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arXiv:1612.04837 · Journal of Physics A Mathematical and Theoretical

Temperature driven quenches in the Ising model: appearance of negative Rényi mutual information

Márton Kormos, Zoltán Zimborás

Published May 4, 2017• 35 citations
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Abstract

Abstract We study the dynamics of the transverse field Ising chain after a local quench in which two independently thermalised chains are joined together and are left to evolve unitarily. In the emerging non-equilibrium steady state the Rényi mutual information with different indices are calculated between two adjacent segments of the chain, and are found to scale logarithmically in the subsystem size. Surprisingly, for Rényi indices α > 2 we find cases where the prefactor of the logarithmic dependence is negative. The fact that the naively defined Rényi mutual information might be negative has been pointed out before, however, we provide the first example for this scenario in a realistic many-body setup. Our numerical and analytical results indicate that in this setup it can be negative for any index α > 2 while it is always positive for α < 2. Interestingly, even for α > 2 the calculated prefactors show some universal features: for example, the same prefactor is also shown to govern the logarithmic time dependence of the Rényi mutual information before the system relaxes locally to the steady state. In particular, it can decrease in the non-equilibrium evolution after the quench.

DOI
10.1088/1751-8121/aa70f6

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Abstract We study the dynamics of the transverse field Ising chain after a local quench in which two independently thermalised chains are joined together and are left to evolve unitarily. In the emerging non-equilibrium steady state the Rényi mutual information with different indices are calculated between two adjacent segments of the chain, and are found to scale logarithmically in the subsystem size. Surprisingly, for Rényi indices α > 2 we find cases where the prefactor of the logarithmic dependence is negative. The fact that the naively defined Rényi mutual information might be negative has been pointed out before, however, we provide the first example for this scenario in a realistic many-body setup. Our numerical and analytical results indicate that in this setup it can be negative for any index α > 2 while it is always positive for α < 2. Interestingly, even for α > 2 the calculated prefactors show some universal features: for example, the same prefactor is also shown to govern the logarithmic time dependence of the Rényi mutual information before the system relaxes locally to the steady state. In particular, it can decrease in the non-equilibrium evolution after the quench.
Abstract We consider a quench in a free-fermion chain by joining two homogeneous half-chains via a defect. The time evolution of the entanglement negativity is studied between adjacent segments surrounding the defect. In case of equal initial fillings, the negativity grows logarithmically in time and essentially equals one-half of the Rényi mutual information with index α = 1/2 in the limit of large segments. In sharp contrast, in the biased case one finds a linear increase followed by the saturation at an extensive value for both quantities, which is due to the backscattering from the defect and can be reproduced in a quasiparticle picture. Furthermore, a closer inspection of the subleading corrections reveals that the negativity and the mutual information have a small but finite difference in the steady state. Finally, we also study a similar quench in the XXZ spin chain via density-matrix renormalization group methods and compare the results for the negativity to the fermionic case.
We study entanglement properties in a nonequilibrium steady state of a free-fermion chain, that emerges after connecting two half-chains prepared at different temperatures. The entanglement negativity and the R\'enyi mutual information between two adjacent intervals scale logarithmically in the system size, with prefactors that we calculate analytically as a function of the bath temperatures. In particular, we show that the negativity and the R\'enyi mutual information with index $\ensuremath{\alpha}=1/2$ are described by different prefactors, and thus the two quantities provide inequivalent information about the state. Furthermore, we show that the logarithmic growth of the negativity during time evolution is also governed by the steady-state prefactor.
