Bulk Viscosity and Conformal Symmetry Breaking in the Dilute Fermi Gas near Unitarity

Kevin Dusling and Thomas Schafer

Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA(Received 18 June 2013; published 19 September 2013)

The dilute Fermi gas at unitarity is scale invariant and its bulk viscosity vanishes. We compute, in the

high temperature limit, the leading contribution to the bulk viscosity when the scattering length is not

infinite. A measure of scale breaking is provided by the ratio ðP� 2=3EÞ=P, where P is the pressure and Eis the energy density. At high temperature this ratio scales as z�=a, where z is the fugacity, � is the thermal

wavelength, and a is the scattering length. We show that the bulk viscosity � scales as the second power of

this parameter, � � ðz�=aÞ2��3.

DOI: 10.1103/PhysRevLett.111.120603 PACS numbers: 05.60.Gg, 67.85.Lm

The dilute Fermi gas at unitarity is a beautiful example ofa scale and conformally invariant many body system. Scaleinvariance implies that thermodynamic properties of thesystem only depend on the dimensionless variable n�3,

where n is the density and � ¼ ½2�@=ðmTÞ�1=2 is the ther-mal de Broglie wavelength. In the high temperature limit,n�3 � 1 and the gas is weakly interacting despite the factthat the two-body scattering length a is tuned to infinity. Inthe low temperature regime n�2 � 1 the gas is stronglycorrelated. It was observed that in this limit the unitaryFermi gas is a very good liquid, characterized by a verysmall shear viscosity � & @n [1–3]. Nearly perfect fluiditywas also observed in the quark gluon plasma produced inheavy collisions at Relativistic Heavy Ion Collider(Brookhaven National Laboratory) and the LHC, and itarises naturally in the context of holographic dualities [4–6].

Scale invariance is broken if the Fermi gas is detunedfrom unitarity and the two-body scattering length is notinfinite. A measure of scale invariance breaking is thedifference P� ð2=3ÞE, where P is the pressure and E isthe energy density. Tan showed that [7,8]

P� 2

3E ¼ @

2C12�ma

; (1)

where C is the contact density. At unitarity and in the hightemperature limit, C ¼ 4�@n2�2 [9]. This implies that½P� ð2=3ÞE�=P� ðn�3Þð�=aÞ. In the present work weaddress the question of how broken scale invariance mani-fests itself in transport properties. The natural quantity toconsider is the bulk viscosity � which vanishes in a scaleinvariant fluid [10–12]. We will show that in the hightemperature limit � scales as the shear viscosity times thesquare of the conformal breaking parameter ðn�3Þð�=aÞ.An analogous relation was derived by Weinberg in thecase of a relativistic gas [13]. He showed that � � �ðc2s �c2=3Þ2, where cs is the speed of sound and c is the speed oflight. In a scale invariant relativistic fluid P ¼ E=3 andc2s ¼ c2=3.

The physical mechanism for generating bulk viscosity ina nonrelativistic gas of structureless particles is subtle. In a

typical nonrelativistic gas, such as air, bulk viscosityarises from rotational and vibrational excitations of theair molecules [14]. In equilibrium, if the gas is com-pressed or expanded internal energy is transferred fromcenter of mass motion to internal degrees of freedom.This transfer requires scattering processes, and if thesereactions are slow, then the system will fall out ofequilibrium. The departure of the pressure from its equi-librium value is related to bulk viscosity. In polyatomicgases bulk viscosity also arises from energy transferbetween different species or from chemical nonequilibra-tion. In systems in which the number of particles is notconserved, such as a gas of phonons, bulk viscosity mayarise from number changing processes. None of thesemechanisms operate in a dilute Fermi gas above thesuperfluid transition.In a relativistic gas bulk viscosity arises from nonzero

particles masses, often combined with number changingprocesses. This is the case, for example, in dilute gases ofquarks and gluons [15], or a dilute gas of pions [16]. In anearly scale invariant gas, such as the quark gluon plasma,masses only arise from interactions and the effectivemass is of the form m� gT, where g is the QCDcoupling constant. In this case bulk viscosity is governedby the scale breaking part of the effective mass,~m2 ¼ ½1� T2ð@=@T2Þ�m2 [15,17]. In QCD, scale breakingarises from the logarithmic running of g with the tempera-ture T. We will show that a similar mechanism operates inthe dilute Fermi gas detuned from unitarity. Bulk viscosityarises from the scale breaking part of a temperature anddensity dependent effective mass. The new ingredientcompared to a relativistic plasma is that the momentumdependence of the effective mass is also crucial. We com-pute this effect to leading order in the fugacity, using aformalism that ensures that the kinetic theory calculationof transport properties is consistent with the virial expan-sion of the equilibrium equation of state [18].Quasiparticle properties.—The effective Lagrangian for

nonrelativistic spin-1=2 fermions interacting via a short-range s-wave potential can be written as [19]

