Interaction Range, Universality and the Upper Critical Dimension

We investigate the universality class of the finite-temperature phase transition of the two-dimensional Ising model with the algebraically decaying ferromagnetic long-range interaction, J_{ij}=|r[over ⃗]_{i}-r[over ⃗]_{j}|^{-(d+σ)}, where d (=2) is the dimension of the system and σ is the decay exponent, by means of the order-N cluster-algorithm Monte Carlo method. In particular, we focus on the upper and lower critical decay exponents, the boundaries between the mean-field-universality, intermediate, and short-range-universality regimes. At the critical decay exponents, it is found that the standard Binder ratio of magnetization at the critical temperature exhibits extremely slow convergence as a function of the system size. We propose more effective physical quantities, namely the combined Binder ratio and the self-combined Binder ratio, both of which cancel the leading finite-size corrections of the conventional Binder ratio. Utilizing these techniques, we clearly demonstrate that in two dimensions, the lower and upper critical decay exponents are σ=1 and 7/4, respectively, contrary to the recent Monte Carlo and renormalization-group studies [M. Picco, arXiv:1207.1018; T. Blanchard et al., Europhys. Lett. 101, 56003 (2013)EULEEJ0295-507510.1209/0295-5075/101/56003].We investigate the universality class of the finite-temperature phase transition of the two-dimensional Ising model with the algebraically decaying ferromagnetic long-range interaction, J_{ij}=|r[over ⃗]_{i}-r[over ⃗]_{j}|^{-(d+σ)}, where d (=2) is the dimension of the system and σ is the decay exponent, by means of the order-N cluster-algorithm Monte Carlo method. In particular, we focus on the upper and lower critical decay exponents, the boundaries between the mean-field-universality, intermediate, and short-range-universality regimes. At the critical decay exponents, it is found that the standard Binder ratio of magnetization at the critical temperature exhibits extremely slow convergence as a function of the system size. We propose more effective physical quantities, namely the combined Binder ratio and the self-combined Binder ratio, both of which cancel the leading finite-size corrections of the conventional Binder ratio. Utilizing these techniques, we clearly demonstrate that in two dimensions, the lower and upper critical decay exponents are σ=1 and 7/4, respectively, contrary to the recent Monte Carlo and renormalization-group studies [M. Picco, arXiv:1207.1018; T. Blanchard et al., Europhys. Lett. 101, 56003 (2013)EULEEJ0295-507510.1209/0295-5075/101/56003].

arXiv:1605.09496Physical review. EJan 23, 201752 citations

Finite-Size Scaling and Universality above the Upper Critical Dimensionality

Erik Luijten, Henk W. J. Blöte

Physical Review LettersMay 6, 199630 citations

On the quantum critical behaviour of a model of structural phase transitions with long-range interaction

Ekaterina S. Pisanova, N. S. Tonchev

The pure quantum limit of an exactly solvable lattice model describing structural phase transitions in an anharmonic crystal with long-range interaction is considered. At the upper quantum critical dimension the free energy density at T = 0 in the neighbourhood of the quantum critical point is exactly calculated in terms of the Lambert W-function. For the three real physical dimensions, the exact results, obtained here, and the asymptotic ones are compared. It is pointed out that the Lambert W-function turns out to be a very effective tool for an exact computation of non-universal characteristics in the upper critical dimensions, especially in a broader neighbourhood of the critical region.The pure quantum limit of an exactly solvable lattice model describing structural phase transitions in an anharmonic crystal with long-range interaction is considered. At the upper quantum critical dimension the free energy density at T = 0 in the neighbourhood of the quantum critical point is exactly calculated in terms of the Lambert W-function. For the three real physical dimensions, the exact results, obtained here, and the asymptotic ones are compared. It is pointed out that the Lambert W-function turns out to be a very effective tool for an exact computation of non-universal characteristics in the upper critical dimensions, especially in a broader neighbourhood of the critical region.

