# Floquet theory of photoinduced topological phase transitions in the organic
salt $\alpha$-(BEDT-TTF)2I3 irradiated with elliptically polarized light

Keisuke Kitayama  Department of Physics, University of Tokyo, Hongo, Bunkyo-
ku, Tokyo 113-8656, Japan  Yasuhiro Tanaka  Department of Applied Physics,
Waseda University, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan  Masao Ogata
Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656,
Japan  Trans-scale Quantum Science Institute, University of Tokyo, Bunkyo-ku,
Tokyo 113-0033, Japan  Masahito Mochizuki  Department of Applied Physics,
Waseda University, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

###### Abstract

We theoretically investigate possible photoinduced topological phase
transitions in the organic salt $\alpha$-(BEDT-TTF)2I3, which possesses a pair
of inclined massless Dirac-cone bands between the conduction and valence bands
under uniaxial pressure. The Floquet analyses of a driven tight-binding model
for this material reveal rich photoinduced variations of band structures,
Chern numbers, and Hall conductivities under irradiation with elliptically
polarized light. The obtained phase diagrams contain a variety of
nonequilibrium steady phases, e.g., the Floquet Chern insulator, Floquet
semimetal, and Floquet normal insulator phases. This work widens a scope of
target materials for research on photoinduced topological phase transitions
and contributes to development of research on the optical manipulations of
electronic states in matters.

##  I 1\. Introduction

Figure 1: (a) Schematic of $\alpha$-(BEDT-TTF)2I3 irradiated with elliptically
polarized light. (b) Two-dimensional (BEDT-TTF)-layer in $\alpha$-(BEDT-
TTF)2I3 with a unit cell composed of four molecular sublattices of A, A′, B
and C and eight kinds of transfer integrals for a tight-binding model of the
BEDT-TTF layer. Dashed rectangle represents the unit cell with $b$ and $a$
being the lattice constant along $x$ and $y$ direction, respectively. (c) Band
dispersion relations of $\alpha$-(BEDT-TTF)2I3 obtained by diagonalizing the
Hamiltonian matrix of the tight-binding model in Eq. (37). There exist a pair
of inclined Dirac-cone bands between the third and fourth bands, where the
Dirac points are located on the Fermi level.

Photoinduced phase transitions have attracted a great deal of research
interest Yonemitsu06 ; Tokura06 ; Aoki14 ; Bukov15 ; Basov17 . The Floquet
theory has been widely exploited for theoretical studies on nonequilibrium
steady states and their phase transitions in photodriven systems Oka09 ;
Kitagawa10 ; Kitagawa11 ; Lindner11 ; Ezawa13 ; Grushin14 ; ZhengW14 ;
Takasan15 ; Mikami16 ; JYZou16 ; Claassen16 ; ZYan16 ; LDu17 ; Ezawa17 ;
Takasan17a ; Takasan17b ; Menon18 ; ChenR18 ; ZhangMY19 ; Kang20 ; Tanaka20 ;
Kitayama20 ; Sato16 ; Takayoshi14 ; Kitamura17 ; Ikeda20 . Emergence of
quantum Hall effects in a tight-binding model on the honeycomb lattice
irradiated with circularly polarized light was theoretically predicted using
the Floquet theory Oka09 ; Kitagawa11 , in which the band structure attains
topological nature similar to that described by the Haldane model Haldane88 .
This topological phase transition manifests photoinduced quantum Hall effects
without external magnetic fields, which was indeed observed by recent optical
experiments in graphene McIver19 .

Since these pioneering works, many studies based on the Floquet theory have
been performed for both electron systems Oka09 ; Kitagawa10 ; Kitagawa11 ;
Lindner11 ; Ezawa13 ; Grushin14 ; ZhengW14 ; Takasan15 ; Mikami16 ; JYZou16 ;
Claassen16 ; ZYan16 ; LDu17 ; Ezawa17 ; Takasan17a ; Takasan17b ; Menon18 ;
ChenR18 ; ZhangMY19 ; Kang20 ; Tanaka20 ; Kitayama20  and spin systems Sato16
; Takayoshi14 ; Kitamura17 ; Ikeda20 . However, most of these studies dealt
with toy models on simple lattices or simple two-dimensional materials such as
graphene Oka09 ; Kitagawa11 ; Kitagawa10 , silicene Ezawa13 , stacked graphene
JYZou16 , transition-metal dichalcogenides Claassen16 ; ZhangMY19 , and black
phosphorene Kang20 . On the other hand, bulk materials have scarcely been
involved in a scope of this kind of research. However, for further development
of this promising research field, it is indispensable to widen the range of
target materials, and, for this sake, theoretical studies on real bulk
materials with complex crystal and electronic structures are highly desired.
In the research on real bulk materials, we expect richer material-specific
photoinduced topological phenomena and physics.

The organic compound $\alpha$-(BEDT-TTF)2I3 is one of the promising candidate
materials for research in this direction Tajima06 . Conduction electrons
confined in a BEDT-TTF layer form a pair of inclined Dirac cones in the band
structure when a uniaxial pressure $P_{a}>2$ kbar is applied along the $a$
axis Katayama06 ; Kobayashi07 ; Kajita14 . Under an ambient pressure, this
material exhibits a charge-ordered ground state and has a band gap at the
Fermi level. This charge order melts above the critical temperature of $T_{\rm
CO}$$\sim$135 K, and a zero-gap semiconducting state with Dirac-cone bands
appears even without uniaxial pressure. A lot of interesting topologically
nontrivial properties and phenomena due to the Dirac-cone bands have been
theoretically proposed for this organic compound to date, e.g., the quantum
Hall effect Kajita14 , the structures of Berry curvature in momentum space
Suzumura11 , and the flux-induced Chern insulator phases Osada17 .

In this paper, we theoretically study possible photoinduced topological phases
in organic compound $\alpha$-(BEDT-TTF)2I3 with a pair of inclined Dirac-cone
bands. The Floquet analyses of a driven tight-binding model for this material
reveals that the irradiation with elliptically polarized light opens a gap at
the Dirac points located on the Fermi level, and the system becomes a
topological insulator. This so-called Floquet Chern insulator phase as a
photoinduced nonequilibrium steady phase is characterized by a quantized
topological number Thouless82 ; Kohmoto85  and conducting chiral edge states
Hao08 . Moreover, we obtain rich phase diagrams as functions of amplitudes,
frequency and ellipticity of the light, which involve Chern insulator,
semimetal, and normal insulator phases as nonequilibrium steady states. We
also discuss the behaviors of Hall conductivity associated with the
photoinduced phase evolution. In fact, the photoinduced phase transitions for
this compound have been theoretically studied in the case of circularly
polarized light Kitayama20 . In the case of elliptically polarized light
studied in the present paper, we obtain richer phase diagrams and phase
evolutions as functions of the light ellipticity. Moreover, we discuss edge
states and the Hall conductivities in the photoinduced states in detail.

As argued in our previous paper Kitayama20 , one advantage of the usages of
organic compounds is that an effective amplitude of light is an order of
magnitude larger than graphene because of much larger lattice constants as
discussed later, which enhances experimental feasibility of the predicted
photoinduced topological phase transitions. Another advantage of
$\alpha$-(BEDT-TTF)2I3 is that the bands near the Fermi level [i.e., four
bands in Fig. 1(c)] relevant to the electronic properties of this material are
well separated from bands above and below them. This specific band structure
provides us a precious opportunity to have a finite window of light frequency
that satisfies the so-called off-resonant condition, with which the Floquet
bands of different photon numbers do not overlap with each other and, thereby,
the nonequilibrium steady phases are well-defined. We will discuss these
aspects later in more detail.

In the following, we formulate the Floquet theory to analyze a time-periodic
Hamiltonian in Sect. 2 and then apply this formalism to the photodriven tight-
binding model for $\alpha$-(BEDT-TTF)2I3 irradiated with elliptically
polarized light in Sect. 3. In Sect. 4, we explain how to calculate the
physical quantities such as Chern numbers, Hall conductivities, and band
structures associated with edge states within the Floquet formalism. In Sect.
5, we show and discuss the results for photoinduced variations of band
structurers, Chern numbers, and Hall conductivities as well as the phase
diagrams. We also discuss the chiral edge states in the Floquet Chern
insulator phase, which manifest the topological nature of this photoinduced
insulating phase. Finally, we summarize this work in Sect. 6.

