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:0803.4255 · Journal of Physics A Mathematical and Theoretical

Alternative description of the 2D Blume–Capel model using Grassmann algebra

Maxime Clusel, Jean-Yves Fortin, V. N. Plechko

Published Sep 11, 2008• 16 citations
arXivHTMLPDF

Abstract

We use Grassmann algebra to study the phase transition in the two-dimensional ferromagnetic Blume-Capel model from a fermionic point of view. This model presents a phase diagram with a second order critical line which becomes first order through a tricritical point, and was used to model the phase transition in specific magnetic materials and liquid mixtures of He$^3$-He$^4$. In particular, we are able to map the spin-1 system of the BC model onto an effective fermionic action from which we obtain the exact mass of the theory, the condition of vanishing mass defines the critical line. This effective action is actually an extension of the free fermion Ising action with an additional quartic interaction term. The effect of this term is merely to render the excitation spectrum of the fermions unstable at the tricritical point. The results are compared with recent numerical Monte-Carlo simulations.

DOI
10.1088/1751-8113/41/40/405004

Related papers

Phase diagram for the spin-3/2 quantum ferromagnetic Blume-Capel model in a transverse crystal field: an approximation at the mean-field level

Carvalho, D. C., J. A. Plascak

We investigate the phase diagram for the spin-$3/2$ ferromagnetic Blume-Capel model in a transverse crystal field using the standard mean-field approximation within the framework of Bogoliubov inequality for free energy. We draw a very rich phase diagram with first- and second-order transition lines; tricritical and tetracritical points; critical endpoint of order 2 and double critical endpoint. Additionally, the behaviour of magnetisation as a function of temperature over a wide range of values of both longitudinal and transverse crystal fields is also analysed. To the best of our knowledge, this quantum spin model has only been studied employing an effective field theory, which in turn was not able to characterise completely the multicritical phenomena in its phase diagram, because that procedure is not based on free energy. Thus, our findings on the phase diagram for the present model are novel as they have been not previously reported.

arXiv:2007.07763•Jul 15, 2020•0 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

Renormalization group analysis of Blume–Capel ferromagnet on two-dimensional triangular lattice

M. N. Najafi

The Blume-Caple model on the triangular lattice is considered in this paper. Using the (perturbative) real-space renormalization group (RSRG) technique we analytically obtain the running of the parameters of the model, i.e., the exchange integral J and the crystal field Δ. The observations show that the first-order calculations are insufficient to capture the tricritical phenomena, and the RG analysis based on second-order expansion of the partition function show qualitative agreement with the previous works. This reveals that the second-order perturbation works appropriately, to obtain the qualitative features of the phase diagram of the model, e.g., a tricritical point which is unstable towards the Ising critical point from one side (which forms the line of second order, separating the ferromagnetic and paramagnetic phases) and the critical point at and from the other side (which forms the line of first-order transition separating ferromagnetic and zero state phases). This analysis shows also that there is another unstable critical point at , with the RG eigenvalue in the Δ-direction (towards S = 0 zero state and paramagnetic phases).

Physica Scripta•Jul 9, 2015•0 citations

Short-time dynamics for the spin- 32 Blume-Capel model

B. C. S. Grandi, W. Figueiredo

We employed Monte Carlo simulations and short-time dynamic scaling to determine the static and dynamic critical exponents for the generalized two-dimensional Blume-Capel model of spin-3/2. We showed that the critical behavior at the second-order phase-transition line between the paramagnetic and ferromagnetic phases is in the same universality class of the two-dimensional Ising model. However, at the double critical end point, which is present in the phase diagram of the model, the critical exponent beta , associated to the order parameter, is different from that of the Ising model.

Physical Review E•Nov 16, 2004•29 citations

Correlation of fluctuations in the Blume-Capel model

K. G. Moter, R. S. Fishman

Using a 1/z expansion we study the correlation of fluctuations in a spin-1 Ising model with anisotropy. Previously, the 1/z expansion has been applied only to the second-order phase transitions of ferromagnets and granular superconductors. By imposing conditions on the free energy F and the parameter x=1-〈${\mathit{S}}_{1\mathit{z}}^{2}$〉, we use the 1/z technique to examine both the first- and second-order parts of the Blume-Capel phase diagram. In agreement with many numerical simulations and experiments on $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures, the value of x at the tricritical point is not shifted from its mean-field value of 2/3 to order 1/z. On the other hand, the tricritical values of the anisotropy parameter A and of the transition temperature ${\mathit{T}}_{\mathit{c}}$ are altered by the correlation of fluctuations. At least to order 1/z, all the derivatives of ${\mathit{T}}_{\mathit{c}}$ are continuous across the tricritical point.

