Single-particle spectral density of the unitary Fermi gas: Novel approach based on the operator product expansion, sum rules and the maximum entropy method

0.52 MB·English

by  Philipp Gubler

RIKEN-QHP-177
Single-particle spectral density of the unitary Fermi gas:
Novel approach based on the operator product expansion,
sum rules and the maximum entropy method
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1 Philipp Gubler,1,2, Naoki Yamamoto,3 Tetsuo Hatsuda,2,4 and Yusuke Nishida5
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2
1ECT*, Villa Tambosi, 38123 Villazzano (Trento), Italy
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A 2RIKEN Nishina Center, Wako, Saitama 351-0198, Japan
9 3Department of Physics, Keio University, Yokohama 223-8522, Japan

] 4RIKEN iTHES Research Group, Wako, Saitama 351-0198, Japan
s
a
g 5Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan
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t
n (Dated: April10,2015)
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u
Abstract
q
.
at Making use of the operator product expansion, we derive a general class of sum rules for the imaginary
m
partofthesingle-particleself-energyoftheunitaryFermigas. Thesumrulesareanalyzednumericallywith
-
d
n thehelpofthemaximumentropymethod,whichallowsustoextractthesingle-particle spectraldensityasa
o
c functionofbothenergyandmomentum. Thesespectraldensitiescontainbasicinformationontheproperties
[

2 oftheunitaryFermigas,suchasthedispersion relationandthesuperfluidpairinggap,forwhichweobtain
v
3 reasonable agreement withtheavailable resultsbasedonquantum Monte-Carlo simulations.
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[e-mail]
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CONTENTS
I. Introduction 3
II. Formalism 7
A. Theoperatorproduct expansion 7
B. TheOPE ofthesingle-particleGreen’sfunctionforgeneral valuesofa 8
C. Three-body scatteringamplitude 9
D. TheOPE ofthesingle-particleGreen’s functionintheunitarylimit 10
E. Derivationofthesumrules 12
F. Choice ofthekernel (w ) 14
K
III. MEManalysisforthespectraldensity 18
A. TheBorel windowand thedefaultmodel 18
B. Thesingle-particlespectral density 19
IV. Summaryand conclusion 23
reg
AppendixA. Numerical solutionofT (k,0;k,0)intheunitarylimit 25
↑
AppendixB. Derivationofthesumrules foragenerickernel 27
AppendixC. Finiteenergy sumrules fortheunitaryFermi gas 34
AppendixD. Themaximumentropymethod 40
References 42
2
I. INTRODUCTION
The unitary Fermi gas, consisting of non-relativistic fermionic particles of two species with
equalmass,hasbeenstudiedintensivelyduringthelastdecade[1–3]. Thegrowinginterestinthis
systemwaspromptedespeciallybytheabilityoftuningtheinteractionbetweendifferentfermionic
species in ultracold atomic gases through a Feshbach resonance by varying an external magnetic
field. This technique allows one to bring the two-body scattering length of the two species to
infinity and therefore makes it possible to study the unitary Fermi gas experimentally. Using
photoemission spectroscopy, the measurement of the elementary excitations of ultracold atomic
gases has in recent years become a realistic possibility [4, 5]. Understanding these elementary
excitationsfromatheoreticalpointofviewishenceimportantandanumberofstudiesdevotedto
thistopichavealreadybeencarriedout[6–8]. Wewillinthisworkproposeanewandindependent
method for computing the single-particle spectral density of the unitary Fermi gas, which makes
useoftheoperatorproductexpansion(OPE).
TheOPE,whichwasoriginallyproposedinthelatesixtiesindependentlybyWilson,Kadanoff
and Polyakov [9–11], has proven to be a powerful tool for analyzing processes related to QCD
(Quantum Chromo Dynamics),for which simpleperturbation theory fails in mostcases. The rea-
son for this is the ability of the OPE to incorporate non-perturbative effects into the analysis as
expectation values of a series of operators, which are ordered according to their scaling dimen-
sions. Perturbative effects can on the other hand be treated as coefficients of these operators (the
“Wilson-coefficients”). TheOPE hasspecificallybeenusedtostudydeepinelasticscatteringpro-
cesses[12]andhasespeciallyplayedakeyroleintheformulationoftheso-calledQCDsumrules
[13, 14].
