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Theory and Data Analysis for Excitations in Liquid 4He Beyond the Roton Minimum

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Theory and Data Analysis for Excitations in Liquid 4He Beyond the Roton Minimum
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    a  r   X   i  v  :  c  o  n   d  -  m  a   t   /   9   8   0   6   0   4   0  v   1   3   J  u  n   1   9   9   8 Theory and data analysis for excitations in liquid  4 He beyond the roton minimum F. Pistolesi ∗ Institut Laue-Langevin, B.P. 156, F-38042 Grenoble Cedex 9, France  (February 1, 2008)The hybridization of the single-excitation branch with the two-excitation continuum in the mo-mentum region beyond the roton minimum is reconsidered by including the effect of the interferenceterm between one and two excitations. Fits to the latest experimental data with our model allowus to extract with improved accuracy the high momentum end of the  4 He dispersion relation. Incontrast with previous results we find that the undamped excitations below two times the rotonenergy survive up to  Q  = 3 . 6 ˚A − 1 due to the attractive interaction between rotons.PACS numbers: 67.40.Db, 61.12.-q Although excitations in superfluid  4 He have beenwidely studied in the last decades (see for instanceRef. [1,2]), the nature of the single particle spectrum ter- mination remains unclear. Forty years ago, Pitaevskiipredicted different kinds of termination depending on thedetailed form of the spectrum at low momenta [3]. In thecase of a decay into pairs of rotons, Pitaevskii theory de-scribes the avoided crossing of the bare one-excitationbranch with the continuum of two excitations. Withinthis picture the low energy pole is repelled by the con-tinuum, so that the spectrum flattens out for large  Q  to-wards 2∆ losing spectral weight (∆ is the roton energy).At the same time, a damped excitation for  ω >  2∆ ap-pears and shifts to higher energies. Neutron scatteringexperiments later suggested that the decay of excitationsinto  pairs of rotons   did actually take place for momen-tum  Q > ∼ 2 . 6 ˚A − 1 [4]. The flattening of the spectrum atan energy of the order of 2∆ being the main feature insupport of Pitaevskii’s picture. Despite the good  quali-tative   agreement between theory and experiment, severalissues addressed by the theory could not be verified dueto insufficient instrumental resolution. In particular, the-ory predicts a singular termination of the spectrum at adefinite value of momentum  Q c  for a repulsive interac-tion ( V   4  >  0) between rotons. In the case of an attrac-tive interaction, hybridization between the roton boundstate and the single quasiparticles is instead expected, asproposed by Zawadowski-Ruvalds-Solana (ZRS) [5], withthe consequence that an undamped quasiparticle peak atan energy slightly below 2∆ should be present [5–8]. To our knowledge it has not yet been possible to distinguishclearly between these two cases.Smith  et al.  [9] performed the first complete experi-mental analysis for 2 . 7  ≤  Q  ≤  3 . 3 ˚A − 1 . Although theyfound indications of a repulsive interaction ( V   4  >  0) usingZRS theory, they pointed out that the experimental find-ing of energies for the quasiparticle peak above 2∆ wasnot accounted for by the theory with reasonable values of the parameters. As a matter of fact, the position of thelow-energy peak was not extracted from the data withZRS theory, but rather by fitting a Gaussian peak ona background of constant slope. The resulting spectrumreached values  above   2∆, in contrast with theoretical pre-dictions. From the theoretical point of view it is not pos-sible to explain in a simple way the presence of   sharp peaks at energies above 2∆, as the corresponding excita-tion should be unstable towards decay into two rotons.Moreover, the experimental finding of a repulsive interac-tion disagrees with different theoretical calculations thatpredict a negative value for  V   4  [10,7]. More recent ex- perimental investigations by F˚ak and coworkers [11,12] concentrate on the interesting temperature dependenceof the dynamical structure factor  S  ( Q ,ω ) and do not ad-dress directly the issue of the quasiparticle energy or theinteraction potential between rotons. In any case theyshow clearly that there is a strong correlation betweenthe low-energy peak and the high energy