Renormalization group fixed points, universal phase diagram, and 1/N expansion for quantum liquids with interactions near the unitarity limit

0.38 MB·English

by  Predrag Nikolic

Renormalization group fixed points, universal phase diagram, and 1/N expansion for
quantum liquids with interactions near the unitarity limit
Predrag Nikoli´c and Subir Sachdev
Department of Physics, Harvard University, Cambridge MA 02138
(Dated: February 6, 2008)
7
0 Ithaslongbeenknownthatparticleswithshort-rangerepulsive interactionsinspatialdimension
0
d = 1 form universal quantum liquids in the low density limit: all properties can be related to
2
those of the spinless free Fermi gas. Previous renormalization group (RG) analyses demonstrated

n that this universality is described by an RG fixed point, infrared stable for d < 2, of the zero
a density gas. We show that for d > 2 the same fixed point describes the universal properties of
J particles with short-range attractive interactions near a Feshbach resonance; the fixed point is now

0 infrared unstable, and the relevant perturbation is the detuning of the resonance. Some exponents
2 are determined exactly, and the same expansion in powers of (d−2) applies for scaling functions
for d < 2 and d > 2. A separate exact RG analysis of a field theory of the particles coupled to

] ‘molecules’ finds an alternative description of the same fixed point, with identical exponents; this
n
approach yieldsa(4−d)expansion which agrees with therecent resultsof NishidaandSon (Phys.
o
Rev. Lett. 97, 050403 (2006)). The existence of the RG fixed point implies a universal phase
c
diagram as a function of density, temperature, population imbalance, and detuning; in particular,
-
r this applies to the BEC-BCS crossover of fermions with s-wave pairing. Our results open the way
p
towards computation of these universal properties using the standard field-theoretic techniques of
u
criticalphenomena,alongwithasystematicanalysisofcorrectionstouniversality. Wealsopropose
s
a 1/N expansion (based upon models with Sp(2N) symmetry) of the fixed point and its vicinity,
.
t and use it to obtain results for thephase diagram.
a
m
- I. INTRODUCTION scattering amplitude, f , takes the form10
d 0
n 1
o f0 = (1)
ν ik
c Thestudyofultracoldatomicgaseshasdrawnrenewed −
[ attention to interacting quantum gases in regimes where where k is the momentum transfer, S = 1+2ikf0, and
3 theirpropertiesareindependentofallmicroscopicdetails ν is the ‘detuning’ across the resonance (it is related to
v oftheinteractionbetweentheatoms. Thishappenswhen the scattering length, a, by ν = 1/a). The resonance
−
6 the scattering cross-section between two atoms reaches occurs at ν = 0, and notice then that S = 1, the uni-
−
0 its unitarity limit. taritylimit. Wewillbeinterestedhereinbroad Feshbach
1 resonances, where Eq. (1) holds over all the momenta of
9
The simplest example of this phenomenon is the di- interest in a quantum gas. For a gas with density n, a
0
lute Bose gas with short-range repulsive interactions in resonance is broad if the (k2) corrections to the de-
6 O
0 spatial dimension d = 1 where, in fact, the properties nominator of Eq. (1) are negligible at the characteristic
/ are generically universal. It is a well-known property momentumn1/3 (weuse ~=1 throughout). Notice then
t
a of quantum-mechanical scattering in one dimension that that the width condition on the Feshbach resonance, is
m for any generic short-range potential, the S matrix for actually a diluteness condition on the quantum gas, and
- two particle scattering reaches its unitarity limit value anyFeshbachresonanceis‘broad’forasufficientlydilute
d of 1 in the limit of zero momentum transfer. However, gas.
n the−strongly interacting Bose gas with S = 1 can be This paper demonstrates that a unified understanding
o reinterpreted1 as a free Fermi gas (the ‘Tonk−s’ gas), al- of the distinct universalproperties of the quantum gases
c
: lowing easy computation of at least the thermodynamic in d = 1 and d = 3 can be obtained in a field-theoretic
v properties. Correlation functions are much harder to renormalizationgroup (RG) analysis. The RG approach
Xi compute, especially at finite temperature and unequal was used previously3,5,11,12 for the repulsive Bose gas in
r times, but much theoretical effort has been expended in d<2,and here we demonstrate that a directgeneraliza-
a this direction.2,3,4,5 Experimental studies of Tonks gas tionapplies toawide classofquantumliquids ingeneral
behavior have also appeared.6 d. WewillusethestructureofanRGfixedpointtoargue
that there is a universal phase diagram as a function of
No generic unitarity limit scattering is obtained in relevant perturbations away from the fixed point: these
d = 3. In this case, we usually have the limit S = 1 for perturbations are the temperature, the detuning away
low momentum, implying that sufficiently slowly mov- from the Feshbachresonance,and the density imbalance
ing particles do not scatter. However, upon fine-tuning for a two-component quantum gas. Our claims of uni-
the strength of an attractive interaction, it is possible versality are stronger than previous ones which applied
to obtain unitarity limit scattering across a Feshbach onlyattheresonance.7,8,9 Inparticular,questionsonthe
resonance.7,8,9 Near a Feshbach resonance, the s-wave nature of the phases, including the existence of exotic
2
gaplesssuperfluidstates,haveauniqueanduniversalan- d < 2
swerforabroadFeshbachresonance. Wewillalsoobtain 0 u*
explicitresultsforthisphasediagram,andforassociated
u
universalscalingfunctions,forthetwo-componentFermi
gas in d= 3 with s-wave pairing using a 1/N expansion (a)
ofa modelwith Sp(2N)symmetry. Inparticular,we ob-
tain results on the equation of state for the normalstate
d > 2
with unbalanced densities and two Fermi surfaces.
u* 0
The form of our universal phase diagram has similari-
tiestoworkbySheehyandRadzihovsky13,14(andrelated u
work by others15) where they departed from the narrow
(b)
resonance limit in an expansion in the resonance width.
