University of Massachusetts Amherst ScholarWorks@UMass Amherst Physics Department Faculty Publication Series Physics 2006 Superglass Phase of 4He M Boninsegni Nikolai Prokof 'ev University of Massachusetts - Amherst, [email protected] Boris Svistunov University of Massachusetts - Amherst, [email protected] Follow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubs Part of the Physics Commons Recommended Citation Boninsegni, M; Prokof'ev, Nikolai; and Svistunov, Boris, "Superglass Phase of 4He" (2006). Physics Review Letters. 1151. Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/1151 This Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Physics Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. Superglass Phase of 4He Massimo Boninsegni1, Nikolay Prokof’ev2,3,4, and Boris Svistunov2,3 1Department of Physics, University of Alberta, Edmonton, Alberta T6G 2J1 2Department of Physics, University of Massachusetts, Amherst, MA 01003 3Russian Research Center “Kurchatov Institute”, 123182 Moscow 4Department of Physics, Cornell University, Ithaca, NY 14850 (Dated: February 2, 2008) We study different solid phases of 4He, by means of Path Integral Monte Carlo simulations based on a recently developed worm algorithm. Our study includes simulations that start off from a high-T gas phase, which is then “quenched” down to T =0.2 K. The low-T properties of the system crucially depend on the initial state. While an ideal hcp crystal is a clear-cut insulator, the disordered system freezes into a superglass, i.e., a metastable amorphous solid featuring off-diagonal long-range order and superfluidity. PACS numbers: 75.10.Jm, 05.30.Jp, 67.40.Kh, 74.25.Dw The remarkable observation by Kim and Chan of a supersolid, (a superglass phase of 4He) is reported in the non-classical moment of inertia in solid 4He [1] has gen- present Letter. To be specific in the definition, by glass erated a new wave of interest in the possible superfluid we mean a spatially disordered (metastable) phase, in- phase of a solid. Supersolidity of 4He is still controver- distinguishable from a solid [9] on a time scale much sial, both at the experimental and theoretical levels. Two shorter than the typical relaxation time, trel, which in of us have recently proven that, irrespective of its mi- turn should be dramatically longer than the inverse De- −1 croscopic structure, any supersolid crystal should con- bye frequency, ωD . Superglass is the term that we use for tain gapless vacancies and/or interstitials [2]. In other such a phase, if it also displays superfluidity. Note that words, any continuous-space supersolid is generically in- our definition of glass does not address the behavior of the > commensurate (i.e., the number of atoms per unit cell is system at time scales t ∼ trel, whereupon it may undergo not an integer) and squeezeable, i.e., by applying pressure structural relaxation into the polycrystalline sample, or it should be possible to squeeze matter from a container simply behave as a very viscous liquid. with supersolid into a buffer volume containing the same In Ref. [10], the idea of glassification of overpressur- supersolid. However, this very experiment has yielded a 4 4 ized liquid He was put forward, in order to explain a negative result for solid He [3]. striking experimental outcome, i.e., the absence of bulk 4 A wealth of numerical studies clearly indicate that He solid nucleation under fast (about 1 µs) acoustic wave is a commensurate (thus insulating) crystal. The finite compression pulses, up to pressures as high as ∼ 160 bar. activation energy of a vacancy computed numerically is The authors conjectured that the glassy phase is normal large, ∼ 15 K, and claimed consistent with the experi- (though the experiment was done at T =0.05 K and the mental observations [4]. The activation energy of an in- adiabatic heating was estimated to be below 0.1 K); the terstitial, ∼ 50 K [5], is significantly larger than that of a absence of superflow towards the nucleation center would vacancy. A simulation study of exchanges in an ideal hcp dramatically suppress the rate of growth of the crystal. crystal [5], yielded indirect evidence that the system is Conceptually, the finding of the present Letter is differ- not superfluid. In sharp contrast, the variational (T =0) ent, but we believe relevant to the interpretation of the calculation of Ref. 6 claims a finite condensate fraction experiment of Ref. [10]. Jamming of structural relaxation 4 in the commensurate He crystal. Thus, additional in- does not per se exclude superfluidity. Crystallization is vestigation is warranted. suppressed by the mere fact that the normal component The experiment of Kim and Chan itself has revealed forms a glassy solid, implying that further evolution to- a number of