Quantization and Fractional Quantization of Currents in Periodically Driven Stochastic Systems I: Average Currents

TL;DR

This paper investigates how average currents in periodically driven stochastic systems become quantized or fractionally quantized at low temperatures, linking these phenomena to topological invariants and providing explicit formulas for analysis.

Contribution

It introduces a quantitative theory of current quantization in Markovian systems under adiabatic driving, connecting it to topological invariants and deriving explicit formulas using Kirchhoff's theorem.

Findings

Average currents become quantized or fractionally quantized at low temperatures.

The theory links current quantization to topological invariants.

Explicit formulas for average currents are derived using Kirchhoff's theorem.

Abstract

This article studies Markovian stochastic motion of a particle on a graph with finite number of nodes and periodically time-dependent transition rates that satisfy the detailed balance condition at any time. We show that under general conditions, the currents in the system on average become quantized or fractionally quantized for adiabatic driving at sufficiently low temperature. We develop the quantitative theory of this quantization and interpret it in terms of topological invariants. By implementing the celebrated Kirchhoff theorem we derive a general and explicit formula for the average generated current that plays a role of an efficient tool for treating the current quantization effects.