Non-Local Currents and the Structure of Eigenstates in Planar Discrete Systems with Local Symmetries

TL;DR

This paper extends the framework of non-local currents to analyze eigenstate structures in planar discrete systems with local symmetries, revealing conditions for local symmetry and eigenstate properties.

Contribution

It introduces a Kirchhoff-type law for non-local currents applicable to all planar discrete Schrödinger systems with local symmetries, including non-uniform connectivities.

Findings

Locally symmetric eigenstates can be supported in specific subsystems.

Conditions for spatially constant non-local currents are derived.

The framework applies to systems with non-uniform connectivity.

Abstract

Local symmetries are spatial symmetries present in a subdomain of a complex system. By using and extending a framework of so-called non-local currents that has been established recently, we show that one can gain knowledge about the structure of eigenstates in locally symmetric setups through a Kirchhoff-type law for the non-local currents. The framework is applicable to all discrete planar Schr\"odinger setups, including those with non-uniform connectivity. Conditions for spatially constant non-local currents are derived and we explore two types of locally symmetric subsystems in detail, closed-loops and one-dimensional open ended chains. We find these systems to support locally similar or even locally symmetric eigenstates.