# Floquet engineering of topological localization transitions and
mobility edges in one-dimensional non-Hermitian quasicrystals

Longwen Zhou<EMAIL_ADDRESS>College of Physics and Optoelectronic
Engineering, Ocean University of China, Qingdao, China 266100

###### Abstract

Time-periodic driving fields could endow a system with peculiar topological
and transport features. In this work, we find dynamically controlled
localization transitions and mobility edges in non-Hermitian quasicrystals via
shaking the lattice periodically. The driving force dresses the hopping
amplitudes between lattice sites, yielding alternate transitions between
localized, mobility edge and extended non-Hermitian quasicrystalline phases.
We apply our Floquet engineering approach to five representative models of
non-Hermitian quasicrystals, obtain the conditions of “photon-assisted”
localization transitions and mobility edges, and find the expressions of
Lyapunov exponents for some models. We further introduce topological winding
numbers of Floquet quasienergies to distinguish non-Hermitian quasicrystalline
phases with different localization nature. Our discovery thus extend the study
of quasicrystals to non-Hermitian Floquet systems, and provide an efficient
way of modulating the topological and transport properties of these unique
phases.

††preprint: APS/123-QED

##  I Introduction

Floquet engineering has enabled the realization of rich dynamical, topological
and transport phenomena in a broad range of physical settings FloExp0 ;
FloExp1 ; FloExp2 ; FloExp3 ; FloExp4 ; FloExp5 ; FloExp6 ; FloExp7 ; FloExp8
; FloExp9 ; FloExp10 ; FloExp11 ; FloExp12 ; FloExp13 ; FloExp14 ; FloExp15 ;
FloExp16 . Some notable examples include the dynamical localization, Floquet
topological phase and discrete time crystal (see DLRev1 ; DLRev2 ; DLRev3 ;
DLRev5 ; FTPRev1 ; FTPRev2 ; FTPRev3 ; FTPRev4 ; FTPRev5 ; DTCRev1 ; DTCRev2 ;
DTCRev3 ; DTCRev4  for reviews and references therein). These discoveries have
broadened the classification of phases of matter in nonequilibrium situations
FTPClass1 ; FTPClass2 ; FTPClass3 , and may lead to applications in emerging
technologies like ultrafast electronics FTPRev1  and topological quantum
computing FTQC1 ; FTQC2 ; FTQC3 .

Non-Hermitian topological matter has been investigated intensively in the past
few years NHRev1 ; NHRev2 ; NHRev3 ; NHTP1 ; NHTP2 ; NHTP3 . Recently, the
Floquet approach was also applied to engineer topological phases in non-
Hermitian systems. Notably, it was found that the interplay between time-
periodic driving fields and gain and loss or nonreciprocal effects could yield
topological phases that are unique to non-Hermitian Floquet systems NHFTP1 ;
NHFTP2 ; NHFTP3 ; NHFTP4 ; NHFTP5 ; NHFTP6 ; NHFTP7 ; NHFTP8 . These
intriguing phases are characterized by large integer or half-integer winding
numbers and degenerate Floquet edge or corner states with real quasienergies
NHFTP1 ; NHFTP2 ; NHFTP4 ; NHFTP5 ; NHFTP6 ; NHFTP7 ; NHFTP8 . Their
topological signatures may further be extracted experimentally from the
dynamical spin textures and quantized displacements of wavepackets NHFTP3 .
Till now, various kinds of non-Hermitian Floquet topological insulators,
superconductors and semimetals with exceptional topological properties have
been found NHFTP1 ; NHFTP2 ; NHFTP3 ; NHFTP4 ; NHFTP5 ; NHFTP6 ; NHFTP7 ;
NHFTP8 ; NHFTP9 ; NHFTP10 ; NHFTP11 ; NHFTP12 ; NHFTP13 ; NHFTP14 ; NHFTP15 ;
NHFTP16 ; NHFTP17 ; NHFTP18 ; NHFTP19 ; NHFTP20 ; NHFTP21 ; NHFTP22 ; NHFTP23
; NHFTP24 ; NHFTP25 ; NHFTP26 ; NHFTP27 ; NHFTP28 ; NHFTP29 ; NHFTP30 .
Meanwhile, much less is known when non-Hermitian Floquet systems are subject
to more complicated correlation effects, such as disorder, nonlinearity and
many-body interactions.

In this work, we employ the idea of Floquet engineering to induce and control
the spectrum, localization, mobility edges and topological transitions in non-
Hermitian quasicrystals. A quasicrystal is a phase of matter with long-range
order in the absence of spatial periodicity. It can be realized in a lattice
with diagonal or off-diagonal spatially quasiperiodic modulations. In Sec. II,
we outline the method of adjusting hopping amplitudes in a tight-binding
lattice by applying high-frequency periodic driving forces, which follows the
idea of engineering dynamical localization in ultracold atoms FloExp0 ;
FloExp1 ; FloExp2 . In Sec. III, we introduce five prototypical models of non-
Hermitian quasicrystal, which can be viewed as non-Hermitian extensions of the
Aubry-André-Harper (AAH) model and the Maryland model. We further apply high-
frequency shaking forces to these models, and obtain driving-induced
transitions between non-Hermitian Floquet quasicrystalline phases with
distinct transport and topological features. Different types of non-Hermitian
quasicrystal phases are also found to emerge alternately with the increase of
the ratio between the amplitude and frequency of the driving field. In Sec.
IV, we summarize our results and discuss future perspectives.

##  II Method

In this section, we introduce our scheme of Floquet engineering to control the
phase transitions in non-Hermitian quasicrystals, which is based on the idea
of realizing dynamical localization in Bose-Einstein condensates DLRev2 . A
schematic illustration of our system and approach is shown in Fig. 1. The
Hamiltonian of the system describes a tight-binding superlattice with onsite
quasiperiodic potential and driving field, i.e.,

$\hat{H}(t)=\sum_{\langle
n,n^{\prime}\rangle}(J_{R}\hat{c}_{n}^{\dagger}\hat{c}_{n^{\prime}}+J_{L}\hat{c}_{n^{\prime}}^{\dagger}\hat{c}_{n})+\sum_{n}[V_{n}+W_{n}(t)]\hat{c}_{n}^{\dagger}\hat{c}_{n},$
(1)

where $n$ is the lattice site index, and $\langle n,n^{\prime}\rangle$
includes site indices with $n^{\prime}>n$. $\hat{c}_{n}^{\dagger}$ and
$\hat{c}_{n}$ denote the creation and annihilation operators of a particle
(either boson or fermion) on the $n$th lattice site. $J_{L}$ ($J_{R}$)
represents the hopping amplitude of the particle from lattice site $n$
($n^{\prime}$) to $n^{\prime}$ ($n$). A quasicrystal model is realized by
setting the superlattice potential $V_{n}$ to a quasiperiodic function of the
lattice index $n$. A non-Hermitian quasicrystal is further obtained by setting
$J_{L}\neq J_{R}^{*}$ (for nonreciprocal hopping) or $V_{n}\neq V_{n}^{*}$
(for gain and loss of particles). The coupling between the driving field and
the particle takes the form of $W_{n}(t)=-nF(t)$, where the driving force
$F(t)=F(t+T)$ varies periodically in time with the driving period $T$ and
driving frequency $\omega=2\pi/T$. All system parameters have been properly
rescaled and given in dimensionless units. We will also set the Planck
constant $\hbar=1$ throughout the discussion.

Figure 1: A schematic illustration of the driven non-Hermitian superlattice.
$J_{L}$ ($J_{R}$) denotes the hopping amplitude from left (right) to right
(left) lattice sites. $V_{n}$ denotes the static potential on the $n$th
lattice site, which introduces quasiperiodicity into the system. $W_{n}(t)$ is
the driving potential, which is proportional to the position index $n$. The
driving force $F(t)$ oscillates periodically in time. The system is made non-
Hermitian if $J_{L}\neq J_{R}^{*}$ or $V_{n}\neq V_{n}^{*}$, corresponding to
the case with nonreciprocal hopping or onsite gain and loss.

In the frame comoving with the lattice, the dynamics of the system is governed
by the time-dependent Schrödinger equation

$i\frac{d}{dt}|\psi(t)\rangle=\hat{H}(t)|\psi(t)\rangle.$ (2)

The driving potential in $\hat{H}(t)$ can be transformed to an oscillating
phase factor in the hopping amplitudes $J_{L,R}$ DLRev2 . To do so, we
consider the following unitary rotation

$\hat{R}(t)=e^{i\sum_{n}f_{n}(t)\hat{c}_{n}^{\dagger}\hat{c}_{n}},$ (3)

where the time-dependent amplitude

$\displaystyle f_{n}(t)=$
$\displaystyle-\int_{t_{0}}^{t}W_{n}(t^{\prime})dt^{\prime}-n\overline{f}$ (4)
$\displaystyle=$ $\displaystyle
n\left(\int_{t_{0}}^{t}F(t^{\prime})dt^{\prime}-\overline{f}\right)\equiv
nf(t).$

Here $\overline{f}$ is chosen to ensure that the one-period integral of
amplitude $\int_{0}^{T}f_{n}(t)dt=0$. Writing the time-evolving state as
$|\psi(t)\rangle=\hat{R}(t)|\psi^{\prime}(t)\rangle$, we obtain the
Schrödinger equation for the state $|\psi^{\prime}(t)\rangle$ in the rotating
frame as

$i\frac{d}{dt}|\psi^{\prime}(t)\rangle=\hat{H}_{{\rm
r}}(t)|\psi^{\prime}(t)\rangle,$ (5)

where the Hamiltonian $\hat{H}_{{\rm r}}(t)$ in the rotating frame reads

$\hat{H}_{{\rm
r}}(t)=\hat{R}^{\dagger}(t)\hat{H}(t)\hat{R}(t)-i\hat{R}^{\dagger}(t)\frac{d\hat{R}(t)}{dt}.$
(6)

Plugging Eqs. (3) and (4) into Eq. (6), we find that $\hat{R}(t)$ commutes
with the onsite terms
$\sum_{n}[V_{n}+W_{n}(t)]\hat{c}_{n}^{\dagger}\hat{c}_{n}$ in $\hat{H}(t)$ and

