We present a high-resolution numerical study of the spin-1 Blume-Capel ferromagnet on the honeycomb lattice ( $z=3$ ). Using the Wang-Landau algorithm to estimate the joint density of states $Ω(E_1,E_2)$ and field-mixing scaling, we map the first-order transition line and locate the tricritical point at $(Δ_t,T_t)=[1.4820(2),0.4022(7)]$ . Independent Metropolis histogram reweighting for $L=36$ and $48$ yields a crossing temperature near $0.4039$ , consistent with the Wang-Landau extrapolation. The field-mixing parameter is $r=-1.585(19)$ , while the coexistence-line slope gives $r=-1.587(5)$ . Normalizing by the zero-temperature field $Δ_0=zJ/2$ , the results for honeycomb, square, and triangular lattices give $r/Δ_0$ between $-1.057$ and $-1.067$ , about 6% larger in magnitude than the mean-field value. Simple-cubic estimates are compatible with this range but do not resolve dimensionality dependence. Finite-size scaling yields $y_t=1.795(7)$ , consistent with the tricritical Ising value $9/5$ , while the fourth-order cumulant scaling agrees with $y_g=4/5$ . Deep in the first-order regime, the cumulant approaches the two-phase limit $U_Q\to2/3$ . Multicanonical simulations up to $L=64$ yield an interface tension exponent $μ=1.265(26)$ , consistent with the tricritical prediction $(d-1)/y_g=5/4$ . We also map the continuous phase boundary, obtaining $T_c(Δ=0)=1.157$ , and identify crossover effects from Ising to tricritical scaling near the tricritical point.We present a high-resolution numerical study of the spin-1 Blume-Capel ferromagnet on the honeycomb lattice ($z=3$). Using the Wang-Landau algorithm to estimate the joint density of states $Ω(E_1,E_2)$ and field-mixing scaling, we map the first-order transition line and locate the tricritical point at $(Δ_t,T_t)=[1.4820(2),0.4022(7)]$. Independent Metropolis histogram reweighting for $L=36$ and $48$ yields a crossing temperature near $0.4039$, consistent with the Wang-Landau extrapolation. The field-mixing parameter is $r=-1.585(19)$, while the coexistence-line slope gives $r=-1.587(5)$. Normalizing by the zero-temperature field $Δ_0=zJ/2$, the results for honeycomb, square, and triangular lattices give $r/Δ_0$ between $-1.057$ and $-1.067$, about 6% larger in magnitude than the mean-field value. Simple-cubic estimates are compatible with this range but do not resolve dimensionality dependence. Finite-size scaling yields $y_t=1.795(7)$, consistent with the tricritical Ising value $9/5$, while the fourth-order cumulant scaling agrees with $y_g=4/5$. Deep in the first-order regime, the cumulant approaches the two-phase limit $U_Q\to2/3$. Multicanonical simulations up to $L=64$ yield an interface tension exponent $μ=1.265(26)$, consistent with the tricritical prediction $(d-1)/y_g=5/4$. We also map the continuous phase boundary, obtaining $T_c(Δ=0)=1.157$, and identify crossover effects from Ising to tricritical scaling near the tricritical point.

arXiv:2610.11068Oct 8, 20260 citations

Finite-Size Scaling at Criticality

WORLD SCIENTIFIC eBooksAug 1, 20000 citations

Anisotropy and universality in finite-size scaling: Critical Binder cumulant of a two-dimensional Ising model

Boris Kastening

We reanalyze transfer-matrix and Monte Carlo results for the critical Binder cumulant U* of an anisotropic two-dimensional Ising model on a square lattice in a square geometry with periodic boundary conditions. Spins are coupled between nearest-neighboring sites and between next-nearest-neighboring sites along one of the lattice diagonals. We find that U* depends only on the asymptotic critical long-distance features of the anisotropy, irrespective of its realization through ferromagnetic or antiferromagnetic next-nearest-neighbor couplings. We modify an earlier renormalization-group calculation to obtain a quantitative description of the anisotropy dependence of U*. Our results support our recent claim towards the validity of universal finite-size scaling for critical phenomena in the presence of a weak anisotropy.We reanalyze transfer-matrix and Monte Carlo results for the critical Binder cumulant U* of an anisotropic two-dimensional Ising model on a square lattice in a square geometry with periodic boundary conditions. Spins are coupled between nearest-neighboring sites and between next-nearest-neighboring sites along one of the lattice diagonals. We find that U* depends only on the asymptotic critical long-distance features of the anisotropy, irrespective of its realization through ferromagnetic or antiferromagnetic next-nearest-neighbor couplings. We modify an earlier renormalization-group calculation to obtain a quantitative description of the anisotropy dependence of U*. Our results support our recent claim towards the validity of universal finite-size scaling for critical phenomena in the presence of a weak anisotropy.

arXiv:1209.0105Physical Review EApr 19, 20139 citations

NONUNIVERSAL FINITE-SIZE EFFECTS NEAR CRITICAL POINTS

V. Dohm

International Conference on Path IntegralsNov 1, 20080 citations

Two temperature scales in the Ising model on the $\{5,4\}$ hyperbolic lattice with free boundaries: susceptibility peak and boundary-induced orderTwo temperature scales in the Ising model on the $\{5,4\}$ hyperbolic lattice with free boundaries: susceptibility peak and boundary-induced order

