Dispersions, weights, and widths of the single-particle spectral function in the normal phase of a Fermi gas

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by  F. Palestini

Dispersions, weights, and widths of the single-particle spectral function
in the normal phase of a Fermi gas
F. Palestini, A. Perali, P. Pieri, and G. C. Strinati
Dipartimento di Fisica, Universita` di Camerino, 62032 Camerino, Italy
The dispersions, weights, and widths of the peaks of the single-particle spectral function in the
presence of pair correlations, for a Fermi gas with either attractive or repulsive short-range inter-
particle interaction, are determined in the normal phase over a wide range of wave vectors, with a
2
twofold purpose. The first one is to determine how these dispersions identify both an energy scale
1
knownasthepseudo-gapneartheFermiwavevector,aswellasanadditionalenergyscalerelatedto
0
2 thecontactC atlargewavevectors. Thesecondoneistodifferentiatethebehaviorsoftherepulsive
gas from the attractive one in terms of crossing versus avoided crossing of the dispersions near the
n
Fermiwavevector. Ananalogywillalsobedrawnbetweentheoccurrenceofthepseudo-gapphysics
a
in a Fermi gas subject to pair fluctuations and the persistence of local spin waves in the normal
J
phaseof magnetic materials.
6
2
PACSnumbers: 67.85.Lm,74.72.kf,32.30.Bv

]
s
a I. INTRODUCTION ∆(k) just in these two intervals, namely, for k kF (ob-
g taining ∆pg) and k kF (obtaining ∆∞). T≈his wave-
- ≫
t Local order of short-range nature in the normal phase vector dependence arises even though the inter-particle
n
of anultra-coldFermigas abovethe superfluid tempera- interactionisofthecontacttype,whichatthemean-field
a
u ture Tc has recently been the subject of intense interest, level below Tc would instead give rise to a wave-vector
q owing to experimental and theoretical advances which independent gap.
t. have hinged on this local order from different perspec- Tothisend,weshallanalyzeindetailthedispersions of
a
tives. thepeaksofthesingle-particlespectralfunctionforvari-
m
The experimentalinterest[1, 2] has mostly focused on ouscouplingsacrosstheBCS-BECcrossoverandtemper-
d- the issue of the pseudo-gap ∆pg, which is a low-energy aturesaboveTc,andshowhowtheycanratheraccurately
n scale that in these systems evolves in temperature with be representedby BCS-like dispersionswith a character-
o continuity out of the pairing gap present in the broken- istic“back-bending”fortheoccupiedstates[1,2]. These
c symmetry (superfluid) phase [3]. On physical grounds, dispersionswillbe obtainedwithinthet-matrixapproxi-
[
this continuous evolution is due to the persistence of mationfor anattractiveinter-particleinteractionfollow-

2 “medium-range”paircorrelations,whicharetheremnant ing the approach of Ref.[7], which was recently applied
v above T of the long-range order below T . toaccountforthe experimentaldataonultra-coldFermi
3 c c
The theoretical interest has been prompted, on the gases [8]. In addition, we will show that the weights of
9
1 otherhand, by the introductionofanumber ofuniversal the two peaks ofthe single-particlespectralfunction can
6 relations due to Tan [4, 5], which are due to the inter- also be described by BCS-like expressions. Determining
6. particle interaction being of the contact type and affect theseweightswillalsobeusefultoobtaintheasymptotic
0 several physical quantities. These universal relations all value of ∆(k) for large k, where tracing the dispersions
1 dependonacoupling-andtemperature-dependentquan- maybecomeill-definedowingtothestrongbroadeningof
1 tity named the contact C, which can in turn be con- the large-k structure of the single-particle spectral func-
:
v veniently expressed in terms of a high-energy scale ∆∞ tion at negative frequencies.
i [6]. The fact that C specifies, in particular, the strength The importance of determining the weights (besides
X
of“short-range”paircorrelationsbetweenoppositespins thanmerelyfocusingontheexistenceofthepseudo-gap)
r
a implies that, in ultimate analysis, the high-energy scale isinlinewiththeemphasisthatwasgivenfromtheearly
∆∞associatedwithC andthelow-energyscale∆pgasso- days of the BCS theory of superconductivity to the role
ciatedwiththepseudo-gapbothoriginatefromthesame ofthe “coherencefactors”. Their presence,infact,made
kind of pair correlations, which remain active above T theBCStheorysoonacceptedasthecorrectone,because
c
even in the absence of long-range order. it was then possible to account for the counter-intuitive
In this paper, we aim at organizing these two energy outcomes of different experiments that could otherwise
scalesintoasinglewave-vectordependentfunction∆(k), not be understood only on the basis of the occurrence of
of which ∆ represents the value about the Fermi wave a gap in the single-particle spectrum [9, 10].
pg
vector kF and ∆∞ its behavior for k much larger than Thecrossedcheck,betweenthedispersionsandweights
k , corresponding to medium- and short-range pair cor- of the two branches of the single-particle spectral func-
F
relations, in the order. In practice, from the numerical tion, will therefore represent a fingerprint of the sur-
calculations it is meaningful to determine the values of vival in the normal phase of typical BCS-like features
2
due to strong pairing fluctuations. Differences, however, Extensive theoretical work on the pseudo-gapissue for a
between the broken-symmetry phase below T and the Fermi gas with attractive interaction was reported more
c
pseudo-gap phase above T will mostly appear in the recently in Refs.[17–22].
