# Topological aspects of periodically driven non-Hermitian Su-Schrieffer-
Heeger model

Vivek M. Vyas  Dibyendu Roy  Raman Research Institute, Bangalore 560080, India

###### Abstract

A non-Hermitian generalization of the Su-Schrieffer-Heeger model driven by a
periodic external potential is investigated, and its topological features are
explored. We find that the bi-orthonormal geometric phase acts as a
topological index, well capturing the presence/absence of the zero modes. The
model is observed to display trivial and non-trivial insulator phases and a
topologically non-trivial Möbius metallic phase. The driving field amplitude
is shown to be a control parameter causing topological phase transitions in
this model. While the system displays zero modes in the metallic phase apart
from the non-trivial insulator phase, the metallic zero modes are not robust,
as the ones found in the insulating phase. We further find that zero modes’
energy converges slowly to zero as a function of the number of dimers in the
Möbius metallic phase compared to the non-trivial insulating phase.

##  I Introduction

In the last few decades, the existence and consequences of topological quantum
numbers in condensed matter systems have been a subject of great interest
Thouless (1998). Unlike symmetry-based Noether conservation laws, the nature
and origin of topological objects and corresponding invariants are
fundamentally different. A macroscopic condensed matter system’s physical
state possessing a non-trivial topological invariant is, by definition, found
to be immune to disorder and external perturbations Bernevig and Hughes
(2013). There has been a lot of activity to exploit this robustness for
diverse quantum applications like lasing and computation Ozawa  _et al._
(2019).

It is a well-known fact that the time evolution of a closed quantum system is
described by a Hermitian Hamiltonian, which gives rise to a unitary time
evolution. Nevertheless, most quantum systems in practice interact with the
external environment, and hence their physics can not always be captured by a
closed system description Breuer and Petruccione (2002). However, solving a
macroscopic quantum system’s dynamics while employing a general open system
approach is usually daunting. In this scenario, a prudent compromise is to use
an effective non-Hermitian Hamiltonian to describe the quantum system at hand,
wherein the non-Hermiticity capture, in essence, the effects like dissipation
arising due to the interaction with the environment Carollo  _et al._ (2003).
In any case, in reality, most quantum condensed matter systems are coupled to
the measuring apparatus in some way, which in turn can give rise to non-
Hermiticity in the system, as was shown in the case of the Kitaev chain
recently Kawabata  _et al._ (2019, 2018); Bondyopadhaya and Roy .

While the topological aspects of closed quantum systems have been the main
focus for a long time Thouless (1998); Bernevig and Hughes (2013), of late,
the study of topological properties of non-Hermitian Hamiltonians has gained
significant attention Chiu  _et al._ (2016); Ozawa  _et al._ (2019); Gong  _et
al._ (2018). It is now understood that the topological structure of the non-
Hermitian system is much richer and diverse than their Hermitian counterparts
Berry (2004); Zhou and Lee (2019); Xiong (2018); Leykam  _et al._ (2017). In
this context, the concepts of bulk-boundary correspondence, topological
invariants, and geometric phases have been a subject of scrutiny for a while
Leykam  _et al._ (2017); Yao and Wang (2018); Kunst  _et al._ (2018); Lieu
(2018); Kawabata  _et al._ (2019); Jin and Song (2019); Ghatak and Das (2019);
Borgnia  _et al._ (2020); Xiao  _et al._ (2020); Helbig  _et al._ (2020).

In condensed matter systems, the bedrock underneath the topological aspects is
a discrete spatial translational symmetry in the system, giving rise to the
Brillouin zone in the $k$-space Bernevig and Hughes (2013); Thouless (1998).
The topological invariant like Chern number and the geometric phase, e.g., the
Pancharatnam-Zak phase Vyas  _et al._ (2019), owe their genesis to this
framework. Interestingly, there is an ingenious way of generalizing the same
mathematical structure in the temporal domain by driving the system externally
by a periodic potential Gómez-León and Platero (2013); Goldman and Dalibard
(2014). Such driving can effectively add another dimension to the spatially
periodic lattice; thus, a driven lattice system’s topological properties are
fundamentally different from the undriven case Kitagawa  _et al._ (2010);
Lindner  _et al._ (2011). This feature is seen in several studies on these
systems, and various techniques to correctly capture the topological aspects
of such systems, based on the Floquet analysis, have been developed Roy and
Harper (2017); Yao  _et al._ (2017); Zhou and Gong (2018a); Zhou (2019); Zhou
and Pan (2019); Wu and An (2020); Zhou (2020). With the advent of precision
cold atoms and quantum materials experiments, topological aspects of such a
driven system have been explored in different experimental setups Jiang  _et
al._ (2011); Fläschner  _et al._ (2016); Sato  _et al._ (2019); McIver  _et
al._ (2020).

As noted earlier, the non-Hermitian systems display a richer topological
structure than the Hermitian ones; one naturally wonders what happens if such
a system is periodically driven. What is the proper topological invariant
capturing the topological phases of such a system, and can these phases of the
system all be probed by tuning the driving potential; are some questions that
immediately arise. In this work, we study the topological properties of a non-
Hermitian extension of the celebrated Su-Schrieffer-Heeger (SSH) model driven
by an external AC electromagnetic field and answer the above queries.

We begin by investigating a non-Hermitian version of the SSH model, possessing
chiral invariance. Constructing a proper gauge-invariant geometric phase viz.
bi-orthonormal generalization of Pancharatnam-Zak phase, which is acquired by
a Bloch state as it traverses a circuit in the Brillouin zone, we find that
the system exhibits three distinct topological phases (a) trivial insulator,
(b) non-trivial insulator, and (c) Möbius metallic phase. In the two insulator
phases, the model displays a band gap in the energy spectrum’s real part. The
geometric phase is $0$ and $\pi$ for the trivial and non-trivial insulator
phases, respectively. When the model is considered with an open boundary
condition (OBC), it is found that the non-trivial insulating phase displays
doubly degenerate zero modes as in the Hermitian case. In the Möbius metallic
phase, we observe that the two bands of the energy spectrum’s real part are
merged into one, giving rise to a metallic behavior; at the same time, the
state space displays Möbius strip topology as explained later. Owing to this
non-trivial topology, a proper definition of the geometric phase requires the
Bloch state to traverse two circuits in the Brillouin zone, and the bi-
orthonormal geometric phase hence acquired, is found to be $\pi$. While the
system in this metallic phase does display the existence of doubly degenerate
zero modes, as one would expect, they, in turn, are found to be embedded in
the band continuum and hence are not robust. The lack of robust zero modes in
the Möbius metallic phase indicates that the topological notion of bulk-
boundary correspondence in this model, as found in its Hermitian counterpart,
is strictly speaking absent.

We further explore the scaling of zero-mode energy in the open chains as a
function of the number of dimers $N$ in the non-trivial insulating and Möbius
metallic phase. We find that the two phases display different scaling behavior
as a function of $N$ when we correctly implement the OBC using a large $N$
limit. The energy of zero modes converges slowly to zero with $N$ in the
Möbius metallic phase compared to the non-trivial insulating phase.

Figure 1:  A schematic diagram of the driven non-Hermitian SSH model. The red
and green circles denote $A$ and $B$ sublattice sites, respectively. The
lattice constant is $a$, and two sites in an unit cell are separated by
distance $b$. The intracell hopping amplitude is $v$, whereas the intercell
amplitudes are $we^{i\textbf{A}(t)}$ and $we^{\theta-i\textbf{A}(t)}$
respectively for to and fro tunnelling in the presence of a time-periodic
vector potential $\textbf{A}(t)$ and a non-Hermiticity measure $\theta$.

