# Quantum phase transitions and disorder-based filter in a Floquet system

Balaganchi A. Bhargava  IFW Dresden and Würzburg-Dresden Cluster of Excellence
ct.qmat, Helmholtzstrasse 20, 01069 Dresden, Germany  Sanjib Kumar Das  IFW
Dresden and Würzburg-Dresden Cluster of Excellence ct.qmat, Helmholtzstrasse
20, 01069 Dresden, Germany  Ion Cosma Fulga  IFW Dresden and Würzburg-Dresden
Cluster of Excellence ct.qmat, Helmholtzstrasse 20, 01069 Dresden, Germany

###### Abstract

Two-dimensional periodically-driven topological insulators have been shown to
exhibit numerous topological phases, including ones which have no static
analogue, such as anomalous Floquet topological phases. We study a two
dimensional model of spinless fermions on a honeycomb lattice with periodic
driving. We show that this model exhibits a rich mixture of weak and strong
topological phases, which we identify by computing their scattering matrix
invariants. Further, we do an in-depth analysis of these topological phases in
the presence of spatial disorder and show the relative robustness of these
phases against imperfections. Making use of this robustness against spatial
disorder, we propose a filter which allows the passage of only edge states,
and which can be realized using existing experimental techniques.

##  I Introduction

In recent years, a great amount of interest has been devoted to the field of
topological insulators Thouless  _et al._ (1982); Bernevig and Zhang (2006);
Bernevig  _et al._ (2006); Hasan and Kane (2010); Qi and Zhang (2011).
Generally, topological insulators are defined as having a gapped bulk
spectrum, but supporting lower dimensional gapless states, which can reside on
surfaces or edges of the system. The existence of the gapless states is a
consequence of the well known bulk-boundary correspondence principle, meaning
the number of those boundary states will be dictated by a topological number,
which is calculated from the bulk eigenstates of the system. This topological
number, also known as a topological invariant, characterizes the nontrivial
nature of the bulk, which is deeply connected to the symmetries of the system.
In this context, many earlier pioneering works have also classified the
topological insulators based on the dimension and symmetries of the system,
leading to the so called periodic table of topological phases Schnyder  _et
al._ (2008); Kitaev (2009); Ryu  _et al._ (2010); Chiu  _et al._ (2016).

Recently, the understanding of topology has also been extended to
periodically-driven systems, which feature rich topological phases that have
no counterparts in static topological systems. The periodically-driven systems
which realize topological phases are called Floquet topological insulators
(FTI) Kitagawa  _et al._ (2010); Lindner  _et al._ (2011); Cayssol  _et al._
(2013); Lindner  _et al._ (2013); Rudner  _et al._ (2013); Lababidi  _et al._
(2014); Leykam  _et al._ (2016); Bomantara  _et al._ (2016); Yao  _et al._
(2017); Oka and Kitamura (2019). The discovery of FTIs has enriched the
earlier periodic table of topological insulators Else and Nayak (2016); Roy
and Harper (2017). FTIs have also been realized in multiple experimental
setups, such as in photonic waveguide lattices Rechtsman  _et al._ (2013);
Mukherjee  _et al._ (2017); Maczewsky  _et al._ (2017), coupled ring
resonators Guo  _et al._ (2020), and ultracold atoms Wintersperger  _et al._
(2020).

One of the unique features of FTIs is the possibility of so-called anomalous
topological phases, in which the bulk bands are topologically trivial, meaning
that the system can host robust extended states even when _all_ states in the
bulk of the system are localized. When the localization of bulk states is a
consequence of disorder, these systems are called anomalous Floquet Anderson
insulators (AFAI) Titum  _et al._ (2016); Nathan  _et al._ (2017); Kundu  _et
al._ (2020). A study of some of the localization properties leading to the
formation of AFAIs has been established using a periodically driven Kitaev
model Kitaev (2006); Fulga  _et al._ (2019). More recently, the generation of
an AFAI phase in the presence of spatial disorder has been studied in a two-
dimensional Chern insulator driven by onsite potential kicks Liu  _et al._
(2020) or by an oscillating electric field Zhang  _et al._ (2021). However, a
detailed study of how the strength and type of spatial disorder impact the
totality of the phase diagram of an FTI is missing. So, it is naturally of
interest to understand the stability and robustness of FTI phases in the
presence of disorder.

In this article we address this question of robustness of FTI on a honeycomb
lattice, a model which we define in section II. Then, in section III we
explore the phases present in the model. We calculate the invariants in each
of the phases. In section IV, a systematic study of the robustness of the
system to spatial disorder is presented. To achieve this, we provide a way to
construct the Floquet operator in real space, analytically, even when the
system is disordered. Using the insight gained with respect to the relative
robustness of the different topological phases, in section V we propose a
filter that can be realized in photonic crystals. This filter can be used to
improve the signal corresponding to the propagating topological modes by
removing the spurious contributions of the bulk states. We conclude in section
VI.

##  II The Model

We consider a system of spinless fermions on a honeycomb lattice with a time-
dependent hopping amplitude. In addition to the nearest-neighbor hoppings, we
consider a hopping across each hexagonal plaquette, see Fig. 1. The
Hamiltonian is given by

${\cal
H}=\mu\sum_{i,\alpha}c^{\dagger}_{i,\alpha}c^{\phantom{{\dagger}}}_{i,\alpha}+\sum_{i,j,\alpha,\beta}J_{(i,j)_{\alpha,\beta}}(t)(c^{\dagger}_{i,\alpha}c^{\phantom{{\dagger}}}_{j,\beta}+{\rm
h.c.}),$ (1)

where $(i,j)_{\alpha,\beta}$ denotes a pair of sites connected by a hopping
$J$, and $t$ is time. The Latin indices denote the unit cell of the triangular
lattice, while the Greek indices denote the sublattice degree of freedom,
corresponding to two sites per unit cell. $c^{\dagger}_{i,\alpha}$ and
$c^{\phantom{{\dagger}}}_{i,\alpha}$ are the usual fermionic creation and
annihilation operators. We consider four types of hopping throughout the
lattice, $J_{1,2,3,4}$, as shown in Fig. 1. The hopping amplitude $J(t)$ is a
function of time and has the following form in one complete driving period,
$T$:

