Dynamical windows for real-time evolution with matrix product states

Abstract

We propose the use of a dynamical window to investigate the real-time evolution of quantum many-body systems in a one-dimensional lattice. In a recent paper [Phien et al., Phys. Rev. B 86, 245107 (2012)], we introduced infinite boundary conditions in order to investigate real-time evolution of an infinite system under a local perturbation. This was accomplished by restricting the update of the tensors in the matrix product state to a finite region known as a window, with left and right boundaries held at fixed positions. Here we consider instead the use of a dynamical window, where the positions of left and right boundaries are allowed to change in time. In this way, all computational efforts can be devoted to the space-time region of interest, which leads to a remarkable reduction in simulation costs. For illustrative purposes, we consider two applications in the context of the spin-1 antiferromagnetic Heisenberg model in an infinite spin chain: one is an expanding window, with boundaries that are adjusted to capture the expansion in time of a local perturbation of the system; the other is a moving window of fixed size, where the position of the window follows the front of a propagating wave.

Infinite boundary conditions for response functions and limit cycles within the infinite-system density matrix renormalization group approach demonstrated for bilinear-biquadratic spin-1 chains

“…The structure factor is considerably increased in the region where the singlemagnon branch enters. While no strong features appear at the upper boundary of the continuum, the van Hove Numerical results for the dynamic spin structure factor (9) in the bilinear-biquadratic spin-1 chain (1) at θ = 0 and θ = π/5 in the Haldane phase. The MPS simulations were done with infinite boundary conditions and, during the time evolution, truncation thresholds were set to λ 2 trunc = 10 −10 and λ 2 trunc = 10 −8 for θ = 0 and θ = π/5, respectively.…”

“…While we work with windows of fixed size, it is also possible to reduce computation costs somewhat by starting from a small window around site 0 and expand it during the time evolution in accordance with the spreading of correlations [9]. Usually, the resulting gains should, however, be minor.…”

“…If we apply a local perturbation to an initial uniform iMPS and evolve the state in time, Lieb-Robinson bounds [7] guarantee that the perturbation only has a significant effect on sites within a causal cone, a finite spatial region that grows linearly with time. We can hence simulate the time evolution using a finite heterogeneous window with infinite boundary conditions, a technique that has been used before to study quantum quenches [8,9]. The difference in our approach is that we shift the heterogeneous windows of two iMPS relative to each other to evaluate the response functions for all required distances between perturbation and measurement.…”

Time evolution within a comoving window: scaling of signal fronts and magnetization plateaus after a local quench in quantum spin chains

“…For Method II (Renormalized Update), applied to the left boundary, we adapt the algorithm of Cazalilla and Marston [59] (Method II is similar to the algorithm introduced in [50,51], see [53]) and construct a renormalized representation for H L + h , +1 to approximate the evolution of the left part and the left junction bond ( , + 1), such that all changes in the left part are compressed into the boundary matrix A s +1 , and the matrices L s j≤ remain unchanged.…”

Enhanced repulsively bound atom pairs in topological optical lattice ladders

“…One possible way to do this is through measurements of the time-dependent onsite particle number after a local quench, in this case after the application of the number operator on a single unit cell. To this end we have calculated the dynamical structure factors for each phase (within the weakly interacting and isolated flat band limit) using an MPS algorithm for time evolving infinite systems after a local perturbation [86,87].…”