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Improving matrix product state methods for long-range interactions

Mingru Yang

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

We study one dimensional models of diatomic molecules where both the electrons and nuclei are treated as quantum particles, going beyond the usual Born-Oppenheimer approximation. The continuous system is approximated by a grid which computationally resembles a ladder, with the electrons living on one leg and the nuclei on the other. To simulate DMRG efficiently with this system, a three-site algorithm has been implemented. We also use a compression method to treat the long-range interactions between charged particles. We find that 1D diatomic molecules with spin-1/2 nuclei in the spin-triplet state will unbind when the mass of the nuclei reduces to only a few times larger than the electron mass, while the molecule with nuclei in the singlet state always binds, given the two electrons in their singlet state in both cases. \nWe propose an improved scheme to do the time dependent variational principle (TDVP) in finite matrix product states (MPS) 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 that 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.\n \nWe apply our improved TDVP method to investigate the spin squeezing dynamics of the XXZ model of $1/r^\\alpha$ interaction in two dimension. Comparing with the spin squeezing parameter and other observables obtained from discrete truncated Wigner approximation (DTWA), we verify the validity of DTWA and unveil the potential for this method to study dynamics of large-scale spin systems. Our results confirm the existence of large collective regime when $\\alpha > 2$, which can be a guide for future experimental realizations. Combined with the purification method, our improved TDVP method is proved to be also useful to study the thermalization of the long-range interacting system.

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We study one dimensional models of diatomic molecules where both the electrons and nuclei are treated as quantum particles, going beyond the usual Born-Oppenheimer approximation. The continuous system is approximated by a grid which computationally resembles a ladder, with the electrons living on one leg and the nuclei on the other. To simulate DMRG efficiently with this system, a three-site algorithm has been implemented. We also use a compression method to treat the long-range interactions between charged particles. We find that 1D diatomic molecules with spin-1/2 nuclei in the spin-triplet state will unbind when the mass of the nuclei reduces to only a few times larger than the electron mass, while the molecule with nuclei in the singlet state always binds, given the two electrons in their singlet state in both cases. \nWe propose an improved scheme to do the time dependent variational principle (TDVP) in finite matrix product states (MPS) 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 that 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.\n \nWe apply our improved TDVP method to investigate the spin squeezing dynamics of the XXZ model of 1/r^\\alpha interaction in two dimension. Comparing with the spin squeezing parameter and other observables obtained from discrete truncated Wigner approximation (DTWA), we verify the validity of DTWA and unveil the potential for this method to study dynamics of large-scale spin systems. Our results confirm the existence of large collective regime when alpha>2\\alpha > 2alpha>2, which can be a guide for future experimental realizations. Combined with the purification method, our improved TDVP method is proved to be also useful to study the thermalization of the long-range interacting system.
We study one-dimensional (1D) models of diatomic molecules where both the electrons and nuclei are treated as quantum particles, going beyond the usual Born-Oppenheimer approximation. The continuous system is approximated by a grid which computationally resembles a ladder, with the electrons living on one leg and the nuclei on the other. To simulate density-matrix renormalization group efficiently with this system, a three-site algorithm has been implemented. We also use a compression method to treat the long-range interactions between charged particles. We find that 1D diatomic molecules with spin-1/2 nuclei in the spin triplet state will unbind when the mass of the nuclei reduces to only a few times larger than the electron mass, while the molecule with nuclei in the singlet state always binds.
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.
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.
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.
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.
Electronic and/or vibronic coherence has been found by recent ultrafast spectroscopy experiments in many chemical, biological, and material systems. This indicates that there are strong and complicated interactions between electronic states and vibration modes in realistic chemical systems. Therefore, simulations of quantum dynamics with a large number of electronic and vibrational degrees of freedom are highly desirable. Due to the efficient compression and localized representation of quantum states in the matrix-product state (MPS) formulation, time-evolution methods based on the MPS framework, which we summarily refer to as tDMRG (time-dependent density-matrix renormalization group) methods, are considered to be promising candidates to study the quantum dynamics of realistic chemical systems. In this work, we benchmark the performances of four different tDMRG methods, including global Taylor, global Krylov, and local one-site and two-site time-dependent variational principles (1TDVP and 2TDVP), with a comparison to multiconfiguration time-dependent Hartree and experimental results. Two typical chemical systems of internal conversion and singlet fission are investigated: one containing strong and high-order local and nonlocal electron-vibration couplings and the other exhibiting a continuous phonon bath. The comparison shows that the tDMRG methods (particularly, the 2TDVP method) can describe the full quantum dynamics in large chemical systems accurately and efficiently. Several key parameters in the tDMRG calculation including the truncation error threshold, time interval, and ordering of local sites were also investigated to strike the balance between efficiency and accuracy of results.