Role of Defects in Possible Superfluidity of Spatially Ordered Helium Keola Wierschem

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2010 Role of Defects in Possible Superfluidity of Spatially Ordered Helium Keola Wierschem

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ROLE OF DEFECTS IN POSSIBLE SUPERFLUIDITY OF SPATIALLY ORDERED

HELIUM

KEOLA WIERSCHEM

A Dissertation submitted to the Department of Physics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

Degree Awarded: Summer Semester, 2010 The members of the committee approve the dissertation of Keola Wierschem defended on June 29, 2010.

Efstratios Manousakis Professor Directing Dissertation

Wei Yang University Representative

Nicholas Bonesteel Committee Member

David Van Winkle Committee Member

Mark Riley Committee Member

Approved:

Mark Riley, Chair, Department of Physics

Joseph Travis, Dean, College of Arts and Sciences

The Graduate School has veriﬁed and approved the above-named committee members.

ii To my parents, for their constant support and encouragement.

iii ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Efstratios Manousakis, for his help, guidance, and support. I would also like to thank Wei Yang, Nick Bonesteel, David Van Winkle, and Mark

Riley for agreeing to serve on my doctoral supervisory committee.

I would like to acknowledge the support recieved from the Department of Physics and the Center for Materials Research and Technology. Additionally, some of the calculations presented in this dissertation were performed on the High Performance Cluster at Florida

State University.

iv TABLE OF CONTENTS

List of Tables ...... vii List of Figures ...... viii Abstract ...... xiii

1 Introduction 1 1.1 Solid helium in three dimensions ...... 3 1.2 Order in two dimensions ...... 5 1.2.1 Kosterlitz-Thouless phase transitions ...... 5 1.2.2 KTHNY two-stage melting theory ...... 6 1.3 Phase diagram of helium in two dimensions ...... 7

2 Computational Methods 11 2.1 Classical Monte Carlo ...... 11 2.1.1 Metropolis Monte Carlo ...... 12 2.1.2 Cell lists ...... 12 2.2 Path integral Monte Carlo ...... 13 2.2.1 The Hamiltonian ...... 15 2.2.2 The density matrix ...... 15 2.2.3 The worm algorithm ...... 18 2.2.4 Calculating observables ...... 21 2.3 Geometric defect analysis ...... 22 2.3.1 Delaunay triangulation ...... 24 2.4 Order parameters ...... 25 2.4.1 Correlation functions ...... 26 2.4.2 Finite size scaling ...... 27

3 Solid Helium with Defects and Impurities in Two and Three Dimensions 28 3.1 Results for two dimensions ...... 28 3.1.1 Distribution functions ...... 29 3.1.2 Energy ...... 35 3.1.3 Superﬂuid order ...... 37 3.2 Results for three dimensions ...... 39 3.3 Discussion ...... 43

v 4 Melting Transition of Lennard-Jones Particles in Two Dimensions 45 4.1 Results ...... 46 4.1.1 Defect excitation energy ...... 47 4.1.2 Signatures of a phase transition ...... 48 4.1.3 Order parameters ...... 53 4.1.4 Correlation functions ...... 61 4.1.5 Distribution functions ...... 64 4.1.6 Critical exponents ...... 69 4.2 Discussion ...... 72

5 Melting and Superﬂuidity of Solid Helium in Two Dimensions 74 5.1 High-density isochore ...... 75 5.1.1 Defect fraction ...... 75 5.1.2 Order parameters ...... 75 5.1.3 Correlation functions ...... 77 5.1.4 Defect distribution functions ...... 80 5.1.5 Finite size scaling ...... 83 5.2 Isotherm ...... 85 5.3 The ρ = 0.0761 A˚−2 isochore ...... 86 5.3.1 Superﬂuidity in ordered helium ...... 88 5.4 Discussion and outlook ...... 89

6 Summary 91 Bibliography ...... 93 Biographical Sketch ...... 97

vi LIST OF TABLES

1.1 Superﬂuid transition temperatures at varying densities, as determined by Ceper- ley and Pollock (CP) [42], Peters and Alder (PA) [43], and Gordillo and Ceper- ley (GC) [44]...... 7

3 3.1 Energy of the various systems studied. N3 is the number of He atoms in the 4 system, and N4 is the number of He atoms in the system...... 37

3.2 Superﬂuid fractions for the various cases of solid helium considered. Results are shown from calculations of 5,000 iterations in trivial parallel fashion after extended thermalization of 13,000 iterations...... 39

3.3 Energy (in Kelvin) of the various systems studied in three dimensions. N3 is 3 4 the number of He atoms in the system, and N4 is the number of He atoms in the system...... 41

3.4 Superﬂuid fractions for the various cases of solid helium considered...... 41

4.1 Defect activation energy for various system sizes and temperatures, as computed using the Arrhenius law. The numbers in parentheses are the uncertainty of the trailing digits...... 48

5.1 Superfluid fractions for the various densities considered for N = 90 atoms at 2 K. Due to the small number of superfluid configurations (a maximum of ten out of 10,000 configurations have non-zero winding numbers), the error bars could be substantially larger than presented...... 86

vii LIST OF FIGURES

1.1 The experimental apparatus (left) and data (right) of Kim and Chan, taken from their paper in Nature [1]. The torsional oscillator (left) is used to determine the resonant period, P, as a function of temperature for a sample of solid 4He in Vycor glass (right). Data is shown for the empty cell, as well as helium with varying concentrations of 3He impurties. The resonant period for each line is shifted by some constant amount, P*, for clarity...... 3

1.2 Phase diagram for two dimensional helium, taken from Gordillo and Ceper- ley [44]. Note that at temperatures below 1 K, the superﬂuid phase borders the solid phase...... 9

2.1 Flow chart for the worm algorithm...... 17

2.2 Illustrative example of a swap move in the worm algorithm. (Left) The particle positions, or beads, at each time slice before the swap move. Each world line is represented by ﬁlled black circles (the beads) connected by a solid line. One of the world lines is the worm. Also shown are empty circles that represent the beads to be created by the swap move. (Right) The system after the swap move has been accepted. Both world lines have been merged into a single world line, and Ira has changed places...... 20

2.3 An illustrative sketch of disclinations due to (a) a ﬁve-coordinated particle and (b) a seven-coordinated particle. Taken from Reference [64]...... 22

2.4 An illustrative sketch of a dislocation formed by a bound pair of disclinations consisting of a ﬁve-coordinated particle and a seven-coordinated particle. Taken from Reference [9]...... 23

4 3.1 The radial form of the pair distribution function for pairs of He atoms, g44(r). The organizational structure of the 4He atoms does not change in the presence of a substitutional 3He impurity. However, when an interstitial 3He impurity 4 or an interstitial He atom is present, we can see that g44(r) becomes less peaked at the nearest-neighbor distances, and subsequently enhanced in the interstitial regions...... 30

viii 3.2 Snapshot of a space-time conﬁguration of the 3He interstitial solid after thermalization. Red crosses represent the 4He atoms at each imaginary time slice, while blue circles represent the 3He impurity. Although initially placed interstitially, the 3He impurity has since relaxed onto a lattice site, thereby pushing a 4He atom into the interstitial region...... 31

4 3.3 Contour plots of the pair distribution function for pairs of He atoms, g44(x, y). Top: the pure solid. Bottom: the substitutional solid...... 32

3.4 Diﬀerence between the pair distribution function of the pure solid and the 3 interstitial He solid. We have subtracted g44(x, y) for the interstitial solid from g44(x, y) for the pure solid to get δg44(x, y)...... 33

3.5 Contour plots of the pair distribution function (PDF) in the 3He interstitial 4 solid. Top: g44, the PDF for pairs of He atoms. Bottom: g34, the PDF for pairs consisting of the 3He impurity and a 4He atom...... 34

3.6 Potential energy of a 3He atom placed either substitutionally (blue line) or interstitially (green line) into triangular solid 4He. After a brief relaxation, both energy values remain close except for occasional “blips” in the potential energy of the initially interstitial 3He atom...... 35

3.7 Snapshot of a space-time conﬁguration of the 3He interstitial solid in a “blip” of elevated potential energy that appears after thermalization. Red crosses represent the 4He atoms at each imaginary time slice, while blue circles represent the 3He impurity. The 3He atom can be seen to be at a region of local disorder...... 36

3.8 The one-body density matrix, n(r). Although no difference is observed between the pure and substitutional solids, both interstitial solids clearly show an enhancement of n(r) at large distances. The interstitial solids also display superfluid behavior, which was not observed in either the pure or substitutional solids. The superfluid fraction (fs = ρs/ρ) of both interstitial solids is shown, with the digits in parentheses indicating the uncertainty of the trailing digits of fs...... 38

4 3.9 The radial form of the pair distribution function for pairs of He atoms, g44(r). Although g44(r) looks the same for all systems, subtracting g44(r) for the interstitial solid from g44(r) for the pure solid to get ∆g44(r), we can see a clear diﬀerence. Note that the axis for ∆g44(r) is shown on the right-hand side of the ﬁgure...... 40

3.10 Potential energy of a 3He atom placed either substitutionally (blue line) or interstitially (green line) into HCP solid 4He. After a brief relaxation, the two energy values are virtually indistinguishable...... 42

3.11 The one-body density matrix, n(r)...... 42

ix 4.1 Fraction of defects, fd, as deﬁned by the fraction of non-six-coordinated particles in the Delaunay triangulation, f = 1 N /N. The rapid rise in f from d − 6 d near zero to almost 25% is a possible sign that dislocation and/or disclination unbinding is occurring...... 49

4.2 Temperature derivative of the defect fraction, as calculated using the ﬁnite difference method. The rate of increase of the defect fraction as the temperature is raised peaks near T = 0.9...... 50

4.3 The presence of a peak in the speciﬁc heat is indicative of a phase transition. Interestingly, the peak near T=0.9 appears to lessen in magnitude as the system size is increased...... 51

4.4 The distribution function shows ordering at low temperatures, as shown above for T=0.50, while at higher temperatures, such as T=2.00 shown above, there is a loss of order over moderate length scales...... 52

4.5 The second moment of the translational (top) and bond orientational (bottom) order parameters...... 54

4.6 Contour plots of ΨG~ (left column) and Ψ6 (right column) on the complex plane for T=0.70 (top row), T=0.90 (middle row), and T=1.10 (bottom row). . . . 55

4.7 The pair distribution function for 7-coordinated particles, g77(r). The peak for T=0.70 extends to 50...... 56 ∼ 4.8 The Delaunay triangulation for N = 1600 particles at T=0.70. Defects are shown in red...... 57

4.9 The Delaunay triangulation for N = 1600 particles at T=0.90. Defects are shown in red...... 58

4.10 The Delaunay triangulation for N = 1600 particles at T=1.10. Defects are shown in red...... 59

4.11 The pair distribution function for pairs consisting of one 5-coordinated particle and one 7-coordinated particle, g (r). The peak for T=0.70 extends to 150. 60 57 ∼

4.12 Example of fitting the bond orientational correlation function, C6, to the form shown in Equation 4.4. The data is for N = 25600 particles at T = 0.97. The critical exponents are fixed at their maximum values, η = 0.33 and η6 = 0.25. The extracted correlation lengths are ξ = 7.40 0.19 and ξ = 32.6 0.7. . . 62 ± 6 ± 4.13 Correlation lengths of the bond orientational order parameter as determined by fitting the bond orientational correlation function to the form mentioned in the text...... 63

x 4.14 Example of fitting the pair distribution function, g(r), to the form shown in Equation 4.14. The data is for N = 25600 particles at T = 0.97. The critical exponent η is fixed at its maximum value, 1/3. The extracted correlation length is ξ = 8.09 0.04...... 66 ± 4.15 Correlation lengths of the translational order parameter as determined by fitting the pair distribution function to the form mentioned in the text. The range of the fit is from 2ξ to 4ξ, with η fixed at its maximum value of 1/3. At T=0.80, the fitting range is from ξ to 2ξ...... 67

4.16 Pair distribution function, g(r), for the N=25600 and N=102400 particle systems at T = 0.92 in the ﬁtting range used to determine ξ...... 68

4.17 Anomalous dimensionality of (top) the translational order parameter and (bottom) the bond orientational order parameter. Our current results are shown in both figures as red circles. Shown for comparison are the results of Wierschem and Manousakis [84] (green squares, bottom figure) and Udink and van der Elsken [79] (blue triangles, both figures). In the top figure, the dashed and dotted lines represent the lower and upper bounds of η at Tm, according to KTHNY theory; in the bottom figure, the dashed line represents the predicted value of η6 at Ti...... 70

2 4.18 Scaling of < Ψ6 > with system size L, shown on a logarithmic plot. Results for T=0.84 are shown as blue circles, while data collected at T=0.92 is represented by red squares. In both cases, the data for the three smaller systems was ﬁt to the equation ln < Ψ2 >= η ln L + const, and the result is plotted as the 6 − 6 dotted and dashed lines (the solid line is the KTHNY value of η6 at Ti). In 2 both cases, the value of ln < Ψ6 > of the largest system size (N=102400) is reasonably close to the value expected from scaling...... 71

5.1 The defect fraction as a function of temperature. The melting transition appears rather broad, occurring somewhere in the range of 3 K to 4 K, where the defect fraction for the two larger systems exhibits the fastest increase. Spline ﬁts are shown as guides to the eye...... 76

5.2 Second moment of the translational (top) and bond orientational (bottom) order parameters. Spline ﬁts are shown as guides to the eye...... 78

5.3 Top panel: pair distribution function g(r) for N = 100 atoms. Bottom panel:

translational correlation function CG~ (r) for the same system...... 79

5.4 Bond orientational correlation function C6(r) for N = 100 atoms at extremal (top) and intermediate (bottom) temperatures. Notice that in the top panel, the vertical axis extends to 0.01 in order to show all of C (r) at 5 K. . . . . 81 − 6

5.5 Pair distribution function for pairs of same-charged disclinations g77(r) (top) and opposite-charged disclinations g57(r) (bottom) for N = 100 atoms. . . . 82

xi 5.6 Critical exponents of the translational and bond orientational order parameters, η and η6. Also shown are the KTHNY values of the critical exponents at the two melting temperatures. The dotted line is the maximal value of η at Tm (η(Tm) < 1/3), while the dashed line is both the minimal value of η at Tm (η(Tm) > 1/4) as well as the value of the bond orientational critical exponent at Ti, η6(Ti) = 1/4...... 84

5.7 Second moment of the translational and bond orientational order parameters along the T = 2 K isotherm...... 85

5.8 Second moment of the translational and bond orientational order parameters for the density 0.0761 A˚−2. At T=1 K, ﬁnite-size scaling gives critical exponents of η = 0.33 0.02 and η = 0.11 0.04, putting this temperature just ± 6 ± above the solid-hexatic phase boundary...... 87

5.9 Second moment of the translational and bond orientational order parameters (vertical axis on left) for N = 64 atoms on the 0.0761 A˚−2 isochore. Shown on the same graph is the superﬂuid density (vertical axis on right)...... 88

xii ABSTRACT

A path integral Monte Carlo investigation of the role of defects in ordered 4He systems has been conducted. We find that interstitial defects in two- and three-dimensional solid 4He can lead to a small but non-zero superfluid fraction, while still maintaining the original solid ordering. Furthermore, a 3He impurity atom initially placed as an interstitial defect is found to relax onto the solid lattice through the promotion of a 4He atom to the interstitial space. We also study the melting transition of pure 4He in two dimensions, where thermally excited defects give rise to unique phases with quasi-long-range order. Finite-size scaling techniques, initially applied to a classical system of Lennard-Jones particles, are found to be able to distinguish two separate melting temperatures. Additionally, coexistence of superfluid and diagonal order was observed in several of our finite-sized samples, raising the possibility that such real- and momentum-space ordering could be held simultaneously by quantum systems in two dimensions.

xiii CHAPTER 1

INTRODUCTION

In 2004, Kim and Chan reported the observation of nonclassical rotational inertia (NCRI) in samples of solid helium in Vycor glass [1] and bulk annular form [2]. As NCRI is a result of superfluid flow [3], these observations raised the exciting possibility that two completely different types of order, namely solid and superfluid, could coexist in the same phase. Soon after these initial discoveries of a possible supersolid phase, Rittner and Reppy[4] were able to reproduce the original observations of Kim and Chan. However, they also reported a reduction and even elimination of the supersolid signature by crystal annealing[4]. Later work showed that rapid freezing of 4He samples from the normal liquid phase resulted in supersolid signals as large as 20% of the sample, suggesting that defects in solid 4He are responsible for the observed supersolid properties of 4He [5]. In this dissertation, we will attempt to understand the role of defects in ordered phases of 4He. To this end, we investigate the properties of bulk solid 4He in two and three dimensions in the presence of defects and impurities.

