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The Journal of Chemical Physics

Delocalized Davydov D1 Ansatz for the Holstein polaron

Jin Li Sun, Liwei Duan, Yang Zhao

Published May 7, 2013• 23 citations
Publisher

Abstract

An efficient, yet very accurate trial wave function, constructed from projecting the well-known Davydov D1 Ansatz onto momentum eigenstates, is employed to study the ground state properties of the generalized Holstein Hamiltonian with simultaneous diagonal and off-diagonal coupling. Ground-state energies have been obtained with a precision matching that of the computationally much more demanding density-matrix renormalization group method. The delocalized D1 Ansatz lowers the ground-state energies at the Brillouin zone boundary significantly compared with the Toyozawa and Global-Local Ansätze in the weak coupling regime, while considerable improvement is demonstrated to have been achieved over the entire Brillouin zone in the strong coupling regime. Unique solutions are obtained with the delocalized D1 for different initial conditions when the transfer integral is 20 times the phonon frequency at the zone center, implying the absence of any self-trapping discontinuity. The scaled correlation variance is found to fit satisfactorily well with the predictions of the perturbation theories.

DOI
10.1063/1.4802961

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An efficient, yet very accurate trial wave function, constructed from projecting the well-known Davydov D1 Ansatz onto momentum eigenstates, is employed to study the ground state properties of the generalized Holstein Hamiltonian with simultaneous diagonal and off-diagonal coupling. Ground-state energies have been obtained with a precision matching that of the computationally much more demanding density-matrix renormalization group method. The delocalized D1 Ansatz lowers the ground-state energies at the Brillouin zone boundary significantly compared with the Toyozawa and Global-Local Ansätze in the weak coupling regime, while considerable improvement is demonstrated to have been achieved over the entire Brillouin zone in the strong coupling regime. Unique solutions are obtained with the delocalized D1 for different initial conditions when the transfer integral is 20 times the phonon frequency at the zone center, implying the absence of any self-trapping discontinuity. The scaled correlation variance is found to fit satisfactorily well with the predictions of the perturbation theories.
We describe a variational method to solve the Holstein model for an electron coupled to dynamical, quantum phonons on an infinite lattice. The variational space can be systematically expanded to achieve high accuracy with modest computational resources (12-digit accuracy for the one-dimensional polaron energy at intermediate coupling). We compute ground-state and low-lying excited-state properties of the model at continuous values of the wave vector k in essentially all parameter regimes. Our results for the polaron energy band, effective mass, and correlation functions compare favorably with those of other numerical techniques, including the density-matrix renormalization-group technique, the global-local method, and the exact diagonalization technique. We find a phase transition for the first excited state between a bound and unbound system of a polaron and an additional phonon excitation. The phase transition is also treated in strong-coupling perturbation theory.
We present selected results for the Holstein molecular crystal model in one space dimension as determined by the Global–Local variational method, including complete polaron energy bands, ground state energies, and effective masses. We juxtapose our results with specific comparable results of numerous other methodologies of current interest, including quantum Monte Carlo, cluster diagonalization, dynamical mean field theory, density matrix renormalization group, semiclassical analysis, weak-coupling perturbation theory, and strong-coupling perturbation theory. Taken as a whole, these methodologies are mutually confirming and provide a comprehensive and quantitatively accurate description of polaron properties in essentially any regime. In particular, this comparison confirms the Global–Local variational method as being highly accurate over a wide range of the polaron parameter space, from the nonadiabatic limit to the extremes of high adiabaticity, from weak coupling through intermediate coupling to strong coupling.