We discover the same phases and transitions between
them, except the possible non-uniform phases which are FIG.1: Theexact RGflow of Eq.(2),as discussed in Ref.5.
nota focus ofthis paper. The mainadvance inour work Here u is a measure of the short-range two-body interaction
is a systematic treatment of the broad resonance limit, between the particles, and theRG applies in thelimit of low
which is experimentally relevant, and in which univer- densityofeitherBose,Fermi,orBose-Fermiquantumliquids.
(a) For d < 2, the infrared stable fixed point at u = u∗ > 0
sality applies. Universality has also been explored using
describes quantum liquids of either bosons or fermions with
a different RG analysis in Ref. 16: they do not give a
repulsive interactions which are generically universal in the
“criticalphenomena”interpretationofthe results,which
low density limit. In d = 1 this fixed point is described by
we claim below yield the most direct interpretation and
thespinlessfreeFermigas(‘Tonks’gas),forallstatisticsand
computation tools for identifying universal behavior. spin of the constituent particles. (b) For d > 2, the infrared
The previous study3,5,11,12 of the dilute Bose gas be- unstable fixed point at u = u∗ < 0 describes the Feshbach
gan with an RG analysis of the field theory of the zero resonancewhichobtainsforthecaseofattractiveinteractions.
density gas17 at chemical potential µ = 0, considered as The relevant perturbation (u−u∗) corresponds to the the
a quantum critical point between the vacuum state for detuningfrom the resonant interaction.
µ<0, and the finite density ground state for µ>0. In-
teractions between a pair of particles created out of the
vacuum are characterizedby a coupling u (defined more The flows are shown in Fig 1. Now there is an infrared
preciselyinthebodyofthepaper),andthiscouplingwas unstable fixed point at a u∗ <0. We show here that this
shown to obey the exact field-theoretic RG equation fixed point describes the Feshbach resonance of particles
with attractive interactions. The unitarity of the scat-
du u2 tering at this fixed point has also been pointed out inde-
=(2 d)u (2)
dℓ − − 2 pendently inRef. 19. The relevantperturbation(u u∗)
−
is proportional to the detuning, ν, of the resonance. By
under a rescaling of length scales by a factor of eℓ, an elementary computation of the eigenvalue of linear
and a dynamic exponent z = 2. As we will review perturbations from the u = u fixed point of Eq. (2) we
∗
later, this RG equation applies equally to two-body in- therefore determine the exact scaling dimension
teractionsbetweenparticleswitharbitrarystatistics and
masses. As shown in Fig. 1, for d < 2, this RG equa- dim[ν]=d 2 (3)
−
tion has an infrared stable fixed point at u >0. It was
∗
argued5,11,12 that this fixed point interaction which de- The universalproperties of the quantum liquid with ν =
6
scribesunitarityscattering. Thechemicalpotential,µ,is 0 and µ = 0 can now be determined by essentially the
6
a relevant perturbation at this fixed point, and its scal- samerenormalizedtheoryearlierusedford<2,andnow
ing dimension, and RG flow can also be determined ex- becomes an expansion in (d 2). Indeed, as we will see
−
actly. A standard18 field-theoretic analysis then allowed in Section II, the earlier5 (2 d) expansions for scaling
−
computation5 of all universal properties of the µ = 0 functions for d < 2, apply unchanged at resonance for
6
Bose gas in d<2, in an expansion which can be carried d > 2. A shortcoming of this (d 2) expansion for the
−
outtoallordersinu =2(2 d). Thedegreeofdifficulty attractive Fermi gas is that the pairing amplitude is of
∗
−
of this computation is the same for thermodynamic and order exp(1/u ) exp( 1/(d 2)), and so all effects
∗
∼ − −
dynamic correlations. Practically speaking, this expan- associated with superconductivity are non-perturbative.
sionisthenusefulforthosephysicalpropertieswhichare Regardlessoftheabilitytodeterminescalingfunctions
difficulttoaccessbythefermionizationmethodind=1. in the (d 2) expansion, the flow equation (2), and its
−
In some cases, an exact computation is possible both in associated exponents are exact, and allow us to deduce
the fermionization approach, and also by resumming all some exact scaling forms associated with the RG fixed
ordersinthe(2 d)expansion,andexactagreementwas point in d > 2. For the two-component Fermi gas, we
obtained.3 A rev−iew of this approach to Bose and Fermi have already noted two relevant perturbations at the
quantum liquids in d<2 appears in Section II. fixed point, the detuning ν and the chemical potential
Let us now examine the RG equation (2) for d > 2. µ. For general d, we define the overall scale of ν by the
3
the zerodensityvacuumfromthe lowdensitysuperfluid.
The RG fixed point is a multicritical point on this line,
m and the whole phase diagram is constructed as a theory
ofrelevantperturbationsfromthispoint. Notethatthere
is no phase transition at finite density within the super-
Superfluid
fluid,onlyasmoothcrossoverbetweentheBCSandBEC
BCS limits. However,thereisaquantumcriticalpointatzero
detuning (at “resonance”) on the phase boundary, and
we use this critical point as a point of departure to de-
BEC scribe the entire phase diagram. We will present further
n
results on this phase diagram in the body of the paper.