In our ﬁrst study, we seek to determine the eﬀect of adding an interstitial defect to the bulk solid phase of 4He in three dimensions. Motivated by the dependence of the supersolid signal and onset temperature on the concentration of 3He impurities in the experiments of

Kim and Chan [1, 6], we also include substitutional and interstitial 3He atoms in this study.

Our goal is not necessarily to explain the observation of supersolid behavior in experiment, but rather to understand the eﬀect of adding 3He and 4He atomic defects to solid 4He.

1 The 2D solid is also investigated for ease of visualization and for possible comparison with lattice model theories [7].

In our second study, we focus on thermally activated defects that are naturally present in two-dimensional systems. In two dimensions, true solid ordering cannot exist above absolute zero. However, there is a solid-like phase with quasi-long-range translational order.

Additionally, there may be a hexatic fluid phase with quasi-long-range bond orientational order. Whether or not such a phase exists in two-dimensional helium systems, and whether this phase could possess superfluid ordering, are questions of great interest. Therefore, we investigate this possibility by studying the melting of solid helium in two dimensions with no added defects. In order to be confident in our analysis of the melting transition of helium in two dimensions, we first study the classical phase transition of Lennard-Jones particles in two dimensions.

Section 1.1 provides a background on the general topic of defects in solid helium, both in two and three dimensions. Section 1.2 reviews the peculiar properties of order in two dimensions, while Section 1.3 discusses the application of these properties to two-dimensional helium. In Chapter 2 we give an account of the calculational methods utilized in our research. In Chapter 3 we present the results of a study of defects and impurities in solid 4He, both in two and three dimensions. In Chapter 4 we present the results of a study of classical two-dimensional Lennard-Jones particles, whereby we try to determine if such a system follows the two-stage continuous melting transition predicted by the Kosterlitz-Thouless-

Halperin-Nelson-Young (KTHNY) theory [8, 9, 10]. Chapter 5 focuses on the melting of solid helium in two dimensions. No defects are added to the systems studied in this chapter.

However, there are thermally activated defects that naturally arise as the solid melts into the ﬂuid phase. Finally, we summarize our results in Chapter 6.

2 Figure 1.1: The experimental apparatus (left) and data (right) of Kim and Chan, taken from their paper in Nature [1]. The torsional oscillator (left) is used to determine the resonant period, P, as a function of temperature for a sample of solid 4He in Vycor glass (right). Data is shown for the empty cell, as well as helium with varying concentrations of 3He impurties. The resonant period for each line is shifted by some constant amount, P*, for clarity.

1.1 Solid helium in three dimensions

In 1956, Penrose and Onsager showed that there can be no Bose-Einstein condensation in a perfect crystal at absolute zero[11]. Later, Andreev and Lifshitz proposed that defects in a quantum crystal may form a Bose-Einstein condensate at low temperature[12], Chester showed that if a Jastrow wave-function correctly describes the solid phase of helium, it will be supersolid [13], and Leggett suggested that if a supersolid phase exists, it should display NCRI [14]. These theoretical speculations led to many experimental searches for a supersolid phase in 4He [15]. However, it was not until the initial experiments of Kim and

Chan [1, 2] that NCRI was observed in solid 4He.

In 2004, Kim and Chan observed a drop in the resonance period of their torsional oscillator apparatus for samples of solid 4He in Vycor glass [1]. This drop in resonance period is a sign of nonclassical rotational inertia [3], and is interpreted as evidence of a

3 supersolid phase in helium. In Figure 1.1, we show the torsional oscillator setup and results from the initial experiment by Kim and Chan, as published in the journal Nature [1]. In the left panel, the torsional oscillator apparatus is shown. Helium is inserted into a torsion bob, which contains the Vycor disk, through a ﬁlling line in the torsion rod. The blocked capillary method is used to solidify the resulting helium sample in the Vycor glass, and a lock-in ampliﬁer is used to determine the resonant period of vibration. The resonant period, P , is related to the moment of inertia, I, of the system by P = 2π I/G, where G is the torsional spring constant of the torsion rod. At low temperatures, Pp(and therefore

I) is observed to decrease, as shown in the right panel of Figure 1.1. The decrease in I is interpreted as resulting from a fraction of the solid helium sample becoming supersolid, and thereby no longer responding to rotations. Later in 2004, Kim and Chan also reported similar evidence for a supersolid phase in bulk solid helium in an annulus [2].

Soon after these initial discoveries of a possible supersolid phase, Rittner and Reppy reported the reduction and even elimination of NCRI in solid helium through crystal annealing[4], and later showed that rapid quenching of 4He samples from the normal liquid phase to the solid phase resulted in very large NCRI signals [5]. Given that the solid helium samples in these experiments were grown using the blocked capillary method, which is known to produce polycrystalline samples [2], it is likely that the observed NCRI is due to defects.

The importance of defects in supersolidity is also bolstered by theoretical arguments that a defect-free solid can never be superﬂuid[16]. This claim is supported by path integral

Monte Carlo calculations of solid helium in the low-temperature limit [17, 18, 19]. Vacancies are a natural candidate for defects in solid helium. However, vacancies have been shown to phase-separate in solid helium [20, 21].

Superﬂuid grain boundaries have been proposed as a possible explanation of NCRI in solid helium [22, 23]. Indeed, superﬂuidity along grain boundaries has been observed in samples of solid helium [24], as well as in exact quantum simulations of solid helium [25].

Yet torsional oscillator measurements on large crystals of 4He grown at constant pressure or

4 constant temperature (as opposed to growth at constant volume using the blocked capillary method) also display NCRI [26], so grain boundary superﬂuidity is unable to fully explain supersolid behavior in solid helium.

There have been proposals that quantum dislocations could play a role in supersolidity in solid helium [27, 7]. Quantum simulations of a screw dislocation have found that the core of the screw dislocation is superﬂuid [28], and experiments on the shear modulus of solid helium in the temperature regime of supersolidity have observed interesting results related to the pinning of dislocations by 3He impurities [29]. The driving mechanism of the observed NCRI in solid 4He is yet to be precisely determined. However, we can say with some conﬁdence that it will ultimately be intimately related to defects in solid helium.

1.2 Order in two dimensions

It has long been theorized that long range order cannot exist in two dimensions at any non-zero temperature. This was ﬁrst argued by Landau and by Peierls, using an argument based on the energy of spin waves with large wavelengths [30, 31, 32]. This was later rigorously proven by Mermin and Wagner using Bogoliubov’s inequality [33]. For two dimensional crystals, Mermin showed that for a large class of particle interactions (including the Lennard-Jones potential) there can be no long range order in two dimensions [34].

However, it is known that spin systems in two dimensions will experience a divergence of the spin susceptibility at non-zero temperature [35], and a similar phase transition occurs for two dimensional crystals [36].

1.2.1 Kosterlitz-Thouless phase transitions

Kosterlitz and Thouless proposed a theory of this phase transition in two dimensions where a high-temperature disordered phase gives way to an algebraically ordered phase at a lower, non-zero temperature [8]. The idea is that while there are spin waves that destroy the ordering in two dimensions at any temperature, there can still be order locally, but

5 additionally there exists vortices that will disrupt the order even locally if they are present at a high enough density. These vortices cost an energy proportional to the natural logarithm of the system size, but because their entropy is also thusly proportioned, there exists a temperature at which the free energy favors their formation. Above this temperature there will be unbound vortices throughout the system, while below this temperature they must exist in oppositely charged pairs, so that their energy is ﬁnite. This leads to a loss of all order at high temperatures, and causes a divergence of the susceptibility at low temperatures.

The Kosterlitz-Thouless (KT) phase transition is a defect-moderated continuous phase transition of inﬁnite order. The critical behavior of systems undergoing this type of phase transition are not the same as those for the typical continuous phase transition. Most notable is that the divergence of the susceptibility grows exponentially as the critical region is approached, instead of the usual power-law type divergence [37].

The KT theory has been applied to the superfluid phase transition of thin helium films with success [38]. However for two dimensional crystals it was found that two such KT phase transitions are required to completely melt the crystal into an isotropic fluid. This was shown by Halperin and Nelson [39, 9] and Young [10]. Just as vortices cause the KT transition for spin systems (XY models) and superfluids (helium), there are defects driving the KTHNY melting theory. First, the translational order is broken by an unbinding of dislocations. This also leads to a loss of long range orientational order, but the orientational susceptibility becomes infinite. Second, the orientational order is broken by an unbinding of disclinations.

1.2.2 KTHNY two-stage melting theory

Melting in two dimensions is theorized to be a defect mediated phase transition [8], driven by the unbinding of dislocations and disclinations at two separate temperatures [9].

The dislocations unbind at the lower melting temperature, Tm, and translational order becomes completely short ranged. At the upper melting temperature, Ti, the remaining (orientational) order is lost, as the unbinding of disclinations leads to an isotropic ﬂuid

6 Table 1.1: Superﬂuid transition temperatures at varying densities, as determined by Ceperley and Pollock (CP) [42], Peters and Alder (PA) [43], and Gordillo and Ceperley (GC) [44].

−2 ρ (A˚ ) Tc (K) Authors 0.0432 0.72 0.02 CP ± 0.0508 0.86 0.02 GC ± 0.0576 0.82 0.02 PA ± 0.0592 0.90 0.03 GC ± 0.0720 0.78 0.02 PA ± state. This two-stage melting transition is described by KTHNY theory [8, 9, 10].

If we apply the KTHNY theory to the melting transition of helium in two dimensions, it is an interesting question to ask where does the KT transition to the superﬂuid phase occur.

Might there be a region of the phase diagram where the superﬂuid phase transition occurs at the same or lower temperature as the upper melting transition of the KTHNY theory? To attempt to answer these questions, we would like to study the melting transition of helium in two dimensions. However, while the KTHNY theory has been extensively tested using both computational simulations and experimental setups, no consensus view has been reached on its validity for a wide range of systems and interactions, and there are competing theories of the melting transition in two dimensions that invoke a ﬁrst-order phase transition.

1.3 Phase diagram of helium in two dimensions

Following the work of Kosterlitz [37] on the critical scaling properties of the XY univer- sality class, Nelson and Kosterlitz [38] predicted a universal jump in the superfluid density of two-dimensional liquid helium as the superfluid transition temperature, Tc, is approached from below. Namely, if the superfluid density, ρs(T ), is plotted as a function of temperature,

2 2 the value at the transition temperature must lie on a line of slope 2m kB/¯h π. This universal jump criterion was observed in two dimensional helium ﬁlms the following year [40, 41].

7 Later, exact computational simulations of two-dimensional helium using the path integral Monte Carlo method used the universal jump criterion together with the Kosterlitz-

Thouless recursion relations to determine the superﬂuid transition temperature in the thermodynamic limit using results from a ﬁnite-sized system. Working at a density of

0.0432A˚−2, near the equilibrium density (0.0432A˚−2) of a prior Green’s Function Monte

Carlo investigation of the ground state properties of two-dimensional helium [45], Ceper- ley and Pollock found a transition temperature of T = 0.72 0.02K, in good agreement c ± with experimentally determined values [42]. In Table 1.1 we show the superﬂuid transition temperature of helium in two dimensions at diﬀerent densities, as calculated by previous path integral Monte Carlo studies. Notice that there is a tendency for Tc to increase with

−2 density, until a peak value is reached around 0.0592 A˚ and Tc begins to decline just before the melting transition.

A recent path integral ground state (PIGS) study has shown that there is no BEC in two-dimensional solid helium at T=0 K [46]. The possibility of having an hexatic phase of two-dimensional helium that also displays superfluidity is of great interest to the study of phase transitions. According to a theoretical analysis using Landau-Ginzburg theory to describe the superfluid and hexatic states, coexistence of superfluidity and hexatic order is possible [47]. Furthermore, a diffusion Monte Carlo study has recently observed a metastable superfluid phase with hexatic order [48]. Therefore, if there exists a stable hexatic phase in the melting transition of two-dimensional helium, it seems very likely that this phase could also be superfluid.

In Figure 1.2 we show the phase diagram for two dimensional helium, as reported by

Gordillo and Ceperley [44]. In the lower half of the diagram, the solid phase is bordered by the superfluid phase. At intermediate densities, there could exist an hexatic phase, which in turn could possibly have superfluid ordering, in addition to the orientational order such a phase possesses by definition. Krishnamachari and Chester [49] have studied the liquid and solid phases of two-dimensional helium in the ground state using the shadow wave function

(SWF) method. After looking at the behavior of order parameters and other observables

8 Figure 1.2: Phase diagram for two dimensional helium, taken from Gordillo and Ceperley [44]. Note that at temperatures below 1 K, the superﬂuid phase borders the solid phase.

9 across the melting transition they conclude that the melting transition is probably weakly

ﬁrst order [49].

10 CHAPTER 2

COMPUTATIONAL METHODS

In this chapter, we discuss the computational methods employed in our numerical studies of condensed matter systems. Starting with a description of the classical Monte Carlo method, and in particular the Metropolis algorithm for importance sampling, we proceed to the path integral Monte Carlo (PIMC) method for quantum systems. The worm algorithm for PIMC calculations is described in detail. This is followed by an explanation of the defect analysis technique used in our studies of two-dimensional melting transitions. Finally, we deﬁne translational and bond orientational order parameters, and discuss their expected form across a two-stage continuous melting transition [9].

2.1 Classical Monte Carlo

We use classical Monte Carlo methods with the Metropolis importance sampling algorithm [50] in order to obtain the properties of systems of particles in thermodynamic equilibrium. Monte Carlo integration converges slowly, but the rate of convergence is independent of the dimensionality of the integral. In statistical mechanics, where we are interested in many degrees of freedom, the Monte Carlo method is clearly advantageous. Because the

Boltzmann factor makes the integrand exponentially small for high energy configurations, much computational efficiency may be gained by using importance sampling, whereby we generate configurations based on the Boltzmann weight. In this method, no re-weighting of the integrand is necessary, and we avoid time generating impractical configurations.

11 2.1.1 Metropolis Monte Carlo

The Metropolis algorithm for Monte Carlo [50] describes how to generate Markov chains of particle configurations in such a way as to converge to the proper Boltzmann distribution in the limit of long chains [51]. First, let us assume that we have a system of N particles, each with a well-defined position in a simulational cell. The total potential energy of this configuration we denote as Uold. Now, let us propose a new position for one of these particles. To do this, we draw a random point in a box (two dimensions) or cube (three dimensions) centered around the current particle position. The width of this box/cube is a parameter that we are free to vary, and is generally chosen so that between 30% and 50% of proposed moves are accepted. Next we compute the total potential energy of this new configuration,

Unew, where all but one particle are in their original positions. Then we accept the new conﬁguration with probability

−β(Unew−Uold) Pold→new = min 1, e . (2.1) h i

Basically, if the new configuration has a lower potential energy, Unew < Uold, then the new configuration is automatically accepted. Otherwise, the new configuration is accepted with a probability that is the ratio of the two Boltzmann factors, exp( βU ) and exp( βU ). − new − old If the new configuration is not accepted, then the old configuration is kept. It can be shown that this method of importance sampling satisfies the detailed balance and ergodicity required to converge to the Boltzmann ensemble [51].

