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arXiv:2509.13140 · The Journal of Chemical Physics

From higher-order moments to time correlation functions in strongly correlated systems: A DMRG-based memory kernel coupling theory

Yunhao Liu, Wenjie Dou

Published Dec 1, 2025• 3 citations
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Abstract

We introduce a hybrid approach for computing dynamical observables in strongly correlated systems using higher-order moments. This method integrates memory kernel coupling theory (MKCT) with the density matrix renormalization group (DMRG), extending our recent work on MKCT [Liu et al., Phys. Rev. Lett. 135, 148001(2025)] to strongly correlated systems. The method establishes that correlation functions can be derived from the moments. Within our framework, operators and wavefunctions are represented as matrix product operators (MPOs) and matrix product states (MPSs), respectively. Crucially, the repeated application of the Liouville operator is achieved through an iterative procedure analogous to the DMRG algorithm itself. We demonstrate the effectiveness and efficiency of MKCT-DMRG by computing the spectral function of the Hubbard model. Furthermore, we successfully apply the method to compute the electronic friction in the Hubbard-Holstein model. In all cases, the results show excellent agreement with time-dependent DMRG (TD-DMRG) benchmarks. The advantage of MKCT-DMRG over TD-DMRG is computational efficiency, which avoids expensive real-time propagation in TD-DMRG. These findings establish MKCT-DMRG as a promising and accurate framework for simulating challenging dynamical properties in strongly correlated quantum systems.