##  II 2\. Floquet Theory for Photodriven Systems

In this section, we introduce a general formalism of the Floquet theory for
photo-irradiated systems. The photodriven systems are described by the time
dependent Schrödinger equation,

$\displaystyle
i\hbar\frac{\partial}{\partial\tau}\ket{\Psi(\tau)}=H(\tau)\ket{\Psi(\tau)}.$
(1)

Here the Hamiltonian $H(\tau)$ is time-periodic as $H(\tau)=H(\tau+T)$ with a
temporal period $T(=2\pi/\omega)$ of light. In such systems, the wavefunction
$\ket{\Psi(\tau)}$ can be written in the form,

$\displaystyle\ket{\Psi(\tau)}=e^{-i\varepsilon\tau/\hbar}\ket{\Phi(\tau)},$
(2)

where $\ket{\Phi(\tau)}$ and $\varepsilon$ are referred to as the Floquet
state and the quasienergy, respectively. The Floquet state satisfies
$\ket{\Phi(\tau)}=\ket{\Phi(\tau+T)}$. This theorem is called Floquet theorem
and can be regarded as a temporal version of the Bloch theorem for spatially
periodic systems. We substitute Eq. (2) into Eq. (1) and perform the Fourier
transformations. Eventually, the time dependent Schrödinger equation is
rewritten in the form,

$\displaystyle\sum_{m=-\infty}^{\infty}\mathcal{H}_{nm}\ket{\Phi_{\nu}^{m}}=\varepsilon^{n}_{\nu}\ket{\Phi_{\nu}^{n}},$
(3)

where

$\displaystyle\mathcal{H}_{nm}=H_{n-m}-m\omega\delta_{n,m}.$ (4)

Here the Fourier coefficients $H_{n}$ and $\ket{\Phi_{\nu}^{n}}$ are defined
as

$\displaystyle H_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}H(\tau)e^{in\omega\tau}d\tau,$ (5)
$\displaystyle\ket{\Phi_{\nu}^{n}}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}\ket{\Phi_{\nu}(\tau)}e^{in\omega\tau}d\tau.$
(6)

The integer $n$ corresponds to the number of photons, and
$\ket{\Phi_{\nu}^{-n}}$ is regarded as a state dressed with $n$ photons. After
the projection to the zero-photon subspace, we obtain the effective
Hamiltonian $H_{\mathrm{eff}}$ Mikami16 ,

$\displaystyle
H_{\mathrm{eff}}=H_{0}+\sum_{n=1}^{\infty}\frac{[H_{-n},H_{n}]}{n\hbar\omega}+O\left(\frac{t^{3}}{\hbar^{2}\omega^{2}}\right).$
(7)

Here $t$ denotes a typical value of the transfer integrals. This effective
Hamiltonian is valid when the light frequency $\omega$ is high enough. It is
noted that the following approximated form, in which only the first term of
the series expansion is taken, is often used in the high-frequency limit,

$\displaystyle H_{\mathrm{eff}}\sim H_{0}+\frac{[H_{-1},H_{1}]}{\hbar\omega}.$
(8)

Now we consider the following tight-binding model that describes a lattice
electron system,

$\displaystyle H=\sum_{i,j}t_{ij}c^{\dagger}_{i}c_{j},$ (9)

where $i$ and $j$ denote the lattice sites. The symbol $c^{\dagger}_{i}$
($c_{i}$) denotes the creation (annihilation) operator of an electron at $i$th
site, while $t_{ij}$ represents the transfer integral between $i$th and $j$th
sites. When the system is irradiated with light, the transfer integrals attain
Peierls phases due to the time-dependent vector potential of light
electromagnetic field. A general form of the vector potential caused by the
elliptically polarized light is given by,

$\displaystyle\bm{A}(\tau)=(A_{x}\sin(\omega\tau+\phi),A_{y}\sin(\omega\tau)).$
(10)

This vector potential gives a time-dependent electric field

$\displaystyle\bm{E}(\tau)$ $\displaystyle=$
$\displaystyle-\frac{d\bm{A}(\tau)}{d\tau}$ (11) $\displaystyle=$
$\displaystyle-(A_{x}\omega\cos(\omega\tau+\phi),A_{y}\omega\cos(\omega\tau)).$

The transfer integrals with a time-dependent Peierls phase are given by

$\displaystyle t_{ij}(\tau)$ $\displaystyle=$ $\displaystyle
t_{ij}\exp\left[-i\frac{e}{\hbar}\bm{A}(\tau)\cdot(\bm{r}_{i}-\bm{r}_{j})\right]$
(12) $\displaystyle=$ $\displaystyle
t_{ij}\exp\left[-i\frac{e}{\hbar}A_{x}(x_{i}-x_{j})\sin(\omega\tau+\phi)\right.$
$\displaystyle\hskip
42.67912pt\left.-i\frac{e}{\hbar}A_{y}(y_{i}-y_{j})\sin(\omega\tau)\right].$

Here we introduce the coordinates $\bm{r}_{i}=(x_{i},y_{i})$ for $i$th site.
The Fourier coefficients $H_{n}$ are calculated using Eq. (5) as

$\displaystyle
H_{n}=\sum_{i,j}t_{ij}J_{n}(\mathcal{A}_{ij})e^{-in\theta_{ij}}c_{i}^{\dagger}c_{j},$
(13)

where $J_{n}$ is the $n$th Bessel function, and $\mathcal{A}_{ij}$ and
$\theta_{ij}$ are respectively defined as

$\displaystyle\mathcal{A}_{ij}=\frac{e}{\hbar}[A_{x}^{2}(x_{i}-x_{j})^{2}+A_{y}^{2}(y_{i}-y_{j})^{2}$
$\displaystyle\hskip
56.9055pt+2A_{x}A_{y}(x_{i}-x_{j})(y_{i}-y_{j})\cos\phi]^{1/2}$ (14)
$\displaystyle\theta_{ij}=\tan^{-1}\left[\frac{A_{x}(x_{i}-x_{j})\sin\phi}{A_{x}(x_{i}-x_{j})\cos\phi+A_{y}(y_{i}-y_{j})}\right].$
(15)

The second term of $H_{\rm eff}$ in Eq. (7) is calculated as

$\displaystyle\sum_{n=1}^{\infty}\frac{[H_{-n},H_{n}]}{n\hbar\omega}$
$\displaystyle=\sum_{\left<\left<i,k\right>\right>}\sum_{n=1}^{\infty}\frac{2i}{n\hbar\omega}\sum_{j}\mathrm{Im}(\chi_{ij}^{-n}\chi_{jk}^{n})(c^{\dagger}_{i}c_{k}-c^{\dagger}_{k}c_{i}).$
(16)

Here we define

$\displaystyle\chi_{ij}^{n}\equiv
t_{ij}J_{n}\left(\mathcal{A}_{ij}\right)e^{-in\theta_{ij}}.$ (17)

This term describes indirect electron hoppings between $i$th and $k$th sites
via the in-between $j$th site connected by two transfer integrals $t_{ij}$ and
$t_{jk}$. On the other hand, the first term $H_{0}$ of $H_{\rm eff}$ in Eq.
(7) is given by,

$\displaystyle
H_{0}=\sum_{i,j}t_{ij}J_{0}(\mathcal{A}_{ij})c_{i}^{\dagger}c_{j}.$ (18)

This term describes the direct electron hoppings between $i$th and $j$th sites
connected by a single transfer integral $t_{ij}$, which are renormalized by
the Bessel function $J_{0}(\mathcal{A}_{ij})$.

##  III 3\. Floquet Theory for the Photodriven $\alpha$-(BEDT-TTF)2I3

We employ a tight-binding model to describe the electronic structure of the
BEDT-TTF layer in $\alpha$-(BEDT-TTF)2I3:

$\displaystyle
H=\sum_{i,j}^{N}\sum_{\alpha,\beta}t_{i\alpha,j\beta}c^{\dagger}_{i\alpha}c_{j\beta}.$
(19)

Here $i$ and $j$ denote the unit cells, while $\alpha$ and $\beta$ denote the
molecule sites A, A′, B, and C. The symbol $c^{\dagger}_{i\alpha}$
($c_{i\alpha}$) denotes the creation (annihilation) operator of an electron at
molecule site ($i$, $\alpha$), while $t_{i\alpha,j\beta}$ represents the
transfer integral between the ($i,\alpha$) and ($j,\beta$) sites.