c
widths of the peaks of the single-particle spectral func- Finally, in the present paper a similarity will be high-
tion, which are much broader above than below T as lighted between the pseudo-gap physics resulting from
c
expected in the absence of a truly long-range order. pairing fluctuations above T and the persistence of spin
c
Inthiscontext,weshallalsoresumeanargumentthat wavesoverlimited spatialregionsin the normalphaseof
wasraisedinRef.[11],accordingto whichthe occurrence ferromagnetic (or anti-ferromagnetic)materials. Besides
of the above mentioned back-bending for k k should being of heuristic value for envisagingthe local orderas-
F
≫
notreflectperse thepresenceofapseudo-gapforkabout sociatedwiththepseudo-gap,thisanalogyevidenceshow
k . This is because the structure at large k, which is the currentdebate aboutthe occurrenceofa pseudo-gap
F
related to the contact C, can be found even in a normal in an ultra-cold Fermi gas retraces a similar debate that
gas with repulsive inter-particle interaction. wentonforsometimeaboutthepersistenceofspinwaves
Accordingly, we shall argue that the main differences, in magnetic materials.
between the features of the single-particle spectral func- The paper is organized as follows. In Section II, the
tion for a Fermi gas with repulsive or attractive interac- dispersions, weights, and widths of the peaks in the
tioninthenormalphase,appearactuallyforkaboutk . single-particlespectralfunctionfortheattractivecaseare
F
Specifically,anavoided crossing resultsinthedispersions studied in detail, to determine how the energy scale as-
of the two peaks of the single-particle spectral function sociated with the pseudo-gap about k evolves for large
F
in the attractive case, while a crossing occurs in the re- k toward the energy scale associated with the contact
pulsive case. In the attractive case, the energy spreadof C. In Section III, the single-particle spectral function
theavoidedcrossingisdirectlyrelatedtothepseudo-gap for the repulsive case is contrasted with that for the
energy scale ∆ . On physical grounds, this difference attractive case, to bring out the issue of the crossing
pg
between avoided crossing and crossing is a consequence vs avoided-crossing of the dispersion relations about k
F
of particle-hole mixing, which survives at a local level in which clearly differentiates between the two cases. In
the attractive case when passing from below to above T SectionIV, ananalogyis drawnbetweenthe pseudo-gap
c
but is absent in the repulsive case. physics and the persistence of spin waves in magnetic
The overallpurpose here, therefore, is not to establish materials,anda suggestionis made foranadditionalex-
specificcriteriafortheexistenceofapseudo-gapphasein perimental evidence for the occurrence of a pseudo-gap.
a Fermi gaswith attractive interaction. Rather, we shall TheAppendixgivesanalyticdetailsaboutthetreatment
beinterestedinframingthetotalamountofinformation, of pair fluctuations in the repulsive case to obtain the
which can be extracted from the single-particle spectral single-particle spectral function over a wide range of k.
function of an interacting Fermi gas subject to pairing
fluctuationsaboveT ,intoaunifiedpicturewhereanalo-
c
gies and differences with respect to a simple BCS-like II. THE ATTRACTIVE CASE: PSEUDO-GAP VS
CONTACT
description below T can be emphasized.
c
It is, nevertheless, relevant to provide at this point an
(albeitconcise)overviewofthemajorrelevantworkdone In this Section, we consider a homogeneous Fermi gas
previously by several groups on the issue of the pseudo- with an attractive interaction v δ(r r′) of short-range
0
−
gap,alsoto recallhowthis concepthaddevelopedinthe between opposite spin atoms with equal populations,
context of a Fermi gas with attractive interaction. The whose strengthv canbe eliminated in favorofthe scat-
0
predictionfortheexistenceofapseudo-gapinthenormal tering length a via the relation:
F
phase of stronglyinteractingultra-coldFermions wasin-
troduced in Ref.[12] within a two-channelfermion-boson m 1 k0 dk m
= + . (1)
model and in Ref.[13] within a single-channel fermion 4πa v (2π)3 k2
F 0 Z
model, before the observation of superfluidity in these
gases. Theseworkswere,inturn,basedonearlierstudies Here, m is the particle mass, k a wave vector, and k a
0
which applied the physics of the BCS-BEC crossover to wave-vector cutoff which can be let while v 0
0
→ ∞ →
the high-temperature cuprate superconductors. In that in order to keep a at a desired value (we set ¯h = 1
F
context,initialworkinterpretedthenormalstateofasu- throughout).
perfluid in the crossover regime between BCS and BEC Since v < 0, a can be positive as well as negative,
0 F
as a phase of uncorrelated pairs [14] or as a spin-gap and the dimensionless interaction parameter (k a )−1
F F
phase [15]. Later, it was shown that this phase reflects rangesfrom(k a )−1 < 1inthe weak-coupling(BCS)
F F
a normal phase pseudo-gap, which displays peculiar fea- regime,to (k a )−1 >∼+−1 in the strong-coupling(BEC)
F F
tures in the fermionic spectral function that reflect the regime, across the un∼itary limit where a diverges and
F
presence of a pairing gap in the superfluid phase [16][7]. (k a )−1 = 0. In practice, the BC|S-B|EC crossover
F F
3
region of most interest is limited to the interval 1 < 2 1
(k a )−1 <+1. − ∼
F F 1 (k a )-1=-1
In the su∼perfluid phase well below T a description of F F
theBCS-BECcrossoverresultsalreadycatthe mean-field 0 0.5
-1
level, while in the normal phase above T inclusion of
c
pairing fluctuations is required to get physically mean- -2 0
iitnnhtgeofucalhaprresaseucutldteosr.-isgtaPicpaiBarbiCnoSgveflmTuecca,tnua-asfitediolidsncseu,nsisenerdgpynaegrxtaitpc.ublaerl,owtuTrnc spersions - 101 (kFaF)-1=0 0.5 weights
di
-2 0
A. Mean-field description below T
c
2 (k a )-1=1
F F
The simplest description of the BCS-BEC crossover 0 0.5
results within mean field for temperatures T below Tc, -2
by supplementing the equation for the BCS gap ∆
-4 0
0 0.5 1 1.5 0 0.5 1 1.5 2
dk 1 2f(E ) m m
k k/k k/k
− = (2) F F
(2π)3 2E − k2 −4πa
Z (cid:18) k (cid:19) F
FIG. 1: Dispersion relations (left panels) and corresponding
with the density equation weights (right panels) for three different couplings, as ob-
tained at T =0 within mean field (dashed-dotted lines) and
n = dk f(E ) 1+ ξk at Tc with the inclusion of pairing fluctuations (diamonds).