2.1.2 Cell lists

In simulations using a short-ranged interparticle potential, it is common to introduce a cutoff distance, rcut, beyond which interactions are set to zero. For example, in our simulations of Lennard-Jones particles, we have employed a cutoff distance of rcut = 3σ. The value of the Lennard-Jones potential at this cutoff point is quite small ( 0.005²), and − thus the influence on the total system energy is negligible. Additionally, using a cutoff

12 saves computational time when calculating the potential energy of the system. Instead of computing the Lennard-Jones potential for all N(N 1)/2 particle pairs in the system, − only a fraction of these particle pairs will be within the cutoﬀ distance. However, we must still calculate all N(N 1)/2 interparticle distances in order to know which pairs lie within − the cutoﬀ distance. Further computational time may be saved by using cell lists to record which particles are near one another.

The use of cell lists allows us to compute particle interactions in order N time (as opposed to N 2, as before). Because we have set the particle interactions to be identically zero for all distances greater than 3σ, when calculating particle interactions we need only consider particle pairs whose spacing is less than 3σ. This allows us to construct cell lists whereby we break down the large simulational cell into smaller cells with linear size near 3σ.

Keeping track of which particles are in which cells, we can ﬁnd all the non-trivial particle pairs (those whose interaction energy is non-zero) in a much faster way than by considering all N(N 1)/2 pairs and searching for the non-trivial cases. Instead, we look for non-trivial − pairs of particles that are in the same or neighboring cell. Although this is a very simple trick, it changes the required computational time from order N 2 to order N.

2.2 Path integral Monte Carlo

Path integral Monte Carlo (PIMC) is an exact simulation method, and has been shown to accurately describe the properties of liquid helium through the λ-transition [52], including the onset of superﬂuidity [53]. This method may also be applied to the study of solid helium, and indeed such calculations have already been performed [17, 18, 19]. It is important to remember that PIMC is an exact quantum mechanical treatment, the accuracy of which is limited by the eﬃciency of our sampling algorithms, the robustness of our random numbers, and the accuracy of our particle interactions. A detailed treatment of the application of

PIMC to the simulation of helium systems is given in a comprehensive review article by

D. M. Ceperley [54]. Application of PIMC to ﬁlm geometries can be found in the works of

13 M. Pierce (helium on graphite) [55] and K. Nho (hydrogen on graphite) [56].

R. P. Feynman ﬁrst applied path integral formulism to the λ-transition in 1953 [57].

The path of each helium atom propagates in imaginary time (by analogy to the Schr¨odinger equation), which is proportional to the inverse temperature. The longer the path, the lower the temperature of the atom in question. Superﬂuidity is described by the entanglement of these paths, which becomes easier the longer the paths of the atoms are, and therefore sets in at low temperatures.

PIMC is based upon applying Feynman’s path integral formulism to quantum statistical mechanics and using the Metropolis Monte Carlo [50] algorithm to evaluate the resultant integrals. PIMC allows for the representation of a quantum system at low temperature by a series of connected classical systems at a higher temperature. The fundamental idea behind the path integral method is to write the density matrix, e−βHˆ , as a product of density ˆ β ˆ β ˆ ˆ matrices at a higher temperature, e−βH = e− 2 H e− 2 H = (e−τH )M , where β = 1 is the kBT β inverse temperature, τ = M , and we call M the number of time slices as we are working in imaginary time.

In quantum statistical mechanics, all properties of interest may be derived from the density matrix, ρ. For example, diagonal observables are deﬁned by

1 1 < O >= dROρˆ (R,P R, β), (2.2) Z N! XP Z

1 where Z = N! P dRρ(R,P R; β) is the partition function of the system, and the sum- mation is overP all RN! particle permutations P . The inclusion of particle permutations is of absolute importance, because the 4He atoms in our simulations are indistinguishable bosons. In practice, a permutation occurs when, at the end of a path in imaginary time, the particle labels have changed. This means that in our simulation, we must sample particle permutations as well as particle paths.

14 2.2.1 The Hamiltonian

The Hamiltonian for a system of N 4He atoms is

¯h2 N N Hˆ = 2 + V (r ), (2.3) −2m ∇i ij Xi=1 Xi

2.2.2 The density matrix

In the position basis the density matrix, ρ, becomes

′ ˆ ′ ρ(R, R ; τ) =< R e−τH R >, (2.4) | |

where R = (r1, r2,..., rN ) is a dN dimensional vector specifying the position of all N helium atoms, where d is the spacial dimensionality of the system considered.

In general, the exact N-body density matrix is not known, so we make the following approximation:

N ′ ′ ′ ρ(R, R ; τ) ρ (R, R ; τ) exp[ U(r , r ; τ)], (2.5) ≈ 0 − ij ij i

′ 2 ′ 3N (R R ) ρ (R, R ; τ) = (4πλτ)− 2 exp[ − ], (2.6) 0 − 4λτ

15 2 where λ ¯h . We may now write the partition function as ≡ 2m

M −S(Ri−1,Ri;τ) Z = dR1 dR2 ... dRM e (2.7) Z Z Z iY=1

′ 2 N ′ 3N (R R ) ′ S(R, R ; τ) = ln(4πλτ) + − + U(r , r ,τ), (2.8) 2 4λτ ij ij Xi

Primitive approximation. By careful consideration of the commutation relations

[T,ˆ Vˆ ] between the kinetic (Tˆ) and potential (Vˆ ) energy operators, the following identity may be derived: ˆ ˆ 1 2 ˆ ˆ 1 3 ˆ ˆ ˆ ˆ ˆ e−τ(T +V )+ 2 τ [T,V ]− 6 τ [T,[T,V ]]+··· = e−τT e−τV . (2.9)

To first order in τ this is known as the primitive approximation. By applying the primitive approximation (e−τHˆ e−τTê−τVˆ ) to the density matrix, we find ≈

′ ′ ′ −τV (R~ ) ρ(R,~ R~ ; τ) = ρ0(R,~ R~ ; τ)e , (2.10) where the reader is left to apply the completeness relations showing this is true. Note that in this case the eﬀective potential, U, is identical to the interatomic pair potential, V .

Fourth order propagator. We use an approximation for the density matrix that is accurate to fourth order in τ [61]. By clever treatment of the commutation relations in

Equation 2.9, it can be shown that the expression

2 2 ′ ′ − τV (R~) − τW (R~) ρ(R,~ R~ ; τ) = ρ0(R,~ R~ ; τ)e 3 e 3 (2.11) is accurate to fourth order in τ, where

τ 2¯h2 N W (R~) = V (R~) + V (R~) (2.12) 6m ∇i Xi=1 h i

16 Begin Worm

Call Open

NO Is Open Accepted? End Worm

YES YES

NO Call Advance/Recede Is Close Accepted?

Call Swap Call Close

Figure 2.1: Flow chart for the worm algorithm.

for “odd” time slices, and W (R~) = 0 for “even” time slices. By even and odd we mean that each time slice is sequentially numbered for the purpose of keeping track of world lines in the calculational code, though the “ﬁrst” time slice has no special meaning.

Wiggle updates. The simplest update scheme is to apply the Metropolis [50] algorithm to each bead of each atom. While these updates can help to relax the system for a given configuration, they do not efficiently sample the phase space, and cannot generate permutations. In practice, we use these updates in combination with the worm algorithm to relax the configurations developed by the worm algorithm.

17 2.2.3 The worm algorithm

Because the integral we wish to calculate via Monte Carlo methods contains an averaging over all N! possible permutations of particle paths, we must sample these in our program. This means that we must allow a possibility for particle paths to “cross” by trading endpoints. The worm algorithm [62] is an update scheme for path integral Monte

Carlo that allows for the eﬃcient sampling of particle permutations. A ﬂow chart for the worm algorithm is shown in Figure 2.1 and the components are described below.

While earlier versions of PIMC also contained methods for the proper sampling of particle permutations [54], these were based on the creation of permutation tables. These tables contained weights for 1-, 2-, 3-, and 4-body permutations, which would then be sam- pled according to their respective weights. However, in this method global permutations for systems greater than four atoms in length can only occur through the entanglement of preexisting particle permutations. Such a sampling method can lead to questions about ergodicity [17]. The worm implementation of PIMC avoids these issues [62].

Open/Close. This complementary set of moves allows us to enter into and exit the worm algorithm. To begin with, we select a world line at random, and propose to open it by the removal of m beads, thus creating what are known as Ira and Masha, the head and tail of the worm. This open move is accepted or rejected with some probability, Popen. If the move is rejected, the call to the worm algorithm is complete. If accepted, we proceed to the advance/recede, swap, and close moves. We continue in this order until a close move is accepted. The close move consists of selecting a number of beads m to add to the worm.

If m is less than the Ira-Masha distance, the close move is rejected. If greater than or equal to, m new bead positions are proposed, which will link Ira to Masha (and perhaps beyond).

This move is then accepted or rejected with probability Pclose. If accepted, the new bead positions are updated, and the call to the worm algorithm is complete. If rejected, these proposed bead updates are erased, and we proceed to the advance/recede moves, as before, followed by a swap move and then another call to close. This iterative procedure is repeated

18 until the worm is closed by a successful close move.

Advance/Recede. In the advance and recede moves, m beads are created or destroyed. These extra beads are either added to or removed from Ira. If m is so large as to remove all the beads of the worm, then the recede move is rejected. Similarly, if m is so large as to move Ira past Masha in an advance move, then this move is rejected (at least for simulations in the canonical ensemble). Otherwise, the move is accepted or rejected with probability Padvance or Precede. Whether to attempt an advancement or a recession is determined at random, such that each move is attempted equally, on average.

Swap. The swap move is the heart of the worm algorithm. It allows for the generation of interconnected world lines, which represent the particle permutations that are needed to accurately simulate bosons in the quantum regime. Because of the hard-core like interaction of helium atoms, the swap move is very important, as it allows us to sample paths so that the (eﬀective) hard cores of permuting atoms do not “touch” as they attempt entanglement.

In Figure 2.2 we illustrate a successful swap move between two world lines, one of which

(by deﬁnition) is the worm line. On the left side of Figure 2.2 we show the original positions of the two world lines, and indicate the location of Ira and Masha, the head and tail of the worm line. Individual beads (ﬁlled circles) at successive time slices are connected by solid lines to indicate that they belong to the same world line. In this example, the swap move has chosen to connect Ira to a neighboring world line at a time slice m = 4 imaginary time steps above. The hollow circles are the proposed bead positions for this connecting line.

On the right side of Figure 2.2 we show the outcome of this accepted swap move. The intermediary beads of the world line Ira has connected to have been eliminated, resulting in a single world line with a new Ira spacial location (but on the same time slice as before). At this point, if a successful close move were to occur, we would be sampling a pair permutation in the phase space of diagonal conﬁgurations. It is easy to imagine that successive swap moves, or calls to the worm algorithm, can build up any possible permutations between the indistinguishable bosonic 4He atoms, including permutations that span the length of the

19 M

Masha Masha i+4

i+3

i+2

i+1

Ira Ira i

Figure 2.2: Illustrative example of a swap move in the worm algorithm. (Left) The particle positions, or beads, at each time slice before the swap move. Each world line is represented by ﬁlled black circles (the beads) connected by a solid line. One of the world lines is the worm. Also shown are empty circles that represent the beads to be created by the swap move. (Right) The system after the swap move has been accepted. Both world lines have been merged into a single world line, and Ira has changed places.

20 simulational cell. Indeed, non-zero winding numbers have been observed using the worm algorithm in simulations of systems 4He atoms as large as N = 2500 [62].

2.2.4 Calculating observables

To calculate the energy, we use the thermodynamic estimator, E = ∂ ln Z . When − ∂β applied to the partition function of Equation 2.7, we ﬁnd

3N (∆R)2 dU E = < > + < >, (2.13) 2τ − 4λτ 2 dτ where we have written the sum over time slices into our expectation values.

To observe the supersolid fraction, we have employed the same methods that have been successfully used to calculate the superfluid fraction in liquid helium [53]. The superflowing fraction is related to the winding number of the system, which is a measure of how many times a permutation chain crosses the cell boundaries. Specifically,

ρ < [ N r r ]2 > s = i=1 i − P i , (2.14) ρ P 2dNλβ where d is the dimensionality and rP i is the permuted position of ri, remembering that the permutation could be identity. Thus any permutation that winds around the simulation cell will contribute towards a non-zero superﬂuid fraction.

The one-body density matrix (OBDM), n(~r, ~r‘) =< aˆ†(~r)â(~r‘) >, is defined as the probability to destroy a particle at ~r‘ and create a particle at ~r (â is the particle annihilation operator). More explicitly, the OBDM may be written as [54]

Ω n(~r , ~r‘ ) = d~r d~r ρ(~r , ~r , . . . , ~r , ~r‘ , ~r , . . . , ~r , β), (2.15) 1 1 Z 2 ··· N 1 2 N 1 2 N Z where Ω is the system area (2D) or volume (3D). In the worm algorithm, Ira and Masha represent the annihilation and creation operators, respectively. This allows for rather straight- forward calculation of the OBDM in the oﬀ-diagonal conﬁguration space of the worm algo-

21 Figure 2.3: An illustrative sketch of disclinations due to (a) a ﬁve-coordinated particle and (b) a seven-coordinated particle. Taken from Reference [64].

rithm. To do so, we simply advance Ira to the point at which it resides on the same time slice as Masha. If this (temporary) advance move is accepted, then the conﬁguration ~rI ,

‘ ~rM of Ira and Masha is added to the OBDM, n(~r, ~r ).

2.3 Geometric defect analysis

In two dimensions, the densest packing of particles of uniform size is achieved in a triangular lattice. In such a conﬁguration, each particle has exactly six nearest neighbors.

In practice, thermal and quantum ﬂuctuations will lead to distortions in the lattice, or even destroy it completely. To quantify this, we use the Delaunay triangulation to determine the nearest neighbor network of our particle conﬁgurations. The nearest neighbor network tells us the number of nearest neighbors, or coordination number, of each particle. For a system of particles in a periodic plane, the average coordination number is always six [63].

Particles in a triangularly ordered region will be six-coordinated, while disruptions in the lattice will lead to particles with coordination numbers greater than or less than six.

A defect is deﬁned as any coordination number other than six. These non-six-coordinated atoms may be thought of as disclinations of charge n, their coordination number being 6+n.

22 Figure 2.4: An illustrative sketch of a dislocation formed by a bound pair of disclinations consisting of a ﬁve-coordinated particle and a seven-coordinated particle. Taken from Reference [9].

The most common type of disruption, or defect, is a ﬁve- or seven-coordinated particle.

These may be interpreted as disclinations of charge plus or minus one (see Figure 2.3 for an illustrative sketch). Two oppositely charged disclinations may be thought of as a dislocation

(see Figure 2.4 for an illustrative sketch). More complex arrangements of disclinations are possible, such as dislocation pairs and grain boundary loops, but in our analysis we have only considered individual defects. The defect fraction, f = 1 N /N, is deﬁned as the d − 6 fraction of particles that do not have six neighbors, where N is the number of particles in the system, and N6 is the number of six-coordinated particles in the system.

Remembering that dislocations are made of two bound disclinations of opposite charge, and that dislocations become unbound above the melting point, we can expect the defect fraction to experience a “jump” at the melting point [63]. Additionally, at low temperatures we can expect an energy gap to occur. This leads to an exponential behavior in the defect fraction, f = exp( β∆), where ∆ is the lowest-energy excitation of the system. Because d − the overall disclinicity of the system must be zero, as well as the net Burgers vector of any

23 dislocations, the lowest-energy excitation is a dislocation pair of opposite Burgers vectors.

