Finding Matrix Product State Representations of Highly Excited Eigenstates of Many-Body Localized Hamiltonians

Abstract

A key property of many-body localization, the localization of quantum particles in systems with both quenched disorder and interactions, is the area law entanglement of even highly excited eigenstates of many-body localized Hamiltonians. Matrix Product States (MPS) can be used to efficiently represent low entanglement (area law) wave functions in one dimension. An important application of MPS is the widely used Density Matrix Renormalization Group (DMRG) algorithm for finding ground states of one dimensional Hamiltonians. Here, we describe two algorithms, the Shift and Invert MPS (SIMPS) and excited state DMRG which finds highly-excited eigenstates of many-body localized Hamiltonians. Excited state DMRG uses a modified sweeping procedure to identify eigenstates whereas SIMPS is a shift-inverse procedure that applies the inverse of the shifted Hamiltonian to a MPS multiple times to project out the targeted eigenstate. To demonstrate the power of these methods we verify the breakdown of the Eigenstate Thermalization Hypothesis (ETH) in the manybody localized phase of the random field Heisenberg model, show the saturation of entanglement in the MBL phase and generate local excitations.

Many-Body-Localization Transition in the Strong Disorder Limit: Entanglement Entropy from the Statistics of Rare Extensive Resonances

“…Therefore, in the MBL phase, excited states are somewhat similar to ground states, with efficient representation via the Density Matrix Renormalization Group (DMRG) or matrix product states [9][10][11][12] and tensor networks [13]. The Fisher strong disorder real space RG to construct the ground states of random quantum spin models [14][15][16][17] has been extended into the strong disorder RG procedure for the unitary dynamics [18,19] and into the Real-Space-RG-for-Excited-States (RSRG-X) in order to construct the whole set of excited eigenstates [20][21][22][23][24][25].…”

Efficient Representation of Fully Many-Body Localized Systems Using Tensor Networks

“…However, since the number of eigenstates is exponential in the system size, for large N one can only tackle the eigenstates in a certain energy window using MPS (see, e.g., Refs. [49][50][51]). On the other hand, spectral tensor networks are meant to encode an approximation to all eigenstates at once, which is a desirable property if one aims to calculate dynamical properties of local observables in MBL systems.…”

“…It was suggested that the accuracy of the approximation for a given chain length can be increased by increasing the number of layers (the depth of the quantum circuit). Compared to the methods targeting eigenstates within an energy window [49][50][51][52], this procedure is constructed to efficiently represent all eigenstates with sufficient accuracy, providing access to dynamical properties of local observables.…”

Fixed Points of Wegner-Wilson Flows and Many-Body Localization

“…Concrete analytic [1], numerical [2][3][4][5], and mathematical [6,7] results establish the existence and robustness of many-body localized phases in sufficiently strongly disordered and/or low-dimensional interacting models at finite extensive entropy. While the understanding of the transition between thermal and MBL phases is only beginning to emerge [8][9][10][11][12], several distinct new directions of inquiry related to MBL and the fundamental issue of ergodicity in quantum many-body systems have taken shape. These include the interplay of MBL with spontaneous symmetry breaking and topological order [13][14][15][16], selflocalization (glassiness) in translationally invariant quantum systems [17][18][19][20], and MBL in driven systems [21][22][23].…”

“…These include the interplay of MBL with spontaneous symmetry breaking and topological order [13][14][15][16], selflocalization (glassiness) in translationally invariant quantum systems [17][18][19][20], and MBL in driven systems [21][22][23]. MBL has also stimulated considerable progress in developing tools for describing excited eigenstates of many-body systems [12,[24][25][26][27][28]. MBL has been realized in recent experiments [29,30] and may also have important implications for quantum engineering problems, e.g., quantum computing [31][32][33][34][35].…”