# Quantum phase diagram for two species hardcore bosons in one-dimensional
optical lattices with the resonantly driven Rabi frequency

Peigeng Zhong  Beijing Computational Science Research Center, Beijing 100084,
China  Tao Wang<EMAIL_ADDRESS>Department of Physics, Chongqing
University, Chongqing, 401331, China  Chongqing Key Laboratory for Strongly
Coupled Physics, Chongqing University, Chongqing 401331, China  Shijie Hu
<EMAIL_ADDRESS>Beijing Computational Science Research Center, Beijing
100084, China  Haiqing Lin  Beijing Computational Science Research Center,
Beijing 100084, China  Department of Physics, Beijing Normal University,
Beijing, 100875, China

###### Abstract

We propose an experimental realization of the time-periodically modulated Rabi
frequency and suggest density-dependent hoppings of two species hardcore
bosons in a one-dimensional optical lattice. Distinct from the previous work
[Phys. Rev. Research 2, 013275 (2020)], we study effects in the first
resonance region. In the effective Hamiltonian, the intra-species hopping
occurs only if the density discrepancy of the other species on these sites is
zero, while the inter-species one is allowed once the relevant density
discrepancy becomes nonzero. At integer-$1$ filling, the quantum phase diagram
of the effective Hamiltonian is determined by the perturbation analysis
together with numerical calculations. We find that in the limit of dominant
$J_{1}$, the system becomes a double-degenerate dimerized state, with
spontaneously breaking the translation symmetry. The interplay of $J_{0}$,
$J_{1}$ and the fixed ${\bar{U}}=1$ leads to three BKT transition lines and a
tricritical BKT point. Exact transition lines are obtained by the level
spectroscopic technique. Besides, general physical properties, including the
charge gap, neutral gap, superfluid density and dimerization strength, are
investigated as well.

##  I Introduction

Floquet engineering with controllable time-periodic driving sources Eckardt
(2017); Weitenberg and Simonet (2021) is a powerful tool to tailor quantum
gases in optical lattices. In contrast to the traditional solid state systems,
it relies on extremely high-intensity laser and a sophisticated scheme of fast
measurements McIver  _et al._ (2020). The typical driving frequency ranges
from several to several tens of kilohertz (kHz), which allows us to unveil the
physics in the strong driving regime experimentally Arimondo  _et al._ (2012).
Floquet engineering greatly enriches accessible models and exotic phenomena in
optical lattices, such as generating artificial gauge fields by breaking the
time-reversal symmetry of the driving function Struck  _et al._ (2012);
Jiménez-García  _et al._ (2012); Schweizer  _et al._ (2019); Barbiero  _et
al._ (2019); Lu  _et al._ (2021), realizing anyon physics combing with Raman
assisted tunneling Greschner and Santos (2015); Sträter  _et al._ (2016) and
designing other unconventional Hubbard models Greschner  _et al._ (2014); Zhao
_et al._ (2019).

For example, Bose-Hubbard models with density-dependent correlated hoppings
have been Floquet-engineered in optical lattices by tuning the magnetic field
near a Feshbach resonance point Meinert  _et al._ (2016); Compagno  _et al._
(2020). These models give novel quantum phases Rapp  _et al._ (2012); Wang
_et al._ (2014) and are also essential for implementing dynamical gauge fields
Zohar  _et al._ (2015); Bañuls and Cichy (2020). However, for two species
hardcore bosons that tightly connect to the Fermi-Hubbard model in one
dimension (1D), it was challenging to tune inter-species interaction together
with holding the hardcore constraint. In the previous work [Phys. Rev.
Research 2, 013275 (2020)], we proposed an alternative way of periodically
driving the Rabi frequency, which has been realized in experiments recently Lu
_et al._ (2021). In the case of high-frequency approximation, we found an
integrable point and a “gauge dressed superfluid” phase Wang  _et al._ (2020).
Once the driving frequency resonates with the interaction strength, new
physics insights are concerned in the presence of different models Weitenberg
and Simonet (2021); Coulthard  _et al._ (2017); Görg  _et al._ (2018).

The paper is organized as follows. In section II, we derive the effective
model in the first resonance region according to an existing time-dependent
Hamiltonian in optical lattices. In section III, we do an explicit
perturbation analysis and numerical simulations to characterize all phases and
phase transitions in the allowable parameter region. At last in section IV, we
summarize all results with a brief discussion.

Figure 1: (a) Schematic picture for the experimental realization. Both hopping
amplitudes $J_{0}$ and raw onsite repulsion strength are controlled by the
lattice depth. Two hyperfine states are coupled to an intermediate state
(black dashed line) by the extra Raman laser beams. The effective Rabi
frequency $J_{\Omega}$ is time-periodically modulated. The static magnetic
field is a bit far away from a Feshbach resonance point on the side of
negative scattering length, so the residual inter-species repulsion strength
$U\sim\hbar\omega$. Meanwhile, the dominantly-large intra-species repulsion
maintains the hardcore constraint. (b) Quantum phase diagram of the effective
model at integer-$1$ filling. It consists of three distinct phases: superfluid
(SF), Mott-insulating (MI) and Dimerized (DM) phases. Border-lines SF-MI, SF-
DM and MI-DM are marked by red, orange and blue respectively. A green five-
star indicates a tricritical point of three phases. Two dashed cutting lines
$J_{0}+J_{1}=1$ and $J_{1}=1.5$ are marked for convenience of discussion.

##  II Density-dependent correlated hoppings in the first resonance region

Following the existing scheme Wang  _et al._ (2020, 2021), the full
Hamiltonian for two species hardcore bosons in optical lattices reads

$\displaystyle{\hat{H}}(t)$ $\displaystyle=$
$\displaystyle{\hat{H}}_{T}+{\hat{H}}_{U}+{\hat{H}}_{\Omega}(t)$ (1)

and

$\displaystyle{\hat{H}}_{T}=-J_{0}\sum_{l}\left({\hat{a}}^{\dagger}_{l}{\hat{a}}_{l+1}+{\hat{b}}^{\dagger}_{l}{\hat{b}}_{l+1}+\rm{h.c.}\right)\,,$
$\displaystyle{\hat{H}}_{U}=U\sum_{l}{\hat{n}}^{a}_{l}{\hat{n}}^{b}_{l}\,,$
$\displaystyle{\hat{H}}_{\Omega}(t)=J_{\Omega}(t)\sum_{l}\left(\hat{a}_{l}^{\dagger}\hat{b}_{l}+\rm{h.c.}\right)\,,$

where ${\hat{a}}_{l}$ (${\hat{b}}_{l}$), ${\hat{a}}^{\dagger}_{l}$
(${\hat{b}}^{\dagger}_{l}$) and
${\hat{n}}^{a}_{l}={\hat{a}}^{\dagger}_{l}{\hat{a}}_{l}$
(${\hat{n}}^{b}_{l}={\hat{b}}^{\dagger}_{l}{\hat{b}}_{l}$) are the
annihilation, creation and particle number operators for the species “$a$”
(“$b$”) respectively, $J_{0}$ the hopping coefficient governed by the lattice
depth, $U$ the effective inter-species onsite repulsion strength,
$J_{\Omega}(t)=J^{0}_{\Omega}\cos(\omega t)$ the time-periodically modulated
“Rabi frequency” with the driving frequency of $\omega$ and the index $l$ runs
over whole $L$ lattice sites.

