Local Lagrangian Formalism and Discretization of the Heisenberg Magnet Model

In this paper we develop the Lagrangian and multisymplectic structures of the Heisenberg magnet (HM) model which are then used as the basis for geometric discretizations of HM. Despite a topological obstruction to the existence of a global Lagrangian density, a local variational formulation allows one to derive local conservation laws using a version of N\"other's theorem from the formal variational calculus of Gelfand-Dikii. Using the local Lagrangian form we extend the method of Marsden, Patrick and Schkoller to derive local multisymplectic discretizations directly from the variational principle. We employ a version of the finite element method to discretize the space of sections of the trivial magnetic spin bundle $N = M\times S^2$ over an appropriate space-time $M$. Since sections do not form a vector space, the usual FEM bases can be used only locally with coordinate transformations intervening on element boundaries, and conservation properties are guaranteed only within an element. We discuss possible ways of circumventing this problem, including the use of a local version of the method of characteristics, non-polynomial FEM bases and Lie-group discretization methods.Comment: 12 pages, accepted Math. and Comp. Simul., May 200

Functional renormalization and mean-field approach to multiband systems with spin-orbit coupling: Application to the Rashba model with attractive interaction

The functional renormalization group (RG) in combination with Fermi surface patching is a well-established method for studying Fermi liquid instabilities of correlated electron systems. In this article, we further develop this method and combine it with mean-field theory to approach multiband systems with spin-orbit coupling, and we apply this to a tight-binding Rashba model with an attractive, local interaction. The spin dependence of the interaction vertex is fully implemented in a RG flow without SU(2) symmetry, and its momentum dependence is approximated in a refined projection scheme. In particular, we discuss the necessity of including in the RG flow contributions from both bands of the model, even if they are not intersected by the Fermi level. As the leading instability of the Rashba model, we find a superconducting phase with a singlet-type interaction between electrons with opposite momenta. While the gap function has a singlet spin structure, the order parameter indicates an unconventional superconducting phase, with the ratio between singlet and triplet amplitudes being plus or minus one on the Fermi lines of the upper or lower band, respectively. We expect our combined functional RG and mean-field approach to be useful for an unbiased theoretical description of the low-temperature properties of spin-based materials.Comment: consistent with published version in Physical Review B (2016

Diffusion and jump-length distribution in liquid and amorphous Cu33_{33}Zr67_{67}

Using molecular dynamics simulation, we calculate the distribution of atomic jum ps in Cu$_{33}$Zr$_{67}$ in the liquid and glassy states. In both states the distribution of jump lengths can be described by a temperature independent exponential of the length and an effective activation energy plus a contribution of elastic displacements at short distances. Upon cooling the contribution of shorter jumps dominates. No indication of an enhanced probability to jump over a nearest neighbor distance was found. We find a smooth transition from flow in the liquid to jumps in the g lass. The correlation factor of the diffusion constant decreases with decreasing temperature, causing a drop of diffusion below the Arrhenius value, despite an apparent Arrhenius law for the jump probability

Rogue waves and downshifting in the presence of damping

Recently Gramstad and Trulsen derived a new higher order nonlinear Schrodinger (HONLS) equation which is Hamiltonian (Gramstad and Trulsen, 2011). We investigate the effects of dissipation on the development of rogue waves and downshifting by adding an additonal nonlinear damping term and a uniform linear damping term to this new HONLS equation. We find irreversible downshifting occurs when the nonlinear damping is the dominant damping effect. In particular, when only nonlinear damping is present, permanent downshifting occurs for all values of the nonlinear damping parameter beta. Significantly, rogue waves do not develop after the downshifting becomes permanent. Thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves. We examine the generation of rogue waves in the presence of damping for sea states characterized by JONSWAP spectrum. Using the inverse spectral theory of the NLS equation, simulations of the NLS and damped HONLS equations using JONSWAP initial data consistently show that rogue wave events are well predicted by proximity to homoclinic data, as measured by the spectral splitting distance delta. We define delta(cutoff) by requiring that 95% of the rogue waves occur for delta \u3c delta(cutoff). We find that delta(cutoff) decreases as the strength of the damping increases, indicating that for stronger damping the JONSWAP initial data must be closer to homoclinic data for rogue waves to occur. As a result when damping is present the proximity to homoclinic data and instabilities is more crucial for the development of rogue waves

Vibrational instability, two-level systems and Boson peak in glasses

We show that the same physical mechanism is fundamental for two seemingly different phenomena such as the formation of two-level systems in glasses and the Boson peak in the reduced density of low-frequency vibrational states g(w)/w^2. This mechanism is the vibrational instability of weakly interacting harmonic modes. Below some frequency w_c << w_0 (where w_0 is of the order of Debye frequency) the instability, controlled by the anharmonicity, creates a new stable universal spectrum of harmonic vibrations with a Boson peak feature as well as double-well potentials with a wide distribution of barrier heights. Both are determined by the strength of the interaction I ~ w_c between the oscillators. Our theory predicts in a natural way a small value for the important dimensionless parameter C ~ 10^{-4} for two-level systems in glasses. We show that C ~ I^{-3} and decreases with increasing of the interaction strength I. We show that the number of active two-level systems is very small, less than one per ten million of oscillators, in a good agreement with experiment. Within the unified approach developed in the present paper the density of the tunneling states and the density of vibrational states at the Boson peak frequency are interrelated.Comment: 28 pages, 3 figure

Predicting rogue waves in random oceanic sea states

Using the inverse spectral theory of the nonlinear Schrodinger (NLS) equation we correlate the development of rogue waves in oceanic sea states characterized by the Joint North Sea Wave Project (JONSWAP) spectrum with the proximity to homoclinic solutions of the NLS equation. We find in numerical simulations of the NLS equation that rogue waves develop for JONSWAP initial data that are near NLS homoclinic data, while rogue waves do not occur for JONSWAP data that are far from NLS homoclinic data. We show the nonlinear spectral decomposition provides a simple criterium for predicting the occurrence and strength of rogue waves

Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation

The cubic nonlinear Schrödinger (NLS) equation with periodic boundary conditions is solvable using Inverse Spectral Theory. The nonlinear spectrum of the associated Lax pair reveals topological properties of the NLS phase space that are difficult to assess by other means. In this paper we use the invariance of the nonlinear spectrum to examine the long time behavior of exponential and multisymplectic integrators as compared with the most commonly used split step approach. The initial condition used is a perturbation of the unstable plane wave solution, which is difficult to numerically resolve. Our findings indicate that the exponential integrators from the viewpoint of efficiency and speed have an edge over split step, while a lower order multisymplectic is not as accurate and too slow to compete. © 2006 Elsevier Inc. All rights reserved

Senile plaque calcification of the lamina circumvoluta medullaris in Alzheimer's disease

Vascular calcification is a common phenomenon in the elderly, predominantly appearing in the basal ganglia and in the lamina circumvoluta medullaris of the hippocampus. Calcifications are not an inherent feature of Alzheimer's disease. On the other hand, a rare presenile type of dementia with symmetrical Fahr-type calcifications and numerous neurofibrillary tangles without senile plaques has been described by Kosaka in 1994 and was termed "diffuse neurofibrillary tangles with calcification" (DNTC). We here report a case of Alzheimer's disease with calcifications both in the basal ganglia and in the lamina circumvoluta medullaris of the hippocampus, differing from DNTC by the presence of senile plaques. The calcifications in the hippocampus were not only vascular in nature but also covered amyloid-β- and phosphorylated tau-positive plaque-like structures that were linearly arranged along the dentate fascia in the CA1 sector, an unusual finding of pathogenetic interest