Subsequently, we study the driven non-Hermitian SSH model, minimally coupled
to an external time-periodic vector potential (see Fig. 1). This model
possesses two discrete translation symmetries respectively in space and time
and hence allows us to solve for the Floquet-Bloch states, the spatiotemporal
generalization of Bloch states. Being mainly interested in diabatic dynamics,
we first consider the regime wherein the driving frequency is much larger than
the model’s hopping parameters. Extending the notion of the bi-orthonormal
Pancharatnam-Zak geometric phase to this driven model, we find that this
geometric phase indeed captures the quintessential topological features in
this case as well. Upon solving the Floquet-Bloch eigenvalue problem in this
large-frequency regime, we find that the system, akin to the undriven model,
admits three distinct topological phases: the trivial insulator, non-trivial
insulator, and Möbius metallic phase. We further support these phases’
appearances by showing the presence/absence of zero-quasienergy modes in the
real part of the quasienergy spectra of the driven model with an OBC using a
large $N$ limit. We discover that the system’s topological phase can be
entirely altered by the amplitude of the driving potential, which plays as a
control parameter (see Fig. 4). In the opposite limit, wherein the driving
frequency is much smaller than hopping parameters, the system’s dynamics
essentially is adiabatic. Then topological features of the system do not get
affected by external driving. Finally, we present numerical results for the
driven non-Hermitian SSH model’s quasienergy spectra at an intermediate
frequency. These spectra show that both the non-trivial and the Möbius
metallic phases can reappear when the driving’s amplitude is tuned.

##  II Non-Hermitian SSH model

We here consider a non-Hermitian generalization of the SSH model, a one-
dimensional nearest-neighbor tight-binding model comprising two sublattice
sites within the unit cell (Fig. 1 with $\textbf{A}(t)=0$). In its generality,
this model is defined in terms of four complex numbers $v_{1,2}$ and
$w_{1,2}$, which parameterize the system Hamiltonian Lieu (2018):

$\displaystyle H=$ $\displaystyle\>\sum_{n=1}^{N}\left(v_{1}|n,B\rangle\langle
n,A|+v_{2}|n,A\rangle\langle n,B|\right)$
$\displaystyle+\sum_{n=1}^{N-1}\left(w_{1}|n+1,A\rangle\langle
n,B|+w_{2}|n,B\rangle\langle n+1,A|\right)$
$\displaystyle+l\left(w_{1}|1,A\rangle\langle N,B|+w_{2}|N,B\rangle\langle
1,A|\right).$ (1)

The state $|n,A(B)\rangle$ represents the localized state in the $n^{th}$ unit
cell with sublattice index $A(B)$. Here, the lattice constant is $a$, and the
distance between the two sublattice sites $A$ and $B$ in the unit cell is
given by $b$. The parameter $l$ in the above Hamiltonian is employed to ensure
the validity of appropriate boundary conditions. For example, the choice $l=1$
represents the periodic boundary condition (PBC) wherein the unit cell
translation operator $T(a)$ commutes with $H$: $[T(a),H]=0$. Evidently, for
any other choice of $l$, such discrete translation symmetry is absent. In the
study of topological aspects, when $l=0$ is of particular interest, it
represents the OBC Bernevig and Hughes (2013). For any value of parameters
$v_{1,2}$, $w_{1,2}$ and $l$, this system possesses chiral symmetry, which
means that the Hamiltonian $H$ anticommutes with the operator $\Sigma$,
$\\{\Sigma,H\\}=0$, where:

$\displaystyle\Sigma=\sum_{n=1}^{N}\left(|n,A\rangle\langle
n,A|-|n,B\rangle\langle n,B|\right).$ (2)

This anti-commutativity ensures that the spectrum of $H$, which is in general
complex, is such that for every eigenstate $|\psi\rangle$ with energy $E$,
there exists another eigenstate $\Sigma|\psi\rangle$ with energy $-E$. It
further indicates that a zero energy eigenmode is doubly degenerate.

To understand the impact of non-Hermiticity on this model’s topological
aspects, we consider a special case in this work, wherein $v_{1}=v_{2}=v$,
$w_{1}=w$, $w_{2}=we^{\theta}$, for all real values of $v$, $w$, and $\theta$
with $v,w,\theta\geq 0$. In this parametrization, the value of $\theta$ can be
understood to measure the divergence of the system from the Hermitian
$\theta=0$ case.

###  II.1 k-space analysis

When $l=1$, the existence of discrete translation symmetry allows us to
convert to $k$-space, and work with states $|k_{j},A(B)\rangle$ (with
$k_{j}=2\pi j/Na$ and $j=1,2,\dots,N$), which are the Fourier transform of
localized states $|n,A(B)\rangle$. The Hamiltonian (1) in the $k$-space reads:

$\displaystyle
H=\sum_{k_{j}}\begin{pmatrix}|k_{j},A\rangle,&|k_{j},B\rangle\end{pmatrix}[H(k_{j})]\begin{pmatrix}\langle
k_{j},A|\\\ \langle k_{j},B|\end{pmatrix},$ (3)

where the non-Hermitian matrix $H(k_{j})$ reads:

$\displaystyle H(k_{j})=\begin{pmatrix}0&we^{ik_{j}}+v\\\ we^{\theta-
ik_{j}}+v&0\end{pmatrix}.$ (4)

We have assumed that the distance between the two sublattice sites is
vanishing ($b=0$) and lattice constant $a=1$ for simplicity. We further set
Planck constant $\hbar=1$. Notice that this Hamiltonian displays $k$-space
periodicity since: $H(k=0)=H(k=2\pi)$.

The complex energy spectrum arising from this $k$-space Hamiltonian is given
by $E_{\pm}(k)=\pm E(k)$, where:

$\displaystyle E(k)=\sqrt{(we^{ik}+v)(we^{\theta-ik}+v)},$ (5)

which depict two complex energy bands $\pm E(k)$ in agreement with the chiral
symmetry of the system. The corresponding left and right eigenvectors
respectively are given by:

$\displaystyle\Phi_{\pm}(k)$
$\displaystyle=\frac{e^{i\lambda(k)}}{\sqrt{2}}\begin{bmatrix}1,\pm\frac{E(k)}{(we^{\theta-
ik}+v)}\end{bmatrix},$ (6) $\displaystyle\Psi_{\pm}(k)$
$\displaystyle=\frac{e^{-i\lambda(k)}}{\sqrt{2}}\begin{bmatrix}1\\\
\pm\frac{E(k)}{(we^{ik}+v)}\end{bmatrix}.$ (7)

Here, ${\lambda(k)}$ stands for some arbitrary function of $k$, not
necessarily respecting the periodicity in $k$. These two eigenvectors are
arranged to respect the bi-orthonormality conditions:
$\Phi_{\pm}(k)\cdot\Psi_{\pm}(k)=1$ and $\Phi_{\mp}(k)\cdot\Psi_{\pm}(k)=0$.
This property of bi-orthonormality is a natural generalization of the
orthonormality property displayed by eigenvectors of a Hermitian matrix Lieu
(2018). By employing this different orthonormalisation, we depart from the
usual Dirac’s inner product and normalization definition. The left and right
Bloch eigenstates of $H$ are related to these eigenvectors respectively as:
$\langle\Phi_{\pm}(k)|=\Phi_{\pm,1}(k)\langle k,A|+\Phi_{\pm,2}(k)\langle
k,B|$ and
$|\Psi_{\pm}(k)\rangle=\Psi_{\pm,1}(k)|k,A\rangle+\Psi_{\pm,2}(k)|k,B\rangle$.
Being the eigenstates of a non-Hermitian operator, these states do not respect
the orthonormality, which follows from the Dirac inner product definition:
$\langle\Psi_{\pm}(k)|\Psi_{\mp}(k)\rangle\neq 0$ and so on. As a result we
choose to work with the bi-orthonormality conditions:
$\langle\Phi_{\pm}(k)|\Psi_{\mp}(k)\rangle=0$, and
$\langle\Phi_{\pm}(k)|\Psi_{\pm}(k)\rangle=1$.

