Path Integral Calculations of Vacancies in Solid Helium

Bryan K. Clark a and David M. Ceperley a,b

aDepartment of Physics, University of Illinois at Urbana-Champagin, Urbana, IL 61801, USAbNCSA, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Abstract

We study properties of vacancies in solid 4He using Path Integral Monte Carlo. We find, in agreement with othercalculations, that the energy to create a single vacancy is 11.5K and is monotonic with the number of vacancies. Ifmore the a few percent of the system becomes vacant, we find the system becomes unstable to melting. We show thenumber of exchanges in the system is increased by vacancies and how the underlying lattice is altered by the presenceof 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 of 4He, it exhibitsa number of anomalous properties including super-flow and Bose-Einstein condensation (BEC) in theliquid 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 solid 4He. The physics behind this behavior is stillunclear.

One initially plausible explanation for this effectwas that the wave function that represented theground state of 4He exhibited superfluidity or Bose-Einstein Condensation (BEC). Recent theoreticalcalculations indicate that the bulk, commensurate,equilibrium solid (either in the ground state or at thefinite temperature of the experiments) has neitherODLRO [7,8] nor superfluidity [6]. Many of thesecalculations were done at a fixed particle numbercommensurate with a lattice that has no vacancies inaccordance with the original suggestion of Andreevand Lifshitz[9] and Chester [11]. This leaves open

the possibility that the ground state of Helium haszero point vacancies [12] and that these vacanciesinduce supersolidity. Another possibility is that al-though a single vacancy may be energetically unsta-ble, a gas of vacancies may have lower energy thanthe commensurate crystal [13].

Even if vacancies do not play an important rolein the equilibrium state of solid 4He at sufficientlylow 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 havea number of defects including vacancies, dislocationsand 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 rolethat quantum vacancies play in realistic materials.A classical model of vacancies is insufficient becausethe quantum nature of the system allows for delocal-

Preprint submitted to Elsevier 17 December 2007

ization and Bose condensation of vacancies. In thiswork, we will explore properties of vacancies usingPath Integral Monte Carlo (PIMC). In Section 2 wewill discuss our method. In Section 3 we will dis-cuss the energy costs of introducing vacancies intothe system. In Section 4 we will examine the way inwhich 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 theeffective 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 ofbosonic equilibrium systems by integrating over thedensity matrix. The PIMC calculation’s input arethe interatomic Helium potential V , taken to bethe Aziz 1995 semi-empirical form[10], the numberof atoms, the size of the periodic box, and the tem-perature. We integrate over the density matrix

ρ(R) = 〈R| exp(−βH)|R〉

whereH = −λ∇2 + V (R).

Calculations are done at temperatures from 0.25Kto 1K in a periodic box with a density of 0.02862ptcl/A3. (This corresponds to a pressure of approx-imately 26.7 bars, just above the melting densitywhere quantum effects should be maximized in thecrystal.) The aspect ratio of the simulation box ischosen so that an hcp lattice of 180 lattice sites iscommensurate with the periodic boundary condi-tions. The number of particles in the system is thenset by the number of vacancies desired in the sys-tem. The only effect of these lattice sites is to seedthe initial Monte Carlo configurations and to definethe Wigner-Seitz cells of the system. Calculations re-ported here were done with PIMC++, a C++ codethat 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 stateof solid 4He could be incommensurate. If the systemis incommensurate, then we expect that the crystal

with a vacancy will have a lower free energy than onewithout. At low enough temperature, and for a lowdensity of vacancies, the concentration of vacanciesis given by exp(−β∆E) where ∆E is the energy costof a vacancy. We calculate the energy cost for intro-ducing a vacancy into the system at T=0.5K. Wemake 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 vacancyinto 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 vacancyinto a bosonic crystal is 11.5±1.1K, which is compa-rable with Pollet [1] who gets a value of 13.0± 0.5Kfor the vacancy energy and to a variational estimateof Pederiva et.al [2] of 11.6±2.0K for a slightly differ-ent density. The delocalization caused by turning onthe bose statistics results in a statistically insignifi-cant change in the perfect crystal but approximatelya 2-3 Kelvin drop in the system with a vacancy. Thisis likely a result of the fact that although permuta-tions are minimal in the perfect crystal, the systemwith a vacancy allows for many permutations allow-ing for a drop in the energy.