Vacuum
Given the limitations of the (d 2) expansion for
−
obtaining useful approximations for the scaling func-
tion, the remainder of the paper presents alternative ap-
proachestoanalyzingthefixedpointdescribedabove. In
Section III, we examine another field theory commonly
usedtodescribeaFeshbachresonance: atheoryofmixed
FIG.2: Universalphasediagramatzerotemperature(T =0) ‘atoms’ and ‘molecules’, with the ‘molecule’ correspond-
and balanced densities (h=0) for the two-component Fermi ingtotheboundstateinthevicinityoftheFeshbachres-
gas in d=3. Thevacuumstate(shown hatched)has nopar- onance (the so-called “two-channel” approach14,24). As
ticles, and is present for µ < 0 and ν > 0, or for ν < 0 and argued above, it is useful to perform an RG analysis in
µ<−ν2/(2m), where m is the mass of a fermion. The posi- the limit of zero density, when the physics reduces to
tionoftheν <0phaseboundaryisdeterminedbytheenergy
the quantum mechanics of few-particle scattering. The
ofthetwo-fermion boundstate. Thedensityofparticlesvan-
RG flow equations can now also be determined exactly,
ishescontinuouslyatthesecondorderquantumphasetransi-
and are presented in Section III. They contain a fixed
tion boundary of the superfluid phase, which is indicated by
point, which, at first glance, appears distinct from that
the thin continuous line. The quantummulticritical point at
of Eq. (2). However, a careful analysis of its universal
µ = ν = h = T = 0 (denoted by the filled circle) is the RG
fixed point which is thebasis for the analysis in this paper. properties,suchasscalingdimensionsofoperators,shows
that they are identical to the results quoted above. The
finitedensityfieldtheoryassociatedwiththisfixedpoint
requirementthatthe s-wavescatteringamplitude is pro- becomes weakly coupled when (4 d) is small. Conse-
portionalto1/(ν ( k2)(d−2)/2). Thescalingdimension quently,astandardrenormalizedp−erturbativeexpansion
− −
of µ is exactly 2. There is also a third relevant pertur- now allows us to obtain universal properties in powers
bation at the fixed point: a ‘field’ h conjugate to the of (4 d). The universal results of the (4 d) expan-
− −
density difference of the two species of fermions, which sionareequaltothoseobtainedrecentlyintheimportant
also has scaling dimension 2. From all this information, workofNishidaandSon25,whomotivatedtheexpansion
and the dynamic exponent z = 2, we can deduce one of from different considerations and use a different compu-
our central results, a scaling form for the grand canoni- tationalscheme. The specialrole ofd=2andd=4 was
calfreeenergydensity (fromtheusualthermodynamic also discussed by Nussinov and Nussinov.26 Our method
F
identity, = P, the pressure) at a temperature T: streamlines the expansion (4 d) onto the standard ap-
F − −
proachused extensively for critical phenomena, and also
µ h ν
=(2m)d/2T1+d/2F , , , (4) opens the way to a systematic analysis of corrections to
F F(cid:18)T T (2mT)(d−2)/2(cid:19) scaling by a consideration of irrelevant perturbations.
where m is the mass of one of the particles, and F is Further considerations on the structure of the (4 d)
−
a universal scaling function whose three argumentsFex- expansion shows that it also has some limitations in its
tend over all real values (if the mass of the particles are range of validity. For the two component Fermi gas,
distinct, the scaling function will also depend upon the the naive (4 d) expansion is restricted to being well
−
ratio of the masses). For the two-component Fermi gas within the superfluid state, or well within the normal
this universal scaling function contains all the informa- state with one Fermi surface. A subtle redefinition of
tion on the BEC-BCS crossover.20,21,22,23 The existence parameter scales25,27 is required to obtain an expansion
of this scaling form also implies that there is a universal for the 2 Fermi surface normal state. The latter is one
phase diagram as a function of µ, ν, h, and T in the low of the most interesting new regimes discovered in recent
densitylimit(orequivalently,forabroadresonance). In- experiments28,29, and so an unrestricted analysis here
formation on all the phases and phase transitions in this would be useful.
phasediagramisalsocontainedinEq.(4). Ath=0and With this motivation, in Section IV we introduce a
T = 0, the form of this universal phase diagram can be third approach towards analyzing the finite density field
deducedexactly,andisshowninFig.2ford=3. Thereis theoryattheRGfixedpointofEq.2. Thisisbasedupon
a second-orderquantum phase transition line separating the 1/N expansion, where N is the number of particle
4
species. This expansionholds uniformly forgeneralµ, ν, renormalized dimensionless coupling u by
h, and T. For the two-componentFermi gas, the critical
fixed point has Sp(2N) symmetry, and the expansion is κ2−dZ4
u = u, (8)
based upon that considered earlier in Ref. 30 well away 0 2mS
d
from the Feshbach resonance.
where κ is a renormalization momentum scale, S =
We conclude this introduction by briefly discussing d
2/(Γ(d/2)(4π)d/2)isthestandardphasespacefactor,and
quantum liquids with attractive interactions in d < 2;
Z isthe renormalizationconstant. Theladderdiagrams
these are not directly described by the fixed points dis- 4
lead to the exact result
cussed so far. In such systems, the ‘atomic’ constituents
will bind to form ‘molecules’and the resulting molecules u
could have repulsive interactions, leading eventually to Z4−1 =1− 2 d (9)
−
thequantumliquidofthetypediscussedaboveford<2.
intheminimalsubtractionscheme. Fromthis,weobtain
The crossovers associated with this behavior will be de-
theexactRGβ functionforupresentedearlierinEq.(2),
scribed by an RG analysis which is a combination of
upon rescaling κ by a factor of eℓ.
that presented in Sections II and III—we need to explic-
Nowweincluderelevantoperatorsawayfromthisfixed
itly include the interactions between the atoms and the
point (apart from (u u ) for d > 2). For the single-
molecules (relevant for d < 2), along with 3-point term ∗
−
component Bose gas, there is only one such term, the
allowingforamoleculetodecayintotwoatoms(relevant
chemical potential
for d < 4). We will not present the details of such an
RG analysis here, but note that these considerations are
entirelyconsistentwitha recentanalysisby Gurarie31 in bp = dτddx µψ†ψ . (10)
d=1 using Bethe ansatz methods. S (− )
Z
Iteasytoseethataninsertionof acquiresnooperator
p
S
II. RG FOR A FIELD THEORY OF ATOMS renormalizationat the zero density criticalpoint, and so
from Eq. (7) we conclude that
We begin by considering the theory of the dilute Bose
dim[µ]=2. (11)
gas,fromthe perspective ofazerodensityquantumcrit-
ical point, as discussed in Ref. 5. We consider a zero
Forthe two-componentFermigas,wehaveanadditional
density theory at the chemicalpotential µ=0. This can
relevant perturbation corresponding to the difference in
be viewed as a quantum critical point between the finite
chemical potential of the two species:
densityphasewithµ>0,andtheemptyvacuumground
state for µ < 0. The critical theory of this quantum
critical point of a boson, ψ, is Sfp = dτddx( − µ(ψ1†ψ1+ψ2†ψ2)
Z
∂ 2 u
Scb =Z dτddx(ψ†(cid:18)∂τ − 2∇m(cid:19)ψ+ 20ψ†ψ†ψψ). (5) − h(ψ1†ψ1−ψ2†ψ2)), (12)
The RG analysis of the zero density theory does not
and again we have
depend upon the statistics of the particles, so we also
consider in parallela theory of two-componentfermions, dim[h]=2 (13)
ψ , σ =1,2. The critical theory is
σ
Now let us examine the universal properties of the fi-
∂ 2 nitedensityµ=0theoryinthevicinityofthefixedpoint
Scf =Z dτddx(ψσ†(cid:18)∂τ − 2∇m(cid:19)ψσ +u0ψ1†ψ2†ψ2ψ1).(6) soefpEaqra.t(e2ly).inWte6hewfiollllcoownisnigdesrubthseecctiaosness.d < 2 and d > 2
The single particle self-energy is identically zero at
T = 0 because the vacuum state contains no particles.
