We study the many-body localization of the random-field Heisenberg chain using the nested shift-invert Lanczos method with an iterative linear solver. We use the minimum residual method (MINRES) inside each Lanczos iteration. The memory consumption of the proposed method is only proportional to the dimension of the Hilbert space. We also introduce a preconditioner that takes into account the effects of disorder and interaction in the random-field Heisenberg chain. As a probe of many-body localization transition, we propose a unitary operator called the twist operator, which has a clear interpretation in the real space. We discuss its behavior for thermal and localized eigenstates. We demonstrate the efficiency of the nested iterative shift-invert diagonalization method with the proposed preconditioner for the many-body localization problem and estimate the transition point of the random-field Heisenberg chain more precisely based on the finite-size analysis of the expectation value of the twist operator.
We study the many-body localization of the random-field Heisenberg chain using the nested shift-invert Lanczos method with an iterative linear solver. We use the minimum residual method (MINRES) inside each Lanczos iteration. The memory consumption of the proposed method is only proportional to the dimension of the Hilbert space. We also introduce a preconditioner that takes into account the effects of disorder and interaction in the random-field Heisenberg chain. As a probe of many-body localization transition, we propose a unitary operator called the twist operator, which has a clear interpretation in the real space. We discuss its behavior for thermal and localized eigenstates. We demonstrate the efficiency of the nested iterative shift-invert diagonalization method with the proposed preconditioner for the many-body localization problem and estimate the transition point of the random-field Heisenberg chain more precisely based on the finite-size analysis of the expectation value of the twist operator.
We provide a pedagogical review on the calculation of highly excited eigenstates of disordered interacting quantum systems which can undergo a many-body localization (MBL) transition, using shift-invert exact diagonalization. We also provide an example code at https://bitbucket.org/dluitz/sinvert_mbl. Through a detailed analysis of the simulational parameters of the random field Heisenberg spin chain, we provide a practical guide on how to perform efficient computations. We present data for mid-spectrum eigenstates of spin chains of sizes up to L=26 L = 26 . This work is also geared towards readers with interest in efficiency of parallel sparse linear algebra techniques that will find a challenging application in the MBL problem.
We present a scalable and matrix-free eigensolver for studying two-level quantum spin chain models with nearest-neighbor XX +YY interactions plus Z terms. In particular, we focus on the Heisenberg interaction plus random on-site fields, a model that is commonly used to study the many-body localization (MBL) transition. This type of problem is computationally challenging because the vector space dimension grows exponentially with the physical system size, and the solve must be iterated many times to average over different configurations of the random disorder. For each eigenvalue problem, eigenvalues from different regions of the spectrum and their corresponding eigenvectors need to be computed. Traditionally, the interior eigenstates for a single eigenvalue problem are computed via the shift-and-invert Lanczos algorithm. Due to the extremely high memory footprint of the LU factorizations, this technique is not well suited for large number of spins L, e.g., one needs thousands of compute nodes on modern high performance computing infrastructures to go beyond L = 24. The new matrix-free approach, proposed in this paper, does not suffer from this memory bottleneck and even allows for simulating spin chains up to L = 24 spins on a single compute node. We discuss the OpenMP and hybrid MPI--OpenMP implementations of matrix-free block matrix-vector operations that are the key components of the new approach. The efficiency and effectiveness of the proposed algorithm is demonstrated by computing eigenstates in a massively parallel fashion, and analyzing their entanglement entropy to gain insight into the MBL transition.
We provide a pedagogical review on the calculation of highly excited eigenstates of disordered interacting quantum systems which can undergo a many-body localization (MBL) transition, using shift-invert exact diagonalization. We also provide an example code at https://bitbucket.org/dluitz/sinvert_mbl. Through a detailed analysis of the simulational parameters of the random field Heisenberg spin chain, we provide a practical guide on how to perform efficient computations. We present data for mid-spectrum eigenstates of spin chains of sizes up to L = 26. This work is also geared towards readers with interest in efficiency of parallel sparse linear algebra techniques that will find a challenging application in the MBL problem.
Abstract We study the one-dimensional spin-1/2 Heisenberg chain with Dzyaloshinskii-Moriya interaction in a random magnetic field using exact diagonalization. In order to obtain many-body mobility edge at infinite temperature, we employ a polynomial filtered Lanczos method that can avoid the fill-in problem when implementing the commonly used shift-and-invert transformation. In stark contrast to the original Heisenberg model, although the localized phase always conforms to Poisson statistics, the ergodic phase exhibits the Gaussian unitary ensemble rather than the Gaussian orthogonal ensemble statistics due to the lack of complex conjugation symmetry. The boundary between the ergodic and localized phases is determined by carefully performing finite-size scalings for the level statistics, entanglement entropy and its standard deviation, as well as fluctuations of the total spin of the system. The two phases are also well distinguished by the full delocalization or localization in the Hilbert space wherein the participation entropies present. To indicate the localized phase in experiment, we propose a scheme for realizing the out-of-time-order correlator on a modern nuclear magnetic resonance quantum simulator.
Despite enormous efforts devoted to the study of the many-body localization (MBL) phenomenon, the nature of the high-energy behavior of the Heisenberg spin chain in a strong random magnetic field is lacking consensus. Here, we take a step back by exploring the weak interaction limit starting from the Anderson localized (AL) insulator. Through shift-invert diagonalization, we find that, below a certain disorder threshold h^{*}, weak interactions necessarily lead to an ergodic instability, whereas at strong disorder the AL insulator directly turns into MBL, in agreement with a simple interpretation of the avalanche theory for restoration of ergodicity. We further map the phase diagram for the generic XXZ model in the disorder h-interaction Δ plane. Taking advantage of the total magnetization conservation, our results unveil the remarkable behavior of the spin-spin correlation functions: in the regime indicated as MBL by standard observables, their exponential decay undergoes an inversion of orientation ξ_{z}>ξ_{x}. We find that the longitudinal length ξ_{z} is a key quantity for capturing ergodic instabilities, as it increases with system size near the thermal phase, in sharp contrast to its transverse counterpart.
We propose several techniques to enhance the parallel scalability of a matrix-free eigensolver designed for studying many-body localization (MBL) of quantum spin chain models with nearest-neighbor interactions and on-site disorder. This type of problem is computationally challenging because the dimension of the associated Hamiltonian matrix grows exponentially with respect to the number of spins L, and we need to average over different realizations of the random disorder to obtain relevant statistical behavior. For each disorder realization, we need to compute eigenvalues from different regions of the spectrum and their corresponding eigenvectors. In previous work, the interior eigenstates for a single eigenvalue problem are computed via the shift-and-invert Lanczos algorithm. Due to the extremely high memory footprint of the LU factorizations, this technique is not well suited for large L’s. For example, we need thousands of compute nodes on modern high performance computing infrastructures to go beyond L = 24. The matrix-free approach does not suffer from this memory bottleneck, however, its scalability is limited by a computation and communication load imbalance. To reduce this imbalance and to significantly enhance the scalability of the matrix-free eigensolver, we reorder the matrix and leverage the consistent space runtime, CSPACER. We also show its efficiency in managing irregular communication patterns at scale compared to optimized MPI non-blocking two-sided and one-sided RMA implementation variants. This effort enables us to study MBL for spin chains with a larger number of spins. The efficiency and effectiveness of the proposed algorithm is demonstrated by computing eigenstates on a massively parallel many-core high performance computer.