Supeuidity of at band Bose-Einstein condensates revisited Aleksi Julku1Grazia Salerno2and P aivi T orm a2y 1Center for Complex Quantum Systems Department of Physics and Astronomy

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Supeﬂuidity of ﬂat band Bose-Einstein condensates revisited

Aleksi Julku,1, ∗Grazia Salerno,2and P¨aivi T¨orm¨a2, †

1Center for Complex Quantum Systems, Department of Physics and Astronomy,

Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark

2Department of Applied Physics, Aalto University, P.O.Box 15100, 00076 Aalto, Finland

(Dated: April 11, 2023)

We consider the superﬂuid weight, speed of sound and excitation fraction of a ﬂat band Bose-

Einstein condensate (BEC) within multiband Bogoliubov theory. The superﬂuid weight is calculated

by introducing a phase winding and minimizing the free energy with respect to it. We ﬁnd that the

superﬂuid weight has a contribution arising from the change of the condensate density and chemical

potential upon the phase twist that has been neglected in the previous literature. We also point out

that the speed of sound and the excitation fraction are proportional to orbital-position-independent

generalizations of the quantum metric and the quantum distance, and reduce to the usual quantum

metric (Fubini-Study metric) and the Hilbert-Schmidt quantum distance only in special cases. We

derive a second order perturbation correction to the dependence of the speed of sound on the

generalized quantum metric, and show that it compares well with numerical calculations. Our

results provide a consistent connection between ﬂat band BEC and quantum geometry, with physical

observables being independent of the orbital positions for ﬁxed hopping amplitudes, as they should,

and complete formulas for the evaluation of the superﬂuid weight within the Bogoliubov theory. We

discuss the limitations of the Bogoliubov theory in evaluating the superﬂuid weight.

I. INTRODUCTION

Flat (dispersionless) bands are of interest since interac-

tion eﬀects dominate over kinetic energy and interaction-

driven phenomena such as superconductivity or ferro-

magnetism may become enhanced. Furthermore, as re-

cent work has shown, quantum geometry often plays an

interesting role in ﬂat band systems. For instance in ﬂat

band superconductivity [1–5], an obvious question is how

can a ﬁnite supercurrent exist with the electron eﬀective

mass being inﬁnite in a ﬂat band? This dilemma was

solved in Ref. [3] by proving that the superﬂuid weight

Dscan be divided to single-band (Ds

conv) and multiband

contributions (Ds

geom) so that Ds=Ds

conv +Ds

geom. The

multiband contribution can be non-zero also in a ﬂat

band; in other words, the Cooper pair mass can be ﬁ-

nite despite the inﬁnite eﬀective mass of the electrons [5].

In Ref. [3,4] the multiband contribution was shown to

be connected to quantum geometric concepts such as the

quantum geometric tensor, quantum metric, and Berry

curvature [6,7]. Furthermore, Bose-Einstein condensates

(BECs) have been predicted in ﬂat bands [8,9], and

quantum geometry turns out to be crucial for the stabil-

ity and superﬂuidity of BECs as well [10,11]. Recently, it

has been however pointed out that the results on multi-

band superﬂuid weight Dsof fermionic superconductivity

have been incomplete [12,13]. In this article we solve a

similar caveat related to multiband, especially ﬂat band,

superﬂuidity of BECs.

Superﬂuid weight is deﬁned as the change of free en-

ergy Fupon a phase twist (supercurrent) qintroduced

∗ajulku@phys.au.dk

†paivi.torma@aalto.ﬁ

to the system: Ds∝d2F/dq2|q=0,Ntot , where the to-

tal particle number Ntot is kept constant. In the pres-

ence of time-reversal symmetry (TRS), this reduces to

Ds∝d2Ω/dq2|q=0,Ntot , with Ω being the grand poten-

tial. If one further assumes, as was done in Ref. [3]

where the geometric contribution of superconductivity

was for the ﬁrst time identiﬁed, that the superconduct-

ing order parameter at each orbital is always real, then

µν ∝∂2Ω

∂qµ∂qν|q=0,µ. In Ref. [4], Dswas computed by us-

ing the linear response theory which turned out to coin-

cide with the result of Ref. [3]. From Ds

µν ∝∂2Ω

∂qµ∂qν|q=0,µ,

it was derived that the superﬂuid weight in a ﬂat band is

proportional to the quantum metric integrated over the

momentum space and bounded from below by the Chern

number [3] and Berry curvature [4]. There remained a

problem, though, that the superﬂuid weight is by deﬁni-

tion independent of the positions of the orbitals within

the unit cell (assuming the connectivities, i.e., hoppings

are ﬁxed), while quantum metric and Berry curvature

depend on them.

