Franz Mertens - Academia.edu (original) (raw)
Papers by Franz Mertens
arXiv (Cornell University), Sep 30, 1998
We examine spin vortices in ferromagnetic quantum Heisenberg models with planar anisotropy on two... more We examine spin vortices in ferromagnetic quantum Heisenberg models with planar anisotropy on two-dimensional lattices. The symmetry properties and the time evolution of vortices built up from spin-coherent states are studied in detail. Although these states show a dispersion typical for wave packets, important features of classical vortices are conserved. Moreover, the results on symmetry properties provide a construction scheme for vortex-like excitations from exact eigenstates, which have a well-controlled time evolution. Our approach works for arbitrary spin length both on triangular and square lattices.
Journal of Physics A, Jul 29, 2021
We consider the novel nonlinear model in (1 + 1)-dimensions for Dirac spinors recently introduced... more We consider the novel nonlinear model in (1 + 1)-dimensions for Dirac spinors recently introduced by Alexeeva et al (Ann. Phys., NY 403 198) (ABS model), which admits an exact explicit solitary-wave (soliton for short) solution. The charge, the momentum, and the energy of this solution are conserved. We investigate the dynamics of the soliton subjected to several potentials: a ramp, a harmonic, and a periodic potential. We develop a collective coordinates (CCs) theory by making an ansatz for a moving soliton where the position, rapidity, and momentum, are functions of time. We insert the ansatz into the Lagrangian density of the model, integrate over space and obtain a Lagrangian as a function of the CCs. This Lagrangian differs only in the charge and mass with the Lagrangian of a CCs theory for the Gross–Neveu equation. Thus the soliton dynamics in the Alexeeva–Barashenkov–Saxena (ABS) spinor model is qualitatively the same as in the Gross–Neveu equation, but quantitatively it differs. These results of the CCs theory are confirmed by simulations, i.e. by numerical solutions for solitons of the ABS spinor model, subjected to the above potentials.
Journal of Physics A, Jul 27, 2020
The European Physical Journal A, Feb 1, 1974
European Physical Journal B, Nov 1, 2004
We analyze the diffusive motion of kink solitons governed by the thermal sine-Gordon equation. We... more We analyze the diffusive motion of kink solitons governed by the thermal sine-Gordon equation. We analytically calculate the correlation function of the position of the kink center as well as the diffusion coefficient, both up to second-order in temperature. We find that the kink behavior is very similar to that obtained in the overdamped limit: There is a quadratic dependence on temperature in the diffusion coefficient that comes from the interaction among the kink and phonons, and the average value of the wave function increases with √ t due to the variance of the centers of individual realizations and not due to kink distortions. These analytical results are fully confirmed by numerical simulations.
We investigate the ratchet dynamics of nonlinear Klein-Gordon kinks in a periodic, asymmetric lat... more We investigate the ratchet dynamics of nonlinear Klein-Gordon kinks in a periodic, asymmetric lattice of point-like inhomogeneities. We explain the underlying rectification mechanism within a collective coordinate framework, which shows that such system behaves as a rocking ratchet for point particles. Careful attention is given to the kink width dynamics and its role in the transport. We also analyze the robustness of our kink rocking ratchet in the presence of noise. We show that the noise activates unidirectional motion in a parameter range where such motion is not observed in the noiseless case. This is subsequently corroborated by the collective variable theory. An explanation for this new phenomenom is given.
NATO ASI Series, 1991
We consider a classical 2D Heisenberg model with easy-plane symmetry. Koster-litz and Thouless1 s... more We consider a classical 2D Heisenberg model with easy-plane symmetry. Koster-litz and Thouless1 showed that such a system has a topological phase transition: at low temperatures there exist bound vortex pairs which start to dissociate above a critical temperature T KT Just above T KT we can assume that there are only a few free vortices which move ballistically between their interactions. A model of dynamics built on such a “vortex gas” has been constructed assuming a Gaussian velocity distribution. Here we use effective equations of motion for the collective (center-of-mass) vortex variables and compare these analytical results of vortex-vortex and vortex-anitvortex interactions with molecular dynamics simulations of the full spin system.3 We investigate both ferromagnets (FM) and antiferromagnets (AFM) with an anisotropy parameter λ varying from zero to one.
