Studying dynamics in two-dimensional quantum lattices using tree tensor network states

Abstract

We analyze and discuss convergence properties of a numerically exact algorithm tailored to study the dynamics of interacting two-dimensional lattice systems. The method is based on the application of the time-dependent variational principle in a manifold of binary and quaternary Tree Tensor Network States. The approach is found to be competitive with existing matrix product state approaches. We discuss issues related to the convergence of the method, which could be relevant to a broader set of numerical techniques used for the study of two-dimensional systems.

Time Evolution of ML-MCTDH Wavefunctions II: Application of the Projector Splitting Integrator

“…The PSI has been shown to be stable with respect to almost or fully rank deficient wavefunction parametrizations in its MCTDH 36 and ML-MCTDH formulations, 31,33,34,39 including the important subcategory of matrix product states (maximally layered trees with physical degrees of freedom at each layer). 32,40,41 The stability of the PSI with respect to nearly rankdeficient wavefunction parametrizations can be attributed to the fact that the EOMs solved are always well-conditioned.…”

“…It is possible that there are regimes in which convergence with respect to the regularization parameter cannot be obtained before numerical issues arise in the solution of the EOMs. 31 An alternative strategy for propagating ML-MCTDH wavefunctions is the projector splitting integrator (PSI) approach, [32][33][34] originally presented by Lubich for singlelayer MCTDH. 35,36 This approach employs an alternative representation of the wavefunction when evolving the coefficient tensors, rendering the equations of motion singularity-free.…”

Tensor Network States for Vibrational Spectroscopy

“…A notable exception are TTNSs, for which a DMRG-like optimization strategy has been devised. 56 Moreover, the recently developed time-dependent TTNS variant 66,67 has paved the route towards the optimization of TTNS with imaginary-time propagation algorithms. Although these developments could set the ground for new efficient tensor-based methods in the near future, DMRG currently provides the optimal balance between the complexity of the tensor factorization and of its optimization.…”

Time Evolution of ML-MCTDH Wavefunctions I: Gauge Conditions, Basis Functions, and Singularities

“…Two approaches of this class are the invariant EOMs 33 and the projector splitting integrator (PSI) method. [38][39][40][41][42] The PSI has found success in the simulation of the unitary dynamics of wavefunctions represented using Matrix Product States, [43][44][45][46] and the MCTDH 40 and ML-MCTDH 42,[47][48][49] ansätze. The EOMs that are treated within this approach are non-singular regardless of whether the ML-MCTDH wavefunction is rank-deficient.…”