In practice this is usually two pairs of 5- and 7-coordinated particles.

2.3.1 Delaunay triangulation

For a system of N particles, the Delaunay triangulation is deﬁned as the set of triangles, with the particle positions determining the vertices, that completely maps the system, such that the circumscribed circle of any triangle does not contain within it the vertex of any other triangle. The legs of the triangles represent nearest neighbor bonds between particles: in two dimensions, there are exactly 3N such bonds [63].

The Delaunay triangulation is the dual graph of the Voronoi tessellation. While it is possible to determine the Delaunay triangulation by ﬁrst calculating the Voronoi tessellation

(for which there are advanced computational algorithms available), and then performing a dual conversion, we have found that the Delaunay triangulation is rather simple to determine directly. For a given set of particles, we ﬁrst construct a triangle for each trio of particles, with the particles forming the vertices of the triangle. Next, we simply eliminate all triangles whose circumscribed circle contains any particle in its interior. The triangles that remain after this elimination form the Delaunay triangulation. It is quite obvious that any three particles forming a Delaunay triangle will necessarily be relatively close to one another, or else the circumscribed circle would be so large as to almost certainly contain some other particle in its interior. For a similar reason, triangles with narrow angles are also unlikely to be a part of the Delaunay triangulation.

In the case of determining the nearest neighbor network for particles in thermal equilibrium (and in or near the solid phase), the above restrictions on Delaunay triangulation are even more obvious. Speciﬁcally, as the particles are more or less evenly distributed throughout the system, large triangles are strictly forbidden, and we need only consider triangles whose particles are within some cutoﬀ distance of each other. In our experience, this ends up being from two to three times the lattice spacing.

24 2.4 Order parameters

Let us deﬁne a global order parameter as

1 N Ψ = ψ , (2.16) N i Xi=1 where the sum is over all N particles of the system and ψi is the value of a local order parameter in the neighborhood of the ith particle. In general, both the local and global order parameters can be complex valued. For the translational and bond orientational order parameters considered in this dissertation, this is indeed the case.

The local translational order parameter for a particle located at some position ~r is

i~r·G~ ψG~ = e , (2.17) where G~ is the ﬁrst reciprocal lattice vector of the triangular solid phase. If particles are located at lattice sites, then ψG~ will be the same value for all particles. This in turn leads to the global translational order parameter,

N 1 ~ Ψ = eiG·~rj , (2.18) G~ N jX=1

th having a magnitude equal to unity (here ~rj is the position of the j particle). At zero temperature, particles will be arranged in a perfect triangular solid lattice, and thus < | Ψ > = 1. At non-zero temperatures below the melting temperature, T , there is quasi- G~ | m long-ranged translational order. On the complex plane, < ΨG~ > forms a ring about the origin, but due to averaging over many phase factors, < Ψ > = 0. Finally, in both | G~ | the hexatic ﬂuid and isotropic ﬂuid phases, there is no translational order, and < ΨG~ > is always near zero.

25 The local bond orientational order parameter of a particle is

1 n ψ = e6iθj , (2.19) 6 n jX=1 where the sum is over all n neighbors of the particle, and θ is the angle between the bond connecting an atom to its neighbor and the axis of primary bond orientation. The global bond orientational order parameter is given by [65]

N nj 1 1 6iθjk Ψ6 = e , (2.20) N nj jX=1 kX=1 where N is the total number of particles, nj is the number of nearest neighbors of the jth particle, and θjk is the angle formed between the vector connecting particles j and k and an arbitrary frame of reference. In the perfectly ordered triangular solid phase, all bonds of a single atom will be spaced by π/3 radians, so that we pick up the same phase factor

(relative to the frame of reference) for all atoms, and thus < Ψ > = 1, with < Ψ > | 6 | 6 lying on some point along the unit circle in the imaginary plane. In the low-temperature solid phase in two dimensions, there exists long-range bond orientational order, so that we have < Ψ > > 0, with < Ψ > once again residing at a single point somewhere in the | 6 | 6 imaginary plane. In the hexatic phase, bond orientational order becomes quasi-long-ranged, and < Ψ > is represented by a ring in the imaginary plane, such that < Ψ > = 0 6 | 6 | but < Ψ >= 0. Finally, in the isotropic ﬂuid phase we have both < Ψ > = 0 and 6 6 | 6 | < Ψ6 >= 0.

2.4.1 Correlation functions

Two correlation functions can be defined from the two order parameters. The translational correlation function is defined as C (r) =< ψ∗ (r)ψ (0) > < ψ >2, while the G~ G~ G~ − G~ bond orientational correlation function is defined as C (r) =< ψ∗(r)ψ (0) > < ψ >2. 6 6 6 − 6 2 2 Because the values < ψG~ > and < ψ6 > are not functions of r, their only effect is to shift the correlation functions up or down by a constant amount. In this dissertation, we

26 shall ignore these contributions, as we are interested in the relative decay rates of CG~ (r) and C6(r), not their absolute values.

2.4.2 Finite size scaling

Finite size scaling is a powerful tool for extracting the properties of inﬁnite-sized systems using the known scaling behavior of ﬁnite-sized systems [66, 67]. Here we derive some of the scaling properties of the order parameters considered.

From Equation 2.16 we can deﬁne the thermodynamic average of the second moment of the global order parameter Ψ,

1 N < Ψ2 >= < ψ ψ∗ > . (2.21) N 2 i j i,jX=1

∗ The quantity < ψiψj > is very closely related to the correlation functions deﬁned above, C(r) =< ψ∗(r)ψ(0) > < ψ >2. In the topological solid phase the translational correlation − function decays algebraically, C (r) r−η. Likewise, in the hexatic phase C (r) r−η6 . G~ ∼ 6 ∼

Additionally, < ψG~ > and < ψ6 > are zero in these phases, due to the lack of true long-range order. Replacing the sum over particles in Equation 2.21 by an integral over the system size, 1 < Ψ2 > C(r)d2r, (2.22) ∼ L2 Z where N = L2 and one factor of N has been taken out due to the double sum over lattice indices. Making use of the power-law decay forms for the correlation function mentioned above, we ﬁnd the following scaling relations for the translational and bond orientational order parameters: < Ψ2 > L−η and < Ψ2 > L−η6 . By plotting < Ψ2 > (or < Ψ2 >) G~ ∼ 6 ∼ G~ 6 and L on logarithmic axes, these scaling relations can be used to ﬁnd η (or η6).

27 CHAPTER 3

SOLID HELIUM WITH DEFECTS AND

IMPURITIES IN TWO AND THREE

DIMENSIONS

In this chapter we present the results of a path integral Monte Carlo study of two- and three- dimensional solid 4He. In recent years, a possible supersolid state has been observed in solid

4He samples [1, 2]. Motivated by the curious dependence of the supersolid fraction on 3He impurity concentration [1], we investigate the behavior of solid 4He with 3He impurities.

Calculations have been performed for the pure solid, as well as the solid with added defects and impurities. In both two and three dimensions we ﬁnd that impurity interstitials become localized at a lattice site by promoting a 4He atom to the interstitial space, leading to oﬀ- diagonal long range order. This result is in agreement with a quantum spin model study of

3He impurities in solid 4He that also ﬁnds 3He interstitial impurities to relax onto lattice sites by promotion of 4He atoms to the interstitial space [7].

3.1 Results for two dimensions

Our simulation cell is designed to accommodate a 56-site triangular lattice, and is very nearly square (25.86 A˚ 25.60 A).˚ The density of lattice sites is ﬁxed at 0.0846 A˚−2. This × density is well within the solid phase of two dimensional helium, and is also close to the

28 surface density of hexagonal close packed (HCP) helium near the solidifying density of bulk solid helium in three dimensions.

We study the following two dimensional helium systems. First, there is the pure solid, where there is exactly one 4He atom per lattice site. Next, there is the substitutional solid, where a single 4He atom has been replaced with a 3He isotope. Finally, there is the interstitial solid, where either a 3He or a 4He atom has been added to the pure solid.

All of our simulations begin from an ordered state, with the atoms located at lattice positions and no variation of atomic position in imaginary time. In the case of the interstitial solids, the added 3He or 4He atom is placed precisely at an interstitial site, equidistant from the three nearest lattice sites. The system is then equilibrated through the use of the worm algorithm for path integral Monte Carlo simulations in the canonical ensemble. About 500

Monte Carlo (MC) steps are required to thermalize the systems considered. Each MC step consists of a call to the worm algorithm, as well as a “wiggle” move for every atom at each time slice. Once thermalization has been achieved, 2500 iterations of 500 MC steps each were run to collect statistics for the observables presented here.

3.1.1 Distribution functions

4 How does the impurity atom aﬀect the pair distribution function g44 of the He atoms of the underlying solid? We ﬁnd that when a substitutional impurity is introduced it becomes localized and occupies an ideal lattice position with its own zero-point motion determined by its mass. In Figure 3.1 we present the calculated radial distribution function,

4 g44(r), of pairs of He atoms for the pure, substitutional and interstitial solids. Notice that within the accuracy of this graph we cannot discern the diﬀerence in the g44 distribution function for the cases of the pure solid and the substitutional solid. When an interstitial

3He impurity is present in the 4He solid, we ﬁnd that the impurity becomes localized at a substitutional position, thereby promoting the extra 4He atom to the interstitial band.

This is demonstrated by the snapshot space-time conﬁguration shown in Figure 3.2. Notice that while the initial conﬁguration places the 3He impurity interstitially, in the thermalized

29 2.0

Pure or Substitutional Solid 3 4 1.6 Interstitial He or He

1.2 (r) 44 g 0.8

0.4

0.0 0 2 4 6 8 10 12 14 r [Å]

Figure 3.1: The radial form of the pair distribution function for pairs of 4He 4 atoms, g44(r). The organizational structure of the He atoms does not change in the presence of a substitutional 3He impurity. However, when an interstitial 3He 4 impurity or an interstitial He atom is present, we can see that g44(r) becomes less peaked at the nearest-neighbor distances, and subsequently enhanced in the interstitial regions.

30 25

0 0 5 10 15 20 25

Figure 3.2: Snapshot of a space-time conﬁguration of the 3He interstitial solid after thermalization. Red crosses represent the 4He atoms at each imaginary time slice, while blue circles represent the 3He impurity. Although initially placed interstitially, the 3He impurity has since relaxed onto a lattice site, thereby pushing a 4He atom into the interstitial region.

conﬁguration shown in Figure 3.2 the 3He impurity has become substitutional by promoting an interstitial 4He atom. The interstitial 4He atom has become entangled with other 4He atoms in the lattice through bosonic permutations, and the conﬁguration has non-zero winding number in the y-direction. As a consequence of this fact, g44(r) in Figure 3.1 is less peaked at the lattice positions.

In Figure 3.3 we present contour plots of the distribution function g44(x, y) for pairs of 4He atoms in the pure solid (top) and the substitutional solid (bottom). This function is nearly identical for these two solids, which implies that the introduced substitutional

3He impurity becomes localized and only aﬀects its neighboring atoms. In the case of an

3 interstitial He impurity the difference in the g44 distribution function, as discussed above and shown in Figure 3.1 and Figure 3.4, is significant because the added impurity takes the position of a 4He atom and thus there is an extra 4He atom that necessarily becomes interstitial. In Figure 3.4 we plot δg44, the difference between g44 of the pure solid and the solid with a single interstitial impurity. Notice that the extra atom is truly interstitial

31 12

0 0 2 4 6 8 10 12

Figure 3.3: Contour plots of the pair distribution function for pairs of 4He atoms, g44(x, y). Top: the pure solid. Bottom: the substitutional solid.

32 12 0.4 0.2 0 10 -0.2 8 0 2 6 4 6 4 8 2 10 12 0

Figure 3.4: Diﬀerence between the pair distribution function of the pure solid and 3 the interstitial He solid. We have subtracted g44(x, y) for the interstitial solid from g44(x, y) for the pure solid to get δg44(x, y).

since the g44 is reduced by an amount in the neighborhood of the ideal lattice positions and enhanced in the interstitial space by the same amount. It was veriﬁed through integration in the enhanced regions (or the reduced regions) ﬁnding exactly one extra 4He atom in the interstitial regions.

Our ﬁnding that the interstitial impurity becomes localized at regular lattice sites can be further illustrated by comparing the contour plots of g44 and g34 for the case where we have a 4He solid with an interstitial 3He impurity. In the top panel of Figure 3.5, we

4 present the contour plot of the distribution function g44 for the case of a He solid with an interstitial impurity. In the lower panel of Figure 3.5 is the distribution function g34 for pairs consisting of the impurity atom and one 4He atom. Because the contour plots for both g34 and g44 are identical in shape and in form, we may surmise that the impurity atoms are located at lattice sites. In the interstitial solid, most of the 4He atoms are at lattice positions. At most, only 1 in 56 (for the interstitial 3He solid) or 1 in 57 (for the interstitial

4 He solid) are in the interstitial space. Thus, g44 shows what it looks like to have most pairs of atoms with both atoms on lattice sites ( N 2 such pairs), and a small number of ∼

33 12

0 0 2 4 6 8 10 12

Figure 3.5: Contour plots of the pair distribution function (PDF) in the 3He 4 interstitial solid. Top: g44, the PDF for pairs of He atoms. Bottom: g34, the PDF for pairs consisting of the 3He impurity and a 4He atom.

34 0

-2 3 Substitutional He 3 Interstitial He -4

-6

-8 Potential Energy

-10

-12

-14 0 100 200 300 400 500 600 700 Iteration

Figure 3.6: Potential energy of a 3He atom placed either substitutionally (blue line) or interstitially (green line) into triangular solid 4He. After a brief relaxation, both energy values remain close except for occasional “blips” in the potential energy of the initially interstitial 3He atom. pairs with one atom located interstitially and the other atom at a lattice position ( N ∼ 3 such pairs). Because g34 for the interstitial He solid is nearly identical to g44 for either interstitial solid, it follows that most 3He-4He atomic pairs consist of both atoms at lattice positions. This is only possible if the 3He “interstitial” atom is actually at a lattice site.

3.1.2 Energy

If a 3He atom initially placed in the interstitial region of a triangular solid of 4He atoms relaxes onto a lattice site by promotion of a 4He atom to the interstitial space, this should be seen in the energy values of the simulated atoms. In Figure 3.6 we show the potential energy of a 3He atom in the substitutional and interstitial 3He solids. A short relaxation time can be seen for the interstitial solid, as the 3He atom relaxes onto the lattice. After that, the potential energy of a 3He atom in both systems is almost the same. After 600 iterations, a small bump is seen in the energy of the (initially) interstitial 3He atom. A

35 25

0 0 5 10 15 20 25

Figure 3.7: Snapshot of a space-time conﬁguration of the 3He interstitial solid in a “blip” of elevated potential energy that appears after thermalization. Red crosses represent the 4He atoms at each imaginary time slice, while blue circles represent the 3He impurity. The 3He atom can be seen to be at a region of local disorder.

36 3 Table 3.1: Energy of the various systems studied. N3 is the number of He atoms 4 in the system, and N4 is the number of He atoms in the system.

4 3 3 4 E( He)/N4 E( He)/N3 E( He+ He) PURE 4He 2.785 0.006 155.9 0.3 ± ± SUB. 3He 2.803 0.006 6.43 0.05 160.6 0.3 ± ± ± INT. 3He 3.640 0.008 7.16 0.11 211.0 0.4 ± ± ± INT. 4He 3.618 0.007 206.2 0.4 ± ± snapshot of the atomic conﬁguration at this elevated energy value is shown in Figure 3.7.

The 3He atom is no longer at an equilibrium lattice position, but rather at what appears to be a possible edge dislocation. This is not entirely unexpected, as a 3He atom in solid 4He exhibits a high rate of diﬀusion. Such “blips” in the energy of the 3He in the interstitial solid occur occasionally throughout our simulation, but account for no more than 5% of conﬁgurations.

