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arXiv:1012.5004 · Physical Review B

Free energy in a magnetic field and the universal scaling equation of state for the three-dimensional Ising model

P. Butera, M. Pernici

Published Feb 25, 2011• 17 citations
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Abstract

We have substantially extended the high-temperature and low-magnetic-field (and the related low-temperature and high-magnetic-field) bivariate expansions of the free energy for the conventional three-dimensional Ising model and for a variety of other spin systems generally assumed to belong to the same critical universality class. In particular, we have also derived the analogous expansions for the Ising models with spin $s=1,3/2,\dots{}$ and for the lattice Euclidean scalar-field theory with quartic self-interaction, on the simple-cubic and the body-centered-cubic lattices. Our bivariate high-temperature expansions, which extend through 24th order, enable us to compute, through the same order, all higher derivatives of the free energy with respect to the field, namely, all higher susceptibilities. These data make more accurate checks possible, in critical conditions, both of the scaling and the universality properties with respect to the lattice and the interaction structure and also help to improve an approximate parametric representation of the critical equation of state for the three-dimensional Ising model universality class.

DOI
10.1103/physrevb.83.054433

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Universality classes encompass the analogous thermodynamic behavior of unlike physical systems, at different spatial dimensions d , in the vicinity of their critical point. Critical exponents define these classes, with the Ising model being the outstanding prototype that elucidates the differences from the mean-field category, believed to be valid above a critical dimension only. Here, in apparent striking contradiction to the Ising universality class, we demonstrate that the critical behavior of a finite Ising system of N spins in d = 3 obeys mean-field Landau theory in the vicinity of its critical point, with classical critical exponents. Yet, when expressed in terms of the linear size L of the system, the free energy unveils its proper finite-size scaling form, from which the thermodynamic limit critical temperature T c and the Ising critical exponents ν , γ and β can be identified. We find that the larger the size L , the smaller the mean-field region, shrinking to zero in the thermodynamic limit. These conclusions are achieved via the use of an alternative data analysis of Monte Carlo simulations of a three-dimensional Ising model that allows for the evaluation of the free energy per spin f = f ( T , m ; L ) and of the coexistence curve, or spontaneous magnetization at zero magnetic field, m coex = m ( T ; L ) as functions of temperature T and magnetization per spin m = M / N . Our results suggest a revision of the role of mean-field theory in the elucidation of critical phenomena.

