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Tree Tensor Network Approach to Transport in Heavily-Doped Organic Materials with Coexisting Electron Correlation and Electron-Phonon Coupling

Liqi Zhou, Weitang Li, Xing Bao Gao, Zhigang Shuai

Published Mar 9, 2026• 0 citations
Publisher

Abstract

Heavy doping in organic materials induces complex many-body interactions characterized by the coexistence of strong electron correlation and electron-phonon coupling, arising from their soft structures and low dielectric constants. Simulating charge transport and thermopower in this regime poses a significant computational challenge. In this work, we propose a finite-temperature time-dependent tree tensor network state (TTNS) method to calculate the conductivity and Seebeck coefficient of organic materials modeled by the Holstein-Hubbard Hamiltonian. Our TTNS ansatz features a tailored “bus-branch” structure, where the “bus” captures the electronic entanglement, and the “branches” encode the local electron-phonon correlations. Benchmarks against exact diagonalization (ED) on small clusters demonstrate that this network structure achieves numerically exact results with smaller bond dimensions than the Matrix Product States (MPS). Furthermore, the “branch” topology efficiently isolates the local electron-phonon entanglement, thereby avoiding the globally large bond dimensions required by MPS. These algorithmic advantages enable us to constrain the relative error of transport properties to within 10% across a wide range of parameter regimes. To validate the capability of this framework, we study the complex interplay between Hubbard repulsion and phonon-mediated attraction, successfully illustrating the roles of spin degrees of freedom and effective interactions in thermodynamics and charge transport, and reproducing the correlation-induced anomalous sign change of the Seebeck coefficient. Our work establishes the “bus-branch” TTNS as a powerful and efficient tool for exploring quantum dynamics in strongly correlated electron-phonon systems.

