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Excitations in superfluid 4He and the condensate
PHYSICA[ Physica B 197 (1994) 189-197
ELSEVIER
Excitations in superfluid
4He and the condensate
L. Reatto*, G.L. Masserini, S.A. V i t i e l l o Dipartimento di Fisica, Universitd degli Studi di Milano, 20133 Milano, Italy
Abstract Starting from an accurate representation of the wave function of the ground state and of the phonon-roton excitations based upon the shadow function we present a microscopic dens;ty matrix for superfluid 4He. The roton and maxon energy is computed both at zero and at finite temperature at different densities. The separate contributions of kinetic and potential energy of the roton excitation are computed and we find that a roton lowers the potential energy. The roton contribution to the depletion of the Bose-Einstein condensate n o is computed. Excitation of 5 to 10 rotons is needed for the depletion of one atom from the condensate and no simple relation is found between the T dependence of n o and of g(r). Other quantities we compute as function of T are the static structure factor and the strength of the 'one-phonon' peak of the dynamical structure factor. Comparison of our results with experiment gives some evidence that additional excitations become important above about 2 K.
1. Introduction T h e ground state of superfluid 4He is considered to be rather well understood on the basis both of the variational theory and of exact simulations like the so-called G r e e n ' s function M o n t e Carlo method. The situation is quite different for the excited states and finite temp e r a t u r e properties as shown by the debate [1] o v e r the nature of rotons, on the role of the B o s e - E i n s t e i n condensate and on the dynamical response function of the system. T h e quantitative theory of the excited states in superfluid 4He starts with F e y n m a n (F) and F e y n m a n - C o h e n [2] (FC) representation of the wave function ~q of a state of m o m e n t u m hq. This can be extended in a systematic way with the m e t h o d of correlated basis functions [3] ( C B F ) , but even with the present most elaborate
* Corresponding author.
c o m p u t a t i o n [4] the roton energy is still 15% a b o v e experiment. In addition, only with the F and FC ~q it has been possible [5,6] to construct the finite T density matrix, but not with the m o r e advanced formulations. Recently [7] a new form of wave function (WF) for the excited states has b e e n introduced, the shadow WF. This is built upon the shadow representation [8] of the ground state gt0. In this new trial W F for gr0 correlations between particles arise both directly and indirectly via coupling to subsidiary variables, the shadows. These are thought to represent the effect of q u a n t u m delocalization of the 4He atoms. By switching the d e p e n d e n c e of the density fluctuation factor in the F-WF from variables representing the positions of the 4He atoms to the subsidiary variables, one gets a representation of ~q free of variational p a r a m e t e r s and that turns out to give the best estimate of energy of a roton both at equilibrium and at freezing density. The structure of aFq is simple enough that it turns out
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possible [9] to construct the density matrix based on an ensemble of such excited states. This has opened the way [10] to the computation of microscopic correlations also at temperatures where roton excitations are important. Here we review this theory and present the results of some recent computations that extend the previous ones [10].
energy. This gives the motivation for writing the trial function [7] in the form
q"q(R) = ~b,.(R) f dS K(R, S)$s(S)o q
(1)
where O-q is the density fluctuation of the shadow variables: N
Crq = ~'~ e '°~ ,
(2)
i-1
2. Excited states
In a shadow WF the positions R--{ r l , . . . , r n } of the 4He atoms are coupled by Gaussians to subsidiary variables S ={s I . . . . , Sn}. The positions {ri} are correlated by a Bijl-Jastrow pair factor and {si} are correlated in a similar way. Since the {s~} are integrated out, many-body couplings among the {rz} arise beyond the pair level. A motivation for this representation of qt0 comes from Feynman's path-integral formulation of the density matrix [11]. As is well known, in this formulation each position r~ of a particle becomes associated to a sequence of positions over 'time' slices so that each quantum particle becomes equivalent to a suitable flexible polymer. In this way different quantum particles become correlated not only directly via the action of the interatomic interaction but also indirectly via the correlations between the other monomers corresponding to the different time slices. In a shadow WF the correlations induced by the subsidiary variables have a similar role and now the position s i can be thought to represent the center of mass of the polymer associated to particle i. It is important to recognize that the analogy is particularly appropriate for hard-core particles at high density like in 4He. Under such conditions cross linkings between different polymers, reflecting the Bose symmetry, are not too frequent so that it is meaningful to speak of the center of mass at least in a transient sense. The quantum hole due to delocalization of particles should be present also in the excited states and compared to the ground state it should be essentially unaltered for excitations of low
K(R, S) is a product of Gaussians which couple r i with s i and Or(R) = lq exp[-u(rq)], i
q,s(S) = Iq exp[-us(si/)] • i<[
(3)
If we had the factor N
P, = Z e iqr'
(4)
i-1
in place of O-q in Eq. (1), then qrq would just be the Feynman WF when the ground state is represented by a shadow WF. The modification has far reaching consequences because Eq. (1) already includes effects of backflow [12] and by expanding the intergrand with respect to r i -s~ one finds that Eq. (1) contains terms of all orders in powers of pq. In the ground state the shadow variables induce correlations beyond the pair level to all orders, in a similar way a density fluctuation in the shadow variables induces fluctuations beyond the linear level for the real variables. We have computed the excitation energy eq as the difference between the expectation value of the Hamiltonian H with ~Fq and the similar expectation value corresponding to g0, i.e. Eq. (1) without the factor trq. As interatomic pair potential we have used the form VA(r) of Aziz et al. [13] and the expectation values have been evaluted by the Metropolis Monte Carlo algorithm for a system of 108 particles and periodic boundary conditions. The pseudopotentials u(r) and us(r ) in Eq. (3) derive from a ground state computation. In the first computation [7] of eq the McMillan form (b/r) m was used both for u(r) and us(r ). It has
L. Reatto et al. / Physica B 197 (1994) 189-197
been found that an improved description of the ground state is obtained [14] if the pseudopotential us(r ) between shadows contains an attractive part. A convenient representation of this pseudopotential is
u~(r) = VA(dr )/A where vA is the Aziz potential and d and A are variational parameters. The results we present here correspond to this pseudopotential. In Fig. 1 we show the excitation energy at three densities, in panel (a) at p = 1.97 a t / n m 3 which is 10% below the equilibrium density p¢q, in (b) the result at Peq = 2.18 a t / n m 3 and in (c) at p = 2 . 6 2 a t / n m 3 close to the freezing density. The excitation energy is computed only for a discrete set of wave vectors consistent with the periodicity of the simulation box and we take q in the (1 0 0 ) , (1 1 0) and (1 1 1) directions. It should be noticed that computation of Eq is very demanding because eq is obtained as difference i
i
a
20.0
3[
10.0
x I
0.0
,
I
,
I
b
20.0 v
LU 10.0
0.0
,
I
,
I
20.0
10.0
0.0
I
\
I
0.0
:'
"11[/
I
1.0 2.0 q (A -~)
Fig. 1. Excitation spectrum of liquid 4He as a function of wave vector q: (a) p = 0.gpeq, (b) Pcq" The solid line is a spline fit to neutron-scattering data [15] and (c) p = 1.2pcq. The dotted line is a parabolic fit to experimental data [16].
