Chi-Deuk Yoo- Phenomenology of Supersolids

8/3/2019 Chi-Deuk Yoo- Phenomenology of Supersolids

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PHENOMENOLOGY OF SUPERSOLIDS

By

CHI-DEUK YOO

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2009

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c 2009 Chi-Deuk Yoo

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To my family

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ACKNOWLEDGMENTS

I am greatly indebted to my advisor, Professor Alan T. Dorsey, for his guidance,

encouragement, and patience he demonstrated throughout my work. Without his support

this work would not have been possible.

I would like to thank Professor P. J. Hirschfeld, Professor Y.-S. Lee, Professor K.

Matchev, Professor M. W. Meisel, Professor S. R. Phillpot, and Professor A. Roitberg

for valuable discussion and support. I also thank Professor M. H. W. Chan of the

Pennsylvania State University and Professor H. Kojima of the Rutgers University for

sharing their valuable experimental data.

Finally I thank my parents, Hae-Seun Yoo and Hee-Sook Yoo, for their unflagging

interest, support, and encouragement. Most of all I want to thank my dearest wife, Sae-il,

and children, Dan and Seul, for standing beside me with endless love and trust.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER

1 INTRODUCTION TO SUPERSOLIDS . . . . . . . . . . . . . . . . . . . . . . . 12

1.1 History of Supersolids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 Torsional Oscillator Experiments on 4He Solid by Kim and Chan . . . . . . 131.3 Recent Theoretical and Experimental Works . . . . . . . . . . . . . . . . . 15

2 VISCOELASTIC SOLIDS: ALTERNATIVE EXPLANATION OF NCRI . . . . 28

2.1 Equation of Motion for the Torsional Oscillator . . . . . . . . . . . . . . . 282.2 Properties of Viscoelastic Solids under Oscillatory Motion . . . . . . . . . 30

2.2.1 Infinite Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.2 Finite Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.3 Infinite Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Torsional Oscillator with Viscoelastic Solids . . . . . . . . . . . . . . . . . 342.4 Possible Connection between Anomalies in Shear Modulus and NCRI . . . 37

3 NON-DISSIPATIVE HYDRODYNAMICS OF A MODEL SUPERSOLID . . . . 493.1 Variational Principle in Supersolids . . . . . . . . . . . . . . . . . . . . . . 49

3.1.1 Introduction to the Variational Principle in Continuum Mechanics . 493.1.2 Isotropic Supersolids . . . . . . . . . . . . . . . . . . . . . . . . . . 523.1.3 Anisotropic Supersolids . . . . . . . . . . . . . . . . . . . . . . . . . 583.1.4 Quadratic Lagrangian Density of Supersolids . . . . . . . . . . . . . 62

3.2 Collective Modes and the Density-Density Correlation Function . . . . . . 66

4 DISSIPATIVE HYDRODYNAMICS OF A MODEL SUPERSOLID . . . . . . . 72

4.1 Andreev and Lifshitz Hydrodynamics of Supersolids . . . . . . . . . . . . . 73

4.2 Density-Density Correlation Function and its Detection . . . . . . . . . . . 774.2.1 Normal Fluids and Superfluids . . . . . . . . . . . . . . . . . . . . . 784.2.2 Normal Solids and Supersolids . . . . . . . . . . . . . . . . . . . . . 79

5 DYNAMICS OF TOPOLOGICAL DEFECTS IN SUPERSOLIDS . . . . . . . . 85

5.1 Vortex Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.2 Dislocation Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

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6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

APPENDIX

A CALCULATION OF BACK ACTION TERMS . . . . . . . . . . . . . . . . . . 105

A.1 Infinite Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.2 Finite Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106A.3 Infinite Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

B VARIATIONAL PRINCIPLE IN SUPERSOLIDS WITH THE ROTATIONALVELOCITY OF SUPER COMPONENTS . . . . . . . . . . . . . . . . . . . . . 111

C STATIC CORRELATION FUNCTIONS OF ISOTROPIC SUPERSOLIDS . . . 114

D KUBO FUNCTIONS AND CORRELATION FUNCTIONS . . . . . . . . . . . 115

E CALCULATION OF THE DENSITY-DENSITY CORRELATION FUNCTION 117

F DERIVATION OF AN EFFECTIVE ACTION FOR EDGE DISLOCATIONS . 119

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

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LIST OF TABLES

Table page

2-1 Fitting parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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LIST OF FIGURES

Figure page

1-1 Resonant period change in temperature . . . . . . . . . . . . . . . . . . . . . . . 20

1-2 Resonant period change in temperature for various concentration of

3

Heimpurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1-3 Resonant period and the amplitude of oscillation of annular cell (panel A) andof blocked annular cell (panel B) as a function of temperature . . . . . . . . . . 22

1-4 Annealing effect in the resonant period . . . . . . . . . . . . . . . . . . . . . . . 23

1-5 Annealing effect in the inverse of Q-factor . . . . . . . . . . . . . . . . . . . . . 24

1-6 Specific heat peaks of4He solid with different concentrations of 3He impurities . 25

1-7 Shear modulus of4He solid as a function of temperature . . . . . . . . . . . . . 26

1-8 Shear modulus change for various concentrations of 3He impurities . . . . . . . . 27

2-1 Schematic illustration of TO and geometry of a torsion cell . . . . . . . . . . . . 39

2-2 Effective moment of inertia of an infinite cylinder of viscoelastic solid as afunction of the driving frequency . . . . . . . . . . . . . . . . . . . . . . . . . 40

2-3 Effective damping coefficient of an infinite cylinder of viscoelastic solid as afunction of the driving frequency . . . . . . . . . . . . . . . . . . . . . . . . . 40

2-4 Displacement vector in a half cycle for

1/E. . . . . . . . . . . . . . . . . . 41

2-5 Displacement vector in a half cycle for = 3/E. . . . . . . . . . . . . . . . . . 41

2-6 Displacement vector in a half cycle for = 4/E. . . . . . . . . . . . . . . . . . 42

2-7 Effective moment of inertia of a finite cylinder of viscoelastic solid as a functionof the driving frequency with /E = 1/100 . . . . . . . . . . . . . . . . . . . 42

2-8 Effective damping coefficient of a finite cylinder of viscoelastic solid as afunction of the driving frequency with /E = 1/100 . . . . . . . . . . . . . . 43

2-9 Effective moment of inertia of an infinite annulus of viscoelastic solid as a

function of the driving frequency with /E = 1/1000 . . . . . . . . . . . . . . 432-10 Effective damping coefficient of an infinite annulus of viscoelastic solid as a

function of the driving frequency with /E = 1/1000 . . . . . . . . . . . . . . 44

2-11 F(x) of finite cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2-12 F(x) of infinite annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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2-13 Resonant period of the blocked capillary sample of BeCu TO . . . . . . . . . . . 45

2-14 Inverse of Q-factor of the blocked capillary sample of BeCu TO . . . . . . . . . 46

2-15 Resonant period of the annealed blocked capillary sample of BeCu TO . . . . . 46

2-16 Inverse of Q-factor of the annealed blocked capillary sample of BeCu TO . . . . 47

2-17 Resonant period of the constant temperature sample of BeCu TO . . . . . . . . 47

2-18 Inverse of Q-factor of the constant temperature sample of BeCu TO . . . . . . . 48

4-1 Brillouin spectrum of liquid argon . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4-2 Brillouin spectra of4He superfluid . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4-3 Density-density correlation functions of isothermal and isotropic normal solidsand supersolids) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4-4 Splitting of the Rayleigh peak due to the defect diffusion mode of a normalsolid into the Brillouin doublet of the second sound modes . . . . . . . . . . . . 84

5-1 Cut for an edge dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of the

Requirements for the Degree of Doctor of Philosophy

PHENOMENOLOGY OF SUPERSOLIDS

By

Chi-Deuk Yoo

May 2009

Chair: Alan T. DorseyMajor: Physics

We investigate the phenomenological properties of supersolids - materials that

simultaneously display both crystalline order and superfluidity. To explain the recent

observation in the torsional oscillator experiments on 4He solid by Kim and Chan we

adopt a viscoelastic solid model which is characterized by a frequency-dependent complex

shear modulus. In this model, we found that a characteristic time scale which accounts for

dissipation in solids grows rapidly as the temperature is reduced, and results in a decrease

in the resonant period and a peak in the inverse of Q-factor. We also briefly discuss the

possible relation between the torsional oscillator results and the anomalous increase of

shear modulus obtained by Day and Beamish.