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L ¼c y�i@0þr2

2m

�c þ½ðc�þc Þ�þH:c:�þ 1

C0

j�j2;(2)

where c is the fermion field, � is an auxiliary difermionfield, and �þ is the Pauli spin raising matrix. The couplingconstant C0 is determined by the s-wave scattering lengtha. In dimensional regularization we find C0 ¼ 4�a=m. Athigh temperature the leading term in the thermodynamicpotential is the free-fermion loop. The leading correctionarises from the boson loop. We find

�2 ¼ TXn

Z d3q

ð2�Þ3 log½D�1ði!n; qÞ�; (3)

where!n ¼ 2�nT are bosonic Matsubara frequencies andD�1ð!n; qÞ is the one loop particle-particle polarizationfunction. In the high temperature limit we can computeD�1 at zeroth order in the fugacity z ¼ expð��Þ. We get

D�1ð!; kÞ ¼ m

4�

�im1=2

�!� �k

2þ 2�

�1=2 � 1

a

�; (4)

with �k ¼ k2=ð2mÞ. It is now straightforward to computethe integral over ! and k to leading order in z. On the BCSside a < 0, we get

�2 ¼ffiffiffi2

pTz2

�3expð�B2Þf1� Erfð ffiffiffiffiffiffiffiffiffi

�B2

p Þg; (5)

with � ¼ 1=T and B2 ¼ 1=ðma2Þ. On the Bose-Einsteincondensation side there is an extra bound state contribu-tion. This result can be compared to the virial expansion� ¼ z��3½1þ b2zþOðz2Þ�, where ¼ 2 is the numberof degrees of freedom. The second virial coefficient b2 ¼b02 þ b2 is the sum of the free part b02 ¼ �1=ð4 ffiffiffi

2p Þ,

which arises from quantum statistics, and an interactingcontribution b2. Near unitarity we have

b2 ¼ 1ffiffiffi2

p�1þ 2ffiffiffiffi

�p 1ffiffiffiffiffiffiffiffi

mTp

aþ � � �

�: (6)

This result is valid on both sides of the resonance. We canuse the same methods to compute fermion self-energy atleading order in the fugacity. The leading diagram is theboson loop contribution with the boson propagator given inEq. (4). Near unitarity the real and imaginary parts of theon-shell self-energy are

Re�ðkÞ ¼ � 4ffiffiffi2

pzTffiffiffiffi�

p 1

affiffiffiffiffiffiffiffimT

pffiffiffiffiffiT

�k

sFD

� ffiffiffiffiffi�kT

r �; (7)

Im�ðkÞ ¼ � 2zTffiffiffiffi�

pffiffiffiffiffiT

�k

sErf

� ffiffiffiffiffi�kT

r �; (8)

where FD is Dawson’s integral. In the high tempera-ture limit we find fermion quasiparticles with energy

Ek ¼ E0k þ �Ek, where E0

k ¼ �k and �Ek¼Re�ðkÞ. Thewidth of the quasiparticles is given by �k ¼ �2Im�ðkÞ.Boltzmann equation and conservation laws.—In kinetic

theory the quasiparticles are described by a Boltzmannequation

Dfp ��@

@tþ ~vp � ~rx þ ~F � ~rp

�fpð ~x; tÞ ¼ C½fp�: (9)

Here, ~vp ¼ ~rpEp is the quasiparticle velocity, ~F ¼� ~rxEp is the force term, and C½fp� is the collision term.

The collision term conserves the number of particles aswell as their total momentum and energy. Momentumconservation in individual collisions leads to a conserva-

tion law for the momentum density ~�, _�i ¼ �rjx�ij.

Here, the momentum density is given by ~�ð ~x; tÞ ¼Rd�p ~pfpð ~x; tÞ and the stress tensor is

�ijð ~x; tÞ ¼Z

d�ppivj

pfpð ~x; tÞ

þ ij

�Zd�pEpfpð ~x; tÞ � Eð ~x; tÞ

�; (10)

where E is the energy density and d�p ¼ d3p=ð2�Þ3. Thecrucial property that ensures momentum conservation isthe consistency condition

Ep ¼ Efp

; (11)

familiar from the theory of Fermi liquids.Off-equilibrium bulk stress.—In fluid dynamics the trace

of the stress tensor in the rest frame of the fluid is given by

� � ð1=3Þ�ii ¼ P� �ð ~r � ~VÞ, where ~V is the fluid ve-locity. Computing the bulk viscosity requires calculatingthe dissipative part of the bulk stress �. In kinetic theorywe write

