[Paper Review] Theory of real supersolids

Gustavo Düring, Christophe Josserand+2 more|arXiv (Cornell University)|Oct 6, 2011

Advanced Thermodynamics and Statistical Mechanics3 references5 citations

TL;DR

This paper develops a mean-field theoretical framework for real supersolids using a non-local Gross-Pitaevskii equation to model the coexistence of elastic and superfluid properties in quantum solids. It demonstrates that supersolids can exhibit both superfluidity and elasticity simultaneously, with numerical simulations confirming Bogoliubov-like sound modes and non-classical rotational inertia, providing a consistent model for dynamical and static properties of supersolids in low-temperature quantum crystals.

ABSTRACT

We review the main properties of a supersolid. We describe first the macroscopic equation that satisfies a supersolid based on general arguments and symmetries and show that such solids might exhibit simultaneously or independently both elastic behavior and superfluidity. We then explain why a supersolid state should exist for solids at very low temperature but with a very small superfluid fraction. Finally, we propose a mean-field model, based on the Gross-Pitaevski\uı equation, which presents the general properties expected for a supersolid and should therefore provide a consistent framework to study its dynamical properties.

Motivation & Objective

• To establish a macroscopic theory of supersolids based on symmetries and conservation laws, showing coexistence of elastic and superfluid responses.
• To explain why supersolids may exist at very low temperatures with a small superfluid fraction.
• To develop a mean-field model based on the Gross-Pitaevskii equation that captures key experimental signatures of supersolids, such as non-classical rotational inertia and quantized vortices.
• To reconcile theoretical predictions with experimental observations, particularly in solid helium, by analyzing boundary conditions and mode structure.
• To propose vibrational frequency measurements as a complementary experimental probe for supersolidity, based on superfluid fraction dependence.

Proposed method

• Derives a macroscopic two-fluid model for supersolids using symmetry and conservation principles, distinguishing superfluid and elastic components.
• Applies the non-local Gross-Pitaevskii equation as a mean-field model to describe the quantum ground state of a supersolid with periodic modulation.
• Performs numerical simulations with varying boundary conditions (sliding, fixed) and mean densities to study superfluid and elastic responses.
• Computes the superfluid fraction using Leggett’s formulas and analyzes its impact on the lowest vibrational modes.
• Analyzes the Bogoliubov spectrum to confirm sound wave dispersion and identifies elastic modes independent of superfluidity.
• Compares numerical results with macroscopic predictions, particularly the scaling of the first excitation frequency with the square root of the superfluid fraction.

Experimental results

Research questions

• RQ1 Can a supersolid state simultaneously exhibit elastic and superfluid properties, and how are these responses distinguished in macroscopic equations?
• RQ2 What is the origin of the small superfluid fraction observed in experiments, and how can it be explained within a quantum many-body framework?
• RQ3 How do boundary conditions influence the observed superfluid response and rotational inertia in a supersolid?
• RQ4 Can the vibrational spectrum of a supersolid serve as a direct experimental probe of superfluidity, independent of rotation measurements?
• RQ5 To what extent does the Gross-Pitaevskii model with non-local interactions reproduce the key signatures of supersolids, such as non-classical rotational inertia and sound modes?

Key findings

• Numerical simulations confirm that the lowest-frequency vibrational mode scales as √f^ss, consistent with Bogoliubov theory and indicating superfluid character.
• The second vibrational mode is independent of the superfluid fraction, identifying it as a pure elastic excitation, thus validating the separation of elastic and superfluid responses.
• The model reproduces non-classical rotational inertia via the superfluid fraction computed using Leggett’s formulas, supporting its physical relevance.
• For a mean density ρ̄=2, the superfluid fraction f^ss ranges from 0.017 to 0.038, consistent with experimental observations in solid helium.
• The first resonance frequency f1 for f^ss ≈ 0.032 follows a √f^ss dependence, with a fit of 0.032√f^ss, providing a quantitative link between superfluidity and dynamics.
• The model shows that superfluidity emerges via a first-order transition in 2D and 3D, with a stable ground state in the large Λ limit, supporting the existence of supersolids in realistic systems.