
guarantees that there exists a nonzero eigenvector
$\bar{\boldsymbol{b}}^{(\alpha)}$ satisfying
$F(z_{\alpha})\bar{\boldsymbol{b}}^{(\alpha)}=0$. By taking
$\boldsymbol{b}^{(\alpha)}_{m}=z_{\alpha}^{m-1}\bar{\boldsymbol{b}}^{(\alpha)}$
in Eq. (32a) it is easy to verify that the coefficient $\boldsymbol{C}_{i}$
reduces to
$\boldsymbol{C}_{i}=z_{\alpha}^{i-1}F(z_{\alpha})\bar{\boldsymbol{b}}^{(\alpha)}=0$.
Choosing $\boldsymbol{a}^{(\alpha)}_{m}=0$ in Eq. (32b) one infers the
existence of $\nu$ zero-energy states, living on one of the sublattices.

In exact analogy to the two-band case, the states obtained via the
construction above $-$ inserting
$b_{i,Y}^{(\alpha)}=z_{\alpha}^{i-1}\bar{b}_{Y}^{(\alpha)}$ and
$a_{i,X}^{(\alpha)}=0$ into Eq. (29), with $a_{i,X}^{(\alpha)}$ and
$b_{i,Y}^{(\alpha)}$ the elements of $\boldsymbol{a}^{(\alpha)}_{i}$ and
$\boldsymbol{b}^{(\alpha)}_{i}$ $-$ decay exponentially as
$\exp(i/\xi_{\alpha})$ with localization length
$\xi_{\alpha}=-1/\ln|z_{\alpha}|$, $|z_{\alpha}|<1$. Thus, also condition (ii)
above is satisfied.

Case $T_{-1}\\!\neq\\!0$: Let us denote by $z_{\beta}$ (with
$|z_{\beta}|\\!<\\!1$) and by $\bar{\boldsymbol{b}}^{(\beta)}$ all solutions
to

$\displaystyle\begin{split}\bar{F}(z_{\beta})\bar{\boldsymbol{b}}^{(\beta)}=(T_{-1}+T_{0}z_{\beta}+T_{1}z_{\beta}^{2})\bar{\boldsymbol{b}}^{(\beta)}=0.\end{split}$
(33)

Assuming that all zeros are nondegenerate (cf. the discussion above when
$T_{-1}=0$), their total number is $\bar{N}_{z}$.  The pairs ($z_{\beta}$;
$\bar{\boldsymbol{b}}^{(\beta)}$) define vectors
$\boldsymbol{b}^{(\beta)}_{m}=z_{\beta}^{m-1}\bar{\boldsymbol{b}}^{(\beta)}$
(taking $\boldsymbol{a}^{(\beta)}_{m}=0$) that  are all linearly-independent
among each other because they decay at distinct rates $z_{\beta}^{m-1}$. Given
this, we now aim to construct independent edge states out of
$\boldsymbol{b}^{(\beta)}_{m}$.  Thus, we look at superpositions
$\boldsymbol{b}^{(\alpha)}_{m}=\sum_{\beta=1}^{\bar{N}_{z}}\lambda_{\alpha\beta}\boldsymbol{b}^{(\beta)}_{m}$
and notice that for each such superposition the coefficients
$\boldsymbol{C}_{i\geq
2}=T_{-1}\boldsymbol{b}^{(\alpha)}_{i-1}+{\color[rgb]{0,0,0}T_{0}\boldsymbol{b}^{(\alpha)}_{i}+T_{1}\boldsymbol{b}^{(\alpha)}_{i+1}}$
in Eq. (32a) vanish identically. The existence of zero-energy states then
hinges on the possibility to construct  independent vectors
$\boldsymbol{b}^{(\alpha)}_{m}$ such that
$\boldsymbol{C}_{1}=\sum_{\beta=1}^{\bar{N}_{z}}\lambda_{\alpha\beta}\boldsymbol{v}_{\beta}=0$
with
$\boldsymbol{v}_{\beta}=T_{0}\boldsymbol{b}^{(\beta)}_{1}+T_{1}\boldsymbol{b}^{(\beta)}_{2}$.

To take on this task, from all $\\{\boldsymbol{v}_{\beta}\\}$ we first select
a set of basis vectors $\\{\boldsymbol{v}^{\text{base}}_{\gamma}\\}$ spanning
the space of $\\{\boldsymbol{v}_{\beta}\\}$. Their number cannot exceed $n$
(dimension of the vectors $\boldsymbol{v}_{\beta}$).  Let us refer to the rest
of the vectors in $\\{\boldsymbol{v}_{\beta}\\}$ as
$\\{\boldsymbol{v}^{\text{nonbase}}_{\delta}\\}$. Each non-basis vector
$\boldsymbol{v}^{\text{nonbase}}_{\delta}$ may then be decomposed as
$\boldsymbol{v}^{\text{nonbase}}_{\delta}=\sum_{\gamma}x_{\delta\gamma}\boldsymbol{v}^{\text{base}}_{\gamma}$
with some coefficients $x_{\delta\gamma}$. It follows that for each $\delta$
we can then construct a state from
$\boldsymbol{b}^{(\alpha)}_{m}=\boldsymbol{b}^{(\delta)}_{m}-\sum_{\gamma}x_{\delta\gamma}\boldsymbol{b}^{(\gamma)}_{m}$
which trivially fulfills the condition $\boldsymbol{C}_{1}=0$. Importantly,
the states thus obtained are linearly independent since they are each tied to
a particular component $\boldsymbol{b}^{(\delta)}_{m}$, with each
$\boldsymbol{b}^{(\delta)}_{m}$ having a unique decay rate $z_{\delta}^{m-1}$.

Following the protocol above, the vectors
$\boldsymbol{b}^{(\alpha)}_{m}=\boldsymbol{b}^{(\delta)}_{m}-\sum_{\gamma}x_{\delta\gamma}\boldsymbol{b}^{(\gamma)}_{m}$
are seen to define at least $\nu=\bar{N}_{z}-n=N_{z}-N_{p}$ independent edge
states. This concludes the analysis for this case.

####  II.4.2 $\nu=N_{z}-N_{p}<0$

We approach this case in the same way as when we analyzed negative winding
numbers for two bands in Sec. II.B, essentially relying only on a
“relabelling” of the equations derived for positive winding numbers.

By interchanging the sublattice indices $X$ and $Y$ and flipping the sign of
the index $j$ which labels the hopping amplitudes $t_{j}^{XY}$ in Eq. (28),
the multiband Hamiltonian $H$ in the same equation takes the form

$\displaystyle\begin{split}H^{\prime}&=\sum_{-1\leq j\leq 1}\sum_{i\geq
1}(t^{\prime}_{j})^{XY}|X,i-j\rangle\langle Y,i|+\mbox{H.c.}\\\ \end{split}$
(34)

with $(t^{\prime}_{j})^{XY}=(t^{XY}_{-j})^{*}$ and where $|X,0\rangle$ is a
null state. Recall that repeated indices $X$ and $Y$ are always summed over.
The function $F(z)=T_{-1}z^{-1}+T_{0}+T_{1}z$, associated to $H$, now gets
replaced by a new function $F^{\prime}(z)$, with the property that $\det
F^{\prime}(z)=(\det F(1/z^{*}))^{*}$. By exactly the same argument as for the
two-band case, it follows that if $\det F(z)$ calls for $\nu<0$ then $\det
F^{\prime}(z)$ entails $\nu^{\prime}=-\nu-{N}_{\text{C}}$ with
${N}_{\text{C}}$ the number of zeros of $\det F(z)$ on the unit circle.

