Critical Binder cumulant of two-dimensional Ising models
ABSTRACT The fourth-order cumulant of the magnetization, the Binder cumulant,
is determined at the phase transition of
Ising models on square and triangular lattices, using Monte
Carlo techniques. Its value at
criticality depends sensitively on
boundary conditions, details of the
clusters used in calculating the cumulant, and symmetry of the
interactions or, here, lattice structure. Possibilities to
identify generic critical cumulants are discussed.
- SourceAvailable from: Tzu-Chieh Wei
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- "Even more important to the search for phase transitions is the so-called " Binder cumulant, " first introduced by Kurt Binder in 1981 in a study of the classical Ising Model . In many settings, such as thermal or disordered systems, it is considered to be one of the most accurate and reliable means of detecting a critical point   , and it has since been applied to a wide variety of models        . "
ABSTRACT: We present a numerical scheme for efficiently extracting the higher-order moments and cumulants of various operators on spin systems represented as tensor product states, for both finite and infinite systems, and present several applications for such quantities. For example, the second cumulant of the energy of a state, $\langle \Delta H^2 \rangle$, gives a straightforward method to check the convergence of numerical ground-state approximation algorithms. Additionally, we discuss the use of moments and cumulants in the study of phase transitions. Of particular interest is the application of our method to calculate the so-called Binder's cumulant, which we use to detect critical points and study the critical exponent of the correlation length with only small finite numerical calculations. We apply these methods to study the behavior of a family of one-dimensional models (the transverse Ising model, the spin-1 Ising model, and the spin-1 Ising model in a crystal field), as well as the two-dimensional Ising model on a square lattice. Our results show that in one dimension, cumulant-based methods can produce precise estimates of the critical points at a low computational cost, and show promise for two-dimensional systems as well.
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ABSTRACT: We study universality in three-dimensional Ising spin glasses by large-scale Monte Carlo simulations of the Edwards-Anderson Ising spin glass for several choices of bond distributions, with particular emphasis on Gaussian and bimodal interactions. A finite-size scaling analysis suggests that three-dimensional spin glasses obey universality.Physical Review B 03/2006; 73(22). DOI:10.1103/PhysRevB.73.224432 · 3.74 Impact Factor
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ABSTRACT: Monte Carlo (MC) simulations and finite-size scaling analysis have been carried out to study the critical behavior in a submonolayer two-dimensional gas of repulsive linear $k$-mers on a triangular lattice at coverage $k/(2k+1)$. A low-temperature ordered phase, characterized by a repetition of alternating files of adsorbed $k$-mers separated by $k+1$ adjacent empty sites, is separated from the disordered state by a order-disorder phase transition occurring at a finite critical temperature, $T_c$. The MC technique was combined with the recently reported Free Energy Minimization Criterion Approach (FEMCA), [F. Rom\'a et al., Phys. Rev. B, 68, 205407, (2003)], to predict the dependence of the critical temperature of the order-disorder transformation. The dependence on $k$ of the transition temperature, $T_c(k)$, observed in MC is in qualitative agreement with FEMCA. In addition, an accurate determination of the critical exponents has been obtained for adsorbate sizes ranging between $k=1$ and $k=3$. For $k>1$, the results reveal that the system does not belong to the universality class of the two-dimensional Potts model with $q=3$ ($k=1$, monomers). Based on symmetry concepts, we suggested that the behavior observed for $k=1, 2$ and 3 could be generalized to include larger particle sizes ($k \geq 2$). Comment: 17 pages, 13 figuresPhysical review. B, Condensed matter 06/2006; 74(15). DOI:10.1103/PhysRevB.74.155418 · 3.66 Impact Factor