Combinatorial Solution of One-Dimensional Quantum Systems

TL;DR

This paper presents a combinatorial approach to exactly solve one-dimensional quantum spin systems, deriving thermodynamic properties using Pfaffian methods under free fermion conditions.

Contribution

It introduces a combinatorial framework for solving 1D quantum spin models, specifically the XY model, using Pfaffian techniques and Trotter formula, which is a novel analytical approach.

Findings

Exact expressions for free energy and ground state energy as functions of temperature, couplings, and magnetic fields.

Solution applies to XY models with periodicities p=1 and 2 under free fermion conditions.

Method provides a new analytical tool for studying quantum spin chains.

Abstract

We give a self-contained exposition of the combinatorial solution of quantum mechanical systems of coupled spins on a one-dimensional lattice. Using Trotter formula, we write the partition function as a generating function of a spanning subgraph of a two-dimensional lattice and solve the combinatorial problem by the method of Pfaffians provided the weights satisfy the so-called free fermion condition. The free energy and the ground state energy as a function of the inverse temperature, couplings J and magnetic fields h, for the XY model in a transverse field with period p=1 and 2, is then obtained.