The Complexity of Ferromagnetic Two-spin Systems with External Fields

We consider general d-dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region (beta<<1). Each model is characterized by a single-site a priori spin probability distribution taken to be noneven. We state our results in terms of the parameter alpha=(<&smacr; (4)>-3<&smacr; (2)>(2)-<&smacr; (3)>(2)<&smacr; (2)>(-1))/(<&smacr; (4)>-<&smacr; (2)>(2)-<&smacr; (3)>(2)<&smacr; (2)>(-1)), where &smacr;=s-<s>, and <s(k)> denotes the kth moment of the single-site distribution. Associated with the model is a lattice quantum field theory which is known to contain a particle of mass m approximately ln beta. Assuming <&smacr;(3)> not equal0 we show that for alpha>0, beta small, there exists a bound state with mass below the two-particle threshold 2m. For alpha<0 bound states do not exist. These results are obtained using a Bethe-Salpeter (BS) equation in the ladder approximation in conjunction with a representation for the inverse of the two-point function designed to analyze the spectrum below but close to 2m.

We study the problem of approximating the partition function of the ferromagnetic Ising model with both pairwise as well as higher order interactions (equivalently, in graphs as well as hypergraphs). Our approach is based on the classical Lee–Yang theory of phase transitions, along with a new Lee–Yang theorem for the Ising model with higher order interactions, and on an extension of ideas developed recently by Barvinok, and Patel and Regts that can be seen as an algorithmic realization of the Lee–Yang theory. Our first result is a deterministic polynomial time approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters \(\beta \) (the interaction) and \(\lambda \) (the external field), except for the case \(\left| \lambda \right| =1\) (the “zero-field” case). A polynomial time randomized approximation scheme (FPRAS) for all graphs and all \(\beta ,\lambda \), based on Markov chain Monte Carlo simulation, has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our algorithm does not rely on the “decay of correlations” property, but, as pointed out above, on Lee–Yang theory. This approach extends to the more general setting of the Ising model on hypergraphs of bounded degree and edge size, where no previous algorithms (even randomized) were known for a wide range of parameters. In order to achieve this latter extension, we establish a tight version of the Lee–Yang theorem for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.

The charging of a metal sphere in a weakly ionized collisional plasma in a uniform external electric field is investigated with allowance for the effect of the space charge field and ionization-recombination processes. The sphere charge and the spatial distributions of charged plasma particles are calculated both numerically and analytically (for some particular cases) for the case of a strong external field. It is found that the size of the perturbed plasma region is determined by the external field and the intensity of recombination processes. It is shown that the total electric charge (the sphere charge plus the plasma space charge) is zero in accordance with predictions of the theory of static currents in a conducting medium.

#NFA refers to the problem of counting the words of length n accepted by a non-deterministic finite automaton. #NFA is #P-hard, and although fully-polynomial-time randomized approximation schemes (FPRAS) exist, they are all impractical. The first FPRAS for #NFA had a running time of Õ(n 17 m 17 ε -14 łog(δ -1 )), where m is the number of states in the automaton, δ ∈ (0,1] is the confidence parameter, and ε &gt; 0 is the tolerance parameter (typically smaller than 1). The current best FPRAS achieved a significant improvement in the time complexity relative to the first FPRAS and obtained FPRAS with time complexity Õ((n 10 m 2 + n 6 m 3 )ε -4 łog 2 (δ -1 )). The complexity of the improved FPRAS is still too intimidating to attempt any practical implementation. In this paper, we pursue the quest for practical FPRAS for #NFA by presenting a new algorithm with a time complexity of O(n 2 m 3 łog(nm)ε -2 łog(δ -1 )). Observe that evaluating whether a word of length n is accepted by an NFA has a time complexity of O(nm 2 ). Therefore, our proposed FPRAS achieves sub-quadratic complexity with respect to membership checks.

The transport properties of water through a nanochannel influenced by the direction of an external electric field has been investigated by using molecular dynamics simulations. Water molecules flow unidirectionally across the nanochannel under a uniform external electric field without an osmotic pressure. It is found that the direction of the external field plays an important role in the interactions and dipole orientations of water molecules in the nanochannel, accordingly changing the net water flux dramatically. Most importantly, a critical angle (θC) between the external field and the nanochannel axis is found. The average net water flux increases as θ increases for θ≤θC but decreases sharply to a near-zero value for a further increase of θ. The maximum value of the average net water flux is 7.33 times as high as the value when the electric field is along the nanochannel axis. Our findings are of great practical importance for nanomolecular engineering, which provide a possible strategy for designing novel controllable water nanopumps.

