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

TL;DR

This paper investigates the unusual size-dependent behavior of the energy gap in the one-dimensional quantum XY model, revealing irregular and oscillatory patterns at phase transitions that impact quantum annealing efficiency.

Contribution

It uncovers the anomalous energy gap behavior in the XY model, contrasting with typical power-law decay, and links these findings to correlation functions and quantum annealing.

Findings

The energy gap behaves irregularly at the second order transition in the XX model.

The gap exhibits nontrivial oscillations in the anisotropic model's incommensurate phase.

The anomalous gap behavior influences quantum annealing performance.

Abstract

We re-examine the well-studied one dimensional spin-1/2 $XY$ model to reveal its nontrivial energy spectrum, in particular the energy gap between the ground state and the first excited state. In the case of the isotropic $XY$ model -- the $XX$ model -- the gap behaves very irregularly as a function of the system size at a second order transition point. This is in stark contrast to the usual power-law decay of the gap and is reminiscent of the similar behavior at the first order phase transition in the infinite-range quantum $XY$ model. The gap also shows nontrivial oscillatory behavior for the phase transitions in the anisotropic model in the incommensurate phase. We observe a close relation between this anomalous behavior of the gap and the correlation functions. These results, those for the isotropic case in particular, are important from the viewpoint of quantum annealing where the efficiency of computation is strongly affected by the size dependence of the energy gap.