Superfluid Stiffness and Nonclassical Rotational Inertia of the Dilute Hard-Sphere Bose Gas

WARNING!

Although we have done extensive machine auditing of these proofs, including adversarial passes with multiple models, the proof is too complex for Lean autoformalization at the present time and it has not yet undergone verification by a human expert. Furthermore, the results rest extensively on unrefereed and unformalized work recently released by OpenAI.

Authors’ Note

With the exception of these notes, the entire document is machine-written. We have carefully supervised its preparation to ensure that the proofs are readable and that previous literature is appropriately cited. We are releasing this manuscript on Hexagon to disseminate the results as quickly as possible because they may be of interest to the mathematical community. We welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas.

P. L. was partially supported by NSF grant DMS-2450004.

Contents

10 Proof of the theorem on the superfluid fraction 142

10.1 The scaled setting. 142

10.2 The winding gain. 143

10.3 Choice of the parameters 145

10.4 Proof of the scaled bound. 147

10.5 Return to physical units 149

10.6 Scaling of the winding vector. 151

11 Proof of the theorem on rotational inertia 152

11.1 The container in scaled units. 152

11.2 Rotating the container 155

11.3 Stochastic area of a Brownian bridge 156

11.4 The mean area gain of a surgery 158

11.5 Choice of the parameters 162

11.6 Proof of the theorem. 162

Thermal estimates in a container

A.1 Setting and statement. 167

A.2 The torus estimates in the container. 171

A.3 Open-domain preliminaries. 173

A.4 Local kinetic estimates on full cubes 176

A.5 A lower bound on the free energy. 177

A.6 Insertion, the Fourier frame, and particle-number comparisons 179

A.7 Localization and diagonal tilting 182

A.8 The local penalty and the density in a small field 184

A.9 Shortages at interior cells 186

A.10 Close counts 188

A.11 The slab and free cutoff bounds 189

A.12 Marked colors without wall killing 189

A.13 A trace comparison and the path moment bound 193

A.14 Completion of the joint failure estimate 196

A.15 Euclidean bridge estimates and useful labels 197

Thermal estimates in a larger temperature window

B.1 The window and the order of choices 199

B.2 Statements whose proofs hold at every temperature 201

B.3 Free energies and particle numbers 202

B.4 Spatial estimates 204

B.5 The trajectory moment 206

B.6 The joint certificate bound 209

B.7 The volume threshold. 210

References 212

Introduction

Below a temperature of about two kelvin, liquid helium-4 flows through narrow channels without measurable friction, and part of it does not follow the rotation of its container. A fluid with these properties is called a superfluid. The goal of this paper is to prove two equilibrium forms of superfluidity for a basic model of an interacting Bose gas: hard spheres in three dimensions, at a fixed small density and a fixed positive temperature, in the thermodynamic limit. Recall that identical particles are called bosons if their wave functions are symmetric under permutations of the particle coordinates. The thermodynamic limit means that the particle number and the volume tend to infinity while their ratio tends to a fixed density. We show that the superfluid fraction, defined through the free-energy cost of a twist of the boundary conditions, has a positive lower bound, and that the moment of inertia of the gas in a slowly rotating container is smaller than its classical value by a positive fraction. We prove both statements through Feynman’s representation of the gas as a system of interacting Brownian loops. We will define two geometric quantities of these loops, their winding and their stochastic area, and prove lower bounds on their fluctuations.

Superfluidity and its equilibrium signatures

In 1938, Kapitza [82] and Allen and Misener [5] independently reported that liquid helium-4, cooled below its transition temperature into the superfluid phase, flows through thin slits and capillaries with almost no viscosity. In the same year, London proposed that the transition is a form of Bose–Einstein condensation, in which a positive fraction of the particles occupies a single one-particle state [107]. Einstein had predicted this condensation for an ideal (noninteracting) gas of bosons [46], on the basis of the quantum statistics of Bose [28]. Tisza then suggested that the liquid behaves as a mixture of two components, a normal one that carries viscosity and a superfluid one that does not [142]. Landau developed a hydrodynamics of two interpenetrating fluids in which the normal component is a gas of collective disturbances of the liquid [90, 91]. Andronikashvili measured the normal fraction with closely spaced disks that oscillate about an axis [6]. Viscosity makes the normal component move with the disks, and the effective moment of inertia decreases as the liquid is cooled. The two components describe different responses of the same liquid, not two permanently labeled populations of atoms. Bogoliubov showed that in a weakly interacting Bose gas, the energy of an excitation is approximately proportional to its momentum at small momentum, as for sound waves [27]. Feynman gave an atomic theory of the transition and of the two-fluid model [48–51]. We refer to [95, 96, 123] for modern accounts.

