Frenkel–Kontorova model
[1] The generalized FK model describes a chain of classical particles with nearest neighbor interactions and subjected to a periodic on-site substrate potential.
[2] In its original and simplest form the interactions are taken to be harmonic and the potential to be sinusoidal with a periodicity commensurate with the equilibrium distance of the particles.
Originally introduced by Yakov Frenkel and Tatiana Kontorova [ru] in 1938 to describe the structure and dynamics of a crystal lattice near a dislocation core, the FK model has become one of the standard models in condensed matter physics due to its applicability to describe many physical phenomena.
Physical phenomena that can be modeled by FK model include dislocations, the dynamics of adsorbate layers on surfaces, crowdions, domain walls in magnetically ordered structures, long Josephson junctions, hydrogen-bonded chains, and DNA type chains.
[1] In the continuum-limit approximation the FK model reduces to the exactly integrable sine-Gordon (SG) equation, which allows for soliton solutions.
A simple model of a harmonic chain in a periodic substrate potential was proposed by Ulrich Dehlinger in 1928.
Dehlinger derived an approximate analytical expression for the stable solutions of this model, which he termed Verhakungen, which correspond to what is today called kink pairs.
An essentially similar model was developed by Ludwig Prandtl in 1912/13 but did not see publication until 1928.
[5] The model was independently proposed by Yakov Frenkel and Tatiana Kontorova in their 1938 article On the theory of plastic deformation and twinning to describe the dynamics of a crystal lattice near a dislocation and to describe crystal twinning.
[4] In the standard linear harmonic chain any displacement of the atoms will result in waves, and the only stable configuration will be the trivial one.
For the nonlinear chain of Frenkel and Kontorova, there exist stable configurations beside the trivial one.
For small atomic displacements the situation resembles the linear chain; however, for large enough displacements, it is possible to create a moving single dislocation, for which an analytical solution was derived by Frenkel and Kontorova.
[6] The shape of these dislocations is defined only by the parameters of the system such as the mass and the elastic constant of the springs.
The defining characteristic of solitons/dislocations is that they behave much like stable particles, they can move while maintaining their overall shape.
The generalized FK model treats a one-dimensional chain of atoms with nearest-neighbor interaction in periodic on-site potential, the Hamiltonian for this system is where the first term is the kinetic energy of the
For non-harmonic interactions and/or non-sinusoidal potential, the FK model will give rise to a commensurate–incommensurate phase transition.
In this section we examine in detail the simplest form of the FK model.
[2] The model describes a one-dimensional chain of atoms with a harmonic nearest neighbor interaction and subject to a sinusoidal potential.
The following dimensionless variables are introduced in order to rewrite the Hamiltonian: In dimensionless form the Hamiltonian is which describes a harmonic chain of atoms of unit mass in a sinusoidal potential of period
Thus in the ground state of the chain each minimum of the substrate potential is occupied by one atom.
The nonlinear equations can support new types of localized excitations, which are best illuminated by considering the continuum limit of the FK model.
Applying the standard procedure of Rosenau[7] to derive continuum-limit equations from a discrete lattice results in the perturbed sine-Gordon equation where the function describes in first order the effects due to the discreteness of the chain.
Kinks, or topological solitons, can be understood as the solution connecting two nearest identical minima of the periodic substrate potential, thus they are a result of the degeneracy of the ground state.
have energy of repulsion whereas kink and antikink attract with interaction A breather is which describes nonlinear oscillation with frequency
Furthermore, any collision between all the excitations of the SG equation result in only a phase shift.
For nearly integrable modifications of the SG equation such as the continuum approximation of the FK model kinks can be considered deformable quasi-particles, provided that discreetness effects are small.
Since the properties of kinks are only modified slightly by the discreteness of the primary model, the SG equation can adequately describe most features and dynamics of the system.
The discrete lattice does, however, influence the kink motion in a unique way with the existence of the Peierls–Nabarro (PN) potential
The existence of the PN potential is due to the lack of translational invariance in a discrete chain.
The value of the PN barrier is the difference between the kink's potential energy for a stable and unstable stationary configuration.
