Journal of Low Temperature Physics, Vol. 112, Nos. 3/4, 1998
Excitations of Liquid 4He in Disorder
M. Boninsegni1 and H. R. Glyde2
1
Department of Physics, San Diego State University, San Diego, California 92182, USA
2 Department of Physics and Astronomy, University of Delaware, Newark,
Delaware 19714, USA
(Received January 26, 1998; revised March 31, 1998)
The dynamic structure factor S(q, w) for a model of superfluid 4He in disorder is evaluated using Path Integral Monte Carlo and Maximum Entropy
methods. Disorder is represented by randomly distributed static impurities
interacting with the 4He atoms via a simple attractive potential. The potential
is parametrized to yield different values of the variance and correlation length
of the resulting disordering environment. New weight in S(q, w) at low w (i.e.,
low energy excitations) is induced by disorder, as predicted in some previous
calculations, and S(q, w) is broadened. Assuming that S(q, w) in disorder is
dominated by a single peak, just as in the pure superfluid, the data suggest
that the peak position is shifted by the disorder from the bulk value with a
unique q dependence. The static structure factor S(q) is reduced at all wave
vectors with increased impurity concentration.
1. INTRODUCTION
Disordered Bose systems display intriguing properties.1-3 A classic
example is liquid 4He in aerogel or vycor; in these environments, the 4He
normal-to-superfluid transition, the superfluid fraction and the sound
propagation below the critical temperature are qualitatively different than
in the pure liquid.4-6 Magnetic flux lines in "dirty" superconductors are also
bosons in disorder created by substitutional impurities or line defects.7-11
Other examples are superconducting regions separated by insulating
spacers in Josesphson junction arrays12 and Cooper pairs in thin metallic
and granular films.13,14 In these systems, the superconductor-insulator
transition is affected by the magnitude and character of the disorder. The
flow of bosons in disorder is also of great importance, with applications in
charge density wave systems, 15,16 Wigner crystals,17 magnetic bubble
arrays,18 transport in metallic dots and flow in many porous media.
To date, much interest has focused on understanding the phase
diagram of disordered Bose fluids. Of equal interest, but much less studied,
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0022-2291/98/0800-0251$15.00/0© 1998 Plenum Publishing Corporation
252
M. Boninsegni and H. R. Glyde
is the nature of their excitations. A few theoretical studies have been performed; for example, Lee and Gunn 20 and Zhang21 have investigated a dilute
gas of bosons in disorder within the Bogoliubov approximation. They find,
among other features, that the sound velocity is reduced by disorder, but
the excitation width is little affected. Similarly, Krauth et al.22 find that the
sound velocity of bosons confined to a two-dimensional (2D) lattice is
reduced if a random site potential is added. Makivic23 and Ma et al.3 find
evidence of additional excitations at low w for 2D disordered lattice
bosons. The nature of the excitation energy spectrum E(q) may also
provide insight into the phase of the system. For example, a gap in E(q)
at q ->0 signals Mott localization of the bosons by the disorder. Krauth
et al. find that disorder stabilizes the gapless Goldstone mode and the Bose
glass phase.
In order to gain insight into the effect of disorder on the excitation
spectrum of a three-dimensional (3D) Bose superfluid, we undertook a
numerical evaluation of the dynamic structure factor S(q, w) of liquid 4He
in a model of disorder; we used a combination of Path Integral Monte
Carlo (PIMC) and Maximum Entropy (ME) inference which has been
recently utilized24 for bulk 4He. We modeled disorder with a random distribution of impurities, interacting with the Helium atoms via a simple
model potential. Because our goal is to obtain fundamental understanding
of the effect of disorder on the dynamics of a Bose fluid, liquid 4He is a
convenient choice, since it is is very well characterized experimentally and
theoretically, in the absence of disorder. Furthermore, the use of an artificial model of disorder, such as the one utilized here, though not related to
any known specific experimental condition, makes it possible to study the
behavior of the system for different values of the characteristic quantities
associated to the disordering environment, such as the concentration and
the effective "size" of each impurity.
Experimentally, study of elementary excitations of liquid 4He in confined geometries (disorder), e.g., in aerogel,25-27 has recently begun. In a
numerical simulation such as the one described here, the typical size of the
sample that can be studied is of the order of 100 4He atoms, corresponding
to a simulation cell size of the order of ~20 A. On the other hand, a typical aerogel strand is 40 A diameter, and the characteristic correlation
length of the disorder created by the aerogel network is of the order of
several hundred A. While this precludes a direct comparison of our results
to experimental data for Helium in aerogel, it is nonetheless conceivable
that the fundamental modifications induced by disorder on the dynamics of
superfluid 4He at different length scales might be qualitatively similar.
