Total momentum and thermodynamic phases of quantum systems

TL;DR

This paper investigates the distribution of total momentum in quantum many-body systems, exploring its implications for different phases such as fluids, crystals, and superfluids, and discusses related phase transitions and superfluid flow.

Contribution

It provides a theoretical analysis of total momentum distribution in interacting quantum systems and its role in phase transitions and superfluidity, highlighting implications of Galilean invariance.

Findings

In non-interacting systems, total momentum distribution is normal with variance ~N.

In fluids, total momentum distribution is continuous with probability less than 1.

In fluid-crystal transition, total momentum becomes finite and lattice-distributed.

Abstract

The total momentum of $N$ interacting bosons or fermions in a cube equipped with periodic boundary conditions is a conserved quantity. Its eigenvalues follow a probability distribution, determined by the thermal equilibrium state. While in non-interacting systems the distribution is normal with variance $\sim N$, interaction couples the single-particle momenta, so that the distribution of their sum is unpredictable, except for some implications of Galilean invariance. First, we present these implications which are strong in 1D, moderately strong in 2D, and weak in 3D. Then, we speculate about the possible form of the distribution in fluids, crystals, and superfluids. The existence of phonons suggests that the total momentum can remain finite when $N\to\infty$. We argue that in fluids the finite momenta distribute continuously, but their integrated probability is smaller than 1, because the momentum can also tend to infinity with $N$. In the fluid-crystal transition we expect that the total momentum becomes finite with full probability and distributed over a lattice, and that in the fluid-superfluid transition a delta peak appears only at zero total momentum. Based on this picture, we discuss the superfluid flow in both the frictionless and the dissipative cases, and derive a temperature-dependent critical velocity. Finally, we show that Landau's criterion for excitations in moving superfluids is an in some cases correct result of an erroneous derivation.