Localization of the random XXZ spin chain in fixed energy intervals

Abstract: A Schrödinger operator $H$ is known to exhibit quasi-locality: Matrix elements of analytic functions of $H$ decay exponentially away from the diagonal. Localization for a random Schrödinger operator can be expressed as an extension of this feature to Borel measurable functions with support in the region of localization.

We show that the spin-$\frac12$ XXZ random chain Hamiltonian $H_{XXZ}$ manifests a suitably defined notion of quasi-locality as well. We exploit this property to prove that $H_{XXZ}$ exhibits localization in any fixed energy interval (expressed as quasi-locality for Borel measurable functions of $H_{XXZ}$ with support in the energy interval). Localization is proved in a nontrivial region of the parameter space, which includes weak interaction and strong disorder regimes, and is independent of the size of the system, depending only on the energy interval.

Based on a joint work with Abel Klein, arxiv.org/abs/2210.14873

mathematical physics

Audience: researchers in the discipline

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