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arXiv:2405.01781

Mori generalized master equations offer an efficient route to predict and interpret transport

Srijan Bhattacharyya, Thomas Sayer, Andrés Montoya−Castillo

Published May 2, 2024• 0 citations
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Abstract

Predicting how a material's microscopic structure and dynamics determine its transport properties remains a fundamental challenge. To alleviate this task's often prohibitive computational expense, we propose a Mori-based generalized quantum master equation (GQME) to predict the frequency-resolved conductivity of small-polaron forming systems described by the dispersive Holstein model. Unlike previous GQME-based approaches to transport that scale with the system size and only give access to the DC conductivity, our method requires only one calculation and yields both the DC and AC mobilities. We further show to easily augment our GQME with numerically accessible derivatives of the current to increase computational efficiency, collectively offering computational cost reductions of up to $90\%$, depending on the transport regime. Finally, we leverage our exact simulations to demonstrate the limited applicability of the celebrated and widely invoked Drude-Smith model in small-polaron forming systems. We instead introduce a cumulant-based analysis of experimentally accessible frequency data to infer the microscopic Hamiltonian parameters. This approach promises to provide valuable insights into material properties and facilitate guided design by linking macroscopic terahertz measurements to the microscopic details of small polaron-forming systems.

DOI
10.48550/arxiv.2405.01781

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Predicting how a material's microscopic structure and dynamics determine its transport properties remains a fundamental challenge. To alleviate this task's often prohibitive computational expense, we propose a Mori-based generalized quantum master equation (GQME) to predict the frequency-resolved conductivity of small-polaron forming systems described by the dispersive Holstein model. Unlike previous GQME-based approaches to transport that scale with the system size and only give access to the DC conductivity, our method requires only one calculation and yields both the DC and AC mobilities. We further show to easily augment our GQME with numerically accessible derivatives of the current to increase computational efficiency, collectively offering computational cost reductions of up to 90%90\%90%, depending on the transport regime. Finally, we leverage our exact simulations to demonstrate the limited applicability of the celebrated and widely invoked Drude-Smith model in small-polaron forming systems. We instead introduce a cumulant-based analysis of experimentally accessible frequency data to infer the microscopic Hamiltonian parameters. This approach promises to provide valuable insights into material properties and facilitate guided design by linking macroscopic terahertz measurements to the microscopic details of small polaron-forming systems.
The exact quantum dynamics of lattice models can be computationally intensive, especially when aiming for large system sizes and extended simulation times necessary to converge transport coefficients. By leveraging finite memory times to access long-time dynamics using only short-time data, generalized master equations can offer a route to simulating the dynamics of lattice problems efficiently. However, such simulations are limited to small lattices whose dynamics exhibit finite-size artifacts that contaminate transport coefficient predictions. To address this problem, we introduce a novel approach that exploits finite memory in both time and space to efficiently predict the many-body dynamics of dissipative lattice problems involving short-range interactions. This advance enables one to leverage the short-time dynamics of small lattices to nonperturbatively and exactly simulate arbitrarily large systems over long times. We demonstrate the strengths of this method by focusing on nonequilibrium polaron relaxation and transport in the dispersive Holstein model, successfully simulating lattice dynamics in one and two dimensions free from finite-size effects, thereby reducing the computational expense of such simulations by multiple orders of magnitude. Our method is broadly applicable and provides an accurate and efficient means to investigate nonequilibrium relaxation with microscopic resolution over mesoscopic length and time scales that are relevant to experiments.
The Holstein model is a benchmark model of systems with electron-phonon interaction. However, its electrical transport properties are not yet fully understood. In this work, we performed numerically exact calculations of the imaginary-time current-current correlation function of the Holstein Hamiltonian for a broad range of model parameters. These calculations were performed using a path-integral-based quantum Monte Carlo method. We compared these results with the results obtained under the assumption of conventional band transport, small polaron hopping, and polaron band transport. From this comparison, we identified the regions in parameter space where each of these transport regimes is valid. In some cases when an imaginary-time comparison of current-current correlation functions could not give conclusive results, we complemented it with real-time comparisons or a comparison with the literature data on numerically exact dc mobilities. Overall, we found that the parameter space is almost completely covered by the three mentioned transport regimes.
The generalized quantum master equation (GQME) approach provides a powerful general-purpose framework for simulating the inherently quantum mechanical dynamics of a subset of electronic reduced density matrix elements of interest in complex molecular systems. Previous studies have found that combining the GQME approach with quasiclassical mapping Hamiltonian (QC/MH) methods can dramatically improve the accuracy of electronic populations obtained via those methods. In this paper, we perform a complimentary study of the advantages offered by the GQME approach for simulating the dynamics of electronic coherences, which play a central role in optical spectroscopy, quantum information science, and quantum technology. To this end, we focus on cases where the electronic coherences predicted for the spin-boson benchmark model by direct application of various QC/MH methods are inaccurate. We find that similar to the case of electronic populations, combining the QC/MH methods with the GQME approach can dramatically improve the accuracy of the electronic coherences obtained via those methods. We also provide a comprehensive analysis of how the performance of GQMEs depends on the choice of projection operator and electronic basis and show that the accuracy and feasibility of the GQME approach can benefit from casting the GQME in terms of the eigen-basis of the observable of interest.
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.
Polarons—quantum particles dressed by a cloud of bath excitations—arise whenever a particle interacts strongly with its host medium. Electron and hole polarons, formed when carriers bind to a phonon cloud through electron-phonon (e-ph) interactions, govern charge transport in transition metal oxides, organic semiconductors, and halide perovskites. The concept generalizes to exciton-phonon interactions, giving rise to exciton polarons that produce the large Stokes shifts and broad photoluminescence (PL) lineshapes central to light-emitting applications. A predictive treatment of polarons has remained elusive. Numerically exact methods such as diagrammatic Monte Carlo (DMC) have been restricted to simplified model Hamiltonians, while first-principles approaches based on density functional theory provide material-specific e-ph interactions but treat the e-ph correlation approximately. This thesis bridges that gap by developing first-principles diagrammatic Monte Carlo (FPDMC), which stochastically sums all e-ph Feynman diagrams using accurate ab-initio interactions, yielding numerically exact polaron properties in real materials. Two methodological advances make FPDMC possible. A matrix-product formalism sums over electronic band indices exactly, alleviating the multi-band sign problem in realistic band structures. A data-driven compression of e-ph interactions based on singular-value decomposition (SVD) in the Wannier basis then accelerates the method by more than 10³×. The SVD compression also reveals a hidden low-dimensional structure of quantum interactions, maintaining accuracy across charge transport, spin relaxation, band renormalization, and superconductivity calculations. Extending this philosophy to phonon-phonon interactions via a permanent CP (PCP) decomposition yields 10³–10⁴× compression of anharmonic force constants and orders-of-magnitude speedups in thermal conductivity calculations. Applying FPDMC, we study polarons in LiF, SrTiO₃, and TiO₂, finding excellent agreement with experimental mobilities and resolving the contrasting small- and large-polaron transport in the two TiO₂ polymorphs. To make the microscopic picture concrete, we introduce an e-ph correlation function that visualizes lattice distortion and polaron orbital character. Extending FPDMC to exciton polarons, we obtain first-principles exciton-phonon interactions from finite-momentum Bethe-Salpeter equation calculations and derive a Fermi-golden-rule PL theory. Applied to CrBr₃, FPDMC predicts a broad PL spectrum agreeing with experiment in both peak position and lineshape.