$i\hat{R}^{\dagger}(t)\frac{d\hat{R}(t)}{dt}=\sum_{n}W_{n}(t)\hat{c}_{n}^{\dagger}\hat{c}_{n},$
(7)

which cancels the driving potential
$\sum_{n}W_{n}(t)\hat{c}_{n}^{\dagger}\hat{c}_{n}$, leaving
$\sum_{n}V_{n}\hat{c}_{n}^{\dagger}\hat{c}_{n}$ as the only onsite term in
$\hat{H}_{{\rm r}}(t)$. Furthermore, the hopping terms in $\hat{H}(t)$ under
the rotation $\hat{R}(t)$ becomes

$\displaystyle\hat{R}^{\dagger}(t)\hat{c}_{n}^{\dagger}\hat{c}_{n^{\prime}}\hat{R}(t)=$
$\displaystyle
e^{-inf(t)\hat{c}_{n}^{\dagger}\hat{c}_{n}}\hat{c}_{n}^{\dagger}e^{inf(t)\hat{c}_{n}^{\dagger}\hat{c}_{n}}$
$\displaystyle\times$ $\displaystyle
e^{-in^{\prime}f(t)\hat{c}_{n^{\prime}}^{\dagger}\hat{c}_{n^{\prime}}}\hat{c}_{n^{\prime}}e^{in^{\prime}f(t)\hat{c}_{n^{\prime}}^{\dagger}\hat{c}_{n^{\prime}}}$
$\displaystyle=$ $\displaystyle
e^{i(n^{\prime}-n)f(t)}\hat{c}_{n}^{\dagger}\hat{c}_{n^{\prime}}.$ (8)

Note that this result holds for both bosons and fermions. Therefore, the
Hamiltonian in the rotating frame is

$\displaystyle\hat{H}_{{\rm r}}(t)=$ $\displaystyle\sum_{\langle
n,n^{\prime}\rangle}J_{L}e^{i(n-n^{\prime})f(t)}\hat{c}_{n^{\prime}}^{\dagger}\hat{c}_{n}$
$\displaystyle+$ $\displaystyle\sum_{\langle
n,n^{\prime}\rangle}J_{R}e^{i(n^{\prime}-n)f(t)}\hat{c}_{n}^{\dagger}\hat{c}_{n^{\prime}}+\sum_{n}V_{n}\hat{c}_{n}^{\dagger}\hat{c}_{n},$
(9)

where the oscillating phase factor
$f(t)=\int_{t_{0}}^{t}F(t^{\prime})dt^{\prime}-\overline{f}$. When $f(t)$
varies quickly in time, so that $\hbar\omega\gg|J_{R,L}|$ and
$|V_{n}-V_{n^{\prime}}|$ for $|n-n^{\prime}|=1$, its effect on $\hat{H}_{{\rm
r}}(t)$ can be approximated by the average over a complete driving period $T$
DLRev2 . This corresponds to taking the zeroth order term in the high-
frequency expansion of the Floquet Hamiltonian HFA1 ; HFA2 ; HFA3 . In this
limit, the micromotion operator is approximated by the rotating frame
transformation $\hat{R}(t)$, and the resulting Floquet effective Hamiltonian
reads

$\displaystyle\hat{H}_{{\rm F}}$
$\displaystyle\approx\frac{1}{T}\int_{0}^{T}dt\hat{H}_{{\rm r}}(t)$
$\displaystyle=\sum_{\langle n,n^{\prime}\rangle}\left(J_{L}^{{\rm
eff}}\hat{c}_{n^{\prime}}^{\dagger}\hat{c}_{n}+J_{R}^{{\rm
eff}}\hat{c}_{n}^{\dagger}\hat{c}_{n^{\prime}}\right)+\sum_{n}V_{n}\hat{c}_{n}^{\dagger}\hat{c}_{n},$
(10)

where the “photon-dressed” hopping amplitudes $J_{L}^{{\rm eff}}$ and
$J_{R}^{{\rm eff}}$ are given by

$\displaystyle J_{L}^{{\rm eff}}=$
$\displaystyle\frac{1}{T}\int_{0}^{T}dtJ_{L}e^{i(n-n^{\prime})f(t)},$ (11)
$\displaystyle J_{R}^{{\rm eff}}=$
$\displaystyle\frac{1}{T}\int_{0}^{T}dtJ_{R}e^{i(n^{\prime}-n)f(t)}.$ (12)

The driving field thus introduces a control knob to the hopping amplitudes,
and thus the transport property of the non-Hermitian quasicrystal.

In experiments, one type of driving force that is relatively easy to engineer
has the form of a harmonic function FloExp0 ; DLRev2 , which can be introduced
by shaking the lattice back and forth periodically HFA4 ; HFA5 ; HFA6  and
described as $F(t)=K\cos(\omega t)$, where $K$ is the driving amplitude and
$\omega$ is the driving frequency of the force. The factor $f(t)$ in the
Peierls phase for such a harmonic drive reads $f(t)=K\sin(\omega t)/\omega$
according to Eq. (4). Working out the the integrals in Eqs. (11) and (12) with
the help of the Bessel function expansion
$e^{ix\sin(y)}=\sum_{l=-\infty}^{\infty}{\cal J}_{l}(x)e^{ily}$, we find
$J_{L,R}^{{\rm eff}}=J_{L,R}{\cal J}_{0}\left((n^{\prime}-n)K/\omega\right)$,
which is derived previously in the study of dynamical localization HFA4 . The
Floquet effective Hamiltonian of the system under the high-frequency harmonic
forcing now takes the form

$\displaystyle\hat{H}_{{\rm F}}=$ $\displaystyle\sum_{\langle
n,n^{\prime}\rangle}J_{L}{\cal
J}_{0}\left((n^{\prime}-n)\frac{K}{\omega}\right)\hat{c}_{n^{\prime}}^{\dagger}\hat{c}_{n}$
(13) $\displaystyle+$ $\displaystyle\sum_{\langle
n,n^{\prime}\rangle}J_{R}{\cal
J}_{0}\left((n^{\prime}-n)\frac{K}{\omega}\right)\hat{c}_{n}^{\dagger}\hat{c}_{n^{\prime}}+\sum_{n}V_{n}\hat{c}_{n}^{\dagger}\hat{c}_{n}.$

Since both the $J_{L,R}$ and the Bessel function of first kind ${\cal
J}_{0}\left((n^{\prime}-n)K/\omega\right)$ decays globaly with the increase of
hopping distance $|n^{\prime}-n|$, we may reserve the hopping terms up to
nearest-neighbor sites, yielding the following Floquet effective Hamiltonian

$\hat{H}_{{\rm F}}=\sum_{n}\left[{\cal
J}_{0}\left(\frac{K}{\omega}\right)\left(J_{R}\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+J_{L}\hat{c}_{n+1}^{\dagger}\hat{c}_{n}\right)+V_{n}\hat{c}_{n}^{\dagger}\hat{c}_{n}\right].$
(14)

In the Hermitian limit, we observe that the hopping amplitudes $J_{L,R}$ are
switched off at the zeros of ${\cal J}_{0}(K/\omega)$, which leads to the
well-known phenomena of dynamical localization for any nonvanishing onsite
potential $V_{n}$ FloExp0 . When the system becomes non-Hermitian and $V_{n}$
is quasiperiodic, the competition between the energy scales of intersite
tunneling and onsite trapping could result in localization-delocalization
transitions caused by non-Hermitian effects. The presence of the driving field
further generates an interplay between the field parameters $(K,\omega)$ and
the non-Hermitian terms of the system, which is expected to yield richer phase
diagrams and dynamically controlled transitions between different non-
Hermitian quasicrystalline phases. These judgments will be demonstrated by
explicit examples in the following section.

##  III Results

In this section, we apply the Floquet engineering approach discussed in Sec.
II to five representative models of non-Hermitian quasicrystal, which can be
viewed as non-Hermitian variants of the AAH model AAH1 ; AAH2 ; AAH3  and the
Maryland model MM1 ; MM2 ; MM3 ; MM5 . The explicit forms of these models and
the driving forces are introduced in Subsec. III.1. Without the driving field,
the conditions of localization transition for these models have been derived
in previous studies. We reveal how these conditions are modified in the
presence of high-frequency lattice shaking forces, and obtain analytical
expressions for their Lyapunov exponents if possible, which are now functions
of the driving amplitude and frequency. Our theoretical results are further
verified by numerical calculations presented in Subsecs. III.2–III.6.

###  III.1 Quasicrystal models

Model | Hopping | Onsite | Driving
---|---|---|---
index | amplitude | potential $V_{n}$ | force $W_{n}(t)$
M1 | $J_{L}=J_{R}=J$ | $Ve^{-i2\pi\alpha n}$ | $-nK\cos(\omega t)$
M2 | $V\cos(2\pi\alpha n+i\gamma)$
M3 | $J_{L}=Je^{\gamma},\,J_{R}=Je^{-\gamma}$ | $V\cos(2\pi\alpha n)$
M4 | $J_{L}=J_{R}=J$ | $V\tan(\pi\alpha n+i\gamma)$
M5 | $V/\left(1-\eta e^{i2\pi\alpha n}\right)$
Table 1: Definitions of the five periodically forced non-Hermitian
quasicrystal models, denoted by M1–M5. In the second column, $J_{L}$ and
$J_{R}$ are left-to-right and right-to-left hopping amplitudes between
nearest-neighbor lattice sites. $\gamma$ controls the asymmetry between the
left and right hopping amplitudes in M3. In the third column, $V$ is the
amplitude of the onsite potential, $i\gamma$ is an imaginary phase shift,
$\eta$ controls the non-Hermiticity of M5 and $\alpha$ is an irrational
number. In the fourth column, $n$ is the lattice site index, $K$ is the
driving amplitude and $\omega$ is the driving frequency. All system parameters
are set in dimensionless units.

The AAH model is one of the standard models in the study of localization-
delocalization transitions in one-dimensional (1D) quasiperiodic systems AAH1
; AAH2 ; AAH3 . In the lattice representation, the Hamiltonian of the model
takes the form

$\hat{H}_{0}=\sum_{n}\left[J\left(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+{\rm
h.c.}\right)+V\cos(2\pi\alpha n)\hat{c}_{n}^{\dagger}\hat{c}_{n}\right],$ (15)

where $J$ is the hopping amplitude and $V$ is the amplitude of the onsite
potential. By setting $\alpha$ to be an irrational number, the superlattice
potential $V\cos(2\pi\alpha n)$ becomes quasiperiodic in the lattice index
$n$, and the system described by $\hat{H}_{0}$ forms a 1D quasicrystal. Thanks
to the self-duality property of the AAH quasicrystal, it has been shown that
the eigenstates of $\hat{H}_{0}$ possess a delocalization-to-localization
transition at $V=2J$ in the thermodynamic limit. When $V<2J$, all eigenstates
of $\hat{H}_{0}$ are delocalized and the system is in an extended phase. When
$V>2J$, the correlated disorder becomes strong enough, such that all
eigenstates of $\hat{H}_{0}$ are localized and the system enters an insulating
phase AAH3 .

