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Theoretical and Mathematical Physics

Quantum Problem of Polaron Localization and Justification of the Su—Schrieffer—Heeger Approximation

Vladimir Nikolaevich Likhachev, George A Vinogradov

Published May 1, 2019• 3 citations
Publisher

Abstract

We consider a polaron localized on a trap in a one-dimensional lattice with a harmonic potential of nearest-neighbor interaction. We study the lattice oscillations in the framework of quantum mechanics and regard the polaron energy as a functional of the wave function. We take the electron—phonon interaction into account in the framework of the linear Su—Schrieffer—Heeger approximation. We show that the results do not differ from the adiabatic consideration if the lattice oscillations are described classically. For the polaron ground state, we obtain approximate analytic expressions that agree well with the results of numerical simulation.

DOI
10.1134/s0040577919050088

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We consider a polaron localized on a trap in a one-dimensional lattice with a harmonic potential of nearest-neighbor interaction. We study the lattice oscillations in the framework of quantum mechanics and regard the polaron energy as a functional of the wave function. We take the electron—phonon interaction into account in the framework of the linear Su—Schrieffer—Heeger approximation. We show that the results do not differ from the adiabatic consideration if the lattice oscillations are described classically. For the polaron ground state, we obtain approximate analytic expressions that agree well with the results of numerical simulation.
Abstract The wave function and energy of the ground state of a lattice described by the Su–Schrieffer–Heeger Hamiltonian with a trap were found using three different methods. In the first of these methods, the energy of the ground state was calculated by numerical minimization of the energy functional. In the second, an auxiliary Hamiltonian was constructed, the diagonalization of which determined the energy of the ground state and the wave functions. And in the third, the wave function is completely restored analytically from the asymptotics of the wave function. All the three approaches produced results that agreed with an accuracy of 10–15.
We obtain analytical expressions for the large- and small-radius polarons on the one-dimensional lattice in the TBA approximation. The equations of motion for this model are treated classically for the oscillator subsystem, while a quantum description is used for the electron. The electron-phonon interaction is considered in the linear Su--Schrieffer--Heeger approximation. Good agreement between analytical formulae and accurate numerical simulation is obtained. The dynamics of polaron formation from different initial conditions is considered. Some features of the wave function evolution, governed by the finite lattice length, are elucidated.
We introduce a variational wave function to study the polaron formation when the electronic transfer integral depends on the relative displacement between nearest-neighbor sites giving rise to a nonlocal electron-phonon coupling with optical phonon modes. We characterize the polaron crossover by analyzing ground-state properties such as the energy, the electron-lattice correlation function, the average phonon occupation, and the quasiparticle spectral weight. Variational results are found in good agreement with numerical exact diagonalization of small clusters,and follow the correct perturbative result at weak coupling. We determine the polaronic phase diagram and we find that the tendency towards strong localization is hindered from the pathological sign change of the effective next-nearest-neighbor hopping.
Starting from the Hamiltonian for a dimer which includes all the electronic and electron-phonon terms consistent with a non-degenerate orbital, by a sequence of displacement and squeezing transformation we obtain an effective polaronic Hamiltonian. The renormalized electronic interactions differ from the results of semiclassical or perturbative treatments. The properties of the variationally determined ground state of two particles in the orbital are discussed for variable dimer length in the adiabatic limit
Polarons with different types of electron-phonon coupling have fundamentally different properties. When the dominant interaction is between the electron density and lattice displacement, the momentum of the ground state does not change and the polaron gets exponentially heavy at strong coupling. In contrast, one-dimensional Peierls/Su-Schrieffer-Heeger (PSSH) polarons with interaction originating from displacement-modulated hopping feature a shift of the ground-state momentum to finite values and moderate values of effective mass as coupling is increased [D. J. J. Marchand et al., Phys. Rev. Lett. 105, 266605 (2010)]. Based on Diagrammatic Monte Carlo method, we investigate whether unusual properties of PSSH polarons depend on the type of the displacement-modulated hopping and to what degree they survive in higher dimension. We study two different PSSH models: with bosonic degrees of freedom residing on sites (model A) and bonds (model B) of the two-dimensional square lattice. For model A, we find that in both adiabatic and intermediate regimes, the momentum of the ground state experiences a continuous transition from zero to a finite value as a function of coupling strength. The transition is driven by quadratic instability of the dispersion function, implying that effective mass diverges at the critical point, and then decreases in an anisotropic fashion with increasing coupling. Unexpectedly, for model B, the momentum of the ground state always stays at zero and the effective mass increases monotonously with coupling. The increase is far from exponential and tends to level off at strong interaction, resulting in relatively light polarons. Having light polarons in the strong coupling regime is crucial for the bipolaron mechanism of high-temperature superconductivity [J. Sous, M. Chakraborty, R. V. Krems, and M. Berciu, Phys. Rev. Lett. 121, 247001 (2018).]
We have applied a variational method to study both the Davydov model and the Su, Schrieffer, and Heeger model for the interaction between an electron and the acoustical phonons of a discrete chain. A class of localized solutions has been found for both of these models. These solutions appear in the intermediate values of the adiabaticity \ensuremath{\gamma}\ensuremath{\sim}1 when the particle-phonon coupling constant \ensuremath{\lambda} is near the critical point for the delocalization-localization transition. We have widely discussed the limits of the continuum approximation, stressing the importance of the application of the method to a discrete system if we want to explore the transition from a large mobile polaron to a strong localized heavy polaron. Furthermore, we have considered the connection between localization and transport in the low-density limit and the possible applications of the method to understanding of the physical properties of some systems with low-dimensionality.