$\begin{split}J(t)=\begin{cases}J_{1}=J_{s},&\quad J_{2}=J_{3}=J_{4}=0,\\\
&\text{for}\quad nT<t\leq nT+\frac{T}{4},\\\ \\\ J_{2}=J_{s},&\quad
J_{1}=J_{3}=J_{4}=0,\\\ &\text{for}\quad nT+\frac{T}{4}<t\leq
nT+\frac{T}{2},\\\ \\\ J_{3}=J_{s},&\quad J_{1}=J_{2}=J_{4}=0,\\\
&\text{for}\quad nT+\frac{T}{2}<t\leq nT+\frac{3T}{4},\\\ \\\
J_{4}=J_{d},&\quad J_{1}=J_{2}=J_{3}=0,\\\ &\text{for}\quad
nT+\frac{3T}{4}<t\leq nT+T,\\\ \end{cases}\end{split}$ (2)

where $J_{s}$ and $J_{d}$ are constants, $n\in\mathbb{Z}$, and the onsite term
is set to $\mu=0$.

Figure 1:  The honeycomb lattice. The unit cell is shown using a green
parallelogram. The blue circles and orange circles denote the sublattice $A$
and $B$, respectively. The four kinds of hopping are marked as $J_{1}$,
$J_{2}$, $J_{3}$ and $J_{4}$, which are active in the time slices 1, 2, 3, and
4, respectively. To avoid clutter, only one of the $J_{4}$ hoppings is shown.
The Bravais vectors $\textbf{a}_{1}$ and $\textbf{a}_{2}$ are shown in one of
the hexagons.

The spectrum of the above system in the case of a time-independent hopping can
be obtained exactly, by using periodic boundary conditions along both
directions. We Fourier transform the Hamiltonian Eq. (1) using the following
definitions:

$\begin{split}c_{i,\alpha}&=\frac{1}{\sqrt{2N}}\sum_{\textbf{k}}e^{-i\textbf{k}\cdot\textbf{r}_{i}}c_{\textbf{k},\alpha}\\\
&\text{and}\\\
c^{\dagger}_{i,\alpha}&=\frac{1}{\sqrt{2N}}\sum_{\textbf{k}}e^{i\textbf{k}\cdot\textbf{r}_{i}}c^{\dagger}_{\textbf{k},\alpha}.\end{split}$
(3)

Here, $N$ denotes the total number of unit cells, ${\bf k}$ is the two-
dimensional momentum vector, and
$\textbf{r}_{i}=n_{1}\textbf{a}_{1}+n_{2}\textbf{a}_{2}$, with $n_{1,2}$
integers and $\textbf{a}_{1,2}$ the Bravais vectors (see Fig. 1). The
momentum-space Hamiltonian is

$\displaystyle{\cal
H}=\sum_{\textbf{k}}\begin{pmatrix}c^{\dagger}_{\textbf{k},A}&c^{\dagger}_{\textbf{k},B}\end{pmatrix}H({\bf
k})\begin{pmatrix}c^{\phantom{{\dagger}}}_{\textbf{k},A}\\\
c^{\phantom{{\dagger}}}_{\textbf{k},B}\end{pmatrix},$ (4) $\displaystyle
H({\bf k})=\begin{pmatrix}0&\mathcal{J}({\bf k})\\\ \mathcal{J}^{*}({\bf
k})&0\end{pmatrix},$ (5)

where $\mathcal{J}({\bf
k})=J_{1}+J_{3}e^{ik_{2}}+J_{2}e^{ik_{1}}+J_{4}e^{i(k_{1}+k_{2})}$, and
$k_{j}={\bf a}_{j}\cdot{\bf k}$. The eigenvalues are given by
$E=\pm|\mathcal{J}|$.

Now turning our attention to the driven system, we study the Floquet operator
of the system and its properties. The time evolution of a system from time
$t_{1}$ to time $t_{2}$ is governed by the time-evolution operator given by

$U(t)=\mathcal{T}e^{-i\int_{t_{1}}^{t_{2}}H(t)dt},$ (6)

where $H(t)$ is the time-dependent Hamiltonian, $\mathcal{T}$ denotes time
ordering, and we have set $\hbar=1$. When the system evolves periodically in
time, the time-evolution operator over one driving period is called the
Floquet operator. For the specific time drive that we have, the Floquet
operator is a product of the time-evolution operators in each time slice:

$\begin{split}F=\mathcal{T}e^{-i\int_{0}^{T}H(t)dt}&=e^{-iH_{4}\frac{T}{4}}e^{-iH_{3}\frac{T}{4}}e^{-iH_{2}\frac{T}{4}}e^{-iH_{1}\frac{T}{4}}\\\
&=F_{4}F_{3}F_{2}F_{1}.\end{split}$ (7)

The eigenvalue equation governed by the Floquet operator is

$F\ket{\psi}=e^{-i\epsilon T}\ket{\psi}.$ (8)

Since the Floquet operator is unitary, the eigenvalues are phases. The
$\epsilon$ in Eq. (8) is called the quasi-energy, and it is defined modulo
$\frac{2\pi}{T}$.

With the above form of the Floquet operator, when the quasi-energy is plotted
as a function of momentum, we obtain the Floquet band structure. The Floquet
bands are therefore periodic in both momentum and quasi-energy. The Floquet
band structure is similar to the bands of static systems. In the system that
we study, there are two quasi-energy gaps, at $\epsilon T=0$ and $\epsilon
T=\pi$.

