Excitation spectrum in two-dimensional superfluid ⁴He
In this work we perform an ab-initio study of an ideal two-dimensional sample of
 ⁴He atoms, a model for
 ⁴He
 films adsorbed on several kinds of substrates. Starting from a realistic hamiltonian we face the microscopic study
 of the excitation phonon–roton spectrum o...
Збережено в:
| Опубліковано в: : | Физика низких температур |
|---|---|
| Дата: | 2013 |
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2013
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/118751 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Excitation spectrum in two-dimensional superfluid ⁴He / F. Arrigoni, E. Vitali, D.E. Galli, L. Reatto// Физика низких температур. — 2013. — Т. 39, № 9. — С. 1021–1030. — Бібліогр.: 37 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860262088828518400 |
|---|---|
| author | Arrigoni, F. Vitali, E. Galli, D.E. Reatto, L. |
| author_facet | Arrigoni, F. Vitali, E. Galli, D.E. Reatto, L. |
| citation_txt | Excitation spectrum in two-dimensional superfluid ⁴He / F. Arrigoni, E. Vitali, D.E. Galli, L. Reatto// Физика низких температур. — 2013. — Т. 39, № 9. — С. 1021–1030. — Бібліогр.: 37 назв. — англ. |
| collection | DSpace DC |
| container_title | Физика низких температур |
| description | In this work we perform an ab-initio study of an ideal two-dimensional sample of
⁴He atoms, a model for
⁴He
films adsorbed on several kinds of substrates. Starting from a realistic hamiltonian we face the microscopic study
of the excitation phonon–roton spectrum of the system at zero temperature. Our approach relies on path integral
ground state Monte Carlo projection methods, allowing to evaluate exactly the dynamical density correlation
functions in imaginary time, and this gives access to the dynamical structure factor of the system S(q, ), containing
information about the excitation spectrum E(q), resulting in sharp peaks in S(q, ). The actual evaluation of
S(q, ) requires the inversion of the Laplace transform in ill-posed conditions, which we face via the genetic inversion
via falsification of theories technique. We explore the full density range from the region of spinodal decomposition
to the freezing density, i.e., 0.0321 Å⁻²
– 0.0658 Å⁻². In particular we follow the density dependence
of the excitation spectrum, focusing on the low-wave vector behavior of E(q), the roton dispersion, the strength
of single quasiparticle peak, Z(q), and the static density response function, (q). As the density increases, the
dispersion E(q) at low-wave vector changes from a superlinear (anomalous dispersion) trend to a sublinear (normal
dispersion) one, anticipating the crystallization of the system; at the same time the maxon–roton structure,
which is barely visible at low density, becomes well developed at high densities and the roton wave vector has a
strong density dependence. Connection is made with recent inelastic neutron scattering results from highly ordered
silica nanopores partially filled with
⁴He.
|
| first_indexed | 2025-12-07T18:56:29Z |
| format | Article |
| fulltext |
© F. Arrigoni, E. Vitali, D.E. Galli, and L. Reatto, 2013
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9, pp. 1021–1030
Excitation spectrum in two-dimensional superfluid
4
He
F. Arrigoni, E. Vitali, D.E. Galli, and L. Reatto
Dipartimento di Fisica, Università degli Studi di Milano, 16 Celoria Via, Milano 20133, Italy
E-mail: Davide.Galli@unimi.it; davidemilio.galli@gmail.com
Received March 14, 2013
In this work we perform an ab-initio study of an ideal two-dimensional sample of
4
He atoms, a model for
4
He
films adsorbed on several kinds of substrates. Starting from a realistic hamiltonian we face the microscopic study
of the excitation phonon–roton spectrum of the system at zero temperature. Our approach relies on path integral
ground state Monte Carlo projection methods, allowing to evaluate exactly the dynamical density correlation
functions in imaginary time, and this gives access to the dynamical structure factor of the system S(q, ), contain-
ing information about the excitation spectrum E(q), resulting in sharp peaks in S(q, ). The actual evaluation of
S(q, ) requires the inversion of the Laplace transform in ill-posed conditions, which we face via the genetic in-
version via falsification of theories technique. We explore the full density range from the region of spinodal de-
composition to the freezing density, i.e., 0.0321 Å
–2
– 0.0658 Å
–2
. In particular we follow the density dependence
of the excitation spectrum, focusing on the low-wave vector behavior of E(q), the roton dispersion, the strength
of single quasiparticle peak, Z(q), and the static density response function, (q). As the density increases, the
dispersion E(q) at low-wave vector changes from a superlinear (anomalous dispersion) trend to a sublinear (nor-
mal dispersion) one, anticipating the crystallization of the system; at the same time the maxon–roton structure,
which is barely visible at low density, becomes well developed at high densities and the roton wave vector has a
strong density dependence. Connection is made with recent inelastic neutron scattering results from highly or-
dered silica nanopores partially filled with
4
He.
PACS: 67.25.bh Films and restricted geometries;
67.25.dt Sound and excitations.
Keywords: superfluidity, two-dimensional quantum fluids, elementary excitations, roton.
1. Introduction
Helium exists in two stable isotopes,
4
He and
3
He,
which differ for their nuclear spin:
4
He atoms are bosons
with nuclear spin I = 0, while
3
He atoms are fermions with
nuclear spin I = 1/2. The effective interaction among he-
lium atoms is well described by a hard core potential plus
an attraction arising from zero-point fluctuations in the
charge distribution. The interaction results in a simple
Lennard–Jones-like two-body spherically symmetric po-
tential ( ),v r for which accurate analytical expressions are
known [1]. The hamiltonian of the bulk system reads:
2
2
=1 < =1
ˆ ˆ ˆ= (| |)
2
N N
i i j
i i j
H v
m
r r (1)
where m is the mass of
4
He atoms. Despite its very simple
structure, helium exhibits numerous exotic phenomena in
condensed form, whose theoretical explanation, in some
aspects, is still a big challenge nowadays. Along with the
many fascinating physical features related to the well-
known phenomenon of superfluidity [2], which have been
the object of several theoretical and experimental efforts, a
unique fingerprint of such a system is the spectrum ( )E q
of the elementary excitations.
