[Paper Review] Auxiliary generalized Brillouin zone method in non-Hermitian band theory

Zhesen Yang, Kai Zhang+2 more|arXiv (Cornell University)|Dec 11, 2019

Quantum Mechanics and Non-Hermitian Physics11 citations

TL;DR

This paper introduces the auxiliary generalized Brillouin zone (aGBZ), an algebraic construct derived from the resultant of a non-Hermitian Hamiltonian’s characteristic polynomial, which fully encodes the analytic structure of the generalized Brillouin zone (GBZ). The aGBZ is a minimal analytic set containing all GBZ components—each of which is a piecewise analytic closed loop—enabling systematic analytical computation of the GBZ via algebraic geometry techniques.

ABSTRACT

We provide a systematic and self-consistent method to calculate the generalized Brillouin Zone (GBZ) analytically in one dimensional non-Hermitian systems. In general, a m-band non-Hermitian Hamiltonian is constituted by m distinct sub-GBZs, each of which is a piecewise analytic closed loop. Based on the concept of resultant, we can show that all the analytic properties of the GBZ can be characterized by an algebraic equation, the solution of which in the complex plane is dubbed as auxiliary GBZ (aGBZ). In general, the GBZ is a subset of aGBZ. On the other hand, the aGBZ is the minimal analytic element containing all the informations of GBZ. We also provide a systematic method to obtain the GBZ from aGBZ.

Motivation & Objective

• To develop a systematic and self-consistent method for analytically computing the generalized Brillouin zone (GBZ) in one-dimensional non-Hermitian systems.
• To characterize the analytic structure of the GBZ using algebraic geometry, particularly through the concept of the resultant.
• To identify the auxiliary GBZ (aGBZ) as the minimal analytic set that contains all information about the GBZ.
• To establish a systematic procedure to reconstruct the GBZ from the aGBZ, ensuring completeness and consistency.

Proposed method

• The method uses the resultant of the characteristic polynomial of a m-band non-Hermitian Hamiltonian to derive an algebraic equation in the complex plane.
• The solution set of this equation defines the auxiliary GBZ (aGBZ), which is a closed algebraic curve in the complex plane.
• Each connected component of the aGBZ corresponds to a sub-GBZ of the original system, with the full GBZ being a subset of the aGBZ.
• The method leverages piecewise analyticity and algebraic closure to ensure all GBZ components are captured.
• The procedure allows for the reconstruction of the GBZ by identifying which parts of the aGBZ satisfy the physical constraints of the system.
• The approach is generalizable to any m-band non-Hermitian Hamiltonian in one dimension, relying only on polynomial algebra and complex analysis.

Experimental results

Research questions

• RQ1 How can the generalized Brillouin zone (GBZ) of a one-dimensional non-Hermitian system be computed analytically in a systematic way?
• RQ2 What algebraic structure underlies the analytic properties of the GBZ in non-Hermitian band theory?
• RQ3 How is the auxiliary GBZ (aGBZ) related to the physical GBZ, and can it serve as a complete and minimal representation of the GBZ?
• RQ4 Can the full GBZ be reconstructed from the aGBZ using a well-defined procedure?
• RQ5 What role does the resultant of the characteristic polynomial play in characterizing the GBZ's analytic structure?

Key findings

• The generalized Brillouin zone (GBZ) of a m-band non-Hermitian Hamiltonian consists of m distinct sub-GBZs, each being a piecewise analytic closed loop.
• The auxiliary GBZ (aGBZ) is defined as the solution set of an algebraic equation derived from the resultant of the Hamiltonian’s characteristic polynomial.
• The aGBZ is a minimal analytic set that contains all information about the GBZ, with the GBZ being a subset of the aGBZ.
• The aGBZ provides a complete and self-consistent framework for computing the GBZ analytically, avoiding numerical approximations.
• The method enables the systematic reconstruction of the GBZ from the aGBZ by identifying physical components within the algebraic curve.
• The approach establishes a rigorous algebraic-geometric foundation for non-Hermitian band theory in one-dimensional systems.