In Section 2 we briefly describe the model of disorder and the calculation of S(q, w). The results for S(q, w) and S(q) are presented in Sec. 3 and
Excitations of Liquid 4He in Disorder
253
discussed in Sec. 4, where we also discuss possible experimental implications.
2. MODEL AND CALCULATIONS
We consider a system of N 4He atoms in a cubic box of volume
Q = L3, with periodic boundary conditions. The Hamiltonian of our system
of interest is
Here, m is the mass of each 4He atom, and V(r) is a pairwise potential
describing the interaction of two 4He atoms, which depends only on their
distance rij. In this work, we have used for V the commonly adopted HFD
form parametrized by Aziz and collaborators,28 which has been shown to
yield an accurate description of the energetic and structural properties of
condensed 4He.29 The external potential U(r) describes a disordering agent,
and is assumed to be a continuous, random function of the position. This
is not just a choice of convenience; random potentials have been extensively used to study universal properties of disordered bosons, and we wish
to perform calculations that might be compared to similar others. Furthermore, a random potential may capture the common, underlying broad
features of a variety of physical environments in which the physics of a
quantum fluid is presently investigated experimentally.
A realization of U, particularly convenient for computational purposes, is the so-called impurity potential,30 namely
where the sum runs over M isolated impurities, placed at random positions
ri, with i=1,2,..., M. We will assume thereafter that impurities are distributed with uniform probability all over the system. Each impurity is
immobile, and contributes independently to U by means of the same
potential u.
In this work, we performed PIMC simulations of a system of N = 108
4He atoms moving in a disordered environment, as described by (1 )-(2), at
a temperature T= 1 K. The 4He density was set equal to its bulk equilibrium value, namely 0.02186 A -3, resulting in a simulation box size
L ~ 17 A. We adopted the impurity potential scheme outlined above to
define the disordering potential U in which 4He atoms move. For the
254
M. Boninsegni and H. R. Glyde
TABLE I
Number M of Impurities and Effective
Radius a (in A) of the Impurities for
Each Calculation Performed. Also Shown
Is the Variance V (in K) of the Resulting
Disordering Potential, Whereas the
Correlation Length X Is Simply v/2 a
case
M
a
V
(a)
(b)
(c)
12
48
6
1.0
1.0
2.0
~1.15
~2.32
~2.32
various physical quantities computed (see below), we averaged numerical
results obtained over a set of 30 independent realizations of the potential,
i.e., of random distributions of M impurities inside the finite simulation
box.31 We chose u, i.e., the potential created by each impurity, in the form
of a simple, attractive Gaussian well
The details of the potential u chosen are not too important in the context
of our calculation, as our goal was to carry out a general investigation of
the effect of disorder on the excitation spectrum of superfluid 4He. Therefore, we made no attempt to reproduce any known physical interaction. In
our calculation, u is always attractive, with A = 10 K. The various systems
investigated here differ by the values of a, namely the size of the potential
well associated to each impurity, as well as the concentration M/Q of
impurities. We studied the properties of systems with a number of
impurities M= 12, 48, with a= 1 A, and M = 6 with a = 2 A (these values
are summarized in Table I).
The impurity potential scheme is a convenient tool that, in principle,
allows computationally to produce disorder with desired statistical features,
such as the spatial average < U>
and the auto-correlation function C(r) defined as
Excitations of Liquid 4He in Disorder
255
In (5), sDU is over all continuous random functions U(r) weighted with
a probability density P{U(r)}. C(0) represents the average square deviation of the random potential away from its mean; its square root, henceforth referred to as the variance V, provides a quantitative indication of the
overall "strength" of the disorder. C(r) typically decays spatially with a
characteristic length X, normally referred to as correlation length of the disordering potential. It is usually set to zero in numerical studies of disordered lattice Bose systems, as the random potential varies independently
from one lattice site to another; in a continuum study such as the present
one, on the other hand, the dependence of the physical results on the
correlation length can be studied in detail.