The construction of the edge states is simple, and can be carried out with the
case $\nu>0$ as a template. To see how, we act with $H^{\prime}$ on the
multiband state $|\psi_{\alpha}\rangle$ in Eq. (29), reading off the condition
that $|\psi_{\alpha}\rangle$ is a zero-energy state:

$H^{\prime}|\psi_{\alpha}\rangle=\sum_{-1\leq j\leq 1}\sum_{i\geq
1}\Big{(}(t^{\prime}_{j})^{XY}b^{(\alpha)}_{i,Y}|X,i-j\rangle\\\
+((t^{\prime}_{j})^{XY})^{*}a^{(\alpha)}_{i-j,X}|Y,i\rangle\Big{)}=0,$ (35)

where $a_{0}=0$. This equation implies the following constraints on the
coefficients (for $i\geq 1$):

$\displaystyle C_{X,i}$
$\displaystyle=\sum_{{\color[rgb]{0,0,0}m=i-1}}^{i+1}(t^{\prime}_{m-i})^{XY}b^{(\alpha)}_{m,Y}=0$
(36a) $\displaystyle D_{Y,i}$
$\displaystyle=\sum_{{\color[rgb]{0,0,0}m=i-1}}^{i+1}((t^{\prime}_{i-m})^{XY})^{*}a^{(\alpha)}_{m,X}=0$
(36b)

or, using matrix notation,

$\displaystyle\boldsymbol{C}_{i}$
$\displaystyle=\sum_{{\color[rgb]{0,0,0}m=i-1}}^{i+1}T^{\prime}_{m-i}\boldsymbol{b}^{(\alpha)}_{m}=0$
(37a) $\displaystyle\boldsymbol{D}_{i}$
$\displaystyle=\sum_{{\color[rgb]{0,0,0}m=i-1}}^{i+1}(T^{\prime}_{i-m})^{*}\boldsymbol{a}^{(\alpha)}_{m}=0$
(37b)

These are the same equations as (32a) and (32b) for the case with $\nu>0$, but
with $T$ replaced by $T^{\prime}$. Thus, we can construct the zero-energy edge
states in exactly the same manner as for $\nu>0$, but now using the zeros of
$\det F^{\prime}(z)$ that will give us $\nu^{\prime}=-\nu-N_{\text{C}}$ such
states.

##  III Symmetry class BDI

###  III.1 Two-band gapless BDI systems in 1D: topological invariant and edge
states

The problem of critical edge states in the BDI symmetry class of 1D models was
solved by Verresen et al.VJP  for the case of two bands. Since any model which
belongs to the BDI symmetry class is also contained in AIII $-$ the symmetry
class which we analyzed in the previous section, including the multiband case
$-$ one may think that there is not much to add. Still, it is instructive to
revisit the problem to see how the BDI formalism in Ref. VJP,  fits into the
scheme presented here, taking into account the symmetries which mark out the
BDI class. Beyond chiral invariance, intrinsic also to AIII, these are time-
reversal symmetry, $THT^{-1}=H$, and particle-hole symmetry, $CHC^{-1}=-H$,
with $T^{2}\\!=\\!C^{2}\\!=\\!1$. To avoid any misconception, it is important
to stress that these symmetries are to be considered as accidental when a
Hamiltonian $H$ is placed in the AIII symmetry class, while being enforced on
any perturbation of the same Hamiltonian when placed in BDI. In particular,
this implies that the symmetry-protection of edge states is stronger if $H$ is
placed in AIII rather than in BDI.

Verresen et al. VJP  performed their analysis on a representative BDI
Hamiltonian, $H_{\text{BDI}}$, describing a spinless superconducting wire. In
real space, assuming translational invariance on a chain with unit cells
labeled by $j$, the second-quantized Hamiltonian can be written as ${\cal
H}_{\text{BDI}}=\sum_{n}t_{n}{\cal H}_{n}$, where

$\displaystyle\begin{split}{\cal
H}_{n}&=\frac{i}{2}\sum_{j}\tilde{\gamma}_{j}\gamma_{j+n}\\\
&=-\frac{1}{2}\sum_{j}(c^{\dagger}_{j}c^{\dagger}_{j+n}+c^{\dagger}_{j}c_{j+n}-c_{j}c^{\dagger}_{j+n}-c_{j}c_{j+n}).\\\
\end{split}$ (38)

Here $\gamma_{j}\\!=\\!\frac{1}{2}(c_{j}^{\dagger}+c_{j})$ and
$\tilde{\gamma}_{j}\\!=\\!\frac{i}{2}(c_{j}^{\dagger}-c_{j})$ are Majorana
modes, with $c_{j}$ and $c_{j}^{\dagger}$ fermion operators. Note that time-
reversal symmetry precludes terms of the form $i\gamma_{j}\gamma_{\ell}$ and
$i\tilde{\gamma}_{j}\tilde{\gamma}_{\ell}$ as well as complex hopping
amplitudes in the decomposition of $H_{\text{BDI}}$. As implied by the
notation, each unit cell with index $j$ contains two Majorana modes,
$\gamma_{j}$ and $\tilde{\gamma}_{j}$.

To make contact with our generic first-quantized AIII Hamiltonian in Eq. (1)
we can introduce Nambu spinors
$\Psi^{\dagger}=(c_{1}^{\dagger}...c_{N}^{\dagger}\,c_{1}...c_{N})$ and
extract the corresponding Bogoliubov-de-Gennes  (BdG) Hamiltonian, call it
$H_{\text{BdG}}$, from ${\cal
H}_{\text{BDI}}=\Psi^{\dagger}H_{\text{BdG}}\Psi$. By this, one obtains

$H_{\text{BdG}}=-\frac{1}{4}\sum_{j,n}t_{n}\left(\tau_{z}\otimes|j\rangle\langle
j+n|+i\tau_{y}\otimes|j\rangle\langle j+n|\right)+\mbox{H.c.},$ (39)

which acts on basis states $|\tau\rangle\otimes|j\rangle$, with
$|\tau=1\rangle=(10)^{T}$ for a particle state and
$|\tau\\!=\\!-\\!1\rangle=(01)^{T}$ for a hole state, and where the state
$|j\rangle=(0,....,1,....0)^{T}$ corresponds to the $j$:th site of the chain.
$\tau_{y}$ and $\tau_{z}$ are Pauli matrices.

Particle-hole symmetry is built into $H_{\text{BdG}}$, with $C=\tau_{x}{\cal
K}$ (where $\tau_{x}$ is the Pauli $x$-matrix and ${\cal K}$ the complex-
conjugation operator), as is also time-reversal symmetry with $T={\cal K}$,
with both operators here written in a $k$-space representation. Here
$T^{2}=C^{2}=1$ as must be since the model is spinless. The chiral symmetry
operator $S$ is given by $\tau_{x}$ in the chosen basis. We can make a
rotation to a “chiral basis” where $S$ is diagonal by simply acting with
$U=(\mathbb{1}+\tau_{y})/\sqrt{2}$ on the basis states. As a result,
$\tau_{z}\rightarrow\tau_{x}$ and $\tau_{y}\rightarrow\tau_{y}$ and therefore
$H_{\text{BdG}}\rightarrow-\frac{1}{4}\sum_{j,n}t_{n}(\tau_{x}\otimes|j\rangle\langle
j+n|+i\tau_{y}\otimes|j\rangle\langle j+n|)+\mbox{H.c.}$, now with chiral
symmetry operator $S=\tau_{z}$. Rewriting $H_{\text{BdG}}$ in terms of the
eigenstates of $\tau_{z}$, call them $|A\rangle$ and $|B\rangle$, one obtains
$H_{\text{BdG}}=-\frac{1}{2}\sum_{j,n}t_{n}|A,j\rangle\langle
B,j+n|)+\mbox{H.c.}$, of the very same form as our Eq. (1) (up to an
immaterial prefactor of $-1/2$). It follows that the $f(z)$ function derived
for the BDI symmetry class in Ref. VJP,  is given by the same expression as in
our Eq. (5), with the crucial difference that the parameters $t_{n}$ are now
constrained to be real. According to the fundamental theorem of algebra, this
restricts the zeros of $f(z)$ to be real or to come in complex-conjugate
pairs. Apart from this, the overall picture does not change. As worked out by
Verresen et al. VJP , the number of topological edge states is determined by
the zeros and poles inside the unit disk of the $f(z)$ function, just as for
AIII.

