Competing instabilities at long length scales in the one-dimensional Bose-Fermi-Hubbard model at commensurate llings Janik Sch onmeier-Kromer1 2and Lode Pollet1 2 3

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Competing instabilities at long length scales in the one-dimensional

Bose-Fermi-Hubbard model at commensurate ﬁllings

Janik Sch¨onmeier-Kromer1, 2 and Lode Pollet1, 2, 3

1Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC),

Ludwig-Maximilians-Universit¨at M¨unchen, Theresienstr. 37, M¨unchen D-80333, Germany

2Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M¨unchen, Germany

3Wilczek Quantum Center, School of Physics and Astronomy,

Shanghai Jiao Tong University, Shanghai 200240, China

We study the phase diagram of the one-dimensional Bose-Fermi-Hubbard model at unit ﬁlling for

the scalar bosons and half ﬁlling for the S= 1/2 fermions using quantum Monte Carlo simulations.

The bare interaction between the fermions is set to zero. A central question of our study is what

type of interactions can be induced between the fermions by the bosons, for both weak and strong

interspecies coupling. We ﬁnd that the induced interactions can lead to competing instabilities

favoring phase separation, superconducting phases, and density wave structures, in many cases at

work on length scales of more than 100 sites. Marginal bosonic superﬂuids with a density matrix

decaying faster than what is allowed for pure bosonic systems with on-site interactions, are also

found.

PACS numbers: 03.75.Hh, 67.85.-d, 64.70.Tg, 05.30.Jp

I. INTRODUCTION

Due to their clean and fully controllable yet versatile

setup, quantum gases in optical lattices have proven to

represent an ideal candidate to realize a quantum simula-

tor for classically incomputable many-body problems in

condensed matter theory [1,2]. For monoatomic bosonic

gases trapped in a three-dimensional (3D) optical lattice

the theoretically predicted [3,4] quantum phase transi-

tion from a Mott insulator to a superﬂuid was experi-

mentally proven to exist [5].

The interplay between bosons and fermions is ubiqui-

tous in nature. In conventional superconductors phonons

mediate an attractive interaction between the electrons.

Mixtures of 3He and 4He caught attention already in the

1940s and showed such eﬀects as a very low solubility

and the Pomeranchuk eﬀect [6–8]. Every Bose-Fermi sys-

tem has however its own types of interactions and char-

acteristics, and requires a separate study. In the ﬁeld

of ultracold atoms, several degenerate Bose-Fermi mix-

tures have been cooled to degeneracy over the past 20

years [9–17]. Induced interactions between the fermions

mediated by the bosons leading to a fermionic pair su-

perﬂuid attracted theoretical interest shortly after the

ﬁrst condensates were experimentally realized [18–21]. In

more recent years, experimental eﬀorts focused on the

interactions between a bosonic and fermionic pair super-

ﬂuid [22–25]. Experimental loading of a fermionic 40K

and bosonic 87Rb mixture in an optical lattice [26] can

lead to stronger localization and interaction eﬀects such

as phase separation, spin or charge density waves and

supersolids [27–43].

In this work we examine the ground state phase dia-

gram of the one-dimensional Bose-Fermi-Hubbard model

at unit ﬁlling for the scalar bosons and half ﬁlling for

the S= 1/2 fermions. The bare fermions are taken to

be free particles. As the system size and inverse temper-

ature increase, the induced interactions grow stronger.

Nevertheless, those induced interactions leading to pair

ﬂow [44] are expected to be weak for weak inter-species

coupling [45]. In this Bardeen-Cooper-Schrieﬀer regime

the pair size is very large, and therefore very large system

sizes are required. For stronger inter-species coupling,

localization eﬀects are stronger and a competition be-

tween instabilities towards superﬂuids and density wave

structures is expected. It turns out that the renormal-

ization ﬂow can still extend to 100s of lattice sites, with

misleading behavior on intermediate length scales. The

purpose of this work is hence to map out the physics of

induced interactions and competing instabilities for the

one-dimensional Bose-Fermi-Hubbard model. It is one of

the few sign-positive models for mixtures where unbiased

numerical simulations can be carried out.

II. MODEL AND PHASE DIAGRAM

We study the one-dimensional Bose-Fermi-Hubbard

model with on-site interactions,

H=−tFX

hi,ji,σ=↑,↓c†

i,σcj,σ + h.c.

