Quantum model of interacting ``strings'' on the square lattice

TL;DR

This paper introduces a quantum model extending the 1D XY-spin chain to a 2D square lattice, analyzing string states and solving the eigenvalue problem for specific configurations, revealing integrable and non-integrable cases.

Contribution

It generalizes the XY-spin chain to two dimensions and provides solutions for string eigenstates, highlighting differences between free-fermion and self-interacting systems.

Findings

Eigenvalues for single-string states with fixed ends

Equivalence of certain string cases to free-fermion models

Identification of a non-integrable self-interacting string system

Abstract

The model which is the generalization of the one-dimensional XY-spin chain for the case of the two-dimensional square lattice is considered. The subspace of the ``string'' states is studied. The solution to the eigenvalue problem is obtained for the single ``string'' in cases of the ``string'' with fixed ends and ``string'' of types (1,1) and (1,2) living on the torus. The latter case has the features of a self-interacting system and looks not to be integrable while the previous two cases are equivalent to the free-fermion model.