Dynamic structure factor of liquid 4He across the normal-superfluid transition
aa r X i v : . [ c ond - m a t . o t h e r] M a r Dynamic structure factor of liquid He across the normal-superfluid transition
G. Ferr´e and J. Boronat ∗ Departament de F´ısica, Universitat Polit`ecnica de Catalunya,Campus Nord B4-B5, E-08034, Barcelona, Spain
We have carried out a microscopic study of the dynamic structure factor of liquid He acrossthe normal-superfluid transition temperature using the path integral Monte Carlo method. Theill-posed problem of the inverse Laplace transform, from the imaginary-time intermediate scatteringfunction to the dynamic response, is tackled by stochastic optimization. Our results show a quasi-particle peak and a small and broad multiphonon contribution. In spite of the lack of strength in thecollective peaks, we clearly identify the rapid dropping of the roton peak amplitude when crossingthe transition temperature T λ . Other properties such as the static structure factor, static response,and one-phonon contribution to the response are also calculated at different temperatures. Thechanges of the phonon-roton spectrum with the temperature are also studied. An overall agreementwith available experimental data is achieved. PACS numbers: 67.25.dt,02.70.Ss,02.30.Zz
I. INTRODUCTION
The most relevant information on the dynamics of aquantum liquid is contained in the dynamic structure fac-tor S ( q , ω ), which is experimentally measured by meansof inelastic neutron scattering. Probably, superfluid Hehas been the most deeply studied system from both the-ory and experiment and a great deal of information aboutit is nowadays accessible. For many years, liquid He wasthe only quantum fluid showing Bose-Einstein conden-sation and superfluidity until the discovery of the fullyBose-Einstein condensate in 1995.
Therefore, the num-ber of measures of S ( q , ω ) at different temperatures andmomentum transfer has been continuously growing, withmore refined data along the years. The emergenceof strong quasi-particle peaks going down the normal-superfluid transition ( T λ = 2 .
17 K) has been associatedwith the superfluidity of the system by application of theLandau criterium. Much of the interest on the dynamicsof strongly-correlated liquid He is then related to theeffects on the dynamics of this second-order λ -transition.In the limit of zero temperature, the richest and mostaccurate microscopic description of the dynamic responseof liquid He has been achieved by progressively moresophisticated correlated basis function (CBF) theory. The development of this theory has been stimulated bythe continuous improvement of the experimental reso-lution in inelastic neutron scattering. Recently, Camp-bell et al. have incorporated three-body fluctuationsin an extended CBF approach and proved a remarkableimprovement of both the excitation spectrum and full S ( q , ω ), with features not so clearly seen before and thatare in nice agreement with the most recent experimen-tal data. On the other hand, the most accurate toolsto deal with ground-state properties are the quantumMonte Carlo (QMC) methods. In the case of bosons as He, these methods are able to produce essentially ex-act results for its equation of state and structure prop-erties which are in close agreement with experimentaldata. Importantly, QMC methods are not restricted to the limit of zero temperature and are equally powerfulto introduce the temperature as a variable through thesampling of the statistical density matrix, implementedby the path integral Monte Carlo (PIMC) method. QMC methods simulate quantum systems usingimaginary-time dynamics since they are intended forachieving the lowest-energy state. Therefore, having noaccess to real-time evolution one looses the possibility ofgetting the dynamic structure factor by a Fourier trans-form of the intermediate scattering function F ( q , t ), as ithappens in simulations of classical systems using Molec-ular Dynamics. Quantum simulations are able to sam-ple this time-dependent function but in imaginary time τ , F ( q , τ ), and from it to get the dynamic responsethrough an inverse Laplace transform. But it is wellknown that this inverse transform is an ill-posed prob-lem. This means, at the practical level, that the alwaysfinite statistical error of QMC data makes impossible tofind a unique solution for the dynamic structure factor.Inverse problems in mathematical physics are a long-standing topic in which elaborated regularization tech-niques have been specifically developed. Focusing onthe inversion of QMC data, to extract the dynamic re-sponse, several methods have been proposed in the lastyears. Probably, the most used approach is the Max-imum Entropy (ME) method which incorporates somea priori expected behavior through an entropic term. This method works quite well if the response is smoothbut it is not able to reproduce responses with well-definedpeaks. In this respect, other methods have recentlyproved to be more efficient than ME. For instance, theaverage spectrum method (ASM), the stochastic op-timization method (SOM), the method of consistentconstraints (MCC), and the genetic inversion via falsi-fication of theories (GIFT) method have been able torecover sharp features in S ( q , ω ) which ME smoothedout. All those methods are essentially stochastic op-timization methods using different strategies and con-straints. It is also possible to work out the inverse prob-lem without stochastic grounds by using the Moore-Penrose pseudoinverse and a Tikhonov regularization. Other approaches try to reduce the ill-conditioned char-acter of this inverse problem by changing the kernel fromthe Laplace transform to a Lorentz one. Finally, thecomputation of complex-time correlation functions hasbeen recently realized in simple problems and proved tobe able to severely reduce the ill-nature of the Laplacetransform. In this paper, we use the PIMC method to estimatethe dynamic response of liquid He in a range of tem-peratures covering the normal-superfluid transition at T λ = 2 .