###  III.2 $\boldsymbol{2n}$-band gapless BDI systems in 1D:
topological invariant and edge states

In Ref. VJPSM, , Verresen et al. presents a formal proof that the topological
invariant for a critical translational invariant system in the BDI symmetry
class is independent of the choice of unit cell. While intuitively clear that
this must be the case, the proof primarily serves as a consistency check of
the construction of an $f(z)$ function when the unit cell is enlarged. If we
restrict the generic AIII multiband Hamiltonian in Eq. (1) to BDI by requiring
invariance under time-reversal and particle-hole symmetry where
$T=\mathbb{1}\otimes\mathbb{1}\mathcal{K}$ and
$C=\tau_{x}\otimes\mathbb{1}\mathcal{K}$ in a $k$-space representation our
$d(z)$ function in Eq. (12), now restricted by these symmetries, can be shown
to be identical to that of Ref. VJPSM, , denoted by $f(z)$ in Eq. (S8) of that
same reference. Note that the unit matrix multiplying ${\cal K}$ in the
expressions for $T$ and $C$ acts in the $n$-dimensional space of internal
states contained in the unit cell (not counting the Nambu degree of freedom
with one particle- and one hole-state). The necessary calculation is
straightforward but long-winded, and for this reason we omit it here.
Moreover, the result is entirely expected: Our construction of critical edge
states for multiband AIII models ensures the existence of exponentially
localized zero-energy edge states also for critical multiband models in the
BDI symmetry class, with the same counting as before, $\nu=N_{z}-N_{p}$.

##  IV Symmetry class CII

A model which is placed in symmetry class CII is protected by all three
symmetries $S,\,T,$ and $C$ of the tenfold way Chiu2016 , analogous to BDI,
but now with $T^{2}=C^{2}=-1$. Thus, this is the appropriate symmetry class to
use for chiral-invariant spinful fermion systems with topological edge states
protected by time-reversal and particle-hole symmetry. We address this problem
similar to how we approached the BDI class in the previous section, where we
analyzed how the functions $f(z)$ (for two bands, Eq. (5)) and $d(z)$ (for the
general multiband case, Eq. (12)) get restricted by the additional symmetries
$T$ and $C$. Here, we have to proceed with some care.

First, we cannot satisfy the constraints $T^{2}=C^{2}=-1$ for a two-band
model. The reason is simple. For these constraints to be satisfied, the
symmetry operators $T$ and $C$, both being anti-unitary, have to be
represented by a product of a Hermitian (and unitary) matrix with purely
imaginary elements and the complex conjugation operator ${\cal K}$. There is
only one such $2\times 2$ matrix, the Pauli $y$-matrix, and hence, with $T$
and $C$ being distinct operators, there is no faithful representation adequate
for two bands.

Secondly, turning to more than two bands, one still has to be wary: There are
multiple choices for the representation of the symmetries and they may
influence the $d(z)$ function in Eq. (12) differently. Here, for ease and
clarity, we limit our discussion to the simplest multiband case, $2n=4$, with
conventional realizations of the time-reversal and particle-hole symmetries;
see below.

###  IV.1 Topological invariant for four-band gapless
CII systems in 1D

For the purpose of connecting the representative four-band CII models to
possible physical realizations (for examples, see e.g. Refs. Zhao2014, ;
Prakash2018, ), we shall begin by identifying the possible varieties of
spinful superconducting chains that satisfy the required symmetries. The case
of four bands is easy to picture, thinking in terms of a Bogoliubov-de-Gennes
construction using Nambu spinors as in our discussion of the BDI symmetry
class, but now with an added spin-1/2 degree of freedom which doubles the
number of bands. We take the time-reversal symmetry operator to be of
conventional type for spin-1/2 systems,
$T=\mathbb{1}\otimes\sigma_{y}\mathcal{K}$ (in a $k$-space representation),
with the first (second) factor acting in the particle-hole (spin) space, and
with ${\cal K}$ acting in both,  yielding $T^{2}=-1$. The particle-hole
symmetry is decoupled from the spin degree of freedom and therefore its
symmetry operator has to be $C=\tau_{y}\otimes\mathbb{1}\mathcal{K}$ (in the
same representation) in order to fulfill the requirement $C^{2}=-1$. Here
$\tau_{y}$ is also a Pauli $y$-matrix, but acting in the particle-hole space
(cf. the text after Eq. (39)). The chiral symmetry operator $S$ is fixed by
$S=T^{-1}C$. In the following we shall explore how the restrictions implied by
the $T$ and $C$ symmetries map out the possible spinful chains belonging to
the CII class. With an eye to parallels with the BDI symmetry class as studied
in Ref. VJP, , we will work in a representation with Majorana modes. Different
from Ref. VJP, , the Majoranas will now carry spin-1/2.

####  IV.1.1  Spinful Majorana chains

To construct the possible CII Majorana chains we must analyze what types of
spinful Majorana bilinears that survive the restrictions from time-reversal
and particle-hole symmetry. Our strategy is to first construct all real-space
first-quantized Hamiltonian matrices $H$ allowed by these symmetries and then
use a Nambu representation to go to second-quantization, and from there, to a
representation in spinful Majoranas,  defined by
$\gamma_{j,\sigma}=\frac{1}{2}(c^{\dagger}_{j,\sigma}+c_{j,\sigma})$ and
$\tilde{\gamma}_{j,\sigma}=\frac{i}{2}(c^{\dagger}_{j,\sigma}-c_{j,\sigma})$,
with $c^{\dagger}_{j,\sigma}$ and $c_{j,\sigma}$ fermion operators at lattice
site $j$ with spin $\sigma=\uparrow,\downarrow$. As an outcome we obtain the
spinful Majorana chains

$\displaystyle\begin{split}{\cal
H}^{\text{CII}}=\sum_{n=-\Lambda}^{\Lambda}\\!(t_{n}{\cal
H}^{\text{CII}}_{n}+\bar{t}_{n}\bar{\cal H}^{\text{CII}}_{n}),\end{split}$
(40)

with

$\displaystyle\begin{split}&{\cal
H}^{\text{CII}}_{n}=i\sum_{j,\sigma}\tilde{\gamma}_{j,\sigma}\gamma_{j+n,\sigma}\end{split}$
(41)

and

$\displaystyle\begin{split}&\bar{\cal
H}^{\text{CII}}_{n}=i\sum_{j}(\tilde{\gamma}_{j,\uparrow}\gamma_{j+n,\downarrow}-\tilde{\gamma}_{j,\downarrow}\gamma_{j+n,\uparrow})\\\
\end{split}$ (42)

serving as bases for all spin-preserving and spin-flipping Majorana chains,
respectively. The amplitudes $t_{n}$ and $\bar{t}_{n}$ are real, with
$[-\Lambda,\Lambda]$ the range of couplings between the Majoranas. The
associated hopping matrix in spin space, constructed as in Sec. II.B, is
readily read off from Eqs. (40) - (42):

$\displaystyle\begin{split}T_{n}=\begin{pmatrix}t_{n}&\bar{t}_{n}\\\
-\bar{t}_{n}&t_{n}\end{pmatrix}.\end{split}$ (43)

For details, see the Appendix.

