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arXiv:2507.19265 · Physical review. B./Physical review. B

Eliminating leading and subleading corrections to scaling in the three-dimensional XY universality class

Martin Hasenbusch B

Published Nov 14, 2025• 6 citations
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Abstract

We study the $(q+1)$-state clock model on the simple cubic lattice by using Monte Carlo simulations. In addition to the nearest-neighbor coupling we consider a next-to-next-to-nearest-neighbor coupling. For a certain range of the parameters, the phase transition of the model shares the XY universality class. Leading corrections to scaling are studied by using finite-size scaling of dimensionless quantities, such as the Binder cumulant ${U}_{4}$. The spatial unisotropy, which causes subleading corrections, is studied by computing the exponential correlation length ${\ensuremath{\xi}}_{\text{exp}}$ in the high-temperature phase for different directions. In the case of the $q$-state clock model it turns out that by tuning the ratio of the two coupling constants, we can eliminate either leading or subleading corrections to scaling. These points on the critical line are close to each other. Hence in the improved model, where leading corrections to scaling vanish, also subleading corrections are small. By using a finite-size scaling analysis of our high statistics data we obtain $\ensuremath{\eta}=0.03816(2)$ and ${y}_{t}=1/\ensuremath{\nu}=1.48872(5)$ as estimates of the critical exponents.

