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arXiv:1710.09378 · Physical review. B./Physical review. B

Time-dependent variational principle in matrix-product state manifolds: Pitfalls and potential

Benedikt Kloss, Yevgeny Bar Lev, David R. Reichman

Published Jan 26, 2018• 87 citations
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Abstract

We study the applicability of the time-dependent variational principle in matrix-product state manifolds for the long time description of quantum interacting systems. By studying integrable and nonintegrable systems for which the long time dynamics are known we demonstrate that convergence of long time observables is subtle and needs to be examined carefully. Remarkably, for the disordered nonintegrable system we consider the long time dynamics are in good agreement with the rigorously obtained short time behavior and with previous obtained numerically exact results, suggesting that at least in this case, the apparent convergence of this approach is reliable. Our study indicates that, while great care must be exercised in establishing the convergence of the method, it may still be asymptotically accurate for a class of disordered nonintegrable quantum systems.

DOI
10.1103/physrevb.97.024307

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We study the applicability of the time-dependent variational principle in matrix-product state manifolds for the long time description of quantum interacting systems. By studying integrable and nonintegrable systems for which the long time dynamics are known we demonstrate that convergence of long time observables is subtle and needs to be examined carefully. Remarkably, for the disordered nonintegrable system we consider the long time dynamics are in good agreement with the rigorously obtained short time behavior and with previous obtained numerically exact results, suggesting that at least in this case, the apparent convergence of this approach is reliable. Our study indicates that, while great care must be exercised in establishing the convergence of the method, it may still be asymptotically accurate for a class of disordered nonintegrable quantum systems.
The projection of time-dependent variational principle (TDVP) for matrix product states enables us to perform long-time simulations of one-dimensional quantum systems with the conservation of the total energy and the norm of wave functions. We compare long-time dynamics after a quantum quench simulated by TDVP with those by the exact diagonalization method in order to evaluate the performance of TDVP. We show that in a nonintegrable model the projection of TDVP clearly improves the long-time behaviors of global observables included in the Hamiltonian, such as the kinetic and interaction energies. In contrast, this projection can lead to larger error for other observables than that caused by the truncation of states.
We propose an improved scheme to do the time-dependent variational principle (TDVP) in finite matrix product states (MPSs) for two-dimensional systems or one-dimensional systems with long-range interactions. We present a method to represent the time-evolving state in a MPS with its basis enriched by state averaging with global Krylov vectors. We show that the projection error is significantly reduced so precise time evolution can still be obtained even if a larger time step is used. Combined with the one-site TDVP, our approach provides a way to dynamically increase the bond dimension while still preserving unitarity for real-time evolution. Our method can be more accurate and exhibit slower bond dimension growth than the conventional two-site TDVP.
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Combining the time-dependent variational principle (TDVP) algorithm with the parallelization scheme introduced by Stoudenmire and White for the density matrix renormalization group (DMRG), we present the first parallel matrix product state (MPS) algorithm capable of time evolving one-dimensional (1D) quantum lattice systems with long-range interactions. We benchmark the accuracy and performance of the algorithm by simulating quenches in the long-range Ising and XY models. We show that our code scales well up to 32 processes, with parallel efficiencies as high as 8686%86. Finally, we calculate the dynamical correlation function of a 201-site Heisenberg XXX spin chain with 1/r21/{r}^{2}1/r2 interactions, which is challenging to compute sequentially. These results pave the way for the application of tensor networks to increasingly complex many-body systems.
Time evolution of tensor-network states is a central tool for studying nonequilibrium dynamics in strongly correlated quantum systems. In the past decade, using the time-dependent variational principle (TDVP) discretized by projector-splitting has become the gold standard for evolving matrix product states (MPS) with long-range Hamiltonians. More recently, a conceptually different class of discretizations of the TDVP, the class of Basis Update and Galerkin (BUG) integrators, has been developed within the framework of dynamical low-rank approximation, offering an alternative, more flexible approach. In this work, we formulate two BUG integrators explicitly for MPS. The BUG integrators result in "single-site" MPS integrators that are robust with respect to small singular values, avoid backward-in-time substeps, and naturally allow for the adaptive choice of bond dimensions. We consider the augmented BUG integrator of first order and the midpoint BUG integrator of second order and clarify their relation to the projector-splitting discretization of the TDVP. Using representative models of strongly correlated quantum systems, like the XY spin model with long-range interaction and two-dimensional lattice models of fermions and spins of various lengths, we perform a systematic numerical comparison between BUG and projector-splitting time discretizations of the continuous-time TDVP, focusing on scalability, numerical complexity, and accuracy. We find that BUG integrators, due to their reduced numerical cost and the flexibility of the concept, provide a competitive and, in certain regimes, advantageous alternative to the well-established time-stepping by projector-splitting in TDVP, especially in models with large local dimensions, and in cases where observables are extracted with high temporal resolution.
Understanding the emergent system-bath correlations in non-Markovian and non-perturbative open systems is a theoretical challenge that has benefited greatly from the application of Matrix Product State (MPS) methods. Here, we propose an autonmously adapative variant of the one-site Time-Dependent-Variational-Principle (1TDVP) method for many-body MPS wave-functions in which the local bond-dimensions can evolve to capture growing entanglement 'on the fly'. We achieve this by efficiently examining the effect of increasing each MPS bond-dimension in advance of each dynamic timestep, resulting in an MPS that can dynamically and inhomogeneously restructure itself as the complexity of the dynamics grows across time and space. This naturally leads to more efficient simulations, oviates the need for multiple convergence runs, and, as we demonstrate, is ideally suited to the typical, finite-temperature 'impurity' problems that describe open quantum system connected to multiple environments.