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The dynamic structure factor S(k, omega ) for liquid 4He is studied with the use of perturbation theory in a correlated basis generated by the Feynman-Cohen (FC) excitation operator acting on the interacting ground state. We consider the coupling of one FC excitation to two FC excitations and sum the important contributions to all orders in perturbation theory. S(k, omega ) is calculated for many values of k in the interval (0.8 A[ring -1], 4.5 A[ring -1)]. We find a low-energy delta -function peak which corresponds to the phonon-maxon-roton spectrum and report its energy e(k) and Z(k) as a function of k. ``Two-quasiparticle'' peaks are also found and they may be identified as roton-roton, maxon-maxon, and maxon-roton contributions. At high k, most of the strength of S(k, omega ) is distributed in the neighborhood of quasifree scattering energy omega [approx. equal] [h-bar] 2k2/2m, and the results are comparable with results of the impulse approximation. Semiquantitative agreement is found between the calculated S(k, omega ) and that inferred from neutron inelastic scattering experiments.
©1986 The American Physical Society.
PACS: 67.40.-w
Experimentally, the density fluctuations of liquid sup 4He are studied by means of neutron inelastic scattering. In a linear response regime the probability for scattering is proportional to the dynamic liquid structure function S(k, omega ).
The measured S(k, omega ) in the range of momentum transfers 0<k <wig2 A ang^{-1}has a well defined first peak, a broad structure at intermediate energies, and a high energy tail. The first peak corresponds to one quasiparticle excitation, and the intermediate energy structure and the high energy tail are attributed to multiquasiparticle excitations. For very high momentum transfers the measured S(k, omega ) has a broad peak in the vicinity of the free particle energy hbar^{2}k^{2}/2m. At intermediate momentum transfers (k app 2.5 A ang^{-1}), S(k, omega ) has complex structure.
The origin, the energy dispersion curve e(k), and the strength Z(k) of the one quasiparticle peak (first peak) are more or less understood.sup 1,2
There is evidence from Raman scattering experimentssup 3 that in the long wavelength limit there is a bound state of two paired rotons with opposite momenta. Zawadowski, Ruvalds, and Solanasup 4 (ZRS) have suggested that the intermediate energy structure of S(k, omega ) may be explained by considering the excitation of roton roton bound states. However, Svensson et al.^{5}find no indication that it is necessary to invoke the existence of two roton bound states to explain the intermediate energy structure in their neutron scattering data. In addition, Hasting and Halleysup 6 have shown that no ZRS model can provide a quantitative account of the intermediate energy structure of S(k, omega ).
Jacksonsup 7 has studied S(k, omega ) with microscopic theory, using a set of nonorthogonal correlated multiphonon states defined by
|n rangle = rho ( k vec_{1}) rho ( k vec_{2}) ... rho ( k vec_{n}) |0 rangle , (1.1)
Where rho ( k vec ) is the k vec Fourier transform of the density operator and |0 rangle is the ground state. He considered the one- and two phonon contribution to S(k, omega ), calculated the one particle Green's function in perturbation theory, and related the spectral function to S(k, omega ). In this calculation the peaks of S(k, omega ) appear at much higher energies than do the experimental ones.
The interplay between the one quasiparticle and two quasiparticle contributions to S(k, omega ) has also been examinedsup 8 using its various omega moments. Within the framework of this approach, using the sum rules, one can place constraints on the multiquasiparticle contribution.
Recently we studiedsup 2 the one quasiparticle excitations in liquid sup 4He using perturbation theory in the Feynman Cohensup 9 (FC) basis. A nonorthogonal set of correlated basis functions (CBF's) is generated by the FC excitation operator:
rho_{B}( k vec ) == ^sum from i=1 to N ^ e^{{i}k vec cdot r vec_{i}}( 1+i ^sum from {j ( != i)} to N ^ k vec cdot r vec_{ij}eta (r_{ij}) right ) . (1.2)
eta (r) is the backflow velocity potential determined in Ref. 2 by means of variational calculations. The CBF n-quasiparticle state is defined as
|n rangle == rho_{B}( k vec_{1}) rho_{B}( k vec_{2}) ... rho_{B}( k vec_{n}) |0 rangle . (1.3)
The diagonal elements of the Hamiltonian matrix in this basis constitute the unperturbed part H_{0_}and the off diagonal elements form the perturbation H_{I_.}The zeroth order (unperturbed) one quasiparticle states and energies are the FC variational results. In Ref. 2 we carried out a detailed variational calculation of the one quasiparticle spectrum using the FC ansatz in conjunction with the hypernetted chain (HNC) technique. We found that the FC states are accurate for the phonons but the maxon and roton energies are app20% too high. We calculated the second order corrections due to the coupling of one FC quasiparticle state to two FC quasiparticle states. These corrections improve the agreement with the experiment significantly; the theoretical e(k) is within 5% of the experimental spectrum and the calculated strength Z(k) is within app30% of the experimental strength.
In this paper we calculate the one- and two quasiparticle contribution to S(k, omega ). A perturbative expansion of the density correlation function is developed in the correlated basis (1.3). We consider only the coupling of one FC excitation to two FC excitations and sum the important contributions to all orders in perturbation theory. The S(k, omega ) is trivially obtained from the density correlation function.