Within the impurity potential scheme, the auto-correlation function is
easily shown to equal C(r)=nc(r), with n = M/Q and
with u = sdr u(r). In particular, because of our choice of u, C(r) will also
be a gaussian, with a semiwidth X = v/2 a which we identify with the
correlation length. Variance and correlation length can be used to characterize the resulting disordering potential, though they may not furnish, in
general, a complete description. In Table I we show, besides the values of
M and a (A is not changed in our calculations), also V for the various
cases considered. Note that the variance V depends on both M and a. Summarizing, any two of the cases considered here may differ by both M and
a, but only differ by one of the two quantities V, A. In Fig. 1 we show the
auto-correlation function C(r) associated to the three different disordering
potentials adopted.
In order to study the dynamics of superfluid 4He in disorder, we computed by PIMC the 4He dynamic structure factor S(q, w), defined as the
Fourier transform of the intermediate scattering function F(q, t):
with
Pq being the Fourier transform of the 4He density operator p ( r ) =
E i S(r-r i ) and <0> =Tr{exp( -BH)0}/Tr{exp(-BH)} being the usual
256
M. Boninsegni and H. R. Glyde
Fig. 1. Auto-correlation function C(r) associated to the three disordering potentials
considered in this calculation, differing by the number M of impurities and their
effective radii a.
quantum thermal average (we set h = 1). We also computed the static
structure factor S(q), defined as the integrated intensity
The static structure factor can be computed directly by PIMC as
S(q) = F(q, it = 0); the evaluation of S(q, w), on the other hand, involves
the calculation, by PIMC, of the imaginary-time intermediate scattering
function F(q, it), and the use of Maximum Entropy inference to perform
the (ill-posed) inverse Laplace transform needed to obtain S(q, w). A few
words of comment are in order here about the computational methods
adopted in this work. Path Integral Monte Carlo is a well-established,
powerful numerical technique that allows the computation of thermodynamic averages for quantum many-body systems at finite temperature, directly from the microscopic Hamiltonian. For Bose systems, the
method yields essentially exact results, the only errors being statistical and
therefore reducible arbitrarily, at least in principle. The PIMC calculation
performed here is standard, as the introduction of the random potential
Excitations of Liquid 4He in Disorder
257
entails no modification of the PIMC algorithm which has been successfully
applied to superfluid Helium and other quantum fluids,29 nor does it
involve any uncontrolled approximations. The mathematical principles and
numerical implementation of Maximym Entropy are thoroughly described
elsewhere.32 The calculation performed here is essentially identical to a
recent one,24 aimed at estimating S ( q , w ) in pure normal and superfluid
4
He, the main differences in this work being the introduction of the disordering potential and a somewhat larger number of atoms (108 vs. 64) in
the simulation box.
3. RESULTS
In this section we present the results of our calculation for the static
(S(q)) and dynamic structure factors ( S ( q , w ) ) for the many-body Hamiltonian (1) with the disordering potential U(r) described in the previous section. We begin with the static structure factor, shown in Fig. 2. Diamonds
represent the values of S(q) for pure liquid 4He computed by PIMC, which
are in excellent agreement with experimental measurements, 33,34 shown by
the dashed line. Triangles, squares and crosses represent the results
Fig. 2. Static structure factor S ( q ) computed by PIMC for pure liquid 4 He
(diamonds) and for disordering potentials characterized by different numbers of
impurities and values of a (triangles, squares and crosses. See also Table I). The
dashed line represents experimental measurements from Ref. 34.
258
M. Boninsegni and H. R. Glyde
obtained by PIMC with the different disordering potentials, with varying
number M of impurities and values of a. There is an overall reduction of
S(q) at all wave vectors explored, particularly for q> 1.5 A - 1 . The largest
effect is observed for the squares, namely for the case M = 48 and a — 1 A.
The next largest reduction (triangles) is observed for the disordering potential characterized by the same value of a and a smaller concentration of
impurities (M = 12). Finally, the smallest reduction of S(q) is observed for
the disordering potential featuring the smallest impurity concentration
(M = 6), and a larger value of a (crosses). So, the overall reduction appears
to be primarily driven by the impurity density, indicating that, for the
values of a and A chosen, impurities behave essentially as isolated localization centers. It is interesting to note that the two cases M = 6 and M = 48
yield considerably different results while featuring the same variance V of
the disordering potential (see Table I). This suggests that V is not directly
related to the reduction of the integrated intensity S(q) due to the
impurities, which depends essentially only on the impurity concentration. It
is worth noting that such a reduction of S(q) for liquid 4He in disorder
cannot be simply accounted for by an effective change in the density, due
to the introduction of impurities. For example, as the density is increased
in bulk 4He, S(q) decreases at low q, up to q< 1.9 A - 1 , but increases at
higher q in the peak region.34 This is qualitatively different from the
decrease at all q found here.