DOI
10.1063/5.0300474

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We introduce a hybrid approach for computing dynamical observables in strongly correlated systems using higher-order moments. This method integrates memory kernel coupling theory (MKCT) with the density matrix renormalization group (DMRG), extending our recent work on MKCT [Liu et al., Phys. Rev. Lett. 135, 148001(2025)] to strongly correlated systems. The method establishes that correlation functions can be derived from the moments. Within our framework, operators and wavefunctions are represented as matrix product operators (MPOs) and matrix product states (MPSs), respectively. Crucially, the repeated application of the Liouville operator is achieved through an iterative procedure analogous to the DMRG algorithm itself. We demonstrate the effectiveness and efficiency of MKCT-DMRG by computing the spectral function of the Hubbard model. Furthermore, we successfully apply the method to compute the electronic friction in the Hubbard-Holstein model. In all cases, the results show excellent agreement with time-dependent DMRG (TD-DMRG) benchmarks. The advantage of MKCT-DMRG over TD-DMRG is computational efficiency, which avoids expensive real-time propagation in TD-DMRG. These findings establish MKCT-DMRG as a promising and accurate framework for simulating challenging dynamical properties in strongly correlated quantum systems.
We introduce a general procedure for computing higher-order moments of correlation functions in open quantum systems, extending the scope of our recent work on Memory Kernel Coupling Theory (MKCT) [Liu et al., arXiv:2407.01923 (2024)]. This approach is demonstrated for arbitrary system-bath coupling that can be expressed as a polynomial, HSB=V̂(α0+α1q̂+α2q̂2+⋯) , where we show that the recursive commutators of a system operator obey a universal hierarchy. Exploiting this structure, the higher-order moments are obtained by evaluating the expectation values of the system and bath operators separately, with bath expectation values derived from the derivatives of a generating function. We further apply MKCT to compute the dipole autocorrelation function for the spin-boson model with both linear and quadratic coupling, achieving agreement with the hierarchical equations of motion approach. Our findings suggest a promising path toward accurate dynamics for complex open quantum systems.
Dynamical observables can often be described by time correlation functions (TCFs). However, efficiently calculating TCFs for complex quantum systems is a significant challenge, which generally requires solving the full dynamics of the systems. This Letter presents the memory kernel coupling theory (MKCT), a general formalism for evaluating TCFs. The MKCT builds upon Mori's memory kernel formalism for TCFs. Our theory further decomposes the memory kernel into auxiliary kernels. Rapid decay of auxiliary kernels allows us to truncate the coupled equations of motion with high accuracy. Notably, only higher-order moments are sufficient as the input for obtaining TCFs. While this formalism is general, we carry out the numerical demonstration for typical open quantum systems-the spin-boson model and the single impurity Anderson model.
Dynamical observables can often be described by time correlation functions (TCFs). However, efficiently calculating TCFs for complex quantum systems is a significant challenge, which generally requires solving the full dynamics of the systems. This Letter presents the memory kernel coupling theory (MKCT), a general formalism for evaluating TCFs. The MKCT builds upon Mori's memory kernel formalism for TCFs. Our theory further decomposes the memory kernel into auxiliary kernels. Rapid decay of auxiliary kernels allows us to truncate the coupled equations of motion with high accuracy. Notably, only higher-order moments are sufficient as the input for obtaining TCFs. While this formalism is general, we carry out the numerical demonstration for a typical open quantum system--the spin-boson model.
Thanks to the high compression of the matrix product state (MPS) form of the wave function and the efficient site-by-site iterative sweeping optimization algorithm, the density matrix normalization group (DMRG) and its time-dependent variant (TD-DMRG) have been established as powerful computational tools in accurately simulating the electronic structure and quantum dynamics of strongly correlated molecules with a large number (10 1–2 ) of quantum degrees of freedom (active orbitals or vibrational modes). However, the quantitative characterization of the quantum many-body behaviors of realistic strongly correlated systems requires a further consideration of the interaction between the embedded active subsystem and the remaining correlated environment, e.g., a larger number (10 2–3 ) of external orbitals in electronic structure or infinite condensed-phase phononic modes in nucleus dynamics. To this end, we introduced three new post-DMRG and TD-DMRG approaches, namely (1) DMRG2sCI-MRCI and DMRG2sCI-ENPT by the reconstruction of selected configuration interaction (sCI) type of compact reference function from DMRG coefficients and the use of externally contracted MRCI (multireference configuration interaction) and Epstein–Nesbet perturbation theory (ENPT), without recourse to the expensive high order n -electron reduced density matrices ( n -RDMs). (2) DMRG combined with RR-MRCI (renormalized residue-based MRCI), which improves the computational accuracy and efficiency of internally contracted (ic) MRCI by renormalizing the contracted bases with small-sized buffer environment(s) of a few external orbitals as probes based on quantum information theory. (3) HM (hierarchical mapping)-TD-DMRG in which a large environment is reduced to a small number of renormalized environmental modes (which accounts for the most vital system–environment interactions) through stepwise mapping transformation. These advances extend the efficacy of highly accurate DMRG/TD-DMRG computations to the quantitative characterization of the electronic structure and quantum dynamics in realistic strongly correlated systems coupled with large environments and are reviewed in this paper.
The generalized quantum master equation provides a powerful framework for non-Markovian dynamics of open quantum systems. However, the accurate and efficient evaluation of the memory kernel remains a challenge. In this work, we introduce a comprehensive tensorial extension to the Memory Kernel Coupling Theory (MKCT) to overcome this bottleneck. By elevating the original scalar formalism to a tensorial framework, the extended MKCT enables the calculation of general expectation values and cross-correlation functions. We demonstrate the numerical accuracy and efficiency of this method across multiple benchmark systems: capturing transient populations and coherences in the spin-boson model, resolving the excitonic absorption spectrum of the Fenna-Matthews-Olson complex, and simulating charge mobility in one-dimensional lattice models. These successful applications establish the tensorial MKCT as a highly efficient tool for investigating complex dynamics in open quantum systems.
Recent advances in experimental techniques have enabled precise characterization of fundamental nonadiabatic and even electronic dynamics in molecules, on ultrafast time scales reaching femtosecond to attosecond resolution. However, accurate theoretical simulation of these ultrafast chemical dynamics processes in large systems remains challenging, largely due to the overwhelming number of degrees of freedom (DoFs) and the pronounced many-body correlations. In recent years, by leveraging efficient decomposition schemes for high-dimensional wave function and operator tensors, the time-dependent density matrix renormalization group (TD-DMRG) has emerged as a powerful and accurate quantum dynamics method for simulating nonadiabatic and electron dynamics in large chemical systems, in conjunction with realistic electron/exciton-vibration/phonon models or ab initio quantum chemistry many-electron Hamiltonians. This review outlines the fundamentals of TD-DMRG for chemical dynamics, covering matrix product state/operator (MPS/MPO) frameworks and algorithms from ground-state calculations to time evolution. We discuss thermal/environmental effects and compare TD-DMRG with other tensor network methods such as multiconfiguration time-dependent Hartree (MCTDH) and multilayer MCTDH (ML-MCTDH). Demonstrated applications include simulations of pyrazine absorption, singlet fission in rubrene crystal, and charge migration in chloroacetylene cation. These show TD-DMRG’s capability for modeling complex ultrafast processes from femtoseconds to attoseconds with controlled accuracy.