Localization Properties of the XY Spin Chain. A review of mathematical results with an eye toward Many-Body Localization

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.

arXiv commentFinal version, to appear in Annalen der Physik; title modified, introduction expanded, references added, new Lemma 8.1

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• Out of Time Ordered Correlators and Entanglement Growth in the Random Field XX Spin Chain
• Slow growth of out-of-time-order correlators and entanglement in integrable disordered systems
• Low-energy Fock-space localization for attractive hard-core particles in disorder
• Lieb-Robinson bounds for open quantum systems with long-ranged interactions
• Manifestations of dynamical localization in the disordered XXZ spin chain
• How much delocalisation is needed for an enhanced area law of the entanglement entropy?
• Bounds on the entanglement entropy of droplet states in the XXZ spin chain
• The exponential decay of eigenfunctions for tight binding Hamiltonians via landscape and dual landscape functions
• Subdiffusion in one-dimensional Hamiltonian chains with sparse interactions
• Out of Time Order Correlations in the Quasi-Periodic Aubry-André model
• Experimental tests of Lieb-Robinson bounds
• Stability of the enhanced area law of the entanglement entropy
• Slow propagation of information on the random XXZ quantum spin chain
• Slow propagation in some disordered quantum spin chains
• Entanglement bounds in the XXZ quantum spin chain
• Entanglement of a class of non-Gaussian states in disordered harmonic oscillator systems
• Localization in the Disordered Holstein model
• Probing unitary two-time correlations in a neutral atom quantum simulator
• Droplet localization in the random XXZ model and its manifestations
• On the Regime of Localized Excitations for Disordered Oscillator Systems
• Localization in the random XXZ quantum spin chain
• Bounded light cone and robust topological order out of equilibrium
• Many-body localization in the droplet spectrum of the random XXZ quantum spin chain
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• Decay of Complex-time Determinantal and Pfaffian\ Correlation Functionals in Lattices
• Stochastic process emerged from lattice fermion systems by repeated measurements and large-time limit