Motion without friction concerns the dynamics of the fluid, and a circulating current can last for a long time without being an equilibrium state. We study two equilibrium criteria for superfluidity: the responses of the equilibrium state to a twist and to a slow rotation, two perturbations whose effect on a classical fluid in equilibrium is trivial. We describe them for $N$ particles of mass $m$ at temperature $T$ in a region of side length $L$, and we write $\rho$ for the number density and $\hbar$ for the reduced Planck constant; Boltzmann’s constant is set equal to one. For a Hamiltonian $H$ (the energy operator) at temperature $T$, we define the free energy by $F=-T\log Z$, where $Z=\operatorname{Tr}e^{-H/T}$ is the partition function.

The first perturbation is a twist. Let the particles move on the torus $\Lambda_L=(\mathbb{R}/L\mathbb{Z})^3$, and require the many-body wave function $\psi$ to satisfy

where $e_1=(1,0,0)$ and the real number $\theta$ is the twist angle. Equivalently, the particles obey periodic boundary conditions and feel a constant vector potential of magnitude $\hbar\theta/L$ in the direction $e_1$. Let $F(\theta)$ denote the free energy with the boundary condition (1.1). In classical statistical mechanics we can remove a constant vector potential by shifting the momentum variables. Then $F$ does not depend on $\theta$. In a superfluid, a gradient of the phase of the wave function corresponds to a velocity of the superfluid component. The twist imposes the phase gradient $\theta/L$. The superfluid component then moves with velocity $v_{\mathrm{s}}=\hbar\theta/(mL)$ relative to the box while the normal component stays at rest, and the free energy increases by the kinetic energy of this motion:

Here $\rho_{\mathrm{s}}$ is the number density of the superfluid component. Fisher, Barber, and Jasnow defined the helicity modulus through the second-order free-energy response to a slowly varying phase, and related it to the superfluid density in this way [52]. Solving (1.2) for $\rho_{\mathrm{s}}$, we obtain the superfluid fraction

The second perturbation is a slow rotation. Place the particles in a container and rotate it with a small angular velocity $\omega$ about an axis with direction $n$. In the frame of the container, the Hamiltonian $H$ becomes $H-\omega\mathcal{L}_n$, where $\mathcal{L}_n$ is the total angular momentum about the axis; its formula for our model is given in § 4.2. Let $F^{\mathrm{rot}}(\omega)$ be the free energy of $H-\omega\mathcal{L}_n$. The moment of inertia of the equilibrium state is $I = -\partial_{\omega}^{2}F^{\operatorname{rot}}(0)$. A classical fluid in equilibrium rotates rigidly with its container, and for a classical system $I$ equals

where $r_j$ is the distance of particle $j$ from the axis and $\langle\cdot\rangle$ is the equilibrium average. In an axisymmetric vessel and in the absence of vortices (circulation around line defects), the superfluid component can remain at rest under slow rotation. In this case we have $I < I_{\mathrm{cl}}$. Hess and Fairbank observed this effect in helium cooled through the transition in a slowly rotating cylinder [70]. Leggett proposed the nonclassical rotationalal inertia fraction $1-I/I_{\mathrm{cl}}$ as a criterion for superfluid behavior, in the context of solids [93]; see also [136] for the moment of inertia of a trapped Bose gas.

