Magnetism: Ising spins

For conventional Ising-spin systems, Pising(o ) takes on the expected simple forms namely, either Puingiq) = < (0) in the (high-temperature, zero magnetization) paramagnetic phase or the double-peaked Pising(o ) = 5(q + M ) + 6 q — M ) for temperatures below the Curie critical temperature, T < Tc. 

This tells us immediately that, just as for Ising spins, we have a spontaneous magnetization and that there is an effective phase transition for T = 1 stored patterns will only be stable for temperatures T < 1. 

Leeuwen theorem. Marathe and Dhar study an Ising spin in a time-varying magnetic field. They verify the Crooks FR and JE, and confirm that time reversibility is required for the transient ES FR. They also observe that the steady state ES FR is not obeyed, confirming that this is an asymptotic result and indicating that observation times were insufficient for convergence. 

For the number of shells in both structures, each lattice is related to the radius (R) of the nanoparticle [27-29]. Therefore, the value of R contains a number of shells and the size of a nanoparticle increases as the number of shells increases. The shells (R) and their numbers are only bounded to the nearest-neighbour pair exchange interactions (J) between spins. To provide the magnetization of the whole particle, each of the spin sites, which stand for the atomistic moments in the nanoparticle, are described by Ising spin variables that take on the values S1-= l, 0. For a core/surface (C/S) morphology, all spins in the nanoparticle are organized in three components that are core (C, filled circles), interface (or core-surface) (CS) and surface (S, empty circles) parts. The number of spins in these parts within the C/S-type nanoparticle are denoted byNc, Ncs and Ns, respectively. But, the total number of spins (N) in a C/S nanoparticle covers only C and S spin numbers, i.e. N =NC + Ns. On the other hand, the numbers of spin pairs for C, CS and S regions in 2D are defined by N [,=(N (.y(. /2)-Ncs,... 

P. Werner, K. Volker, M. Troyer, and S. Chakravarty (2005) Phase diagram and critical exponents of a dissipative Ising spin chain in a transverse magnetic field. Phys. Rev. Lett. 94, p. 047201... 

Fig. 69. (a) Part of the body-centered cubic lattice ordered in the B2 structure (left part) and in the Dtp structure (right part). Left part shows assignment of four sublattices a, b, c and d, In the B2 structure (cf. also fig. 66a), the concentrations of A atoms are the same at the a and c sublatticcs, but differ from the concentrations of the b, d sublattices, while in the DOj structure the concentration of the b sublattice differs from that of the d sublatlice, but both differ from those of the a, c sublattices (which are still the same). In terms of an Ising spin model, these sublattice concentrations translate into sublattice magnetizations mu, mu, mc, m,i, which allow to define three order parameter components / = ma + mL- — mu — m,/, fa = m - mc + mu — m,j, and fa = -ma + m., + mu — nij. 

In a computational study, the values of Ja and J e can be obtained by mapping the total energies of different long-range ordered magnetic arrangements onto an Ising spin Hamiltonian of the form... 

Just as the magnetic state energies can be mapped onto an Ising spin Hamiltonian, so may the energies of various crystal field split states be mapped onto the Kanamori Hamiltonian [89]. For an isolated metal site in which bonding and magnetic interactions are neglected, this takes the form... 

E. Average Squared Magnetization of an Open Chain of Ising Spins... 

Fig. 18. Magnetic phase diagram for the SK model (EA model for Ising spins with infinite-range couplings). J and / denote the width and mean of the exchange distribution. P = paramagnet FM = ferromagnet SG = spin glass. F is a ferromagnetic phase viith replica symmetry breaking, i.e. irreversibility ( mixed phase ) and is separated from FM by an AT line.

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