In recent years, it was noted that the OPE can also be applied to strongly coupled non-
relativisticsystemssuch as theunitary Fermi gas [15–27]. Initially,theOPE was used to rederive
some of the Tan-relations [28–30] in a natural way [15] and, for instance, to study the dynamic
structure factor of unitary fermions in the large energy and momentum limit [20, 23]. Further-
more, the OPE for the single-particle Green’s function of the unitary Fermi gas was computed
by one of the present authors [25] up to operators of momentum dimension 5, from which the
single-particle dispersion relation was extracted. As the OPE is an expansion at small distances
and times (or large momenta and energies), the result of such an analysis can be expected to
give the correct behavior in the large momentum limit and is bound to become invalid at small
3
momenta. Theanalysis of[25]confirmed this,but inadditionsomewhatsurprisinglyshowedthat
the OPE is valid for momenta as small as the Fermi momentum k , where the OPE still shows
F
goodagreement withtheresultsobtainedfromquantumMonte-Carlosimulations[7].
The purpose of this paper is to extend this analysis to smaller momenta, by making use of the
techniques of QCD sum rules, which have traditionally been employed to study hadronic spectra
from theOPE applied toGreen’s functionsin QCD. Ourgeneral strategygoesas follows:
Step 1: Construct OPE
•
At first, we need to obtain the OPE for the single-particle Green’s function (k ,k) in
0
G↑
the unitary limit, which can be rewritten as an expansion of the single-particle self-energy
S (k ,k). The subscript here represents the spin-up fermions. The main work of this
0
↑ ↑
step has already been carried out in [25]. S (k ,k) can be considered to be an analytic
0
↑
functiononthecomplexplaneoftheenergy variablek ,withtheexceptionofpossiblecuts
0
and poles on the real axis. Considering the OPE at T = 0, with equal densities for both
fermionic species (n =n ) and taking into account operators up to momentum dimension
↑ ↓
5, the only parameters appearing in the OPE are the Bertsch parameter and the contact
density, which are by now well known from both experimental measurements [31–33] and
theoreticalquantumMonte-Carlo calculations[34, 35].
Step 2: Derivesumrules
•
FromthefactthattheOPEisvalidatlarge k andtheanalyticpropertiesoftheself-energy,
0
| |
a general class of sum rules for ImS (w ,k) can be derived. In contrast to the complex k ,
0
↑
w hereisarealparameter. Thesesumrulesarerelationsbetweencertainweightedintegrals
ofImS (w ,k)andcorrespondinganalyticalexpressionsthatcanbeobtainedfromtheOPE
↑
result(for detailsseeSection II):
¥
DOPE(M,k)= dw (w ,M)ImS (w ,k). (1)
↑ ¥ K ↑
Z−
The kernel (w ,M) here must be an analytic function that is real on the real axis of w
K
and falls off to zero quickly enough at w +¥ , while M is some general parameter that
→
characterizes the form of the kernel. In the practical calculations of this paper, we will use
theso-calledBorel kernelsoftheform (w ,M)=(w /M)ne w 2/M2.
n −
K
Step 3: Extract ImS (w ,k) via MEM and obtain ReS (w ,k) from the Kramers-Kro¨nig
• ↑ ↑
relation
4
As a next step, we use the maximum entropy method (MEM) to extract the most probable
form of ImS (w ,k) from the sum rules, following an approach proposed in [36] for the
↑
QCD sumrulecase.
It should be mentioned here that this method is somewhat different from the analysis pro-
cedure most commonly employed in QCD sum rule studies, where the spectral function
(which corresponds to ImS here) is parametrized using a simple functional ansatz with a
↑
small number of parameters which are then fitted to the sum rules. This method has tradi-
tionallyworkedwellifsomesortofpriorknowledgeonthespectralfunctionisavailableand
assumptions on its form can thus be justified. On the other hand, in cases where one does
not really know what specific form the spectral function can be expected to have, sum rule
analyses based on (potentially incorrect) assumptions on the spectral shape always involve
thedangerofgivingambiguousandevenmisleadingresults. MEMisthereforeourmethod
of choice, as it allows us to analyze the sum rules without making any strong assumption
onthefunctionalform ofthespectralfunctionandhencemakes itpossibletopickthemost
probablespectralshapeamongan infinitelylargenumberofchoices.