It can be checked that these Bloch eigenstates indeed obey the Bloch condition
of periodicity modulo an overall phase:
$T(a)|\Psi_{\pm}(k)\rangle=e^{ik}|\Psi_{\pm}(k)\rangle$. The non-trivial
content of the Bloch states is contained in the cell periodic Bloch states
$|u_{\pm}(k)\rangle(\equiv T(a)|u_{\pm}(k)\rangle)$, that can be readily found
from the Bloch states by applying the momentum translation operator
$\mathcal{T}(-k)$, which in this case reads:

$\displaystyle|u_{\pm}(k)\rangle$
$\displaystyle=\Psi_{\pm,1}(k)|0,A\rangle+\Psi_{\pm,2}(k)|0,B\rangle,$ (8)
$\displaystyle\langle\tilde{u}_{\pm}(k)|$
$\displaystyle=\Phi_{\pm,1}(k)\langle 0,A|+\Phi_{\pm,2}(k)\langle 0,B|.$ (9)

Evidently the above right and left cell periodic Bloch states solve the
eigenvalue problem for the Hamiltonian $H_{k}=\mathcal{T}(-k)H\mathcal{T}(k)$,
so that $H_{k}|u_{\pm}(k)\rangle=\pm E(k)|u_{\pm}(k)\rangle$, and
$\langle\tilde{u}_{\pm}(k)|H_{k}=\pm E(k)\langle\tilde{u}_{\pm}(k)|$.

The fact that the matrix $H(k)$ is periodic in $k$-space may lead one to
naively believe that the energy spectrum and the average
$\langle\tilde{u}_{\pm}(k)|\mathcal{O}|u_{\pm}(k)\rangle$ of some generic
operator $\mathcal{O}$ respecting PBC, is always a periodic function of $k$,
and hence return to its initial value after $2\pi$ circuit in the $k$-space.
However, this is generally not true and happens only in the region $\kappa>1$
or $\kappa<e^{-\theta}$, where $\kappa=w/v$. In this region, the energy $E(k)$
is an analytic function of $k$, and the band structure of the system is
generically given by Fig. (2(a)). While the energy is in general complex, we
can infer that the non-Hermitian system behaves as a band insulator owing to a
gap in the energy spectrum’s real part. Further, the eigenvectors
$|\Psi_{\pm}(k+2\pi)|^{2}=|\Psi_{\pm}(k)|^{2}$ and
$|\Phi_{\pm}(k+2\pi)|^{2}=|\Phi_{\pm}(k)|^{2}$ in this region. Therefore, the
band structure as well as the state space of the system have a cylinder
topology when $\kappa>1$ or $\kappa<e^{-\theta}$.

From the pioneering work of Pancharatnam Pancharatnam (1956); Mukunda and
Simon (1993); Vyas  _et al._ (2019), we now understand that the notion of
geometric phase can be purely defined kinematically in terms of a cyclic
overlap of states. In the case of the lattice model, the geometric phase
referred to as Pancharatnam-Zak phase $\nu$, can be defined corresponding to a
given band in terms of (Dirac normalized) cell periodic Bloch states
$|u_{j}\rangle\equiv|u(k_{j})\rangle$:

$\displaystyle\Delta=\langle u_{0}|u_{N}\rangle\langle
u_{N}|u_{N-1}\rangle\cdots\langle u_{2}|u_{1}\rangle\langle
u_{1}|u_{0}\rangle,$

with $\nu=\text{Arg}\>\Delta$. It is evident that $\nu$ is phase acquired by
the Bloch state as it completes a circuit in the Brillouin zone.

While the above definition of the geometric phase properly holds in the case
of Hermitian models, it is not satisfactory for the present non-Hermitian
case. The main reason for this is that the Bloch states living in different
bands are not orthogonal regarding Dirac’s definition of the inner product.

Given this scenario, we here generalize the above definition of Pancharatnam-
Zak phase to bi-orthonormal setup, which is given by $\gamma$ where
$\gamma=\text{Arg}\>D$ with:

$\displaystyle
D=\langle\tilde{u}_{0}|u_{N}\rangle\langle\tilde{u}_{N}|u_{N-1}\rangle\cdots\langle\tilde{u}_{2}|u_{1}\rangle\langle\tilde{u}_{1}|u_{0}\rangle.$

In the continuum limit, this phase for both the bands takes the form:

$\displaystyle\gamma_{\pm}(2\pi)$
$\displaystyle=\text{Arg}\>(\langle\tilde{u}_{\pm}(0)|u_{\pm}(2\pi)\rangle)$
$\displaystyle+i\int_{0}^{2\pi}dk\>\langle\tilde{u}_{\pm}(k)|\partial_{k}|u_{\pm}(k)\rangle.$
(10)

By a careful choice of function $\lambda(k)$ (which amounts to fixing a
gauge), the argument term can be made to vanish, in which case, this
Pancharatnam-Zak phase becomes the so called complex geometric phase studied
in the literature Garrison and Wright (1988); Lieu (2018). It must be
mentioned that unlike the complex geometric phase, this geometric phase is a
proper geometric construct, is gauge invariant (which implies its
insensitivity to any choice of $\lambda(k)$) and hence immune to any gauge
redefinitions of the type: $|u_{\pm}(k)\rangle\rightarrow
e^{i\Lambda(k)}|u_{\pm}(k)\rangle$,
$\langle\tilde{u}_{\pm}(k)|=e^{-i\Lambda(k)}\langle\tilde{u}_{\pm}(k)|$, for
some function $\Lambda(k)$.

The above gauge-invariant definition of the bi-orthonormal geometric phase
allows us to extend the results obtained in Ref. Lieu (2018) for the complex
geometric phase. It is known that owing to the chiral symmetry, the complex
geometric phase is real and is quantized in the units of $\pi$. On evaluating
(II.1) using (8) and (9), we immediately find that:

$\displaystyle\gamma_{\pm}(2\pi)$
$\displaystyle=\pi,\quad\text{when}\quad\kappa>1,$
$\displaystyle=0,\quad\text{when}\quad\kappa<e^{-\theta}.$

(a) For $v=1$, $\theta=0.5$, and $w=-0.1+e^{-\theta}$. Topologically the band
structure is like a cylinder with the curves $\pm E(k)$ defining the edges.

(b) For $v=1$, $\theta=0.5$, and $w=0.1+e^{-\theta}$. Topologically the band
structure is like a Möbius strip with the curve $E(k)$ defining the edge.

Figure 2: Plot of real and imaginary parts of energy eigenvalues $E_{\pm}(k)$
as a function of $k$. Here, the red continuous and dot-dash curves
respectively denote the real and imaginary part of the upper band’s energy;
whereas the blue dotted and dashed curves respectively indicate the real and
imaginary part of the lower band’s energy.

In the parameter region $e^{-\theta}<\kappa<1$, we find that the energy
eigenvalue $E(k)$ as a function of $k$ is not analytic/single valued, but
rather becomes _double valued_ , that has been studied since a while in
various systems Berry (2004); Xiong (2018); Leykam  _et al._ (2017). As a
result we observe that the two limits do not agree: $E(k\rightarrow\pi)\neq
E(k\rightarrow-\pi)$. We readily see that in such a case:
$E_{-}(k+2\pi)=E_{+}(k)$, and $E_{-}(k+4\pi)=E_{-}(k)$. This fact is neatly
brought out in the Fig. (2(b)). If we were to follow the trajectory of the
complex energy in this figure starting from the lower band, then it is evident
that after one circuit in $k$-space, we reach the upper band. It is only after
yet another circuit that we return to the initial starting point. This is a
clear evidence of the double valued nature of the energy eigenvalues. Since
the energy eigenvalues change from lower to upper band after completing a
circuit in $k$-space, it also follows that the eigenvectors must follow the
same route: $|\Psi_{\pm}(k+2\pi)|^{2}=|\Psi_{\mp}(k)|^{2}$ and
$|\Psi_{\pm}(k+4\pi)|^{2}=|\Psi_{\pm}(k)|^{2}$. These results indicate that
the state space of right eigenvectors is like a Möbius strip, with the edge of
the Möbius strip being the states $\Psi_{\pm}(k)$.

