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arXiv:2105.09781 · Physical review. B./Physical review. B

Restoring isotropy in a three-dimensional lattice model: The Ising universality class

Martin Hasenbusch B

Published Jul 26, 2021• 36 citations
arXivHTMLPDF

Abstract

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 $D$ 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 $D$ 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.

DOI
10.1103/physrevb.104.014426

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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 \ensuremathν=0.62998(5)\ensuremath{\nu}=0.629\phantom{\rule{0.16em}{0ex}}98(5)\ensuremathν=0.62998(5) and \ensuremathη=0.036284(40)\ensuremath{\eta}=0.0362\phantom{\rule{0.16em}{0ex}}84(40)\ensuremathη=0.036284(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 compute an improved action for the Ising universality class in three dimensions that has suppressed leading corrections to scaling. It is obtained by tuning models with two coupling constants. We studied three different models: the $\ifmmode\pm\else\textpm\fi{}1$ Ising model with nearest-neighbor and body diagonal interaction, the spin-1 model with states $0,\ifmmode\pm\else\textpm\fi{}1$, and nearest-neighbor interaction, and ${\ensuremath{\varphi}}^{4}$ theory on the lattice (Landau-Ginzburg model). The remarkable finite-size scaling properties of the suitably tuned spin-1 model are compared in detail with those of the standard Ising model. Great care is taken to estimate the systematic errors from residual corrections to scaling. Our best estimates for the critical exponents are $\ensuremath{\nu}=0.6298(5)$ and $\ensuremath{\eta}=0.0366(8)$, where the given error estimates take into account the statistical and systematic uncertainties.
Using transfer-matrix extended phenomenological renormalization-group methods, we study the critical properties of the spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths J→=(J′,J′,J)\mathop J\limits^ \to = (J',J',J)J→​=(J′,J′,J) . The universality of both fundamental critical exponents y t and y h is confirmed. It is shown that the critical finite-size scaling amplitude ratios U=Aχ(4)Aκ/Aχ2,Y1=Aκ′′/AχU = A_{\chi ^{(4)} } A_\kappa /A_\chi ^2 ,Y_1 = A_{\kappa ''} /A_\chiU=Aχ(4)​Aκ​/Aχ2​,Y1​=Aκ′′​/Aχ​ , and Y2=Aκ(4)/Aχ(4)Y_2 = A_{\kappa ^{(4)} } /A_{\chi ^{(4)} }Y2​=Aκ(4)​/Aχ(4)​ are independent of the lattice anisotropy parameter Δ=J′/J. For the Y 2 invariant of the three-dimensional Ising universality class, we give the first quantitative estimate Y 2≈2.013 (shape L×L×∞, periodic boundary conditions in both transverse directions).
We study the critical behavior of a generalized icosahedral model on the simple cubic lattice. The field variable of the icosahedral model might take 1 of 12 vectors of unit length which are given by the normalized vertices of the icosahedron as value. Similar to the Blume-Capel model, where in addition to $\ensuremath{-}1$ and 1, as in the Ising model, the spin might take the value 0, we add in the generalized model (0,0,0) as allowed value. There is a parameter DDD that controls the density of these voids. For a certain range of DDD, the model undergoes a second-order phase transition. On the critical line, O(3)\text{O}(3)O(3) symmetry emerges. Furthermore, we demonstrate that within this range, similar to the Blume-Capel model on the simple cubic lattice, there is a value of DDD, where leading corrections to scaling vanish. We perform Monte Carlo simulations for lattices of a linear size up to L=400L=400L=400 by using a hybrid of local Metropolis and cluster updates. The motivation to study this particular model is mainly of technical nature. Less memory and CPU time are needed than for a model with O(3)\text{O}(3)O(3) symmetry at the microscopic level. As the result of a finite-size scaling analysis we obtain $\ensuremath{\nu}=0.711\phantom{\rule{0.16em}{0ex}}64(10)$, $\ensuremath{\eta}=0.037\phantom{\rule{0.16em}{0ex}}84(5)$, and $\ensuremath{\omega}=0.759(2)$ for the critical exponents of the three-dimensional Heisenberg universality class. The estimate of the irrelevant renormalization group eigenvalue that is related with the breaking the O(3)\text{O}(3)O(3) symmetry is ${y}_{\mathrm{ico}}=\ensuremath{-}2.19(2)$.
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 ${\ensuremath{\xi}}_{\text{exp}}$ 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 $\ensuremath{\eta}=0.03816(2)$ and ${y}_{t}=1/\ensuremath{\nu}=1.48872(5)$ as estimates of the critical exponents.