Abstract
We consider an integrable system of two one-dimensional fermionic chains connected by a link. The hopping constant at the link can be different from that in the bulk. Starting from an initial state in which the left chain is populated while the right is empty, we present time-dependent full counting statistics and the Loschmidt echo in terms of Fredholm determinants. Using this exact representation, we compute the above quantities as well as the current through the link, the shot noise and the entanglement entropy in the large time limit. We find that the physics is strongly affected by the value of the hopping constant at the link. If it is smaller than the hopping constant in the bulk, then a local steady state is established at the link, while in the opposite case all physical quantities studied experience persistent oscillations. In the latter case the frequency of the oscillations is determined by the energy of the bound state and, for the Loschmidt echo, by the bias of chemical potentials.
Highlights
The extension of the derivation to the case of finite temperatures (or, as a matter of fact, any finite-entropy initial state given by the single particle density ρ(E)) results in modifying the kernel according to X (φ, φ ) → ρ(E(φ))X (φ, φ ) ρ(E(φ )), which leads to the Fredholm determinant formula for full-counting statistics (FCS) (8) with the kernel (9)
In this paper we studied the non-equilibrium evolution of domain-wall type initial conditions in the free-fermionic one-dimensional system
Our main result is a representation of the full counting statistics and the return amplitude in the form of Fredholm determinants with integrable kernels
Summary
Quantum non-equilibrium dynamics of one-dimensional systems is a fascinating subject that nowadays attracts a lot of attention. After long time evolution from such a state a non-vanishing flow can still be present and the system relaxes to a non-equilibrium steady state This setup is the most relevant for transport properties and can be treated within the generalized hydrodynamics approach [15,16,17]. The presence of bound state provides oscillating long time behavior of the FCS, which results in an alternating non-vanishing current though the defect, the variance and the entropy rate. The frequency of these oscillations is given by the energy of the bound state. The appendix contains technical results and provides some reference materials for the Bessel functions
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