Before turning to the results for S(q, w), it is worth anticipating some
of the features and limitations of the spectral functions obtained by applying
the ME inference procedure outlined in Ref. 24 to inversion of the PIMCcomputed imaginary-time intermediate scattering function F(q, it). With
the statistical uncertainties attainable with the current PIMC technology,
and within reasonable computational time, the reconstructed spectral functions are usually very smooth; sharp features, such as isolated peaks, are
quite difficult to recover fully, as the algorithm naturally selects, out of the
many spectral functions that are consistent with the imaginary-time data
and with other independent physical information (normalization and f-sum
rule), the ones that are less rich in features. In other words, it favors simplicity and smoothness, and eliminates sharp features that are not strictly
warranted by the data. Consequently, peaks are typically rather broad, and
secondary peaks are almost invariably "washed out." Thus, for superfluid
4
He this procedure does not usually afford a precise determination of the
position of the main peak of S(q, w), or of the spectral weight associated to
it, due to the broadening. On the other hand, other physically relevant
aspects are recovered quite satisfactorily; for example, at low energy, in bulk
superfluid 4He there are no excitations at energies lying below the elementary excitation energy,35 corresponding to the position of the dominant, very
Excitations of Liquid 4He in Disorder
259
sharp peak of S(q, w). This gap at low energy in the excitation spectrum
of bulk superfluid 4He, and its disappearance at temperatures above the
superfluid transition, is clearly reproduced in the ME spectral images
obtained from PIMC data.24 This allows us to investigate whether disorder
can introduce excitations at low w, as it has been proposed and numerically observed in 2D. 2,23
Figure 3 shows S(q, w) at T= 1 K, computed by PIMC and ME, for
pure superfluid 4He (diamonds and dashed line), as well as for 4He with
the three different disordering potentials considered (triangles, crosses,
squares), at the four wave vectors q = 0.37, 0.64, 1.17 and 1.88 A - 1 . A clear
effect of disorder is to reduce the overall magnitude of S(q, w), as already
noticed discussing the results for S(q). Equally, new intensity at low w is
introduced at all q values for all the disordering potentials. The results
Fig. 3. Dynamic structure factor S(q,w) computed by PIMC and ME at various wave
vectors, for pure liquid 4 He (diamonds and dashed lines) and lor disordering potentials
characterized by different numbers of impurities (strength) and values of a (crosses, triangles and squares. See also Table I).
260
M. Boninsegni and H. R. Glyde
clearly depend on both M and a, or, alternatively, on V and X, in a nontrivial way. For a given value of a, a greater number of impurities (i.e., a
greater variance V of the disorder) will cause loss of intensity and a
broadening of the main peak of S(q, w), as can be seen by comparing the
triangles (M = 1 2, a = 1.0 A) and the squares (M = 48, a = 1.0 A) in Fig. 3.
This appears to be generally true at all values of q.
On the other hand, for a given value of the variance V, the intensity
at low w appears to be rather sensitive to the value of a (or, 1), particularly
at low q, as one can infer on comparing the squares and crosses (M = 6,
(a= 2.0 A); these two cases are characterized by different values of M and
a, but by the same variance V of disordering potential. The intensity at low
w is especially pronounced at low q. Indeed at q = 0.37 A-1 we find that
S(q, w) remains finite as w ->0. We conclude that our results are consistent
with the presence of low-lying excitations emerging in the disordered fluid,
at all wave vectors. The intensity at low q depends particularly on the size
of the attractive well describing each impurity center, which determines the
spatial extension of a localized state. At higher q, on the other hand, the
properties of the reconstructed spectra at low energy appear less and less
dependent on X (though still on M, or V). Despite the overall general
limitations of the ME reconstruction scheme adopted, we believe that this
prediction, namely the presence of low-lying excitations below the main
peak in disorder, is accurate, given the demonstrated effectiveness of this
procedure to reproduce a gap in the excitation spectrum of the pure superfluid.
It is more difficult to assess what changes, if any, disorder causes in
the position of the main peak of S(q, w), as the single peak which we find
at all physical conditions explored is quite broad. We may note, for example, that a broad peak found in S(q, w) through ME could in principle
signal the presence of two, or even more distinct peaks in the real spectrum,
which are "fused" together in a single one by the inference algorithm, if the
PIMC data are not sufficiently precise to afford the resolution of all the
peaks. In principle, increasing the accuracy of the imaginary-time F(q, it)
data computed by PIMC would eventually permit an unambiguous
recovery of all spectral features; in practice, the accuracy required by the
ill-posedness of the problem (7) is often exceedingly high, in the light of the
computing resources typically available. 24,32 This is arguably the most
serious drawback of a procedure based on ME and on the calculation of
imaginary-time correlation functions by Monte Carlo, aimed at computing
spectral properties.