OnceImS (w ,k) is obtainedfromtheMEManalysisofthesumrules, itis asimplematter
↑
tocomputeReS (w ,k)by theKramers-Kro¨nig relation,
↑
1 ¥ ImS (w ,k)
ReS (w ,k)= p P dw ′ w ↑ w′ . (2)
↑ − Z−¥ − ′
Step 4: Computesingle-particlespectraldensity
•
Fromthereal andimaginarypartsoftheself-energy,thesingle-particlespectraldensitycan
then beobtainedas,
1 1
A (w ,k)= Im , (3)
↑ −p w +i0+ e k S (w +i0+,k)
− − ↑
wheree isdefined as e =k2/(2m),withmbeing thefermion mass.
k k
Theabovestepsareshownoncemoreinpictorialformin Fig. 1.
As a result of the above procedure, we find a two-peak structure in the imaginary part of the
self-energy, the two peaks moving from the origin (w =0) to positive and negative directions of
the energy with increasing momentum k . Translated to the single-particle spectral density, this
| |
leads to a typical superfluid BCS-Bogoliubov-like dispersion relation with both hole and particle
branches and anonzero gapvalue.
5
Step1: Imk0 regionofbadOPE
ConstructOPEforthe convergence
self-energyS (k ,k),which
0
isvalidatlarge k0 Rek0
| |
Imk0
Step2:
Derivesumrulesfor DOPE(M,k)=
ImS (w ,k)ontherealaxis Rek0 dw (w ,M)ImS (w ,k)
fromS OPE(k ,k)atlarge k K
0 0
| | R
MEM
ImS (w ,k)
ReS (w ,k) −
K-K
Step3:
relation
ExtractImS (w ,k)viaMEM
andobtainReS (w ,k)bythe w
Kramers-Kro¨nigrelation
w
A(w ,k)=
1 1
− p Im w ek S (w ,k)
− −
A(w ,k) (cid:2) (cid:3)
Step4:
Computethesingle-particle
spectraldensityA(w ,k)from
S (w ,k)
w
FIG.1. Stepsforcomputingthesingle-particle spectraldensityfromtheOPEofthesingle-particle Green’s
function ofafermionicoperator.
The paper is organized as follows. In Section II, we discuss the OPE of the single-particle
Green’s function and explain how it can be rewritten as an expansion of the single-particle self-
energy. Next, we outline the derivation of the sum rules from the OPE. In Section III, the MEM
analysis results of the sum rules are shown and the consequent final form of the single-particle
spectraldensityandthedispersionrelationarepresented. ThespectraldensityisvisualizedinFig.
5asadensityplotandthedetailednumericalpropertiesofthedispersionrelationaredescribedin
Table II. Section IV is devoted to the summary and conclusions of the paper. For the interested
reader, we provide in the appendices detailed accounts of the relevant calculations, which were
6
needed forthiswork.
II. FORMALISM
A. Theoperatorproductexpansion
The operator product expansion (OPE) is based on the observation that a general product of
non-localoperators can beexpandedas aseries oflocaloperators. Thiscan beexpressedas
Oi(x+12y)Oj(x−12y)=(cid:229) WOk(y)Ok(x). (4)
k
Here,wehaveusedtheabbreviations(x)=(x ,x)and(y)=(y ,y)forthefour-dimensionalvec-
0 0
tors. WO (y) aretheWilson-coefficients,which onlydependon therelativetimeand distanceyof
k
thetwooriginaloperators. Theoperatorsontheright-handsideofEq.(4)areorderedaccordingto
their scaling dimensions D , in ascending order. This expansion works for small time differences
k
D D D D D D
(or small distances), as the Wilson coefficients behave as ( y0 ) k− i− j ( y k− i− j) and be-
| | | |
causetheoperatorswithlargerscalingdimensionsarethusspuppressedbyhigherpowersof y0
| |
( y ). FouriertransformingEq.(4),theabovestatementistranslatedintoenergy-momentumsppace,
| |
where the OPE is a good approximation in the large energy or momentumlimitas operators with
largerscaling dimensionsare suppressedby higherpowersof1/ k (1/ k ).