Excitations in
4
He bulk systems have been extensively
investigated after Landau’s original conjecture [3] about
the phonon–roton dispersion relation ( )E q and its connec-
tion with the definition of superfluidity in terms of a criti-
cal velocity. In 1953 Feynman showed that the shape of the
phonon–roton spectrum can be justified on a quantum me-
chanical basis, relying on Bose statistics together with
hard-core interactions [4]. Moreover, he suggested that the
excitation spectrum of superfluid
4
He may be investigated
by inelastic neutron scattering experiments. This was rea-
lized only almost one decade later [5], beautifully confirm-
ing the original Landau’s guess. Actually, within the first
Born approximation, the differential cross section in a
thermal neutron scattering experiment on a sample of
4
He
atoms, apart from kinematical factors, is provided by the
dynamical structure factor:
ˆ ˆ1
ˆ ˆ( , ) = e e e
2
t t
i H i H
i tS q dt
N
q q (2)
F. Arrigoni, E. Vitali, D.E. Galli, and L. Reatto
1022 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9
where the brakets indicate a ground state or a thermal av-
erage, Ĥ is the Hamiltonian of the helium system (1), and
1
ˆ ˆ= exp ( ),
N
i
i
iq q r îr being the position operator of the
ith
4
He atom, is the local density operator in Fourier space.
Sharp peaks in of ( , )S q provide the spectrum of the
elementary excitations of the system.
On the theoretical side, a systematic effort has been
devoted to pursue an accurate description of the elemen-
tary excitations of the system. The original idea of Feyn-
man–Cohen [6] of introducing back–flow correlations to
improve variational excited states wave functions (later
on extended by the correlated–basis–functions strategy
[7]), has flown into the excited states shadow wave func-
tions (SWF) technique [8]; SWF reproduced the experi-
mental bulk dispersion relation ( )E q up to a high accura-
cy level [9] and even confirmed [9,10] the physical
picture of a roton as a microscopic smoke ring [11]. A
further turn in the study of excited states of superfluid
4
He was given by the advent of exact simulation methods
for interacting Bose particles. It is not yet possible to per-
form a direct exact computation of excited states due to
the sign problem. However, it is possible to extract dy-
namical properties from exact correlation functions in
imaginary time [12]. This has been worked out by path
integral Monte Carlo (PIMC) at finite temperature [13] or
by ground state Monte Carlo [14,15] at T = 0 K. Indeed,
such functions contain information on excited states of
the system. In particular the density correlation function
is related to ( , )S q by an inverse Laplace transform.
Due to discretization and statistical noise, the mathemati-
cal problem of extracting ( , )S q is ill-posed, but power-
ful inversion methods have been introduced recently
[15,16] and reliable results on the excitation spectrum of
superfluid
4
He have been obtained [15,17,18].
Bosons in two dimensions (2D) are of great theoretical
interest because the standard scenario of superfluidity as-
sociated with Bose-Einstein condensation (BEC) is not
appropriate. In fact, in 2D and in almost 2D systems the
order parameter, i.e., the condensate wave function, ( ),r
vanishes at any finite temperature for a bulk system. The
notion of long range order is replaced by that of topologi-
cal long range order [19] with correlation function of the
local order parameter decaying algebraically very slowly
to zero. Notwithstanding a vanishing order parameter, a
superfluid response is theoretically predicted up to a tem-
perature where vortex and antivortex pairs unbind. These
predictions have been beautifully confirmed by experi-
ments [20]. Therefore a 2D Helium system is an interest-
ing microscopic model for quasi-two-dimensional many-
body quantum systems [21,22]: helium films on suitable
substrates. For most substrates the interaction potential
between the helium atoms and the substrate is much
stronger (as it is the case of He–graphite interaction) than
the He–He interaction and the helium atoms are adsorbed
in a well-defined layer structure. Typically, only the first or
the first two layers are strongly influenced by the details of
the helium–substrate interaction. Several different physical
realizations of substrates have been investigated, both in
experimental and in theoretical works. For many substrates
the closest He atoms to the substrate are disordered and
localized, they form what the experimentalists call a
“dead layer”. Beyond that the first layer of mobile atoms
are superfluid and can be well represented by a strictly
2D model. The experimental study of ( , )S q of this film
has shown the existence of elementary excitations with a
phonon–maxon–roton structure [23]. A favorite substrate
for adsorption studies is graphite because it offers rather
extended regions of perfectly flat basal planes. At first
sight this might be considered as an ideal situation for us-
ing the 2D model. This is not so for the first adsorbed layer
because the adsorption potential is strongly corrugated.
The consequence of the corrugation is that at low tempera-
ture the
4
He atoms form an ordered structure, either a tri-
angular lattice that is commensurate with the substrate or,
at higher coverage, an incommensurate triangular solid
[24]. Experimentally no evidence has been found for su-
perfluidity in the first adsorbed layer on graphite. Super-
fluidity has been found only in additional layers for which
the 2D model can be used as a reasonable approximation.
Computation of the spectrum of elementary excitations
of
4
He is of interest on one hand to uncover the depen-
dence of rotons on the dimensionality of the system. On
the other hand, this theoretical input is useful for the inter-
pretation of scattering experiments from adsorbed
4
He.
