Path integral calculations of vacancies in solid Helium

Abstract

We study properties of vacancies in solid 4He using Path Integral Monte Carlo. We find, in agreement with other calculations, that the energy to create a single vacancy is 11.5 K and is monotonic with the number of vacancies. If more then a few percent of the system becomes vacant, we find the system becomes unstable to melting. We show the number of exchanges in the system is increased by vacancies and how the underlying lattice is altered by the presence of a vacancy. We also examine the efficacy of using a tight binding Hamiltonian to describe the vacancy in the crystal, show that vacancies are attractive, and find values for the effective mass and inter-vacancy attraction.

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PathIntegralCalculationsofVacanciesinSolidHelium
Bryan K. Clarkaand David M. Ceperleya,b
aDepartment of Physics, University of Illinois at Urbana-Champagin, Urbana, IL 61801, USA
bNCSA, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Abstract
We study properties of vacancies in solid4He using Path Integral Monte Carlo. We find, in agreement with other
calculations, that the energy to create a single vacancy is 11.5K and is monotonic with the number of vacancies. If
more the a few percent of the system becomes vacant, we find the system becomes unstable to melting. We show the
number of exchanges in the system is increased by vacancies and how the underlying lattice is altered by the presence
of a vacancy. We also examine the efficacy of using a tight binding hamiltonian to describe the vacancy in the crystal,
show that vacancies are attractive, and find values for the effective mass and inter-vacancy attraction.
Key words:
PACS:
1. Introduction
Because of the quantum nature of4He, it exhibits
a number of anomalous properties including super-
flow and Bose-Einstein condensation (BEC) in the
liquid phase, the ability to stay liquid at zero tem-
perature, and a large zero point motion in the solid.
Recent work by Kim and Chan[5] have also found in-
dications of non-classical rotational inertia (NCRI)
in solid4He. The physics behind this behavior is still
unclear.
One initially plausible explanation for this effect
was that the wave function that represented the
ground state of4He exhibited superfluidity or Bose-
Einstein Condensation (BEC). Recent theoretical
calculations indicate that the bulk, commensurate,
equilibriumsolid(eitherinthegroundstateoratthe
finite temperature of the experiments) has neither
ODLRO [7,8] nor superfluidity [6]. Many of these
calculations were done at a fixed particle number
commensuratewithalatticethathasnovacanciesin
accordance with the original suggestion of Andreev
and Lifshitz[9] and Chester [11]. This leaves open
the possibility that the ground state of Helium has
zero point vacancies [12] and that these vacancies
induce supersolidity. Another possibility is that al-
though a single vacancy may be energetically unsta-
ble, a gas of vacancies may have lower energy than
the commensurate crystal [13].
Even if vacancies do not play an important role
in the equilibrium state of solid4He at sufficiently
low temperatures, it is plausible they may be preva-
lent in experiments at higher temperatures. In fact,
recent experimental results have indicated that an-
nealing of the crystal results in an elimination or re-
duction of the supersolid signal [14]. It is a reason-
able conjecture that the pre-annealed crystals have
a number of defects including vacancies, dislocations
and grain boundaries, while the post-annealed crys-
tal has fewer of these “non-equilibrium” defects.
Hence, the understanding of vacancies in solid he-
lium is of great interest. Beyond these considera-
tions, there has been little work calculating the role
that quantum vacancies play in realistic materials.
A classical model of vacancies is insufficient because
the quantum nature of the system allows for delocal-
Preprint submitted to Elsevier17 December 2007

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ization and Bose condensation of vacancies. In this
work, we will explore properties of vacancies using
Path Integral Monte Carlo (PIMC). In Section 2 we
will discuss our method. In Section 3 we will dis-
cuss the energy costs of introducing vacancies into
the system. In Section 4 we will examine the way in
which an introduction of a vacancy distorts the un-
derlying solid lattice. In Section 6 we discuss the re-
lation of the vacancy system to tight binding hamil-
tonians. Finally in Sections 7 and 8 we calculate the
effective mass of the vacancy and the attraction be-
tween vacancies, respectively.