As long as $\mu=0$, the system obeys particle-hole symmetry (PHS). This can be
seen as follows. The Hamiltonian $H({\textbf{k})}$ in Eq. (5) belongs to class
BDI Altland and Zirnbauer (1997) and is particle-hole symmetric:

$\mathcal{P}^{-1}H(\textbf{k})\mathcal{P}=-H(-\textbf{k}),$ (9)

where $\mathcal{P}=\sigma_{z}\mathcal{K}$ is the PHS operator, $\sigma_{z}$ is
the third Pauli matrix in sublattice space, and $\mathcal{K}$ denotes complex
conjugation. The manifestation of this in the Floquet operator can be worked
out as

$\mathcal{P}^{-1}F(\textbf{k})\mathcal{P}=F(-\textbf{k}).$ (10)

As a result, in the Floquet band structure, one can find a state at
$(-\epsilon,-{\bf k})$ corresponding to a state at $(\epsilon,{\bf k})$.

Figure 2:  Left: The analytically obtained phase diagram for the Floquet
system, showing the bulk gap closing lines as $J_{s}$ and $J_{d}$ are varied.
There are a total of six different topological phases, separated by families
of straight lines, as indicated in Eq. (11). The red lines correspond to
$k_{1,2}=0$, the blue lines correspond to $k_{1,2}=\pi$ and the green lines
correspond to $k_{1,2}=\mp\pi/2$ gap closing points in the Brillouin zone.
Right: Band structures for all the six phases in a ribbon geometry: infinite
along ${\bf a}_{1}$, and having a width of 50 unit cells along ${\bf a}_{2}$.
$k$ denotes the dimensionless momentum along the infinite direction of the
ribbon. Each panel has has been labeled according to the phase that it
corresponds to. Panels $\mathfrak{A}$, $\mathfrak{B}$, $\mathfrak{C}$,
$\mathfrak{D}$, $\mathfrak{E}$, and $\mathfrak{F}$ are plotted at
$(J_{s}T/4,J_{d}T/4)=(1,6)$, $(0.85,5)$, $(1,1.55)$, $(0.85,3)$, $(1.5,3)$,
and $(0.5,1)$. The colors in the band structures show where the states are
localized along the $\textbf{a}_{2}$ direction. The green color corresponds to
a state that is present in the bulk. The red color corresponds to a state on
the bottom edge and blue corresponds to a state on the top edge.

##  III Phase Diagram

By varying the values of $J_{s}$ and $J_{d}$, the Floquet system realizes a
series of topological phases. Between them, topological phase transitions are
signaled by the closing and reopening of either (or both) of the $\epsilon
T=0$ and $\epsilon T=\pi$ bulk gaps. We show these phases and phase
transitions in Fig. 2. All our numerical results are obtained using the kwant
code Groth  _et al._ (2014), and are available as supplemental material files
at Bhargava  _et al._ (2021).

Due to PHS, the bulk gaps of the Floquet operator given by Eq. (7) closes
whenever $\epsilon T=n\pi$ for some ${\bf k}$, with $n\in\mathbb{Z}$. We find
that these gap closings occur only at $k_{1,2}=0,\pm\pi/2,\pm\pi$, when

$\begin{split}(J_{d}+3J_{s})\frac{T}{4}&=n\pi\quad\text{for}\quad(k_{1},k_{2})=(0,0),\\\
(J_{d}-J_{s})\frac{T}{4}&=n\pi\quad\text{for}\quad(k_{1},k_{2})=(\pi,\pi),\\\
(J_{d}+J_{s})\frac{T}{4}&=n\pi\quad\text{for}\quad(k_{1},k_{2})=(-\pi/2,\pi/2).\end{split}$
(11)

Therefore, the gap closings are families of straight lines in the $J_{s}$ –
$J_{d}$ plane. Also, note that the phase diagram is periodic in both the
hopping variables $J_{s}$ and $J_{d}$:

$F(J_{s},J_{d})=F(J_{s},J_{d}+\frac{8\pi}{T})=F(J_{s}+\frac{8\pi}{T},J_{d}).$
(12)

Each of the phases in the phase diagram is characterized by a set of strong
and weak topological indices. Since the characterization of the phases
involves the study of edge states, we now shift to a ribbon geometry. We
consider a system which is infinite along the $\textbf{a}_{1}$ direction, but
has a finite width along $\textbf{a}_{2}$.

The strong index is a $\mathds{Z}$ invariant and measures the net number of
chiral edges states on a given edge at a particular quasi-energy gap. Since
there are two quasi-energy gaps, there are two strong indices for each phase.
We denote them $\mathcal{W}_{0}$ and $\mathcal{W}_{\pi}$, corresponding to the
strong index at the $\epsilon T=0$ and $\epsilon T=\pi$ quasi-energy gaps,
respectively. These invariants are related to the Chern numbers of each of the
two bulk bands by the relation Rudner  _et al._ (2013)

$C_{\pm}=\pm(\mathcal{W}_{\pi}-\mathcal{W}_{0}),$ (13)

where $C_{+}$ is the Chern number of the band at $\epsilon T>0$ in our
convention, and $C_{-}$ is the Chern number of the band at $\epsilon T<0$.

The weak indices, on the other hand, are $\mathds{Z}_{2}$ invariants which
measure the parity of the number of edge states at each of the particle-hole
symmetric momentum and quasi-energy points, $\epsilon T=0,\pi$ and $k=0,\pi$,
where $k$ is the dimensionless ribbon momentum. These can be in general
different for a ribbon along the $\textbf{a}_{1}$ and the $\textbf{a}_{2}$
directions, although this does not happen for our model Fulga  _et al._
(2019). Since there are two particle-hole symmetric momenta and quasi-energies
each, there are in total four weak indices. We denote them by $\nu_{k,\epsilon
T}$, with $k\in\\{0,\pi\\}$ and $\epsilon T\in\\{0,\pi\\}$. An odd parity of
the number of edge states is indicated by $\nu_{k,\epsilon T}=-1$, whereas an
even parity means $\nu_{k,\epsilon T}=+1$.