By going to $k$-space via a Fourier transform and performing an analytic
continuation to the entire complex plane, one can now use Eq. (43) to define
an $F$-matrix by $F(z)=\sum_{n}T_{n}z^{n}$, in exact analogy to how it was
done in Sec. II.B for the AIII symmetry class. We can again use the
construction $d(z)=\det F(z)$ for extracting the winding number
$\nu=N_{z}-N_{p}$, with $d(z)$ well defined also at criticality when the gap
is closed (and $d(z)$ has one or several zeros on the unit circle).

Any first-quantized Hamiltonian which can be put in symmetry class CII can
also be put in AIII (however, with a stronger protection of its topological
edge states  since now there are many more types of perturbations against
which the states are protected, having removed the constraints of time-
reversal and particle-hole symmetry). One thus expects that the bulk-boundary
correspondence that we derived in Sec. II for AIII should still be valid. In
other words, one anticipates that the winding number $\nu=N_{z}-N_{p}$
obtained for a 1D model in symmetry class CII correctly counts its number of
topological edge states, also at criticality. In fact, when discussing the
multiband problem for symmetry class BDI in Sec III we relied precisely on
this line of argument, grounded in the work by Verresen et al. VJP  on two-
band models. While the argument is expected to be valid also for CII, the
absence of results for two-band models in CII $-$ since none exists in this
symmetry class! $-$ may call for a closer look. To this we turn next. However,
rather than proving the existence of edge states by an explicit construction
for CII, we shall again exploit the bulk-boundary correspondence for critical
edge states in AIII. To make the argument formally sound we will make the
connection between the CII and AIII winding numbers mathematically manifest,
taking off from the four-band Majorana chain in Eq. (42).  This will clarify
why the CII winding numbers can take only even integer values, whereas there
is no such restriction for AIII Chiu2016 . Let us here point out that the
generalization to $2n$ bands with $n>2$ becomes cumbersome to handle in the
second-quantized Majorana representation. Instead, it is preferable to start
directly with a first-quantized single-particle CII Hamiltonian and proceed as
for the multiband BDI systems in Sec. III.B, exploiting the results in Sec.
II.D for multiband single-particle models in the AIII symmetry class. Again,
the calculation is straightforward but long-winded. Since the existence of
edge states is fully expected given the analysis of the four-band case, we
omit it here.

####  IV.1.2 Connection between CII and AIII winding numbers

We begin by writing the general CII Hamiltonian in Eq. (42) on first-quantized
form, reversing the Nambu procedure from above and by this extracting the
corresponding BdG Hamiltonian $H^{\text{CII}}_{\text{BdG}}$, call it simply
$H$:

$\displaystyle\begin{split}H=\sum_{j,n}\big{(}&t_{n}(\tau_{z}\otimes\mathbb{1}-i\tau_{x}\otimes\sigma_{y})\\\
&+\bar{t}_{n}(i\tau_{z}\otimes\sigma_{y}+\tau_{x}\otimes\mathbb{1})\big{)}\otimes|j\rangle\langle
j+n|+\text{H.c.}\end{split}$ (44)

We then perform a sequence of unitary transformations acting in the particle-
hole and spin spaces, with the transformations given by $U=U_{3}U_{2}U_{1}$
with
$U_{1}=\frac{1}{2}(\mathbb{1}+\tau_{z})\otimes\mathbb{1}+\frac{i}{2}(\mathbb{1}-\tau_{z})\otimes\sigma_{y},U_{2}=\frac{1}{\sqrt{2}}(\mathbb{1}\otimes\mathbb{1}+i\tau_{y}\otimes\mathbb{1})$,
and
$U_{3}=\frac{1}{\sqrt{2}}(\mathbb{1}\otimes\mathbb{1}+i\mathbb{1}\otimes\sigma_{x})$.
As a result, $H\rightarrow UHU^{-1}=H^{\prime}$, with

$\displaystyle\begin{split}H^{\prime}=&\sum_{j,n}\big{(}t_{n}(\tau_{x}\otimes\mathbb{1}+i\tau_{y}\otimes\mathbb{1})\\\
&+i\bar{t}_{n}(\tau_{x}\otimes\sigma_{z}+i\tau_{y}\otimes\sigma_{z})\big{)}\otimes|j\rangle\langle
j+n|+\text{H.c.}\end{split}$ (45)

The chiral symmetry operator $S$ becomes diagonal in the rotated basis and
takes the form $S=\sum_{j}\tau_{z}\otimes\mathbb{1}|j\rangle\langle j|$ (as is
easily verified by checking the identity $SH^{\prime}S^{-1}=-H^{\prime}$).
Moreover, $H^{\prime}$ is diagonal in spin space and can be decomposed as

$\displaystyle\begin{split}H^{\prime}=\sum_{j,n}\big{(}&(t_{n}+i{\color[rgb]{0,0,0}\bar{t}_{n}})|A,\uparrow,n\rangle\langle
B,\uparrow,j+n|\\\
&+(t_{n}-i{\color[rgb]{0,0,0}\bar{t}_{n}})|A,\downarrow,j\rangle\langle
B,\downarrow,j+n|\big{)}+\text{H.c.},\end{split}$ (46)

where $|A,\sigma,j\rangle$ and $|B,\sigma,j\rangle$ are particle and hole
states, respectively with spin $\sigma=\uparrow,\downarrow$, attached to the
unit cell with index $j$. But this is nothing but two copies, labeled by
$\uparrow$ and $\downarrow$, of the general two-band AIII Hamiltonian in Eq.
(1)! It follows immediately that the winding number, is derivable from
$d(z)=\det F(z)=\det\sum_{n}T^{\prime}_{n}z^{n}$, with

$\displaystyle\begin{split}T_{n}^{\prime}=\begin{pmatrix}t_{n}+i\bar{t}_{n}&0\\\
0&t_{n}-i\bar{t}_{n}\end{pmatrix}.\end{split}$ (47)

We can explicitly rewrite $d(z)=\det F(z)=f^{2}(z)+g^{2}(z)$ with
$f(z)=\sum_{n}t_{n}z^{n}$ and $g(z)=\sum_{n}\bar{t}_{n}z^{n}$. This result is
in perfect agreement with the one obtained above using a Majorana
representation, but now we can directly refer to the bulk-edge correspondence
proved for class AIII. This establishes that the winding number
$\nu=N_{z}-N_{p}$ for symmetry class CII in 1D correctly counts the number of
topological edge states, also at criticality.  As mandated by the tenfold way
for the 1D CII symmetry classChiu2016 , $\nu$ is restricted to even integers:
Factorizing $d(z)=(f(z)+ig(z))(f(z)-ig(z))$, complex zeros are seen to come in
conjugate pairs, with real zeros coming with even multiplicities since $t_{n}$
and $\bar{t}_{n}$ are real. (The poles trivially come with even
multiplicities.)

###  IV.2 Edge states in four-band gapless
CII systems in 1D

Let us finally address the character of the zero-energy edge states, showing
that they are Majoranas.