191
of total energies that are about 100 times larger of %. Typical runs consist of several 106 attempted moves per particle. We discuss first the result at Peq- A t smaller q we see the phonon liner behavior followed by the maxon at about the correct q with an energy 1 6 . 4 - + 0 . 5 K which is 2 . 4 K above experiment [7]. The roton energy is 10.2---0.5K, 1 . 6 K above experiment, and the position is at a q vector perhaps displaced by 0.1 A-~ to larger values, but it is difficult to locate precisely the minimum due to statistical fluctuations. These results compared with previous ones obtained with a different shadow pseudopotential for the ground state show an improvement of 1.3 K for the maxon energy. The present results have also a smaller variance than the previous ones. These facts confirm the strong sensitivity of the maxon energy on the form of the ground state. In the roton region, although we cannot definitively exclude agreement with our old result due to the statistical uncertainties, we get now an energy 0.4 K higher. On the other hand it confirms that the simple Feynman like shadow WF gives a result which is competitive with the best results [4] obtained with CBF. The density dependence of E q from our computation has the correct behavior, a velocity of sound and maxon energy increasing with p and a lowering of the roton energy. C o m p a r e d w i t h Peq, at p = 1.2Pe q the Erot drops by 1.6 K to 8.6-+ 0.5 K, a change similar to the experimentally observed drop [16] of erot by 1.4 K at 24 bar corresponding to p -- 1.18Pe q. AS p increases the displacement of the roton minimum to larger q is somewhat weaker than experiment. We present also the first result for eq at a density lower of the equilibrium value. At p = 0.9Pe q the roton energy increases by 0.7 K and the maxon decreases by 1.5 K so that the energy difference between maxon and roton is reduced to about 4 K from 6.3 K at Peq" Planned experiments in expanded liquid 4He should be able to test this prediction. Our computations have shown an interesting feature of a roton: the energy erot has a negative contribution Urot from potential energy and correspondingly the kinetic energy is larger of erot.
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L. Reatto et al. / Physica B 197 (1994) 1 8 9 - 1 9 7
A t Peq we find Urot = - 1 . 8 K and at 1.2Peq, ttro t - 2 . 2 K. This is an effect of short-range correlations as discussed below and indicates the strong anharmonic character of a roton. For a harmonic excitation like a long-wavelength phonon the energy has equal contributions from potential and from kinetic energy. Recently we have also extended [18] the computation of the excitation energy at a finite temperature by considering a single excitation added on top of thermally excited rotons. We find a significant lowering of erot at finite T, similar to what is observed experimentally.
3. Density matrix and finite temperature properties In the shadow representation of excited states a multiple excited state is obtained by replacing in Eq. (1) the o-q factor by a product of O-q, one for each of the excitations. This WF is an exact eigenstate of the total momentum corresponding to /'tot = hEqnqq where nq is the number of q excitations. If Eqnq is finite the excitation energy is simply Eqnqeq. Assuming as an approximation that this is true also when the number of excitations is of order N, we have obtained [9] an explicit expression for the density matrix that should be accurate when the average number of excitations Eq(nq)/N ~ 1 is small. It reads (U'lP~ln > =
~O,(R')~Or(R)f dS' dS
× qJ~(S')~b~(S)K(R',S')K(R, S) T
× I] e x p [ - x 1 i
T
× [I e x p [ - x 2 i,j
t T (sij) - X l (sij)] r
Isi -
sjII/QT.
(5)
Doubling of the shadow variables is due to the fact that PT is related to the square of wave functions. QT is the normalization constant, the T dependent factors xT and x2v are functions of the density fluctuations spectrum of shadows in the ground state and of %. From Eq. (5) we see that excited states induce in PT extra correlations of the pair product form in the shadow variables,
but the mathematical structure is very similar to the P e n r o s e - R e a t t o - C h e s t e r form [5] of PT. At low temperature only low-energy excitations, phonons and rotons, play a role in PxPhonons induce long-range correlations that we cannot handle directly due to the small size of our simulation box. On the other hand the effect of thermally excited long-wavelength phonons is well known from other theories. Here we focus on the effects due to rotons so that Eq is approximated by the roton parabola eq = A + h2(q2 T qrot) /2bt" Since eq enters the expression for X1 T and Xz as a counting factor, for better accuracy we have taken eq from experiment. At Peq we have used [16] qrot = 1.92"~-1 , /z =0.136mile and a T dependent A(T) = A(O) + ~A(T) with [16] A ( 0 ) = 8.608K and a roton energy shift ~A(T) as given by the phenomenological theory of Bedell, Pines and Zawadowski [19], which is in good agreement with experiment. With this PT we have computed the energy E(T) and a number of correlation functions. We present here the results of some new computations with better statistics than the earlier results. The interest is mainly in the correlation functions. The energy E(T) + (H)T for our approximate PT should just be equal to the Landau expression E(T) = E o + ~qEq(nq)/N, where (nq) is the Bose-Einstein distribution. However, the direct computation of E(T) = (H) T is a very severe test of the consistency of the expression of PT and of our simulation. We find very good agreement between E(T) computed in the two different ways. For instance at T = 2.1 K direct computation gives E(T) - E 0 = 1.017 -0.007 K, a value to be compared with E ( T ) E 0 = 1.04 + 0 . 0 4 K estimated from the Landau expression, in which our result for the roton energy is used. The structure factor S(q) corresponding to the ground state at the three densities of our computation is shown in Fig. 2. Notice that the present W F does not include the long-range correlations due to the zero point motion of phonons so that S(q) does not have the correct linear q behavior as q ~ 0. Experimentally [20,21] it is known that the main peak of S(q) becomes higher and sharper with increasing T. This is exactly what
L. Reatto et al. / Physica B 197 (1994) 189-197
193
i
Pl
0.040 ¢,i II I---
,~ 1 . 0 09
it
a
0,020
03
0.000
-0.020 /
0.0 0.0
,
I
,
2.0 q (A -1)
I
4.0
Fig. 2. Structure factor S(q) at T = 0, the ground state at three densities: dots for p = 0.9peq; at Peq the solid line shows our results and the plus signs show the experimental data [21], the dashed line displays the results for p = 1.2pe q.