This compound exhibits a charge-ordered ground state at ambient pressure. When
a uniaxial pressure larger than 2 kbar is applied along the $a$ axis, this
charge order melts, and eventually the peculiar band structure with a pair of
inclined Dirac cones emerges. In this study, we consider the situation that
the uniaxial pressure of $P_{a}$=4 kbar is applied. The applied pressure
modulates the transfer integrals $t_{i\alpha,j\beta}$ in Eq. (19) which are
given by eight kinds of transfer integrals shown in Fig. 1 (b). The $P_{a}$
dependencies of the eight transfer integrals are deduced theoretically by
interpolation as Kobayashi04 ,

$\displaystyle t_{a1}=\tilde{t}_{a1}=-0.028(1.00+0.089P_{a})\;{\rm eV},$ (20)
$\displaystyle t_{a2}=0.048(1.0+0.167P_{a})\;{\rm eV},$ (21) $\displaystyle
t_{a3}=-0.020(1.0-0.025P_{a})\;{\rm eV},$ (22) $\displaystyle
t_{b1}=0.123\;{\rm eV},$ (23) $\displaystyle
t_{b2}=0.140(1.0+0.011P_{a})\;{\rm eV},$ (24) $\displaystyle
t_{b3}=-0.062(1.0+0.032P_{a})\;{\rm eV},$ (25) $\displaystyle
t_{b4}=-0.025\;{\rm eV}.$ (26)

For $P_{a}$=4 kbar, the values are $t_{a1}=\tilde{t}_{a1}=-0.038$ eV,
$t_{a2}=0.080$ eV, $t_{a3}=-0.018$ eV, $t_{b1}=0.123$ eV, $t_{b2}=0.146$ eV,
$t_{b3}=-0.070$ eV, and $t_{b4}=-0.025$ eV.

Using the Fourier transforms,

$\displaystyle
c_{i\alpha}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{\bm{k}}c^{\dagger}_{\bm{k}\alpha}e^{i\bm{k}\cdot\bm{r}_{i\alpha}},$
(27)

we rewrite the tight-binding Hamiltonian in the momentum representation,

$\displaystyle
H=\sum_{\bm{k}}(c^{\dagger}_{\bm{k}A}\,c^{\dagger}_{\bm{k}A^{\prime}}\,c^{\dagger}_{\bm{k}B}\,c^{\dagger}_{\bm{k}C})\hat{H}(\bm{k})\left(\begin{array}[]{c}c_{\bm{k}A}\\\
c_{\bm{k}A^{\prime}}\\\ c_{\bm{k}B}\\\ c_{\bm{k}C}\end{array}\right),$ (32)

with

$\displaystyle\hat{H}(\bm{k})=\left(\begin{array}[]{cccc}0&A_{2}(\bm{k})&B_{2}(\bm{k})&B_{1}(\bm{k})\\\
A_{2}^{*}(\bm{k})&0&B_{2}^{*}(\bm{k})&B_{1}^{*}(\bm{k})\\\
B_{2}^{*}(\bm{k})&B_{2}(\bm{k})&0&A_{1}(\bm{k})\\\
B_{1}^{*}(\bm{k})&B_{1}(\bm{k})&A_{1}(\bm{k})&0\end{array}\right),$ (37)

where

$\displaystyle A_{1}(\bm{k})=2t_{a1}\cos\left(\frac{k_{y}a}{2}\right),$ (38)
$\displaystyle A_{2}(\bm{k})=t_{a2}e^{ik_{y}a/2}+t_{a3}e^{-ik_{y}a/2},$ (39)
$\displaystyle
B_{1}(\bm{k})=t_{b1}e^{i(k_{x}b/2+k_{y}a/4)}+t_{b4}e^{-i(k_{x}b/2-k_{y}a/4)},$
(40) $\displaystyle
B_{2}(\bm{k})=t_{b2}e^{i(k_{x}b/2-k_{y}a/4)}+t_{b3}e^{-i(k_{x}b/2+k_{y}a/4)}.$
(41)

Here $b$ and $a$ are the lattice constants along the $x$ and $y$ axes,
respectively [see Fig. 1(b)]. In the following calculations, we use
experimental values of the lattice constants $a$=0.9187 nm and $b$=1.0793 nm
Mori12 . By diagonalizing the Hamiltonian matrix in Eq. (37), we obtain the
band dispersion relations shown in Fig. 1(c), which have a pair of inclined
massless Dirac cones between the third and fourth bands. Because the electron
filling of this compound is $n_{e}=3/4$, the Fermi level is located between
these two bands, and the Dirac points are on the Fermi level.

We consider a situation that this system is irradiated with elliptically
polarized light described by Eq. (10). Here we introduce dimensionless
quantities $\mathcal{A}_{b}$ and $\mathcal{A}_{a}$ as,

$\displaystyle\mathcal{A}_{a}=\frac{eaA_{y}}{\hbar},\quad\quad\mathcal{A}_{b}=\frac{ebA_{x}}{\hbar}.$
(42)

Amplitudes of the two electric-field components $E_{a}^{\omega}$ and
$E_{b}^{\omega}$ for the elliptically polarized light are given by,

$\displaystyle
E_{a}^{\omega}=A_{y}\omega=\frac{\mathcal{A}_{a}\hbar\omega}{ea},\quad\quad
E_{b}^{\omega}=A_{x}\omega=\frac{\mathcal{A}_{b}\hbar\omega}{eb}.$ (43)

For the Hamiltonian in the momentum representation, the effects of irradiation
with elliptically polarized light can be treated simply by replacing the
momenta $k_{x}$ and $k_{y}$ as

$\displaystyle k_{x}\quad\rightarrow\quad
k_{x}+\mathcal{A}_{b}\sin(\omega\tau+\phi),$ (44) $\displaystyle
k_{y}\quad\rightarrow\quad k_{y}+\mathcal{A}_{a}\sin(\omega\tau).$ (45)

It is straightforward to obtain the Fourier coefficients of the Hamiltonian
$H_{n}$ for $\alpha$-(BEDT-TTF)2I3 as in Eq. (13). The transfer integrals
$t_{i\alpha,j\beta}$ in Eqs. (20)-(26) are renormalized as

$\displaystyle t_{a1}\quad\rightarrow\quad t_{a1}J_{n}(\mathcal{A}_{a}/2),$
(46)
$\displaystyle\tilde{t}_{a1}\quad\rightarrow\quad\tilde{t}_{a1}J_{-n}(\mathcal{A}_{a}/2),$
(47) $\displaystyle t_{a2}\quad\rightarrow\quad
t_{a2}J_{-n}(\mathcal{A}_{a}/2),$ (48) $\displaystyle
t_{a3}\quad\rightarrow\quad t_{a3}J_{n}(\mathcal{A}_{a}/2),$ (49)
$\displaystyle t_{b1}\quad\rightarrow\quad
t_{b1}J_{-n}(\mathcal{A}_{+})e^{-in\theta_{+}},$ (50) $\displaystyle
t_{b2}\quad\rightarrow\quad t_{b2}J_{n}(\mathcal{A}_{-})e^{+in\theta_{-}},$
(51) $\displaystyle t_{b3}\quad\rightarrow\quad
t_{b3}J_{n}(\mathcal{A}_{+})e^{-in\theta_{+}},$ (52) $\displaystyle
t_{b4}\quad\rightarrow\quad t_{b4}J_{-n}(\mathcal{A}_{-})e^{+in\theta_{-}}.$
(53)

where

$\displaystyle\mathcal{A}_{\pm}=\frac{1}{4}\sqrt{4\mathcal{A}_{b}^{2}+\mathcal{A}_{a}^{2}\pm
4\mathcal{A}_{a}\mathcal{A}_{b}\cos\phi},$ (54)
$\displaystyle\theta_{\pm}=\tan^{-1}\left(\frac{2\mathcal{A}_{b}\sin\phi}{\pm
2\mathcal{A}_{b}\cos\phi+\mathcal{A}_{a}}\right).$ (55)

By using the renormalizations, the Fourier coefficients of the Hamiltonian
$H_{n}$ is explicitly given in the matrix form,