In a related study, we employ a variational principle together with Galilean covariance

and thermodynamic relations to obtain the non-dissipative hydrodynamics and an

effective Lagrangian density for supersolids. We study the mode structure of supersolids

by calculating the second and fourth sound speeds due to defect propagation. We also

calculate the density-density correlation function of a model supersolid using the

hydrodynamics of Andreev and Lifshitz, and propose a light scattering experiment to

measure the density-density correlation function (which is related to the intensity of

scattered light). We find that the central Rayleigh peak of the defect diffusion mode of a

normal solid in the density-density correlation function splits into an additional Brillouin

doublet due to the longitudinal second sound modes in supersolid phase.

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Finally, we study the dynamics of vortices and dislocations in supersolids by using

the derived Lagrangian for supersolids. We obtain the effective actions for vortices and

dislocations in two-dimensional isotropic supersolids emphasizing the differences from the

dynamics in superfluids and solids. As a result we obtain the frequency-dependent inertial

masses for slowly moving vortices and dislocations.

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CHAPTER 1INTRODUCTION TO SUPERSOLIDS

1.1 History of Supersolids

After Kapitza [1], and Allen and Misener [2] simultaneously discovered the

superfluidity of He II at a temperature around 2.17 K in 1938, there were theoretical

speculations about the coexistence of crystalline order and superfluidity in matter. In 1956

Penrose and Onsager studied the possibility of such a supersolid phase of matter using the

density matrix formalism, and concluded that supersolids could not exist [3]. However, in

1969 several novel theoretical proposals for a supersolid phase were made. Andreev and

Lifshitz proposed the possibility of a condensation of zero-point defects in a 4He solid.

Every solid contains defects: vacancies, interstitials, and so on. Classically these defects

are considered to be objects localized at the lattice sites. However, at low temperatures,

due to quantum fluctuations, defects in 4He solid vibrate from the lattice sites and become

mobile. They called these quantum excitations zero-point defectons. They generalized

the two-fluid model developed by Landau for superfluids to solids with defects, and

obtained a new collective mode (the propagation of defects) at zero temperature [ 4].

One year later, Chester suggested that a system of interacting bosons can exhibitboth crystalline order and Bose-Einstein condensation at the same time [5]. Also, Leggett

suggested that the non-classical rotational inertia (NCRI) of liquid He II may be

observed in the solid phase. NCRI of He II can be explained by the two fluid model.

When a vessel containing He II is rotated, the absence of viscosity of the superfluid part

causes only the normal part to be dragged by the wall of container. Thus, the effective

moment of inertia of He II is less than that of normal helium liquid. Leggett predicted an

supersolid fraction s/ of 3104, and a simple and direct experiment of rotating a solidwas suggested to detect it [6].

Thereafter, Saslow [7] and Liu [8] improved Andreev and Lifshitzs calculation of

hydrodynamic modes. In Ref. [9], the supersolid fraction for a fcc lattice as a function

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of the ratio between the localization parameter and the lattice constant was obtained.

Fernandez et al. obtained another upper bound for supersolid fraction for the hcp lattice

of 0.3 [10]. Other feasible systems that can exhibit supersolidity were also proposed

theoretically: vortex crystals in type-II superconductors [11], Wigner crystals formed by

excitons in electron-hole bilayers [12], and cold atoms in optical lattices [13].

However, experimental searches for possible signatures of supersolidity of solid 4He

were not successful prior to 2004. These include sound speed experiments [14], mass flow

experiments [15], and torsional oscillator (TO) experiments [16, 17]. The early experiments

searching for the supersolid phase are summarized in Ref. [18].

1.2 Torsional Oscillator Experiments on 4He Solid by Kim and Chan

In 2004, Kim and Chan reported two TO experiments that may have shown the

superfluid phase of 4He solid [19, 20]. Both experiments observed drops of resonant period

in the solid phase of 4He, which might be an indication of the NCRI proposed by Leggett.

The TO used by Kim and Chan consist of a Be-Cu torsion bob and a Be-Cu torsional

rod which also was used to introduce 4He into the torsion bob. In Ref. [19], the torsion

bob contained a porous medium (Vycor glass), while in Ref. [20] the experiment was

performed with bulk 4He confined in an annular channel. They used pressures of 62 bar

and 51 bar for 4He in porous media and bulk 4He, respectively. They then electrically

drove the oscillator and measured the resonant period at a fixed temperature. The

characteristic dependence of resonant period on the temperature is shown in Figs. 1-1

and 1-2A. The drop of resonant period occurred below the critical temperature around

Tc = 175 mK for in porous media and Tc = 250 mK for bulk helium.

The resonant period of a TO without dissipation is given by

P = 2

I/k, (11)

where I is the moment of inertia of the torsion bob with 4He, and k is the torsional spring

constant. Using Eq. 11 Kim and Chan interpreted their results of the decrease in the

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resonant period as a change in the moment of inertia of the 4He solid, assuming that the

torsion constant k remained constant. This would mean that a part of the mass of 4He

solid is decoupled from the oscillatory motion, indicating the supersolid phase. The NCRI

fraction (NCRIF) which is defined as the ratio of the superfluid density s to the total

density of a bose solid, is related to the relative change in the total moment of inertia

[21],

s(T)

=

I(T0) I(T)I(T0) Iempty =

P(T0) P(T)P(T0) Pempty , (12)

where T0 is the onset temperature, and Iempty the moment of inertia of the empty TO. The

largest observed NCRIFs are about 0.5 % and 1.7% for 4He in Vycor glass and bulk 4He,

respectively.Kim and Chan also performed several control experiments to support their

interpretation. First, they investigated on the effect of the critical velocity, and found

that, in both experiments, the drop in period decreases with increasing rim velocity. Kim

and Chan estimated the critical velocity, at which the NCRI disappears, to be 300 m/s

for 4He in the Vycor glass and 420 m/s for bulk 4He (Figs. 1-1 and 1-3A). Second, they

repeated the same experiment with solid 3He, which is a fermionic solid; consequently, no

Bose condensation is possible. With solid 3He they did not observed any change in the

resonant period. However, an important and interesting feature is found that increasing

the concentration of 3He impurities in solid 4He increases the onset temperature and

broadens the change in the resonant period (Fig. 1-2). Third, taking advantage of the cell

geometry of the bulk 4He sample they inserted a barrier into the annulus channel around

which superflow is blocked, and found that the resonant period is significantly reduced

(Fig. 1-3B). Kim and Chan concluded that this is due to the disruption of superflow

around the annulus. Finally, they measured the amplitude, which is related to dissipation,

and observed a broad minimum over the range of temperatures where the resonant

period dropped. The minimum in the amplitude has the same trend as the rim velocity,

Fig. 1-3A.

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Two years after the initial reports several groups replicated the TO result [2226].

Rittner and Reppy reported first the annealing effect on TO results [22] and the sample

preparation effect [24]. Both signatures of TO experiments - a drop in the resonant period

and a peak in the inverse of Q-factor - disappeared by annealing the 4He solid (Figs.

1-4 and 1-5). In Ref. [24] they reported the effects of quenching the sample by cooling

it rapidly, the effect of which is to make a solid of poor quality with a large number of

defects. They reported NCRIFs as high as 20%. Based on their results, Rittner and

Reppy suggested that extended defects such as dislocations and grain boundaries play an

important role in understanding the TO results.

On the other hand, Aoki et al. studied the frequency dependence of the NCRI using

a double torsional oscillator. They added another dummy cell concentrically above the

torsion bob with solid 4He. This allowed them to investigate the NCRI of the same sample

with two different frequencies: the resonant frequency of the out-of-phase mode was a little

more than a twice that of the in-phase mode. They found no frequency dependence to

the onset temperature of NCRI. In addition to this, Aoki et al. found a hysteresis that

depends on the rim velocity: at T = 19 mK the NCRIF increased upon lowering the rim

velocity, but saturated at the maximum value as the rim velocity was again increased.

1.3 Recent Theoretical and Experimental Works

Kim and Chans results have revived both theoretical and experimental interest in

supersolids [27]. Reviews on both recent theoretical and experimental works can be found

in Refs. [28] and [21]. Saslow has suggested that one should use a three-fluid model,

instead of the two-fluid model of Landau, to correctly describe the supersolid. In his

model the mass density and the mass current contain an additional term consisting of thelattice velocity and the lattice mass density [29]. Ceperley and Bernu calculated exchange

frequencies in perfect bulk hcp 4He using the Path Integral Monte Carlo (PIMC) method

and concluded that superfluidity would not be observed in a perfect crystal [30]. Prokofev

and Svistunov have found a similar conclusion that zero-point defects were necessary

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for 4He solid to be a supersolid by using a coarse-graining procedure [31]. However,

Boninsegni et al. obtained a large activation energy of 13 K for vacancies and of 23 K

for interstitials, and suggested that point defects are unlikely to be present in the low

temperature range of experimental conditions [32].