In the previous work, our attention was centered on the non-resonance region,
where the driving frequency must be much larger than $J_{0}$ and $U$, but not
too large to avoid “photon-assisted hopping” between energy bands of the
optical lattices Wang  _et al._ (2020). In this work, we further investigate
rich quantum phases and phase transitions in the first resonance region
$\hbar\omega\sim U$, which is achieved by moving the static magnetic field a
bit further away from the Feshbach resonance point on the side of negative
scattering length. Meanwhile, $J^{0}_{\Omega}/\hbar$ is of several kHz, the
same as $\omega$.

We are willing to restate three concerns. Firstly, by using an acousto-optic
modulator (AOM) to change both the amplitude and polarization angle of the
pump laser Eklund  _et al._ (1975); Sapriel  _et al._ (1979), the setup can
effectively generate the time-periodically modulated Rabi frequency in
practice Wang  _et al._ (2020). Secondly, the initial phase of the time-
periodically modulated Rabi frequency is irrelevant to the stroboscopic
measurement at an integer number of periods, which has already been realized
and well controlled in experiments Eckardt (2017). Thirdly, although the
system usually heats up to the infinite temperature in a long time due to the
absence of energy conservation D’Alessio and Rigol (2014), a prethermal stage
still survives Bukov  _et al._ (2016); Rubio-Abadal  _et al._ (2020). Various
theoretical and experimental studies have exhibited that it is long enough for
stabilizing and measuring the effects governed by the effective Hamiltonian
Boulier  _et al._ (2019); Wintersperger  _et al._ (2020). Even in the
resonance region Görg  _et al._ (2018), the heating rate is exponentially
suppressed in the Floquet Bosonic optical lattices Messer  _et al._ (2018).

According to the Floquet theory Rapp  _et al._ (2012); Arimondo  _et al._
(2012); Wang  _et al._ (2014); Greschner  _et al._ (2014); Meinert  _et al._
(2016); Eckardt (2017) for a time-periodically driven system
${\hat{H}}(t)={\hat{H}}(t+T)$ with a period of $T=2\pi/\omega$, the dynamics
of the system is depicted by the effective Hamiltonian and corresponding kick
operators, which can be determined from a nearly-degenerate perturbation
method Eckardt (2017); Wang  _et al._ (2020). By adopting standard operations
(Appendix A), we obtain the effective Hamiltonian

$\displaystyle{\hat{H}}_{e}$ $\displaystyle=$
$\displaystyle-\sum_{l}\left[J_{0}({\hat{H}}^{aa}_{l,l+1}+{\hat{H}}^{bb}_{l,l+1})+J_{1}({\hat{H}}^{ab}_{l,l+1}+{\hat{H}}^{ba}_{l,l+1})\right]$
(2) $\displaystyle+{\bar{U}}\sum_{l}{\hat{n}}^{a}_{l}{\hat{n}}^{b}_{l}$

with $J_{1}=J_{0}{\cal J}_{1}(2K_{\Omega})$, the first order of the first kind
Bessel function ${\cal J}_{1}(x)$, a dimensionless parameter
$K_{\Omega}=J^{0}_{\Omega}/\hbar\omega$, a renormlized onsite repulsion
strength ${\bar{U}}=U-\hbar\omega$ and four distinct hopping terms

$\displaystyle{\hat{H}}^{aa}_{l,l+1}$ $\displaystyle\\!=\\!$
$\displaystyle{\hat{a}}^{\dagger}_{l}{\hat{a}}_{l+1}[{\hat{n}}^{b}_{l}{\hat{n}}^{b}_{l+1}+(1-{\hat{n}}^{b}_{l})(1-{\hat{n}}^{b}_{l+1})]\\!+\\!{\rm
h.c.}\,,$ $\displaystyle{\hat{H}}^{bb}_{l,l+1}$ $\displaystyle\\!=\\!$
$\displaystyle{\hat{b}}^{\dagger}_{l}{\hat{b}}_{l+1}[{\hat{n}}^{a}_{l}{\hat{n}}^{a}_{l+1}+(1-{\hat{n}}^{a}_{l})(1-{\hat{n}}^{a}_{l+1})]\\!+\\!{\rm
h.c.}\,,$ $\displaystyle{\hat{H}}^{ab}_{l,l+1}$ $\displaystyle\\!=\\!$
$\displaystyle{\hat{a}}^{\dagger}_{l}{\hat{b}}_{l+1}[{\hat{n}}^{b}_{l}(1-{\hat{n}}^{a}_{l+1})+(1-{\hat{n}}^{b}_{l}){\hat{n}}^{a}_{l+1}]\\!+\\!{\rm
h.c.}\,,$ $\displaystyle{\hat{H}}^{ba}_{l,l+1}$ $\displaystyle\\!=\\!$
$\displaystyle{\hat{b}}^{\dagger}_{l}{\hat{a}}_{l+1}[{\hat{n}}^{a}_{l}(1-{\hat{n}}^{b}_{l+1})+(1-{\hat{n}}^{a}_{l}){\hat{n}}^{b}_{l+1}]\\!+\\!{\rm
h.c.}\,.$ (3)

Due to the properties of ${\cal J}_{1}$, the intra-species hopping in the
effective Hamiltonian occurs only if the density discrepancy of the other
species on these sites is zero, while the inter-species one is allowed once
the relevant density discrepancy becomes nonzero. Hereafter we choose
${\bar{U}}=1$ as unit since two out of three parameters are free in the model.
Additionally, we only consider the case of integer-$1$ filling, namely
$N_{a}+N_{b}=L$ and $N_{a/b}=\langle\sum_{l}{\hat{n}}^{a/b}_{l}\rangle$, where
more interesting phenomena and a rich phase diagram are anticipated.

