Critical indices for models with long-range interaction

R.T. Scalettar

Xue-Jia Yu, Chengxiang Ding, Limei Xu

Based on large-scale density matrix renormalization group techniques, we investigate the critical behaviors of quantum three-state Potts chains with long-range interactions. Using fidelity susceptibility as an indicator, we obtain a complete phase diagram of the system. The results show that as the long-range interaction power α increases, the critical points f_{c}^{*} shift towards lower values. In addition, the critical threshold α_{c}(≈1.43) of the long-range interaction power is obtained for the first time by a nonperturbative numerical method. This indicates that the critical behavior of the system can be naturally divided into two distinct universality classes, namely the long-range (α<α_{c}) and short-range (α>α_{c}) universality classes, qualitatively consistent with the classical ϕ^{3} effective field theory. This work provides a useful reference for further research on phase transitions in quantum spin chains with long-range interaction.

P. M. Blekher, Ya. G. Sinaǐ

M.A. Gouker, Fereydoon Family

A new large-cell Monte Carlo renormalization group following the approach of Reynolds et al. is developed to study site percolation with long-range interaction. The results show that site percolation with a long, but finite, range of interactions has the same correlation-length exponent ${\ensuremath{\nu}}_{P}$ as ordinary percolation. However, as the interaction range tends to infinity, ${\ensuremath{\nu}}_{P}$ approaches its mean-field value of $\frac{1}{2}$ indicating that infinite-range percolation exhibits classical critical behavior.

Taro Nakada, Per Arne Rikvold, Takashi Mori +2 more

Recently, it has been found that an effective long-range interaction is realized among local bistable variables (spins) in systems where the elastic interaction causes ordering of the spins. In such systems, we generally expect both long-range and short-range interactions to exist. In the short-range Ising model, the correlation length diverges at the critical point. In contrast, in the long-range interacting model the spin configuration is always uniform and the correlation length is zero. As long as a system has nonzero long-range interactions, it shows criticality in the mean-field universality class, and the spin configuration is uniform beyond a certain scale. Here we study the crossover from the pure short-range interacting model to the long-range interacting model. We investigate the infinite-range model (Husimi-Temperley model) as a prototype of this competition, and we study how the critical temperature changes as a function of the strength of the long-range interaction. This model can also be interpreted as an approximation for the Ising model on a small-world network. We derive a formula for the critical temperature as a function of the strength of the long-range interaction. We also propose a finite-size scaling form for the spin correlation length at the critical point, which is finite as long as the long-range interaction is included, though it diverges in the limit of the pure short-range model. These properties are confirmed by extensive Monte Carlo simulations.