Dai et al. [13] propose that the energy cost forintroducing a single vacancy might be energeticallycostly, 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. Theenergy cost for introducing vacancies is positive andmonotonic in the number of vacancies up until thepoint where the system begins to destabilize andmelt. This is evidence that the normal equilibriumcrystal is commensurate and there is no indicationthat a non-zero finite concentration of vacancies isenergetically favorable in the ground state of 4He.During the PIMC calculation, if more than 2% (4

out of 180 sites) were vacant, the system collapsedinto a liquid-like phase. Such unstable systems aremarked with arrows on figure 1. The stability of thecrystal was determined by monitoring the structurefactor for k-vectors close to the hcp reciprocal lat-tice vector. The structure factor is shown in fig. 2 forcalculations of 4 and 5 vacancies. It is seen that theaddition of one vacancy causes the maximum valueof 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 theinternal energy, with the density held fixed. The arrows in-

dicate the onset of melting as signalled by the loss of Braggpeaks.

Fig. 2. Structure factor for a system at 2K with either 4 (redstars) or 5 (black circles) vacancies in the system. Points atthe 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 solid4He. As these vacancies are injected into the sys-tem, there are macroscopic jumps in the pressure ofthe system which indicates a restructuring or melt-ing/refreezing of the crystal, consistent with what isseen in our simulation.

4. Lattice Distortion due to a Vacancy

If a hole is introduced into the lattice and thelattice does not relax, there is a unique lattice sitewhose Wigner-Seitz cell is devoid of atoms. How-ever, with zero point motion, the lattice will relaxand the missing density will be distributed in neigh-boring lattice sites. The size of this distortion is aproperty of the quantum vacancy. If we average longenough, since our system has translation invariance,the density will be uniformly spread through thesimulation cell. In order to say something about thesize of the vacancy at finite temperature, we proceedas follows. Let ρNk =

∑i exp(ik · ri) be the Fourier

transform of the instantaneous density for a systemwith N atoms. Then

ρ(r) =1

(2π)3

∫d3k exp(−ik · r)ρk.

The structure factor is

S(k) =1N〈|ρk|2〉.

An estimate of the density of a vacancy is obtainedby subtracting the density of a perfect crystal fromthat of a single vacancy. Let us assume that the k-space density of the vacancy is δρ ≡ ρN − ρN−1.Then

|ρN−1k |2 − |ρNk |2 = |δρk|2 − 2ReρNk δρk.

We note that Re[ρNk δρk] can be neglected since ρNkis small for k not a reciprocal lattice vector and hasa random phase. Therefore, we have

δρk ≈ [(N − 1)SN−1k −NSNk ]1/2

where we have assumed that the vacancy is centeredat the origin and is hence real. Figure (3) shows thevalue of δρ(r) for a quantum system of 4He. Note thevacancy has a “negative presence” from 1.6 to 3.2 A;i. e. an increase in density surrounding the vacancycaused by other particles expanding and encroach-ing upon the vacant area. If we define the vacancyby the missing density (i.e. ignoring the extra den-sity that is added onto the system), the primary siteof the vacancy has 9% of the total density of the va-cancy, the 12 nearest neighbors contain 32% of thedensity and the second nearest neighbors containthe 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. Because4He has a large zero point motion, though, looking

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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 paircorrelation function of a perfect helium crystal.

Fig. 4. Shown is a snapshot of a 2d projection from the 3dhcp crystal. The x’s (blue) are the location of the latticesites. The dots (blue) are the centroids of the paths of aperfect crystal. The +’s (red) are the centroids of the paths

of a crystal with a single vacancy. Many particles in thesystem 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 asnapshot 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 atomsdrift significantly away from their respective latticesites

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 singlevacancy.

fects the cyclic permutations that are generated bythe method. In the path integral method, paths per-muting with each other are how Bose statistics areimplemented and their presence reflects exchange.Figure 5 shows the number of permutations of agiven size. When the system is commensurate, thereare only a few 2,3, and 4 particle permutations. Onthe other hand, when a vacancy is introduced intothe system, permutations up to size 14 exist and thenumber of exchanges increases drastically.

5. Vacancy-Interstitial Definition

To further explore properties of the vacancy, weneed 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 asa Wigner-Sietz cell that is empty and has no doublyoccupied nearest neighbors. In ref. [15] vacancies arelocated by removing pairs of close particles and lat-tice sites from the system in a “greedy” fashion; thelast remaining lattice site is designated as the loca-tion of the vacancy. Alternatively, one may definea vacancy in the following way. First, one matcheseach particle to at most one lattice site such that thesum of the distance squared between particles andlattice sites is minimized. Suppose that there are Mlattice sites and N < M helium atoms and let ri bethe 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,j is 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 χP is minimized. 1 We chooseto use this last definition because it finds the globalminimum of χ, not necessarily a local one. It allowsus to produce a path for the vacancy as a functionof imaginary time. Where fluctuations might allowa Monte Carlo configuration to have varying num-ber of vacancies, this definition always uniquely de-fines the number of vacancies equal to the numberof sites minus the number of atoms. The presenceof a nearby double occupation, does not exclude thedesignation 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 modelwhose Hamiltonian is:

H = t∑〈i,j〉

cic†j + Vin

∑i,j∈plane

cic†i cjc

†j

+ Vout∑

i,j 6∈plane

cic†i cjc

†j (2)

having a nearest neighbor hopping and a nearestneighbor interaction term. In this model, two vacan-cies cannot occupy the same site. Here ci creates avacancy at lattice site i and ti,j is the hopping ma-trix element between nearest neighbor sites (i, j). Inan hcp lattice the hopping can be different in andout of the basal plane: tin and tout. Here Vin andVout are nearest neighbor interactions.

However, there are limitations to this model. Letus now contrast the picture of what is going on inthe 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 siteto another lattice site, it “loses” memory of where ithas come from. In the continuum case, though, thevacancy may “hop” from one lattice site to anotherlattice site by moving a fraction of the inter-particle

1 This problem is the linear sum assignment problem and is

solved exactly in order M3 operations with the Hungarian

method [18]. The vacancies are then located at lattice sitesZPN+1 . . . ZPM

. Interstitials (i.e. M < N) can be defined

by interchanging the role of atoms and lattice sites in theabove description. The definition di,j as the squared distancerather 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 (topdashed 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 thishopping there is a resident memory of where it hascome from and so it is significantly less costly to hopback to its former lattice site than to hop to anothernearest neighbor. In figure (6) we plot the log of theprobability the vacancy has moved from 0 to r inimaginary time 2τ = 0.05. We show model systemswith effective masses ranging from 0.1mHe to 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 artifactof how the vacancy is defined, we believe that thisgives strong evidence that the wide band gap hamil-tonian is a poor representation of a vacancy in solidHelium.

7. The Effective Mass of a Vacancy

One aspect unique to a quantum crystal is theability 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 differentimaginary times. The hopping rate is given by t andis inversely proportional to the effective mass of thevacancy. In this section, we show that the vacancyhas an effective mass of approximately 0.15 timesthat of a Helium atom. Naively, an effective masssmaller than a helium atom seems unusual becauseit would seem that moving to a neighboring latticesite 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 asliding window.

currently in that site. This reasoning is misleading,though, and a result of ignoring that the system isactually on a continuum as opposed to a lattice. Inthe continuum, a Helium atom need move only frac-tion of a lattice site for the vacancy to swap ontoanother lattice site. Below we will use two differenttechniques to calculate the effective mass of the va-cancy.

7.1. Measuring via Fk(τ)

One method for establishing the effective mass fora vacancy is through the imaginary time dynamicstructure factor Fk(τ) Note that

δFk(τ) = FNk (τ)− FN−1k (τ)

where Fk(τ) = ρk(0)ρ−k(τ). Figure 7 shows δFk(τ)for τ ∈ {0.1, 0.2, 0.3, 0.4}. Defining λτk by δFk(τ) =δSk exp(−λτkk2τ) we can calculate the effectivemass of the vacancy. For each k, we fit the valuesof λτk/λ4He versus τ to a line for a range of τ ∈(0.025, 0.25), a sampling of which is seen in figure8. Figure 9 plots the slope of these lines (calculatedby doing a chi-squared fit) versus k2. By averagingthe value of this mass at large k, we find an effectivemass that is 0.15 times that of a helium atom.

7.2. Effective mass from the tight binding model

In the previous subsection, we calculated theeffective mass of the vacancy by looking at the dy-namic structure factor. In this subsection, we willestimate 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∗/m4He as a function of k2. The y-axis

is in units of the 4He mass. The dotted line is a guide tothe eye indicating the average of the large k values.

the imaginary time position-position correlationfunctions in both the continuum (with the exactHamiltonian) and the lattice model (with an ef-fective Hamiltonian as described in Eq. (2) ). Al-though there is reason to believe that this latticemodel can not accurately represent the contin-uum system, it should, nonetheless, be an effec-tive approximation at small τ and small r. To getthe imaginary time correlation functions for themodel system, we diagonalize the model hamilto-nian and calculate the correlation functions as afunction of the eigenfunctions and eigenvalues ofthe 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 analysisof Fk(τ)