$\displaystyle\hat{H}_{n}(\bm{k})=\left(\begin{array}[]{cccc}0&A_{2,n}(\bm{k})&B_{2,n}(\bm{k})&B_{1,n}(\bm{k})\\\
A_{2,-n}^{*}(\bm{k})&0&B_{2,-n}^{*}(\bm{k})&B_{1,-n}^{*}(\bm{k})\\\
B_{2,-n}^{*}(\bm{k})&B_{2,n}(\bm{k})&0&A_{1,n}(\bm{k})\\\
B_{1,-n}^{*}(\bm{k})&B_{1,n}(\bm{k})&A_{1,-n}^{*}(\bm{k})&0\end{array}\right)$
(60)

with

$\displaystyle
A_{1,n}(\bm{k})=\tilde{t}_{a1}\,e^{ik_{y}a/2}J_{-n}(\mathcal{A}_{a}/2)+t_{a1}\,e^{-ik_{y}a/2}J_{n}(\mathcal{A}_{a}/2),$
$\displaystyle
A_{2,n}(\bm{k})=t_{a2}\,e^{ik_{y}a/2}J_{-n}(\mathcal{A}_{a}/2)+t_{a3}\,e^{-ik_{y}a/2}J_{n}(\mathcal{A}_{a}/2),$
$\displaystyle
B_{1,n}(\bm{k})=t_{b1}\,e^{i(k_{x}b/2+k_{y}a/4)}J_{-n}(\mathcal{A}_{+})e^{-in\theta_{+}}$
$\displaystyle\quad\quad+t_{b4}\,e^{-i(k_{x}b/2-k_{y}a/4)}J_{-n}(\mathcal{A}_{-})e^{+in\theta_{-}},$
(64) $\displaystyle
B_{2,n}(\bm{k})=t_{b2}\,e^{i(k_{x}b/2-k_{y}a/4)}J_{n}(\mathcal{A}_{-})e^{+in\theta_{-}}$
$\displaystyle\quad\quad+t_{b3}\,e^{-i(k_{x}b/2+k_{y}a/4)}J_{n}(\mathcal{A}_{+})e^{-in\theta_{+}}.$
(65)

In the present study, we examine the case of $\phi=\pi/2$, for which we have

$\displaystyle\mathcal{A}_{+}=\mathcal{A}_{-}=\frac{1}{4}\sqrt{4\mathcal{A}_{b}^{2}+\mathcal{A}_{a}^{2}}\equiv\mathcal{A},$
(66)
$\displaystyle\theta_{+}=\theta_{-}=\tan^{-1}\left(\frac{2\mathcal{A}_{b}}{\mathcal{A}_{a}}\right)\equiv\theta.$
(67)

In this case, the ellipses with major and minor axes are parallel to the
crystallogrophic $a$ and $b$ axes.

In the present study, we adopt two different approaches. One approach is to
directly solve the eigenequation in Eq. (3) by restricting the number of
photon as $|m|\leq 16$ where the size of the Floquet Hamiltonian matrix
$\mathcal{H}_{nm}(\equiv H_{n-m}-m\omega\delta_{n,m})$ in Eq. (4) is
132$\times$132 because $\hat{H}_{n}$ is a 4$\times$4 matrix. The other
approach is to solve the eigenequation for $H_{\rm eff}$ in Eq. (7) by
restricting the summation over $n$ within $n\leq 16$. We have confirmed that
the results do not alter between these two approaches even quantitatively
except for a few points in the phase diagrams. The former method provides
finer and more precise structures of the phase diagrams particularly in the
low-frequency region of $\hbar\omega/t<1$ because the latter method is based
on the effective Hamiltonian valid in the high-frequency limit. We also note
that for smaller frequencies $\omega$, a larger size $|m|$ of the Floquet
matrix is required, typically of the order of $W/\hbar\omega$, where $W$ is
the band width Mikami16 . Since we adopt $|m|\leq 16$ for $\alpha$-(BEDT-
TTF)2I3 with $W\sim 0.75$ eV, the obtained results are trustable for
$\hbar\omega\geq$ 0.05 eV. Thus, in the following, we mainly discuss the
results obtained by the former method unless otherwise noted.

##  IV 4\. Calculations of Physical Quantities

###  IV.1 4.1 Chern number

The Chern number of the $\nu$th band $N_{\rm Ch}^{\nu}$ ($\nu$=1,2,3,4) is
related to the Berry curvature $B_{z}^{\nu}(\bm{k})$,

$\displaystyle N_{\rm Ch}^{\nu}=\frac{1}{2\pi}\int\int_{\rm
BZ}\;B_{z}^{\nu}(\bm{k})dk_{x}dk_{y},$ (68)

where the Berry curvature $B_{z}^{\nu}(\bm{k})$ at each $\bm{k}$ point is
given by

$\displaystyle B_{z}^{\nu}(\bm{k})=$ $\displaystyle
i\sum_{(m,\mu)}\frac{\bra{\Phi_{\nu}^{n}(\bm{k})}\frac{\partial\hat{\mathcal{H}}}{\partial
k_{x}}\ket{\Phi_{\mu}^{m}(\bm{k})}\bra{\Phi_{\mu}^{m}(\bm{k})}\frac{\partial\hat{\mathcal{H}}}{\partial
k_{y}}\ket{\Phi_{\nu}^{n}(\bm{k})}-{c.c.}}{[\varepsilon^{m}_{\mu}(\bm{k})-\varepsilon^{n}_{\nu}(\bm{k})]^{2}}.$
(69)

Here $\hat{\mathcal{H}}$ denotes the Floquet Hamiltonian matrix, whereas
$\varepsilon^{n}_{\nu}(\bm{k})$ and $\ket{\Phi_{\nu}^{n}(\bm{k})}$ are,
respectively, the eigenenergies and the corresponding eigenvectors of Eq. (3)
with $\nu=1,2,3,4$ and $|m|\leq 16$. The summation is taken over $m$ and $\mu$
where $(m,\mu)\neq(n,\nu)$; “$c.c.$” denotes the complex conjugate of the
first term of the numerator. Note that $B_{z}^{\nu}$ is independent of the
photon number $n$. In this work, the Chern numbers are calculated using a
numerical technique proposed by Fukui $et$ $al.$ in Ref. Fukui05 .

###  IV.2 4.2 Hall conductivity

The Hall conductivity $\sigma_{xy}$ is a physical quantity sensitive to the
topological properties of electronic states and thus can be exploited to
identify topological phases. This quantity is associated with the Berry
curvatures of bands $B_{z}^{\nu}(\bm{k})$ as

$\displaystyle\sigma_{xy}=\frac{2e^{2}}{h}\int\int_{\rm
BZ}\;\frac{dk_{x}dk_{y}}{2\pi}\sum_{\nu}n_{\nu}(\bm{k})B_{z}^{\nu}(\bm{k}).$
(70)

The factor 2 comes from the spin degeneracy. Here $n_{\nu}(\bm{k})$ is the
nonequilibrium distribution function for the $\nu$-th Floquet band in the
photodriven steady states Aoki14 ; Tsuji08 ; Tsuji09 ,

$\displaystyle
n_{\nu}(\bm{k})=\frac{\braket{\Phi_{\nu}^{0}(\bm{k})}{\hat{N}_{\bm{k}}(\varepsilon_{\nu}^{0}(\bm{k}))}{\Phi_{\nu}^{0}(\bm{k})}}{\braket{\Phi_{\nu}^{0}(\bm{k})}{\hat{A}_{\bm{k}}(\varepsilon_{\nu}^{0}(\bm{k}))}{\Phi_{\nu}^{0}(\bm{k})}}.$
(71)

This quantity is calculated using the Floquet–Keldysh formalism Aoki14 ;
Tsuji08 ; Tsuji09 , which combines the Keldysh Green’s function technique
Jauho94 ; Mahan00  with the Floquet theory. The quantities $\hat{A}_{\bm{k}}$
and $\hat{N}_{\bm{k}}$ are given by,

$\displaystyle\hat{A}_{\bm{k}}(\varepsilon)=\frac{i}{2\pi}\left(\hat{G}^{\rm
R}(\bm{k},\varepsilon)-\hat{G}^{\rm A}(\bm{k},\varepsilon)\right),$ (72)
$\displaystyle\hat{N}_{\bm{k}}(\varepsilon)=-\frac{i}{2\pi}\hat{G}^{<}(\bm{k},\varepsilon).$
(73)

The lesser Green’s function $\hat{G}^{<}$ and the lesser self-energy
$\hat{\Sigma}^{<}$ are calculated, respectively, by,