Although $|J_{1}/J_{0}|<1$ because of $|{\cal J}_{1}|<1$, its regime could be
extended if the driving forms are chosen appropriately. For instance, one can
implement the above proposal in a two-leg Wannier-Stark ladder, where the
energy difference between the nearest-neighboring sites along the ladder
resonates with the driving frequency $\omega$ Eckardt and Holthaus (2007). In
the setup, $J_{0}$ and $J_{1}$ are dressed by ${\cal J}_{1}$ and the second
order of the first kind Bessel function ${\cal J}_{2}$ separately, so we study
the full parameter space of $|J_{0}/J_{1}|$ in the following sections.

Before going to the details of the quantum phase diagram, let us discuss the
symmetry related to the controllable parameters $J_{0}$, $J_{1}$ and
${\bar{U}}$. First of all, positive and negative $J_{0/1}$ are linkable under
the transformation ${\cal
W}^{a/b}_{l}=\exp\left(i\pi{\hat{n}}^{a/b}_{l}\right)$, which yields
${\hat{a}}_{l}\rightarrow-{\hat{a}}_{l}$ and
${\hat{b}}_{l}\rightarrow-{\hat{b}}_{l}$ respectively. For instance, in order
to reverse the sign of $J_{0/1}$ at the same time, we can apply ${\cal
W}^{a/b}_{l}$ to all even sites. Secondly, the effective Hamiltonian does not
have the particle-hole symmetry for each species in principle. Under the
partial particle-hole transformation with
${\hat{a}}^{\dagger}_{l}\rightarrow{\hat{a}}_{l}$ for each site, the hopping
terms remain invariant but ${\hat{H}}_{\bar{U}}$ becomes
${\hat{H}}_{-{\bar{U}}}+{\bar{U}}\sum_{l}{\hat{n}}^{b}_{l}$. In the quantum
phase diagram discussed later, phases always have zero polarization
$N_{a}=N_{b}$ and the chemical potential term
${\bar{U}}\sum_{l}{\hat{n}}^{b}_{l}$ is a trivial constant actually. Thus, the
positive and negative $\bar{U}$ regions are symmetrically equivalent. In the
work, we are merely concerned with the region where both $J_{0/1}$ and
${\bar{U}}$ are positive.

##  III Quantum phase diagram

###  III.1 Perturbation analysis

Once $J_{1}=0$, the hopping between neighboring double occupied and empty
sites is not allowed Wang  _et al._ (2020). The Hamiltonian no longer
distinguishes between double occupied and empty sites, we can denote both of
them with pseudo-spin up $|\\!\\!\uparrow\rangle$. Similarly, the hopping
between neighboring single occupied sites is forbidden regardless of whether
they are species “$a$” or “$b$”, so both can be denoted with pseudo-spin down
$|\\!\\!\downarrow\rangle$. The resulting Hamiltonian at integer-$1$ filling
is equivalent to an exactly integrable spin-$1/2$ XY model

$\displaystyle{\hat{H}}_{J_{0}}=\sum_{l}\left[-J_{0}({\hat{S}}^{+}_{l}{\hat{S}}^{+}_{l+1}+{\rm
h.c.})+\frac{\bar{U}}{2}({\hat{S}}^{z}_{l}+1/2)\right]\ ,$ (4)

where ${\hat{S}}^{\pm/z}$ are the ordinary operators for spin-$1/2$. $\bar{U}$
provides a Zeeman Splitting between $|\\!\\!\uparrow\rangle$ and
$|\\!\\!\downarrow\rangle$. For ${\bar{U}}>4J_{0}$, the system is saturated
with only single occupied sites and has a finite charge gap, which corresponds
to the Mott-insulating (MI) phase. When ${\bar{U}}\leq 4J_{0}$, the
groundstate is in a gapless $xy$ phase without superfluid (SF) response
indicated by a vertical line $J_{1}=0$ in Fig. 1 (b).

We now turn to the case of $J_{0}=0$. When $J_{1}=0$ as well, the system falls
into a clean MI state purely made up of single-occupied sites. Given a small
$J_{1}$, cooperative hoppings are allowed, where the annihilation of a double
occupancy at site-$l$ is followed by creating one at site-$(l+1)$, namely
$|a\rangle_{l}|a\rangle_{l+1}\rightarrow|0\rangle_{l}|ab\rangle_{l+1}$ and
vice verse. By the second order perturbation, the above hoppings are
generalized into an effective Hamiltonian (Appendix B)

$\displaystyle{\hat{H}}^{(2)}_{\text{MI}}\\!=\\!-\frac{J^{2}_{1}}{\bar{U}}\sum_{l}\left[2({\hat{S}}^{+}_{l}{\hat{S}}^{+}_{l+1}\\!+\\!{\rm
h.c.})\\!-\\!4{\hat{S}}^{z}_{l}{\hat{S}}^{z}_{l+1}+1\right]\ .$ (5)

Different from the Hamiltonian (4), the spin states $|\\!\\!\uparrow\rangle$
and $|\\!\\!\downarrow\rangle$ here correspond to the single occupation
$|a\rangle$ and $|b\rangle$ respectively. We further write a new set of bases
following $|\\!\\!\uparrow\rangle\rightarrow-|\\!\\!\downarrow\rangle$ and
$|\\!\\!\downarrow\rangle\rightarrow-|\\!\\!\uparrow\rangle$ at all odd sites.
Accordingly, the effective hamiltonian goes back to a regular form of the
antiferromagnetic Heisenberg chain

$\displaystyle{\hat{H}}^{(2)}_{\text{MI}}=\frac{J^{2}_{1}}{\bar{U}}\sum_{l}\left(4{\hat{\mathbf{S}}}_{l}\cdot{\hat{\mathbf{S}}}_{l+1}-1\right)\
.$ (6)

In the light of the Bethe Ansatz solution Lieb and Wu (1968), we know the
energy per site $e^{(2)}_{MI}=-4(\ln 2)(J^{2}_{1}/{\bar{U}})$ in the
thermodynamical limit (TDL).

Usually, a regular chain of two species interacting hardcore bosons favors
establishing a counter-flow SF order in the MI phase Kuklov and Svistunov
(2003); Altman  _et al._ (2003); Kuno  _et al._ (2013); Venegas-Gomez  _et
al._ (2020). However, the $J_{1}$ density-dependent hoppings in the
Hamiltonian (2) give a non-local quadrupole order Zvyagin and Zvyagina (2019);
Gen  _et al._ (2019), which easily breaks into a polarized fluid under the
species-dependent perturbation.