Z (2π)3 (cid:20) k (cid:18) Ek(cid:19) Energies are in units of EF.
ξ
k
+ (1 f(E )) 1 . (3)
k
− − E
(cid:18) k(cid:19)(cid:21)
when including pairing fluctuations within the t-matrix
Here, E = ξ2+∆2 with ξ = k2/(2m) µ and
f(E) = (keE/(kBT)k+1)−1 is the Fkermi function−(µ being approximation. It is still possible to identify two peaks
p inA(k,ω)forgivenk overanextendedrangeofcoupling
the fermionic chemical potential and k the Boltzmann
B and temperature, by locating their positions and deter-
constant). Note that the mean-field gap ∆ does not de-
mining their weights and widths. It is found that the
pend on k = k owing to the short-range nature of the
| | positions of these peaks can be rather well represented
inter-particle interaction.
by a BCS-like dispersion of the form (4), provided the
Whenone looksatthe structuresofthe single-particle
mean-field ∆ is replaced by a “pseudo-gap” value ∆
pg
spectralfunctionA(k,ω)withintheBCSapproximation,
that remains finite above T . This finding was explicitly
c
at a given k two sharp peaks appear centered at the fre-
demonstrated in Ref.[7] only for the dispersion relations
quency values
nottoofarfromk andontheBCSsideofthecrossover.
F
As an example, we report in Fig. 1 the dispersion re-
ω = E (4)
k
± lations and weights of the two peaks of A(k,ω) at T for
c
with weights (1 ξ /E )/2, respectively [23]. three couplings acrossthe BCS-BEC crossover,obtained
k k
The dispersio±n relations (4) and the corresponding accordingto the t-matrix approximation(diamonds) [7].
weights are shown in Fig. 1 for three characteristic cou- In all cases, the similarity with the corresponding values
plingsacrosstheBCS-BECcrossoveratzerotemperature obtained for these quantities within mean field at T =0
(dashed-dottedlines). Here, E =k2/(2m)is the Fermi (dashed-dotted lines) appears striking.
F F
energy with k =(3π2n)1/3. Marked differences appear instead for the widths of
F
the peaks of A(k,ω), when passing from the mean-field
B. Pairing fluctuations above T description below T where they are delta-like, to the
c c
t-matrix description (at and) above T where they are
c
The abovepicture gets somewhatmodified whenpair- broad and overlapping. This is shown in Fig. 2, where
ing fluctuations beyond mean field are considered below A(k,ω) is plotted versusω at unitarity for the wavevec-
T [24]. It is, however,above T that inclusion of pairing tor where the maximum of the lower dispersion relation
c c
fluctuationsaltersmostlythebehaviorofA(k,ω)fromits occurs (that is, 0.76k within mean field at T = 0 and
F
trivial mean-field description with ∆ = 0, whereby only 0.91k with the inclusion of pairing fluctuations at T ).
F c
a single sharp peak of unit weight survives consistently ThispicturealsoevidenceshowarealgapatT =0trans-
with a Fermi-liquid picture [9]. forms into a pseudo-gap at T , through a partial filling
c
InthecontextoftheBCS-BECcrossover,anon-trivial of the spectral function in the region between the two
behaviorofthespectralfunction(atand)aboveT results peaks. Inthepresentpaperweshalldwellextensivelyon
c
4
presenceofapseudo-gap,incaseswhenthe twopeaksof
1
A(k,ω) stronglyoverlapjust in the rangeofk where the
0.8 two branches of the dispersion come close to each other
F (cf. Fig. 1). In these cases, in fact, a strict definition
E 0.6
ω) of the pseudo-gap as a depression of the spectral weight
A(k, 0.4 just in this range of k would lead one to conclude that
pseudo-gapphenomenawereabsentinthesingle-particle
0.2
excitations, while they still appear clearly over a more
0 extended range of k.
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
ω/E
F C. Inputs from experiments on ultra-cold Fermi
atoms
FIG. 2: Single-particle spectral function A(k,ω) versus ω at
unitarity, within mean field at T = 0 (dashed-dotted lines) The original motivation for looking at A(k,ω) has
andwiththeinclusionofpairingfluctuationsatT (fullline), been the issue of “preformed pairs” in high-temperature
c
for the wave vector where the maximum of the lower disper- (cuprate) superconductors, before the occurrence of the
sion relation occurs in both cases (see text). BCS-BEC crossover was explicitly demonstrated with
ultra-cold Fermi atoms (cf., e.g., Ref.[25]).