continuum as Q  increases from 2.3 to 3.6 ˚A − 1 [12], thus indicating thathybridization takes place. We further recall that thishybridization is expected as direct consequence of Bosecondensation as explained in details in Ref. [1].From the theoretical point of view, we recall that thevalidity of Pitaevskii and ZRS theories is restricted to asmall region around 2∆. Indeed Pitaevskii in his src-inal paper [3] exploited the logarithmic divergence ap-pearing in the two-roton response function [ F  o ( Q ,ω ) = i    dω ′ 2 π    d 3 Q ′ (2 π ) 3 G  (  p −  p ′ ) G  (  p ′ ), where  G  − 1 (  p ) =  ω  + i 0 + − ω ( Q ),  ω ( Q ) is the measured spectrum and  p  = ( Q ,ω )] tosolve  exactly   the many-body equations. This elegant the-ory provides explicit expressions for the Green and thedensity-density correlation functions, valid  only   in thesmall energy range where the singularity dominates. Thisfact leads to problems in data analysis when  Q  grows sothat the bare excitation energy  ω o ( Q ) takes values above2∆. In fact, the signal around 2∆ strongly decreases with Q  and spectral weight shifts to higher energies following ω o ( Q ). To understand the correlation between the highenergy part of the spectrum and the one-excitation con-tribution, it is necessary to extend the validity of thetheory to a wider range of energies in order to describeproperly the continuum contribution to  S  ( Q ,ω ). It thenbecomes crucial to consider the effect of the direct excita-tion of two quasiparticles by the neutron and its interplaywith the one-quasiparticle excitations usually considered.The aim of the present paper is to construct such an1  extension of the theory to describe experimental data for S  ( Q ,ω ) at very low temperatures. Since excitations in 4 He are stable (Γ( Q ) /ω ( Q )  ≈  10 − 2 for rotons at 1 . 3 Kwith Γ the half width of the excitation), it is an excel-lent approximation to write the Hamiltonian directly interms of the creation and destruction operators  b † and  b of these excitations: H  o  =  p ω o ( p ) b † p b p  + V  3  p , q  b † p b † q b p + q  + cc  .  (1)We consider for the moment only the  V  3  interaction thatinduces the hybridization of the single with the doubleexcitation. In general this vertex will be a function of two momenta, for instance the total momentum of thetwo particles and the momentum of one of them. We ne-glect from the outset this dependence on momentum asit is expected to be smooth in the region of interest andless important than the frequency dependence retainedin the following. Since  S  ( Q ,ω ) is the imaginary part of the density-density response function [ χ ( Q ,ω )] it is con-venient to express the density operator  ρ p  in terms of the b  and  b † field operators. In general this will be an infiniteseries in the  b -fields, by retaining only the one and twoquasi-particle terms we obtain: ρ p  =  α ( p )  b † p  + b − p   +  q γ  ( p , q ) b † p + q b q +  q β  ( p , q )  b † p + q b †− q  + b q b − p − q  .  (2)Eq. (2) gives the most general second order form for  ρ p in terms of   b  and  b † that fulfills parity, time-reversal, and ρ † p  =  ρ − p  transformation properties. For the same in-variances the three functions introduced  α ,  β  , and  γ   arebound to be real. Although the above expression for  ρ p is quite general it can be obtained microscopically withina particular approximation scheme [13]. 000000000000000000000000000000000000000000000000000000000000000111111111111111111111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111111111111111 000000000000000000000000000000000000000000000000000000000000000111111111111111111111111111111111111111111111111111111111111111 = β  = = V  3  == α  = G o = o = F  0000000000000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111111111111111 0000000000000000000000000000000011111111111111111111111111111111 000000000000000000000000000000000000111111111111111111111111111111111111 = + t  = F  = + ++ += χ 000000000000000000000000000000000000000000000111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111111111111111 = + F   == + G  = FIG. 1. Full lines stands for the exact Green function,dashed boxes for the sum of all one-particle irreducible di-agrams, and the black box for the sum of all diagrams withtwo lines closing at the two ends.  