we find with our PT. In Fig. 3(a) our result for AS(q,T = 2.1), AS(q,T) = S(q,T) - S(q,T = 0), is compared with experiment [20] (average of the data at T = 2.07 and 2.12 K). The present result for AS(q, T) is similar to the one obtained [22] with the simpler P e n r o s e - R e a t t o - C h e s t e r density matrix. This is consistent with the result of a detailed study [23] that has shown that FC backflow has a limited effect on S(q; T). In Fig. 3(b) we present the quantity Y ( q ; T ) = AS(q, T)/Nrot(T), where Nrot(T ) is the number of thermally excited rotons per atom at temperature T. The computation has been performed at T=I.IK, 1 . 4 K and 2 . 1 K where Nrot(T ) is equal to 0.00097, 0.0061 and 0.092, respectively. We see that at low temperature AS(q; T) scales very accurately with Nro t. So the low T limit of Y(q; T) represents the change of S(q; T) due to a single roton. The related quantity
Ag(r; T)INrot(T), where Ag(r; T) = g(r; T) - g(r; T = 0), is plotted in Fig. 4 and clearly shows that a roton induces stronger short-range order. Y(q; T) at higher T is larger than the low-T result. This nonlinear behavior of AS in Nro t is a manifestation of r o t o n - r o t o n interaction and indicates that it is easier to excite a roton if one is already present, because of a cooperative effect in the induced change of short-range order.
0.20
>0.00
-0.20
0.5
i
t
1.5
2.5
q (A-1) Fig. 3. Variation of the structure function with the temperature as a function of q. (a) The solid line is the computed AS(q; T = 2.1) and the dots show the average of experimental data [21] obtained at 2.07 and 2.12K. (b) Y(q; T): the scaled variation of the structure factor with respect to the number of rotons at temperature 1.1 (dots), 1.4 (solid line) and 2 . 1 K (plus signs). Notice that AS in (a) has been computed by extending the simulated g(r) at large r with a damped oscillating tail. No tails have been used for Y and this is the origin of the oscillations of Y at small and large q.
F r o m PT we have computed also the oneparticle off-diagonal density matrix and from this we deduce the Bose-Einstein condensate no(T ) and the m o m e n t u m distribution. As expected no(T ) decreases with increasing T. At Peq the ground state we are using has [10] n ( 0 ) = 7.8% and this is reduced to 5.8% at T = 2.17K. Since in our computation n r o t ( T = 2 . 1 7 ) =0.127 we conclude that with our density matrix there is one 4He atom depleted from the condensate for every - 6 thermally excited rotons. At p = 1.2peq this n u m b e r rises to - 1 0 rotons per depleted atom from the condensate. This indicates the profound difference from the Bogoliubov regime in which there is a one-to-one correspondence between depletion of the condensate and the n u m b e r of excitations. T o view a roton as an atom taken out of the condensate is in complete contradiction with the present theory and we
L. Reatto et al. / Physica B 197 (1994) 189-197
194
1.5 0.05
s"
1.0
'
'
'
'
0.03
'4
J
~*
II
I-..E. JC
0.5
0.01
-0.01
0.0 b
-0.03 -0.0
0.10
L2 t3"}
,
J 0.01
,
L 0.03
, 0.05
h (r; T=2.10)
z p"
, -0.01
...,.
o.oo
Fig. 5. h(r; T = 0 ) versus h(r; T = 2 . 1 ) for the range 6 , ~ < r < 13 ~, at poq. The best straight line has a slope B = 0.898 --+ 0.009. The dashed line has a slope 0.927 corresponding to the
:4 -0.10
best fit of experimental data [21].
-0.20
--
'
0.0
i
i
5.0
10.0
15.0
r (A)
Fig. 4. The radial distribution function at poq and its T variation scaled with the number of rotons. (a) g(r;T = 0); (b) Ag(r; T = 1.4)/N~o, is shown by dots, Ag(r; T = 2.1)/N,o' is displayed by plus signs. believe it is a wrong picture of what happens in superfluid 4He. T h e increasing short-range order with T has attracted large attention [20,21] because of the hypotized relation [24] between the T dependence of g(r) and no(T ). Based on a certain decomposition of the two-particle density matrix Hyland, Rowlands and Cummings ( H R C ) were led to the following relation for h(r)=-g(r)- 1, when r > 5 ,~:
h(r; T) = A(T)h(r; Tx) ,
(6)
A(T) = [1 - n o ( T ) ] 2 .
(7)
Relation (6) implies that the zeros of h(r; T) do not change with T and that h(r; T)/h(r; T ' ) should be independent of r. In other words a plot of h(r.; T) versus h(r; T') should give a straight line. Experiments [20, 21] display such behavior and also our results indicate that Eq. (6) is followed rather accurately. In Fig. 5 such a plot is shown for T = 2 . 1 K and T = 0 K . It is clear that Eq. (6) is a good representation of our
results but due to the high accuracy of the present computation, our results indicate the presence of some small deviations from it. The deviations are below the level of accuracy of the available measured S(q), but new measurements with higher accuracy might be able to show if such deviations are present. Since from our PT we are able to separately determine no(T ) and A(T), we have been able to perform [10] the first microscopic test of the H R C relation (7). We find that A(T) is considerably different from [ 1 - n0(T)] 2, the T dependence of A(T) being stronger than that implied by no(T ) . The reason for the failure of relation (7) is that the 'uncondensed' part of the two-body density matrix in the H R C decomposition has a strong T dependence which is neglected by H R C . The final quantity we discuss is the strength Z(q; T) of the single-excitation peak in the dynamical structure factor S(q,~o). In the Feynman approximation Z(q; T)= S(q; T) and this implies that the q mode is a pure density excitation, all the strength of S(q, w) is in the single excitation peak and there is no background. This is correct in the q--~ 0 limit, but not in the m a x o n - r o t o n region where Z(q; T) is substantially smaller [15] than S(q; T). Also, in this q region the roton peak broadens and loses intensity [25,26] as T---~ Tx. With our WF and density matrix we are able to compute Z(q; T)
L. Reatto et al. / Physica B 197 (1994) 189-197 1.5
i
C/~
,
i
+
~ 0.5
0.0
,
~
,
+. +.,
\
110
210 310 q (A-~)
4.0
Fig. 6. The strength Z(q; T) of the single excitation peak as a function of q at Peq for T equal to 0 (dots) and 2.1 K (plus signals). The solid line shows the structure factor S(q) at T = 0 and the cross signs that at T = 2.1 K. and the result at T = 0 and 2.1 K are shown in Fig. 6. Our W F describes an almost pure density fluctuation at small q but this is not so in the maxon and roton region, the multiphonon background having a weight of 20% in the maxon region and of 10% in the roton region. At finite T the multiphonon background increases and for a roton it is about 20% at 2.1 K. This trend is consistent with the observed weakening of the single-roton peak in S(q, to). A direct comparison with experiment is not possible due to the substantial width of the roton peak for T > 2 K and to the presence of an overlapping multip h o n o n background. In our theory we have no information on the lifetime of excitations. At p = 1.2peq the multiphonon background is 8% f o r a r o t o n at T = 0 K a n d 15% at T = l . 8 5 K .