Excitations for 2D
4
He have been studied by correlated
basis function theory [25]. As far as we know, the only
existing ab initio quantum Monte Carlo (QMC) calculation
of excitations in 2D
4
He has been performed with varia-
tional theory using shadow wave functions [26]. As men-
tioned above exact QMC techniques are able to give access
to estimations of ( , )S q via exact calculations of dynami-
cal correlation functions in imaginary time. The path
integral ground state (PIGS) method [27] and in particular
the shadow path integral ground state (SPIGS) method
[28,29] together with the genetic inversion via falsification
of theories (GIFT) method [15] have been applied to bulk
4
He systems [15,17], to adsorbed
4
He systems [18] and
even to a pure 2D
3
He system [30] (via a quite sophisti-
cated novel strategy). Here we apply such approaches to
address the calculation of dynamical properties of a pure
2D
4
He system at zero temperature.
The article is structured as follows: in the next section we
sketch the methodology; in Sec. 3 we present and discuss
the results and our conclusions are in Sec. 4. In the Appen-
dix we give the results of a variational computation of the
ground state properties of
4
He in 2D based on SWF that are
a byproduct of the exact SPIGS computation of Sec. 3.
Excitation spectrum in two-dimensional superfluid
4
He
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 1023
2. Methodology
We focus thus on a strictly 2D collection of N struc-
tureless spinless bosons at zero temperature. The hamil-
tonian operator is (1). We let 0 ( ) be the ground state
of ˆ ,H where we use the notation 1= ( , , ).Nr r . The
basic relation underlying QMC projection methods is the
following:
0
ˆ( ) exp( ) ( )lim TH (3)
where ( )T is any many-body wave function with non-
zero overlap on 0 ( ) . The operator ˆexp( )H can be
seen as the evolution operator, ˆexp( ( / ) ),i t H written for
imaginary times; in this way, 0 ( ) turns out to be the
limit of the imaginary time evolution of ( ),T with
playing the role of imaginary time. A Trotter decomposition:
ˆ ˆ
e = (e ) , =H H M
M
(4)
together with an (analytical or numerical) approximation
for the imaginary time propagator:
ˆ
| e | = ( , , ) ( )H m (5)
where the order m depends on the approximation, allows
to build up an approximate expression for the ground state
wave function of the form:
0 1 1( ) { } ( , , ) ( , , ) ( )i M M T Md
(6)
where we have omitted an overall normalization factor.
Any expectation value of an operator diagonal in coordi-
nate representation (or of the Hamiltonian operator):
0 0
ˆ| |O (7)
is expressed as a multidimensional average of a function
( )O over a probability density of the form:
2
1 1 2
=1
1
({ }) = ( ) ( , , ) ( )
M
i T i i T M
i
p (8)
which can be sampled using Metropolis algorithm. The re-
sults can be considered exact in the sense that the errors aris-
ing from approximations can be reduced under the level of
the statistical noise via a suitable choice of the time step
and the total projection time = .M Of course this also
assumes that the results, for large enough , are indepen-
dent on the choice of .T This has been verified [31], even
by starting with T of a liquid for the solid phase or of a
solid for a liquid phase one finds convergence to the correct
result. Notwithstanding this, a judicious choice of T is
important to accelerate convergence of T to 0 , i.e., a
smaller value of is needed, and to reduce the variance of
the results. What has been described here is the PIGS me-
thod, or the SPIGS method if T is a SWF.
This calculation scheme can be straightforwardly gene-
ralized to evaluate dynamical imaginary time correlation
functions:
ˆ ˆ †
0 0
ˆ ˆ| e e | .H HO O (9)
The particular choice:
ˆ ˆ
0 0ˆ ˆ( , ) = | e e |H HF q qq (10)
provide the intermediate scattering function in imaginary
time, which is related to the dynamical structure factor by
the relation:
0
( , ) = e ( , ).F q d S q (11)
Thus, the estimation of ( , )S q requires to invert the
integral relation (11) in ill-posed conditions, since ( , )F q
is known only on a discrete and finite set of instants
(typically = ,n = 0, , )n n and is affected by a sta-
tistical uncertainty arising from the stochastic Monte Carlo
calculation. Despite the well-known difficulties related to
the inversion of the Laplace transform in ill-posed condi-
tions, the evaluation of ( , )S q starting from the QMC
estimation of ( , )F q (10) has been proved to be fruitful
for several bosonic systems using a recent technique GIFT
[15]. GIFT is a statistical inversion method: it samples a
suitable space of spectral functions looking for models
compatible with the QMC data ( , )F q via a stochastic
search scheme relying on genetic algorithms.
3. Simulation details and results
In our simulations of
4
He in 2D we have used as intera-
tomic potential ( )v r the 1979 Aziz potential [1] and N =
= 120 number of atoms with periodic boundary conditions.
As propagator ( , , ) we have used the pair–
product approximation [32] with =1/160 K
–1
, a value
that we have verified to be small enough for the adopted
propagator. As projection time we have used = 1.1 K
–1
and typical length of the simulation is 3 10
6
Monte Carlo
steps (MCS); ( , )F q has been computed over the range
(1–90) . A typical run starts from a triangular lattice con-
figuration which quickly “melts”, when the density is not
too large, in few thousand MCS leading to disordered con-
figurations allowing to simulate the liquid phase without
memory of the starting point. When the density is large
enough the system remains in an ordered state as shown by
the presence in the static structure factor ( )S q of sharp
Bragg peaks corresponding to triangular solid
4
He. Only in
the density range of the liquid–solid transition one gets
convergence to two different states depending on the initial
configuration: starting from a disordered configuration the
system remains disordered whereas it remains ordered
when started from the ordered configuration. The energies
F. Arrigoni, E. Vitali, D.E. Galli, and L. Reatto
1024 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9
of the two states take different values, the lowest one
represents the stable phase and the higher energy one
represents a metastable state for a single phase state.
