The Blume-Capel Model as a φ6 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
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 UL 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 λ/J=0.5 , resulting in a critical temperature at Tc/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 λ , which tunes the amount the degrees of freedom deviate from their discrete values ±1 . The limit of large λ recovers the Onsager value Tc/J=2.269 . The red circles are Tc/J while the black triangles are normalized by ⟨ϕ2⟩ . The value of Tc at the smallest λ/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(ϕ) for four values of λ in Eq. 1 , the soft-spin variant of the Ising model. With increasing λ the wells at ϕ=±1 become deeper. (b) shows the potential of Eq. 3 , the soft-spin variant of the Blume–Capel model, for fixed vacancy potential δ=1 and varying α , the parameter which tends to confine ϕ to ϕ=0,±1 . (c) shows fixed α=16 at several δ . Increasing δ favors vacancies ( ϕ=0 ). In both (b) and (c), the lower curvature of the ϕ=0 well relative to the ϕ=±1 wells is evident. This blurs the mapping to the Blume–Capel limit. See text.
Figure 4: Phase diagram of the ϕ6 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 Δ in order to spread out the different α 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 Tc/J≃2.269 to which the critical temperature of the discrete spin Blume–Capel model converges as δ→−∞ . Error bars are smaller than the markers.
Figure 5: Convergence of the position of the soft-spin tricritical point Tt/J (panel (a)) and Δt/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: δc 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 δ=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 Uρ of the density restricted to the zero-skew curve for both flows and MCMC. Upper panels: Temperature locations Tt of the tricritical points.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Figure A1: Tc(λ) for the ϕ4 model within mean field theory. Although Tc/⟨ϕ2⟩Tc , is equal to 4J for all λ , the un-normalized Tc shown here has a non-trivial λ 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 – δ/J plane of the continuous spin version of the Blume–Capel model for α/J=1,2,4,8 . The positions of the tricritical points are shown as solid circle markers.
α/J
Tt/J
δt/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.
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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
Learned field transformations may help address ubiquitous critical slowing down and signal-to-noise problems in lattice field theory. In the context of an annealed sequence of distributions, field transformations are defined by integrating flow fields that exactly solve a local transport problem. These proceedings discuss a new Monte Carlo approach to evaluating these flow fields, which can then be used directly in such contexts or as a means of generating unbiased training data for machine learning approaches. By defining the Monte Carlo estimator using coupled Langevin noise, the statistical noise in the required integrals is significantly mitigated. Demonstrations of the method include a U(1) transport problem and an SU(N) glueball correlator.
Michael S. Albergo, Gurtej Kanwar
Society of Fellows, Harvard University 78 Mount Auburn St, Cambridge, MA United States · Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh, EH9 3FD Edinburgh, United Kingdom
Generating independent, equilibrium samples of molecular systems at scale remains a central obstacle in computational statistical mechanics. Boltzmann Generators address this by pairing a generative model with importance sampling to obtain consistent samples from the target distribution. We introduce Normalizing Flow Flow Maps (NF2M), which combines the strengths of recent stochastic flow maps with the tractability of classic normalizing flows to build a Boltzmann Generator. Unlike most methods, which correct the generative model only at the end, NF2M reweighs each denoising transition as generation proceeds, avoiding wasted compute on trajectories that are ultimately discarded. At each denoising step, a conditional normalizing flow proposes clean configurations given the current noisy state (a simpler task than sampling directly from the target) and its exact likelihood enables correcting each proposal toward the true denoising transition of the target Boltzmann distribution. This is in contrast to most existing methods, whose likelihoods are approximate or expensive to evaluate, undermining the statistical reliability of the correction. We establish consistency of the corrected transitions and bound how approximation errors propagate through the sampling chain. We evaluate NF2M on peptide systems, demonstrating improved sampling efficiency and sample quality.
Louis Grenioux, RuiKang OuYang, Luhuan Wu
Center for Computational Mathematics, Flatiron Institute, New York, NY, USA · Department of Engineering, University of Cambridge, Cambridge, UK · Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD, USA