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High momentum response of liquid 3He F. Mazzanti,1 A. Polls,2 J. Boronat,3 and J. Casulleras3 1 Departament d’Electrònica, Enginyeria i Arquitectura La Salle, Pg. Bonanova 8, Universitat Ramon Llull, E-08022 Barcelona, Spain 2 Departament d’Estructura i Constituents de la Matèria, 4 Diagonal 645, Universitat de Barcelona, 0 E-08028 Barcelona, Spain 0 and 2 3 Departament de F´ısica i Enginyeria Nuclear, Universitat Politècnica n de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain a (Dated: February 2, 2008) J Afinal-state-effectsformalismsuitabletoanalyzethehigh-momentumresponseofFermiliquidsis 9 presentedandusedtostudythedynamicstructurefunctionofliquid3He. Thetheory,developedas anaturalextensionoftheGersch-Rodriguezformalism, incorporatestheFermistatisticsexplicitely ] h through a new additive term which depends on the semi-diagonal two-body density matrix. The c use of a realistic momentum distribution, calculated using the diffusion Monte Carlo method, and e the inclusion of this additive correction allows for a good agreement with available deep-inelastic m neutron scattering data. - t PACSnumbers: 67.55.-s,61.12.Bt a t s . t Inelasticneutronscatteringisthemostefficienttoolto ferred momenta are not high enough. Therefore, FSE a explore the structure and dynamics of quantum liquids which take into account the interactions of the struck m 4 3 Heand HesincethedynamicstructurefunctionS(q,ω) atom with the medium can not be disregarded. - d isreadilyobtainedfromthedoubledifferentialscattering The search for an unambiguous experimental signa- n cross-section[1]. The range of momenta q transferred to ture of n0 in liquid 4He using DINS has originated a o the system determines the kind of microscopic informa- great deal of theoretical and experimental work for the c tionthatcanbeextracted. Themostinterestingregimes last two decades. At present, theoretical predictions [ correspond to low and high q’s. At low q, the scatter- [6,7,8,9,10]forboththeFSEandtheIAprovideasat- 1 ing data allows for the determination of the low-energy isfactory description of the experimental measurements, v excitation spectrum. In the opposite limit, known as withanoverallagreementonthevalueofthecondensate 5 3 deep-inelastic neutron scattering (DINS), q is so high fraction, n0 ∼9% at the equilibrium density. It is worth 1 that single-particle properties of the system become ac- noticing thatFSE insuperfluid 4He are enhanceddue to 1 cessible. n0 and therefore, even at the highest momenta achieved 0 It is well known that in the q → ∞ limit, S(q,ω) ap- in the laboratory, FSE play a fundamental role. Com- 4 0 proaches the impulse approximation (IA). The only in- paratively, few works have been devoted to the analysis / gredienttocalculatetheresponseinIAisthemomentum of the high-q response of liquid 3He. The main reasons t a distribution n(k), a fundamental function in the study underlyingthissituationare,fromtheexperimentalside, m of 4He, 3He, and the 4He-3He mixture. The boson and the large neutron absorption of 3He, and from the the- - fermion quantum statistics of 4He and 3He, respectively, oretical one, the difficulties the Fermi statistics of 3He d introduce significant differences in their corresponding introducesin the quantummany-bodycalculations. The n o momentum distributions. Liquid 4He presents a macro- most accurate data have been reported by Azuah et al. c scopic occupation of the zero-momentum state, charac- [11], andmore recentlyby Senesiet al. [12], butonly the