4. Discussion The introduction of shadow WF has led to a rather comprehensive theory of liquid nile. This gives an excellent description of the ground state, a rather accurate description of the excited states and a density matrix that at last gives the possibility of computing microscopic correlation properties of superfluid 4He at the level of L a n d a u theory for macroscopic properties. The fact that without variational parameters for the excited states we get the roton energy at all
195
densities to within 1-1.5 K is remarkable. We believe this is due to the fact that in this formulation the backflow implicit in the W F is in the exponential form, like in the original [2] FC WF. Up to now it has not been possible to treat this exponential form and the W F has always been linearized. No such problem arises in our formulation. The form of backflow with our W F is determined by the intershadow pseudopotential of the ground state. By allowing the presence of an additional backflow term specific to the excitation it should be possible to bring theory closer to experiment. The strength Zq of the single excitation peak in S(q, to) is significantly lower than S(q), and this indicates that in our description a roton is not a simple density fluctuation. Other important features that emerge from our computations are that a roton induces an increased local order and this is accompanied by a negative contribution to the internal energy. In addition it appears that 4He atoms of all momenta participate in the excitation of a roton and the depletion of the condensate is much smaller than the n u m b e r of excitations. Our computation of the roton energy at finite temperature indicates the importance of r o t o n roton interaction and an important development of our density matrix will be a self-consistent treatment of excitations at finite T. Both the T dependence of Zq and of condensate no(T ) have the correct behavior. However, it does not signal the strong T evolution as T~ is approached, like the vanishing of n o and the strong change of shape of S(q, to). All this could be due to the effects of r o t o n - r o t o n interaction that are not included in a simple renormalization of the roton energy as we have used. These effects, however, should have only a small effect on short-range correlations. A more likely explanation is the presence of additional excitations in addition to phonons and rotons, like vortices as advocated [27] by Williams. Such excitations [28] are particularly effective in destroying phase coherence and therefore in depressing n o . However, vortices should not modify very much the local order because only the small vortex core region should be affected. On the other hand the
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L. Reatto et al. / Physica B 197 (1994) 189-197
presence of vortices can offer an explanation to s o m e puzzling features of rotons. A roton in the flow field vs of a vortex get a modified energy and if the size of a roton is small c o m p a r e d with the scale of variation of v~ [29] we have
shown also here for strong coupling effects in roton excitations, and this should be t a k e n into account in any theory of superfluid 4He.
Acknowledgements
eq eq + hvs'q.
(8)
In the presence of a statistical distribution of vortices, the roton energy becomes distributed over a range depending on vs so that there is an additional broadening mechanism to the roton linewidth. It is interesting that the situation is rather different in the case of neutron scattering c o m p a r e d with the R a m a n case. For neutrons the roton p e a k in S(q, to) is directly affected by the distribution of roton energy as given by Eq. (8). In the R a m a n response two rotons are emitted with almost opposite values of the wave vector. This means that the total energy of the two rotons is not affected by vS and one expects to see a R a m a n linewidth narrower than the one inferred from S(q, to). This is exactly what is o b s e r v e d experimentally [30]. We have conside r e d a very simple model of this effect elsewhere [18]. In our theory we have well-defined roton excitations, that are not simple density fluctuations, and a condensate. The roton appears at the same time in the density response function, as apq is not orthogonal to pql/f0, and in the single-particle G r e e n ' s function G(q, to), because ~q is an a p p r o x i m a t e eigenstate of H. We cannot answer the question if a roton would exist and would show up in S(q, to) if the system had no condensate because we do not know a natural way in three dimensions to construct a Bose fluid ground state without a condensate. O u r roton has the character both of a quasiparticle and of a collective excitation. We find it not very useful to try to establish the 'true' nature of a roton. This is a question to which one is led on the basis of perturbation theory. In the p h o n o n region, it is well known that strong coupling effects change the condensate n o, in the Bogoliubov spectrum and in the weight of G(q, to) in S(q, to) into the total density p. T h e r e is ample evidence from variational theory as
We would like to acknowledge useful discussions with G.V. Chester and M . H . Kalos. This work was partially supported by Consorzio I N F M and by Consiglio Nazionale delle Ricerche under Progetto Finalizzato 'Sistemi Informatici e Calcolo Parallelo'.
References [1] See contributions by C.E. Campbell, H.R. Glyde, A. Griffin, W.G. Stirling, E.C. Svensson and others, in: Excitations in Two-Dimensional and Three-Dimensional Quantum Fluids, eds. A.G.F. Wyatt and H.J. Lauter (Plenum Press, New York, 1991). [2] R.P. Feynman and M. Cohen, Phys. Rev. 102 (1956) 1189. [3] See, for instance, E. Feenberg, Theory of Quantum Fluids (Academic Press, New York, 1969). A specific study of rotons is contained in: D.K. Lee and F.J. Lee, Phys. Rev. B 11 (1975) 4318; C.C. Chang and C.E. Campbell, Phys. Rev. B 13 (1976) 3779. [4] E. Manousakis and V.R. Pandharipande, Phys. Rev. B 30 (1984) 5062. [5] O. Penrose, in: Proceedings of the International Conference on Low Temperature Physics, ed. J.R. Dillinger (Wisconsin University Press, Madison, 1958) p. 117; L. Reatto and G.V. Chester, Phys. Rev. 155 (1967) 88. [6] S. Battaini and L. Reatto, Phys. Rev. B 28 (1983) 1263. [7] W. Wu, S.A. Vitiello, L. Reatto and M.H. Kalos, Phys. Rev. Lett. 67 (1991) 1446. [8] S.A. Vitiello, K. Runge and M.H. Kalos, Phys. Rev. Lett. 60 (1988) 1970; S.A. Vitiello, K.J. Runge, G.V. Chester and M.H. Kalos, Phys. Rev. B 42 (1990) 228. [9] G.L. Masserini, L. Reatto and S.A. Vitiello, Europhys. Lett. 21 (1993) 679. [10] G.L. Masserini, L. Reatto and S.A. Vitiello, Phys. Rev. Lett. 69 (1992) 2098. [I1] R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965). [12] W. Wu, S.A. Vitiello and L. Reatto, in: Monte Carlo Methods in Theoretical Physics, eds. S. Caracciolo and A. Fabrocini (Giardini Editore, Pisa, 1991) 197. [13] R.A. Aziz, V.P.S. Nain, K.S. Carley, W.L. Taylor and G.T. McConville, J. Chem. Phys. 70 (1979) 4330. [14] T. Mac Farland, S.A. Vitiello and L. Reatto, J. Low Temp. Phys. 8 (1992) 433.