Zero-energy states have support on only one sublattice in a basis where the
chiral symmetry operator $S$ is represented by a diagonal matrix. This
transpired from our analysis in Sec II, and holds quite general AsbothBook .
Knowing that $S$ is indeed diagonal in the rotated spin-diagonal basis of
$H^{\prime}$, we can therefore consider, without loss of generality, a zero-
energy state, call it $|\phi\rangle$, with support on the $A$ sublattice only.
When writing Eq. (46) we associated $A$ with a particle state, $(1\,0)^{T}$
[with sublattice $B$ being associated with a hole state, $(0\,1)^{T}$]. In
this notation, and leaving out the spatial part of $|\phi\rangle$, we thus
write $|\phi\rangle=(1\,0)^{T}\otimes(a\,b)^{T}=(a\,b\,0\,0)^{T}$, with
$(a\,b)^{T}$ a spinor in spin space with amplitudes $a$ and $b$. Particle-hole
symmetry implies that $C|\phi\rangle$ is also a zero-energy state, where, in
the rotated basis, $C=\tau_{z}\otimes\sigma_{x}\mathcal{K}$. We can always
construct two new zero-energy states by adding and subtracting $\phi$ and
$C|\phi\rangle$, in this way obtaining
$|\phi\rangle_{\pm}=(u_{1}^{\pm}\,u_{2}^{\pm}\,0\,0)^{T}$ with
$u_{1}^{\pm}=a\pm b^{\ast}$ and $u_{2}^{\pm}=b\pm a^{\ast}$. We now go back to
the original basis by carrying out the inverse transformations
$|\phi\rangle_{\pm}\rightarrow|\psi\rangle_{\pm}=U^{-1}|\phi\rangle_{\pm}=U_{1}^{\dagger}U_{2}^{\dagger}U_{3}^{\dagger}|\phi\rangle_{\pm}=\frac{1}{2}(u^{\pm}_{1}\\!-\\!iu^{\pm}_{2}\
\ u^{\pm}_{2}\\!-\\!iu^{\pm}_{1}\ \ -u^{\pm}_{2}\\!-\\!iu^{\pm}_{1}\ \
u^{\pm}_{1}\\!+\\!iu^{\pm}_{2})^{T}$. Exploiting the Nambu spinor
representation with
$\Psi^{\dagger}_{\text{A}}=(c^{\dagger}_{\uparrow}\,c^{\dagger}_{\downarrow}\,c_{\uparrow}\,-\\!c_{\downarrow})$
(with $\Psi^{\dagger}_{\text{A}}$ acting in the composite particle-hole and
spin space in any unit cell on the $A$ sublattice), we then express the
$|\psi\rangle_{\pm}$ states in second-quantization as

$\displaystyle\begin{split}&\frac{1}{2}\Psi^{\dagger}_{A}(u^{\pm}_{1}\\!-\\!iu^{\pm}_{2}\
u^{\pm}_{2}\\!-\\!iu^{\pm}_{1}\ -\\!u^{\pm}_{2}\\!+\\!iu^{\pm}_{1}\
u^{\pm}_{1}\\!-\\!iu^{\pm}_{2})^{T}|0\rangle\\\ =&\
\frac{1}{2}\big{(}(u^{\pm}_{1}\\!-\\!iu^{\pm}_{2})\,c_{\uparrow}^{\dagger}\,+\,(u^{\pm}_{2}\\!-\\!iu^{\pm}_{1})\,c_{\downarrow}^{\dagger}\\\
-&\,(u^{\pm}_{2}\\!-\\!iu^{\pm}_{1})\,c_{\downarrow}\,-\,(u^{\pm}_{1}\\!-\\!iu^{\pm}_{2})\,c_{\uparrow}\big{)}\,|0\rangle,\end{split}$
(48)

with the composite particle-hole and spin space spanned by
$c^{\dagger}_{\uparrow}|0\rangle=(1000)^{T},\,c^{\dagger}_{\downarrow}|0\rangle=(0100)^{T},\,c_{\uparrow}|0\rangle=(0010)^{T}$,
and $c_{\downarrow}|0\rangle=(0001)^{T}$. Reading off from Eq. (48), one
immediately identifies the mode operators which correspond to these states:

$\displaystyle\begin{split}d_{\pm}&=(u^{\pm}_{1}-iu^{\pm}_{2})c_{\uparrow}^{\dagger}+(u^{\pm}_{2}-iu^{\pm}_{1})c_{\downarrow}^{\dagger}\\\
&-(u^{\pm}_{2}-iu^{\pm}_{1})c_{\downarrow}-(u^{\pm}_{1}-iu^{\pm}_{2})c_{\uparrow}.\end{split}$
(49)

Using that $u^{\pm}_{1}\\!=\\!\pm(u_{2}^{\pm})^{\ast}$, it follows that
$d_{\pm}\\!=\\!\pm id_{\pm}^{\dagger}$, implying that the zero–energy states
thus constructed are Majorana states.

##  V Robustness of the edge states: a numerical test

In the previous sections we showed that the boundary states in unperturbed
critical 1D multiband chains are connected to a topological number $\nu$,
generalizing the ordinary winding number for gapped systems to critical
systems in chiral symmetry classes AIII, BDI, and CII. In gapped systems the
edge states are robust to any perturbation from uncorrelated disorder which
respects the relevant symmetries and leaves the bulk gap open. Here we
numerically verify on a case study that the edge states of a CII critical
chain also exhibit robustness to disorder although the gap is now closed. By a
symmetry analysis we identify the chiral symmetry as solely responsible for
the protection of the states. A possible mechanism of such protection was
proposed in Ref. Verresen2020,  for the case of two-band models, but its
generalization to the multiband case has remained somewhat unclear and
requires a more thorough investigation.

Figure 1:  Number of disorder realizations $M$ (linear scale) corresponding to
(a) energy and (b) participation ratio (logarithmic scales) of the four edge
states of the gapless CII Hamiltonian in Eq. (50). The computations cover a
total of $10^{3}$ realizations of each type of the following on-site
uncorrelated disorder: proportional to $\tau_{x}\otimes\mathbf{1}$ (blue, $T$
and $C$ preserving), $\tau_{y}\otimes\sigma_{y}$ (light red, $C$ breaking),
and $\tau_{y}\otimes\mathbf{1}$ (green, $T$ breaking). (Partial) coloring of
bins by dark red corresponds to superposition of light red and green. Each
realization of disorder is obtained by summing up on-site perturbations with
random amplitudes $\in[-1,1]$. The number of unit cells is $N=100$. Figure 2:
Number of disorder realizations $M$ (linear scale) corresponding to (a) energy
and (b) participation ratio (logarithmic scales) of the four edge states of
the gapless CII Hamiltonian in Eq. (50). The computations cover a total of
$10^{3}$ realizations of each type of the following on-site uncorrelated
disorder: proportional to $\tau_{x}\otimes\sigma_{y}$ (red) and
$\tau_{z}\otimes\sigma_{y}$ (light blue), with both perturbations breaking $T$
and $C$ but preserving $S$. (Partial) coloring of bins by dark blue
corresponds to superposition of red and light blue. Each realization of
disorder is obtained by summing up on-site perturbations with random
amplitudes $\in[-1,1]$. The number of unit cells is $N=100$.