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We have substantially extended the high-temperature and low-magnetic-field (and the related low-temperature and high-magnetic-field) bivariate expansions of the free energy for the conventional three-dimensional Ising model and for a variety of other spin systems generally assumed to belong to the same critical universality class. In particular, we have also derived the analogous expansions for the Ising models with spin s=1,3/2,…s=1,3/2,\dots{}s=1,3/2,… and for the lattice Euclidean scalar-field theory with quartic self-interaction, on the simple-cubic and the body-centered-cubic lattices. Our bivariate high-temperature expansions, which extend through 24th order, enable us to compute, through the same order, all higher derivatives of the free energy with respect to the field, namely, all higher susceptibilities. These data make more accurate checks possible, in critical conditions, both of the scaling and the universality properties with respect to the lattice and the interaction structure and also help to improve an approximate parametric representation of the critical equation of state for the three-dimensional Ising model universality class.
High-temperature expansions are presently the only viable approach to the numerical calculation of the higher susceptibilities for the spin and the scalar-field models on high-dimensional lattices. The critical amplitudes of these quantities enter into a sequence of universal amplitude ratios that determine the critical equation of state. We have obtained a substantial extension, through order 24, of the high-temperature expansions of the free energy (in presence of a magnetic field) for the Ising models with spin s≥1/2 and for the lattice scalar-field theory with quartic self-interaction on the simple-cubic and the body-centered-cubic lattices in four, five, and six spatial dimensions. A numerical analysis of the higher susceptibilities obtained from these expansions yields results consistent with the widely accepted ideas, based on the renormalization group and the constructive approach to Euclidean quantum field theory, concerning the no-interaction ("triviality") property of the continuum (scaling) limit of spin-s Ising and lattice scalar-field models at and above the upper critical dimensionality.
The high-temperature expansion coefficients of the ordinary and the higher susceptibilities of the spin-1/2 nearest-neighbor Ising model are calculated exactly up to the 20th order for the general d-dimensional (hyper)simple-cubical lattices. These series are analyzed to study the dependence of critical parameters on the lattice dimensionality. Using the general d expression of the ordinary susceptibility, we have more than doubled the length of the existing series expansion of the critical temperature in powers of 1/d.
We construct a single reduced free-energy generator for the simple-cubic three-dimensional Ising model by finite gluing of complementary local relations. With the convention Φ=+log⁡Z/V\Phi=+\log Z/VΦ=+logZ/V, the final object is Φ3(K,H,L)=Φ0(K)+ΔΦH(K,H)+ΦFSS(K,H,L),\Phi_3(K,H,L)=\Phi_0(K)+\Delta\Phi_H(K,H)+\Phi_{\rm FSS}(K,H,L),Φ3​(K,H,L)=Φ0​(K)+ΔΦH​(K,H)+ΦFSS​(K,H,L), from which M=∂HΦ3M=\partial_H\Phi_3M=∂H​Φ3​, χ=∂H2Φ3\chi=\partial_H^2\Phi_3χ=∂H2​Φ3​, and UK=∂KΦ3U_K=\partial_K\Phi_3UK​=∂K​Φ3​ are generated. The construction covers the high- and low-temperature regimes, the critical region, finite magnetic field, strong-field saturation, periodic-film finite size, and simultaneous finite HHH and finite LLL. The materials and operations are separated explicitly. A diagnostic dimensional stage uses d=1,2,4d=1,2,4d=1,2,4 anchors in μc=−log⁡tanh⁡Kc\mu_c=-\log\tanh K_cμc​=−logtanhKc​ to transport critical coordinates and positive film amplitudes toward d=3d=3d=3. The practical 3D generator then protects established 3D values of KcK_cKc​, critical exponents, selected amplitude ratios, and scalar zero-field critical data. The zero-field term joins truncated simple-cubic high- and low-temperature series through a singular critical patch with C2C^2C2 matching. The inner magnetic increment joins an O(K2)O(K^2)O(K2) high-temperature field expansion, a minimal 3D parametric critical free energy fixed only by the basic 3D anchors, and a low-temperature/cluster continuation using fixed quintic switches in KKK. A recovered z4z^4z4 strong-field expansion is integrated to an absolute free energy. The final inner-to-strong crossover is performed by constructing a monotone beta-CDF magnetization whose endpoint values and integral reproduce the two endpoint free energies, and then integrating that derivative; thus convexity is enforced at the free-energy level rather than by clipping observables. The periodic-film excess is built from dimensionally weighted 2D and leading 4D film surfaces in common correlation-length coordinates, with an HHH-independent residual stabilizer that preserves the H=0H=0H=0 axis. A previously reconstructed full 3D parametric equation-of-state benchmark and the full 3D finite-size scaling function are withheld from construction. The pure dimensional stage gives Kc(3)=0.22301549K_c(3)=0.22301549Kc​(3)=0.22301549 and sub-percent estimates of β\betaβ, δ\deltaδ, and ν\nuν. After 3D calibration, the final implementation passes 29/29 stated structural and numerical checks. In the withheld low-field window ∣K−Kc∣≤0.01|K-K_c|\le0.01∣K−Kc​∣≤0.01, H≤0.1H\le0.1H≤0.1, the magnetization has RMS relative error 1.21%1.21\%1.21% and maximum error 4.29%4.29\%4.29%. The result is not an exact lattice solution; it is a finite, auditable reconstruction in which the input anchors, switch intervals, matching conditions, numerical tolerances, and validation data are kept distinct.