L. Reatto et al. / Physica B 197 (1994) 189-197
[15] R.J. Donnelly, J.A. Donnelly and R.H. Hills, J. Low Temp. Phys. 44 (1981) 471. [16] W.G. Stirling, in Ref. [1], p. 25. [17] R.A. Cowley and A.D.B. Woods, Can. J. Phys. 49 (1991) 177. [18] L. Reatto, S.A. Vitiello and G.L. Masserini, in: Proceedings of the Conference on The Role of the Condensate and Vorticity in Dense Bose Systems, Minneapolis, August, 1993 (to be published). [19] K. Bedell, D. Pines and A. Zawadowski, Phys. Rev. B 29 (1984) 102. [20] H.N. Robkoff, D.A. Ewen and R.B. Hallock, Phys. Rev. Lett. 43 (1979) 2006; F.H. Wirth and R.B. Hallock, Phys. Rev. B 35 (1987) 89. [21] V.F. Sears and E.C. Svensson, Phys. Rev. Lett. 43 (1979) 2009; E.C. Svensson, V.F. Sears, A.D.B. Woods and P. Martel, Phys. Rev. B 21 (1980) 3638. [22] G. Gaglione, G.L. Masserini and L. Reatto, Phys. Rev. B 23 (1981) 1129.
197
[23] L. Oddi, P.A. Whitlock and L. Reatto, J. Low Temp. Phys., in press. [24] G.J. Hyland, G. Rowlands and F.W. Cummings, Phys. Lett. 31A (1970) 465. [25] A.D.B. Woods and E.C. Svensson, Phys. Rev. Lett. 41 (1978) 974. [26] E.F. Talbot, H.R. Glyde, W.G. Stirling and E.C. Svensson, Phys. Rev. B 38 (1988) 11229; W. Stirling and H.R. Glyde, Phys. Rev. B 41 (1990) 4224. [27] G.A. Williams, Phys. Rev. Lett. 68 (1992) 2054. [28] Here by vortices we mean vortex rings of diameter large compared with the interatomic distance. The identification [27] of rotons with the smallest vortex ring is not essential to the theory of Williams and we prefer to distinguish the two excitations because we do not find very strong arguments in favor of their identification. [29] W.I. Glaberson, D.M. Strayer and R.J. Donnelly, Phys. Rev. Lett. 21 (1968) 1740. [30] K. Ohbayashi, in Ref. [1], p. 77.
ELSEVIER
Excitations in superfluid
4He and the condensate
L. Reatto*, G.L. Masserini, S.A. V i t i e l l o Dipartimento di Fisica, Universitd degli Studi di Milano, 20133 Milano, Italy
Abstract Starting from an accurate representation of the wave function of the ground state and of the phonon-roton excitations based upon the shadow function we present a microscopic dens;ty matrix for superfluid 4He. The roton and maxon energy is computed both at zero and at finite temperature at different densities. The separate contributions of kinetic and potential energy of the roton excitation are computed and we find that a roton lowers the potential energy. The roton contribution to the depletion of the Bose-Einstein condensate n o is computed. Excitation of 5 to 10 rotons is needed for the depletion of one atom from the condensate and no simple relation is found between the T dependence of n o and of g(r). Other quantities we compute as function of T are the static structure factor and the strength of the 'one-phonon' peak of the dynamical structure factor. Comparison of our results with experiment gives some evidence that additional excitations become important above about 2 K.
1. Introduction T h e ground state of superfluid 4He is considered to be rather well understood on the basis both of the variational theory and of exact simulations like the so-called G r e e n ' s function M o n t e Carlo method. The situation is quite different for the excited states and finite temp e r a t u r e properties as shown by the debate [1] o v e r the nature of rotons, on the role of the B o s e - E i n s t e i n condensate and on the dynamical response function of the system. T h e quantitative theory of the excited states in superfluid 4He starts with F e y n m a n (F) and F e y n m a n - C o h e n [2] (FC) representation of the wave function ~q of a state of m o m e n t u m hq. This can be extended in a systematic way with the m e t h o d of correlated basis functions [3] ( C B F ) , but even with the present most elaborate
* Corresponding author.
c o m p u t a t i o n [4] the roton energy is still 15% a b o v e experiment. In addition, only with the F and FC ~q it has been possible [5,6] to construct the finite T density matrix, but not with the m o r e advanced formulations. Recently [7] a new form of wave function (WF) for the excited states has b e e n introduced, the shadow WF. This is built upon the shadow representation [8] of the ground state gt0. In this new trial W F for gr0 correlations between particles arise both directly and indirectly via coupling to subsidiary variables, the shadows. These are thought to represent the effect of q u a n t u m delocalization of the 4He atoms. By switching the d e p e n d e n c e of the density fluctuation factor in the F-WF from variables representing the positions of the 4He atoms to the subsidiary variables, one gets a representation of ~q free of variational p a r a m e t e r s and that turns out to give the best estimate of energy of a roton both at equilibrium and at freezing density. The structure of aFq is simple enough that it turns out
0921-4526/94/$07.00 (~ 1994 Elsevier Science B.V. All rights reserved SSDI: 0921-4526(93)E0466-T
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possible [9] to construct the density matrix based on an ensemble of such excited states. This has opened the way [10] to the computation of microscopic correlations also at temperatures where roton excitations are important. Here we review this theory and present the results of some recent computations that extend the previous ones [10].
