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epl draft Universal dynamic structure factor of a strongly correlated Fermi gas H. Hu1 (a) and X.-J. Liu1 (b) 1 1 ARC Centre of Excellence for Quantum-Atom Optics, 1 Centre for Atom Optics and Ultrafast Spectroscopy, 0 2Swinburne University of Technology, Melbourne 3122, Australia n a J PACS 03.75.Hh–Static properties of condensates; thermodynamical, statistical and structural 6 properties PACS 03.75.Ss–Degenerate Fermi gases ] s PACS 05.30.Fk–Fermion systems and electron gas a g Abstract. - Universality of strongly interacting fermions is a topic of great interest in diverse - t fields. Here we investigate theoretically the universal dynamic density response of resonantly n interactingultracoldFermiatomsinthelimitofeitherhightemperatureorlargefrequency: (1)At a hightemperature,weusequantumvirialexpansiontoderiveuniversal,non-perturbativeexpansion u functionsof dynamicstructurefactor; (2) At large momentum,we identify that thesecond-order q expansion function gives the Wilson coefficient used in the operator expansion product method. . t Thedynamicstructurefactoristhereforedeterminedbyitssecond-orderexpansionfunctionwith a m anoverallnormalization factor givenbyTan’scontact parameter. Weshowthatthespinparallel and antiparallel dynamic structure factors have respectively a tail of the form ω−5/2 for d- ω ,decayingslowerthanthetotaldynamicstructurefactorfoundpreviously(∼ω±−7/2). Our →∞ ∼ n predictionsfordynamicstructurefactorathightemperatureorlargefrequencyaretestableusing o Bragg spectroscopy for ultracold atomic Fermigases. c [ 1 v 4 3 Introduction. – The study of strongly interacting momenta and frequencies. 1fermions has brought together very different areas of Exact results are very valuable for strongly interacting 1physics - neutron stars, quark-gluon plasmas, high tem- fermions, due to their non-perturbative, strongly corre- . 1peraturesuperconductors,andcoldatoms-which,atfirst lated nature. There is no small interaction parameter to 0sight, have little in common. There is, however, an im- controltheaccuracyoftheories. Inspecificcases,ab-initio 1portant generic idea of fermionic universality behind [1]: calculationsarepossibleusingMonteCarlomethods[7–9]. 1 all strongly interacting, dilute Fermi gases should behave However,ingeneralthisapproachsuffersfromthefermion : videntically, depending only on scaling factors equal to the signproblem[10]. In2008,Tanderivedasetofexact,uni- Xiaverage particle separation and/or thermal wavelength, versalrelations for two-component(spin-1/2)Fermi gases but not on the details of the interaction. Recent manipu- with a large s-wave scattering length a [11]. These uni- r alation of ultracold Fermi gases of 6Li and 40K atoms near versal relations all involve a many-body parameter called abroadcollisional(Feshbach)resonanceprovidesanideal the contact , which measures the density of pairs within avenuetounderstandthis fermionicuniversality[2–5]. To short distanIces. Tan’s relations can be understood us- date, universal thermodynamics of strongly interacting ingtheshort-distanceand/orshort-timeoperatorproduc- fermions has been clearly demonstrated [6], by measuring tion expansion (OPE) method [12–14], which separates the static equation of state. The purpose of this Letter in a natural way few-body from many-body physics. For is to show that universality appears in dynamical proper- the “interaction” DSF, ∆Sσσ′(q,ω,T) Sσσ′(q,ω,T) tiesaswell. Wederiveexact resultsforuniversaldynamic S(1)(q,ω,T),theOPEpredictsthat(q ≡ andω −) structure factor (DSF) at high temperatures or at large σσ′ →∞ →∞ [13,14] (a)[email protected] (b)[email protected] ∆Sσσ′(q,ω,T) Wσσ′(q,ω,T) , (1) ≃ I p-1 H. Hu and X.