Force balance in thermal quantum many-body systems from Noether's theorem

TL;DR

This paper derives an exact force balance relation in thermal quantum many-body systems using Noether's theorem, linking invariance of free energy under spatial deformations to local force equilibrium.

Contribution

It introduces a novel application of Noether's theorem to quantum many-body systems, establishing a local force balance sum rule from invariance properties of free energy.

Findings

Derived a local force balance sum rule from free energy invariance.

Showed that the average external force vanishes in homogeneous shifts at equilibrium.

Linked spatial invariance to physical force equilibrium in quantum systems.

Abstract

We address the consequences of invariance properties of the free energy of spatially inhomogeneous quantum many-body systems. We consider a specific position-dependent transformation of the system that consists of a spatial deformation and a corresponding locally resolved change of momenta. This operator transformation is canonical and hence equivalent to a unitary transformation on the underlying Hilbert space of the system. As a consequence, the free energy is an invariant under the transformation. Noether's theorem for invariant variations then allows to derive an exact sum rule, which we show to be the locally resolved equilibrium one-body force balance. For the special case of homogeneous shifting, the sum rule states that the average global external force vanishes in thermal equilibrium.