math.MP — Mathematical Physics

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

Contributed by P. M. Aronow, Patrick Lopatto

We study the three-dimensional Bose gas of hard spheres of diameter $a$ at density $\rho$ and temperature $T$, in the thermodynamic limit with $a$, $\rho$, and $T$ fixed. When the gas parameter $\rho a^3$ is sufficiently small and $0<T<T_*$, we prove two equilibrium signatures of superfluidity. First, in a periodic box, the superfluid fraction, defined through the free-energy cost of twisted boundary conditions, has thermodynamic lower limit at least $c\,(\rho a^3)^{3/8}$, uniformly in $T$ in this interval. Second, in dilations of an arbitrary nonempty bounded open container with hard walls, the nonclassical rotational inertia fraction $1-I/I_{\mathrm{cl}}$ has thermodynamic lower limit at least $c'(\rho a^3)^{3/8}$ for every fixed rotation axis, where $I$ is the moment of inertia, $I_{\mathrm{cl}}$ its classical value, and $c'>0$ depends only on the container and the axis. In both results the derivative of the free energy is taken at finite volume, before the thermodynamic limit. The temperature $T_*$ is of order $(\rho a^3)^{1/3}/\log(1/(\rho a^3))$ times the critical temperature of the ideal Bose gas. We also bound the moment generating function of the winding vector of the Feynman–Kac paths by that of a centered Gaussian, at every density and positive temperature. Combining this upper bound with the first result, we determine the order of the winding fluctuations in the regime above. Further, for $0<\varepsilon\le1$, $\rho a^3\le(\varepsilon/100)^3$, and $T\ge(1+\varepsilon)4\pi\rho^{2/3}$, about $1.9(1+\varepsilon)$ times the critical temperature of the ideal Bose gas, the superfluid fraction in a periodic box tends to zero exponentially in its side length. By identities of Pollock and Ceperley and of Sindzingre, Klein, and Ceperley, we can express the two responses as multiples of the variances of the winding and of the signed projected area of the paths. We bound these variances from below with a variance inequality of Hammersley–Chapman–Robbins type, applied to random rewirings of the paths in a short imaginary-time interval along closed lattice routes. The rewirings adapt route comparisons developed in OpenAI's preprints on Bose–Einstein condensation in dilute Bose gases, and we extend the thermal estimates of the positive-temperature hard-sphere preprint in that series to higher temperatures. We average the rewirings over randomly translated closed lattice walks whose independent copies rarely approach each other. This makes the changed law close to the original law in chi-square divergence. Using conditional independence between disjoint imaginary-time intervals, we obtain bounds uniform in temperature. We also transfer the thermal estimates to arbitrary containers.