2. Path Integral Monte Carlo
Path Integral Monte Carlo (PIMC) is a numer-
ical technique that exactly calculates properties of
bosonic equilibrium systems by integrating over the
density matrix. The PIMC calculation’s input are
the interatomic Helium potential V , taken to be
the Aziz 1995 semi-empirical form[10], the number
of atoms, the size of the periodic box, and the tem-
perature. We integrate over the density matrix
ρ(R) = ?R|exp(−βH)|R?
where
H = −λ∇2+ V (R).
Calculations are done at temperatures from 0.25K
to 1K in a periodic box with a density of 0.02862
ptcl/˚ A3. (This corresponds to a pressure of approx-
imately 26.7 bars, just above the melting density
where quantum effects should be maximized in the
crystal.) The aspect ratio of the simulation box is
chosen so that an hcp lattice of 180 lattice sites is
commensurate with the periodic boundary condi-
tions. The number of particles in the system is then
set by the number of vacancies desired in the sys-
tem. The only effect of these lattice sites is to seed
the initial Monte Carlo configurations and to define
theWigner-Seitzcellsofthesystem.Calculationsre-
ported here were done with PIMC++, a C++ code
that implements the algorithms described in Ceper-
ley[3].
3. Energy of Adding Vacancies
We start by calculating the energy cost of intro-
ducing a single vacancy into an equilibrium crystal.
Anderson et al.[12] argue that the equilibrium state
of solid4He could be incommensurate. If the system
is incommensurate, then we expect that the crystal
with a vacancy will have a lower free energy than one
without. At low enough temperature, and for a low
density of vacancies, the concentration of vacancies
is given by exp(−β∆E) where ∆E is the energy cost
of a vacancy. We calculate the energy cost for intro-
ducing a vacancy into the system at T=0.5K. We
make the distinction between the energy cost of in-
troducing a vacancy into a quantum “boltzmannon”
crystal (where there are no permutations that im-
plement Bose statistics) and introducing a vacancy
into a bosonic quantum crystal. While at zero tem-
perature, statistics should not matter, at finite tem-
perature the internal energy will depend on statis-
tics.
We find the energy cost of introducing a vacancy
into a bosonic crystal is 11.5±1.1K, which is compa-
rable with Pollet [1] who gets a value of 13.0±0.5K
for the vacancy energy and to a variational estimate
ofPederivaet.al[2]of11.6±2.0K foraslightlydiffer-
ent density. The delocalization caused by turning on
the bose statistics results in a statistically insignifi-
cant change in the perfect crystal but approximately
a 2-3 Kelvin drop in the system with a vacancy. This
is likely a result of the fact that although permuta-
tions are minimal in the perfect crystal, the system
with a vacancy allows for many permutations allow-
ing for a drop in the energy.
Dai et al. [13] propose that the energy cost for
introducing a single vacancy might be energetically
costly, but that the energy might drop for a non-
zero concentration of vacancies. To test this, we ex-
amine the energy of the system as a function of va-
cancy concentration. This is shown in figure 1. The
energy cost for introducing vacancies is positive and
monotonic in the number of vacancies up until the
point where the system begins to destabilize and
melt. This is evidence that the normal equilibrium
crystal is commensurate and there is no indication
that a non-zero finite concentration of vacancies is
energetically favorable in the ground state of4He.
During the PIMC calculation, if more than 2% (4
out of 180 sites) were vacant, the system collapsed
into a liquid-like phase. Such unstable systems are
marked with arrows on figure 1. The stability of the
crystal was determined by monitoring the structure
factor for k-vectors close to the hcp reciprocal lat-
tice vector. The structure factor is shown in fig. 2 for
calculations of 4 and 5 vacancies. It is seen that the
addition of one vacancy causes the maximum value
of S(k) to drop from roughly 40 to less than 10.