Physically, the two bands are not distinct or disjoint, but in fact are
connected/merged into one another in a sense that the system can be
continuously changed to go from one band to another. Thus, the system in this
regime behaves as a metal, and the band indices $\pm 1$ are redundant since
the bands are merged together. Owing to the Möbius strip nature, we need to
extend the notion of circuit in $k$-space as a loop from $k=0$ to $k=4\pi$. In
fact after one $2\pi$ circuit, we have $\Psi_{\mp}(2\pi)=\Psi_{\pm}(0)$, so
that $\Phi_{\pm}(0)\cdot\Psi_{\pm}(2\pi)=0$. As a result, we can not define
geometric phase using relation (II.1) since the initial and final states are
orthogonal. Instead, we are required to consider $4\pi$ circuit geometric
phase:

$\displaystyle\gamma(4\pi)$
$\displaystyle=\text{Arg}\>(\langle\tilde{u}_{\pm}(0)|u_{\pm}(4\pi)\rangle)$
$\displaystyle+i\int_{0}^{4\pi}dk\>\langle\tilde{u}_{\pm}(k)|\partial_{k}|u_{\pm}(k)\rangle.$
(11)

We find from explicit calculation $\gamma(4\pi)=\pi$ through out this region
of $e^{-\theta}<\kappa<1$. This result indicates that although the system is a
gapless metal but it is in a topologically non-trivial phase in this parameter
region.

###  II.2 Real space analysis

It is a well known fact that in the presence of PBC, when the Hermitian SSH
model displays non-trivial value of geometric phase, the same in the presence
of OBC displays doubly degenerate zero-energy modes protected from the band
gap. Such a result is often referred to as the bulk-boundary correspondence,
and in the non-Hermitian lattice systems this concept has off late generated a
lot of interest Kunst  _et al._ (2018); Jin and Song (2019); Borgnia  _et al._
(2020); Leykam  _et al._ (2017).

In order to investigate whether such a correspondence exist in the model at
hand, let us consider the real space Hamiltonian (1) without assuming anything
about the form of $l$. The right eigenvalue problem for this Hamiltonian is
expressible as:

$\displaystyle H|\psi_{\zeta}\rangle=E_{\zeta}|\psi_{\zeta}\rangle,$ (12)

where the eigenvalues and eigenvectors are indexed by quantum number $\zeta$.
This eigenvalue problem can also be alternatively viewed as a matrix
eigenvalue problem:

$\displaystyle\sum_{s^{\prime}=A,B;n^{\prime}=1}^{N}\langle
n,s|H|n^{\prime},s^{\prime}\rangle\langle
n^{\prime},s^{\prime}|\psi_{\zeta}\rangle=E_{\zeta}\langle
n,s|\psi_{\zeta}\rangle,$ (13)

where we use the completeness property of localized states $|n,s\rangle$, with
site index $n=1,2,\cdots,N$, and sublattice index $s=A,B$. Alternatively this
can be written as:

$\displaystyle\begin{bmatrix}0&v&0&0&\cdots&0&lw\\\
v&0&we^{\theta}&0&\cdots&0&0\\\ 0&w&0&v&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
lwe^{\theta}&0&\cdots&\cdots&0&v&0\end{bmatrix}\begin{bmatrix}\langle
1,A|\psi_{\zeta}\rangle\\\ \langle 1,B|\psi_{\zeta}\rangle\\\ \langle
2,A|\psi_{\zeta}\rangle\\\ \vdots\\\ \langle
N,B|\psi_{\zeta}\rangle\end{bmatrix}=E_{\zeta}\begin{bmatrix}\langle
1,A|\psi_{\zeta}\rangle\\\ \langle 1,B|\psi_{\zeta}\rangle\\\ \langle
2,A|\psi_{\zeta}\rangle\\\ \vdots\\\ \langle
N,B|\psi_{\zeta}\rangle\end{bmatrix}.$ (14)

Evidently the trace of $H$ is $0$, which can be understood to arise from the
chiral symmetry of the model. The determinant of $H$ can be readily found from
the above matrix, and it reads:

$\displaystyle\text{Det}\;H=(-1)^{N}\left({v}^{{N}}+(-1)^{N-1}le^{{N\theta}}{w}^{{N}}\right)$
$\displaystyle\times\left({v}^{{N}}+(-1)^{N-1}l{w}^{{N}}\right).$ (15)

This expression, which holds for any value of $l$ and $N$, is obtained from
mathematical induction and can be straight away checked using symbolic
computation.

Let us consider the Hermitian case ($\theta=0$) for a moment. It is well known
fact that the determinant of a matrix is a product of its eigenvalues. In the
PBC case, we analytically know that the system admits two eigenvalue bands. In
the OBC case with $l=0$, it is known that the system admits apart from the two
bands, two midgap states with energy close to zero. In order to pull out the
behaviour of these energy levels near zero energy, we study the ratio of two
determinants respectively, of the system with OBC made up of $N+1$ dimers, and
PBC with $N$ dimers:

$\displaystyle
r=\frac{\text{Det}H\lvert_{OBC,N+1}}{\text{Det}H\lvert_{PBC,N}}.$ (16)

The ratio $r$ is thus:

$\displaystyle r=E_{+,0}E_{-,0},$ (17)

where $E_{\pm,0}$ are the two energy levels closest to zero. It is evident
that when the system does not admit any zero-energy modes, this ratio will be
equal to $-1$; and if zero-energy modes exists then the ratio must tend to $0$
as $N\rightarrow\infty$. It is clear that when $\kappa<1$, the power
$\kappa^{N}$ converges to $0$ as $N\rightarrow\infty$. Thus the ratio
$r\rightarrow-1$ when $N\rightarrow\infty$ while $\kappa<1$. On the other hand
when $\kappa>1$ the object $\kappa^{N}$ becomes larger and larger as $N$
grows, thus the ratio can now be written as an exponential
$r\simeq-\kappa^{-2N}$. This is in agreement with the earlier known results
Bernevig and Hughes (2013).

Extending the above treatment to non-Hermitian system ($\theta\neq 0$),
considering $N$ to be odd and $v=1$ for specificity but without losing
generality, we immediately see from (II.2) and (16) that the ratio reads:

$\displaystyle r=-\frac{1}{(1+\kappa^{N})(1+(\kappa e^{\theta})^{N})}.$ (18)

As observed earlier, when $\kappa>1$, this ratio displays exponential drop
$r\simeq-e^{-N\theta}\kappa^{-2N}$ in the large $N$ limit. When
$e^{-\theta}<\kappa<1$, we find that the ratio reads:

$\displaystyle r\simeq-e^{-N\theta}\kappa^{-N}.$ (19)

The ratio in this case also shows an exponential drop albeit slower than the
insulator case viz. $\simeq-e^{-N\theta}\kappa^{-2N}$. Interestingly we see
that while the zero modes exist in both non-trivial insulating and metallic
phases, their scaling as a function of system size has a characteristic
difference: the drop is sharper in the insulating phase in comparison to the
metallic phase.

Figure 3: Plots showing an agreement between analytically obtained energy
spectrum with PBC (top) and numerically obtained energy spectrum for $N=300$
with OBC (bottom). The real part of energy is plotted as a function of
$\kappa$ for $\theta=0.5$. Apart from the zero modes, the two spectra closely
match.

We notice that the scaling of zero-mode energies $r$ as a function of $N$ is
explicitly dependent upon the non-Hermiticity parameter $\theta$. As a result,
the drop in the zero-mode energies is not uniform, and systems with different
$\theta$ values show different scaling behavior. This dependence of zero-mode
energy scaling, being an explicit function of $\theta$, is an unpleasant
feature. We wonder if we can get a better understanding of this scaling
phenomenon. Often in several condensed matter systems, the technique of large
$N$ expansion is employed to understand better the underlying physics Imbo and
Sukhatme (1985); Jevicki and Sakita (1981). In particular, this technique has
found immense applications in the study of quantum phase transitions Sachdev
(2011) and is known to be a non-perturbative technique.

