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Entanglement Dynamics of Disordered Quantum XY Chains

Abdul-Rahman, Houssam, Nachtergaele, Bruno, Sims, Robert, Stolz, Günter

Published Jan 1, 2015• 20 citations
OpenAlex

Abstract

We consider the dynamics of the quantum XY chain with disorder under the\ngeneral assumption that the expectation of the eigenfunction correlator of the\nassociated one-particle Hamiltonian satisfies a decay estimate typical of\nAnderson localization. We show that, starting from a broad class of product\ninitial states, entanglement remains bounded for all times. For the XX chain,\nwe also derive bounds on the particle transport which, in particular, show that\nthe density profile of initial states that consist of fully occupied and empty\nintervals, only have significant dynamics near the edges of those intervals,\nuniformly for all times.

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We consider the dynamics of the quantum XY chain with disorder under the\ngeneral assumption that the expectation of the eigenfunction correlator of the\nassociated one-particle Hamiltonian satisfies a decay estimate typical of\nAnderson localization. We show that, starting from a broad class of product\ninitial states, entanglement remains bounded for all times. For the XX chain,\nwe also derive bounds on the particle transport which, in particular, show that\nthe density profile of initial states that consist of fully occupied and empty\nintervals, only have significant dynamics near the edges of those intervals,\nuniformly for all times.
We review several aspects of Many‐Body Localization‐like properties exhibited by the disordered XY chains: localization properties of the energy eigenstates and thermal states, propagation bounds of Lieb‐Robinson type, decay of correlation functions, absence of particle transport, bounds on the bipartite entanglement, and bounded entanglement growth under the dynamics. We also prove new results on the absence of energy transport and Fock space localization. All these properties are made accessible to mathematical analysis due to the exact mapping of the XY chain to a system of quasi‐free fermions given by the Jordan‐Wigner transformation. Motivated by these results we discuss conjectured properties of more general disordered quantum spin and other systems as possible directions for future mathematical research.
We consider the isotropic or anisotropic XY spin chain in the presence of a transversal random magnetic field, with parameters given by random variables. It is shown that eigenfunction correlator localization of the corresponding effective one-particle Hamiltonian implies a uniform area law bound in expectation for the bipartite entanglement entropy of all eigenstates of the XY chain, i.e., a form of many-body localization at all energies. Here, entanglement with respect to arbitrary connected subchains of the chain can be considered. Applications where the required eigenfunction correlator bounds are known include the isotropic XY chain in random field as well as the anisotropic chain in strong or strongly disordered random field.
We review recent results on many-body localization for two explicitly analyzable models of many-body quantum systems, the XY spin chain in transversal magnetic field as well as interacting systems of harmonic quantum oscillators. In both models the presence of disorder leads to dynamical localization in the form of zero-velocity Lieb-Robinson bounds and to exponential decay of ground state correlations. Moreover, for oscillator systems one can also show exponential decay of thermal states as well as an area law bound for the entanglement entropy of ground and thermal states. The key fact which allows a rigorous analysis of these models is that they are given by many-body Hamiltonians which can be reduced to effective single particle Hamiltonians.
We study the non-equilibrium dynamics of a disordered quantum system consisting of harmonic oscillators in a [Formula: see text]-dimensional lattice. If the system is sufficiently localized, we show that, starting from a broad class of initial product states that are associated with a tiling (decomposition) of the [Formula: see text]-dimensional lattice, the dynamical evolution of entanglement follows an area law in all times. Moreover, the entanglement bound reveals a dependency on how the subsystems are located within the lattice in dimensions [Formula: see text]. In particular, the entanglement grows with the maximum degree of the dual graph associated with the lattice tiling.
We study the unitary relaxation dynamics of disordered spin chains following a sudden quench of the Hamiltonian. We give analytical arguments, corroborated by specific numerical examples, to show that the existence of a stationary state depends crucially on the spectral and localization properties of the final Hamiltonian, and not on the initial state. We test these ideas on integrable one-dimensional models of the Ising or XY class, but argue more generally on their validity for more complex (nonintegrable) models.
We study out-of-time ordered correlations C(x,t)C(x,t)C(x,t) and entanglement growth in the random-field XX model with open boundary conditions using the exact Jordan-Wigner transformation to a fermionic Hamiltonian. For any nonzero strength of the random field, this model describes an Anderson insulator. Two scenarios are considered: a global quench with the initial state corresponding to a product state of the N\'eel form, and the behavior in a typical thermal state at $\ensuremath{\beta}=1$. As a result of the presence of disorder, the information spreading as described by the out-of-time correlations stops beyond a typical length scale ${\ensuremath{\xi}}_{\text{OTOC}}$. For $|x|<{\ensuremath{\xi}}_{\text{OTOC}}$, information spreading occurs at the maximal velocity vmax=J{v}_{\mathrm{max}}=Jvmax​=J and we confirm predictions for the early-time behavior of $C(x,t)\ensuremath{\sim}{t}^{2|x|}$. For the case of the quench starting from the N\'eel product state, we also study the growth of the bipartite entanglement, focusing on the late- and infinite-time behavior. The approach to a bounded entanglement is observed to be slow for the disorder strengths we study.