-J. Liu where (σ,σ′) =↑,↓ denote the spin, Sσσ′ and Sσ(1σ)′ are respectively the DSF of interacting and non-interacting ~ Fermigasesatthesamechemicalpotentialµ andtemper- 6 q = 1/3 2 ~ q = 1 ature T, and Wσσ′ are the temperature-dependent Wil- ) son coefficients that rely only on few-body physics. At ~ hightemperatures,quantumvirialexpansionprovidesan- ~ q, 1 ( 4 other rigorous means to bridge few-body and many-body ~ S2 physics [15–18]. It was shown that static thermodynamic 0 properties of a strongly correlated Fermi gas can be ex- -2 0 2 4 6 panded non-perturbatively in fugacity using some univer- 2 q~ = 1 sal, temperature-independent virial coefficients [5,17,18], ~ which are exactly calculable from few-fermion solutions. q = 3 Both OPE and virial expansion give useful insight into 0 the challenging many-body problem. However, their con- -5 0 5 10 15 20 ~ nection is yet to be understood. In this Letter, we investigate theoretically the univer- Fig. 1: (Color on-line) Universal second orderexpansion func- sal dynamic properties of a strongly correlated Fermi gas tion of DSF at q˜= 1/3, 1, and 3. The inset shows the rapid in the limit of either high temperature or large momen- convergenceof∆S˜2(q˜,ω˜)atsmall¯hω0/(kBT)(thicklines)and, tum/frequency. In the former limit, we show that the ∆S˜↑↑,2 (thinsolidline)and∆S˜↑↓,2 (thindashedline)atq˜=1. dynamic structure factor can be virial expanded in fu- gactiy, using some universal, temperature-implicit virial Here we consider the expansion functions of a homo- expansion functions. We derive, for the first time, these geneous Fermi gas in the unitarity limit and emphasize universal virial expansion functions for spin parallel and their universal aspect, which is not known so far. As the antiparallel DSFs of a homogeneous Fermi gas in the res- scatteringlengthdiverges,allmicroscopicscalesofthe in- onance (unitarity) limit, where the scattering length di- teraction are absent [1]. For this few-body problem, the verges(a ). In the latter limit of largemomentum, →±∞ only energy scale is kBT and length scale is the thermal we show that the Wilson coefficient in Eq. (1) is given by wavelength λ h/(2πmk T)1/2. Dimensional analysis B the second-orderexpansionfunction. Therefore, the large ≡ leads to, momentum DSF is universally determined by the second order virial expansion, together with a many-body pref- V actor - the contact. Our results can be easily examined ∆Sσσ′,n(q,ω,T)= k Tλ3∆S˜σσ′,n(q˜,ω˜), (3) B using Bragg spectroscopy for ultracold Fermi gases of 6Li or 40K atoms [19]. where V is the volume, q˜ = [¯h2q2/(2mkBT)]1/2, ω˜ = ¯hω/(kBT),and∆S˜σσ′,nisadimensionlessexpansionfunc- Universal expansion function of DSF. – Virial tion. The temperature T is now implicit in the variables expansion is a powerful tool for studying the high- q˜ and ω˜. This universal form implies a simple relation temperature properties of ultracold atomic Fermi gases between the trapped and homogeneous expansion func- [17]. It expresses any physical quantities of interest as an tion. In a shallow harmonic trap, V (r) = m(ω2x2 + Trap x expansion in fugacity with some coefficients or functions, ω2y2 +ω2z2)/2, where ω (ω ω ω )1/3 0, the sys- y z 0 ≡ x y z → since the fugacity z exp(µ/kBT) 1 is a controllable tem may be viewed as a collection of many cells with a ≡ ≪ smallparameterathightemperatures. Fortheinteraction local chemical potential µ(r)=µ V (r) and fugacity Trap − DSF, the expansion is given by [20] z(r)=zexp[ V (r)/k T],so that the trapped DSF is Trap B − ∆Sσσ′(q,ω,T)=z2∆Sσσ′,2+z3∆Sσσ′,3+ , (2) gOivweinngbyto∆tSheσ(Tσur′anpi)v(eqrs,aωl,qT˜-)a=nd ω˜d-rd[∆epSeσnσd′e(nqc,eω,inT,trh)e/Vex]-. ··· R pansion functions, the spatial integration can be easily where ∆Sσσ′,n(q,ω,T) (n = 2,3, ) is the n-th expan- performed, giving