We develop a hierarchical equations of motion (HEOM) approach to compute real-time single-particle correlation functions and thermodynamic properties of the Holstein model at finite temperature. We exploit the conservation of the total momentum of the system to formulate the momentum-space HEOM whose dynamical variables explicitly keep track of momentum exchanges between the electron and phonons. Our symmetry-adapted HEOM enable us to overcome the numerical instabilities inherent to the commonly used real-space HEOM. The HEOM method is then used to study the spectral function and thermodynamic quantities of chains containing up to ten sites. The HEOM results compare favorably to existing literature. To provide an independent assessment of the HEOM approach and to gain insight into the importance of finite-size effects, we devise a quantum Monte Carlo (QMC) procedure to evaluate finite-temperature single-particle correlation functions in imaginary time and apply it to chains containing up to twenty sites. QMC results reveal that finite-size effects are quite weak, so that the results on 5 to 10-site chains, depending on the parameter regime, are representative of larger systems. A detailed comparison between the HEOM and QMC data place our HEOM method among reliable methods to compute real-time finite-temperature correlation functions in parameter regimes ranging from low- to high-temperature, and weak- to strong-coupling regime.
We develop a hierarchical equations of motion (HEOM) approach to compute real-time single-particle correlation functions and thermodynamic properties of the Holstein model at finite temperature. We exploit the conservation of the total momentum of the system to formulate the momentum-space HEOM whose dynamical variables explicitly keep track of momentum exchanges between the electron and phonons. Our symmetry-adapted HEOM enable us to overcome the numerical instabilities inherent to the commonly used real-space HEOM. The HEOM method is then used to study the spectral function and thermodynamic quantities of chains containing up to ten sites. The HEOM results compare favorably to existing literature. To provide an independent assessment of the HEOM approach and to gain insight into the importance of finite-size effects, we devise a quantum Monte Carlo (QMC) procedure to evaluate finite-temperature single-particle correlation functions in imaginary time and apply it to chains containing up to twenty sites. QMC results reveal that finite-size effects are quite weak, so that the results on 5 to 10-site chains, depending on the parameter regime, are representative of larger systems. A detailed comparison between the HEOM and QMC data place our HEOM method among reliable methods to compute real-time finite-temperature correlation functions in parameter regimes ranging from low- to high-temperature, and weak- to strong-coupling regime.
Numerically "exact" methods addressing the dynamics of coupled electron-phonon systems have been intensively developed. Nevertheless, the corresponding results for the electron mobility μdc are scarce, even for the one-dimensional (1d) Holstein model. Building on our recent progress on single-particle properties, here we develop the momentum-space hierarchical equations of motion (HEOM) method to evaluate real-time two-particle correlation functions of the 1d Holstein model at a finite temperature. We compute numerically "exact" dynamics of the current-current correlation function up to real times sufficiently long to capture the electron's diffusive motion and provide reliable results for μdc in a wide range of model parameters. In contrast to the smooth ballistic-to-diffusive crossover in the weak-coupling regime, we observe a temporally limited slow-down of the electron on intermediate time scales already in the intermediate-coupling regime, which translates to a finite-frequency peak in the optical response. Our momentum-space formulation lowers the numerical effort with respect to existing HEOM-method implementations, while we remove the numerical instabilities inherent to the undamped-mode HEOM by devising an appropriate hierarchy closing scheme. Still, our HEOM remains unstable at too low temperatures, for too strong electron-phonon coupling, and for too fast phonons.
It is generally accepted that the dynamical mean field theory gives a good solution of the Holstein model, but only in dimensions greater than two. Here, we show that this theory, which becomes exact in the weak coupling and in the atomic limit, provides an excellent, numerically cheap, approximate solution for the spectral function of the Holstein model in the whole range of parameters, even in one dimension. To establish this, we make a detailed comparison with the spectral functions that we obtain using the newly developed momentum-space numerically exact hierarchical equations of motion method, which yields electronic correlation functions directly in real time. We crosscheck these conclusions with our path integral quantum Monte Carlo and exact diagonalization results, as well as with the available numerically exact results from the literature.
