# Time-reversal invariant finite-size topology

R. Flores-Calderon  Max Planck Institute for the Physics of Complex Systems,
Nöthnitzer Strasse 38, 01187 Dresden, Germany  Max Planck Institute for
Chemical Physics of Solids, Nöthnitzer Strasse 40, 01187 Dresden, Germany
Roderich Moessner  Max Planck Institute for the Physics of Complex Systems,
Nöthnitzer Strasse 38, 01187 Dresden, Germany  Ashley M. Cook  Max Planck
Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187
Dresden, Germany  Max Planck Institute for Chemical Physics of Solids,
Nöthnitzer Strasse 40, 01187 Dresden, Germany

###### Abstract

We report finite-size topology in the quintessential time-reversal (TR)
invariant systems, the quantum spin Hall insulator (QSHI) and the three-
dimensional, strong topological insulator (STI): previously-identified helical
or Dirac cone boundary states of these phases hybridize in wire or slab
geometries with one open boundary condition for finite system size, and
additional, topologically-protected, lower-dimensional boundary modes appear
for open boundary conditions in two or more directions. For the quasi-one-
dimensional (q(2-1)D) QSHI, we find topologically-protected, quasi-zero-
dimensional (q(2-2)D) boundary states within the hybridization gap of the
helical edge states, determined from q(2-1)D bulk topology characterized by
topologically non-trivial Wilson loop spectra. We show this finite-size
topology furthermore occurs in 1T’-WTe2 in ribbon geometries with sawtooth
edges, based on analysis of a tight-binding model derived from density-
functional theory calculations, motivating experimental investigation of our
results. In addition, we find quasi-two-dimensional (q(3-1)D) finite-size
topological phases occur for the STI, yielding helical boundary modes
distinguished from those of the QSHI by a non-trivial magneto-electric
polarizability linked to the original 3D bulk STI. Finite-size topological
phases therefore exhibit signatures associated with the non-trivial
topological invariant of a higher-dimensional bulk, clearly distinguishing
them from previously-known topological phases. Finally, we find the q(3-2)D
STI also exhibits finite-size topological phases, finding the first signs of
topologically-protected boundary modes of codimension greater than $1$ due to
finite-size topology. Finite-size topology of four or higher-dimensional
systems is therefore possible in experimental settings without recourse to
thermodynamically large synthetic dimensions.

##  I Introduction

Figure 1: Schematic of the finite-size TRI systems studied. From left to
right, a) QSHI wire, b) slab of 3D TI and c) 3D TI wire. Blue and red cones
are schematic of the gap openings of the 3D TI due to the hybridization of the
the Dirac cones. Similarly blue and red helical edge states get hybridized
(yellow/purple) in the QSHI wire and the finite-size quasi-1D edge states
(blue and red) get hybridized (yellow/purple) for the 3D TI wire. Topological
edge states (pink) are present as quasi-0D modes or quasi-1D modes polarized
in spin, for wire or slab configurations respectively. Figure 2: Schematic of
the finite-size process for topological insulators. From left to right a $D$
dimensional phase gets shrunk in one direction $x_{D}$ to give rise to a quasi
$D-1$ dimensional phase. This phase can be further shrunk in a remaining
$x_{D-1}$ direction so that $x_{1},x_{2},\dots x_{D-2}$ are still periodic
directions and now a quasi $D-2$ dimensional phase is realized.

The discovery of the first topological insulator (TI), the quantum spin Hall
insulator (QSHI) in HgTe quantum wells [1, 2] heralded a paradigm shift in
condensed matter physics towards broad study of topological phases of matter.
Understanding and characterization of topology is now central to the field,
with major applications ranging from fault-tolerant quantum computing [3, 4]
to unconventional superconductivity [5]. Consequently, searching for novel,
experimentally-accessible topological systems is a major theme of the last few
decades [6, 7, 8, 2, 9, 10, 11, 12, 13]. These efforts usually target
experimental confirmation of a hallmark of topological phases known as bulk-
boundary correspondence: a non-trivial topological invariant of the system
bulk is associated with topologically-robust, gapless boundary states. While
it has long been understood that a $D$-dimensional bulk topology yields
$\left(D-1\right)$-dimensional gapless boundary states for most topological
phases [14], the recent discovery of additional bulk-boundary correspondence
even in the canonical phases, known as finite-size topology [15], shows this
foundational aspect of topological physics is richer than previously-thought.
If a system is characterized by a topological invariant computed in the
$D$-dimensional infinite bulk, but is finite in size and thin in one direction
as illustrated in Fig. 1 (for the QSHI $D=2$ while for the 3D TI $D=3$ ), such
that topologically-protected boundary states interfere with one another, this
quasi-$(D-1)$\- or q$(D-1)$-dimensional bulk is characterized by an additional
topological invariant. When this additional invariant takes non-trivial
values, open boundary conditions in a second direction yield an additional set
of quasi-$(D-2)$-dimensional, topologically-protected boundary states
localized on this boundary of the quasi-$(D-1)$-dimensional system. As these
quasi-$(D-2)$-dimensional states are localized on the boundary in
correspondence with a non-trivial value for a topological invariant of the
quasi-$(D-1)$-dimensional bulk, and robust against local perturbations
respecting the symmetries protecting the topological phase in the
$D$-dimensional infinite bulk, they constitute previously-unidentified
topological phases of matter.

Following the previous thinning process we end up with a q$(D-1)$-dimensional
bulk with topological edge states in one less dimension, the situation is then
just like at the start of the program, but with $D$ replaced by $D-1$. Thus
one may think of applying the thinning process once again, now thinning the
$x_{D-1}$ dimension and hybridizing the previous q$(D-2)$ edge states. We then
arrive at a q$(D-2)$ dimensional bulk with q$(D-2-1)$ dimensional edge states
which again can be subjected to the same procedure. The general process is
illustrated in Fig. 2, while Fig. 1 c) shows the specific case of the 3D TI
q$(3-2)$ bulk. We note that this procedure could in principle be applied until
there are no more number of dimensions to thin down.

Although theoretical discovery of the Chern insulator [16] preceded
theoretical prediction of the TR-invariant QSHI derived from it [17, 18],
experimental confirmation of the QSHI [2] occurred within one year of the
prediction, while more than two decades passed for the Chern insulator [19].
This reflects a broader trend in the field, of TR-invariant topological
insulators being confirmed experimentally more quickly and easily than TR-
symmetry-broken topological insulators reliant on engineering particular
magnetic orders [20, 21, 2]. Following this idea in order to more rapidly
observe finite-size topology in experiment, we study the time-reversal
invariant finite-size topology of the QSHI and the strong TI (STI), by
considering these systems in geometries as shown in Fig. 1. We also note that,
due to the vast experimental studies in TI ultra-thin films [22, 23, 24], Van
der Waals heterostructures [25, 26, 27, 28], and transition-metal
dichalcogenides in particular given their large spin-orbit coupling [29, 30],
there may already be signs of finite-size topology in previous experiments.
Past work, for instance, indicates few-layer 1T’-MoTe2 is semi-metallic [31],
while the monolayer is predicted to be a quantum spin Hall insulator [32],
suggesting the few-layer topology derives from the Weyl semimetal phase of the
three-dimensional bulk, while the monolayer topological phase has a distinct
origin due to a strictly two-dimensional bulk.

