PHYSICAL REVIEW A 81, 043617 (2010)

Low-energy density correlation function of a degenerate unitary Fermi gas from the ε expansion

Andrei KryjevskiDepartment of Physics, North Dakota State University, Fargo, North Dakota 58108, USA

(Received 11 February 2010; published 19 April 2010)

Using the ε expansion proposed in Y. Nishida and D. T. Son [Phys. Rev. Lett. 97, 050403 (2006)], we calculatethe density correlation function and the dynamic structure function of the unitary Fermi gas at zero temperatureto next-to-leading order in ε for excitation energies below the quasiparticle threshold.

DOI: 10.1103/PhysRevA.81.043617 PACS number(s): 03.75.Ss

I. INTRODUCTION

Cold dilute gas made of two fermion species (which will belabeled as ↑,↓) with two-body interactions characterized by

n a3 � 1, n r30 � 1, (1)

where n is the number density, a is the s wave scattering length,and r0 is the effective range of the potential, named “unitaryFermi gas,” is one of the simplest examples of a stronglyinteracting Fermi system. Yet, the unitary Fermi gas is only aslight idealization of dilute neutron matter (a � −18 fm, r0 �2.6 fm in the 1S0 channel) that may exist in the crustof a neutron star. Also, it has been studied extensively inthe experiments on trapped cold atomic gases with tunableinteractions using the Feshbach resonance technique [1–5](see Ref. [6] for a recent review of experiments). From thetheoretical point of view, the unitary Fermi gas lacks anyintrinsic scale and is expected to exhibit universal properties.At the same time, a theoretical description of the unitaryFermi gas is difficult due to the apparent absence of a smalldimensionless parameter which could be used in a perturbativeexpansion.

Inspired by ideas of Nussinov and Nussinov [7], ananalytical technique similar to the ε expansion in the theoryof critical phenomena has been recently proposed [8]. Theidea is to describe the three-dimensional unitary Fermi gasusing perturbation theory in the dimensionality of space, d,around d = 4 and d = 2. The rational for such an approachis that the unitary Fermi gas simplifies to the noninteractingFermi gas in d = 2, and the Bose-Einstein condensate ofnoninteracting spin 0 dimers in d = 4 [7,9]. Comparison withthe results of Monte Carlo simulations suggests that alreadyat the next-to-leading order (NLO) in ε the expansion aroundd = 4 may be a useful tool in the description of this system[8]. While subsequent investigation raised doubts about theapparent convergence of the series in ε [10], one obviousway to check the usefulness of the technique is to makepredictions for various observable quantities at NLO. If thetrend suggested by the initial NLO results of Refs. [8,9,11]were to hold, it would serve as a strong encouragement forfurther investigation.

As a step in this direction, in this article we report theresults of a calculation of the density correlation function of thedegenerate unitary Fermi gas in d = 4 − ε spatial dimensionsto the NLO in ε for the energies below quasiparticle threshold.The resulting expression is shown in Eq. (15). Expressions forthe low-energy dynamic structure function and static structurefactor are presented in Eqs. (17) and (18). The simplest way

to make predictions for d = 3 is to set ε = 1 in the NLOexpressions (15), (17), and (18).

II. THE LAGRANGIAN AND DEFINITIONS

Due to universality of unitary Fermi gas, we choose towork with the local four-Fermi interaction with a couplingconstant c0 tuned to reproduce infinite scattering length. TheLagrangian density of the unitary Fermi gas in the presence ofthe source coupled to density, ρ(x) [where x = (t, x)], is (hereand below h = 1)

L = ψ†[i∂t + ∇2

2m+ µ + ρ(x)

]ψ + c0 n↑ n↓, (2)

where ψ = (ψ↑, ψ↓)T is a spin-1/2 fermion field and ni =ψ

†i ψi, with i = {↑,↓} are particle density operators and

µ is the particle number chemical potential which in thiscalculation is assumed to be the same for both pairing species.The theory is defined in d = 4 − ε spatial dimensions, so that,for example, the action is

S =∫

dtdxdL.

