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Physical Review B

Two-site small polaron quantum states: Optimization of the dressing fraction

Vincent Pouthier

Published Jun 24, 2009• 7 citations
Publisher

Abstract

A modified Lang-Firsov transformation is used to study the exciton-phonon interaction in a two-site system embedded in a one-dimensional lattice. It describes an exciton partially dressed by a virtual phonon cloud and depends on a single variational parameter, the so-called dressing fraction, whose optimization is achieved by using both the standard Bogoliubov inequality and its improved version defined by Decoster [J. Phys. A 37, 9051 (2004)]. The optimization procedure is applied to build a phase diagram in the parameter space which defines the different states of the exciton depending on the adiabaticity, the coupling strength, and the temperature. It is shown that the two Bogoliubov inequalities yield different variational principles that give rise to two optimal dressing fractions. Special attention is thus paid to characterize their differences in the nonadiabatic limit, where the exciton evolves continuously from a partially dressed state in the weak-coupling limit to a fully dressed state in the strong-coupling limit, and in the adiabatic limit where a self-trapped transition takes place.

DOI
10.1103/physrevb.79.214304

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A modified Lang-Firsov transformation is used to study the exciton-phonon interaction in a two-site system embedded in a one-dimensional lattice. It describes an exciton partially dressed by a virtual phonon cloud and depends on a single variational parameter, the so-called dressing fraction, whose optimization is achieved by using both the standard Bogoliubov inequality and its improved version defined by Decoster [J. Phys. A 37, 9051 (2004)]. The optimization procedure is applied to build a phase diagram in the parameter space which defines the different states of the exciton depending on the adiabaticity, the coupling strength, and the temperature. It is shown that the two Bogoliubov inequalities yield different variational principles that give rise to two optimal dressing fractions. Special attention is thus paid to characterize their differences in the nonadiabatic limit, where the exciton evolves continuously from a partially dressed state in the weak-coupling limit to a fully dressed state in the strong-coupling limit, and in the adiabatic limit where a self-trapped transition takes place.
Polaronic features of dopant-induced charge carriers are manifest in recent experiments on ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$Srx{\mathrm{Sr}}_{\mathit{x}}Srx​CuO4+y{\mathrm{CuO}}_{4+\mathit{y}}CuO4+y​ and ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$Nix{\mathrm{Ni}}_{\mathit{x}}Nix​CuO4+y{\mathrm{CuO}}_{4+\mathit{y}}CuO4+y​. To study the formation of hole (bi)polarons in such systems exhibiting strong Coulomb interactions, the Holstein t-J model is examined by means of a variational diagonalization technqiue for a wide range of phonon frequencies and electron-phonon (EP) couplings on finite square lattices, up to 18 effective sites in size. Including static displacement field, polaron, and squeezing effects, our approach allows for the description of nearly free polarons in the weak-coupling limit as well as for adiabatic Holstein polarons and nonadiabatic Lang-Firsov polarons in the case of strong EP interactions. At low doping levels and low phonon frequencies we demonstrate that antiferromagnetic spin correlations and EP interactions reinforce each other to the effect of lowering the threshold for polaronic self-localization in a strongly distorted lattice. This is contrasted with the nonadiabatic regime, where we observe the formation of Lang-Firsov polarons with moderate polaronic mass enhancement in a nearly undistorted lattice. In the case of two doped holes, the hole binding energy is analyzed in detail and we find that hole binding is enhanced as a dynamical (static) effect of a rather weak (extremely strong) EP interaction for delocalized (self-trapped) polarons. At quarter-filling we notice, in the adiabatic regime, a sequence of transitions as the EP coupling increases from nearly free mobile polarons to a polaronic superlattice and finally to a charge-separated state. The ordering of polarons disappears at higher phonon frequencies.
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.
Recent development of theoretical method for dissipative quantum transport has achieved notable progresses in the weak or strong electron-phonon coupling regime. However, a generalized theory for dissipative quantum transport in a wide parameter space had not been established. In this work, a variational polaron theory for dissipative quantum transport in a wide range of electron-phonon coupling is developed. The optimal polaron transformation is determined by the optimization of the Feynman-Bogoliubov upper bound of free energy. The free energy minimization ends up with an optimal mean-field Hamiltonian and a minimal interaction Hamiltonian. Hence, second-order perturbation can be applied to the transformed system, resulting in an accurate and efficient method for the treatment of dissipative quantum transport with different electron-phonon coupling strength. Numerical benchmark calculation on a single site model coupled to one phonon mode is presented.
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.