arXiv subjects

Henk W. J. Blöte

Publications and source records attributed to Henk W. J. Blöte.

At least 19 recordsLinked to original sources

arXiv2026

Ising models on the hydrogen peroxide and other lattices

We perform a Monte Carlo analysis of the Ising model on many three-dimensional lattices. By means of finite-size scaling we obtain the critical points and determine the scaling dimensions. As expected, the critical exponents agree with the three-dimensional Ising universality class for all models. The irrelevant field, as revealed by the correction-to-scaling amplitudes, appears to be relatively large. Combining the Monte Carlo results for the hydrogen peroxide lattice with those for five other three-dimensional lattices, we obtain a set of data covering a wide range of the irrelevant temperature field. This is helpful in the determination of the parameters describing the corrections to scaling. As a consequence, new results are obtained for the universal parameters describing Ising criticality in three dimensions, with reduced error margins in comparison with earlier Monte Carlo analyses. The critical exponents describing the thermodynamic singularities are determined by the temperature renormalization exponent $y_t = 1.58693 (9)$ and the magnetic renormalization exponent $y_h = 2.48178 (5)$. The corrections to scaling are governed by the irrelevant exponent $y_1 = -0.821 (5)$.

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arXiv2019

The three-state Potts model on the centered triangular lattice

We study phase transitions of the Potts model on the centered-triangular lattice with two types of couplings, namely $K$ between neighboring triangular sites, and $J$ between the centered and the triangular sites. Results are obtained by means of a finite-size analysis based on numerical transfer matrix calculations and Monte Carlo simulations. Our investigation covers the whole $(K, J)$ phase diagram, but we find that most of the interesting physics applies to the antiferromagnetic case $K<0$, where the model is geometrically frustrated. In particular, we find that there are, for all finite $J$, two transitions when K is varied. Their critical properties are explored. In the limits $J\to \pm \infty$ we find algebraic phases with infinite-order transitions to the ferromagnetic phase.

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arXiv2018

Equivalent-neighbor percolation models in two dimensions: crossover between mean-field and short-range behavior

We investigate the influence of the range of interactions in the two-dimensional bond percolation model, by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges, as expressed by the number $z$ of equivalent neighbors. We also consider the $z \to \infty$ limit, i.e., the complete graph case, where percolation bonds are allowed between each pair of sites, and the model becomes mean-field-like. All investigated models with finite $z$ are found to belong to the short-range universality class. There is no evidence of a tricritical point separating the short-range and long-range behavior, such as is known to occur for $q=3$ and $q=4$ Potts models. We determine the renormalization exponent describing a finite-range perturbation at the mean-field limit as $y_r \approx 2/3$. Its relevance confirms the continuous crossover from mean-field percolation universality to short-range percolation universality. For finite interaction ranges, we find approximate relations between the coordination numbers and the amplitudes of the leading correction terms as found in the finite-size scaling analysis.

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arXiv2016

Special transitions in an O($n$) loop model with an Ising-like constraint

We investigate the O($n$) nonintersecting loop model on the square lattice under the constraint that the loops consist of ninety-degree bends only. The model is governed by the loop weight $n$, a weight $x$ for each vertex of the lattice visited once by a loop, and a weight $z$ for each vertex visited twice by a loop. We explore the $(x,z)$ phase diagram for some values of $n$. For $0 1$, the O($n$)-like transition line appears to be absent. Thus, for $z=0$, the $(n,x)$ phase diagram displays a line of phase transitions for $n\le 1$. The line ends at $n=1$ in an infinite-order transition. We determine the conformal anomaly and the critical exponents along this line. These results agree accurately with a recent proposal for the universal classification of this type of model, at least in most of the range $-1 \leq n \leq 1$. We also determine the exponent describing crossover to the generic O($n$) universality class, by introducing topological defects associated with the introduction of `straight' vertices violating the ninety-degree-bend rule. These results are obtained by means of transfer-matrix calculations and finite-size scaling.