�½f0p þ fp� � �½f0p� þ � � �0 þ �; (12)

where f0p is the equilibrium distribution function and fpis an off-equilibrium correction induced by the bulk flow

( ~r � ~V). In order to compute � we use Eq. (10), andfunctionally expand E and Ep. Details are provided in the

Supplemental Material [20]. Using the consistency condi-tions we can show that

�¼

3

Zd�pfp

�~p� ~rpþ2�

@

@�þ2T

@

@T�2

��Ep:

(13)

This result has a simple interpretation as the shift in thepressure due to the scale breaking part of the quasiparticleenergy. In particular, if �Ep has the scale invariant form

Ep � zTgð�p=TÞ with an arbitrary function gðxÞ, then �

vanishes independently of the structure of the distributionfunction fp.

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Solution of the Boltzmann equation.—We determine fpby solving the Boltzmann equation using the standardChapman-Enskog procedure. We write

fpð ~x; tÞ ¼ f0pð ~x; tÞ�1� c p

T

�; (14)

where f0pð ~x; tÞ is the equilibrium distribution function com-

puted in a fluid with local velocity ~Vð ~x; tÞ, temperatureTð ~x; tÞ, and chemical potential �ð ~x; tÞ. In the case ofbulk stress the off-equilibrium factor has the form

c p ¼ �Bð ~pÞ ~r � ~V.

Streaming term.—On the left-hand side of theBoltzmann equation the leading term arises from thestreaming operator D defined in Eq. (9) acting onf0pð ~x; tÞ. This leads to a fairly complicated expression,

which can be simplified using thermodynamic identitiesand the Euler equation (see Supplemental Material [20]).At leading order in the fugacity we find

T

f0Df0jbulk¼

�� c2TcV

h�mc2sþ�2

3�� c2T

cV

��p

þ1

3

�~p� ~rpþ2�

@

@�þ2T

@

@T�2

��Ep

�~r� ~V:(15)

Here, � is the thermal expansion coefficient, h is theenthalpy per particle, cT;s is the speed of sound at constant

temperature or entropy per particle, and cV is the specificheat at constant volume. We observe that the bulk viscosityterm in the Boltzmann equation depends on the same scalebreaking part of the quasiparticle energy that also appearsin the bulk stress, Eq. (13). There are a number of simpleconsistency checks for Eq. (15). In a noninteracting gas,�Ep ¼ 0 and cP=cV ¼ 5=3. Using these values we find

that the coefficient of the bulk stress vanishes. This resultcan be found in standard textbooks on kinetic theory [21].We also find that the bulk stress vanishes for a general scaleinvariant equation of state characterized by a temperatureindependent second virial coefficient. In order to computethe streaming term near unitarity, we use the second virialcoefficient given in Eq. (6) and the quasiparticle self-

energy in Eq. (7). We get ðT=f0ÞDf0 � Xpð ~r � ~VÞ, with

Xp ¼ 2ffiffiffi2

p9

ffiffiffiffi�

p zT

affiffiffiffiffiffiffiffimT

p(�pT

� 9

2þ 6

ffiffiffiffiffiffiT

�p

sFD

ffiffiffiffiffiffi�pT

r !): (16)

This result satisfies two nontrivial sum rules,

Zd�pf

0pXp ¼ 0;

Zd�pf

0p�pXp ¼ 0; (17)

which follow from the conservation of particle number andenergy at leading order in z. Clearly, these sum rules canonly be satisfied if the quasiparticle energy is consistentwith the equation of state.

Collision term.—At leading order in the fugacity thecollision term is dominated by two-body collisions. Inthe case of bulk stress the linearized collision operator isgiven by

C½f0p þ fp� �f0pTCL½�BðpÞ�ð ~r � ~VÞ; (18)

with

CL½�Bðp1Þ� ¼Z �Y4

i¼2

d�i

�wð1; 2; 3; 4Þf0p2

½�Bðp1Þ

þ �Bðp2Þ � �Bðp3Þ � �Bðp4Þ�: (19)

The transition rate wð1; 2; 3; 4Þ is given by

wð1; 2; 3; 4Þ ¼ ð2�Þ43

�Xi

~pi

�

�Xi

Ei

�jAj2; (20)

and the scattering amplitude is

jAj2 ¼ 16�2

m2

a2

a2q2 þ 1; (21)

where ~q ¼ ð1=2Þð ~p2 � ~p1Þ. To leading order in z we canapproximate the quasiparticle energy by the noninteractingresult Ep ’ �p. Conservation of particle number and en-

ergy then leads to the sum rules given in Eq. (17). In orderto compute �B to leading order in 1=a, we can also use thescattering amplitude in the unitary limit. We solve thelinearized Boltzmann equation

