Many-body Localization and Local information
|Many-body Localization and Local information
|NAISS Medium Compute
|Jens Bardarson <firstname.lastname@example.org>
|Kungliga Tekniska högskolan
|2023-12-21 – 2025-01-01
Static and dynamic properties of many-body quantum systems are intrinsically hard to investigate. The reason is that quantum entanglement, generated by the presence of many interacting degrees of freedom, generally requires resources that grow exponentially with the system size. This prevents exact simulations for large system sizes and intermediate and long-time scales on classical computers. Many open questions in many-body quantum physics remain in this domain. To overcome this barrier, we have developed various, radically different, approximate algorithms whose underlying idea is to represent quantum states with less than exponential (in system-size) degrees of freedom, keeping track only of the most relevant ones. Our project uses two kinds of algorithms.
(i) Tensor Networks. The presence of disorder can exponentially suppress long-range entanglement in many-body quantum systems, inducing a Many-Body Localized (MBL) phase where quantum states admit a compressed representation using tensor networks. We have developed two algorithms using tensor networks to study the entanglement statistics of disordered quantum spin chains in (or near) the MBL phase. The first (xDMRG) calculates excited eigenstates of spin chain models by adapting the Density Matrix Renormalization Group (DMRG) algorithm, which is normally used for ground states. The second (fLBIT) indirectly simulates the dynamics of MBL systems, similar to the Time-Evolving Block Decimation (TEBD) algorithm. Both algorithms rely heavily on tensor contractions and singular-value decompositions.
(ii) Local-Information Algorithms. In a first project, we have developed a new time-evolution algorithm that systematically discards long-range entanglement. This allows us to approximate the dynamics of effectively infinite interacting quantum systems with an error that is controlled by the length scale above which entanglement is disregarded. This algorithm is based on recent insights by members of our group and their collaborators. Specifically, our code solves the von Neumann equation (vNE) that describes the time evolution of density matrices by using the locality of the Hamiltonians we investigate. Instead of solving the vNE for the entire system, it decomposes the whole system into smaller subsystems and solves the corresponding vNEs in parallel. Thus, this algorithm is particularly well suited for parallelization via OpenMP and MPI. It is also highly versatile and can be applied to various physical contexts. In a second project, we use the concept of local information for investigating the entanglement properties of eigenvectors of disordered Hamiltonians. This project relies heavily on diagonalization of big matrices.