The average value of the hamiltonian operator as a
function of the density provides the equation of state of the
system. We have computed the energy in a quite large
range of densities both in the liquid and in the solid phase
and the results are shown in Fig. 1 and listed in Tables 3
and 4 of the Appendix. The SPIGS results agree within the
statistical uncertainty with the result of a Green function
Monte Carlo (GFMC) computation [21]. One can also see
that the SWF variational results follow quite closely the
exact SPIGS results both in the liquid and in the solid
phase, thus confirming the accuracy of the SWF as in the
3D case. In order to determine the melting and freezing
densities, the energies have been fitted with a third degree
polynomial in density in the solid phase, and a fourth de-
gree polynomial in the liquid phase. We write the fitting
function as
2 3 4
0 0 0
0
0 0 0
( ) =lE E A B C (12)
in the liquid phase, where a minimum energy 0E at the
equilibrium density 0 is present. The last term, which is
not typical in literature, turned out to be necessary in order
to obtain a good fit in the whole density range here consi-
dered. On the other hand, in the solid phase we use the
expression:
2 3( ) = .sE (13)
The obtained fitting parameters, together with their statis-
tical uncertainties, are listed in Table 1 of the Appendix.
The interpolation curves, depicted in Fig. 1, are truncated
in the coexistence region, delimited by the melting and
freezing densities m and .f m and f have been
estimated using the Maxwell construction, and they are
given in Table 2 of the Appendix.
In Fig. 2 we show some quantities like the pressure p,
the chemical potential , the compressibility , and the
sound velocity sv in the liquid and in the solid phase; such
quantities have been obtained from 0( )E via the expres-
sions:
2 0 ( )
( ) = ,
E
p 0( ) = ( ) ( )/ ,E p
1
( )
( ) =
p
and 2 0 ( )1
( ) =s
E
v
m
.
In the solid phase sv represents the velocity of the lon-
gitudinal sound mode.
In Fig. 3 we show the SPIGS result for the static struc-
ture factor 0 0ˆ ˆ( ) = | |S q q q for a density close to
the equilibrium one and at a density close to freezing. It is
evident the emergence of more structure as the density
increases towards the freezing density. Moreover, we em-
phasize the linear behavior of ( )S q for 0q which ma-
nifests itself at very small wave vectors. This is due to the
zero–point motion of long wavelength phonons [33].
Fig. 1. Variational energies in the liquid () and in the solid ()
phase. Exact energies in the liquid () and in the solid () phase.
The curves are the interpolated equations of state, and are trun-
cated in the coexistence region.
Fig. 2. Thermodynamical properties derived from the equation of
state as functions of the density; liquid phase (solid line); solid
phase (dashed line): pressure p (a); chemical potential (b);
compressibility (c); sound velocity sv (d).
Excitation spectrum in two-dimensional superfluid
4
He
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 1025
We have computed the dynamical correlation functions
for imaginary time ( , )F q at six densities, and namely
0.0321, 0.0421, 0.04315, 0.049, 0.0536, and 0.0658 Å
–2
in
the liquid phase. From ( , ),F q the GIFT method allows to
reconstruct the dynamical structure factor of the sample,
( , )S q . An example of ( , )F q and of the reconstructed
( , )S q is shown in Fig. 4. ( , )S q in general has a sharp
peak in and this defines the energy ( )E q of the excita-
tion for the given wave vector q. In addition there is a
much broader peak at larger energy and this represents the
so-called multiphonon contribution to ( , ).S q The ele-
mentary excitation peak in the reconstructed ( , )S q has a
finite width. This width can have two different origins. As
discussed in Ref. 15, even if the system has an infinitely
long-lived excitation the peak in the reconstructed ( , )S q
has a finite width because the inversion method can only
identify the excitation energy with a certain uncertainty
due to the limited and noisy information on ( , ).F q In this
case the full width at half maximum (FWHM) can be taken
as a measure of statistical uncertainty of the excitation
energy. Under certain circumstances even at T = 0 K an
elementary excitation acquires a finite lifetime when it can
decay into two or more excitations. This happens, for in-
stance, for the maxon excitation in superfluid
4
He in 3D at
large pressure when the maxon energy is larger than twice
the roton energy. In this case the excitation peak has an
intrinsic finite linewidth and its FWHM is a measure of the
inverse life-time of the excitation. Under such circums-
tances we expect that the width of the reconstructed
( , )S q has also a contribution of intrinsic origin due to
such physical processes, even if it is difficult to quantify
precisely how large this intrinsic contribution is from the
overall FWHM.
Fig. 3. SPIGS estimations of the static structure factor ( )S q ()
and strenght of the single particle peak ( )Z q () at the different
densities , Å
–2
: 0.04315 (a), 0.0536 (b), and 0.0658 (c).
Fig. 4. An example of QMC evaluation of an imaginary time
correlation function ( , ),F q defined in (10). We have plotted the
-dependence of ( , )F q for a given wave vector q (see the le-
gend) in logarithmic scale to show the asymptotic single expo-
nential behavior governed by the elementary excitation energy
(a). Reconstructed ( , ):S q one can see the sharp elementary
excitation peak together with the higher energy broad multipho-
non contribution (b).