Since finite-size topological phases occur for the Kitaev chain and Chern
insulator [15], non-trivial finite-size topology is expected for TR-invariant
systems of the QSHI and STI given concrete relationships between Hamiltonians
for these topological phases: the Kitaev chain Hamiltonian may be used to
construct the Chern insulator Hamiltonian, if many chains are coupled forming
a 2D system [33], and a Chern insulator Hamiltonian and its time-reversed
partner are the basis of Hamiltonians for the QSHI[17, 18]. We find that FST
extends to these TRI topological phases. As Hamiltonians for TR-invariant
topological phases are used to construct Hamiltonians for other topological
phases, these results also reveal that a larger set of topological phases
harbor FST: a Weyl semimetal phase[34] Hamiltonian may be constructed from
magnetically-doped STI and trivial insulator thin films stacked alternatingly,
while a stack of QSHIs corresponds directly to the weak 3D TI [35, 36].
Topological crystalline phases may furthermore be constructed, for instance,
as Chern insulators within mirror subsectors or with the Chern insulator bulk
confined to a mirror-invariant plane of a three-dimensional Brillouin zone
[37]. More generally, topological crystalline phases are characterized by
considering symmetry-protection by crystalline point group symmetries in
addition to the internal symmetries of the ten-fold way. On a technical level
this is accomplished by expressing the Hamiltonian in a block diagonal form
using the additional symmetry, in each sub-sector internal symmetries are
still present and thus can be analyzed by classification schemes obtained from
the ten-fold way [38, 39].

In this manuscript, we first, section II, characterize finite-size topology in
a QSHI wire. We start by considering a thin QSHI system with one open thin
dimension and one infinite periodic dimension. The energy and Wilson loop
spectra of this q(2-1)D system reveal that the non-trivial zones in phase
space are a subset of the original 2D bulk topological regions. Furthermore
opening boundary conditions again in the remaining periodic direction shows
the presence of edge states localized on the q(2-2)D boundaries. We end this
section by studying the response to on-site disorder and perturbations, where
the robustness of the edge states indicates a link to the original 2D bulk
gap. Afterwards in section III, we consider a more realistic and
experimentally accessible system $1T^{\prime}$ $\text{WTe}_{2}$ in the QSHI
phase. We find similarly that this material realizes a finite-size topological
phase with topologically-robust q(2-2)D edge states for a sawtooth ribbon
geometry, the presence of this edge states is again verified to be predicted
by a non-trivial Wilson loop spectrum. Extending our analysis to the 3D case
we consider in section IV the STI in a q(3-1)D slab geometry. In this case,
interference between the STI Dirac cone surface states yields q(3-2)D edge
states. We show the Wilson loop spectrum of the q(3-1)D bulk displays
topologically non-trivial signatures in correspondence with these boundary
states, indicating Wilson loop spectra are a robust bulk diagnostic of finite-
size topology. Additionally we compute the magneto-electric polarizability,
which should be trivially zero if the system is just a 2D QSHI, instead we
encounter the response expected for the infinite 3D TI bulk. This central
result allows us to contemplate the idea of detecting topological signatures
of higher dimensional phases, say the 4D TI, in quasi lower dimensional
systems. Finally, we study the case of a STI in a q(3-2)D wire geometry, where
we once again use the Wilson loop indicator to find a novel bulk-boundary
correspondence restricted to a subset of the original 3D topological phase
diagram. In this final case the number of edge states is seen to follow the
number of $\pm\pi$ phases such that only even numbers of distinct edge states
appear. In section VI we summarize our results and present some concluding
remarks.

##  II QSHI wire

As a starting point of our analysis we consider a QSHI first considering the
canonical Bernevig-Hughes-Zhang Hamiltonian for HgTe quantum wells [1] where
we also add a Rashba-type spin orbit coupling. Thus the Hamiltonian in
momentum space has the form [40]:

$\displaystyle h(k_{x},k_{y})=$ $\displaystyle(u+2t(\cos k_{x}+\cos
k_{y}))\sigma_{z}+\sin{k_{y}}\ \sigma_{y}$ (1) $\displaystyle+$
$\displaystyle\sin{k_{x}}\ s_{z}\sigma_{x}+c\ s_{x}\sigma_{y},$

where $s_{i},\sigma_{i}$ are Pauli matrices in spin and orbital space
respectively. For simplicity we omit the identity in spin space and denote the
tensor product by placing two matrices next to each other. The real number $u$
corresponds to a staggered potential, $t$ to a hopping parameter and $c$ is
the spin orbit coupling that breaks $s_{z}$ spin symmetry. The phase diagram
for this Hamiltonian includes both a region in which the QSHI phase is
realized and a region in which the Dirac semimetal (DSM) phase is realized, as
discussed in reference [40] and plotted in Fig. 3 a) and b) . In the following
analysis, we first consider the QSHI regime, and then that of the DSM.

Next we consider what happens if we open boundary conditions (OBC) in the $x$
direction for a small number of lattice sites $N$. Since the helical edge
modes of the QSHI are not completely localized at the boundary, but instead
decay exponentially into the bulk [41], these boundary states interfere in
systems of finite-width. The lattice second quantized Hamiltonian with open
boundary conditions in the $\hat{x}$-direction and periodic boundary
conditions in the $\hat{y}$ direction is:

$\displaystyle\hat{H}$
$\displaystyle=\sum_{k,n}\Psi_{k,n}^{\dagger}\left((u+2t\cos{k})\sigma_{z}+\sin{k}\
\sigma_{y}+c\ s_{x}\sigma_{y}\right)\Psi_{k,n}$
$\displaystyle+\Psi_{k_{y},n+1}^{\dagger}\left(t\
\sigma_{z}+\frac{i}{2}s_{z}\sigma_{x}\right)\Psi_{k,n}+\text{h.c.}\ ,$ (2)

Figure 3:  a) Direct gap heat plot of the 2D bulk hamiltonian eq. (1) as a
function of potential $u$ and spin-orbit coupling constant $c$, b) Topological
phase diagram of the 2D bulk , showing the QSHI phase (yellow) and DSM gapless
phase (blue) of the 2D bulk. c) Quasi-1D dispersion for PBC in $y$ and OBC
(PBC) in $x$ with $N=6$ sites. The parameters for the gap closing with OBC in
$x$ are $u=0.76,c=0.8,t=1/2$ d) Spectrum for PBC in $y$ as a function of the
staggered potential $u$ and $c=0.8,t=1/2$.

where $k\equiv k_{y}$ and $n$ runs over the $N$ sites of the open $x$
direction. Here, $\Psi_{k,n}$ are four component spinor fermion operators
acting on the spin and orbit degrees of freedom. We first examine the spectrum
of Eq. 2 for small $N$ on the order of a few lattice constants. We first
consider the spectrum for a particular point in phase space as shown in Fig. 3
c) for a small system with $N=6$ with non-trivial 2D bulk invariant, as shown
in Fig. 3 b). Comparing the dispersion for the system with periodic boundary
conditions in each direction (black lines) to that of the system with open
boundary conditions only in the $\hat{x}$-direction (red lines), we see the
periodic system is gapped, while the system with open boundary conditions is
instead gapless. While gapless boundary states are expected due to the non-
trivial bulk topology, the gaplessness in this case is not topologically-
robust: the gapless boundary modes interfere in finite-size systems to open a
hybridization gap in general. Under certain conditions, however, the boundary
modes interfere destructively, corresponding to a fine-tuned gapless state
when hybridization matrix elements pass through zero. Examining the spectrum
for the q(2-1)D bulk as a function of $u$ as shown in Fig. 3 d) , we see this
more general pattern of finite interference gaps, with a discrete set of $u$
corresponding to gap-closings and destructive interference between the helical
boundary modes. We will show these gap-closings can correspond to topological
phase transitions, and some of these gapped regions host finite-size
topological phases.