Now we implement the OBC in our problem by employing the large $N$ technique,
and we consider $lN=\text{constant}~{}(=1)$ in which case the determinant at
hand reads:

$\displaystyle\text{Det}\;H=(-1)^{N}\left({v}^{N}+(-1)^{N-1}\frac{1}{N}e^{{N\theta}}{w}^{{N}}\right)$
$\displaystyle\left({v}^{{N}}+(-1)^{N-1}\frac{1}{N}{w}^{{N}}\right),$ (20)

and the ratio now reads (for non-zero but small $\theta$ and $w$ around 1):

$\displaystyle r\simeq-\frac{(1-\frac{1}{N}\kappa^{N})(1-\frac{1}{N}(\kappa
e^{\theta})^{N})}{(1+\kappa^{N})(1+(\kappa e^{\theta})^{N})}.$ (21)

When $\kappa>1$, which corresponds to the non-trivial insulator phase, both
the exponentials $\kappa^{N}$ and $(\kappa e^{\theta})^{N}$ grow indefinitely
as $N\rightarrow\infty$, and so the ratio can be approximated as:

$\displaystyle r\simeq-\frac{(-\frac{1}{N}\kappa^{N})(-\frac{1}{N}(\kappa
e^{\theta})^{N})}{(\kappa^{N})((\kappa e^{\theta})^{N})}=-\frac{1}{N^{2}}.$
(22)

This shows that the zero-mode energies are real and they fall as
$\frac{1}{N}$, that is, $E_{+,0}=\frac{1}{N}$ and $E_{-,0}=-\frac{1}{N}$. We
note that unlike the usual OBC condition $l=0$, here the fall of zero-mode
energies is primarily independent of the non-Hermiticity parameter $\theta$ or
$\kappa$.

When the system is in the metallic phase, which means $e^{-\theta}<\kappa<1$,
we find that the exponential $\kappa^{N}$ converges to $0$ as
$N\rightarrow\infty$, while the exponential $(\kappa e^{\theta})^{N}$ grows
indefinitely. As a result, the ratio can be written as:

$\displaystyle r\simeq-\frac{(1-\frac{1}{N}(\kappa
e^{\theta})^{N})}{(1+(\kappa e^{\theta})^{N})}=\frac{1}{N}.$ (23)

The above implies that the zero-mode energies are imaginary, and they fall as
$\frac{1}{\sqrt{N}}$, that is, $E_{+,0}=\frac{i}{\sqrt{N}}$ and
$E_{-,0}=-\frac{i}{\sqrt{N}}$. Thus, the zero-mode energies in the metallic
phase converge to zero slower as $\frac{1}{\sqrt{N}}$ rather than the
$\frac{1}{N}$ in the insulator phase, but the once again the scaling property
is independent of the Hamiltonian parameters. The imaginary energy of zero
modes as well as their slower convergence to zero with system size in the
metallic phase further indicate the non-robustness of these modes there.

The different scaling of zero-mode energy as a function of $N$ in the two
topological phases displays a kind of universality since the scaling property
is sensitive only to the topology and not the actual values of the Hamiltonian
parameters. Apparently this scaling behaviour is not specific to the choice of
Hamiltonian parameters as $v_{1}=v_{2}=v$, $w_{1}=w$, $w_{2}=we^{\theta}$, but
rather is a feature displayed by general non-Hermitian chiral Hamiltonian (1)
for any complex choices of $v_{1,2}$ and $w_{1,2}$, as is shown in the
Appendix.

The above discussion shows that the implementation of OBC employing a large
$N$ limit captures the topological features of the model, the zero-energy
modes, which begin right from the metallic Möbius phase itself and extend to
the non-trivial insulator phase as well: $\kappa>e^{-\theta}$. Nevertheless,
the physical significance of zero modes, which are embedded in the gapless
continuum of energy levels in the metallic phase, is diminished in the Möbius
region. The physical reason is that there is no gap to distinguish these modes
from the other infinitesimally close by levels. As a result, these modes are
not robust compared to the zero modes in the non-trivial insulator region. The
plots in Fig. (3) of the real part of the spectrum for both PBC and OBC show a
close agreement, modulo the zero modes, validating the above-presented
analysis. The above treatment also shows that the topological properties of
the system with OBC manifested as zero-energy modes are well captured by the
non-trivial value of the bi-orthonormal geometric phase in both insulator and
metallic phases. The topological phase structure of this model can be
summarized as:

  1. A.

Trivial insulator phase: $\kappa<e^{-\theta}$, wherein $\gamma(2\pi)=0$. The
bands have cylinder topology.

  2. B.

Non-trivial insulator phase: $\kappa>1$, wherein $\gamma(2\pi)=\pi$. The bands
have cylinder topology, and robust zero modes exist.

  3. C.

Möbius metallic phase: $e^{-\theta}<\kappa<1$, $\gamma(2\pi)$ is not defined,
but $\gamma(4\pi)=\pi$. The bands are merged into Möbius strip topology, and
zero modes exist but are not robust.

The above analysis further indicates that there is an absence of bulk-boundary
correspondence in the non-Hermitian SSH model, when we consider the Möbius
metallic phase as topologically non-trivial, but the zero modes there are not
robust.

##  III Driven Non-Hermitian SSH model

We now discuss a scenario wherein the non-Hermitian SSH model, defined by (1),
is driven by an external electromagnetic field represented by a vector
potential $\textbf{A}(t)$. The minimal Hamiltonian that depicts such a driven
model can be written as Gómez-León and Platero (2013):

$\displaystyle H_{D}(t)$
$\displaystyle=\>\sum_{n=1}^{N}\left(v|n,B\rangle\langle
n,A|+v|n,A\rangle\langle
n,B|\right)+\sum_{n=1}^{N-1}\left(we^{i\textbf{A}(t)}|n+1,A\rangle\langle
n,B|+we^{\theta-i\textbf{A}(t)}|n,B\rangle\langle n+1,A|\right)$
$\displaystyle+l\left(we^{i\textbf{A}(t)}|1,A\rangle\langle
N,B|+we^{\theta-i\textbf{A}(t)}|N,B\rangle\langle 1,A|\right).$ (24)

In this discussion, we shall assume that the external electromagnetic field is
sinusoidally varying in time: $\textbf{A}(t)=A_{0}\sin\Omega t$, with a period
$T=\frac{2\pi}{\Omega}$, where $A_{0}$ is the amplitude of the vector
potential and $\Omega$ is its frequency. This provides us with an interesting
physical setup wherein the system possesses two independent discrete
translational symmetries (assuming the PBC):

$\displaystyle[H_{D}(t),T(a)]=0,\quad\text{and}\quad[H_{D}(t),U(T)]=0.$

Here, $U(T)$ represents the discrete time translation operator for time period
$T$: $|\varphi(t+T)\rangle=U(T)|\varphi(t)\rangle$, where $|\varphi(t)\rangle$
is some generic state.

Owing to the fact that the system at hand is periodically driven in time, we
wish to find the analogues of Bloch eigenstates in this case, albeit which
solve the time-dependent Schrodinger equation as also respect the Bloch
property spatially and temporally when the PBC is imposed in both real space
and time. The states obeying the Bloch property in the temporal domain are
well studied in literature and are referred to as Floquet states Shirley
(1965); Goldman and Dalibard (2014); Gómez-León and Platero (2013). Let us
denote the Floquet states obeying the spatial Bloch property as
$|Y_{\alpha}(k,t)\rangle$ which are defined as:

$\displaystyle\left(H_{D}(t)-i\partial_{t}\right)|Y_{\alpha}(k,t)\rangle=0,$
(25) $\displaystyle
T(a)|Y_{\alpha}(k,t)\rangle=e^{ik}|Y_{\alpha}(k,t)\rangle,$ (26)
$\displaystyle
U(T)|Y_{\alpha}(k,t)\rangle=e^{-i\varepsilon_{\alpha}(k)T}|Y_{\alpha}(k,t)\rangle.$
(27)

Here, the quasienergy $\varepsilon_{\alpha}(k)$ is the temporal analogue of
wave vector $k$, and $\alpha$ is the band index. Akin to the wavevector, the
quasienergy is also ambiguous up to an addition of integer multiples of
$\Omega$ Shirley (1965).

As observed earlier in the undriven SSH model, the geometric phase and the
topological aspects associated with the system are captured by the cell
periodic Bloch states. Here, we can correspondingly define the cell periodic
Floquet-Bloch states $|u_{\alpha}(k,t)\rangle$ as
$|Y_{\alpha}(k,t)\rangle=e^{-i\varepsilon_{\alpha}(k)t}\mathcal{T}(k)|u_{\alpha}(k,t)\rangle$,
which solve the eigenvalue problem:

$\displaystyle\left(\mathcal{T}(-k)H_{D}(t)\mathcal{T}(k)-i\partial_{t}\right)|u_{\alpha}(k,t)\rangle=\varepsilon_{\alpha}(k)|u_{\alpha}(k,t)\rangle,$
(28)

and obey: $T(a)|u_{\alpha}(k,t)\rangle=|u_{\alpha}(k,t)\rangle$, and
$U(T)|u_{\alpha}(k,t)\rangle=|u_{\alpha}(k,t)\rangle$.

