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

Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs

Several recently developed multisymplectic schemes for Hamiltonian PDEs have been shown to preserve associated local conservation laws and constraints very well in long time numerical simulations. Backward error analysis for PDEs, or the method of modified equations, is a useful technique for studying the qualitative behavior of a discretization and provides insight into the preservation properties of the scheme. In this paper we initiate a backward error analysis for PDE discretizations, in particular of multisymplectic box schemes for the nonlinear Schrodinger equation. We show that the associated modified differential equations are also multisymplectic and derive the modified conservation laws which are satisfied to higher order by the numerical solution. Higher order preservation of the modified local conservation laws is verified numerically.Comment: 12 pages, 6 figures, accepted Math. and Comp. Simul., May 200

On the heterogeneous character of water's amorphous polymorphism

In this letter we report {\it in situ} small--angle neutron scattering results on the high--density (HDA) and low-density amorphous (LDA) ice structures and on intermediate structures as found during the temperature induced transformation of HDA into LDA. We show that the small--angle signal is characterised by two $Q$ regimes featuring different properties ($Q$ is the modulus of the scattering vector defined as $Q = 4\pi\sin{(\Theta)}/\lambda_{\rm i}$ with $\Theta$ being half the scattering angle and $\lambda_{\rm i}$ the incident neutron wavelength). The very low--$Q$ regime ($< 5\times 10^{-2}$ \AA $^{-1}$) is dominated by a Porod--limit scattering. Its intensity reduces in the course of the HDA to LDA transformation following a kinetics reminiscent of that observed in wide--angle diffraction experiments. The small--angle neutron scattering formfactor in the intermediate regime of $5 \times 10^{-2} < Q < 0.5$ \AA$^{-1}$ HDA and LDA features a rather flat plateau. However, the HDA signal shows an ascending intensity towards smaller $Q$ marking this amorphous structure as heterogeneous. When following the HDA to LDA transition the formfactor shows a pronounced transient excess in intensity marking all intermediate structures as strongly heterogeneous on a length scale of some nano--meters

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

Scalable and Energy-Efficient Millimeter Massive MIMO Architectures: Reflect-Array and Transmit-Array Antennas

Hybrid analog-digital architectures are considered as promising candidates for implementing millimeter wave (mmWave) massive multiple-input multiple-output (MIMO) systems since they enable a considerable reduction of the required number of costly radio frequency (RF) chains by moving some of the signal processing operations into the analog domain. However, the analog feed network, comprising RF dividers, combiners, phase shifters, and line connections, of hybrid MIMO architectures is not scalable due to its prohibitively high power consumption for large numbers of transmit antennas. Motivated by this limitation, in this paper, we study novel massive MIMO architectures, namely reflect-array (RA) and transmit-array (TA) antennas. We show that the precoders for RA and TA antennas have to meet different constraints compared to those for conventional MIMO architectures. Taking these constraints into account and exploiting the sparsity of mmWave channels, we design an efficient precoder for RA and TA antennas based on the orthogonal matching pursuit algorithm. Furthermore, in order to fairly compare the performance of RA and TA antennas with conventional fully-digital and hybrid MIMO architectures, we develop a unified power consumption model. Our simulation results show that unlike conventional MIMO architectures, RA and TA antennas are highly energy efficient and fully scalable in terms of the number of transmit antennas.Comment: submitted to IEEE ICC 201

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

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