A. d<2
Consequently, we can perform an RG transformation
with the exact dynamic critical exponent z = 2, and
obtain the exact scaling dimension The fixed point of Eq. (2) has repulsive interactions
with u =0.
∗
dim[ψ]=dim[ψ ]=d/2. (7) The propertiesofthe Bosegasatthis fixedpointwere
σ
alreadyconsideredinRefs.5,11. Asanexample,wenote
Despite the trivial nature of the vacuum at T = 0, thegrandcanonicalfreeenergydensityatT =0,i.e. the
there isa non-trivialRGtransformationofthe twobody pressure P. From Eq. (4), we deduce that this obeys
interaction u . This comes entirely from the ladder di-
0
agram of repeated two-particle scattering. Defining a P =Cbµ(2mµ)d/2 (14)
d
5
where Cb is a universalnumber. A (2 d) expansionfor ofEq.(2). Inadditiontothe‘elementary’particles,ψ ,
Cb was pdresented in Ref. 5, and the le−ading result is5 wewillalsoallowfora‘composite’molecule,Φ. Forg1e,n2-
d
erality,wewillnowallowthemassesoftheψ particlesto
σ
1 2ln2 1
Cb =S + − + (2 d) . (15) bepossiblydistinct,m1 andm2. ByGalileaninvariance,
d d 4(2 d) 16 O − the mass of the ‘composite’ particle is m + m . The
(cid:18) − (cid:19) 1 2
statisticsofthetwoψ canbearbitrary—theycanbebe
The corresponding universal properties of the two- σ
Fermi-Fermi, Bose-Fermi32, or Bose-Bose33. The ‘com-
component Fermi gas at the repulsive u = u are also
∗ posite’ particle Φ is then, naturally, a boson, fermion, or
easy to work out. For simplicity, we present results only
a boson respectively.
at T = 0 and h = 0. Then, the result (14) still applies,
Our analysis will find a ‘critical’ fixed point describ-
butwitha differentuniversalconstantCf. This iseasily
d ing the unitarity limit theory. All exponents and scaling
computed from the perturbation theory for the pressure
dimensions associated with the fixed point will turn out
of the two-component Fermi gas to the first order in u :
0 to be identicalto those obtained for a seemingly distinct
√2mµ ddk k2 fixedpointinSectionII,suggestingthatthetwotheories
P = 2 µ are in fact identical.
(2π)d 2m −
Z0 (cid:18) (cid:19) AsinSectionII,itisusefultobeginwithanRGanaly-
2
√2mµ ddk sisofthezerodensity‘critical’theory,whichispresented
+u + (u2) (16)
0"Z0 (2π)d# O 0 icnalSpeoctiniotnarIeIIcAo.nsTidheerepderitnuSrbeacttiioonnsIIaIwBa.y from the criti-
The universal result is obtained by evaluating (16) at
u=u consistently to order (2 d).
∗
−
A. Zero density critical theory
1 (2 d)
Cf =S − + (2 d)2 . (17)
d d 2 − 8 O −
(cid:18) (cid:19) The action of the critical theory is now
Ofcourse,inboththeBoseandFermicasesconsidered
in this subsection, we can obtain exact results in d = 1.
∂ 2
Tthheefrtehee,rsmpoindlyesnsa,mFiecrsmoifgabso.thNomteodthelastaerveeneqtuhievathleenrtmtoo- Sc = Z dτddx(ψσ† (cid:18)∂τ − 2∇mσ(cid:19)ψσ
dynamicsofthedilutetwo-componentFermigasisgiven ∂ 2
by the spinless Fermi gas. From this, we easily obtain +Φ† ∇ +r0c Φ
∂τ − 2(m +m )
(cid:18) 1 2 (cid:19)
2
Cb =Cf = (18)
1 1 3π −g0 Φ†ψ1ψ2+ψ2†ψ1†Φ +...) (19)
(cid:16) (cid:17)
B. d>2 Here r is a cutoff-dependent bare “mass” term which
0c
hastobetunedtoplacethetheoryatitscriticalpointi.e.
ThefixedpointofEq.(2)isnowatu = 2(d 2)<0. describe the unitarity limit scattering. In dimensional
∗
The Bose gas with such attractive inter−action−s is un- regularization,r0c =0. Atree-levelscalinganalysiswith
stable, and further interactions are needed to stabilize z =2 shows that
thetheory: wewillthereforenotconsideritfurtherhere.
The Fermi gas is stable and has universal properties dim[ψ ]=d/2 ; dim[Φ]=d/2 ; dim[g ]=(4 d)/2
σ 0
−
that can be computed for small (d 2). Indeed, the (20)
−
expansion described in Section IIA applies unchanged All other interactions are irrelevant for d >2. The scal-
alsoford>2. Inparticular,thepressureisstillgivenby ingdimensionofg naturallysuggeststhatwe performa
0
Eq. (17), and this agrees with a recent result27 obtained renormalizationgroupanalysisasanexpansionin(4 d).
