Anomalous behavior of the energy gap in the one-dimensional quantum XY model

TL;DR

This paper investigates the unusual non-monotonic behavior of the energy gap in a one-dimensional quantum XY model at a first-order quantum phase transition, revealing irregular gap scaling that impacts quantum annealing efficiency.

Contribution

It demonstrates the anomalous, irregular size dependence of the energy gap in the 1D quantum XY model at first-order transitions, contrasting typical exponential gap closing.

Findings

Energy gap behaves non-monotonically with system size

Gap closing can be polynomial, exponential, or factorial

Implications for quantum annealing efficiency

Abstract

We show for the one-dimensional quantum $XY$ model with $s=1/2$ that the energy gap between the ground and first excited states behaves anomalously as a function of the system size at the first-order quantum phase transition. Although it is generally the case that the energy gap closes exponentially at a quantum first-order transition, the gap in the present model behaves non-monotonically as a function of the system size, apparently very irregularly in some cases. This property of the gap is similar to that of the infinite-range quantum $XY$ model, in which the gap closes polynomially, exponentially, or even factorially fast depending on the choice of the series of system sizes toward the thermodynamic limit. This observation is surprising in consideration of the apparent maturity of our understanding of the one-dimensional quantum $XY$ model. Our result is also important from the viewpoint of quantum annealing, where the rate of gap closing determines the efficiency of computation.