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arXiv:1809.09821 · Physica Scripta

An interplay of topology and quantized geometric phase for two different symmetry-class Hamiltonians

Rahul S, Ranjith Kumar R, Y R Kartik, Amitava Banerjee, Sujit Sarkar

Published Apr 29, 2019• 6 citations
arXivHTMLPDF

Abstract

Abstract Study of symmetry, topology and geometric phase can reveal many new and interesting results on the topological states of matter. Here we present a completely new and interesting result of symmetry, topology and quantization of geometric phase along with the physical explanation for two different symmetry classes. We present a detailed study of the auxiliary space for two different symmetry classes of Hamiltonians. We show explicitly that the origin of the auxiliary space inside the curve is only a necessary condition but it is not a sufficient condition for the topological state. One of the most interesting results is that same symmetry-class Hamiltonians show different behavior in topology and quantized geometric phase.

DOI
10.1088/1402-4896/ab1d7b

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Abstract Study of symmetry, topology and geometric phase can reveal many new and interesting results on the topological states of matter. Here we present a completely new and interesting result of symmetry, topology and quantization of geometric phase along with the physical explanation for two different symmetry classes. We present a detailed study of the auxiliary space for two different symmetry classes of Hamiltonians. We show explicitly that the origin of the auxiliary space inside the curve is only a necessary condition but it is not a sufficient condition for the topological state. One of the most interesting results is that same symmetry-class Hamiltonians show different behavior in topology and quantized geometric phase.
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The analysis of geometric phases associated with level crossing is reduced to the familiar diagonalization of the Hamiltonian in the second quantized formulation. A hidden local gauge symmetry, which is associated with the arbitrariness of the phase choice of a complete orthonormal basis set, becomes explicit in this formulation (in particular, in the adiabatic approximation) and specifies physical observables. The choice of a basis set which specifies the coordinate in the functional space is arbitrary in the second quantization, and a subclass of coordinate transformations, which keeps the form of the action invariant, is recognized as the gauge symmetry. We discuss the implications of this hidden local gauge symmetry in detail by analyzing geometric phases for cyclic and noncyclic evolutions. It is shown that the hidden local symmetry provides a basic concept alternative to the notion of holonomy to analyze geometric phases and that the analysis based on the hidden local gauge symmetry leads to results consistent with the general prescription of Pancharatnam. We however note an important difference between the geometric phases for cyclic and noncyclic evolutions. We also explain a basic difference between our hidden local gauge symmetry and a gauge symmetry (or equivalence class) used by Aharonov and Anandan in their definition of generalized geometric phases.
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.
The analysis of geometric phases is briefly reviewed by emphasizing various gauge symmetries involved. The analysis of geometric phases associated with level crossing is reduced to the familiar diagonalization of the Hamiltonian in the second quantized formulation. A hidden local gauge symmetry becomes explicit in this formulation and specifies physical observables; the choice of a basis set which specifies the coordinates in the functional space is arbitrary in the second quantization, and a sub-class of coordinate transformations, which keeps the form of the action invariant, is recognized as the gauge symmetry. It is shown that the hidden local symmetry provides a basic concept which replaces the notions of parallel transport and holonomy. We also point out that our hidden local gauge symmetry is quite different from a gauge symmetry used by Aharonov and Anandan in their definition of non-adiabatic phases.
All the geometric phases are shown to be topologically trivial by using the second quantized formulation. The exact hidden local symmetry in the Schrödinger equation, which was hitherto unrecognized, controls the holonomy associated with both of the adiabatic and nonadiabatic geometric phases. The second quantized formulation is located in between the first quantized formulation and the field theory, and thus it is convenient to compare the geometric phase with the chiral anomaly in field theory. It is shown that these two notions are completely different.