17 K. The inversion method from imaginarytime to energy is carried out via the simulated annealingmethod, which is a well-known stochastic multidimen-sional optimization method widely used in physics andengineering. Our method is rather similar to the GIFTone but changing the genetic algorithm by simulatedannealing. The GIFT method was applied to the studyof the dynamic response of liquid He at zero temperatureand proved to work much better than ME, producing arather sharp quasi-particle peak and also some structureat large energies, corresponding to multiparticle excita-tions. The temperature dependence of S ( q , ω ) has beenmuch less studied. Apart from a quantum-semiclassicalestimation of the response at high q , the only reportedresults where obtained by combining PIMC and the MEmethod which worked well in the normal phase but notin the superfluid part. Therefore, the significant effectof the temperature on the dynamics of the liquid throughthe λ transition was lost. We show that the improvementon the inversion method leads to a significantly better de-scription of S ( q , ω ) in all the temperature range studied,with reasonable agreement with experimental data.The rest of the paper is organized as follows. A shortdescription of the PIMC method and a discussion of theinversion method used is contained in Sec. II. In Sec. III,we report the results achieved for the dynamic response,excitation spectrum, phonon strength and lowest energy-weighted sum rules across the transition. Finally, themain conclusions and a summary of the main results arecontained in Sec. IV. II. METHOD
The thermal density matrix of a quantum system isgiven by ˆ ρ = e − β ˆ H Z , (1)where β = 1 / ( k B T ), k B is the Boltzmann constant, and Z = Tr( e − β ˆ H ) is the partition function. The knowledgeof ˆ ρ allows for the calculation of the expected value ofany operator ˆ O , h ˆ O i = Tr(ˆ ρ ˆ O ) , (2) which in coordinate representation turns to h ˆ O i = Z d R ρ ( R , R ; β ) O ( R ) , (3)with R = { r , . . . , r N } for an N -particle system. Deepin the quantum regime, i.e. at very low temperature, theestimation of the density matrix for a many-body systemis obviously a hard problem. However, the convolutionproperty of ˆ ρ , ρ ( R , R M +1 ; β ) = Z d R . . . d R M M Y j =1 ρ ( R j , R j +1 ; τ ) , (4)with M an integer and τ = β/M , shows how to buildthe density matrix at the desired temperature T froma product of density matrices at a higher temperature M T . If the temperature is large enough, one is able towrite accurate approximations for ˆ ρ and thus the quan-tum density matrix can be calculated, as stated by theTrotter formula e − β ( ˆ K + ˆ V ) = lim M →∞ (cid:16) e − τ ˆ K e − τ ˆ V (cid:17) M . (5)In Eq. (5), we have considered a Hamiltonian ˆ H = ˆ K + ˆ V ,with ˆ K and ˆ V the kinetic and potential operators, re-spectively. In the limit of high temperature the sys-tem approaches the classical regime where e − β ( ˆ K + ˆ V ) = e − β ˆ K e − β ˆ V . This factorization, called primitive approx-imation, is however not accurate enough to simulate aquantum liquid as He because the number of requiredterms ( beads ) M is too large. To make our PIMC sim-ulations of superfluid He reliable, we use a fourth-ordertime-step ( τ ) approximation due to Chin, following theimplementation discussed in Ref. 29. Liquid He is aBose liquid and thus we need to sample not only particlepositions but permutations among them. To this end, weuse the worm algorithm. In the present work, we are mainly interested in calcu-lating the intermediate scattering function F ( q , τ ), de-fined as F ( q , τ ) = 1 N h ˆ ρ q ( τ ) ˆ ρ † q (0) i , (6)with ˆ ρ q ( τ ) = P Ni =1 e i q · r i the density fluctuation oper-ator. The