Leggett’s bound for amorphous solids Giulio Biroli, Bryan Clark, Laura Foini, Francesco Zamponi To cite this version: Giulio Biroli, Bryan Clark, Laura Foini, Francesco Zamponi. Leggett’s bound for amorphous solids. Physical Review B: Condensed Matter and Materials Physics (1998-2015), American Physical Society, 2011, 83, pp.094530. 10.1103/PhysRevB.83.094530. cea-02524971 HAL Id: cea-02524971 https://hal-cea.archives-ouvertes.fr/cea-02524971 Submitted on 30 Mar 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Leggett’s bound for amorphous solids Giulio Biroli,1 Bryan Clark,2 Laura Foini,3, 4 and Francesco Zamponi3 1Institut de Physique Th´eorique (IPhT), CEA, and CNRS URA 2306, F-91191 Gif-sur-Yvette, France 2Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA 3LPTENS, CNRS UMR 8549, associ´ee `al’UPMC Paris 06, 24 Rue Lhomond, 75005 Paris, France. 4 SISSA and INFN, Sezione di Trieste, via Bonomea 265, I-34136 Trieste, Italy We investigate the constraints on the superfluid fraction of an amorphous solid following from an upper bound derived by Leggett. In order to accomplish this, we use as input density profiles generated for amorphous solids in a variety of different manners including by investigating Gaus- sian fluctuations around classical results. These rough estimates suggest that, at least at the level of the upper bound, there is not much difference in terms of superfluidity between a glass and a crystal characterized by the same Lindemann ratio. Moreover, we perform Path Integral Monte Carlo simulations of distinguishable Helium 4 rapidly quenched from the liquid phase to very lower temperature, at the density of the freezing transition. We find that the system crystallizes very quickly, without any sign of intermediate glassiness. Overall our results suggest that the experimen- tal observations of large superfluid fractions in Helium 4 particles after a rapid quench correspond to samples evolving far from equilibrium, instead of being in a stable glass phase. Other scenarios and comparisons to other results on the super-glass phase are also discussed. I. INTRODUCTION behavior. The natural and still open question is why freezing in an amorphous density profile should enhance Recent experiments on solid He4 by Kim and Chan [1– superfluidity compared to the crystalline case, which in- 3] raised, among many others, the important question stead is thought to show zero or very small condensate of whether disorder can foster the formation of super- fractions [11, 12]. Superfluidity is related to exchange, fluidity in solid samples. Following earlier theoretical which is a local process and depends mostly on the local analyses [4–6], Ritner and Reppy [7, 8] showed that fast neighborhood of a particle. Thus, one might expect, con- quenches produce disordered samples with a change in trary to the findings discussed above, that dense glasses the moment of inertia that corresponds to an extremely should have a fraction of superfluid density comparable high fraction of superfluid density, on the order of 20%. to the one of crystals at the same particle density. In- In addition, the role of He3 impurities [3] suggest that deed, a theoretical investigation of the superglass phase disorder must play an important role in the experiments. in a simplified (and yet realistic) model of interacting Other studies [9] suggest that the role of disorder is not bosons found an extremely small condensate fraction in to enhance the superfluid fraction but instead to induce the superglass phase [13]. Clearly, the relation between non-equilibrium states in the sample that modify the mo- disorder and superfluidity deserves further investigation, ment of inertia as a function of temperature and fre- in order to reach a better microscopic understanding of quency. Consequently, in spite of a long series of theoret- superfluidity in amorphous solids and to explain the nu- ical and experimental studies, the relationship between merical and experimental results. disorder and superfluidity in quantum solids is still not The main difficulty in the numerical investigation of clear. this problem comes from the fact that the glass phase Here we want to focus on one particular proposal that (if any) is always expected to be metastable with respect was put forward by Boninsegni et al. [6]: the possibility to the crystal phase, which is the true equilibrium phase of a bulk long-lived metastable glass phase of He4. These of solid Helium. In a classical system, it is reasonably authors performed a Path Integral Monte Carlo (PIMC) straightforward to get properties of a metastable phase numerical simulation of Helium 4 