$\displaystyle\hat{G}^{<}(\bm{k},\varepsilon)=\hat{G}^{\rm
R}(\bm{k},\varepsilon)\;\hat{\Sigma}^{<}(\varepsilon)\;\hat{G}^{\rm
A}(\bm{k},\varepsilon),$ (74)
$\displaystyle\hat{\Sigma}^{<}(\varepsilon)=\frac{\hat{\Sigma}^{\rm
A}+\hat{\Sigma}^{\rm K}(\varepsilon)-\hat{\Sigma}^{\rm R}}{2}.$ (75)

The retarded, advanced and Keldysh Green’s functions $\hat{G}^{\rm R}$,
$\hat{G}^{\rm A}$, and $\hat{G}^{\rm K}$ are calculated by solving the
following Dyson equation,

$\displaystyle\left(\begin{array}[]{cc}\hat{G}^{\rm
R}(\bm{k},\varepsilon)&\hat{G}^{\rm K}(\bm{k},\varepsilon)\\\ 0&\hat{G}^{\rm
A}(\bm{k},\varepsilon)\end{array}\right)^{-1}$ (78)
$\displaystyle=\left(\begin{array}[]{cc}[\hat{G}^{\rm
R0}(\bm{k},\varepsilon)]^{-1}&0\\\ 0&[\hat{G}^{\rm
A0}(\bm{k},\varepsilon)]^{-1}\end{array}\right)-\left(\begin{array}[]{cc}\hat{\Sigma}^{\rm
R}&\hat{\Sigma}^{\rm K}(\varepsilon)\\\ 0&\hat{\Sigma}^{\rm
A}\end{array}\right),$ (83)

where $\hat{\Sigma}^{\rm R}$, $\hat{\Sigma}^{\rm A}$ and $\hat{\Sigma}^{\rm
K}$ are matrices of the retarded, advanced, and Keldysh self-energies,
respectively. Each of the matrices is composed of $(2m_{\rm
max}+1)\times(2m_{\rm max}+1)$ block matrices where we set $m_{\rm max}=16$ in
the present study, while the size of each block matrix is 4$\times$4\. The
matrix components of $\hat{G}^{\rm R0}$, $\hat{G}^{\rm A0}$,
$\hat{\Sigma}^{\rm R}$, $\hat{\Sigma}^{\rm A}$ and $\hat{\Sigma}^{\rm K}$ are
given respectively by

$\displaystyle[\hat{G}^{\rm
R0}(\bm{k},\varepsilon)]^{-1}_{n\nu,m\mu}=\varepsilon\delta_{n,m}\delta_{\nu,\mu}-\mathcal{H}_{n\nu,m\mu}(\bm{k}),$
(85) $\displaystyle[\hat{G}^{\rm
A0}(\bm{k},\varepsilon)]^{-1}_{n\nu,m\mu}=\varepsilon\delta_{n,m}\delta_{\nu,\mu}-\mathcal{H}_{n\nu,m\mu}(\bm{k}),$
(86) $\displaystyle[\hat{\Sigma}^{\rm
R}]_{n\nu,m\mu}=-i\Gamma\delta_{n,m}\delta_{\nu,\mu},$ (87)
$\displaystyle[\hat{\Sigma}^{\rm
A}]_{n\nu,m\mu}=i\Gamma\delta_{n,m}\delta_{\nu,\mu},$ (88)
$\displaystyle[\hat{\Sigma}^{\rm
K}(\varepsilon)]_{n\nu,m\mu}=-2i\Gamma\tanh\left[\frac{\varepsilon-\mu+m\omega}{2k_{\rm
B}T_{\rm hr}}\right]\delta_{n,m}\delta_{\nu,\mu}.$

Here the symbol $\hat{M}_{n\nu,m\mu}$ denotes the $(\nu,\mu)$th component of
the $(m,n)$th block matrix $\hat{M}_{nm}$ ($4\times 4$), and the block matrix
$\hat{\mathcal{H}}_{n,m}$ constituting the Floquet Hamiltonian is given by Eq.
(4). We assume that the system is coupled to a heat reservoir at temperature
$T_{\rm hr}$ with a dissipation coefficient $\Gamma$ where we set $\Gamma$=0.1
eV for the calculations. For simplicity, we neglect the frequency-dependence
and the momentum-dependence of the dissipation coefficient $\Gamma$, for which
the dissipations appear only in the diagonal components of the self-energy
matrices.

###  IV.3 4.3 Chiral edge states

Figure 2: Monolayer nanoribbon-shaped system of $\alpha$-(BEDT-TTF)2I3 having
AA′ and BC edges parallel to the $y$ ($a$) axis used for calculation of the
band dispersion relations including those of the edge states. The open
boundary conditions are imposed in the $x$ direction, whereas the periodic
boundary conditions in the $y$ direction. The numberings of the unit cells
along the $x$ and $y$ axes are also presented.

In addition to the Chern numbers, the emergence of conductive chiral edge
states is another feature of the topologically nontrivial Chern insulator
phase. The band dispersion relations including those associated with the edge
states are calculated for systems having edges by imposing the open boundary
conditions in the $x$ direction and the periodic boundary conditions in the
$y$ direction and vice versa. Here we explain how to perform the calculations
by taking a system having AA′ and BC edges, both of which are parallel to the
$y$ axis (see Fig. 2). The positions of unit cells are numbered in ascending
order from the left most ($i$=1) to the right most ($i=N_{x}$) along the $x$
axis, whereas from the bottom ($j$=1) to the top ($j=N_{y}$) along the $y$
axis. Because we impose the periodic boundary conditions in the $y$ direction,
we can perform the Fourier transformation for the creation and annihilation
operators with respect to the $y$ coordinate,

$\displaystyle
c^{\dagger}_{ij\alpha}=\frac{1}{\sqrt{N_{y}}}\sum_{k_{y}}c^{\dagger}_{k_{y}i\alpha}e^{ik_{y}y_{j,\alpha}},$
(90)

where $i$ and $j$ are the integer coordinates of unit cells along the $x$ and
$y$ axes (i.e., $1\leq i\leq N_{x}$ and $1\leq j\leq N_{y}$), respectively.
The index $\alpha$ specifies a molecule site among four in the unit cell
($\alpha$=1,2,3,4). The coordinate $y_{i,\alpha}$ is the $y$ coordinate of the
$\alpha$th molecule site in the (1,$j$)th unit cell. Then we can solve the
eigenvalue problem for the nanoribbon-shaped system shown in Fig. 2 using the
basis
$\left(c_{k_{y},A,1}^{\dagger},c_{k_{y},A^{\prime},1}^{\dagger},\cdots,c_{k_{y},B,N_{x}}^{\dagger},c_{k_{y},C,N_{x}}^{\dagger}\right)$
with

$\displaystyle
k_{y}=\frac{2\pi}{N_{y}a}n_{y},\left(n_{y}=-\frac{N_{y}}{2}+1,\cdots,\frac{N_{y}}{2}\right).$
(91)

##  V 5\. Results

###  V.1 5.1 Phase diagrams

Figure 3: (a) Phase diagram of photodriven $\alpha$-(BEDT-TTF)2I3 in the plane
of the electric-field amplitudes $E_{a}^{\omega}$ and $E_{b}^{\omega}$ of
applied elliptically polarized light. (b, c) Color maps of (b) the Chern
number of the highest (fourth) band $N_{\rm Ch}^{4}$ and (c) the band gap
$E_{\rm gap}$ in the plane of $E_{a}^{\omega}$ and $E_{b}^{\omega}$. (d-f)
Typical band structures for (d) the Floquet semimetal phase
($E_{a}^{\omega}=8$ MV/cm and $E_{b}^{\omega}=3$ MV/cm), (e) the Floquet Chern
insulator phase ($E_{a}^{\omega}=11$ MV/cm and $E_{b}^{\omega}=11$ MV/cm), and
(f) the Floquet normal insulator phase ($E_{a}^{\omega}=5$ MV/cm and
$E_{b}^{\omega}=8$ MV/cm). The light frequency is fixed at $\hbar\omega=0.5$
eV. Figure 4: Phase diagrams of photodriven $\alpha$-(BEDT-TTF)2I3 in the
plane of $E_{a}^{\omega}$ and $E_{b}^{\omega}$ of applied elliptically
polarized light for various light frequencies, (a) $\hbar\omega=0.6$ eV, (b)
$\hbar\omega=0.7$ eV, and (c) $\hbar\omega=0.8$ eV.

