Slow semiclassical dynamics of a two-dimensional Hubbard model in disorder-free potentials

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Slow semiclassical dynamics of a two-dimensional Hubbard model

in disorder-free potentials

Aleksander Kaczmarek, Adam S. Sajna∗

Department of Theoretical Physics, Faculty of Fundamental Problems of Technology,

Wrocław University of Science and Technology, 50-370 Wrocław, Poland

The quench dynamics of the Hubbard model in tilted and harmonic potentials is discussed within

the semiclassical picture. Applying the fermionic truncated Wigner approximation (fTWA), the dy-

namics of imbalances for charge and spin degrees of freedom is analyzed and its time evolution is

compared with the exact simulations in one-dimensional lattice. Quench from charge or spin density

wave is considered. We show that introduction of harmonic and spin-dependent linear potentials

suﬃciently validates fTWA for longer times. Such an improvement of fTWA is also obtained for the

higher order correlations in terms of quantum Fisher information for charge and spin channels. This

allows us to discuss the dynamics of larger system sizes and connect our discussion to the recently

introduced Stark many-body localization. In particular, we focus on a ﬁnite two-dimensional system

and show that at intermediate linear potential strength, the addition of a harmonic potential and

spin dependence of the tilt, results in subdiﬀusive dynamics, similar to that of disordered systems.

Moreover, for speciﬁc values of harmonic potential, we observed phase separation of ergodic and non-

ergodic regions in real space. The latter fact is especially important for ultracold atom experiments

in which harmonic conﬁnement can be easily imposed, causing a signiﬁcant change in relaxation

times for diﬀerent lattice locations.

I. INTRODUCTION

The search for robust quantum many-body systems

which show no thermalization or whose thermalization is

very slow, has become a focus of a number of theoretical

and experimental investigations (see e.g. [1–10] and ref-

erences therein). The best known example in closed sys-

tems that show robust non-ergodic behavior is the many-

body localized (MBL) phase [11–14]. MBL systems are

considered as potential models for quantum memory de-

vices [2, 15] and are relevant for quantum computational

problems [16]. MBL behavior comes from the interplay

of a disorder and interactions and such systems have al-

ready been realized experimentally on many platforms

like ultracold atoms in optical lattices, trapped ions and

superconducting qubits [17–21]. However, it has been

recently shown that MBL features can also be observed

in the systems without quenched disorder but showing a

linear and weak harmonic potential [22]. Another possi-

bility is to add a weak disorder potential to a tilted lat-

tice [23]. Such a phenomenon has been named the Stark

many-body localization (SMBL) and some of its features

have already been investigated experimentally [24–27].

Focusing on the one-dimensional dynamical behavior

of SMBL we have to mention the non-decaying charac-

ter of the imbalance function [22, 23, 28], the appear-

ance of logarithmic-in-time growth of entanglement en-

tropy, quantum Fisher information (QFI) and quantum

mutual information [22, 25, 28–32], non-ergodic behav-

ior of the squared width of the excitation [33] and aver-

age participation ratio which is directly related to the

return probability [23]. For two-dimensional systems,

∗Electronic address: adam.sajna@pwr.edu.pl

much less is known about a possible SMBL behavior. It

seems that the absence of rare regions can lead to non-

ergodic behavior in the thermodynamic limit [23]. How-

ever, strongly non-ergodic polarized regions [34], which

can lead to the SMBL phase in the thermodynamic limit

of one-dimensional systems, are less relevant in two di-

mensions. Therefore the existence of SMBL in higher

dimensional systems can be questioned [35]. This con-

clusion is consistent with the experimental observation

that the presence of defects in polarized regions can lead

to subdiﬀusive behavior [27]. Moreover, going beyond

the linear potential e.g. by adding harmonicity to the

lattice, can lead to various dynamical types of behavior

depending on the lattice location. Such an analysis, for

one-dimensional systems, has recently been given in the

context of SMBL [29–31] leaving two-dimensional sys-

tems unexplored.

