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arXiv:1610.01939 · Annalen der Physik

Localization properties of the disordered XY spin chain

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

Published Mar 24, 2017• 43 citations
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Abstract

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.

DOI
10.1002/andp.201600280

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This dissertation presents new results on two problems concerning the dynamics of certainclasses of interacting many-body systems in quantum mechanics. Chapter 1 is an introduction,\nwhich includes mathematical preliminaries and a summary of the results.\nIn Chapter 2 we introduce the notion of transmission time to study the dynamics of disordered\nquantum spin chains and prove results relating its behavior to many-body localization properties.\nWe also study two versions of the so-called Local Integrals of Motion (LIOM) representation of\nspin chain Hamiltonians and their relation to dynamical many-body localization. We prove that\nuniform-in-time dynamical localization expressed by a zero-velocity Lieb-Robinson bound implies\nthe existence of a LIOM representation of the dynamics as well as a weak converse of this statement.\nWe also prove that for a class of spin chains satisfying a form of exponential dynamical localization,\nsparse perturbations result in a dynamics in which transmission times diverge at least as a power\nlaw of distance, with a power for which we provide lower bound that diverges with increasing\nsparseness of the perturbation.\nIn Chapter 3 we introduce a class of UV-regularized two-body interactions for fermions in Rd\nand prove a Lieb-Robinson estimate for the dynamics of this class of many-body systems. As a\nstep toward this result, we also prove a propagation bound of Lieb-Robinson type for Schrodinger\noperators. We apply the propagation bound to prove the existence of infnite-volume dynamics as\na strongly continuous group of automorphisms on the CAR algebra.
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The study of disordered systems in statistical mechanics is a topic that garners strong interest from the mathematical and physics communities. In this dissertation, we present two problems dealing with some type of strongly disordered interacting particle system. In particular, we study properties related to transport of energy in these systems. The first model deals with the spreading of wave packets in a 1D lattice governed by the disordered discrete non-linear Schrödinger (DNLS) equation. We study the rate of decay of the maximum of a wave packet initially supported around the origin, and prove slower than power law spreading. The second model deals with many-body localization (MBL), a phenomenon in which certain strongly disordered interacting particle systems fail to reach thermal equilibrium. We prove the vanishing of the heat conductivity for a robust 1D spin chain model. This result serves to constitute a first step toward a rigorous proof of many-body localization.
The Lieb-Robinson correlation function can be used to characterize the propagation of quantum information in Ising chains. Considerable work has been done to establish bounds on this correlation function in various circumstances. To actually calculate the value of the correlation function directly typically requires a state space which grows exponentially with system size, and so is intractable for all but relatively small systems. We employ a recently-developed method that enables direct calculation of the Lieb--Robinson correlation function and which scales linearly with system size. This enables the computation for systems with more than a thousand qubits, revealing the propagation of quantum information down the chain. We extend this technique to the problem of Ising chains with randomly disordered coupling strengths. Increasing disorder causes localization of the quantum correlations and halts propagation of quantum information down the chain. We see light-cone isocontours therefore becoming vertical. The decay of correlations down the chain is characterized by a correlation length which in the paramagnetic phase increases quadratically with mean coupling strength and inverse quadratically with disorder level. The correlation length peaks at the phase transition. We map our general expression for the correlation function in the Ising system onto the probability density in the single particle tight-binding Su-Schrieffer-Heeger model for a specific initial condition.
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.