Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.20982

The Blume-Capel Model as a $ϕ^6$ Field Theory: A Monte Carlo and Flow Model Study

Ejaaz Merali, Ryan Larson, Ehsan Khatami, Richard T. Scalettar

Published Oct 14, 2026• 0 citations
arXivHTMLPDF

Abstract

We report Monte Carlo simulations of a continuous spin variant of the Blume-Capel model. We compute the phase diagrams in the crystal-field-temperature plane, as a function of the softness of the spins. We show that normalizing flows, a class of generative models capable of direct sampling without autocorrelation, are effective samplers across the phase diagram, including in the first-order transition region and in the vicinity of the tricritical point. Finite-size scaling analyses of flow samples locate the first- and second-order transitions as well as the tricritical point with accuracy comparable to canonical and extended-ensemble Monte Carlo.

DOI
10.48550/arxiv.2610.20982

Related papers

Critical and tricritical singularities from small-scale Monte Carlo simulations: the Blume–Capel model in two dimensions

Leïla Moueddene, Nikolaos G. Fytas, Yurij V. Holovatch +2 more

Abstract We show that accurate insights into the critical properties of the Blume–Capel model at two dimensions can be deduced from Monte Carlo simulations, even for small system sizes, when one analyses the behaviour of the zeros of the partition function. The phase diagram of the model displays a line of second-order phase transitions ending at a tricritical point, then a line of first-order transitions. We concentrate on critical and tricritical properties and compare the accuracy achieved via standard finite-size scaling of thermodynamic quantities with that from the zeros analysis. This latter analysis showcases spectacular precision, even for systems as small as 64 spins. We also show that the zeros are very sensitive to subtle crossover effects.

arXiv:2401.02720•Journal of Statistical Mechanics Theory and Experiment•Feb 9, 2024•30 citations

Tricriticality and the Blume-Capel model: A Monte Carlo study within the microcanonical ensemble

Markus Deserno

The microcanonical partition function ${\ensuremath{\Omega}}_{\ensuremath{\alpha},N}(E)$ of a three-dimensional ferromagnetic simple cubic Blume-Capel model is calculated for several coupling ratios $\ensuremath{\alpha}$ close to and on the first-order side of the tricritical point. To this end a single Monte Carlo simulation of a suitably extended partition function for systems of $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ spins with $L\ensuremath{\in}{8,10,12,14,18}$ was performed. A finite system analog of the latent heat is introduced in order to define a tricritical point in finite systems as well. An empirical scaling of its coordinates to the thermodynamic limit yields ${\ensuremath{\alpha}}^{(t)}=2.84479\ifmmode\pm\else\textpm\fi{}0.00030$ for the tricritical coupling ratio, and ${k}_{B}{T}^{(t)}/J=1.4182\ifmmode\pm\else\textpm\fi{}0.0055$ for the tricritical transition temperature. The results are compared with values obtained on fcc lattices.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Nov 1, 1997•75 citations

Tricriticality and finite-size scaling in the triangular Blume-Capel ferromagnet

Dimitrios Mataragkas, Alexandros Vasilopoulos, Nikolaos G. Fytas +1 more

We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.

arXiv:2411.11689•Physical Review Research•Feb 27, 2025•13 citations

Parallel multicanonical study of the three-dimensional Blume-Capel model

Johannes Zierenberg, Nikolaos G. Fytas, Wolfhard Janke E

We study the thermodynamic properties of the three-dimensional Blume-Capel model on the simple cubic lattice by means of computer simulations. In particular, we implement a parallelized variant of the multicanonical approach and perform simulations by keeping a constant temperature and crossing the phase boundary along the crystal-field axis. We obtain numerical data for several temperatures in both the first- and second-order regime of the model. Finite-size scaling analyses provide us with transition points and the dimensional scaling behavior in the numerically demanding first-order regime, as well as a clear verification of the expected Ising universality in the respective second-order regime. Finally, we discuss the scaling behavior in the vicinity of the tricritical point.

arXiv:1502.07214•Physical Review E•Mar 18, 2015•38 citations

Finite-size scaling study of the two-dimensional Blume-Capel model

Paul D. Beale

The phase diagram of the two-dimensional Blume-Capel model is investigated by using the technique of phenomenological finite-size scaling. The location of the tricritical point and the values of the critical and tricritical exponents are determined. The location of the tricritical point (${T}_{t}$=0.610\ifmmode\pm\else\textpm\fi{}0.005, ${D}_{t}$=1.9655\ifmmode\pm\else\textpm\fi{}0.0010) is well outside the error bars for the value quoted in previous Monte Carlo simulations but in excellent agreement with more recent Monte Carlo renormalization-group results. The values of the critical and tricritical exponents, with the exception of the leading thermal tricritical exponent, are in excellent agreement with previous calculations, conjectured values, and Monte Carlo renormalization-group studies.