In this work, we focus on the disorder-free quantum

evolution of the weakly polarized initial states and point

out dynamical similarities with disordered systems in one

and two dimensions. We give an approximate descrip-

tion of the quench dynamics from density waves with a

short wavelength which evolve under a wide range of tilt

strength (density waves with a short wavelength corre-

spond to the weakly polarized initial states which can be

more easily delocalized [35]). In contrast to the recent

studies of quantum dynamics in two dimensions [27, 35]

we mostly assume that the ﬁeld gradient is applied at

an irrational angle in order to remove the equipotential

directions [23]. In particular, we show that a ﬁnite two-

dimensional lattice system with relatively weak harmonic

potential and suﬃciently strong tilt exhibits subdiﬀusive

dynamical behavior similar to that known for disordered

systems [36, 37]. We achieve this by analyzing the quan-

tum dynamics of the Hubbard model which can be di-

rectly experimentally realized [9, 19, 27, 38–41]. In our

arXiv:2210.01082v2 [cond-mat.stat-mech] 17 Oct 2022

numerical study, we exploit fermionic truncated Wigner

approximation (fTWA) to deal with system of larger sizes

[37, 42–46]. Such an analysis is possible because fTWA

gives a reliable description in the parameter space in

which together with the tilt potential, a harmonic poten-

tial has been added to the lattice and a spin dependence

of the linear ﬁeld has been taken into account. The im-

portance of the spin-dependent local potential has been

previously linked to the full MBL in the disordered Hub-

bard system because it is responsible for the localization

of the spin degrees of freedom [47]. Here we observe a

similar eﬀect for spin dynamics on a tilted lattice and

demonstrate that the prediction of fTWA dynamics is

highly enhanced in this limit.

To discuss the dynamics of a Hubbard model on the

tilted lattice we focus our analysis on the imbalance and

QFI for charges and spins. Both observables are related

to the on-site density measurements and are experimen-

tally accessible [17, 19, 20, 24, 38, 40, 48]. Imbalance and

QFI were chosen because both are well-established indi-

cators of non-ergodicity. Moreover QFI can distinguish

the Wannier-Stark localization from SMBL through a

logarithmic-in-time type growth in the SMBL phase [25].

In this work, we show that in two dimensions QFI ex-

hibits a slow logarithmic-like growth which is similar to

the QFI behavior of disordered systems [17, 37, 49–51]

and recently studied tilted triangular ladder [25]. More-

over, we discuss the way in which harmonic potential

together with spin-dependent tilt causes a change in the

charge imbalance decay from diﬀusive to subdiﬀusive be-

havior for intermediate strength of linear potential. In-

terestingly for spins we show that the decay of imbalance

is even more pronounced and changes from superdiﬀusive

to subdiﬀusive behavior. It is worth stressing that due to

the approximation made in studying dynamical behavior,

we cannot conclude about a possibility of a transition to

SMBL phase in two dimensions. However, we can in-

dicate certain dynamical features which are diﬃcult to

handle by other computational methods.

Finally, the fTWA method also enables us to discuss

the appearance of phase separation of ergodic and non-

ergodic long-lived phases in a two-dimensional lattice,

which is an extension of previous theoretical studies per-

formed for one-dimensional lattices [29–31].

The manuscript is constructed as follows. In Sec. II,

the fTWA method is shortly discussed. In Sec. III,

the benchmark of fTWA method against exact diago-

nalization (ED) in one-dimensional Hubbard system is

provided together with the mean square error analysis

(MSE) for imbalances and QFI. It is realized for the

charge and spin density wave initial conditions and the

roles of harmonic and spin-dependent linear potentials

are described. The two-dimensional analysis of the many-

body dynamics in tilted lattices is given in Sec. IV. The

paper ends with a summary of the obtained results (Sec.

V).