The Maryland model forms another paradigm in the study of Anderson
localization and quantum chaos MM1 ; MM2 ; MM3 ; MM5 . It is an integrable
model and can be mapped to a Floquet system. The Hamiltonian of the Maryland
model takes the form

$\hat{H}_{\rm M}=\sum_{n}\left[J\left(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+{\rm
h.c.}\right)+V\tan(\pi\alpha n)\hat{c}_{n}^{\dagger}\hat{c}_{n}\right],$ (16)

where $n$ is the lattice index, $J$ is the hopping amplitude, $V$ is the
amplitude of the onsite potential, and a quasicrystal is formed once $\alpha$
takes an irrational value. Since the potential $V_{n}$ depends on a tangent
function, which is not bounded, all states of the system are localized and are
found to have an energy-dependent Lyapunov exponent (inverse localization
length) $\lambda(E)={\rm
arccosh}[(\sqrt{(2J+E)^{2}+V^{2}}+\sqrt{(2J-E)^{2}+V^{2}})/4|J|]$ MM1 .
Besides, the Maryland model can also be employed to study the topoloigcal
nature of integer quantum Hall effects MM4 .

In recent years, the impact of non-Hermiticity on localization transitions in
AAH-type and Maryland-type quasicrystals have been explored MM3 ; LonghiQC1 ;
LonghiQC2 ; LonghiQC3 ; LonghiQC4 ; ChenQC1 ; ChenQC2 ; ChenQC3 ; ChenQC4 ;
ChenQC5 ; ChenQC6 ; CaiQC1 ; NHQC1 ; NHQC2 ; NHQC3 ; NHQC4 ; NHQC5 ; NHQC6 ;
NHQC7 ; NHQC8 ; NHQC9 ; NHQC10 . The non-Hermitian effects are introduced by
either setting the onsite quasiperiodic potential to be non-Hermitian MM3 ;
LonghiQC1 ; LonghiQC2 ; LonghiQC3 , or making the hopping amplitudes between
adjacent lattice sites to be nonreciprocal LonghiQC4 ; ChenQC1 . In these
models, it was found that the non-Hermitian terms could induce ${\cal
PT}$-transitions of the energy spectrum from real to complex (or the
opposite), together with localization-delocalization transitions of the
eigenstates. These transitions could further be characterized by the quantized
change of winding numbers of the spectrum around certain base points of the
complex energy plane MM3 ; LonghiQC1 ; LonghiQC2 ; LonghiQC3 ; LonghiQC4 ;
ChenQC1 .

In the following subsections, we apply the Floquet engineering approach to
induce and control phase transitions in five non-Hermitian quasicrystal
models. The time-dependent Hamiltonians of these models take the general form
of Eq. (1), with the explicit expressions of hopping amplitudes, onsite
potential and driving force given in Table 1. We will refer to these five
models as M1–M5 for simplicity. In the high-frequency limit, the Floquet
effective Hamiltonians of these models then take the formalism of Eq. (14),
where the hopping amplitudes are controlled by the ratio of the driving
amplitude and driving frequency $K/\omega$ through the Bessel function ${\cal
J}_{0}(K/\omega)$. Since the localization-delocalization transitions in these
models are originated from the competition between the energy scales of
hopping $J$ and onsite potential $V$ AAH3 , the “photon-dressed” hopping
amplitude $J_{L,R}{\cal J}_{0}(K/\omega)$ provides a flexible knob to tune the
transitions between different non-Hermitian quasicrystalline phases of these
systems, as will be demonstrated below. Note in passing that the models M1–M5
do not possess the translation symmetry. They also do not have the chiral
symmetry due to the absence of sublattice or spin structures. Meanwhile, the
effective Hamiltonians of M1–M5 all possess the $\cal{PT}$-symmetry, which is
relevant for the characterization of their phase transitions.

###  III.2 M1: Floquet spectrum, localization transition and topological
invariant

Following Table 1, the time-dependent Hamiltonian of the quasicrystal M1 reads

$\displaystyle\hat{H}_{1}(t)=$
$\displaystyle\sum_{n}J\left(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+{\rm
h.c.}\right)$ $\displaystyle+$ $\displaystyle\sum_{n}\left[Ve^{-i2\pi\alpha
n}-nK\cos(\omega t)\right]\hat{c}_{n}^{\dagger}\hat{c}_{n},$ (17)

which is non-Hermitian due to the complex onsite potential $Ve^{-i2\pi\alpha
n}$ and quasiperiodic when $\alpha$ is irrational. Following the procedure of
Sec. II, we find the Floquet effective Hamiltonian of the system under high-
frequency driving as

$\hat{H}_{1{\rm F}}=\sum_{n}\left[J{\cal
J}_{0}\left(\frac{K}{\omega}\right)\left(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+{\rm
h.c.}\right)+Ve^{-i2\pi\alpha n}\hat{c}_{n}^{\dagger}\hat{c}_{n}\right].$ (18)

The hopping strength now depends on the ratio between the amplitude and
frequency of the driving field, and can thus be controlled dynamically.
Meanwhile, the effective Hamiltonian $\hat{H}_{1{\rm F}}$ possesses the ${\cal
PT}$-symmetry as $V_{n}=V_{-n}^{*}$, which means that its Floquet spectrum
could be real in certain parameter regions.

Phase | Extended | Localized
---|---|---
Condition | $|V|<|J{\cal J}_{0}(K/\omega)|$ | $|V|>|J{\cal J}_{0}(K/\omega)|$
IPR |  $\simeq 0$ for all states |  $>0$ for all states
Lyapunov exponent | $\lambda=\ln\left|\frac{V}{J{\cal J}_{0}(K/\omega)}\right|\begin{cases}<0&{\rm Extended}\\\ >0&{\rm Localized}\end{cases}$
Quasienergy | $E(k)=2J{\cal J}_{0}(K/\omega)\cos k$ | $E(k)=2J{\cal J}_{0}(K/\omega)\cos(k-i\lambda)$
Winding number | $w=\int_{0}^{2\pi}\frac{d\theta}{2\pi i}\partial_{\theta}\ln\det[\hat{H}^{\prime}_{1{\rm F}}(\theta)-E_{0}]=\begin{cases}0&{\rm Extended}\\\ -1&{\rm Localized}\end{cases}$
Table 2: Summary of the results for the non-Hermitian Floquet quasicrystal M1.
$J$ is the nearest-neighbor hopping amplitude, $V$ is the amplitude of onsite
non-Hermitian quasiperiodic potential, $K$ is the driving amplitude, $\omega$
is the driving frequency, and ${\cal J}_{0}(K/\omega)$ denotes the Bessel
function of first kind. $\lambda$ denotes the Lyapunov exponent Note1 . The
parameter $k$ fills uniformly the range of $[0,2\pi)$.
$\hat{H}^{\prime}_{1{\rm F}}(\theta)$ is obtained from $\hat{H}_{1{\rm F}}$
after taking the discrete Fourier transformation and introducing a phase twist
$e^{-i\theta}$ to its top-right corner matrix element in momentum
representation. $E_{0}$ is the base quasienergy.

The eigenvalue equation of $\hat{H}_{1{\rm F}}$ is $\hat{H}_{{\rm
1F}}|\psi\rangle=E|\psi\rangle$. Inserting the single-particle state
$|\psi\rangle=\sum_{n}\psi_{n}\hat{c}_{n}^{\dagger}|0\rangle$ into the
equation, we obtain the eigenvalue equation in the lattice representation as

$J{\cal
J}_{0}\left(\frac{K}{\omega}\right)(\psi_{n+1}+\psi_{n-1})+Ve^{-i2\pi\alpha
n}\psi_{n}=E\psi_{n}.$ (19)

Here $E\mod 2\pi$ corresponds to the quasienergy of $\hat{H}_{1{\rm F}}$. For
a lattice of length $L$ and under the periodic boundary condition (PBC), one
can take a discrete Fourier transformation from lattice to momentum
representations by expanding the amplitude $\psi_{n}$ as

$\psi_{n}=\frac{1}{\sqrt{L}}\sum_{m=1}^{L}\varphi_{m}e^{-i2\pi\alpha mn}.$
(20)

The transformed eigenvalue equation in momentum space then reads

$V\varphi_{n-1}+2J{\cal J}_{0}\left(\frac{K}{\omega}\right)\cos(2\pi\alpha
n)\varphi_{n}=E\varphi_{n}.$ (21)

Following the method outlined in Ref. LonghiQC2 , it can be readily shown that
when $\alpha$ takes irrational values, the non-Hermitian Floquet quasicrystal
M1 possesses a ${\cal PT}$-transition and a localization-delocalization
transition under the condition

$|V|=|J{\cal J}_{0}(K/\omega)|.$ (22)

When $|V|<|J{\cal J}_{0}(K/\omega)|$, the hopping process dominates and all
Floquet eigenstates of $\hat{H}_{1{\rm F}}$ are extended with vanishing
Lyapunov exponents (diverging localization lengths). The quasienergy
dispersion of $\hat{H}_{1{\rm F}}$ in this case takes the form of
$E(k)=2J{\cal J}_{0}\left(K/\omega\right)\cos(k)\in\mathbb{R}$, where
$k\in[0,2\pi)$. When $|V|>|J{\cal J}_{0}(K/\omega)|$, the quasiperiodic
potential dominates, and all Floquet eigenstates of $\hat{H}_{1{\rm F}}$ are
localized with the same Lyapunov exponent (inverse localization length)
$\lambda=-\ln|J{\cal J}_{0}(K/\omega)/V|>0$. The Floquet spectrum of the
system in this case has the form of $E(k)=2J{\cal
J}_{0}\left(K/\omega\right)\cos(k-i\lambda)\in\mathbb{C}$, where
$k\in[0,2\pi)$. The extended and localized phases could be further
distinguished by a topological winding number of the Floquet spectrum. With
the eigenvalue equation (21), we find the matrix elements of the system’s
Floquet effective Hamiltonian $\hat{H}^{\prime}_{1{\rm F}}$ in momentum space
to be