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arXiv2015

Short-range correlations in percolation at criticality

We derive the critical nearest-neighbor connectivity $g_n$ as $3/4$, $3(7-9p_c^{tri})/[4(5-4p_c^{tri})]$, and $3(2+7p_c^{tri})/[4(5-p_c^{tri})]$ for bond percolation on the square, honeycomb and triangular lattice respectively, where $p_c^{tri}=2\sin(π/18)$ is the percolation threshold for the triangular lattice; and confirm these values via Monte Carlo simulations. On the square lattice, we also numerically determine the critical next-nearest-neighbor connectivity as $g_{nn}=0.687\;500\;0(2)$, which confirms a conjecture by Mitra and Nienhuis in J. Stat. Mech. P10006 (2004), implying the exact value $g_{nn}=11/16$. We also determine the connectivity on a free surface as $g_n^{surf}=0.625\;000\;1(13)$ and conjecture that this value is exactly equal to $5/8$. In addition, we find that at criticality, the connectivities depend on the linear finite size L as $\sim L^{y_t-d}$, and the associated specific-heat-like quantities $C_n$ and $C_{nn}$ scale as $\sim L^{2y_t-d} \ln (L/L_0)$, where $d$ is the lattice dimensionality, $y_t=1/ν$ the thermal renormalization exponent, and $L_0$ a non-universal constant. We provide an explanation of this logarithmic factor in the theoretical framework reported recently by Vasseur et al. in J. Stat. Mech. L07001 (2012).

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arXiv2014

Completely packed O($n$) loop models and their relation with exactly solved coloring models

We explore the physical properties of the completely packed O($n$) loop model on the square lattice, and its generalization to an Eulerian graph model, which follows by including cubic vertices which connect the four incoming loop segments. This model includes crossing bonds as well. Our study of the properties of this model involve transfer-matrix calculations and finite-size scaling. The numerical results are compared to existing exact solutions, including solutions of special cases of a so-called coloring model, which are shown to be equivalent with our generalized loop model. The latter exact solutions correspond with seven one-dimensional branches in the parameter space of our generalized loop model. One of these branches, describing the case of nonintersecting loops, is already known to correspond with the ordering transition of the Potts model. We find that another exactly solved branch, which describes a model with nonintersecting loops and cubic vertices, corresponds with a first-order Ising-like phase transition for $n>2$. For $1 2$ this branch is the locus of a first-order phase boundary between a phase with a hard-square lattice-gas like ordering, and a phase dominated by cubic vertices. The first-order character of this transition is in agreement with a mean-field argument.

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arXiv2013

Ising-like transitions in the O($n$) loop model on the square lattice

We explore the phase diagram of the O($n$) loop model on the square lattice in the $(x,n)$ plane, where $x$ is the weight of a lattice edge covered by a loop. These results are based on transfer-matrix calculations and finite-size scaling. We express the correlation length associated with the staggered loop density in the transfer-matrix eigenvalues. The finite-size data for this correlation length, combined with the scaling formula, reveal the location of critical lines in the diagram. For $n>>2$ we find Ising-like phase transitions associated with the onset of a checkerboard-like ordering of the elementary loops, i.e., the smallest possible loops, with the size of an elementary face, which cover precisely one half of the faces of the square lattice at the maximum loop density. In this respect, the ordered state resembles that of the hard-square lattice gas with nearest-neighbor exclusion, and the finiteness of $n$ represents a softening of its particle-particle potentials. We also determine critical points in the range $-2\leq n\leq 2$. It is found that the topology of the phase diagram depends on the set of allowed vertices of the loop model. Depending on the choice of this set, the $n>2$ transition may continue into the dense phase of the $n \leq 2$ loop model, or continue as a line of $n \leq 2$ O($n$) multicritical points.

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arXiv2013

Percolation in the canonical ensemble

We study the bond percolation problem under the constraint that the total number of occupied bonds is fixed, so that the canonical ensemble applies. We show via an analytical approach that at criticality, the constraint can induce new finite-size corrections with exponent y_{can}=2y_t-d both in energy-like and magnetic quantities, where y_t=1/ν is the thermal renormalization exponent and d is the spatial dimension. Furthermore, we find that while most of universal parameters remain unchanged, some universal amplitudes, like the excess cluster number, can be modified and become non-universal. We confirm these predictions by extensive Monte Carlo simulations of the two-dimensional percolation problem which has y_{can}=-1/2.

