A one-dimensional Ising model with a potential falling off like r -( sigma +1) is considered as a limit of a set of Gaussian models. The spin-spin correlation function is derived, and the critical indices alpha , beta , nu , eta are calculated.
"Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified."
Critical properties of Dyson's hierarchical Ising model in one dimension with 'potential' falling off like r -(1+ sigma ) are examined in the range 0< sigma <1 where a phase transition is known to occur. A new exact renormalisation group recursion relation is derived for a 'dual' spin probability density function. Together with a scaling-type assumption the authors obtain eta =2- sigma and delta =(1+ sigma )/(1- sigma ) for all 0< sigma <1. Independent evidence, however, suggests that delta =3 for 0< sigma <or=1/2, along with other classical critical exponents in this region. In the non-classical region 1/2< sigma <1 accurate numerical values are obtained for the critical exponents nu and gamma and various Delta sigma = sigma -1/2 expansions to third order in Delta sigma .
The spin-spin correlation function of a one-dimensional random Ising model decays in a manner inversely proportional to the square of the partition function of the part of the system between the spins. At low temperatures and strong coupling there are logarithmic corrections to the scaling power-law dependence of the correlation length on the magnetic field. The correlation of the boundary spins as a function of the system size is evaluated analytically and numerically. At low temperatures and strong coupling this function has a crossover between different behaviors and its Fourier transform oscillates with wave vector.
Simple but rigorous derivations are presented for the inequalities2−η≤dγ′/(2β+γ′)≤dγ′/(2−α′),2−η≤d(δ−1)/(δ+1),γ≤(2−η)ν.(Proofs of the first two inequalities have been given by Buckingham and Gunton.1) Here d is the dimensionality and α′, β, γ, γ′ and δ are the exponents characterizing a ferromagnet near its critical point,2 while η and ν describe the decay of the corresponding spin-spin correlation functions2 Γ2(r)=〈S0zSrz〉, according to ∑ |r|<RΓ2(r)∼R2−η(T=Tc,H=0,R→∞),κ1∼(T−Tc)ν,(T→Tc,H=0),where the inverse range of correlation κ1 may be defined by κ1−2=[Σrr2Γ2(r)]/[ΣrΓ2(r)]. The corresponding exponent ηE for the energy-energy correlation function (essentially Γ4(r, a)−[Γ(a)]2, where Γ4(r, a)=〈S0zSazSrzSr+az〉) is proved to satisfy 2−ηE≤dα′,2−ηE≤dαc/(1+ζ+αc),where the specific heat CM at T=Tc diverges with the magnetization as M−αc while the energy derivative | ∂U/∂M |Tc varies as Mζ. (Mean Field or classical values are αc=0, ζ=1.) The proofs are based on the ``intuitively obvious'' properties: (a) positivity, namely, Γ1=〈S0z〉≥0,Γ(r)−Γ12≥0,Γ4(r, a)−[Γ2(a)]2≥0, and (b) monotonic increase of Γ1, Γ2 and Γ4 under increase of magnetic field and decrease of temperature. These properties are known3 to be rigorously valid for Ising models of arbitrary spin, lattice structure and ferromagnetic coupling (Jij≥0). Their validity for real magnets together with the experimental observation δ≤4.7 leads to the significant conclusion η≥0.05.
Understanding the low-temperature pure state structure of spin glasses remains an open problem in the field of statistical mechanics of disordered systems. Here we study Monte Carlo dynamics, performing simulations of the growth of correlations following a quench from infinite temperature to a temperature well below the spin-glass transition temperature T_{c} for a one-dimensional Ising spin-glass model with diluted long-range interactions. In this model, the probability P_{ij} that an edge {i,j} has nonvanishing interaction falls as a power law with chord distance, P_{ij}∝1/R_{ij}^{2σ}, and we study a range of values of σ with 1/2<σ<1. We consider a correlation function C_{4}(r,t). A dynamic correlation length that shows power-law growth with time ξ(t)∝t^{1/z} can be identified in the data and, for large time t, C_{4}(r,t) decays as a power law r^{-α_{d}} with distance r when r≪ξ(t). The calculation can be interpreted in terms of the maturation metastate averaged Gibbs state, or MMAS, and the decay exponent α_{d} differentiates between a trivial MMAS (α_{d}=0), as expected in the droplet picture of spin glasses, and a nontrivial MMAS (α_{d}≠0), as in the replica-symmetry-breaking (RSB) or chaotic pairs pictures. We find nonzero α_{d} even in the regime σ>2/3 which corresponds to short-range systems below six dimensions. For σ<2/3, the decay exponent α_{d} follows the RSB prediction for the decay exponent α_{s}=3-4σ of the static metastate, consistent with a conjectured statics-dynamics relation, while it approaches α_{d}=1-σ in the regime 2/3<σ<1; however, it deviates from both lines in the vicinity of σ=2/3.