A tensor network approach to many-body localization

Anushya Chandran, Juan Carrasquilla, Isaac H. Kim +2 more

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.

Christian Groß

Zohreh Gholami, Zahra Noorinejad, Mohsen Amini +1 more

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.

Matthias Rupp

Qiyu Liu, Jan-Niklas Herre, Dante Marvin Kennes +1 more

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.

Elmer V. H. Doggen, I. V. Gornyi, A. D. Mirlin +1 more