DOI
10.26434/chemrxiv.15000730/v1

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Heavy doping in organic materials induces complex many-body interactions characterized by the coexistence of strong electron correlation and electron-phonon coupling, arising from their soft structures and low dielectric constants. Simulating charge transport and thermopower in this regime poses a significant computational challenge. In this work, we propose a finite-temperature time-dependent tree tensor network state (TTNS) method to calculate the conductivity and Seebeck coefficient of organic materials modeled by the Holstein-Hubbard Hamiltonian. Our TTNS ansatz features a tailored “bus-branch” structure, where the “bus” captures the electronic entanglement, and the “branches” encode the local electron-phonon correlations. Benchmarks against exact diagonalization (ED) on small clusters demonstrate that this network structure achieves numerically exact results with smaller bond dimensions than the Matrix Product States (MPS). Furthermore, the “branch” topology efficiently isolates the local electron-phonon entanglement, thereby avoiding the globally large bond dimensions required by MPS. These algorithmic advantages enable us to constrain the relative error of transport properties to within 10% across a wide range of parameter regimes. To validate the capability of this framework, we study the complex interplay between Hubbard repulsion and phonon-mediated attraction, successfully illustrating the roles of spin degrees of freedom and effective interactions in thermodynamics and charge transport, and reproducing the correlation-induced anomalous sign change of the Seebeck coefficient. Our work establishes the “bus-branch” TTNS as a powerful and efficient tool for exploring quantum dynamics in strongly correlated electron-phonon systems.
Organic/polymeric materials are of emerging importance for thermoelectric conversion. The soft nature of these materials implies strong electron-phonon coupling, often leading to carrier localization. This poses great challenges for the conventional Boltzmann transport description based on relaxation time approximation and band structure calculations. In this work, combining Kubo formula with finite-temperature time-dependent density matrix renormalization group (FT-TD-DMRG) in the grand canonical ensemble, we developed a nearly exact algorithm to calculate the thermoelectric power factor PF=α^2 σ, where α is the Seebeck coefficient and σ is electrical conductivity, and apply the algorithm to organic materials modeled by Holstein Hamiltonian with electron-phonon coupling. Our algorithm can provide a unified description covering the weak coupling limit described by bandlike Boltzmann transport to the strong coupling hopping limit.
Tensor network states are an indispensable tool for the simulation of strongly correlated quantum many-body systems. In recent years, tree tensor network states (TTNS) have been successfully used for two-dimensional systems and to benchmark quantum simulation approaches for condensed matter, nuclear, and particle physics. In comparison to the more traditional approach based on matrix product states (MPS), the graph distance of physical degrees of freedom can be drastically reduced in TTNS. Surprisingly, it turns out that, for large systems in D>1D>1D>1 spatial dimensions, MPS simulations of low-energy states are nevertheless more efficient than TTNS simulations. With a focus on D=2D=2D=2 and 3, the scaling of computational costs for different boundary conditions is determined under the assumption that the system obeys an entanglement (log-)area law, implying that bond dimensions scale exponentially in the surface area of the associated subsystems.
Light-harvesting in nature is known to work differently than conventional man-made solar cells. Recent studies found electronic excitations, delocalised over several chromophores, and a soft, vibrating structural environment to be key schemes that might protect and direct energy transfer yielding increased harvest efficiencies even under adversary conditions. Unfortunately, testing realistic models of noise assisted transport at the quantum level is challenging due to the intractable size of the environmental wave function. I developed a powerful tree tensor network states (TTNS) method that finds an optimally compressed explicit representation of the combined electronic and vibrational quantum state. With TTNS it is possible to simulate exciton-phonon quantum dynamics from small molecules to larger complexes, modelled as an open quantum system with multiple bosonic environments. After benchmarking the method on the minimal spin-boson model by reproducing ground state properties and dynamics that have been reported using other methods, the vibrational quantum state is harnessed to investigate environmental dynamics and its correlation with the spin system. To enable simulations of realistic non-Born-Oppenheimer molecular quantum dynamics, a clustering algorithm and novel entanglement renormalisation tensors are employed to interface TTNS with ab initio density functional theory (DFT). A thereby generated model of a pentacene dimer containing 252 vibrational normal modes was simulated with TTNS reproducing exciton dynamics in agreement with experimental results. Based on the environmental state, the (potential) energy surfaces, underlying the observed singlet fission dynamics, were calculated yielding unprecedented insight into the super-exchange mediated avoided crossing mechanism that produces ultrafast and high yield singlet fission. This combination of DFT and TTNS is a step towards large scale material exploration that can accurately predict excited states properties and dynamics. Furthermore, application to biomolecular systems, such as photosynthetic complexes, may give valuable insights into novel environmental engineering principles for the design of artificial light-harvesting systems.
We present a comprehensive theoretical study of charge transport in organic materials on the basis of the one-dimensional dispersive Holstein-Peierls model. The hierarchical equations of motion in tensor-train format (HEOM-TT) method is employed to accurately describe both local Holstein-type and nonlocal Peierls-type electron-phonon couplings in the presence of low-frequency dispersive phonons. Linear response formulation of carrier mobility is obtained by employing the generating functional method. We demonstrate the capability of the method by calculating current correlation functions, diffusivities, and mobilities for various transport regimes. Our results provide new insights into the mechanisms underlying charge transport in organic semiconductors and establish HEOM-TT as a powerful tool for studying complex vibronic quantum dynamics and transport properties in condensed matter systems.
Evaluation of the charge transport property of organic semiconductors requires exact quantum dynamics simulation of large systems. We present a numerically nearly exact approach to investigate carrier transport dynamics in organic semiconductors by extending the non-Markovian stochastic Schrödinger equation with complex frequency modes to a forward-backward scheme and by solving it using the matrix product state (MPS) approach. By utilizing the forward-backward formalism for noise generation, the bath correlation function can be effectively treated as a temperature-independent imaginary part, enabling a more accurate decomposition with fewer complex frequency modes. Using this approach, we study the carrier transport and mobility in the one-dimensional Peierls model, where the nonlocal electron-phonon interaction is taken into account. The reliability of this approach was validated by comparing carrier diffusion motion with those obtained from the hierarchical equations of motion method across various parameter regimes of the phonon bath. The efficiency was demonstrated by the modest virtual bond dimensions of MPS and the low scaling of the computational time with the system size.
Tree tensor network states (TTNS) decompose the system wavefunction to the product of low-rank tensors based on the tree topology, serving as the foundation of the multi-layer multi-configuration time-dependent Hartree method. In this work, we present an algorithm that automatically constructs the optimal and exact tree tensor network operators (TTNO) for any sum-of-product symbolic quantum operator. The construction is based on the minimum vertex cover of a bipartite graph. With the optimal TTNO, we simulate open quantum systems, such as spin relaxation dynamics in the spin-boson model and charge transport in molecular junctions. In these simulations, the environment is treated as discrete modes and its wavefunction is evolved on equal footing with the system. We employ the Cole-Davidson spectral density to model the glassy phonon environment and incorporate temperature effects via thermo-field dynamics. Our results show that the computational cost scales linearly with the number of discretized modes, demonstrating the efficiency of our approach.