The XX--model with boundaries. Part I: Diagonalization of the finite chain

TL;DR

This paper analyzes the diagonalization of the finite XX quantum chain with arbitrary boundary terms, extending previous work to include non-hermitian boundaries and deriving spectral and wave-function solutions.

Contribution

It introduces a method to diagonalize the XX chain with general boundary conditions, including non-hermitian cases, and provides explicit solutions for certain analytically solvable scenarios.

Findings

Derived the spectrum and wave-functions for specific boundary conditions.

Identified cases where the secular equation can be solved analytically.

Expressed one- and two-point functions in terms of Pfaffians.

Abstract

This is the first of three papers dealing with the XX finite quantum chain with arbitrary, not necessarily hermitian, boundary terms. This extends previous work where the periodic or diagonal boundary terms were considered. In order to find the spectrum and wave-functions an auxiliary quantum chain is examined which is quadratic in fermionic creation and annihilation operators and hence diagonalizable. The secular equation is in general complicated but several cases were found when it can be solved analytically. For these cases the ground-state energies are given. The appearance of boundary states is also discussed and in view to the applications considered in the next papers, the one and two-point functions are expressed in terms of Pfaffians.