# A new signature of quantum phase transitions from the numerical range

## TL;DR

This paper links quantum phase transitions to the geometry of the numerical range, showing how discontinuities in quantum state inference relate to boundary smoothness and energy level crossings.

## Contribution

It establishes a geometric criterion for quantum phase transitions based on the differential geometry of the numerical range boundary.

## Key findings

- Discontinuities in the inference map occur at smooth boundary crossings of energy levels.
- Discontinuities are associated with $C^1$-smooth boundary points of the numerical range.
- Stronger discontinuities occur at $C^2$-smooth non-analytic boundary points.

## Abstract

The ground state energy of a finite-dimensional one-parameter Hamiltonian and the continuity of a maximum-entropy inference map are discussed in the context of quantum critical phenomena. The domain of the inference map is a convex compact set in the plane, called the numerical range. We study the differential geometry of its boundary in relation to the ground state energy. We prove that discontinuities of the inference map correspond to $C^1$-smooth crossings of the ground state energy with a higher energy level. Discontinuities may appear only at $C^1$-smooth points of the boundary of the numerical range considered as a manifold. Discontinuities exist at all $C^2$-smooth non-analytic boundary points and are essentially stronger than at analytic points or at points which are merely $C^1$-smooth (non-exposed points).