### II.1 Periodic system

To characterize the finite-size topological phases of this time-reversal
invariant system, we now re-interpret the original model with OBC in the
$\hat{x}$-direction as a q(2-1)D bulk, and characterize topology of this
q(2-1)D bulk system similarly to characterization of a $d$-dimensional bulk.
We therefore first compute a phase diagram for the minimum direct gap over the
Brillouin zone of the q(2-1)D bulk as a function of $u,c$ for fixed hopping
$t=1/2$, shown in Fig.4 a),b) for $N=6$ and $N=7$ layers in the
$\hat{x}$-direction, respectively. A dome forms in the phase diagram,
consisting of a set of curved, stripe-like regions of finite minimum direct
gap separated by lines along which the q(2-1)D minimum direct bulk gap is
zero, with these lines intersecting to form a checkerboard-like pattern at
larger values of $c$. As the number of lattice sites in the
$\hat{x}$-direction increases, the number of gap-closing lines increases while
the regions of finite minimum direct gap decrease in size. This pattern is
consistent with a picture of gap-closings due to interference between the
helical boundary modes of the QSHI: the boundary modes in this q(2-1)D system
possess a standing wave character, and the gap-closing lines correspond to
hybridisation matrix elements passing through zero with tuning of system
parameters. With increasing system size, this interference pattern becomes
denser as the difference in wavelength between the oscillatory components of
the helical boundary modes generically decreases.

Figure 4: Quasi-(2-1)D minimum direct bulk gap for a) $N=6$, b) $N=7$ and
$t=1/2$. Plot of the number of $\pm\pi$ phases $N_{\pm\pi}$ (red is 2, black
0) in the Wilson loop eigenvalues for c) $N=6$, d) $N=7$

It is particularly interesting to compare these phase diagrams for the q(2-1)D
bulk with the counterpart phase diagram of the 2D bulk shown in Fig. 3 a),b) ,
which reveals that different kinds of topological phases of the 2D bulk (and
corresponding different gapless boundary states) yield different interference
patterns as a function of $u$ and $c$. Notably, the checkerboard region of the
phase diagram corresponds to the DSM phase region of the corresponding phase
diagram for the 2D bulk, revealing that the DSM phase is generally gapped out
in the q(2-1)D regime, and exhibits more complex interference pattern than
does the QSHI.

As the 2D minimum direct bulk gap remains finite over the region of the phase
diagram where we observe this interference pattern between helical boundary
modes of the QSHI, and the 2D minimum direct bulk gap remains closed due to
topologically-protected band-touchings of the DSM, topological invariants of
the 2D bulk do not change within these regions. However, as subsets of each of
these regions possess a finite minimum direct gap in the q(2-1)D spectrum, it
is possible to further characterize the topology of the q(2-1)D system if
suitable topological invariant(s) are identified. To further characterize
finite-size topology of this quasi-1D TRI system with Wilson loop spectra, we
compute the Wilson loop eigenvalues [42], which distinguish between
topologically-distinct phases of matter as they characterize holonomy in a
system due to parallel transport through non-contractible loops in the BZ [43,
44, 45].

The Wilson loop spectra for the q(2-1)D system are computed by integrating the
Berry connection over the remaining $k\equiv k_{y}$ momentum coordinate, using
the following expression: [42]

$\displaystyle\displaystyle\mathcal{W}=\mathcal{P}e^{\textstyle-\int_{-\pi}^{\pi}dkA(k)},$
(3)

where $A(k)$ is the non-Abelian Berry connection over the occupied bands and
$\mathcal{P}$ is the path ordering operator. Since we compute the Wilson
matrix for a tight binding system we discretize Eq. (3). The set of Wilson
loop eigenvalue phases is the Wannier charge center spectrum characterizing
polarization. In a topologically non-trivial phase, Wannier charge center(s)
are fixed to value(s) of $\pm\pi$, so we compute the number of these non-
trivial phases as $N_{\pm\pi}$. The phase diagrams characterizing $N_{\pm\pi}$
vs. $u$ and $c$ are shown in Fig. 4 c), d) for systems with $N=6$ or $N=7$
layers in the $\hat{x}$-direction, respectively.

These $N_{\pm\pi}$ vs. $c$ and $u$ phase diagrams shown in Fig. 4 c) and d)
reveal alternating regions of $N_{\pm\pi}=0$ and $N_{\pm\pi}=2$, indicating
the system undergoes a variety of topological phase transitions. We even
observe stripe-like regions at smaller $c$, which intersect to form
checkerboard patterns at larger $c$. These lines across which $N_{\pm\pi}$
changes in value are in direct correspondence with lines shown in Fig. 4 a)
and b), respectively, along which the q(2-1)D minimum direct gap goes to zero.
Taken together, these phase diagrams in Fig. 4 reveal a topological phase
transition occurs every time the q(2-1)D minimum direct bulk gap goes to zero.

The phase diagrams for $N=6$ layers differ dramatically from those for $N=7$
layers, reflecting the dependence of this topology on finite-size effects.
From the plots, one can see that, as the number of layers in the
$\hat{x}$-direction increases, the number of topologically-distinct regions
also increases in agreement with the number of lines along which the q(2-1)D
minimum direct bulk gap is zero. The topological phase diagram of the 2D bulk
Hamiltonian (1) studied in Ref. [40] is therefore being further divided into
topologically-distinct regions in the q(2-1)D regime in a strongly
$N$-dependent manner, revealing that topological phase transitions due to
finite-size topology [15], may occur without the minimum direct gap of the 2D
bulk going to zero.

### II.2 Bulk-boundary correspondence and disorder

Having characterized finite-size topology of the q(2-1)D bulk of Hamiltonian
Eq. (1) with open boundary conditions in the $\hat{x}$-direction and periodic
in $\hat{y}$, we now explore the additional bulk-boundary correspondence of
finite-size topological phases. We first study the spectral signatures of this
bulk-boundary correspondence that appear for non-trivial Wilson loop spectra
in accordance with the modern theory of polarization of Ref. [46].
$N_{\pm\pi}\neq 0$ for the q1D bulk corresponds to topologically-protected,
q0D bound states for open boundary conditions in the $\hat{y}$-direction in
addition to open boundary conditions in the $\hat{x}$-direction. With system
size in the $\hat{y}$-direction of $L_{y}$, such a bulk-boundary
correspondence characterized in the q1D bulk by $N_{\pm\pi}$ is clear for
$L_{y}\gg L_{x}$ as shown in Fig. 5 a) b). In this case, one finds in-gap
states close to zero energy within the q(2-1)D bulk gap of the energy
spectrum. The separation in energy between these states decreases
exponentially to zero as a function of $L_{y}$, realizing a four-fold
degenerate manifold of zero-energy states. Such states are not present for
periodic boundary conditions in the $\hat{y}$-direction, further indicating
they appear as a consequence of bulk-boundary correspondence for the finite-
size topological phase.

To further explore the extent to which these in-gap states are due to an
additional bulk-boundary correspondence of finite-size topological phases, we
compute the probability density for these in-gap states. We find these states
are localized at the boundaries of the q1D system as shown in Fig. 5 f).
Probability density peaks at the corners of the system as seen in Fig. 5 f)
and decays over fifteen to twenty unit cells to zero for $x$ approaching the
q(2-1)D bulk. As the system is time-reversal invariant, we also compute the
spin polarization for these q(2-2)D boundary modes. We find the q(2-2)D
boundary modes are spin-polarized in the $\pm\hat{z}$ direction, with a spin
up/down pair localized at each end of the q(2-1)D system.

