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arXiv:1310.4997

Lindblad equation for a non-interacting fermionic system: full-counting statistics

Mariya V. Medvedyeva, Stefan K. Kehrein

Published Oct 18, 2013• 4 citations
arXivHTMLPDF

Abstract

We develop a method of calculating the full-counting statistics for a non-interacting fermionic system coupled to the memory-less reservoirs. The evolution of the system is described by the Lindblad equation. By the basis change the Liouvillian operator is brought to the quadratic form. This allows us a straightforward calculation of any observable in the non-equilibrium steady state. We introduce the counting field in the Lindblad equation which brings us to the generating function and helps us to obtain all cumulants of the charge transport. For the two-site system we give the expression for the generating function. For system longer than two sites we perform numerical investigations which suggest that it in a uniform system the cumulants of order $k$ are independent of the size of the system for system sizes larger $k+1$. The counting statistics from the Lindblad approach does not take into account interference in the reservoirs which gives a decreased noise in comparison with the Green function method which describes phase coherent leads. The current obtained by two methods is the same, which relies on the current conservation. The Fano factors are different (with a linear relation connecting them) and allow to distinguish between memory-less and phase coherent reservoirs.

DOI
10.48550/arxiv.1310.4997

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We develop a method of calculating the full-counting statistics for a non-interacting fermionic system coupled to the memory-less reservoirs. The evolution of the system is described by the Lindblad equation. By the basis change the Liouvillian operator is brought to the quadratic form. This allows us a straightforward calculation of any observable in the non-equilibrium steady state. We introduce the counting field in the Lindblad equation which brings us to the generating function and helps us to obtain all cumulants of the charge transport. For the two-site system we give the expression for the generating function. For system longer than two sites we perform numerical investigations which suggest that it in a uniform system the cumulants of order kkk are independent of the size of the system for system sizes larger k+1k+1k+1. The counting statistics from the Lindblad approach does not take into account interference in the reservoirs which gives a decreased noise in comparison with the Green function method which describes phase coherent leads. The current obtained by two methods is the same, which relies on the current conservation. The Fano factors are different (with a linear relation connecting them) and allow to distinguish between memory-less and phase coherent reservoirs.
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The calculation of the full counting statistics of the charge within a finite interval of an interacting one-dimensional system of electrons is a fundamental, yet as of now, unresolved problem. Even in the noninteracting case, charge counting turns out to be more difficult than anticipated because it necessitates the calculation of a nontrivial determinant and requires regularization. Moreover, interactions in a one-dimensional system are best described using bosonization. However, this technique rests on a long-wavelength approximation and is a priori inapplicable for charge counting due to the sharp boundaries of the counting interval. To mitigate these problems, we investigate the counting statistics using several complementary approaches. To treat interactions, we develop a diagrammatic approach in the fermionic basis, which makes it possible to obtain the cumulant generating function up to arbitrary order in the interaction strength. Importantly, our formalism preserves charge quantization in every perturbative order. We derive an exact expression for the noise and analyze its interaction-dependent logarithmic cutoff. We compare our fermionic formalism with the results obtained by other methods, such as the Wigner crystal approach and numerical calculations using the density-matrix renormalization group. Surprisingly, we show good qualitative agreement with the Wigner crystal for weak interactions, where the latter is in principle not expected to apply.
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Unless constrained by symmetry, measurement of an observable on an ensemble of identical quantum systems returns a distribution of values which are encoded in the full counting statistics. While the mean value of this distribution is important for determining certain properties of a system, the full distribution can also exhibit universal behavior. In this paper we study the full counting statistics of particle number in one-dimensional interacting Bose and Fermi gases which have been quenched far from equilibrium. In particular, we consider the time evolution of the Lieb-Liniger and Gaudin-Yang models quenched from a Bose-Einstein condensate initial state and calculate the full counting statistics of the particle number within a subsystem. We show that the scaled cumulants of the charge in the initial state and at long times are simply related and, in particular, the latter are independent of the model parameters. Using the quasiparticle picture we obtain the full time evolution of the cumulants and find that although their end points are fixed, the finite-time dynamics depends strongly on the model parameters. We go on to construct the scaled cumulant generating functions and from this determine the limiting charge probability distributions at long time which are shown to exhibit distinct, nontrivial, and non-Gaussian fluctuations and large deviations. Published by the American Physical Society 2024