Physical review. B, Condensed matter•Mar 1, 1992•4 citations

Phase Diagram of the Antiferromagnetic Blume-Capel Model on Triangular Lattice

Sojeong Park, Wooseop Kwak

We perform Monte-Carlo simulations of the anti-ferromagnetic (AF) spin-1 Blume- Capel (BC) model and the AF Ising model on triangular lattice. We estimate the exact critical magnetic fields for both models at zero temperature using the Wang-Landau sampling method. We also show the phase diagrams and the critical lines for the models using the joint density functions. We find that the shapes of critical lines for the models are identical, but the phase transitions across the critical lines are different.

Journal of Physics Conference Series•Aug 1, 2016•3 citations
We use Grassmann algebra to study the phase transition in the two-dimensional ferromagnetic Blume-Capel model from a fermionic point of view. This model presents a phase diagram with a second order critical line which becomes first order through a tricritical point, and was used to model the phase transition in specific magnetic materials and liquid mixtures of He3^33-He4^44. In particular, we are able to map the spin-1 system of the BC model onto an effective fermionic action from which we obtain the exact mass of the theory, the condition of vanishing mass defines the critical line. This effective action is actually an extension of the free fermion Ising action with an additional quartic interaction term. The effect of this term is merely to render the excitation spectrum of the fermions unstable at the tricritical point. The results are compared with recent numerical Monte-Carlo simulations.
We investigate the phase diagram for the spin-3/23/23/2 ferromagnetic Blume-Capel model in a transverse crystal field using the standard mean-field approximation within the framework of Bogoliubov inequality for free energy. We draw a very rich phase diagram with first- and second-order transition lines; tricritical and tetracritical points; critical endpoint of order 2 and double critical endpoint. Additionally, the behaviour of magnetisation as a function of temperature over a wide range of values of both longitudinal and transverse crystal fields is also analysed. To the best of our knowledge, this quantum spin model has only been studied employing an effective field theory, which in turn was not able to characterise completely the multicritical phenomena in its phase diagram, because that procedure is not based on free energy. Thus, our findings on the phase diagram for the present model are novel as they have been not previously reported.
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.
The Blume-Caple model on the triangular lattice is considered in this paper. Using the (perturbative) real-space renormalization group (RSRG) technique we analytically obtain the running of the parameters of the model, i.e., the exchange integral J and the crystal field Δ. The observations show that the first-order calculations are insufficient to capture the tricritical phenomena, and the RG analysis based on second-order expansion of the partition function show qualitative agreement with the previous works. This reveals that the second-order perturbation works appropriately, to obtain the qualitative features of the phase diagram of the model, e.g., a tricritical point which is unstable towards the Ising critical point from one side (which forms the line of second order, separating the ferromagnetic and paramagnetic phases) and the critical point at and from the other side (which forms the line of first-order transition separating ferromagnetic and zero state phases). This analysis shows also that there is another unstable critical point at , with the RG eigenvalue in the Δ-direction (towards S = 0 zero state and paramagnetic phases).
We employed Monte Carlo simulations and short-time dynamic scaling to determine the static and dynamic critical exponents for the generalized two-dimensional Blume-Capel model of spin-3/2. We showed that the critical behavior at the second-order phase-transition line between the paramagnetic and ferromagnetic phases is in the same universality class of the two-dimensional Ising model. However, at the double critical end point, which is present in the phase diagram of the model, the critical exponent beta , associated to the order parameter, is different from that of the Ising model.
Using a 1/z expansion we study the correlation of fluctuations in a spin-1 Ising model with anisotropy. Previously, the 1/z expansion has been applied only to the second-order phase transitions of ferromagnets and granular superconductors. By imposing conditions on the free energy F and the parameter x=1-〈S1z2{\mathit{S}}_{1\mathit{z}}^{2}S1z2​〉, we use the 1/z technique to examine both the first- and second-order parts of the Blume-Capel phase diagram. In agreement with many numerical simulations and experiments on $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures, the value of x at the tricritical point is not shifted from its mean-field value of 2/3 to order 1/z. On the other hand, the tricritical values of the anisotropy parameter A and of the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ are altered by the correlation of fluctuations. At least to order 1/z, all the derivatives of Tc{\mathit{T}}_{\mathit{c}}Tc​ are continuous across the tricritical point.
We perform Monte-Carlo simulations of the anti-ferromagnetic (AF) spin-1 Blume- Capel (BC) model and the AF Ising model on triangular lattice. We estimate the exact critical magnetic fields for both models at zero temperature using the Wang-Landau sampling method. We also show the phase diagrams and the critical lines for the models using the joint density functions. We find that the shapes of critical lines for the models are identical, but the phase transitions across the critical lines are different.