Exactly Solvable 1D Quantum Models with Gamma Matrices

TL;DR

This paper introduces exactly solvable one-dimensional quantum models using Gamma matrices, leading to quadratic fermionic Hamiltonians and revealing quantum phase transitions.

Contribution

It presents a novel class of exactly solvable 1D quantum models based on Gamma matrices, generalizing XY and Ising models.

Findings

Models are exactly solvable with quadratic fermionic Hamiltonians

Demonstrates quantum phase transitions in the new models

Uses 4-dimensional Gamma matrices as a specific example

Abstract

In this paper, we write exactly solvable generalizations of 1-dimensional quantum XY and Ising-like models by using $2^d$-dimensional Gamma ($\Gamma$) matrices as the degrees of freedom on each site. We show that these models result in quadratic Fermionic Hamiltonians with Jordan-Wigner like transformations. We illustrate the techniques using a specific case of 4-dimensional $\Gamma$ matrices and explore the quantum phase transitions present in the model.