The formation energy of an interstitial 4He atom is determined by taking the difference in total energy between the interstitial solid and the pure solid. Using the values listed in Table 3.1, we find ∆E = 50.27 0.54 K. If the interstitial 3He solid actually contains ± a substitutional 3He atom, as is suggested by the comparison of 3He energy values in the substitutional and interstitial solids shown in Figure 3.6, and instead a 4He interstitial is present, then the formation energy of an interstitial 4He atom can also be found using the substitutional and interstitial 3He solids. Again using the values in Table 3.1, we find

′ ∆E = 50.41 0.55 K, well within the error bars of our previous value. This is further ± evidence that an initial interstitial 3He atom will relax onto the lattice by promoting a 4He atom to the interstitial band.

3.1.3 Superﬂuid order

We have seen that the substitutional solid looks remarkably like the pure solid, especially from the point of view of a 4He atom. Additionally, the 3He interstitial solid looks very

37 1e+00

Pure 4He 1e−01 3 Substitutional He 3 Interstitial He [fs=0.0208(71)] Interstitial 4He [f =0.0108(56)] 1e−02 s

1e−03 n(r)

1e−04

1e−05

1e−06 0 2 4 6 8 10 12 14 r [Å]

Figure 3.8: The one-body density matrix, n(r). Although no difference is observed between the pure and substitutional solids, both interstitial solids clearly show an enhancement of n(r) at large distances. The interstitial solids also display superfluid behavior, which was not observed in either the pure or substitutional solids. The superfluid fraction (fs = ρs/ρ) of both interstitial solids is shown, with the digits in parentheses indicating the uncertainty of the trailing digits of fs.

much like the 4He interstitial solid, at least from the perspective of the constituent 4He atoms. This is because the 3He impurity in both the substitutional and interstitial cases occupies a lattice site without too great a disturbance to its neighbors.

Now let us look at the superﬂuid or oﬀ-diagonal properties of the systems considered. In

Figure 3.8 we compare the one-body density matrix (OBDM), n(r), for the pure, substitutional, and both the interstitial 3He and interstitial 4He solids. Notice that the substitutional

3He impurity and the pure solid have similar one-body density matrices. In both cases, n(r) decays exponentially across all distances in the simulation cell. In contrast to this behavior, the interstitial 3He solid and the interstitial 4He solid have one-body density matrices that are significantly enhanced at long distances, indicating the presence of off-diagonal long- range order. This result agrees with the fact that winding numbers (and hence superflow) are observed in the interstitial solids, but not in the pure or substitutional solids. This is to be expected, as we have just shown that, from the point of view of a 4He atom, there is little

38 Table 3.2: Superﬂuid fractions for the various cases of solid helium considered. Results are shown from calculations of 5,000 iterations in trivial parallel fashion after extended thermalization of 13,000 iterations.

Solid Type Temp. ρs/ρ Interstitial 3He 0.5 K 0.074 0.018 ± 1.0 K 0.021 0.009 ± Interstitial 4He 0.5 K 0.048 0.014 ± 1.0 K 0.024 0.010 ± difference between the pure and substitutional solids, and also both interstitial solids are quite similar from this viewpoint. The superfluid and off-diagonal properties of the system will be entirely determined by the behavior of the 4He atoms; it is their permutation cycles that make superfluidity and off-diagonal long range order possible.

In Table 3.2 we list the superﬂuid fraction of the interstitial solids. While this is small

(only one or two percent of the system is superﬂuid), it is statistically signiﬁcant. Also, at

1 K, we are above the superfluid transition in two dimensions, so a greater superfluid fraction would be expected as we lowered the system temperature. However, this is computationally quite expensive, but this result for our finite size system still suggests that, at least locally, an interstitial defect or impurity can give rise to supersolid behavior. No superfluid fraction was observed for either the pure or the substitutional solid systems. Notice that these superfluid fractions are very high considering that the simulation was carried out at 1 K.

The reason for these high superfluid fractions is finite size effects. These results for the superfluid fraction are presented in order to make the case that an interstitial 3He impurity has a very similar effect on the superfluid fraction and OBDM as an interstitial 4He atom.

3.2 Results for three dimensions

Here we present the results of our simulational study of bulk solid helium in three dimensions. We simulate 180 4He atoms in an HCP solid lattice. The lattice density is kept

39 2.0 0.03

Pure 4He / Substitutional 3He Interstitial 3He / 4He 0.02 1.6 4 3 Pure He − Interstitial He

0.01 1.2 (r) (r)

0.00 44 44 g g 0.8 ∆ −0.01

0.4 −0.02

0.0 −0.03 0 2 4 6 8 10 r [Å]

Figure 3.9: The radial form of the pair distribution function for pairs of 4He atoms, g44(r). Although g44(r) looks the same for all systems, subtracting g44(r) for the interstitial solid from g44(r) for the pure solid to get ∆g44(r), we can see a clear diﬀerence. Note that the axis for ∆g44(r) is shown on the right-hand side of the ﬁgure.

ﬁxed at 0.0286 A˚−3. As in two dimensions, we study the following systems. The pure solid, with one 4He atom per lattice site. The substitutional solid, where a single 4He atom has been replaced by the lighter isotope, 3He. And the interstitial solid, where an extra helium atom has been placed interstitially, of either the 3He or 4He variety. For each of these systems, data has been collected over roughly 1000 iterations, with each iteration consisting of 500 MC steps, as before.

In Figure 3.9 we show the radial distribution function, g44(r), for pairs of atoms. Due to the small fraction of defects (one defect or impurity atom for 180 atoms at lattice positions), there is no visually discernible diﬀerence between g44(r) for any of the four systems considered. However, we have also included a plot of the diﬀerence between g44(r) for the pure

3 solid and g44(r) for the solid with an interstitial He impurity. As in the two dimensional solid, g44(r) for the interstitial solid is less peaked at the lattice distances than g44(r) for the pure solid. We have veriﬁed numerically that g44(r) is nearly identical for the pure and substitutional solids. Also, both interstitial solids have the same g44(r). We conclude that

40 Table 3.3: Energy (in Kelvin) of the various systems studied in three dimensions. 3 4 N3 is the number of He atoms in the system, and N4 is the number of He atoms in the system.

4 3 3 4 E( He)/N4 E( He)/N3 E( He+ He) PURE 4He -5.209 0.005 -937.6 0.8 ± ± SUB. 3He 0.52 0.07 -5.196 0.005 -929.5 0.9 ± ± ± INT. 3He 0.69 0.07 -5.034 0.005 -905.4 0.9 ± ± ± INT. 4He -5.055 0.005 -915.0 0.9 ± ±

Table 3.4: Superﬂuid fractions for the various cases of solid helium considered.

Solid Type Temp. ρs/ρ Interstitial 3He 1.0 K 0.005 0.003 ± Interstitial 4He 1.0 K 0.010 0.005 ± once again the substitutional 3He impurity sits neatly at a lattice site, while an interstitial

3He impurity quickly relaxes to a lattice position through the promotion of a 4He atom into the interstitial region.

In Table 3.3 we show the energy values of the systems considered. Using the difference in energy between the interstitial 4He solid and the pure 4He solid, we find a formation energy for a 4He interstitial of ∆E = 22.4 1.3 K. Using the energy difference between the ± ′ interstitial 3He solid and the substitutional 3He solid, we find ∆E = 24.1 1.2 K. Both ± values are in statistical agreement with one another, as is expected if the interstitial 3He solid is composed of a substitutional 3He atom and an interstitial 4He atom. As in the two dimensional case, after a brief thermalization, the initially interstitial 3He atom has relaxed onto the lattice. However, as shown in Figure 3.10, no “blips” in the potential energy of the

3He atom in the interstitial solid are seen. This may be due to the fact that dislocation pairs can be freely created in the two dimensional system, and in fact prevent true long-range translational order from occurring at non-zero temperatures.

41 -18

-20 3 Substitutional He 3 Interstitial He -22

-24

-26 Potential Energy

-28

-30

-32 0 100 200 300 400 500 600 700 Iteration

Figure 3.10: Potential energy of a 3He atom placed either substitutionally (blue line) or interstitially (green line) into HCP solid 4He. After a brief relaxation, the two energy values are virtually indistinguishable.

1e+00

Pure 4He 1e−01 Substitutional 3He Interstitial 3He Interstitial 4He 1e−02 n(r) 1e−03

1e−04

1e−05 0 2 4 6 8 10 r [Å]

Figure 3.11: The one-body density matrix, n(r).

42 The one-body density matrix (OBDM) is shown in Figure 3.11. Notice that, as in the

OBDM for two-dimensional helium, both the pure and substitutional solids show exponential decay of n(r). This indicates a lack of off-diagonal long-range order, which is corrobo- rated by an absence of winding numbers in our simulations. In contrast, both interstitial solids show either an evening out, or, in the case of the 4He interstitial, an enhancement of n(r) at long distances. While this result is at odds with our interpretation that an interstitial 3He atom quickly relaxes by promotion of a 4He atom to the interstitial space, we believe that it is an anomaly due to insufficient Monte Carlo observation time. In any case, both interstitial solids show a nonzero superfluid component, as displayed in Table 3.4.

3.3 Discussion

The main conclusion of this chapter is that added interstitial impurities become substitutional by creating interstitial 4He atoms, and this gives rise to a non-zero superfluid response and significant enhancement of the OBDM at long-distances. Let us point out that while it may be tempting to explain the relaxation of the 3He impurity onto the lattice as a result of its greater zero-point motion (it is the lighter of the two helium isotopes, after all), we have simulated some test systems using heavier (fictional) isotopes, and the result remains the same. This points to an explanation in terms of the bosonic permutations that an interstitial 4He atom is capable of generating.

Our results suggest that if interstitial 3He atoms are present in pure solid 4He, they will become localized through the promotion of 4He atoms to the interstitial band, and such a solid could support superﬂuid order. It is not clear that such interstitial defects exist in the 4He solid caused by 3He or other impurities. This is an issue which could depend on the process of the crystal growth. Although solid 3He-4He mixtures are known to phase-

3 separate at low temperatures [68, 69, 70], for dilute concentrations X3 of He atoms in solid

4 He (X3 < 0.1%) the temperature of phase separation is less than 150 mK [71], below the observed onset of supersolid behavior [1, 2].

43 The superfluid response, which was calculated at 1 K, is very large considering the fact that the calculation was done at such a high temperature. This is a finite-size effect but at a much lower temperature the superfluid response is expected to be greater. A calculation of the superfluid density at a significantly lower temperature requires much larger computational time scales in order to be able to accurately sample it. The zero temperature condensate fraction obtained as the asymptotic value (infinite distance value) of the off-diagonal OBDM at zero temperature, is much smaller by an order of magnitude or more. Therefore, there seems to be a large factor relating the superfluid response to the actual condensate fraction.

44 CHAPTER 4

MELTING TRANSITION OF LENNARD-JONES

PARTICLES IN TWO DIMENSIONS

Before proceeding with a quantum simulation study of the melting transition of helium in two dimensions, it is beneficial to determine the nature of the two-dimensional melting transition for a congruent classical system. The reason for this is twofold. First, the melting transition is not a quantum phase transition, so there is reason to believe that results for a the melting transition of a classical system should in principle be applicable to the melting transition of a quantum system. Second, because we are interested in the KTHNY melting scenario for which very strong finite-size effects are expected, study of a classical system will allow for much larger system sizes, due to the extensive computational power required to study quantum statistical systems. The reduced computational requirements of a classical system of particles also allows for more extensive data collection, reducing statistical errors.

The Lennard-Jones potential is a natural choice for the classical system. It is qualitatively very similar to more accurate interatomic helium potentials (such as the Aziz [58] potential), and has even been applied to the study of the ground-state properties of liquid helium in

3D [72].

In addition being a reasonable approximation of the exact helium system, the Lennard-

Jones potential still retains suﬃcient simplicity as to be studied widely. There have been many computational studies of the classical Lennard-Jones system in two dimensions [64].

Even though most of these studies have focused on the determination of the nature of

45 the melting transition, a general consensus has not been established. Some studies have favored a ﬁrst-order transition from solid to liquid [73, 74, 75, 76], as predicted by the grain- boundary theory of Chui [77], while other studies [78, 79, 80, 81] have leaned toward the so-called KTHNY theory of Kosterlitz and Thouless [8], Halperin and Nelson [39, 9], and

Young [10], which predicts that melting in two dimensions occurs via two continuous phase transitions, first from solid to hexatic fluid, and then from hexatic fluid to isotropic fluid.

In an eﬀort to understand the proper scaling behavior expected from the KTHNY theory of melting, we have implemented a classical Monte Carlo study of the two-dimensional melting transition for Lennard-Jones particles.

The Lennard-Jones potential, VLJ , is very easy to compute. For two particles separated by a distance r, σ 12 σ 6 VLJ (r) = 4² . (4.1) "µ r ¶ − µ r ¶ # An attractive inverse sixth power tail is combined with a repulsive inverse twelfth power hard core, with only two parameters: ², the potential well depth, and σ, the hard-core distance. In our calculations, we have set these parameters to unity, so that energy is measured in units of ² and distance is measured in units of σ. Additionally, the Boltzmann constant is set to unity, kB = 1, so that temperature may also be expressed in units of ².

4.1 Results

We have performed our calculations on the High Performance Cluster at Florida State

University (http://www.hpc.fsu.edu), which contains 2500 compute nodes. The proces- ∼ sors on these nodes range in speed from 2.3 GHz to 2.8 GHz, and it takes about 34 hours to perform 1,000,000 Monte Carlo sweeps for N=25600 particles, including calculating observables every 100 Monte Carlo sweeps.

To take advantage of our computational resources, we utilized a trivially parallel Monte

Carlo implementation of 100 “threads”, each with a unique random number seed and initial conﬁguration. Simulations begin from an initial near-ordered conﬁguration (particles are

46 placed in a triangular lattice, with 5% lattice spacing random ﬂuctuations). Statistics for thermodynamic variables are collected by generating averages on each of the 100 parallel threads, then using the central limit theorem, as we have 100 independent means.

Although in our preliminary studies we have computed thermodynamic quantities for a range of densities and temperatures, the eﬀects of critical slowing down near the melting transition and our desire to study the largest possible systems have led us to focus on a single density, 0.873 (all densities in this chapter are in units of particles per σ2). This density was chosen for several reasons. This is a density that could be readily compared to prior numerical simulations of Lennard-Jones melting [79]. Also, we wanted a density that is relatively low, but large enough to avoid the solid-vapor coexistence phase at low temperatures. Strictly speaking, there is a solid phase in the zero temperature limit only at densities of 0.9165 (the density at which the spacing of the triangular lattice is the same ≈ as the position of the Lennard-Jones potential minimum) and above. Below this density there is a solid-vapor coexistence phase. However, the triple point density is roughly 0.82, so higher densities will in general become solid before the onset of melting occurs [64].

We have collected data for systems of 1600, 6400, and 25600 particles over a wide temperature range at a density of 0.873. Additionally, we have simulated a system of

102400 particles for two temperatures at the same density in order to verify our results for the smaller system sizes. To accommodate the expected low temperature triangular solid phase, a periodic simulation cell of proportion 2 : √3 is used. We have computed the energy, the pressure (using the virial estimator), both the translational and orientational order parameters, and their moments, the fraction of defects, and several correlation and distribution functions.

4.1.1 Defect excitation energy

In the KTHNY theory, dislocations are bound at low temperatures, and there is a defect core energy associated with their creation. This leads to an energy gap, and thus using the

−2Ec/kBT Arrhenius law, we expect fd = e , where we have used 2Ec because dislocation

47 Table 4.1: Defect activation energy for various system sizes and temperatures, as computed using the Arrhenius law. The numbers in parentheses are the uncertainty of the trailing digits.

