On the critical parameter δ(=5)

Igor F. Herbut

M. A. Yurishchev

The coordinates of the critical points of spin-S Ising models with coupling constants J and J′ are calculated for 1/2 ≤ S ≤ 13/2. The calculations are performed for several values of S and Δ ≡ J′/J independently by using the phenomenological renormalization-group method or (approximate) self-duality. Numerical results combined with a mean-field analysis show that the critical coupling strength for Δ ∼ 1 (weakly anisotropic lattice) is K c () (Δ) = K c () (1)[1 + a(1 − Δ)], where a = (d − 1)/d is independent of S (d is the space dimension). Both free energy and internal energy are determined at the critical points. An extremum of the critical internal energy is found at Δ* ∈ (0, 1). The parameter Δ* can be used as a criterion that separates quasi-isotropic and quasi-one-dimensional regimes (Δ* < Δ ≤ 1 and Δ < Δ*, respectively). The finite-size scaling amplitudes A s and A e of the inverse spin-spin and energy-energy correlation lengths are estimated. Calculations show that the amplitudes A s and A e are independent of S within the accuracy of the adopted approximations. Moreover, their ratio A e/A s is independent of the anisotropy parameter Δ. These results support the Ising universality hypothesis.

Zou Wei-Ke, Xiang-Mu Kong, Chun-Yang Wang +1 more

With a real-space renormalization-group method, the anisotropic quantum Heisenberg model on three-dimensional diamond-type hierarchical lattice is studied, and the phase diagram and critical properties are obtained. For the ferromagnetic system, it is shown that there is a finite-temperature phase transition for Δ=0 where Δ is the anisotropy parameter. The order parameter and critical exponents are also calculated. For the antiferromagnetic model, we find that the critical temperature is not equal to zero for Δ=0 and there is not reentrant behavior on the critical line.

B. Widom

Kenn Kubo, Satoshi Takada

The S x - S x correlation function of S =1 XXZ chain is calculated at low temperatures by means of the transfer matrix with use of the Trotter formula. Critical properties of the correlation function, such as critical indices γ, η and ν and the mass identified by the inverse correlation length at T =0, are obtained. We conclude that a groundstate with an excitation gap appears for the anisotropy parameter Δ<Δ 1 ≃ -0.3 including the isotropic antiferromagnetic point (Δ=-1). We have obtained η≃1/4 at Δ=Δ 1 in accord with the theoretical prediction by Haldane and den Nijs. Our result, however, shows the monotonic increase of the excitation gap with decreasing Δ for Δ<Δ 1 contrary to Haldane's prediction.

Wolfgang Nolting