After the Hubbard-Stratonovich transformation, the La-grangian density (2) takes the form

L = �†(

i∂t + σ3∇2

2m

)� + �†[µ + ρ(x)]σ3 �

− 1

c0φ∗φ + �†σ+�φ + �†σ−�φ∗, (3)

where φ(x) is an auxiliary difermion field, � is a two-component Nambu-Gor’kov field, � = (ψ↑, ψ

†↓)T , σ± =

12 (σ1 ± iσ2), with σ1,2,3 being the Pauli matrices. In thiscalculation we will concentrate on the unitary regime, a =∞, which in the dimensional regularization corresponds to1/c0 = 0 [8]. It is possible to use the ε expansion technique todescribe this Fermi system near unitarity [12], and the resultsof this article may be extended to the near-unitary regime. Thisis left for future work.

The density correlation function is defined by

S(x) = −i〈�|T ψ†(x)ψ(x) ψ†(0)ψ(0)|�〉= i

δ2

δ ρ(x) δ ρ(0)ln Z[ρ(x)]|ρ=0, (4)

1050-2947/2010/81(4)/043617(4) 043617-1 ©2010 The American Physical Society

ANDREI KRYJEVSKI PHYSICAL REVIEW A 81, 043617 (2010)

where |�〉 is the ground state,

Z[ρ(x)] =∫

Dψ Dψ†Dφ Dφ∗ exp

[i

∫x

L(ψ, φ; ρ(x))]

is the generating functional and x = (t, x).

III. LOW-ENERGY BOSON PROPAGATOROF THE SUPERFLUID PHASE

Let us review some of the low-energy properties of thesuperfluid phase of the unitary Fermi gas which will be relevantfor the density-density correlator calculation [9,10].

For energies below the quasiparticle mass, ∼ φ0, the onlyrelevant degrees of freedom are the bosonic excitations of thesuperfluid condensate. Their Lagrangian is given by∫

x

Leff(φ) = −i Tr ln

(i ∂t + ∇2

2 m+ µ φ0 + g ϕ(x)

φ0 + g ϕ∗(x) i ∂t − ∇2

2 m− µ

),

(5)

where φ0 is the uniform medium order parameter which to theNLO in ε is given by

φ0 = 2 µ0[1 + ε(3 C − 1 + ln 2)], C ≈ 0.14424,(6)

µ0 = µ/ε,

[8], while the coupling

g = (8π2ε)1/2

m

(mφ0

2π

)ε/4

, (7)

was chosen by Nishida and Son [8] in order to give bosons acanonical kinetic term.

Equation (5) may be expanded in powers of φ and itsderivatives as well as in powers of ε. To NLO in ε and tothe second order in p0, εp, we have

Leff = 12�†D−1�, (8)

where �T = (φ, φ∗) and

D−111 (p) = D−1

22 (−p)

= Z(p0 − ε p

2

)

+ 2 µ + ε

φ0

(5 p2

0

24− p0 ε p

6+ 7 ε2

p

120

)

− 3 ε φ0

2+ O

(ε2, ε

p03

φ20

, εε3

p

φ20

),

D−112 (p) = D−1

21 (p) = −ε φ0

2+ ε ε p

8− ε ε2

p

40 φ0− ε p2

0

24 φ0

+O(

ε2, εp03

φ20

, εε3

p

φ20

), (9)

with p = (p0, p), εp = p2/2m,Z = 1 + ε(ln2 − γ )/2, whereγ ≈ 0.57722, is the Euler-Mascheroni constant. To NLO in ε

the low-energy effective Lagrangian only contains quadraticterms in φ. The non-Gaussian terms are suppressed by higherpowers of ε. The situation is somewhat analogous to thechiral perturbation theory of the color-favor-locked phase ofasymptotically dense quark matter where meson interaction

terms are exponentially suppressed [13]. A generic term in theNLO Lagrangian is of the form �†crs εpr

0εsp/

r+s−1�, r +s � 1, crs is O(1); p0, εp � are treated as O(1) in ε

expansion.1

In the ε expansion approach, observables are calculated inthe perturbation theory where fermions interact with bosonicexcitations of the condensate with the coupling g ∝ √

ε of(7). Detailed descriptions of Feynman rules for both zero andfinite temperature cases are presented in Refs. [12] and [9],respectively.