In this context, the interest in the detailed shape of
A(k,ω)aboveT acrosstheBCS-BECcrossoverhascon-
this and related ideas. c
siderably raised lately, after a new measurement tech-
The t-matrix approximationthat we adopt in this pa-
niquewasintroducedtoprobedirectlythesingle-particle
pertoobtainA(k,ω)aboveT correspondstothefollow-
c
excitations of a Fermi gas [1]. Intensity maps were thus
ing choice of the fermionic self-energy [7]:
obtained for the single-particle excitation spectra, relat-
dq ing the single-particle energy to the wave vector. More
Σ(k,ω )= k T Γ (q,Ω )
n − (2π)3 B 0 ν recently,new measurementsperformedoveranextended
Z ν
X temperature rangeaboveT [2]haverevealeda BCS-like
G (q k,Ω ω ) (5) c
0 ν n
× − − dispersion with a characteristic “back-bending” close to
whereω =(2n+1)πk T (ninteger)andΩ =2νπk T k which identifies a pseudo-gap energy scale and per-
n B ν B F
(ν integer)arefermionicandbosonicMatsubarafrequen- sists well above Tc.
cies, in the order, G (k,ω ) = (iω ξ )−1 is the bare Thisfindinggivesusmotivationsforextendingthethe-
0 n n k
−
fermionic single-particle Green’s function, and Γ (q,Ω ) oreticalanalysisofRef.[7]forthedispersionsofthepeaks
0 ν
is the particle-particle ladder given by ofA(k,ω)acrossthe unitaryregion,althoughthe widths
of the peaks can increase considerably with respect to
Γ (q,Ω )−1 = dk k T G (k,ω ) (6) the BCS side, reflecting the fact that quasi-particle exci-
− 0 ν (2π)3 " B 0 n tationsmaybepoorlydefined. Specifically,thecombined
Z n
X m m experimental and theoretical analysis of Ref.[8] suggests
G (q k,Ω ω ) + .
× 0 − ν − n − k2 4πa us to concentrate our efforts in the coupling range ap-
i F proximatelybetween(kFaF)−1 =0and(kFaF)−1 =0.4.
The single-particle spectral function is then obtained
through analytic continuation iω ω+iη to the real
n → D. Emergence of the contact in A(k,ω)
frequency axis (η =0+):
1 ImΣ(k,ω) Yet, it was pointed out [11] that the persistence of
A(k,ω)= .
−π [ω ξk ReΣ(k,ω)]2+[ImΣ(k,ω)]2 the back-bending for large k (≫ kF) is dominated by
− − interactioneffectsthatdonotreflectthepseudo-gapclose
(7)
tok . Rather,itisconnectedwiththeuniversalk−4 tail
The shape of A(k,ω) versus ω thus depends crucially on F
the interplay between ReΣ(k,ω) and ImΣ(k,ω) for the ofthewave-vectordistributionn(k)ofadiluteFermigas,
chosen value of k = k. whose coefficient is given by the Tan’s contact C [4, 5].
Aderivedquantity|o|finterestisthesingle-particleden- This property can be readily verified within the t-
sity of states: matrix approximation that we use to obtain A(k,ω).
When k2/(2m) or ω are much larger than the energy
n
dk scales k T and µ|, in| fact, the self-energy (5) can be
N(ω) = A(k,ω) . (8) B
| |
(2π)3 approximated by:
Z
The averaging that this definition introduces on A(k,ω) 1
over an extended range of k can be of support to the Σ(k,ωn) nfΓ0(k,ωn) ∆2∞G0(k, ωn) . (9)
≃−2 − −
5
Here, by following the dispersions, weights, and widths of the
peaks of A(k,ω) from k = 0, through k k , and up
dk ≈ F
nf =2 (2π)3 nf(k) (10) to k ≫kF, even in cases when these peaks appear quite
broad and overlapping.
Z
On physical grounds, the evolution, from ∆ when
with pg
k kF to ∆∞ when k kF, is expected on the basis
≈ ≫
n (k)=k T eiωnηG (k,ω ) (11) of a (local in space and transient in time) order which is
f B 0 n
established by pair fluctuations above T , in the absence
n c
X
of long-range order (as described by mean field below
is the free density associated with G for given µ, and
0 T ). This is in line with the definition of the contact
c
dq C (and thus of ∆∞) through the short-range behavior
∆2∞ = (2π)3 kBT eiΩνηΓ0(q,Ων) (12) of the pair-correlation function between opposite spins
Z ν [4], which corresponds to k k ; while the pseudo-gap
X F
≫
∆ is expected to depend on pair-correlations that are
isthesquareofthehigh-energyscaleintroducedinRef.[6] pg
established more extensively over medium range, which
that was mentioned in the Introduction. The two terms
corresponds to k k .
onthe right-handside of the approximateexpression(9) ≈ F
Asaconsequence,weexpectthe“pseudo-gapphysics”
originatefromthe singularitiesinthe complex frequency
to be associated with pair correlations which are built
planeofthesingle-particleGreen’sfunctionG andofthe
0 at intermediate distances of the order of k−1, while the
particle-particle ladder Γ0, in the order, once the sum F
“contactphysics”withpaircorrelationsthatsurviveeven
over the Matsubara frequency in the expression (5) of
at smaller distances ( k−1). Both quantities ∆ and
the fermionic self-energy is transformed into a contour ≪ F pg
integral. ∆∞ are thus affected by the same sort of pair corre-
lations in the particle-particle channel, to which the t-
Analytic continuation iω ω + iη to the real fre-
n
→ matrix approximation that we adopt in this paper pro-
quency axis then results into the following approximate
vides an important contribution. This is because the
expression for large k:
particle-particle ladder propagator [given by Eq.(6) in
∆2 ∆2 the attractive case and by Eq.(22) in the repulsive case]
∞ ∞
A(k,ω) 1 δ(ω ξ )+ δ(ω+ξ ) (13)
≃ − 4ξ2 − k 4ξ2 k generalizestoamany-bodyenvironmentthetwo-bodyt-
(cid:18) k(cid:19) k
matrix,whichdescribestwo-bodybindingandscattering
whichpresentsindeedawell-definedstructureattheneg- of unbound particles at the same time.
ative frequency ω = ξk. One obtains correspondingly: Within this approximation, (the square of) ∆∞ is de-
−
fined from Eq.(12) as a wave-vector and frequency av-
+∞ ∆2∞ (m∆∞)2 eraging of the particle-particle ladder propagator. The
n(k)= dωf(ω)A(k,ω) (14) wave-vector and frequency structures of the same prop-
Z−∞ ≃ 4ξk2 ≈ k4 agator give also rise to the pseudo-gap ∆pg, which then
emergesasacharacteristiclow-energyscaleinthesingle-
yielding the relation C = (m∆∞)2 between the contact
particle excitations.