This eigenvalue problem can be well tackled by going over to the temporal
domain Fourier space, by invoking the completeness properties of the Fourier
modes $|m\rangle\equiv e^{-im\Omega t}$, and using the Sambe inner product:
$\langle\langle
A,m|B,m^{\prime}\rangle\rangle=\frac{1}{T}\int_{0}^{T}dt\>e^{i(m-m^{\prime})\Omega
t}\langle A|B\rangle$ Shirley (1965). This allows us to write the above
eigenvalue problem as:

$\displaystyle\sum_{m=-\infty}^{\infty}\langle\langle
m^{\prime}|\mathcal{H}_{k}(t)|m\rangle\rangle\langle\langle
m|u_{\alpha}(k,t)\rangle\rangle=\varepsilon_{\alpha}(k)\langle\langle
m^{\prime}|u_{\alpha}(k,t)\rangle\rangle,$ (29)

where the Floquet-Bloch operator
$\mathcal{H}_{k}(t)\equiv\left(\mathcal{T}(-k)H_{D}(t)\mathcal{T}(k)-i\partial_{t}\right)$.
This problem can be further simplified by going over to the $k$-space, in
which case the matrix elements of the Floquet-Bloch operator
$\mathcal{H}_{k}(t)$ can be straight away evaluated and expressed using the
Bessel functions $J_{m}(x)$ to read:

$\displaystyle\langle\langle
m^{\prime}|\mathcal{H}_{k}(t)|m\rangle\rangle=\begin{pmatrix}|0,A\rangle,&|0,B\rangle\end{pmatrix}[\mathcal{H}_{m^{\prime},m}(k)]\begin{pmatrix}\langle
0,A|\\\ \langle 0,B|\end{pmatrix},$

where

$\displaystyle[\mathcal{H}_{m^{\prime},m}(k)]=$
$\displaystyle\begin{pmatrix}-m^{\prime}\Omega\delta_{m^{\prime},m}&v+we^{ik}J_{m-m^{\prime}}(A_{0})\\\
v+we^{-ik+\theta}J_{m^{\prime}-m}(A_{0})&-m^{\prime}\Omega\delta_{m^{\prime},m}\end{pmatrix}.$
(30)

So by going over to the spatial and temporal Fourier domain, the problem of
finding the quasienergy spectrum for the Floquet-Bloch problem (28) is now
reduced to diagonalizing the infinite dimensional matrix
$[\mathcal{H}_{m^{\prime},m}(k)]$.

In general, in the absence of any other symmetries, analytical diagonalization
of $[\mathcal{H}_{m^{\prime},m}(k)]$ for arbitrary values of parameters is an
arduous task. As a result, we are forced to work with a judicious
approximation, which renders this diagonalization possible and maximally
captures the accurate quasienergy spectrum.

High-frequency driving: We note that the matrix
$[\mathcal{H}_{m^{\prime},m}(k)]$ is dominated by the diagonal elements in the
Fourier space so long as $\Omega>>v,w,e^{\theta},J_{0}(A_{0})$. To the leading
order in this limit, we can approximate this matrix as being diagonal,
neglecting the off-diagonal terms in the Fourier space Gómez-León and Platero
(2013):

$\displaystyle[\mathcal{H}_{m^{\prime},m}(k)]\simeq-m^{\prime}\Omega\delta_{m^{\prime},m}+\mathcal{H}_{0}(k),$
(31)

where

$\displaystyle\mathcal{H}_{0}(k)=\begin{pmatrix}0&v+we^{ik}J_{0}(A_{0})\\\
v+we^{-ik+\theta}J_{0}(A_{0})&0\end{pmatrix}.$ (32)

This shows that the dynamics of each Fourier block is essentially governed by
the same non-trivial matrix $\mathcal{H}_{0}(k)$ while only the diagonal part
changes albeit inconsequentially since the quasienergy is ambiguous upto
addition of term $m^{\prime}\Omega$. As a result, the essential dynamics of
the system in this regime is captured by the non-Hermitian matrix
$\mathcal{H}_{0}(k)$.

Remarkably, we see that the structure of the matrix $\mathcal{H}_{0}(k)$ is
identical to that of $H(k)$, encountered in (4), which governs the dynamics of
the undriven model. The matrix $\mathcal{H}_{0}(k)$ is indeed $H(k)$, albeit
with a redefinition $w\rightarrow\tilde{w}=wJ_{0}(A_{0})$. With this
identification, we have solved the eigenvalue problem for
$\mathcal{H}_{0}(k)$, to find that the quasienergy band structure is:

$\displaystyle\varepsilon_{\pm}(k)=\pm\sqrt{(\tilde{w}e^{ik}+v)(\tilde{w}e^{\theta-
ik}+v)}.$ (33)

Owing to the similarity with the energy band spectrum $E_{\pm}(k)$ of undriven
non-Hermitian SSH model, we can immediately infer that the quasienergy
spectrum $\varepsilon_{\pm}(k)$ also realizes distinct phases depending upon
the value of $\tilde{\kappa}=\tilde{w}/v$. In particular, the quasienergy band
merger takes place when $e^{-\theta}<\tilde{\kappa}<1$, and the system becomes
Möbius metallic in this region, whereas in the other regions the quasienergy
spectrum displays a band gap and behaves as an insulator (see Fig. (1)). It is
interesting to note that the parameter $\tilde{\kappa}$ is a function of
driving potential amplitude $A_{0}$ through the Bessel function
$J_{0}(A_{0})$, which is an oscillating function of $A_{0}$. Due to the non-
monotonic dependence of $\tilde{\kappa}$ on $A_{0}$, the system makes back and
forth transitions between the metallic and insulator phases as $A_{0}$ is
increased monotonically. The last feature is neatly brought out in the Figs.
(5) and (6).

The left and right eigenstates of $\mathcal{H}_{0}(k)$ are also respectively
the same as $\Phi_{\pm}$ and $\Psi_{\pm}$ in (6) and (7), with the replacement
$w\rightarrow\tilde{w}$. This allows us to apply the definition of the bi-
orthonormal geometric phase (II.1) in this driven case as well, which reads:

$\displaystyle\gamma_{\pm}(2\pi)$
$\displaystyle=\text{Arg}\>(\Phi_{\pm}(0)\cdot\Psi_{\pm}(2\pi))$
$\displaystyle+i\int_{0}^{2\pi}dk\>\Phi_{\pm}(k)\cdot\partial_{k}\Psi_{\pm}(k).$
(34)

We note that the Pancharatnam-Zak phase is also quantized in the units of
$\pi$ in the insulator phases of the driven non-Hermitian SSH model. Upon
explicit calculation, we find that the ratio $\tilde{\kappa}$ separates the
trivial and non-trivial insulating phase of the driven model. When
$\tilde{\kappa}>1$, the system is in a non-trivial insulating phase
characterized by $\gamma_{\pm}(2\pi)=\pi$. For $0<\tilde{\kappa}<e^{-\theta}$,
the system behaves as a trivial band insulator, wherein
$\gamma_{\pm}(2\pi)=0$. In the metallic phase, we deduce that the two circuit
Pancharatnam-Zak phase $\gamma_{\pm}(4\pi)$ is the proper geometric invariant
with the value $\pi$. In Fig. (4), we plot the geometric phase as a function
of $A_{0}$, $\theta$ and $\kappa$, bringing out the phase diagram of the
driven non-Hermitian system.