S. Jaroszewicz, Nahuel Daniel Mendez, Maria P. Beccar-Varela +1 more

On the $\{5,4\}$ hyperbolic lattice, the outermost generation holds 73\% of the sites at all system sizes, making macroscopic averages strictly boundary-dependent. We study the ferromagnetic Ising model on this geometry with free boundaries by Monte Carlo simulation, demonstrating that conventional observables cease to identify a single critical scale. Instead, the finite lattices are organized by two distinct temperature scales. The susceptibility maximum identifies the lower transition scale at $T_{c2}=1.4782(15)J/k_{B}$ without extrapolation. However, the order-parameter distribution lacks a conventional fixed point at this scale. A distinct pseudocritical scale emerges instead at $T^{*}\simeq1.67(3)$ inside the boundary-sensitive intermediate phase, where Binder cumulant curves cross and effective exponents are compatible with mean-field criticality. Between $T_{c2}$ and $T^{*}$ , the interior amplifies boundary fluctuations into induced order. Furthermore, because the volume grows exponentially with depth, finite-size scaling must be formulated in the generation index rather than the total number of sites. The lack of asymptotic power-law convergence in the scaling stretch confirms through an independent observable that $T^{*}$ is a finite-size crossover rather than a thermodynamic fixed point. Finally, by applying a coherent field to the outermost generation alone, we recover the upper thermodynamic transition at $T_{pt}=2.81(1)J/k_{B}$ , demonstrating that it remains accessible under an appropriate boundary perturbation.On the $\{5,4\}$ hyperbolic lattice, the outermost generation holds 73\% of the sites at all system sizes, making macroscopic averages strictly boundary-dependent. We study the ferromagnetic Ising model on this geometry with free boundaries by Monte Carlo simulation, demonstrating that conventional observables cease to identify a single critical scale. Instead, the finite lattices are organized by two distinct temperature scales. The susceptibility maximum identifies the lower transition scale at $T_{c2}=1.4782(15)J/k_{B}$ without extrapolation. However, the order-parameter distribution lacks a conventional fixed point at this scale. A distinct pseudocritical scale emerges instead at $T^{*}\simeq1.67(3)$ inside the boundary-sensitive intermediate phase, where Binder cumulant curves cross and effective exponents are compatible with mean-field criticality. Between $T_{c2}$ and $T^{*}$, the interior amplifies boundary fluctuations into induced order. Furthermore, because the volume grows exponentially with depth, finite-size scaling must be formulated in the generation index rather than the total number of sites. The lack of asymptotic power-law convergence in the scaling stretch confirms through an independent observable that $T^{*}$ is a finite-size crossover rather than a thermodynamic fixed point. Finally, by applying a coherent field to the outermost generation alone, we recover the upper thermodynamic transition at $T_{pt}=2.81(1)J/k_{B}$, demonstrating that it remains accessible under an appropriate boundary perturbation.

arXiv:2609.29323Sep 24, 20260 citations

We present a high-resolution numerical study of the spin-1 Blume-Capel ferromagnet on the honeycomb lattice (

z=3

). Using the Wang-Landau algorithm to estimate the joint density of states

Ω(E_1,E_2)

and field-mixing scaling, we map the first-order transition line and locate the tricritical point at

(Δ_t,T_t)=[1.4820(2),0.4022(7)]

. Independent Metropolis histogram reweighting for

L=36

and

48

yields a crossing temperature near

0.4039

, consistent with the Wang-Landau extrapolation. The field-mixing parameter is

r=-1.585(19)

, while the coexistence-line slope gives

r=-1.587(5)

. Normalizing by the zero-temperature field

Δ_0=zJ/2

, the results for honeycomb, square, and triangular lattices give

r/Δ_0

between

-1.057

and

-1.067

, about 6% larger in magnitude than the mean-field value. Simple-cubic estimates are compatible with this range but do not resolve dimensionality dependence. Finite-size scaling yields

y_t=1.795(7)

, consistent with the tricritical Ising value

9/5

, while the fourth-order cumulant scaling agrees with

y_g=4/5

. Deep in the first-order regime, the cumulant approaches the two-phase limit

U_Q\to2/3

. Multicanonical simulations up to

L=64

yield an interface tension exponent

μ=1.265(26)

, consistent with the tricritical prediction

(d-1)/y_g=5/4

. We also map the continuous phase boundary, obtaining

T_c(Δ=0)=1.157

, and identify crossover effects from Ising to tricritical scaling near the tricritical point.

Two temperature scales in the Ising model on the

\{5,4\}

hyperbolic lattice with free boundaries: susceptibility peak and boundary-induced order

On the

\{5,4\}

hyperbolic lattice, the outermost generation holds 73\% of the sites at all system sizes, making macroscopic averages strictly boundary-dependent. We study the ferromagnetic Ising model on this geometry with free boundaries by Monte Carlo simulation, demonstrating that conventional observables cease to identify a single critical scale. Instead, the finite lattices are organized by two distinct temperature scales. The susceptibility maximum identifies the lower transition scale at

T_{c2}=1.4782(15)J/k_{B}

without extrapolation. However, the order-parameter distribution lacks a conventional fixed point at this scale. A distinct pseudocritical scale emerges instead at

T^{*}\simeq1.67(3)

inside the boundary-sensitive intermediate phase, where Binder cumulant curves cross and effective exponents are compatible with mean-field criticality. Between

T_{c2}

and

T^{*}

, the interior amplifies boundary fluctuations into induced order. Furthermore, because the volume grows exponentially with depth, finite-size scaling must be formulated in the generation index rather than the total number of sites. The lack of asymptotic power-law convergence in the scaling stretch confirms through an independent observable that

T^{*}

is a finite-size crossover rather than a thermodynamic fixed point. Finally, by applying a coherent field to the outermost generation alone, we recover the upper thermodynamic transition at

T_{pt}=2.81(1)J/k_{B}

, demonstrating that it remains accessible under an appropriate boundary perturbation.