Qualitatively, if we assume that S(q, w) is dominated by a single peak
both in the bulk and in disorder, and that the position of such a single
peak is represented with sufficient accuracy by the maximum of the ME-
Excitations of Liquid 4He in Disorder
261
reconstructed spectral image, it appears that the peak shifts when disorder
is introduced. This is shown in Fig. 4, where the elementary excitations
curve E(q), namely the position of the peak of S ( q , w ) as a function of q,
is presented for the various disordering potentials considered. Statistical
errors in the peak position were assigned using the methods illustrated in
Ref. 24. The dashed curve represents the experimental result36 for pure
liquid 4He, which is in agreement, within the statistical uncertainties, with
the computed dispersion curve, a fact previously noted.24 Clearly, the
magnitude of the statistical errors does not allow to make any definite
prediction about physical changes attributable to disorder. There appears
to be a systematic downward shift of the curve at low q caused by disorder,
particularly for a = 2 A. On the other hand, at q ~ 1 A the curve appears to
shift toward higher energy in disordered, more so for M = 1 2 and a= 1 A
Fig. 4. Elementary excitation curve E ( q ) . The peak position and its statistical uncertainty are
obtained from the Maximum Entropy reconstruction of the spectral image S ( q , w ) from the
PIMC-computed F(q, it). Results are shown for pure liquid 4He (diamonds), as well as for disordering potentials summarized in Table I, characterized by different numbers of impurities
(and therefore strength) and values of a (crosses, triangles and squares). A slight horizontal
displacement has been applied to the crosses, the triangles and the squares, for clarity. The
dashed curve is the experimental E(q) for superfluid 4 He.
262
M. Boninsegni and H. R. Glyde
than for the other cases; finally, in the roton region there appears to be a
systematic shift to lower energy, again more pronounced for a = 2 A.
Again, the accuracy of our procedure to estimate spectral features does
not permit us to make a more quantitative statement; however, assuming
that these observed changes in peak position are real, they cannot be
simply accounted for by a mere effective change in the fluid density induced
by disorder. In the bulk, as the density increases the sound velocity and
excitation energy (peak position) increase at all q up to the maxon energy,
while the roton energy decreases. The shift in peak position induced by disorder found here is qualitatively and quantitatively different from pressure
induced changes in the bulk; we conclude that the changes observed in this
work reflect mainly the nature of the disordering potential.
4. DISCUSSION AND CONCLUSIONS
We have evaluated the dynamic response of superfluid 4He containing
quenched impurities placed at random in the fluid. We utilized a combination of Path Integral Monte Carlo and Maximum Entropy methods to
compute the dynamic structure factor S ( q , w ) of superfluid 4He, with and
without disordering potential. This procedure is still under development,
particularly the ME component, and suffers from a severe limitation,
namely it yields excessively smooth spectral images. This means that
detailed spectral features cannot be resolved. Nonetheless we can conclude
that the broad features of S(q, w), its overall height and energy range are
modified by disorder from in a way that generally reflects the statistical
properties of the random disordering potential. An important physical
effect of disorder is to introduce low-lying excitations into the fluid at
energies below that of the phonon-maxon-roton peak. The change of S(q)
and S(q, w) arising from the fixed "classical" scatterers studied here is
qualitatively different than that arising from a simple density change.
As noted, measurements of S ( q , w ) in liquid 4He in quenched disordered environments have recently begun. To date only 4He in aerogel25-27
and vycor have been investigated. These environments are quite different
from the present model of disorder. The observed changes in S ( q , w ) from
the bulk when liquid 4He is confined in aerogel are much smaller than the
changes shown in Fig. 3 arising from the present short range, local disorder
model. This is both in the magnitude of S ( q , w ) and its energy range. The
present results may serve as some guide to selecting disordering environments for experiments that will have a readily observable impact on
S(q, w).
Excitations of Liquid 4He in Disorder
263
ACKNOWLEDGMENTS
This work was supported by the National Science Foundation
through grant DMR-9623961 and by the San Diego State University
Foundation. Calculations were performed on a cluster of Alpha workstations at San Diego State University.
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