To investigate the dielectrophoretic motions of multiple particles, we have performed the so-called direct numerical simulations on two-dimensional flows involving inertialess dielectric particles of two to five suspended in a viscous fluid under a uniform external electric field and then compared the results with those of the corresponding magnetophoretic counterparts. For the simulations, the electric field (or the force acting on each particle) is described by the numerical solution of the Maxwell equation (or Gauss’s law), where the smoothed representation technique is employed to tackle the jump of electric conductivity across the particle-fluid interface. The flow field, on the other hand, is described by the solution of the continuity and momentum equations, where one-stage smoothed profile method is employed to satisfy the no-slip condition at the interface. In all the simulations, the particles are initially equi-spaced on a circle with an origin at the center while a uniform electric field is externally imposed. Results show that all particles move with repelling or attracting one another depending on the locally nonuniform electric field formed due to the presence of multiple particles. Consequently, they become clustered largely into two groups, then revolve in the clockwise or counterclockwise direction, and finally get aligned in a line with the field direction. One exception is their initial configuration which is symmetric with respect to the axis perpendicular to the electric-field direction, where all the particles move eternally far away from one another with keeping the symmetry. In addition, it is found that the two-dimensional relative motions of dielectric particles under a uniform external electric field are qualitatively in fairly exact agreement with those of paramagnetic particles suspended in a nonmagnetic fluid under a uniform external magnetic field.

Quantum electrodynamics in an external constant field

Chiral interactions between magnetic moments in ferromagnetic systems with broken inversion symmetry has attracted lots of attention in the recent years. Dzyaloshinskii-Moriya interaction[1] (DMI) present in these systems leads to an ordering of the spins that forms various exotic spin configurations such as chiral domain walls[2] and skyrmion[3]. DMI is an anti-exchange interaction which prefers to cant the neighbouring spins orthogonal to each other that forms whirling structures such as skyrmions. A magnetic skyrmion is a topological object having particle-spin like configuration, which can be stabilized in ferromagnetic systems with DMI. Recently magnetic skyrmions are proposed as promising candidates for non-volatile memory and logic applications due to their nanoscale size, high stability, high density and small driving currents required for their motion. Besides memory applications, skyrmions also hold great potential in the emerging area of magnonics where collective spin excitations (magnons) are used as a medium of information processing. In addition, observation of skyrmions on low Gilbert damping materials may allow the use of the skyrmions for manipulations of magnon transport. Recently skyrmion generation at an antidot is demonstrated using micromagnetic simulations[4].In this work, we present the various routes for obtaining giant tunability in microwave properties based on skyrmion dynamic via micromagnetic simulations using Mumax3. We have investigated microwave properties of magnetic skyrmions in a circular nanodisk having antidot at the centre. An in-plane ac magnetic field excites gyrotropic dynamics with a clockwise and a counter-clockwise mode. We have systematically investigated skyrmion resonant modes as a function of DMI and perpendicular magnetic anisotropy (K). Large variation of resonant modes as a function of small out-of-plane bias magnetic field has been demonstrated.Simulated microwave spectrum of this Néel type skyrmion in an antidot nanodisk has been shown in figure 1(a) and it is compared with an antidot nanodisk having uniform OOP magnetization without any skyrmions. It is to be noted that there is no external bias field applied for these results in fig 1(a). Microwave responses are found in 13-22 GHz frequency range. The most prominent modes are found at 13 GHz and 17 GHz for the disk with a skyrmion and no skyrmion, respectively. Therefore, the modes at 13.1 GHz can be associated to the skyrmion resonances whereas the other higher frequency modes arise from the bulk magnetization (OOP) of the disk. Next, we demonstrate the tunability of the skyrmion resonance modes by applying a small bias field (Hext) along the OOP direction of the nanodisk with a single skyrmion. We have varied the polarity and the magnitude of the OOP field. The results are shown in fig 1(b) where the microwave resonance spectra are shown at different external fields: 0 to 35 mT. A relatively large frequency shift of 3.3 GHz has been observed for a change of 35 mT bias field.To further explore the functionalities, we have investigated skyrmion stabilization with various combinations of the parameters, DMI and K. Néel skyrmion is found to be stabilized at K = 0.25×106 J/m3 and DMI = 1.8×10-3 J/m2. Next, we have varied the K from 0.22 to 0.34 (×106) J/m3 for a fixed value of DMI = 1.8 mJ/m2 and we have varied the parameter DMI from 1.6 mJ/m2 to 1.9 mJ/m2, for a fixed value of K = 0.25×106 J/m3 shown in fig 2(a & b) respectively. Furthermore, a broad microwave tunability of 9.8 GHz has been shown with a variation of magnetic anisotropy of 1×105 J/m3. The manifestation of inertial mass of the skyrmion, its dependence on the resonance responses and the size of the skyrmion are explained by using Thiele’s equation. The inertial mass of the skyrmion is estimated to be in the order of 10-23 kg. We believe these results will have potential implications in the area of skyrmion based nano-scale reconfigurable microwave devices. **

