Quantization of geometric phase with integer and fractional topological characterization in a quantum Ising chain with long-range interaction

Sujit Sarkar

doi:10.1038/s41598-018-24136-1

Sujit Sarkar

Open Access

https://doi.org/10.1038/s41598-018-24136-1

Copy DOI

Abstract

An attempt is made to study and understand the behavior of quantization of geometric phase of a quantum Ising chain with long range interaction. We show the existence of integer and fractional topological characterization for this model Hamiltonian with different quantization condition and also the different quantized value of geometric phase. The quantum critical lines behave differently from the perspective of topological characterization. The results of duality and its relation to the topological quantization is presented here. The symmetry study for this model Hamiltonian is also presented. Our results indicate that the Zak phase is not the proper physical parameter to describe the topological characterization of system with long range interaction. We also present quite a few exact solutions with physical explanation. Finally we present the relation between duality, symmetry and topological characterization. Our work provides a new perspective on topological quantization.

Highlights

We have presented the quantization of geometric phase for both integer and fractionalize topological characterization for this system along with the physical explanations
We have shown that all the quantum critical lines are not topologically equivalent
The most important conclusion is that Zak phase is not the proper physical quantity for the characterization of topological state in the presence of long range interactions

Summary

Introduction

The physics of Berry Phase (geometric phase) has been playing an important role in understanding pivotal findings of quantum condensed-matter systems[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. The motivation of this research presented here is derived from the seminal work of Berry Berry has found this phase along with the dynamical phase of the system for the cyclic evolution of a wave function and, at the same time, this phase is gauge invariant. When the system is given to the cyclic evolution described by a closed curve It is evident from the analytical expression that Berry phase depends on the geometry of the parameter and loop (C) therein. This is the basic origin of geometrical character of the Berry phase. We mention very →briefly the famous example of geometric phase of spin-1/2 particle, which is moving in an external magnetic field B rotating adiabatically under an angle θ around z-axis.

Methods

Results

Conclusion