Second root of dilute Bose-Fermi mixtures

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O. Hryhorchak1and V. Pastukhov∗1

1Professor Ivan Vakarchuk Department for Theoretical Physics,

Ivan Franko National University of Lviv, 12 Drahomanov Street, Lviv, Ukraine

(Dated: October 12, 2022)

We discuss an equilibrium mean-ﬁeld properties of mixtures consisting of bosons and spin-

polarized fermionic atoms with a point-like interaction in an arbitrary dimension 2 < d < 4.

Particularly, we discuss except the standard weak-coupling limit of the system with slightly de-

pleted Bose condensate and almost ideal Fermi gas, the (meta)stable phase with dimers composed

exactly of one boson and one fermion. The peculiarities of the fermion-dimer and the boson-dimer

three-body eﬀective interactions and their impact on the thermodynamic stability of the dilute

Bose-Fermi mixtures are elucidated.

PACS numbers: 67.85.-d

Keywords: Bose-Fermi mixture, contact interaction, three-body problem

I. INTRODUCTION

Mixtures composed of mutually interacting Bose and

Fermi particles attract attention of researches from the

early days of the quantum liquids history. The phase dia-

gram of the paradigmatic example – the 4He-3He mixture

– was extensively studied both theoretically [1, 2] and ex-

perimentally [3, 4]. Developments of last few decades in a

ﬁeld of cooling atoms of diﬀerent species of alkalis [5, 6],

together with the possibility to control the parameters

of their two-body potentials by means of the Feshbach

resonances [7] have lead to the creation of the gaseous

Bose-Fermi mixtures with an arbitrary strong interpar-

ticle interaction [8, 9]. Although a simple perturbative

analysis of the thermodynamic stability of these systems

at a weak coupling was carried out long ago by Saam

[10], and in more recent times of realization of ultracold

quantum gases in Refs. [11–15], the interest to the topic

still does not go down [16–22] in the community.

The present article deals with the dilute Bose-Fermi

mixture with the tightly-bound dimers formed in dimen-

sions between d= 2 and d= 4 at absolute zero. The

previous studies in this context were mostly focused on

the three-dimensional systems with the boson-fermion in-

teraction ﬁne-tuned to either narrow [23] or broad [24–

26] Feshbach resonances. Nonetheless, there are some

quantitative diﬀerences in physics of these two models,

the general features of their properties are quite similar.

Particularly, depending on the interaction strength, the

mean-ﬁeld phase diagram [27] of the composite fermions

at ﬁnite temperatures includes states with and without

the Bose-Einstein condensate. For the imbalanced com-

positions of bosons and fermions, there are two Fermi

surfaces with the manifested [28] two Luttinger theorems

in a phase with an unbroken global symmetry. These

results mostly remain true when the Gaussian ﬂuctua-

tions are taken into account, and the ground state of the

∗e-mail: volodyapastukhov@gmail.com

system possesses a rich phase diagram with a number of

quantum phase transitions [29] of various order.

Likewise the Fermi-polaron problem [30], the t-matrix

approximation was shown to be quite accurate [31] for

the description of the paired Bose-Fermi mixture even

when the concentration of bosons is not small. This

was explicitly veriﬁed in [32] by comparison to the re-

sults of Monte Carlo simulations, and in Refs. [33, 34]

the interaction-induced breakdown of the Bose-Einstein

condensation in the whole temperature range was inves-

tigated within this approach. In condensate phase, the

composite-fermion propagator possesses two poles [35] of

a diﬀerent physical nature. Even richer structure of the

quasiparticle excitations [36] is displayed by the single-

particle fermionic and bosonic spectral functions in the

normal phase, when the Bose condensate is totally de-

pleted by the strong boson-fermion attraction. The com-

mon opinion is that the thermodynamic stability of the

system requires some ﬁnite inter-bosonic repulsion [37],

especially when the number of bosons exceeds number of

fermions.

An important aspect [38] of the Bose-Fermi mixtures

properties, less discussed in the literature, is the atom-

dimer scatterings. From the point of view of the sta-

bility of a mixed state in a dilute system, the eﬀective

fermion-dimer repulsion stabilizes mixture with majority

of fermions and boson-dimer attraction forces the col-

lapse. Another feature of the three-body physics is the

emergence of the universal Eﬁmov eﬀect [39] (see [40] for

recent discussion). The latter is crucial for the thermo-

dynamic properties of the Bose-Fermi mixture by making

a phase with the boson-fermion dimers formed at least

metastable. Here we attempted to address all these ques-

tions in order to explore in detail the stability regions of

dilute systems.