facts pointing to a strongly inhomogeneous wards a lower-energy polycrystal structure necessarily in- scenario of superfluidity, chiefly the contaminating effect volves a chain of exponentially rare quantum-tunnelling 3 or thermoactivation events, rather than a rapid growth arXiv:cond-mat/0512103v6 [cond-mat.stat-mech] 29 Mar 2006 of a small concentration of He, and non-XY behavior of the superfluid density at the critical temperature. The of the supercritical nucleus. Indeed, the boundary be- need of exploring inhomogeneous (metastable) scenar- tween the perfect-crystal nucleus and the superglass, is ios of supersolidity, dictated both by theory and experi- a solid-solid interface which realizes a pronounced local ments, has already resulted in some relevant theoretical energy minimum. Its evolution should therefore imply developments, revealing superfluid interfaces in a lattice either quantum tunnelling (in the T → 0 limit), or ther- solid [7] and a superfluid layer at the boundary between moactivation. 4 the He crystal and a disordered substrate [8]. Our study is based on accurate Path Integral Monte The numerical observation of a metastable disordered Carlo (PIMC) simulations of condensed 4He, making 2 2 hcp 0 glass superglass 1 -2 o -3 0.0359 A -4 0 g(r) hcp glass 10 -6 1 log (n(r)) o -3 hcp crystal 0.0292 A -8 0 0 2 4 6 8 10 12 14 o -10 r(A) 0 5 o 10 r (A) FIG. 1: (Color online). Pair correlation function of the ideal 4 hcp n FIG. 2: (Color online). Single-particle density matrix n(r) He solid and superglass at the near-melting ( =0.0292 4 ˚−3 n ˚−3 for ideal He hcp crystal (filled symbols) and superglass (open A , lower panel) and higher ( =0.0359 A , upper panel) −3 densities. symbols) at the near-melting density n=0.0292 A˚ (squares) and high-density n=0.0359 A˚−3 (circles). Solid lines through filled symbols represent exponential decay. use of a recently developed worm algorithm [11]. This method allows for efficient sampling and accurate deter- mination of the single-particle Green function and su- uration corresponding to an ideal hcp crystal, we con- perfluid density, for systems comprising a relatively large sistently find an exponential decay of n(r) at large dis- number N of particles (of the order of several thousand). tances, with short-range oscillations due to coordination- Specifically, we address the following two issues: (i) Is sphere effects. We observe no change in the results be- it possible to obtain definitive first-principle theoretical tween the temperatures of 0.2 and 1 K, to indicate that 4 evidence that an ideal hcp He crystal is an insulator ? those shown in Fig. 2 are essentially ground state esti- 4 (ii) What happens to a sample of liquid He quenched mates. This result provides a robust confirmation that through the first-order liquid-solid phase transition? an ideal hcp crystal is not a Bose condensate (superfluid). 4 We consider a system of N He atoms (N=216 and This conclusion is consistent with the theoretical expec- 800), at a temperature 0.2 K ≤ T ≤ 1 K, and at the tation that a crystal with finite activation energies for 3 two densities n=0.0292 (0.0359) A˚− , corresponding to vacancies and interstitials will not display superfluidity 4 an ideal hcp He crystal at a pressure of approximately [2], and is in agreement with arguments based on the 32 (155) bars [12]. The sample cell geometry with peri- statistics of exchange cycles observed in the same system odic boundary conditions is designed to fit an ideal hcp [5]. The results and conclusions of Ref. [6] appear there- 4 crystal. We use the standard microscopic model of He, fore to be erroneous, possibly artifacts of the variational based on the Aziz pair potential [13]. approach. In Figs. 1 and 2, we show data for the pair correlation At present, there is no clear understanding of what function g(r) and the single-particle density matrix n(r). crystalline defects dominate in the experimental samples ˚−3 For both the near-melting density of n=0.0292 A and of Ref. 1. It is not known whether individual dislocations, ˚−3 the higher density of n=0.0359 A we study two sam- dislocation sheets and networks, or grain boundaries in ples, differing in one respect only, namely their initial bulk 4He may underlie the experimentally observed su- configurations before equilibration. perfluid response, though model simulations of domain The single-particle density matrix is defined as walls in quantum solids hint at such possibilities [7]. But r r′ ˆ† r ˆ r′ ˆ r n( , ) = h ψ ( ) ψ( ) i where ψ( ) is the particle an- regardless of their nature, in the absence of crystalline r ˆ† r ˆ r 4 nihilation operator,ρ ˆ( ) = ψ ( )ψ( ) is the local He defects no theoretical interpretation seems viable of the density operator, and h...i stands for thermal average.