Hierarchical Mapping for Efficient Simulation of Strong System-Environment Interactions

Abstract

Quantum dynamics (QD) simulation is a powerful tool for interpreting ultrafast spectroscopy experiments and unraveling their microscopic mechanism in out-of-equilibrium excited state behaviors in various chemical, biological, and material systems. Although state-of-the-art numerical QD approaches such as the time-dependent density matrix renormalization group (TD-DMRG) already greatly extended the solvable system size of general linearly coupled exciton–phonon models with up to a few hundred phonon modes, the accurate simulation of larger system sizes or strong system-environment interactions is still computationally highly challenging. Based on quantum information theory (QIT), in this work, we realize that only a small number of effective phonon modes couple to the excitonic system directly regardless of a large or even infinite number of modes in the condensed phase environment. On top of the identified small number of direct effective modes, we propose a hierarchical mapping (HM) approach through performing block Lanczos transformations on the remaining indirect modes, which transforms the Hamiltonian matrix to a nearly block-tridiagonal form and eliminates the long-range interactions. Numerical tests on model spin-boson systems and realistic singlet fission models in a rubrene crystal environment with up to 7000 modes and strong system-environment interactions indicate HM can reduce the system size by 1–2 orders of magnitude and accelerate the calculation by ∼80% without losing accuracy.

New Density Matrix Renormalization Group Approaches for Strongly Correlated Systems Coupled with Large Environments

“…As mentioned in Section 1, the 183-mode singlet fission model, 112 spin-boson model, 108 and singlet fission in realistic rubrene crystals 63 have been studied by HM-TD-DMRG. 75 The 183-mode singlet fission model contains three exciton states: local exciton, charge transfer, and triplet pair (LE, CT, TT). The spin-boson model mimics the transition between the spin-up and spin-down states, where the bosonic environment is continuous and in our study discretized into 2000 modes.…”

“…Under the principles of minimizing the number of modes involved in the embedded subsystem and diminishing the long-range interactions, recently we proposed a hierarchical mapping (HM) which can build and select effective modes for TD-DMRG simultaneously. The first step is performing SVD on the exciton–phonon coupling tensor g : .25ex2ex g i j K infix= prefix∑ α N ex ( N ex + 1 ) / 2 normalΓ i j α S α normalΛ α , K † ≈ ∑ α N dir normalΓ i j α S α normalΛ α , K † infix= prefix∑ α N dir G i j α normalΛ α , K † Note that the nonzero singular values in eq is exactly N ex ( N ex +1)/2, in agreement with HEM.…”

“…As mentioned in Section , the 183-mode singlet fission model, spin-boson model, and singlet fission in realistic rubrene crystals have been studied by HM-TD-DMRG . The 183-mode singlet fission model contains three exciton states: local exciton, charge transfer, and triplet pair (LE, CT, TT).…”

“…In recent years many algorithms have been proposed to accelerate TD-DMRG, such as increasing the basis number adaptively in the single-site algorithm − and utilizing the purification-projection (PP) to restore the U (1)-particle number conservation symmetry in the phononic system , as well as using the graphical processing units (GPUs) to accelerate the heavy tensor contractions, but nowadays it is still impossible to include all environmental phononic modes into TD-DMRG. Rather than including all environmental phononic modes into the embedded active subsystem explicitly, latest developments in TD-DMRG incline for methods based on open quantum systems or developing new algorithms to identify a small number of environmental modes which have strongest couplings with the interested subsystem. − For this purpose, we recently proposed a hierarchical mapping (HM) method which uses QIT to identify a small number of renormalized environmental phononic modes directly coupled to the quantum subsystem followed by successively transforming the vibronic Hamiltonian matrix to a nearly block-tridiagonal form by the block Lanczos algorithm. Numerical tests on model spin-boson systems and realistic singlet fission models in rubrene crystal environment with up to 7000 phononic modes and strong system–environment interactions indicate the HM can reduce the size of full quantum subsystem by 1–2 orders of magnitude and accelerate the calculation by ∼80% with negligible accuracy loss.…”

Encoding a Many-Body Potential Energy Surface into a Grid-Based Matrix Product Operator

“…In recent years, the time evolution algorithm based on the Dirac-Frenkel time-dependent variational principle (TDVP) has been proposed for the wave function in the MPS form. − These contributions enabled efficient simulations of systems involving long–range interactions beyond the nearest–neighbor interactions model. The TDVP-based MPS dynamics ansatz has been applied to various systems, including vibrational (phonon) wave functions, such as the MPS MCTDH (MPS-MCTDH), , the tDMRG (TD-DMRG), , and the tensor-train split-operator KSL (TT-SOKSL), and demonstrated in various systems at both zero and finite temperatures. − The use of matrix product operator (MPO) − as tensor network operator to act on the wave function in the MPS form has facilitated the simulation of a wide range of phonon quantum dynamics. ,− The conventional MPO is designed to handle a sum of products (SOP) of one-body operators such as position operators and second quantization operators. However, in practice, considering the situation where a globally coupled multidimensional potential energy surface (PES) is fitted by SOP, we cannot avoid problems such as overfitting and limited representability.…”