In a recent study [12], it was shown that the expres-

sion Ds

µν ∝∂2Ω

∂qµ∂qν|q=0,µ for fermionic superconductivity

is incomplete within the mean-ﬁeld theory and one has

to carefully take into account the derivatives of the su-

perconducting order parameter. These new terms were

shown to always decrease the superﬂuid weight and in

some cases qualitative diﬀerences between the old results

and the new complete superﬂuid weight expressions could

be revealed. This ﬁnding solved the dilemma related to

the orbital positions, as it was shown that the superﬂuid

weight is actually proportional to the minimal quantum

metric (quantum metric with a minimal trace), a quan-

tity that does not depend on orbital positions for ﬁxed

hopping parameters (i.e. does not depend on the orbitals’

embedding in the Hamiltonian). Furthemore, when the

arXiv:2210.11906v2 [cond-mat.quant-gas] 10 Apr 2023

orbitals are at high-symmetry positions, the quantum

metric is automatically minimal. The new terms appear

only in multiband systems and thus the new revelation

does not aﬀect the usual single-band BCS superconduc-

tors.

The new superﬂuid terms pointed out in Ref. [12] were

derived for fermionic systems. However, the question re-

mains whether similar arguments apply to bosonic multi-

band systems as well. In Refs. [10,11], BEC speed of

sound and excitation fraction (quantum depletion) were

shown to be proportional to the quantum metric and

the quantum distance, respectively, and the superﬂuid

weight was computed for bosonic kagome model with the

linear response theory that coincides with the old expres-

sion Ds

µν ∝∂2Ω

∂qµ∂qν|q=0,µ. Thus it is not completely clear

whether the superﬂuid weight calculations of Ref. [11] are

complete or not, and whether the connection of the BEC

properties to quantum geometric quantities in [10,11]

needs further inspection. In this article we answer these

questions. It turns out that indeed the superﬂuid weight

in the bosonic case also has correction terms that have

been omitted in previous literature, originating from the

dependence of the condensate density n0and chemical

potential µon the phase twist. These are important to

consider in the general case, although we show that their

eﬀect in the examples studied in [10,11] is very small

within the Bogoliubov theory. We also discuss the limi-

tations of the Bogoliubov approach.

We moreover point out that one should be cautious

when evaluating the speed of sound and excitation frac-

tion. They are shown in [11] to be related to generalized

versions of the quantum metric and the quantum dis-

tance; we discuss here that these quantities are indepen-

dent on the orbital positions for ﬁxed hopping parame-

ters, as they should. On the other hand, we emphasize

that the connection to the usual quantum metric and the

Hilbert-Schmidt quantum distance – which do depend on

orbital positions – is valid only under speciﬁc conditions

for the Bloch functions.

In Section II, the theoretical framework of multiband

Bogoliubov theory is introduced. Then, the relation of

the speed of sound and the excitation fraction to the

quantum metric and distance is considered in Section III.

We then compute the superﬂuid weight in Section IV.

In Section Vwe furthermore provide the second order

perturbation result for the speed of sound in case of the

ﬂat band BEC. We ﬁnally conclude in Section VI.