χ (  p ) is the sum of all dia-grams with two density insertions (wiggly lines). It is possible at this point to calculate perturbatively χ ( Q ,ω ) using the explicit form of   ρ p  given in Eq. (2)and the Hamiltonian (1). At zero temperature the con-tribution of the  γ  ( p , q ) term to  χ  vanishes, while themomentum dependence of   α ( p ) and  β  ( p , q ) is neglectedfor the sake of simplicity. The diagrammatic theory forthe model is shown in Fig. 1 together with the definitionof   χ (  p ),  F  o (  p ),  F  t (  p ),  F  (  p ), and  G  o (  p ) = [ ω − ω o ( Q )] − 1 .In this figure dashed boxes represent the sum of all one-particle irreducible diagrams.  F  t  stands for the sum of alldiagrams with two lines closing at the two ends, and  F   isthe self-energy without the two external couplings. Sincethe momentum dependence of the vertices is neglected,the Dyson equations for  G   and  F  t  (see Fig. 1) take thesimple form of a coupled algebraic system: G  (  p ) =  G  o (  p ) + G  o (  p ) V  3  F  (  p ) V  3 G  (  p ) ,  (3a) F  t (  p ) =  F  (  p ) + F  (  p ) V  3 G  o (  p ) V  3 F  t (  p ) .  (3b)By solving Eqs. (3) and substituting  G   and  F  t  into theexpression for  χ (  p ) we finally obtain: χ (  p ) =  α 2 + 2 αβV  3 F  (  p ) + β  2 F  (  p ) G  − 1 o  (  p ) G  − 1 o  (  p ) − V  23  F  (  p ) .  (4)Eq. (4) is the basic expression that we will use in the fol-lowing for the fits to the data [ S  ( Q ,ω ) =  − Im χ ( Q ,ω )].The presence of a  V  4  interaction does not change theabove treatment since no  V  4  vertex can connect  F   to  G  at zero temperature. This implies that the additional dia-grams due to a  V  4  interaction contribute only to  F  . Sincethe explicit calculation of   F   is difficult and in general de-pends on the detailed structure of the vertex functions,we prefer to extract it directly from data by exploitingthe large (energy and momentum) region of validity of Eq. (4). We proceed by noting that in Eq. (4) only  F  and  ω o  (through G  o ) depend on  Q . Explicit evaluation of  F  o  suggests that the momentum dependence of   F   shouldnot be pronounced in the range of interest 2.3-3.2 ˚A − 1 .We thus drop this dependence completely in  F   and fit theresulting function  F  ( ω ) to the data. The  Q -dependenceof   ω o  is not negligible because it drives  ω o ( Q ) throughthe 2∆ line. In this way we are left with only one param-eter dependent on  Q . It is now possible to extract both ω ( Q ) and  F  ( ω ) by fitting Eq. (4) to all sets of data of different momentum  at the same time  . This procedureexploits fully the information contained in the data be-cause it is sensitive to the correlation among sets withdifferent  Q . For this reason we are able to extract infor-mation for  ω ( Q ) and for the function  F  ( ω ) over a scale(slightly) smaller than the instrumental resolution.A few words are necessary to explain how we can findthe complex function  F  ( ω ) from the data. The imag-inary and real part of   F   are related by the relationRe F  ( ω ) =  − 1 /π P     dω ′ Im F  ( ω ) / ( ω − ω ′ ). We thus needonly to parametrizeone of them, we chooseIm F  ( ω ), sinceon this function it is easier to apply the physical con-straint that excitations are kinematically stable for ener-gies smaller than 2∆. This condition reads Im F  ( ω ) = 02  for  ω <  2∆ (the small spectral density between  ω ( Q ) and2∆ can be neglected). A simple way to parametrize thefunction Im F   is then to choose a set of values of   ω  say { ω 1 ,ω 2 ,...,ω N  }  with 2∆ =  ω 1  < ω 2  < ... < ω N   reason-ably spaced and to assign a  free   parameter,  a i , to eachof them. Im F  ( ω ) can then be defined as a cubic splineinterpolation on such a set. The integral to calculatethe real part can be performed analytically so that itsevaluation is fast and reliable also near the logarithmicsingularity at the threshold. FIG. 2. Fit to the data from Ref. [12] with a parametrizedIm f  . In the inset the resulting Im f   is shown compared withIm F  o  averaged over 2 . 3  < Q <  3 . 