Superfluidity is different from Bose–Einstein condensation. For a state $\Gamma$ of $N$ particles, that is, a positive operator of trace one, the one-particle density matrix is $\gamma^{(1)} = N\operatorname{Tr}_{2,\ldots,N}\Gamma$, where the partial trace integrates out the coordinates of the other $N-1$ particles. It is an operator on one-particle wave functions with trace $N$. Penrose and Onsager defined condensation in an interacting system as the existence of an eigenvalue of $\gamma^{(1)}$ of order $N$ [120]; the corresponding eigenvector is the condensate orbital. Yang placed this definition in the framework of off-diagonal long-range order [146]. In a homogeneous gas, this concerns the persistence of the kernel $\gamma^{(1)}(x,y)$ at large separation $\lvert x-y\rvert$. Penrose and Onsager already pointed out that the condensate fraction and the superfluid fraction need not be equal. For the ideal Bose gas in the thermodynamic limit, Barber showed that the helicity modulus is proportional to the condensate density, so that $f_{\mathrm{s}}=1-(T/T_{\mathrm{c}})^{3/2}$ below the critical temperature $T_{\mathrm{c}}$ [10, Eq. (3.27) and Eqs. (4.2)–(4.3)]. Yet the Landau critical velocity of the ideal gas is zero: by Landau’s energetic criterion, a moving obstacle can create excitations in it at any velocity [123, Section 6.1, pp. 65–67]. At zero temperature, where the state is a ground state, that is, a state of minimum energy, Leggett derived conditions for a unit superfluid fraction, involving translation and time-reversal invariance and an additional condition related to off-diagonal long-range order [94, pp. 935–936, conditions (C1)–(C2)]. Interactions, on the other hand, deplete the condensate; the depleted fraction is the fraction of particles outside the condensate orbital. At zero temperature, Lee, Huang, and Yang predicted the leading depleted fraction $8(\rho a^{3})^{1/2}/(3\sqrt{\pi})$ for hard spheres of diameter $a$ [ref-92]. This asymptotic is proved in [113, Corollary 1.2], uniformly over density operators supported on the lowest eigenspace of the Hamiltonian, with the thermodynamic limit taken before the dilute limit $\rho a^{3}\to0$. In a one-dimensional ground-state model with a random potential, Könenberg, Moser, Seiringer, and Yngvason proved that complete condensation in a Gross–Pitaevskii limit can coexist with an arbitrarily small superfluid fraction [87]. In that one-dimensional model the pair-interaction coupling is scaled by $N^{-1}$ in a fixed region. In the three-dimensional Gross–Pitaevskii limits discussed below, it is the scattering length that of order $N^{-1}$ relative to the fixed spatial scale; see § 4.3. The microscopic relation between condensation and superfluidity was analyzed by Hohenberg and Martin [73]; see also [17, 80, 96].

Feynman developed a sum-over-paths formulation of quantum mechanics [47]. In imaginary time, we can describe both responses probabilistically through this formulation [48]. We use units with $\hbar= 1$ and $m = 1/2$, so that the kinetic energy operator is $-\sum_j \Delta_j$, and with inverse temperature $\beta= 1/T$. The paths are parametrized by the time variable $t \in[0,\beta]$ of the heat semigroup $e^{-tH}$, which physicists call imaginary time; it is not the physical time of the dynamics. Let $p_t(x,y)$ be the heat kernel of the Laplacian on $\mathbb{R}^3$, or the periodic heat kernel on the torus. Let $\mathbb{P}_{X,Y}$ be the law of $N$ independent Brownian bridges with generator $\Delta$ in the time interval $[0,\beta]$, on $\mathbb{R}^3$ or on the torus, the $j$-th going from $x_j$ to $y_j$. Since the underlying Brownian motion has generator $\Delta$, each of its coordinates has variance $2t$ at time $t$. For hard spheres of diameter $a$, let $\mathcal{C}$ be the event that $\lvert x_i(t)-x_j(t)\rvert> a$ for all $t \in[0,\beta]$ and all $i \ne j$, where on the torus $\lvert x_i(t)-x_j(t)\rvert$ denotes the torus distance. In a container we use the free Euclidean heat kernel and free Euclidean bridges, and $\mathcal{C}$ also requires the paths to stay inside the container. The Feynman–Kac formula expresses the bosonic partition function as