rise to ··· sion function. The expansion is non-perturbative as the few-body problemcouldbe solvedexactly,no matter how (h¯ω )3 strong the interaction. Virial expansion is anticipated to ∆S˜σσ′,n(q˜,ω˜)=n3/2(k T0)2∆Sσ(Tσr′,anp)(q,ω,T). (4) B work for temperatures down to the superfluid transition, although a nontrivial resummation of expansion terms The (non-universal) correction to the above local density may be required if z 1. In Ref. [20], the lowest second approximation is at the order of O[(¯hω )2/(k T)2]. Eq. ≫ 0 B orderexpansionfunction∆S(Trap) foratrapped Fermigas (4) is vitally important because the calculation of expan- σσ′,2 wascalculatedusingtwo-fermionsolutionsintraps. How- sion functions in harmonic traps is much easier than in ever, the universal aspect of expansion functions was not free space. realized. As a result, for different temperatures/momenta Fig. 1 reports the homogeneous expansion function the expansion functions had to be re-calculated. ∆S˜ = 2[∆S˜ + ∆S˜ ] at three different momenta, 2 ↑↑,2 ↑↓,2 p-2 Universal dynamic structure factor of a strongly correlated Fermi gas using ∆S(Trap) in Ref. [20] as the input. One observes σσ′,2 a quasielastic peak at ω˜ = q˜2/2 or ω = ¯hq2/(4m), as a result of the formation of fermionic pairs. We note that thethirdexpansionfunction∆S˜σσ′,3 or∆Sσ(Tσr′,a3p) canalso be calculated straightforwardly using exact three-fermion solutions [22]. We may derive sum rules that constrain the universal Fig. 2: (Color on-line) Diagrammatic contributions to the expansionfunctions,usingthewell-knownf-sumrulessat- interaction dynamic susceptibility. The self-energy (S) and isfiedbyDSF: −+∞∞ωS↑↑(q,ω,T)dω =Nq2/(4m)[23]and Maki-Thompson (MT) diagrams contribute to ∆χ↑↑(r,τ) and +∞ωS (q,ω,T)dω = 0 [24]. The latter immediately ∆χ↑↓(r,τ), respectively, while the Aslamazo-Larkin diagram −∞ ↑↓ R (AL) contributes to both. The shadow in the diagrams repre- leads to R +∞ sents the contact . The crossed part in the diagram (AL) is ω˜∆S˜ (q˜,ω˜)dω˜ =0. (5) the vertex. I ↑↓,n Z−∞ On the other hand, virial expansion of the total number in the diagrams the shadow part of the vertex function of fermions N implies that Γ(r = 0,τ = 0−) represents the contact . These dia- I +∞ grams are well-known in condensed matter community. ω˜∆S˜↑↑,n(q˜,ω˜)dω˜ =nq˜2∆bn, (6) In the context of calculating the change in conductiv- Z−∞ ity due to conducting fluctuations, the last two diagrams where∆b isthen-thvirialcoefficientandintheunitarity arecalledthe Maki-Thompson(MT)[26]andAslamazov- n limit ∆b =1/√2 and ∆b 0.355 [5,17]. Larkin (AL) contributions [27] respectively, while first di- 2 3 At large momentum, ≃th−e spin-antiparallel static agram gives the self-energy (S) correction. At zero tem- structure factor satisfies the Tan relation [21], perature,we calculate these diagramsinvacuum atµ=0 S↑↓(q,ω,T)dω ≃I/(8¯hq). Thisindicatesavirialexpan- and obtain that W↑T↑=0 =(fS −fAL)/(4π2√m¯hω3/2) and sion of the contact: = 16π2V z2c +z3c + /λ4, WT=0 =(f f )/(4π2√m¯hω3/2), where, R I 2 3 ··· ↑↓ MT − AL where the contact coefficients c are given by, n (cid:2) (cid:3) 1 x/2 ∆S˜ (q˜ 1) +∞∆S˜ (q˜,ω˜)dω˜ = π3/2cn. (7) fS = (1 −x)2 , ↑↓,n ↑↓,n p ≫ ≡ q˜ − Z−∞ 1 1+√2x x2 f = ln − , The expansion of the contact was alternatively obtained MT √2x 1 x | − | using Tan’s adiabatic sweeprelation[18]. In the unitarity 1 1+√2x x2 ltihmeits,amitewlaimssihtoowflnartgheatmco2m=en1t/uπma,ntdhec3sp≃in−-p0a.1ra4l1le[l1s8t].atIinc fAL = 2x 1 x/2"ln2 1 x− −π2Θ(x−1)#, − | − | structurefactorisnearlyunitysothat S (q,ω,T)dω ↑↑ p N/(2¯h) [20,21]. This leads to ≃ x ¯h2q2/(2m¯hω), and Θ is the step function. These R ≡ results agree with the previous calculations by Son and +∞ Thompson [14], although there the spin parallel and an- ∆S˜ (q˜ 