Figure 5: a) Spectrum for OBC (red)in both directions, PBC in $y$ (black) as a
function of the staggered potential $u$ and $c=0.6,t=1/2,N=6$. b) Spectrum for
OBC (red) in both directions, PBC in $y$ (black) as a function of the
staggered potential $u$ and $c=0.8,t=1/2,N=6$ with a particle-hole symmetry
and sublattice symmetry breaking on-site potential $0.1\cos(2\pi
n/N)\sigma_{x}$. c) Disorder-averaged spectrum for $N=6$ sites and $u=1,t=1/2$
as a function of $c$ for 200 uniformly distributed random particle hole
symmetric potentials of strength $\kappa=0.5u$. d) Disorder-averaged spectrum
with the same parameters but for $200$ particle-hole symmetry-breaking
disorder potentials of strength $\kappa=0.2u$. e) 2D minimum direct bulk gap
as a function of $c$ for $u=1$. f)Density profile of q(2-2)D state for the
same number of sites, $c=0.8,u=1.0$ and $L_{y}=300$.

We also study the robustness of the in-gap q(2-2)D boundary states against
symmetry-breaking due to disorder. We introduce disorder in the q(2-1)D system
for OBC as a uniform random potential, with strength $\kappa$. The disorder
average spectrum for $200$ disorder realizations of the form
$\kappa_{n}s_{0}\sigma_{z}$ is shown in Fig. 5 c) as a function of spin-orbit
coupling $c$ for $\kappa_{n}\in(-0.2u,0.2u)$. The q(2-2)D states are present
over a wide range in $c$. Surprisingly, they survive even for disorder
strengths $\kappa_{n}$ greater than the q(2-1)D minimum direct bulk gap for
the QSHI phase. The edge states may move away from zero energy, if the
perturbation breaks particle-hole symmetry, such effect is shown in Fig. 5 b)
as a function of $u$, where an onsite perturbation $0.1\cos(2\pi
n/N)s_{0}\sigma_{x}$ is present. For disorder effects we considered a term of
the form $V_{i}s_{0}\sigma_{0}$, the spectrum is thus shown in Fig. 5 d) where
the edge modes deviate from zero energy, as in the case of an SSH chain with
next nearest neighbours studied in reference [47]. However, the q(2-2)D
boundary modes persist and remain strongly localized, with their pair
degeneracy preserved.

Interestingly, the topological invariant remains fixed at non-trivial values
even when a particle-hole breaking term is present in the Hamiltonian. This
reflects the dependence of this non-trivial topology on the presence of the
topologically-protected boundary states of the higher-dimensional phase, which
require only time-reversal symmetry to remain robust up to the minimum direct
2D bulk gap going to zero. We further find $N_{\pm\pi}$ still predicts the
existence of edge states in the nontrivial region. The validity of the
invariant and bulk-boundary correspondence in the absence of particle hole
symmetry will be further verified for the case of the
$1T^{\prime}$-$\text{WTe}_{2}$ wire. The phase is therefore protected by time-
reversal symmetry alone while particle-hole symmetry is required only to
anchor the in-gap states to zero energy. This analysis parallels the one on
ref. [47] where analogously the invariant and phase are protected by only
inversion symmetry while both inversion and particle-hole symmetry secure the
zero-energy value. For particle-hole-symmetric disorder Fig.5 c) the
perturbation strength is protected not by the q(2-1)D gap but rather the 2D
bulk gap of the system, which, for $c=0.8,u=1.0$, is $\Delta\approx 0.5$ as
seen in Fig. 5 e). If the perturbation breaks time reversal symmetry, however,
the Kramers degeneracy of the q(2-2)D bound states is broken as expected.

We conclude from this analysis that the q(2-1)D wire with spinful time-
reversal symmetry, realized for a system with a 2D bulk and open boundary
conditions in one direction, exhibits finite-size topological phases for
subsets of the topologically non-trivial regions of the 2D bulk topological
phase diagram. In these subsets, bounded by lines along which the minimum
direct q(2-1)D bulk goes to zero rather than the minimum direct 2D bulk gap,
in general, the wire harbors topologically-protected q(2-2)D boundary modes
for open boundary conditions in two directions appearing in Kramers pairs. In
the quasi-1D bulk, these subsets correspond to Wilson loop spectra with some
Wilson loop eigenvalue phases fixed to $\pm\pi$, protected by spinful time-
reversal symmetry. These signatures of non-trivial topology are therefore
associated with time-reversal invariant, q(2-1)D finite-size topological
phases due to interference between topologically-protected gapless boundary
modes resulting from 2D bulk topology, either of the quantum spin Hall
insulator or the 2D Dirac semimetal. In contrast, the ten-fold way
classification scheme of topological phases of matter determines a 1D system
in class AII [48] has trivial topological classification. These results are
therefore evidence of topologically non-trivial phases of matter outside of
the ten-fold way classification scheme.

##  III $1T^{\prime}$-$\text{WTe}_{2}$ wire

Although the previous model is based on the celebrated HgTe quantum wells,
which allowed for the discovery of the first QSHI, it may not be most suitable
for the experimental discovery of finite-size topology. The need for a small
sample size in one direction may be easier to achieve for monolayers such as
the $1T^{\prime}$-$\text{TWe}_{2}$ QSHI [49]. We thus consider the model
studied in reference [50] which combines density-functional theory
calculations, symmetry considerations and fitting to experimental data. Since
the finite-size topology comes from the hybridization of the edge states, we
consider only the lattice termination which results in a Dirac crossing near
the Fermi energy. That is we study the sawtooth $y$ ribbon with open boundary
conditions in the $x$ direction. The $1T^{\prime}$-$\text{WTe}_{2}$
Hamiltonian is presented in the Supplementary Material.

Under such circumstances we expect the Dirac crossing to gap out in general on
larger and larger energy scales as the system size in the $x$ direction
$N_{x}$ decreases. Such a gap opening does occur for the original material
parameters for even $N_{x}$ values ranging from $4$ to $20$. It is worth
noting that the gap still exists even for larger $N_{x}$ but its magnitude
becomes very small compared to other material energy scales. Such a gapped
system may then host topological q(2-2)D edge states if the Wilson loop
spectrum is nontrivial. Specifically, we find that for $N_{x}=8,10,12$ the gap
has a $\pm\pi$ phase in the Wilson loop spectrum. Even in the case where the
spectrum is trivial, for example at $N_{x}=6$, we may still tune the system so
that a gap closing occurs and a nontrivial Wilson loop phase appears. To do
so, we may apply an electric field perpendicular to the monolayer
corresponding to addition of a symmetry-allowed Rashba spin-orbit coupling
term to the Hamiltonian. Since the original model already includes such
couplings, we further include a change of the in-plane SOC parameters by a
quantity $\Delta\lambda_{\text{SOC}}$.

For the case of $N_{x}=6$, we obtain a phase diagram for the number of Wilson
loop eigenvalues with phase $\pm\pi$, $N_{\pm\pi}$, vs. spin-orbit coupling
anisotropy, $\Delta\lambda_{SOC}$, showing a change in $N_{\pm\pi}$ from $0$
to $1$ with increasing $\Delta\lambda_{SOC}$ as shown in Fig. 6 a). Based on
our previous results for a canonical toy model of the QSHI, we expect q(2-2)D
edge states to appear if we open boundary conditions in the
$\hat{y}$-direction as well. Such edge states due to an additional bulk-
boundary correspondence characterized by $N_{\pm\pi}$ of the q(2-1)D bulk do
exist, as shown in Fig. 6 b)corresponding to a line of four-fold degenerate,
in-gap states as a function of $\Delta\lambda_{SOC}$. The four-fold degeneracy
corresponds to a two-fold Kramers degeneracy due to spinful time-reversal
symmetry, and two-fold degeneracy corresponding to boundary modes localized at
the left and right edge as in the previous simple model of eqref. (1). For a
fixed Rashba spin orbit coupling change of
$\Delta\lambda_{\text{SOC}}=0.35eV$, we find that the edge states localize on
each end of the wire as shown in Fig. 6 c).