$\displaystyle[\hat{H}^{\prime}_{1{\rm F}}]_{n,n}$ $\displaystyle=2J{\cal
J}_{0}\left(\frac{K}{\omega}\right)\cos(2\pi\alpha n)\qquad n=1,...,L;$
$\displaystyle{}[\hat{H}^{\prime}_{1{\rm F}}]_{n+1,n}$
$\displaystyle=[\hat{H}^{\prime}_{1{\rm F}}]_{1,L}=V,\qquad n=1,...,L-1.$ (23)

Multiplying the corner matrix element $[\hat{H}^{\prime}_{1{\rm F}}]_{1,L}$ by
a phase factor $e^{-i\theta}$, we obtain the effective Hamiltonian
$\hat{H}^{\prime}_{1{\rm F}}(\theta)$ under the twist boundary condition in
momentum space. The spectral winding number of $\hat{H}^{\prime}_{1{\rm
F}}(\theta)$ can then be obtained following the construction of Ref. ChenQC1 .
Such a winding number counts the number of times the Floquet spectrum of
$\hat{H}^{\prime}_{1{\rm F}}(\theta)$ winds around a base quasienergy $E_{0}$
on the complex plane ${\rm Re}E$-${\rm Im}E$ when $\theta$ changes from zero
to $2\pi$. When the quasienergy spectrum is real (complex), the value of this
winding number is found to be zero ($-1$). This winding number could thus
serve as a topological order parameter to distinguish the extended phase (with
real quasienergies) and localized phase (with complex quasienergies) of the
system. In Table 2, we summarize the key results about the spectrum,
localization transition and topological invariant of the non-Hermitian Floquet
quasicrystal M1. Notably, the condition of localization transition in the
system now depends on the parameter $K/\omega$ of the driving field. The
states of the system can thus be tuned between extended and localized phases
dynamically by varying the ratio $K/\omega$ between the amplitude and
frequency of the driving force.

Figure 2: The maximal imaginary parts of Floquet quasienergies in panel (a),
the minimal values of IPRs in panel (b), and the signs of Lyapunov exponents
in panel (c) of the non-Hermitian Floquet quasicrystal M1. Other system
parameters are set as $\alpha=\frac{\sqrt{5}-1}{2}$, $J=1$, and the length of
lattice is taken as $L=610$. The red dashed line in each figure panel denotes
the boundary between the extended phase (with real quasienergies for all
Floquet eigenstates) and the localized phase (with complex quasienergies for
all Floquet eigenstates), which satisfies the condition $|V|=|J{\cal
J}_{0}(K/\omega)|$ (see Table 2). The topological winding number $w$ of each
phase is denoted explicitly in panel (c).

To demonstrate our theoretical results, we compute the spectrum $E$, inverse
participation ratio (IPR), Lyapunov exponent $\lambda$ and winding number $w$
of the M1 versus the amplitude of onsite potential $V$ and the ratio between
driving amplitude and frequency $K/\omega$. The numerical results are
presented in Fig. 2. In the calculation, we choose the undressed hopping
amplitude $J=1$ and the quasicrystal parameter
$\alpha=\frac{\sqrt{5}-1}{2}\simeq\frac{p}{q}$, with $p,q$ being two adjacent
elements of the Fibonacci sequence ($p<q$). The length of lattice is chosen as
$L=610$, and the base quasienergy is set as $E_{0}=0$. In Fig. 2(a), we show
the maximal imaginary parts of quasienergy $\max|{\rm Im}E|$ versus $V$ and
$K/\omega$. The red dashed line is given by the phase boundary equation
$|V|=|J{\cal J}_{0}(K/\omega)|$. It is clear that $\max|{\rm Im}E|=0$
($\max|{\rm Im}E|>0$) when $|V|<|J{\cal J}_{0}(K/\omega)|$ ($|V|>|J{\cal
J}_{0}(K/\omega)|$). The condition $|V|=|J{\cal J}_{0}(K/\omega)|$ thus
determines the boundary of the ${\cal PT}$-transition between real and complex
quasienergy spectrum of the system.

For any normalized Floquet right eigenstate
$|\psi\rangle=\sum_{n}\psi_{n}\hat{c}_{n}^{\dagger}|0\rangle$ of
$\hat{H}_{1{\rm F}}$, with $\sum_{n=1}^{L}|\psi_{n}|^{2}=1$, we define its IPR
as

${\rm IPR}=\sum_{n=1}^{L}|\psi_{n}|^{4}.$ (24)

The minimum of IPRs over all eigenstates at a fix set of system parameters
under the PBC then yields the $\min({\rm IPR})$. In Fig. 2(b), we present the
$\min({\rm IPR})$ versus $V$ and $K/\omega$, which is found to be $\simeq 0$
when $|V|<|J{\cal J}_{0}(K/\omega)|$ (below the red dashed line) and $>0$
otherwise. The maximum of IPRs of all eigenstates is found to have the same
dependence on $V$ and $K/\omega$. Therefore, the system is indeed in the
extended (localized) phase with all states being delocalized (localized) when
$|V|<|J{\cal J}_{0}(K/\omega)|$ ($|V|>|J{\cal J}_{0}(K/\omega)|$). Finally, we
present the signs of Lyapunov exponent $\lambda=-\ln|J{\cal
J}_{0}(K/\omega)/V|$ and the topological winding numbers $w$ of M1 versus $V$
and $K/\omega$ in Fig. 2(c). It is clear that the winding number $w=0$ in the
extended phase with real quasieneriges, and $w=-1$ in the localized phase with
complex quasienergies. The two regions are separated by the phase boundary
$|V|=|J{\cal J}_{0}(K/\omega)|$ (red dashed line). Therefore, it is verified
that $w$ could be utilized as a topological order parameter to discriminate
the ${\cal PT}$-invariant extended phase and ${\cal PT}$-breaking localized
phase of the non-Hermitian Floquet quasicrystal M1. Note that without the
driving field, the phase boundary between the extended and localized states
reduces to a point at $V=J$. The driving force instead allows the localized
phase to disappear and reappear alternately with the change of the driving
parameter $K/\omega$, which highlights the advantage of Floquet engineering in
the realization and control of phase transitions in non-Hermitian
quasicrystals.

###  III.3 M2: Floquet spectrum, localization transition and topological
invariant

We next consider a driven non-Hermitian AAH quasicrystal with an imaginary
phase shift $i\gamma$, whose Hamiltonian takes the form of

$\displaystyle\hat{H}_{2}(t)=$
$\displaystyle\sum_{n}J\left(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+{\rm
h.c.}\right)$ $\displaystyle+$ $\displaystyle\sum_{n}\left[V\cos(2\pi\alpha
n+i\gamma)-nK\cos(\omega t)\right]\hat{c}_{n}^{\dagger}\hat{c}_{n}.$ (25)

According to the theory presented in Sec. II, the Floquet effective
Hamiltonian of this non-Hermitian quasicrystal M2 is given by

$\displaystyle\hat{H}_{2{\rm F}}=$ $\displaystyle\sum_{n}J{\cal
J}_{0}\left(\frac{K}{\omega}\right)\left(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+{\rm
h.c.}\right)$ $\displaystyle+$ $\displaystyle\sum_{n}V\cos(2\pi\alpha
n+i\gamma)\hat{c}_{n}^{\dagger}\hat{c}_{n}.$ (26)

It is clear that the hopping amplitude is modified by the driving field and
its magnitude could be controlled by the ratio between the amplitude and
frequency of the driving force $K/\omega$. Besides, the $\hat{H}_{2{\rm F}}$
also holds the ${\cal PT}$-symmetry, since its onsite potential
$V_{n}=V\cos(2\pi\alpha n+i\gamma)=V_{-n}^{*}$. The Floquet quasienergy
spectrum of $\hat{H}_{2{\rm F}}$ could then take real values in certain
parameter regions.

Plugging the single-particle state
$|\psi\rangle=\sum_{n}\psi_{n}\hat{c}_{n}^{\dagger}|0\rangle$ into
$\hat{H}_{2{\rm F}}|\psi\rangle=E|\psi\rangle$, we obtain

$J{\cal
J}_{0}\left(\frac{K}{\omega}\right)(\psi_{n+1}+\psi_{n-1})+V\cos(2\pi\alpha
n+i\gamma)\psi_{n}=E\psi_{n},$ (27)

where $E\mod 2\pi$ refers to the quasienergy. Taking the PBC for a finite
lattice of length $L$ and performing the discrete Fourier transformation by
Eq. (20), we find the eigenvalue equation in momentum space to be

$\frac{V}{2}(e^{\gamma}\varphi_{n-1}+e^{-\gamma}\varphi_{n+1})+2J{\cal
J}_{0}\left(\frac{K}{\omega}\right)\cos(2\pi\alpha
n)\varphi_{n}=E\varphi_{n},$ (28)

which can also be viewed formally as a quasiperiodic lattice with
nonreciprocal hopping and “photon-dressed” onsite potential.

Phase | Extended | Localized
---|---|---
Condition | $|V|e^{|\gamma|}<|2J{\cal J}_{0}(K/\omega)|$ | $|V|e^{|\gamma|}>|2J{\cal J}_{0}(K/\omega)|$
IPR |  $\simeq 0$ for all states |  $>0$ for all states
Lyapunov exponent | $\lambda=\ln\left|\frac{Ve^{|\gamma|}}{2J{\cal J}_{0}(K/\omega)}\right|\begin{cases}<0&{\rm Extended}\\\ >0&{\rm Localized}\end{cases}$
Quasienergy | Real | Complex
Winding number | $w=\int_{0}^{2\pi}\frac{d\theta}{2\pi i}\partial_{\theta}\ln\det[\hat{H}_{2{\rm F}}(\theta)-E_{0}]=\begin{cases}0&{\rm Extended}\\\ -1&{\rm Localized}\end{cases}$
Table 3: Summary of the results for the non-Hermitian Floquet quasicrystal M2.
$J$ is the nearest-neighbor hopping amplitude, $V$ is the amplitude of onsite
potential, $\gamma$ is the imaginary part of superlattice phase shift, $K$ is
the driving amplitude, $\omega$ is the driving frequency, and ${\cal
J}_{0}(K/\omega)$ denotes the Bessel function of first kind. $\lambda$ denotes
the Lyapunov exponent Note1 . $H_{2{\rm F}}(\theta)$ is obtained from
$H_{2{\rm F}}$ after setting $V_{n}=V\cos(2\pi\alpha n+i\gamma+\theta/L)$,
with $L$ being the length of lattice. $E_{0}$ is the base quasienergy.