For a numerical test we take a topologically nontrivial critical spinful
Majorana chain with open boundaries, ${\cal
H}^{\text{CII}}_{\\{1,2\\}}=\sum_{n=1}^{2}\\!({\cal
H}^{\text{CII}}_{n}+\bar{\cal H}^{\text{CII}}_{n})$, and study how its four
edge states behave under disorder respecting the various symmetries of the CII
class.  (To confirm the criticality of ${\cal H}^{\text{CII}}_{\\{1,2\\}}$,
insert the nonzero hopping amplitudes of ${\cal
H}^{\text{CII}}_{\\{1,2\\}},t_{1}\\!=\\!\bar{t}_{1}\\!=\\!t_{2}\\!=\\!\bar{t}_{2}\\!=\\!1$,
into Eq. (47) and verify that the resulting $d(z)$ has two zeros on the unit
circle.) As follows from Eq. (44), the Hamiltonian ${\cal
H}^{\text{CII}}_{\\{1,2\\}}$ can be rewritten in first-quantization as

$\displaystyle\begin{split}H_{\\{1,2\\}}=\sum_{n=\\{1,2\\}\atop
j=1,2,...,N}\big{(}&(\tau_{z}\otimes\mathbb{1}-i\tau_{x}\otimes\sigma_{y})\\\
&+\\!(i\tau_{z}\otimes\sigma_{y}\\!+\\!\tau_{x}\otimes\mathbb{1})\big{)}\otimes|j\rangle\langle
j+n|\\!+\\!\text{H.c.},\\\ \end{split}$ (50)

with $|N+1\rangle$ and $|N+2\rangle$ null states. Having placed
$H_{\\{1,2\\}}$ in symmetry class CII, its relevant symmetries are identified
by the time-reversal operator
$T=\sum^{N}_{j=1}\mathbb{1}\otimes\sigma_{y}\,{\cal K}\otimes|j\rangle\langle
j|$, the particle-hole operator
$C=\sum^{N}_{j=1}\tau_{y}\otimes\mathbb{1}\,{\cal K}\otimes|j\rangle\langle
j|$, and the chiral symmetry operator
$S=\sum^{N}_{j=1}\tau_{y}\otimes\sigma_{y}\otimes|j\rangle\langle j|$, where
${\cal K}$ is the complex-conjugation operator.

In analogy to gapped symmetry-protected topological systems we are interested
to see if the edge states of $H_{\\{1,2\\}}$ in Eq. (50) remain localized
under symmetry-preserving locally uncorrelated perturbations and, if so, if
they stay pinned exactly at zero energy. The localization of the edge states
can be quantitatively addressed by calculating the participation ratio (PR)
that is defined by $R=1/\sum_{i}p_{i}^{\,2}$, where $p_{i}$ is the occupation
of the Bogoliubov quasiparticle at site $i$, explicitly obtained by summing up
all probability amplitudes of the corresponding Nambu spinor. The PR
quantifies localization of an eigenstate: A completely localized state has
$R=1$ while a completely delocalized state, such as a plane wave, has $R=N$
(where $N$ measures the extent of the chain, counting the total number of unit
cells). The robustness of an edge state is established if both of the
following conditions are satisfied: its energy level remains at zero and the
PR is of order unity.

In Fig. 1 we have perturbed the edge states with random uncorrelated on-site
disorder proportional to $\tau_{x}\otimes\mathbf{1}$ ($T$ and $C$ preserving),
$\tau_{y}\otimes\sigma_{y}$ ($C$ breaking), and $\tau_{y}\otimes\mathbf{1}$
($T$ breaking). The edge states display robustness when both symmetries are
preserved (implying that also chiral symmetry, $S=T^{-1}C$, is preserved) but
get destroyed once one of the symmetries,  $T$ or $C$, is broken (implying
that also chiral symmetry gets broken Chiu2016 ). Thus, the critical edge
states are seen to be well protected in the CII symmetry class.

It is interesting to inquire what happens if allowing disorder which breaks
both $T$ and $C$ symmetries but leaves chiral symmetry unbroken. This is
tantamount to put the Hamiltonian $H_{\\{1,2\\}}$ in the AIII symmetry class
where any perturbation which respects only chiral symmetry is allowed. One may
maybe object and say that this assignment of symmetry class is impossible:
“Particle-hole symmetry is a ’built-in’ feature of any second-quantized
Hamiltonian expressed in a Nambu spinor basis, implying that the corresponding
first-quantized BdG Hamiltonian (like $H_{\\{1,2\\}}$), as well as any
perturbation thereof, is also particle-hole symmetric.” However, whereas the
use of the Nambu basis does preclude particle-hole symmetry-breaking
perturbations in the second-quantized theory, there is no such constraint on
perturbations of the BdG Hamiltonian. In other words, once the BdG Hamiltonian
matrix has been extracted from the underlying second-quantized theory using a
Nambu basis, this matrix defines a single-particle theory which can be put
into the AIII symmetry class, allowing for perturbations that break particle-
hole symmetry. True, such perturbations do not represent physical symmetry-
breaking perturbations of the original unperturbed theory of a mean-field
superconductor since they cannot be traced back to a second quantized
formulation using the Nambu basis. However, the maneuver allows us to formally
address the question whether critical edge states of a first-quantized
Hamiltonian (like $H_{\\{1,2\\}}$) are robust against perturbations which
respect only chiral symmetry. This is an important issue, independent of
whether such perturbations can be realized in an underlying second-quantized
theory or not.

With this as a backdrop, we now consider perturbations which break both $T$
and $C$ but, different from the cases of Fig. 1, preserve $S$. The results are
displayed in Fig. 2 where we have applied random uncorrelated on-site disorder
proportional to $\tau_{x}\otimes\sigma_{y}$ and $\tau_{z}\otimes\sigma_{y}$
($T$ and $C$ breaking, $S$ preserving), showing that the critical edge states
of $H_{\\{1,2\\}}$ do survive perturbations which respect only chiral
symmetry.

Given these results we conjecture that it is precisely the chiral symmetry $S$
which protects the topological edge states in disordered 1D critical phases.
Numerical examinations of other disorder types and critical chains support
this conjecture unpublished . One should here note that while chiral symmetry
by definition is indeed the only possible protecting symmetry for critical
phases of AIII, it is a priori not evident that it actually fulfills this
role. However, our numerical results provide strong evidence that it does.

##  VI Summary

Building on the work by Verresen et al. VJP  on critical two-band BDI models
in 1D, we have carried out a study of critical multiband models in any of the
1D chiral symmetry classes AIII, BDI, and CII. We use an approach where the
enlarged unit cell responsible for the multiband structure of a model is
further extended until one is left with intercell hopping of fermions only
between nearest-neighbor auxiliary cells. This allows for a transparent and
rigorous analysis of the problem, enabling us to prove the existence of
critical edge states for any 1D multiband model belonging to one of the chiral
symmetry classes. As in the original work in Ref. VJP, , the number of such
edge states is coded by a topological invariant generalizing the winding
number of gapped 1D models, now providing a bulk-boundary correspondence for
all chiral critical phases in 1D.

A numerical case study of a four-band model in the CII symmetry class $-$
perturbing its Hamiltonian by uncorrelated disorder distributions of different
symmetry contents $-$ shows that the robustness of its critical edge states is
protected solely by chiral invariance, with time-reversal and particle-hole
symmetries playing no role. We conjecture that this picture is general, with
chiral symmetry being the sole protecting symmetry of 1D critical topological
edge states not only in the AIII symmetry class (where any perturbation which
respects only chiral symmetry is allowed), but also for any 1D model belonging
to the CII or BDI symmetry class. Put differently, we expect that the subsets
of the CII and BDI symmetry classes composed of models that support
topological critical phases are entirely contained within the AIII symmetry
class, demoting $T$ and $C$ to accidental symmetries when it comes to
protection of these phases.

