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arXiv:cond-mat/9710058 · Physical review. B, Condensed matter

Density-matrix renormalization-group study of the polaron problem in the Holstein model

Eric Jeckelmann, Steven R. White

Published Mar 15, 1998• 244 citations
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Abstract

We propose a density-matrix renormalization-group (DMRG) approach to study lattices including bosons. The key to the approach is an exact mapping of a boson site containing ${2}^{N}$ states to $N$ pseudosites, each with 2 states. The pseudosites can be viewed as the binary digits of a boson level. We apply the pseudosite DMRG method to the polaron problem in the one- and two-dimensional Holstein models. Ground-state results are presented for a wide range of electron-phonon coupling strengths and phonon frequencies on lattices large enough (up to 80 sites in one dimension and up to $20\ifmmode\times\else\texttimes\fi{}20$ sites in two dimensions) to eliminate finite-size effects, with up to 128 phonon states per phonon mode. We find a smooth but quite abrupt crossover from a quasi-free-electron ground state with a slightly renormalized mass at weak electron-phonon coupling to a polaronic ground state with a large effective mass at strong coupling, in agreement with previous studies.

DOI
10.1103/physrevb.57.6376

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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.
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.
The spectral weight functions and the optical conductivity of the Holstein model are studied on a one-dimensional six-site lattice with periodic boundary conditions for three different electron concentrations: a single electron, two electrons of opposite spins, and half filling. A density matrix approach is used to obtain an optimal phonon basis and to truncate the phonon Hilbert space without significant loss of accuracy. This approach allows us to calculate spectral functions for electrons dressed locally by the optimal phonons as well as for bare electrons. We obtain evidence for a smooth crossover from quasifree electrons to a heavy itinerant small polaron (single-electron case) or bipolaron (two-electron case) as the electron-phonon coupling strength increases. At half filling, we observe a crossover from a quasifree-electron ground state to a quasidegenerate Peierls charge-density-wave ground state for a finite electron-phonon coupling. This crossover is marked by an abrupt drop of the Drude weight, which is vanishingly small in the Peierls phase.
This work introduces an efficient numerical tool to calculate thermodynamic and spectral properties for electron-phonon problems at finite temperature in one dimension. Combining a finite-temperature density-matrix renormalization group method with local basis optimization allows one to access system sizes out of reach for other wave-function based approaches. The authors demonstrate the method's capabilities by computing several spectral functions of the Holstein polaron problem. The image shows the electron spectral function for two different system sizes (LLL=21, 101).
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.
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.