II. THEORETICAL FRAMEWORK OF

MULTIBAND BEC

We start by considering a generic multiband Bose-

Hubbard grand-canonical Hamiltonian

H=H0+Hint −µX

iα

c†

iαciα,with (1)

H0=X

iαjβ

tiαjβ c†

iαcjβ (2)

Hint =U

iα

c†

iαc†

iαciαciα.(3)

Here H0is the kinetic and Hint is the interaction Hamil-

tonian. Moreover, ciα is a bosonic annihilation operator

for the αth sublattice (in the following, we use terms or-

bital and sublattice interchangeably) within the ith unit

cell, tiα,jβ is the kinetic hopping term, µis the chemical

potential and U > 0 is repulsive on-site interaction. The

sublattice index αruns from 1 to M, where Mis the

number of lattice sites per unit cell (i.e the number of

sublattices). For example in a honeycomb lattice one has

M= 2 and kagome geometry M= 3. One can introduce

the Fourier transform

ciα =1

√NX

eik·riα ckα,(4)

where Nis the number of unit cells and riα =ri+rα

with ribeing the spatial coordinate of unit cell iand rα

the coordinate of a site belonging to sublattice αwithin a

unit cell. One can then recast the non-interacting Hamil-

tonian H0as

H0=X

c†

kαHαβ(k)ckβ≡X

c†

kH(k)ck,(5)

where H(k) is a M×Mmatrix and ckis a M×1 vector

such that [H(k)]αβ =Hαβ (k) and [ck]α=ckα. The

hopping elements in the momentum space read

Hαβ(k) = X

rαβ

t(rαβ)e−irαβ ·k,(6)

where rαβ are all possible vectors connecting a lattice site

of sublattice index αto sites residing in sublattice βand

t(rαβ) are the corresponding hopping terms. This form

follows from the translational invariance of the hopping

Hamiltonian.

One can diagonalize H(k) as H(k)|un(k)i=

n(k)|un(k)i, where n(k) are the eigenenergies and |unki

are the corresponding periodic parts of the Bloch states

(nis the band index). The Bloch band energies are or-

dered in the ascending order, i.e. 1(k)≤2(k)≤... ≤

M(k) for all k. Explicitly, one has

H0=X

c†

kU(k)D(k)U†(k)ck≡X

γ†

kD(k)γk.(7)

Here D(k) is a diagonal matrix containing the energies of

the Bloch bands, i.e. [D(k)]nn =n(k), and the columns

of U(k) contain the corresponding Bloch functions, i.e.

[U(k)]αn =hα|un(k)i. The annihilation operators for

the Bloch states are expressed as γk=U†(k)ckso that

[γk]n=γknwhere γknis the annihilation operator for

the Bloch state of momentum kwithin the nth Bloch

band.

As we are dealing with equilibrium physics, it is plau-

sible to assume that Bose-Einstein condensation takes

place within the lowest Bloch band at momentum kc

so that the bosons condense at the Bloch state |φ0i ≡

|u1(kc)iat the energy 0≡1(kc) with the corresponding

BEC wavefunction being ψ0(riα) = exp(ikc·riα)hα|φ0i.

Note that here we assume that the condensate takes place

within a single momentum. If this was not the case, the

unit cell can often be expanded such that all the conden-

sate momenta are folded back to a single momentum, i.e.

to k= 0. In the weak-coupling regime, the condensate

Bloch state and its momentum kccan be solved by mini-

mizing the corresponding mean-ﬁeld energy as a function

of kc[9,11].

In the case of a dispersive band and weak interaction

regime, the condensation can be usually assumed to take

place within the Bloch state of the lowest kinetic en-

ergy. For example for a square lattice this would mean

kc= 0. However, here our primary focus is to consider

a BEC taking place within a ﬂat band for which all the

momentum states have the same kinetic energy and thus

the BEC emerges in the Bloch state that minimizes the

repulsive on-site interaction energy. Consequently, parti-

cles try to distribute as uniformly as possible among all

the sublattices [9–11]. In the following, we thus assume

the uniform condensate density condition |hα|φ0i|2=1

This is analogous to the uniform pairing condition often

assumed in fermionic ﬂat band superconductivity stud-

ies [3,14]. This condition is important and interesting

since it requires a local (point group) symmetry of a spe-

ciﬁc type to be enforced [13].