function F ( q , τ ) is the Laplace transform ofthe dynamic structure factor S ( q , ω ) which satisfies thedetailed balance condition, S ( q , − ω ) = e − βω S ( q , ω ) , (7)relating the response for negative and positive energytransfers ω . Taking into account Eq. (7), one gets F ( q , τ ) = Z ∞ dω S ( q , ω )( e − ωτ + e − ω ( β − τ ) ) . (8)The intermediate scattering function is periodic with τ ,as it can be immediately seen from Eq. (8): F ( q , β − F ( q , t ) t (K −1 )q = 0.43 Å −1 q = 1.24 Å −1 q = 1.91 Å −1 FIG. 1. (Color online) Intermediate scattering function com-puted for He at saturated vapor pressure ( ρ = 0 . − )and T = 1 . q . τ ) = F ( q , τ ). Therefore, it is necessary to sample thisfunction only up to β/ F ( q , τ ) at the discrete points inwhich the action at temperature T is decomposed (Eq.4).In Fig. 1, we show the characteristic behavior of F ( q , τ ) for three different q values at T = 1 . T / T →
0. Theinitial point at τ = 0 corresponds to the zero energy-weighted sum rule of the dynamic response, which in turnis the static structure factor at that specific q value, m = S ( q ) = Z ∞−∞ dω S ( q , ω ) . (9)With the PIMC results for F ( q , τ ), the next step is tofind a reasonable model for S ( q , ω ) having always in mindthe ill-conditioned nature of this goal. In our scheme, weassume a step-wise function, S m ( q, ω ) = N s X i =1 ξ i Θ( ω − ω i ) Θ( ω i +1 − ω ) , (10)with Θ( x ) the Heaviside step function, and ξ i and N s parameters of the model. As our interest relies on thestudy of homogeneous translationally invariant systems,the response functions depend only of the modulus q .Introducing S m ( q, ω ) in Eq. (8), one obtains the corre- S ( q , w ) ( K − ) w ( K )Normal averageAverage (cut) c Average c Average (cut) −5 −4 −3 −2 −1
10 20 30 40 50 60
FIG. 2. (Color online) Dynamic structure factor at T = 1 . q = 0 .
76 ˚A − usingdifferent averaging methods. Inset shows same data but usinga log scale in the y -axis. sponding model for the intermediate scattering function, F m ( q, τ ) = N s X i =1 ξ i (cid:20) τ (cid:0) e − τω i − e − τω i +1 (cid:1) (11)+ 1 β − τ (cid:16) e − ( β − τ ) ω i − e − ( β − τ ) ω i +1 (cid:17)(cid:21) Written in this way, the inverse problem is convertedinto a multivariate optimization problem which tries toreproduce the PIMC data with the proposed model, Eq.11. To this end, we use the simulated annealing methodwhich relies on a thermodynamic equilibration procedurefrom high to low temperature according to a predefinedtemplate schedule. The cost function to be minimizedis the quadratic dispersion, χ ( q ) = N p X i =1 [ F ( q, τ i ) − F m ( q, τ i )] , (12)with N p the number of points in which the PIMC estima-tion of the intermediate scattering function is sampled.Eventually, one can also introduce as a denominator ofEq. (12) the statistical errors coming from the PIMCsimulations. However, we have checked that this is notaffecting so much the final result since the size of theerrors is rather independent of τ .The optimization leading to S ( q, ω ) is carried outover a number N t of independent PIMC calculations of F ( q, τ ). Typically, we work with a population N t = 24and for each one we perform a number N a = 100 of inde-pendent simulated annealing searches. The mean averageof these N a optimizations is our prediction for the dy-namic response for a given F ( q, τ ). We also register the S ( q , w ) ( K − ) a) This workMaximum Entropy 0 0.05 0.1 0.15 0.2 10 20 30 40 50 60 S ( q , w ) ( K − ) w ( K )b) This workMaximum Entropy FIG. 3. (Color online) Comparison between the present re-sults for the dynamic structure factor and those obtained inRef. 27 using the maximum entropy method for q = 0 . − (a) and 1 .
