Energy gap at first-order quantum phase transitions: An anomalous case

TL;DR

This paper investigates the diverse behaviors of the energy gap closing at first-order quantum phase transitions in the infinite-range quantum XY model, revealing complex dependencies on system parameters and challenging existing assumptions.

Contribution

It demonstrates that the energy gap can close at polynomial, exponential, or factorial rates, depending on the transverse field and system size choices, highlighting the lack of a universal rule.

Findings

Energy gap closing rates vary widely at first-order transitions.

The behavior depends on whether the transverse field is rational.

No universal relation between gap closing and transition order.

Abstract

We show that the rate of closing of the energy gap between the ground state and the first excited state, as a function of system size, behaves in many qualitatively different ways at first-order quantum phase transitions of the infinite-range quantum XY model. Examples include polynomial, exponential and even factorially fast closing of the energy gap, all of which coexist along a single axis of the phase diagram representing the transverse field. This variety emerges depending on whether or not the transverse field assumes a rational number as well as on how the series of system size is chosen toward the thermodynamic limit. We conclude that there is no generically applicable rule to relate the rate of gap closing and the order of quantum phase transitions as is often implied in many studies, in particular in relation to the computational complexity of quantum annealing in its implementation as quantum adiabatic computation.