We first discuss a nonequilibrium phase diagram of photodriven $\alpha$-(BEDT-
TTF)2I3 under irradiation with elliptically polarized light. Because the
crystal structure of $\alpha$-(BEDT-TTF)2I3 is not symmetric with respect to
the interchange of the $a$ and $b$ axes, a rich phase diagram is obtained in
the plane of electric-field amplitudes $E_{a}$ and $E_{b}$ for the
elliptically polarized light. Figure 3(a) presents a phase diagram in the
plane of $E_{a}^{\omega}$ and $E_{b}^{\omega}$ when the light frequency is
fixed at $\hbar\omega=0.5$ eV. The phase diagram contains three phases, that
is, the Floquet semimetal phase, the Floquet Chern insulator phase, and the
Floquet normal insulator phase. These phases are classified by calculating the
Chern number $N_{\rm Ch}$ [Fig. 3(b)] and the band gap $E_{\rm gap}$ [Fig.
3(c)] as well as the quasienergy band structures [Figs. 3(d)-(f)]. The band
gap $E_{\rm gap}$ is defined by,

$E_{\rm
gap}=\min[\varepsilon_{4}^{0}(\bm{k})]-\max[\varepsilon_{3}^{0}(\bm{k})],$
(92)

where $\min[\varepsilon_{4}^{0}(\bm{k})]$ is the minimum quasienergy of the
fourth band, while $\max[\varepsilon_{3}^{0}(\bm{k})]$ is the maximum
quasienergy of the third band in the zero-photon subspace (i.e., $n$=0). When
$E_{\rm gap}$ is negative (positive), the phase is classified into the Floquet
semimetal (insulator) phase. We further classify the insulator phases into two
kinds, that is, the topologically non-trivial Floquet Chern insulator phase
and the topologically trivial Floquet normal insulator phase according to the
Chern number $N_{\rm Ch}$, which is defined as a sum of the Chern numbers of
bands below the Fermi level,

$N_{\rm Ch}=\sum_{\nu=1}^{3}N_{\rm Ch}^{\nu}=-N_{\rm Ch}^{4}.$ (93)

Namely, the band structures of the photoinduced Floquet phases in Figs.
3(d)-(f) are respectively characterized by $E_{\rm gap}$ and $N_{\rm Ch}$ as
summarized in Table 1.

Floquet phase | Band gap $E_{\rm gap}$ | Chern number $N_{\rm Ch}$
---|---|---
Semimetal | $E_{\rm gap}\leq 0$ | $N_{\rm Ch}=0$
Chern insulator | $E_{\rm gap}>0$ | $N_{\rm Ch}\neq 0$
Normal insulator | $E_{\rm gap}>0$ | $N_{\rm Ch}=0$
Table 1: Classification of the photoinduced phases in $\alpha$-(BEDT-TTF)2I3
according to the band gap $E_{\rm gap}$ and the Chern number $N_{\rm Ch}$.

As explained in Sect. 3, the present calculations are performed using the
Floquet Hamiltonian matrix $\mathcal{H}_{nm}$ in Eq. (4) after we truncate the
matrix size to $|m|\leq 16$, instead of the effective Floquet Hamiltonian
$H_{\rm eff}$ in Eq. (7). Note that we obtain the Floquet normal insulator
phase characterized by $E_{\rm gap}>0$ and $N_{\rm Ch}=0$ when both amplitudes
(i.e., $E_{a}^{\omega}$ and $E_{b}^{\omega}$) and frequency $\hbar\omega$ are
low. This phase, however, cannot be reproduced by calculations with the
effective Floquet Hamiltonian $H_{\rm eff}$, which indicates that the Floquet
normal insulator phase is realized by effects which are not incorporated in
the $1/\omega$ expansion in the high-frequency limit.

Figures 4(a)-(c) present obtained nonequilibrium phase diagrams of photodriven
$\alpha$-(BEDT-TTF)2I3 for various light frequencies of (a) $\hbar\omega=0.6$
eV, (b) $\hbar\omega=0.7$ eV, and (c) $\hbar\omega=0.8$ eV. In addition to
these frequency points, we have studied the phase diagrams at several
frequency points although they are not presented here. We find that the
Floquet normal insulator phase appears when $\hbar\omega\lesssim 0.75$ eV, but
its area in the phase diagram gets smaller as the light frequency
$\hbar\omega$ increases and eventually disappears for $\hbar\omega\gtrsim
0.75$ eV. These results indicate that we can selectively realize Floquet
phases even with rather weak light amplitudes by tuning the light frequency
$\hbar\omega$. Namely, we can have an opportunity to observe the Floquet
semimetal phase for $\hbar\omega\sim 0.6$ eV, the Floquet normal insulator
phase for $\hbar\omega\sim 0.7$ eV, and the Floquet Chern insulator phase for
$\hbar\omega\sim 0.8$ eV with a rather weak light fields $E_{a}^{\omega}$ and
$E_{b}^{\omega}$ even less than 1 MV/cm. This fact increases the experimental
feasibility, and we expect that the phase diagrams obtained here are helpful
for designing the experiments.

In the present study, we consider a situation with sufficiently small
dissipations where the system can reach a nonequilibrium steady state which
evolves time-periodically synchronizing with the ac electric fields of light
even when the amplitudes $E_{a}^{\omega}$ and $E_{b}^{\omega}$ are
infinitesimally small. In this ideal situation, our theoretical treatment
based on the Floquet theorem is justified, and the obtained phase diagrams are
valid even in the limit of $E_{a}^{\omega}\rightarrow 0$ and
$E_{b}^{\omega}\rightarrow 0$, where infinitesimally small gaps are opened at
the Dirac points and the system becomes the insulator or semimetal. (Note that
the point at $E_{a}^{\omega}=0$ and $E_{b}^{\omega}=0$ is a singular point
where the gaps are exactly closed). On the contrary, the dissipation effect
necessarily exists in real materials which hinders the realization of
nonequilibrium steady state when the light electric fields are weak and thus
makes the Floquet description inappropriate. Therefore, the areas around
$E_{a}^{\omega}\sim 0$ and $E_{b}^{\omega}\sim 0$ in the present phase
diagrams must be modified when we take the dissipation effect into account.
We, however, expect that overall features of the phase diagram hold even in
real systems with finite dissipations. The effect of dissipations on the
threshold amplitude of light electric fields for the predicted photoinduced
phase transitions should be clarified in the future by a more elaborate
theory.

###  V.2 5.2 Chiral edge states

Figure 5: Quasienergy band structures with chiral edge-state bands for systems
with vertical edges on which the open boundary conditions are imposed in the
$x$ ($b$) direction. (a) Those for a steady system with AA′-BC edges before
light irradiation. (b-d) Those for driven systems with (b) AA′-BC edges (c)
AA′-BC edges, and (d) BC-BC edges where the periodic boundary conditions are
imposed in the $y$ ($a$) direction. (e-f) Systems with (e) AA′-BC edges, (f)
AA′-AA′ edges, and (g) BC-BC edges used for the calculations. The amplitudes
and frequency of light are chosen to be $E_{a}^{\omega}=8$ MV/cm,
$E_{b}^{\omega}=8$ MV/cm, and $\hbar\omega=0.7$ eV. Figure 6: Quasienergy band
structures with chiral edge-state bands for systems with horizontal edges on
which the open boundary conditions are imposed in the $y$ ($a$) direction. (a)
Those for a steady system with AB-A′C edges before light irradiation. (b-d)
Those for driven systems with (b) AB-A′C edges (c) AB-AB edges, and (d)
A′C-A′C edges where the periodic boundary conditions are imposed in the $x$
($b$) direction. (e-f) Systems with (e) AB-A′C edges, (f) AB-AB edges, and (g)
A′C-A′C edges used for the calculations. The amplitudes and frequency of light
are chosen to be $E_{a}^{\omega}=8$ MV/cm, $E_{b}^{\omega}=8$ MV/cm, and
$\hbar\omega=0.7$ eV.

