Construction of many-body-localized models where all the eigenstates are matrix-product-states

Abstract

The inverse problem of 'eigenstates-to-Hamiltonian' is considered for an open chain of N quantum spins in the context of many-body-localization. We first construct the simplest basis of the Hilbert space made of 2 N orthonormal matrix-product-states (MPS), that will thus automatically satisfy the entanglement area-law. We then analyze the corresponding N local integrals of motions (LIOMs) that can be considered as the local building blocks of these 2 N MPS, in order to construct the parent Hamiltonians that have these 2 N MPS as eigenstates. Finally we study the matrix-product-operator form of the diagonal ensemble density matrix that allows to compute long-time-averaged observables of the unitary dynamics. Explicit results are given for the memory of local observables and for the entanglement properties in operator-space, via the generalized notion of Schmidt decomposition for density matrices describing mixed states.

Optimal parent Hamiltonians for time-dependent states

“…This matrix encodes the covariances of the local operators in the stationary state and its kernel contains all the possible local parent Hamiltonians. This approach has been extended in several ways, as the search for a stationary parent Hamiltonian associated to a state after a quantum quench 11 , to open quantum systems governed by a Lindblad dynamics 12,13 and the determination of a parent Hamiltonian of a Matrix Product State 14 . The non-uniqueness of the parent Hamiltonian is reflected in the degeneracy of the kernel of the QCM.…”