81 ˚A − (b). Both results are calculated at SVPand T = 1 . mean value of χ (Eq. 12) of the N a optimizations. As anexample, the mean value of χ in a simulation with dataat T = 1 . q = 1 .
91 ˚A − is 2 . · − , with mini-mum and maximum values of 2 . · − and 3 . · − ,respectively. At this same temperature, N p = 41 andthe number of points of the model S ( q, ω ) (Eq. 11) is N s = 150.With the outcome for the N t series we have tried dif-ferent alternatives to get the final prediction. We cantake just the statistical mean of the series or a weightedmean, in which the weight of each function is the in-verse of its corresponding χ , to give more relevance tothe best-fitted models. Additionally, we have also triedto make both of these estimations but selecting the 20%best functions according to its χ . In Fig. 2, we plot theresults obtained following these different possibilities. Allthe results are quite similar, with minor differences; onlyat large energies we can observe that the weighted mean gives slightly more structure (see inset in Fig. 2). Also,the effect of selecting the best χ models seems to be notmuch relevant.In Fig. 3, we compare the results obtained for the dy-namic structure factor at T = 1 . The ME results aresignificantly broader, mainly at the lowest q value, andwith only smooth features. This broadening is probablya result of the entropic prior used in those estimations,which seems to favor smooth solutions. In the figure, wecan observe that the position of the ME peak is coinci-dent with ours but the ME solution lacks of any structurebeyond the quasi-particle peak (see Appendix for addi-tional comparisons between ME and our stochastic opti-mization procedure). In our estimation, we do not useany prior information in the search of optimal reconstruc-tions and thus it is free from any a priori information ex-cept that the function is positive definite for any energy.Moreover, the simulated annealing optimization leads todynamic responses that fulfil the energy-weighted sumrules m and m , m = Z ∞−∞ dω ωS ( q, ω ) = ~ q m , (13)without imposing them as constraints in the cost function(Eq. 12). Also, the m − sum-rule, related to the staticresponse, is in agreement with experiment (see next Sec-tion). III. RESULTS
We have performed PIMC calculations of liquid Hefollowing the SVP densities, from T = 0 . and the number ofparticles in the simulation box, under periodic boundaryconditions, is N = 64. In some cases we have used alarger number of particles ( N = 128) without observingany significant change in F ( q, τ ). The number of terms M (Eq. 4) is large enough to eliminate any bias comingfrom the path discretization; we used τ = 0 . − .We compare our result for the dynamic response inthe superfluid phase with experimental data from Ref.5 in Fig. 4. The theoretical peak is located around anenergy which is very close to the experimental one butit is still broader than in the experiment. However, thestrength (area) of this peak is in good agreement with theexperimental one, as we will comment later. The quasi-particle peak disappears in the normal phase, above T λ ,as we can see in Fig. 5. In this figure, we compare ourresults at T = 4 K with experimental outcomes at thesame T . In this case, we see that both the position ofthe peak and its shape is in an overall agreement withthe experiment.One of the main goals of our study has been the studyof the effect of the temperature on the dynamics of liquid He. In Fig. 6, we report results of S ( q, ω ) in a range S ( q , w ) ( K − ) w ( K )This work T=1.2 K, q=1.76 Å −1 Expt T=1.3 K, q=1.7 Å −1 FIG. 4. (Color online) Dynamic structure factor at T = 1 .
2K and q = 1 .
76 ˚A − compared with experimental data ( T =1 . K , q = 1 . − ). S ( q , w ) ( K − ) w ( K )This work q=1.39 (Å −1 )Expt q=1.40 (Å −1 ) FIG. 5. (Color online) Dynamic structure factor at T = 4 .
0K and q = 1 .
40 ˚A − . The experimental data is from Ref. 32. of temperatures from T = 0 . q = 0 .