−tBX

hi,jib†

ibj+ h.c.+U

i(nB

i−1)

+VX

i(nF

i,↑+nF

i,↓),(1)

where b†

icreates a soft-core boson on site iand c†

i,σ a

hard-core boson on site iwith spin σ=↑,↓. Particles

can hop between nearest neighbors hi, ji; the hopping

amplitudes are tFfor the hard-core bosons and tBfor

the soft-core bosons. Those amplitudes are chosen equal,

arXiv:2210.10113v2 [cond-mat.quant-gas] 3 Feb 2023

FIG. 1. Phase diagram of the model Eq. 1at unit bosonic

and fermionic half ﬁlling with all hopping amplitudes set to

1, as obtained from quantum Monte Carlo simulations for

V= 1,2,3,4,5,6 and 8. Lines are a guide to the eye. Left

(bosons): The red area demarcates the area of bosonic super-

ﬂuidity; the green area is the uniform bosonic Mott inslulator,

and phase separation is shown in light red. For comparison

with the right panel, we show the blue dashed line which

corresponds to ηCDW = 1.0 (i.e., the exponent controlling

the (fermionic) charge density wave (CDW) correlation func-

tion) on a length scale L= 70, with lower values than 1

above the line. Right (fermions): Dominant superconduct-

ing ﬂuctuations (SC) are found in the yellow area. The dot-

ted green line is the crossover on our length scales towards

quasi free fermionic behavior (purple area). In the white area

Luttinger liquid behavior is seen with dominant charge cor-

relations above the blue dashed line. The dotted and lower

dashed black lines are estimates for phase separation; the up-

per dashed line is a leading order strong coupling argument

separating uniform systems from ones with charge density

waves. Those lines are only informative, but are approached

asymptotically for larger values of V. The cyan dotted line

is where the numerically observed boundary of phase separa-

tion. Further explanations are given in a separate paragraph

in Sec. II in the text.

and set as the energy unit, tB=tF=t= 1. We work

at unit ﬁlling for the soft-core bosons and half ﬁlling for

the hard-core bosons. The on-site density-density inter-

actions are constant over the lattice. Its amplitude for

the intra-species soft-core bosonic interaction is U > 0,

and the amplitude for the inter-species interactions is V,

whose sign is irrelevant at half ﬁlling. There is no bare

intra-species interaction between the hard-core bosons.

The lattice spacing is set to one, a= 1, and the length

of the chain is L. The Jordan-Wigner transformation

between hard-core bosons and fermions requires an odd

number of fermions if we use periodic boundary condi-

tions; we will use the language of bosons and fermions in

this sense below. The model is simulated using a straight-

forward extension of the worm algorithm [46] presented

in Ref. [47,48].

The parameters are chosen such that there is no

apparent small parameter, i.e., in a regime where

numerics are necessary. Contemporary Density Matrix

Renormalization Group (DMRG) studies put a strong

cutoﬀ on the bosonic occupation number in order to

keep the local Hilbert space tractable and focused on

superﬂuid-insulator transitions [49–51]. Our main result

is summarized in the phase diagram shown in Fig. 1,

valid in the thermodynamic limit. However, monitoring

the competing instabilities that develop on length scales

consisting of hundreds of sites can not be read oﬀ in full

from this phase diagram. This competition is explained

in the detailed analysis in the sections below. As we

will see, the simulations are notoriously hard, and, in

certain cases, autocorrelation times that exceed one

million Monte Carlo steps are observed, indicative of

strong metastabilities and competing phases, perhaps

in the vicinity of ﬁrst order transitions. This inevitably

leads to a few uncertainties in the phase diagram, which

can only be resolved if better algorithms are devised and

better computer hardware is available.

To set the ideas, we brieﬂy explain the phase diagram

shown in Fig. 1. The phase diagram is dominated by two

big areas: for VUthe system separates into a bosonic

and a fermionic system, and for UVthe bosons form

a uniform n= 1 Mott insulator above which the fermions

are quasi free. We focus on the channel-like region in be-

tween. The red solid line demarcates the area of uniform

bosonic superﬂuidity. It extends to remarkably large val-

ues of Uand Vand probably closes in a cuspy way for

a value of Vclose to V= 6. Mesoscopic superﬂow is

found for V= 8, U = 14 as well but, extrapolating our

results, it vanishes around L≈500. In the tip of this

region, the bosonic superﬂuid is marginal with a den-

sity matrix decaying faster than what is allowed for the

pure bosonic system. We found no indications of bosonic

supersolid behavior (i.e., concomitant bosonic superﬂu-

idity and density waves breaking the lattice symmetry).

The dotted black line is the weak coupling argument for

phase separation (see Sec. III A), which is found to exist

everywhere in the lower parts of the phase diagram. The

dashed black lines are the leading order strong coupling

predictions (see Sec. III B) separating phase separation,

a structure with density wave character, and a uniform

system. We suspect that phase separation extends up to

the black and cyan dotted lines. In the fermionic sec-

tor, the fermions are insulating below the full green line

and show pair ﬂow above and to the left of it, at least

up to the system sizes that we can simulate. Above (be-

low) the dashed blue line the decay of the charge density

wave (CDW) correlations is slower (faster) than for free

fermions. Whether the charge density waves can sponta-

neously break the lattice symmetry is impossible to say

based on our system sizes for V≤6, but for V= 8

we see some strong indications of that for U= 13. The

green dotted line indicates a crossover scale where, on

our length scales, the up and down particles are so weakly

coupled on top of a uniform bosonic Mott insulator that

they can be considered quasi-free. For exponentially low

temperatures, pair ﬂow is expected everywhere in the

upper part of the phase diagram.