Journal of Physics Conference SeriesNov 1, 20102 citations

Universality in the critical two-dimensional? 4-model

B. Mehlig, Bruce M. Forrest

The European Physical Journal BFeb 1, 19928 citations

The $6-ε$ Expansion for Long-Range Lee--Yang and Percolation CriticalityThe $6-ε$ Expansion for Long-Range Lee--Yang and Percolation Criticality

Zhiyi Li, Kun Chen, Zhijie Fan +1 more

The crossover from long-range (LR) to short-range (SR) criticality in percolation has remained unsettled because previous renormalization-group (RG) analysis within the $ε'=3σ-d$ expansion fixes the anomalous dimension at $η=2-σ$ , whereas SR percolation has $η_{\rm SR}<0$ near $d=6$ . Sak's matching condition then places the crossover above $σ=2$ , outside the regime in which the LR interaction dominates. In spatial dimension $d=6-ε$ , we formulate a perturbative expansion for the LR $ϕ^3$ field theory and perform a one-loop RG analysis throughout the perturbatively accessible nonclassical regime $0<δ<ε/3$ , where $δ= 2-σ$ . We derive the one-loop corrections to the critical exponents $η$ and $ν$ , which acquire nontrivial dependence on $ε$ and $δ$ . They reduce to their mean-field values at the LR upper critical line and continuously recover the SR $6-ε$ results as $σ\to2$ . These results support a crossover threshold $σ_*=2$ and remove the apparent discontinuity of $η$ between the LR and SR values within this framework. The same approach also yields the anomalous and edge exponents of the LR Lee--Yang universality class and $q$ -state Potts universality classes with $q<2$ .The crossover from long-range (LR) to short-range (SR) criticality in percolation has remained unsettled because previous renormalization-group (RG) analysis within the $ε'=3σ-d$ expansion fixes the anomalous dimension at $η=2-σ$, whereas SR percolation has $η_{\rm SR}<0$ near $d=6$. Sak's matching condition then places the crossover above $σ=2$, outside the regime in which the LR interaction dominates. In spatial dimension $d=6-ε$, we formulate a perturbative expansion for the LR $ϕ^3$ field theory and perform a one-loop RG analysis throughout the perturbatively accessible nonclassical regime $0<δ<ε/3$, where $δ= 2-σ$. We derive the one-loop corrections to the critical exponents $η$ and $ν$, which acquire nontrivial dependence on $ε$ and $δ$. They reduce to their mean-field values at the LR upper critical line and continuously recover the SR $6-ε$ results as $σ\to2$. These results support a crossover threshold $σ_*=2$ and remove the apparent discontinuity of $η$ between the LR and SR values within this framework. The same approach also yields the anomalous and edge exponents of the LR Lee--Yang universality class and $q$-state Potts universality classes with $q<2$.

arXiv:2608.15120Aug 15, 20260 citations

The

6-ε

Expansion for Long-Range Lee--Yang and Percolation Criticality

The crossover from long-range (LR) to short-range (SR) criticality in percolation has remained unsettled because previous renormalization-group (RG) analysis within the

ε'=3σ-d

expansion fixes the anomalous dimension at

η=2-σ

, whereas SR percolation has

η_{\rm SR}<0

near

d=6

. Sak's matching condition then places the crossover above

σ=2

, outside the regime in which the LR interaction dominates. In spatial dimension

d=6-ε

, we formulate a perturbative expansion for the LR

ϕ^3

field theory and perform a one-loop RG analysis throughout the perturbatively accessible nonclassical regime

0<δ<ε/3

, where

δ= 2-σ

. We derive the one-loop corrections to the critical exponents

η

and

ν

, which acquire nontrivial dependence on

ε

and

δ

. They reduce to their mean-field values at the LR upper critical line and continuously recover the SR

6-ε

results as

σ\to2

. These results support a crossover threshold

σ_*=2

and remove the apparent discontinuity of

η

between the LR and SR values within this framework. The same approach also yields the anomalous and edge exponents of the LR Lee--Yang universality class and

q

-state Potts universality classes with

q<2