Our work may open a path towards a more comprehensive study of symmetry
protection of multiband topological phases at criticality, including those of
non-translational invariant models in higher dimensions and artificially
generated phases from Floquet topological engineering Harper .  The
classification of periodically driven (Floquet) systems at criticality is here
a particularly promising direction for exploration: Our multiband analysis
should be quite useful for treating critical time-periodic systems within
Floquet theory – a theory that is intrinsically multiband due to the
repetition of frequency zones Shirley ; Sambe . In fact, the Floquet systems
represented within frequency domain and truncated at sufficiently large
frequency index are mathematically equivalent to time-independent multiband
systems Shirley ; Sambe ; Rudner . Any Floquet chiral system is then
anticipated to satisfy the time-independent chiral relation after the
truncation, and combined with our results this shows the existence of the
topological edge states. Importantly, these arguments are expected to hold
also at the closed induced gap (known as the ’anomalous gap’) corresponding to
the existence of anomalous edge states AsbothFL ; Rudner , having no analog in
static systems, also at criticality.

Another important topic to explore is the effect of interactions in multiband
critical models. There are already some results VJP ; Verresen2019 ;
Verresen2020 ; VJPSM  on the robustness of critical topological edge modes
against interactions, using density matrix renormalization group (DMRG) and
effective field theory methods, but so far only for two bands. The extension
to more bands is technically challenging, but we expect that our present work
will be of use also here.

In conclusion, further theoretical work and the growing backdrop of relevant
multiband experimental systems Lutchyn ; Stanescu ; Setiawan ; Samokhin ;
Mizushima ; Mendl ; Maslowski  hold up the prospect of some very interesting
developments.

## Acknowledgments

We thank Eddy Ardonne, Abhishodh Prakash, Christian Spånslätt, and Ruben
Verresen for discussions and communications. This work was supported by the
Swedish Research Council under Grant No. 621-2014-5972 and by the Knut and
Alice Wallenberg foundation under grant No. 2017.0157.

##
APPENDIX:
CII spinful Majorana chains

To obtain all possible CII spinful Majorana chains we first construct all
real-space first-quantized Hamiltonian matrices $H$ allowed by the proper
time-reversal and particle-hole symmetries and then use a Nambu representation
to go to second quantization with complex fermions and, from there, to a
representation in spinful Majoranas.

Recall that the symmetry conditions are $THT^{-1}=H$ and $CHC^{-1}=-H$, with
$T^{2}=C^{2}=-1$. In a real-space representation (with $H$ defined on a chain
with $N$ unit cells) we have that

$T=\sum^{N}_{j=1}\mathbb{1}\\!\otimes\\!\sigma_{y}\,\mathcal{K}\otimes|j\rangle\langle
j|,\ \ \
C=\sum^{N}_{j=1}\tau_{y}\\!\otimes\\!\mathbb{1}\,\mathcal{K}\otimes|j\rangle\langle
j|,$ (A1)

with $j$ running over all unit cells, and where $\tau_{\alpha}$ and
$\sigma_{\alpha}$ are Pauli matrices when $\alpha=x,y,z$ and equal to the
$2\times 2$ unit matrix $\mathbb{1}$ when $\alpha=0$. The operator ${\cal K}$
effects complex conjugation. For the four-band case, with the real-space
Hamiltonian given by a Hermitian $4N\times 4N$ matrix, we make the observation
that any such matrix can be written as a linear combination over real numbers
of matrices

$H_{j,j+n}=(i)^{m}\tau_{\alpha}\otimes\sigma_{\beta}\otimes|j\rangle\langle
j\\!+\\!n|+\text{H.c.},\ \ m=0,1,$ (A2)

with $j$ constrained by $|j+n|\leq N$. Examining all 32 such matrices (for any
fixed $j$ and $n$), we find that 8 of them respect the $T$ and $C$ symmetries.
Writing out only the parts which act in the spin- and particle-hole spaces
(suppressing the common $|j\rangle\langle j\\!+\\!n|$ factors and the
Hermitian conjugate to save space), these are:

(a) $\,\,\tau_{x}\otimes\mathbb{1}$; (b) $\tau_{z}\otimes\mathbb{1}$; (c)
$\,\tau_{y}\otimes\sigma_{x};\,\,\,\,\,\mbox{(d)}\ \
\tau_{y}\otimes\sigma_{z}$;
(e) $i\mathbb{1}\otimes\sigma_{x}$; (f) $i\mathbb{1}\otimes\sigma_{z}$; (g)
$i\tau_{x}\otimes\sigma_{y}$; (h) $i\tau_{z}\otimes\sigma_{y}$.

To construct the symmetry-respecting second-quantized Hamiltonians ${\cal
H}_{j,j+n}$ corresponding to (a) - (h), we introduce four-component real-space
Nambu spinors for the $N$ unit cells:
$\Psi^{\dagger}_{j}=(c^{\dagger}_{j,\uparrow}\ c^{\dagger}_{j,\downarrow}\
c_{j,\downarrow}\,-c_{j,\uparrow})$ and $\Psi_{j}\\!=\\!(c_{j,\uparrow}\
c_{j,\downarrow}\ c^{\dagger}_{j,\downarrow}\,-c^{\dagger}_{j,\uparrow})^{T}$,
$j=1,2,...,N$, yielding the $4N$-component Nambu spinors for the full lattice,
$\Psi^{\dagger}=(\Psi^{\dagger}_{1}...\Psi^{\dagger}_{j}....\Psi^{\dagger}_{j+n}...\Psi^{\dagger}_{N})$
and $\Psi=(\Psi_{1}...\Psi_{j}....\Psi_{j+n}...\Psi_{N})^{T}$ respectively.
Inserting the expressions for $H_{j,j+n}$ implied by (a) - (h), cf. Eq.
(A2), into the prescription ${\cal
H}_{j,j+n}=(1/2)\Psi^{\dagger}H_{j,j+n}\Psi$, we find that only four of them,
corresponding to (a), (b), (g), and (h), give a nonzero second-quantized
Hamiltonian. In the other four cases, the Hermiticity of $H_{j,j+n}$ when
combined with the fermion algebra cancels out the resulting second-quantized
expressions, signaling an incompatibility with fermion statistics. Listing the
four surviving contributions, one finds, from Eq. (A2) and the table (a)-(h)
above,

$\displaystyle{\cal H}_{j,j+n}^{(a)}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{1}{2}\Psi^{\dagger}H^{(a)}_{j,j+n}\Psi$
$\displaystyle\\!=\\!$
$\displaystyle\frac{1}{2}\Psi^{\dagger}(\tau_{x}\otimes\mathbb{1})\otimes(|j\rangle\langle
j\\!+\\!n|+|j+n\rangle\langle j|)\Psi$ $\displaystyle\\!=\\!$
$\displaystyle\frac{1}{2}\Psi^{\dagger}_{j}(\tau_{x}\otimes\mathbb{1})\Psi_{j+n}+\frac{1}{2}\Psi^{\dagger}_{j+n}(\tau_{x}\otimes\mathbb{1})\Psi_{j}$
$\displaystyle\\!=\\!$ $\displaystyle
c_{j,\downarrow}c_{j+n,\uparrow}-c_{j,\uparrow}c_{j+n,\downarrow}-c^{\dagger}_{j,\downarrow}c^{\dagger}_{j+n,\uparrow}+c^{\dagger}_{j,\uparrow}c^{\dagger}_{j+n,\downarrow},$

and similarly for the other three cases,

$\displaystyle{\cal H}_{j,j+n}^{(b)}$ $\displaystyle\\!=\\!$
$\displaystyle\sum_{\sigma=\uparrow,\downarrow}(c_{j,\sigma}^{\dagger}c_{j+n,\sigma}+c^{\dagger}_{j+n,\sigma}c_{j,\sigma}),$
(A4) $\displaystyle{\cal H}_{j,j+n}^{(g)}$ $\displaystyle\\!=\\!$
$\displaystyle\sum_{\sigma=\uparrow,\downarrow}(c_{j,\sigma}c_{j+n,\sigma}-c^{\dagger}_{j,\sigma}c^{\dagger}_{j+n,\sigma}),$
(A5) $\displaystyle{\cal H}_{j,j+n}^{(h)}$ $\displaystyle\\!=\\!$
$\displaystyle
c^{\dagger}_{j,\uparrow}c_{j+n,\downarrow}\\!-\\!c^{\dagger}_{j,\downarrow}c_{j+n,\uparrow}\\!-\\!c^{\dagger}_{j+n,\uparrow}c_{j,\downarrow}\\!+\\!c^{\dagger}_{j+n,\downarrow}c_{j,\uparrow}.$

In Eqs. (A4) and ( APPENDIX:  CII spinful Majorana chains) an immaterial
constant (from anticommuting the fermion operators) has been dropped.