On the experimental side, several results that are unfavorable to the supersolid

interpretation are reported. Day et al. performed experiments of mass flow through small

capillaries with solid 4He in Vycor glass [33] and in bulk [34]. One expects a persistent

mass flow in a supersolid; however, they detected no mass flow in either experiments. On

the contrary, a mass flow in solid 4He was detected by Sasaki et al. [35] and by Ray and

Hallock [36]. Sasaki et al. observed a flow in 4He solid on the melting curve with grain

boundaries (a poor quality crystal). For a single crystal (good quality crystal) no flow

was detected. Initially they suggested that the flow took place through grain boundaries,

but further experiments showed that mass could flow along the channel between a grain

boundary and a wall [37]. Ray and Hallock injected superfluid through one line into a

cell filled with solid 4He, and detected a change on the other line. They have found a

mass flow in solid 4He at pressure off the melting curve [36]. Similarly, superfluidity in

grain boundaries [38] and in screw dislocations [39] was studied using PIMC simulations.

Pollet et al. found that superfluid is formed within grain boundaries in solid 4He at

temperature around 0.5 K. Boninsegni et al. found superfludity along the core of a screw

dislocation in a 4He solid at zero temperature. On the other hand, there are two neutron

scattering experiments to measure the condensate fraction [40] and to detect changes in

the Debye-Waller factor [41]. Neither experiment showed any evidence for the existence of

a supersolid phase in solid4

He.It is well known that the transition of 4He from the normal fluid to superfluid is

a second order phase transition accompanied with a -anomaly in the specific heat at

the transition temperature, e.g. see Ref. [42]. Dorsey et al. suggested that there should

be a -anomaly in the specific heat if the supersolid transition is of second order [43].

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They also found that the -anomaly will be smeared out due to the inhomogeneity of

solid. Therefore, finding a feature (possibly a cusp) in the specific heat would support

the supersolid interpretation of the observed NCRI. Following the TO experiments,

Clark and Chan carried out a measurement of the specific heat in solid 4He [44]. They

measured down to a temperature of about 80 mK, and did not observed any signature.

In contrast to this null result, Lin et al. reported a peak in the specific heat at about

T = 75 mK [45]. In this second experiment a silicon sample cell was used instead of an

aluminum cell; silicon has a smaller heat capacity and higher thermal conductivity at

low temperatures than aluminum. Thus, they could measure the specific heat down to

a temperature about 30 mK, and they observed deviations from the T3 Debye specific

heat. Figure 1-6 shows the observed peaks in specific heat of solid 4He with various

concentrations of 3He impurities after subtracting the contributions of the empty cell,

phonons, and 3He impurities. In addition, the constant specific heat term, which might

be due to the mobility of 3He impurities, was also found in the 10 p.p.m. and 30 p.p.m

samples (the inset of Fig. 1-6). It is found that the height of peaks is about 20 J mol1

K1, and does not depend on concentration of 3He impurities. Lin et al. estimated the

NCRIF to be about 0.06 % which is comparable to one of their TO results [45]. Finally

they concluded that the observed peaks in specific heat measurements are another possible

signature of the supersolid phase transition, in addition to their TO results.

Another interesting and important experiment on solid 4He was performed by Day

and Beamish [46]. They measured directly the shear modulus at low frequencies and

amplitudes using two piezoelectric transducers filled with solid 4He. One transducer was

used to apply a shear stress while the other detects the induced shear deformation. Theyfound that the shear modulus of solid 4He increased by about 10% upon lowering the

temperature (Fig. 1-7). Day and Beamish explained their observation using the adsorption

(desorption) of 3He impurities into (from) a dislocation network. The adsorbed 3He

impurities pin dislocations at low temperatures, increasing the shear modulus. If 3He

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impurities evaporate from dislocations by thermal fluctuations, the dislocations become

mobile, reducing the shear modulus. The anomalous increase in the shear modulus of 4He

solid manifested surprisingly similar behaviors to the TO results, such as the dependence

on the maximum amplitude of applied shear stress and on the concentration of 3He

impurities. In Fig. 1-8 the reduced changes of the shear modulus for different values of 3He

impurity concentration are shown as a function of temperature, and in comparison with

the reduced NCRI. Moreover, Day and Beamish also studied the resonance in the cavity.

They monitored the resonant frequency and the Q-factor, and found similar behavior to

the TO experiments: the resonant frequency increased as the temperature was lowered,

accompanied with a peak in the inverse Q-factor. The two measurements are very similar,

suggesting that they are closely related. This would mean that dislocations and grain

boundaries present in solids might be responsible for both the shear anomaly and the

NCRI [47].

On the other hand, several alternative explanations for the observed NCRI of 4He

solid have been proposed as well. Dash and Wettlaufer gave an argument that there exists

a thin layer of liquid helium between the wall and the helium solid, and they showed

that the slippage between them could be responsible for the NCRI [ 48]. Nussinov at el.

proposed a glass model for the 4He solid [49]. In their model, solid 4He was assumed to be

in glassy phase at low temperatures, and they studied its effect on the TO experiments.

Remarkably they could find the reasonable agreement with the experiment of Ref. [ 22].

Finally, Huse and Khandker proposed a phenomenological two-fluid model for a supersolid

with a temperature dependent coupling constant [50] to explain the TO results.

To summarize, the existence of a supersolid phase in solid4

He has remainedcontroversial, both theoretically and experimentally. The explanation for the observed

TO experiments is not complete, opening possibilities ranging from a supersolid transition

to a possibly already known mechanical effect such as dislocation unbinding. In this work

we will focus on

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finding an alternative model to explain the TO result, such as a viscoelastic solidmodel;

deriving the hydrodynamic equations for a supersolid, and obtaining thehydrodynamic mode and new sound speeds;

proposing another method to detect the supersolid phase. We believe that lightscattering measurements can give some important information about detecting thesecond sound modes;

studying the dynamics of vortices and dislocations in a supersolid.

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Figure 1-1. Resonant period change in temperature. The empty cell data and film dataare shifted up by 4,260 ns and 3290 ns, respectively. P = 971, 000 ns.

Reprinted by permission from Macmillan Publishers Ltd: Nature [E. Kim andM. H. W. Chan, Nature 427, 225 (2004)], copyright (2004).

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Figure 1-2. Resonant period change in temperature for various concentration of 3He

impurities. P = 971, 000 ns. Reprinted by permission from MacmillanPublishers Ltd: Nature [E. Kim and M. H. W. Chan, Nature 427, 225 (2004)],copyright (2004).

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Figure 1-3. Resonant period and the amplitude of oscillation of annular cell (panel A)and of blocked annular cell (panel B) as a function of temperature. is theresonant period at T = 300 mK. From E. Kim and M. H. W. Chan, Science305, 1941 (2004). Reprinted with permission from AAAS. Copyright (2004) byAAAS.

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Figure 1-4. Annealing effect in the resonant period. P = 5.428053 ms. Reprinted figure 3with permission from A. S. C. Rittner and J. D. Reppy, Phys. Rev. Lett. 97,165301 (2006). Copyright (2006) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.97.165301).

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Figure 1-5. Annealing effect in the inverse of Q-factor. Reprinted figure 4 with permissionfrom A. S. C. Rittner and J. D. Reppy, Phys. Rev. Lett. 97, 165301 (2006).Copyright (2006) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.97.165301).

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Figure 1-6. Specific heat peaks of 4He solid with different concentrations of 3Heimpurities. Open squares are of 10 p.p.m., blue triangles 0.3 p.p.m., and redcircles 1 p.p.b. The inset shows the data of the 10 p.p.m. sample beforesubtracting a constant term (dotted line) of 59 J mol1 K1. Reprinted bypermission from Macmillan Publisher Ltd: Nature [X. Lin, A. C. Clark, andM. H. W. Chan, Nature 449, 1025 (2007)], copyright (2007).

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Figure 1-7. Shear modulus of4He solid as a function of temperature. Data are shifted forbetter clarity. Reprinted by permission from Macmillan Publisher Ltd: Nature[J. Day and J. Beamish, Nature 450, 853 (2007)], copyright (2007).

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Figure 1-8. Shear modulus change for various concentrations of 3He impurities. Reprintedby permission from Macmillan Publisher Ltd: Nature [J. Day and J. Beamish,Nature 450, 853 (2007)], copyright (2007).

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CHAPTER 2VISCOELASTIC SOLIDS: ALTERNATIVE EXPLANATION OF NCRI

2.1 Equation of Motion for the Torsional Oscillator

In this section we investigate alternatives to the supersolid explanation for the

observed decrease in the resonant period of TO experiments. We believe that it is

worthwhile to speculate on these possibilities for several reasons. First, the NCRI is

accompanied by an increase in the damping, signaled by a peak in the inverse of quality

factor. A true supersolid transition in three dimensions will not involve any change in

damping. Second, the supersolid interpretation of the TO results is not supported by other

experiments as described in Chapter 1. These include the lack of pressure driven mass

flow, an anomalous increase of shear modulus with its dependence on the concentration

of 3He impurities similar to NCRIF [46], and annealing effects of NCRI related to sample

preparation [22, 24]. These effects suggest that the elastic properties of solid 4He must be

well understood because they might be responsible for the NCRI.