Note that the above invariants are not all independent. They can be related to
each other by the relation $\nu_{0,\epsilon T}\nu_{\pi,\epsilon
T}=(-1)^{\mathcal{W}_{\epsilon T}}$ Ran (2010). Therefore, there are four
independent indices characterizing each phase: $\mathcal{W}_{0}$,
$\mathcal{W}_{\pi}$, $\nu_{k=\pi,\epsilon T=0}$, and $\nu_{k=\pi,\epsilon
T=\pi}$. Also note that, unlike the strong index, the weak invariants require
PHS. Perturbations which break this symmetry will thus convert the weak phases
into trivial ones. Full details of the strong and weak indices for Floquet
systems can be found in Ref. Fulga and Maksymenko (2016).

We describe each of the phases in detail. We have labeled them as
$\mathfrak{A}$, $\mathfrak{B}$, $\mathfrak{C}$, $\mathfrak{D}$,
$\mathfrak{E}$, and $\mathfrak{F}$. Representative band structures of these
phases are shown in Fig. 2.

Phases $\mathfrak{A}$ and $\mathfrak{B}$ are strong phases, which host chiral
edge states only in the $\epsilon T=0$ gap, while there are no topologically
protected edge modes in the $\epsilon T=\pi$ gap. Phase $\mathfrak{B}$ has
edge modes with an opposite direction of propagation compared to those phase
$\mathfrak{A}$.

Phases $\mathfrak{C}$ and $\mathfrak{D}$ have edge states present at both the
$\epsilon T=0$ and $\epsilon T=\pi$ gaps. The edge states at $\epsilon T=0$
are counter-propagating, whereas those at $\epsilon T=\pi$ are chiral.
Therefore, these phases show a combination of strong and weak topology.
Similar to $\mathfrak{A}$ and $\mathfrak{B}$, the chiral edge modes in the
$\epsilon T=\pi$ gap of phases $\mathfrak{C}$ and $\mathfrak{D}$ propagate in
opposite directions.

Phase $\mathfrak{E}$ is called an anomalous phase, because it hosts
topologically protected chiral edges states at both the quasi-energy gaps,
even though the Chern number is zero for both of the bulk bands. Due to this
feature, there can be no static counterpart of this phase. At a resonant
hopping strength of $J_{s}T/4=\pi/2$ and $J_{d}=0$, the bulk bands in the
phase become completely flat.

The phase $\mathfrak{F}$ is a weak phase. This phase has trivial strong
indices and nontrivial weak indices which are protected by PHS and translation
symmetry. It has a pair of counter-propagating edge states on each edge at
$\epsilon T=0$, and no edge states in the quasi-energy gap at $\epsilon
T=\pi$.

By direct inspection of the band structures in each phase, we can enumerate
their topological invariants, see Tab. 1.

Phase | $\mathcal{W}_{\epsilon T=0}$ | $\mathcal{W}_{\epsilon T=\pi}$ | $\nu_{k=\pi,\epsilon T=0}$ | $\nu_{k=\pi,\epsilon T=\pi}$
---|---|---|---|---
$\mathfrak{A}$ | $-1$ | $0$ | $-1$ | $+1$
$\mathfrak{B}$ | $+1$ | $0$ | $-1$ | $+1$
$\mathfrak{C}$ | $0$ | $+1$ | $-1$ | $+1$
$\mathfrak{D}$ | $0$ | $-1$ | $-1$ | $+1$
$\mathfrak{E}$ | $-1$ | $-1$ | $-1$ | $+1$
$\mathfrak{F}$ | $0$ | $0$ | $-1$ | $+1$
Table 1:  The table of invariants. The table lists all the topological
invariants of different phases present in the system. Each phase is
characterized by a set of strong and weak invariants. Four invariants uniquely
determine each phase.

##  IV Effects of disorder

Topological phases are in general robust to disorders in the system. In the
presence of disorder, the bulk bands of a two-dimensional system may become
localized, provided their Chern numbers vanish. However, as the strength of
the disorder is increased, the bulk may become delocalized, signaling a
transition away from the topological phase.

We consider a system consisting of $L\times L$ unit cells, indexed by pairs of
integers $(n_{1},n_{2})$, and impose periodic boundary conditions along the
${\bf a}_{1}$ direction. We define the scattering matrix at any quasi-energy
$\epsilon$ by attaching absorbing terminals to the top and bottom boundaries,
$n_{2}=1$ and $n_{2}=L$. The expression for the scattering matrix is

$S(\epsilon)=P\big{[}1-e^{i\epsilon}F(1-P^{T}P)\big{]}^{-1}e^{i\epsilon}FP^{T}$
(14)

where the superscript $T$ stands for matrix transpose, and we have omitted the
time period multiplying $\epsilon$ to avoid confusion. $P$ is a projection
operator onto the absorbing terminals,

$\begin{split}P=\begin{cases}1\quad\text{if}\quad n_{2}\in\\{1,L\\},\\\
0\quad\text{otherwise.}\end{cases}\end{split}$ (15)

The scattering matrix has the form

$S(\epsilon)=\begin{pmatrix}\mathfrak{r}&\mathfrak{t}\\\
\mathfrak{r}^{\prime}&\mathfrak{t}^{\prime}\end{pmatrix},$ (16)

where $\mathfrak{t}^{(\prime)}$ and $\mathfrak{r}^{(\prime)}$ are the so-
called transmission and reflection blocks. Their elements are the probability
amplitudes for states to be transmitted between the two terminals, or
backscattered in the same terminal, respectively. From either of the
transmission blocks in the scattering matrix, we can compute the total
quasiparticle transmission as

$G=\text{Tr}(\mathfrak{t}^{\dagger}\mathfrak{t}).$ (17)

In addition, the scattering matrix can be used to compute the strong and weak
topological invariants, even in the presence of disorder. We briefly review
how this is done in App. A.