Figure 6:  a) Energy spectrum (eV) as a function of the change in Rashba spin
orbit coupling $\Delta\lambda_{\text{SOC}}$ for a $N_{x}=6$ saw-tooth
terminated $1T^{\prime}$-$\text{TWe}_{2}$ wire. b) Energy spectrum (eV) for
the system with $N_{x}=12$ as a function of $\Delta\lambda_{\text{SOC}}$
showing edge states at zero field. c) Number of Wilson loop spectrum $\pm\pi$
phases for $N_{x}=6$ as a function of $\Delta\lambda_{\text{SOC}}$ d) Wave
function probability density for one edge state at
$\Delta\lambda_{\text{SOC}}=0.35eV$ and $N_{x}=6,N_{y}=200$ for the same saw-
tooth terminated $1T^{\prime}$-$\text{TWe}_{2}$ wire.

##  IV 3D TI slab

We now study the finite size topology of the quintessential 3D TI in symmetry
class AII [48, 39]. The classification in this case is dictated by four
$\mathbb{Z}_{2}$ topological invariants $(\nu_{0};\nu_{1},\nu_{2},\nu_{3})$
[35, 36] , where the last three invariants are related to the translational
invariance of the system classifying the weak TI phase (WTI) while the
$\nu_{0}$ parameter classifies the strong TI phase (STI). In the following, we
study the STI phase with trivial lower-dimensional invariants corresponding to
$(1;000)$, which is realized in the model Bloch Hamiltonian [51]:

$\displaystyle
H(\bm{k})=-2\lambda\sum_{\mu}\sin{k_{\mu}}\sigma_{z}s_{\mu}+\sigma_{x}s_{0}(M-t\sum_{\mu}\cos{k_{\mu}}),$
(4)

where the Pauli matrices $\\{\sigma_{i}\\}$,$\\{s_{j}\\}$ act on orbital and
spin degrees of freedom respectively, $\lambda$ is a spin orbit coupling
parameter that breaks spin conservation, $M$ is an onsite staggered potential,
and $t$ is a nearest-neighbor hopping integral. In the following, we take all
energies to be in units of $t$ by setting $t=1$, and consider the regime in
which $1<M<3$ and $\lambda$ positive, for which the model realizes the desired
strong TI phase. First, we specialize to the case of a thin slab in the
$\hat{x}$-direction of $N$ layers. In that case, we expect in analogy to the
QSHI that the Dirac cones from the upper and lower surfaces interfere due to
the thin bulk and hybridize to open a gap even for open boundary conditions.

The hybridization gap, just as in the previous 2D case, is expected to
sometimes protect a non-trivial finite size topological phase. To study this,
we once again characterize topology using the Wilson loop spectrum, now as a
function of $k_{y}$ or $k_{z}$ with OBC in $x$ to characterize topology first
of a q(3-1)D bulk, in regions of the phase diagram where the 3D bulk
corresponds to the strong TI. As the isotropy of the Hamiltonian Eq. (4)
suggests, there is no difference between the Wilson loop eigenvalues of
$W(k_{y})$ and $W(k_{z})$, thus we consider only $W(k_{z})$, where each Wilson
loop matrix is now defined as:

$\displaystyle\displaystyle\mathcal{W}(k_{y})=\mathcal{P}e^{\textstyle-\int_{-\pi}^{\pi}dk_{z}A_{z}(k_{y},k_{z})},$
(5)
$\displaystyle\displaystyle\mathcal{W}(k_{z})=\mathcal{P}e^{\textstyle-\int_{-\pi}^{\pi}dk_{y}A_{y}(k_{y},k_{z})}.$
(6)

A typical Wannier charge center spectrum vs. the remaining momentum component,
$k_{z}$, is plotted in Fig. 7, which shows the spectral flow a) characteristic
of a TR invariant topological insulator for some values and a trivial spectrum
b) for others within the $(1;000)$ 3D bulk phase classification.

The non-trivial spectral flow only appears for certain parameter regimes:
these regions can be distinguished in a systematic way by counting the number
of fixed $\pm\pi$ phases in the Wilson loop spectrum, as in the previous case.
This regions of parameter space which have an energy gap are again the
”bubbles” observed in the QSHI case. We plot the phase diagram as a function
of the model parameters in Fig. 8 a),b) for $N=5,6$ layers, respectively. The
phase diagram changes dramatically for each value of $N$, the number of
layers, indicative of the finite-size topology [15]. The pattern of trivial
and nontrivial regions is entirely contained with the non-trivial region of
the 3D bulk topological phase diagram for Hamiltonian (4), indicating the 3D
minimum direct bulk gap remains finite during these topological phase
transitions of the q(3-1)D bulk. We remark that the sudden changes in color
near $\lambda=0$ seem to be just an artifact of the numerical precision when
approaching the STI gap-closing in the 3D bulk.

One of the consequences of the system being in a topologically non-trivial
regime, according to the Wilson loop spectra of the q(3-1)D bulk, is an
additional bulk-boundary correspondence: opening boundary conditions in a
second direction in these regions of phase space, we find topologically-
protected q(3-2)D states that now are localized at the edges of the slab, as
shown in Fig. 8 c). These q(3-2)D states appear within the q(3-1)D bulk gap
similarly to the case of q(2-2)D boundary states appearing in the q(2-1)D bulk
gap of the QSHI. A topological phase diagram for the q(3-1)D system with open-
boundary conditions in the $\hat{y}$-direction as a function of $M$ is also
shown in Fig. 8 d), demonstrating the direct correspondence of the
topologically-protected boundary modes with the topologically non-trivial
regions of the q(3-1)D bulk in Fig. 8 a). This indicates that $N_{\pm\pi}$,
the number of $\pm\pi$ phases in the Wilson loop eigenvalue spectrum,
characterizes finite-size topological phases resulting from interference of
the topologically-protected Dirac cones of the 3D TI, in addition to
characterizing finite-size topological phases due to interference between the
helical boundary modes of the QSHI.

Figure 7: a) Wilson loop nontrivial eigenvalue spectrum as a function of
$k_{z}$ for OBC in $x$ and PBC in $y,z$. The parameters are
$M=2.0,\lambda=0.2$ and $N=6$ layers. b) Wilson loop trivial eigenvalue
spectrum as a function of $k_{z}$ for OBC in $x$ and PBC in $y,z$. The
parameters are $M=1.6,\lambda=0.1$ and $N=6$ layers.

While the finite-size topological phase in the q(3-1)D system exhibits helical
boundary modes analogous to those of the QSHI, this topological phase is not
just a QSHI. Notably, the 3D minimum direct gap remains finite in this region
of the phase diagram where non-trivial finite-size topological phases occur,
so the 3D bulk is still in the topological phase $(1;000)$. The finite-size
topological phase therefore exhibits signatures associated with a non-trivial
intrinsically 3D topological invariant. This is indicated by adding
perturbations to the system to probe the magneto-electric polarizability of
the system, which depends on the intrinsically 3D topological invariant, a
connection identified in previous work by Essin  _et al._ [52].

We then compare the result to a q(3-1)D stack of 2D QSHI in the $x$ direction
with the same perturbation to demonstrate the finite-size topological phase of
the q(3-1)D system is distinct from a QSHI. The type of perturbations we
consider to determine the magnetoelectric polarizability in these two cases
are TRS-breaking terms in the form of weak Zeeman field in only the uppermost
and lowest layers i.e. $V=\bm{\kappa}\cdot\bm{s}$ with
$\mid\bm{\kappa}\mid\approx 0.1$. This could correspond to ferromagnetically-
ordered magnetic dopants in just these layers. This situation is illustrated
schematically in Fig. 9 a). We first consider a Zeeman field oriented in the
$yz$-plane and labeled $\bm{\kappa}_{\parallel}$ as shown in Fig. 9 a) for
both the $(1;000)$ system (q(3-1)D STI) and the $(0;001)$ system (q2D WTI, or
stack of QSHIs). In this case, these systems react similarly, their spectra
gapping out as shown in Fig.9 b). This is expected, as the perturbation breaks
TR symmetry.