2 ˚A − 1 and properly scaled. We take advantage of a symmetry in Eq. (4) to definea dimensionless function  f  ( ω ) =  λF  ( ω ), constrained bythe normalization condition   ω N  2∆  dω Im f  ( ω ) =  ω N   −  2∆.We thus define  g 3  =  V  3 /λ 1 / 2 , and ˜ β   =  β/ ( λ 1 / 2 α ). In thisway  α , ˜ β  ,  g 3 , and the  N   −  1 parameters that define  f  are independent and can be fitted to the data. We setIm f  ( ω ) =  a N   for  ω > ω N   and subtract out the infiniteconstant that would appear in Re F  ( ω ).We thus fitted Eq. (4) (convolved with the known in-strumental resolution) to experimental data of Ref. [12](at 1.30 K) by minimizing numerically  χ 2 for the param-eters:  α , ˜ β  ,  g 3 ,  ω ( Q 1 ),  ... ,  ω ( Q n ),  a 1 ,  ... ,  a N  − 1  ( N   = 15for the fit presented). The minimization procedure hasbeen performed with different standard routines and theresult turned out to be independent of the choice of   { ω i } and the initial value of   a i . A typical starting point forIm f   is simply  a i  = 1 for all  i . The resulting fit is shownin Fig. 2. Good agreement between theory and experi-ment is obtained with a reduced  χ 2 ( χ 2 R ) of nearly 4, thusindicating that even if we are leaving a large freedom in f   the agreement is significant.Our main result is summarized by the dispersion rela-tion for the undamped excitation shown in Fig. 3, whereit is compared with the one reported in Ref. [14]. We findthat the model can  quantitatively   explain that the peakposition of   S  ( Q,ω ) is slightly larger than 2∆ for  Q >  2 . 6˚A − 1 . This originates from a mixing within the instru-mental resolution of the contribution of the sharp peakat energy  ω Q  slightly smaller than 2∆ with that of thecontinuum of two rotons excitations starting at 2∆. Ongeneral grounds the continuum should depend stronglyon  ω  near 2∆, as for  ω >  2∆ there are much more statesavailable to decay into. For these reasons, the assump-tion of a background of constant slope, used to obtain thevalues of Ref. [14], is not valid in this momentum region.Our procedure exploits instead the theoretical model for χ (  p ) to extract the value of   ω Q . In this way we can findthe final part of the dispersion relation with improved ac-curacy and it turns out that data agree with a dispersionrelation for the excitations always below 2∆. FIG. 3. Dispersion relation found in the present work (solidline), and in Ref. [4,14] (dashed line). The inset shows thecomplete dispersion relation. The importance of the fitted parameter ˜ β   has beenchecked by studying the function  χ 2 (˜ β  ), where all theother parameters are properly modified to minimize  χ 2 for each value of  ˜ β  . The confidence region for ˜ β  ,  i.e. the values of  ˜ β   such that  χ 2 (˜ β  ) /χ 2min  <  1 . 5, turns outto be  − 0 . 17  <  ˜ β <  − 0 . 01 meV − 1 / 2 with a best value˜ β   =  − 0 . 06 meV − 1 / 2 . This implies that the direct ex-citations of two quasiparticles by the neutron gives asmall but  sizable   contribution to  S  ( Q ,ω ). Concerningthe other parameters we find  α 2 = 1 . 4 and  g 3  = 0 . 8meV 1 / 2 .The shape of Im f   found by the fit has two mainfeatures: a clear peak at  ω  ≈  2 meV and a “quasi-divergent” behavior at the threshold. The peak is dueto the maxon-roton van Hove singularity. This can be3  verified by comparing the function Im F  o ( Q,ω ), averagedover 2 . 3  ≤  Q  ≤  3 . 2 ˚A − 1 and properly scaled, with thefitted Im f  ( ω ) (see the inset of Fig. 2). Hence it is clearthat the peak corresponds in shape and position to themaxon-roton singularity. It is remarkable that no traceof the peak is apparent in any of the experimental plots.It is only by exploiting the correlation between plots withdifferent  Q  that we have been able to extract this infor-mation. On this basis, using the fitted parameters it isalso possible to predict that a peak and its shape shouldbe observable with a resolution of 0 . 1 meV (to be com-pared with 0 . 5 meV of Fig. 2).The quasi-singularity at threshold can be understoodas an interaction effect, namely a signature of the attrac-tive interaction between rotons. As a matter of fact, inthe small region of energy near the threshold we can ap-ply Pitaevski-ZRS [3,5,6] theory to evaluate  F  ( ω ) thatreads: F  ( Q ,ω ) = 4 ρ  ln 2∆ − ωD  1 − g 4  ln 2∆ − ωD  − 1 ,  (5)where  ρ  = − Im F  o ( Q ,ω  = 2∆ + ) / 2 π  is the threshold den-sity of states,  g 4  = 2 V  4 ρ , and  D  is a cutoff that canbe set to 1 meV as changes in  D  can be easily reab-sorbed in small variations of   ω o ,  g 3 , and  g 4 . One canthus readily verify that with a negative value of  g 4  Eq. (5)gives a Im F   that for  ω  →  2∆ + reproduces qualitativelyIm f   of Fig. 2. To verify quantitatively this fact andto find an estimate of   g 4  we have repeated the datafit using Eq. (5) to parametrize  f  ( ω ), (instead of thespline parametrization) and setting a cutoff in energy at2∆+0.2 meV, in such a way to apply Eq. (5)  only   whereit is supposed to hold. The resulting  χ 2 R  = 1 . 