DOI
10.1103/1d4g-483z

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We study the (q+1)(q+1)(q+1)-state clock model on the simple cubic lattice by using Monte Carlo simulations. In addition to the nearest-neighbor coupling we consider a next-to-next-to-nearest-neighbor coupling. For a certain range of the parameters, the phase transition of the model shares the XY universality class. Leading corrections to scaling are studied by using finite-size scaling of dimensionless quantities, such as the Binder cumulant U4{U}_{4}U4​. The spatial unisotropy, which causes subleading corrections, is studied by computing the exponential correlation length \ensuremathξexp{\ensuremath{\xi}}_{\text{exp}}\ensuremathξexp​ in the high-temperature phase for different directions. In the case of the qqq-state clock model it turns out that by tuning the ratio of the two coupling constants, we can eliminate either leading or subleading corrections to scaling. These points on the critical line are close to each other. Hence in the improved model, where leading corrections to scaling vanish, also subleading corrections are small. By using a finite-size scaling analysis of our high statistics data we obtain \ensuremathη=0.03816(2)\ensuremath{\eta}=0.03816(2)\ensuremathη=0.03816(2) and yt=1/\ensuremathν=1.48872(5){y}_{t}=1/\ensuremath{\nu}=1.48872(5)yt​=1/\ensuremathν=1.48872(5) as estimates of the critical exponents.
We study a generalized clock model on the simple cubic lattice. The parameter of the model can be tuned such that the amplitude of the leading correction to scaling vanishes. In the main part of the study, we simulate the model with Z8{Z}_{8}Z8​ symmetry. At the transition, with increasing length scale, O(2)O(2)O(2) symmetry emerges. We perform Monte Carlo simulations using a hybrid of local Metropolis and cluster algorithms of lattices with a linear size up to L=512L=512L=512. The field variable requires less memory and the updates are faster than for a model with O(2)O(2)O(2) symmetry at the microscopic level. Our finite-size scaling analysis yields accurate estimates for the critical exponents of the three-dimensional XYXYXY-universality class. In particular, we get $\ensuremath{\eta}=0.03810(8),\ensuremath{\nu}=0.67169(7)$, and $\ensuremath{\omega}=0.789(4)$. Furthermore, we obtain estimates for fixed point values of phenomenological couplings and critical temperatures.
We study a generalized Blume-Capel model on the simple cubic lattice. In addition to the nearest-neighbor coupling there is a next-to-next-to-nearest-neighbor coupling. In order to quantify spatial anisotropy, we determine the correlation length in the high-temperature phase of the model for three different spatial directions. It turns out that the spatial anisotropy depends very little on the dilution or crystal-field parameter DDD of the model and is essentially determined by the ratio of the nearest-neighbor and the next-to-next-to-nearest-neighbor coupling. This ratio is tuned such that the leading contribution to the spatial anisotropy is eliminated. Next we perform a finite-size scaling (FSS) study to tune DDD such that also the leading correction to scaling is eliminated. Based on this FSS study, we determine the critical exponents $\ensuremath{\nu}=0.629\phantom{\rule{0.16em}{0ex}}98(5)$ and $\ensuremath{\eta}=0.0362\phantom{\rule{0.16em}{0ex}}84(40)$, which are in nice agreement with the more accurate results obtained by using the conformal bootstrap method. Furthermore, we provide accurate results for fixed-point values of dimensionless quantities such as the Binder cumulant and for the critical couplings. These results provide the groundwork for broader studies of universal properties of the three-dimensional Ising universality class.
We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter DDD on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter DDD, where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for D=0.641D=0.641D=0.641 and 0.655 on lattices of a linear size up to L=360L=360L=360 we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.
We investigate several three-dimensional lattice models believed to be in the Ising universality class by means of Monte Carlo methods and finite-size scaling. These models include spin-1 / 2 models with nearest-neighbor interactions on the simple-cubic and on the diamond lattice. For the simple cubic lattice, we also include models with third-neighbor interactions of varying strength, and some "equivalent-neighbor" models. Also included are a spin-1 model and a hard-core lattice gas. Separate analyses of the numerical data confirm the Ising-like critical behavior of these systems. On this basis, we analyze all these data simultaneously such that the universal parameters occur only once. This leads to an improved accuracy. The thermal, magnetic, and irrelevant exponents are determined as y(t)=1.5868(3), y(h)=2.4816(1), and y(i)=-0.821(5), respectively. The Binder ratio is estimated as Q=<m(2)>(2)/<m(4)>=0.62 341(4).
We study the cubic fixed point for N=3N=3N=3 and 4 by using finite-size scaling applied to data obtained from Monte Carlo simulations of the NNN-component ${\ensuremath{\phi}}^{4}$ model on the simple cubic lattice. We generalize the idea of improved models to a two-parameter family of models. The two-parameter space is scanned for the point, where the amplitudes of the two leading corrections to scaling vanish. To this end, a dimensionless quantity is introduced that monitors the breaking of the O(N)O(N)O(N) invariance. For N=4N=4N=4, we determine the correction exponents ${\ensuremath{\omega}}_{1}=0.763(24)$ and ${\ensuremath{\omega}}_{2}=0.082(5)$. In the case of N=3N=3N=3, we obtain Y4=0.0142(6){Y}_{4}=0.0142(6)Y4​=0.0142(6) for the renormalization group exponent of the cubic perturbation at the O(3)O(3)O(3)-invariant fixed point, while the correction exponent ${\ensuremath{\omega}}_{2}=0.0133(8)$ at the cubic fixed point. Simulations close to the improved point result in the estimates $\ensuremath{\nu}=0.7202(7)$ and $\ensuremath{\eta}=0.0371(2)$ of the critical exponents of the cubic fixed point for N=4N=4N=4. For N=3N=3N=3, at the cubic fixed point, the O(3)O(3)O(3) symmetry is only mildly broken and the critical exponents differ only by little from those of the O(3)O(3)O(3)-invariant fixed point. We find $\ensuremath{-}0.000\phantom{\rule{0.16em}{0ex}}01⪅{\ensuremath{\eta}}_{\text{cubic}}\ensuremath{-}{\ensuremath{\eta}}_{O(3)}⪅0.000\phantom{\rule{0.16em}{0ex}}07$ and ${\ensuremath{\nu}}_{\text{cubic}}\ensuremath{-}{\ensuremath{\nu}}_{O(3)}=\ensuremath{-}0.000\phantom{\rule{0.16em}{0ex}}61(10)$.
We study the phase transition of the classical XYXYXY model on a diamond lattice by Monte Carlo simulations using the Wolff cluster algorithm. Finite-size scaling (FSS) analysis of the Binder cumulant and the second-moment correlation length ratio ξ2nd/Lξ_{2\rm nd}/Lξ2nd​/L yields Tc=1.30036(1)T_c = 1.30036(1)Tc​=1.30036(1) and ν=0.671(6)ν= 0.671(6)ν=0.671(6). Data collapse of both quantities confirms the three-dimensional XYXYXY universality class.