Figure 3:  Phase diagram for $V_{h}T/4=0.2$ (a) and $V_{h}T/4=1.0$ (b). The
largest of the transmissions at $\epsilon T=0$ and $\epsilon T=\pi$ is plotted
as a function of $J_{s}$ and $J_{d}$. Each point is obtained by averaging over
$100$ disorder configurations. (c) Phase diagram for hopping disorder strength
$V_{h}T/4$ versus $J_{s}T/4$. This is plotted on the path
$J_{d}T/4=J_{s}T/4+2$. Same as in panels (a) and (b), this plot shows larger
of the transmissions at $\epsilon T=0$ and $\epsilon T=\pi$. Each point is
obtained by averaging over $200$ disorder configurations. All the panels have
been obtained for a system of size of $24\times 24$ unit cells with periodic
boundary conditions along the $\textbf{a}_{1}$ direction.

###  IV.1 Analytic Floquet operator

From Eq. (14) we see that computing the scattering matrix involves writing the
Floquet operator in real space. To do this, one has to write down the time-
evolution operator in each time slice separately and multiply them. The task
of computing the time-evolution operators in general for a Hamiltonian in real
space requires numerically exponentiating a large Hermitian matrix, which can
be time consuming. However, for the specific kind of time drive that we have,
one can write down an analytic form of the time-evolution operator in real
space for each time slice. Hence, we can construct the Floquet operator in
real space by not resorting to numerical exponentiation. This method
significantly improves the efficiency of constructing the Floquet operator and
hence is useful to do simulations of large systems. The details for
constructing the analytic Floquet operator are given in Appendix B. In the
next two subsections we study the effects of disorder in hoppings and the
effects of disorder in the onsite term. We see how each of the phases behaves
for various strengths of hopping as well as onsite disorders.

###  IV.2 Hopping Disorder

We add disorder in the hoppings as

$\begin{split}J_{s}\frac{T}{4}\rightarrow(J_{s}+\delta_{h})\frac{T}{4},\\\
J_{d}\frac{T}{4}\rightarrow(J_{d}+\delta_{h})\frac{T}{4},\end{split}$ (18)

where $\delta_{h}$ is random number drawn independently for each hopping from
the uniform distribution $[-V_{h},V_{h})$, with $V_{h}$ the disorder strength.
Note that hopping disorder preserves PHS, meaning that the weak topological
phases remain well defined even as disorder is introduced. This is because, as
shown in Refs. Fulga  _et al._ (2014); Fulga and Maksymenko (2016), weak
topological phases can survive the disorder-induced breaking of translation
symmetry, provided that the ensemble of disordered systems still preserves
this symmetry on average.

Figure 4:  Plot of quasi-energy versus bulk transmission. Weak anti-
localization can be observed at particle-hole symmetric quasi-energies
$\epsilon T=0$ and $\epsilon T=\pi$, where $G$ increases as $\log L$ with
system size (inset). The transmission values in both the scaling plots and the
quasi-energy versus $G$ are obtained by averaging over 1000 disorder
configurations. All the calculations use $JsT/4=1$, $J_{d}T/4=3$, and
$V_{h}T/4=1.5$.

In Fig. 3 we plot the transmission as a function of hopping strengths $J_{s}$
and $J_{d}$ for two different disorder strengths. We see that $G$ increases at
places where there is a bulk gap closing line in the clean limit. This
increase in the transmission indicates the existence of delocalized bulk
states at the quasi-energies $\epsilon T=0$, $\epsilon T=\pi$, or both. This
shows that a topological phase transition has occurred at those parameter
values.

When the strength of disorder is small, we see that all the phases remain as
in the clean limit [see Fig. 3(a)]. We have also verified this by calculating
the invariants in each phase. For larger disorder strengths, however, we see
that many phases are lost [Fig. 3(b)]. Phases $\mathfrak{A}$ and
$\mathfrak{B}$ become delocalized at most of the regions in their parameter
space except for a small region near to phases $\mathfrak{E}$ and
$\mathfrak{F}$. A similar property can seen for phases $\mathfrak{C}$ and
$\mathfrak{D}$.

Figure 5:  Phase diagram for onsite disorder strength $V_{o}T/4=0.2$ (a) and
$V_{o}T/4=1.0$ (b). The largest of the transmissions at $\epsilon T=0$ and
$\epsilon T=\pi$ is plotted as a function of $J_{s}$ and $J_{d}$. (c) The
right most panel shows the transmission along the line $J_{d}T/4=J_{s}T/4+2$
as a function of onsite disorder strength $V_{o}T/4$. All the panels have been
obtained for a system of size $24\times 24$ unit cells with periodic boundary
conditions along the $\textbf{a}_{1}$ direction. In panels (a) and (b), each
point has been averaged over 100 disorder configurations, whereas in panel (c)
each point has been averaged over 200 disorder configurations.

Phases $\mathfrak{E}$ and $\mathfrak{F}$, on the other hand, remain localized
for some regions of the hopping parameters even for relatively large
disorders. This is particularly true for phase $\mathfrak{E}$, which still is
present in a large region of the $J_{s}$ – $J_{d}$ plane even at disorder
strengths for which almost all other phases have disappeared [see Fig. 3(b)].
We trace this robustness to the large bulk gap of this phase in the clean
limit, close to its so-called “resonant driving point,”
$\frac{J_{s}T}{4}=\pi/2$ and $\frac{J_{d}T}{4}=0$. There, the bulk bands are
completely flat and positioned at $\epsilon T=\pm\pi/2$, such that the bulk
gaps around $\epsilon T=0,\pi$ each occupy one half of the full Floquet quasi-
energy zone.