F. Arrigoni, E. Vitali, D.E. Galli, and L. Reatto
1026 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9
The integral over all of ( , )S q is equal to the static
structure factor ( ).S q An important information is con-
tained in the strength ( )Z q of the elementary excitation
peak, i.e., the integral of ( , )S q limited to the main peak.
The ratio ( )/ ( )Z q S q gives the probability that in the scat-
tering process there is emission of a single elementary ex-
citation whereas 1 ( )/ ( )Z q S q gives the probability of
emission of other excitations, the so-called multiphonon
processes. The behavior of ( )Z q is shown in Fig. 3 at
three densities.
In Fig. 5 we show the obtained dispersion relation
( )E q for four values of the density. The reported bar
represents the FWHM of the main peak in ( , ).S q
At the lowest density, 0.0321 Å
–2
, which is near the
spinodal decomposition, the excitation spectrum shows a
large flat region, and a very weak roton minimum. At small
wave vector the spectrum shows an anomalous dispersion,
i.e., a positive curvature. At the density 0 = 0.04315 Å
–2
close to equilibrium the phonon–maxon–roton structure
starts to be visible but maxon energy does not differ by
more than 10% with respect to the roton energy. As the
density further increases the maxon–roton region becomes
more and more prominent until, at the highest density
0.0658 Å
–2
, near the freezing point, the maxon energy is
about three times the roton energy. At the larger density
the peaks in the maxon region are quite broadened, as it is
evident from the error bars in Fig. 5; we believe that in
this case the linewidth largely represents an intrinsic effect
due to the fact that a maxon can decay into two rotons
because its energy is more than twice the roton energy.
This fact is known experimentally [34] and theoretically
[9,10] in 3D superfluid
4
He at density in the region of
freezing. In Fig. 6 we plot the energy and the wave vector
of roton and of maxon as function of density. It can be
noticed that the roton energy in 2D (from 5.5 to 3.8 K
depending on density) is significantly below the value in
3D (from 8.6 K at equilibrium to 7.2 K at freezing densi-
ty). It can also be noticed that the roton wave vector has a
significant density dependence while the maxon wave
vector is almost density independent.
In Fig. 5 we show also the Feynman spectrum,
2 2( ) = /2 ( ),FE q q mS q obtained using our estimation of
( ).S q Feynman dispersion relation is accurate, as it is well
known, only in the low-wave vectors region. The discre-
pancy between ( )E q and ( )FE q increases with the densi-
ty: ( )FE q is more than twice ( )E q near the freezing
point. We notice also that the present ( )E q is in good
agreement with the variational result at the equilibrium
density obtained using SWF in Ref. 8. At larger density the
variational roton energy is about 1 K above the present
result. In Fig. 7 we show more details about the low q
behavior of the dispersion relation at four considered den-
sities. It is apparent that the phononic dispersion is superli-
near for the two lowest densities, and becomes sublinear at
larger densities up to the freezing point. This is qualitative-
ly similar to what happens in superfluid
4
He in 3D.
With respect to the strength of the quasiparticle peak
( ),Z q at all densities ( ) ( )Z q S q at small q, i.e., the col-
lective excitation peak almost exhausts the f-sum rule and
multiphonon contributions are negligible. At equilibrium
density the roton peak has about 2/3 of the full integrated
intensity and 1/3 is due to multiphonon contribution. This
multiphonon contribution is larger than in 3D and we
attribute this to the fact that equilibrium density in 2D is
Fig. 5. Excitation spectrum from GIFT reconstructions of SPIGS evaluations of imaginary time correlation functions in the liquid phase
(), together with Feynman spectrum (), at four densities as shown in the legends.
Excitation spectrum in two-dimensional superfluid
4
He
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 1027
rather low where short range order is not very pronounced.
Only near freezing the multiphonon contribution of the
roton is small (of order of 20%) as in 3D superfluid
4
He.
Studies of the elementary excitations of
4
He in re-
stricted geometry have been performed by inelastic neutron
scattering on
4
He confined in a number of nanopore mate-
rials. Of special relevance is a recent study [35] of
4
He in
smooth cylindrical silica pores of diameter of about 28 Å.
When the pores are filled with
4
He experiment shows the
presence of phonon–maxon–roton excitations with a dis-
persion relation very similar to that of bulk
4
He. Such exci-
tations are interpreted as propagating in the central part of
the pore. An additional roton excitation at smaller energy
is present and this is interpreted as roton confined in a
compressed layer close to the cylinder wall. When the
pores are only partially filled with
4
He the compressed
layer rotons are still present, whereas the bulk-like pho-
non–maxon–roton branch disappears. In its place there is a
modified phonon–maxon–roton brach with a decreased
energy of the maxon (11 K instead of 14 K in the bulk) and
a roton energy only 2 K below the maxon (the energy dif-
ference beyween maxon and roton in bulk
4
He is about 5 K
at equilibrium density). In addition this new roton is found
at a shifted wave vector, at 1.78 Å
–1
in place of 1.92 Å
–1
of the bulk one. This modified maxon–roton branch has
been interpreted as propagating in a thin film inside the
unfilled pore and connection has been made with the exci-
tations in 2D
4
He as computed in Ref. 26. Indeed some
similarity between the dilute layer modes of experiment
and the present results for
4
He in 2D is present, such as
the reduced energy difference between maxon and roton
and a reduced wave vector .Rq On the other hand, some
significant difference is present. For instance, we find
1.75Rq Å
–1
at a density close to freezing but here the
roton energy is about 4 K, less than half the value of the
dilute layer mode. Of course there is a difference between
the present mathematical 2D system and the finite curva-
ture of the
4
He film in an unfilled pore of the experiment.
It is unclear if this might be the origin of that difference
for the roton energy.