energy. This gives the motivation for writing the trial function [7] in the form
q"q(R) = ~b,.(R) f dS K(R, S)$s(S)o q
(1)
where O-q is the density fluctuation of the shadow variables: N
Crq = ~'~ e '°~ ,
(2)
i-1
2. Excited states
In a shadow WF the positions R--{ r l , . . . , r n } of the 4He atoms are coupled by Gaussians to subsidiary variables S ={s I . . . . , Sn}. The positions {ri} are correlated by a Bijl-Jastrow pair factor and {si} are correlated in a similar way. Since the {s~} are integrated out, many-body couplings among the {rz} arise beyond the pair level. A motivation for this representation of qt0 comes from Feynman's path-integral formulation of the density matrix [11]. As is well known, in this formulation each position r~ of a particle becomes associated to a sequence of positions over 'time' slices so that each quantum particle becomes equivalent to a suitable flexible polymer. In this way different quantum particles become correlated not only directly via the action of the interatomic interaction but also indirectly via the correlations between the other monomers corresponding to the different time slices. In a shadow WF the correlations induced by the subsidiary variables have a similar role and now the position s i can be thought to represent the center of mass of the polymer associated to particle i. It is important to recognize that the analogy is particularly appropriate for hard-core particles at high density like in 4He. Under such conditions cross linkings between different polymers, reflecting the Bose symmetry, are not too frequent so that it is meaningful to speak of the center of mass at least in a transient sense. The quantum hole due to delocalization of particles should be present also in the excited states and compared to the ground state it should be essentially unaltered for excitations of low
K(R, S) is a product of Gaussians which couple r i with s i and Or(R) = lq exp[-u(rq)], i
q,s(S) = Iq exp[-us(si/)] • i<[
(3)
If we had the factor N
P, = Z e iqr'
(4)
i-1
in place of O-q in Eq. (1), then qrq would just be the Feynman WF when the ground state is represented by a shadow WF. The modification has far reaching consequences because Eq. (1) already includes effects of backflow [12] and by expanding the intergrand with respect to r i -s~ one finds that Eq. (1) contains terms of all orders in powers of pq. In the ground state the shadow variables induce correlations beyond the pair level to all orders, in a similar way a density fluctuation in the shadow variables induces fluctuations beyond the linear level for the real variables. We have computed the excitation energy eq as the difference between the expectation value of the Hamiltonian H with ~Fq and the similar expectation value corresponding to g0, i.e. Eq. (1) without the factor trq. As interatomic pair potential we have used the form VA(r) of Aziz et al. [13] and the expectation values have been evaluted by the Metropolis Monte Carlo algorithm for a system of 108 particles and periodic boundary conditions. The pseudopotentials u(r) and us(r ) in Eq. (3) derive from a ground state computation. In the first computation [7] of eq the McMillan form (b/r) m was used both for u(r) and us(r ). It has
L. Reatto et al. / Physica B 197 (1994) 189-197
been found that an improved description of the ground state is obtained [14] if the pseudopotential us(r ) between shadows contains an attractive part. A convenient representation of this pseudopotential is
u~(r) = VA(dr )/A where vA is the Aziz potential and d and A are variational parameters. The results we present here correspond to this pseudopotential. In Fig. 1 we show the excitation energy at three densities, in panel (a) at p = 1.97 a t / n m 3 which is 10% below the equilibrium density p¢q, in (b) the result at Peq = 2.18 a t / n m 3 and in (c) at p = 2 . 6 2 a t / n m 3 close to the freezing density. The excitation energy is computed only for a discrete set of wave vectors consistent with the periodicity of the simulation box and we take q in the (1 0 0 ) , (1 1 0) and (1 1 1) directions. It should be noticed that computation of Eq is very demanding because eq is obtained as difference i
i
a
20.0
3[
10.0
x I
0.0
,
I
,
I
b
20.0 v
LU 10.0
0.0
,
I
,
I
20.0
10.0
0.0
I
\
I
0.0
:'
"11[/
I
1.0 2.0 q (A -~)
Fig. 1. Excitation spectrum of liquid 4He as a function of wave vector q: (a) p = 0.gpeq, (b) Pcq" The solid line is a spline fit to neutron-scattering data [15] and (c) p = 1.2pcq. The dotted line is a parabolic fit to experimental data [16].
191
of total energies that are about 100 times larger of %. Typical runs consist of several 106 attempted moves per particle. We discuss first the result at Peq- A t smaller q we see the phonon liner behavior followed by the maxon at about the correct q with an energy 1 6 . 4 - + 0 . 5 K which is 2 . 4 K above experiment [7]. The roton energy is 10.2---0.5K, 1 . 6 K above experiment, and the position is at a q vector perhaps displaced by 0.1 A-~ to larger values, but it is difficult to locate precisely the minimum due to statistical fluctuations. These results compared with previous ones obtained with a different shadow pseudopotential for the ground state show an improvement of 1.3 K for the maxon energy. The present results have also a smaller variance than the previous ones. These facts confirm the strong sensitivity of the maxon energy on the form of the ground state. In the roton region, although we cannot definitively exclude agreement with our old result due to the statistical uncertainties, we get now an energy 0.4 K higher. On the other hand it confirms that the simple Feynman like shadow WF gives a result which is competitive with the best results [4] obtained with CBF. The density dependence of E q from our computation has the correct behavior, a velocity of sound and maxon energy increasing with p and a lowering of the roton energy. C o m p a r e d w i t h Peq, at p = 1.2Pe q the Erot drops by 1.6 K to 8.6-+ 0.5 K, a change similar to the experimentally observed drop [16] of erot by 1.4 K at 24 bar corresponding to p -- 1.18Pe q. AS p increases the displacement of the roton minimum to larger q is somewhat weaker than experiment. We present also the first result for eq at a density lower of the equilibrium value. At p = 0.9Pe q the roton energy increases by 0.7 K and the maxon decreases by 1.5 K so that the energy difference between maxon and roton is reduced to about 4 K from 6.3 K at Peq" Planned experiments in expanded liquid 4He should be able to test this prediction. Our computations have shown an interesting feature of a roton: the energy erot has a negative contribution Urot from potential energy and correspondingly the kinetic energy is larger of erot.
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L. Reatto et al. / Physica B 197 (1994) 1 8 9 - 1 9 7
A t Peq we find Urot = - 1 . 8 K and at 1.2Peq, ttro t - 2 . 2 K. This is an effect of short-range correlations as discussed below and indicates the strong anharmonic character of a roton. For a harmonic excitation like a long-wavelength phonon the energy has equal contributions from potential and from kinetic energy. Recently we have also extended [18] the computation of the excitation energy at a finite temperature by considering a single excitation added on top of thermally excited rotons. We find a significant lowering of erot at finite T, similar to what is observed experimentally.
3. Density matrix and finite temperature properties In the shadow representation of excited states a multiple excited state is obtained by replacing in Eq. (1) the o-q factor by a product of O-q, one for each of the excitations. This WF is an exact eigenstate of the total momentum corresponding to /'tot = hEqnqq where nq is the number of q excitations. If Eqnq is finite the excitation energy is simply Eqnqeq. Assuming as an approximation that this is true also when the number of excitations is of order N, we have obtained [9] an explicit expression for the density matrix that should be accurate when the average number of excitations Eq(nq)/N ~ 1 is small. It reads (U'lP~ln > =
~O,(R')~Or(R)f dS' dS
× qJ~(S')~b~(S)K(R',S')K(R, S) T
× I] e x p [ - x 1 i
T
× [I e x p [ - x 2 i,j
t T (sij) - X l (sij)] r
Isi -
sjII/QT.