−
while this paper was in preparation. Of course, as noted ItturnsoutthatitispossibletoperformanRGanaly-
inSection I,this expansionknowsnothing aboutpairing sis exactly, to allordersin(4 d). Mostof the Feynman
between the fermions,as sucheffects are27 exponentially diagramsvanishbecause of th−e zero density of the parti-
small in 1/(2 d). cles. Theonlynon-zerorenormalizationisawavefunction
−
renormalization of the composite particle Φ. The wave-
functionrenormalizationiscomputedfromtheΦGreen’s
III. RG FOR A FIELD THEORY OF ATOMS function
AND MOLECULES
1
G (k,ω)=
Thissectionwillusethepopular“two-channel”formu- Φ iω+k2/(2(m +m ))+r Σ (k,ω)
1 2 0c Φ
lation of the Feshbach resonance14,24 to obtain an alter- − − (21)
native field theoretic description of universal fixed point For the above theory, at T =0, the exact expression for
6
the self energy is the Feshbachresonance. The attractiveflow towardsthe
g = g fixed point above corresponds to the (k2) cor-
∗
dΩ ddp 1 rections to the scattering amplitude we discusOsed below
Σ (k,ω) = g2
Φ 0 2π (2π)d( iΩ+p2/(2m )) Eq. (1): the system can be assumed to be exactly at
Z Z − 2
1 g = g∗ for a broad Feshbach resonance, while the irrel-
evant flows towards g = g have to be included for a
× ( i(ω Ω)+(k p)2/(2m )) ∗
− − − 1 narrowresonance. Asstatedearlier,werestrictourcom-
= g2Γ(1−d/2) 2m1m2 d/2 putations here to the broad resonance.
0 (4π)d/2 (m +m ) The criticalexponents areto be evaluatedatthe fixed
(cid:20) 1 2 (cid:21)
point of Eq. (26). From this, we obtain the exact results
k2 d/2−1
iω+ . (22)
×(cid:20)− 2(m1+m2)(cid:21) η =η(g∗)=(4−d) ; dim[Φ]=(d+η)/2=2. (28)
In the last equation, we have dropped a constant cut-
This (along with Galilean invariance)implies that the Φ
souffbdtreapcetniodnesntatreernmecwehssiachrycafonrcedls>ag4a)i.nsWter0nco(wadddeitfiinoenaal Green’s function behaves like G−Φ1 ∼[−iω+k2/(2(m1+
m ))]1 η/2, which is, of course, the result already ob-
2 −
renormalized coupling g by
tained in Eq. (22).
At this point, the above RG fixed point appears to
g =κ(4 d)/2 2m1m2 −d/4 1 g. (23) bear little direct relation to the fixed point discussed in
0 −
(cid:20)(m1+m2)(cid:21) √ZΦSd Section II. In the following subsection we show that the
spectrum of eigenoperators of the two fixed points are
whereZ allowsforanon-trivialwavefunctionrenormal-
Φ identical, indicating that they describe the same univer-
ization of the Φ field. It is easy to check that additional
sal physics.
self-energyandvertexcorrectionsvanishinthezeroden-
sity theory at T = 0, and so there is no wavefunction
renormalizationfor the ψ , nor an independent coupling
σ B. Perturbations to finite density
constant renormalization of g . In terms of g, we can
0
expand the self energy to leading order in (4 d), and
− Havingobtainedthecriticaltheory ,wecannowlook
obtain c
S
at all possible perturbations, and classify them accord-
g2 k2 ing to their renormalization group eigenvalues. It turns
Σ (k,ω)= iω+ (24)
Φ −Z (4 d) − 2(m +m ) outthatthereare3relevantperturbations,andallother
Φ − (cid:20) 1 2 (cid:21) perturbations are irrelevant. Anticipating the RG com-
Cancellingpolesintheminimalsubtractionscheme34,we putation, we can arrangethese perturbations into eigen-
obtain the exact value of Z : operators of the RG transformationas
Φ
g2
ZΦ =1− (4−d) (25) Sp = Z dτddx(−µ(ψ1†ψ1+ψ2†ψ2+2Φ†Φ)
Using Eq.(23), we can compute the exactβ function for
g: +δΦ†Φ−h(ψ1†ψ1−ψ2†ψ2)) (29)
dg (4 d) g3
= − g . (26) We can interpret the three relevant perturbations as the
dℓ 2 − 2
chemical potential, µ, a measure of the detuning away
Also,fromEq.(25), weobtainforthe anomalousdimen- from unitary scattering, δ, and the ‘magnetic’ field, h,
sion of the composite field Φ: which breaks the symmetry between ψ1 and ψ2. It is a
simple matter to relate δ to the detuning ν in Eq. (1).
η(g)=g2. (27) The T matrix for scattering between ψ and ψ is equal
1 2
to g2G . We havetoevaluate this on-shell,for incoming
0 Φ
For d < 4, the RG flow equation Eq. (26) has an particles with momenta k and k , which then requires
1 2
infrared stable fixed point at g∗ = (4−d). At first Gφ(k = k1 +k2,ω = −i(k12/(2m1)+k22/(2m2))) in the
sight, this may seem puzzling because in Section II we presence of a non-zero δ. Using Eq. (22) in d = 3, and
p
describedtheFeshbachresonancebyaninfraredunstable comparing the result with Eq. (1), we conclude
fixed point. However,this is just a matter of our restric-
tion here to a particularcriticalmanifold. We will see in 2π(m +m ) δ
1 2
ν = (30)
SectionIIIB,whereweconsiderthefullparameterspace m m g2
1 2 0
of the theory, that there is indeed an additional relevant
operator here which corresponds to the relevant pertur- TheRGflowequationsfortheµ,ν,hperturbationsare
bation of Section II, and this is the detuning away from computedusingthestandardoperatorinsertionmethods.
7
From this we find the exact flow equations whichisthegeneralizationoftheantisymmetric(4 d)αβ
−
tensorforSU(2)=Sp(2). Itiseasytoworkouttheexact
∼
dµ
β function for this case, and we find
= 2µ
dℓ
dν
= (2 η(g))ν dg (4 d)
dℓ − = − g Ng3 (34)
dℓ 2 −
dh
= 2h (31)
dℓ
Note that the fixed point value of g = g is of order
∗
Using the fixed point value of η in Eq. (28), we see that 1/N. This implies that a 1/N expansion is possible at
the fixed point values of these scaling dimensions are any value of (4 d).