At the first step of this chapter, we start describing the dynamics of an empty TO.

Let us assume that the torsion bob is completely rigid. The rigid body motion of the

torsion bob results in a constant moment of inertia Iosc. The torsion rod is assumed tobe massless with a spring constant k that provides a restoring force proportional to the

angular displacement. Then the mode frequency of the undamped harmonic oscillator is

empty =

k/Iosc. Let Mext(t) be the external torque applied on the torsion bob producing

angular displacements (t) from the equilibrium position. Since the motion of the TO is

along the axis of the torsion bob, (t) is sufficient to describe the dynamics of the TO. If

the motion is damped, the equation of motion for small of the torsion bob is

Iosc

d2

dt2+ osc

d

dt+ k

(t) = Mext(t), (21)

where osc is the dissipation coefficient. With damping the resonant frequencies become

=

2empty 2osc

4I2osc. (22)

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Now lets fill the torsion cell with solid 4He. If solid 4He is perfectly rigid, it is

expected that the total moment of inertia of the TO will be increased by the moment of

inertia of solid 4He (IHe) with a no-slip boundary condition between the solid4He and the

torsion cell. In this case, the mode frequency of undamped harmonic oscillator becomes

0 =

k

Itot, (23)

where Itot = Iosc + IHe. Therefore, the principal effect of loading solid4He in the torsion

cell is to simply increase the total moment of inertia by a constant IHe. However, if we

assume that solid 4He is not completely rigid, we need to be careful in describing the

dynamics of the TO with solid 4He. In fact, angular displacements of the torsion bob

induce a shear stress in the solid 4He, and drag it along into motion. The generated stress

causes elastic shear deformations to propagate with a finite velocity throughout the solid

4He. The deviation from the rigid body motion results in an effective moment of inertia

which depends upon the driving frequency. Moreover, damping processes present in every

solid produce an effective frequency-dependent damping coefficient.

As discussed in Chapter 1, the modification of Eq. 21 for a TO with solid 4He was

made first by Nussinov et al. [49] by adding the back action torque M(t) due to solid 4He:Iosc

d2

dt2+ osc

d

dt+ k

(t) = Mext(t) M(t). (24)

The motion of solid 4He affects the dynamics of the TO by exerting back again an torque

M(t) on the torsion bob. Following Nussinov et al. [49] the torque exerted by the solid

4He on the TO is taken to be related to the angular displacement through a linear

response function asM(t) =

dtg(t, t)(t). (25)

The response function g(t) is referred to as the back action term by Nussinov et al. [49].

Let us assume time translational symmetry so that g(t, t) depends on t t. Fourier

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transforming Eq. 25, we obtain

g() = M()/(). (26)

Additionally, we assume that the response of (t) to the external torque Mext(t) is linear

and related by the susceptibility (t); in Fourier space, we have

1() = Iosc2 iosc + k g(), (27)

where we have used Eq. 26. The zeros of the susceptibility, 1() = 0, give the resonant

period

P =2

[], (28)

and the quality factor of the TO

Q1 = 2[][] . (29)

Therefore, if the back action g(), and therefore the torque, depend on the temperature

T, the back action term contains all the information of the dynamics of the TO filled with

solid 4He. In the following section we calculate the back action terms by modeling solid

4

He as a viscoelastic solid.2.2 Properties of Viscoelastic Solids under Oscillatory Motion

In this section we study the general dynamical properties of a viscoelastic solid under

shear oscillation. A viscoelastic solid is a material that possesses both elastic and viscous

properties: its response to external disturbances becomes liquid-like or solid-like depending

upon the perturbing frequency. For simplicity we only consider isotropic viscoelastic solids

without pressure gradients.

In TO experiments, the shapes of torsion cells vary from a simple cylindrical cell

to complicated ones such as a blocked annulus. In this work we consider, for simplicity,

torsion cells with cylindrical symmetry. In Fig. 2-1 we show the geometry of a finite

cylinder torsion cell of radius R and height h. When a shear stress is applied to a

viscoelastic solid, due to the cylindrical symmetry, the only displacement is in the

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azimuthal direction (u), and its magnitude depends on the distances from and along

the axis of oscillation. In this geometry the divergence of the displacement field vanishes.

For large driving frequencies, viscoelastic solids respond as viscous fluids to the shear

stress, and the Navier-Stokes equation for the velocity field v(t) is suitable to describe its

dynamics

tv =

2r +

1

rr 1

r2+ 2z

v, (210)

where is the mass density and is the shear viscosity. By contrast, the elastodynamic

equation for the displacement u(t) describes the solid-like dynamics for small frequencies

2t u = 2r +1

rr 1

r2+ 2zu, (211)

where is the shear modulus. Equation 211 predicts the transverse sound speed of

cT =

/. Identifying v(t) = tu(t), and combining these two equations together we

have the Voigt model, also called the Kelvin model, of viscoelastic solids [51, 52]

2t u = ( + t)

2r +

1

rr 1

r2+ 2z

u. (212)

From Eq. 212 we also identify a relaxation time defined as = /. When

1, a viscoelastic solid responds elastically whereas for 1 it responds viscously.Equation 212 can be generalized to a viscoelastic solid model of a form like Eq. 211 with

a frequency dependent complex shear modulus () with the following properties: for

small

lim0

() = 0, (213)

lim0

[()]

=

0, (214)

and for large

lim

[()] = , (215)

lim

()

= i. (216)

The shear modulus of the Voigt model of viscoelastic solids is () = i.

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We study the dynamical properties of viscoelastic solids by solving Eq. 212 for

the displacement field, given an oscillatory boundary condition. From the obtained

displacement field we calculate the shear stresses (r and/or z) on the boundary, and

the torque M(t). Then the effective description of a system in oscillatory motion can be

investigated by analyzing the effective moment of inertia

Ieff() = 120

M(t)exp(it)

, (217)

and the effective damping coefficient

eff() = 10

M(t) exp(it)

, (218)

where 0 is the initial angular displacement. The viscoelastic solid model predicts that, for

small , eff() vanishes and Ieff() becomes the moment of inertia of rigid body IRB. In

the following we present the results of calculation for three different geometries: infinite

cylinder, finite cylinder and infinite annulus (Appendix A for details).

2.2.1 Infinite Cylinder

When an infinite cylinder of viscoelastic solid of radius R is oscillating with a

frequency , its effective moment of inertia and the effective damping coefficient are

Ieff() = IRB

H

E

1 i

, (219)

eff() = IRB

H

E

1 i

, (220)

where IRB = R4h/2,

H(x) =4J2(x)

xJ1(x), (221)

and a characteristic time E = R/cT. In Figs. 2-2 and 2-3 we show the effective moment

of inertia and the effective damping coefficients, respectively. There are elastic resonances

that appear as peaks in the damping coefficients and an effective moment of inertia that

increases and then decreases to a negative value. To understand the negative effective

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moment of inertia consider the displacement field with = 0,

u = R0J1(r/cT)

J1(E)exp(it). (222)

The resonant frequencies are given by the zeros of J1(E) in Eq. 222. In Fig. 2-4 we

show how the normalized displacement vector evolves during a half cycle when 1/E.In such a limit, the motion becomes purely like a rigid body motion (straight lines in Fig.

2-4). Increasing from 0, the deviation of the amplitude of displacement field from the

rigid body motion becomes apparent, developing an effectively larger out-of-phase motion

with the applied shear stress (Fig. 2-5). As a result, elastic solids have an effectively

larger moment of inertia than the rigid body. When passes through the first resonant

frequency, the direction of the displacement field changes. Figure 2-5 is a schematic

showing the in-phase motion, thus the apparent moment of inertia becomes negative. We

estimate the first resonance frequency 1 3.83cT/R = 0.2 MHz for cT 300 m/s andR 0.5 cm. In experimental conditions, driving frequencies ( 1 KHz) are well below thefirst resonance frequency to observe elastic resonances. The effect of viscosity (for large )

is to smear out the resonances.

2.2.2 Finite Cylinder

The effective moment of inertia for a finite cylinder of viscoelastic solid of radius R

and height h is

Icyleff ()

IRB=

8

2

m=1

1

(2m 1)2 2R2

h22m

H(im)

+

8R2

h2

m=1

1

2m

, (223)

where m =

(2m 1)22R2/h2 22E/(1 i) and IRB = R4h/2. On the other

hand, its effective damping coefficient is

fin cyleff ()

IRB=

8

2

m=1

1

(2m 1)2 2R2

h22m

H(im)

+

8R2

h2

m=1

1

2m

. (224)

Figures 2-7 and 2-8 illustrate the effective moment of inertia and the effective damping

coefficient of finite cylinder, respectively. The presence of the top and bottom of the finite

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cylinder makes the mode structure of the elastic oscillation more complicated because

there is one more degree of freedom along the z-axis. Consequently, the position of the

resonant frequencies changes for different dimensions of the cylinder. Increasing the height

h or the radius R results in decreasing the resonant frequency.