Lecture Notes in Physics
The 2-dimensional anisotropic Heisenberg model with XY-or easy-plane symmetry bears non-planar vo... more The 2-dimensional anisotropic Heisenberg model with XY-or easy-plane symmetry bears non-planar vortices which exhibit a localized structure of the z-components of the spins around the vortex center. In order to study the dynamics of these vortices under thermal fluctuations we use the Landau-Lifshitz equation and add white noise and Gilbert damping. Using a collective variable theory we derive an equation of motion with stochastic forces which are shown to represent white noise with an effective diffusion constant. We compare ...
Zeitschrift für Physik A Hadrons and nuclei, 1969
A short derivation of the Random Phase Approximation of angular momentum waves in solid Ortho-H2 ... more A short derivation of the Random Phase Approximation of angular momentum waves in solid Ortho-H2 is presented. Methods and results of papers on the same topic are compared.
![Research paper thumbnail of Erratum: Nonlinear Dirac equation solitary waves in external fields [Phys. Rev. E86, 046602 (2012)]](https://attachments.academia-assets.com/115311637/thumbnails/1.jpg)
](Physical review, May 10, 2016
Journal of Physics A: Mathematical and Theoretical, 2019
Journal of Physics A, Jan 5, 2016
Franz G. Mertens Physikalisches Institut, Universität Bayreuth, D–95440 Bayreuth, Germany (Dated:... more Franz G. Mertens Physikalisches Institut, Universität Bayreuth, D–95440 Bayreuth, Germany (Dated: November 11, 2002) Abstract The Levinson theorem for two–dimensional scattering is generalized for potentials with inverse square singularities. By this theorem, the number of bound states in a given m–th partial wave is related to the phase shift and the singularity strength of the potential. For the m–wave phase shift the asymptotic behaviour is calculated for short wavelengths.
Journal of Applied Physics, 2015
A spin-polarized electrical current leads to a variety of periodical magnetic structures in nanos... more A spin-polarized electrical current leads to a variety of periodical magnetic structures in nanostripes. In the presence of the Ørsted field, which always assists an electrical current, the basic types of magnetic structures, i.e., a vortex-antivortex crystal and cross-tie domain walls, survive. The Ørsted field prevents saturation of the nanostripe and a longitudinal domain wall appears instead. Possible magnetization structures in stripes with different geometrical and material properties are studied numerically and analytically.
Physics Letters A, 1985
For certain integrable gas and lattice models in one dimension the Bethe ansatz leads to a formal... more For certain integrable gas and lattice models in one dimension the Bethe ansatz leads to a formal description of the system as a spinless Fermi gas. The particle-hole structure of the zero-temperature excitations is generalized to finite temperatures. Dispersion and thermal distribution of the excitations are calculated, and tested for the Toda lattice.
Physical Review-Section E-Statistical Nonlinear and Soft Matter Physics, 2005
Journal of Physics A: Mathematical and Theoretical, 2019
Soliton dynamics in the damped and parametrically driven nonlinear Dirac equation, in 1 + 1 dimen... more Soliton dynamics in the damped and parametrically driven nonlinear Dirac equation, in 1 + 1 dimension with scalar–scalar self-interaction is analysed. The considered parametric force has the spatial period . A variational approach using collective coordinates for studying the time dependent response of the solitary waves to this parametric force is developed. The dynamical equations for the collective coordinates are also obtained by an alternative method, namely the method of moments. The soliton dynamics depends crucially on the competition between two length scales: the spatial period and the width of the soliton l s . For the soliton oscillates in an effective potential, while for it moves uniformly as a free particle. The transition between these two regimes occurs when is comparable to the soliton width. This match enhances the soliton instabilities so that even small values of the perturbation are enough to modify drastically the soliton shape and destroy it for long times.