When $J_{1}\gg{\bar{U}}$ and $J_{0}=0$, we suppose that a pure DM state is a
product of the local dimers
$|\phi_{l,l+1}\rangle=B[4J_{1}(|ab\rangle_{l}|0\rangle_{l+1}+|0\rangle_{l}|ab\rangle_{l+1})+(\bar{U}+A)(|a\rangle_{l}|a\rangle_{l+1}+|b\rangle_{l}|b\rangle_{l+1})]$
on all even bonds. And the average energy per site is
$e^{(0)}_{DM}=({\bar{U}}-A)/4$ with $A=\sqrt{16J^{2}_{1}+{\bar{U}}^{2}}$ and
$B=1/(2\sqrt{A^{2}+{\bar{U}}A})$. According to the details in Appendix B, we
estimate the modification of average energy up to the second order, that is
$e^{(1)}_{\text{DM}}=0$ and
$e^{(2)}_{\text{DM}}=(J^{2}_{1}/4)[64J^{4}_{1}+{\left({\bar{U}}+A\right)}^{4}]/{[16J^{2}_{1}+{\bar{U}}({\bar{U}}+A)]}^{2}/({\bar{U}}-A)$.

The competition between the MI and DM phases leads to a critical point,
roughly indicated by
$e^{(2)}_{\text{MI}}(J^{c}_{1})=e^{(0)}_{\text{DM}}(J^{c}_{1})+e^{(2)}_{\text{DM}}(J^{c}_{1})$,
which suggests $J^{c}_{1}\approx 0.3$. In comparison with $J^{c}_{1}\approx
0.5$ determined by numerical simulations later as shown in Fig. 1 (b), the
underestimated value above-mentioned is caused by the limited perturbation
order near the transition point.

Now we do a similar analysis on the charge gap in the MI phase keeping
$J_{0/1}\ll{\bar{U}}$. And the signal of the gap closure implies the SF-MI
transition. Adding an atom “$a$” to the clean MI state immediately raises the
repulsion energy. Meanwhile, the added atom can also move over long distances.
In the perturbation theory (Appendix B), the relevant projection operator onto
the clean MI state reads

$\displaystyle{\hat{\cal P}}_{\text{MI}+}=\sum_{l}{\hat{\cal
P}}^{(d)}_{l}\sum_{\\{r,q\neq l\\}}\prod_{r}{\hat{\cal
P}}^{(a)}_{p}\prod_{q}{\hat{\cal P}}^{(b)}_{q},$ (7)

with operators ${\hat{\cal P}}^{(d)}_{l}={\hat{n}}^{a}_{l}{\hat{n}}^{b}_{l}$,
${\hat{\cal P}}^{(a/b)}_{l}={\hat{n}}^{a/b}_{l}(1-{\hat{n}}^{b/a}_{l})$, $l$
running over all double occupied sites, $p$ and $q$ giving a complete set of
combinations of the single occupied sites “$a$” or “$b$”. Only up to the first
order, the $J_{1}$ terms contribute none at all, while $J_{0}$ allows the
exchange between a double occupation and a single one at neighboring sites.
The resulting effective Hamiltonian is

$\displaystyle{\hat{H}}^{(1)}_{\text{MI}+}=-J_{0}\sum_{l}{\hat{\cal
P}}^{(s)}_{l}\left[{\hat{H}}^{aa,1}_{l,l+1}+{\hat{H}}^{bb,1}_{l,l+1}\right]\
,$ (8)

with the single-occupation projection operator ${\hat{\cal
P}^{(s)}}_{l}=\sum_{r,q\neq
l,l+1}\prod_{r}{\hat{P}}^{(a)}_{r}\prod_{q}{\hat{P}}^{(b)}_{q}$,
${\hat{H}}^{aa,1}_{l,l+1}={\hat{a}}^{\dagger}_{l}{\hat{a}}_{l+1}{\hat{n}}^{b}_{l}{\hat{n}}^{b}_{l+1}+{\rm
h.c.}$ and
${\hat{H}}^{bb,1}_{l,l+1}={\hat{b}}^{\dagger}_{l}{\hat{b}}_{l+1}{\hat{n}}^{a}_{l}{\hat{n}}^{a}_{l+1}+{\rm
h.c.}$. In such a complicated Hamiltonian, we find that the groundstate has
highly-degenerate manifolds corresponding to various combinations of the
single occupied sites regardless of “$a$” or “$b$”. For simplicity, we just
choose the case where all sites are occupied by “$b$” except for the site $l$.
So the Hamiltonian becomes

$\displaystyle{\hat{H}}^{(1)}_{\text{MI}+}=-J_{0}\sum_{l}\left({\hat{a}}^{\dagger}_{l}{\hat{a}}_{l+1}+{\rm
h.c.}\right),$ (9)

which yields a groundstate energy $e^{(1)}_{\text{MI}+}=-2J_{0}+{\bar{U}}$ in
the TDL. Likewise, removing an atom gives the kinetic energy
$e^{(1)}_{\text{MI}-}=-2J_{0}$. Consequently, we get the charge gap
$\Delta_{c}=e^{(1)}_{\text{MI}+}+e^{(1)}_{\text{MI}-}-2e^{(2)}_{\text{MI}}=-4J_{0}+{\bar{U}}+8(\ln
2)(J^{2}_{1}/{\bar{U}})$. The SF-MI borderline $\Delta_{c}=0$ or equivalently
$J^{c}_{0}={\bar{U}}/4+2(\ln 2)(J^{2}_{1}/{\bar{U}})\propto J^{2}_{1}$ gives
the asymptotical behavior near $J_{1}=0$ in Fig. 1 (b).

At the end of the section, we roughly estimate the borderline from the SF to
DM phases for the case of large $J_{0/1}$ but relatively small ${\bar{U}}$. As
$J_{0}\gg J_{1}$, the system recovers integrability and the groundstate energy
is roughly $-2J_{0}/\pi$. In contrast, following equations for
$e^{(0/2)}_{\text{DM}}$ as $J_{0}\ll J_{1}$, the groundstate energy gives
$-69J_{1}/64$. As a result, the SF-DM borderline is
$J_{0}/J_{1}=69\pi/128\approx 1.69$.

###  III.2 Numerical results

Beyond the above perturbation analysis, we look forward to an elaborate
description of the quantum phase diagram systematically through numerical
simulations.

