Algebraic approach and Berry phase of a Hamiltonian with a general $SU(1,1)$ symmetry

TL;DR

This paper analyzes a Hamiltonian with $SU(1,1)$ symmetry, diagonalizes it using group transformations, and computes the Berry phase for its time-dependent form, advancing understanding of geometric phases in such systems.

Contribution

It introduces a method to diagonalize a general $SU(1,1)$-structured Hamiltonian and calculates its Berry phase, providing new insights into geometric phases in systems with this symmetry.

Findings

Successfully diagonalized the Hamiltonian using $SU(1,1)$ and $SU(2)$ transformations.

Derived an explicit expression for the Berry phase in the general $SU(1,1)$ case.

Enhanced understanding of geometric phases in systems with $SU(1,1)$ symmetry.

Abstract

In this paper we study a general Hamiltonian with a linear structure given in terms of two different realizations of the $SU(1,1)$ group. We diagonalize this Hamiltonian by using the similarity transformations of the $SU(1,1)$ and $SU(2)$ displacement operators performed to the $su(1,1)$ Lie algebra generators. Then, we compute the Berry phase of a general time-dependent Hamiltonian with this general $SU(1,1)$ linear structure.