Following the method developed in Ref. LonghiQC1 , it is straightforward to
show that the non-Hermitian Floquet quasicrystal M2 could undergo a ${\cal
PT}$-transition and a localization-delocalization transition when its
parameters satisfy the condition

$|V|e^{|\gamma|}=|2J{\cal J}_{0}(K/\omega)|,$ (29)

which is clearly dependent on the parameters of the driving field $K$ and
$\omega$. If $|V|e^{|\gamma|}<|2J{\cal J}_{0}(K/\omega)|$, the hopping energy
overwhelms the quasiperiodic onsite disorder, and the system sits in an
extended phase. All Floquet eigenstates in this phase have real quasieneriges
and delocalized profiles (with vanishing IPRs) throughout the lattice. If
$|V|e^{|\gamma|}>|2J{\cal J}_{0}(K/\omega)|$, the non-Hermitian onsite
potential governs the behavior of the system and drives it into a localized
phase. The Floquet states of the system in this phase are all localized (with
finite IPRs) and have complex quasienergies. These localized eigenstates are
found to share the common Lyapunov exponent $\lambda=-\ln|2J{\cal
J}_{0}(K/\omega)/(Ve^{|\gamma|})|$, whose value is also controlled by the
ratio between the amplitude and frequency of the driving force. To construct a
topological winding number for M2, we add a real phase shift to the
quasiperiodic onsite potential LonghiQC1  by setting $\cos(2\pi\alpha
n+i\gamma)\rightarrow\cos(2\pi\alpha n+i\gamma+\theta/L)$ in Eq. (26), where
$L$ is the length of lattice. We denote the phase-shifted effective
Hamiltonian by $\hat{H}_{2{\rm F}}(\theta)$, and the number of times that its
quasienergy spectrum winds around a base point $E_{0}$ on the complex plane
defines a topological invariant of the system, which is equal to $-1$ when the
spectrum is complex, and zero otherwise. Therefore, the non-Hermitian Floquet
quasicrystal M1 possesses an extended phase with real spectrum and winding
number $w=0$, and a localized phase with complex spectrum and winding number
$w=-1$. By tuning the ratio $K/\omega$ between the amplitude and frequency of
the driving force, the system could also roam alternately between the ${\cal
PT}$-invariant extended phase and the ${\cal PT}$-breaking localized phase.
The topological and transport properties of the non-Hermitian quasicrystal M2
could thus be controlled by the periodic driving field. We summarize the key
results about the spectrum, phases and topological features of M2 in Table 3.

Figure 3: The maximal imaginary parts of Floquet quasienergies in panel (a),
the minimal values of IPRs in panel (b), and the signs of Lyapunov exponents
in panel (c) of the non-Hermitian Floquet quasicrystal M2. Other system
parameters are set as $\alpha=\frac{\sqrt{5}-1}{2}$, $J=2$, $V=1$, and the
length of lattice is taken as $L=610$. The red dashed line in each figure
panel denotes the boundary between the extended phase (with real quasienergies
for all Floquet eigenstates) and the localized phase (with complex
quasienergies for all states), which satisfies the condition
$|V|e^{|\gamma|}=|J{\cal J}_{0}(K/\omega)|$ (see Table 3). The topological
winding number $w$ of each phase is denoted explicitly in panel (c).

To verify our theoretical predictions, we compute the quasienergy spectrum,
IPR, Lyapunov exponent and winding number of the non-Hermitian Floquet
quasicrystal M2 numerically, with results reported in Fig. 3. In the
calculation, we choose the system parameters $(J,V)=(2,1)$ and the
quasiperiodic parameter $\alpha=\frac{\sqrt{5}-1}{2}\simeq\frac{p}{q}$, where
$p,q$ are two neighboring elements of the Fibonacci sequence ($p<q$). The
length of lattice is set as $L=610$, and the base quasienergy is chosen to be
$E_{0}=0$. In Fig. 3(a), we show the maximum of the imaginary parts of Floquet
spectrum versus the parameter of driving force $K/\omega$ and imaginary part
of phase shift $\gamma$. The red-dashed line highlights the boundary of ${\cal
PT}$-transition, which is determined by Eq. (29). The numerical results
confirm that the Floquet spectrum of the system is indeed real when
$|V|e^{|\gamma|}<|2J{\cal J}_{0}(K/\omega)|$ (below the red-dashed line), and
complex in other circumstances. In Fig. 3(b), we show the minimum of IPRs,
which behaves similarly as the maximum of IPRs versus $K/\omega$ and $\gamma$
at each given set of system parameters. The results indicate that the IPRs of
all states approach zero when $|V|e^{|\gamma|}<|2J{\cal J}_{0}(K/\omega)|$
(below the red-dashed line), and become finite if $|V|e^{|\gamma|}>|2J{\cal
J}_{0}(K/\omega)|$. Therefore, the condition in Eq. (29) defines the boundary
between the ${\cal PT}$-invariant extended phase and the ${\cal PT}$-breaking
localized phase of the non-Hermitian Floquet quasicrystal M2. Finally, we show
the signs of Lyapunov exponent $\lambda$ and the topological winding numbers
$w$ versus $K/\omega$ and $\gamma$ in Fig. 3(c), in which we observe
$\lambda<0$ ($\lambda>0$) for all states in the extended (localized) phase
with $w=0$ ($w=-1$), as shown by the blue (yellow) regions below (above) the
phase boundary (i.e., the red-dashed line satisfying Eq. (29)). Therefore, the
winding number $w$ works as a topological invariant to distinguish the ${\cal
PT}$-breaking localized phase and ${\cal PT}$-invariant extended phase of the
system. Similar to the case of M1, the driving force now allows one to change
the system over a series of transitions between localized and delocalized
phases. This further demonstrates the generality of our Floquet engineering
approach to the design and control of non-Hermitian quasicrystalline phases in
different model systems.

###  III.4 M3: Floquet spectrum, localization transition and topological
invariant

Phase | Extended | Localized
---|---|---
Condition | $|V|e^{-|\gamma|}<|2J{\cal J}_{0}(K/\omega)|$ | $|V|e^{-|\gamma|}>|2J{\cal J}_{0}(K/\omega)|$
IPR |  $\simeq 0$ for all states |  $>0$ for all states
Lyapunov exponent | $\lambda=\ln\left|\frac{Ve^{-|\gamma|}}{2J{\cal J}_{0}(K/\omega)}\right|\begin{cases}<0&{\rm Extended}\\\ >0&{\rm Localized}\end{cases}$
Quasienergy | Complex | Real
Winding number | $w=\int_{0}^{2\pi}\frac{d\theta}{2\pi i}\partial_{\theta}\ln\det[H_{3{\rm F}}(\theta)-E_{0}]=\begin{cases}-1&{\rm Extended}\\\ 0&{\rm Localized}\end{cases}$
Table 4: Summary of results for the non-Hermitian Floquet quasicrystal M3. $J$
is the symmetric part of nearest-neighbor hopping amplitude, $\gamma$ controls
the asymmetry between left and right hopping amplitudes, $V$ is the amplitude
of onsite quasiperiodic potential, $K$ is the driving amplitude, $\omega$ is
the driving frequency, and ${\cal J}_{0}(K/\omega)$ denotes the Bessel
function of first kind. $\lambda$ denotes the Lyapunov exponent Note1 .
$H_{3{\rm F}}(\theta)$ is obtained from $H_{3{\rm F}}$ after taking the
periodic boundary condition and setting its top-right (bottom-left) corner
matrix element to be $J_{L}{\cal J}_{0}(K/\omega)e^{-i\theta}$ ($J_{R}{\cal
J}_{0}(K/\omega)e^{i\theta}$). $E_{0}$ is the base quasienergy.

In the thrid part of this section, we consider a nonreciprocal variant of the
AAH model LonghiQC4 ; ChenQC1 , which is subject to a periodically modulated
driving force. The time-dependent Hamiltonian of the model takes the form

$\displaystyle\hat{H}_{3}(t)=$
$\displaystyle\sum_{n}J\left(e^{-\gamma}\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+e^{\gamma}\hat{c}_{n+1}^{\dagger}\hat{c}_{n}\right)$
$\displaystyle+$ $\displaystyle\sum_{n}\left[V\cos(2\pi\alpha n)-nK\cos(\omega
t)\right]\hat{c}_{n}^{\dagger}\hat{c}_{n},$ (30)

which is non-Hermitian if $\gamma\neq 0$. Following the steps of Sec. II, we
obtain the Floquet effective Hamiltonian of this non-Hermitian quasicrystal M3
under the high-frequency approximation as

$\displaystyle\hat{H}_{3{\rm F}}=$ $\displaystyle\sum_{n}{\cal
J}_{0}\left(\frac{K}{\omega}\right)J\left(e^{-\gamma}\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+e^{\gamma}\hat{c}_{n+1}^{\dagger}\hat{c}_{n}\right)$
$\displaystyle+$ $\displaystyle\sum_{n}V\cos(2\pi\alpha
n)\hat{c}_{n}^{\dagger}\hat{c}_{n}.$ (31)

The hopping amplitude is now controlled by the amplitude $K$ and frequency
$\omega$ of the driving field. Applying $\hat{H}_{3{\rm F}}$ to the eigenstate
$|\psi\rangle=\sum_{n}\psi_{n}\hat{c}_{n}^{\dagger}|0\rangle$, we further
obtain the eigenvalue equation in lattice representation

${\cal
J}_{0}\left(\frac{K}{\omega}\right)J(e^{\gamma}\psi_{n-1}+e^{-\gamma}\psi_{n+1})+V\cos(2\pi\alpha
n)\psi_{n}=E\psi_{n}.$ (32)

That this equation takes the same form as Eq. (28) under the exchange of
system parameters $V\leftrightarrow 2J{\cal J}_{0}(K/\omega)$, which implies
that the non-Hermitian Floquet quasicrystals M2 and M3 are dual to each other
concerning their spectrum and states. Furthermore, taking the Fourier
transformation of Eq. (32) with the help of Eq. (20), we obtain

