Analytical solutions for a boundary driven XY chain

TL;DR

This paper derives analytical solutions for the normal modes of boundary-driven XY spin chains with non-number-conserving Hamiltonians, reducing the problem to solving a scalar trigonometric equation, and provides explicit solutions for the Ising case.

Contribution

It introduces a method to diagonalize the master equation for boundary-driven XY chains by transforming it into a tridiagonal bordered Toeplitz matrix problem, enabling explicit solutions.

Findings

Normal modes can be obtained by diagonalizing an LxL tridiagonal bordered Toeplitz matrix.

Explicit analytical solutions are derived for the Ising chain case.

The approach simplifies the analysis of boundary-driven non-conserving fermionic systems.

Abstract

We study non-interacting fermionic systems dissipatively driven at their boundaries, focusing in particular on the case of a non-number-conserving Hamiltonian, which for example describes an $XY$ spin chain. We show that despite the lack of number conservation, it is possible to convert the problem of calculating the normal modes of the master equations and their corresponding rapidities, into diagonalizing simply an $L\times L$ tridiagonal bordered $2-$Toeplitz matrix, where $L$ is the size of the system. Such structure of matrix allows us to further reduce the problem into solving a scalar trigonometric non-linear equation for which we also show, in the case of an Ising chain, exact analytical explicit, and system size independent, solutions.