0
| | | |
For the above expansion to work in the context of a non-relatpivisticatomic gas, certain condi-
tionshavetobesatisfied. Firstly,itisimportantthatthepotentialranger oftheatomicinteraction
0
is much smaller than all other length scales of the system, so that the detailed form of the inter-
action becomes irrelevant. Furthermore, the energy or momentum scale at which the system is
probed needs to be much larger than the corresponding typical scales of the system. Hence, for
the OPE to be a useful expansion, the following separation of scales must hold, which must be
satisfiedby either1/ k or1/ k :
0
| | | |
p r0 1/ k0 , 1/ k a ,ns−1/3,l T. (5)
≪ | | | |≪| |
p
1/3
Here, a is the s-wave scattering length between spin-up and -down fermions, n−s the mean
interparticle distance of both fermionic species, and l 1/√mT the thermal de Broglie wave
T
∼
length. In other words, k or k have to be large enough so that for example an expansion in
0
| | | |
1/(a k0 ), ns1/3/ k0pand 1/(l T k0 ) is valid, whilethey shouldbe still small enough not to
| | | | | |
probeptheactual strpuctureoftheindipvidualatoms.
7
Inpractice,wewillinthisworktakethezero-rangelimitr 0,studythesystematvanishing
0
→
temperature T =0 and will in the course of the derivation of the sum rules take the unitary limit
a ¥ . Furthermore, forstudyingthedetailed momentumdependenceofthespectral-density,we
→
will in the following discussion make use of an expansion in 1/ k , but not in 1/ k . k will
0
| | | | | |
insteadalways bekeptat theorderofFermi-momentumofthestupdiedsystem.
B. TheOPEofthesingle-particle Green’sfunctionforgeneralvaluesofa
Inthispaper, wewillemploytheOPE ofthesingle-particleGreen’sfunction,whichwas com-
puted in [25]. Let us here briefly recapitulate this result and discuss its form rewritten as an
expansionoftheself-energy S (k ,k). Thestartingpointofthecalculationis
0
↑
i
i (k) dyeiky T[y (x+ y)y †(x y)] = , (6)
G↑ ≡Z h ↑ 2 ↑ −2 i k0−e k−S ↑(k)
where k should be understood as (k) = (k ,k). The OPE for (k) can then be carried out, as
0
G↑
discussed in detail in [25]. If translational and rotational invariance holds, all sorts of currents
vanish and the OPE expression (taking into account terms up to momentum dimension 5) can be
simplifiedasfollows:
¶
OPE(k)=G(k) G2(k)A(k)n C G2(k) A(k) C G2(k)Treg(k,0;k,0)
G↑ − ↓−4p ma ¶ k0 −m2 ↑
(7)
G2(k) ¶ A(k)+m (cid:229) 3 ¶ 2 A(k) dq q2 r (q) C .
− ¶ k0 3 i=1¶ ki2 Z (2p )32m ↓ −q4
h i h i
Here, G(k)is thefree fermion propagator,
1
G(k)= , (8)
k e
0 k
−
A(k)represents thetwo-bodyscatteringamplitudebetween spin-upand-downfermions,
4p 1
A(k)= , (9)
m k2 mk 1/a
4 − 0−
q
reg
andT (k,p;k ,p)standsfortheregularizedthree-bodyscatteringamplitudeofaspin-upfermion
′ ′
↑
withinitial(final)momentumk(k )andadimerwithinitial(final)momentump(p ). “regularized”
′ ′
means that infrared divergences originally appearing in the three-body scattering amplitude have
been subtracted(see SectionsIII Cand IIIFof[25]):
dq m2
reg
T (k,0;k,0) T (k,0;k,0) A(k) . (10)
↑ ≡ ↑ − Z (2p )3 q4
8
Furthermore, r s (q) is the momentum distribution function of spin-s fermions, n the density of
↓
spin-downfermionsand theso-calledcontact density[28–30].
C
Comparing Eq.(7) with the definition of the self-energy of Eq.(6), one can easily find an ex-
pression for S (k), which (again up to terms with momentum dimension 5) is consistent with the
↑
OPE ofthesingle-particleGreen’sfunction:
¶
S OPE(k)= A(k)n C A(k) C Treg(k,0;k,0)
↑ − ↓−4p ma¶ k0 −m2 ↑
(11)
¶ A(k)+m (cid:229) 3 ¶ 2 A(k) dq q2 r (q) C .