II. FTWA FOR THE HUBBARD MODEL IN

DISORDERED-FREE POTENTIALS

Before we deﬁne the semiclassical dynamics within

fTWA we begin with writing the Hubbard Hamiltonian

in terms of the creation ˆc†

iσ and annihilation ˆciσ operators

H=−X

ij, σ

Jij ˆc†

iσ ˆcjσ +UX

ˆni↑ˆni↓+X

i,σ

∆(i, σ)ˆniσ,(1)

where the operator ˆc†

iσ (ˆciσ) creates (annihilates)

fermionic particle at position iwith spin σ∈ {↑,↓},

ˆniσ = ˆc†

iσ ˆciσ is the density operator, Jij is the hop-

ping energy, ∆(i, σ)is the spin-dependent on-site po-

tential and Uis the on-site interaction energy between

two spin species. Throughout this work it is assumed

that Jij is non-zero for the nearest neighbour sites only

for which we set Jij =J. Then, instead of solving the

Schrödinger equation, approximated quantum dynamics

in fTWA is obtained by equating Hamilton equations of

motion with the addition of quantum ﬂuctuation encoded

in the initial conditions through the Wigner function W

[42, 52]. Equations of motion for the Hubbard take the

form [37, 42]

idρmσnσ

dt =−X

(Jnkρmσ,kσ −Jkmρkσ,n σ)

+ρmσnσ [∆(n, σ)−∆(m, σ) + U(ρn−σn−σ−ρm−σm−σ)] ,

(2)

where ρmσnσ are phase space variables corresponding to

fermionic bilinears ˆ

Enσ

mσ =ˆc†

nσ ˆcmσ −ˆc†

mσ ˆcnσ/2(ρnσmσ

are obtained by the Wigner-Weyl quantization procedure

[42]). Here the so-called ρrepresentation of Hamiltonian

Hwas used [37, 42]. In order to obtain the expectation

value of a given observable, e.g. ˆ

O, trajectories are sam-

pled from the initial Wigner function W(ρ0)and summed

up according to the following procedure

Dˆ

O(t)EfTWA

≈ZOW(ρ(t))W(ρ0)dρ0=hOW(t)icl ,(3)

where OWis a Weyl symbol of ˆ

O,ρ(t) =

{ρiσjσ0(t) : i, j ∈ {1,2, ..., N}, σ, σ0∈ {↑,↓}},Nis the

number of sites, ρ0=ρ(t= 0). Initial conditions en-

coded in the Wigner function W(ρ0)are obtained by ap-

proximating W(ρ0)as multivariate Gaussians and read-

ing oﬀ its ﬁrst and second moments from matching the

semiclassical and quantum expectation values [42].

Except for non-interacting systems, fTWA gives an ac-

curate description of general systems only in the early

times [52]. However, in the next section, we numerically

show that slight modiﬁcation of the linear potential leads

to the improvement of the long-time fTWA predictions.

In one-dimensional systems, we consider the following

form of the onsite potential

∆(j, σ)=∆1(δσ↓+Aδσ↑)j+ ∆2(j−j0)2,(4)

where ∆1(∆2) is the strength of linear (harmonic) po-

tential, Aintroduce a spin dependence to the linear po-

tential for any A6= 1. In this work a weak spin depen-

dence (A= 0.9) is considered as in the recent experiment

by S. Scherg et al. [9]. In Sec. IV we assume a two-

dimensional system and then the potential is modiﬁed

correspondingly.

Throughout the paper, the interaction strength is set

to U/J = 1 and open boundary conditions are assumed.

III. THE ROLE OF HARMONIC POTENTIAL

AND SPIN DEPENDENCE OF THE LINEAR

FIELD

To benchmark the fTWA method, we compare the re-

sults of semiclassical simulations with those of ED in

a ﬁnite one-dimensional system at half-ﬁlling (8 lattice

sites are investigated). The role of harmonic potential

and spin dependence of the linear ﬁeld is stressed by us-

ing the imbalance functions and QFI. We chose these

quantities because they are accessible experimentally in

trapped atoms and ions experiments and are useful in

a discussion of ergodicity breaking in diﬀerent systems

[17, 19, 20, 24, 38, 40, 48].

The imbalance function measures the distribution of

charges (densities) and spin degrees of freedom at a given

time. Assuming that the system starts from a charge

density wave (CDW) where the even sites are doubly oc-

cupied and the odd ones are empty, the imbalance ICis

deﬁned as

IC=1

NDˆ

CeE−Dˆ

CoE,(5)

with

Ce/o =X

i∈even/odd sites

ˆci,(6)

where ˆci= ˆni↑+ ˆni↓is the local charge density, Nis the

number of fermions, ˆ

Ceand ˆ

Coare the operators of the

total charge on even and odd sites, respectively.