In a non-equilibrium many-body system, the quantum information dynamics between non-complementary regions is a crucial feature to understand the local relaxation towards statistical ensembles. Unfortunately, its characterization is a formidable task, as non-complementary parts are generally in a mixed state. We show that for integrable systems, this quantum information dynamics can be quantitatively understood within the quasiparticle picture for the entanglement spreading. Precisely, we provide an exact prediction for the time evolution of the logarithmic negativity after a quench. In the space-time scaling limit of long times and large subsystems, the negativity becomes proportional to the Rényi mutual information with Rényi index α = 1 / 2 . We provide robust numerical evidence for the validity of our results for free-fermion and free-boson models, but our framework applies to any interacting integrable system.
We investigate the non-equilibrium dynamics of the transverse field quantum Ising chain evolving from an inhomogeneous initial state given by joining two macroscopically different semi-infinite chains. We obtain integral expressions for all two-point correlation functions of the Jordan--Wigner Majorana fermions at any time and for any value of the transverse field. Using this result, we analytically compute the profiles of various physical observables in the space-time scaling limit and show that they can be obtained from a hydrodynamic picture based on ballistically propagating quasiparticles. Going beyond the hydrodynamic limit, we analyze the approach to the non-equilibrium steady state and find that the leading late time corrections display a lattice effect. We also study the fine structure of the propagating fronts which are found to be described by the Airy kernel and its derivatives. Near the front we observe the phenomenon of energy back-flow where the energy locally flows from the colder to the hotter region.
In this thesis, we study the properties of the transverse field Ising (TFI) chain in several non-equilibrium setups. One such setup is finite-time quantum quench—quench in which a Hamiltonian parameter is continuously changed during a finite-time, as opposed to the usual instantaneous change in the sudden quenches. A different type of quantum quench, which does not involve changing a Hamiltonian parameter, can be realised by the so-called partitioning protocol. In this protocol, two independent chains are prepared at different temperatures and then coupled into a single system and evolved as such. Another possibility to take the system out of equilibrium is to introduce a driving—for example, by suddenly and continuously changing a Hamiltonian parameter over long times. We discuss the general treatment of each of these situations, and apply them to several quench and driving protocols in the TFI chain. We analyse the behaviour of several observables, such as the total energy, spin correlation functions and Loschmidt echo, during and after quenches and during driving. While we choose the specific protocols to display the dependence of observables on the protocol features, we also investigate the universal behaviours of the observables in the late-time limit. Isolated quenched TFI chains do not relax at late times, however, local observables in these systems do—they relax to a special stationary state described by the generalised Gibbs ensemble (GGE). A similar situation happens in the case of periodic driving of the TFI chain—the late time state of the driven TFI chain, at least as far as the local observables are concerned, it is a periodic steady state described by the periodic Gibbs ensemble (PGE). In this thesis, we observe the relaxation behaviour for various quench and driving protocols and observables. In addition to studying statistical ensembles in the context of relaxation of out-of-equilibrium systems, we describe a machine learning algorithm based on the generalised Gibbs ensemble. We first describe a traditional machine learning algorithm, the Boltzmann machine, and its quantum version. Based on our knowledge of the relaxation of the out-of-equilibrium TFI chain, we modify the existing Boltzmann machine algorithm to include the many GGE charges and test it on the commonly used MNIST dataset. Rather than learning a complicated energy function and using the Boltzmann distribution, we then use the relatively simple TFI Hamiltonian and learn the effective temperatures of the GGE and find reasonably low error rates, while learning a comparatively low number of parameters.
A bstract In this paper, we consider the dynamics of Rényi negativities after a quantum quench in the free-boson chain with homogeneous dissipation. Initially we prepare the system in the squeezed thermal state, and then let it evolves under the tight-binding bosonic Hamiltonian with local linear dissipation. We use the Lindblad equation to solve the time evolution of the covariance matrix, from which one can obtain the time dependence of Rényi negativities. We are interested in the weak dissipation hydrodynamic limit where a quasi-particle picture emerges. In this limit, exact results of non-equilibrium dynamics of Rényi negativities can be obtained using the stationary phase method. We consider the Rényi negativities between both adjacent and disjoint regions in a infinite chain. We numerically test our analytical predictions and perfect matches have found.