Correspondence between the XXZ model in roots of unity and the one-dimensional quantum Ising chain with different boundary conditions

TL;DR

This paper establishes an exact correspondence between the XXZ model at roots of unity with specific boundary conditions and the quantum Ising chain with various boundary conditions, linking them through the minimal model LM(3,4).

Contribution

It demonstrates a precise mapping between the XXZ model at roots of unity and the quantum Ising chain with different boundary conditions via quantum group reduction.

Findings

Exact correspondence between XXZ model and quantum Ising chain for various boundary conditions.

Identification of the minimal model LM(3,4) as the underlying theory.

Connection established at anisotropy Δ=√2/2 with specific site numbers.

Abstract

We consider the integrable XXZ model with special open boundary conditions that renders its Hamiltonian ${SU(2)}_q$ symmetric, and the one-dimensional quantum Ising model with four different boundary conditions. We show that for each boundary condition the Ising quantum chain is exactly given by the Minimal Model of integrable lattice theory $LM(3, 4)$. This last theory is obtained as the result of the quantum group reduction of the XXZ model at anisotropy $\Delta=(q + q^{-1})/2=\sqrt{2}/2$, with a number of sites in the latter defined by the type of boundary conditions.