We calculate S(k, omega ) for k in the interval (0.8 A ang^{-1,}4.5 A ang^{-1}) and find that the one quasiparticle excitation is well defined at all values of k. It exhausts more than half of the strength of S(k, omega ) for 0<k <wig2 A ang^{-1;}however, its strength Z(k) becomes very small for k greater than 3 A ang^{-1.}We also find smaller peaks at intermediate energies that can be identified as simultaneous excitations of two rotons or two maxons and one maxon plus one roton. For 2 A ang^{-1}<wig k <wig3 A^{-1,}a two peak structure is found for S(k, omega ); the first peak is the continuation of the ``one quasiparticle'' branch, while the second is close to the Feynman phonon energy of hbar^{2}k^{2}/2mS(k). For k >wig 3 A ang^{-1}most of the strength is distributed in the neighborhood of hbar^{2}k^{2}/2m. For k>4 A ang^{-1}the calculated S(k, omega ) is comparable to that obtained using the momentum distribution and the impulse approximation. The main features of the experimental S(k, omega ), over a wide range of k and omega, are semiquantitatively explained by this microscopic theory.
The theoretical calculation of the density correlation and dynamic liquid structure functions using CBF perturbation theory is discussed in Sec. II, and the results are given in Sec. III. II. THEORY
The density correlation function is defined as
D(k, omega ) == langle2 0 left | rho_{{k}vec}^{dag}1 over {H-E_{0}- omega -i eta} rho_{{k}vec} right | 0 rangle2 , (2.1)
where E_{0_}is the ground state energy and we take the limit eta ->0. It is knownsup 10 that
S(k, omega )= 1 over pi ^Im [D(k, omega )] . (2.2)
We write the many body Hamiltonian H as
H=H_{0}+H_{I} , (2.3)
where
langle n|H_{0}|m rangle == delta_{nm}langle n|H|n rangle , (2.4)
langle n|H_{I}|m rangle == (1- delta_{nm}) langle n|H|m rangle , (2.5)
and |m rangle ,|n rangle ,^.^.^. are orthonormalized FC states (1.3). On performing a Goldstone type perturbative expansion for D(k, omega ) we obtain
D(k, omega ) = ^sum from {n( != 0)} ^ (-1)^{n}langle2 0 left | rho_{{k}vec}^{dag}1 over {H_{0}-E_{0}- omega -i eta} ( H_{I}1 over {H_{0}-E_{0}- omega -i eta} right )^{n}rho_{{k}vec} right | 0 rangle2 . (2.6)
For the moment, we allow only for one- or two FC quasiparticle intermediate states | k vec rangle and | l, m vec rangle. These are obtained by orthonormalizing the following FC states:
|1 rangle = rho_{B}( k vec ) |0 rangle , (2.7)
|2 rangle = rho_{B}( l) rho_{B}( m vec ) |0 rangle , (2.8)
| k vec rangle =|1 rangle / (N_{11})^{1/2} , (2.9)
| l, m vec rangle = ( |2 rangle - N_{12}over N_{11}| 1 rangle right ) slash2 (N_{22})^{1/2} , (2.10)
N_{ij}== langle i|j rangle . (2.11)
The coupling of one FC quasiparticle to two FC quasiparticle states is real and is defined as
a(k,l,m) == langle k vec |H-E_{0}| l, m vec rangle = langle l, m vec |H-E_{0}| k vec rangle .
The terms of the series (2.6) are represented by diagrams such as those of Figs. 1(a) and 1(b). A single line labeled k arrow denotes a one FC quasiparticle propagator: (2.13) G_{1^{(0)}(k,}omega ) = 1 over {e_{B}(k)- omega -i eta} , (2.12)
where e_{B}(k)_ is the FC one quasiparticle energy, i.e.,
e_{B}(k) == langle k vec |H-E_{0}| k vec rangle , (2.14)
and a double line labeled {l arrow , m arrow} denotes the propagator of the two FC quasiparticle state
G_{2^{(0)}(l,m,}omega ) == 1 over {e_{B}(l) +e_{B}(m)- omega -i eta} . (2.15)
The vertices are given in Figs. 1(c)ndash1(e). We define
xi_{1}(k) == langle 0| rho_{{k}vec}^{dag}| k vec rangle = langle k vec | rho_{{k}vec} |0 rangle , (2.16)
and
xi_{2}(k,l,m) == langle 0| rho_{{k}vec}^{dag}| l, m vec rangle = langle l, m vec | rho_{{k}vec} |0 rangle . (2.17)
Note that both xi_{1_}and xi_{2_}are real. In this notation, for example, the contribution of Fig. 1(a) is given by
xi_{1}(k) G_{1^{(0)}(k,}omega )a(k,l,m) G_{2^{(0)}(l,m,}omega ) xi_{2}(k,l,m) , L (2.18)
and a summation over the two quasiparticle momenta l , m vec is assumed. We can have the following three types of terms.
(i) First, there are terms having langle 0| rho_{{k}vec}^{dag}| k vec rangle_ on the left and langle k vec | rho_{{k}vec} |0 rangle_ on the right. The contribution of such terms is denoted by D_{11}(k, omega )_, and it is the sum of the terms in Fig. 2(a). We obtain
D_{11}(k, omega ) = {xi_{1^{2}(k)}}over {e_{B}(k) + SIGMA_{0}(k, omega ) - omega -i eta} , (2.19)
where
SIGMA_{0}(k, omega ) =- cf12 ^sum from {l, m vec} ^ |a(k,l,m)|^{2}G_{2^{(0)}(l,m,}omega ) . (2.20)
Here, we define SIGMA_{0}(k, omega )_ as the self energy insertion (2.20); we reserve the notation SIGMA (k, omega ) for the next, improved approximation.