1) ∆S˜ (q˜,ω˜)dω˜ =n∆b . (8) ↑↑,n ≫ ≡ ↑↑,n n tiparallel DSFs were not treated separately. At small Z−∞ q2/ω,we findthatthe spinparallelandantiparallelDSFs For the second expansion function, ∆S˜σσ′,2, we have have the tail checked numerically that all the above mentioned sum ¯h1/2q2 rules are strictly satisfied. WT=0 = WT=0 = . (9) ↑↑ − ↑↓ 12π2m3/2ω5/2 Wilson coefficient of DSF. – We now turn to the largemomentum/frequencylimits,wheretheOPEEq. (1) Thispredictionshowsthatforω the spindependent →∞ is assumed to be applicable. It is clear from the equation DSFs decay an order slower in magnitude than the total thattheWilsoncoefficientdeterminesentirelytheDSFat DSF. The latter is proportional to q4/ω7/2, as shown in large (q,ω) as far as the many-body contact is known. Refs. [14] and [28]. The faster decay in the total dynamic At T=0, the Wilson coefficient Wσσ′ can be calcu- structure factor is due to the cancellation of the leading lated using Feynman diagrams [14] for dynamic suscepti- terms in WT=0 and WT=0. ↑↑ ↑↓ bility χσσ′(r,τ)= Tτρˆσ(r,τ)ρˆσ′(0,0) , as Sσσ′(q,ω)= Itisnotclearhowtoobtainthe finitetemperatureWil- −h i Imχσσ′(q,ω)/[π(1 e−h¯ω/kBT)]. Inthelimitof(q,ω) son coefficientusing diagrammatictechnique, since the fi- − − → , the contributing diagrams to χσσ′(q,ω) are sketched nite temperature diagrams involve many-body process in ∞ in Fig. 2 [14]. Diagrammatically, the contact may be medium. However,Eq. (1) gives the hint. It has a strong identified as the vertex function at short distance and constraint on the expansion functions of DSF. As Wσσ′ time [18,25]: = m2Γ(r = 0,τ = 0−)/¯h4. Therefore, involves only the few-body physics and hence does not I − p-3 H. Hu and X.-J. Liu 0.5 0.06 (a) 0.4 T/TF = 0.225 (a) (b) x) 0.4 ( 0.04 f ) ) 0.2 S(q, 0.3 (q, 3 S T 3 0.02 kB0.2 kTB 0.0 0.5 0 1 2 3 0.00 (b) Son & Thompson 0.1 1 0.2 ~ q=10 (x) 5 0.0 2 -0.02 q = 5kF f 3 0.0 0.5 1/.0R 1.5 2.0 0.0 0.5 1/.0R 1.5 2.0 0.0 Fig. 4: (Color on-line) Universal total DSF (a) and spin- antiparallel(b)DSFatT/T =0.225,0.5,1,and2,calculated F -0.2 usingEq. (12). Weusea pair fluctuation theory todetermine 0 1 ~ ~2 2 3 thefugacity z and contact [18]. 1/x = /q I Fig. 3: (Color on-line) fσσ′ =√m¯hω3/2(z2/I2)∆Sσσ′,2 at q˜= 3, 5, and 10. With increasing momentum and/or frequency, In Fig. 3 we check the validity of Eq. (11) at T = 0, fσσ′ approaches smoothly to the T = 0 result by Son and by calculating √m¯hω3/2(z2/ 2)∆Sσσ′,2 at different mo- I Thompson [14]. menta. With decreasing temperature T or increasing q˜ q/√T, it approaches gradually to √m¯hω3/2WT=0. ∝ σσ′ This confirms numerically that Eq. (11) holds at zero contain the fugacity z, a count of the term zn on both temperature. Moreover, at high temperatures where the sides of Eq. (1) leads to fugacityissmall,toagoodapproximationwehavethe in- ∆Sσσ′,n(q,ω,T) = (cn/c2)∆Sσσ′,2(q,ω,T), (10) teractionDSF∆Sσσ′ ≃z2∆Sσσ′,2. AsthecontactI ≃I2 Wσσ′(q,ω,T) = z2/I2 ∆Sσσ′,2(q,ω,T),(11) athtehliigmhitTo,fitlaisrgterifvrieaqluteonccyo,nfithrme tEaiql.ω(−151/)2. oNfoWteTt=h0ata,nind ↑↑ whereI2 =z216π2Vc2/λ4 is(cid:0)the co(cid:1)ntactupto the second W↑T↓=0isfairlyevidentinthesecondordervirialexpansion order expansion [18]. Therefore, the Wilson coefficient is functions. givenbythe secondexpansionfunction. This resultis ob- UniversalDSFatlarge(q,ω).