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arXiv:1402.1480 · Journal of Mathematical Physics

A geometric approach to the Landauer-Büttiker formula

R. Ben Sâad, Claude‐Alain Pillet

Published Jun 19, 2014• 12 citations
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Abstract

We consider an ideal Fermi gas confined to a geometric structure consisting of a central region – the sample – connected to several infinitely extended ends—the reservoirs. Under physically reasonable assumptions on the propagation properties of the one-particle dynamics within these reservoirs, we show that the state of the Fermi gas relaxes to a steady state. We compute the expected value of various current observables in this steady state and express the result in terms of scattering data, thus obtaining a geometric version of the celebrated Landauer-Büttiker formula.

DOI
10.1063/1.4879238

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On Landauer–Büttiker formalism from a quantum quench

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Abstract We study transport in the free fermionic one-dimensional systems subjected to arbitrary local potentials. The bias needed for the transport is modeled by the initial highly non-equilibrium distribution where only half of the system is populated. Additionally to that, the local potential is also suddenly changed when the transport starts. For such a quench protocol we compute the full counting statistics (FCS) of the number of particles in the initially empty part. In the thermodynamic limit, the FCS can be expressed via the Fredholm determinant with the kernel depending on the scattering data and Jost solutions of the pre-quench and the post-quench potentials. We discuss the large-time asymptotic behavior of the obtained determinant and observe that if two or more bound states are present in the spectrum of the post-quench potential the information about the initial state manifests itself in the persistent oscillations of the FCS. On the contrary, when there are no bound states the asymptotic behavior of the FCS is determined solely by the scattering data of the post-quench potential, which for the current (the first moment) is given by the Landauer–Büttiker formalism. The information about the initial state can be observed only in the transient dynamics.

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We consider an ideal Fermi gas confined to a geometric structure consisting of a central region – the sample – connected to several infinitely extended ends—the reservoirs. Under physically reasonable assumptions on the propagation properties of the one-particle dynamics within these reservoirs, we show that the state of the Fermi gas relaxes to a steady state. We compute the expected value of various current observables in this steady state and express the result in terms of scattering data, thus obtaining a geometric version of the celebrated Landauer-Büttiker formula.
Using the formalism of Ford-Kac-Mazur we study nonequilibrium transport in fermionic systems connected to general particle and thermal reservoirs. Several exact results on electrical and thermal transport in one-dimensional disordered non-interacting fermionic wires are obtained. Landauer’s formula is shown to follow for a special choice of reservoirs. For general reservoirs the currents in the system depend on the spectral properties of the reservoirs and Landauer’s formula needs modification. Some experimental implications are discussed.
Electron conduction in one-dimensional system is investigated explicitly taking into account the reservoirs to which 1D channel is connected at its ends. We calculate the current driven by the chemical potential difference of two reservoirs, using the linear response theory by Kubo. At 0K, the conductance of 1D channel is expressed in terms of one-electron Green's functions even in the presence of the interaction, in contrast to the theory based on Kubo formula, in which two-electron Green's functions are essentially involved. The formalism in this paper provides a new interpretation of Landauer formula. As regards the interaction effects, the conductance of Fermi liquid without potential scattering is 2 e 2 / h , while that of Tomonaga-Luttinger liquid deceases with power low as the length of 1D channel increases.
Abstract We study transport in the free fermionic one-dimensional systems subjected to arbitrary local potentials. The bias needed for the transport is modeled by the initial highly non-equilibrium distribution where only half of the system is populated. Additionally to that, the local potential is also suddenly changed when the transport starts. For such a quench protocol we compute the full counting statistics (FCS) of the number of particles in the initially empty part. In the thermodynamic limit, the FCS can be expressed via the Fredholm determinant with the kernel depending on the scattering data and Jost solutions of the pre-quench and the post-quench potentials. We discuss the large-time asymptotic behavior of the obtained determinant and observe that if two or more bound states are present in the spectrum of the post-quench potential the information about the initial state manifests itself in the persistent oscillations of the FCS. On the contrary, when there are no bound states the asymptotic behavior of the FCS is determined solely by the scattering data of the post-quench potential, which for the current (the first moment) is given by the Landauer–Büttiker formalism. The information about the initial state can be observed only in the transient dynamics.