In the phase diagram, the transition from the SF to MI phases breaks neither a
continuous nor a discrete symmetry and is clearly of the BKT type in the $1$D
quantum system. Nevertheless, both the MI-DM and SF-DM transitions take place
by breaking the translation symmetry, so the criticality depends on the
scaling dimensions of relevant operators. From a trial (Appendix C), the
dimerization strength does not follow the regular finite-size scaling with a
polynomial function and shows us a clear behavior of the imbalanced data
collapse instead. Therefore we believe that both of them are also of the BKT
type.

Even though it is very interesting to understand the mechanism by microscopic
theories, e.g. bosonization, we only exploit the level spectroscopy technique
to determine the BKT transition lines here. Level spectroscopy technique,
based on the renormalization group analysis and the symmetry consideration,
was proposed in history to overcome the difficulties of determining the BKT
phase transitions Nomura and Kitazawa (1998); Matsuo and Nomura (2006); Ueda
and Oshikawa (2021). The BKT transition points have been proven to be related
to the crossings of representative excited energy levels following the quantum
field theory Nomura and Kitazawa (1998); Matsuo and Nomura (2006); Ueda and
Oshikawa (2021).

Figure 2: BKT transition points determined by the level spectroscopy
technique. Each quasi-critical point is marked by the intersection of relevant
representative states for a certain $L$: $E_{\text{MI}}$ (violet),
$E_{\text{DM}}$ (green) and $E_{\text{SF}}$ (red) for MI, DM and SF
respectively. (a) Take an example for fixed $L=14$ and $J_{1}/{\bar{U}}=0.8$,
their intersections give quasi-critical points located at
$J^{c}_{0}/{\bar{U}}=0.403(5)$ for the MI-DM transition and $0.526(5)$ for the
SF-MI transition. (b) Scaling of borderlines of SF phase for $J_{1}/{\bar{U}}$
from $0.2$ to $2.0$, where $L\leq 36$ and interval of physical parameters
$\delta J_{0/1}/{\bar{U}}\geq 0.005$. Values of exponent $b$ by fitting (blue
lines) are listed in TABLE 1.

Under the periodic boundary condition (PBC), the $s$-th ($0$, $1$, $\ldots$)
lowest-lying level is marked by the number of atoms $N=N_{a}+N_{b}$, momentum
$k$ and parity $P$, the energy of which is referred to as $E_{s}(N,k,P)$ in
ascending order. The groundstate $E_{\text{GS}}=E_{0}(L,0,+1)$ at integer-$1$
filling always has zero momentum and even parity. In the DM phase, a higher-
energy state $E_{\text{DM}}=E_{0}(L,\pi,+1)$ has a small gap for a finite $L$
and gradually becomes degenerate with the groundstate in the TDL. In the MI
phase, the state $E_{\text{MI}}=E_{1}(L,0,+1)$ describes the gapless
excitation with zero neutral gap $\Delta_{n}=E_{\text{MI}}-E_{\text{GS}}$ in
the TDL. Once $J_{0}=0$, it forms a triplet excitation together with states
$E_{2}(L,0,+1)$ and $E_{0}(L,\pi,-1)$. In the SF phase, the states with one
atom more or less form the charge excitations. A bit specially, because the
${\bar{U}}$ term does not have the particle-hole symmetry, we use the average
value $E_{\text{SF}}=\min_{k,P}[E_{0}(L+1,k,P)+E_{0}(L-1,k,P)]/2$ in practice.

In the first step, we obtain all representative levels by the exact
diagonalization (ED) and DMRG methods, followed by getting the quasi-critical
points $J^{c}_{0}(L)$ for a finite $L$. We take the SF-MI and SF-DM
transitions as examples. For $L\leq 36$, the minimum spacing of parameters
$J_{0/1}/{\bar{U}}$ is equal to $0.005$. In DMRG, the truncated dimensions
$m\leq 4096$ and the number of sweeps is enough for convergence accuracy. In
the vicinity of quasi-critical points, we adopt a cubic interpolation of the
raw data before accurately determining the places of crossings.

In the second step, we use data extrapolation to estimate the true critical
points in the TDL following a polynomial function of $1/L^{b}$. The values of
exponent $b$ in Fig. 2 are listed below.

Table 1: Values of exponent $b$ fitting in Fig. 2 $J_{1}/{\bar{U}}$ | $0.2$ | $0.4$ | $0.6$ | $0.8$ | $1$
---|---|---|---|---|---
$b$ | $0.476986$ | $0.459362$ | $0.584881$ | $0.633980$ | $0.710094$
$J_{1}/{\bar{U}}$ | $1.2$ | $1.4$ | $1.6$ | $1.8$ | $2$
$b$ | $0.634221$ | $0.649848$ | $0.658709$ | $0.677195$ | $0.690281$

In the DM phase, each dimer $|\phi_{l,l+1}\rangle$ is almost frozen and can
not move easily. $J_{0}$ term tends to destroy a local dimer and form a
running $ab$ pair in the chain. Therefore, between the SF and DM phases, it
expects to create a paired SF (PSF) with a finite charge gap $\Delta_{c}$ but
a zero neutral gap in principle. In that case, as $J_{0}$ increases,
$E_{\text{DM}}$ would first intersect with the PSF state
$E_{\text{PSF}}=E_{1}(L,0,+1)$ and then $E_{\text{SF}}$. In fact, we find that
$E_{\text{DM}}$ just has single crossing with $E_{\text{SF}}$. So there is no
evidence of the appearance of the PSF phase at all. Besides, a tricritical
point marked by a five star in Fig. 1 (b), where all representative states
merge, is trivial without any new emerging symmetry.

Figure 3: (a) Charge gap $\Delta_{c}$ ($+$), neutral gap $\Delta_{n}$
($\times$), superfluid density $\rho_{s}$ ($\bullet$) and dimerization
strength $\cal D$ ($\blacklozenge$) as a function of $J_{1}/{\bar{U}}$ along a
line $J_{1}+J_{2}={\bar{U}}$. Left dashed and right dot-dashed lines indicate
two BKT transition points $J^{c}_{1}=0.438(4)$ and $0.63(2)$ determined by
level spectroscopy technique previously. Scaled charge gap $L\Delta_{c}$ (b),
scaled neutral gap$L\Delta_{n}$ (d), superfluid density $\rho_{s}$ and scaled
dimerization $\sqrt{L}{\cal D}$ (e) for various $L$ are zoomed in near
transition points. In details, $\Delta_{c}$, $\Delta_{n}$ and ${\cal D}$ are
obtained under OBC for $L=50$ (red), $100$ (orange), $150$ (green) and $200$
(blue) respectively. However, $\rho_{s}$ is calculated under PBC for $L=10$
(orange), $20$ (green) and $30$ (blue) respectively.