2.2.3 Infinite Annulus

In the case of an infinite annular viscoelastic solid of inner radius Ri and outer radius

R, the effective moment of inertia is

Iinf anneff () =

2hR2

qE

AJ2(qER) + BN2(qER)

2hR2i

qE

AJ2(qERi) + BN2(qERi)

, (225)

and the effective dampting coefficient is

inf anneff () =

2hR2

qE

AJ2(qER) + BN2(qER)

2hR2i

qE

AJ2(qERi) + BN2(qERi)

, (226)

where IRB = h(R4

R4i )/2 and

A =RiN1(qER) RN1(qERi)

J1(qERi)N1(qER) J1(qER)N1(qERi) , (227)

B =RiJ1(qER) RJ1(qERi)

N1(qERi)J1(qER) N1(qER)J1(qERi) . (228)

In Figs. 2-9 and 2-10 we show the effective moment of inertia and the effective damping

coefficients of the infinite annulus of viscoelastic solid, respectively. We observe the same

trend of resonant frequencies as in the previous cases that resonant frequencies decrease as

the gap of the annulus is increased.

2.3 Torsional Oscillator with Viscoelastic Solids

Now we study the TO dynamics by modeling solid 4He as a viscoelastic solid.

Our approach is to assume that solid 4He at low temperatures, in the range that TO

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experiments are done, becomes an isotropic viscoelastic solid which has only two complex

elastic moduli dependent on frequency. This is a simplification: solid 4He has a hcp

crystalline structure, and has five independent elastic moduli.

In Section 2.2, we investigated the dynamical response of an isotropic viscoelastic

solid to oscillatory disturbances by studying its effective moment of inertia and effective

damping coefficient. Our main conclusion is that both quantities change with the driving

frequency and a relaxation time . Hence, the resonant frequency and the quality factor

of the TO depend on the relaxation time as well.

As explained in Section 2.1, to get the resonant period and the quality factor for each

case, we need to find the poles of Eq. 27 employing the back action term of a viscoelastic

solid with a given geometry [Appendix A for the calculation of g()]. But under all

experimental conditions, the wavelength of the transverse sound mode is much larger than

the typical dimensions of torsion cells (|qER|, |qEh| 1). In this regime, the back actionterms, Eqs. A8, A29 and A38, can be cast into a single form

g() = 2IHe +R24IHeF(h/R)

24(1 i) , (229)

where a function F(x) is defined such that for an infinite cylinder Finf cyl = 1, for an

infinite annulus

Finf ann =1

R2(R Ri)2(R + Ri)2

R2 + R2i, (230)

and for a finite cylinder

Ffin cyl(x) = 192x2

4

m=1

1

(2m 1)4 H

(2m 1)x

i

, (231)

where H(x) H(x) 1. In Figs. 2-11 and 2-11 we show Finf ann(R/Ri) and Ffin cyl(h/R),respectively. Note that 0 Finf ann, Ffin cyl 1.

Now we are in a position to calculate the resonant period and the inverse of the

quality factor of the TO. With the back action term, Eq. 229, the susceptibility of

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Eq. 27 reduces to

1() Itot2 iosc + k R24IHeF(h/R)

24(1 i) . (232)

For simplicity we assume that there is no damping from the oscillator ( osc = 0). In fact,

this assumption is acceptable because for a high quality TO (Q 106) the damping of theoscillator (osc) can be neglected. Since IHe/Iosc 103 under experimental conditions, thecontribution due to viscoelasticity can be treated as a perturbation from the rigid body

motion, whose resonant frequency is 0 given in Eq. 23. We expand the poles about 0

such that 1( = 0 + 1) = 0 with 1 0. Then we obtain

1 = R2

IHeF(h/R)48Itot

2

0(1 i0) . (233)

In the viscoelastic solid model, the resonant period and the inverse of quality factor of TO

are, using Eqs. 28 and 29,

P P P0 R20IHeF(h/R)

24Itot

1

1 + 220, (234)

and

Q1 R2

2

0IHeF(h/R)24Itot

01 + 220

, (235)

where P0 = 2/0. Equations 234 and 235 are our central results in this chapter.

To interpret these results, first notice that when passes through 1/0, there would

be a peak in Q1 and a drop in the resonant period whose sizes are given by

Q1max = P/P0 =R220IHeF(h/R)

48Itot. (236)

Therefore, the viscoelastic model predicts that Q1max/(P/P0) = 1. However, the TO

experiments showed that the maximum dissipation Q1max is lower than (P/P0). The

typical experimental value varies around 0.1 [24, 53] reaching up to 0.65 [22] and down

to 0.01 [54]. This implies that we can only fit either the peak in Q1max or the decrease

in P taking the shear modulus as a fitting parameter. Other quantities such as the

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radius and moment of inertia are controllable by the experimental setup. Second, since

the sizes of Ffin cyl and Finf ann are less than the unity (Figs. 2-11 and 2-12), the reduction

of dimension of the system from the infinite cylinder geometry results in decreased period

shifts and sizes of peak in Q

1. Consequently, in fitting the experimental data, the shear

modulus (fitting parameter) of the finite cylinder turns out to be greater than that of

the infinite cylinder by a factor of 1/F. Third, following Nussinov et al. [49], we take the

relaxation time as

= 0 exp

E0

kBT

, (237)

and use 0 and E0 as fitting parameters. As the temperature is lowered below E0/kB, the

relaxation time becomes larger than 1/0, and solid4

He behaves like a viscous fluid. InFigs. 2-13 - 2-18 we show change in the resonant period and in the inverse of quality factor

measured in Clark et al. [53] and their fitting using Eqs. 234 and 235, respectively. We

have chosen to fit Q1max using the infinite cylinder model. In Table 2-1 we list the shear

modulus, the activation energy E0, and 0 used in fitting. As mentioned earlier, the change

in the resonant period of the viscoelastic model only accounts for about 10% of what is

actually observed. This might imply that the unexplained part is caused by a supersolid

transition. Moreover, it is useful to study the dynamical effect of dislocations on the

shear modulus of normal solids because the motion of dislocations can change the elastic

properties of regular crystals, and determine the characteristic time of the viscoelastic

solid model (e.g., Ref. [55]).

2.4 Possible Connection between Anomalies in Shear Modulus and NCRI

In this section, we now briefly discuss a possible relation between TO experiments and

shear modulus experiments. In Section 2.3, we have studied how the viscoelastic model

explains the observed signatures in the resonant period and Q1 of TO experiments. We

have shown that the relaxation time that decreases exponentially in temperature is

responsible for TO results with a constant shear modulus of solid 4He. However, Day

and Beamish reported about 10% increase of shear modulus at the same temperature

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range that TO experiments are performed [46]. We focus only on elasticity because in the

range of the employed frequency to measure the shear modulus the viscosity of solid 4He

could not be probed simultaneously. As we discussed in Chapter 1, Day and Beamish also

investigated the resonance effect in the cavity where the apparatus to measure the shear

modulus was embedded. They found behaviors that resemble the NCRI. However, the

relation between the dissipation peak observed in the cavity resonance and the viscosity

present in the gap between two transducers is not yet clear.

Let us examine first the effective moment of inertia of an infinite cylinder. For small

, we have Iinf cyl() IRB[1 + (2R2/24)] without viscosity. The second term is thecorrection to the rigid body value due to a finite shear modulus. Since the correction term

in Iinf cyl() is inversely proportional to , at a fixed frequency Iinf cyl() decreases as ()

increases. Therefore an increase of shear modulus will enhance the change in resonant

period. This could be a connection between the increasing shear modulus observed by Day

and Beamish and NCRI.

We calculate the actual change in the resonant period induced by an increase of shear

modulus. The resonant period of TO for an elastic solid of a finite shear modulus can be

easily obtained by setting to zero in Eq. 234. The result is

P() = P0

1 +

R220IHeF(h/R)

48Itot

. (238)

One can verify this result finding the zeros of Eq. 232 without and . Lets consider a

small change in shear modulus from 0. Then the fractional change in P() due to

becomes

P

P(0) R220IHeF(h/R)

48Itot

0 . (239)

For Ref. [24], Eq. 239 predicts only 0.8105 % decrease change in the resonant periodwhereas the measured change is about 2.6103 %. Therefore, we find that an increasingshear modulus accompanies a decrease in the resonant period of a TO; however, the elastic

solid model does not account for all the change observed in experiments.