88 ˚A − . At this low q value, the behavior with T is not much different forthe superfluid and normal phases, a feature which is alsoobserved in neutron scattering data. We observe a pro-gressive broadening of the peak with T which appearsalready below T λ and continues above it. Even at thehighest temperature T = 4 K, we identify a collectivepeak corresponding to a sound excitation. The main dif-ference between both regimes is that the quasi-particleenergy below T λ is nearly independent of T whereas, inthe normal phase, this energy decreases in a monotonous S ( q , w ) ( K − ) w ( K ) T=0.8 KT=1.2 KT=2.0 KT=2.3 KT=2.8 KT=3.2 KT=3.6 K FIG. 6. (Color online) Dynamic structure factor of liquid Hefor q = 0 .
88 ˚A − at different temperatures. S ( q , w ) ( K − ) w ( K ) T=0.8 KT=1.2 KT=2.0 KT=2.3 KT=2.8 KT=3.2 KT=3.6 K FIG. 7. (Color online) Dynamic structure factor of liquid Hefor q = 1 .
91 ˚A − at different temperatures. way.Near the roton, the dependence of the dynamic re-sponse with T is significantly different. In Fig. 7, wereport results of S ( q, ω ) at q = 1 .
91 ˚A − at differenttemperatures across T λ . The most relevant feature isthe drop of the quasi-particle peak for T > T λ . In thesuperfluid phase, the peak remains sharp with a nearlyconstant energy. Just crossing the transition (in our datafor T ≥ . R o t o n e n er gy s h i ft D * − D ( T ) ( K ) T (K) ( D * − D (0)) + 19 (cid:214)‘ T e − D (T)/T [K] This workExpt
FIG. 8. (Color online) Temperature dependence of the rotonenergy. The experimental points and suggested fit are fromRef. 33 with ∆(0) = 8 .
62 K . In the fit, ∆ ∗ > ∆( T ) standsfor an arbitrary energy value. with this picture since we observe as the resulting super-fluid density, derived from the winding number estimator,goes to zero at T λ , in agreement with the disappearanceof the roton excitation in S ( q, ω ).Our results for the temperature dependence of the ro-ton energy ∆( T ) are shown in Fig. 8. For temperatures T < . T ) is practically constant around a value8 .
60 K, in agreement with experiment. . For larger tem-peratures, still in the superfluid part, this energy gapstarts to decrease with the largest change around thetransition temperature. For temperatures T > . T ) flattens but then one reallycan not continue speaking about the roton mode. In thesame figure we report experimental results for the rotonenergy in the superfluid phase. At same temperature,our results agree well with the experimental ones whichshow some erratic behavior around T ≃ T . Stillin the same figure, we report the fit used in Ref. 33,that is based on the roton-roton interaction derived fromLandau and Khalatnikov theory. This law seems to beright only at the qualitative level, with significant devi-ation with our results and still larger discrepancies withthe experimental values.The results obtained for S ( q, ω ) in the present calcu-lation are summarized in Fig. 9 as a color map in themomentum-energy plane. In the superfluid phase, thephonon-roton curve is clearly observed, with the higheststrength of the quasi-particle peak located in the rotonminimum, in agreement with experiment. The multipar-ticle part above the single-mode peak is also observed butwithout any particular structure. At T = 2 K the rotonpeak is still observed but some intensity starts to appear −10010203040 w ( K ) T = 1.2 K T = 2.0 K0.5 1.0 1.5 2.0 2.5q (Å −1 )−10010203040 w ( K ) T = 2.5 K 0.5 1.0 1.5 2.0 2.5q (Å −1 ) 0 0.05 0.1 0.15 0.2 0.25 0.3T = 3.0 K FIG. 9. (Color online) Color map of the dynamic response inthe momentum-energy plane at different temperatures, belowand upper T λ . below it, At T = 2 . T = 0 . . T = 1 . q > . − the dynamic response that we obtain from the recon-struction of the imaginary-time intermediate scatteringfunction is rather broad and one can not distinguish thedouble peak structure observed in experiments. Also, no-tice that the energies corresponding to q . . − arenot accessible in our simulations since our minimum q min value is restricted to be 2 π/L , with L the length of thesimulation box. At T = 2 K, very close to the superfluidtransition temperature, we observe as the energies of themaxon and roton modes significantly decrease whereasthe phonon part is less changed. When the temperatureis above the transition, we can observe that the maxi-mum of the peaks, now much broader, seem to collapseagain in a common curve around the maxon. Instead, inthe roton it seems that the energy could increase againat the largest temperature. This latter feature is quiteunexpected and could be a result of our difficulty in thelocalization of the maximum in a rather broad dynamicresponse. The overall description on the evolution of thephonon-roton spectrum with T is in close agreement with w ( K ) q (Å −1 )Expt. T=1.2 KT=0.8T=1.2T=2.0T=2.5T=3.0 FIG. 10. (Color online) Phonon-roton spectrum of liquid Heat different temperatures. The line corresponds to experimen-tal data at T = 1 . . Straight lines at small q standfor the low q behavior, ω = cq with c the speed of sound, attemperatures T = 0 .