(ii) Second, terms having langle 0| rho_{{k}vec}^{dag}| k vec rangle_ on the left and langle l, m vec | rho_{{k}vec} |0 rangle_ on the [Fig. 2(b)] and vice versa [Fig. 2(c)] are denoted by D_{12}(k, omega )_ and D_{21}(k, omega )_, respectively. Their contribution is given by
D_{12}(k, omega ) = {xi_{1}(k) theta (k, omega )} over {e_{B}(k) + SIGMA_{0}(k, omega ) - omega -i eta} , (2.21)
where
theta (k, omega ) =- cf12 ^sum from {l, m vec} ^ xi_{2}(k,l,m) G_{2^{(0)}(l,m,}omega )a(k,l,m) , L (2.22)
and
D_{21}(k, omega )=D_{12}(k, omega ) . (2.23)
(iii) Finally we have the terms of Fig. 2(d) with langle 0| rho_{{k}vec}^{dag}| l, m vec rangle_ on the left and langle lprime , m vec prime | rho_{{k}vec} |0 rangle_ on the right. Their contribution is given by
D_{22}(k, omega ) = mark cf12 ^sum from {l, m vec} ^ | xi_{2}(k,l,m)|^{2}G_{2^{(0)}(l,m,}omega )
lineup + {theta^{2}(k, omega )} over {e_{B}(k)+ SIGMA_{0}(k, omega )- omega -i eta} . (2.24)
In this approximation
D(k, omega ) =D_{11}(k, omega ) +2 D_{12}(k, omega ) +D_{22}(k, omega ) , (2.25)
and we can combine the xi_{1^{2_,}2xi}sub 1 theta, and theta^{2}terms to obtain
D(k, omega ) ={[ zeta_{0}(k, omega )]^{2}}over {e_{B}(k)+ SIGMA_{0}(k, omega )- omega -i eta} +D_{0^{sprime}(k,}omega ) , L (2.26)
where
zeta_{0}(k, omega ) == xi_{1}(k) + theta (k, omega ) (2.27)
and
D_{0^{sprime}(k,}omega ) = cf12 ^sum from {l, m vec} ^ | xi_{2}(k,l,m) |^{2}G_{2^{(0)}(l,m,}omega ) . (2.28)
By using Eq. (2.2) the dynamic structure function is simply given by
S(k, omega ) = mark pi^{-1}[2 ^Re [ zeta_{0}(k, omega )] ^ Im [ zeta_{0}(k, omega ) ] "{" e_{B}(k)+ Re [ SIGMA_{0}(k, omega )]- omega "}" - Im [ SIGMA_{0}(k, omega )] ("{" Re [ zeta_{0}(k, omega ) ] "}"^{2}- "{" Im [ zeta_{0}(k, omega )] "}"^{2})] L
lineup times ([e_{B}(k) + Re [ SIGMA_{0}(k, omega )] - omega ]^{2}+ "{" Im [ SIGMA_{0}(k, omega )] "}"^{2})^{-1}+Im [ D_{0^{sprime}(k,}omega )] , (2.29)
where
Re [ zeta_{0}(k, omega )] = xi_{1}(k) - cf12 ^sum from {l, m vec} ^ xi_{2}(k,l,m)a(k,l,m) ^ Re [ G_{2^{(0)}(l,m,}omega )] , (2.30)
Im [ zeta_{0}(k, omega ) ] =- cf12 ^sum from {l, m vec} ^ xi_{2}(k,l,m)a(k,l,m) ^Im [G_{2^{(0)}(l,m,}omega ) ] . (2.31)
Similarly, in calculating the real and imaginary parts of SIGMA_{0}(k, omega )_ and D_{0^{sprime}(k,}omega )_ we take the real and imaginary part of G_{2^{(0)}(l,m,}omega )_ inside the sums of Eqs. (2.20) and (2.28), respectively. Here
Re [G_{2^{(0)}(l,m,}omega )] = P1 over {e_{B}(l) +e_{B}(m)- omega} , (2.32)
Im [G_{2^{(0)}(l,m,}omega )] = pi delta (omega -e_{B}(l)-e_{B}(m) ) . (2.33)
The symbol P stands for the Cauchy principal value. In this approximation the one quasiparticle pole of D(k, omega ) is at omega =e(k) given by
e(k)=e_{B}(k) + SIGMA_{0}(k,e(k) ) , (2.34)
and if the Im[ SIGMA_{0}(k, omega )]_=0 and Im[ zeta_{0}(k, omega )]_=0, the residue of the pole is
Z(k)= ( {zeta_{0^{2}(k,}omega )} over {1- {partial SIGMA_{0}(k, omega )} over {partial omega}} right )_{{omega}=e(k)} , (2.35)
and the contribution from the pole to S(k, omega ) is Z(k) delta (omega -e(k) ).
We note that the expression (2.34) for e(k) is the Brillouin Wigner perturbation formula used in Ref. 2. Moreover, the expression (2.35) is also identical to the expression for the Z(k) used in Ref. 2; namely,
Z(k)= {|langle 0| rho_{{k}vec}^{dag}| psi ( k vec ) rangle |^{2}}over{langle psi ( k vec ) | psi ( k vec ) rangle} , (2.36)
where psi ( k vec ) is the perturbed one quasiparticle state given by
| psi ( k vec ) rangle = | k vec rangle - cf12 ^sum from {l, m vec} ^ a(k,l,m) G_{2^{(0)}(l,m,}omega ) | l, m vec rangle . L (2.37)
Notice that the denominator of (2.35) is identically equal to the norm of psi ( k vec ).