In this paper, we study a form of the Holstein molecular crystal model in which the influence of lattice vibrations on the transfers of electronic excitations between neighboring sites (off-diagonal coupling) is taken into account. Using the Toyozawa Ansatz and the Lanczos algorithm, the Holstein Hamiltonian with two types of off-diagonal coupling is studied focusing on a number of analyticity issues in the ground state. For finite-sized lattices and antisymmetric coupling, a sequence of discontinuities are found in the polaron energy dispersion, the size of the ground-state phonon cloud, and the linearized von Neumann entropy used to quantify the quantum entanglement between the exciton and the phonons in the ground state. Such behavior is accompanied by a shift of the ground-state crystal momentum from zero to nonzero values as the coupling strength is increased. In the thermodynamic limit, all discontinuities associated with antisymmetric coupling vanish except the one corresponding to the initial departure of the ground-state wavevector from the Brillouin zone center. For the case of symmetric off-diagonal coupling, a smooth crossover is found to exist in all parameters regimes.
Based on a recently developed variational method, we explore the properties of the Holstein polaron on an infinite lattice in D dimensions, where 1< D< 4.1<~D<~4.1< D< 4. The computational method converges as a power law, so that highly accurate results can be achieved with modest resources. We present the most accurate ground state energy (with no small parameter) to date for polaron problems, 21 digits for the one-dimensional (1D) polaron at intermediate coupling. The dimensionality effects on polaron band dispersion, effective mass, and electron-phonon (el-ph) correlation functions are investigated in all coupling regimes. It is found that the crossover to large effective mass of the higher-dimensional polaron is much sharper than the 1D polaron. The correlation length between the electron and phonons decreases significantly as the dimension increases. Our results compare favorably with those of the quantum Monte Carlo, dynamical mean-field theory, density-matrix renormalization-group, and Toyozawa variational methods. We demonstrate that the Toyozawa wave function is qualitatively correct for the ground-state energy and the two-point electron-phonon correlation functions, but fails for the three-point functions. Based on this finding, we propose an improved Toyozawa variational wave function.
Polaron ground states and finite-temperature dynamics remain challenging to simulate because existing methods struggle to combine nonperturbative accuracy, systematic improvability, and scalability from models to materials-specific Hamiltonians. We introduce a translationally invariant variational coupled-cluster (CC) theory for polarons, termed delocalized CC (dCC), with closed-form energies at cost as low as O(N3)\mathcal{O}(N^3)O(N3) and no phonon-number cutoff. dCC accurately describes the ground states of the one- and two-dimensional Holstein and Su--Schrieffer--Heeger (optical and bond) models and the Fr{ö}hlich model, in close agreement with density matrix renormalization group (DMRG) and diagrammatic Monte Carlo benchmarks. A projected tangent-space response formalism built on the same ansatz yields electron-addition spectral functions and optical conductivities at zero and finite temperature. The resulting spectra agree well with DMRG, Lanczos, and neural-network quantum-state benchmarks while retaining a physically interpretable excitation hierarchy, and extend to two-dimensional lattices at finite temperature beyond the practical reach of these methods. The same framework applies directly to \textit{ab initio} electron--phonon matrix elements, yielding LiF electron- and hole-polaron binding energies that match state-of-the-art many-body calculations. These results establish dCC as a unified variational framework for polaron ground states and dynamics, from model systems to real materials.
The dynamics of a coupled electron–boson system is investigated by employing a multitude of the Davydov D 1 trial states, also known as the multi-D 1 Ansatz, and a second trial state based on a superposition of the time-dependent generalized coherent state (GCS Ansatz ). The two Ansätze are applied to study population dynamics in the spin-boson model and the Holstein molecular crystal model, and a detailed comparison with numerically exact results obtained by the (multilayer) multiconfiguration time-dependent Hartree method and the hierarchy equations of motion approach is drawn. It is found that the two methodologies proposed here have significantly improved over that with the single D 1 Ansatz, yielding quantitatively accurate results even in the critical cases of large energy biases and large transfer integrals. The two methodologies provide new effective tools for accurate, efficient simulation of many-body quantum dynamics thanks to a relatively small number of parameters which characterize the electron–nuclear wave functions. The wave-function-based approaches are capable of tracking explicitly detailed bosonic dynamics, which is absent by construct in approaches based on the reduced density matrix. The efficiency and flexibility of our methods are also advantages as compared with numerically exact approaches such as QUAPI and HEOM, especially at low temperatures and in the strong coupling regime.