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arXiv:2412.17770 · Physical review. B./Physical review. B

Precursors to Anderson localization in the Holstein model: Quantum and quantum-classical solutions

Petar Mitrić, V. Dobrosavljević, D. Tanasković

Published Apr 7, 2025• 10 citations
arXivHTMLPDF

Abstract

We calculate the frequency-dependent mobility of the Holstein polaron in one dimension near the adiabatic limit using the method based on dynamical quantum typicality, as well as the quantum-classical method. The agreement between fully quantum and quantum-classical solutions is very good. The most prominent feature is the appearance of a zero-frequency peak in the mobility, in addition to the displaced peak associated to the precursors of Anderson localization. The zero-frequency peak cannot be obtained within the phenomenological transient localization approach, which is often used in a semiquantitative description of charge transport in quasi-one-dimensional organic semiconductors.

DOI
10.1103/physrevb.111.l161105

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We calculate the frequency-dependent mobility of the Holstein polaron in one dimension near the adiabatic limit using the method based on dynamical quantum typicality, as well as the quantum-classical method. The agreement between fully quantum and quantum-classical solutions is very good. The most prominent feature is the appearance of a zero-frequency peak in the mobility, in addition to the displaced peak associated to the precursors of Anderson localization. The zero-frequency peak cannot be obtained within the phenomenological transient localization approach, which is often used in a semiquantitative description of charge transport in quasi-one-dimensional organic semiconductors.
We solve the disordered Holstein model via the density-matrix renormalization group method to investigate the combined roles of electron-phonon coupling and disorder on the localization of a single charge or exciton. The parameter regimes chosen, namely the adiabatic regime, $\ensuremath{\hbar}\ensuremath{\omega}/4{t}_{0}={\ensuremath{\omega}}^{\ensuremath{'}}<1$, and the large polaron regime, $\ensuremath{\lambda}<1$, are applicable to most conjugated polymers. We show that as a consequence of the polaron effective mass diverging in the adiabatic limit (defined as ${\ensuremath{\omega}}^{\ensuremath{'}}\ensuremath{\rightarrow}0$ subject to fixed $\ensuremath{\lambda}$) self-localized, symmetry-breaking solutions are predicted by the quantum Holstein model for infinitesimal disorder, in complete agreement with the predictions of the Born-Oppenheimer Holstein model. For other parts of the (${\ensuremath{\omega}}^{\ensuremath{'}}$, $\ensuremath{\lambda}$) parameter space, however, self-localized Born-Oppenheimer solutions are not expected. If ${\ensuremath{\omega}}^{\ensuremath{'}}$ is not small enough and $\ensuremath{\lambda}$ is not large enough, then the polaron is predominately localized by Anderson disorder, albeit more than for a free particle, because of the enhanced effective mass. Alternatively, for very small electron-phonon coupling ($\ensuremath{\lambda}\ensuremath{\ll}1$) the disorder-induced localization length is always smaller than the classical polaron size, $2/\ensuremath{\lambda}$, so that disorder always dominates. We comment on the implication of our results on the electronic properties of conjugated polymers.
We investigate the transport properties of the Holstein model using the numerically exact quantum typicality (QT) approach. Roughly speaking, QT exploits the fact that even a single, randomly chosen pure state can effectively represent the full statistical ensemble in a high-dimensional Hilbert space. This allows us to compute frequency-dependent mobilities, representative of the thermodynamic limit, that are well converged with respect to all numerical parameters. Our results are compared against other numerically exact methods, and used to analyze the contribution of vertex corrections to frequency-dependent mobility. The promising accuracy and efficiency of the QT approach suggest its applicability to a broader class of Hamiltonians.
A new numerical method is proposed for determining the low-frequency dynamics of the charge carrier coupled to the deformable quantum lattice. As an example, the polaron band structure is calculated for the one-dimensional Holstein model. The adiabatic limit on the lattice, which cannot be reached by other approaches, is investigated. In particular, an accurate description is obtained of the crossover between quantum small adiabatic polarons, pinned by the lattice, and large adiabatic polarons, moving along the continuum as classical particles. It is shown how the adiabatic contributions to the polaron dispersion, involving spatial correlations over multiple lattice sites, can be treated easily in terms of coherent states.
We study Holstein polarons in three-dimensional anisotropic materials. Using a variational exact diagonalization technique we provide highly accurate results for the polaron mass and polaron radius. With these data we discuss the differences between polaron formation in dimensions one and three and at small and large phonon frequencies. Varying the anisotropy we demonstrate how a polaron evolves from a one-dimensional to a three-dimensional quasiparticle. We thereby resolve the issue of polaron stability in quasi-one-dimensional substances and clarify to what extent such polarons can be described as one-dimensional objects. We finally show that even the local Holstein interaction leads to an enhancement of anisotropy in charge-carrier motion.
A variational technique is developed to investigate the polaronic features of the Holstein molecular-crystal model. It is based on a linear superposition of Bloch states that describe large and small polaron wave functions. It is shown that this method provides a very good description of the regime characterized by intermediate values of the electron-phonon coupling constant (the so-called intermediate polaron) for any value of the adiabatic parameter ${\ensuremath{\omega}}_{0}/t.$ The polaron ground state energy in one and two dimensions is calculated and successfully compared with the best estimates available providing a clear physical interpretation of the intermediate polaron. The band structure, the spectral weight of the ground state and the lattice displacement associated to the polaron are also calculated and discussed. The method has the advantage of requiring very little computational effort.
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.