The above analysis of the driven non-Hermitian SSH model was performed while
committing to PBC. Nevertheless, we wonder if the non-trivial insulating phase
has a manifestation in zero-quasienergy eigenmodes in the OBC case of this
driven model. To investigate this possibility, we consider the Hamiltonian in
(24), where we correctly implement the OBC by employing a large $N$ limit as
$l=1/N$. Akin to the real space analysis of undriven SSH model, we now discuss
the Floquet eigenvalue problem of the above Hamiltonian in real space:

$\displaystyle\left(H_{D}(t)-i\partial_{t}\right)|\psi_{\mu}\rangle=\varepsilon_{\mu}|\psi_{\mu}\rangle.$
(35)

Using the completeness property of the Fourier modes $|m\rangle$ and the
localized states $|n,s\rangle$, and employing the Sambe inner product, this
Floquet problem reads:

$\displaystyle\sum_{m^{\prime},n^{\prime},s^{\prime}}\langle\langle m|\langle
n,s|\left(H_{D}(t)-i\partial_{t}\right)|n^{\prime},s^{\prime}\rangle|m^{\prime}\rangle\rangle\langle\langle
m^{\prime}|\langle
n^{\prime},s^{\prime}|\psi_{\mu}\rangle\rangle=\varepsilon_{\mu}\langle\langle
m|\langle n,s|\psi_{\mu}\rangle\rangle.$ (36)

Similar to the real space analysis of undriven model, the object
$\langle\langle m|\langle
n,s|\left(H_{D}(t)-i\partial_{t}\right)|n^{\prime},s^{\prime}\rangle|m^{\prime}\rangle\rangle$
can be expressed in the form of a matrix, keeping only finite number of
Fourier modes (call $M$), and then be numerically diagonalized to yield the
quasienergy spectrum.

Figure 4:  Geometric phase $\gamma$ in a driven non-Hermitian SSH model as a
function of vector potential’s amplitude $(A_{0})$ and ratio $\kappa~{}(\equiv
w/v)$ for non-Hermiticity parameter $\theta=0.75$ (a), and the same as a
function of $A_{0}$ and $\theta$ for $\kappa=1.5$ (b). Here the blue region
represents the trivial insulating phase ($\gamma(2\pi)=0$), while the yellow
region indicates the non-trivial insulating phase ($\gamma(2\pi)=\pi$). The
green region denotes the Möbius metallic phase ($\gamma(4\pi)=\pi$).

In Fig. (5), we compare quasienergy spectrum obtained using the above
procedure for the driven Hermitian $(\theta=0)$ and non-Hermitian $(\theta\neq
0)$ SSH model. While the quasienergy spectrum is real for the Hermitian model,
it is complex for the non-Hermitian model. Thus, we plot the real part of the
quasienergy spectrum for the driven non-Hermitian SSH model. The Möbius strip
phase appears and reappears only in the driven non-Hermitian model, and it is
absent in the driven Hermitian SSH model. In Fig. (6), we then compare the
above quasienergy spectrum with the one obtained analytically from approximate
description of (33) using PBC, considering a lattice of $N=70$ and number of
Fourier modes $M=61$. Apart from the zero modes for the spectrum obtained with
OBC, we observe a good agreement between the spectra in the two cases. The
phase diagrams in Fig. (4) show that the topological features of the OBC
spectra found in Figs. (5,6) are well captured by the geometric phase
relations obtained using PBC earlier.

Figure 5: Quasienergy spectrum versus amplitude of vector potential $(A_{0})$
of a driven Hermitian SSH model (top) and a driven non-Hermitian SSH model
(bottom) at high-frequency driving and with OBC. The Möbius strip phase
appears and reappears only in the driven non-Hermitian model. All parameters
are shown at the headings.  Figure 6: Plots showing an agreement between
analytically obtained quasienergy spectrum with PBC (top) and numerically
obtained quasienergy spectrum with OBC (bottom). The real part of quasienergy
is plotted as a function of amplitude of vector potential $(A_{0})$ for a
driven non-Hermitian SSH model at high-frequency driving. Apart from the zero
modes for the spectrum with OBC, the two spectra closely match. All parameters
are shown at the headings.

Low-frequency driving: In the above discussion, it is assumed that the driving
frequency $\Omega$ is much larger than the other parameters in the model. In
the opposite limit, if the driving frequency $\Omega\ll v,w$, then it implies
that the rate of change of Hamiltonian
$\partial_{t}H_{D}(t)=\Omega\partial_{\varphi}H_{D}(\varphi)$ (where
dimensionless variable $\varphi=\Omega t$) is much smaller than unity. It is
known that in such a scenario, the dynamics is adiabatic Garrison and Wright
(1988) and the system evolves through the instantaneous Bloch states
$|\Psi_{\pm}(k,A(\varphi))\rangle$, such that
$H_{D}(\varphi)|\Psi_{\pm}(k,A(\varphi))\rangle=E_{\pm}(k,A(\varphi))|\Psi_{\pm}(k,A(\varphi))\rangle$
where $|\Psi_{\pm}(k,A(\varphi))\rangle=|\Psi_{\pm}(k+A(\varphi))\rangle$ and
$E_{\pm}(k,A(\varphi))=E_{\pm}(k+A(\varphi))$. The corresponding cell periodic
instantaneous Bloch states are then given by:
$|u_{\pm}(k,A(\varphi))\rangle=\mathcal{T}(-k)|\Psi_{\pm}(k,A(\varphi))\rangle$,
which solve the right eigenvalue problem:
$H_{k+A(\varphi)}|u_{\pm}(k,A(\varphi))\rangle=E_{\pm}(k,A(\varphi))|u_{\pm}(k,A(\varphi))\rangle$,
where

$\displaystyle
H_{k+A(\varphi)}=\begin{pmatrix}|0,A\rangle,&|0,B\rangle\end{pmatrix}[H(k+A(\varphi))]\begin{pmatrix}\langle
0,A|\\\ \langle 0,B|\end{pmatrix},$ (37)

and $H(k+A(\varphi))$ reads:

$\displaystyle H(k+A(\varphi))=\begin{pmatrix}0&we^{i(k+A(\varphi))}+v\\\
we^{\theta-i(k+A(\varphi))}+v&0\end{pmatrix}.$ (38)

From here it is evident that given an initial state with a fixed wavevector
say $k_{0}$, the net effect of time evolution due to $A(\varphi)$ is same as
changing $k_{0}$ to $k_{0}+A(\varphi)$. Thus the state of the system at any
instant of time is expressible as:
$|u_{\pm}(k_{0},A(\varphi))\rangle\equiv|u_{\pm}(k_{0}+A(\varphi))\rangle$ Zak
(1989); Vyas  _et al._ (2019). As the parameter $\varphi$ changes, the state
effectively sweeps the $k$-space so as to acquire the geometric phase
$\gamma_{\pm}$ solely depending upon the ratio $\kappa$, as shown in the
earlier section. Thus, we see that the driving potential, in this case, is
unable to alter the system’s topological structure as in the high-frequency
case.

Intermediate-frequency driving: Finally, we here discuss the intermediate
frequency of driving when $\Omega\approx v,w$. In this regime, the exact
analytical solution of the eigenvalue problem in (28) is intractable, and the
approximate approaches employed in the earlier two frequency regimes fail.
This is because the matrix $[\mathcal{H}_{m^{\prime},m}(k)]$ as defined in
(30) does not get simplified, since different Fourier blocks in
$[\mathcal{H}_{m^{\prime},m}(k)]$ get strongly coupled to each other in the
intermediate-frequency regime. Nevertheless, we can numerically study the
quasienergy spectrum of the driven SSH model in this regime applying PBC and
OBC with $l=1/N$.

In Fig. (7), we plot the quasienergy spectrum of the driven Hermitian SSH
model for PBC and OBC. We immediately notice crossing and avoided-crossing of
two quasienergy modes within the band gap in the PBC case. We further observe
that these modes manifest as zero-quasienergy edge modes in the OBC case. The
occurrence of such a phenomenon is well known in the literature, albeit in the
context of higher-dimensional lattice models having non-trivial topology,
e.g., the Qi-Wu-Zhang (QWZ) model, which are well explored Asbóth  _et al._
(2016). The above features of higher-dimensional models are found to survive
and get altered in the presence of non-Hermiticity at the intermediate
frequency of driving, which we display in Fig. (8). While the size of the bulk
band gap and the number of crossing and avoided-crossing of quasienergy modes
inside a bulk band gap increase with higher non-Hermiticity, the overlap of
quasienergy bands indicating the Möbius metallic-like phase is also clearly
visible in the non-Hermitian model. The above observations suggest that the
topological phases in this intermediate regime, as captured by the level
crossings and edge modes, are different from those found in high and low-
frequency regimes. From the study of higher-dimensional lattice models, it is
well established that such topological features are not described by geometric
phase as defined in (II.1), but are rather captured by higher-dimensional
topological invariants like Chern number Bernevig and Hughes (2013); Asbóth
_et al._ (2016); Zhou  _et al._ (2014).