Figure 8:  Phase diagram from wilson loop spectrum for a slab with a) $N=5$,
b) $N=6$, black-0, yellow-2 $\pm\pi$ phases. c)Quasi-1D dispersion for OBC in
$x$ $N=6$, PBC in $z$ and OBC (PBC) in $y$ as a function of $k_{z}$. The
parameters are $M=1.8,\lambda=0.1$. d) Spectrum for OBC in $x,y$ as a function
of $M$, for discretized values of $k_{z}$ with $N=5$ and $100$ sites in the
remaining directions, $\lambda=0.1$.

However, the responses of the two systems are strikingly distinct if we
instead consider an applied Zeeman field oriented along the $\hat{x}$-axis, or
field $\bm{\kappa}_{\perp}$ as shown in Fig. 9 a). In the case of the
$(1;000)$ system (q(3-1)D STI), two of the q1D boundary states gap out,
leaving two gapless boundary modes remaining. They constitute chiral boundary
modes of a QHE on each surface as shown in Fig. 9 c), corresponding to a
layer-dependent Hall conductivity $\sigma_{xy}$ as shown in Fig. 9 d). This is
a known manifestation of the quantized magnetoelectric polarizability of the
STI, resulting from the non-trivial value of the strong invariant, these
results can be directly compared with past work for systems with a 3D bulk
rather than q(3-1)D bulk [52]. Notably, the layer-dependent Hall conductivity
exhibits this non-trivial response only for topologically non-trivial finite-
size topological phases as characterized by $N_{\pm\pi}$. This agrees with the
response theory of previously-studied finite-size topological phases, and
reflects the fact that only occupied states of the non-trivial bubbles carry
Berry phase contributions to the underlying 3D topological invariant.

In constrast, there is no topological magnetoelectric effect and no QHE in the
surface layers of the QSHI stack. Instead, the states maintain their
degeneracy and helicity, for small fields, which is consistent when we view
the field as parallel to the edge surfaces hosting the helical boundary states
of the QSHI layers. Therefore, although the spectrum of q(3-1)D STI slab
appears to be similar to the spectrum of a QSHI stack, its response to TRS-
breaking perturbations reveals that it is a finite-size topological phase
arising from $(1;000)$ topology of the 3D bulk.

Figure 9:  a) Diagram of perturbations and edge states (pink) on a q(3-1)D STI
slab, parallel field (purple) and perpendicular field (black), we consider the
perturbations for OBC in $x,y$ and PBC in $z$. b) Spectrum of STI slab with
OBC in $x$ and $y$, $N_{x}=5$, $N_{y}=100$. A constant Zeeman field ordering
$\kappa=0.1$ parallel to the slab top and bottom layers is present. The
parameters are $M=1.8,\lambda=0.1$. c) Same parameters as in c) but now with a
perpendicular field the system exhibits a QHE.d) Layer conductivity (Chern
number) for $N_{x}=6$, $M=2.0$,$\lambda=0.2$ .

##  V 3D TI wire

We finally consider a q(3-2)D wire geometry for the STI, demonstrating that
finite-size topology arises from interference between topologically-protected
boundary states of finite-size topological phases, as well as interference
between the topologically-protected boundary states of topological phases in
the ten-fold way. We therefore consider the same Hamiltonian Eq. (4) as
considered in the previous section, but now with open boundary conditions in
$\hat{x}$\- and $\hat{y}$-directions and periodic boundary conditions in the
$\hat{z}$-direction, with the number of lattice sites in the $\hat{x}$\- and
$\hat{y}$-directions, $N_{x}$ and $N_{y}$, respectively, each much less than
$N_{z}$, the number of lattice sites in the $\hat{z}$-direction (so
$N_{x},N_{y}\ll N_{z}$).

According to the ten-fold way classification scheme for topological phases of
matter, this is an effectively 1D system in class AII, which is topologically-
trivial [48, 38, 39]. Here, we show finite-size topological phases are
nonetheless possible in this system, first characterizing finite-size topology
of the q1D bulk through analysis of Wilson loop spectra, and then by
demonstrating an additional bulk-boundary correspondence yielding q(3-3)D
topologically-protected boundary modes in the q1D wire, ultimately resulting
from interference between the topologically-protected Dirac cone surface
states of the STI.

Figure 10:  a) Phase diagram for the same finite size parameters but PBC in z,
red reflects 4 $\pm\pi$ phases, yellow 2, blue 0. with $N_{x}=N_{y}=4$ b) Heat
plot of direct gap for the same system parameters. c),d) and e),f) same as
before, but for $N_{x}=4,N_{y}=5$ and $N_{x}=N_{y}=6$ respectfully.

For the q(3-2)D bulk of the STI with periodic boundary conditions only in the
$\hat{z}$-direction, we compute Wilson loop spectra by integrating over this
one good momentum component, similarly to the Wilson loop spectra calculations
for the q(3-2)D bulk of the QSHI in Section II. The topological phase diagrams
of the STI q(3-2)D bulk are then determined by computing the number of
eigenvalues in the Wilson loop spectrum with $\pm\pi$ phases, or $N_{\pm\pi}$.
Although in the previous cases the spectrum only had two $\pm\pi$ phases or
none corresponding to nontrivial and trivial regions, we find for the 3D TI
wire that some regions of the phase diagram have four $\pm\pi$ phase
eigenvalues as shown in red in Fig. 10 a) for $N_{x}=N_{y}=4$. In the case of
$N_{x}=N_{y}=6$, the phase diagram changes again and now there are not only
two and four $\pm\pi$ phases but also six $\pm\pi$ phases as shown in blue in
Fig. 10 e). The change of invariant is consistent with a gap closing of the
q(3-2)D bulk as shown in Fig.10 b),f) where the gap closings coincide with the
change of number of $\pm\pi$ phases in the Wilson loop spectrum.

Figure 11:  a) Spectrum for OBC in x,y,z as a function of $M$, with
$N_{x},N_{y}=4$ and $N_{z}=50$ sites in the remaining directions with
$\lambda=0.1$. b) Same plot but for $N_{x}=4,N_{y}=5$ and $N_{z}=100$.

As in previous cases, we find the topological phase diagrams for the q(3-2)D
bulk of the STI depend strongly on system size. In the case of
$N_{x}=4,N_{y}=5$, we find that the region with four $\pm\pi$ phases occurs
for smaller parameter regions and is furthermore shifted in $M$ and skewed
relative to the corresponding regions in the $N_{x}=N_{y}=4$, reflecting the
difference between $x$ and $y$ directions in phase space. The skewness is also
more prominent as we increase the number of sites as seen for $N_{x}=N_{y}=6$
in Fig. 10 e). In this case, even though the topologically non-trivial regions
diminish in size, they are more strongly skewed. The deformation is such that,
starting in a trivial region according to $N_{\pm\pi}$ near $M=1.75$, with
almost zero $\lambda$, we can drive the system into a topologically non-
trivial regime as determined by $N_{\pm\pi}$ by increasing the spin-orbit
coupling to $\lambda\approx 0.1$. A comparison of the phase diagrams in Fig.
10, indicates that, as the number of sites increases, the regions present
originally in $N_{x}=N_{y}=4$ remain for $N_{x}=N_{y}=6$ although now shifted
to greater $M$ and reduced in size over phase space. We notice also that the
new regions appear from the left.