Temperature N=1600 N=6400 N=25600 0.50 1.49946(30) 1.4919(19) 1.4873(14) 0.55 1.50267(85) 1.4884(31) 1.4778(22) 0.60 1.4996(14) 1.4635(43) 1.4252(31) 0.65 1.4872(26) 1.3908(59) 1.3480(29) 0.70 1.4543(30) 1.2795(49) 1.2835(15) 0.75 1.3640(54) 1.2027(19) 1.2390(26)

pairs are the lowest energy excitation (isolated dislocations are forbidden). In Table 4.1 we show the defect activation energy as calculated by the Arhhenius law at low temperatures.

Taking the low temperature limit, we ﬁnd E = 1.49 0.01. c ±

4.1.2 Signatures of a phase transition

According to the KTHNY theory, disclinations remain very tightly bound below Tm.

Above Tm, the disclinations are screened from one another by the presence of free dislocations yet remain bound, albeit by a weaker logarithmic binding [9]. Thus we expect a proliferation of defects to occur around Tm, and to continue growing until somewhere above

Ti, where a saturation should occur. In Figure 4.1 we show the average defect fraction as a function of temperature. At low temperature, there are very few defects, while at high temperature there is a considerable fraction of the system that is defected. In between, there is a region of rapidly increasing defect fraction, from T = 0.8 to T = 1.0. This can be seen quantitatively by calculating the ﬁrst derivative (see Figure 4.2),

df f (T + ∆T ) f (T ∆T ) d = d − d − + O (∆T )2 . (4.2) dT 2∆T ³ ´

Additionally, we can see some size dependence in the region 0.6 < T < 1.0, although this seems to be an issue mostly for comparisons of the smallest system size (N=1600) to the

48 0.25

N=1600 0.2 N=6400 N=25600 N=102400

0.15 d f

0.1

0.05

0 0.5 0.6 0.7 0.8 0.9 1 1.1 T

Figure 4.1: Fraction of defects, fd, as deﬁned by the fraction of non-six-coordinated particles in the Delaunay triangulation, f = 1 N /N. The rapid rise in f from d − 6 d near zero to almost 25% is a possible sign that dislocation and/or disclination unbinding is occurring.

49 0.5

N=1600 0.4 N=6400 N=25600 N=102400

0.3 /dT d df 0.2

0.1

0 0.6 0.8 1 1.2 1.4 1.6 1.8 T

Figure 4.2: Temperature derivative of the defect fraction, as calculated using the ﬁnite diﬀerence method. The rate of increase of the defect fraction as the temperature is raised peaks near T = 0.9.

50 1.6

N=1600 1.4 N=6400 N=25600 N=102400

1.2 v c

0.8

0.6 0.6 0.8 1 1.2 1.4 1.6 1.8 2 T

Figure 4.3: The presence of a peak in the speciﬁc heat is indicative of a phase transition. Interestingly, the peak near T=0.9 appears to lessen in magnitude as the system size is increased.

larger system sizes.

The specific heat per particle at constant volume, cv, can be calculated from the energy fluctuations, 1 E2 E 2 cv = − h2 i , (4.3) N k®BT where E is the total energy of an N particle system. We have calculated the specific heat and show it as a function of temperature in Figure 4.3. One can see a broad peak in the specific heat per particle. According to the KTHNY theory, there should be an essential singularity in the specific heat at both Tm and Ti [9]. However, it is not clear whether this will be visible above background contributions to the specific heat. Either way, the peak in specific heat points to a rearrangement of order in the systems studied.

Also, if we look at the distribution function (Figure 4.4), we see ordering at low tem-

51 5

4 T=0.50 3 g(r) 2

0 0 2 4 6 8 10 12 14 16 18 20 r

3 T=2.00

2 g(r)

0 0 2 4 6 8 10 12 14 16 18 20 r

Figure 4.4: The distribution function shows ordering at low temperatures, as shown above for T=0.50, while at higher temperatures, such as T=2.00 shown above, there is a loss of order over moderate length scales.

52 peratures, and ﬂuid behavior at high temperatures. We could also expect to see solid-like behavior in the structure factor at low temperatures, and liquid-like behavior at high temperatures, but we have not calculated the structure factor at this time.

Overall, it is clear that there is a phase transition occurring, with a disordered ﬂuid state at high temperatures and an ordered state at low temperatures. Between these two regimes there is an intermediate state. Whether this intermediate state is due to phase-separation, and thus the result of a ﬁrst order transition, or is the hexatic phase of KTHNY theory, is the question we would like to answer.

4.1.3 Order parameters

In the top panel of Figure 4.5 we show the second moment of the translational order parameter, Ψ2 . There appears to be a transition from a translationally ordered phase at G~ low temperatures to a disordered phase at higher temperatures. In the ordered phase there is a clear relation between Ψ2 and system size. We will explore this relation in a later G~ section, but for now let us point out that this ﬁnite-size scaling relation begins to break down above T=0.60.

Also shown in Figure 4.5 is the second moment of the bond orientational order parame-

2 ter, Ψ6 (bottom panel). At low temperatures there is substantial bond orientational order. Below T 0.70 there is very little dependence of Ψ2 on system size. As the temperature is ∼ 6 2 increased, Ψ6 begins to show a marked dependence on system size as well as a steep decline in value as we approach the high temperature disordered phase.

The main prediction of Halperin and Nelson [39, 9] and Young [10] is that if two- dimensional melting is the result of dislocation unbinding, as proposed by Kosterlitz and

Thouless [8], then a second unbinding transition (of disclinations) is required to reach an isotropic fluid state. This implies the presence of a novel hexatic fluid phase. In Figure 4.6 we show ΨG~ and Ψ6 on the complex plane for three different temperatures. At T=0.70

(top row), we see a “ring” of values for ΨG~ , while Ψ6 is localized in a small region away from the origin. This is consistent with the presence of long-range bond orientational order

53 0.6

N=1600 0.5 N=6400 N=25600 N=102400

0.4 > 2 G 0.3 Ψ <

0.2

0.1

0 0.5 0.6 0.7 0.8 0.9 1 1.1 T 0.7

N=1600 0.6 N=6400 N=25600 N=102400 0.5

0.4 > 2 6 Ψ < 0.3

0.2

0.1

0 0.5 0.6 0.7 0.8 0.9 1 1.1 T

Figure 4.5: The second moment of the translational (top) and bond orientational (bottom) order parameters.

54 1 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

1 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

1 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

Figure 4.6: Contour plots of ΨG~ (left column) and Ψ6 (right column) on the complex plane for T=0.70 (top row), T=0.90 (middle row), and T=1.10 (bottom row).

55 10

T=0.70 8 T=0.90 T=1.10

6 (r) 77 g 4

0 0 2 4 6 8 10 r

Figure 4.7: The pair distribution function for 7-coordinated particles, g77(r). The peak for T=0.70 extends to 50. ∼

( Ψ > 0), while the ring of Ψ values is expected for quasi-long-range translational |h 6i| G~ order. At T=0.70 (middle row), ΨG~ is now clustered about the origin, indicating a lack of translational order. Interestingly, Ψ6 now shows a ring of values about the origin, indicating quasi-long-range order. This is exactly what is expected of the hexatic ﬂuid phase. Finally, at T=1.10 (bottom row) we see that both order parameters are distributed about the origin, indicating an isotropic ﬂuid phase of no order.

In Figure 4.7 we show the pair distribution function for pairs of 7-coordinated particles, g77(r). A sharp peak is observed at low temperatures (T=0.70), indicating that dislocations are tightly bound. This is demonstrated in Figure 4.8, where we see that defects occur in quadruplets consisting of two 5-coordinated and two 7-coordinated particles. At higher temperatures (T=0.90 and T=1.10) in Figure 4.7, the peak in g77(r) is greatly diminished, and disclinations become ﬁrst weakly bound (T=0.90) and then completely unbound (T=1.10).

56 35

0 0 5 10 15 20 25 30 35 40 45

Figure 4.8: The Delaunay triangulation for N = 1600 particles at T=0.70. Defects are shown in red.

57 35

0 0 5 10 15 20 25 30 35 40 45

Figure 4.9: The Delaunay triangulation for N = 1600 particles at T=0.90. Defects are shown in red.

58 35

0 0 5 10 15 20 25 30 35 40 45

Figure 4.10: The Delaunay triangulation for N = 1600 particles at T=1.10. De- fects are shown in red.

59 25

T=0.70 20 T=0.90 T=1.10

15 (r) 57 g 10

0 0 2 4 6 8 10 r

Figure 4.11: The pair distribution function for pairs consisting of one 5-coordinated particle and one 7-coordinated particle, g57(r). The peak for T=0.70 extends to 150. ∼

This eﬀect can also be seen in Figure 4.9 and Figure 4.10, where isolated dislocations are seen.

The pair distribution function for disclinations, g57(r), is shown in Figure 4.11. While the sharp peak at low (T=0.70) and intermediate (T=0.90) temperature is expected, the peak at T=1.10, while quite lower, is still very substantial. This indicates that disclinations have not become unbound, and indeed it is difficult to find isolated disclinations in the snapshot configurations presented in Figure 4.10. When isolated disclinations do occur, they are still next-nearest neighbors with at least one other disclination of opposite charge.

60 4.1.4 Correlation functions

We have used a particle-centric definition of the bond orientational correlation function, so in our calculations of C6(r) there will be an influence from g(r), the pair distribution function. In the limit of perfect bond orientational ordering (ψ6 = 1 everywhere), C6(r) and g(r) will be equivalent. We fit the g(r) portion of C6(r) using a damped oscillator. The periodic form is captured by using sin(kr + δ), where k is expected to be near the first reciprocal lattice vector in magnitude ( 6) and δ is just a phase-shift parameter. The size ∼ of the oscillations are expected to decay exponentially in the fluid phase, and algebraically in the hexatic phase, so we add also a power law, ending up with a term sin(kr+δ)r−ηe−r/ξ.

The asymptotic form of C (r) is exp( r/ξ ) [9]. At shorter distances, however, a 6 ∼ − 6 power law decay comes into play, such that as ξ6 diverges as Ti is approached from above, we ﬁnd C (r) r−η6 at T and below, where η (T ) = 1/4. Additionally, we observe 6 ∼ i 6 i oscillations in C6(r) that we believe are due to the contribution of g(r); these oscillations seem to decay with an exponential envelope. Thus, we propose the following ﬁtting form for the bond orientational correlation function for distances r much less than the system size L:

−η6 −r/ξ6 −η −r/ξ C6(r) = Ar e + B sin(kr + δ)r e . (4.4)

We observe an upturn in C6(r) as r approaches L/2. At temperatures closer to melting (larger correlation lengths), the upturn occurs further from L/2. Let us assume a periodic form for the correlation function:

C (r) = A r−η6 e−r/ξ6 + (L r)−η6 e−(L−r)/ξ6 . (4.5) 6 − h i

Neglecting the power-law term, the upturn is expected to occur when the L r terms are − a signiﬁcant fraction of the r terms. Thus,

1 ξ R = L 6 ln(x), (4.6) 2 − 2

61 0.055

0.05

0.045 (r) 6 C 0.04

0.035

0.03

32 34 36 38 40 42 44 46 48 r

Figure 4.12: Example of ﬁtting the bond orientational correlation function, C6, to the form shown in Equation 4.4. The data is for N = 25600 particles at T = 0.97. The critical exponents are ﬁxed at their maximum values, η = 0.33 and η6 = 0.25. The extracted correlation lengths are ξ = 7.40 0.19 and ξ = 32.6 0.7. ± 6 ±

where R is the distance at which the (L r) terms are a fraction x of the r terms. Using − x = 0.05, or 5%, this leads to L 3 R = ξ . (4.7) 2 − 2 6

An example ﬁt is shown in Figure 4.12.

In Figure 4.13 we show ξ6(T ) as determined by ﬁtting C6(r) in the range ξ6 < r <

R. These values were ﬁt to the KTHNY form of the expected divergence of ξ6 as Ti is approached from above: ξ (T ) = A exp(b/tν), where t = (T T )/T and ν = 1/2. This ﬁt 6 − i i gives a value for Ti near 0.89.

62 40

A=0.503765 b=1.27619 30 Ti=0.887682 6

ξ 20

0 1 1.2 1.4 1.6 1.8 2 T

Figure 4.13: Correlation lengths of the bond orientational order parameter as determined by ﬁtting the bond orientational correlation function to the form mentioned in the text.

63 4.1.5 Distribution functions

The pair distribution function tells us the likelihood of ﬁnding two particles at some distance (and possibly orientation) from one another. We have have calculated the pair distribution function for particles, as well as distribution functions for the defects deﬁned by the Delaunay triangulation.

Let us look closely at the deﬁnition of the translational correlation function,

C (~r) =< ψ∗ (~r)ψ (~0) > < ψ∗ (~r) >< ψ (~0) > . (4.8) G~ G~ G~ − G~ G~

For ~r diﬀerent from zero, this can be shown to be proportional to the pair distribution function, ~ C (~r = 0) = e−iG·~r [g(~r) 1] . (4.9) G~ 6 −

The reason we have to avoid the ~r = 0 term is that while the distribution function is identically zero when ~r = 0, the correlation function is not. Rearranging the terms, we have an expression for the pair distribution function in terms of the translational correlation function, ~ g(~r = 0) = 1 + eiG·~rC (~r). (4.10) 6 G~

Now let us integrate over the angular terms to get the radial distribution function,

1 2π g(r) = g(~r)dφ. (4.11) 2π Z0

If CG~ (~r) has some angular dependence, this integration can be very diﬃcult. However, in the disordered phase, there should be no angular dependence for r >> ξ6, as ξ6 is the correlation length of bond orientational (that is, angular) order. In this case, CG~ (~r) should be purely radial, and we have

2π 1 iG~ ·~r g(r) = 1 + e CG~ (r) dφ. (4.12) 2π 0 Z h i

64 iG~ ·~r The integration of e will give us a zeroth-order Bessel function of the ﬁrst kind, J0(Gr), and using the KTHNY form of the translational correlation function, C (r) exp( r/ξ)r−η, G~ ∼ − we wind up with the following form for the radial pair distribution function (in the high temperature limit):

−r/ξ −η g(r) = 1 + AJ0(Gr)e r , (4.13) where A is some amplitude.

The G~ that we use here is the same as in the definition of the translational order parameter, namely we use the first reciprocal lattice vector of the idealized triangular lattice that is commensurate with our simulation cell. For the density considered (ρσ2 = 0.873), this means G 6.3σ−2. Since we are already requiring r >> ξ , we can use the asymptotic ≈ 6 expansion of J , namely J (x) 2/πx cos(x π/4). Thus in practice we fit g(r) in the 0 0 ∼ − disordered phase to the following form,p

g(r) = 1 + A cos(kr + δ)e−r/ξr−η−1/2. (4.14)

An example ﬁt is shown in Figure 4.14.

In Figure 4.15 we show the correlation length of translational order as calculated by

ﬁtting g(r) to the above form. Results are shown for the N=25600 and N=102400 particle systems. Clearly, ξ remains ﬁnite even as the orientational correlation length diverges.

However, there are some discrepancies in our values of ξ. At T=0.92, the value of ξ extracted from the N=25600 particle system does not agree with the value for N=102400 particles.

Looking at the pair distribution function, g(r), for these two systems in Figure 4.16, we can see that their values do not agree in the range over which we have ﬁt g(r) to the form in

Equation 4.14. Some of this difference may be attributable to finite size effects. Additionally, there is also the possibility that the N=102400 particle system has not fully thermalized.