Xp ¼ CL½�BðpÞ� (22)

by expanding �BðpÞ in generalized Laguerre (Sonine)polynomials:

�BðpÞ ¼XNi¼2

ciL1=2i

��pT

�: (23)

Restricting the sum to terms of order i � 2 ensures that thesum rules are satisfied. We solve for ci by taking momentsof the linearized Boltzmann equation. The simplest case isN ¼ 2. We find

�BðpÞ ¼ffiffiffiffi�

p64

z

affiffiffiffiffiffiffiffimT

p�15� 20

��pT

�þ 4

��pT

�2�; (24)

and, using Eq. (13),

� ¼ 1

24ffiffiffi2

p���3

�z�

a

�2: (25)

We observe that �B is first order in the conformal breakingparameter (z�=a) whereas the bulk viscosity is secondorder. The expansion in Laguerre polynomials convergesrapidly. We can write � ¼ k��3ðz�=aÞ2, where k is apure number. For N ¼ 2, 3, 4, we find k¼f9:378;9:739;9:771g10�3. The N ¼ 3 term gives a 4%correction, and the N ¼ 4 term leads to a 0.3% shift.

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Outlook.—The result in Eq. (25) can be written in theform

�

n¼ 1

9ffiffiffiffiffiffiffi2�

p 1

ðkFaÞ2�TF

T

�5=2

; (26)

where kF ¼ ð3�2nÞ1=3 and TF ¼ k2F=ð2mÞ are defined interms of the local density. We first address the question ofwhether or not current experiments are sensitive to a bulkviscosity in this range. Measurements of the shear viscosityusing collective modes are sensitive to values as small as�=n ’ 0:1. The bulk viscosity grows with 1=ðkFaÞ andTF=T, so part of the issue is how far one can extrapolateour result in these two variables. We know that the bulkviscosity vanishes for both jkFaj ! 1 and jkFaj ! 0. Thismeans that at fixed T=TF the bulk viscosity has a maximumat some finite value of (kFa). Independent of the location ofthis maximum, we also know that in typical experimentshydrodynamics breaks down for jkFaj * 1 [22]. As afunction of T=TF we expect the bulk viscosity to have amaximum near the phase transition, T � Tc ’ 0:167ð13ÞTF

[23]. In the case of shear viscosity, we know that kinetictheory is remarkably accurate down to temperatures as lowas T � 2Tc; see, for example, Ref. [11]. Using jkFaj � 1and T � 2Tc in Eq. (26), we conclude that �=n could be aslarge as 0.5, within the range accessible in experiment.

Another interesting issue concerns the frequency depen-dence of the bulk viscosity. Taylor and Randeria proved thesum rule [24,25]

1

�

Zd!�ð!Þ ¼ 1

72�ma2@C@a�1

��������s=n: (27)

Using the virial expansion we find

1

�

Zd!�ð!Þ � T��3

�z�

a

�2: (28)

We conclude that the kinetic theory result is consistentwith this sum rule provided the width of the transportpeak is less than T. Note that the width of the shear peakis ��1

R ¼ P=�� zT � T [26]. The high frequency tail ofthe bulk viscosity was determined in [27]:

�ð!Þ ¼ C36�

ffiffiffiffiffiffiffiffim!

p 1

1þ a2m!: (29)

In the high temperature limit this implies

�ð!Þ � ��3

�z�

a

�2�T

!

�3=2

; (30)

showing that the transport peak and the high frequency tailcan match smoothly if the transport peak is broad, with awidth of order T. On the other hand, if the transport peak isnarrow, then the sum rule is saturated by the continuumcontribution, and the spectral function must have twopeaks, a transport peak at ! ¼ 0 and a continuum peakat !� T.

There are a number of issues that remain to beaddressed. Based on the discussion above it would beinteresting to compute the frequency dependence of thebulk viscosity in kinetic theory. It would also be interestingto generalize our calculation to a two-dimensional Fermigas. In two dimensions scale invariance is always broken,but experiments indicate that the bulk viscosity is verysmall [28,29]. Finally, it would be interesting to constructa complete quasiparticle model of the dilute Fermi gas nearunitarity, including the effects of both the real and imagi-nary parts of the self-energy.We thank Paulo Bedaque and John Thomas for useful

discussions. This work was supported in part by the U.S.Department of Energy Grant No. DE-FG02-03ER41260.

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