Correspondingly, for the spin degrees of freedom, the

imbalance function IScan be deﬁned in the following way

IS=1

NDˆ

SeE−Dˆ

SoE,(7)

with

Se/o =X

i∈even/odd sites

ˆsi,(8)

where ˆsi= ˆni↑−ˆni↓is the local spin magnetization, ˆ

and ˆ

Soare the operators of the total spin magnetizaton (z

component) on even and odd sites, respectively. In order

to study the dynamics of the spin degrees of freedom we

chose the initial spin density wave (SDW), i.e. even (odd)

sites containing fermions with spins up (down).

Moreover, to eﬃciently discuss a quantitative diﬀer-

ence between fTWA and ED, the mean square error

(MSE) is analyzed, given by the formula

MSE(IC/S ) = 1

Ns+ 1

j=0 IED

C/S (j∆t)−IfTWA

C/S (j∆t)2

(9)

where ∆t= 0.01/J is the time step after which data are

numerically collected, Ns∆t= 300/J is the total time of

simulations, Cand Sindices correspond to the charge

and spin channel, respectively. Correspondingly, IED

C/S

and IfTWA

C/S stand for the imbalances calculated by using

the ED and fTWA methods.

In Fig. 1 we plot the time dependences of the imbal-

ances ICand ISin the fTWA and ED simulations. We

ﬁrst focus on the role of spin dependence of the linear

potential. It is easily seen that for a spin-independent

potential, A= 1 (see Fig. 1 a and d), delocalization of

spin degrees of freedom takes place (Fig. 1 d). A similar

behavior was previously observed in the context of the

spin-independent disordered systems [47, 54–58]. In our

simulations, this happens at times of the order of O(tJ)

and makes the fTWA to completely fail to describe the

many-body quantum dynamics in the intermediate and

large linear potential strength limit (see also the growth

of MSE(IS) function in Fig. 2 b). In Fig. 1 e, we show

that introduction of a weak spin dependence of the lin-

ear potential, i.e. A= 0.9, forbids spin delocalization

within the analyzed times and recovers the approximate

predictability of fTWA.

Having established an eﬃcient description of the spin

channel, we focus on the role of harmonic potential in

our semiclassical dynamics by setting ∆2/J = 0.5(see,

Fig. 1 c and f). Then the situation is reversed to that

of the spin channel. We observe enhancement of fTWA

prediction in the charge channel which is explicitly seen

in MSE(IC) for intermediate and large linear potential

strength (see, Fig. 2 a).

In our studies we also look at the QFI which is a higher

order correlation function in comparison to imbalance

(QFI is proportional to the variance of ˆ

Ce−ˆ

Coor ˆ

Se−ˆ

So).

For pure initial states analyzed here, i.e. for CDW and

SDW, the corresponding normalized QFI for charges fC

and spins fShas the form [59–62]

fC=4

Nˆ

Ce−ˆ

Co2−Dˆ

Ce−ˆ

CoE2,(10)

fS=4

Nˆ

Se−ˆ

So2−Dˆ

Se−ˆ

SoE2.(11)

Similarly as in the imbalance case we focus on the three

regimes: (i) with spin-independent tilt (A= 1) and with-

out a harmonic potential (∆2= 0), see Fig. 3 a and d,

(ii ) with spin-dependent tilt (A= 0.9) and without a

harmonic potential (∆2= 0), see Fig. 3 b and e, (iii )

with spin-dependent tilt (A= 0.9) and with a harmonic

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Slowsemiclassicaldynamicsofatwo-dimensionalHubbardmodelindisorder-freepotentialsAleksanderKaczmarek,AdamS.SajnaDepartmentofTheoreticalPhysics,FacultyofFundamentalProblemsofTechnology,WrocªawUniversityofScienceandTechnology,50-370Wrocªaw,PolandThequenchdynamicsoftheHubbardmodelintiltedandharmonicpot...

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Slow semiclassical dynamics of a two-dimensional Hubbard model in disorder-free potentials

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