The remainder of the paper is structured as follows.

In Sec. III we present analytical arguments in the limit-

ing cases of the phase diagram such as weak and strong

inter-species coupling, arguments based on bosonization,

and the type of induced interactions in the random phase

approximation for various parameter regimes. This is fol-

lowed by Sec. IV where we highlight some speciﬁcs of our

quantum Monte Carlo algorithm, and we list in Sec. Vall

relevant quantities that are computed in the simulations

and used for analyzing the phases. In Sec. VI we system-

atically go through the phase diagram for various values

of Vand discuss the obtained Monte Carlo results. In

particular, renormalization ﬂows as a function of the sys-

tem size allow one to monitor the competing instabilities,

and infer the structure of the phase diagram. Note that

in this paper we refer to a renormalization ﬂow always

as a ﬂow in the system size, unlike in the setting of the

renormalization group. We conclude in Sec. VII.

III. SIMPLE ANALYTICAL CONSIDERATIONS

IN LIMITING CASES

A. Weak interspecies coupling, V≤2t

For weak values of Uthe bosons are well described by a

Luttinger liquid with a linear spectrum. To set the ideas,

we can for U≤2.5 use the Bogoliubov approximation,

which is very accurate for this range of U-values [52].

Integrating out the Bogoliubov quasiparticles results in

a total action consisting of the one for the bare fermions

(including a shift to µFby n0Vwhich we do not mention

because we stay at fermionic half ﬁlling) and an induced

part exp(−S) with Sgiven by

S=n0V2

2Zβ

dτ1dτ2X

i,j,σ,σ0

nσ

i(τ1)D0(i−j, τ1−τ2)nσ0

j(τ2).

(2)

Here, n0is the quasi-condensate and the kernel D0(x, τ)

is given by

D0(x, τ) = Zdk

2πeikx eEkτ+eEk(β−τ)

eβEk−1

k

.(3)

Here, Ekis the Bogoliubov dispersion, and k≥0 is

the bare bosonic dispersion shifted by 2t. This kernel is

peaked at τ= 0 (and τ=β) in imaginary time. The

instantaneous limit can be taken when the bosons are

fast compared with the fermions. The velocity of the bare

fermions at half ﬁlling (kF=π/2) is vF= 2tsin(kF) =

2t. The instantaneous approximation is hence reasonable

for low values of Vwhen Uis not too small; in particular,

close to the bosonic Mott transition it is valid. In the

static approximation we have

static(x) = Zdk

2πeikx 2k

p2

k+ 2kn0U.(4)

This induced interaction is spin agnostic, attractive

on-site (x= 0), but repulsive for x > 0 and scaling

as ∼1/x2, which is short-range in 1D but can be

rather large for next and next-next-nearest neighbor

interactions. If we nevertheless only keep the local part,

then we arrive at an attractive Hubbard model whose

phase diagram is known [53]. In particular, we expect

dominant superconducting pair ﬂuctuations, especially

for weak Vin this BCS-like regime, but the pairing

gap is exponentially weak in the induced interaction

and scales exponentially in pV2/U . Note that this

approach neglects the back-coupling of the fermions

on the bosons, i.e., underestimates CDW structures,

which, given the non-local form of the static interaction

could still be important. Furthermore, the induced

pair-superﬂuidity is in competition with a tendency to

phase separate, which at the mean-ﬁeld level, is found

when V2> U [20,28,36]. At U= 0 the bosons occupy a

number of sites scaling as L1/3, the fermions occupy the

rest. In the thermodynamic limit coupling free bosons

with spinless fermions via Vis hence always unstable,

with a separation line given by U=V2/π. However,

in case of induced interactions, the previous arguments

do not immediately apply and the criterion for phase

separation can be written as ∂µσ/∂nσ<0 [20], which

is numerically hard to use however. Since the gain in

energy due to quasi-condensation is rather small in

practice, we expect that the criterion found above is a

very good upper bound and it is indicated by the dotted

black line in Fig. 1. Simulations in close proximity to

phase separation are diﬃcult, and we saw no indications

of being able to signiﬁcantly improve on the behavior

U=V2/π (which is also close to the bosonization

prediction, see Sec. III D).