In this calculation, the results for the one quasiparticle contribution are identical to those of Ref. 2. In Ref. 2 we have seen that the differences between the variational (FC) and the experimental one quasiparticle spectrum are essentially removed by the inclusion of the second order correction given by Eq. (2.34). However, this approximation does not describe well the two quasiparticle contribution to S(k, omega ), because the two quasiparticle propagator [Eq. (2.15)] has poles at the FC energies. This is a consequence of our approximation to include only one- and two FC quasiparticle intermediate states. To obtain the two quasiparticle contribution at reasonable energies we must consider self energy insertions in the propagation of the double lines. On including some of the simpler self energy insertions shown in Fig. 3, the two quasiparticle propagator is modified as
G_{2}(l,m, omega ) = 1 over {e_{B}(l)+ SIGMA_{0}(l,^ omega -e_{B}(m) )+e_{B}(m) + SIGMA_{0}(m,^ omega -e_{B}(l) )- omega -i eta} , (2.38)
and, hence, the quantities SIGMA_{0}(k, omega )_, zeta_{0}(k, omega )_, and D_{0^{sprime}(k,}omega )_ entering in the expression for D(k, omega ) [Eq. (2.26)] are, respectively, modified as
SIGMA (k, omega ) =- cf12 ^sum from {l, m vec} ^ |a(k,l,m)|^{2}G_{2}(l,m, omega ) , (2.39)
zeta (k, omega ) = xi_{1}(k) - cf12 ^sum from {l, m vec} ^ xi_{2}(k,l,m) G_{2}(l,m, omega )a(k,l,m) ,
(2.41) D prime (k, omega ) = cf12 ^sum from {l, m vec} ^ | xi_{2}(k,l,m)|^{2}G_{2}(l,m, omega ) . (2.40)
S(k, omega ) in this approximation is still given by Eq. (2.29) by replacing the real and imaginary part of SIGMA_{0}(k, omega )_, zeta_{0}(k, omega )_, and D prime (k, omega ) by those of Eqs. (2.39)ndash(2.41). The real and imaginary parts of the above quantities come from the real and imaginary parts of G_{2}(l,m, omega )_ given by
Re [G_{2}(l,m, omega )] = {e_{2}(l,m, omega )- omega} over {[e_{2}(l,m, omega ) - omega ]^{2}+ GAMMA_{2^{2}(l,m,}omega )} , L (2.42)
Im [G_{2}(l,m, omega )] =- {GAMMA_{2}(l,m, omega )} over {[e_{2}(l,m, omega )- omega ]^{2}+ GAMMA_{2^{2}(l,m,}omega )} , L
where e_{2}(l,m, omega ) = mark e_{B}(l) + Re [ SIGMA_{0}(l,^ omega -e_{B}(m) )] (2.43)
lineup + e_{B}(m)+ Re [ SIGMA_{0}(m,^ omega -e_{B}(l) )] , (2.44)
GAMMA_{2}(l,m, omega ) = mark Im [ SIGMA_{0}(l,^ omega -e_{B}(m) )]
lineup + Im [ SIGMA_{0}(m, ^ omega -e_{B}(l) )] . (2.45)
If GAMMA_{2}(l,m, omega )_=0, then of course
Re [G_{2}(l,m, omega )] =P 1 over {e_{2}(l,m, omega )- omega} , (2.46)
Im [G_{2}(l,m, omega ) ] = pi delta (omega -e_{2}(l,m, omega ) ) . (2.47)
To calculate S(k, omega ) we need the matrix elements (i) of the Hamiltonian, i.e., e_{B}(k)_ and a(k,l,m), (ii) of the unit operator, i.e., N_{ij_,}and (iii) of the density operator, i.e., langle 0| rho_{{k}vec}^{dag}|1 rangle_ and langle 0| rho_{{k}vec}^{dag}|2 rangle_. All of them have been calculated in Ref. 2 and we use them here.
The numerical calculation of the real and imaginary parts of the quantities entering in the expression of S(k, omega ) is explained in the Appendix.
Using the two particle propagator with the self energy insertions, we have calculated S(k, omega ) for various values of k in the interval 0.8 A ang^{-1}<k<4.5 A ang^{-1.}For smaller values of k the total strength of S(k, omega ) is almost exhausted by the one phonon peak at energy e(k)= hbar ck and strength Z(k)-S(k)=( hbar /2mc)k, where c is the sound velocity. In general we find the following three different kinds of peaks in S(k, omega ).
(i) There is a well defined delta function peak at energy e(k) for all values of k in the above interval. S (k,^ omega <e(k) )is zero. This peak corresponds to the excitation of one quasiparticle out of the condensate. Its dispersion e(k) gives the phonon maxon roton endon spectrum (the excitations at k >wig2.4 A ang^{-1}are often called endons). The strength Z(k) of this peak becomes very small for large k (>3 A ang^{-1}).
(ii) There are peaks which presumably can be identified as a result of simultaneous excitation of two quasiparticles.
(iii) At high k we find that the main peak in the calculated S(k, omega ) is a somewhat broad peak in the vicinity of hbar^{2}k^{2}/2m as may be expected from the impulse approximation and sum rules. We call this peak the quasifree peak. On the other hand we divide the range of k into the following three regions.
(i) The region of k <wig2.2 A ang^{-1,}where the main peak is the one quasiparticle peak and peaks of secondary importance are the two quasiparticle excitations.