We now follow two dashed cutting lines in Fig. 1 (b) to browse the properties
of phases. The dashed line $J_{0}+J_{1}=1$ or equivalently $J_{0}=1-J_{1}$
goes through three distinct phase regions. For a small $J_{1}$ close to the
$J_{0}$ axis, the system stays in the SF phase because of $J_{0}>{\bar{U}}/4$
and the charge gap closes following a function of $1/L$, which means that
$L\Delta_{c}$ collapses to a constant as $J_{1}/{\bar{U}}<0.44(5)$ shown in
Fig. 3 (a) and (b) respectively Blöte  _et al._ (1986); Affleck (1988);
Rossini and Fazio (2012). Under a twisting angle $\theta$ at edges, the system
has a finite energy response $\delta
E_{\text{GS}}(\theta)=E_{\text{GS}}(\theta)-E_{\text{GS}}(0)$. In the SF
phase, the second order differential $\rho_{s}=\delta
E_{\text{GS}}(\theta)/\theta^{2}$, so called the “superfluid density”
Prokof’ev and Svistunov (2000); Masaki-Kato  _et al._ (2019), remains finite
as $\theta$ decreases towards infinitesimal. In Fig. 3 (d), $\rho_{s}$ is
finite and invariant with respect to $L$ in the deep SF region. While near the
SF-MI transition, $\rho_{s}$ decreases as $L$ grows and expects to vanish in
the TDL. Besides, in the SF phase region, the lowest-lying neutral excitation
$\Delta_{n}=2\Delta_{c}$, considered as the second order effect of the charge
excitations correspondingly, also scales of $1/L$ and vanishes in the TDL.

Differently, when the system entries the MI phase region, the charge gap
$\Delta_{c}$ becomes finite and the scaled value $L\Delta_{c}$ expects to be
divergent in TDL. So in Fig. 3 (b), the single line starts to split into
diverse ones at the SF-MI transition point $J_{1}=0.44(5)$. However, the
neutral gap remains closed, which means that a pair of neighboring $a$ can be
flipped to a pair of neighboring $b$ or vice versa without costing energy.
Likewise, the scaled neutral gap $L\Delta_{n}$ is a constant shown in Fig. 3
(c).

Figure 4:  (a) Charge gap $\Delta_{c}$ ($+$), neutral gap $\Delta_{n}$
($\times$), superfluid density $\rho_{s}$ ($\bullet$) and dimerization
strength $\cal D$ ($\blacklozenge$) as a function of $J_{1}/{\bar{U}}$ along a
line $J_{1}/{\bar{U}}=1.5$. The dashed line indicates the BKT transition point
$J^{c}_{0}=1.28(4)$ determined by level spectroscopy technique previously.
Scaled charge gap $L\Delta_{c}$ (b), scaled neutral gap$L\Delta_{n}$ (d),
superfluid density $\rho_{s}$ and scaled dimerization $\sqrt{L}{\cal D}$ (e)
for various $L$ are zoomed in near transition points. In details,
$\Delta_{c}$, $\Delta_{n}$ and ${\cal D}$ are obtained under OBC for $L=50$
(red), $100$ (orange), $150$ (green) and $200$ (blue) respectively. However,
$\rho_{s}$ is calculated under PBC for $L=10$ (orange), $20$ (green) and $30$
(blue) respectively.

In the DM phase, both the charge and neutral gaps are finite. For an even
number of sites with the PBC or an odd number with the open boundary condition
(OBC), the groundstate is double-degenerate due to the broken translation
symmetry. As $J_{1}>0.63(5)$ in Fig. 3 (a), the finite strength discrepancy
between even and odd bonds is characterized by the nonzero dimerization
strength ${\cal
D}=|\langle{\hat{h}}_{2l,2l+1}\rangle-\langle{\hat{h}}_{2l-1,2l}\rangle|\neq
0$ with the local Hamiltonian
${\hat{h}}_{l,l+1}=-J_{0}({\hat{H}}^{aa}_{l,l+1}+{\hat{H}}^{bb}_{l,l+1})-J_{1}({\hat{H}}^{ab}_{l,l+1}+{\hat{H}}^{ba}_{l,l+1})$
for bond-$l$. But in the MI and SF regions, $\sqrt{L}{\cal D}$ turns a
constant in the TDL shown in Fig. 3 (e). Referring to the Peierls theory about
the DM phase Haldane (1982); White and Affleck (1996); Hu  _et al._ (2014),
changes in the behavior of $\sqrt{L}{\cal D}$ imply the MI-DM transition, the
place of which matches the value determined by the level spectroscopy
technique previously.

Following the other dashed line $J_{1}=1.5$, the groundstate transits from the
DM phase to the SF phase with the closure of both charge and neutral gaps, as
shown in Fig. 4 (a), which was also verified by the scaling behaviors of
$\Delta_{c}$, $\Delta_{n}$, $\rho_{s}$ and $\cal D$ in Fig. 4 (b)-(e).
Moreover, the strong finite-size effects near $J^{c}_{0}=1.28(4)$ result in an
interesting SF-DM coexistence region $0.6\lesssim J_{0}<J^{c}_{0}$ for
$L=200$, indicated by both finite $\rho_{s}$ and $\cal D$. It vanishes in the
TDL but may exist in other lattice models for the ultra-cold atoms.

##  IV Summary and Discussion

Our work provides a way of Floquet engineering a new example in the family of
correlated-hopping Bose-Hubbard models. We propose to load two species
hardcore bosons into 1D optical lattices with the time-periodically modulated
Rabi frequency. In the first resonance region, the inter-species and intra-
species hoppings of the effective Hamiltonian are adjusted independently by
tuning the driving amplitude. Particularly, the intra-species hopping occurs
only if the density discrepancy of the other species on these sites is zero,
while the inter-species one is allowed once the relevant density discrepancy
becomes nonzero. In the effective model, $J_{0}$ is only related to the
lattice depth, while the ratio of $J_{0}$ to $J_{1}$ can be adjusted by the
driving frequency $\omega$. In addition, the static magnetic field controls
the renormalized repulsion strength $\bar{U}$.

