Physical origins of ruled surfaces on the reduced density matrices geometry

TL;DR

This paper investigates the physical origins of ruled surfaces on the geometry of reduced density matrices in quantum many-body systems, revealing they can indicate either symmetry-breaking or gapless phases, with a finite size scaling method to distinguish these origins.

Contribution

It introduces a finite size scaling approach to differentiate ruled surfaces caused by symmetry-breaking from those due to gapless systems in quantum models.

Findings

Ruled surfaces can signal symmetry-breaking or gapless phases.

Finite size scaling effectively distinguishes the origin of ruled surfaces.

Application to the two-mode XY model confirms the method's validity.

Abstract

The reduced density matrices (RDMs) of many-body quantum states form a convex set. The boundary of low dimensional projections of this convex set may exhibit nontrivial geometry such as ruled surfaces. In this paper, we study the physical origins of these ruled surfaces for bosonic systems. The emergence of ruled surfaces was recently proposed as signatures of symmetry-breaking phase. We show that, apart from being signatures of symmetry-breaking, ruled surfaces can also be the consequence of gapless quantum systems by demonstrating an explicit example in terms of a two-mode Ising model. Our analysis was largely simplified by the quantum de Finetti's theorem---in the limit of large system size, these RDMs are the convex set of all the symmetric separable states. To distinguish ruled surfaces originated from gapless systems from those caused by symmetry-breaking, we propose to use the finite size scaling method for the corresponding geometry. This method is then applied to the two-mode XY model, successfully identifying a ruled surface as the consequence of gapless systems.