(ii) The region k >wig2.7 A ang^{-1,}where the main peak is the ``quasifree'' peak and there are other secondary peaks of two quasiparticle excitations at lower energies.
(iii) In the narrow interval 2.2 A ang^{-1}<wig k <=2.7 A ang^{-1}the strength of the one quasiparticle peak decreases very fast, while that in the quasifree region increases. At k=2.5 A ang^{-1}we have two peaks of comparable strength; the low energy one is on the continuation of the phonon maxon roton curve, while that at high omega is close to the quasifree.
The D(k, omega ) has a pole at omega =e(k) where the Im[ SIGMA (k,e(k) )]=0. This gives the so called one quasiparticle contribution Z(k) delta (omega -e(k) )to S(k, omega ). The e(k) is obtained by solving the Eq. (2.34) with the full SIGMA instead of the SIGMA_{0_,}and the Z(k) from Eq. (2.35) with zeta and SIGMA instead of zeta_{0_}and SIGMA_{0_.}The calculated energies e(k) for the values of k<3.0 A ang^{-1}are shown by the + signs in Fig. 4. The open circles are the experimental data and the solid line (from Ref. 2) is obtained by solving Eq. (2.34) using SIGMA_{0_.}Figure 5 presents the calculated Z(k<3.0 A ang^{-1}). [+ signs for the full calculation, solid curve (from Ref. 2) for calculation with zeta_{0_}and SIGMA_{0_,}and the dashed curve for the experimental datasup 11.] We see from Figs. 4 and 5 that the effect of the self energy insertions in G_{2_}is not too significant for the one quasiparticle contribution.
The total strength of S(k, omega ) is given by S(k):
int^ d omega ^S(k, omega ) =S(k) . (3.1)
The contribution to the above sum rule from the one quasiparticle excitation is Z(k); the rest, S(k)-Z(k), comes from two or more quasiparticle contributions. S(k) is also plotted in Fig. 5 (dashed dotted curve). We can see from this figure that in the range k <wig2.3 A ang^{-1}the one quasiparticle excitation gives the major contribution to S(k, omega ). For k >wig2.5 A ang^{-1,}however, the contribution of this excitation is relatively small.
The f sum rule
int^ d omega ^ omega S(k, omega ) = {hbar^{2}k^{2}}over 2m , (3.2)
is satisfied by the exact S(k, omega ). The one quasiparticle contribution to it is given by Z(k)e(k). If one assumes that the one quasiparticle contribution exhausts the sum rules (3.1) and (3.2), then Z(k)=S(k) and e(k) is given by the Bijl Feynman energy hbar^{2}k^{2}/2mS(k).
We find several peaks which may be identified as two quasiparticle excitations. These never become the major structures of S(k, omega ). For 0.8 A ang^{-1}<wig k <wig 2.0 A ang^{-1}they provide intermediate energy structure in the range 14 K <wig omega <wig30 K; and contribute 10 ndash 20^% of the overall strength. The results are shown in Figs. 6ndash9. The vertical lines in these figures show the one quasiparticle delta function contribution. The strength Z(k) is given in a box on the line. The strength of the rest of the contribution is given in a box under the two quasiparticle peaks, and the total strength S(k) is given along with the value of k in the top right hand corner. The results of our calculations with the G_{2^{(0)_}are}shown by dashed lines in Figs. 6 and 7. We note that as expected the self energy insertions in G_{2_}have a significant effect on the energies of the two quasiparticle peaks.
In order to understand the origin of the various peaks in this interval, we study the two quasiparticle density of states (DOS) defined as
rho_{2}(k, omega ) = 1 over N 1 over 2 ^sum from {l, m vec} ^ delta_{{k}vec , l+ m vec} delta (omega -e(l)-e(m) ) . (3.3)
It is instructive to understand the structure of the DOS. The one quasiparticle DOS diverges at the roton minimum and maxon maximum. Therefore, we expect that combinations of the above extrema to give large contributions to the two quasiparticle DOS and study them in detail.
a. Two rotons. We consider two rotons on the roton sphere as shown in Fig. 10(a) with center of mass momentum k vec = l+ m vec (| l| =| m vec | =k_{r_,}k_{r_}is the location of the roton minimum) and total energy omega =2 omega_{r_,}where omega_{r_}denotes the roton energy. As long as k remains finite ( !=0) there is only one angle theta for which k vec = l+ m vec; namely, that with costheta = k bhat cdot {lhat} =k/2k_{r_.}If k=0, theta can take any value from 0 to pi. In this case, rho_{2}(k, omega )_ diverges. In the case k !=0 (k<2k_{r})_, however, the phase space is limited and the two roton DOS is finite at omega =2 omega_{r_.}
b. Two maxons. The analysis here is identical to the two roton case, i.e., for k=0 there is a singularity at omega =2 omega_{m_,}where omega_{m_}is the maxon energy, but for k !=0 (k <2k_{m_;}k_{m_}is the maxon momentum) the DOS at omega =2 omega_{m_}is finite.