– Inthislimit,using tainedbyapplyingtheOPEandvirialexpansionmethod. Eqs. (1) and (11) the DSF is approximated by Asaresult,inprincipleitshouldbevalidattemperatures above the superfluid transition. However, we may expect λ4 thatitholdsatalltemperatures,asboththe Wilsoncoef- Sσσ′ ≃Sσ(1σ)′(q,ω,T)+ 1I6πV ∆Sσσ′,2(q,ω,T). (12) ficientandsecondexpansionfunctionareirrelevanttothe many-body pairing in the superfluid phase. The many- This approximate DSF should be quantitatively accurate body effect enters through the many-body parameter of forsufficientlylargemomentumandfrequency. Itholdsat contact only. alltemperaturesandthemany-bodyeffectisrespectedby Eq. (11) is the main result of this Letter. At a first the contact. However, the momentum q should be larger glance, it may be a suprising result. However, it could be than a critical momentum q max(λ−1,k ) in order c F ≫ understood from the proportionality shown in Eq. (10). to validate the use of the OPE equation (1). Here, k F On the other hand, a rigorous proof of Eq. (10) at large isthe Fermiwavevector. Quantitatively,anestimate ofq c q and ω justifies the use of the OPE method. We note requiresthecalculationof∆Sσσ′,n>2 andtheexamination that, for the two-component Fermi gas, our result shows ofEq. (10). We note thatEq. (12)maynotbe reliable at thatthe Wilsoncoefficientis determinedby the two-body small frequency, ω 0. As a result, the structure factor ∼ physics solely. This is in agreement with previous work. sum-rulesmaynotbestrictlysatisfied. Wenotealsothat, For example, in Ref. [13] a two-body scattering ampli- inthelimitofhightemperatureswherethecontact , 2 I ≃I tude was used to calculate the zero temperature Wilson the prefactor of the ∆Sσσ′,2 term is z2. Therefore, the coefficient for the rf-spectroscopy of a strongly interact- approximateDSF reduces back to the virialexpansionup ing Fermi gas. However, in general cases where three or tothe secondorder. Inthis high-T limit, Eq. (12)isvalid four-body physics come into play, we anticipate that the for arbitrary q and ω. Wilson coefficient should be related to the higher order We present in Fig. 4 the temperature dependence of virial expansion function. In that case, new universal re- the approximation DSF of a normal, homogeneous uni- lations with new many-body “contact” parameters would tary Fermi gas at q = 5k , by assuming that q 5k . F c F ∼ appear. Themany-bodycontactandfugacityarecalculatedbyus- p-4 Universal dynamic structure factor of a strongly correlated Fermi gas ing a strong-coupling pair fluctuation theory [18], which [3] Steward J. T., Gaebler J. P., Regal C. A. and Jin is shown to be accurate for describing the unitary equa- D. S., Phys. Rev. Lett., 97 (2006) 220406. tion of state. Close to the superfluid transition temper- [4] LuoL.,ClancyB.,JosephJ.,KinastJ.andThomas ature, a quasielastic peak clearly emerges at the recoil J. E., Phys. Rev. Lett., 98 (2007) 080402. energy for pairs, ω = h¯q2/(4m), in agreement with the [5] Nascimbne S., Navon N., Jiang K. J., Chevy F. and Salomon C.,Nature, 463 (2010) 1057. low-temperature experimental observation [19]. [6] Hu H., Drummond P. 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Phys., 323 be determined independently using the f-sum ruleI[21]. (2008) 2971; Ann. Phys., 323 (2008) 2987. These predictions break down below the critical momen- [12] Braaten E. and Platter L., Phys. Rev. Lett., 100 tum q . Note that, by tuning the transferred momentum (2008) 205301. c in Bragg beams, the value of q max(λ−1,k ) might [13] Braaten E., Kang D. and Platter L., Phys. Rev. c ≫ F Lett., 104 (2010) 223004. be determined experimentally. [14] SonD.T.andThompsonE.G.,Phys.Rev.A,81(2010) Conclusion. – We have studied the finite tempera- 063634. ture dynamic structure factor of a homogeneous unitary [15] HoT.-L.andMuellerE.J.,Phys.Rev.Lett.,92(2004) 160404. Fermi gas, using quantum virial expansion and operator [16] Horowitz C. J. and Schwenk A., Phys. Lett. B, 638 productexpansion. The universalsecondorderexpansion (2006) 153. functionhasbeencalculatedandrelatedtotheWilsonco- [17] Liu X.-J., Hu H. and Drummond P. D., Phys. Rev. efficient at large momentum q. 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