Probability Conservation and Localization in a One-Dimensional Non-Hermitian System

• URL: http://arxiv.org/abs/2310.00830v1
• Date: Mon, 2 Oct 2023 00:32:18 GMT
• Title: Probability Conservation and Localization in a One-Dimensional Non-Hermitian System
• Authors: Yositake Takane, Shion Kobayashi, and Ken-Ichiro Imura
• Abstract summary: We consider transport through a non-Hermitian conductor connected to a pair of Hermitian leads. In a typical non-Hermitian system, the continuity of probability and probability current is broken at a local level. We derive a global probability conservation law that relates $R_rm I$ and $R_rm T$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: We consider transport through a non-Hermitian conductor connected to a pair of Hermitian leads and analyze the underlying non-Hermitian scattering problem. In a typical non-Hermitian system, such as a Hatano--Nelson-type asymmetric hopping model, the continuity of probability and probability current is broken at a local level. As a result, the notion of transmission and reflection probabilities becomes ill-defined. Instead of these probabilities, we introduce the injection rate $R_{\rm I}=1-|{\cal R}|^2$ and the transmission rate $R_{\rm T}=|{\cal T}|^2$ as relevant physical quantities, where ${\cal T}$ and ${\cal R}$ are the transmission and reflection amplitudes, respectively. In a generic non-Hermitian case, $R_{\rm I}$ and $R_{\rm T}$ have independent information. We provide a modified continuity equation in terms of incoming and outgoing currents, from which we derive a global probability conservation law that relates $R_{\rm I}$ and $R_{\rm T}$. We have tested the usefulness of our probability conservation law in the interpretation of numerical results for non-Hermitian localization and delocalization phenomena.

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