c. One maxon plus one roton. If k_{r}-k_{m}<k<k_{r}+k_{m_}(i.e., 0.8 A ang^{-1}<wig k <wig3 A ang^{-1}), a simultaneous excitation of one roton and one maxon can contribute to the two quasiparticle DOS in the vicinity of omega = omega_{r}+ omega_{m_.}There are a lot of degenerate states which have the above energy. We can demonstrate this as follows. We put a particle at the roton minimum and another at the maxon peak, so we create the state |A rangle =|k_{r}, k_{m}rangle_. Next we move the first particle a little away from the roton minimum and at the same time we let the maxon move to a neighboring state so that the energy of the new two particle state B [Fig. 10(b)] is the same as that of the state A. If k_{r}-k_{m}<k< k_{r}+k_{m_,}we can always choose the directions of the momenta of the new state to satisfy the momentum conservation. Hence, there is a large (ultimately infinite) number of two quasiparticle states in the neighborhood of omega = omega_{r}+ omega_{m_.}In fact, if one approximates the spectrum in the vicinity of the roton and maxon by two parabolas, it is easy to show that there is a logarithmic singularity in the two quasiparticle DOS for omega app omega_{r}+ omega_{m_:}
rho_{2}(k, omega ) prop - ln | omega - omega_{r}- omega_{m}| . (3.4)
The results of the numerically calculated DOS are shown in Fig. 11. We used the experimental dispersion curve with a momentum cutoff such that l and m are both less than 2.4 A ang^{-1}[e(2.4 A ang^{-1})=15.5 K]. The inclusion of states with higher momenta (greater than 2.4 A ang^{-1}) does not modify the rho_{2}(k, omega )_ for omega <=24 K, but it will of course alter the DOS for higher omega. We can see that the main peak is at the maxon plus roton energy omega apeq22.5 K.
In Fig. 12 we present the matrix elements
a_{rr}(k)=a(k,^ | l|=k_{r},^| m vec |=k_{r}) ,
a_{mm}(k)=a(k,^ | l| =k_{m},^| m vec | =k_{m}) ,
and
a_{mr}(k)=a (k,^| l| =k_{m},^| m vec | =k_{r})
for the cases of roton roton (rr), maxon maxon (mm), and maxon roton (mr) states, respectively. The contribution of these states to S(k, omega ) is proportional to the product of the DOS and the matrix element a.
The solid curve in Fig. 6 shows the calculated S(k, omega ) for k=0.825 A ang^{-1.}The two quasiparticle contribution has various peaks. The main peak occurs at omega apeq25 K and it is due to a one maxon plus one roton contribution. Two other peaks, one at omega apeq20 K and the other at omega apeq28.5 K, are due to roton roton and maxon maxon contributions, respectively. Those three peaks come in our calculation at somewhat higher energies from the ones that can be found by summing the corresponding one quasiparticle energies, because our two quasiparticle propagator (2.38) has the pole at the solution of the following equation:
e_{B}(l)+ SIGMA_{0}(l, ^ omega -e_{B}(m) )+e_{B}(m) + SIGMA_{0}(m,^ omega -e_{B}(l) )= omega . L
The solutions of the above equation for l,m being the roton roton, maxon maxon, and maxon roton momenta are systematically higher than e(l)+e(m). We note that in principle the maxon roton peak in S(k, omega ) should be singular. However, in the results of numerical calculations shown in the figures it is not singular, due to crude resolution of numerical methods discussed in the Appendix.
At k=1.125 A ang^{-1}the maxon roton peak dominates the two quasiparticle contribution to S(k=1.125,^ omega ) (Fig. 7). In S(k=1.525,^ omega ) (Fig. 8) the maxon roton and roton roton peaks are of comparable strengths, while at k=1.925 A ang^{-1}there is no maxon roton peak in the S(k=1.925,^omega ) (Fig. 9), but there is a strong maxon maxon and a relatively weaker roton roton peak.
We can understand these changes from the behavior of the a(k,l,m) and the DOS. In the neighborhood of k app1 A ang^{-1}the matrix elements a_{rr_,}a_{mm_,}and a_{mr_}have comparable magnitudes and so, as a result of the structure of the DOS (Fig. 11), the peak with higher intensity in S(k, omega ) is the maxon roton. For k app1.5 A ang^{-1}the matrix element |a_{rr}|_ is approximately twice as large as the |a_{mr}|_ and |a_{mn}|_ and so the roton roton peak has comparable intensity with the maxon roton peak even though it is not there in the DOS. At k app 1.9 ndash2.0 A ang^{-1}the a_{mr_}is very small, while the a_{rr_}is significant and the a_{mm_}is very big.
The S(k=0.825 A ang^{-1},^ omega ) has another peak at omega apeq15 K. This is due to two phonon states which are in the neighborhood of omega apeq ck. The multiple peak structure around 16ndash18 K of S(k=1.525,^ omega ) is presumably due to roton phonon and maxon phonon states.
In Fig. 13 we compare the calculated S(k=0.825,^ omega ) with the experimentalsup 5 S(k, omega ) at k=0.8 A ang^{-1.}The lowest peak of the experimental S(k, omega ) has to have zero width, but it is broadened by the instrumental resolution. In fact, the experimental resolution is essentially given by the width of this peak. The measured S(k=0.8,^ omega ) has a second broader and much weaker peak in the two quasiparticle energy region at omega apeq23 K, and a high energy tail. The theoretical S(k, omega ) has more structure in the two quasiparticle energy region and the main peak is at the maxon roton frequency omega apeq25 K. A much better instrumental resolution is necessary to see this structure in experiments. A crude numerical broadening of the theoretical S(0.825, omega ) with the present experimental resolution removes most of the structure. The theoretical two quasiparticle peaks appear at somewhat higher omega (8 ndash 9^% higher). This disagreement may be removed by the inclusion of self energy corrections which involve four FC quasiparticle states. The inclusion of these states make the numerical calculation much more complicated and we leave them out in this work. The high energy tail of the observed S(k, omega ) is presumably due to three and more quasiparticle excitations omitted in this study.