This instability may be relevant to a form of crys-
tal destabilization seen in the experiment on solid
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Fig. 1. Energy difference as a function of vacancy number nv
in a system of ns = 180 lattice sites for 2K (black circles),
1K (red squares) and 0.5K (blue triangles). E(nv,T) is the
internal energy, with the density held fixed. The arrows in-
dicate the onset of melting as signalled by the loss of Bragg
peaks.
Fig. 2. Structure factor for a system at 2K with either 4 (red
stars) or 5 (black circles) vacancies in the system. Points at
the same value of k are in different directions.
4He by Toennies et al.[4] In their experimental sys-
tem they “inject vacancies” through a hole in solid
4He. As these vacancies are injected into the sys-
tem, there are macroscopic jumps in the pressure of
the system which indicates a restructuring or melt-
ing/refreezing of the crystal, consistent with what is
seen in our simulation.
4. Lattice Distortion due to a Vacancy
If a hole is introduced into the lattice and the
lattice does not relax, there is a unique lattice site
whose Wigner-Seitz cell is devoid of atoms. How-
ever, with zero point motion, the lattice will relax
and the missing density will be distributed in neigh-
boring lattice sites. The size of this distortion is a
property of the quantum vacancy. If we average long
enough, since our system has translation invariance,
the density will be uniformly spread through the
simulation cell. In order to say something about the
size of the vacancy at finite temperature, we proceed
as follows. Let ρN
transform of the instantaneous density for a system
with N atoms. Then
1
(2π)3
k=?
iexp(ik · ri) be the Fourier
ρ(r) =
?
d3kexp(−ik · r)ρk.
The structure factor is
S(k) =
1
N?|ρk|2?.
An estimate of the density of a vacancy is obtained
by subtracting the density of a perfect crystal from
that of a single vacancy. Let us assume that the k-
space density of the vacancy is δρ ≡ ρN− ρN−1.
Then
|ρN−1
k
|2− |ρN
k|2= |δρk|2− 2ReρN
kδρk] can be neglected since ρN
kδρk.
We note that Re[ρN
is small for k not a reciprocal lattice vector and has
a random phase. Therefore, we have
k
δρk≈ [(N − 1)SN−1
where we have assumed that the vacancy is centered
at the origin and is hence real. Figure (3) shows the
value of δρ(r) for a quantum system of4He. Note the
vacancy has a “negative presence” from 1.6 to 3.2˚ A;
i. e. an increase in density surrounding the vacancy
caused by other particles expanding and encroach-
ing upon the vacant area. If we define the vacancy
by the missing density (i.e. ignoring the extra den-
sity that is added onto the system), the primary site
of the vacancy has 9% of the total density of the va-
cancy, the 12 nearest neighbors contain 32% of the
density and the second nearest neighbors contain
the rest of the density.
One can also recognize the distortion of the lat-
tice by examining the particle locations with re-
spect to their corresponding lattice sites. Because
4He has a large zero point motion, though, looking
k
− NSN
k]1/2
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Fig. 3. The solid line (blue) represents the value of δρ(r) for
the vacancy. Values below 0 represent an increase in density
or a “negative” vacancy. The dotted line (red) is the pair
correlation function of a perfect helium crystal.
Fig. 4. Shown is a snapshot of a 2d projection from the 3d
hcp crystal. The x’s (blue) are the location of the lattice
sites. The dots (blue) are the centroids of the paths of a
perfect crystal. The +’s (red) are the centroids of the paths
of a crystal with a single vacancy. Many particles in the
system with a vacancy have migrated significantly off their
respective lattice sites.
at a snapshot of any imaginary time slice on the He-
lium “path” is very noisy. Instead, one can look at a
snapshot of the centroids of the paths (i.e. the cen-
ter of mass of each polymer) as shown in Figure 4;
compared with the commensurate crystal, in the sit-
uation with a single vacancy the surrounding atoms
drift significantly away from their respective lattice
sites
The presence of a vacancy also significantly af-
Fig. 5. Number of paths permuting onto each other to form
a cycle of a given size. The dots (red) are for the perfect
crystal and the x’s (blue) are for the system with a single
vacancy.
fects the cyclic permutations that are generated by
the method. In the path integral method, paths per-
muting with each other are how Bose statistics are
implemented and their presence reflects exchange.