The topological phase diagram and corresponding q(3-2)D minimum direct bulk
gap phase diagram for $N_{x}=N_{y}=6$ are shown in Fig. 10 e), f),
respectively. While there are similarities between results for this system
size and the smaller ones, there is a topological region for which six Wilson
loop eigenvalues have phases fixed to $\pm\pi$. The regions of greater
$N_{\pm\pi}$ appear to be subsets of regions with lesser $N_{\pm\pi}$: the
blue region is contained within a red region, and red regions are contained
within yellow regions. The states again localize at the boundaries of the wire
and states occur in Kramers pairs. These results indicate the number of
$\pm\pi$ phases in the Wilson loop spectrum, $N_{\pm\pi}$, corresponds to half
the number of zero energy edge states $N(E=0)$ in the q(3-2)D STI system with
OBC in all directions. We can verify this relation appears to hold for the
q(2-1)D QSHI wire and the q(3-1)D STI slab as well. As $N_{\pm\pi}>2$ occurs
for the q1D STI wire, this comes to suggest an integer classification
$2\mathbb{Z}$ for the q(3-2)D STI finite-size topological phases, to be
explored in greater detail in future work.

Based on the topological phase diagrams for the q(3-2)D bulk, we now check for
a finite-size topological bulk-boundary correspondence in this geometry by
opening boundary conditions in the $\hat{z}$-direction, searching for
topologically-protected boundary modes localized at the ends of the q(3-2)D
wire. In analogy to the q(2-1)D QSHI wire, we study the non-trivial number of
$\pm\pi$ eigenvalues in the Wilson loop spectrum, Fig. 11 a),b). The system
with $N_{\pm\pi}=4$ phases has now eight edge states within the q(3-2)D bulk
gap. These states occur in Kramers pairs, with each state in a given Kramers
pair localized at the same edge There are, however, differences between these
states observable in the probability density distributions. We show the
probability densities as a function of layer index in each of the $\hat{z}$\-
and $\hat{y}$-directions, respectively,for four of the in-gap states, in Figs.
12 a) and b), respectively. The corresponding probability densities as a
function of layer in the $\hat{z}$-direction and $\hat{y}$-direction for the
other four in-gap states are shown in Figs. 12 c) and d), respectively. We see
that the second set of four are distinguished from the first four by their
localization: the second set of four are pushed inwards from the edge in both
the $\hat{z}$\- and $\hat{y}$-direction relative to the first four in-gap
states. We find similar physics for $N_{x}=N_{y}=6$ in the $N_{\pm\pi}=6$
phase: there are 12 q(3-3)D edge states at zero energy within the q(3-2)D bulk
gap. This change in localization suggests that there is a distinction between
edge states which may give way to distinct phases not distinguished by just
the parity of the number of edge states.

Figure 12:  a) Probability density plot for one q(3-3)D edge mode as a
function of wire length $z$ with $M=1.25,\lambda=0.1$,$N_{x}=N_{y}=4,N_{z}=80$
b) same edge state as a function of site index $y$ , c) Probability density
for another quasi-0D edge mode within the same model parameters as a function
of wire length d) now as a function of site index $y$.  Figure 13: a)
Schematic diagram of a 4D topological phase consisting of some $x,y,z$ real
space directions and an $L_{z}$ orbital degree of freedom which gets thinned
in the orbital direction. In this case the quasi-3D slab spectrum from the 4D
bulk results from the hybridization of the original 3D boundary modes. b) The
previous q(4-1)D slab possess an additional bulk-boundary correspondence when
additional open boundary conditions in the $z$ direction are considered. This
results in q(4-2)D boundary modes.

##  VI Concluding remarks

In this work, we have studied finite-size topology in time-reversal invariant
systems, emerging from the hybridization of helical boundary modes in QSHIs
and of Dirac cones in the strong TI. In the case of the QSHI, we find the
helical boundary modes generically interfere to realize regions in phase space
where the q(2-1)D bulk spectrum (periodic boundary conditions in one direction
and open boundary conditions in the other) is gapped. These regions are
separated from one another by critical points at which the q(2-1)D minimum
direct bulk gap is zero. We characterize the topology of these gapped regions
by computing Wilson loop spectra, finding topologically non-trivial gapped
phases of the q(2-1)D bulk corresponding to a non-trivial number of Wilson
loop eigenvalues with phase fixed to $\pm\pi$.

For open boundary conditions in each direction and a q(2-1)D wire geometry,
these Wilson loop eigenvalues with phase $\pm\pi$ correspond to topologically-
protected q(2-2)D boundary modes localized at the ends of the wire. These
q(2-2)D boundary modes occur in Kramers pairs and are robust against disorder
respecting spinful time-reversal symmetry, maintaining a fourfold degeneracy
for particle-hole symmetric disorder, and splitting into doubly-degenerate
Kramers pairs for disorder breaking particle-hole symmetry. In these cases,
the in-gap, q(2-2)D modes are still topologically-robust in that they must
correspond to time-reversal invariant charge transfer in an aperiodic Thouless
pump from valence bands to conduction bands, and this connectivity between
q(2-1)D bulk valence and conduction bands is observed in topological phase
diagrams.

We first observe this finite-size topology of the QSHI for a canonical
Hamiltonian describing HgTe quantum wells, but also find the finite-size
topological phase occurs in a tight-binding model for 1T’-WTe2 ribbons with
sawtooth edges derived from density functional theory calculations, thus
potentially relevant to experiment. In the case of the strong topological
insulator protected by time-reversal symmetry, we find finite-size topological
phases both for q(3-1)D slab geometries and q(3-2)D wire geometries. Wilson
loop spectra are used to characterize the topology of the q(3-1)D and q(3-2)D
bulk: the winding of the Wilson loop eigenvalue phases characterizes the
q(3-1)D topology, similarly to characterization of 2D topological phases in
the bulk, while the q(3-2)D wire topology in this case is also characterized
by the number of $\pm\pi$ Wilson loop eigenvalue phases as in the case of the
q(2-1)D QSHI. For open boundary conditions in two directions, the q(3-1)D STI
slab exhibits helical boundary modes in the finite-size topological phase, but
also exhibits signatures of the magneto-electric polarizability of the STI,
distinguishing this finite-size topological phase from the QSHI. In the case
of the q(3-2)D wire, q(3-3)D boundary modes occur for open boundary conditions
in all three directions, similarly to those of the q(2-1)D QSHI. However,
results indicate that topological classification for the q(3-2)D STI is
integer rather than $\mathbb{Z}_{2}$, with unusual localization of the
quasi-0D topological boundary modes. Importantly, these results show that
finite-size topology yields topologically-protected boundary modes of
codimension greater than $1$.

We close by pointing out an intriguing possible extension of the work to a
system with dimension $D>3$. For example, a four-dimensional topological phase
is expected in a q(4-1)D setting, as pictured schematically in Fig. 13 for a
small system size in some fourth dimension. These extra non-spatial dimensions
could come from physical degrees of freedom, typically considered for three-
dimensional systems, such as a $p$ orbital degree of freedom as considered in
Fig. 13 a). One could now imagine an infinite 4D bulk defined by the
$\hat{x}$-, $\hat{y}$-, $\hat{z}$-, and $\hat{L}_{z}$ (angular momentum)-axes.
For non-trivial 4D bulk topological invariant, open boundary conditions in the
$\hat{L}_{z}$-direction and a large system size in the
$\hat{L}_{z}$-direction, three-dimensional topologically-protected boundary
modes are realized. For the physical scenario of small system sizes in the
$\hat{L}_{z}$-direction as shown in Fig. 13 a), these three-dimensional
boundary modes could interfere to realize a topologically non-trivial FST
phase, such that additional topologically-protected boundary modes are
realized when boundary conditions are additionally opened in a second real-
space direction, such as the $\hat{z}$-direction as shown in Fig. 13 b). This
raises the possibility of topologically-protected boundary states for systems
in one to three-dimensions reflecting higher-dimensional, $D>3$ bulk topology.