at relatively high den- or a glass, because one can easily simulate the physical arXiv:1011.2863v2 [cond-mat.dis-nn] 31 Mar 2011 sity (ρ 0.03 A˚−3), where the system was very quickly dynamics of the system by solving Newton’s equations quenched∼ from the equilibrium liquid phase at high T to of motion [14]. In contrast, the real-time dynamics of a low temperature T = 0.2 K, at which the HCP solid quantum systems is not accessible numerically because phase is stable. They reported the observation of a phase of the sign problem, and calculating properties involv- which is structurally similar to the liquid, and with a frac- ing glassy quantum system is problematic. Previous nu- tion of superfluid density as high as 60%; this phase was merical work of Boninsegni et al. [6] has looked at the observed to last for a large number of Monte Carlo sweeps fraction of superfluid density of a quenched Helium 4 via before the system eventually freezes into the equilibrium directly calculating it for a system whose PIMC dynam- ordered solid. Boninsegni et al. labeled this the “super- ics slowly equilibrates. More recently, a quantum version glass” phase. Actually, the experimental protocols used of the Mode-Coupling Theory of dynamics in glasses has to solidify Helium likely produce very disordered solids, been developed and compared with Path Integral Molec- possibly glasses. In fact the experiments in [10] showed ular Dynamics (PIMD) simulations [15], obtaining ac- evidences of very slow dynamics, the hallmark of glassy curate informations on the glass transition in quantum 2 hard spheres. However, in this study exchange effects density ρs by: were neglected and therefore superfluidity could not be 2 investigated. Therefore, for the moment path integral ρs 1 ∂ Emin(ω) = lim 2 simulations are not conclusive. ρ ω→0 I0 ∂ω Here we approach the problem in a different way. In 2 one of the first works on supersolidity, Leggett showed where ρ is the particle density and I0 = NmR the how one can derive an upper bound for the fraction of su- classical moment of inertia. From this expression it is perfluid density of a generic many-body system in which clear that upper bounds on the superfluid density can be translational invariance is broken, by means of a varia- obtained by using variational wavefunctions that in the tional computation [16]. The output of Leggett’s compu- ω 0 limit tend to the wavefunction for a non-rotating → tation is a formula that needs as only input the average container. Leggett used a variational wavefunction of density profile of the solid. This formula has been ap- the form Ψ(~r1, , ~rN )=Ψ0(~r1, , ~rN ) exp[i ϕ(~ri)], ··· ··· i plied to Helium crystals, and the aim of this work is to where Ψ0 is the ground state wavefunction for the non- P use it to study the amorphous solid. At present, there is rotating case and φ = i ϕ(~ri) a sum of phases satisfying not yet any reliable first principle computation or experi- the condition ϕ(θ)= ϕ(θ+2π) 2πmR2ω/~ [16, 17]. The P − mental measurements of the density profile of amorphous bound can be improved by including two-body correla- Helium 4. We endeavor to generate robust estimates of tions [18]. Defining it using a number of different techniques, in particular by investigating a model of zero-point Gaussian fluctuations ρ(~r)= d~r d~r Ψ (~r , , ~r ) 2 δ(~r ~r ), (1) 1 ··· N | 0 1 ··· N | − i around classical configurations, and PIMC simulations Z i without exchange (which should be closer to the classical X dynamics). Checking whether these techniques all give which is the density profile in the ground state, one finds roughly similar orders for the bound is a way to assess that the variational estimation of Emin(ω) reads: the robustness of our result. In the following, we will ~2 denote the fraction of superfluid density by “superfluid E (ω)= E + d~r[ ϕ(~r)]2ρ(~r), (2) fraction” and we always refer to Leggett’s upper bound min 0 2m ∇ to this quantity, unless otherwise specified. Z The rest of this paper is organized as follows. In sec- where E0 is the ground state energy in the non-rotating tion II, we discuss how to adapt Leggett’s bound to an case. amorphous solid. In section III A, we compute the bound Because of the assumption that the thickness of the for a profile made of Gaussian fluctuations around a clas- cylinder is much smaller than the radius, one can simplify sical configuration, and compare the results for an amor- the problem even further by “unrolling” the annulus and phous and an ordered solid, while in section III B we dis- consider the system inside a parallelepiped of length L = cuss previous numerical computations [6].