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arXiv2010

Crossover phenomena involving the dense O($n$) phase

We explore the properties of the low-temperature phase of the O($n$) loop model in two dimensions by means of transfer-matrix calculations and finite-size scaling. We determine the stability of this phase with respect to several kinds of perturbations, including cubic anisotropy, attraction between loop segments, double bonds and crossing bonds. In line with Coulomb gas predictions, cubic anisotropy and crossing bonds are found to be relevant and introduce crossover to different types of behavior. Whereas perturbations in the form of loop-loop attractions and double bonds are irrelevant, sufficiently strong perturbations of these types induce a phase transition of the Ising type, at least in the cases investigated. This Ising transition leaves the underlying universal low-temperature O($n$) behavior unaffected.

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arXiv2010

Berezinskii-Kosterlitz-Thouless-like percolation transitions in the two-dimensional XY model

We study a percolation problem on a substrate formed by two-dimensional XY spin configurations, using Monte Carlo methods. For a given spin configuration we construct percolation clusters by randomly choosing a direction $x$ in the spin vector space, and then placing a percolation bond between nearest-neighbor sites $i$ and $j$ with probability $p_{ij} = \max (0,1-e^{-2K s^x_i s^x_j})$, where $K > 0$ governs the percolation process. A line of percolation thresholds $K_{\rm c} (J)$ is found in the low-temperature range $J \geq J_{\rm c}$, where $J > 0$ is the XY coupling strength. Analysis of the correlation function $g_p (r)$, defined as the probability that two sites separated by a distance $r$ belong to the same percolation cluster, yields algebraic decay for $K \geq K_{\rm c}(J)$, and the associated critical exponent depends on $J$ and $K$. Along the threshold line $K_{\rm c}(J)$, the scaling dimension for $g_p$ is, within numerical uncertainties, equal to $1/8$. On this basis, we conjecture that the percolation transition along the $K_{\rm c} (J)$ line is of the Berezinskii-Kosterlitz-Thouless type.

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arXiv2009

Conducting-angle-based percolation in the XY model

We define a percolation problem on the basis of spin configurations of the two dimensional XY model. Neighboring spins belong to the same percolation cluster if their orientations differ less than a certain threshold called the conducting angle. The percolation properties of this model are studied by means of Monte Carlo simulations and a finite-size scaling analysis. Our simulations show the existence of percolation transitions when the conducting angle is varied, and we determine the transition point for several values of the XY coupling. It appears that the critical behavior of this percolation model can be well described by the standard percolation theory. The critical exponents of the percolation transitions, as determined by finite-size scaling, agree with the universality class of the two-dimensional percolation model on a uniform substrate. This holds over the whole temperature range, even in the low-temperature phase where the XY substrate is critical in the sense that it displays algebraic decay of correlations.

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arXiv2009

Specific heat of the simple-cubic Ising model

We provide an expression quantitatively describing the specific heat of the Ising model on the simple-cubic lattice in the critical region. This expression is based on finite-size scaling of numerical results obtained by means of a Monte Carlo method. It agrees satisfactorily with series expansions and with a set of experimental results. Our results include a determination of the universal amplitude ratio of the specific-heat divergences at both sides of the critical point.

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arXiv2009

Phase transitions in self-dual generalizations of the Baxter-Wu model

We study two types of generalized Baxter-Wu models, by means of transfer-matrix and Monte Carlo techniques. The first generalization allows for different couplings in the up- and down triangles, and the second generalization is to a $q$-state spin model with three-spin interactions. Both generalizations lead to self-dual models, so that the probable locations of the phase transitions follow. Our numerical analysis confirms that phase transitions occur at the self-dual points. For both generalizations of the Baxter-Wu model, the phase transitions appear to be discontinuous.