v: terized by its condensate fraction n0; n(k) of liquid 3He, firstone wascarriedoutatthe equilibriumdensity. FSE i consideredas a normalFermi liquid, shows a discontinu- in 3He havebeen takeninto accountby Moroniet al. [8] X ity at the Fermi momentum kF [2]. Nowadays, different using the bosonic formalism of Carraro and Koonin [6]. r theoretical calculations of n(k) ranging from variational Their results [8] show less strength at the peak than the a theory, based on the (Fermi-)hypernetted-chain equa- experimental S(q,ω), pointing to possible limitations of tions((F)HNC),tothemoreexactdiffusionMonteCarlo the formalism when applied to a Fermi liquid. On the (DMC) method are in an overallquantitative agreement other hand, an analysis of the experimental data based [3, 4, 5]. However, a direct comparison with experimen- on cumulant expansions [1, 13] has revealed significant tal data is not possible due to instrumental resolution differencesbetweentheexperimental andtheoretical3He effects (IRE) and, more fundamentally, to final state ef- momentum distributions at equilibrium density [11]. fects (FSE). From the theoretical side, the problem is We present in this letter results for the high-q re- that the IA does not account completely for the scat- sponse of 3He using a theoretical formalism that incor- tering in most of the DINS experiments since the trans- poratesexplicitlyandconsistentlytheFermistatisticsto 2 the FSE. The inputs required are the momentum dis- sponding to a system of spinless bosons with the mass, 3 B tribution n(k) and the semi-diagonal two-body density density, and interatomic potential of He. ρ is positive N matrix ρ2(r1,r2;r1′,r2). Both are obtained from micro- and then it can be used as the starting point of a cumu- scopictheory: n(k)fromaDMCcalculation,andρ2from lant expansion of the response function. In terms of ρBN, variationalFHNC.TheresultsobtainedforS(q,ω)repro- a convenient decomposition of ρ turns out to be N duce the experimentaldata better than previousestima- ttihoenFs,SpEo.inting to non-negligible Fermi contributions to ρN(r1,...,rN;r1′)=ρ1(r11′)(cid:20)ρB1(1r11′) As long as 4He is concerned, most theories introduce FSE as a convolution in energies, which turn into an al- ×ρBN(r1,...,rN;r1′)(cid:21)+∆ρN(r1,...,rN;r1′) , (2) gebraic product in time representation. Hence, FSE are included by means of a new function R(q,t) which mul- ρB1(r11′) being the one-body density matrix extracted tiplies SIA(q,t) to obtain the total response, S(q,t) = fromρBN. Inthethermodynamiclimit,ρB1(r11′)factorizes SIA(q,t)R(q,t). Thisconvolutiveapproachisclearlyam- fromρB,andthusthefirstterminEq. (2)correspondsto N biguous when applied to fermions because, contrary to an artificialN-body density matrix containing fermionic the Bose case, SIA(q,t) has an infinite number of nodes, correlations between points 1 and 1′ only. a fact that leads to a singular definition of R(q,t). In Inserting ρ (2) in S(q,t) (1), the 3He response be- N order to overcome this serious drawback that emerges comes from a direct translation of the FSE theories for bosons to fermions, we have used an alternative formulation S(q,t)=SIA(q,t)R(q,t)+∆S(q,t) , (3) that can be considered a natural extension of Gersch- Rodriguez theory [14] to fermionic systems. In previous with SIA(q,t) the exact 3He IA, and R(q,t) the Gersch- works [15, 16], we have applied this theory to evaluate Rodriguez FSE function calculated with the bosonic FSE in 4He-3He mixtures, but there the fermionic cor- semi-diagonaltwo-body density matrix ρB2(r1,r2;r1′), rections are much smaller due to the low 3He concentra- 1 tions. R(q,t) = exp − drρB(r,0;r+s) Athighq,thedensity-densitycorrelationfactorS(q,t) (cid:20) ρB1(r11′)Z 2 can be well approximated by i s × 1−exp ds′∆V(r,s′) .