# The A-Cycle Problem In XY model with Ring Frustration

## TL;DR

This paper rigorously analyzes the XY model with ring frustration, revealing two unique gapless phases by diagonalizing the model without approximations, thus addressing the previously neglected 'a-cycle' problem.

## Contribution

It introduces a precise solution to the XY model with ring frustration, explicitly treating the 'a-cycle' problem that was previously approximated or ignored.

## Key findings

- Discovery of two peculiar gapless phases
- Exact diagonalization of the XY model with ring frustration
- Clarification of the 'a-cycle' problem in XY models

## Abstract

Traditionally, the transverse spin-1/2 XY model is mapped to a fermionic "c-cycle" problem, where the prior periodic boundary condition is applied to the fermionic chain and the additional boundary term has been neglected. However, the "a-cycle" problem (the original problem without any approximation) has not been treated seriously up to now. In this paper, we consider the XY model with ring frustration and diagonalize it without any approximation with the help of parity constraint. Then two peculiar gapless phases have been found.