Next, we rewrite Eqs. ( APPENDIX:  CII spinful Majorana chains)-( APPENDIX:
CII spinful Majorana chains) in terms of spinful Majorana operators, defined
by

$\gamma_{j,\sigma}=\frac{1}{2}(c^{\dagger}_{j,\sigma}+c_{j,\sigma}),\ \ \
\tilde{\gamma}_{j,\sigma}=\frac{i}{2}(c^{\dagger}_{j,\sigma}-c_{j,\sigma}).$
(A7)

By this we obtain

$\displaystyle{\cal H}_{j,j+n}^{(a)}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{i}{2}\\!(\\!\tilde{\gamma}_{j,\uparrow}\gamma_{j+n,\downarrow}\\!\\!-\\!\tilde{\gamma}_{j,\downarrow}\gamma_{j+n,\uparrow}\\!-\\!\tilde{\gamma}_{j+n,\downarrow}\gamma_{j,\uparrow}\\!+\\!\tilde{\gamma}_{j+n,\uparrow}\gamma_{j,\downarrow}\\!)$
$\displaystyle{\cal H}_{j,j+n}^{(b)}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{i}{2}\sum_{\sigma=\uparrow,\downarrow}(\tilde{\gamma}_{j,\sigma}\gamma_{j+n,\sigma}+\tilde{\gamma}_{j+n,\sigma}\gamma_{j,\sigma}),$
(A9) $\displaystyle{\cal H}_{j,j+n}^{(g)}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{i}{2}\sum_{\sigma=\uparrow,\downarrow}(\tilde{\gamma}_{j,\sigma}\gamma_{j+n,\sigma}-\tilde{\gamma}_{j+n,\sigma}\gamma_{j,\sigma}),$
(A10) $\displaystyle{\cal H}_{j,j+n}^{(h)}$ $\displaystyle\\!=\\!$
$\displaystyle\frac{i}{2}(\\!\tilde{\gamma}_{j,\uparrow}\gamma_{j+n,\downarrow}\\!-\\!\tilde{\gamma}_{j,\downarrow}\gamma_{j+n,\uparrow}\\!+\\!\tilde{\gamma}_{j+n,\downarrow}\gamma_{j,\uparrow}\\!-\tilde{\gamma}_{j+n,\uparrow}\gamma_{j,\downarrow}).$

By summing Eqs. (A9) and (A10) and then summing over all unit cells, we obtain
a basis $\\{{\cal H}_{n}^{\text{CII}}\\}$ for all four-band spin-preserving
CII Majorana chains:

${\cal
H}^{\text{CII}}_{n}=i\sum_{j,\sigma}\tilde{\gamma}_{j,\sigma}\gamma_{j+n,\sigma}.$
(A12)

Analogously, by summing Eqs. ( APPENDIX:  CII spinful Majorana chains) and (
APPENDIX:  CII spinful Majorana chains) and then again summing over all unit
cells, one obtains a basis $\\{\bar{\cal H}_{n}^{\text{CII}}\\}$ for all four-
band spin-flipping CII Majorana chains:

$\bar{\cal
H}^{\text{CII}}_{n}=i\sum_{j}(\tilde{\gamma}_{j,\uparrow}\gamma_{j+n,\downarrow}-\tilde{\gamma}_{j,\downarrow}\gamma_{j+n,\uparrow}).\\\
$ (A13)

It follows that any spinful Majorana chain can be constructed from Eqs. (A12)
and (A13),

${\cal H}^{\text{CII}}=\sum_{n=-\Lambda}^{\Lambda}\\!(t_{n}{\cal
H}^{\text{CII}}_{n}+\bar{t}_{n}\bar{\cal H}^{\text{CII}}_{n}),$ (A14)

with real constants $t_{n}$ and $\bar{t}_{n}$, and where $[-\Lambda,\Lambda]$
is the range of couplings between the Majoranas.

For the purpose of verifying that ${\cal H}^{\text{CII}}$ has the desired CII
symmetries, we introduce the time-reversal and particle-hole symmetry
operators ${\cal T}$ and ${\cal C}$, respectively, acting on the Nambu spinors
$\Psi_{j}=\\!(c_{j,\uparrow}\ c_{j,\downarrow}\ c^{\dagger}_{j,\downarrow}\
-c^{\dagger}_{j,\uparrow})^{T}$ introduced above and constructed from the
unitary parts of the corresponding $T$ and $C$ operators Chiu2016  in Eq.
(A1),

${\cal T}\psi_{k}{\cal T}^{-1}=(U_{T})_{k}^{\ell}\psi_{\ell},\ \ {\cal
T}i{\cal T}^{-1}=-i,$ (A15)

and

${\cal C}\psi_{k}{\cal
C}^{-1}=(U^{\ast}_{C})_{k}^{\ell}\psi_{\ell}^{\dagger}.$ (A16)

Here $U_{T}=\mathbb{1}\otimes\sigma_{y}$ and $U_{C}=\tau_{y}\otimes\mathbb{1}$
(after having suppressed the extraneous cell index that labels the Nambu
spinors), with
$\psi_{1}=c_{\uparrow},\psi_{2}=c_{\downarrow},\psi_{3}=c^{\dagger}_{\downarrow},$
and $\psi_{4}=-c^{\dagger}_{\uparrow}$ their common elements in the composite
spin and particle-hole space. It follows from Eqs. (A7), (A15), and (A16) that

$\gamma_{\uparrow}\rightarrow-i\gamma_{\downarrow},\ \
\gamma_{\downarrow}\rightarrow i\gamma_{\uparrow},\ \
\tilde{\gamma}_{\uparrow}\rightarrow-i\tilde{\gamma}_{\downarrow},\ \
\tilde{\gamma}_{\downarrow}\rightarrow i\tilde{\gamma}_{\uparrow}$ (A17)

under a ${\cal T}$-transformation, while

$\gamma_{\uparrow}\rightarrow i\gamma_{\downarrow},\ \
\gamma_{\downarrow}\rightarrow-i\gamma_{\uparrow},\ \
\tilde{\gamma}_{\uparrow}\rightarrow-i\tilde{\gamma}_{\downarrow},\ \
\tilde{\gamma}_{\downarrow}\rightarrow i\tilde{\gamma}_{\uparrow}$ (A18)

under a ${\cal C}$-transformation. Reinserting the cell index, one confirms
from Eqs. (A12), (A13), (A17) and (A18) that

${\cal T}{\cal H}^{\text{CII}}{\cal T}^{-1}={\cal H}^{\text{CII}},\ \ \ {\cal
C}{\cal H}^{\text{CII}}{\cal C}^{-1}={\cal H}^{\text{CII}},$ (A19)

as required.

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