We test the Gluing Method on the square-lattice two-dimensional Ising model by reconstructing global free-energy generators from complementary local relations. The emphasis is kept explicit throughout: the inputs used as \emph{gluing material} are separated from quantities obtained \emph{after gluing} and from data reserved for validation. At zero magnetic field, Kramers--Wannier duality is absorbed into the coordinate τ=log⁡sinh⁡2K\tau=\log\sinh 2Kτ=logsinh2K, so that the reduced free energy decomposes as Φ(τ)=τ/2+G(∣τ∣)\Phi(\tau)=\tau/2+G(|\tau|)Φ(τ)=τ/2+G(∣τ∣). The gluing materials are then only a critical logarithmic local expansion and a low-order high-temperature outer expansion; the low-temperature sector is generated by duality rather than fitted independently. Positive flat weights preserve the prescribed critical asymptotics to every finite algebraic order. With one crossover parameter fixed from three scalar anchors, the resulting bulk free energy agrees with the exact Onsager benchmark at a maximum relative error of 1.915×10−51.915\times10^{-5}1.915×10−5 on the tested interval. Internal energy and specific heat are derived from the same glued generator. Zero-field finite-size corrections are organized separately as one-variable scaling functions multiplying L−2,L−4,…L^{-2},L^{-4},\ldotsL−2,L−4,…. At nonzero field in the thermodynamic limit, the gluing materials are the established high-temperature, critical, and low-temperature local expansions of the universal Ising magnetic scaling free energy Φmag(η)\Phi_{\rm mag}(\eta)Φmag​(η), η=m/∣h∣8/15\eta=m/|h|^{8/15}η=m/∣h∣8/15. The low-temperature input contains the ∣h∣|h|∣h∣ cusp associated with spontaneous symmetry breaking. Mapping the glued scaling free energy back to the square lattice through nonlinear scaling fields yields a single generator for the magnetic free-energy increment, magnetization, and susceptibility. The construction reproduces high-temperature linear response, the critical isotherm M∝∣H∣1/15M\propto |H|^{1/15}M∝∣H∣1/15, and the two one-sided limits M(H→0±)=±M0M(H\to0^\pm)=\pm M_0M(H→0±)=±M0​ without separately fitting magnetization branches. Finally, finite size and finite field are combined in the scaling variables xt=Lmx_t=Lmxt​=Lm and xh=L15/8hx_h=L^{15/8}hxh​=L15/8h. The two protected one-dimensional axes, a central mixed-response expansion, a low-temperature spectral-rounding relation, and a massive outer description are glued into a two-variable scaling surface Yglue(xt,xh)\mathcal Y_{\rm glue}(x_t,x_h)Yglue​(xt​,xh​). The resulting lattice free energy closes in the compact form ΦLglue(K,H)=Φlatglue(K,H)+Yglue(Lm,L15/8h)−Ybulk(Lm,L15/8h)L2+O(L−4)\Phi_L^{glue}(K,H)=\Phi_{lat}^{glue}(K,H)+\frac{\mathcal{Y}_{glue}(Lm,L^{15/8}h)- Y_{bulk}(Lm,L^{15/8}h)}{L^2}+O(L^{-4})ΦLglue​(K,H)=Φlatglue​(K,H)+L2Yglue​(Lm,L15/8h)−Ybulk​(Lm,L15/8h)​+O(L−4) where m=m(r,H)m=m(r,H)m=m(r,H) and h=h(r,H)h=h(r,H)h=h(r,H) are the nonlinear scaling fields and YbulkY_{\rm bulk}Ybulk​ is the thermodynamic singular scaling surface expressed in the same variables. Thus all finite-size/finite-field information is carried by the excess Yglue−Ybulk\mathcal Y_{\rm glue}-Y_{\rm bulk}Yglue​−Ybulk​, while the thermodynamic lattice free energy remains the previously constructed Φlatglue\Phi_{\rm lat}^{\rm glue}Φlatglue​. On eleven transfer-matrix points withheld from the construction, the blind free-energy test gives RMS absolute error 3.90×10−43.90\times10^{-4}3.90×10−4 and maximum absolute error 9.55×10−49.55\times10^{-4}9.55×10−4 in Y\mathcal YY. Differentiation of the same free energy produces finite-size magnetization and susceptibility; magnetization is especially stable on the low-temperature side, while the second derivative is more sensitive in crossover regions.
Universality classes encompass the analogous thermodynamic behavior of unlike physical systems, at different spatial dimensions ddd, in the vicinity of their critical point. Critical exponents define these classes, with the Ising model being the outstanding prototype that elucidates the differences from the mean-field category, believed to be valid above a critical dimension only. Here, in apparent striking contradiction to the Ising universality class, we demonstrate that the critical behavior of a finite Ising system of NNN spins in d=3d = 3d=3 obeys mean-field Landau theory in the vicinity of its critical point, with classical critical exponents. Yet, when expressed in terms of the linear size LLL of the system, the free energy unveils its proper finite-size scaling form, from which the thermodynamic limit critical temperature TcT_cTc​ and the Ising critical exponents ννν, γγγ and βββ can be identified. We find that the larger the size LLL, the smaller the mean-field region, shrinking to zero in the thermodynamic limit. These conclusions are achieved via the use of an alternative approach to collect data from a Monte Carlo simulation of a three-dimensional Ising model that allows for the evaluation of the free energy per spin f=f(T,m;L)f = f(T,m;L)f=f(T,m;L) and of the coexistence curve, or spontaneous magnetization at zero magnetic field, mcoex=m(T;L)m_{\rm coex} = m(T;L)mcoex​=m(T;L) as functions of temperature TTT and magnetization per spin m=M/Nm = M/Nm=M/N. Our results suggest a revision of the role of mean-field theory in the elucidation of critical phenomena.
Universality classes encompass the analogous thermodynamic behavior of unlike physical systems, at different spatial dimensions d , in the vicinity of their critical point. Critical exponents define these classes, with the Ising model being the outstanding prototype that elucidates the differences from the mean-field category, believed to be valid above a critical dimension only. Here, in apparent striking contradiction to the Ising universality class, we demonstrate that the critical behavior of a finite Ising system of N spins in d = 3 obeys mean-field Landau theory in the vicinity of its critical point, with classical critical exponents. Yet, when expressed in terms of the linear size L of the system, the free energy unveils its proper finite-size scaling form, from which the thermodynamic limit critical temperature T c and the Ising critical exponents ν , γ and β can be identified. We find that the larger the size L , the smaller the mean-field region, shrinking to zero in the thermodynamic limit. These conclusions are achieved via the use of an alternative data analysis of Monte Carlo simulations of a three-dimensional Ising model that allows for the evaluation of the free energy per spin f = f ( T , m ; L ) and of the coexistence curve, or spontaneous magnetization at zero magnetic field, m coex = m ( T ; L ) as functions of temperature T and magnetization per spin m = M / N . Our results suggest a revision of the role of mean-field theory in the elucidation of critical phenomena.