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arXiv2009

Percolation and critical O($n$) loop configurations

We study a percolation problem based on critical loop configurations of the O($n$) loop model on the honeycomb lattice. We define dual clusters as groups of sites on the dual triangular lattice that are not separated by a loop, and investigate the the bond-percolation properties of these dual clusters. The universal properties at the percolation threshold are argued to match those of Kasteleyn-Fortuin random clusters in the critical Potts model. This relation is checked numerically by means of cluster simulations of several O($n$) models in the range $1\leq n \leq 2$. The simulation results include the percolation threshold for several values of $n$, as well as the universal exponents associated with bond dilution and the size distribution of the diluted clusters at the percolation threshold. Our numerical results for the exponents are in agreement with existing Coulomb gas results for the random-cluster model, which confirms the relation between both models. We discuss the renormalization flow of the bond-dilution parameter $p$ as a function of $n$, and provide an expression that accurately describes a line of unstable fixed points as a function of $n$, corresponding with the percolation threshold. Furthermore, the renormalization scenario indicates the existence, in $p$ versus $n$ diagram, another line of fixed points at $p=1$, which is stable with respect to $p$.

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arXiv2008

Tricritical O(n) models in two dimensions

We show that the exactly solved low-temperature branch of the two-dimensional O($n$) model is equivalent with an O($n$) model with vacancies and a different value of $n$. We present analytic results for several universal parameters of the latter model, which is identified as a tricritical point. These results apply to the range $n \leq 3/2$, and include the exact tricritical point, the conformal anomaly and a number of scaling dimensions, among which the thermal and magnetic exponent, the exponent associated with crossover to ordinary critical behavior, and to tricritical behavior with cubic symmetry. We describe the translation of the tricritical model in a Coulomb gas. The results are verified numerically by means of transfer-matrix calculations. We use a generalized ADE model as an intermediary, and present the expression of the one-point distribution function in that language. The analytic calculations are done both for the square and the hexagonal lattice.

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arXiv2008

Critical properties of a dilute O($n$) model on the kagome lattice

A critical dilute O($n$) model on the kagome lattice is investigated analytically and numerically. We employ a number of exact equivalences which, in a few steps, link the critical O($n$) spin model on the kagome lattice to the exactly solvable critical $q$-state Potts model on the honeycomb lattice with $q=(n+1)^2$. The intermediate steps involve the random-cluster model on the honeycomb lattice, and a fully packed loop model with loop weight $n'=\sqrt{q}$ and a dilute loop model with loop weight $n$, both on the kagome lattice. This mapping enables the determination of a branch of critical points of the dilute O($n$) model, as well as some of its critical properties. For $n=0$, this model reproduces the known universal properties of the $θ$ point describing the collapse of a polymer. For $n\neq 0$ it displays a line of multicritical points, with the same universal properties as a branch of critical behavior that was found earlier in a dilute O($n$) model on the square lattice. These findings are supported by a finite-size-scaling analysis in combination with transfer-matrix calculations.

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arXiv2006

Geometric properties of two-dimensional O(n) loop configurations

We study the fractal geometry of O($n$) loop configurations in two dimensions by means of scaling and a Monte Carlo method, and compare the results with predictions based on the Coulomb gas technique. The Monte Carlo algorithm is applicable to models with noninteger $n$ and uses local updates. Although these updates typically lead to nonlocal modifications of loop connectivities, the number of operations required per update is only of order one. The Monte Carlo algorithm is applied to the O($n$) model for several values of $n$, including noninteger ones. We thus determine scaling exponents that describe the fractal nature of O($n$) loops at criticality. The results of the numerical analysis agree with the theoretical predictions.

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arXiv2005

Exact characterization of O(n) tricriticality in two dimensions

We propose exact expressions for the conformal anomaly and for three critical exponents of the tricritical O(n) loop model as a function of n in the range $-2 \leq n \leq 3/2$. These findings are based on an analogy with known relations between Potts and O(n) models, and on an exact solution of a 'tri-tricritical' Potts model described in the literature. We verify the exact expressions for the tricritical O(n) model by means of a finite-size scaling analysis based on numerical transfer-matrix calculations.

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