− ¶ k0 3 i=1¶ ki2 Z (2p )32m ↓ −q4
h i h i
Assuming the considered system to be spin symmetric [r (q) = r (q)], the integral of the mo-
↑ ↓
mentum distribution function appearing in the above equation can be evaluated by one of the
Tan-relations[28–30],
(cid:229) dq q2 r s (q) C = + C , (12)
(2p )32m −q4 E 4p ma
s = , Z
↑↓ h i
where istheenergy densityofthesystem. Wehenceget,
E
¶
S OPE(k)= A(k)n C A(k) C Treg(k,0;k,0)
↑ − ↓−4p ma¶ k0 −m2 ↑
(13)
1 ¶ m (cid:229) 3 ¶ 2
A(k)+ A(k) + C .
−2 ¶ k 3 ¶ k2 E 4p ma
0 i=1 i
h i(cid:16) (cid:17)
Among the various terms appearing in Eq.(13), the most involved piece to evaluate is the three-
reg
bodyscatteringamplitudeT (k,0;k,0),whichwillbestudiednextina separatesubsection.
↑
C. Three-bodyscattering amplitude
reg
The difficulty in obtaining T (k,0;k,0) stems from the fact that this scattering amplitude by
↑
itself does not solve a closed integral equation and therefore can not be computed directly. We
thus have to use T (k,0;p,k p) with a more general momentum dependence, which will, for
↑ −
simplicityofnotation,fromnowonbedenotedasT (k;p). T (k;p)satisfiesthefollowingintegral
↑ ↑
equation(notethatweforthemomentwork withthenon-regularizedversionoftheamplitude):
dq dq
0
T (k;p)=G( p)+i T (k;q)G(q)A(k q)G(k p q)
↑ − (2p )4 ↑ − − −
Z
1
=
− p +e (14)
0 p
dq 4p T (k;e ,q)
q
−Z (2p )3 12 3q2−2q·k+k2−4mik0−1a (p+q2−k)2 +↑ mp0+q22 −mk0.
p
9
In goingtothesecondand thirdlines,theintegraloverq isperformed and thusq isfixed toe .
0 0 q
Next,setting p =e providesaclosed equation,
0 p
m dq 4p T (k;e ,q)
T↑(k;e p,p)=−p2 −Z (2p )3 12 3q2−2q·k+k2−4mk0−1a (p+q2−k)↑2 +pq2+2q2 −mk0, (15)
whichneedstobesolvednumericaplly. ThetechnicaldetailsofthissteparepresentedinAppendix
A. OncetheaboveequationissolvedandT (k;e ,p)hashencebeenobtained,onecanextractthe
p
↑
desired amplitudeT (k;k)from Eq.(14)by setting p=k:
↑
1 dq 4p T (k;e ,q)
q
T (k;k)= ↑
↑ −k0+e k −Z (2p )3 12 3q2−2q·k+k2−4mk0−1a q2
1 dq 4p m
p
= +
−k0+e k Z (2p )3 12 3q2−2q·k+k2−4mk0−1a q4 (16)
dq p 4p T (k;e q,q)+qm2
↑ .
− (2p )3 1 3q2 2q k+k2 4mk 1 q2
Z 2 − · − 0−a
p reg reg
Finally, returning to the regularized scattering amplitude T (k,0;k,0) = T (k;k) [defined in
↑ ↑
Eq.(10)], weget,
reg
T (k;k)
↑
2
dq m
=T (k;k) A(k)
↑ − (2p )3 q2
Z (cid:18) (cid:19)
1 dq 4p 4p m (17)
= +
−k0+e k Z (2p )3"12 3q2−2q·k+k2−4mk0−1a − 12 k2−4mk0−1a#q4
dq p4p T (k;e q,q)+qm2p
↑ .
− (2p )3 1 3q2 2q k+k2 4mk 1 q2
Z 2 − · − 0−a
p
D. TheOPEofthesingle-particle Green’sfunctionintheunitarylimit
So far, we have studied theOPE for arbitrary values of the s-wave scattering length a between
the two spin degrees of freedom (which should however be kept large enough for the conditions
of a valid OPE to apply). One could in principle continue with these general expressions, derive
sum rules for nonzero a 1 values and analyze them according to our strategy outlined in the
−
introduction.
In order to provide a clear account of the proposed method, we will however not do this here
butconcentrateontheunitarylimit(a ¥ ), whichconsiderablysimplifiesmanyoftheequations
→
needed to derive the sum rules, but already exhibits all non-trivial technical difficulties that will
10