(5)
Doubling of the shadow variables is due to the fact that PT is related to the square of wave functions. QT is the normalization constant, the T dependent factors xT and x2v are functions of the density fluctuations spectrum of shadows in the ground state and of %. From Eq. (5) we see that excited states induce in PT extra correlations of the pair product form in the shadow variables,
but the mathematical structure is very similar to the P e n r o s e - R e a t t o - C h e s t e r form [5] of PT. At low temperature only low-energy excitations, phonons and rotons, play a role in PxPhonons induce long-range correlations that we cannot handle directly due to the small size of our simulation box. On the other hand the effect of thermally excited long-wavelength phonons is well known from other theories. Here we focus on the effects due to rotons so that Eq is approximated by the roton parabola eq = A + h2(q2 T qrot) /2bt" Since eq enters the expression for X1 T and Xz as a counting factor, for better accuracy we have taken eq from experiment. At Peq we have used [16] qrot = 1.92"~-1 , /z =0.136mile and a T dependent A(T) = A(O) + ~A(T) with [16] A ( 0 ) = 8.608K and a roton energy shift ~A(T) as given by the phenomenological theory of Bedell, Pines and Zawadowski [19], which is in good agreement with experiment. With this PT we have computed the energy E(T) and a number of correlation functions. We present here the results of some new computations with better statistics than the earlier results. The interest is mainly in the correlation functions. The energy E(T) + (H)T for our approximate PT should just be equal to the Landau expression E(T) = E o + ~qEq(nq)/N, where (nq) is the Bose-Einstein distribution. However, the direct computation of E(T) = (H) T is a very severe test of the consistency of the expression of PT and of our simulation. We find very good agreement between E(T) computed in the two different ways. For instance at T = 2.1 K direct computation gives E(T) - E 0 = 1.017 -0.007 K, a value to be compared with E ( T ) E 0 = 1.04 + 0 . 0 4 K estimated from the Landau expression, in which our result for the roton energy is used. The structure factor S(q) corresponding to the ground state at the three densities of our computation is shown in Fig. 2. Notice that the present W F does not include the long-range correlations due to the zero point motion of phonons so that S(q) does not have the correct linear q behavior as q ~ 0. Experimentally [20,21] it is known that the main peak of S(q) becomes higher and sharper with increasing T. This is exactly what
L. Reatto et al. / Physica B 197 (1994) 189-197
193
i
Pl
0.040 ¢,i II I---
,~ 1 . 0 09
it
a
0,020
03
0.000
-0.020 /
0.0 0.0
,
I
,
2.0 q (A -1)
I
4.0
Fig. 2. Structure factor S(q) at T = 0, the ground state at three densities: dots for p = 0.9peq; at Peq the solid line shows our results and the plus signs show the experimental data [21], the dashed line displays the results for p = 1.2pe q.
we find with our PT. In Fig. 3(a) our result for AS(q,T = 2.1), AS(q,T) = S(q,T) - S(q,T = 0), is compared with experiment [20] (average of the data at T = 2.07 and 2.12 K). The present result for AS(q, T) is similar to the one obtained [22] with the simpler P e n r o s e - R e a t t o - C h e s t e r density matrix. This is consistent with the result of a detailed study [23] that has shown that FC backflow has a limited effect on S(q; T). In Fig. 3(b) we present the quantity Y ( q ; T ) = AS(q, T)/Nrot(T), where Nrot(T ) is the number of thermally excited rotons per atom at temperature T. The computation has been performed at T=I.IK, 1 . 4 K and 2 . 1 K where Nrot(T ) is equal to 0.00097, 0.0061 and 0.092, respectively. We see that at low temperature AS(q; T) scales very accurately with Nro t. So the low T limit of Y(q; T) represents the change of S(q; T) due to a single roton. The related quantity
Ag(r; T)INrot(T), where Ag(r; T) = g(r; T) - g(r; T = 0), is plotted in Fig. 4 and clearly shows that a roton induces stronger short-range order. Y(q; T) at higher T is larger than the low-T result. This nonlinear behavior of AS in Nro t is a manifestation of r o t o n - r o t o n interaction and indicates that it is easier to excite a roton if one is already present, because of a cooperative effect in the induced change of short-range order.
0.20
>0.00
-0.20
0.5
i
t
1.5
2.5
q (A-1) Fig. 3. Variation of the structure function with the temperature as a function of q. (a) The solid line is the computed AS(q; T = 2.1) and the dots show the average of experimental data [21] obtained at 2.07 and 2.12K. (b) Y(q; T): the scaled variation of the structure factor with respect to the number of rotons at temperature 1.1 (dots), 1.4 (solid line) and 2 . 1 K (plus signs). Notice that AS in (a) has been computed by extending the simulated g(r) at large r with a damped oscillating tail. No tails have been used for Y and this is the origin of the oscillations of Y at small and large q.
F r o m PT we have computed also the oneparticle off-diagonal density matrix and from this we deduce the Bose-Einstein condensate no(T ) and the m o m e n t u m distribution. As expected no(T ) decreases with increasing T. At Peq the ground state we are using has [10] n ( 0 ) = 7.8% and this is reduced to 5.8% at T = 2.17K. Since in our computation n r o t ( T = 2 . 1 7 ) =0.127 we conclude that with our density matrix there is one 4He atom depleted from the condensate for every - 6 thermally excited rotons. At p = 1.2peq this n u m b e r rises to - 1 0 rotons per depleted atom from the condensate. This indicates the profound difference from the Bogoliubov regime in which there is a one-to-one correspondence between depletion of the condensate and the n u m b e r of excitations. T o view a roton as an atom taken out of the condensate is in complete contradiction with the present theory and we
L. Reatto et al. / Physica B 197 (1994) 189-197
194
1.5 0.05
s"
1.0
'
'
'
'
0.03
'4
J
~*
II
I-..E. JC
0.5
0.01
-0.01
0.0 b
-0.03 -0.0
0.10
L2 t3"}
,
J 0.01
,
L 0.03
, 0.05
h (r; T=2.10)
z p"
, -0.01
...,.
o.oo
Fig. 5. h(r; T = 0 ) versus h(r; T = 2 . 1 ) for the range 6 , ~ < r < 13 ~, at poq. The best straight line has a slope B = 0.898 --+ 0.009. The dashed line has a slope 0.927 corresponding to the
:4 -0.10
best fit of experimental data [21].
-0.20
--
'
0.0
i
i
5.0
10.0
15.0
r (A)
Fig. 4. The radial distribution function at poq and its T variation scaled with the number of rotons. (a) g(r;T = 0); (b) Ag(r; T = 1.4)/N~o, is shown by dots, Ag(r; T = 2.1)/N,o' is displayed by plus signs. believe it is a wrong picture of what happens in superfluid 4He. T h e increasing short-range order with T has attracted large attention [20,21] because of the hypotized relation [24] between the T dependence of g(r) and no(T ). Based on a certain decomposition of the two-particle density matrix Hyland, Rowlands and Cummings ( H R C ) were led to the following relation for h(r)=-g(r)- 1, when r > 5 ,~:
h(r; T) = A(T)h(r; Tx) ,
(6)
A(T) = [1 - n o ( T ) ] 2 .