(4) S(q,t) = 1 eiωvqqs dr1...drNρN(r1,...,rN;r1+s) (cid:20) (cid:18)vq Z0 (cid:19)(cid:21)(cid:21) N! Z The new additive term ∆S(q,t) in Eq. (3) is a con- i N s sequence of ∆ρN introduced in Eq. (2). The leading ×exp ∆V(r1j,s), (1) contribution to the FSE at high q depends on the semi- vq Xj=2Z0 diagonal two-body density matrix, and then ∆ρ2 is re-   quired for the calculation of ∆S(q,t). A cluster expan- with ωq = q2/2m, vq = q/m, s = vqt, and ∆V(r,u) = sionin the frameworkof the FHNC formalismallows for V(r−u)−V(r). Equation (1) is derived assuming that an estimation of ∆ρ2 according to the following struc- the atom struck by the neutron recoils in a medium of ture, non-moving 3He atoms. S(q,t) is still hard to evaluate usingEq. (1)sinceitimpliesanintegrationoverthecom- 1∆ρ2(r1,r2;r1′) = ρ1(r11′)G(r1,r2;r1′) plete semi-diagonal N-body density matrix of the sys- ρ tem, which is essentially the square of the ground-state +ρ1D(r11′)F(r1,r2;r1′) . (5) wave function. In 4He, a truncated cumulant expansion of Eq. (1) leads to the Gersch-Rodriguez expression for The form factors G(r1,r2;r1′) and F(r1,r2;r1′) can be the FSE [14]. Recently, this formalism has proven its expressed in terms of auxiliary functions defined in the efficiency by reproducing 4He DINS data with high ac- FHNC theory, and ρ1D(r11′) is positive everywhere and curacy[7]. Onthe contrary,the nodalstructure ofρ in similar to a bosonic one-body density matrix [17, 18] . N a Fermi system prevents from a straightforwardapplica- Therefore, ρ1D(r11′) can be used as the basis of a cumu- tion of these methods. lant expansion by simply adding and subtracting it to 3 In order to extend the FSE theory to He, one in- ∆ρ . The resulting additive correction ∆S(q,t) is, to N troduces an auxiliary N-body density matrix ρB corre- the lowest order, N 1 1 i s ∆S(q,t)=eiωqs/vq ρ1D(r11′) exp − dr∆ρ2(r,0;r+s) 1−exp ds′∆V(r,s′) −1 . (6) ρ (cid:26) (cid:20) ρ1D(s)Z (cid:20) (cid:18)vq Z0 (cid:19)(cid:21)(cid:21) (cid:27) 3 4 Finally, a Fourier transform of S(q,t) provides the dy- He [7]. Certainly, short-rangecorrelations,which domi- namic structure function S(q,ω). Furthermore,the scal- natetheFSE,arealreadycontainedintheJastrow-Slater ingpropertiesoftheIA,intermsoftheWestvariable[19] approximation. Accordingly, the diagrammatic analysis Y =mω/q−q/2,suggestsas usualto write the response for 3He has been performed at the two-body level, thus (q/m)S(q,ω) as a Compton profile, making the analysis easier. TheFSE broadeningfunctionR(q,Y)atρ0 andamo- J(q,Y)= dY′J(Y′)R(q,Y −Y′)+∆J(q,Y) , (7) mentum transfer q = 19.4 ˚A−1 is shown in Fig. 1. This Z valueofqhasbeenusedthroughoutthisworksinceitcor- J(Y)=1/(2π2ρ) ∞ dkkn(k) being the IA. respondstothemomentumreportedinthe experimental |Y| data by Azuah et al. [11]. R(q,Y) has been calculated The microscoRpic functions entering the high- in the bosonic approximation and then its structure is momentum response J(q,Y) (7) are the one- and similarto the FSE function of4He [7]. When q increases the semi-diagonal two-body density matrices of the R(q,Y)narrowsandsharpens,becomingadeltafunction actual system and its bosonic counterparts. The most in the q →∞ limit. relevant quantity is the one-body density matrix, or The 3He additive correction ∆J(q,Y) = equivalently the momentum distribution, which is used 3 (q/m)∆S(q,Y) at the same density and momen- to evaluate J(Y). We have estimated the He momentum transfer is also shown in Fig. 1. This function, tum distribution using the DMC methodology that has which introduces fermionic correlations to the FSE, recently proved to be very accurate in the calculation of the 3He equation of state at zero temperature [20, 21]. presents a shape that is entirely different from that At the equilibrium density ρ0 = 0.273 σ−3 (σ = 2.556 of R(q,Y). The strong oscillations that appear in the ˚A), n(k) is well parameterized by region Y ≈ ±kF modify the shape of the IA response around these points. Furthermore, a central peak a0−a3x3 x≤1 centered at Y = 0 enhances the strength of the total n(x)=(cid:26)(b0+b1x+b2x2)e−btx x>1 , (8) response at the origin. Out of this region (|Y| >∼ kF), ∆J(q,Y) is much smaller and its correction to the with x=k/kF, kF being the Fermi momentum. The set response becomes negligible. Further analysis indicates of parameters that best fit n(k) (8) is reported in Table that ∆J(q,Y) decays to zero in the high momentum I. The kinetic energy per particle, related to the second transfer limit. This fact, together with the limiting moment of n(k), is 12.3 K and the discontinuity of n(k) condition R(q →∞,Y)→δ(Y), indicates that the total atthe FermisurfaceisZ =0.236. Thevalue ofZ,which response asymptotically approaches J(Y) when q →∞. defines the strength of the quasi-particle pole, is rather The final result of the 3He response at q = 19.4 ˚A−1 small, indicating that the system is strongly correlated. is shown in Fig. 2. We compare our results with the Ontheotherhand,thetailofthemomentumdistribution DINS data of Azuah et al. [11] because their data cor- extends up to high momenta generatingsignificanthigh- respond to densities around the equilibrium density ρ0, energy wings in J(q,Y). The present n(k) is in overall and also because their experimental setup produces a agreement with the DMC one from Ref. [3]. rathernarrowinstrumentalresolutionfunction. Morere- The semi-diagonal two-body density matrix ρ2, and cent data [12] are focused at higher liquid densities and theauxiliarybosonicfunctionsρB1 andρB2,havebeenob- to the solid phase. In this experiment, the momentum tained inthe frameworkofthe FHNC andHNC theories transfer is much larger (q ∼ 90 to 120 ˚A−1) produc- using a Jastrow-Slater variational wave function. It is well known that this trial wave function is not accurate enoughifthemainobjectiveistogetagoodupper-bound 2 tleotttehre.eInnefragcyt.,wTehihsaivsensohtohwonwienvperretvhieouasimwoorfktthheaptraesJeanst- oY) (A) 1.5 trowwavefunctioncanefficientlyaccountfortheFSEin J(q, 1 ∆ & 0.5 Y) a0 0.481319 R(q, 0 a3 0.0842956 -0.5 b0 1.39056 -2 -1 0 1 2 b1 0.157930 Y (Å-1) b2 0.0829832 bt 2.31398 FIG. 1: FSE broadening function R(q,Y) (solid line) and additive correction ∆J(q,Y) (dashed line) at q = 19.4 ˚A−1 and ρ0. ∆J(q,Y) is multiplied by a factor 20 to fit into the TABLE I:Parameters of n(k) (8) at ρ0. scale. 4 viously achieved in the study of the high-q response of 4 liquid He. The two key features behind the present re- 0.6 0.6 sults are, on one hand, the use of a realistic momentum 0.5 distribution provided by the DMC method, and on the other, the explicit introduction of Fermi corrections in 0.4 0.4 the FSE. The latter effect is estimated at the lowest or- oY) (A) 0.3-0.5 0 danerdbautbeittsteirncklnuosiwolnedaglleowofssfporecaifiscigmnieficchaanntisimmpsrionvfleumeennct- J(q, 0.2 ing the FSE in liquid 3He. This workhas beenpartiallysupportedby