While we have tried to ensure that the data for this largest system is completely thermalized, it can be very diﬃcult to distinguish between stable and metastable states. In either case, we can not self-consistently ﬁt all the data to the KTHNY form, ξ = A exp(b/tν), so instead

65 1.02

1.01

1 g(r)

0.99

0.98 16 18 20 22 24 26 28 30 32 r

Figure 4.14: Example of ﬁtting the pair distribution function, g(r), to the form shown in Equation 4.14. The data is for N = 25600 particles at T = 0.97. The critical exponent η is ﬁxed at its maximum value, 1/3. The extracted correlation length is ξ = 8.09 0.04. ±

66 60

A=0.0017 b=6.9 50 N=25600 t =0.62 m N=102400

(T) 30 ξ

0 0.8 0.85 0.9 0.95 1 1.05 1.1 T

Figure 4.15: Correlation lengths of the translational order parameter as determined by fitting the pair distribution function to the form mentioned in the text. The range of the fit is from 2ξ to 4ξ, with η fixed at its maximum value of 1/3. At T=0.80, the fitting range is from ξ to 2ξ.

67 1.006 N=102400 N=25600 1.004

1.002

1.000 g(r)

0.998

0.996

0.994

36 38 40 42 44 46 48 50 52 54 56 r

Figure 4.16: Pair distribution function, g(r), for the N=25600 and N=102400 particle systems at T = 0.92 in the ﬁtting range used to determine ξ.

68 we have made the ﬁt for only the N=25600 data. The result of a ﬁt with Tm = 0.62 and

ν = 0.36963 is shown in Figure 4.15, along with the fit parameters A and b. If Tm is allowed to vary as well, the result of the non-linear least squares fitting we utilize is to reduce the value of Tm significantly (to at least 0.45), while the parameter A gets smaller and b gets larger. The value of χ2, however, is only reduced by 10%. ∼ Several experimental investigations [82, 83] have used the decay of the envelope of g(r) to extract ξ. The resulting values of ξ appear not to diverge across the melting transition, so perhaps there is some shortfall in using g(r) to get ξ at low temperature. For instance,

Murray and Van Winkle observe a ﬁnite peak in ξ, while for ξ6 a divergence is seen to occur [82].

4.1.6 Critical exponents

In the topological solid phase, the scaling form for the second moment of the translational order parameter is < Ψ2 > L−η, where L is the (linear) system size and η is a critical G~ ∼ exponent. In the hexatic ﬂuid phase, a similar relation holds for bond orientational order,

< Ψ2 > L−η6 . By plotting < Ψ2 > (or < Ψ2 >) and L on a log-log plot, we can ﬁnd η 6 ∼ G~ 6 (or η6).

According to the KTHNY theory of melting, the critical exponents η and η6 will have speciﬁc values at melting. The translational critical exponent is bounded at lower melting

1 1 temperature: 4 < η(Tm) < 3 . Additionally, the bond orientational critical exponent grows from zero at T to 1/4 at T , and is related to the translational correlation length: η (T ) m i 6 ∼ ξ−2(T ) [9].

In Figure 4.17 we show the extracted values of η and η6, the critical exponents of translational and bond orientational order. In both panels, we show our results as red circles. In the top panel, we can see that η crosses the KTHNY melting value in the temperature range 0.6 < T < 0.65. In the bottom panel we show the critical exponent of bond orientational order, η6. This exponent crosses the KTHNY melting value (see dashed line) at a temperature near 0.89, in close agreement with the value for Ti derived from the

69 0.6

0.5

0.4 (T) η 0.3

0.2

0.1 0.5 0.6 0.7 0.8 0.9 1 T

0.5

0.4

0.3 (T) 6 η 0.2

0.1

0 0.5 0.6 0.7 0.8 0.9 1 T

Figure 4.17: Anomalous dimensionality of (top) the translational order parameter and (bottom) the bond orientational order parameter. Our current results are shown in both figures as red circles. Shown for comparison are the results of Wierschem and Manousakis [84] (green squares, bottom figure) and Udink and van der Elsken [79] (blue triangles, both figures). In the top figure, the dashed and dotted lines represent the lower and upper bounds of η at Tm, according to KTHNY theory; in the bottom figure, the dashed line represents the predicted value of η6 at Ti.

70 -0.5

-1 > 2 6

Ψ -1.5 ln <

-2 T=0.80 T=0.84 T=0.92

-2.5 3 3.5 4 4.5 5 5.5 6 ln L

2 Figure 4.18: Scaling of < Ψ6 > with system size L, shown on a logarithmic plot. Results for T=0.84 are shown as blue circles, while data collected at T=0.92 is represented by red squares. In both cases, the data for the three smaller systems was ﬁt to the equation ln < Ψ2 >= η ln L + const, and the result is plotted 6 − 6 as the dotted and dashed lines (the solid line is the KTHNY value of η6 at Ti). 2 In both cases, the value of ln < Ψ6 > of the largest system size (N=102400) is reasonably close to the value expected from scaling.

divergence of the correlation length ξ6.

Also shown in Figure 4.17 are the algebraic exponents reported by Udink and van der

Elsken [79]. In both panels, we can see that their values cross the KTHNY melting zone at higher temperatures than our values. We believe this disagreement may be due to insuﬃcient thermalization time in their study, as this could lead to artiﬁcially low values of the critical exponents.

2 To check that our results are not limited by system size, in Figure 4.18 we plot ln < Ψ6 > by ln L for all system sizes at the two temperatures where we have results for the N=102400

71 system. Results of linear least squares ﬁts to the three smallest system sizes (used to generate the data for Figure 4.17) are shown as a dotted blue line (for data at T=0.80), a dashed red line (for data at T=0.84), and a long-dashed green line (for data at T=0.92).

For the higher temperature, the N=102400 data falls directly on this line, within error bars. At T=0.84, however, the N=102400 data indicates that a smaller value for η6 at this temperature may be necessary. This could either be due to the (presumably) large translational correlation lengths at this temperature, which would invalidate results for small system sizes, or perhaps a very long relaxation time. Either way, from our data it is clear that by T=0.92 the KTHNY value of η6 at Ti has been well passed.

4.2 Discussion

We have shown that several key predictions from the KTHNY theory of two-stage continuous melting are seen in the classical system of Lennard-Jones (LJ) particles in two dimensions. Among these are the behavior of the critical exponents η and η6, the exponential divergence of the correlation length ξ6, and the unbinding of dislocations. In particular, we have shown that the finite-size scaling (FSS) relation for the second moments of the order parameters is a reliable way to determine the critical exponents of the system, and that the extracted values η and η6 are applicable to systems of larger size than used in their determination. Additionally, a fit of the exponential divergence of ξ6 yielded a value of Ti that was consistent with that determined from the KTHNY crossing point of the critical exponent η6, indicating that this latter method should be a reliable way to determine Ti for system sizes too small to extract extensive values of ξ6. While these results are qualitatively in agreement with a past study of the same isochore using FSS of LJ systems, quantitatively our results are in disagreement about location of the melting zone (Udink and van der Elsken find T 0.83 and T 0.93 [79]). We conclude that although the pre- m ≈ i ≈ cise determination of the melting temperatures can be quite difficult, the FSS techniques employed in both studies are useful in differentiating Tm from Ti. In the next chapter, we

72 hope to apply these techniques to the very interesting problem of two-dimensional melting of a quantum system, namely 4He.

73 CHAPTER 5

MELTING AND SUPERFLUIDITY OF SOLID

HELIUM IN TWO DIMENSIONS

In this chapter we present the results of a path integral Monte Carlo study of the melting transition of two-dimensional 4He. The goal of this study is to determine whether the melting transition of a quantum ﬂuid such as helium can be described by the so-called

KTHNY theory of Kosterlitz, Thouless, Halperin, Nelson, and Young [8, 9, 10]. We are also interested in the behavior of the superﬂuid density across the melting transition. In particular, if there is a hexatic phase of two dimensional helium, we would like to know if this phase is superﬂuid.

Path integral Monte Carlo simulations have been carried out across the melting transition. Two isochores (0.0761 A˚−2 and 0.0846 A˚−2) have been studied, as well as an isotherm

(T=2 K). Systems of up to 100 helium atoms have been considered. Due to the computational complexity of simulating quantum systems, larger system sizes were beyond the reach of our calculational power. In an eﬀort to remedy this shortcoming, we have performed a

ﬁnite-size scaling analysis of our data.

We work in units where the Boltzmann constant has been set to unity (kB = 1), so that all energy values can be referred to by their temperature equivalent. All energies and temperatures are given in Kelvin (K), while distances are given in Angstroms (A).˚ Data has been collected over 10,000 iterations. Each iteration consists of 100 calls to the worm algorithm. Each call to the worm algorithm is followed by a “wiggle” move, in which each

74 bead of every 4He atom is updated with a call to the Metropolis algorithm. Additionally, every 100 iterations we have recorded the system conﬁguration. We use these conﬁgurations when calculating the translational and bond orientational order parameters, as well as the defect fraction and any correlation functions presented. M time steps are used to simulate our systems at an inverse temperature β = Mτ. We have chosen τ −1 to be at least 320 K, as we have determined that this is a high enough temperature to ensure that the energy as well as structural properties of the system have converged to their proper values.

5.1 High-density isochore

In this section we present results along the isochore 0.0846 A˚−2. In order to study the

ﬁnite-size scaling of the observables considered, three system sizes have been utilized. Each of these three systems is of the proportion 2 : √3 to accommodate the low-temperature triangular lattice solid phase. The three system sizes are of 36, 64, and 100 helium atoms.

5.1.1 Defect fraction

According to the KTHNY theory of melting [8, 9, 10], dislocations become unbound from one another as the solid melts. Thus, the number of defects should experience a

“jump” at the melting point [63]. This can be seen in Figure 5.1, where we have plotted the defect fraction, fd, as a function of temperature for each system size. Below 2.6 K, the defect fraction for each system size is nearly constant. However, from 3 K to 4 K there is a noticeable increase in fd. This suggests that the low-temperature solid phase is melting into a high-temperature liquid phase. In order to determine if this is occurring through two continuous phase transitions, as is possible in the KTHNY picture, we look next at order parameters for the two types of order present in the low-temperature solid phase.

5.1.2 Order parameters

In KTHNY theory, melting proceeds via two continuous phase transitions. The ﬁrst, occurring at some temperature Tm, is from a topological solid phase to an hexatic ﬂuid

75 0.4

0.35 N=36 N=64 N=100 0.3 d

f 0.25

0.2

0.15

0.1 2 3 4 5 T

Figure 5.1: The defect fraction as a function of temperature. The melting transition appears rather broad, occurring somewhere in the range of 3 K to 4 K, where the defect fraction for the two larger systems exhibits the fastest increase. Spline ﬁts are shown as guides to the eye.

76 phase. Above Tm, all translational ordering is destroyed, while some algebraic bond orientational ordering remains. The second melting transition occurs at some higher temperature,

Ti > Tm, where the hexatic ﬂuid melts into an isotropic ﬂuid, and all remaining orientational order is lost. Thus, we can use the translational order parameter to determine the location of Tm, and the bond orientational order parameter to determine the location of Ti.

In Figure 5.2 we show the second moments of the translational order parameter and the bond orientational order parameter. Both are expected to scale as 1/N in the high temperature limit, and they appear to do so. Notice, however, that the low temperature values are well short of the completely ordered value of 1. This is actually expected for such quantum systems, as even at absolute zero temperature quantum ﬂuctuations will be present and the perfect order of a triangular lattice will never be achieved [49]. At low temperatures there is a transition to an ordered phase. This transition appears to occur somewhere between 3 K and 4 K, with a tendency for larger system sizes to have a lower transition temperature. Comparing the two order parameters shown in Figure 5.2, both transitions appear to occur together. While this is possible within the context of KTHNY theory, this would mean there is no hexatic phase. We will use ﬁnite-size scaling methods later in this chapter to try more carefully to determine whether or not this is the actual case.

5.1.3 Correlation functions

Now let us examine the behavior of the correlation functions across the melting transition. While we have calculated these for all three system sizes, we only show the results for N = 100 atoms. The results for this system size are the most informative, as it is the long-distance behavior of the correlation functions that we are most interested in.

In Figure 5.3, we show the pair distribution function (top) and the translational correlation function (bottom) at 2 K and 5 K. At 2 K there is clearly translational order. While

CG~ (r) appears to approach a constant value, the peaks in g(r) show a slight tendency to decay. In an inﬁnite-sized system, both CG~ (r) and the envelope of g(r) are expected to

77 0.4

N=36 N=64 N=100 0.3 > 2 G 0.2 Ψ <

0.1

0 2 3 4 5 T 0.25

N=36 N=64 0.2 N=100

0.15 > 2 6 Ψ < 0.1

0.05

0 2 3 4 5 T

Figure 5.2: Second moment of the translational (top) and bond orientational (bottom) order parameters. Spline ﬁts are shown as guides to the eye.

78 2

T=2.0 K T=5.0 K

1.5

1 g(r)

0.5

0 0 4 8 12 16 r 1.2

T=2.0 K T=5.0 K

0.8 (r) G C 0.4

0 4 8 12 16 r

Figure 5.3: Top panel: pair distribution function g(r) for N = 100 atoms. Bottom panel: translational correlation function CG~ (r) for the same system.

79 display a power-law (or algebraic) decay in the low-temperature solid phase above absolute zero, but this is not always seen in ﬁnite-sized systems. At 5 K, CG~ (r) quickly decays to zero, with oscillations that arise due to azimuthal averaging. Also, the envelope of g(r) shows clear signs of decay, as expected in the ﬂuid phase.

At low temperatures the bond orientational correlation function, C6(r), approaches a constant value. In contrast, in the high temperature liquid phase, C6(r) decays rapidly to zero. This can be seen in the top panel of Figure 5.4, where we show C6(r) at 2 K and at and 5 K. In the bottom panel, C6(r) is displayed for temperatures between 3 K and

4 K. In this region, it becomes diﬃcult to determine whether C6(r) is approaching some constant value, or if it is slowly decaying to zero. In the hexatic phase, for example, C6(r) is expected to decay algebraically to zero. At any rate, we can clearly see that the system is orientationally ordered at low temperature, but not at high temperature.

5.1.4 Defect distribution functions

Recall that (6 + n)-coordinated atoms represent disclinations of charge n, and that oppositely charged disclinations can bind together to form a dislocation. Additionally, in the low temperature solid phase, dislocations become bound into pairs with opposite Burgers vectors. Thus, the pair distribution function (PDF) between pairs of disclinations of the same charge is in eﬀect the PDF between dislocations. For simplicity, we consider only singly charged disclinations, as these are by far the majority of defects. In the top panel of

Figure 5.5 we plot g77(r), the PDF for pairs of 7-coordinated atoms. At low temperatures, we can see that dislocations are bound, as indicated by the sharp peak in g77(r) at 2 K. By 5 K, this peak is completely gone, as dislocations are unbound in the ﬂuid phase. Also shown is g77(r) at 3.2 K, the temperature where the peak ﬁrst develops. A similar result is obtained for g55(r), the PDF for pairs of 5-coordinated atoms (not shown).

In the bottom panel of Figure 5.5 we show g57(r), the PDF between pairs consisting of one 5-coordinated atom and one 7-coordinated atom. In this case, we are looking at the eﬀective PDF between disclinations. At low temperature (2 K), there is a very high

80 0.6

T=2.0 K 0.5 T=5.0 K

0.4 (r)

6 0.3 C

0.2

0.1

0 0 4 8 12 16 r 0.5

T=3.2 K T=3.4 K 0.4 T=3.6 K T=3.8 K

0.3 (r) 6 C

0.2

0.1

0 0 4 8 12 16 r

Figure 5.4: Bond orientational correlation function C6(r) for N = 100 atoms at extremal (top) and intermediate (bottom) temperatures. Notice that in the top panel, the vertical axis extends to 0.01 in order to show all of C (r) at 5 K. − 6

81 4

T=2.0 K T=3.2 K T=5.0 K 3

(r) 2 77 g

0 0 4 8 12 16 r 8

T=2.0 K T=3.2 K T=5.0 K 6

(r) 4 57 g

0 0 4 8 12 16 r

Figure 5.5: Pair distribution function for pairs of same-charged disclinations g77(r) (top) and opposite-charged disclinations g57(r) (bottom) for N = 100 atoms.