Finally, we obtained from the –1-moment of ( , )S q
also the static density response function, ( ),q which is
shown in Fig. 8. As in 3D ( )q is dominated by a peak at
the roton wave vector. One can notice that at the equili-
brium density ( )q has an enhancement at small q which
is absent in 3D [15]. This is another manifestation that the
ground state of
4
He in 2D is at low density where atoms
are not strongly coupled as in 3D.
Fig. 6. Density dependence of the wave vector and of the energy
of the maxon ( ,Mq ( ))ME q and the roton ( ,Rq ( ))RE q modes.
Lines are guides to the eye.
Fig. 7. Small wave vectors behavior of the estimated dispersion
relation ( )E q for the different densities , Å
–2
: 0.0321 (),
0.04315 (), 0.0536 () and 0.0658 (). The dotted straight lines
represent the linear behavior from which the ( )E q significantly
deviate.
Fig. 8. Density response function extracted from the dynamical
structure factor at the different densities , Å
–2
: 0.04315 (),
0.0536 () and 0.0658 ().
F. Arrigoni, E. Vitali, D.E. Galli, and L. Reatto
1028 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9
4. Conclusions
We have presented the first ab initio QMC computation
of the excitation spectrum of superfluid
4
He in 2D starting
from exact density correlation functions in imaginary time
and using advanced inversion methods to infer the dynam-
ical structure factor ( , ).S q We find well defined excita-
tions in the full density range of the superfluid but signifi-
cant differences are present with respect to
4
He in 3D. In
3D the excitation spectrum over the full density range from
the equilibirum density to the freezing one has a well de-
fined phonon–maxon–roton structure with the maxon
energy ME larger by at least 50% than the roton energy
R . In 2D the excitation spectrum evolves with density
from maxon and roton almost coaleshing in a plateau at
density close to the spinodal to a well defined maxon–
roton structure at density above the equilibrium one with
/M RE becoming as large as 3 at freezing. At the same
time the wave vector Rq of the roton has a strong density
dependence whereas that of the maxon is almost density
independent. This strong evolution with density of the
shape of the excitation spectrum is due to the large density
range of existence of the fluid in 2D, the freezing density is
more than twice the spinodal density while in 3D the freez-
ing density is only 60% larger than the spinodal one. At the
2D equilibrium density the maxon–roton structure is rather
weak with the maxon energy only 10% larger than the ro-
ton energy. This is due to the low value of the equilibrium
density so that the amount of short range order is rather
small. At the same time in the phonon region there is a
strong anomalous dispersion (i.e., ( )E q has a positive cur-
vature). As a consequence of the shape of ( ),E q over an
extended region of q and of density, the elementary excita-
tions are expected to have a finite lifetime even at T = 0 K,
because they can decay into other excitations. We find
evidence for this finite lifetime from our computation but
the present method does not allow to quantify this.
It has been suggested [35] that the excitation spectrum
of 2D
4
He might be relevant for the interpretation of the
excitations of
4
He partially filling smooth cylindrical silica
pores as measured by inelastic neutron scattering. We in-
deed find some similarity between our results and the ex-
perimental ones. However we find a strong disagreement
in the value of the roton energy which is well beyond the
uncertainty of the present theory. This discrepancy might
be due to a curvature effect that is present in the pore but
not in the present computation. It will be interesting to ex-
tend the present computation to the case of a pore geome-
try; present developments of QMC techniques are such that
this is a feasible project.
This work has been supported by Regione Lombardia
and CINECA Consortium through a LISA Initiative (La-
boratory for Interdisciplinary Advanced Simulation) 2012
grant [http://www.hpc.cineca.it/services/lisa], and by a
grant “Dote ricerca”: FSE, Regione Lombardia.
Appendix
In Tables 1 and 2 we give the fitting parameters of the
energy as function of density with expressions (12) and (13).
In our implementation of the QMC projection tecnhique, as
T we use a SWF. Such a wave function, introduced by
Vitiello et al. [36], is known to provide a very accurate de-
scription of the condensed phases of
4
He [37]: it has explicit
pair correlations between the coordinates of the atoms as
well as indirect many-body correlations via some auxiliary
shadow variables, denoted 1= ( , , ),Ns s which are in-
tegrated over:
2
< <
( ) = exp ( ) ( ) | | .
N N N
T r ij s ij i i
i j i j i
u r v s c dr s
(A.1)
The pseudo-potentials are chosen to be a generalized
McMillian form ( ) = ( / ) ,m
r ij iju r b r whereas the one for the
shadow variables is chosen of the Aziz rescaled form
( ) = ( ).s ij ijv s v s This SWF has the same form used by
Grisenti and Reatto [26] but as power m we have used m = 6
because this values improves the energy compared to m = 5
used in [37]. We have optimized the trial wave function
(A.1) varying the remaining variational parameters b, ,
and c through variational Monte Carlo simulations for vari-
ous densities. Notice that the form of T is the same for the
liquid and for the solid, only the variational parameters take
different values. The optimized SWF is used as trial wave
function for exact simulations at the same densities: the ex-
act technique is named SPIGS method [28,29].