Physical review. B, Condensed matter•Feb 1, 1986•119 citations

Pseudo Transitions in the Finite-Size Blume-Capel Model

Shi, Lei, Wei Liu, Li, Xiang +4 more

This article investigates the pseudo transitions of the Blume-Capel model on two-dimensional finite-size lattices. By employing the Wang-Landau sampling method and microcanonical inflection point analysis, we identified the positions of phase transitions as well as higher-order phase transitions. Through Metropolis sampling and canonical ensemble analysis, we determined the geometric characteristics of the system at these transition points. When the crystal field parameter $D$ exceeds 1.965, crossing the tricritical point, no third-order dependent phase transition is observed. However, a fourth-order independent transition was identified in the high-temperature region, and through Metropolis sampling analysis, a phase transition from the ordered paramagnetic phase to the disordered paramagnetic phase was confirmed, enhancing the phase diagram. Furthermore, the positions of the third-order phase transition obtained from both microcanonical and canonical analyses are consistent and mutually corroborative. We speculate that third-order dependent transitions vanish in the presence of strong first-order phase transitions.

arXiv:2411.01743•Nov 4, 2024•0 citations
The Blume-Capel Model as a ϕ6ϕ^6ϕ6 Field Theory: A Monte Carlo and Flow Model Study
We report Monte Carlo simulations of a continuous spin variant of the Blume-Capel model. We compute the phase diagrams in the crystal-field-temperature plane, as a function of the softness of the spins. We show that normalizing flows, a class of generative models capable of direct sampling without autocorrelation, are effective samplers across the phase diagram, including in the first-order transition region and in the vicinity of the tricritical point. Finite-size scaling analyses of flow samples locate the first- and second-order transitions as well as the tricritical point with accuracy comparable to canonical and extended-ensemble Monte Carlo.
Abstract We show that accurate insights into the critical properties of the Blume–Capel model at two dimensions can be deduced from Monte Carlo simulations, even for small system sizes, when one analyses the behaviour of the zeros of the partition function. The phase diagram of the model displays a line of second-order phase transitions ending at a tricritical point, then a line of first-order transitions. We concentrate on critical and tricritical properties and compare the accuracy achieved via standard finite-size scaling of thermodynamic quantities with that from the zeros analysis. This latter analysis showcases spectacular precision, even for systems as small as 64 spins. We also show that the zeros are very sensitive to subtle crossover effects.
The microcanonical partition function ${\ensuremath{\Omega}}_{\ensuremath{\alpha},N}(E)$ of a three-dimensional ferromagnetic simple cubic Blume-Capel model is calculated for several coupling ratios $\ensuremath{\alpha}$ close to and on the first-order side of the tricritical point. To this end a single Monte Carlo simulation of a suitably extended partition function for systems of $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ spins with $L\ensuremath{\in}{8,10,12,14,18}$ was performed. A finite system analog of the latent heat is introduced in order to define a tricritical point in finite systems as well. An empirical scaling of its coordinates to the thermodynamic limit yields ${\ensuremath{\alpha}}^{(t)}=2.84479\ifmmode\pm\else\textpm\fi{}0.00030$ for the tricritical coupling ratio, and ${k}_{B}{T}^{(t)}/J=1.4182\ifmmode\pm\else\textpm\fi{}0.0055$ for the tricritical transition temperature. The results are compared with values obtained on fcc lattices.
We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.
We study the thermodynamic properties of the three-dimensional Blume-Capel model on the simple cubic lattice by means of computer simulations. In particular, we implement a parallelized variant of the multicanonical approach and perform simulations by keeping a constant temperature and crossing the phase boundary along the crystal-field axis. We obtain numerical data for several temperatures in both the first- and second-order regime of the model. Finite-size scaling analyses provide us with transition points and the dimensional scaling behavior in the numerically demanding first-order regime, as well as a clear verification of the expected Ising universality in the respective second-order regime. Finally, we discuss the scaling behavior in the vicinity of the tricritical point.
The phase diagram of the two-dimensional Blume-Capel model is investigated by using the technique of phenomenological finite-size scaling. The location of the tricritical point and the values of the critical and tricritical exponents are determined. The location of the tricritical point (Tt{T}_{t}Tt​=0.610\ifmmode\pm\else\textpm\fi{}0.005, Dt{D}_{t}Dt​=1.9655\ifmmode\pm\else\textpm\fi{}0.0010) is well outside the error bars for the value quoted in previous Monte Carlo simulations but in excellent agreement with more recent Monte Carlo renormalization-group results. The values of the critical and tricritical exponents, with the exception of the leading thermal tricritical exponent, are in excellent agreement with previous calculations, conjectured values, and Monte Carlo renormalization-group studies.
This article investigates the pseudo transitions of the Blume-Capel model on two-dimensional finite-size lattices. By employing the Wang-Landau sampling method and microcanonical inflection point analysis, we identified the positions of phase transitions as well as higher-order phase transitions. Through Metropolis sampling and canonical ensemble analysis, we determined the geometric characteristics of the system at these transition points. When the crystal field parameter DDD exceeds 1.965, crossing the tricritical point, no third-order dependent phase transition is observed. However, a fourth-order independent transition was identified in the high-temperature region, and through Metropolis sampling analysis, a phase transition from the ordered paramagnetic phase to the disordered paramagnetic phase was confirmed, enhancing the phase diagram. Furthermore, the positions of the third-order phase transition obtained from both microcanonical and canonical analyses are consistent and mutually corroborative. We speculate that third-order dependent transitions vanish in the presence of strong first-order phase transitions.