The Chern insulator is characterized by existence of chiral edge states as
well as the Chern number $N_{\rm Ch}$. Therefore, we next investigate the edge
states for systems having edges to validate the predicted photoinduced Chern
insulator phase. We examine several monolayer systems of $\alpha$-(BEDT-
TTF)2I3 with different edges. Figures 5(a) and (b) present the calculated band
dispersion relations before and during the light irradiation, respectively,
for a system with vertical edges along the $y$ axis, that is, the AA′ and BC
edges at left and right ends of the sample [Fig. 2 and Fig. 5(e)]. As
explained in Sect. 4, the open boundary conditions are imposed in the $x$
direction, while the periodic boundary conditions are imposed in the $y$
direction. Note that we obtain the band dispersion relations including those
associated with the edge states by diagonalizing the Floquet effective
Hamiltonian. Here the amplitudes and frequency of the elliptically polarized
light are chosen to be $E_{a}^{\omega}=8$ MV/cm, $E_{b}^{\omega}=8$ MV/cm, and
$\hbar\omega=0.7$ eV, respectively, where the Chern insulator is realized. The
band dispersion relations in Figs. 5(a) and (b) show appearance of the bands
associated with the edge states. The Dirac points are degenerate before light
irradiation in Fig. 5(a), whereas they become gapped under light irradiation
in Fig. 5(b). Importantly, the edge-state bands connect the lower valence band
and the upper conduction band, which clearly evidences that the predicted
photoinduced phase assigned to the Chern insulator phase is indeed
topologically nontrivial. We also examine other systems with different
vertical edges. Figure 5(c) presents the band dispersion relations for a
system having two AA′ edges [Fig. 5(f)], whereas Fig. 5(d) presents those for
a system having two BC edges [Fig. 5(g)]. We again obtain the band dispersions
associated with the chiral edge states connecting the upper and lower bands of
the gapped Dirac cones.

We also calculate the band dispersion relations for systems with horizontal
edges [see Figs. 6(a)-(d)]. Systems used for the calculations are presented in
Figs. 6(e)-(g). Specifically, we examine the systems with (e) AB-A′C edges,
(f) AB-AB edges, and (g) A′C-A′C edges at upper and bottom ends of the system.
The open boundary conditions are imposed in the $y$ direction, whereas the
periodic boundary conditions are imposed in the $x$ direction. The amplitudes
and frequency of light are again set to be $E_{a}^{\omega}=8$ MV/cm,
$E_{b}^{\omega}=8$ MV/cm, and $\hbar\omega=0.7$ eV, respectively. The Dirac
points are degenerate before the light irradiation [Fig. 6(a)], but the
degeneracy is lifted by the photoinduced gap during the light irradiation
[Figs. 6(b)-(d)]. Although the band dispersions associated with the edge
states depend on the edge species, they always connect the lower valence band
and the upper conduction band separated by the photoinduced band gap at the
Dirac points, again indicating that the corresponding photoinduced Chern
insulator phase is indeed topologically nontrivial.

###  V.3 5.3 Hall conductivities

Figure 7: Calculated $E_{a}^{\omega}$ dependencies of physical quantities for
a fixed light frequency of $\hbar\omega$=0.5 eV. (a1)-(a3) Chern number
$N_{\rm Ch}$ (a1), band gap $E_{\rm gap}$ (a2), and Hall conductivity
$\sigma_{xy}$ (a3) for $\alpha$-(BEDT-TTF)2I3 irradiated with elliptically
polarized light as functions of $E_{a}^{\omega}$ when $E_{b}^{\omega}=3$
MV/cm, for which the system is always in the semimetallic phase. (b1)-(b3)
Those when $E_{b}^{\omega}=8$ MV/cm, for which the system exhibits a phase
transition from the normal insulator phase to the semimetallic phase.
(c1)-(c3) Those when $E_{b}^{\omega}=12$ MV/cm, for which the system exhibits
a phase transition from the Chern insulator phase with $N_{\rm Ch}=+1$ to the
semimetallic phase. Note that the calculations are hypothetically performed
even for unrealistically large light amplitudes $E_{a}^{\omega}$ to discuss
inherent behaviors of the physical quantities. Figure 8: Calculated
$E_{a}^{\omega}$ dependencies of physical quantities for a fixed light
frequency of $\hbar\omega$=0.7 eV. (a1)-(a3) Chern number $N_{\rm Ch}$ (a1),
band gap $E_{\rm gap}$ (a2), and Hall conductivity $\sigma_{xy}$ (a3) for
$\alpha$-(BEDT-TTF)2I3 irradiated with elliptically polarized light as
functions of $E_{a}^{\omega}$ when $E_{b}^{\omega}=2$ MV/cm, for which the
system exhibits a phase transition from the normal insulator phase to the
semimetallic phase. (b1)-(b3) Those when $E_{b}^{\omega}=4.5$ MV/cm, for which
the system exhibits two phase transitions from the normal insulator phase to
the Chern insulator phase with $N_{\rm Ch}=+1$ to the semimetallic phase.
(c1)-(c3) Those when $E_{b}^{\omega}=7$ MV/cm, for which the system exhibits a
phase transition from the Chern insulator phase with $N_{\rm Ch}=+1$ to the
semimetallic phase. Note that the calculations are hypothetically performed
even for unrealistically large light amplitudes $E_{a}^{\omega}$ to discuss
inherent behaviors of the physical quantities. Figure 9: Calculated electronic
structure in the momentum space for the Floquet normal insulator phase of
$\alpha$-(BEDT-TTF)2I3 irradiated with elliptically polarized light when
$\hbar\omega=0.7$ eV, $E_{a}^{\omega}=5$ MV/cm and $E_{b}^{\omega}=4.5$ MV/cm.
(a) Floquet band structure in the zero-photon subspace ($n$=0). (b) Berry
curvatures $B_{z}^{\nu}(\bm{k})$ of the respective bands with $\nu$=1, 2, 3
and 4. (c) Band occupations $n_{\nu}(\bm{k})$ of the respective bands.

We finally study the Hall conductivity in the photoinduced nonequilibrium
phases. First, we examine the case where the light frequency is
$\hbar\omega=0.5$ eV [see also the phase diagram in Fig. 3(a)]. Figure 7
presents the calculated Chern number $N_{\rm Ch}$, band gap $E_{\rm gap}$, and
Hall conductivity $\sigma_{xy}$ as functions of $E_{a}^{\omega}$ for several
values of $E_{b}^{\omega}$, that is, (a1)-(a3) $E_{b}^{\omega}=3$ MV/cm,
(b1)-(b3) $E_{b}^{\omega}=8$ MV/cm, and (c1)-(c3) $E_{b}^{\omega}=12$ MV/cm.
The Chern number $N_{\rm Ch}$ and the band gap $E_{\rm gap}$ are used to
identify the phases. We find that

  * •

The Floquet Chern insulator phase with $N_{\rm Ch}=1$ has a nearly quantized
Hall conductivity $\sigma_{xy}=2e^{2}/h$ at low temperatures, which is
equivalent to the Chern insulator phase in equilibrium.

  * •

The Floquet normal insulator phase also has a nearly quantized Hall
conductivity $\sigma_{xy}=2e^{2}/h$ despite $N_{\rm Ch}$ is zero.

  * •

The Floquet semimetal phase has a finite Hall conductivity that gradually
decreases as the system becomes distant from the Floquet normal insulator
phase [Fig. 7(b3)] and the Floquet Chern insulator phase [Fig. 7(c3)] with
increasing $E_{a}^{\omega}$.

Here the second finding is surprising because the quantized Hall conductivity
in the Floquet normal insulator phase is in sharp contrast with a vanishing
Hall conductivity in the equilibrium normal insulator, which indicates that
the Floquet normal insulator phase is not really insulating. This finite and
quantized Hall conductivity is observed not only at the small frequency of
$\hbar\omega=0.5$ eV, but also at $\hbar\omega=0.7$ eV as seen in Fig. 8 [see
also the phase diagram in Fig. 4(b)]. Note that the Floquet normal insulator
phase appears only when the light frequency is smaller than 0.75 eV, and hence
the frequency of $\hbar\omega=0.7$ eV is almost as high as this threshold
frequency. Figures 8(a3) and (b3) indeed show that the Hall conductivity in
the Floquet normal insulator phase has a quantized value
$\sigma_{xy}=2e^{2}/h$ especially at low temperatures, although they rapidly
decrease and the deviation from this quantized value is enhanced as
temperature increases because of the small band gap [see Figs. 8(a2) and
(b2)].

