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

Wahl, TB, Arijeet Pal, Steven H. Simon

Published May 8, 2017• 57 citations
OpenAlex

Abstract

We propose a tensor network encoding the set of all eigenstates of a fully many-body localized system in one dimension. Our construction, conceptually based on the ansatz introduced in Phys. Rev. B 94, 041116(R) (2016), is built from two layers of unitary matrices which act on blocks of \nℓ \n contiguous sites. We argue that this yields an exponential reduction in computational time and memory requirement as compared to all previous approaches for finding a representation of the complete eigenspectrum of large many-body localized systems with a given accuracy. Concretely, we optimize the unitaries by minimizing the magnitude of the commutator of the approximate integrals of motion and the Hamiltonian, which can be done in a local fashion. This further reduces the computational complexity of the tensor networks arising in the minimization process compared to previous work. We test the accuracy of our method by comparing the approximate energy spectrum to exact diagonalization results for the random-field Heisenberg model on 16 sites. We find that the technique is highly accurate deep in the localized regime and maintains a surprising degree of accuracy in predicting certain local quantities even in the vicinity of the predicted dynamical phase transition. To demonstrate the power of our technique, we study a system of 72 sites, and we are able to see clear signatures of the phase transition. Our work opens a new avenue to study properties of the many-body localization transition in large systems.

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We propose a tensor network encoding the set of all eigenstates of a fully many-body localized system in one dimension. Our construction, conceptually based on the ansatz introduced in Phys. Rev. B 94, 041116(R) (2016), is built from two layers of unitary matrices which act on blocks of \nℓ \n contiguous sites. We argue that this yields an exponential reduction in computational time and memory requirement as compared to all previous approaches for finding a representation of the complete eigenspectrum of large many-body localized systems with a given accuracy. Concretely, we optimize the unitaries by minimizing the magnitude of the commutator of the approximate integrals of motion and the Hamiltonian, which can be done in a local fashion. This further reduces the computational complexity of the tensor networks arising in the minimization process compared to previous work. We test the accuracy of our method by comparing the approximate energy spectrum to exact diagonalization results for the random-field Heisenberg model on 16 sites. We find that the technique is highly accurate deep in the localized regime and maintains a surprising degree of accuracy in predicting certain local quantities even in the vicinity of the predicted dynamical phase transition. To demonstrate the power of our technique, we study a system of 72 sites, and we are able to see clear signatures of the phase transition. Our work opens a new avenue to study properties of the many-body localization transition in large systems.
Subsystems of strongly disordered, interacting quantum systems can fail to thermalize because of the phenomenon of many-body localization (MBL). In this paper, we explore a tensor network description of the eigenspectra of such systems. Specifically, we will argue that the presence of a complete set of local integrals of motion in MBL implies an efficient representation of the entire spectrum of energy eigenstates with a single tensor network, a spectral tensor network. Our results are rigorous for a class of idealized systems related to MBL with integrals of motion of finite support. In one spatial dimension, the spectral tensor network allows for the efficient computation of expectation values of a large class of operators (including local operators and string operators) in individual energy eigenstates and in ensembles.
Abstract A novel method has been devised to compute the Local Integrals ofMotion~(LIOMs) for a one-dimensional many-body localized system. Inthis approach, a class of optimal unitary transformations is deducedin a tensor-network formalism to diagonalize the Hamiltonian of thespecified system. To construct the tensor network, we utilize theeigenstates of the subsystems' Hamiltonian to attain the desiredunitary transformations. Subsequently, we optimize the eigenstatesand acquire appropriate unitary localized operators that willrepresent the LIOMs tensor network. The efficiency of the method wasassessed and found to be both fast and almost accurate. In frameworkof the introduced tensor-network representation, we examine how theentanglement spreads along the considered many-body localized systemand evaluate the outcomes of the approximations employed in thisapproach. The important and interesting result is that in theproposed tensor network approximation, if the length of the blocksis greater than the length of localization, then the entropy growthwill be linear in terms of the logarithmic time. Also, it has beendemonstrated that, the entanglement can be calculated by onlyconsidering two blocks next to each other, if the Hamiltonian hasbeen diagonalized using the unitary transformation made by theprovided tensor-network representation.
Eigenstates of fully many-body localized (FMBL) systems are described by quasilocal operators ${\ensuremath{\tau}}_{i}^{z}$ (l-bits), which are conserved exactly under Hamiltonian time evolution. The algebra of the operators ${\ensuremath{\tau}}_{i}^{z}$ and ${\ensuremath{\tau}}_{i}^{x}$ associated with l-bits $({\mathbf{\ensuremath{\tau}}}_{i})$ completely defines the eigenstates and the matrix elements of local operators between eigenstates at all energies. We develop a nonperturbative construction of the full set of l-bit algebras in the many-body localized phase for the canonical model of MBL. Our algorithm to construct the Pauli algebra of l-bits combines exact diagonalization and a tensor network algorithm developed for efficient diagonalization of large FMBL Hamiltonians. The distribution of localization lengths of the l-bits is evaluated in the MBL phase and used to characterize the MBL-to-thermal transition.
The phenomenon of many-body localization generalizes Anderson localization to interacting systems. Understanding this conceptually novel phenomenon requires a study of many-body eigenstates at finite energy densities. This is a very challenging task since the most efficient numerical methods such as, e.g., the density matrix renormalization group method, can only access the ground state and low lying excitations. In this work, the authors introduce a unitary tensor network based variational method that approximately finds all many-body eigenstates of fully localized Hamiltonians and scales polynomially with system size. The usefulness of their approach is demonstrated by considering the Heisenberg chain in a strongly disordered magnetic field.
We develop variational continuous unitary transformations (VCUTs), which integrate Wegner-Wilson flow equations with tensor network techniques to approximately diagonalize many-body localized (MBL) Hamiltonians. The diagonalizing unitary is represented as a matrix product operator whose bond dimension controls the accuracy. For the disordered Heisenberg chain, VCUTs accurately reproduce the full spectrum across the ergodic-to-MBL crossover at small system sizes and scale to L = 48 sites. Beyond eigenenergies, the method can track the spatial entanglement structure of the diagonalizing unitary U ( l ) at each flow step, enabling identification of local integrals of motion deep in the MBL phase.