arXiv (Cornell University), Sep 30, 1998
We examine spin vortices in ferromagnetic quantum Heisenberg models with planar anisotropy on two... more We examine spin vortices in ferromagnetic quantum Heisenberg models with planar anisotropy on two-dimensional lattices. The symmetry properties and the time evolution of vortices built up from spin-coherent states are studied in detail. Although these states show a dispersion typical for wave packets, important features of classical vortices are conserved. Moreover, the results on symmetry properties provide a construction scheme for vortex-like excitations from exact eigenstates, which have a well-controlled time evolution. Our approach works for arbitrary spin length both on triangular and square lattices.
Journal of Physics A, Jul 29, 2021
We consider the novel nonlinear model in (1 + 1)-dimensions for Dirac spinors recently introduced... more We consider the novel nonlinear model in (1 + 1)-dimensions for Dirac spinors recently introduced by Alexeeva et al (Ann. Phys., NY 403 198) (ABS model), which admits an exact explicit solitary-wave (soliton for short) solution. The charge, the momentum, and the energy of this solution are conserved. We investigate the dynamics of the soliton subjected to several potentials: a ramp, a harmonic, and a periodic potential. We develop a collective coordinates (CCs) theory by making an ansatz for a moving soliton where the position, rapidity, and momentum, are functions of time. We insert the ansatz into the Lagrangian density of the model, integrate over space and obtain a Lagrangian as a function of the CCs. This Lagrangian differs only in the charge and mass with the Lagrangian of a CCs theory for the Gross–Neveu equation. Thus the soliton dynamics in the Alexeeva–Barashenkov–Saxena (ABS) spinor model is qualitatively the same as in the Gross–Neveu equation, but quantitatively it differs. These results of the CCs theory are confirmed by simulations, i.e. by numerical solutions for solitons of the ABS spinor model, subjected to the above potentials.
Journal of Physics A, Jul 27, 2020
The European Physical Journal A, Feb 1, 1974
European Physical Journal B, Nov 1, 2004
We analyze the diffusive motion of kink solitons governed by the thermal sine-Gordon equation. We... more We analyze the diffusive motion of kink solitons governed by the thermal sine-Gordon equation. We analytically calculate the correlation function of the position of the kink center as well as the diffusion coefficient, both up to second-order in temperature. We find that the kink behavior is very similar to that obtained in the overdamped limit: There is a quadratic dependence on temperature in the diffusion coefficient that comes from the interaction among the kink and phonons, and the average value of the wave function increases with √ t due to the variance of the centers of individual realizations and not due to kink distortions. These analytical results are fully confirmed by numerical simulations.
We investigate the ratchet dynamics of nonlinear Klein-Gordon kinks in a periodic, asymmetric lat... more We investigate the ratchet dynamics of nonlinear Klein-Gordon kinks in a periodic, asymmetric lattice of point-like inhomogeneities. We explain the underlying rectification mechanism within a collective coordinate framework, which shows that such system behaves as a rocking ratchet for point particles. Careful attention is given to the kink width dynamics and its role in the transport. We also analyze the robustness of our kink rocking ratchet in the presence of noise. We show that the noise activates unidirectional motion in a parameter range where such motion is not observed in the noiseless case. This is subsequently corroborated by the collective variable theory. An explanation for this new phenomenom is given.
NATO ASI Series, 1991
We consider a classical 2D Heisenberg model with easy-plane symmetry. Koster-litz and Thouless1 s... more We consider a classical 2D Heisenberg model with easy-plane symmetry. Koster-litz and Thouless1 showed that such a system has a topological phase transition: at low temperatures there exist bound vortex pairs which start to dissociate above a critical temperature T KT Just above T KT we can assume that there are only a few free vortices which move ballistically between their interactions. A model of dynamics built on such a “vortex gas” has been constructed assuming a Gaussian velocity distribution. Here we use effective equations of motion for the collective (center-of-mass) vortex variables and compare these analytical results of vortex-vortex and vortex-anitvortex interactions with molecular dynamics simulations of the full spin system.3 We investigate both ferromagnets (FM) and antiferromagnets (AFM) with an anisotropy parameter λ varying from zero to one.