In a neutron scattering experiment, if the momentum transfer k is very high, the neutron ``sees'' the individual helium atoms distributed with the microscopic momentum distribution n(p). In this case the dynamic structure factor may be approximated by the impulse approximation (IA) L (3.6) S_{roman}IA (k, omega ) = ^int^ {d^{3}p} over {(2 pi )^{3}rho} n(p) delta ( omega - k^{2}over 2m - {p vec cdot k vec} over m right ) , (3.5)
where n(p) is normalized as
int^ {d^{3}p} over {(2 pi )^{3}rho} n(p)=1 . (3.7)
The structure of n(p) is as follows:
n(p)=n_{c}N delta_{p,0}+n(p != 0) , (3.8)
where n_{c_}is the condensate fraction of particles. In Ref. 12 we calculated n(p) using the variational ground state wave function. Now we can use the n(p) and (3.6) to calculate S_{roman}IA (k, omega )_ and compare the results with the calculated S(k, omega ) for high k. We obtain
S_{roman}IA (k, omega ) = mark n_{c}delta ( omega - k^{2}over 2m right )
lineup + m over {4 pi^{2}rho} 1 over k ^int_{{P}sub min}^{inf}^ dp^pn(p != 0) , (3.9)
P_{min}= left | k over 2 - {m omega} over k right | . (3.10)
The second part of Eq. (3.9) peaks when P_{min_=0,}i.e., at hbar^{2}k^{2}/2m ( == omega_{roman}QF )_. The first term is a delta function peak also at omega = omega_{roman}QF_. Its strength equals the fraction n_{c_}of the particles in the p=0 condensate.
The impulse approximation is expected to be valid for very high k. Its validity has been criticizedsup 13 for hard core liquids.
Before we compare our results for high k, at which the dominant peak of S(k, omega ) is the ``quasifree,'' we discuss the calculated S(k, omega ) starting from the value of k where a peak appears in the neighborhood of hbar^{2}k^{2}/2m, for the first time.
As we have already discussed, in the interval 2.2 A ang^{-1}<wig k <wig2.7 A ang^{-1}the strength of the one quasiparticle peak is decreasing rapidly, and this is a signal that another peak has to appear elsewhere to compensate for the rest of the strength S(k)-Z(k). In fact, this is the case in practice. The calculated S(k=2.475,^ omega ) shown in Fig. 14 has two main peaks. A delta function peak at the energy omega apeq14 K and strength Z=0.34, and a broad peak with some structure in the region of omega_{roman}QF_=37 K. In Figs. 14 to 18 the value of omega_{roman}QF_ is given in the top right corner. The dashed line in Fig. 14 gives the experimental S(k=2.5,^ omega ) taken from Fig. 6 of Ref. 1 and normalized in order to satisfy the sum rule (3.1). The two peak structure is present also in the experimental data. The theoretical and experimental strength of the peaks agree to a reasonable accuracy. The energies of the peaks of the calculated S(k=2.475,^ omega ) appear at somewhat different energies from those of the experimental peaks.
The calculated S(k, omega )= at k=2.975 and 3.475 A ang^{-1}(Figs. 15 and 16) has broad structures in the region of omega = omega_{roman}QF_. These contain most of the strength. However, there are one or two significant peaks at lower omega. The contribution of the ``one quasiparticle'' excitation has become insignificant.
The calculated S(k=3.975 A ang^{-1},^ omega ) and S(k=4.475 A ang^{-1},^ omega ) are compared with the impulse approximation results in Figs. 17 and 18. The S_{roman}IA (k, omega )_ (dashed dotted line) has a delta function peak at hbar^{2}k^{2}/2m with strength n_{c_}and another broad peak at the same place. In S(k, omega ) (solid line) we find a sharp peak at hbar^{2}k^{2}/2m and a broad structure in the neighborhood of this peak and at lower energies. The dashed curve in Fig. 18 shows the experimental S(k, omega ) taken from Ref. 11. This curve has a broad peak at a lower value of omega apeq100 K. Therefore the experimental data must have a very long energy tail in order to obtain an average energy omega=121 K required from the sum rule (3.2). It appears that k=4.5 A ang^{-1}is still low for the impulse approximation to be valid.
In Table I we give the calculated values of Z(k) and the total integral I_{0}(k)_ of the calculated S(k, omega ) to be compared with S(k). We also give the one quasiparticle contribution to the sum rule (3.2) and I_{1}(k)_\p = int^d omega ^ omega S(k, omega ) to be compared with hbar^{2}k^{2}/2m. We find that the sum rule (3.1) is violated by 7ndash 29^% and the sum rule (3.2) by 30ndash 47^%. To correct for this presumably one needs to consider higher contributions in the S(k, omega ). Some of the neglected vertices and terms are given in Figs. 19(a)ndash19(d). Their contributions become more and more important as one tries to satisfy higher omega moments of S(k, omega ). In the spirit of this theory the next terms to be included should be those with three phonon states. The new vertices at this level are shown in Fig. 19(a) and some of the new D(k, omega ) contributions are illustrated in Fig. 19(b). The still higher level of accuracy should include four phonon states illustrated in Figs. 19(c) and 19(d). Hopefully the contribution of the various diagrams to S(k, omega ) decreases rapidly as the number of interacting phonons increases. Nevertheless, the present calculation with only one- and two phonon states seems to give a semiquantitative description of the S(k, omega ) over a wide range of k and omega.
The authors wish to thank Professor David Pines and Dr. W. Stirling for many valuable discussions. This work was supported by the U.S. Department of Energy, Divison of Materials Sciences under Contract No. DE AC02 76ER01198.