We now take into account the quantum ﬂuctuations

around the BEC by writing the bosonic annihilation op-

erators as

ciα =√n0ψ0(riα) + δciα =√n0eikc·riα hα|φ0i+δciα,

(8)

where n0is the condensation density, i.e. number of

condensed bosons per unit cell and δciα describes the

ﬂuctuations on top of the condensate. By Fourier trans-

forming, one ﬁnds ckα=√N n0hα|φ0iδk,kc+δckαand

γkn=√Nn0δk,kcδn,1+δγkn.

As we are considering a system of weakly interacting

Bose-condensed gas, we treat the Hamiltonian within the

multiband Bogoliubov approximation by neglecting the

interaction terms that are higher than quadratic order

in the ﬂuctuations δckαand δc†

kαwith k6=kcand fur-

thermore ignoring the anomalous oﬀ-diagonal self-energy

contribution. As a result, we write our Bogoliubov

Hamiltonian as (for details, see Appendix A)

N= (0−µ)n0+Un2

α|hα|φ0i|4

kαβ

(Hαβ(k)−µδαβ )δc†

kαδckβ

2NX

kαh4n0|hα|φ0i|2δc†

kαδckα+

n0hα|φ0i2δc†

kαδc†

2kc−kα+ h.c.i.(9)

Here the ﬁrst line is constant and describes the conden-

sate while second line is the kinetic energy of the ﬂuctu-

ations and the last two lines are the interaction Hamil-

tonian written in the Bogoliubov approximation.

The chemical potential µcan be solved by demanding

that the Hamiltonian terms linearly proportional to the

ﬂuctuation of the BEC, i.e. δγkc1, vanish. Thus, by

plugging Eq. (8) to (1) and demanding that linear terms

in δγkc1are zero, one ﬁnds (see Appendix A)

µ=0+Un0X

α|hα|φ0i|4.(10)

With uniform BEC condition, this reduces to µ=0+

Un0

M, i.e. to the expression used in Ref. [10,11]. By sub-

stituting this to Eq. (9), one obtains H=Hc+HB, where

Hcis a constant and HBis the quantum ﬂuctuation con-

tribution describing the quasi-particle excitations on top

of the BEC, i.e.

HB=1

0Ψ†

kHB(k)Ψk,(11)

where HB(k) is a 2M×2Mmatrix given by

HB(k) = H(k)−µeﬀ ∆

∆∗H∗(2kc−k)−µeﬀ,,

Ψk= [δck1, δck2, ..., δckM, δc†

2kc−k1, ..., δc†

2kc−kM]T,

[∆]αβ =δα,βUn0hα|φ0i2,

µeﬀ = (0−Un0

M)δα,β.(12)

The primed sum in Eq. (11) indicates that all the oper-

ators within the sum are for non-condensed states only,

i.e. k6=kcand 2kc−k6=kc.

To obtain the excitation energies from HB, one needs

to take care of the bosonic commutation rules by solv-

ing the eigenstates of L(k)≡σzHB(k), where σzis

the Pauli matrix in the particle-hole space [15]. One

ﬁnds the excitation energies for each momentum kto

be EM(k)≥...E2(k)≥E1(k)≥0≥ −E1(2kc−k)≥

... −EM(2kc−k). Here positive (negative) energies

describe quasi-particle (-hole) excitations. The quasi-

particle and -hole states are labelled as |ψ+

m(k)iand

|ψ−

m(k)isuch that

L(k)|ψ+

m(k)i=Em(k)|ψ+

m(k)i(13)

L(k)|ψ−

m(k)i=−Em(2kc−k)|ψ−

m(k)i.(14)

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SupeuidityofatbandBose-EinsteincondensatesrevisitedAleksiJulku,1,GraziaSalerno,2andPaiviTorma2,y1CenterforComplexQuantumSystems,DepartmentofPhysicsandAstronomy,AarhusUniversity,NyMunkegade120,DK-8000AarhusC,Denmark2DepartmentofAppliedPhysics,AaltoUniversity,P.O.Box15100,00076Aalto,Finland(Dated:...

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Supeuidity of at band Bose-Einstein condensates revisited Aleksi Julku1Grazia Salerno2and P aivi T orm a2y 1Center for Complex Quantum Systems Department of Physics and Astronomy

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