−
identical to those of the three relevantperturbations ob- Formally, the 1/N expansion is obtained by dropping
tained in Section II for a field theory of atoms alone. thebarequadratictermsinthecriticaltheoryforΦ. This
Universalcorrelationsofthefinitedensityquantumliq- is because, as we have seen in Eq, (22), the self energy
uid can now be computed as an expansion in the non- contributionsto the Φ Green’sfunctions aremore singu-
linear coupling g = g∗. Because g∗2 is of order (4 d), lar than the bare terms, and dominate in the universal
−
thisleadstoanexpansioninpowersof(4 d). Indeedre- scaling limit.
−
sultsfromsuchanexpansionhavealreadybeenpresented
It is now possible to explicitly write down the theory
in Ref. 25.
which generates the universal properties to all orders in
As we noted in Section I, the resulting (4 d) expan-
− 1/N. Henceforth, all momentum integrals are implicitly
sion does have some limitations. All fluctuation correc-
evaluated in dimensional regularization, and so we set
tions to the bare action + are of order(4 d), and
areconsequentlyunableStoc chSapngethemean-fiel−dground r0c = 0 at the outset. We use the action Sc +Sp in its
Sp(2N) invariant form, rescale Φ Φ/g , and finally
state obtained when µ, ν, and h are of order unity. This → 0
integrate the ψ fermions out completely. This yields
has the consequence that a direct expansion in powers α
a theory for the single complex field Φ alone, with the
of (4 d) is only able to describe the fully gapped su-
− effective action
perfluid, or the normal ground state with a single Fermi
surface. States with 2 Fermi surfaces, or normal states
atT >0areonlyaccessedafteraredefinitionofparame- δ
= N ddxdτ Φ(x,τ)2
mteertshcoadlesin25t,r2o7d,uacneddairnetmhoerfeoclloonwvienngiesnetcltyioanc.cessedbythe SΦ (g02 Z | |
∂ 2
Trln h+ ∇ µ τ3
− ∂τ − −2m −
(cid:18) (cid:19)
IV. 1/N EXPANSION FOR THE 2N h
COMPONENT FERMI GAS +Re[Φ]τ1+Im[Φ]τ2 (35)
)
i
Following, Ref. 30, we consider fermions ψ with α =
α
1...2N whichhaveanSp(2N)invariantinteractionwith Here we have rescaled δ Nδ, and τ1,2,3 are the 2 2
the boson Φ. The critical theory for this case is (for → ×
matrices in Nambu space. With the explicit factor of N
simplicity we take the fermions to have equal mass m):
in front of the effective action for Φ, a 1/N expansion
can be generated for all physical properties.
∂ 2
Sc = Z dτddx(ψσ†(cid:18)∂τ − 2∇m(cid:19)ψσ froTmhethaebouvseuaalppflruocatcuhatmioanys-anbootuat-pBpCeaSr tvheeroyrydiofffertehnet
∂ 2 paired Fermi gas. However, our formal key observation,
+Φ† ∇ +r0c Φ which follows from the existence of the RG fixed point
∂τ − 4m
(cid:18) (cid:19) described in Section II and III, is that upon evaluation
− g20 Φ†Jαβψαψβ +H.c. ) (32) otofmalaltiincatellgyraolbstuanindetrhedismcaelninsigonfuanlcrteigounlsaraiszsaotcioiant,ewdewaituh-
(cid:0) (cid:1) crossoversacross the Feshbach resonance.
where αβ is the Sp(2N) invariant tensor The effective action (35) leads immediately to an ex-
J
plicit N = result for the grand potential which obeys
1 the univers∞al scaling form in Eq. (4). In principle, this
1 result can be obtained for an arbitrary spatial depen-
− 
1 dence of Φ, which would then allow for the spatially
= 1  (33) modulated ‘FFLO’ phases (magnetized superfluids with
J  − 
 ...  CbyoothpeerZpeaeimrsaant-sapfilintiFteermmoimweanvteu-mvecktFo↑rs−kkF↓adnedtekrmin)3e5d;
 ...  this extension is discussed briefly in AppFe↑ndix AF.↓We
 
 
8
restrictourattentionhere toaconstantΦ, inwhichcase where the dimensionally regularized term can be evalu-
ated directly in d = 3. It can now be checked that the
(0) mν d3p above expression is independent of Λ, and is most easily
F = Φ2
N 4π | | − (2π)3" evaluatedatΛ=0. Theprocedureabovealsoshowsthat
Z the subtractionis only needed for the Φ2 term, but not
T ln 1+e (√(p2/(2m) µ)2+Φ2 h)/T (36) for any of the higher powers in Φ. | |
− − | | −
+Tln(cid:16)1+e (√(p2/(2m) µ)2+Φ2+h)/T(cid:17)
− − | | The phase diagram is determined by minimizing
(cid:16) (cid:17) Eq. (37) with respect to Φ. The resulting free energy
p2 2 p2
+ µ + Φ2 µ . obeysEq.(4),andsoyieldsauniversalphasediagramas
s(cid:18)2m − (cid:19) | | −(cid:18)2m − (cid:19)# afunctionofµ,h,ν andT. Wewillpresentresultsforas-
pects of this phase diagram,along with 1/N corrections,
The superscript on denotes that expression above is in the subsections below.
F
divergentinthe ultraviolet. As discussedabove,the mo-
mentum integralhas to be evaluated in dimensional reg-
ularization, by a process of analytic continuation from
a convergent result for d < 2 to d > 2. We do this
by adding and subtracting mΦ2/(p2 + Λ2) from the
| |
integrand, where Λ is an arbitrary momentum scale.