C and ∆∞, which will be extensively used below.
Consistentlywiththisphysicalpictureoflocallyestab-
Inpractice,thestructureofA(k,ω)fornegativerealω
lished pair correlations, we expect ∆ to survive above
at large k is not delta-like but spreads over a sizable fre- pg
quency range. This difference from the approximate re- Tcoveramorelimitedtemperaturerangethan∆∞,since
thermal fluctuations act first to destroy the order estab-
sult(13)stemsfromthenon-commutativitybetweentak-
lished over intermediate distances. Our analysis about
ing the analytic continuationandperforming the large-k
the characteristic features of the spectral functions will
expansionoftheself-energy,asrecalledintheAppendix.
accordingly be extended over a meaningful temperature
Nevertheless,the actualstructureofA(k,ω)fornegative
interval above T .
ω preserves the same total area ∆2 /(4ξ2) found above c
∞ k
in the expression (13). F. Working procedures
E. Connecting the two energies ∆pg and ∆∞ Working experience on the single-particle spectral
function suggestsus to identify the low-energyscale∆
pg
With these premises, it seems natural to frame the intherange0<k <2kF,whilethehigh-energyscale∆∞
low-energy scale ∆pg and the high-energy scale ∆∞ into can be extrac∼ted ∼with sufficient accuracy already from
a unified physical picture, in which they emerge from the not too extreme range 2k < k < 4k . In a more
F F
A(k,ω)inthetwodistinctrangesofwavevectorsk k extreme range of k (> 4k ), in∼fact∼, it would become
F F
and k k , respectively. To this end, we shall ex≈tend quite difficult to deter∼mine A(k,ω) for large negative ω.
F
≫
to the unitary limit and beyond the analysis that was Owing to the shape of the ω-structures of A(k,ω) for
limited in Ref.[7] to the BCS side of the unitary region, given k, different strategies need to be adopted in the
6
above two ranges of wave vectors. Namely, in the range (17)fromunitycanthusbetakenasatestforthevalidity
0 < k < 2k the shape of the two peaks of A(k,ω) per- of the effective BCS description that we are attempting
F
m∼its,in∼practice,tobothfollowtheirdispersionsforvary- to establish in the normal phase close to T .
c
ingk andidentify theirweightsintermsofthetotalarea Concerning,finally,thewidths ofthe peaksofA(k,ω),
they comprise. In the range 2k < k < 4k , on the theywillbe conventionallydeterminedasthe fullwidths
F F
other hand, the structure of A(k,ω)∼atne∼gative frequen- at half maximum. Whenever necessary, however, these
ciesissobroadthatonlyitstotalareacanbe reasonably values will be compared with the numerical values ob-
identified. tained alternatively by a two-Lorentzianfit to the peaks
of A(k,ω), a procedure which is of course more reliable
Range 0<k<2k :
F the more these peaks are separated in frequency.
∼ ∼
Let’s first consider the range 0 < k < 2k . Here, the The BCS-like expressions (15), (16), and (17) are
F
dispersions that we are able to det∼ermi∼ne independently meant to test the persistence of a BCS-like description
in the normal phase of a Fermi gas with an attractive
forthetwopeaksatpositiveandnegativefrequenciesare
pairing interaction. In addition, the presence in Eq.(15)
fitted, respectively, by the BCS-like expressions:
ofthe“Luttingerwavevectors”kL(±) highlightsthe per-
ω(±)(k) = ±vu 2km2 − k2L2m(±)!2 + ∆2pg(±) (15) spwiashrtaettincwcleeoeuoxlfdciabtnaetaiuonFndesr,emrwlyih-ilinicqghuriFdeepdrrmeessicernsiutpsrtfitaohcneeolfafostrthertehFmeernsmainnigtglaoes-f
u
t if attractive pairing interactions above Tc would not be
whereadifferentpseudo-gapenergy∆pgisintroducedfor considered [8]. Nevertheless, the large widths associated
theupper(+)andlower( )branches. Thefittingidenti- with the peaks of A(k,ω) represent per se an evidence
−
fies,inaddition,thelocationskL(+) forthe“up-bending” that a Fermi-liquid description above Tc does not apply
of the upper branch and kL(−) for the “down-bending” in the presence of pairing fluctuations.
of the lower branch [26]. The fittings are carried out by
Range 2k <k<4k :
a χ2-analysis. We have found that k is consistently F F
L(+) ∼ ∼
smaller than kL(−) in all situations we have examined. In this range, the structure at negative frequencies in
Because the peaks at negative and positive frequen- A(k,ω)becomes so spreadandbroadthatit is meaning-
cies are in general broad and partially overlapping with less to determine its dispersionnumerically and then try
eachother,wehavechosentodetermineoperativelytheir to fit it by an expression similar to (15). In this case,
weights at a given k by integrating A(k,ω) from up however, it remains meaningful to determine the total
−∞
to ω = 0 and from ω = 0 up to + , in the order. We area of the broad structure at negative ω over a chosen
∞
then fit these values obtained for several k by the BCS- meshofk andthenmakeaχ2-fittothesevaluesthrough
like expressions: the following expression which is inspired by Eq.(13):
v(k)2 = 12 1 − 2km2 −2km2µe−ff 2µe+ff ∆2pg(−) (16) v(k)2large = 4 2km2∆−2larµgelarge 2 (18)
 q(cid:0) (cid:1)  where ∆large and µlarge are(cid:0)fitting param(cid:1)eters to be de-
for the lower branch at negative frequencies, and termined in this range of “large” k.
