Geometric phase of the one-dimensional Ising chain in a longitudinal field

TL;DR

This paper investigates the geometric phase of a one-dimensional Ising chain in a longitudinal field, revealing nonzero Berry phases under certain conditions and exploring topological phase transitions using fermionic transformations and mean-field theory.

Contribution

It provides a theoretical analysis of the Berry phase in the Ising chain with a longitudinal field, addressing the topological phase transition and characterizing the ground state properties.

Findings

Nonzero Berry phases exist at zero longitudinal field in specific parameter regimes.

Theoretical characterization of topological phase transition in the longitudinal field case.

Ground state energy and wave function are derived using fermionic and mean-field methods.

Abstract

For the one-dimensional Ising chain with spin-$1/2$ and exchange couple $J$ in a steady transverse field(TF), an analytical theory has well been developed in terms of some topological order parameters such as Berry phase(BP). For a TF Ising chain, the nonzero BP which depends on the exchange couple and the field strength characterizes the corresponding symmetry breaking of parity and time reversal(PT). However, there seems to exist a topological phase transition for the one-dimensional Ising chain in a longitudinal field(LF) with the reduced field strength $\epsilon$. If the LF is added at zero temperature, researchers believe that the LF also could influence the PT-symmetry and there exists the discontinuous BP. But the theoretic characterization has not been well founded. This paper tries to aim at this problem. With the Jordan-Wigner transformation, we give the four-fermion interaction form of the Hamiltonian in the one-dimensional Ising chain with a LF. Further by the method of Wick's theorem and the mean-field theory, the four-fermion interaction is well dealt with. We solve the ground state energy and the ground wave function in the momentum space. We discuss the BP and suggest that there exist nonzero BPs when $\epsilon=0$ in the paramagnetic case where $J<0$ and when $-1<\epsilon<1$, in the diamagnetic case where $J>0$.