arXiv:2210.05231v1 [cond-mat.quant-gas] 11 Oct 2022

II. CONTACT TWO-BODY POTENTIAL IN

ARBITRARY d < 4

Let us brieﬂy overview the main ideas behind the con-

tact (δ-like) interaction between particles. This concept

is quite useful when the range of a two-body potential

is small and one is interested in the universal low-energy

physics of the system. Then the contact interaction can

be presented in a twofold way: the ﬁrst one stipulates the

introduction of a large ultraviolet (UV) cutoﬀ Λ (which is

a reminiscent of a range of the realistic two-body poten-

tial), while the second one explicitly appeals to the (typ-

ically non-hermitian) δ-pseudo-potential. Below, while

considering the many-body behavior of the system, the

former way will be of great practical use, therefore let us

ﬁrst consider a fermion of mass mfand boson of mass

mbin a very large volume Ldin ddimensions with the

periodic boundary conditions imposed. The appropriate

Schroedinger equation for the relative motion of two par-

ticles reads

−~2

2mr∇2ψ(r) + gΛδΛ(r)ψ(r) = Eψ(r),(2.1)

where mr=mfmb

mf+mband gΛare the reduced mass and the

bare coupling, respectively, and the UV-cutoﬀ-dependent

delta-function should be understood as the following in-

tegral δΛ(r) = 1

LdP|p|<Λeipr in the momentum space.

Being interested in the spherically symmetric (s-wave)

bound states with energy E=B=−~2

2mra2, we can

write down the non-normalized wave function

ψ(r) = 1

LdX

|p|<Λ

eipr

p2+a−2.(2.2)

The requirement of comparability with the Eq. (2.1)

yields the condition on gΛ

g−1

Λ+1

LdX

|p|<Λ

2mr/~2

p2+a−2= 0.(2.3)

An introduction of the renormalized (‘observable’) cou-

pling constant

g−1=g−1

Λ+1

LdX

|p|<Λ

2mr

~2p2,(2.4)

enables to eliminate the explicit dependence of gΛon the

UV cutoﬀ in dimensions below d < 4. In a higher ds

the wave function (2.2) is not square normalizable. By

plugging (2.4) in (2.3) and computing the sum over wave

vector in the limit of free space (L→ ∞), one relates a

positive gs to the width of a binding energy of two atoms

in vacuum

g−1=−Γ(1 −d/2)

(2π)d/2mr

~2d/2|B|d/2−1.(2.5)

The simplicity of potential energy in Eq. (2.1) also allows

the exact calculations of the scattering states. Particu-

larly, the computed to all orders of perturbation theory

energy shift E=g/Ld+O(1/L2d) of the lowest level

(p=0) reveals the nature of the contact two-body inter-

action: depending on a sign of the renormalized coupling,

the potential behaves either like the repulsive (g > 0)

or the attractive (g < 0) one (recall that a single s-

wave bound state appears for positive gs, which may be

somewhat misleading). Another important lesson to be

learned from the above analysis is that the renormalized

coupling gshould be identiﬁed as the physical one, i.e.

determines the strength of the two-body interaction and

all observables.

III. STANDARD WEAK-COUPLING

CONSIDERATION

Although below only limit of zero temperature will be

considered, for the discussion of the thermodynamic limit

of the Bose-Fermi mixture we adopt the Euclidean-time

path integral approach with an action

S=Zdx b∗{∂τ−ξb}b−gb,Λ

2Zdx |b|4

+Zdx f∗{∂τ−ξf}f−gΛZdx f∗f|b|2,(3.6)

where x= (τ, r) is the short-hand notation for posi-

tion r∈Ldand an imaginary time τ∈[0, β) (with

the inverse temperature β→ ∞ in the ﬁnal formulas);

ξb,f =εb,f −µb,f =−~2∇2

2mb,f −µb,f and µb,f are the chemi-

cal potentials that ﬁx the average densities nb,f of bosons

and fermions in the system. The boson-boson bare cou-

pling gb,Λcan be related to its ‘observable’ counterpart

gbin a way described in the previous section. It is also

understood that Grassmann f(x) and complex b(x) are

deﬁned in such a way that all the Fourier amplitudes

with |p|>Λ equal zero identically, i.e. the UV cutoﬀ

is setted up only on ﬁelds’ spatial dependence. The fur-

ther consideration in this section, except a modiﬁcations

associated with the presence of a fermion ﬁeld, will be

held in a spirit of the Hugenholtz and Pines paper [41]

and Ref. [42]. The ﬁrst step of their approach, which

is well-suited for the calculation of thermodynamics of

bosons at weak coupling gb→0, is the separation of the

condensate modes b(x) = b0+˜

b(x) with the constraint

RLddr˜

b= 0. After this, the number of the interaction

vertices increases

S=βLdµb|b0|2+Rdx ˜

b∗{∂τ−ξb}˜

b−βLdgb,Λ

2|b0|4

−gb,Λ

2Rdx n4|b0|2|˜

b|2+ (b∗

0)2˜

b2+ (˜

b∗)2b2

+2b∗

0˜

b∗˜

b2+ 2(˜

b∗)2˜

bb0+|˜

b|4o+Rdx f∗{∂τ−ξf}f

−gΛRdx f∗fn|b0|2+b∗

0˜

b+˜

b∗b0+|˜

b|2o,(3.7)

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SecondrootofdiluteBose-FermimixturesO.Hryhorchak1andV.Pastukhov11ProfessorIvanVakarchukDepartmentforTheoreticalPhysics,IvanFrankoNationalUniversityofLviv,12DrahomanovStreet,Lviv,Ukraine(Dated:October12,2022)Wediscussanequilibriummean-eldpropertiesofmixturesconsistingofbosonsandspin-polarizedfermio...

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