Session P16: Solid Helium: Theory

Simulating the Melting Transition of Helium in Two Dimensions

We study the melting behavior of helium in two dimensions with the path integral Monte Carlo method. Two dimensional melting theory predicts two melting transitions: solid to hexatic and hexatic to isotropic liquid, described by a loss of translational and orientational order, respectively. We calculate the translational and orientational order parameters, and use finite size scaling to determine the two melting transitions in the thermodynamic limit. We also study the superfluid/normal phase boundary of 2D helium relative to the above mentioned two stage melting boundaries.   

Superfluid networks with mesoscopic structure as models of supersolid 4He

One proposal for understanding supersolidity is that grain boundaries and/or defect lines in solid 4He may support superfluidity. To understand the consequences of this proposal, we carry out simulations of the XY model with mesoscale structure corresponding to grain boundaries and/or defect lines. In the absence of disorder, we find a sharp phase transition unlike the gradual transition seen in experiments on supersolids. However, with disorder we find results that are qualitatively similar to the experiments.   

Absence of Dislocation Quantum Roughening in Solid $^4$He

Dislocations in quantum crystals are shown to be smooth at zero temperature because of the effective Coulomb-type interaction between kinks induced by exchange of bulk phonons. We provide heuristic Kosterlitz-Thouless and Renormalization Group arguments against quantum roughening and confirm them by Monte Carlo simulations of the effective model of edge dislocation moving in its gliding plane –- a quantum string (or classical membrane in $d=2$) subjected to periodic Peierls potential and Coulomb-type interaction. Simulations of such Sine-Gordon type action have been conducted in the Villain approximation in terms of the J-current formulation. Renormalized stiffness as a function of the long-range interaction strength $C$ and dislocation length $L$ is shown to be described by a master curve $F(C \ln L)$, where $F(x) \to 0$, as $x \to \infty$. We also discuss a mechanism of suppression of superfluidity along the dislocation core by thermal kinks and show that it leads to locking in of the mechanical and superfluid responses at finite temperature, which is consistent with the recent experiment of Day and Beamish (Nature {\bf 450}, 853 (2007)).   

Classical roughening of dislocations and the effect of shear modulus softening in solid $^4$He.

We propose that shear modulus $\mu(T)$ softening with increasing temperature $T$ observed by Day and Beamish [1] is due to a crossover experienced by dislocations from quantum smooth to classically rough state in the Peierls potential. Quantum dislocation is described by the Sine-Gordon model in dimensions $d=1+1$ with long-range interactions between kinks (induced by exchanging bulk phonons). Monte Carlo simulations of this model show that finite $T$ response on external stress can fit well the data $\mu(T)$ [1] for the parameters typical for $^4$He. We compare this model with the one proposed in Ref. [1]: the $^3$He impurities boiling off from the dislocations. Good fit of $\mu(T)$ cannot be achieved within this model for realistic values of the dislocation densities and relative fractions of $^3$He atoms. [1] J. Day and J. Beamish, Nature {\bf 450}, 853(2007)   

Quantum Glass in Solid He?

Recent discovery of a possible supersolid state by Kim and Chan has stimulated an active debate about true nature of a low temperature state of solid $^{ 4}$He. We will discuss possible glassy component that could be present in solid $^{ 4}$He. We will focus on i) the role of tunneling systems (TS) as a component that freezes out at lowest temperatures and ii) interactions between TS. We will address possible quantum effects and the role of TS statistics in solid $^{4}$He vs solid $^{ 3}$He-$^{ 4}$He mixtures. Implications for the torsional oscillator and for thermodynamics will be discussed as well.   

The glassy response of torsion oscillators of solid $^{4}$He

We have calculated the glassy response of a torsional oscillator filled with solid $^{4}$He assuming a phenomenological glass model. Making only a few assumptions about the distribution of glassy relaxation times in a small subsystem of otherwise rigid solid $^{4}$He, we can account for the bulk of the magnitude of the observed period shift and dissipation peak as reported in several torsion oscillator experiments. The glass model places stringent constraints on dynamic and thermodynamic responses of solid $^{4}$He and the magnitude of a possible supersolid phase. We also discuss the implications for a superglass state proposed recently by the Cornell group.   

A `Superglass' State in Solid $^{4}$He

We study the relaxation dynamics of both the resonance frequency $f(T)$ and the dissipation rate $D(T)=Q^{-1}(T)$ of a torsional oscillator (TO) containing solid $^{4}$He. Abruptly at the temperature $T*$ characteristic of the proposed supersolid phase, the relaxation times within $f(T)$ and $D(T)$ begin to increase precipitously together. Moreover, for all $T [Preview Abstract]

Viscoelastic Behavior of Solid $^4$He

We model the torsional oscillator experiments by using the Kelvin-Voigt model of viscoelasticity for solid $^4$He~[1]. With this model we find that a relaxation time which grows rapidly as the temperature is lowered can produce both a peak in the inverse of $Q$-factor and a decrease in the resonant period of the torsional oscillator. We also identify two different regimes of the relaxation in temperature: the activation energy is found to be about 260 mK at high temperatures and 18.6 mK at low temperatures. By using the derived relaxation time we fit to the torsional oscillator result obtained by Clark {\it et al}.~[2]. We find that the viscoelastic solid model provides a good agreement with the observed dissipation; however, it only accounts for a part of the measured resonant period shift, suggesting a possibility of the onset of superfluidity in solid $^4$He. \newline \newline \noindent[1] C.-D. Yoo and A. T. Dorsey, arXiv:0810.2525. \newline \noindent[2] A. C. Clark, J. T. West, and M. H. W. Chan, Phys. Rev. Lett. {\bf 99}, 135302 (2007).   

Binding energy of $^3$He to dislocations in solid $^4$He

Recent heat capacity experiments on solid $^4$He [1] show a peak in the specific heat which is interpreted as the signature of the supersolid transition. We pursue an alternative explanation for the heat capacity feature in which $^3$He impurities desorb from dislocations in solid $^4$He; the peak temperature scales with the binding energy of $^3$He to dislocations in $^4$He. Within a continuum elastic model for solid $^4$He, we make quantum mechanical estimates for the binding energy, using a combination of variational and numerical methods. We find for a short distance cut-off of one lattice constant of $^4$He, the binding energy is about 70 mK for edge and 60 mK for a screw dislocation.\newline\newline \noindent [1] X. Lin, A. C. Clark, and M. H. W. Chan, Nature {\bf 449}, 1025 (2007).   

Specific heat due to the binding of $^3$He impurities to dislocations in solid $^4$He

A statistical lattice model is used to study the binding of $^3$He impurities to dislocations in solid $^4$He. By considering a chemical equilibrium between the $^3$He atoms in the bulk and those adsorbed onto the dislocations, we are able to calculate the equilibrium thermodynamic properties of the system. The specific heat, as expected, exhibits a Schottky bump whose attributes depend on parameters like the binding energy and the concentrations of $^3$He atoms as well as defect sites. The calculated specific heat for typical values of these parameters shows a close match with experiment \footnote{X. Lin, A. C. Clark, and M. H. W. Chan, Nature {\bf 449}, 1025 (2007).}, the peak magnitude being of the order of 10 $\mu$J mol$^{-1}$ K$^{-1}$ and peak being located at around 50 mK. We show that the essential features of our model are independent of the exact lattice structure and derive an expression to estimate the shift in peak position from the binding energy value, which is an effect of the chemical potential.