To understand the origin of this quantized Hall conductivity in the Floquet
normal insulator phase, we examine the electronic structure of the Floquet
normal insulator phase for $\hbar\omega=0.7$ eV, $E_{a}^{\omega}=5$ MV/cm and
$E_{b}^{\omega}=4.5$ MV/cm. Figure 9(a) shows the calculated band structure in
the zero-photon subspace ($n$=0). The Chern numbers $N^{\nu}_{\rm Ch}$ for
respective bands ($\nu$=1, 2, 3 and 4) are evaluated as $N^{1}_{\rm Ch}=-2$,
$N^{2}_{\rm Ch}=+1$, $N^{3}_{\rm Ch}=+1$ and $N^{4}_{\rm Ch}=0$. The Berry
curvatures $B_{z}^{\nu}(\bm{k})$ of the respective bands in the momentum space
show momentum-resolved contributions to the Chern number from each band [see
Fig. 9(b)]. Importantly, the plot of $B_{z}^{\nu}(\bm{k})$ for $\nu$=4
indicates that positive and negative contributions from different momentum
points cancel out resulting in the vanishing Chern number of $N^{4}_{\rm
Ch}=0$ in the fourth band. This Berry-curvature structure is a key to
understanding the observed quantized Hall conductivity. Figure 9(c) shows the
nonequilibrium band occupations $n_{\nu}(\bm{k})$ for the respective bands
calculated using Eq. (71). In the equilibrium case where the Fermi-
distribution function governs the band occupations, the lower three bands
($\nu$=1,2,3) are fully occupied, while the highest band ($\nu$=4) is almost
unoccupied at low temperatures because the electron filling of $\alpha$-(BEDT-
TTF)2I3 is 3/4. Thereby, we expect the Chern number $N_{\rm Ch}$ and the Hall
conductivity $\sigma_{xy}$ vanish in the equilibrium normal insulator phase.
On the other hand, the band occupations in the nonequilibrium system deviate
from those in equilibrium governed by the Fermi-distribution function. The
topmost panel in Fig. 9(c) indicates that the fourth band is partially
occupied at the momenta around $(k_{x},k_{y})=(\pm\pi,0)$ where the Berry
curvature $B_{z}^{4}(\bm{k})$ has positive peaks, whereas the bottom panel
indicates that the first band is partially unoccupied at the momenta around
$(k_{x},k_{y})=(\pm\pi,0)$ where the Berry curvature $B_{z}^{1}(\bm{k})$ has
negative peaks.

The emergence of these positive and negative peaks in the Berry curvature
$B_{z}^{\nu}(\bm{k})$ for the $\nu$=4 band as well as the emergence of the
Floquet normal insulator phase at lower light frequencies can be understood by
the hybridization between the topmost band ($\nu=4$) in the zero-photon
subspace ($n=0$) and the bottom band ($\nu=1$) in the one-photon-absorbed
subspace ($n=-1$) located above the zero-photon subband set. This band
hybridization gives rise to the positive peaks of the Berry curvature at the
band crossing points, which offset the negative contributions from the Dirac
points on the band labeled with $(n,\nu)=(0,4)$, resulting in the vanishing
Chern number and the emergence of the Floquet normal insulator phase. Because
the energy spacing between the subband sets of $n=0$ and $n=-1$ is governed by
the light frequency $\hbar\omega$, such a band hybridization occurs when the
light frequency $\omega$ is small as compared with the bandwidth $W$ (i.e.,
$\hbar\omega<W$). Conversely, the band hybridization does not occur when
$\hbar\omega>W$. Because the bandwidth of $\alpha$-(BEDT-TTF)2I3 in the
equilibrium is approximately 0.75 eV, the Floquet normal insulator phase does
not appear when $\hbar\omega\geq 0.8$ eV.

##  VI 6\. Summary and Discussion

We theoretically predicted rich photoinduced phases and phase transitions in
organic conductor $\alpha$-(BEDT-TTF)2I3 under irradiation with elliptically
polarized light. This compound possesses a pair of tilted Dirac cones in the
band structure, and the Dirac points are located on the Fermi level under a
uniaxial pressure $P_{a}$($>2$ kbar). By analyzing a photodriven tight-binding
model using the Floquet theory, we obtained rich nonequilibrium phase diagrams
in the plane of light amplitudes that contain the Floquet semimetal phase, the
Floquet Chern insulator phase, and the Floquet normal insulator phases as
nonequilibrium steady states. We demonstrated that all the three phases are
accessible with a rather weak light amplitude of less than 1 MV/cm by tuning
the light frequency. More specifically, the Floquet semimetal phase, the
Floquet normal insulator phase, and the Floquet Chern insulator phase are
accessible with weak laser light when $\hbar\omega\sim$0.6 eV,
$\hbar\omega\sim$0.7 eV, $\hbar\omega\sim$0.8 eV, respectively. We also
revealed that the predicted Floquet Chern insulator phase is characterized by
both a quantized topological invariant (i.e., the Chern number) and chiral
edge states, which indicates that this phase is indeed topologically
nontrivial. In addition, the Floquet normal insulator phase classified
according to the band structure and the band Chern numbers turned out to
exhibit a quantized Hall conductivity because of partial occupations of the
fourth band in the nonequilibrium.

The present organic conductor $\alpha$-(BEDT-TTF)2I3 has several advantages
for experimental study on the photoinduced phase-transition phenomena as
discussed in Ref. Kitayama20 . First, this material is composed of large
molecules and thus has relatively large lattice constants ($a$=0.9187 nm and
$b$=1.0793 nm) Mori12 . These lattice constants are a few times larger than
the lattice constant in graphene ($a$=0.246 nm), which effectively enhance the
dimensionless laser amplitude $\mathcal{A}_{a}=eaE^{\omega}/\hbar\omega$ and
$\mathcal{A}_{b}=ebE^{\omega}/\hbar\omega$. Second, because the bands
associated with the BEDT-TTF layer are well separated from upper and lower
band sets by energy gaps Kino06 , we have a finite window of light frequency
where the BEDT-TTF bands in the zero-photon subspace overlap neither the bands
in the one-photon-absorbed ($n=-1$) subspace nor the bands in the one-photon-
emitted ($n=+1$) subspace. This is an experimentally suitable situation called
off-resonant situation, where the electron occupations of the Floquet bands
are well approximated by the Fermi-Dirac distribution function in equilibrium,
and, thereby, the photoinduced phases become well-defined.

As we argued above, the present organic compound $\alpha$-(BEDT-TTF)2I3 is an
interesting material because we can access various kinds of photoinduced
phases with a weak light field by tuning the light frequency. At the same
time, our results indicate that richer photoinduced evolutions of the phases
can be observed when we apply a much stronger light field. Needless to say,
samples of the organic materials might be damaged or even broken under
continuous application of an intense light field. Note that it is difficult to
evaluate to what extent a sample of this organic compound can endure the
intense laser, but still we can argue the feasibility of experiments in the
light of recently reported experimental achievements. Some experiments of
photoinduced phase transitions have been successfully performed for
$\kappa$-type (BEDT-TTF)-based organic salts using a few-cycle-pulse laser
light or one-cycle-pulse laser light as intense as $E^{\omega}=$16 MV/cm
Uchida15 ; Uchida16 . In addition, recent experiments demonstrated that a
small-number-cycle pulse or even a less-than-10-cycle pulse, instead of
continuous-wave photoexcitation, is enough to realize Floquet phases or
nonequilibrium steady phases Kawakami18 ; Tachizaki19 ; Kawakami20 .
Considering these experimental achievements, experimental explorations of the
predicted photoinduced phases and phase transitions in $\alpha$-(BEDT-TTF)2I3
are feasible or, at least, worth trying. The short duration of light
irradiation is also convenient to avoid or suppress the heating effect,
although the time scale of heating is uncertain and its theoretical evaluation
is important, which is left for a future study. We expect that the present
work widens a range of target materials for research on the photoinduced phase
transitions and contribute to development of the optical control of electronic
states in matters.

##  VII Acknowledgment

KK is supported by the World-leading Innovative Graduate Study Program for
Materials Research, Industry, and Technology (MERIT-WINGS) of the University
of Tokyo. MM is supported by JSPS KAKENHI (Grants No. 16H06345, No. 19H00864,
No. 19K21858 and No. 20H00337), CREST, the Japan Science and Technology Agency
(Grant No. JPMJCR20T1), a Research Grant in the Natural Sciences from the
Mitsubishi Foundation, and a Waseda University Grant for Special Research
Projects (Project No. 2020C-269 and No. 2021C-566). YT is supported by JSPS
KAKENHI (Grants No. 19K23427 and No. 20K03841). MO is supported by JSPS
KAKENHI (Grant No. 18H01162).

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