The above analysis shows that large disorders can delocalize the bulk.
However, this does not give a complete picture of the behavior of phases as a
function of disorder strength $V_{h}$. To get a better understanding of the
robustness of phases to disorder, we choose a path $J_{d}T/4=J_{s}T/4+2$ in
the parameter space and plot the transmission at the gap closing quasi-
energies $\epsilon T=0$ and $\epsilon T=\pi$ as $J_{s}$ and $V_{h}$ are
varied. Note that the path chosen covers all the phases as one varies
$J_{s}T/4$ from $0$ to $4$. Therefore, we can track the behavior of all the
phases as a function of the disorder strength. A plot of the transmission is
shown in Fig. 3(c). The bulk becomes delocalized above a certain critical
disorder strength, which is different for different phases. As one can notice,
the phases $\mathfrak{E}$ and $\mathfrak{F}$ can withstand a relatively large
disorder strengths.

To understand the nature of the phases once the transition has happened, we
plot the transmission as a function of quasi-energy and system size in Fig. 4.
We see that at the particle hole symmetric quasi-energies $\epsilon T=0$ and
$\epsilon T=\pi$, the transmission increases as a function of system size.
This peak in the bulk transmission is due to weak anti-localization. We
perform a scaling analysis at these particle hole symmetric quasi-energies,
which shows that the system behaves similar to a thermal metal at $\epsilon
T=0$ and $\epsilon T=\pi$, since $G\propto\log(L)$ Cho and Fisher (1997);
Kagalovsky  _et al._ (1999); Chalker  _et al._ (2001); Evers and Mirlin
(2008); Medvedyeva  _et al._ (2010); Fulga  _et al._ (2020) (see inset of Fig.
4). In contrast, away from the particle-hole symmetric quasi-energies, the
transmission is a decreasing function of system size, a consequence of weak
localization.

###  IV.3 Onsite Disorder

Next we study the effects of onsite disorder, which is added as

$\mu\rightarrow 0+\delta_{o}\frac{T}{4}.$

The random number $\delta_{o}$ is drawn independently for each site from the
uniform distribution of $[-V_{o},V_{o})$, with $V_{o}$ the onsite disorder
strength. Unlike the hopping disorder considered previously, onsite disorder
breaks PHS. This means that all weak phases will become trivial as soon as
disorder is turned on, and the only remaining topology of the model will be
due to its strong topological invariants.

In Fig. 5 we plot the largest of the two bulk transmissions at the quasi-
energies $\epsilon T=0$ and $\epsilon T=\pi$ for various values of $J_{s}$ and
$J_{d}$. Phase $\mathfrak{F}$, which is a weak phase, is lost as soon as
disorder is introduced, as discussed above.

The phases $\mathfrak{A}$, $\mathfrak{B}$, $\mathfrak{C}$, and $\mathfrak{D}$
are robust against small disorder strengths, since they are characterized by
nonzero strong invariants. However, all of them are eventually destroyed as
disorder strength is progressively increased.

Phase $\mathfrak{E}$, the anomalous phase, is robust against a large disorder
strength, as can be seen in Fig. 5. As the strength of disorder is increased,
the boundaries separating different topological phases move in the $J_{s}$ –
$J_{d}$ plane. However the middle region of $\mathfrak{E}$ remains localized
even for a large disorder, as seen in Fig. 5(c). Since the bulk bands in this
phase have a zero Chern number, in phase $\mathfrak{E}$ all bulk states are
localized by disorder irrespective of their quasi-energy. This is the
signature of an AFAI.

One prominent feature of Fig. 5 is the presence of large finite-size effects.
Since onsite disorder breaks PHS, no thermal metal-like phases are possible
Evers and Mirlin (2008), and we expect that the system should always show a
localized bulk, with the exception of sharp lines where the strong invariant
changes. Instead, we observe that there are wide regions in parameter space
where the finite system shows a significant transmission, and these regions
become larger as the strength of disorder is increased. To show that these
regions are indeed finite-size effects, we perform a scaling analysis, as
shown in Fig. 6.

Figure 6: Scaling of the bulk transmission across a phase transition, using
$J_{d}T/4=J_{s}T/4+2$ and $V_{o}T/4=0.2$. Each point is obtained by averaging
over 1000 disorder configurations, and the lines are guides to the eye.

As the system size is increased, we indeed observe the expected behavior. The
transmission is a decreasing function of $L$ for all values of $J_{s}$ except
at the phase transition point, even though its values remain relatively large
for the system sizes we can reach.

##  V A filter

Recent experiments in photonic crystals have been able to simulate Floquet
systems with a great amount of tunability Mukherjee  _et al._ (2017);
Maczewsky  _et al._ (2017). These experiments hence provide a way to realize
many of the phases that we have obtained in previous sections. The photonic
crystal consists of an array of one-dimensional optical waveguides, through
which light can propagate. Each waveguide plays the role of a site in the
tight-binding model, whereas the distance between waveguides controls the
hopping strength. The position along the longitudinal direction of the
waveguides is the effective time variable of the Floquet system: as light
propagates, the distance between adjacent waveguides changes, which emulates a
time-periodic hopping strength.

In principle, such photonic crystal platforms allow to freely control both the
onsite potential and the hopping of the Floquet system they simulate, by
adjusting the radius of each waveguide and the distance between them,
respectively. However, selectively accessing bulk states and edge states can
prove difficult in practice. This is because in conventional experiments, the
light injected in one or several waveguides will generally populate a mixture
of bulk and edge modes, even if these occur at different quasi-energies. Thus,
potential applications of photonic crystals, such as using topological edge
states for signal processing Chong (2013), may be hindered by the spurious
contribution of the propagating bulk states.