7 for thefit gives evidence that the theory works  quantitatively  in this region and we get for the interaction parameter V  4  =  g 4 / (2 ρ ) ≈− 4 . 7 meV ˚A 3 ( g 4  = − 0 . 15). The bound-state energy that we obtain from Eq. (5) is indeed verysmall  E  B  =  D  exp { 1 /g 4 } ≈  1 . 3 µeV  . This second fitgives also an additional estimate of the dispersion re-lation that agrees with the previous one and confirmsthat an undamped state is present up to  Q  = 3 . 6 ˚A − 1 .The confidence region for  g 4  is  − 0 . 22  < g 4  <  − 0 . 08, re-stricting ˜ β   to its confidence region already found, whichindicates that the interaction is definitely  attractive  .Some final comments are in order. First, while Fig. 2shows the fit for 2 . 3 ≤ Q ≤ 3 . 2, the inclusion of the addi-tional set of data for  Q  = 3 . 6 increases slightly  χ 2 R  (sincethe momentum range over which we are assuming thevertices to be momentum-independent may be too large)but it remains in any case a good fit to data. For thisreason we report in Fig. 3 the value for  ω ( Q ) obtainedin this way. Second, we studied the cutoff dependence of  χ 2 R  when  f  ( ω ) is parametrized according to Eq. (5) andwe found that it is very weak up to  ω  = 2 meV wherethe effect of the maxon-roton peak becomes important.Thus use of the Pitaevskii-ZRS theory for  F   in our ex-pression (4) does not give a good description of the dataif the cutoff in energy is removed. This indicates that themaxon-roton structure and, in general, the whole shapeof Im f   plays a  crucial   role in determining  S  ( Q ,ω ).In conclusion, we presented a theory for  S  ( Q ,ω ) thattakes into account both one- and two-quasiparticle exci-tations by the neutron. The theory reproduces the exper-imental result over a large range of energy and momen-tum. We have thus been able to extract the final part of the spectrum dispersion relation in  4 He. Our theory canbe regarded as an extension of the Pitaevskii-ZRS theorytaking into account the effect of two-particle excitations.Moreover the range of validity is enlarged as we make nohypothesis on the  F   function, but we extract it directlyfrom data. In the region where Pitaevskii-ZRS theoryholds we have used it to parametrize  F   in Eq. (4) and wefound that the interaction potential among rotons ( V  4 )is attractive in this momentum region.I am indebted to P. Nozi`eres for suggesting this prob-lem to me and for many discussions. I gratefully acknowl-edge B. F˚ak for discussions and critical reading of themanuscript. I acknowledge B. F˚ak and J. Bossy for let-ting me use their data prior to publication. I also thankN. Cooper, N. Manini, A. W¨urger, P. Pieri, G.C. Strinati,and H.R. Glyde for useful discussions. ∗ electronic address: pistoles@ill.fr.[1] A. Griffin,  Excitations in a Bose-Condensed Liquid  (Cambridge University Press, Cambridge, UK, 1993).[2] H. R. Glyde,  Excitations in Liquid and Solid Helium   (Ox-ford University Press, Oxford, UK, 1994).[3] L. P. Pitaevskii, Sov. Phys.-JETP  36 , 830 (1959).[4] R. A. Cowley and A. D. B. Woods, Can. J. Phys.  49 , 177(1971).[5] J. Ruwalds and A. Zawadowski, Phys. Rev. Lett.  25 , 333(1970). A. Zawadowski, J. Ruvalds, and J. Solana, Phys.Rev. B  5 , 399 (1972).[6] L. P. Pitaevskii, JETP Lett.  12 , 82 (1970).[7] K. Bedell, D. Pines, and A. Zawadowski, Phys. Rev. B 29 , 102 (1984).[8] K. J. Juge and A. Griffin, J. Low. Temp. Phys.  97 , 105(1994).[9] A. J. Smith  et al. , J. Phys. C  10 , 543 (1977).[10] D. K. Lee, Phys. Rev.  162 , 134 (1967).[11] B. F˚ak and K. H. Andersen, Phys. Lett. A  160 , 469(1991). B. F˚ak, L. P. Regnault, and J. Bossy, J. LowTemp. Phys.  89 , 345 (1992).[12] B. F˚ak and J. Bossy, J. Low Temp. Phys. in press (1998).[13] Within the standard notation for the  u p  and  v p  func-tions we find in the Bogolubov approximation:  α ( p ) = n 1 / 2 o  ( u p  −  v p ),  β  ( p , q ) =  − u p + q v q , and  γ  ( p , q ) = u p + q u q  + v q v p + q , with  n o  the condensate density.[14] R. J. Donnelly, J. A. Donnelly, and R. N. Hills, J. LowTemp. Phys.  44 , 471 (1981). 4
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