Superficially, to calculate density correlator to the NLO inε it should be sufficient to use the leading order (LO) bosonpropagator, which is proportional to (p0 − εp/2)−1. However,this expression does not have the superfluid mode pole atp0 = cs |p| as |p| → 0, where cs is the speed of sound, whichwe know should be present in the spectrum. It has been shownthat to capture the right form of the superfluid mode excitationenergy it is sufficient to use boson propagator with momentumindependent O(ε) boson self-energy corrections resummed[9,10]. The relevant NLO boson propagator is then

D(p)11 = p0 + εp2

p20 − εp

( εp4 + µ

) + iδ, δ = 0+. (10)

From the NLO boson propagator (8), (9) using (6) one gets

c2s = µ

2 m

(1 + ε

4

), (11)

sound velocity to NLO which will be used later.

IV. THE DENSITY CORRELATION FUNCTION

To NLO in ε the density-density correlation function willconsist of the contributions shown in Fig. (1): the one bosonexchange diagram which produces O(1/ε) contribution, andthe O(1) one fermion loop diagram.

The dashed line in Fig. (1) is the dressed boson propagatordiscussed in the previous section [Eq. (10)]. The thin solid linerepresents fermion propagator

G(p0, p) = 1

p20 − E2

p + iδ

(p0 + εp − µ −φ0

−φ0 p0 − εp + µ

),

(12)δ = 0+,

where Ep = [(εp − µ)2 + φ20]1/2. The thick solid line is the

source ρ(x) coupled to the particle density. The coupling g isgiven in Eq. (7). For energies below twice the quasiparticle gapthe only contribution to the imaginary part of density-densitycorrelator will come from the one boson exchange diagram.

The one boson exchange diagram and the one loop graphcontributions are given by

Sboson(k) = −∫

p, q

g2 tr[G(p) σ3 G(p + k) σ−]D(k)11

× tr[G(q) σ3 G(q − k) σ+] + (k ↔ −k), (13)

1In the εp � µ regime it may be more appropriate to treat εp ∼ ε

[14]. However, as we will see later the density correlator satisfiesseveral sum rules using the power counting scheme assumed here.Full resolution of this issue is deferred to future work.

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LOW-ENERGY DENSITY CORRELATION FUNCTION OF A . . . PHYSICAL REVIEW A 81, 043617 (2010)

O(1/ε)

O(1)

gg

ρ(x)

ρ(x) ρ(0)

ρ(0)

FIG. 1. Feynman diagrams relevant for the calculation of thedensity correlation function to NLO in ε. Thin solid lines depictpropagating fermions in the superfluid background (lines with squarescorrespond to the anomalous propagators), dashed line is the dressedboson propagator, thick solid line is the source coupled to the particledensity. The O(1/ε) diagram contributes to the imaginary part of thedensity-density correlator below quasiparticle threshold, ω < 2 ,

where ∼ φ0 is the quasiparticle mass.

and

S1 loop(k) = −i

∫p

tr[G(p) σ3 G(p − k) σ3], (14)

respectively, and∫k

≡ ∫dd+1k/(2 π )d+1. The fermion loop

integrals are calculated in d = 4 − ε dimensions with subse-quent expansion to the second order in k0, εk. Matrices σ±, σ3

have been defined below Eq. (3). It was argued in Ref. [8] thatµ was to be counted as O(ε). To the NLO µ ∼ ε is omittedeverywhere in (13) except in the denominator of D(k)11.

The result normalized to the NLO particle density, n, is

1

nS(p0, ε p)

=2 ε p

(1 − ε

4

) − ε6 µ0

(6 p2

0 + ε2p

) + O(ε2, εε3

p

/µ2

0

)p2

0 − ε p(µ + ε p

4

) + iδ + O(εε pµ)

− ε

4 µ0

(1 − ε p

6 µ0+ p2

0

24 µ20

+ ε2p

120 µ20

)

+O(

ε2, εp03

µ40

, εε3

p

µ40

),

0 � p0 � 2, δ = 0+, (15)

where

n =(m µ0

2π

)d/2 4

ε

(1 + ε

[6C − γE

2+ 2 ln 2 − 7

4

]),

(16)γE ≈ 0.57722, C ≈ 0.14424,

where µ0 = µ/ε ∼ φ0 and /µ0 � 2(1 − 0.345ε) is the NLOquasiparticle mass [8] and the NLO expression for the uniformmedium superfluid order parameter (6) has been used. Theboson exchange contribution to (15) is not fully expandedin the powers of p0, εp to preserve the pole structure. The

dynamic structure factor, σ (ω, εp), is, then

σ (p0, ε p) ≡ −Im S(p0, ε p)