GrantsNo. BFM2002-01868and BFM2002-00466from DGI (Spain) 0 and Grants No. 2001SGR-00064 and 2001SGR-00222 from the Generalitat de Catalunya. -4 -2 0 2 4 6 o -1 Y (A ) FIG.2: 3Heresponseatq=19.4˚A−1andρ0(solidline),folded totheIREfunction(dottedline). Thepointswitherrorbars [1] H. R. Glyde, Excitations in Liquid and Solid Helium are the experimental data from Ref. [11]. The instrumental (Oxford UniversityPress, Oxford, 1994). resolution function [11] hasbeen dividedbya factor 10 tofit [2] MomentumDistributions,ed.byR.SilverandP.E.Sokol into the scale. The inset shows J(q,Y) near the peak with (Plenum Press, New York,1989). (solid line) and without (dashed line) the additivecorrection [3] S.Moroni,G.Senatore,andS.Fantoni,Phys.Rev.B55, ∆J(q,Y). 1040 (1997). [4] E.Manousakis,V.R.Pandharipande,andQ.N.Usmani, Phys. Rev.B 31, 7022 (1985). [5] A. Fabrocini and S. Rosati, Nuovo Cimento D 1, 567 ing a simultaneous decrease of the FSE corrections and (1982); 1, 615 (1982) . a widening of the IRE. The IRE function estimated by [6] C. Carraro and S. E. Koonin, Phys. Rev.Lett. 65, 2792 Azuah et al. [11] is shown in Fig. 2 scaled by a factor (1990). 0.1. The theoretical response J(q,Y), shown in the fig- [7] F. Mazzanti, J. Boronat, and A.Polls, Phys.Rev.B 53, ure, has been foldedwith the experimentalIRE function 5661 (1996). I(q,Y) to make a direct comparisonpossible. The kinks [8] S. Moroni, S. Fantoni, and A. Fabrocini, Phys. Rev. B of J(q,Y) at Y =±kF present in the IA, are completely 58,11607 (1998). [9] A.S. Rinat et al.,Phys.Rev.B 57, 5347 (1998). washed out by the succesive folding with R(q,Y) and [10] H.R. Glyde, R.T. Azuah, and W.G. Stirling, Phys. Rev. I(q,Y), although ∆J(q,Y) introduces additional struc- B 62, 14337 (2000). turearoundthosepoints. Themostremarkablefeatureis [11] R.T.Azuahet al.,J.LowTemp.Phys.101,951(1995). the enhancement of the strength of the response around [12] R. Senesi et al.,Phys. Rev.Lett. 86, 4584 (2001). thepeakduetothenewaddititvetermintroducedinthe [13] H.R. Glyde, Phys.Rev.B 50, 6726 (1994). present approach (see inset in Fig. 2). This small but [14] H. A. Gersch and L. J. Rodriguez, Phys. Rev. A 8, 905 significantincreaseofstrengthallowsforthefirsttimeto (1973). reproduce the available 3He DINS experimental data. [15] F. Mazzanti, Phys.Lett. A 270, 204 (2000). [16] F. Mazzanti, A.Polls, andJ. Boronat, Phys.Rev.B 63, To summarize, we have presented a FSE formalism 054521 (2001). suitable to study the dynamic structure function of a [17] M. L. Ristig and J. W. Clark, Phys. Rev. B 41, 8811 Fermi liquid like liquid 3He at large momentum trans- (1990). fer. The method is a natural extension of the Gersch- [18] A. Polls and F. Mazzanti, in Introduction to Modern Rodriguez theory for bosons. According to the present MethodsofQuantumMany-BodyTheoryandtheirAppli- actions ed.byA.Fabrocini,S.FantoniandE.Krotscheck formulation, J(q,Y) results from the convolution of the (World Scientific, Singapore, 2002). IAwithapurelybosonicFSEbroadeningfunction,which [19] G. B. West, Phys.Rep. C 18, 263 (1975). incorporates the short-range correlations induced by the [20] J. Casulleras and J. Boronat, Phys. Rev. Lett. 84, 3121 interatomic potential, plus an additive correction term (2000). thattakesintoaccountFermistatisticseffectsintheFSE. [21] J. Boronat, in Microscopic Approaches to Quantum Liq- The results obtained are in good agreement with DINS uids in Confined Geometries, ed. by E. Krotscheck and data, comparable for the first time to the accuracy pre- J. Navarro(World Scientific, Singapore, 2002).