Parallel transport dynamics for mixed quantum states with applications to time-dependent density functional theory

TL;DR

This paper extends parallel transport dynamics to mixed quantum states to improve the efficiency of real-time density functional theory simulations, enabling larger time steps and better accuracy for complex quantum systems.

Contribution

It generalizes PT dynamics from pure to mixed states, providing a gauge choice that allows larger time steps in quantum simulations beyond the adiabatic regime.

Findings

PT dynamics error is bounded by commutators involving Hamiltonians and density matrices.

PT-IM method is effective for large spectral radius Hamiltonians.

Numerical results show improved efficiency and accuracy in rt-TDDFT simulations.

Abstract

Direct simulation of the von Neumann dynamics for a general (pure or mixed) quantum state can often be expensive. One prominent example is the real-time time-dependent density functional theory (rt-TDDFT), a widely used framework for the first principle description of many-electron dynamics in chemical and materials systems. Practical rt-TDDFT calculations often avoid the direct simulation of the von Neumann equation, and solve instead a set of Schr\"odinger equations, of which the dynamics is equivalent to that of the von Neumann equation. However, the time step size employed by the Schr\"odinger dynamics is often much smaller. In order to improve the time step size and the overall efficiency of the simulation, we generalize a recent work of the parallel transport (PT) dynamics for simulating pure states [An, Lin, Multiscale Model. Simul. 18, 612, 2020] to general quantum states. The PT dynamics provides the optimal gauge choice, and can employ a time step size comparable to that of the von Neumann dynamics. Going beyond the linear and near adiabatic regime in previous studies, we find that the error of the PT dynamics can be bounded by certain commutators between Hamiltonians, density matrices, and their derived quantities. Such a commutator structure is not present in the Schr\"odinger dynamics. We demonstrate that the parallel transport-implicit midpoint (PT-IM) method is a suitable method for simulating the PT dynamics, especially when the spectral radius of the Hamiltonian is large. The commutator structure of the error bound, and numerical results for model rt-TDDFT calculations in both linear and nonlinear regimes, confirm the advantage of the PT dynamics.