Solving the time-independent Schr\"odinger equation for chains of coupled excitons and phonons using tensor trains

TL;DR

This paper introduces a tensor-train based numerical method to efficiently solve the Schrödinger equation for coupled exciton-phonon systems, enabling analysis of quantum states and phenomena like self-trapping with reduced computational resources.

Contribution

The authors develop a tensor-train approach with an integrated Wielandt deflation technique to solve high-dimensional eigenproblems in exciton-phonon chains, improving efficiency and enabling new quantum state analyses.

Findings

Tensor-train ranks grow marginally with chain length in homogeneous systems.

The method confirms the dependence of wavepacket width on exciton-phonon coupling.

The approach enables direct study of mutual self-trapping phenomena.

Abstract

We demonstrate how to apply the tensor-train format to solve the time-independent Schr\"{o}dinger equation for quasi one-dimensional excitonic chain systems with and without periodic boundary conditions. The coupled excitons and phonons are modeled by Frenkel-Holstein type Hamiltonians with on-site and nearest-neighbor interactions only. We reduce the memory consumption as well as the computational costs significantly by employing efficient decompositions to construct low rank tensor-train representations, thus mitigating the curse of dimensionality. In order to compute also higher quantum states, we introduce an approach which directly incorporates the Wielandt deflation technique into the alternating linear scheme for the solution of eigenproblems. Besides systems with coupled excitons and phonons, we also investigate uncoupled problems for which (semi-)analytical results exist. There, we find that in case of homogeneous systems the tensor-train ranks of state vectors only marginally depend on the chain length which results in a linear growth of the storage consumption. However, the CPU time increases slightly faster with the chain length than the storage consumption because the alternating linear scheme adopted in our work requires more iterations to achieve convergence for longer chains and a given rank. Finally, we demonstrate that the tensor-train approach to the quantum treatment of coupled excitons and phonons makes it possible to directly tackle the phenomenon of mutual self-trapping. We are able to confirm the main results of the Davydov theory, i.e., the dependence of the wavepacket width and the corresponding stabilization energy on the exciton-phonon coupling strength, though only for a certain range of that parameter. In future work, our approach will allow calculations also beyond the restrictions of the Frenkel-Holstein type Hamiltonians.