Spin-wave corrections to single-domain behavior in finite ferromagnetic systems

The perturbation expansion is obtained for a free electron gas interacting with a nuclear or ionic spin system via a ferromagnetic s—f‐interaction. No divergent terms appear in the expansion if the unperturbed ground state of the system is assumed to consist of a polarized electron gas and a ferromagnetic spin system. This assumption is made plausible by considering the diagonal part of the perturbation and minimizing the resultant energy with respect to both the polarization of the electron and the structure of the spin system. In this way an energy up to second order in J is also obtained, and this is lower than the R.K.K.Y. result for a ferromagnetic spin system. No corrections to the magnetization of the spin system appear up to first order in J.

This paper concerns the analysis of the Shapley value in matching games. Matching games constitute a fundamental class of cooperative games which help understand and model auctions and assignments. In a matching game, the value of a coalition of vertices is the weight of the maximum size matching in the subgraph induced by the coalition. The Shapley value is one of the most important solution concepts in cooperative game theory. After establishing some general insights, we show that the Shapley value of matching games can be computed in polynomial time for some special cases: graphs with maximum degree two, and graphs that have a small modular decomposition into cliques or cocliques (complete k-partite graphs are a notable special case of this). The latter result extends to various other well-known classes of graph-based cooperative games. We continue by showing that computing the Shapley value of unweighted matching games is #P-complete in general. Finally, a fully polynomial-time randomized approximation scheme (FPRAS) is presented. This FPRAS can be considered the best positive result conceivable, in view of the #P-completeness result.

Boundary-element calculations for amplification of effects of low-frequency electric fields in a doublet-shaped biological cell

We describe a systematic method for complete enumeration of configuration classes (CCs) of the spin-1/2 Ising model in the energy-magnetization plane. This technique is applied to the antiferromagnetic (AF) Ising model in an external magnetic field on the square lattice, which is simulated using the tomographic entropic sampling (TES) algorithm. We estimate the number of configurations, Ω(E,m,L), and related microcanonical averages, for all allowed energies E and magnetizations m for L = 10 to 30, with ΔL = 2. With prior knowledge of the CCs, we can be sure that all allowed classes are sampled during the simulation. Complete enumeration of CCs also enables us to use the final estimate of Ω(E,m,L) to obtain good initial estimates, Ω0(E,m,L′), for successive system sizes (L′ &gt; L) through a two-dimensional interpolation. Using these results we calculate canonical averages of the thermodynamic quantities of interest as continuous functions of temperature T and external field h. In addition, we determine the critical line in the h-T plane using finite-size scaling analysis, and compare these results with several approximate theoretical expressions.

The deformation of polydimethylsiloxane (PDMS) spheres under uniform external electric field was studied experimentally and analytically. In the experiments, ∼1 mm diameter PDMS spheres with base-to-curing-agent mixing ratios of 10:1 and 60:1 were exposed to uniform external electric field with varying magnitudes and poling durations. The spheres elongate along the electric field direction. For a given electric field strength, the sphere deformation is initially a time function but reaches a terminal strain value over a certain time period. This terminal strain value is larger for stronger external electric fields and larger PDMS mixing ratio spheres. At this state, the material is no longer poled and the surface charge distribution remains constant. In the analysis, an expression for the sphere deformation is obtained by modeling the PDMS as a linear elastic solid and solving the Navier equation along with Maxwell’s equations for boundary conditions. The analysis takes into account the surface charge distribution and predicts well the experimental trends.

In this paper, we study the energy levels of an electron moving in a thin film. This film is considered as a two-dimensional electron gas which is under the influence of a uniform external magnetic field B and a uniform external electric field E. Here, the magnetic field is perpendicular to the film. Also, in this paper, we have selected the Landau gauge, because this gauge is useful for working in rectangular geometries.