Table 1. Values of the fit parameters for fitting functions (12)
and (13) of the variational equation of state
E0, K –0.753(3) , K –15.2 28%
0, Å
–2
0.0393(2) , K Å
2
765 20%
A, K 1.39(6) , K Å
4
–13311 13%
B, K 0.7(3) , K Å
6
80497 8.5%
C, K 0.93(15)
f, Å
–2
0.0677 m, Å
–2
0.0721
Table 2. Values of the fit parameters for fitting functions (12)
and (13) of the exact equation of state
E0, K –0.862(1) , a.u. –25.1
0, Å
–2
0.0430(1) , K Å
2
765 20%
A, K 2.00(3) , K Å
4
–13311 13%
B, K 2.1(1) , K Å
6
80497 8.5%
C, K 0.52(14)
f, Å
–2
0.0674 m, Å
–2
0.0701
Excitation spectrum in two-dimensional superfluid
4
He
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9 1029
For reference purpose we give the optimal values of the
SWF variational parameters in Tables 3 and 4 for the liq-
uid and solid phases, respectively. The values of the exact
and of the variational energy are also given in that tables.
1. R.A. Aziz, V.P.S. Nain, J.S. Carley, W.L. Taylor, and G.T.
McConville, J. Chem. Phys. 70, 4330 (1979).
2. A.J. Leggett, Rev. Mod. Phys. 71, S318 (1999).
3. L.D. Landau, J. Phys. USSR 5, 71 (1941); J. Phys. USSR 11,
91 (1947).
4. R.P. Feynman, Phys. Rev. 90, 1116 (1953).
5. D.G. Henshaw and A.D.B. Woods, Phys. Rev. 121, 1266
(1961).
6. M. Cohen and R.P. Feynman, Phys. Rev. 120, 1189 (1956).
7. E. Manousakis and V.R. Pandharipande, Phys. Rev. B 30,
5062 (1984).
8. W. Wu, S.A. Vitiello, L. Reatto, and M.H. Kalos, Phys. Rev.
Lett. 67, 1446 (1991).
9. D.E. Galli, E. Cecchetti, and L. Reatto, Phys. Rev. Lett. 77,
5401 ( 1996).
10. L. Reatto and D.E. Galli, Int. J. Mod. Phys. B 13, 607
(1999).
11. R.P. Feynman, Statistical Mechanics, W.A. Benjamin Inc.,
New York (1972).
12. R.N. Silver, D.S. Sivia, and J.E. Gubernatis, Phys. Rev. B 41,
2380 (1990).
13. M. Boninsegni and D.M. Ceperley, J. Low Temp. Phys. 104,
339 (1996).
14. S. Baroni and S. Moroni, Phys. Rev. Lett. 82, 4745 (1999).
15. E. Vitali, M. Rossi, L. Reatto, and D.E. Galli, Phys. Rev.
B 82, 174510 (2010).
16. A.W. Sandvik, Phys. Rev. B 57, 10287 (1998); O.F.
Syljuåsen, Phys. Rev. B 78, 174429 (2008).
17. M. Rossi, E. Vitali, L. Reatto, and D.E. Galli, Phys. Rev.
B 85, 014525 (2012).
18. M. Nava, D.E. Galli, M.W. Cole, and L. Reatto, J. Low
Temp. Phys., DOI 10.1007/s10909-012-0770-9 (2012).
19. J.M. Kosterlitz and D.J. Thouless, J. Phys. C 5, 124 (1972).
20. D.J. Bishop, and J.D. Reppy, Phys. Rev. Lett. 40, 1727
(1978).
21. P.A. Whitlock, G.V. Chester, and M.H. Kalos, Phys. Rev.
B 38, 2418 (1988).
22. B.E. Clernents, E. Krotscheck, and C.J. Tymczak. Phys. Rev.
B 53 12253 (1996).
Table 3. Optimal values of the variational parameters for the SWFs at various densities in the liquid phase, along with the values of
the energy, computed both with variational and exact methods. The results are compared with previous Green function Monte Carlo
results GFMCE of reference [21]
, Å
–2
b, Å c, Å
–2
, K Evar, K ESPIGS, K EGFMC, K
0.0310 –0.750(3)
0.0321 2.700 0.710 0.042 0.918 –0.710(1) –0.765(2) –0.78(2)
0.0332 –0.787(3)
0.0358 2.715 0.720 0.042 0.920 –0.739(2) –0.816(2) –0.81(1)
0.0421 2.720 0.710 0.044 0.920 –0.747(2) –0.862(2) –0.85(3)
0.04315 –0.861(2)
0.0478 2.720 0.700 0.044 0.920 –0.675(2) –0.835(3)
0.0490 –0.817(2)
0.0536 2.720 0.700 0.043 0.900 –0.516(3) –0.704(2) –0.67(3)
0.0600 2.715 0.660 0.042 0.870 –0.189(3) –0.404(2)
0.0658 2.710 0.645 0.042 0.840 –0.295(2) –0.065(3) –0.01(4)
0.0719 2.710 0.535 0.042 0.820 1.057(3) –0.798(3) 0.82(4)
Table 4. Optimal values of the variational parameters for the SWFs at various densities in the solid phase, along with the values of
the energy, computed both with variational and exact methods. The results are compared with previous Green function Monte Carlo
results GFMCE of reference [21].
, Å
–2
b, Å c, Å
–2
, K Evar, K ESPIGS, K EGFMC, K
0.0690 0.441(5)
0.0740 2.705 0.470 0.183 0.810 1.186(2) 0.951(2)
0.0765 2.710 0.500 0.184 0.835 1.504(3) 1.292(3) 1.30(2)
0.0835 2.700 0.650 0.182 0.860 2.789(3) 2.579(2) 2.78(7)*
0.0905 2.710 0.700 0.183 0.880 4.733(2) 4.483(2) 4.91(3)*
0.0975 2.705 0.800 0.182 0.900 7.520(3) 7.223(3)
Comment: values with “*” are computed near the given density.
http://dx.doi.org/%2010.1007/s10909-012-0770-9
F. Arrigoni, E. Vitali, D.E. Galli, and L. Reatto
1030 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 9
23. H.J. Lauter, H. Godfrin, and P. Leiderer, J. Low Temp. Phys.
87, 425 (1992).