Studying charge transport in models with nonlocal carrier-phonon interaction is difficult because it requires finite-temperature real-time correlation functions of mixed carrier-phonon operators. Focusing on models with discrete undamped phonon modes, we show that such correlation functions can be retrieved from the hierarchical equations of motion (HEOM), although phonons have been integrated out. Our procedure relies on the explicit expression of HEOM auxiliaries in terms of phonon creation and annihilation operators. It reveals that the auxiliaries describe multiphonon-assisted carrier transitions induced by genuine many-phonon correlations, from which lower-order correlations are subtracted according to the finite-temperature Wick's theorem. Applying the procedure to our recently developed momentum-space HEOM method featuring a specific hierarchy closing, we compute the numerically exact dynamical mobility of a carrier within the one-dimensional Peierls model. The carrier mobility at moderate temperatures decreases with increasing interaction, whereas high temperatures see the opposite trend, reflecting the prevalence of the phonon-assisted current over the purely electronic band current. The pronounced finite-size effects and HEOM instabilities delimit the range of applicability of our approach to moderate interactions, moderate to high temperatures, and not too fast phonons. Importantly, this range comprises the values relevant for charge transport in crystalline organic semiconductors, and we present and discuss the corresponding numerically exact results in a companion paper (arXiv:2501.05055).
The hierarchical equations of motion (HEOM) method has become one of the most popular methods for the studies of the open quantum system. However, its applicability to systems at ultra-low temperatures is largely restrained by the enormous computational cost, which is caused by the numerous exponential functions required to accurately characterize the non-Markovian memory of the reservoir environment. To overcome this problem, a Fano spectrum decomposition (FSD) scheme has been proposed recently [Cui et al., J. Chem. Phys. 151, 024110 (2019)], which expands the reservoir correlation functions using polynomial-exponential functions and hence greatly reduces the size of the memory basis set. In this work, we explicitly establish the FSD-based HEOM formalisms for both bosonic and fermionic environments. The accuracy and efficiency of the FSD-based HEOM are exemplified by the calculated low-temperature dissipative dynamics of a spin-boson model and the dynamic and static properties of a single-orbital Anderson impurity model in the Kondo regime. The encouraging numerical results highlight the practicality and usefulness of the FSD-based HEOM method for general open systems at ultra-low temperatures.
Hierarchical equations of motion(HEOM) provide exact dynamics of open quantum systems coupled to harmonic baths, but their computational cost becomes prohibitive for systems with many independent local environments. In this work, we develop a symmetry-adapted HEOM formalism to significantly reduce the computational cost for the permutationally invariant Holstein-Tavis-Cummings (HTC) model. The method removes redundant information in two stages. First, all auxiliary density operators (ADOs) related only by relabeling identical molecules and their bath channels are replaced by a single canonical occupation-pattern representative. Second, molecules with the same local hierarchy occupation produce repeated matrix elements within each representative, allowing only the distinct complex variables to be propagated instead of the full (N + 1) × (N + 1) ADO matrices. The resulting matrix-free equations are evaluated using precomputed connections and molecular multiplicities. At fixed hierarchy depth L and number of bath correlation exponentials m, the number of canonical representatives becomes independent of the ensemble size for N ≥ L and the number of unique variables saturates for N ≥ L + 2. The formulation easily extends to multiple-exponential bath decompositions, arbitrary initial density operators, static disorders, and cavity loss. Our benchmarks reproduce conventional HEOM dynamics while requiring far fewer propagated variables and substantially less memory.
Hierarchical Equations of Motion (HEOM) in the Tensor-Train (TT) representation is applied to study the charge-transfer dynamics in organic semiconductors (OSCs). The theoretical formulation as well as the basic computational aspects of HEOM-TT are discussed in detail. Charge transfer in OSCs is modeled using dissipative polaronic models that incorporate the effects of both high- and low-frequency molecular vibrations, and it is simulated in a fully quantum and nonperturbative manner, which has not been studied intensively. The capability of treating complex electron-vibrational systems is examined by analyzing and comparing the numerical behavior of the time-dependent variational approach and the time-Alternating Minimal Energy methods and by calculating the current autocorrelation function and diffusivity across various models. Our results indicate that the HEOM-TT framework offers a robust tool for the detailed analysis of complex polaronic systems, suggesting its potential for broader applications.