# How “Berry Phase” Analysis of Non-Adiabatic Non-Hermitian Systems Reflects Their Geometry

## Abstract

**:**

## 1. Overview

**H**Ψ = EΨ, where

**H**is the Hamiltonian operator, Ψ is the “wavefunction” and E represents the eigenvalues of the operator, the point being that real eigenvalues are only guaranteed for an Hermitian operator. The topical review of Ghatak and Das [16] concentrates on non-Hermitian systems but underlines the general applicability of the ideas.

## 2. The Geometry of Irreversibility

**η**is defined by the relation between the electric field (

**E**) and the displacement (

**D**) vectors:

**E**=

**η ∙ D**. Then, the symmetric part of

**η**is the anisotropy tensor and the antisymmetric part is determined by the optical activity vector. For a dichroic crystal, the anisotropy tensor is complex (for a transparent crystal it is real). By formally considering all the possibilities (only touched on here), Berry and Dennis [37] show how the strange behaviour of such “simple” systems can be elegantly modelled in the general case.

**D**can be represented in a type of “Schrödinger equation”:

**M**∙

**D**= λ

**D**

**M**. This matrix is real symmetric for a transparent nonchiral crystal but complex non-Hermitian for a dichroic chiral crystal. Examination of Figure 2 illustrates discontinuities of various sorts in the phase solutions of this equation in a general case. It is the behaviour of these discontinuities that characterises the system behaviour.

## 3. Other examples of Berry phase treatments of Irreversibility

^{14}N nuclear spin may be “polarised by optical excitation”. The qubit readout is from the fluorescence generated by the excitation. Ma et al. [59] demonstrate that this device is capable of pumping charge, and they interpret it as a “generalised Thouless pump” after Thouless [60], who first showed how such a charge pump would work. Takahashi et al. [61] explains that such “geometrical pumping” is a topological phenomenon to be interpreted in terms of the Berry phases of the system and that it is not restricted to adiabatic processes but is quite general.

## 4. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Caustics produced by a glass of water (wikimedia commons; © Heiner Otterstedt, Jan.2006; CCBY-SA3.0).

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Jeynes, C.
How “Berry Phase” Analysis of Non-Adiabatic Non-Hermitian Systems Reflects Their Geometry. *Entropy* **2023**, *25*, 390.
https://doi.org/10.3390/e25020390

**AMA Style**

Jeynes C.
How “Berry Phase” Analysis of Non-Adiabatic Non-Hermitian Systems Reflects Their Geometry. *Entropy*. 2023; 25(2):390.
https://doi.org/10.3390/e25020390

**Chicago/Turabian Style**

Jeynes, Chris.
2023. "How “Berry Phase” Analysis of Non-Adiabatic Non-Hermitian Systems Reflects Their Geometry" *Entropy* 25, no. 2: 390.
https://doi.org/10.3390/e25020390