Then, the subtracting integrand is explicitly convergent
ford=3,while the compensatingtermcanbe evaluated
analytically by dimensional regularization. This yields
A. Superconductor to normal transition with
mν mΛΓ(1 d/2) d3p increasing T
F = − Φ2
N (cid:18)4π − (4π)d/2 (cid:19)| | −Z (2π)3"
Tln 1+e (√(p2/(2m) µ)2+Φ2 h)/T (37)
− − | | −
At sufficiently high T, it is alwaysthe case that Φ =
+T ln(cid:16)1+e−(√(p2/(2m)−µ)2+|Φ|2+h)/T(cid:17) 0. In this case, we can expand SΦ about Φ = 0 fhorian
arbitraryspatialandtemporaldependence ofΦ. Tofirst
(cid:16)p2 2 p2 (cid:17) mΦ2 order in 1/N, we need to expand the action for Φ to
+ µ + Φ2 µ | | . fourth order:
s(cid:18)2m − (cid:19) | | −(cid:18)2m − (cid:19)− p2+Λ2#
ddk
Φ
SN = T (2π)dK2(k,ωn)Φ†(k,ωn)Φ(k,ωn)
Xωn Z
1 3 ddk
i
+ 2 T (2π)d K4(ki,ωin)Φ†(k1,ω1n)Φ†(k2,ω2n)Φ(k3,ω3n)
!
iY=1 Xωin Z
Φ(ek +k k ,ω +ω ω ) (38)
1 2 3 1n 2n 3n
× − −
Here
mν ddp 1 f[(p+k)2/(2m) µ h] f[p2/(2m) µ+h] m
K (k,ω )= − − − − − (39)
2 n 4π − (2π)d iω +p2/(2m)+(p+k)2/(2m) 2µ − p2
Z (cid:18) − n − (cid:19)
where f(ǫ) = 1/(eǫ/T +1) is the Fermi function, and we have performed the dimensional regularization with the
subtraction of the m/p2 term. Also, we will only need K when two of its arguments have zero momenta and
4
frequency, in which case it equals K (k,ω ), given by
4 n
e
1 ddp
K (k,ω ) = I (k p)2/(2m) µ h,p2/(2m) µ+h,ω
4 n 2 (2π)d − − − − n
Z (cid:20)
(cid:0) (cid:1)
+I (k p)2/(2m) µ+h,p2/(2m) µ h,ω (40)
n
− − − −
(cid:21)
(cid:0) (cid:1)
9
where
tanh(ǫ /(2T))+tanh(ǫ /(2T))
1 2
I(ǫ ,ǫ ,ω ) =
1 2 n 4ǫ (ǫ +ǫ iω )2
2 1 2 n
−
tanh(ǫ /(2T))+tanh(ǫ /(2T)) (ǫ /T)sech2(ǫ /(2T))
1 2 2 2
+ −
8ǫ2(ǫ +ǫ iω )
2 1 2− n
tanh(ǫ /(2T)) tanh(ǫ /(2T))
1 2
− (41)
− 8ǫ2(ǫ ǫ iω )
2 1− 2− n
Now we lowerthe temperature to critical temperature were obtained explicitly. The summation over these
T = T at which there is an onset of superconductivity. leading terms was performed analytically, along with
c
To order 1/N, the condition for T is given by the required time-splitting convergence factors required
c
for canonical bosons. After subtraction of these lead-
2T ddk K (k,ω ) ing terms, the summationoverthe remaining subleading
4 n
K (0,0)+ =0 (42)
2 N (2π)dK (k,ω ) termswasperformednumerically,andconvergedrapidly.
Xωn Z 2 n UsingthefirsttworesultsinEq.(44)andthecondition
Eq. (42) we obtain the critical chemical potential:
We will restrict our analysis of the consequences of
Eq. (42) to the balanced case at unitarity, h =0, ν =0,
µ 2.785
for which precise Monte Carlo results are also available. =1.50448+ + (1/N2). (45)
Thiswillallowustotesttheaccuracyofthe 1/N expan- T T=Tc N O
(cid:12)
sion. The total(cid:12)(cid:12)density of fermions, ρ, is givento order 1/N
Wetabulatetheresultsofsomeintegralsneededforthe by
1/N computation. At N = , the transitionis obtained
∞
by the condition ρ d3k k2 T d
= 2f µ ln[K (k,ω )]
K2(0,0)=0 µ =1.504476695 (43) N Z 8π3 " (cid:18)2m − (cid:19)− N Xωn dµ 2 n #
⇒ T T=Tc (46)
(cid:12) As is conventional, we relate ρ to a “Fermi energy”, ε ,
(cid:12) F
All numericalvalues tabulate(cid:12)d beloware for the value of by
µ in Eq. (43) and for h=0, ν =0.
(3π2ρ/N)2/3
dK2(0,0) = 0.018671(2m)3/2T 1/2 εF = 2m (47)
−
− dµ
d3kK (k,ω ) Using the third, fourth, and fifth equations in Eq. (44),
T 4 n = 0.0263(2m)3/2T1/2 the value of µ in Eq. (45) and the relations Eq. (46) and
8π3K (k,ω )
Xωn Z 2 n (47) we obtain
d3k k2
2 8π3f 2m −µ = 0.096549(2mT)3/2 εF =2.01424+ 5.317 + (1/N2) (48)
Z (cid:18) (cid:19) T T=Tc N O
2 d d3kf k2 µ = 0.056179(2m)3/2T1/2 Finally,(cid:12)(cid:12)(cid:12)the pressure P is the negative of the grand
dµ 8π3 2m −
Z (cid:18) (cid:19) potential, and so at order 1/N
d3k d
T ln[K (k,ω )] = 0.2258(2mT)3/2
− Xωn Z 8π3dµ 2 n NP = T d8π3k3 2ln(1+e−(k2/2m−µ)/T)
2T d3k ln(1+e (k2/2m µ)/T) = 0.13188(2m)3/2T5/2 Z "
8π3 − − T
Z ln[K (k,ω )] . (49)
d3k − N 2 n #
−T 8π3 ln[K2(k,ωn)] = 0.1357(2m)3/2T5/2 Xωn
Xωn Z UsingthethirdandthelasttworelationsinEq.(44),the
(44) value of µ in Eq. (45), and Eq. (49), we obtain
We make a few remarks about the numerical tech- P/N 0.4046
=0.13188+ + (1/N2) (50)
niques used to obtain the results in Eq. (44). All fre- (2m)3/2T5/2 N O
quencysummationswereevaluatedontheimaginaryfre- (cid:12)T=Tc
(cid:12)
quencies. For allof the expressions,the leading terms at Although th(cid:12)(cid:12)e 1/N corrections in Eqs. (45), (48) and
large ω , such as terms of order ln(ω ), 1/ω , 1/ω3/2, (50), are quite large,direct evaluationof the expressions
n n n n
10
at N = 1 yields values that are in reasonable agree- thenegativedetuning impliesthereisamolecularbound
ment with recent Monte Carlo results36, which obtained state10 at energy ν2/m. So we can expect a non-zero
−
µ/T = 3.247, ε /T = 6.579, and P/(2m3/2T5/2) = density of particles for 2µ> ν2/m. This reasoning led
c F c c −
0.776. The leading N = contributions in the expan- tothephasediagramshowninFig.2. Wewillseeshortly
sionarethe contributiono∞ffermionloops,whilethe1/N that this threshold value of µ is indeed obtained in the
correctionscomefromthebosonloops: thedistinctphys- N = theory.