(7)
Relation (6) implies that the zeros of h(r; T) do not change with T and that h(r; T)/h(r; T ' ) should be independent of r. In other words a plot of h(r.; T) versus h(r; T') should give a straight line. Experiments [20, 21] display such behavior and also our results indicate that Eq. (6) is followed rather accurately. In Fig. 5 such a plot is shown for T = 2 . 1 K and T = 0 K . It is clear that Eq. (6) is a good representation of our
results but due to the high accuracy of the present computation, our results indicate the presence of some small deviations from it. The deviations are below the level of accuracy of the available measured S(q), but new measurements with higher accuracy might be able to show if such deviations are present. Since from our PT we are able to separately determine no(T ) and A(T), we have been able to perform [10] the first microscopic test of the H R C relation (7). We find that A(T) is considerably different from [ 1 - n0(T)] 2, the T dependence of A(T) being stronger than that implied by no(T ) . The reason for the failure of relation (7) is that the 'uncondensed' part of the two-body density matrix in the H R C decomposition has a strong T dependence which is neglected by H R C . The final quantity we discuss is the strength Z(q; T) of the single-excitation peak in the dynamical structure factor S(q,~o). In the Feynman approximation Z(q; T)= S(q; T) and this implies that the q mode is a pure density excitation, all the strength of S(q, w) is in the single excitation peak and there is no background. This is correct in the q--~ 0 limit, but not in the m a x o n - r o t o n region where Z(q; T) is substantially smaller [15] than S(q; T). Also, in this q region the roton peak broadens and loses intensity [25,26] as T---~ Tx. With our WF and density matrix we are able to compute Z(q; T)
L. Reatto et al. / Physica B 197 (1994) 189-197 1.5
i
C/~
,
i
+
~ 0.5
0.0
,
~
,
+. +.,
\
110
210 310 q (A-~)
4.0
Fig. 6. The strength Z(q; T) of the single excitation peak as a function of q at Peq for T equal to 0 (dots) and 2.1 K (plus signals). The solid line shows the structure factor S(q) at T = 0 and the cross signs that at T = 2.1 K. and the result at T = 0 and 2.1 K are shown in Fig. 6. Our W F describes an almost pure density fluctuation at small q but this is not so in the maxon and roton region, the multiphonon background having a weight of 20% in the maxon region and of 10% in the roton region. At finite T the multiphonon background increases and for a roton it is about 20% at 2.1 K. This trend is consistent with the observed weakening of the single-roton peak in S(q, to). A direct comparison with experiment is not possible due to the substantial width of the roton peak for T > 2 K and to the presence of an overlapping multip h o n o n background. In our theory we have no information on the lifetime of excitations. At p = 1.2peq the multiphonon background is 8% f o r a r o t o n at T = 0 K a n d 15% at T = l . 8 5 K .
4. Discussion The introduction of shadow WF has led to a rather comprehensive theory of liquid nile. This gives an excellent description of the ground state, a rather accurate description of the excited states and a density matrix that at last gives the possibility of computing microscopic correlation properties of superfluid 4He at the level of L a n d a u theory for macroscopic properties. The fact that without variational parameters for the excited states we get the roton energy at all
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densities to within 1-1.5 K is remarkable. We believe this is due to the fact that in this formulation the backflow implicit in the W F is in the exponential form, like in the original [2] FC WF. Up to now it has not been possible to treat this exponential form and the W F has always been linearized. No such problem arises in our formulation. The form of backflow with our W F is determined by the intershadow pseudopotential of the ground state. By allowing the presence of an additional backflow term specific to the excitation it should be possible to bring theory closer to experiment. The strength Zq of the single excitation peak in S(q, to) is significantly lower than S(q), and this indicates that in our description a roton is not a simple density fluctuation. Other important features that emerge from our computations are that a roton induces an increased local order and this is accompanied by a negative contribution to the internal energy. In addition it appears that 4He atoms of all momenta participate in the excitation of a roton and the depletion of the condensate is much smaller than the n u m b e r of excitations. Our computation of the roton energy at finite temperature indicates the importance of r o t o n roton interaction and an important development of our density matrix will be a self-consistent treatment of excitations at finite T. Both the T dependence of Zq and of condensate no(T ) have the correct behavior. However, it does not signal the strong T evolution as T~ is approached, like the vanishing of n o and the strong change of shape of S(q, to). All this could be due to the effects of r o t o n - r o t o n interaction that are not included in a simple renormalization of the roton energy as we have used. These effects, however, should have only a small effect on short-range correlations. A more likely explanation is the presence of additional excitations in addition to phonons and rotons, like vortices as advocated [27] by Williams. Such excitations [28] are particularly effective in destroying phase coherence and therefore in depressing n o . However, vortices should not modify very much the local order because only the small vortex core region should be affected. On the other hand the
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presence of vortices can offer an explanation to s o m e puzzling features of rotons. A roton in the flow field vs of a vortex get a modified energy and if the size of a roton is small c o m p a r e d with the scale of variation of v~ [29] we have
shown also here for strong coupling effects in roton excitations, and this should be t a k e n into account in any theory of superfluid 4He.
Acknowledgements
eq eq + hvs'q.
(8)
In the presence of a statistical distribution of vortices, the roton energy becomes distributed over a range depending on vs so that there is an additional broadening mechanism to the roton linewidth. It is interesting that the situation is rather different in the case of neutron scattering c o m p a r e d with the R a m a n case. For neutrons the roton p e a k in S(q, to) is directly affected by the distribution of roton energy as given by Eq. (8). In the R a m a n response two rotons are emitted with almost opposite values of the wave vector. This means that the total energy of the two rotons is not affected by vS and one expects to see a R a m a n linewidth narrower than the one inferred from S(q, to). This is exactly what is o b s e r v e d experimentally [30]. We have conside r e d a very simple model of this effect elsewhere [18]. In our theory we have well-defined roton excitations, that are not simple density fluctuations, and a condensate. The roton appears at the same time in the density response function, as apq is not orthogonal to pql/f0, and in the single-particle G r e e n ' s function G(q, to), because ~q is an a p p r o x i m a t e eigenstate of H. We cannot answer the question if a roton would exist and would show up in S(q, to) if the system had no condensate because we do not know a natural way in three dimensions to construct a Bose fluid ground state without a condensate. O u r roton has the character both of a quasiparticle and of a collective excitation. We find it not very useful to try to establish the 'true' nature of a roton. This is a question to which one is led on the basis of perturbation theory. In the p h o n o n region, it is well known that strong coupling effects change the condensate n o, in the Bogoliubov spectrum and in the weight of G(q, to) in S(q, to) into the total density p. T h e r e is ample evidence from variational theory as
We would like to acknowledge useful discussions with G.V. Chester and M . H . Kalos. This work was partially supported by Consorzio I N F M and by Consiglio Nazionale delle Ricerche under Progetto Finalizzato 'Sistemi Informatici e Calcolo Parallelo'.
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