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Localization for random block operators related to the XY spin chain

Jacob Chapman, Günter Stolz

Published Aug 17, 2016• 19 citations
OpenAlex

Abstract

We study a class of random block operators which appear as effective one-particle Hamiltonians for the anisotropic XY quantum spin chain in an exterior magnetic field given by an array of i.i.d. random variables. For arbitrary non-trivial single-site distribution of the magnetic field, we prove dynamical localization of these operators at non-zero energy.

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We study a class of random block operators which appear as effective one-particle Hamiltonians for the anisotropic XY quantum spin chain in an exterior magnetic field given by an array of i.i.d. random variables. For arbitrary non-trivial single-site distribution of the magnetic field, we prove dynamical localization of these operators at non-zero energy.
Ever since the introduction of the Anderson model in 1958, physicists and mathematicians alike have been interested in the effects of disorder on quantum mechanical systems. For example, it is known that transport is suppressed for an electron moving about in a random environment, which follows from localization results proven for the Anderson model. Quantum spin systems provide a relatively simple starting point when one is interested in studying many-body systems. Here we investigate a random block operator arising from the anisotropic xy-spin chain model. Allowing for arbitrary nontrivial single-site distributions, we prove a zero-velocity Lieb-Robinson bound under the assumption of dynamical localization at all energies. After a preliminary study of basic properties and location of the almost-sure spectrum of this random block operator, we apply a transfer matrix formalism and prove contractivity and irreducibility properties of the Furstenberg group and, in particular, positivity of Lyapunov exponents at all nonzero energies. Then in the general setting of random block Jacobi matrices, we establish a Thouless formula, and under contractivity and irreducibility assumptions, we conclude dynamical localization via multiscale analysis by proving a Wegner estimate and an initial length scale estimate. Finally we apply our general results to prove localization for the special case of the Ising model, and we discuss a critical energy that arises.
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.
The spin-1/2 quantum anisotropic XY spin chain in a transverse random magnetic field parallel to the z axis is numerically studied by means of the density-matrix renormalization group. The dependence of the spontaneous magnetization and the energy gap on both the strength of the random magnetic field and the factor of anisotropy is determined. The critical line for the order-disorder phase transition is obtained and the resulting phase diagram is drawn. Our results are compatible with the fact that models with different factors of anisotropy fall within the same universality class as the quantum Ising model in a transverse random field.
The spin-1/2 quantum anisotropic XY\mathrm{XY}XY spin chain in a transverse random magnetic field parallel to the z axis is numerically studied by means of the density-matrix renormalization group. The dependence of the spontaneous magnetization and the energy gap on both the strength of the random magnetic field and the factor of anisotropy is determined. The critical line for the order-disorder phase transition is obtained and the resulting phase diagram is drawn. Our results are compatible with the fact that models with different factors of anisotropy fall within the same universality class as the quantum Ising model in a transverse random field.
Abstract We study the many-body localization (MBL) properties of the Heisenberg XXZ spin- 12\frac 1221​ chain in a random magnetic field. We prove that the system exhibits localization in any given energy interval at the bottom of the spectrum in a nontrivial region of the parameter space. This region, which includes weak interaction and strong disorder regimes, is independent of the size of the system and depends only on the energy interval. Our approach is based on the reformulation of the localization problem as an expression of quasi-locality for functions of the random many-body XXZ Hamiltonian. This allows us to extend the fractional moment method for proving localization, previously derived in a single-particle localization context, to the many-body setting.
We observe signatures of disorder-induced order in one-dimensional XY\mathit{XY}XY spin chains with an external site-dependent uniaxial random field within the XY\mathit{XY}XY plane. We numerically investigate signatures of a quantum phase transition at T=0T=0T=0, in particular, an upsurge of the magnetization in the direction orthogonal to the external magnetic field and the scaling of the block entropy with the amplitude of this field. Also, we discuss possible realizations of this effect in ultracold atomic experiments.