Lecture Notes in Physics
The 2-dimensional anisotropic Heisenberg model with XY-or easy-plane symmetry bears non-planar vo... more The 2-dimensional anisotropic Heisenberg model with XY-or easy-plane symmetry bears non-planar vortices which exhibit a localized structure of the z-components of the spins around the vortex center. In order to study the dynamics of these vortices under thermal fluctuations we use the Landau-Lifshitz equation and add white noise and Gilbert damping. Using a collective variable theory we derive an equation of motion with stochastic forces which are shown to represent white noise with an effective diffusion constant. We compare ...
Zeitschrift für Physik A Hadrons and nuclei, 1969
A short derivation of the Random Phase Approximation of angular momentum waves in solid Ortho-H2 ... more A short derivation of the Random Phase Approximation of angular momentum waves in solid Ortho-H2 is presented. Methods and results of papers on the same topic are compared.
Physical review, May 10, 2016
Journal of Physics A: Mathematical and Theoretical, 2019
Journal of Physics A, Jan 5, 2016
Franz G. Mertens Physikalisches Institut, Universität Bayreuth, D–95440 Bayreuth, Germany (Dated:... more Franz G. Mertens Physikalisches Institut, Universität Bayreuth, D–95440 Bayreuth, Germany (Dated: November 11, 2002) Abstract The Levinson theorem for two–dimensional scattering is generalized for potentials with inverse square singularities. By this theorem, the number of bound states in a given m–th partial wave is related to the phase shift and the singularity strength of the potential. For the m–wave phase shift the asymptotic behaviour is calculated for short wavelengths.
Journal of Applied Physics, 2015
A spin-polarized electrical current leads to a variety of periodical magnetic structures in nanos... more A spin-polarized electrical current leads to a variety of periodical magnetic structures in nanostripes. In the presence of the Ørsted field, which always assists an electrical current, the basic types of magnetic structures, i.e., a vortex-antivortex crystal and cross-tie domain walls, survive. The Ørsted field prevents saturation of the nanostripe and a longitudinal domain wall appears instead. Possible magnetization structures in stripes with different geometrical and material properties are studied numerically and analytically.
Physics Letters A, 1985
For certain integrable gas and lattice models in one dimension the Bethe ansatz leads to a formal... more For certain integrable gas and lattice models in one dimension the Bethe ansatz leads to a formal description of the system as a spinless Fermi gas. The particle-hole structure of the zero-temperature excitations is generalized to finite temperatures. Dispersion and thermal distribution of the excitations are calculated, and tested for the Toda lattice.
Physical Review-Section E-Statistical Nonlinear and Soft Matter Physics, 2005
Journal of Physics A: Mathematical and Theoretical, 2019
Soliton dynamics in the damped and parametrically driven nonlinear Dirac equation, in 1 + 1 dimen... more Soliton dynamics in the damped and parametrically driven nonlinear Dirac equation, in 1 + 1 dimension with scalar–scalar self-interaction is analysed. The considered parametric force has the spatial period . A variational approach using collective coordinates for studying the time dependent response of the solitary waves to this parametric force is developed. The dynamical equations for the collective coordinates are also obtained by an alternative method, namely the method of moments. The soliton dynamics depends crucially on the competition between two length scales: the spatial period and the width of the soliton l s . For the soliton oscillates in an effective potential, while for it moves uniformly as a free particle. The transition between these two regimes occurs when is comparable to the soliton width. This match enhances the soliton instabilities so that even small values of the perturbation are enough to modify drastically the soliton shape and destroy it for long times.