In practice, it is convenient to set µ at the corre-
large
1 k2 µ spondingvalueofthe thermodynamicchemicalpotential
u(k)2 = 1 + 2m − eff (17)
2  k2 µ 2 + ∆2  µ from the outset (thus leaving ∆large as the only fit-
2m − eff pg(+) tingparameter). Thisisbecauseinthecouplingrangeof
for the upperbrancqh (cid:0)at positive(cid:1) frequencies. Here, interest µlarge is small enough that it becomes meaning-
less to extract it from the denominator of Eq.(18) where
∆pg(±) are the pseudo-gap energies already determined k2/(2m) dominates in this range of k.
fromthe fitting (15)to the dispersions,andµ is a new
eff We will check whether the value of ∆ determined
parameter common to the two branches which is deter- large
in this way coincides with the value of ∆∞ obtained in-
mined from the position where the weights cross. We
dependently by the expression (12), and how it differs
shall find that the value of √2mµ is intermediate be-
tweenkL(+) andkL(−) obtainedfroemffthedispersions(15) nfreoamr tthheerveagliuoen∆ofpgt(h−e)boafctkh-ebepnsdeuindgo-ogfatpheoblotawienredbraabnocvhe.
of the two branches, consistently with enforcing a com-
mon value µeff in Eqs.(16) and (17). G. Results for dispersions, weights, and widths
Note thatthe expressions(16)and(17)do notrequire
a priori thesumu(k)2+v(k)2 tobeunity,asitwouldbe Wepasstodeterminethequantitiesofinterestaccord-
thecaseforthecoherencefactorsinastrictBCSdescrip- ing to the procedures outlined above. We shall specif-
tion. Deviations of the sum of the expressions (16) and ically consider the two coupling values (k a )−1 = 0
F F
7
4
3 ω/EF
2 -2 -1 0 1 2
s 1.2
n 1
o
ersi 0
disp --21 ω)EF 0.8
-3 k,
A( 0.4
-4
0.8
0
hts 0.6 2.5
g
wei 0.4 2
s
0.2 dth 1.5
wi
1
0
0 0.5 1 1.5 2 0.5
k/kF
0
0 0.5 1 1.5 2
k/kF
FIG.3: Dispersions(upperpanel)andcorrespondingweights
(lower panel) at unitarity and T =T . Circles (squares) and
c
full (dashed) lines represent the results of the numerical cal- FIG. 4: Unitarity limit and T = T . Upper panel: A(k,ω)
c
culationandoftheBCS-likefitsforthelower(upper)branch vsω forthewavevectorsk=(0.6,0.7,0.8,0.9,1.0,1.1,1.2)k
F
at negative and positive frequencies, respectively. Energies corresponding to the peaks at negative frequency from top
are in unitsof EF. to bottom (here full and dashed lines alternate to help the
analysis of the figure). Lower panel: Widths (in units of
E ) of the peaks at negative (circles) and positive (squares)
F
frequencies.
and (k a )−1 = 0.25 as representatives of the coupling
F F
range where pseudo-gap phenomena are expected to be
maximal. Two representative temperatures will also be
several wave vectors, while in the lower panel the cor-
considered for each coupling.
responding widths of the peaks at negative and positive
Figure 3 shows the dispersion relations and weights of
frequenciesarereportedoverawidersetofwavevectors.
the two peaks of A(k,ω) at T when (k a )−1 = 0 in
c F F In all cases, the widths are rather large (being com-
the range 0 k 2k , as obtained from the numerical
F parable to E ) and show strong deviations from what
≤ ≤ F
calculationbasedonEq.(7)andfromthefitsobtainedac-
would be expected for a Fermi liquid picture, according
cording to Eqs.(15)-(17). In this case the fitting param-
to which they should acquire a minimum value at about
eters are kL(−) = 0.78kF and ∆pg(−) = 0.83EF for the k . These deviations from a Fermi liquid picture are
F
lower branch, and k =0.62k and ∆ =0.74E
L(+) F pg(+) F of course expected for a Fermi gas with attractive inter-
for the upper branch, while µ = 0.41E is quite close
eff F action, taking further into account that at unitarity the
to the correspondingvalue of the thermodynamic poten-
valueofT isaconsiderablefractionoftheFermitemper-
c
tial µ = 0.365E obtained within the t-matrix approxi-
F ature T whereas a Fermi-liquid description holds only
F
mation. [The results of the numerical calculations have
for T T [27].
F
already been reported in the central panels of Fig. 1, ≪
The above analysis of pseudo-gap phenomena around
although with the different purpose of comparing them
k is expected to remain meaningful for temperatures
F
with the mean-field description at T =0.]
largerthanT , but not exceeding the pair-breakingtem-
c
In this case the BCS-like fits are excellent for the dis- perature scale T∗ where a “preformed-pair scenario” is
persions and quite good for the weights, especially near bound to fade away. In particular,atunitarity the value
the value k = 0.64kF where the weights exchange with of T∗ (as estimated by the mean-field critical temper-
one another. Note also that, while the numerical values ature) is about twice the value of T given by the t-
c
of the weights for each value of k are specular to each matrix approximation we are considering [28][13]. As
other about one half, the fitted values are not always so a representative case of a temperature above T , Fig. 5
c
indicating deviations from their sum being unity. showsthe dispersionsandweightsobtainedfromA(k,ω)
ThissuccessofaBCS-likeinterpretationforthedisper- at unitarity and T = 1.2T . The fitting parameters are
c
sions and weights of the peaks should be complemented now kL(−) = 0.99kF and ∆pg(−) = 0.48EF for the lower
by the further information about their widths. This is branch, and k = 0.69k and ∆ = 0.62E for
L(+) F pg(+) F
done in Fig. 4, where in the upper panel the shape of the upper branch, while µ = 0.48E (to be compared
eff F
A(k,ω) at T and (k a )−1 = 0 is shown explicitly for with the thermodynamic value µ=0.39E ).
