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
86. Finally, we calculate the dynamical correlation function of a 201-site Heisenberg XXX spin chain with
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 dynamical structure factor of the Babujan–Takhtajan antiferromagnetic spin-1 chain is computed numerically at zero temperature and zero magnetic field, using the higher spin generalization of an algebraic Bethe ansatz-based method previously used for spin-1/2 integrable chains. This method, which consists in the explicit construction of eigenstates and the summation of the Lehmann representation of the correlator, is particularly challenging to implement here in view of the presence of strongly deviated string solutions to the Bethe equations. We show that a careful treatment of these deviations makes it possible to obtain perfect saturation of sum rules for small system sizes, and extremely good saturation for large system sizes where the dynamical structure factor is computed by including all two-spinon and four-spinon contributions. The real-space spin–spin correlation, obtained by Fourier transforming our results, displays asymptotics fitting predictions from conformal field theory.
We present a matrix product state (MPS) algorithm to approximate ground states of translationally invariant systems with periodic boundary conditions. For a fixed value of the bond dimension
D of the MPS, we discuss how to minimize the computational cost to obtain a seemingly optimal MPS approximation to the ground state. In a chain with
N sites and correlation length $\ensuremath{\xi}$, the computational cost formally scales as $g(D,\ensuremath{\xi}/N){D}^{3}$, where $g(D,\ensuremath{\xi}/N)$ is a nontrivial function. For $\ensuremath{\xi}\ensuremath{\ll}N$, this scaling reduces to
D3, independent of the system size
N, making our method
N times faster than previous proposals. We apply the algorithm to obtain MPS approximations for the ground states of the critical quantum Ising and Heisenberg spin-
1/2 models as well as for the noncritical Heisenberg spin-
1 model. In the critical case, for any chain length
N, we find a model-dependent bond dimension
D(N) above which the polynomial decay of correlations is faithfully reproduced throughout the entire system.
We show that recursively generated Chebyshev expansions offer numerically efficient representations for calculating zero-temperature spectral functions of one-dimensional lattice models using matrix product state (MPS) methods. The main features of this Chebyshev matrix product state (CheMPS) approach are as follows: (i) it achieves uniform resolution over the spectral function's entire spectral width; (ii) it can exploit the fact that the latter can be much smaller than the model's many-body bandwidth; (iii) it offers a well-controlled broadening scheme that allows finite-size effects to be either resolved or smeared out, as desired; (iv) it is based on using MPS tools to recursively calculate a succession of Chebyshev vectors $|{t}_{n}\ensuremath{\rangle}$, (v) the entanglement entropies of which were found to remain bounded with increasing recursion order
n for all cases analyzed here; and (vi) it distributes the total entanglement entropy that accumulates with increasing
n over the set of Chebyshev vectors $|{t}_{n}\ensuremath{\rangle}$, which need not be combined into a single vector. In this way, the growth in entanglement entropy that usually limits density matrix renormalization group (DMRG) approaches is packaged into conveniently manageable units. We present zero-temperature CheMPS results for the structure factor of spin-
21 antiferromagnetic Heisenberg chains and perform a detailed finite-size analysis. Making comparisons to three benchmark methods, we find that CheMPS (a) yields results comparable in quality to those of correction-vector DMRG, at dramatically reduced numerical cost; (b) agrees well with Bethe ansatz results for an infinite system, within the limitations expected for numerics on finite systems; and (c) can also be applied in the time domain, where it has potential to serve as a viable alternative to time-dependent DMRG (in particular, at finite temperatures). Finally, we present a detailed error analysis of CheMPS for the case of the noninteracting resonant level model.
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.
The density-matrix renormalization group (DMRG) algorithm can be adapted to the calculation of dynamical correlation functions in various ways which all represent compromises between computational efficiency and physical accuracy. In this paper we reconsider the oldest approach based on a suitable Lanczos-generated approximate basis and implement it using matrix product states (MPS) for the representation of the basis states. The direct use of matrix product states combined with an ex post reorthogonalization method allows us to avoid several shortcomings of the original approach, namely the multitargeting and the approximate representation of the Hamiltonian inherent in earlier Lanczos-method implementations in the DMRG framework, and to deal with the ghost problem of Lanczos methods, leading to a much better convergence of the spectral weights and poles. We present results for the dynamic spin structure factor of the spin-1/2 antiferromagnetic Heisenberg chain. A comparison to Bethe ansatz results in the thermodynamic limit reveals that the MPS-based Lanczos approach is much more accurate than earlier approaches at minor additional numerical cost.