24. M.W. Cole, D.R. Frankl, and D.L. Goodstein, Rev. Mod.
Phys. 53, 199 (1981).
25. B.E. Clements, H. Forbert, E. Krotscheck, H.J. Lauter, M.
Saarela, and C.J. Tymczak, Phys. Rev. B 50, 6958 (1994).
26. R.E. Grisenti, and L. Reatto, J. Low Temp. Phys. 109, 477
(1997).
27. A. Sarsa, K.E. Schmidt, and W.R. Magro, J. Chem. Phys.
113, 1366 (2000).
28. D.E. Galli and L. Reatto, Mol. Phys. 101, 1697 (2003).
29. D.E. Galli, and L. Reatto, J. Low Temp. Phys. 136, 343
(2004).
30. M. Nava, A. Motta, D.E. Galli, E. Vitali, and S. Moroni,
Phys. Rev. B 85, 184401 (2012).
31. M. Rossi, M. Nava, L. Reatto, and D.E. Galli, J. Chem.
Phys. 131, 154108 (2009).
32. D.M. Ceperley, Rev. Mod. Phys. 67, 279 (1995).
33. L. Reatto and G.V. Chester, Phys. Rev. 155, 88 (1967).
34. R.A. Cowley and A.D.B. Woods, Can. J. Phys. 49, 177
(1971); A.D.B. Woods and R.A. Cowley, Rep. Prog. Phys.
36, 1135 (1973).
35. T.R. Prisk, N.C. Das, S.O. Diallo, G. Ehlers, A.A. Podlesnyak,
N. Wada, S. Inagaki, and P.E. Sokol, arXiv:1211.0350.
36. S.A. Vitiello, K.J. Runge, and M.H. Kalos, Phys. Rev. Lett.
60, 1970 (1988).
37. B. Krishnamachari and G.V. Chester, Phys. Rev. B 61, 9677
(2000).
|
| id | nasplib_isofts_kiev_ua-123456789-118751 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T18:56:29Z |
| publishDate | 2013 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Arrigoni, F. Vitali, E. Galli, D.E. Reatto, L. 2017-05-31T06:31:49Z 2017-05-31T06:31:49Z 2013 Excitation spectrum in two-dimensional superfluid ⁴He / F. Arrigoni, E. Vitali, D.E. Galli, L. Reatto// Физика низких температур. — 2013. — Т. 39, № 9. — С. 1021–1030. — Бібліогр.: 37 назв. — англ. 0132-6414 PACS: 67.25.bh,67.25.dt https://nasplib.isofts.kiev.ua/handle/123456789/118751 In this work we perform an ab-initio study of an ideal two-dimensional sample of
 ⁴He atoms, a model for
 ⁴He
 films adsorbed on several kinds of substrates. Starting from a realistic hamiltonian we face the microscopic study
 of the excitation phonon–roton spectrum of the system at zero temperature. Our approach relies on path integral
 ground state Monte Carlo projection methods, allowing to evaluate exactly the dynamical density correlation
 functions in imaginary time, and this gives access to the dynamical structure factor of the system S(q, ), containing
 information about the excitation spectrum E(q), resulting in sharp peaks in S(q, ). The actual evaluation of
 S(q, ) requires the inversion of the Laplace transform in ill-posed conditions, which we face via the genetic inversion
 via falsification of theories technique. We explore the full density range from the region of spinodal decomposition
 to the freezing density, i.e., 0.0321 Å⁻²
 – 0.0658 Å⁻². In particular we follow the density dependence
 of the excitation spectrum, focusing on the low-wave vector behavior of E(q), the roton dispersion, the strength
 of single quasiparticle peak, Z(q), and the static density response function, (q). As the density increases, the
 dispersion E(q) at low-wave vector changes from a superlinear (anomalous dispersion) trend to a sublinear (normal
 dispersion) one, anticipating the crystallization of the system; at the same time the maxon–roton structure,
 which is barely visible at low density, becomes well developed at high densities and the roton wave vector has a
 strong density dependence. Connection is made with recent inelastic neutron scattering results from highly ordered
 silica nanopores partially filled with
 ⁴He. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Сверхтекучесть в низкоразмерных системах и в условиях ограниченной геометрии Excitation spectrum in two-dimensional superfluid ⁴He Article published earlier |
| spellingShingle | Excitation spectrum in two-dimensional superfluid ⁴He Arrigoni, F. Vitali, E. Galli, D.E. Reatto, L. Сверхтекучесть в низкоразмерных системах и в условиях ограниченной геометрии |
| title | Excitation spectrum in two-dimensional superfluid ⁴He |
| title_full | Excitation spectrum in two-dimensional superfluid ⁴He |
| title_fullStr | Excitation spectrum in two-dimensional superfluid ⁴He |
| title_full_unstemmed | Excitation spectrum in two-dimensional superfluid ⁴He |
| title_short | Excitation spectrum in two-dimensional superfluid ⁴He |
| title_sort | excitation spectrum in two-dimensional superfluid ⁴he |
| topic | Сверхтекучесть в низкоразмерных системах и в условиях ограниченной геометрии |
| topic_facet | Сверхтекучесть в низкоразмерных системах и в условиях ограниченной геометрии |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/118751 |
| work_keys_str_mv | AT arrigonif excitationspectrumintwodimensionalsuperfluid4he AT vitalie excitationspectrumintwodimensionalsuperfluid4he AT gallide excitationspectrumintwodimensionalsuperfluid4he AT reattol excitationspectrumintwodimensionalsuperfluid4he |