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Figure 2-1. Schematic illustration of TO and geometry of a torsion cell.

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-8

-6

-4

-2

0

2

4

6

8

10

0 2 4 6 8 10 12 14 16 18 20

Ieff

/IRB

E

/E = 1

/E = 5

/E = 1/10

/E = 1/100

Figure 2-2. Effective moment of inertia of an infinite cylinder of viscoelastic solid as afunction of the driving frequency .

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14 16 18 20

effE

/IRB

E

/E = 1

/E = 5

/E

= 1/10

/E = 1/100

Figure 2-3. Effective damping coefficient of an infinite cylinder of viscoelastic solid as afunction of the driving frequency .

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-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

u(r)/R

0

r / R

t = 0

t = / 4

t = / 2

t = 3 / 4

t = /

Figure 2-4. Displacement vector in a half cycle for 1/E.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

u(r)/R

0

r / R

t = 0

t = / 4

t = / 2t = 3 / 4

t = /

Figure 2-5. Displacement vector in a half cycle for = 3/E.

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-10

-8

-6

-4

-2

0

2

4

6

8

10

0 0.2 0.4 0.6 0.8 1

u(r)/R

0

r / R

t = 0

t = / 4

t = / 2

t = 3 / 4

t = /

Figure 2-6. Displacement vector in a half cycle for = 4/E.

-4

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10

Ieff

/IRB

E

h / R = 0.5

h / R = 1

h / R = 2

Figure 2-7. Effective moment of inertia of a finite cylinder of viscoelastic solid as afunction of the driving frequency with /E = 1/100.

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0

5

10

15

20

25

30

35

40

45

0 1 2 3 4 5 6 7 8 9 10

effE

/IRB

E

h / R = 0.5

h / R = 1

h / R = 2

Figure 2-8. Effective damping coefficient of a finite cylinder of viscoelastic solid as afunction of the driving frequency with /E = 1/100.

-60

-40

-20

0

20

40

60

0 10 20 30 40 50 60 70 80 90 100

Ieff

/IRB

E

Ri / R = 0.9

Ri / R = 0.7

Ri/ R = 0.5

Figure 2-9. Effective moment of inertia of an infinite annulus of viscoelastic solid as afunction of the driving frequency with /E = 1/1000.

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0

100

200

300

400

500

600

700

800

900

0 10 20 30 40 50 60 70 80 90 100

effE

/IRB

E

Ri / R = 0.9

Ri / R = 0.7

Ri / R = 0.5

Figure 2-10. Effective damping coefficient of an infinite annulus of viscoelastic solid as afunction of the driving frequency with /E = 1/1000.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20

F

h/R

Figure 2-11. F(x) of finite cylinder.

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0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

F

Ri / R

Figure 2-12. F(x) of infinite annulus.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0

5

10

15

P[

ns]

T [K]

Clark et al.infinite cylinder modelfinite cylinder model

Figure 2-13. Resonant period of the blocked capillary sample of BeCu TO. Experimentaldata were adapted with permission from A. C. Clark, J. T. West, and M. H.W. Chan, Phys. Rev. Lett. 99, 135302 (2007). Copyright (2007) by theAmerican Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302).

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(

Q

1)

106

T [K]


Figure 2-14. Inverse of Q-factor of the blocked capillary sample of BeCu TO.Experimental data were adapted with permission from A. C. Clark, J. T.West, and M. H. W. Chan, Phys. Rev. Lett. 99, 135302 (2007). Copyright(2007) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

1

2

3

4

5

6

7

8

9

10

P[

ns]

T [K]


Figure 2-15. Resonant period of the annealed blocked capillary sample of BeCu TO.Experimental data were adapted with permission from A. C. Clark, J. T.West, and M. H. W. Chan, Phys. Rev. Lett. 99, 135302 (2007). Copyright(2007) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302).

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

1

2

3

4

5

6

7

8

9

10

(

Q

1)

107

T [K]


Figure 2-16. Inverse of Q-factor of the annealed blocked capillary sample of BeCu TO.Experimental data were adapted with permission from A. C. Clark, J. T.West, and M. H. W. Chan, Phys. Rev. Lett. 99, 135302 (2007). Copyright(2007) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

1

2

3

4

5

6

P[

ns]

T [K]


Figure 2-17. Resonant period of the constant temperature sample of BeCu TO.Experimental data were adapted with permission from A. C. Clark, J. T.West, and M. H. W. Chan, Phys. Rev. Lett. 99, 135302 (2007). Copyright(2007) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302).

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.5

1

1.5

2

2.5

3

3.5

4

(

Q

1)

107

T [K]


Figure 2-18. Inverse of Q-factor of the constant temperature sample of BeCu TO.

Experimental data were adapted with permission from A. C. Clark, J. T.West, and M. H. W. Chan, Phys. Rev. Lett. 99, 135302 (2007). Copyright(2007) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302).

Table 2-1. Fitting parameters.

sample [g/cm s2] 0 [s] E0/kB [mK]blocked capillary 0.34 108 2.86 158

annealed blocked capillary 0.75 108 0.21 395constant temperature 1.53 108 3.72 166

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CHAPTER 3NON-DISSIPATIVE HYDRODYNAMICS OF A MODEL SUPERSOLID

3.1 Variational Principle in Supersolids

In this chapter we investigate the hydrodynamics of a supersolid whose properties

are governed by conservation laws and broken symmetries. As we discussed in Chapter 1,

the hydrodynamic equations of motion for supersolids are first derived by Andreev and

Lifshitz [4]. In their model point defects, such as interstitials and vacancies, in bosonic

solids undergoes a Bose-Einstein condensation and a crystal becomes a supersolid. The

hydrodynamics of supersolids was studied further by Saslow [7] and Liu [8].

We use the variational principle to derive a Lagrangian for the supersolid and the

non-dissipative hydrodynamic equations of motion. The variational principle was often

used in the literature to obtain the hydrodynamics of various continuum systems: normal

fluids [5658], superfluids [56, 5963], normal solids [57, 64], liquid crystals [65], and so on.

Let us start this section by giving a plain example of variational principle applied to ideal

fluids to show the simplicity of the method.

3.1.1 Introduction to the Variational Principle in Continuum Mechanics

We write a Lagrangian density L for isentropic ideal fluids, which are irrotational,inviscid, and incompressible, in the Eulerian description1 as follows

LIF = 12

v2 UIF(), (31)

where is the mass density, v the velocity field, and UIF the internal energy density. The

internal energy density satisfies the thermodynamic relation

dUIF = d, (32)

1 The Eulerian description of the motion of continuum medium is a field descriptionwith a coordinate system fixed in space. In this Eulerian description, the properties offlow are functions of both space and time. An alternative is the Lagrangian description inwhich the motion of individual particles is traced as a function of time.

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where the chemical potential per unit mass. All the dynamical variables are taken to be

functions of the position x and time t.

The variational principle states that the equations of motion can be derived by

minimizing the action S of a Lagrangian density L

S =

dt

dx L, (33)

with respect to all the dynamical variables i.e., and v in this example. However,

the variation of the action with the Lagrangian density in Eq. 31 with respect to the

dynamical variables and v does not provide us with the right equations of motion - they

are not independent and restricted by side conditions such as conservation laws. The easy

way to see this is to take the variation of the action of Eq. 31 with respect to v: the

resulting equation of motion is a trivial and irrelevant one (v = 0). In order to overcome

this problem, the side constraints must be included into the Lagrangian density. As one

knows, for fluids the total mass is conserved, and this conservation law is expressed in the

continuity equation

t + i(vi) = 0. (34)

Since we are considering isentropic ideal fluids, the mass conservation is the only

side condition to be taken into account. In fact the momentum is conserved as well.

Nonetheless, the momentum conservation law is not a side condition, but the byproduct

of the variational principle. The continuity equation is incorporated into the Lagrangian

density Eq. 31 by using a Lagrange multiplier :

LIF =

1

2

v2

UIF() + t + i(vi). (35)

Then the equations of motion are obtained by taking variations of the action with respect

to and v. With Eq. 35 we obtain

vi i = 0, (36)

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1

2v2 UIF

D

Dt= 0, (37)

where D/Dt t + vii. The other trivial equation of motion is the continuity equationwhich one obtains by taking the variation with respect to the Lagrange multiplier .

Equation 36 implies that

vi = i, (38)

and this confirms that there is no vorticity ( vs = 0) for ideal fluids, as expected.We can express Eq. 37 in more familiar form by taking its gradient. Since in thermal

equilibrium the change in the chemical potential per unit mass is related to the change

in the pressure P by the Gibbs-Duhem relation as follows

d = dP, (39)

we obtain

DviDt

= iP. (310)

Equation 310 is the Euler equation for ideal fluids without external forces. Consequently,

we derived the hydrodynamics describing ideal fluids using the variational principle: the

continuity equation and the Euler equation.

