Repository: Freie Universität Berlin, Math Department

Solving the time-independent Schrödinger equation for chains of coupled excitons and phonons using tensor trains

Gelß, P. and Klein, R. and Matera, S. and Schmidt, B. (2022) Solving the time-independent Schrödinger equation for chains of coupled excitons and phonons using tensor trains. J. Chem. Phys., 156 (02). 024109.

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Abstract

We demonstrate how to apply the tensor-train format to solve the time-independent Schrödinger equation for quasi one-dimensional excitonic chain systems with and without periodic boundary conditions. The coupled excitons and phonons are modeled by Fröhlich-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 wave packet 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 validity regime of that theory and/or beyond the restrictions of the Fröhlich-Holstein type Hamiltonians.

Item Type:Article
Subjects:Physical Sciences > Physics > Chemical Physics
Physical Sciences > Physics > Mathematical & Theoretical Physics > Quantum Mechanics
Physical Sciences > Physics > Mathematical & Theoretical Physics > Computational Physics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics > Geophysical Fluid Dynamics Group
ID Code:2619
Deposited By: BioComp Admin
Deposited On:01 Oct 2021 12:06
Last Modified:17 Jan 2022 11:07

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