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arXiv:1911.01926 · SciPost Physics

Fredholm determinants, full counting statistics and Loschmidt echo for domain wall profiles in one-dimensional free fermionic chains

Oleksandr Gamayun, Oleg Valentinovich Lychkovskiy, Jean-Sébastien Caux

Published Mar 5, 2020• 61 citations
arXivHTMLPDF

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.

DOI
10.21468/scipostphys.8.3.036

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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.
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.
We consider free-fermion chains where full and empty parts are connected by a transition region with narrow surfaces. This can be caused by a linear potential or by time evolution from a step-like initial state. Entanglement spectra, entanglement entropies and fluctuations are determined for subsystems either in the surface region or extending into the bulk. In all cases there is logarithmic behaviour in the subsystem size, but the prefactors in the surface differ from those in the bulk by 3/2. A previous fluctuation result is corrected and a general scaling formula is inferred from the data.
We consider the non-equilibrium physics induced by joining together two tight-binding fermionic chains to form a single chain. Before being joined, each chain is in a many-fermion ground state. The fillings (densities) in the two chains might be different. We present a number of exact results, focusing on two-point correlators and the Loschmidt echo (return probability). For the non-interacting case, we identify through an exact derivation the regime in which a semiclassical ansatz is valid. We present a number of analytical results beyond semiclassics, such as the approach to the non-equilibrium steady state and the appearance of Tracy-Widom distributions at the front of the light cone. The light cone behavior is quantified through a series expansion in time, and this description is shown to be valid for interacting systems as well. Results on the Loschmidt echo, presented for finite and zero interactions, illustrate that the physics is different from both local and global quenches.
The full counting statistics is studied for a one-dimensional system of noninteracting fermions with and without disorder. For two unbiased LLL site lattices connected at time t=0t=0t=0, the charge variance increases as the natural logarithm of ttt, following the universal expression $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\approx}\frac{1}{{\ensuremath{\pi}}^{2}}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$. Since the static charge variance for a length lll region is given by $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\approx}\frac{1}{{\ensuremath{\pi}}^{2}}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}l$, this result reflects the underlying relativistic or conformal invariance and dynamical exponent z=1z=1z=1 of the disordered free lattice. With disorder and strongly localized fermions, we have compared our results to a model with a dynamical exponent $z\ensuremath{\ne}1$ and a model for entanglement entropy based upon dynamical scaling at the infinite disorder fixed point (IDFP). The latter scaling, which predicts $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\propto}\mathrm{ln}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$, appears to better describe the charge variance of disordered one-dimensional fermions. When a bias voltage is introduced, the behavior changes dramatically, and the charge and variance become proportional to ${(\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t)}^{1/\ensuremath{\psi}}$ and lnt\mathrm{ln}\phantom{\rule{0.16em}{0ex}}tlnt, respectively. The exponent $\ensuremath{\psi}$ may be related to the critical exponent, characterizing spatial/energy fluctuations at the IDFP.
We study the melting of a domain wall in free-fermion chains, where the periodic variation of the hopping amplitudes gives rise to a band structure. It is shown that the entanglement grows logarithmically in time, and the prefactor is proportional to the number of filled bands in the initial state. For a dimerized chain the particle density and current are found to have the same expressions as in the homogeneous case, up to a rescaling of the velocity. The universal contribution to the entropy profile is then doubled, while the non-universal part can be extracted numerically from block-Toeplitz matrices.
Abstract We study the quench dynamics in free fermionic systems in the prototypical bipartitioning protocol obtained by joining two semi-infinite subsystems prepared in different states, aiming at understanding the interplay between quantum coherences in space in the initial state and transport properties. Our findings reveal that, under reasonable assumptions, the more correlated the initial state, the slower the transport is. Such statement is first discussed at qualitative level, and then made quantitative by introducing proper measures of correlations and transport ‘speed’. Moreover, it is supported for fermions on a lattice by an exact solution starting from specific initial conditions, and in the continuous case by the explicit solution for a wider class of physically relevant initial states. In particular, for this class of states, we identify a function, that we dub transition map , which takes the value of the stationary current as input and gives the value of correlation as output, in a protocol-independent way. As an aside technical result, in the discrete case, we give an expression of the full counting statistics in terms of a continuous kernel for a general correlated domain wall initial state, thus extending the recent results in Moriya et al (2019 J. Stat. Mech. 2019 063105) on the one-dimensional XX spin chain.