As we showed in the analysis above, the variational principle provides us with

the Lagrangian density for a continuum system; Eq. 35 is the Lagrangian density for

isentropic ideal fluids. Moreover, as long as the boundary contributions are negligible in

the action, the analysis above is equivalent to that with a Lagrangian density

LIF = t 12

(i)2 UIF(). (311)

We have integrated by parts the action of Eq. 35, and replaced the velocity with

using Eq. 38. This Lagrangian density also can be derived from the time-dependent

Gross-Pitaevskii Lagrangian density for the superfluid which is

LSF = i2

[t t] 12m

(ij)(ij) g

2()2, (312)

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where = h/2 with h the Planck constant, and m the mass of the superfluid component.

Taking =

nei with the number density n, and the velocity vi = (/m)i, the

Lagrangian density Eq. 312 can be written as follows:

LSF = i2

tn nt 2

2mn(i)

2 28mn

(in)2 n g

2n2. (313)

Since the first term is a total derivative of an analytical field, it does not contribute to the

dynamics and can be neglected. Hence we can identify that = mn and = (/m);

therefore, we have shown that the time-dependent Gross-Pitaevskii Lagrangian density is

the same as the Lagrangian density for isentropic ideal fluids Eq. 311 with a particular

form of the internal potential energy density.

3.1.2 Isotropic Supersolids

Let us now turn on the variational principle applied for supersolids which have

both crystalline order and superfluidity. In this part we start with the simple case of an

isotropic supersolid whose Lagrangian density in the Eulerian description is given by

LSS = 12

svs2 +

1

2( s)vn2 USS(, s, s , Rij), (314)

where s is the density of super-components, the total density, vs the velocity of

super-components, vn the velocity of normal components, s the entropy density, and

Rij Rj/xi, (315)

the deformation tensor with R and x being Lagrangian and Eulerian coordinates,

respectively. The first two terms in the Lagrangian density are the kinetic energy

densities of the super-component and normal components, and the last term is theinternal potential energy density. Because of the isotropy, the Lagrangian density of

an isotropic supersolid is very similar to the Lagrangian density of superfluid, e.g.

the Lagrangian density of a superfluid used by Zisel (Eq. 2.1 in Ref. [59]). The only

exception is that USS depends on the deformation tensor. In contrast to fluids, the elastic

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energy which is proportional to the Eulerian strain tensor (ij RikRjk)/2 arises forsolids. In linear elasticity, the Eulerian strain tensor turns into a more familiar form

(ij RikRjk)/2 = (iuj + jui)/2 iukjuk/2 with the displacement vector u.

Given the Lagrangian density, Eq. 314, the total energy density for supersolids is

defined as the sum of the kinetic energy densities and the potential energy density

ESS =1

2svs

2 +1

2( s)vn2 + USS(, s, s , Rij). (316)

This total energy density can be related to the energy density measured in the frame

where the super-component is at rest as follows:

ESS = 12vs2 + ( s)(vni vsi)vsi + . (317)

As is a Galilean invariant, it must depend on Galilean invariant quantities [ 66]. Similar

to the two-fluid model for superfluid [66], has a thermodynamic relation [4]

d = T ds + d ikdRik + (vni vsi)d[( s)(vni vsi)], (318)

with ik the stress tensor. We now can obtain the thermodynamic relation for ESS by

differentiating Eq. 317 and replacing Eq. 318 for d. The result is

dESS = T ds ikdRik (vni vsi)vnids +

+1

2(2vn

2 2vnivsi + vs2)

d

+svsidvsi + ( s)vnidvni. (319)

This thermodynamic relation was used by Saslow [7] and by Liu [8] in deriving the

hydrodynamics of a supersolid provided that the chemical potential per unit mass is given

by

Saslow, Liu = vnivsi + vs2/2. (320)

Finally from Eq. 316 and Eq. 319, we get the thermodynamic relation for USS:

dUSS = T ds +

+

1

2(vni vsi)2

d 1

2(vni vsi)2ds ikdRik. (321)

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Note that the difference between a superfluid and an isotropic supersolid is the dependence

of the potential energy on the deformation tensor Rij.

As illustrated in Section 3.1.1, the dynamical variables of a supersolid in the

Lagrangian density, Eq. 314, are not independent each other. The side conditions

relating the dynamical variables must be included in the Lagrangian density in order to

derive the correct equations of motion. There are two important conservation laws for a

supersolid: the mass conservation law and the entropy conservation law. Once again the

momentum conservation law is not a side condition to be imposed, but a consequence of

the variational principle. The conservation of mass is expressed by the continuity equation

t + i

svsi + ( s)vni = 0, (322)with the total current

ji = svsi + ( s)vni. (323)

For the entropy conservation we have

ts + i(svni) = 0. (324)

One should note that in the equation for the conservation of entropy, Eq. 324, only the

velocity of the normal component are involved because the entropy is carried solely by the

normal component. In addition to the two conservation laws above there is one more side

condition, called Lins constraint, to be included [56]:

DnRiDt

= 0, (325)

where Dn/Dt t + vnii. Lins constraint for solids is an expression of the fact that theLagrangian coordinates, (i.e., the initial positions of particles) do not change along the

paths of the normal component. The same condition is also used for an isentropic normal

fluid to generate vorticity whereas the entropy conservation equation produces vorticity for

the normal fluid [56]. Lets incorporate these constraints into the Lagrangian density using

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Lagrangian multipliers , and . Then we have

LSS = 12

svs2 +

1

2( s)vn2 U(, s,s ,Rij) +

t + i

svsi + ( s)vni

+

ts + i(svni)

+ i

t(sRi) + j(sRivnj)

, (326)

where we have used the Lins constraint combined with the conservation equation of

entropy making it in a form of continuity equation. It is also possible to use the continuity

equation, instead of the conservation equation of entropy, combining the Lins constraint.

We are now in a position to get the equations of motion. We take the variations of

the action with respect to all the dynamical variables. We obtain

1

2vn

2 USS

DnDt

= 0; (327)

s1

2vs

2 12

vn2 USS

s (vsi vni)i = 0; (328)

sDn

Dt+ Ri

DniDt

+USS

s= 0; (329)

vsisvsi si = 0; (330)

vni( s)vni ( s)i si sRjij = 0; (331)

Ris

DniDt

j

USSRji

= 0. (332)

Obviously the variations with respect to the Lagrange multipliers recover the imposed

constraints, Eqs. 322 through 325. In the following we show that the derived equations

of motion can be rearranged to recover the hydrodynamics of a supersolid developed by

Andreev and Lifshitz [4], by Saslow [7], and by Liu [8].

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We obtain first the Clebsch potential representation [57] for the velocity of the

super-component, from Eq. 330,

vsi = i, (333)

and the velocity of the normal component, from Eq. 331,

vni vsi =s

s

i + Rjij

. (334)

With these representations we find first that

vs = 0. (335)

Consequently the velocity of the super-component is irrotational as one expects for asuperfluid without vortices: vs is only longitudinal. On the other hand, we also find that

vorticity can be generated for vn:

vn =

s

s

+

sRi

s

i. (336)

In fact it is also possible to include systematically the transverse part of vs by introducing

a second Lins constraint (Appendix B).

The use of Eq. 333 in Eq. 328 corroborates one of the thermodynamic relations

given in Eq. 321,

USSs

= 12

(vni vsi)2. (337)

Taking the gradient of Eq. 327 and using the Clebsh representation of vs, Eq. 333, we

get the Josephson equation for vs

Dsvsi

Dt = iUSS iUSSs = i, (338)

where Ds/Dt t + vsii. In deriving Eq. 338, we have used Eq. 337 and vsjivsj =vsjjvsi because vs is irrotational.

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The Euler equation for vn can also be derived. We take Dn/Dt of Eq. 331, and use

the derived equations of motion to eliminate the Clebsch potentials of the relative velocity.

The resulting equation is the Euler equation for vn:

( s) DnvniDt

= ( s)i

USS

( s)i

USSs

j

USSRjk

Rik

si

USSs

1

2( s)i(vnj vsj)2

ts + j(svsj)

vsi

t( s) + j

( s)vnj

vni

= ( s)i jjkRik siT 12

( s)i(vnj vsj)2

t( s) + j( s)vnj(vni vsi), (339)where we have used an useful identity of the convective derivative,

D(aib)

Dt= ib

Da

Dt+ ai

Db

Dt

ajbivj. (340)

Finally the Josephson equation and the Euler equation can be put together into the

momentum conservation equation,

t

svsi + ( s)vni

= k

svsivsk + ( s)vnivnk

T s n(vnj vsj)2

ik kjRij

,

(341)

because jkiRjk + jjkiRk = j(jkRik). In deriving Eq. 341 we have used the mass

conservation equation and the thermodynamic equation for .