where $X=(x_1,\ldots,x_N)$ and $\pi X=(x_{\pi(1)},\ldots,x_{\pi(N)})$; see Proposition 3.5 and Lemma 3.6 for the representation and the trace normalization, and [48, 61–63, 133] for background. The sum over permutations expresses the symmetry of bosonic wave functions. After normalization, the integrand in (5) defines a probability measure $\mu$ on permutations and paths, which we call the Bose path measure. Since the path of particle $j$ ends where the path of particle $\pi(j)$ begins, the $N$ paths join into closed loops; a cycle of $\pi$ of length $k$ gives one loop of duration $k\beta$. In this way the Bose gas becomes a gas of Brownian loops with a hard-core interaction at equal times.

On the torus, we lift each loop to $\mathbb{R}^3$. After one full turn, its displacement is $L$ times an integer vector, which we call its winding vector. Let $W \in\mathbb{Z}^3$ be the sum of the winding vectors of all loops. Pollock and Ceperley observed that the twist (1) multiplies the weight of each configuration by the phase $e^{\mathrm{i}\theta W_1}$, the phase acquired by closed loops in a constant vector potential, so that $Z(\theta)/Z(0)=\mathbb{E}_{\mu}e^{\mathrm{i}\theta W_1}$ [124]. Since $\mathbb{E}_{\mu}W_1=0$ by symmetry, differentiating twice at $\theta=0$ yields $F''(0)=T\operatorname{Var}_{\mu}(W_1)$. In our units, (3) becomes

This is the winding-number formula of Pollock and Ceperley [124]. For rotation, the Hamiltonian $H-\omega\mathcal{L}_n$ differs from the Hamiltonian of particles in a uniform magnetic field along $n$ by a centrifugal term proportional to $\omega^{2}\sum_j r_j^{2}$, which accounts for $I_{\mathrm{cl}}$. The magnetic field multiplies the weight of each configuration by $e^{i\omega A_n}$, where the stochastic area $A_n$ is the sum over loops of the signed areas of their projections onto the plane orthogonal to $n$. We obtain

which is the area formula of Sindzingre, Klein, and Ceperley [135]; see Ceperley’s review [33] for both formulas and their use in path-integral Monte Carlo simulations.

Superfluidity, in the two senses above, is the statement that the winding (or the area) of an interacting system of Brownian loops has macroscopic fluctuations. By (6), the superfluid fraction has a positive lower bound in the thermodynamic limit if and only if the normalized variance $\operatorname{Var}_{\mu}(W_1)/(\beta\rho L)$ has a positive lower limit. In a container of linear size $L$, the classical moment of inertia is at most a constant times $NL^2$. By (7), the nonclassical rotational inertia fraction then has a positive lower limit whenever $\operatorname{Var}_{\mu}(A_n)/(\beta\rho L^5)$ does. Bose statistics is essential here. For noninteracting distinguishable particles, only the identity permutation occurs, each loop has duration $\beta$, and the probability that a single loop winds around the torus is of order $e^{-L^2/(4\beta)}$; the superfluid fraction then tends to zero. At fixed inverse temperature, long permutation cycles provide a natural mechanism for winding on the scale of the container, since a cycle through $k$ particles is a loop of duration $k\beta$. Feynman related the superfluid transition to the appearance of such cycles [48]; see also [33, Section III.F]. The present paper proves that, for the dilute hard-sphere gas at low positive temperature, the winding and the area have fluctuations of the required size.

The model and the main results

We use units in which $\hbar=1$, the particle mass is $m=1/2$, and Boltzmann’s constant is one, so that $\hbar^2/(2m)=1$ and temperature is measured in units of energy. The dictionary to general $\hbar$ and $m$ is given in Remark 1.10. We fix the diameter $a>0$ of the hard spheres, which is the smallest allowed distance between two particle centers. It is also the scattering length of the interaction: the zero-energy scattering solution outside the excluded ball is $1-a/|x|$, and its coefficient $a$ is called the scattering length. Throughout, $T>0$ is the temperature and $\beta=1/T$.