c F F F
8
4
3 ω/EF
2 -2 -1 0 1 2
s 1.2
n 1
o
ersi 0
disp --21 ω)EF 0.8
-3 k,
A( 0.4
-4
0.8
0
hts 0.6 3.5
weig 0.4 3
2.5
s
0.2 dth 2
wi 1.5
0 1
0 0.5 1 1.5 2
0.5
k/kF
0
0 0.5 1 1.5 2
k/kF
FIG. 5: Dispersions (upperpanel) and weights (lower panel)
at unitarity and T =1.2T . Conventions are as in Fig. 3.
c
FIG.6: UnitaritylimitandT =1.2T . Upperpanel: A(k,ω)
c
vsωforthesamewavevectorsasintheupperpanelofFig.4.
Lowerpanel: Widths(inunitsofE )ofthepeaksatnegative
F
Note that in this case the analysis of the dispersion (circles) and positive (squares) frequencies.
of the lower branch had to be interrupted over a non-
negligible intervalofk aboutk , because inthis interval
F
the structure ofA(k,ω)atnegativefrequenciesis almost 2
completely masked by the stronger structure at positive
frequencies. This represents a signal that pseudo-gap )F 1.5
k
phenomena are beginning to fade away at this temper- m
ature. We have nevertheless performed a BCS-like fit ω)/( 1
N(
to the part of the dispersion that can still be clearly 2
identified, as shown by the full curve in the upper panel π2 0.5
of Fig. 5. By our procedure no problem instead arises
0
in identifying the corresponding weights reported in the -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
lower panel of Fig. 5, which follow again a BCS-like dis- ω/E
F
persionalthoughwith less accuracythan those shownat
T in Fig. 3.
c
FIG.7: Densityofstatesperspincomponentvsω calculated
The corresponding shapes of A(k,ω) vs ω for a cho-
at unitarity within the t-matrix approximation for the tem-
sen set of k across k are shown explicitly in the upper
F peratures: T = Tc (full line), T = 1.2Tc (long-dashed line),
panel of Fig. 6, from which one can appreciate the phe- T = 1.4T (short-dashed line), and T = 1.65T (dot-dashed
c c
nomenonmentionedabove,whenthestructureofA(k,ω) line). In the present case, T∗ ≈ 2T . The dotted line shows
c
for the lower branchbecomes a shoulder attached to the the corresponding mean-field result when T = 0. The non-
structureoftheupperbranch. Thecorrespondingbroad- interacting value mkF/(2π2) of N(ω = 0) at T = 0 is used
for normalization.
enings of these two structures are reported in the lower
panelofFig.6,whichreinforcesourconclusionaboutthe
non-Fermi-liquid nature of the system.
Aquestionnaturallyarises,aboutwhetherornotthese inition (8) of the single-particle density of states N(ω).
profiles of A(k,ω) still allow one to identify the presence Figure 7 shows a plot of N(ω) vs ω at unitarity for sev-
ofapseudo-gapinthecrucialrangeofwavevectorsabout eral temperatures at and above T [29]. For increasing
c
kF. AstheupperpanelofFig.5shows,infact,anoverall T, the depression of N(ω) near ω = 0 well survives for
BCS-like fit to the lower branch can be attempted even T = 1.2T at which the dispersion of the lower branch
c
inthiscase,becausethetwostructuresofA(k,ω)remain neark inFig.5hadtobeinterrupted,andprogressively
F
distinct from each other away from kF. disappears for temperatures somewhat below the pair-
The relevance of this restricted interval about k can breaking temperature scale T∗. That the depression of
F
be strongly reduced by averaging the profiles of A(k,ω) densityofstatessurvivesattemperatureshigherthanthe
over all wave vectors, in the way it is done in the def- crossovertemperaturewherethepseudo-gapfeaturesdis-
9
4
3 ω/EF
2 -2 -1 0 1 2
s 1.5
n 1
o
ersi 0
disp --21 ω)EF 1
-3 k,
A( 0.5
-4
0.8
0
hts 0.6 2.5
g
wei 0.4 2
s
0.2 dth 1.5
wi
1
0
0 0.5 1 1.5 2 0.5
k/kF
0
0 0.5 1 1.5 2
k/kF
FIG. 8: Dispersions (upperpanel) and weights (lower panel)
atT =T forthecoupling(k a )−1 =0.25. Conventionsare
c F F
as in Fig. 3. FIG.9: Coupling(k a )−1 =0.25andT =T . Upperpanel:
F F c
A(k,ω)vsω for thesame wavevectors as in theupperpanel
of Fig. 4. Lower panel: Widths (in units of E ) of thepeaks
F
at negative (circles) and positive (squares) frequencies.
appear in the spectral function was previously discussed
in Refs.[16, 18]. At about T∗, N(ω) for ω = 0 coincides
(withinafewpercent)withitsnon-interactingvalueeval-
4
uated at the same temperature and chemical potential, 3
indicating that all effects of pairing have faded away at 2
s
ω =0 (althoughthey willpersist athigher temperatures on 1
for ω E , indicating the survival of the “contact” ersi 0
≪ − F sp -1
even at quite high temperatures [30]). The density of di
-2
states obtained within mean field at zero temperature is -3
also reported for comparison in Fig. 7, and shows two -4
sharp peaks located at ∆ with ∆=0.69E .