$\frac{V}{2}(\varphi_{n+1}+\varphi_{n-1})+2J{\cal
J}_{0}\left(\frac{K}{\omega}\right)\cos(2\pi\alpha
n-i\gamma)\varphi_{n}=E\varphi_{n},$ (33)

which again shares the same form with Eq. (27). Putting together, we find that
under the PBC, there is indeed a duality relation between the M2 and M3, in
the sense that the localization properties of the M2 in momentum space is
consistent with the localization properties of the M3 in position space under
the PBC. With the help of this duality relation, we can immediately write down
the phase boundary condition that determines the ${\cal PT}$-transition and
localization-delocalization transition in the M3 under the PBC, i.e.,

$|V|=|2J{\cal J}_{0}(K/\omega)|e^{|\gamma|}.$ (34)

In parallel with the discussions of Subsec. III.3, if $|V|<|2J{\cal
J}_{0}(K/\omega)|e^{|\gamma|}$, the M3 is found to be in a ${\cal
PT}$-breaking extended phase (dual to the localized phase of M2) with complex
quasienergies and vanishing IPRs for all Floquet eigenstates. When
$|V|>|2J{\cal J}_{0}(K/\omega)|e^{|\gamma|}$, the M3 is in a ${\cal
PT}$-invariant localized phase (dual to the extended phase of M2), where all
Floquet eigenstates have real quasienergies, finite IPRs, and the same
positive Lyapunov exponent $\lambda=-\ln|2J{\cal
J}_{0}(K/\omega)e^{|\gamma|}/V|$ in the thermodynamic limit
$L\rightarrow\infty$. Note that due to the dependence of phase boundary
condition Eq. (34) and Lyapunov exponent on the ratio between the amplitude
and frequency of the driving force $K/\omega$, transitions between phases with
different spectrum and transport features in the M3 can also be induced by the
driving field.

To define a spectral winding number, we take the twist boundary condition
ChenQC1  by adding phase factors to the corner matrix elements of
$\hat{H}_{3{\rm F}}$ in the lattice representation, i.e.,

$[\hat{H}_{3{\rm F}}]_{1,L}=Je^{(\gamma-i\theta)},\qquad[\hat{H}_{3{\rm
F}}]_{L,1}=Je^{(i\theta-\gamma)},$ (35)

where $L$ is the length of lattice and $\theta\in[0,2\pi)$. The spectrum of
the phase-dependent Hamiltonian $\hat{H}_{3{\rm F}}(\theta)$ could now possess
a winding number around certain base quasienergy $E_{0}$ on the complex plane
when $\theta$ changes over a cycle from zero to $2\pi$. Such a winding number
could be nonzero only if the spectrum of $\hat{H}_{3{\rm F}}(\theta)$ is
complex. Therefore, we expect a quantized winding number to appear for the
${\cal PT}$-breaking extended phase, and a vanishing one for the ${\cal
PT}$-invariant localized phase of the system. Such a winding number can then
be used as a topological invariant to distinguish the two non-Hermitian
Floquet quasicrystalline phases of M3 with distinct spectrum and transport
nature. We summarize the main results about the M3 in Table 4 for the ease of
reference.

Figure 4: The maximal imaginary parts of quasienergies in panel (a), the
minimal values of IPRs in panel (b), and the signs of Lyapunov exponents in
panel (c) of the non-Hermitian Floquet quasicrystal M3. Other system
parameters are set as $\alpha=\frac{\sqrt{5}-1}{2}$, $J=1$, $V=1$, and the
length of lattice is taken as $L=610$. The red dashed line in each figure
panel denotes the boundary between the extended phase (with complex
quasienergies for all states) and the localized phase (with real
quasienergies), which satisfies the condition $|V|e^{-|\gamma|}=|J{\cal
J}_{0}(K/\omega)|$ (see Table 4). The topological winding number $w$ of each
phase is denoted explicitly in panel (c).

To confirm our theoretical predictions, we present the Floquet spectrum, IPR,
Lyapunov exponent and winding number of the M3 versus the imaginary part of
phase shift $\gamma$ and the ratio between the amplitude and frequency of
driving field $K/\omega$ in Fig. 4. Other system parameters are set as $J=V=1$
in the calculation. The quasicrystal parameter
$\alpha=\frac{\sqrt{5}-1}{2}\simeq\frac{p}{q}$, with $p,q$ being adjacent
elements of the Fibonacci sequence ($p<q$). The length of lattice is chosen to
be $L=610$, and the base quasienergy is set as $E_{0}=0$. In Figs. 4(a)–(c),
the red dashed lines refer to the phase boundary satisfying the Eq. (34). We
observe that the maximum of imaginary parts of quasienergy $\max|{\rm Im}E|$
vanishes below the red dashed line and taking finite values above it, as shown
in Fig. 4(a). The system thus undergoes a ${\cal PT}$-transition when
$|V|e^{-|\gamma|}$ increases from below $|2J{\cal J}_{0}(K/\omega)|$ to above
it, through which the Floquet spectrum changes from real to complex. In Fig.
4(b), we show the minimum of IPRs over all Floquet eigenstates at each given
set of system parameters. The maximum of IPRs versus $K/\omega$ and $\gamma$
are found to have the same pattern. Therefore, all states of the system are
extended in the ${\cal PT}$-breaking regime (with complex spectrum), and
localized in the ${\cal PT}$-invariant regime (with real spectrum). In Fig.
4(c), we show the signs of Lyapunov exponent ${\rm sgn}(\lambda)$ and winding
numbers $w$ of different phases. We obtain identical and positive (negative)
$\lambda$ for all states below (above) the phase boundary line, implying a
phase with localized (delocalized) states at all quasienergies. Moreover, we
find the winding number $w=0$ and $w=-1$ in the localized and delocalized
phases, respectively, and a quantized change of $w$ across the phase boundary
(red dashed line). The driven non-Hermitian quasicrystal M3 thus possesses a
${\cal PT}$-breaking extended phase with topological winding number $w=-1$, in
which all Floquet eigenstates are delocalized with complex quasienergies, and
a ${\cal PT}$-invariant localized phase with topological winding number $w=0$,
in which all Floquet eigenstates are localized with real quasienergies. As the
phase boundary is controlled by the amplitude and frequency of the driving
force, the system can also be tuned dynamically to go through transitions
between different non-Hermitian quasicrystalline phases following our Floquet
engineering scheme.

###  III.5 M4: Floquet spectrum, localization transition, mobility edge and
topological invariant

Phase | Localized | Mobility Edge | Extended
---|---|---|---
Condition | $\gamma<\gamma_{1}$ | $\gamma_{1}<\gamma<\gamma_{2}$ | $\gamma_{2}<\gamma$
IPR |  $>0$ for all states |  $>0$ & $\simeq 0$ coexist |  $\simeq 0$ for all states
Lyapunov exponent | $\lambda(E)={\rm arccosh}\left[\frac{\sqrt{[2J{\cal J}_{0}(K/\omega)+E_{{\rm r}}]^{2}+(V-E_{{\rm i}})^{2}}+\sqrt{[2J{\cal J}_{0}(K/\omega)-E_{{\rm r}}]^{2}+(V-E_{{\rm i}})^{2}}}{4|J{\cal J}_{0}(K/\omega)|}\right]\begin{cases}=0&{\rm Extended}\\\ >0&{\rm Localized}\end{cases}$
Quasienergy | $E=\begin{cases}\pm\sqrt{[2J{\cal J}_{0}(K/\omega)\cos(\beta+i\gamma)]^{2}+V^{2}\cot^{2}(\beta+i\gamma)}&{\rm Localized\,\,States}\\\ 2J{\cal J}_{0}(K/\omega)\cos\beta+iV&{\rm Extended\,\,States}\end{cases}$
Winding number | $w_{1,2}=\int_{0}^{\pi}\frac{d\theta}{2\pi i}\partial_{\theta}\ln\det[H_{4{\rm F}}(\theta)-E_{1,2}]=\begin{cases}(1,1)&{\rm Localized}\\\ (0,1)&{\rm Mobility\,\,Edge}\\\ (0,0)&{\rm Extended}\end{cases}$
Table 5: Summary of results for the non-Hermitian Floquet quasicrystal M4. $J$
is the nearest-neighbor hopping amplitude, $V$ is the amplitude of onsite
potential, $\gamma$ is the imaginary part of superlattice phase shift, $K$ is
the driving amplitude, $\omega$ is the driving frequency, and ${\cal
J}_{0}(K/\omega)$ denotes the Bessel function of first kind. The phase
variable $\beta\in[-\pi,\pi)$. $H_{4{\rm F}}(\theta)$ is obtained from
$H_{4{\rm F}}$ after setting $V_{n}=V\tan(\pi\alpha n+i\gamma+\theta/L)$, with
$L$ being the length of lattice. $E_{1}=iV$ and $E_{2}=2J{\cal
J}_{0}(K/\omega)+iV$ are two base quasienergies with respect to which the two
winding numbers $(w_{1},w_{2})$ are defined.

To further reveal the usefulness of the Floquet engineering scheme, we now
apply it to a non-Hermitian quasicrystal that could possess energy-dependent
mobility edges. In the lattice representation, the Hamiltonian of the system
describes a non-Hermitian extension of the Maryland model MM1 ; MM2 ; MM3
subjecting to an oscillating force, which takes the form

$\displaystyle\hat{H}_{4}(t)=$
$\displaystyle\sum_{n}J\left(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+{\rm
h.c.}\right)$ $\displaystyle+$ $\displaystyle\sum_{n}\left[V\tan(\pi\alpha
n+i\gamma)-nK\cos(\omega t)\right]\hat{c}_{n}^{\dagger}\hat{c}_{n}.$ (36)

Following the theory presented in Sec. II, the Floquet effective Hamiltonian
of this non-Hermitian quasicrystal M4 reads

$\displaystyle\hat{H}_{4{\rm F}}=$ $\displaystyle\sum_{n}J{\cal
J}_{0}\left(\frac{K}{\omega}\right)\left(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+{\rm
h.c.}\right)$ $\displaystyle+$ $\displaystyle\sum_{n}V\tan(\pi\alpha
n+i\gamma)\hat{c}_{n}^{\dagger}\hat{c}_{n},$ (37)

in which the fast driving field generates a modulation to the hopping
amplitude through the factor ${\cal J}_{0}(K/\omega)$. Expressing the Floquet
eigenstate as $|\psi\rangle=\sum_{n}\psi_{n}\hat{c}_{n}^{\dagger}|0\rangle$,
we obtain the eigenvalue equation

$J{\cal
J}_{0}\left(\frac{K}{\omega}\right)(\psi_{n+1}+\psi_{n-1})+V\tan(\pi\alpha
n+i\gamma)\psi_{n}=E\psi_{n},$ (38)

where $E\mod 2\pi$ refers to the quasienergy of $|\psi\rangle$.