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Supplemental material for “Time-reversal invariant finite-size topology”
R. Flores-Calderon,1,2 Roderich Moessner,1 and Ashley M. Cook1,2,∗

1Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse
38, 01187 Dresden, Germany
2 Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Strasse 40,
01187 Dresden, Germany
∗Electronic address<EMAIL_ADDRESS>
(Dated: )

##  S1 Tight-binding model for $1T^{\prime}$-$\text{WTe}_{2}$

In the main text we discussed a realistic model for realizing finite-size
topology in a $1T^{\prime}-\text{WTe}_{2}$ monolayer. We adapted the finite-
size calculation from the model explored and derived in reference [50]. The
model predicts that the four bands closest to the Fermi level are composed
mainly of contributions from two $3d_{x^{2}-y^{2}}$ type orbitals centered at
W and two $5p_{x}$ type orbitals centered at a subset of Te. With this
information at hand and some experimental fitting the authors construct a
minimal tight binding model that includes also a spin orbit interaction given
by:

$\displaystyle H_{\text{WTe}_{2}}(\bm{k})=$ $\displaystyle
s_{0}\Big{(}{\left[\frac{\mu_{p}}{2}+t_{px}\cos\left(ak_{x}\right)+t_{py}\cos\left(bk_{y}\right)\right]\Gamma_{1}^{-}}+\left[\frac{\mu_{d}}{2}+t_{dx}\cos\left(ak_{x}\right)\right]\Gamma_{1}^{+}+t_{dAB}e^{-ibk_{y}}\left(1+e^{iak_{x}}\right)e^{i\mathbf{k}\cdot\Delta_{1}}\Gamma_{2}^{+}$
$\displaystyle+t_{pAB}\left(1+e^{iak_{x}}\right)e^{i\mathbf{k}\cdot\Delta_{2}}\Gamma_{2}^{-}+t_{0AB}\left(1-e^{iak_{x}}\right)e^{i\mathbf{k}\cdot\Delta_{3}}\Gamma_{3}-2it_{0x}\sin\left(ak_{x}\right)\left[e^{i\mathbf{k}\cdot\Delta_{4}}\Gamma_{4}^{+}+e^{-i\mathbf{k}\cdot\Delta_{4}}\Gamma_{4}^{-}\right]$
$\displaystyle+t_{0ABx}\left(e^{-iak_{x}}-e^{2iak_{x}}\right)e^{i\mathbf{k}\cdot\Delta_{3}}\Gamma_{3}+\text{
H.c.
}\Big{)}+{\left[\left(\lambda_{dx}^{z}s_{z}+\lambda_{dx}^{y}s_{y}\right)\sin\left(ak_{x}\right)\right]\Gamma_{5}^{+}}+\left[\left(\lambda_{px}^{z}s_{z}+\lambda_{px}^{y}s_{y}\right)\sin\left(ak_{x}\right)\right]\Gamma_{5}^{-}$
$\displaystyle-i\lambda_{0AB}^{y}s_{y}\left(1+e^{iak_{x}}\right)e^{i\mathbf{k}\cdot\Delta_{3}}\Gamma_{6}-i\left(\lambda_{0}^{z}s_{z}+\lambda_{0}^{y}s_{y}\right)\left(e^{i\mathbf{k}\cdot\Delta_{4}}\Gamma_{4}^{+}-e^{-i\mathbf{k}\cdot\Delta_{4}}\Gamma_{4}^{-}\right)-i\left(\lambda_{0}^{z}s_{z}+\lambda_{0}^{y}s_{y}\right)$
$\displaystyle\times\left(e^{-ibk_{y}}e^{i\mathbf{k}\cdot\Delta_{4}}\Gamma_{4}^{+}-e^{ibk_{y}}e^{-i\mathbf{k}\cdot\Delta_{4}}\Gamma_{4}^{-}\right)+\text{H.c.,
}$ (S1)

,where $s_{i}$ are Pauli matrices representing the spin degree of freedom and
they defined the gamma matrices as:

$\displaystyle\Gamma_{0}$ $\displaystyle=\tau_{0}\sigma_{0}$ (S2)
$\displaystyle\Gamma_{1}^{\pm}$
$\displaystyle=\frac{\tau_{0}}{2}\left(\sigma_{0}\pm\sigma_{3}\right)$ (S3)
$\displaystyle\Gamma_{2}^{\pm}$
$\displaystyle=\frac{1}{4}\left(\tau_{1}+i\tau_{2}\right)\left(\sigma_{0}\pm\sigma_{3}\right)$
(S4) $\displaystyle\Gamma_{3}$
$\displaystyle=\frac{1}{2}\left(\tau_{1}+i\tau_{2}\right)i\sigma_{2}$ (S5)
$\displaystyle\Gamma_{4}^{\pm}$
$\displaystyle=\frac{1}{4}\left(\tau_{0}\pm\tau_{3}\right)\left(\sigma_{1}+i\sigma_{2}\right)$
(S6) $\displaystyle\Gamma_{5}^{\pm}$
$\displaystyle=\frac{\tau_{3}}{2}\left(\sigma_{0}\pm\sigma_{3}\right)$ (S7)
$\displaystyle\Gamma_{6}$
$\displaystyle=\frac{1}{2}\left(\tau_{1}+i\tau_{2}\right)\sigma_{1}$ (S8)

,where $\tau_{i},\sigma_{i}$ are Pauli matrices acting in sublattice and
orbital degrees of freedom respectively. Finally the constants from the
previous Hamiltonian that reproduce the experimental results are:

$\displaystyle\begin{array}[]{lccc}\mu_{p}&-1.75\mathrm{eV}&\lambda_{0AB}^{y}&0.011\mathrm{eV}\\\
\mu_{d}&0.74\mathrm{eV}&\lambda_{0}^{y}&0.051\mathrm{eV}\\\
t_{px}&1.13\mathrm{eV}&\lambda_{0}^{z}&0.012\mathrm{eV}\\\
t_{dx}&-0.41\mathrm{eV}&\lambda_{0}^{\prime y}&0.050\mathrm{eV}\\\
t_{pAB}&0.40\mathrm{eV}&\lambda_{0}^{\prime z}&0.012\mathrm{eV}\\\
t_{dAB}&0.51\mathrm{eV}&\lambda_{px}^{y}&-0.040\mathrm{eV}\\\
t_{0AB}&0.39\mathrm{eV}&\lambda_{px}^{z}&-0.010\mathrm{eV}\\\
t_{0ABx}&0.29\mathrm{eV}&\lambda_{dx}^{y}&-0.031\mathrm{eV}\\\
t_{0x}&0.14\mathrm{eV}&\lambda_{dx}^{z}&-0.008\mathrm{eV}\\\
t_{py}&0.13\mathrm{eV}&&\\\ a&3.477\mbox{\AA}&b&6.249\mbox{\AA}\\\
\mathbf{r}_{Ad}&(-0.25a,0.32b)&\mathbf{r}_{Bp}&(0.25a,0.07b)\\\
\mathbf{r}_{Ap}&(-0.25a,-0.07b)&\mathbf{r}_{Bd}&(0.25a,-0.32b)\end{array}$
(S22)

Finally we extended the model with an inclusion of a perpendicular electric
field in the weak limit where the effect manifests itself as a change of
Rashba spin orbit interaction, in the previous Hamiltonian this is modelled as
a replacement $\lambda^{i}\rightarrow\lambda^{i}+\Delta\lambda$, with $i=x,y$.