∞
ical origins of these contributions indicates that it is not For h = 0, we will limit our analysis here to phases
6
meaningfulto comparetheir valuesas a testof the accu- with Φ spatially uniform in the N = theory. At
h i ∞
racyofthe expansion. Allsubsequent1/N contributions N = and T = 0, the free energy density in Eq. (37)
∞
are corrections to either the boson or fermion loops, and at Λ=0 reduces to:
theagreementwiththeMonteCarloresultsisanencour-
mν d3p p2 2
agingsignalthatsuchhigherordercorrectionsareindeed F = Φ2 h µ + Φ2
small. N 4π | | − (2π)3" −s 2m − | | 
Z (cid:18) (cid:19)
 
p2 2
1. Relationship to other work θ h µ + Φ2 (51)
×  −s 2m − | | 
(cid:18) (cid:19)
The structure of our leading 1/N corrections bears  p2 2 p2 mΦ2
somesimilaritytotherecentworkofHaussmannet al37. + µ + Φ2 µ | | ,
Theyconsiderthesinglefermionloop(ourN = terms) s(cid:18)2m − (cid:19) | | −(cid:18)2m − (cid:19)− p2 #
∞
and the single boson loop (our 1/N corrections) at an
where h > 0 was assumed without loss of generality. At
equal footing. Thus their approach can be viewed as
h = 0, we can check for the transition in Fig. 2 by ex-
effectively similar to a self-consistent 1/N computation. panding Eq. (51) to order Φ2 for µ<0. This yields
However,their approachbreaksdown near T , and leads | |
c
to unphysical results. Near T , the value of K (0,0) ap- mν d3p 1 m
c 2 F = Φ2 +...
proaches 0, and is also allowed to become negative in N | | 4π − 8π3 2(p2/(2m)+ µ) − p2
a self-consistent approach. Indeed, a direct attempt to (cid:20) Z (cid:18) | | (cid:19)(cid:21)(52)
obtain Tc by solving Eq. (42) self-consistently for arbi- Evaluating the integral, we observe that the co-efficient
trary N leads immediately to this difficulty: we obtain of Φ2 changes sign at µ= ν2/(2m), thus establishing
K2(0,0) < 0 already for T > Tc, and the fluctuation the| lo|cation of the phase bo−undary in Fig. 2 at N = .
propagatorin the secondterm has a negative ‘mass’and It is also easy to see that there are no corrections∞to
so is obtained by performing a functional integral with the positionof this phase boundary to allordersin 1/N:
an unphysical inverted Gaussian weight. However, orga- the Φ propagator has spectral weight only at positive
nizingtheperturbationtheoryorder-by-orderin1/N (as frequencies at the phase boundary (including a pole at
we have done) avoids this difficulty, and leads to a well- ω = k2/(4m)), and this implies that all loop corrections
defined andcontrolledexpansionfor the thermodynamic vanish.
properties at Tc, with all boson loop propagatorsalways Let us now turn to h = 0. The superfluid phase in
obeying Re[K2(k,ωn)] 0. Fig. 2 has a gap, equal t6o Φ (‘spin’ excitations), and
≥ | |
consequently a non-zero h Φ has no effect on the
| | ≤ | |
superfluid phase. The density of the particles remains
B. Phase diagram at zero temperature balanced in this regime, and depends only upon the val-
ues of µ and ν.
We will now present some results on the universal We show a finite h phase diagram in Fig. 3, obtained
phase diagram at T = 0. This is determined by 3 by determining the global minimum of the N = free
∞
parameters: µ, ν, and h. However, these parameters energy in Eq. (51) with respect to variations in a space-
are dimensionful, and the phase diagram can only de- independent Φ — a remarkable amount of information
pend only upon two independent ratiosofthese parame- emerges from the minimization of this single function.
ters whichhavezeroscalingandengineeringdimensions. Thestructureofthisuniversalphasediagramhassimilar-
FromEq.(4), wecandeduce thatthe phase diagramde- itiestothatobtainedintheresonance-widthexpansion14.
pends upon the ratios of µ, h, and ν2/(2m). The sign Becausethe physicsisinvariantunderthe changeofsign
of h is immaterial, while those of µ and ν are certainly of h, we plot the phases as a function of the dimension-
significant, and one or more of these parameters may be less parameters µ/h and ν/ 2mh In addition to the
| | | |
zero;consequentlywecannotexplorethecompletephase conventional gapped superfluid state with h < Φ and
p | | | |
diagram in a single two-dimensional plot. balanced densities, there is also a more exotic superfluid
First, let us begin with some generalconsiderationsat phaseinthephasediagram. TheLuttinger theoremsde-
h = 0. As long as the detuning, ν, is positive, it will scribed in Ref. 38, require that any ground state with
pay to have particles in the ground state of the grand anunbalanceddensity must have atleast one Fermi sur-
canonical ensemble only for µ > 0. However, for ν < 0, face (as long as the ‘spin’ rotation symmetry about the