Figure 5 shows the number of permutations of a
given size. When the system is commensurate, there
are only a few 2,3, and 4 particle permutations. On
the other hand, when a vacancy is introduced into
the system, permutations up to size 14 exist and the
number of exchanges increases drastically.
5. Vacancy-Interstitial Definition
To further explore properties of the vacancy, we
need to specify where the vacancy is located by as-
signing it to a lattice site. A variety of such defini-
tions exist. Galli and Reatto [2] define a vacancy as
a Wigner-Sietz cell that is empty and has no doubly
occupied nearest neighbors. In ref. [15] vacancies are
located by removing pairs of close particles and lat-
tice sites from the system in a “greedy” fashion; the
last remaining lattice site is designated as the loca-
tion of the vacancy. Alternatively, one may define
a vacancy in the following way. First, one matches
each particle to at most one lattice site such that the
sum of the distance squared between particles and
lattice sites is minimized. Suppose that there are M
lattice sites and N < M helium atoms and let ribe
the instantaneous position of atom i. Define
?
χP=
i=1,M
di,Pi
(1)
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Here P is a permutation of the integers {1,...,M},
di,jis the squared distance between atom i and lat-
tice site j and define di,j= 0 for i > N. We then de-
termine P such that χPis minimized.1We choose
to use this last definition because it finds the global
minimum of χ, not necessarily a local one. It allows
us to produce a path for the vacancy as a function
of imaginary time. Where fluctuations might allow
a Monte Carlo configuration to have varying num-
ber of vacancies, this definition always uniquely de-
fines the number of vacancies equal to the number
of sites minus the number of atoms. The presence
of a nearby double occupation, does not exclude the
designation of a site as being vacant.
6. Mapping onto a tight-binding model
The simplest model for the energetics of a di-
lute gas of vacancies is a tight binding lattice model
whose Hamiltonian is:
?
+Vout
i,j?∈plane
having a nearest neighbor hopping and a nearest
neighbor interaction term. In this model, two vacan-
cies cannot occupy the same site. Here cicreates a
vacancy at lattice site i and ti,jis the hopping ma-
trix element between nearest neighbor sites (i,j). In
an hcp lattice the hopping can be different in and
out of the basal plane: tinand tout. Here Vinand
Voutare nearest neighbor interactions.
However, there are limitations to this model. Let
us now contrast the picture of what is going on in
the lattice system with what is going on in the con-
tinuum system. In the lattice system, when the va-
cancy “hops” in imaginary time from one lattice site
to another lattice site, it “loses” memory of where it
has come from. In the continuum case, though, the
vacancy may “hop” from one lattice site to another
lattice site by moving a fraction of the inter-particle
H = t
?i,j?
cic†
j+Vin
?
?
i,j∈plane
cic†
icjc†
j
cic†
icjc†
j
(2)
1This problem is the linear sum assignment problem and is
solved exactly in order M3operations with the Hungarian
method [18]. The vacancies are then located at lattice sites
ZPN+1...ZPM. Interstitials (i.e. M < N) can be defined
by interchanging the role of atoms and lattice sites in the
above description. The definition di,jas the squared distance
rather than some other metric means that the hypervolumes
are bounded by hyperplanes, not by curved surfaces.
Fig. 6. Log of the probability the vacancy is on a lattice
site r in the basal plane after time 2τ = 0.05. Dashed lines
represent model systems with effective masses of 0.1 (top
dashed line), 1(middle dashed line) and 10 (bottom dashed
line) times the mass of Helium. Solid line is PIMC data.
