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

cond-mat.stat-mechOct 14, 2026

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

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

Organizations: Department of Physics and Astronomy, University of California, Davis, CA 95616, USA · Department of Physics and Astronomy, San José State University, San José, CA 95192 USA

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.

Figures & tables

Figure 1: Critical temperatures are found using the crossings of the Binder parameter

U_{L}

for different linear system sizes

L

. In this representative example for the continuous spin variant of the Ising model [ Eq. 1 ] on the square lattice, we chose

\lambda/J=0.5

, resulting in a critical temperature at

T_{c}/J=3.641

Figure 2: Critical temperatures of the continuous spin variant of the Ising model [ Eq. 1 ] on the square lattice as a function of the parameter

\lambda

, which tunes the amount the degrees of freedom deviate from their discrete values

\pm 1

. The limit of large

\lambda

recovers the Onsager value

T_{c}/J=2.269

. The red circles are

T_{c}/J

while the black triangles are normalized by

\langle\phi^{2}\rangle

. The value of

T_{c}

at the smallest

\lambda/J=0.5

corresponds to the Binder crossing of Fig. 1 . Where error bars aren’t visible they lie inside of the point markers.

Figure 3: (a) The potential

V(\phi)

for four values of

\lambda

in Eq. 1 , the soft-spin variant of the Ising model. With increasing

\lambda

the wells at

\phi=\pm 1

become deeper. (b) shows the potential of Eq. 3 , the soft-spin variant of the Blume–Capel model, for fixed vacancy potential

\delta=1

and varying

\alpha

, the parameter which tends to confine

\phi

\phi=0,\pm 1

. (c) shows fixed

\alpha=16

at several

\delta

. Increasing

\delta

favors vacancies (

\phi=0

). In both (b) and (c), the lower curvature of the

\phi=0

well relative to the

\phi=\pm 1

wells is evident. This blurs the mapping to the Blume–Capel limit. See text.

Figure 4: Phase diagram of the

\phi^{6}

theory of Eq. 3 . The inset zooms in on the region near the tricritical points, where we’ve plotted against the Blume–Capel crystal field

\Delta

in order to spread out the different

\alpha

curves for legibility. Lines for the continuous transition are provided as guides to the eye, while the first order curve is actually made of many densely packed measurements, with only a few markers displayed for legibility. The vertical dashed line is the Ising

T_{c}/J\simeq 2.269

to which the critical temperature of the discrete spin Blume–Capel model converges as

\delta\rightarrow-\infty

. Error bars are smaller than the markers.

Figure 5: Convergence of the position of the soft-spin tricritical point

T_{t}/J

(panel (a)) and

\Delta_{t}/J

(panel (b)) to the discrete Blume–Capel (BC) limits. Error bars are smaller than the markers of the zoomed out plots and are only displayed in the inset. Literature value is from [ 38 ] .

Figure 7:

\delta_{c}

for the first order transition through the

T=0.3J

slice. Extrapolation to the infinite size limit for the flows gives an almost flat curve as expected. MCMC on the other hand is dominated by noise at these scales, resulting in a steeper line.

Figure 8: Upper panels: Binder cumulants of the magnetization taken along the

\delta=1

slice through the second order phase transition estimated by flow models (error bars are smaller than the markers) along with the estimated critical temperature in the thermodynamic limit. The displayed MCMC critical temperature was extracted from a fit using the same temperature grid and sizes for the sake of fair comparison. Lower panels: deviation between the flow and MCMC estimates. The gray ribbon shows one standard error of the MCMC estimate, the thick error bars on the markers are single standard error bars of the flow’s estimate, while the thin bar is the combined error bar following the standard quadrature rule.

Figure 9: Lower panels: Crossings of the Binder cumulant

U_{\rho}

of the density restricted to the zero-skew curve for both flows and MCMC. Upper panels: Temperature locations

T_{t}

of the tricritical points.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure A1:

T_{c}(\lambda)

for the

\phi^{4}

model within mean field theory. Although

T_{c}/\langle\phi^{2}\rangle_{T_{c}}

, is equal to

4J

for all

\lambda

, the un-normalized

T_{c}

shown here has a non-trivial

\lambda

dependence. Interestingly, it has the same shallow minimum found in the (exact) Monte Carlo results of Fig. 2 .

Figure A2: Phase diagram in the

T/J

–

\delta/J

plane of the continuous spin version of the Blume–Capel model for

\alpha/J=1,2,4,8

. The positions of the tricritical points are shown as solid circle markers.

$\alpha/J$	$T_{t}/J$	$\delta_{t}/J$
$1$	$0.683$	$1.857$
$2$	$0.753$	$1.758$
$4$	$0.845$	$1.615$
$8$	$0.959$	$1.431$
$16$	$1.086$	$1.236$
$32$	$1.204$	$1.078$

Appendix

Table A1: Locations of the tricritical points of the soft-spin Blume-Capel predicted by Mean Field Theory.

Phase Transitions

Phase transitions are abrupt changes in collective behavior as temperature, density, noise, or other control parameters vary.

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Diffusion-warm sampling of the XY model enables fast thermalization at scale

Sehmimul Hoque, Roger Melko, Pooya Ronagh

Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada · Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada · Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada+1

Markov Chain Monte Carlo Diffusion Sampling

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We introduce a novel technique for scalable sampling of spin-system states with continuous symmetries using diffusion models. By applying our approach to the XY model, a fundamental continuous-spin model in condensed matter physics, we show that our technique addresses the shortfalls of the Markov chain Monte Carlo (MCMC) in generalization to varying system sizes. More specifically, we show that training a temperature-conditioned diffusion model on smaller-size XY model lattices enables the generation of accurate samples in larger lattice sizes. By tracking physically important observables of the model, such as spin correlations, our experiments demonstrate that diffusion sampling followed by a few MCMC steps reduces the thermalization time by an order of magnitude relative to the standard MCMC with random initialization. Our study provides valuable insight as to how generative models can be used to study continuous-state condensed matter systems at scale.

Abstract

Phase Transitions

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