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

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.

22 pages, 7 figures

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• Nonadiabatic Control of Geometric Pumping
• Geometrical Formulation of Adiabatic Pumping as a Heat Engine
• Gauge freedom in observables and Landsbergs nonadiabatic geometric phase: pumping spectroscopy of interacting open quantum systems
• No-pumping theorem for many particle stochastic pumps
• Geometric Fluctuation Theorem
• Non-adiabatic effect in quantum pumping for a spin-boson system
• Quantization and Fractional Quantization of Currents in Periodically Driven Stochastic Systems II: Full Counting Statistics
• Unification and new extensions of the no-pumping theorems of stochastic pumps
• Fluctuation Relations For Adiabatic Pumping
• Hybrid models of molecular machines and the no-pumping theorem
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• Stochastic Dynamics of Extended Objects in Driven Systems II: Current Quantization in the Low-Temperature Limit
• The validity of the no-pumping theorem in systems with finite-range interactions between particles
• Pumping conductance, the intrinsic anomalous Hall effect, and statistics of topological invariants
• Geometric criterion of topological phase transition for non-Hermitian systems
• Geometric pumping induced by shear flow in dilute liquid crystalline polymer solutions