8, 1 .
2, and 3 . experimental data. .The static structure factor S ( q ) is the zero energy-weighted sum rule of the dynamic response (Eq. 9).This function can be exactly calculated using the PIMCmethod as it is the value of the imaginary-time inter-mediate scattering function at τ = 0. In Fig. 11, weshow results of S ( q ) for the range of analyzed tempera-tures. The effect of the temperature on the position andheight of the main peak is quite small, in agreement withthe x-ray experimental data from Ref. 37. We observea small displacement of the peak to larger q values anda simultaneous decrease of the height when T increases.These effects can be mainly associated to the decrease ofthe density along SVP when the temperature grows. Forvalues q . . − we do not have available data dueto the finite size of our simulation box. Therefore, wecan not reach the zero momentum value which is relatedto the isothermal compressibility χ T through the exactrelation S ( q = 0) = ρk B T χ T , (14)with k B the Boltzmann constant and ρ the density. Therequirement of this condition produces that S ( q ) startsto develop a minimum around q ≃ . − when T in-creases. Our results also show this feature but for larger T ( ∼ . ∼ q .From the dynamic structure factor, we can calculatethe static response function χ ( q ) since this is directly S ( q ) q (Å −1 )T=0.8 KT=1.2 KT=2.0 KT=2.8 KT=3.2 KT=3.6 KT=4.0 KExpt T=3.6 K FIG. 11. (Color online) Static structure factor S ( q ) at dif-ferent temperatures across T λ . The results have been shiftedvertically a constant value to make its reading easier. Thedashed line stands for experimental data from Ref. 37. Shorthorizontal lines at q = 0 correspond to the value (Eq. 14)obtained from PIMC. − c ( q ) / r q (Å −1 )ExptT=1.2 KT=2.0 KT=2.5 KGIFT at GS FIG. 12. (Color online) Static response function at T =1 .
2, 2 .
0, and 2 . and experimental data obtainedat T = 1 . related to the 1 /ω sum rule through the relation χ ( q ) = − ρ Z ∞−∞ dω S ( q, ω ) ω = − ρm − . (15)The dominant contribution to the m − sum rule is thequasi-particle peak and thus it is less sensitive to themulti-phonon part. In Fig. 12, we report the results Z ( q ) q (Å −1 )Expt T=1.2 KT=1.2 KT=2.0 K FIG. 13. (Color online) One-phonon contribution to the dy-namic response, Z ( q ), at different temperatures. Experimen-tal results at T = 1 . obtained for χ ( q ) at temperatures 1 .
2, 2 .