Note that it is qualitatively inconsistent with the model.
distance so that the missing density now mainly in-
habits a neighboring Wigner-Seitz cell. After this
hopping there is a resident memory of where it has
come from and so it is significantly less costly to hop
back to its former lattice site than to hop to another
nearest neighbor. In figure (6) we plot the log of the
probability the vacancy has moved from 0 to r in
imaginary time 2τ = 0.05. We show model systems
with effective masses ranging from 0.1mHeto 10mHe
and see that they all have a different qualitative be-
havior from the actual vacancy in the PIMC data.
Although we can’t rule out that this is an artifact
of how the vacancy is defined, we believe that this
gives strong evidence that the wide band gap hamil-
tonian is a poor representation of a vacancy in solid
Helium.
7. The Effective Mass of a Vacancy
One aspect unique to a quantum crystal is the
ability for it to delocalize. In path integrals, the sig-
nature of delocalization of the vacancy is the va-
cancy’s presence on different lattice sites at different
imaginary times. The hopping rate is given by t and
is inversely proportional to the effective mass of the
vacancy. In this section, we show that the vacancy
has an effective mass of approximately 0.15 times
that of a Helium atom. Naively, an effective mass
smaller than a helium atom seems unusual because
it would seem that moving to a neighboring lattice
site would require pushing the Helium atom that is
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Fig. 7. ρk(0)ρk(τ) for τ = 0.1,0.2,0.3,0.4 (respectively de-
scending). Curves have been smoothed through the use of a
sliding window.
currently in that site. This reasoning is misleading,
though, and a result of ignoring that the system is
actually on a continuum as opposed to a lattice. In
the continuum, a Helium atom need move only frac-
tion of a lattice site for the vacancy to swap onto
another lattice site. Below we will use two different
techniques to calculate the effective mass of the va-
cancy.
7.1. Measuring via Fk(τ)
One method for establishing the effective mass for
a vacancy is through the imaginary time dynamic
structure factor Fk(τ) Note that
δFk(τ) = FN
where Fk(τ) = ρk(0)ρ−k(τ). Figure 7 shows δFk(τ)
for τ ∈ {0.1,0.2,0.3,0.4}. Defining λτ
δSkexp(−λτ
mass of the vacancy. For each k, we fit the values
of λτ
(0.025,0.25), a sampling of which is seen in figure
8. Figure 9 plots the slope of these lines (calculated
by doing a chi-squared fit) versus k2. By averaging
the value of this mass at large k, we find an effective
mass that is 0.15 times that of a helium atom.
k(τ) − FN−1
k
(τ)
kby δFk(τ) =
kk2τ) we can calculate the effective
k/λ4Heversus τ to a line for a range of τ ∈
7.2. Effective mass from the tight binding model
In the previous subsection, we calculated the
effective mass of the vacancy by looking at the dy-
namic structure factor. In this subsection, we will
estimate the effective mass of the vacancy using
Fig. 8. Sample of 5 large k-values (jagged lines) as a function
of τ. The slopes of these lines should be λk. The dotted line
is the best fit to these slopes.
Fig. 9. Values of m∗/m4Heas a function of k2. The y-axis
is in units of the4He mass. The dotted line is a guide to
the eye indicating the average of the large k values.
