The two-point spin-spin correlation function of the weakly disordered Baxter model in the phase transition point is calculated. The corresponding critical exponent appears to be zero, i.e. the decay of the correlation function is slower than the power law: R - eta . Together with the previous result that the specific heat critical exponent of the weakly disordered Baxter model is also universal: alpha -0, it gives the complete set of universal critical exponents which describe the phase transition. The critical exponents of the weakly disordered Baxter model coincide with those of the weakly disordered Ising model.
The two-point spin-spin correlation function of the weakly disordered Baxter model in the phase transition point is calculated. The corresponding critical exponent appears to be zero, i.e. the decay of the correlation function is slower than the power law: R - eta . Together with the previous result that the specific heat critical exponent of the weakly disordered Baxter model is also universal: alpha -0, it gives the complete set of universal critical exponents which describe the phase transition. The critical exponents of the weakly disordered Baxter model coincide with those of the weakly disordered Ising model.
The critical behavior of the disordered two-dimensional antiferromagnetic Potts model with the number of spin states q= 3 on a triangular lattice with disorder in the form of nonmagnetic impurities is studied by the Monte Carlo method. The critical exponents for the susceptibility γ, magnetization β, specific heat α, and correlation radius ν are calculated in the framework of the finite-size scaling theory at spin concentrations p = 0.90, 0.80, 0.70, and 0.65. It is found that the critical exponents increase with the degree of disorder, whereas the ratios and do not change, thus holding the scaling equality ν2β+νγ=d . Such behavior of the critical exponents is related to the weak universality of the critical behavior characteristic of disordered systems. All results are obtained using independent Monte Carlo algorithms, such as the Metropolis and Wolff algorithms.
We report extensive Monte Carlo simulations of disordered Ising systems in the ferromagnetic region with concentrations of magnetic sites between p=1.0 and 0.5. The magnetization, the susceptibility, and the caloric properties have been studied in the critical region. The critical exponents \ensuremath{\beta} and \ensuremath{\gamma} as well as the universal amplitude A of the magnetization and the ratio C+/${\mathit{C}}_{\mathrm{\ensuremath{-}}}$ of the susceptibility amplitudes have been determined with high precision. In addition to the cusplike specific heat, we have also measured the magnetization-energy correlation function \ensuremath{\Gamma}, which is a divergent thermal quantity. The corresponding critical exponent \ensuremath{\zeta}, which is related to the other exponents by \ensuremath{\zeta}=1-\ensuremath{\beta}=(\ensuremath{\alpha}+\ensuremath{\gamma})/2, has been determined. We have found that all quantities show power-law behavior within the temperature range of our simulation. All critical exponents change continuously with dilution. Even in the range of weak dilution (p\ensuremath{\ge}0.8), the effective critical exponents are concentration dependent and are clearly different from their pure system values. In the strongly diluted regime the critical exponents gradually reach new asymptotic behavior at p=0.5--0.6 with values of \ensuremath{\beta}=0.335\ifmmode\pm\else\textpm\fi{}0.01 and \ensuremath{\gamma}=1.49\ifmmode\pm\else\textpm\fi{}0.02. The exponent \ensuremath{\alpha} of the specific heat becomes -0.17\ifmmode\pm\else\textpm\fi{}0.04, which corresponds to a cusplike singularity. We conclude that disorder profoundly changes the critical behavior for weakly as well as strongly disordered spin systems.
Abstract We investigate the first two moments of the critical internal energy E in a weakly disordered two-dimensional Baxter’s eight-vertex model as a function of the system size L , evaluated at the pseudo-critical point. Disorder is introduced via an equivalent representation of the pure eight-vertex model in terms of two ferromagnetic Ising models coupled by a four-spin interaction of strength g 0 , where the Ising couplings consist of a uniform ferromagnetic part J > 0 supplemented by weak Gaussian spatial disorder. In the critical regime, the model is formulated in terms of interacting Grassmann–Majorana spinor fields with quartic interactions and analyzed, for small positive g 0 , using a combination of replica and renormalization-group methods. We also run extensive numerical simulations measuring the critical internal energy. Our results show that its relative variance increases with L and approaches a finite constant as L → ∞ for both ± g 0 . Hence, fluctuations remain relevant independently of the sign of g 0 (and thus of the specific-heat exponent), implying a lack of self-averaging of both the critical internal energy and the free energy. Consequently, reliable estimates of these quantities require averaging over many disorder realizations. In addition, we numerically confirm earlier predictions concerning the absence of self-averaging of the critical internal energy in the disordered Ising model.
The spin-spin correlation function in the ground state of some two-dimensional fully frustrated spin models are investigated. The correlations are long ranged and decay as $\frac{1}{{r}^{\ensuremath{\eta}}}$, where r is the distance between the two spins and $\ensuremath{\eta}=\frac{1}{2}$. This result is true for both the square and triangular fully frustrated lattices, thus suggesting that these models form a universality class. It is shown that the fully frustrated Ising model on a square lattice can be mapped exactly into a special Baxter model. The spin-spin correlation function of this Baxter model can be calculated exactly. The ground state of the fully frustrated square Ising model corresponds to the free fermion point of the F model and therefore, through duality, to the decoupling point of the critical Baxter line. The relationship between the F model above the critical temperature and the Gaussian model above the multicritical point is discussed.
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.
There are two independent critical exponents that describe the behavior of systems near their critical point. However, at the critical point only the exponent η, which describes the decay of the correlation function, is usually discussed. We emphasize that there is a second independent exponent η′ that describes the decay of the fourth-order correlation function. The exponent η′ is related to the exponents determining the behavior of thermodynamic functions near criticality via a fluctuation-response equation for the specific heat. We also discuss a scaling law for η′.