rvac(τ) is the position of the vacancy at imaginarytime slice τ . We separately calculate correlationfunctions for sites in the basal plane and on the ABlattice. Once we determine the hopping parameterby fitting the correlation function, we can map ontoa mass. To do this, we rewrite the Laplacian as a12 point formula on the hcp lattice and find thatλ∇2 = λ/(2a2) = t giving us that m∗ = 1/(4a2t)where a is the nearest neighbor distance. We fit thehopping parameter t in two ways. To begin with, wefit the entire correlation function for τ = 0.025 byminimizing the chi-squared difference of the corre-lation function of our model lattice system and thecontinuum path integral results. This gives an effec-tive mass of approximately 0.08 mHe4 . We shouldnote that although this is the best fit, it does not fitthe data particularly well. Because there is reasonto believe that the data at large r is an especiallybad fit to the model (since at small τ it shouldn’thop there), we also fit only the ratio between near-est neighbor basal plane hopping (this cancels outnormalization effects influencing our fit). Doing thisgives us an effective mass of 0.11 mHe4 . These valuesare in reasonable agreement with the calculative us-ing Fk(τ). See figure 10 for a comparison of our ourmodel 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 insolid 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 parametermodel fits for the in-plane (AB) sites.

strong attractive interaction will cause clusters ofvacancies (or voids) to form. Unlike voids in met-als, which might contain a gas, these voids couldcollapse. The void formation would stymie the roleof vacancies in promoting superflow as there wouldbe domains of perfect bulk crystal which have beenshown to be a normal solid[6]. On the other hand, arepulsive or very weak attractive interaction mightallow a gas of delocalized vacancies to permeate thesolid Helium background. Although previous calcu-lations [15] have looked at attraction of multiplevacancies, to date only variational calculations [16]have looked at 2 vacancies and no one has calcu-lated the vacancy-vacancy interaction term. This isparticularly 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 thedivacancy is bound at 1K since the probability of be-ing arbitrarily far away is still non-zero. To quantifythis attraction, we fit the parameters to our tightbinding model. For our model of 2 vacancies, wemake a change of basis to the center of mass coordi-nates on the lattice. We then calculate g (|r|)for thebasal plane and AB sites in the lattice model andpath integral calculation and choose parameters Vin,Vout that minimize the squared difference betweenthese results). Fitting these two parameters we getVin/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 ableto calculate the energy spectrum of the system asshown in figure 12. There appears to be only a single(s-state) bound eigenvalue. The binding energy of adi-vacancy is approximately 0.47t. Using our valuefor the hopping parameter t = 1.6K we find thebinding energy is 0.75 K. Hence, we expect at lowtemperatures, any free vacancies, would form bounddi-vacancy states.

This work was performed with computational re-sources at NCSA. We acknowledge support of NSFgrants DMR-04-04853 and DMR-03 25939ITR.

References

[1] M. Boninsegni, A. B. Kuklov, L. Pollet, N. V. Prokof’ev,

B. V. Svistunov, and M. Troyer, Phys. Rev. Lett. 97,080401 (2006).

[2] F. Pederiva, G. V. Chester, S. Fantoni and L. Reatto,Phys. Rev. B 56, 5909 (1997).

[3] D. M. Ceperley, Rev. Mod. Phys. 67, 279 (1995).[4] G. Benedek, F. Dalfovo, R. E. Grisenti, M. Ksz, and J.

P. Toennies, Phys. Rev. Lett. 95, 095301 (2005).[5] E. Kim and M. H. Chan, Science 305, 1941 (2004).[6] D. M. Ceperley and B. Bernu, Phys. Rev. Letts. 93,

155303 (2004).[7] B. K. Clark and D. M. Ceperley , Phys. Rev. Lett. 96,

105302 (2006)[8] M. Boninsegni, N. Prokof’ev, and B. Svistunov, Phys.

Rev. Lett. 96, 105301 (2006).[9] A. F. Andreev and I. M. Lifshitz, Sov. Phys. JETP 29,

1107 (1969).[10] R. A. Aziz, A. R. Janzen, and M. R. Moldover, Phys.

Rev. Lett. 74, 1586 (1995).[11] G. V. Chester, Phys. Rev. A 2, 256 (1970).[12] P. W. Anderson, W. F. Brinkman, and David A. Huse

Science 310 (5751), 1164.

[13] X. Dai, M. Ma, and F.-C. Zhang, Phys. Rev. B 72,

132504 (2005).

[14] A. S. C. Rittner and J. D. Reppy, Phys. Rev. Lett. 97,165301 (2006)

[15] N. Prokof ev and B. Svistunov, Phys. Rev. Lett. 94,

155302 (2005).[16] B. Chaudhuri, F. Pederiva, and G. V. Chester, Phys.

Rev. B 60, 5 (1999)

[17] P. Remeijer, S. C. Steel, R. Jochemsen, G. Frossati, andJ. M. Goodkind, Fiz. Nizk. Temp. 23 586-597.

[18] Burkard, R. E. and U. Derigs, 1980 “Assignment andMatching Problems” Springer-Verlag Berlin.

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