Chirality, Causality, and Fluctuation-Dissipation Theorems in Nonequilibrium Steady States

Abstract

Edges of some quantum Hall liquids and a number of other systems exhibit chiral transport: excitations can propagate in one direction only, e.g., clockwise. We derive a family of fluctuationdissipation relations in non-equilibrium steady states of such chiral systems. The theorems connect nonlinear response with fluctuations far from thermal equilibrium and hold only in case of chiral transport. They can be used to test chiral or non-chiral character of the system. PACS numbers: 05.70. Ln, 05.40.Ca, 73.43.Cd According to the causality principle, past events influence the future but the future has no effect on the past. This principle has no general counterpart in terms of the spatial separation of events: consequences of some events can be felt in every point after a sufficient wait time. A spatial version of the causality principle emerges in low-energy effective theories of some many-body systems. The best known example is the integer quantum Hall effect (QHE): low-energy excitations are confined to the edges and can propagate only clockwise or counterclockwise [1]. This can lead to a situation in which earlier events affect only those future events that occur "downstream". Similar chiral transport is possible in a number of other systems: some fractional quantum Hall liquids [1], interfaces of topological insulators, superconductors and ferromagnets [2,3], surface states in 3D QHE and so on. The simplest example comes from the statistical mechanics models of traffic [4]: chiral transport is possible on a network of one-way roads as long as no traffic jams form.In this paper we explore consequences of the extended causality principle in chiral systems. Causality is crucial for linear response theory. One of its celebrated results is the fluctuation-dissipation theorem (FDT). We show that a family of generalized FDTs holds in chiral sys-(a) tems. While the usual FDT applies in thermal equilibrium only, our theorems are also valid in non-equilibrium steady states. The simplest relation [5][6][7] of such sort was derived for the exactly-solvable chiral Luttinger liquid model with a single impurity. We have recently found an FDT-type relation between the current noise and nonlinear conductance in a general chiral system in a non-equilibrium steady state in a three-terminal geometry [8]. In this paper we prove a much more general result: we express nonlinear responses of the currents of various conserved quantities, such as the electric current and thermal current, in terms of the second and higher order cumulants of the statistical distributions of the currents in a nonequilibrium steady state in a multi-terminal system with an arbitrary number of terminals. The generalization is achieved due to a much simpler approach. The result for the chiral Luttinger liquid model follows from its technically difficult exact solution [7]. A more general result [8] was obtained with a simpler but still rather subtle method, generalizing the equilibrium Kubo formalism. In this paper we use a completely different trick based o...

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