A uniform area law for the entanglement of eigenstates in the disordered XY chain

Abstract

Abstract. 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. MSC: 81P40, 82B44

Localization properties of the disordered XY spin chain

“…[11] and [12]. For the disordered XY chain a uniform area law for all eigenstates was shown in [2], see (29) in Section 6. As we will describe, this is based on the fact, shown in [48], that eigenstates of disordered free Fermion systems in arbitrary dimension d satisfy an area law bound.…”

“…Here we will discuss a broad range of results of this type. In particular, we will describe recent localization results in the disordered XY chain and closely related disordered free Fermion systems from [1,2,29,48,55]. In the last two subsections we will also include several previously unpublished results, where we will provide detailed proofs.…”

“…This is true, for example, if the random variables ν j have continuous distribution, see Appendix A in [2].…”

“…Using that a j , a * j , a * j a j and a j a * j generate the local operators at site j, this leads to the claim (14) 2 . More details on this proof are in [29]; see also the review [54].…”

“…In fact, this result extends to any finite number of quenches in the initial state, with bounds not depending on the number of quenches. The proofs of (29) and (30) in [2] and [1] rely heavily on the fact that the eigenstates ρ of H are quasi-free, as already used in the previous section, so that they are entirely determined by their two-point function with respect to the fermionic modes c j , c * j , summarized in the correlation matrix…”