As shown in the example of ideal fluids, the Lagrangian density for the isotropic

supersolid can also be reduced in a compact form by using the Clebsh representation of vs.

Neglecting the boundary contributions, the Lagrangian density Eq. 326 is equivalent to

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the following Lagrangian density:

LSS = 12

svs2 +

1

2( s)vn2 USS(, s,s ,Rij) t

svsi + ( s)vni

i

sDn

Dt sRiDni

Dt . (342)

The next step is to use Eqs. 329 through 331 to replace vs, and . Then we obtain

the Lagrangian density for an isotropic supersolid

LSS = t 12

(i)2 +

1

2( s) (vni i)2 f(, s, T , Rij), (343)

where f = USS sT. This Lagrangian density used in the derivation of the hydrodynamics

of a supersolid has many other applications, including the calculation of correlation

functions and collective modes, and the study of the dynamics of dislocations and vortices.

3.1.3 Anisotropic Supersolids

Having completed the variational principle in an isotropic supersolid, let us extend

the analysis to an anisotropic supersolid. Because of the anisotropy, the density of the

super-component becomes a tensor sij which is another distinction of solids from fluids.

Then the Lagrangian density for an anisotropic supersolid in Eulerian description is given

by

LSS = 12

sijvsivsj +1

2(ij sij)vnivnj USS(, sij, s , Rij). (344)

The internal potential energy satisfies the thermodynamic relation

dUSS = T ds +

+

1

2(vni vsi)2

d ikdRik 1

2(vni vsi)(vnj vsj)dsij . (345)

The derivation of this thermodynamic relation is identical to the calculation of Eq. 321

for isotropic supersolids.

The constraints to be imposed for the anisotropic supersolid are

conservation of mass

t + i

sijvsj + (ij sij)vnj

= 0; (346)

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conservation of entropyts + i(svni) = 0; (347)

Lins constraintDnRi

Dt

= 0. (348)

We incorporate these constraints into the Lagrangian density using Lagrange multipliers ,

and . Then we have

LSS = 12

sijvsivsj +1

2(ij sij)vnivnj USS(, sij , s , Rij) +

ts + i(svni)

+

t + i

sijvsj + (ij sij)vnj

+ i

t(sRi) + j(sRivnj)

. (349)

In the above Lagrangian density Lins constraint was introduced after combining it withthe equation of the entropy conservation. We calculate the equations of motion as follow

1

2vn

2 USS

t vnii = 0; (350)

sij1

2vsivsj

1

2vnivnj USS

sij 1

2(vsj vnj)i

1

2(vsi vni)j = 0; (351)

sDn

Dt+ Ri

DniDt

+USS

s= 0; (352)

vsisij(vsj j) = 0; (353)

vnii + Rjij =

1

s(ij sij)(vnj vsj); (354)

Ri DniDt

1s

j

USSRji

= 0. (355)

Additionally, the variations with respect to the Lagrange multipliers just reproduce the

imposed side conditions, Eqs. 346 through 348. First, the variation with respect to

vsi leads again to Eq. 333. Next, the derivation of the Josephson equation, the Euler

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equation for vn, and the momentum conservation equation for an anisotropic supersolid is

analogous to that for an isotropic supersolid. The results are: the Josephson equation

tvsi =

i

1

2

ivs2, (356)

the Euler equation

DnDt

(ij sij)(vnj vsj)

= si

USS

s

iRjk

USSRkj

(ij sij)(vnj vsj)kvnk(jk sjk)(vnk vsk)ivnj , (357)

and the equation of the conservation of momentum

t

(ij sij)vnj + sijvsj

j

Rikjk

+ j

vsjvsi + vsipj + vnjpi

i

T s (vnj vsj)pj

= 0,

(358)

where

pi (ij sij)(vnj vsj). (359)

Note that this momentum conservation equation agrees with the non-dissipativemomentum current derived by Andreev and Lifshitz (Eq. 12 in Ref. [4]) except for the

nonlinear strain term that they neglected. Moreover, the derived momentum conservation

equation also can be mapped into Eq. 4.16 of Saslow [7] by taking vs as a Galilean

velocity, and Eq. 3.40 of Liu [8] with the vanishing super thermal current with the

chemical potential given by Eq. 320.

Similar to the isotropic supersolid case, the equivalent Lagrangian density of

anisotropic supersolids to Eq. 349 is

LSS = t 12

sijij +1

2(ij sij)vnivnj

(ij sij)vnji f(, sij, T , Rij), (360)

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where f USS T s. There are several works that have derived a Lagrangian density forsupersolid at zero temperature, that employ different methods. Son derived the Andreev

and Lifshitz nondissipative hydrodynamics and a Lagrangian density by using Galilean

invariance and symmetry arguments [67]. First, we can also make the connection to the

Lagrangian density derived by Son [67] by replacing vn with the displacement vector R

and the strain tensor Rij using the inverted Lins constraint,

vni = R1ji tRj, (361)

where R1ji xi/Rj and RijR1jk = ik. Josserand et al. used the homogenization methodstarting from the time-dependent Gross-Pitaevskii equation [68]. On the other hand, Ye

proposed a Lagrangian density for a supersolid introducing an arbitrary phenomenological

coupling constant between elasticity and superfluidity in the Gross-Pitaevskii equation

[69].

At this point it is worthwhile to speculate about the possible connection between the

derived Lagrangian density and the time-dependent Gross-Pitaevskii Lagrangian density

coupled to an elastic field. In extending the superfluid Gross-Pitaevskii equation Eq. 312,

to a supersolid one must require Galilean invariance. The covariant form of the gradient of

the order parameter wave function is

ii ii mtuj + ikujk, (362)

where m is the mass of the bosonic particle. However, the use of the covariant gradient

results in the coupling constant s between vs and vn while from the interaction term

in the Lagrangian density, Eq. 360, the coupling constant is (ij sij) which is setby conservation laws and Galilean invariance. Unfortunately the connection between the

variational principle and the Gross-Pitaevskii Lagrangian density is still unclear.

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3.1.4 Quadratic Lagrangian Density of Supersolids

The quadratic Lagrangian is interesting because it gives us the linearized

hydrodynamics from which collective modes can be obtained. We study the dynamics

of a supersolid by considering small fluctuations from a thermal equilibrium state (or

at T = 0). Unless we specify otherwise, in the remaining part of this chapter thermal

fluctuations are excluded. We first consider small fluctuations in densities, denoted with ,

from constant equilibrium values, subscripted by zero, as

= 0 + , (363)

and

sij = s0ij + sij. (364)

For the velocity of the super-component, we take a variation in the Clebsch potential so

that we have its non vanishing gradient and the time derivative: i and t. In addition,

we assume that the Lagrangian coordinates differ from the Euler coordinates by a small

displacement vector field u:

R = x

u. (365)

The deformation tensor then becomes

Rij = ij wij, (366)

where wij iuj. From the inverted Lins constraint, Eq. 361, we obtain the linearrelation between the velocity of the normal component and the displacement vector,

vni = tui. (367)

Therefore, we found that in linear elasticity the velocity of the normal component is given

by the time derivative of the displacement vector. It becomes clear that Lins constraint

is the hydrodynamic equation for the elastic variables that arises from the translational

broken symmetries of solids rather than the conservation of the initial positions.

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We now expand the Lagrangian density up to second order in small fluctuations ,

sij , and u:

LquadSS =

0t

0ijwij

0

t

1

2

0(i)2

wij

wij

1

2

wij

()2

+1

2n0ij (tui i) (tuj j)

1

2

ijwlk

wijwlk, (368)

where n0ij 0ij s0ij , and we have used

f

= +

1

2(vni vsi)2 , (369)

and

fwij = ij. (370)

In the expansion we have dropped constants which do not contribute to the equations of

motion. Additionally the terms proportional to f/sij are neglected because they are of

higher order:

f

sij=

USSsij

= 12

(vni vsi)2 0. (371)

As a result, the dependence on sij is absent in the quadratic expansion of the

Lagrangian density, Eq. 368. Often the first two terms in Eq. 368 are neglected because

they are total derivatives, and do not contribute to the equations of motion as long as

boundary contributions are not important. This is true as long as topological defects,

such as vortices or dislocations, are not present in the supersolid. In Chapter 5 we will

show that the first term is responsible for the Magnus force acting on vortices [ 70] while

the second term gives the Peach-Koehler force on a dislocation [ 71, 72]. Since we are not

considering any topological defects in this chapter, we neglect these terms for now.The linearized equations of motion are obtained taking the variations of the action of

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Chi-Deuk Yoo- Phenomenology of Supersolids

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