the imaginary time position-position correlation
functions in both the continuum (with the exact
Hamiltonian) and the lattice model (with an ef-
fective Hamiltonian as described in Eq. (2) ). Al-
though there is reason to believe that this lattice
model can not accurately represent the contin-
uum system, it should, nonetheless, be an effec-
tive approximation at small τ and small r. To get
the imaginary time correlation functions for the
model system, we diagonalize the model hamilto-
nian and calculate the correlation functions as a
function of the eigenfunctions and eigenvalues of
the system for a given value of the inverse temper-
ature β. We use the correlation functions D(r) =
?(rvac(0) − rvac(τ))δ(r − (rvac(0) − rvac(τ))? where
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Fig. 10. Comparison of the PIMC data (blue dots (basal
plane) and x’s (AB sites)) with the model (dotted lines) for
a hopping parameter of t = 1.6K as found in in the analysis
of Fk(τ)
rvac(τ) is the position of the vacancy at imaginary
time slice τ. We separately calculate correlation
functions for sites in the basal plane and on the AB
lattice. Once we determine the hopping parameter
by fitting the correlation function, we can map onto
a mass. To do this, we rewrite the Laplacian as a
12 point formula on the hcp lattice and find that
λ∇2= λ/(2a2) = t giving us that m∗= 1/(4a2t)
where a is the nearest neighbor distance. We fit the
hopping parameter t in two ways. To begin with, we
fit the entire correlation function for τ = 0.025 by
minimizing the chi-squared difference of the corre-
lation function of our model lattice system and the
continuum path integral results. This gives an effec-
tive mass of approximately 0.08 mHe4. We should
note that although this is the best fit, it does not fit
the data particularly well. Because there is reason
to believe that the data at large r is an especially
bad fit to the model (since at small τ it shouldn’t
hop there), we also fit only the ratio between near-
est neighbor basal plane hopping (this cancels out
normalization effects influencing our fit). Doing this
gives us an effective mass of 0.11 mHe4. These values
are in reasonable agreement with the calculative us-
ing Fk(τ). See figure 10 for a comparison of our our
model with our continuum answer with a t of 1.6K.
8. Vacancy Attraction
Beyond the effective mass, another critical ingre-
dient to understanding the behavior of vacancies in
solid Helium is the vacancy-vacancy interaction. A
Fig. 11. Pair correlation functions of 2 vacancies at 1 K.
The dots (red) are for AB sites. The x’s (yellow) are for the
in-plane sites. The blue (black) lines are the best parameter
model fits for the in-plane (AB) sites.
strong attractive interaction will cause clusters of
vacancies (or voids) to form. Unlike voids in met-
als, which might contain a gas, these voids could
collapse. The void formation would stymie the role
of vacancies in promoting superflow as there would
be domains of perfect bulk crystal which have been
shown to be a normal solid[6]. On the other hand, a
repulsive or very weak attractive interaction might
allow a gas of delocalized vacancies to permeate the
solid Helium background. Although previous calcu-
lations [15] have looked at attraction of multiple
vacancies, to date only variational calculations [16]
have looked at 2 vacancies and no one has calcu-
lated the vacancy-vacancy interaction term. This is
particularly important because it allows us to calcu-
late the relevant interaction terms in our tight bind-
ing hamiltonian. As can be seen from the vacancy-
vacancy correlation function in figure 11, our cal-
culations clearly show the vacancies are attractive.
Although there is a noticeable attraction, the pair
correlation functions does not seem to indicate the
divacancy is bound at 1K since the probability of be-
ing arbitrarily far away is still non-zero. To quantify
this attraction, we fit the parameters to our tight
binding model. For our model of 2 vacancies, we
make a change of basis to the center of mass coordi-
nates on the lattice. We then calculate g (|r|)for the
basal plane and AB sites in the lattice model and
path integral calculation and choose parameters Vin,
Voutthat minimize the squared difference between
these results). Fitting these two parameters we get
Vin/t = 7.3 ± 0.5 and Vout/t = 4.5 ± 0.5. Given the
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Fig. 12. Eigenvalue energies (as a multiple of the hopping
parameter t). All values below 0 are bound. We note that
there is a single bound state in the system.
value of these lattice parameters, we are then able
to calculate the energy spectrum of the system as
shown in figure 12. There appears to be only a single
(s-state) bound eigenvalue. The binding energy of a
di-vacancy is approximately 0.47t. Using our value
for the hopping parameter t = 1.6K we find the
binding energy is 0.75 K. Hence, we expect at low
temperatures, any free vacancies, would form bound
di-vacancy states.
This work was performed with computational re-
sources at NCSA. We acknowledge support of NSF
grants DMR-04-04853 and DMR-03 25939ITR.
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