Davydov’s Soliton

The authors present a certain type of exact solutions for the two nonlinear equations by making an ansatz for the solution.

https://iopscience.iop.org/article/10.1088/0305-4470/23/17/021/pdf

Exact solutions for two nonlinear equations. I

Exact solutions of nonlinear equations

Following the Dirac-Frenkel time-dependent variational principle, dynamics of a one-dimensional Holstein polaron is probed by employing the Davydov ${\text{D}}_{2}$ ansatz with two sets of variational parameters, one for each constituting particle in the exciton-phonon system, and a simplified variant of the Davydov ${\text{D}}_{1}$ ansatz, also known as the $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\text{D}}$ ansatz, with an additional set of phonon displacement parameters. A close examination of variational outputs from the two trial states reveals fine details of the polaron structure and intricacies of dynamic exciton-phonon interactions. Superradiance coherence sizes, speeds of exciton-induced phonon wave packets, linear optical absorption, and polaron energy compositions are also included in the study.

/pdf/dynamics-of-a-one-dimensional-holstein-polaron-with-the-36m9mwuxnh.pdf

Dynamics of a one-dimensional Holstein polaron with the Davydov ansätze

We propose an extension to the Davydov D2 Ansatz in the dynamics study of the Holstein molecular crystal model with diagonal and off-diagonal exciton-phonon coupling using the Dirac-Frenkel time-dependent variational principle. The new trial state by the name of the "multi-D2 Ansatz" is a linear combination of Davydov D2 trial states, and its validity is carefully examined by quantifying how faithfully it follows the Schrodinger equation. Considerable improvements in accuracy have been demonstrated in comparison with the usual Davydov trial states, i.e., the single D1 and D2 Ansatze. With an increase in the number of the Davydov D2 trial states in the multi-D2 Ansatz, deviation from the exact Schrodinger dynamics is gradually diminished, leading to a numerically exact solution to the Schrodinger equation.

/pdf/polaron-dynamics-with-a-multitude-of-davydov-d2-trial-states-26dp8xbvyj.pdf

Polaron dynamics with a multitude of Davydov D2 trial states

Classical and quantum approach to Davydov's soliton theory.

Soliton excitations in the anisotropic Heisenberg chain are studied in the classical limit by means of the Holstein-Primakoff boson representation, taking into account the complete series. The results obtained are equivalent to those obtained by the classical treatment of spins. The connection with different calculations using limited numbers of terms is established, as well as the relation to the generalised coherent-states method.

The classical limit for the Holstein-Primakoff representation in the soliton theory of Heisenberg chains

We have studied the spin-wave dispersion at low temperatures and the transition temperature ${(T}_{N})$ of the spin-$\frac{1}{2}$ antiferromagnet and high-${T}_{C}$ parent ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}.$ The values of the in-plane exchange parameters (including first, second, and third nearest neighbors) are determined by an accurate fit to the recently experimentally observed in-plane spin-wave spectrum, obtained by the high-resolution inelastic neutron scattering performed on ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}$ [Phys. Rev. Lett. 86, 5377 (2001)]. The analysis of the N\'eel temperature shows that the in-plane spin anisotropy $(\ensuremath{\eta})$ is much more significant than the three dimensionality, since ${T}_{N}$ of the three-dimensional (3D)-antiferromagnet depends rather weakly on the value of the interlayer coupling $({\ensuremath{\lambda}}_{\ensuremath{\perp}}).$ We obtain that the N\'eel temperature of the 3D-antiferromagnet varies only weakly within the very wide interval of ${\ensuremath{\lambda}}_{\ensuremath{\perp}}$ and the N\'eel temperature of the anisotropic 2D $(\ensuremath{\eta}\ensuremath{\ne}0,$ ${\ensuremath{\lambda}}_{\ensuremath{\perp}}=0)$ antiferromagnet does not differ from the 3D value for the same $\ensuremath{\eta}.$ These conclusions are valid for both tetragonal and orthorhombic structures. However, $\ensuremath{\eta}$ dependence of ${T}_{N}$ is essentially different: for $\ensuremath{\eta}=0,$ ${T}_{N}$ of the tetragonal structure becomes 0, while ${T}_{N}$ of the orthorhombic structure remains finite. These results are valid within the frame of the Tyablikov approximation.

Spin-wave dispersion and transition temperature in the cuprate antiferromagnet La 2 CuO 4

It is shown that the Holstein-Primakoff boson representation of spin operators leads to the same soliton excitations, in the anisotropic Heisenberg chain, as the classical treatment of spins, if one retains the terms up to products of six Bose operators.

Solitons in the anisotropic Heisenberg chain in the Holstein-Primakoff representation

It is shown that the application of both the spin-coherent states and the Holstein-PrimakofF representation with boson-coherent states leads to the same classical limit for the spin-operator average values, if the proper procedure for normal ordering is applied. A general expression for the boson expansion is derived, and the first quantum correction is obtained. Consequences of the study of nonlinear excitations in magnetic systems are presented.

S. D. Stojanović

Papers

Classical and quantum approach to Davydov's soliton theory.

The classical limit for the Holstein-Primakoff representation in the soliton theory of Heisenberg chains

Spin-wave dispersion and transition temperature in the cuprate antiferromagnet La 2 CuO 4

Solitons in the anisotropic Heisenberg chain in the Holstein-Primakoff representation

Relation between spin-coherent states and boson-coherent states in the theory of magnetism.