82 peak in g57(r), indicating that disclinations are strongly bound. However, this peak remains quite substantial even at 5 K. This implies that disclinations have not become unbound.

However, the defect fraction is very large at 5 K (fd > 0.35), so this peak is probably due to the fact that, on average, two of every six nearest neighbors are defects. Additionally, the aversion between same-charged disclinations, seen in the top panel of Figure 5.5, means that defected neighbors are most likely of opposite disclinicity.

According to KTHNY theory, oppositely charged disclinations become bound below Ti.

These bound pairs of disclinations form dislocations. Below Tm, these dislocations become bound. Although it should be possible to estimate the values of Tm and Ti by looking at the above PDFs, these values are still affected by strong finite size effects.

5.1.5 Finite size scaling

The second moment of the translational order parameter in the low temperature solid phase obeys the scaling relation < Ψ2 > L−η, where L is the linear size of a finite G~ ∼ simulational system. In the hexatic phase, the second moment of the bond orientational order parameter obeys a similar scaling relation, < Ψ2 > L−η6 . The quantities η and η 6 ∼ 6 are critical exponents of the KTHNY theory of melting, and may be determined from this scaling relation. In particular, if the natural logarithm of < Ψ2 > (or < Ψ2 >) is plotted G~ 6 as a function of the natural logarithm of L, a simple linear least squares fit can be used to determine the slope of the resulting line, which happens to be η (or η ). − − 6 In Figure 5.6, we plot the critical exponents of the translational and bond orientational order parameters. Even at the lowest temperatures we have considered, η is already within the range of KTHNY values at Tm. By 2.4 K the upper bound is exceeded, but it is not until 2.8 K that the error bars of η are also well outside this range. We will discuss how to estimate lower and upper bounds for Tm from this data in the next paragraph, but for now let us note that 2.8 K is also the first temperature for which η6 passes its KTHNY melting value, η6(Ti) = 1/4. In this case, the statistical uncertainty clearly places η6 above the melting value. Because the value of η6 at T=2.6 K is clearly below this limit, we may

83 1

η(T) 0.8 η6(T)

0.6

0.4

0.2

0 2 2.2 2.4 2.6 2.8 3 T

Figure 5.6: Critical exponents of the translational and bond orientational order parameters, η and η6. Also shown are the KTHNY values of the critical exponents at the two melting temperatures. The dotted line is the maximal value of η at Tm (η(Tm) < 1/3), while the dashed line is both the minimal value of η at Tm (η(Tm) > 1/4) as well as the value of the bond orientational critical exponent at Ti, η6(Ti) = 1/4.

84 0.3

0.25 2 <ΨG > 2 <Ψ6 > 0.2

0.15

0.1

0.05

0 0.07 0.075 0.08 0.085 -2 Density in Å

Figure 5.7: Second moment of the translational and bond orientational order parameters along the T = 2 K isotherm.

safely infer that 2.6 K < Ti < 2.8 K. Also shown in Figure 5.6 is a linear least squares fit of η(T ). From the intersection of this line with the lower (η = 1/4) and upper (η = 1/3) bounds from KTHNY theory, we determine a lower bound for Tm of 1.97 K and an upper bound for Tm of 2.27 K, leading to a total estimate of T = 2.12 0.15 K. This value is now used in a linear least squares fit m ± of η6(T ) for the four values shown with T > 2.3 K. The values at T = 2.0 K and T = 2.2 K are neglected due to large finite-size errors for η6 near Tm. The crossing of this line with

η6(Ti) = 1/4 gives an estimate of the hexatic to isotropic ﬂuid transition temperature, T = 2.75 0.10 K. i ±

5.2 Isotherm

In the ρ=0.0846 A˚−2 isochore studied above, the melting transition occurs well above the superﬂuid transition of liquid helium in two dimensions. At the equilibrium density, liquid

85 Table 5.1: Superfluid fractions for the various densities considered for N = 90 atoms at 2 K. Due to the small number of superfluid configurations (a maximum of ten out of 10,000 configurations have non-zero winding numbers), the error bars could be substantially larger than presented.

−2 Density in A˚ ρs/ρ 0.0677 0.00036 0.00016 ± 0.0698 0.00038 0.00015 ± 0.0719 0.00043 0.00017 ± 0.0761 0.00038 0.00017 ± 0.0778 0.00001 0.00001 ± helium becomes superfluid around 0.72 K [42]. Thus, if we are to study the interplay of two dimensional melting with the superfluid transition, we need to find an isochore with a lower melting transition. To this end we study the T = 2 K isotherm for N = 90 atoms in search of a density that is still in the liquid phase, but that will solidify at lower temperatures. Our results are shown in Figure 5.7. Both the translational and the bond orientational order parameters point to a melting transition between ρ = 0.075 A˚−2 and ρ = 0.08 A˚−2.

Curiously, a non-zero superfluid fraction is observed for five of the lowest densities considered, and is reported in Table 5.1. In the thermodynamic limit there is no superfluidity at 2 K, even in the liquid phase. Thus, these values are the result of the finite-size of our simulational system. However, it is still interesting to ask what happens to the superfluid fraction as the temperature is lowered into the solid phase. To do this, we choose a density that is right at the melting transition, and yet still displays non-zero winding numbers. The

ρ = 0.0761 A˚−2 isochore meets both of these criteria.

2 5.3 The ρ = 0.0761 A˚− isochore

In order to observe melting closer to the superﬂuid transition we study the ρ = 0.0761 A˚−2 isochore for N = 36, N = 64, and N = 100 atoms. As the temperature is lowered along this density, the order parameters show a sharp increase at T = 2 K, as shown in Figure 5.8.

86 0.3

0.25 N=36 N=64 N=100 0.2 > 2 G 0.15 Ψ <

0.1

0.05

0 0 1 2 3 4 5 6 T 0.16

N=36 N=64 0.12 N=100 > 2 6 0.08 Ψ <

0.04

0 0 1 2 3 4 5 6 T

Figure 5.8: Second moment of the translational and bond orientational order parameters for the density 0.0761 A˚−2. At T=1 K, ﬁnite-size scaling gives critical exponents of η = 0.33 0.02 and η = 0.11 0.04, putting this temperature just ± 6 ± above the solid-hexatic phase boundary.

87 0.25 0.002

2 <Ψ > 0.2 G ρ /ρ 2 s 0.0015 <Ψ6 >

0.15

0.001

0.1

0.0005 0.05

0 0 0 1 2 3 4 5 6 T

Figure 5.9: Second moment of the translational and bond orientational order parameters (vertical axis on left) for N = 64 atoms on the 0.0761 A˚−2 isochore. Shown on the same graph is the superﬂuid density (vertical axis on right).

Both order parameters appear to change from a liquid-like behavior to a solid-like behavior at the same temperature. However, ﬁnite-size scaling indicates that T=1 K is in the hexatic

ﬂuid phase, with the isotropic ﬂuid forming at some temperature between 1 K and 2 K.

5.3.1 Superﬂuidity in ordered helium

In Figure 5.9, we superimpose the superﬂuid fraction over the order parameters for the

N = 64 atom system to see the behavior of ρs/ρ as the liquid begins to solidify. Beginning at T=2.4 K, while still in the liquid phase, the superﬂuid fraction becomes non-zero. At

T=2.2 K, still in the liquid phase, ρs/ρ has increased in value. Then, at T=2 K, just as the order parameters begin their rapid increase, we see a sudden drop in ρs/ρ. As the temperature is lowered to 1 K, the superﬂuid fraction again begins to increase.

We ﬁnd that only a small percentage of the 10,000 iterations contribute to the superﬂuid fraction, with at most 34 having non-zero winding numbers. One may wonder if these

88 individual superfluid configurations are actually ordered. To answer this, we have looked at a superfluid configuration at T=1 K, and we find that the “snapshot” of atomic positions appears no different than any other configuration. Also, the order parameters are only

10% lower than their average values, and are still much larger than those of typical liquid ∼ configurations. Thus we can be confidant that this finite-size system does posses both solid and superfluid ordering.

Because these results are above the superfluid transition temperature of liquid helium in two dimensions, the superfluid fractions observed here must be due to the finite size of our system. Additionally, no superfluid fraction is obtained for N = 100 atoms. However, the behavior seen in Figure 5.9 is still interesting in its own right, as we see the possibility of the liquid helium phase becoming superfluid before the solid transition occurs. Also of interest is that the superfluid fraction displays re-entrant behavior, as the onset of translational and bond orientational order first dampens the superfluid behavior, before the lower temperatures compensate and the superfluid fraction increases again. If there were to be a hexatic superfluid or even a supersolid phase in two-dimensional helium, this is what we believe the phase transition from the superfluid phase would look like.

5.4 Discussion and outlook

We have simulated the melting transition of two-dimensional helium using path integral

Monte Carlo. Finite-size scaling (FSS) along a high density (ρ = 0.0846 A˚−2) isochore was able to distinguish two separate continuous melting temperatures: T = 2.12 0.15 K and m ± T = 2.75 0.10 K. However, no superfluid behavior was observed. This led us to study the i ± 2 K isotherm in search of a liquid density that would freeze at lower temperatures. We chose the ρ = 0.0761 A˚−2 isochore, which displayed some superfluidity even at 2 K due to finite- size effects. This allowed us to observe the behavior of the superfluid fraction across the melting transition, even though this transition was still above the superfluid transition for liquid helium in two dimensions. What we saw was that as the system began to order, the

89 superfluid fraction dropped, but that as the temperature was lowered further, it began to increase. Whether or not such an occurrence is possible for systems in the thermodynamic limit is uncertain. However, we have observed the coexistence of diagonal (translational and bond orientational) and off-diagonal (superfluid) order in a finite-sized system.

The calculations presented in this chapter were run on the nodes of a Beowulf computer cluster. The nodes have a processing speed of 2 GHz, and results can take upwards of ∼ three months to generate. Due to limitations in time and computational power, larger system sizes and lower simulation temperatures were not attainable in the present work.

One immediate direction for future work would be to extend the current calculations to larger system sizes in order to verify that the FSS obtained in this chapter are in agreement with data for larger system sizes. Also desirable is to investigate the melting transition along isochores very close to the zero temperature transition density, ρ 0.07 A˚−2, where ≈ the melting temperature may be low enough to support a superfluid phase at Ti, Our results indicate that in such a case there could also be a hexatic superfluid phase below Ti, and possibly even a supersolid phase below Tm. Another promising area of research is in the study of experimental realizations of two- dimensional helium. These include the study of helium films on surfaces such as graphite.

The path integral Monte Carlo method has already been successfully applied to the first four layers of helium on graphite [55, 85, 86, 87]. Although the first layer is strongly affected by corrugations in the substrate potential, the second layer is a reasonable approximation of a purely two dimensional system. Thus, this is a natural system to look to for possible coexistence of superfluid and solid order. Torsional oscillator experiments of helium films on graphite have reported an anomalous temperature dependence of the superfluid fraction in the second layer film [88]; however, a recent reevaluation of the second layer of helium on graphite found no evidence for a supersolid phase [89].

90 CHAPTER 6

SUMMARY

Inspired by the experiments of Kim and Chan [1, 2], we have examined the behavior of solid 4He in the presence of 3He impurities. We have performed our calculations on solid helium in two and three dimensions. By looking at the pair distribution functions, as well as snapshot configurations of the two-dimensional system, we have found that interstitial impurities quickly relax to lattice positions through the promotion of 4He atoms to the interstitial band. This leads to a small, but non-zero, superfluid fraction in the finite systems that we have studied. Although these insterstitial configurations are metastable, due to their large energy of formation, it is intriguing to ask whether such metastable mechanisms could be responsible for the superfluidity observed in recent experiments on solid helium, particularly considering the evidence from annealing studies that such systems are indeed metastable [4].

Next, we simulated a classical system of Lennard-Jones (LJ) particles in two dimensions, focusing our study of the melting transition to a single isochore above the triple point. The translational and bond orientational order parameters showed two distinct behaviors across the melting transition. Furthermore, a ﬁnite-size scaling (FSS) analysis according to the

KTHNY theory of melting led to the determination of two melting temperatures (T 0.62 m ≈ and T 0.9), with a presumably hexatic ﬂuid phase between them. i ≈ Finally, we reported on our study of the melting transition of helium in two dimensions, where we searched for evidence of two-stage KTHNY melting and any possible coexistence of

91 spatial (translational or bond orientational) order and superfluidity. We studied the melting transition along an isotherm (T=2 K) and two isochores (0.0761 A˚−2 and 0.0846 A˚−2). In all cases, we observed the translational and bond orientational order parameters to “cross over” from solid to liquid behavior at more or less the same density (for the isotherm) or temperature (for the isochores). However, using FSS scaling based on the KTHNY theory of melting, we were able to determine two separate melting temperatures. Whether or not the intermediate phase is a hexatic fluid, or is a phase-separated region of solid-fluid coexistence, is beyond the scope of our current study, and thus we are left with the possibility of a first- order melting transition. Regardless of the phase behavior in the thermodynamic limit, we have observed a superfluid component in the ordered phase of a finite size system. Even though this was seen above the superfluid transition temperature of two-dimensional liquid helium systems of infinite size, it is quite interesting that above the melting transition, where thermally activated defects and fluctuations have destroyed any solid ordering, superfluid ordering should persist. This leads us to speculate that perhaps, at very low temperatures, systems near to the melting density could exhibit a supersolid or superhexatic phase in the thermodynamic limit. One direction for future work on this topic is to study the superfluid behavior of two-dimensional helium across the melting transition of an isochore just above the zero temperature transition density.

Thus, to summarize our findings, we have seen that interstitial 3He impurities can lead to a metastable superfluid phase of solid 4He. Furthermore, we have observed a superfluid phase of two-dimensional solid helium, albeit for finite-sized systems. Future work on this area looks promising in regards to the search for a supersolid phase. We also have detected a two-stage melting transition in systems of Lennard-Jones particles in two dimensions, as is predicted by the KTHNY theory of melting.

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96 BIOGRAPHICAL SKETCH

Keola Wierschem was born in the morning, April 1, 1981 in Honokaa, Hawaii. He is a natural born citizen of the United States of America, and a card-carrying member of the

American Physical Society (APS).

Education

Ph. D. in Physics, 2010, Florida State University • Advisor: Professor E. Manousakis.

Thesis title: “Role of Defects in Possible Superﬂuidity of Spatially Ordered Helium”.

M. S. in Physics, 2006, Florida State University •

B. S. in Physics, Magna cum laude, 2003, Florida State University •

Publications

“Complex bursting in pancreatic islets: a potential glycolytic mechanism”, K. Wier- • schem and R. Bertram, J. Theor. Biol. 228, 513 (2004)

“Monolayer charged quantum ﬁlms: a quantum simulation study”, K. Wierschem and • E. Manousakis, Int. J. Mod. Phys. B 20, 2667 (2006)

“Simulation of two-dimensional melting of Lennard-Jones solid”, K. Wierschem and • E. Manousakis, Physics Procedia 3, 1515 (2010)

“Path integral Monte Carlo study of defects and impurities in solid helium”, K. Wier- • schem and E. Manousakis, to be submitted.

97 Conference Presentations

“Quantum Monte Carlo studies of charged monolayer Bose ﬂuids”, K. Wierschem and • E. Manousakis, APS March Meeting 2006, Baltimore, MD

“Defects and Impurities in Solid Helium”, K. Wierschem and E. Manousakis, APS • March Meeting 2007, Denver, CO

“Simulating the Melting Transition of Helium in Two Dimensions”, K. Wierschem • and E. Manousakis, APS March Meeting 2008, New Orleans, LA

“Simulation of 2D Melting of Lennard-Jones Solid”, K. Wierschem and E. Manousakis, • Center for Simulational Phyiscs Workshop 2009, University of Georgia, Athens, GA