Neighbor Ising Model Research Articles

Hamiltonian studies of the two-dimensional axial next-nearest neighbour Ising (ANNNI) model. II. Finite-lattice mass gap calculations

The quantum Hamiltonian analogue of the two-dimensional ANNNI model is investigated by finite-lattice mass gap methods. By using lattice sizes capable of simulating systems of varying modulation, the authors are able to show the existence of a modulated phase between the paramagnetic and (2,2) antiphase regions. The modulation on the incommensurate to paramagnetic boundary is shown to vary and this variation is calculated as a function of the anisotropy. In addition, they find evidence for an XY-like transition from the incommensurate to the paramagnetic phase and perhaps a non-universal transition from the paramagnetic phase to the antiphase.

Statistical mechanics of the ripple phase in lipid bilayers

The essential features of lipid molecule interactions responsible for the ripple phase in lecithin bilayers are identified and incorporated into a general statistical mechanical model. As a first approximation, the model is reduced to the axial next-nearest neighbor Ising (ANNNI) model. This model exhibits an intermediate ripple phase, over a narrow temperature range, between a high-temperature (chain melted) disordered phase and a low-temperature (chain tilted) ordered phase. In this way the ripples are found to arise from an interplay between the attractive van der Waals forces and the action of the headgroup to hinder level packing of the lipid molecules.

Finite-size behaviour of the two-dimensional ANNNI model

The two-dimensional axial next-nearest neighbour Ising (ANNNI) model of finite size with periodic boundary conditions is studied by the Monte Carlo method. The model shows an interesting finite size dependence in connection with its oscillatory correlations pretending for finite systems a Lifshitz point in one part of the phase diagram, while the infinite system appears to display one in another part of the phase diagram.

The mean field approximation for a model with competing interactions

The axial next-nearest neighbor Ising (ANNNI) model is analysed in the mean field approximation (MFA), and results are compared to numerical and exact low temperature calculations. A model is presented in which MFA is exact.

Low temperature analysis of the axial next-nearest neighbour Ising model near its multiphase point

The general, axial next-nearest neighbour Ising (or ANNI) model inddimensions consists of (d— 1)-dimensional layers of spins, si= + 1, with nearest-neighbour ferromagnetic coupling, J0> 0, within layers but competing ferromagnetic, and antiferromagnetic,J2< 0, first- and second-neighbour axial coupling between layers. By systematic low temperature expansions in powers of e-2Jo/kBT, extended to all orders where necessary, it is shown ford> 2 that in the vicinity of amultiphasepoint atT= 0 and k = —J2/J1= 1/2 there occurs an infinite sequence of distinct, commensurate modulated phases characterized by wavevectors + l)< z for j = 0, 1, 2, .... The phase boundaries, kj(T), and corresponding free energies and interfacial tensions are derived explicitly for low temperatures.

Exactly soluble Ising models exhibiting multiphase points

A class of decorated spin 1/2 Ising models is introduced: all bonds of a hypercubic lattice parallel to one axis are decorated by n spins. Within each decorated bond, nearest neighbor ferromagnetic interactions compete with next-nearest neighbor antiferromagnetic interactions. An exact dedecoration transformation reduces the model to an anisotropic Ising model which is exactly soluble in two dimensions (d = 2) and for which accurate series expansion estimates are available for d = 3. Phase diagrams are presented for d = 2 and 3 : all exhibit a multiphase point at which many distinct, spatially modulated commensurate phases coexist. The variation of the characteristic wavevector and other properties are compared and contrasted with those of the axial next-nearest neighbor Ising (ANNNI) models.

An infinity of commensurate phases in a simple Ising system: The ANNNI model (invited)

The ANNNI or axial next-nearest neighbor Ising model in d dimensions consists of (d−1)-dimensional layers of Ising spins, si = ±1, with nearest neighbor ferromagnetic couplings, J0≳0, within layers but competing ferromagnetic, J1, and antiferromagnetic, J2 = −κJ1<0, axial couplings between spins in first- and second-neighboring layers. It is the simplest model expected to display spatially modulated magnetically ordered states with wavevectors nontrivially related to the lattice spacing a, as observed in various real systems. The ground state is ferromagnetic for κ<1/2; of (2, 2), or 〈2〉, antiphase character (two layers ’’up’’, two ’’down’’, periodically) for κ≳1/2; but infinitely degenerate for κ = 1/2. Systematic low-temperature expansions may be generated about all possible ground states and carried to all orders where necessary. For d≳2 they show that (T,κ) = (0,1/2) is a multiphase point from which spring an infinite sequence of distinct, spatially modulated commensurate phases, 〈2j−13〉, with wavevectors q̄ = πj/(2j+1)a for j = 1,2, ⋅⋅⋅ . The free energy, phase boundaries, and corresponding domain wall energies can be given explicitly at low temperatures. On approaching the ’’melting’’ line, κ∞(T), of the 〈2〉 phase the wavevector varies quasicontinuously as 1/‖1n[κ∞(T)−κ]‖, displaying a ’’devil’s top step’’ just as the 〈2〉 phase locks in.