= π n

(2 ε p

(1 − ε

4

) − ε6 µ0

(6 p2

0 + ε2p

)2 p0

)

× δ

(p0 −

√ε p

(µ + ε p

4

)), (17)

where n is given by Eq. (16). And the inelastic form factor(per unit volume), Sinel(εp), in this approximation is given by

Sinel(εp) ≡ − 1

π n

∫ ∞

0dω Im S(ω, εp)

= 2√

εp

εp + 4 µ

(1 − ε

4− 5 ε εp

24µ0

). (18)

V. SUM RULE CHECKS

Let us check the correlation function (15) against severalidentities for a density correlator. Using S(k) of (15) we findthat (i) the dispersion relation

S(ω = 0, εk) = 2

π

∫ ∞

0dω

Im S(ω, εk)

ω(19)

is satisfied to the NLO in ε and to O(ε2k ).

(ii) The compressibility sum rule

S(ω = 0, εk = 0) = − n

m c2s

, (20)

where n is the equilibrium density given by (16) and cs is thespeed of sound given by (11), is satisfied to the NLO in ε. Notethat the left-hand side of (20) contains only the LO expressionfor cs . One also observes that the 1 − ε/4 coefficient at εp inthe numerator of the fraction in (15) is necessary to satisfy(20) to NLO.

(iii) The energy-weighted sum rule which for a unit volumeof the medium is

− 1

πn

∫ ∞

0dω ω Im S(ω, εp) = εp (21)

for any εp. From S(ω, εp) of (15) we get

− 1

πn

∫ ∞

0dω ω Im S(ω, εp) = εp

[(1 − ε

4

)− 5 ε εp

24µ0

].

(22)

Let us emphasize that sum rule (21) is sensitive to the dynamicsof quasiparticles at ω > 2 which we have not included in thecalculation. We see that the sum rule is satisfied at LO inε. At NLO the εp coefficient is reduced to 1 − ε/4 insteadof 1 prescribed by (21) which is due to the omission ofthe quasiparticle contribution above the 2 threshold. Also,higher-order terms suppressed by powers of εp/µ0 ∼ εp/

appear. This is natural within the framework of the low-energy effective theory. So we see that by including only thecondensate excitation contribution to the NLO Im S(p) andomitting the rest of the (heavy) intermediate states we do notviolate the energy-weighted sum rule (21) too badly.

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ANDREI KRYJEVSKI PHYSICAL REVIEW A 81, 043617 (2010)

These checks indicate that the density-density correlatorgiven by Eq. (15) reproduces response properties of themedium at low energy reasonably well.

VI. CONCLUSIONS

We have computed the density correlation function of theunitary Fermi gas, S(ω, εp), to NLO in the ε expansion forenergies below quasiparticle threshold, ω � 2, and from itderived expressions for low-energy dynamic structure functionand static structure factor.

The correlation function at low energy is dominated by theexchange of a single boson (see Fig. 1). This is as expectedsince the low-energy dynamics of the system is dominated bythe superfluid modes which are coupled derivatively.

The simplest extrapolation to d = 3 may be made by simplysetting ε = 1 in the NLO expressions (15), (17), and (18).

As is well known (see, e.g., Ref. [15]), the dynamicstructure function is proportional to the inelastic-scatteringcross section of any external probe coupled to density and

may be measured experimentally. Based on the calculation,one predicts a significant change in the dynamic structurefunction at the quasiparticle threshold, ω = 2, which mayhelp experimental measurement of the quasiparticle gap.

Future work will include the calculation of the dynamicstructure function for energies above quasiparticle thresholdand of the spin response function relevant for the descriptionof neutrino coupling to the dilute neutron matter (e.g.,Refs. [16,17]).

ACKNOWLEDGMENTS

The author is grateful to C. J. Horowitz for suggestingthe problem and numerous helpful discussions and to SanjayReddy for pointing out significance of the boson exchange dia-gram of Fig. 1. Also the author thanks Yusuke Nishida and DamSon for useful conversations. This work is supported in part bythe US Department of Energy grants DE-FG02-87ER40365,DE-FG02-05ER41375 (OJI), and DE-FG52-08NA28921 andby the National Science Foundation grant PHY-0555232.

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