The bosons undergo a transition from a Luttinger liq-

uid to a Mott insulator, which, for V= 0, is found at

U= 3.25(5) [54]. Turning on V, and having established

that the bosons are fast with respect to the fermions, the

renormalization ﬂow of the bosons is understood to be

already strong on the system sizes that we can simulate,

whereas the fermions remain nearly free. Neglecting the

backaction of the bosons on the free fermions, the bosonic

Mott transition shifts upwards quadratically with V2

and proportionality factor D() = p1−(/2t)2/(πt) at

= 0. This prediction for the location of the bosonic

to superﬂuid Mott insulator is a reasonable approxima-

tion for low values of V1. As can be seen in Fig. 1,

it is even not too bad at V= 2 but slightly overesti-

mates the critical point because not all fermions can be

considered non-interacting at the length scales of strong

bosonic renormalization. When the bosons are deep in

the uniform Mott insulator, we can approximate their

dispersion by E(k) = ∆ + k2

2m∗where ∆ is the gap (of

order Uin the pure bosonic system) and m∗the eﬀec-

tive mass for particle and hole excitations. The kernel is

now DMott(x, τ ) = Rdk

2πeikxe−∆τto leading order in the

ground state. For ∆ &2twe hence cannot expect to see

any induced interaction, i.e., the bosonic and fermionic

systems decouple and the bosons remain in a uniform

n= 1 Mott insulator and the fermions are quasi-free

(this is the upper part of the phase diagram in Fig. 1).

For ∆ 2tinduced interactions remain possible. How-

ever, in the weak Vlimit and recalling the exponential

weakness of the pairing gap, we can only see such pair-

ing ﬂuctuations when ∆ is very close to 0, which in turn

requires very big system sizes to distinguishes pair ﬂow

from two correlated superﬂuids.

B. Strong interspecies coupling, V4t

FIG. 2. Results for the phase separation (OPS, dashed

lines in the left part of the ﬁgure) and charge-density wave

(OCDW =SCDW (k=π)/L, solid lines in the right part of the

ﬁgure) from exact diagonalization (Lanczos calculations) for

V= 20. The black dotted lines are the strong coupling pre-

dictions separating a regime of phase separation (left) from

a CDW phase (middle) and a uniform phase (right). Note

that the calculations did not converge for L= 8 in the range

26 < U < 29.

FIG. 3. Same as in Fig. 2but for V= 8.

FIG. 4. Same as in Fig. 2but for V= 12.

Since the system phase separates when UV, we

focus on the regime where both couplings are strong,

U, V 4t. We can map the system onto an eﬀective

model as follows. First, turning oﬀ the hopping am-

plitudes, we consider product states that minimize the

potential energy subject to our ﬁlling constraints. We

can ﬁll Msites with 2 bosons, Nsites with one fermion

and one boson, and L−M−Nsites with up and down

fermions. Sites which contain more than 2 particles per

site are higher in energy and are discarded. The energy

of this conﬁguration is Eg=MU +NV , subject to the

constraint of particle number conservation, 2M+N=L.

Eliminating N, we minimize Eg=M(U−2V). Hence,

for V < U/2 (this is the upper black dashed line in Fig. 1)

we expect that every site contains one boson (i.e., we

have a homogeneous bosonic Mott insulator with unit

density) and one fermion on average. As there is no in-

teraction between the fermions and the potential energy

between bosons and fermions is constant (this case is

equivalent to the Hartree approximation), the fermions

delocalize on top of the Mott insulator and behave as

(nearly) free fermions. From the arguments in the previ-

ous paragraph we expect this behavior when the bosonic

gap is large, while pair ﬂow is expected for a small bosonic

gap. For V > U/2 we must have that M=L/2, i.e., half

of the system is doubly occupied with bosons and the

other half with fermionic doublons (molecules). Even

though all conﬁgurations are equally likely, we expect

that the kinetic terms will select either the fully phase

separated regime or a phase with charge density wave

order.

Second, we analyze the eﬀect of the hopping terms

on those ground states to see which phase can lower its

energy the most, and is hence preferred when we lower U

away from U= 2V. In second order perturbation theory

there is only a diagonal virtual exchange term (no ﬂips

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摘要：

Competinginstabilitiesatlonglengthscalesintheone-dimensionalBose-Fermi-HubbardmodelatcommensuratellingsJanikSchonmeier-Kromer1,2andLodePollet1,2,31DepartmentofPhysicsandArnoldSommerfeldCenterforTheoreticalPhysics(ASC),Ludwig-Maximilians-UniversitatMunchen,Theresienstr.37,MunchenD-80333,Germany2...

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Competing instabilities at long length scales in the one-dimensional Bose-Fermi-Hubbard model at commensurate llings Janik Sch onmeier-Kromer1 2and Lode Pollet1 2 3

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