Following the mapping between the Maryland model and an equivalent Floquet
problem MM1 ; MM2 ; MM3 , it can be shown that this non-Hermitian quasicrystal
M4 possesses two phase transitions at $\gamma=\gamma_{1}$ and
$\gamma=\gamma_{2}$, with

$\displaystyle\gamma_{1}$ $\displaystyle=\frac{1}{2}{\rm
arcsinh}\left(V/|J{\cal J}_{0}(K/\omega)|\right),$ (39)
$\displaystyle\gamma_{2}$ $\displaystyle=\frac{1}{2}{\rm
arccosh}\sqrt{\Delta+\sqrt{\Delta^{2}-1}},$ (40)

where $\Delta\equiv(2+[V/J{\cal J}_{0}(K/\omega)]^{2})/2$. When the imaginary
part of phase shift $\gamma<\gamma_{1}$, all eigenstates of the system are
localized with quasienergies

$E=\pm\sqrt{[2J{\cal
J}_{0}(K/\omega)\cos(\beta+i\gamma)]^{2}+V^{2}\cot^{2}(\beta+i\gamma)},$ (41)

where $\beta\in[0,\pi]$ and ${\rm Im}E>0$. These quasienergies form a closed
loop around $E_{1}=iV$ on the complex plane, whose size and shape can be
controlled by the ratio between the amplitude and frequency of the driving
force.

At $\gamma=\gamma_{1}$, the system undergoes a transition from a localized
phase to a phase with quasienergy-dependent mobility edges, and stays in this
mobility edge phase when $\gamma\in(\gamma_{1},\gamma_{2})$. The spectrum of
the system in this phase can be separated into two parts. Within the range of
$(-2|\cos\beta_{0}|,2|\cos\beta_{0}|)$ on the complex energy plane, where

$\beta_{0}=\arccos\left(\cos(2\gamma)\sqrt{1-\frac{V^{2}}{[J{\cal
J}_{0}(K/\omega)\sin(2\gamma)]^{2}}}\right),$ (42)

the imaginary part of the spectrum is pinned at $V$, and $E$ takes the form of

$E=2J{\cal
J}_{0}(K/\omega)\cos\beta+iV,\quad\beta\in[-|\beta_{0}|,|\beta_{0}|].$ (43)

It is clear that the two ends of the spectrum along the real axis are
controlled by the driving field, which could merge into one when $K/\omega$ is
tuned to the zeros of the Bessel function ${\cal J}_{0}$. Since only the
states whose quasienergies satisfy Eq. (43) have delocalized profiles in the
mobility edge phase, the driving field provides us with a convenient knob to
engineer and control the mobility edges of the non-Hermitian Maryland model.
Beyond the range of $(-2|\cos\beta_{0}|,2|\cos\beta_{0}|)$, the spectrum of
the system is still described by Eq. (41), which now arranges into two loops
around $E_{2}^{\pm}=\pm 2J{\cal J}_{0}(K/\omega)+iV$.

At $\gamma=\gamma_{2}$, the system experiences a second transition from the
mobility edge phase to an extended phase, in which all eigenstates are
delocalized and their quasienergies are described by Eq. (43) for
$\gamma>\gamma_{2}$, with $\beta\in[-\pi,\pi)$. The spectrum now collapses
into a line ending at $E_{2}^{\pm}$, and the loops described by Eq. (41) all
vanish. The ending points of the spectrum of delocalized states could again be
controlled by the driving field.

Following the Avila’s global theory MM3 ; Avila1 ; Avila2 , the system is
found to have a quasienergy-dependent Lyapunov exponent $\lambda(E=E_{{\rm
r}}+iE_{{\rm i}})$ in the thermodynamic limit, where $E_{{\rm r}}$ and
$E_{{\rm i}}$ are the real and imaginary parts of $E$ [see Table 5 for the
expression of $\lambda(E)$]. In the extended and localized phases, one has
$\lambda(E)=0$ and $\lambda(E)>0$ for all states, respectively. In the
mobility edge phase, we have $\lambda(E)=0$ for eigenstates with $E_{{\rm
r}}\in(-2|\cos\beta_{0}|,2|\cos\beta_{0}|)$, and $\lambda(E)>0$ otherwise. As
expected, the Lyapunov exponents, and therefore the localization nature of the
Floquet states can be controlled by the ratio between the amplitude and
frequency of the driving field $K/\omega$.

Figure 5: The the minimal (maximal) values of IPRs in panel (a) [(b)], and the
minimum (maximum) of Lyapunov exponents in panel (c) [(d)] of the non-
Hermitian Floquet quasicrystal M4. Other system parameters are set as
$\alpha=\frac{\sqrt{5}-1}{2}$, $J=2$, $V=1$, and the length of lattice is
taken as $L=610$. The red dashed and dotted lines in each figure panel denote
the boundaries between the extended phase, mobility edge phase and localized
phase, respectively, which satisfy the conditions in Eqs. (39) and (40).
Figure 6: The topological phase diagram of the non-Hermitian Floquet
quasicrystal M4. System parameters are chosen as
$\alpha=\frac{\sqrt{5}-1}{2}$, $J=2$, $V=1$, and the length of lattice is
taken as $L=377$. Each region with a uniform color corresponds to a phase,
whose winding numbers $(w_{1},w_{2})$ are denoted explicitly therein. The red
dashed and dotted lines are phase boundaries satisfying Eqs. (39) and (40),
respectively.

Finally, following the routines of Subsecs. III.2–III.4, we could construct a
pair of spectral winding numbers $(w_{1},w_{2})$ to characterize the
topological nature of transitions between different phases (see Table 5 for
their definitions). In the localized phase, both the base energies $E_{1}=iV$
and $E_{2}=2J{\cal J}_{0}(K/\omega)+iV$ are encircled once by the spectrum of
$H_{4{\rm F}}(\theta)$ when $\theta$ changes over a cycle from zero to $\pi$,
and the winding numbers $(w_{1},w_{2})=(1,1)$. In the mobility edge phase,
only $E_{2}=2J{\cal J}_{0}(K/\omega)+iV$ is encircled once by the spectral of
$H_{4{\rm F}}(\theta)$ for $\theta\in[0,\pi]$, and the winding numbers
$(w_{1},w_{2})=(0,1)$. In the extended phase, none of the bases energies
$E_{1,2}$ are encircled by the spectral and we have $(w_{1},w_{2})=(0,0)$.
Therefore, these winding numbers could be employed to distinguish phases with
different transport properties in the M4. For the ease of reference, we
summarize the main results of the M4 in Table 5.

To verify our theoretical results, we present numerical calculations of the
IPRs, Lyapunov exponents and winding numbers of the non-Hermitian Floquet
quasicrystal M4 versus $K/\omega$ and $\gamma$ in Figs. 5 and 6. We find that
states with the minimum of IPRs $\simeq 0$ and Lyapunov exponent
$\lambda_{\min}\simeq 0$ start to emerge when $\gamma$ passes over
$\gamma_{1}$ from below, as shown in Figs. 5(a) and 5(c). This means that some
states of the system become delocalized after the first transition at
$\gamma=\gamma_{1}$ (red dashed line). When $\gamma$ further increases to
$\gamma_{2}$ (red dotted line), a second transition happens and all states
become extended with max(IPR) $\simeq 0$ and $\lambda_{\max}\simeq 0$ after
the transition ($\gamma>\gamma_{2}$), as shown in Figs. 5(b) and 5(d). The
boundaries between the three distinct quasicrystal phases are precisely
determined by Eqs. (39) and (40). Besides, the system could also enter and
leave the localized, mobility edge and extended phases alternately with the
increase of $K/\omega$, and the regime of mobility edge is modified at
different values of $K/\omega$. Therefore, both the localization-
delocalization transitions and mobility edges in the non-Hermitian Maryland
model M4 could be controlled by the driving field.

In Fig. 6, we compute the winding numbers $(w_{1},w_{2})$ of M4 following
their definitions in Table 5. We observe that $(w_{1},w_{2})=(1,1)$ for the
localized phase (below the red dashed line), $(w_{1},w_{2})=(0,1)$ for the
mobility edge phase (between the red dashed and dotted lines), and
$(w_{1},w_{2})=(0,0)$ for the extended phase (above the red dotted line).
These results confirm that the winding numbers $(w_{1},w_{2})$ could be
utilized to distinguish quasicrystal phases of the non-Hermitian Floquet
Maryland model with different transport nature, and characterize the
transitions between them. Notably, the winding numbers $(w_{1},w_{2})$ could
also change with the increase of the ratio between the amplitude and frequency
of the forcing $K/\omega$, which means that the topological properties of the
non-Hermitian Maryland model can be engineered by the driving field.

