Signatures of real-space geometry, topology, and metric tensor in quantum transport in periodically corrugated spaces

TL;DR

This paper investigates how real-space geometry, topology, and metric variations influence quantum transport in periodically corrugated two-dimensional spaces, revealing emergent phenomena and effective potentials affecting conductance.

Contribution

It introduces a framework for analyzing quantum transport on manifolds with periodically modulated metrics, highlighting the impact of topology and geometry on quantum behavior.

Findings

Formation of energy bands due to metric modulation

Calculation of quantum conductance using S-matrix approach

Topology affects global properties like Aharonov-Bohm phase

Abstract

The motion of a quantum particle constrained to a two-dimensional non-compact Riemannian manifold with non-trivial metric can be described by a flat-space Schroedinger-type equation at the cost of introducing local mass and metric and geometry-induced effective potential with no classical counterpart. For a metric tensor periodically modulated along one dimension, the formation of bands is demonstrated and transport-related quantities are derived. Using S-matrix approach, the quantum conductance along the manifold is calculated and contrasted with conventional quantum transport methods in flat spaces. The topology, e.g. whether the manifold is simply connected, compact or non-compact shows up in global, non-local properties such as the Aharonov-Bohm phase. The results vividly demonstrate emergent phenomena due to the interplay of reduced-dimensionality, particles quantum nature, geometry, and topology.