Correlation inequalities for Ising ferromagnets with symmetries

We derive new correlation inequalities for even Ising ferromagnets whose interaction is invariant under some symmetry transformation and satisfies a growth condition. The recent results of Schrader [1] and Messager and Miracle-Sol [2] for the nearest neighbour (n.n.) Ising model reappear as a special case. In addition we obtain monotonicity of 〈σ0σ j 〉 under translation ofj perpendicular to diagonal hyperplanes and the inequality 〈σ0σ j 〉≧\(\left\langle {\sigma _0 \sigma _{\left( {\sum {\left| {j_\nu } \right|,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{0} } } \right)} } \right\rangle \) for n.n. and other interactions.

Critical exponents and elementary particles

Particles are shown to exist for a.e. value of the mass in single phase φ4 lattice and continuum field theories and nearest neighbor Ising models. The particles occur in the form of poles at imaginary (Minkowski) momenta of the Fourier transformed two point function. The new inequalitydm2/dσ≦Z, where σ=m02 is a bare mass2 andZ is the strength of the particle pole, is basic to our method. This inequality implies inequalities for critical exponents.

Evidence for long-range interactions in DNA. Analysis of melting curves of block polymers d(C15A15)-d(T15G15),d(C20A15)-d(T15G20), and d(T15G20).

AbstractThe influence of one DNA region on the stability of an adjoining region (telestability) was examined. Melting curves of three block DNA's, d(C15A15)·d(T15G15), d(C20A15)·d(T15G20), and d(C20A10)·d(T10G20) were analyzed in terms of the nearest neighbor Ising model. Comparisons of predicted and experimental curves were made in 0.01 M and 0.1 M sodium ion solutions. The nearest neighbor formalism was also employed to analyze block DNA transition in the presence of actinomycin, a G·C specific molecule. The results show that nearest neighbor base‐pair interaction cannot predict the melting curves of the block DNA's. Adjustments in theoretical parameters to account for phosphate repulsion assuming a B conformation throughout the DNA's do not alter this conclusion. Changes in the theoretical parameters, which provide good overall agreement, are consistent with a substantial stabilization of the A·T region nearest the G·C block. The melting temperature T A·T for the average A·T pari in d(C20A10)·d(T10G20), with 10 A·T pairs, appears to be 4°C greater than TA·T for d(C15A15)·d(T15G15) and d(C20A15)·d(T15G20), both with 15 A·T pairs. Actinomycin bound to the G·C end effectively stabilizes the A·T end by 9°C. These results indicate a long‐range contribution to the interactions governing DNA stability. A possible mechanism for these interactions will be discussed.

“Clusters” in the Ising model, metastable states and essential singularity

Various possibilities for the definition of “clusters” which are used in theories of critical phenomena and nucleation are discussed for the case of nearest neighbor Ising models. Using a two coordinate description in terms of a contour (of “size” s) around ℓ reversed spins, it is shown that scaling assumptions for the cluster concentration p(ℓ, s) imply that the critical behavior cannot be attributed to fully “ramified” clusters as suggested by Domb. Monte Carlo results for p(ℓ, s) are also presented and are shown to be consistent with scaling. For large ℓ, a crossover to geometric behavior is found, and again interpreted in terms of scaling. Relating the “clusters” to fluctuations of a coarsegrained order parameter, the arguments of Andreev in favor of an essential singularity at the coexistence curve below the critical temperature are recovered. The stability limit of the metastable states, which can thus be defined in terms of dynamic considerations only, is obtained for the whole temperature range from computer simulations.

Nearest neighbor ising model with added local 4-spin interaction

The effects of adding a local 4-spin term to the nearest neighbor Ising model are studied. The relevant parameter isx, the ratio of the four- to the two-spin interaction strength. We have evaluated the first six terms of the high temperature susceptibility expansion and found them to be consistent with the value 1.75 of the critical exponent γ. The critical exponent α is shown to remain zero to first order inx, provided that the free energy is analytic inx. We also demonstrate the equivalence of the model examined to an elastic spin system with pure two-spin interactions.

Perturbational Approach to the Two-Dimensional Next Nearest Neighbor Ising Problem

The Feynman diagram technique is applied to the Ising model with the next nearest neighbor interaction according to the suggestions by Hurst and Gibberd. The partition func­ tion in the lowest order approximation is evaluated. The critical temperature is evaluated in this approximation, and is shown to be exact when either the nearest or the next nearest neighbor interaction is very small. The rules for perturbation calculation are discussed and some simple diagrams are investigated. neighbor Ising model,5) and Slater's model of ferroelectric,7) the exponent has additional quartic terms. The partition function of the last case cannot be solved exactly, but the technique of many-body theory is applicable to obtaining an approximate solution. In this paper, we apply their method to the rectangular next nearest neighbor Ising model. We derive an expression for the partition function in § 2. In § 3, the partition function in the lowest order approximation is calculated. The critical temperature is evaluated in this approximation, and is shown to be exact when either the nearest or the next nearest neighbor interaction is very small. The physical meaning of this approximation is also discussed. Correction to this approximation can be obtained by expanding the partition function in the pertur­ bation series by using the Feynman diagram technique. In § 4, we evaluate the Green function which is necessary for this perturbation calculation. In § 5, we present the rules for calculating the terms of the perturbation series, and in­ vestigate some simple diagrams.