The quantum phase diagram at integer-$1$ filling is studied by the
perturbation analysis and numerical simulations. On condition that both
$J_{0}$ and $J_{1}$ are very small, the groundstate favors the MI phase with a
finite charge gap and a zero neutral gap. As $J_{1}=0$, an integrable model
emerges with a highly-degenerate incoherent state. Furthermore, small $J_{1}$
lifts the degeneracy in the presence of the SF phase with zero charge and
neutral gaps as well as finite superfluid density. A double-degenerate DM
state breaks the translation symmetry as $J_{1}/J_{0}={\cal
J}_{1}[2K_{\Omega}]>0.5$ allowable in the new setup. In the DM phase, both
charge and neutral gaps keep open. The interplay of $J_{0}$, $J_{1}$ and the
fixed ${\bar{U}}=1$ leads to three BKT transition lines and a tricritical BKT
point, the exact places of which can be determined by the level spectroscopic
technique.

Many future works could be done based on the effective model. For example, it
is worth studying quantum scars near the integrable point. Also in a two-leg
Wannier-Stark ladder, the frozen dimers in the DM phase would be released and
their collective breathing tends to form a resonating valence bond state in a
proper parameter space, which certainly deserves more systematic studies in
the future.

##  V Acknowledgement

We thank Xuefeng Zhang, and Xiaoqun Wang for the grateful discussion. T. W.
acknowledges funding from the National Science Foundation of China under
Grants No. 11804034, No. 11647165, Grant No. 12147102, and China Postdoctoral
Science Foundation Funded Project No. 2020M673118. P. G. Z., and S. H.
acknowledge funding from the National Science Foundation of China under Grants
No. 12174020. P. G. Z., S. H., and H. L. further acknowledge support from
Grant NSAF-U1930402. The computations were performed on the Tianhe-2JK at the
Beijing Computational Science Research Center (CSRC) as well as Quantum Many-
body I cluster, at the School of Physics and Astronomy, Shanghai Jiaotong
University.

##  Appendix A Effective Hamiltonian in the first resonance region

At the preliminary step of the standard operations, we should remove extra
energy scales $J^{0}_{\Omega}$ and $U$ by applying a time-dependent rotation
Eckardt (2017), both of which are comparable with $\hbar\omega$. The rotation
${\hat{V}}(t)={\hat{V}}_{\omega}(t){\hat{V}}_{\Omega}(t)$ consists of two
commuting parts

$\displaystyle{\hat{V}}_{\omega}(t)$ $\displaystyle=$
$\displaystyle\exp\left[-i\omega
t\sum_{l}{\hat{n}}^{a}_{l}{\hat{n}}^{b}_{l}\right]\,,$
$\displaystyle{\hat{V}}_{\Omega}(t)$ $\displaystyle=$
$\displaystyle\exp\left[-i{\tilde{K}}_{\Omega}(t)\sum_{l}\left(\hat{a}_{l}^{\dagger}\hat{b}_{l}+\rm{h.c.}\right)\right]$
(10)

with dimensionless parameters ${\tilde{K}}_{\Omega}(t)=K_{\Omega}\sin(\omega
t)$ and $K_{\Omega}=J^{0}_{\Omega}/\hbar\omega$. After rotation, the
Hamiltonian is divided into two parts
${\hat{H}}^{r}(t)={\hat{H}}^{r}_{T}(t)+{\hat{H}}_{\bar{U}}$, where

$\displaystyle\hat{H}^{r}_{T}(t)$ $\displaystyle=$ $\displaystyle-
J_{0}\sum_{l}\left[{\hat{J}}^{b}_{l,l+1}{\hat{a}}^{\dagger}_{l}{\hat{a}}_{l+1}+{\hat{J}}^{a}_{l,l+1}{\hat{b}}^{\dagger}_{l}{\hat{b}}_{l+1}\right.$
(11)
$\displaystyle\left.+{\hat{Q}}^{ba}_{l,l+1}{\hat{a}}^{\dagger}_{l}{\hat{b}}_{l+1}+{\hat{Q}}^{ab}_{l,l+1}{\hat{b}}^{\dagger}_{l}{\hat{a}}_{l+1}+{\rm
h.c.}\right]\,$

with the emergent density-dependent “gauge fields”
${\hat{J}}^{a/b}_{l,l+1}=\cos[2{\tilde{K}}_{\Omega}(\hat{n}^{a/b}_{l}-\hat{n}^{a/b}_{l+1})]\exp[i\omega
t(\hat{n}^{a/b}_{l}-\hat{n}^{a/b}_{l+1})]$ and
${\hat{Q}}^{ab/ba}_{l,l+1}=-i\sin[2{\tilde{K}}_{\Omega}(\hat{n}^{a/b}_{l}-\hat{n}^{b/a}_{l+1})]\exp[i\omega
t(\hat{n}^{a/b}_{l}-\hat{n}^{b/a}_{l+1})]$ as well as the renormalized inter-
species onsite repulsion
${\hat{H}}_{\bar{U}}={\bar{U}}\sum_{l}{\hat{n}}^{a}_{l}{\hat{n}}^{b}_{l}$.
Obviously, microscopic hopping processes are sorted into two groups: the
intra-species and inter-species hoppings between two neighboring sites. In
contrast to the previous work Wang  _et al._ (2020), we can see the modulated
terms ${\hat{J}}^{a/b}_{l,l+1}$ and ${\hat{Q}}^{ab/ba}_{l,l+1}$ in Eq. (11)
are not pure phases. The involved cosine/sine function conserves time-reversal
symmetry and belongs to the even/odd sector respectively. We found that the
extra symmetry would help hopping processes select the distinct order of the
first-kind Bessel function.

Furthermore, the time-periodic function ${\hat{H}}_{r}(t)$ can be expanded
into a Fourier series
${\hat{H}}_{r}=\sum^{+\infty}_{-\infty}{\hat{H}}^{(n)}_{r}\exp(i\omega t)$.
The $0$-th order term or the effective Hamiltonian is given by simply
averaging over a period, that is,

$\displaystyle{\hat{H}}_{e}={\hat{H}}^{(0)}_{r}=\frac{1}{T}\int^{T}_{0}dt{\hat{H}}_{r}(t)=\sum_{l}{\hat{h}}_{l,l+1}+{\hat{H}}_{\bar{U}}.$
(12)

In the derivation, we use a relation of ${\cal J}_{-1}(x)=-{\cal
J}_{1}(x)={\cal J}_{1}(-x)$. For the intra-species/inter-species hoppings
governed by ${\hat{J}}^{a/b}_{l,l+1}$/${\hat{Q}}^{ab/ba}_{l,l+1}$, the time-
average of the cosine/sine term in a period leads to identity/${\cal J}_{1}$
separately.