We show how to overcome these issues by designing a disorder-based filter. In
the previous sections we analyzed the properties of all the phases in the
presence of disorder, showing that the anomalous phase, phase $\mathfrak{E}$,
remains robust to both hopping disorder and onsite disorder. All of the the
bulk states are localized for large disorder strengths, even as the edges in
finite a sample remain extended. This property can be made use of to build
filters that allow transmission only of the edges, regardless of whether bulk
states are populated or not.

We show the action of the filter in a system of $30\times 30$ unit cells in
phase $\mathfrak{E}$ ($J_{s}T/4=1.5$, $J_{d}T/4=3$). Disorder is present in
the central region of the system, in the interval $n_{2}\in[12,20]$, where
$n_{2}$ labels the coordinate of the unit cells along the ${\bf a}_{2}$
direction. Hence, the disordered region divides the entire system into three
regions: the central, filter region, the bottom region, labeled A, in the
interval $n_{2}\in[1,11]$, and the top region, labeled B, in the interval
$n_{2}\in[21,30]$ (see Fig. 7).

To simulate the initialization procedure commonly used in photonic crystals,
we choose a wavefunction in the region A such that it is spread entirely on
the bottom most edge. This initial wavefunction is time-evolved by acting on
it with the full Floquet operator. It begins propagating through both the bulk
as well as the edge of the system, since it is a mixture of both bulk and edge
modes. However, as it reaches the disordered region, all of the bulk states
are stopped and only the edge states continue to propagate.

Figure 7:  Comparison of the probability density $|\psi|^{2}$ of the evolving
state after $t=0$, $50T$, $100T$, $150T$, and $200T$ (from left to right) with
and without filter for a system of size $L\times W=30\times 30$. The top
panels show the system without filter. The bottom panels show the system with
filter. The bottom left most panel shows the regions A and B. The region in
between the red lines in the bottom panels shows the disordered region which
acts as a filter. The supplemental material contains two movies of this time
evolution Bhargava  _et al._ (2021).

The disordered region filters out the signals carried by the bulk states and
allows signals to pass only along the edges. We quantify the effectiveness
with which the bulk states are filtered out by defining the filter coefficient
$\eta_{t}$ after an amount of time $t$ as

$\eta_{t}=\sum_{r\in\text{bulk of region B}}|\psi(r)|^{2}.$ (19)

Here, the bulk of region B denotes all the sites in this region, except for
the last two unit cells along the boundary of the sample in region B.

When we have an ideal filter, $\eta_{t}=0$, because the filter allows only the
edges to transmit. For the simulation that we have carried out with
$V_{h}T/4=0.2$ and $V_{o}T/4=1.0$, we find that $\eta_{t=200T}\approx 0.004$.
On the other hand, when no filter is present, the probability of finding the
initial state in the bulk of region B increases, leading to
$\eta_{t=200T}\approx 0.30$ (see Fig. 8). Thus, even for a relatively narrow
disordered region, the filter has an efficiency of roughly two orders of
magnitude.

Figure 8:  Filter coefficient $\eta_{t}$ as a function of $t$ in the anomalous
phase with parameters
$\left(J_{s}T/4,J_{d}T/4\right)\equiv\left(1.5,3\right)$. The red curve shows
the coefficient when the filter is present. In this case the disorder
strengths are $\left(V_{h}T/4,V_{o}T/4\right)=\left(0.2,1\right)$. The black
curve shows the coefficient when the filter is not present.

Note that one of the preconditions for an efficient filter is that the system
realizes an anomalous topological phase, since this guarantees both chiral
edge modes as well as a disorder-induced localization of all bulk states. If
the system would instead host a Floquet Chern insulating phase, such as phase
$\mathfrak{A}$, then some of the bulk modes would necessarily propagate
through the disordered region. This is because bands with nonzero Chern
numbers cannot be fully localized by disorder: there must exist at least one
delocalized bulk state which “carries the Chern number.”

##  VI Conclusions

In this work, we have considered a topological insulator belonging to the
symmetry class BDI. The time drive we consider breaks the time-reversal
symmetry in the system and hence only particle-hole symmetry is preserved. In
the clean limit, this system possesses in total of six different Floquet
topological phases. Each of these phases is characterized by four independent
topological invariants, two strong indices and two weak indices. Among these
six phases there is an anomalous phase as well as a weak phase that has
trivial strong indices.

After the classification of the clean system we introduced disorder. First, we
studied the hopping disorder which preserves the particle-hole symmetry. An
in-depth analysis shows that all the phases are robust to these disorders,
i.e. the invariants remain robust. It was shown that the anomalous phase can
withstand more disorder than other phases. At large hopping disorder, weak
anti-localization leads to a thermal metal-like phase.

Next, onsite disorder was studied. The onsite disorder breaks the particle-
hole symmetry in the problem. Hence, the weak phase becomes trivial. However,
all the other phases were shown to be robust to onsite disorder. Again, the
anomalous phase was shown to be the most robust phase.

The anomalous phase is the most robust phase, a fact that can be used in
making filters. In this anomalous Floquet Anderson insulating phase, the
presence of disorder fully localizes the bulk states, with edge states
remaining extended along the edges. We made use of this property and simulated
a filter that allowed signals to pass only through the edges. We also
calculated its filter coefficient to quantify the effectiveness of the filter.

Disorder in real systems is ubiquitous, resulting in unequal hopping strength
between neighboring sites on a lattice. Earlier experiments Rechtsman  _et
al._ (2013); Maczewsky  _et al._ (2017); Mukherjee  _et al._ (2017) with
optical waveguide arrays have opened up the possibility to mimic such Floquet
systems. One way of manipulating the hopping would be to vary the distance
between different waveguide arrays. The onsite potential can be changed by
varying the radius of the waveguides. Motivated by these experimental
platforms, our proposal for a disordered based filter is feasible to realize
in photonic systems.