###  III.6 M5: Floquet spectrum, localization transition, mobility edge and
topological invariant

Phase | Extended | Mobility Edge | Localized
---|---|---|---
Condition | $E_{r}^{\max}<E_{c}$ | $E_{r}^{\min}<E_{c}<E_{r}^{\max}$ | $E_{c}<E_{r}^{\min}$
IPR |  $\simeq 0$ for all states |  $>0$ & $\simeq 0$ coexist |  $>0$ for all states
Mobility edge | $E_{c}=|J{\cal J}_{0}(K/\omega)(\eta+1/\eta)|$
Quasienergy | Real | Complex
Winding number | $w=\int_{0}^{2\pi}\frac{d\theta}{2\pi i}\partial_{\theta}\ln\det[H_{5F}(\theta)-E_{c}^{+}]=\begin{cases}1&{\rm Mobility\,\,edge}\\\ 0&{\rm{\rm Otherwise}}\end{cases}$
Table 6: Summary of results for the non-Hermitian Floquet quasicrystal M5. $J$
is the nearest-neighbor hopping amplitude, $\eta$ controls the non-
Hermiticity, $K$ is the driving amplitude, $\omega$ is the driving frequency,
and ${\cal J}_{0}(K/\omega)$ denotes the Bessel function of first kind.
$H_{5{\rm F}}(\theta)$ is obtained from $H_{5{\rm F}}$ after setting
$V_{n}=V/[1-\eta e^{i(2\pi\alpha n+\theta/L)}]$, with $V$ being the amplitude
of onsite potential and $L$ the length of lattice. $E_{c}^{+}=E_{c}+i0^{+}$ is
the base quasienergy with respect to which the winding number $w$ is defined.
Figure 7: The real part of quasienergies and IPRs of the non-Hermitian Floquet
quasicrystal M5. Other system parameters are set as
$\alpha=\frac{\sqrt{5}-1}{2}$, $J=2.5$, $V=1$, and the length of lattice is
taken as $L=610$. The color bars in panels (a), (c) and (d) are taken from
zero to one. In panel (a), the real part of quasienergies ${\rm Re}E$ of all
states are plotted with respect to $K/\omega$ at $\eta=0.5$, and the color of
each state represents its IPR. The red dashed and magenta dotted lines show
the mobility edge $E_{c}$ and winding number $w$ versus $K/\omega$. The phase
diagram in panel (b) is obtained by evaluating ${\rm
sign}(E_{c}-E_{r}^{\max})+{\rm sign}(E_{c}-E_{r}^{\min})$, which is equal to
$2$ (yellow regions), $0$ (green regions) and $-2$ (blue regions) for the
extended, mobility edge and localized phases, respectively. The winding number
is $w=1$ for the mobility edge phase and $w=0$ otherwise. The maximum and
minimum of IPRs over all states are shown in panels (c) and (d), respectively.
The mobility edge phase (localized phase) sets in when the maximum (minimum)
of IPRs starts to deviate from zero.

To give another transparent view on how the mobility edge is modified by the
driving field, we consider again a variant of the driven non-Hermitian AAH
model, whose Hamiltonian in lattice representation takes the form

$\displaystyle\hat{H}_{5}(t)=$
$\displaystyle\sum_{n}J\left(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+{\rm
h.c.}\right)$ $\displaystyle+$ $\displaystyle\sum_{n}\left[V/(1-\eta
e^{i2\pi\alpha n})-nK\cos(\omega t)\right]\hat{c}_{n}^{\dagger}\hat{c}_{n}.$
(44)

Here the parameter $\eta$ controls the amount of non-Hermiticity, and the
system is quasiperiodic when $\alpha$ is irrational. In the high-frequency
limit, we obtain the Floquet effective Hamiltonian of this M5 as

$\displaystyle\hat{H}_{5{\rm F}}=$ $\displaystyle\sum_{n}J{\cal
J}_{0}\left(K/\omega\right)\left(\hat{c}_{n}^{\dagger}\hat{c}_{n+1}+{\rm
h.c.}\right)$ $\displaystyle+$ $\displaystyle\sum_{n}V/(1-\eta e^{i2\pi\alpha
n})\hat{c}_{n}^{\dagger}\hat{c}_{n},$ (45)

where the hopping amplitude is controlled by the ratio between the amplitude
and frequency of the driving force. Acting $\hat{H}_{5{\rm F}}$ on the state
$|\psi\rangle=\sum_{n}\psi_{n}\hat{c}_{n}^{\dagger}|0\rangle$, we find the
eigenvalue equation

$J{\cal
J}_{0}\left(\frac{K}{\omega}\right)(\psi_{n+1}+\psi_{n-1})+\frac{V}{1-\eta
e^{i2\pi\alpha n}}\psi_{n}=E\psi_{n},$ (46)

whose solutions yield the quasienergy $E$ and Floquet eigenstates of the
system. Note that since the onsite potential amplitude $V_{n}=V^{*}_{-n}$, the
system Hamiltonian also possesses the ${\cal PT}$-symmetry and its spectrum
could be real in certain parameter regime.

Following the method outlined in Ref. LonghiQC3 , the non-Hermitian Floquet
quasicrystal M5 is found to have a quasienergy-dependent mobility edge at
$E=E_{c}$, where

$E_{c}=|J{\cal J}_{0}(K/\omega)(\eta+1/\eta)|.$ (47)

Let us denote $E_{r}^{\max}$ and $E_{r}^{\min}$ as the maximum and minimum of
the real parts of quasienergies over all states at a fixed set of system
parameters. Then if $E_{r}^{\max}<E_{c}$, all eigenstates of the system are
delocalized and the system sits in an extended phase. When
$E_{r}^{\min}<E_{c}<E_{r}^{\max}$, we find the states with ${\rm Re}E<E_{c}$
(${\rm Re}E>E_{c}$) to be extended (localized). A mobility edge thus emerges
at $E=E_{c}$, and the system resides in the mobility edge phase under the
condition $E_{r}^{\min}<E_{c}<E_{r}^{\max}$. When $E_{c}<E_{r}^{\min}$, the
system enters a phase in which all eigenstates are localized, and the mobility
edge vanishes. By tuning the driving field parameters $K$ and $\omega$, the
system could undergo alternate transitions among the extended, mobility edge
and localized phases. The Floquet engineering scheme can again be utilized to
dynamically control the mobility edges and transport nature of the non-
Hermitian Floquet quasicrystal M5. We summarize our key results about this
model in Table 6.

To demonstrate our theoretical findings, we compute the spectrum and IPRs of
M5, as presented in Fig. 7. Note that in Fig. 7(b), we evaluated a winding
number $w$ of the quasienergy with respect to the mobility edge $E_{c}$ (see
Table 6 for its definition), and find that $w=1$ for the mobility edge phase
and $w=0$ otherwise. This winding number can thus be used as a topological
order parameter to distinguish the mobility edge phase and other phases of the
system, and to signify the transitions bewteen them. The numerical results
shown in Fig. 7 are all consistent with our theoretical predictions about the
non-Hermitian Floquet quasicrystal M5. Combining these observations with the
results obtained in Subsecs. III.2–III.5, we conclude that the Floquet
engineering strategy could indeed be exploited as a useful means to control
and modulate the spectrum, topological and transport properties in a broad
class of non-Hermitian quasicrystals.

##  IV Summary and discussion

In this work, we apply Floquet driving fields to engineer ${\cal
PT}$-breaking, localization-to-delocalization, topological transitions and
mobility edges in non-Hermitian quasicrystals. Following the well-known scheme
of dynamical localization, we utilize high-frequency shaking forces to control
the hopping amplitudes of non-Hermitian quasiperiodic lattices, and observing
alternate transitions between extended, localized and mobility edge phases
with the change of driving field parameters. We implement our scheme in five
prototypical models of non-Hermitian quasicrystal, and obtain the conditions
of their driving-assisted ${\cal PT}$-transitions, localization-delocalization
transitions and mobility edges. We further introduced winding numbers of the
quasienergy to distinguish non-Hermitian Floquet quasicrystals with different
spectrum and localization features, and establishing the topological phase
diagrams of these intriguing phases. Our results thus extend the study of non-
Hermitian quasicrystals to periodically driven systems, and further uncover
the usefulness of the Floquet engineering approach in the design and control
of non-Hermitian quasicrystals with distinctive topological and transport
properties.

In theory, the critical exponent of localization length (inverse Lyapunov
exponent) could characterize the critical nature of delocalization-
localization transitions ChayesIneq . For the original AAH model (see Eq.
(15)) we have $\lambda=\ln(V/2J)$, and $\lambda^{-1}\propto|V-V_{c}|^{-1}$
around the critical point $V_{c}=2J$ of the transition. The critical exponent
is thus $\nu=1$. Referring to the tables 2–4, we find that close to the
critical points of models M1–M3, the localization length scales as
$\lambda^{-1}\propto|V-V_{c}|^{-1}$,
$\lambda^{-1}\propto|\gamma-\gamma_{c}|^{-1}$ and
$\lambda^{-1}\propto|(K/\omega)-(K/\omega)_{c}|^{-1}$. Therefore, the critical
exponent of these models is also $\nu=1$. In Ref. ChayesIneq , it was found
that for 1D systems with independent bond or site disorder, the critical
exponent satisfies the Chayes’ inequality $\nu\geq 2$. For 1D quasicrystals,
since the disorder introduced by the quasiperiodic potential can be viewed as
long-range correlated, the Chayes’ inequality may not hold. Our results thus
provide examples for the violation of the Chayes’ inequality in 1D non-
Hermitian Floquet quasicrystals.

Besides discrete lattices, localization transitions in quasicrystals have also
been explored in continuum models ContinuumQC1 ; ContinuumQC2 ; ContinuumQC3 .
A generic observation for continuum models is that mobility edges will emerge
beyond the tight-binding limit. Since we focused on the high-frequency region
of the driving field in our study, the resulting systems are described by
effective Floquet Hamiltonians with modified hopping amplitudes. Therefore, we
expect that with long-range hoppings or in continuum versions of the models
M1–M3, mobility edges may also appear, and the physical mechanism is similar
to that discovered in Refs. ContinuumQC1 ; ContinuumQC2 ; ContinuumQC3 . In
the meantime, the transition of the spectrum from real to complex in continuum
versions of M1–M3 would be related to the transition from an extended or a
localized phase to a mobility edge phase. Due to the possible presence of
mobility edges, the energy-independent Lyapunov exponents shown in tables 2–4
may not be valid for continuum models, and should be reconsidered case by
case.

In future work, it would be interesting to apply our scheme to the engineering
of non-Hermitian quasicrystals with other types of mobility edges, long-range
hoppings, many-body effects and in higher spatial dimensions. In the present
study, the experimentally established high-frequency approach DLRev2  is
applied to control the tunneling amplitudes, and we expect it to be valid
under the condition listed below Eq. (9). Interestingly, our initial results
(in preparation) suggest that driving fields could induce transitions among
extended, localized and mobility edge phases at generic frequencies. The
results presented in Figs. 2–7 indicate that this general feature has already
been captured qualitatively under the high-frequency approximation. Under
moderate driving frequencies, quasicrystalline phases that are unique to non-
Hermitian Floquet systems may also emerge, which certainly deserve more
thorough explorations.

###### Acknowledgements.

L.Z. is supported by the National Natural Science Foundation of China (Grant
No. 11905211), the China Postdoctoral Science Foundation (Grant No.
2019M662444), the Fundamental Research Funds for the Central Universities
(Grant No. 841912009), the Young Talents Project at Ocean University of China
(Grant No. 861801013196), and the Applied Research Project of Postdoctoral
Fellows in Qingdao (Grant No. 861905040009).

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