0, and 2 . q the effect of T is negligiblebut around the peak, q ≃ − , is really large. In thesuperfluid regime, the height of the peak clearly increaseswith T , a feature that has not been reported previouslyneither from theory nor from experiment. At T = 2 . at T = 1 . q but with less strength in thepeak. Results from QMC at zero temperature from Ref.21 are in an overall agreement with ours at the lowest T ,but somehow ours have a slightly higher peak.The dynamic response of liquid He is usually writtenas the sum of two terms, S ( q, ω ) = S ( q, ω ) + S m ( q, ω ) , (16)where S ( q, ω ) stands for the sharp quasi-particle peakand S m ( q, ω ) includes the contributions from scatteringof more than one phonon (multiphonon part). The in-tensity (area) below the sharp peak is the function Z ( q )which we report in Fig. 13. Our results are comparedwith experimental data at T = 1 . T increases, as reported in experiments. IV. CONCLUSIONS
We have carried out PIMC calculations of liquid Hein a wide range of temperatures across the normal-superfluid transition T λ to calculate the imaginary-timeintermediate scattering function F ( q, τ ). From thesefunctions one can in principle access to the dynamicresponse S ( q, ω ) through an inverse Laplace transform.But this is an ill-conditioned problem that can not besolved to deal with a unique solution. In recent works, it has been shown that the use of stochastic optimiza-tion tools can produce results with a richer structurethan previous attempts relying on the maximum entropymethod. We have adopted here the well-known sim-ulated annealing technique to extract the dynamic re-sponse, without any a priori bias in the search in orderto get a result as unbiased as possible. In spite of thelack of any constraint in the cost function, we have veri-fied that the three lowest energy-weighted sum rules aresatisfactorily satisfied giving us some confidence on thereliability of our algorithm.The results of the dynamic response are still notenough sharp in the quasi-particle peaks of the super-fluid phase but the position of the peaks and the area be-low them are in nice agreement with experimental data.Interestingly, our results show clearly the signature ofthe transition in the roton peak, whose amplitude dropsrapidly for
T > T λ . The effect of the temperature onthe phonon-roton spectrum, static structure factor, andstatic response has been also studied.The difficulties of extending correlated perturbativeapproaches to finite T have lead to a really unexploreddynamics of superfluid liquid He, at least from a micro-scopic approach. With the present work, which can beconsidered an extension and improvement of a previouswork based on the maximum entropy method, we haveshown that the combination of PIMC and stochastic re-construction is able to produce a satisfactory descriptionof the quantum dynamics at finite temperature. We arealso convinced that in the near future we can improveeven more the present results. In this respect, one ofthe more promising avenues could be the estimation ofcomplex-time correlation functions, instead of the merelyimaginary ones, which can reduce the ill-posed characterof the inversion problem due to its non-monotonic struc-ture. ACKNOWLEDGMENTS
This research was supported by MICINN-Spain GrantNo. FIS2014-56257-C2-1-P. S ( q , w ) ( K − ) w ( K )ME (Ref. 27), q = 0.75 Å −1 Sim. an. (Ref. 27), q = 0.62 Å −1 Sim. an. present work, q = 0.62 Å −1 FIG. 14. (Color online) Comparison between the dynamicresponse obtained with ME and our stochastic optimizationmethod using intermediate scattering data from Ref. 27. S ( q , w ) ( K − ) w ( K )Sim. annealing, q = 0.62 Å −1 ME, q = 0.62 Å −1 FIG. 15. (Color online) Comparison between the dynamicresponse obtained with ME and our stochastic optimizationmethod using our intermediate scattering data.
APPENDIX
In Fig. 3, we have compared results for S ( q, ω ) derivedfrom our stochastic optimization method and results re-ported in Ref. 27 using ME. As the intermediate scatter-ing function F ( q, τ ) used in both estimations is differentand used by different authors it could happen that thedifferences observed in Fig. 3 were due more to the dif-ferences between the calculated imaginary-time responsethan to the inversion method itself. To clarify this point,we report in this Appendix results of two additinal com-parisons.In Fig. 14, we report results for S ( q, ω ) at q = 0 . − using our imaginary-time data and stochastic opti-mization. In the figure, we also show the dynamic re-sponse that we have obtained by applying our inversionmethod to the imaginary-time data reported in Ref. 27.Finally, the figure also shows the ME results reported inRef. 27 but for a slightly different q value since resultsfor q = 0 .
62 ˚A − are not given in that paper. As one cansee, starting from their published data and applying ourmethod the results compare favorably with our response S ( q, ω ). Therefore, the different quality of the input datais so small that no effect is observed.In order to make a more clear comparison betweenboth inversion methods we show in Fig. 15 results forthe dynamic response using our data for F ( q, τ ). At thesame q value than in Fig. 14, we report results obtaindedwith stochastic optimization and using the ME method.The results are similar to the ones shown in Fig. 14 andlead to the same conclusion, that is, the ME method gen-erates smoother functions than our method. This conclu-sion is in agreement with a similar analysis reported byVitali et al. .0 ∗ [email protected] S. W. Lovesey,
Condensed Matter Physics: dynamic cor-relations (Benjamin-Cummings, San Francisco, 1986). Henry R. Glyde,
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