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The Journal of Chemical Physics

Polaronic discontinuities induced by off-diagonal coupling

Yuyu Zhang, Liwei Duan, Qing‐Hu Chen, Yang Zhao

Published Jul 17, 2012• 8 citations
Publisher

Abstract

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.

DOI
10.1063/1.4733986

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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.
We present a detailed numerical study of the one-dimensional Holstein model with a view to understanding the self-trapping process of electrons or excitons in crystals with short-range particle-lattice interactions. Applying a very efficient variational Lanczos method, we are able to analyze the ground-state properties of the system in the weak- and strong-coupling, adiabatic and nonadiabatic regimes on lattices large enough to eliminate finite-size effects. In particular, we obtain the complete phase diagram and comment on the existence of a critical length for self-trapping in finite (closed) one-dimensional systems. In order to characterize large and small polaron states we calculate self-consistently the lattice distortions and the particle-phonon correlation functions. In the strong-coupling case, two distinct types of small polaron states are shown to be possible according to the relative importance of static displacement field and dynamic polaron effects. Special emphasis is on the intermediate-coupling regime, which we also study by means of direct diagonalization, preserving the full dynamics and quantum nature of phonons. The crossover from large to small polarons shows up in a strong decrease of the kinetic energy accompanied by a substantial change in the optical absorption spectra. We show that our numerical results in all important limiting cases reveal excellent agreement with both analytical perturbation theory predictions and very recent density matrix renormalization group data.
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.
The paper deals with the ground and the first excited state of the polaron in the one-dimensional Holstein model. Various variational methods are used to investigate both the weak-coupling and strong-coupling case, as well as the crossover regime between them. Two of the methods, presented here, introduce interesting elements to the understanding of the nature of the polaron. Reliable numerical evidence is found that, in the strong-coupling regime, the ground and the first excited state of the self-trapped polaron are well described within the adiabatic limit. The lattice vibration modes associated with the self-trapped polarons are analyzed in detail, and the frequency softening of the vibration mode at the central site of the small polaron is estimated. It is shown that the first excited state of the system in the strong-coupling regime corresponds to the excitation of the soft phonon mode within the polaron. In the crossover regime, the ground and the first excited state of the system can be approximated by the anticrossing of the self-trapped and the delocalized polaron state. In this way, the connection between the behavior of the ground and the first excited state is qualitatively explained.
We describe quantum entanglement inherent to the polaron ground states of coupled electron-phonon (or, more generally, particle-phonon) systems based on a model comprising both local (Holstein-type) and nonlocal (Peierls-type) couplings. We study this model using a variational method supplemented by the exact numerical diagonalization on a system of finite size. By way of subsequent numerical diagonalization of the reduced density matrix, we determine the particle-phonon entanglement as given by the von Neumann and linear entropies. Our results are strongly indicative of the intimate relationship between the particle localization/delocalization and the particle-phonon entanglement. In particular, we find a compelling evidence for the existence of a nonanalyticity in the entanglement entropies with respect to the Peierls-coupling strength. The occurrence of such nonanalyticity---not accompanied by an actual quantum phase transition---reinforces analogous conclusion drawn in several recent studies of entanglement in the realm of quantum-dissipative systems. In addition, we demonstrate that the entanglement entropies saturate inside the self-trapped region where the small-polaron states are nearly maximally mixed.
We present numerical exact results for the polaronic band structure of the Holstein molecular crystal model in one and two dimensions. The use of a direct Lanczos diagonalization technique, preserving the full dynamics and quantum nature of phonons, allows us to analyze in detail the renormalization of both quasiparticle bandwidth and dispersion by the electron-phonon interaction. For the two-dimensional case some of our exact data are compared with the results obtained in the framework of a recently developed finite-cluster strong-coupling perturbation theory.
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.