Figure 7: Quasienergy spectrum versus amplitude of vector potential $(A_{0})$
of a driven Hermitian SSH model at intermediate-frequency driving with PBC
(top) and OBC (bottom). Two quasienergy modes display crossing and avoided-
crossing with $A_{0}$ within a bulk band gap in the PBC case. All parameters
are shown at the headings. Figure 8: Real part of quasienergy spectrum versus
amplitude of vector potential $(A_{0})$ of a driven non-Hermitian SSH model at
intermediate-frequency driving with PBC (top) and OBC (bottom). While the size
of bulk band gap and the number of crossing and avoided-crossing of
quasienergy modes inside a bulk gap increase with higher non-Hermiticity, the
overlap of quasienergy bands is also clearly visible for the non-Hermitian
model. All parameters are shown at the headings.

##  IV Outlook

In this paper, the topological aspects of a non-Hermitian generalization of
the SSH model periodically driven by an external electromagnetic field are
studied. A bi-orthonormal conception of the Pancharatnam-Zak geometric phase
is constructed for the undriven model in the PBC framework. We have found that
this geometric phase correctly captures the model’s topological phase
structure, which comprises trivial and non-trivial insulating phases and an
exotic Möbius metallic phase. We also see the zero-energy modes when the model
is investigated with OBC. We note that implementing an OBC in this model
requires care, and when implemented using the large $N$ limit, consistent
results are obtained. We further discover that while the metallic phase is a
topologically non-trivial phase, it does not support robust zero-energy modes,
as displayed by the non-trivial insulating phase.

Subsequently, the topological phase structure of the driven non-Hermitian SSH
model is studied using the Floquet approach. It is found that this model also
admits trivial and non-trivial insulator phases and the Möbius metallic phase.
Interestingly, the driving potential’s strength acts as a control parameter
guiding the system through the various topological phases. We observe that the
bi-orthogonal geometric phase corresponding to the Floquet-Bloch states, in
this case, acts as a topological index characterizing the different
topological phases at high and low-frequency driving. Nevertheless, we could
not resolve the nature of the topological phases at the intermediate-frequency
driving, which might require a further generalization of the bi-orthogonal
geometric phase to higher dimensions.

The present work unifies and extends the earlier works on Hermitian geometric
phases due to Pancharatnam Pancharatnam (1956), Zak Zak (1989), and the
results on the non-Hermitian geometric phase of Garrison and Wright Garrison
and Wright (1988) and others Lieu (2018). Although several earlier studies
define various topological invariants, which can capture and classify the
topological phase structure of the driven non-Hermitian model, the bi-
orthonormal geometric phase considered here is a straightforward elementary
construction accomplishing this task. There have been earlier works dealing
with topological aspects of driven non-Hermitian lattice models, wherein
different kinds of non-Hermiticity and driving mechanisms were employed Asbóth
_et al._ (2014); Jin and Song (2019); Zhou and Gong (2018b); Dal Lago  _et
al._ (2015); Goldman and Dalibard (2014); Kunst  _et al._ (2018). We hope that
the approach presented here will shed light on topological aspects in such
models as well.

Acknowledgments. We thank Prof. M. V. Berry for discussion at the initial
stages of the work. This research is funded by the Department of Science and
Technology, India via the Ramanujan Fellowship, and the Ministry of
Electronics $\&$ Information Technology (MeitY), India under grant for “Centre
for Excellence in Quantum Technologies” with Ref. No. 4(7)/2020-ITEA.

## Appendix

For the general non-Hermitian Hamiltonian, wherein no assumption is made about
the choice of parameters $v_{1,2}$ and $w_{1,2}$, it is straightforward to see
that the determinant of $H$ (defined in (1)) reads:

$\displaystyle\text{Det}\>H=(-1)^{N}(v_{1}^{N}+(-1)^{N-1}lw_{2}^{N})(v_{2}^{N}+(-1)^{N-1}lw_{1}^{N}).$
(39)

The ratio $r$ for the general case is then given by:

$\displaystyle
r=\frac{\text{Det}\>H\lvert_{OBC,N+1}}{\text{Det}\>H\lvert_{PBC,N}}\simeq-\frac{(v_{1}^{N}-\frac{1}{N}w_{2}^{N})(v_{2}^{N}-\frac{1}{N}w_{1}^{N})}{(v_{1}^{N}+w_{2}^{N})(v_{2}^{N}+w_{1}^{N})}.$
(40)

Her,e we have assumed $N$ to be odd for specificity, and it can be checked
that the end results are not affected by this assumption. Further analysis can
be simplified by working with real parameters $\rho_{1,2}$ and $\phi_{1,2}$,
defined as: $\frac{w_{1}}{v_{2}}=\rho_{1}e^{i\phi_{1}}$ and
$\frac{w_{2}}{v_{1}}=\rho_{2}e^{i\phi_{2}}$. The ratio $r$ now reads:

$\displaystyle
r\simeq-\frac{(1-\frac{1}{N}\rho_{1}^{N}e^{iN\phi_{1}})(1-\frac{1}{N}\rho_{2}^{N}e^{iN\phi_{2}})}{(1+\rho_{1}^{N}e^{iN\phi_{1}})(1+\rho_{2}^{N}e^{iN\phi_{2}})}.$
(41)

This expression for $r$ immediately shows the phase diagram of this general
model and the zero-mode energy scaling. It is clear that the system depending
upon the Hamiltonian parameters exhibits three different phases classified as:

  * (a)

Trivial insulator phase: when $\rho_{1}<1$ and $\rho_{2}<1$, no zero mode
exists;

  * (b)

Non-trivial insulator phase: when $\rho_{1}>1$ and $\rho_{2}>1$, zero modes
exist, and their energies fall as $\frac{1}{N}$;

  * (b)

Metallic phase: when $\rho_{1}<1$ & $\rho_{2}>1$ or when $\rho_{1}>1$ &
$\rho_{2}<1$, zero modes exist, and their energies fall as
$\frac{i}{\sqrt{N}}$.

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---|---|---|---|---|---|---|---|---|---|---|---
0.12703 | 0.11096 | 0.10803 | 0.11061 | 0.11526 | 0.12055 | 0.12592 | 0.13112 | 0.13609 | 0.14086 | 1.97898 | 2.11389
2.04462 | 0.53361 | 0.15189 | 0.13275 | 0.14324 | 0.15783 | 0.17263 | 0.18659 | 0.19934 | 0.46801 | 2.16097 | 2.43498
2.97688 | 1.21611 | 0.25912 | 0.18733 | 0.21085 | 0.24633 | 0.28181 | 0.31464 | 0.34411 | 0.76764 | 2.37859 | 2.72843
3.49092 | 2.05810 | 0.40965 | 0.26923 | 0.30785 | 0.36634 | 0.42335 | 0.47498 | 0.52054 | 1.08621 | 2.61033 | 2.99299
3.81426 | 2.90145 | 0.60903 | 0.36654 | 0.41701 | 0.49778 | 0.57601 | 0.64609 | 0.70726 | 1.42199 | 2.86607 | 3.23862
4.05861 | 3.51932 | 0.89918 | 0.49440 | 0.55437 | 0.65817 | 0.75809 | 0.84651 | 0.92272 | 1.82150 | 3.14369 | 3.47779
4.21761 | 3.93500 | 1.34703 | 0.63635 | 0.69930 | 0.82514 | 0.94646 | 1.05279 | 1.14338 | 2.27052 | 3.42705 | 5.48892
4.43936 | 4.24694 | 2.52282 | 0.81571 | 0.87183 | 1.01942 | 1.16261 | 1.28696 | 1.39157 | 2.85635 | 3.72076 | 6.38169
4.47214 | 4.45733 | 4.15105 | 1.03388 | 1.06507 | 1.23320 | 1.39845 | 1.54079 | 1.65894 | 3.74137 | 4.02277 | 6.83677

Table 26: Results for Case 4.