The two quasiparticle DOS is defined by Eq. (3.3). If we consider the following representation of the delta function:
delta_{epsilon}(x) = 1 over {sqrt pi epsilon} e^{{-(x/}epsilon )^{2}} , (A1)
where delta (x) is the limit of the above distribution for epsilon ->0, the DOS may be written as
rho_{2}(k, omega ) = ^lim from {epsilon -> 0} ^ ( 1 over {8 pi^{2}rho k} ^int^ dl^l ^int_{|k-l|^{k+l}^}dm^m delta_{epsilon}(omega -e(l)-e(m) )right ) . (A2)
In practice we calculate rho_{2_}at a chosen value of epsilon and test the convergence of the lim_{{epsilon}-> 0}_ by recalculating the rho_{2_}with an epsilon that is smaller by approximately a factor of 2. The numerical integral is calculated with a grid spacing h for the l and m mesh. As epsilon is reduced, it is necessary to decrease h to assure accuracy of the numerical methods. The limit epsilon ->0 is achieved when the results are stable with respect to variations in epsilon and the grid spacing h of the integration. The results of the integral (A2) for k=0.885 A ang^{-1}and omega=20 K are shown in Table II. We notice that the convergence is achieved for epsilon apeq0.4 K and h=0.025 A ang^{-1.}
Another representation of the delta function may be the following:
delta_{epsilon}(x) = 1 over pi epsilon over {epsilon^{2}+x^{2}} . (A3)
A calculation of rho_{2}(k, omega )_ with the representation (A3) shows that the convergence is much slower and the required values of epsilon and h are smaller. Therefore, the form (A1) is more convenient for practical reasons.
From Eqs. (2.20) and (2.32)ndash(2.33) we may write
SIGMA_{0}(k, omega ) = Re [ SIGMA_{0}(k, omega ) ] +i^ Im [ SIGMA_{0}(k, omega ) ] , (A4)
Im [ SIGMA_{0}(k, omega )] =- pi over 2 ^sum from {l, m vec} ^ |a(k,l,m)|^{2}delta (omega -e_{B}(l) -e_{B}(m) ) , (A5)
Re [ SIGMA_{0}(k, omega ) ] = 1 over pi ^int_{0^{inf}^}d omega prime {Im [ SIGMA_{0}(k, omega prime )]} over {omega prime - omega} . (A6)
We calculate the Im[ SIGMA_{0}(k, omega )]_ using the expression
Im [ SIGMA_{0}(k, omega ) ]=- 1 over {8 pi^{2}rho k} ^int_{0^{inf}^}dl^l ^int_{|k-l|^{k+l}^}dm^m|a(k,l,m)|^{2}delta_{epsilon}(omega -e_{B}(l) -e_{B}(m) ) , (A7)
where delta_{epsilon}(x)_ is the representation (A1) and epsilon=0.4 K.
The real part of SIGMA_{0}(k, omega )_ may be written as
Re [ SIGMA_{0}(k, omega ) ] = - cf12 ^sum from {l, m vec} ^ {|a(k,l,m) |^{2}}over {e_{B}(l) +e_{B}(m)- omega} CTHETA (|e_{B}(l) +e_{B}(m) - omega |- eta )+ 1 over pi ^int_{{|}omega prime - omega | < eta} ^ d omega prime {Im [ SIGMA_{0}(k, omega prime )]} over {omega prime - omega} . (A8)
For small eta the second term is approximated by
1 over pi 2 eta {partial Im [ SIGMA_{0}(k, omega )]} over {partial omega} . (A9)
The value eta=0.4 K is used in the calculations. The contribution of the second term of Eq. (A8) is very small for this value of eta, and we have verified that the results do not change when eta is reduced to 0.2 K.
The real and imaginary parts of SIGMA (k, omega ) are given as follows:
left cpile { Re [ SIGMA (k, omega )] above Im [ SIGMA (k, omega )] } right } =- 1 over {8 pi^{2}rho k} ^int_{0^{inf}^}dl^l ^int_{|k-l|^{k+l}^}dm^m|a(k,l,m)|^{2}{ cpile { times Re [ G_{2}(l,m, omega )] , above times Im [ G_{2}(l,m, omega )] , } (A10)
where if GAMMA_{2}(l,m, omega )_ [Eq. (2.45)] is greater than epsilon, the Re[G_{2}(l,m, omega )]_ and Im[G_{2}(l,m, omega )]_ are given by Eqs. (2.42) and (2.43). If GAMMA_{2}(l,m, omega )_ is less than epsilon we use Eqs. (2.46) and (2.47) for the real and imaginary part of G_{2}(l,m, omega )_. The following representations are used for the principal values and delta function:
Re [G_{2}(l,m, omega )] = 1 over {e_{2}(l,m, omega )- omega} { 1- exp [ - ( {e_{2}(l,m, omega )- omega} over epsilon right )^{2}right ] right } , (A11)
Im [G_{2}(l,m, omega )] = pi { 1 over {sqrt pi epsilon} ^ exp [ - ( {e_{2}(l,m, omega )- omega} over epsilon right )^{2}right ] right } . (A12)
We use the value of epsilon=0.2 K for calculating SIGMA (k, omega ), and the integrals over l and m are done with grid spacing h=0.0125 A ang^{-1.}The integrals zeta (k, omega ) and D prime (k, omega ) [Eqs. (2.40) and (2.41)] are also evaluated in a similar fashion.
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