F 0.8
±
It is important to extend the above analysis past the
unitarity limitto the BEC sideofthe crossover(but still ghts 0.6
before the pseudo-gap turns into a real gap associated wei 0.4
with the binding energy of the composite bosons which 0.2
formintheBEClimit). Tothisend,Fig.8showsthedis-
0
persions and weights at T for the coupling (k a )−1 = 0 0.5 1 1.5 2
c F F
0.25,togetherwiththecorrespondingBCS-likefitswhere k/kF
now kL(−) = 0.77kF and ∆pg(−) = 1.09EF for the lower
branch, and k = 0.28k and ∆ = 0.91E for
L(+) F pg(+) F FIG.10: Dispersions(upperpanel)andweights(lowerpanel)
the upper branch, while µeff = 0.09EF (which in this atT =1.4T forthecoupling(k a )−1 =0.25. Conventions
c F F
case almost coincides with the thermodynamic value). are as in Fig. 3.
Comparedwith Fig. 3, the dispersions have now become
quite flatinthe range0<k <k , while the twoweights
F
cross at smaller value of k. ∼At even stronger couplings,
the lower dispersion down-bends and the upper disper- When the coupling increases toward the BEC
sion up-bends already at k =0 while the weights always regime,thepair-breakingtemperatureT∗ increasesmore
remain well separated from each other for all k (as it markedlythanT [28][13]andpairingfluctuationsareac-
c
is shown in the lower panels of Fig. 1 for the coupling cordingly expected to affect A(k,ω) over a progressively
(kFaF)−1 =1.0). wider temperature range above Tc. We then report in
For completeness, Fig. 9 shows the corresponding Fig. 10 the dispersions and widths of the two branches
shapes of A(k,ω) across k (upper panel) as well as the of A(k,ω) for the coupling (k a )−1 = 0.25 and the
F F F
broadenings of two structures of A(k,ω) (lower panel). higher temperature T = 1.4T . Again, a signal that the
c
10
1.2
ω/EF k=0.6kF
-2 -1 0 1 2
1.5 0.8

F 1 0.4
E
ω)
k,
A( 0.5 0
k=0.9kF
0 F 0.8
E
4 ω)
k,
A( 0.4
3
s
h
widt 2 0
1 k=1.2kF
0.8
0
0 0.5 1 1.5 2
k/kF 0.4
FIG. 11: Coupling (k a )−1 = 0.25 and T = 1.4T . Upper 0
F F c -2 -1 0 1 2
panel: A(k,ω)vsω forthesamewavevectorsasintheupper ω/EF
panel of Fig. 4. Lower panel: Widths (in units of E ) of the
F
peaksatnegative(circles) andpositive(squares)frequencies.
FIG. 12: Two-Lorentzian fits (dashed lines) of A(k,ω) vs
ω (full lines) at unitarity and T = T , when k/k =
c F
(0.6,0.9,1.2) from top to bottom.
pseudo-gap is beginning to fade away emerges from the
analysisofthedispersionforthelowerbranch,whichhas
to be interrupted about kF. The fitting parameters are spectral function A(k,ω), which were also shown in the
now kL(−) = 1.03kF and ∆pg(−) = 0.60EF for the lower same figures. It may also be of use, however,to organize
branch,andkL(+) =0.25kF and∆pg(+) =0.86EF forthe thespectraofA(k,ω)forarangeofk andω intoasingle
upperbranch,whileµeff =0.09EF (tobecomparedwith intensityplot. ThisisdoneinFig.13(a)forthe sameset
thethermodynamicvalueµ=0.14EF). Thecorrespond- oftemperaturesandcouplingsconsideredintheprevious
ing shapes of A(k,ω) across kF and the broadenings of figures. SimilarintensityplotswerepresentedinRefs.[18]
two structures of A(k,ω) are shown, respectively, in the and [19]. Note that the log scale, used here like in the
upper and lower panels of Fig. 11. experimental works [1, 2], makes the back-bending more
Beginning with Fig. 2, we have often emphasized that evidentwhencomparedwiththeintensityplotspresented
one of the major characteristics of the two structures of in Refs.[18] and [19]. For the sake of comparison with
A(k,ω)(atand)aboveT istheirsubstantialbroadening, those references, we also report in Fig. 13(b) the same
c
which may hinder in practice a straightforward identifi- intensity plots in a linear scale.
cation of the pseudo-gap about kF in cases when these Thus far we have concentrated our attention to the
structures strongly overlap with each other. In these range 0 < k < 2k where the pseudo-gap physics man-
F
cases, however, one may resort to a two-Lorentzian fit ifests its∼elf. ∼We pass now to discuss the more asymp-
of the two structures of A(k,ω) which helps separating totic range 2k < k < 4k where the contact physics
F F
them. This is shown Fig. 12 where the dashed lines rep- emerges. To this∼end,∼we adopt the procedure outlined
resent the two Lorentzians. For instance, by this type in sub-section II-F and determine the parameter ∆
large
of fit our previous estimates for the weight (0.29) and from the expression (18) with µ fixed at the corre-
large
width (0.93EF) of the structure of A(k,ω) at k =0.9kF sponding value of the chemical potential.
corresponding to the lower branch are replaced by 0.28 Figure14showstheweightsofthestructureofA(k,ω)
and0.77EF,intheorder. Comparabledeviationsareob- at negative ω for the values of coupling and tempera-
tained in the other cases. These results thus confirm the tureconsideredsofar,asdeterminednumerically(circles)
validity of our previous analysis where the weights and over a mesh of values of k in the range 2k k 4k
F F
≤ ≤
widths were extracted from A(k,ω) in a simpler fashion. andthenfitted(fulllines)intermsoftheexpression(18).
The numerical values of the dispersions, weights, and These fits are alsocomparedwith anexpressionof the
widthsthatwerereportedinthepreviousfigureswereall form (18), where now the low-energy scale ∆pg(−) that
obtained from the detailed profiles of the single-particle was previously determined in the range 0<k < 2k re-
F
∼ ∼