###### Acknowledgements.

The authors would like to thank Ulrike Nitzsche for technical support. This
work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy through the Würzburg-
Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter –
_ct.qmat_ (EXC 2147, project-id 390858490), as well as through the DFG grant
FU 1253/1-1.

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##  Appendix A Scattering matrix invariants

We determine the strong and weak topological invariants from the two-terminal
scattering matrix defined in the main text. To this end, we apply twisted
periodic boundary conditions to the $L\times W$ unit cell system along the
${\bf a}_{1}$ direction, setting:

$\ket{1,n_{2}}=e^{i\phi}\ket{L,n_{2}},$ (20)

where, as before, $\ket{n_{1},n_{2}}$ labels a state positioned on the unit
cell with coordinates $(n_{1},n_{2})$, $n_{1,2}\in\mathbb{Z}$. The phase
$\phi$ determines the kind of boundary conditions present. For $\phi=0$ we
have the periodic boundary conditions and for $\phi=\pi$ we have anti-periodic
boundary conditions.

Since the Floquet operator is now a function of $\phi$, the scattering matrix
and its reflection block $\mathfrak{r}$ also depend on this twist angle. This
enables us to determine the strong topological index as Fulga and Maksymenko
(2016):

$\mathcal{W}_{\epsilon}=\frac{1}{2\pi
i}\int_{0}^{2\pi}d\phi\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{d}{d\phi}\log\det\mathfrak{r}(\phi,\epsilon).$ (21)

The weak indices, on the other hand, are given by the sign of the determinant.
These can be computed by

$\begin{split}\nu_{k=0,\epsilon}&=\text{sign}\det\mathfrak{r}(\phi=0,\epsilon),\\\
\nu_{k=\pi,\epsilon}&=\text{sign}\det\mathfrak{r}(\phi=\pi,\epsilon).\\\
\end{split}$ (22)

##  Appendix B Analytic Floquet operator

The time-evolution operators in each driving step of the Floquet operator Eq.
(7) can be determined analytically, even in the most general case in which
both the hoppings and the onsite potential are random. For the purpose of
illustration, we show how this is done in the case of a single plaquette (Fig.
9). The Hamiltonian in the time interval $0<t<T/4$ for random complex hoppings
and a random onsite terms is given by

$H_{1}=\begin{pmatrix}\mu_{A2}&0&0&0&J_{s}&0\\\
0&\mu_{A3}&0&0&0&J^{\prime}_{s}\\\ 0&0&\mu_{A4}&0&0&0\\\ 0&0&0&\mu_{B1}&0&0\\\
J^{*}_{s}&0&0&0&\mu_{B2}&0\\\ 0&J^{\prime*}_{s}&0&0&0&\mu_{B3}\end{pmatrix}.$
(23)

Figure 9: A single plaquette, showing the nonzero onsite terms and hopping
terms. In the time interval $0<t<T/4$, only the hoppings marked $J_{s}$ and
$J_{s}^{\prime}$ are active.

With an appropriate basis transformation, we can write the above Hamiltonian
in a block diagonal form as

$\tilde{H}_{1}=\begin{pmatrix}\mu_{B1}&0&0&0&0&0\\\ 0&\mu_{A2}&J_{s}&0&0&0\\\
0&J_{s}^{*}&\mu_{B2}&0&0&0\\\ 0&0&0&\mu_{A3}&J^{\prime}_{s}&0\\\
0&0&0&J_{s}^{\prime*}&\mu_{B3}&0\\\ 0&0&0&0&0&\mu_{A4}\end{pmatrix}.$ (24)

Each of the $2\times 2$ blocks have the form

$h=\begin{pmatrix}\mu_{A}&J\\\ J^{*}&\mu_{B}\end{pmatrix},$ (25)

where $J$ is the hopping strength and $\mu_{A}$ and $\mu_{B}$ are the onsite
terms. Since the exponential of a block diagonal matrix is a block diagonal
matrix consisting of the exponentials of the individual blocks, it is enough
to see what happens to the exponential of single block:

$\begin{split}&\exp\left[-i\begin{pmatrix}\mu_{A}&J\\\
J^{*}&\mu_{B}\end{pmatrix}\frac{T}{4}\right]=e^{-i\frac{\mu_{A}+\mu_{B}}{2}\times\frac{T}{4}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \times\\\
&\begin{pmatrix}\cos(|\textbf{p}|)-i\frac{\mu}{|\textbf{p}|}\sin(|\textbf{p}|)&-i\frac{|JT/4|}{|\textbf{p}|}\sin(|\textbf{p}|)e^{i\phi}\\\
-i\frac{|JT/4|}{|\textbf{p}|}\sin(|\textbf{p}|)e^{-i\phi}&\cos(|\textbf{p}|)+i\frac{\mu}{|\textbf{p}|}\sin(|\textbf{p}|)\end{pmatrix},\end{split}$
(26)

where $\mu=(\mu_{A}-\mu_{B})T/8$, $\phi=\arg(JT/4)$ and
$\textbf{p}=(|JT/4|\cos(\phi),-|JT/4|\sin(\phi),\mu/2)$. The time-evolution
operator $\tilde{F}_{1}$ is obtained by replacing the blocks with the above
exponentiated blocks. However, since this time-evolution operator is not in
the original basis, one can do a reverse basis transformation to obtain
$F_{1}$ in the original basis of Eq. (23).

Note that this matrix can be constructed explicitly for different hopping
strengths. A matrix such as the one above is constructed for each time slice
and multiplied to obtain the complete Floquet operator. Since this form of
constructing the Floquet operator does not require numerical exponentiation,
this method can speed up the process of finding the Floquet operator. Also,
this is an efficient method when disorder is present. The strengths of hopping
$J_{s}$ and the onsite terms $\mu_{A},\mu_{B}$ can be independently varied.
Hence, the Floquet matrix can be constructed efficiently.