##  Appendix B Perturbation theory

To obtain the low-energy effective Hamiltonian $\hat{H}_{e}$ in a subspace
${\cal H}_{0}$, we follow the strong-coupling expansion method suggested by
Takahashi Takahashi (1977). For a system described by a Hamiltonian
$\hat{H}=\hat{H}_{0}+\hat{V}$, the expansion takes the form

$\displaystyle\hat{H}_{e}$
$\displaystyle=\mathcal{P}_{0}\hat{H}_{0}\mathcal{P}_{0}+\hat{H}^{(1)}+\hat{H}^{(2)}+\cdots$
(13)

where

$\displaystyle\begin{aligned}
\hat{H}^{(1)}=\mathcal{P}_{0}{\hat{V}}\mathcal{P}_{0}\ ,\
\hat{H}^{(2)}=\mathcal{P}_{0}{\hat{V}}\frac{\mathcal{P}_{0}-1}{E_{0}-\hat{H}_{0}}{\hat{V}}\mathcal{P}_{0}\
,\end{aligned}$ (14)

with the projection operator $\mathcal{P}_{0}$ onto $H_{0}$ and the zeroth
order energy $E_{0}$.

Firstly, let us consider the case of zero $J_{0}$ but small
$J_{1}\ll{\bar{U}}$. At integer-$1$ filling, the groundstate for the non-
perturbed $\hat{H}_{0}\equiv\hat{H}_{\bar{U}}$ is $2^{L}$-fold degenerate with
$E_{0}=0$. Similar to the Hubbard model, all the odd-order perturbation terms
vanish Takahashi (1977). So we obtain the effective Hamiltonian (5) up to the
second order using the following identities

$\displaystyle\mathcal{P}_{0}\hat{a}_{l}^{\dagger}\hat{b}_{l}\mathcal{P}_{0}$
$\displaystyle=\hat{S}_{l}^{+}\
,\quad\mathcal{P}_{0}\hat{a}_{l}^{\dagger}\hat{a}_{l}\mathcal{P}_{0}=\frac{1}{2}+\hat{S}_{l}^{z}\
,$ (15)
$\displaystyle\mathcal{P}_{0}\hat{a}_{l}\hat{b}_{l}^{\dagger}\mathcal{P}_{0}$
$\displaystyle=\hat{S}_{l}^{-}\
,\quad\mathcal{P}_{0}\hat{b}_{l}^{\dagger}\hat{b}_{l}\mathcal{P}_{0}=\frac{1}{2}-\hat{S}_{l}^{z}\
.$

Figure 5:  A finite-size analysis of the dimerization strength $\cal D$ at
$J_{1}/{\bar{U}}=1.5$ following the standard scaling theory with polynomial
functions. For $L=50$ (red), $100$ (orange), $150$ (green) and $200$ (blue),
the best fitting to the raw data suggests a critical point $J^{c}_{0}=1.7724$
with exponents $a=0.2213$ and $b=0.1988$.

As $J_{0}=0$ and $J_{1}\gg{\bar{U}}$, we decouple odd and even bonds for
generating the dimerized manifold naturally. The Hamiltonian
$\hat{H}_{0}\equiv-
J_{1}\sum_{\mathrm{even}\,l}[\hat{H}_{l,l+1}^{ab}+\hat{H}_{l,l+1}^{ba}]+\hat{H}_{\bar{U}}$
for all even bonds is exactly solvable. The corresponding groundstate wave
function $|\psi_{0}\rangle=\otimes_{\mathrm{even}\,l}|\phi_{l,l+1}\rangle$ is
a product of local dimers
$|\phi_{l,l+1}\rangle=B[4J_{1}(|ab\rangle_{l}|0\rangle_{l+1}+|0\rangle_{l}|ab\rangle_{l+1})+(\bar{U}+A)(|a\rangle_{l}|a\rangle_{l+1}+|b\rangle_{l}|b\rangle_{l+1})]$
with $A=\sqrt{16J_{1}^{2}+\bar{U}^{2}}$ and $B=1/(2\sqrt{A^{2}+\bar{U}A})$.
The resulting energy is $E_{0}=L\left(\bar{U}-A\right)/2$. We treat the
Hamiltonian $\hat{V}\equiv-
J_{1}\sum_{\mathrm{odd}\,l}[\hat{H}_{l,l+1}^{ab}+\hat{H}_{l,l+1}^{ba}]$ for
all odd bonds as the perturbation. It is easy to know that the projection of
${\hat{V}}$ onto the subspace $\mathcal{H}_{0}$ in ${\hat{H}}^{(1)}$ is always
null. The leading-order term comes from the second-order perturbation. At
last, the second-order perturbation term ${\hat{H}}^{(2)}$ is simplified by
$\langle\psi|1/(E_{0}-\hat{H}_{0})|\psi\rangle\approx
1/[E_{0}-\langle\psi|\hat{H}_{0}|\psi\rangle]$ and the energy modification is
$E_{0}^{(2)}\approx\langle\psi_{0}|{\hat{V}}^{2}|\psi_{0}\rangle/E_{0}=LJ_{1}^{2}[64J_{1}^{4}+{({\bar{U}}+A)}^{4}]/[2(A^{2}+{\bar{U}}A)^{2}({\bar{U}}-A)]$.

##  Appendix C Finite-size analysis

In this section, we check the scaling behavior of dimerization strength $\cal
D$ in the vicinity of the SF-DM transition and attempt to find out whether it
follows the ordinary finite-size scaling theory with polynomial functions.

For example, we fix $J_{1}/{\bar{U}}=1.5$ and complete the standard finite-
size analysis in Fig. 5. We can see that all curves are nearly overlapping on
the SF side while their difference remains finite on the DM side, so called
the imbalanced data collapse, which was considered as a signal of the BKT
transition in many previous works Pai and Pandit (2005); Ceccarelli  _et al._
(2013); Dalmonte  _et al._ (2015); Sun  _et al._ (2015); Wang  _et al._
(2021). In addition, near the suggested critical point $J^{c}_{0}=1.7724$, the
system still behaves like the SF phase, which has vanishing $\Delta_{c}$,
$\Delta_{n}$ and $\cal D$ as well as a convergent $\rho_{s}$ independent of
the system size $L$ in Fig. 4. Further confirmed by the unusual critical
exponents $a=0.2213$ and $b=0.1988$ in Fig. 5, we suppose that the SF-DM
transition is of the BKT type.

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