Exactly Solvable 1D Quantum Models with Gamma Matrices

Abstract

In this paper, we write exactly solvable generalizations of 1-dimensional quantum XY and Ising-like models by using 2 d -dimensional Gamma (Γ) 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 Γ matrices and explore the quantum phase transitions present in the model.