# Relevant Analytic Spontaneous Magnetization Relation for the Face-Centered-Cubic Ising Lattice

## Abstract

**:**

## 1. Introduction

## 2. Methods and Results

#### 2.1. Expansion of the Callen–Suzuki Identity

#### 2.2. Derivation of the Analytic Relation for the Spontaneous Magnetization

#### 2.3. Spontaneous Magnetization through MC Simulation

## 3. Discussion and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Stanley, H. Introduction to Phase Transitions and Critical Phenomena; International Series of Monographs on Physics; Oxford University Press: New York, NY, USA, 1971. [Google Scholar]
- Ising, E. Contribution to the Theory of Ferromagnetism. Z. Phys.
**1925**, 31, 253–258. [Google Scholar] [CrossRef] - Smits, J.; Stoof, H.T.C.; van der Straten, P. Spontaneous breaking of a discrete time-translation symmetry. Phys. Rev. A
**2021**, 104, 023318. [Google Scholar] [CrossRef] - Onsager, L. Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition. Phys. Rev.
**1944**, 65, 117–149. [Google Scholar] [CrossRef] - Yang, C.N. The Spontaneous Magnetization of a Two-Dimensional Ising Model. Phys. Rev.
**1952**, 85, 808–816. [Google Scholar] [CrossRef] - Baxter, R.J. Onsager and Kaufman’s Calculation of the Spontaneous Magnetization of the Ising Model: II. J. Stat. Phys.
**2012**, 149, 1164–1167. [Google Scholar] [CrossRef][Green Version] - Houtappel, R. Order-disorder in hexagonal lattices. Physica
**1950**, 16, 425–455. [Google Scholar] [CrossRef] - Husimi, K.; Syôzi, I. The statistics of honeycomb and triangular lattice. I. Prog. Theor. Phys.
**1950**, 5, 177–186. [Google Scholar] [CrossRef] - Syozi, I. The statistics of honeycomb and triangular lattice. II. Prog. Theor. Phys.
**1950**, 5, 341–351. [Google Scholar] [CrossRef] - Newell, G.F. Crystal Statistics of a Two-Dimensional Triangular Ising Lattice. Phys. Rev.
**1950**, 79, 876–882. [Google Scholar] [CrossRef] - Temperley, H. Statistical mechanics of the two-dimensional assembly. Proc. R. Soc. Lond. A Math. Phys. Sci.
**1950**, 202, 202–207. [Google Scholar] [CrossRef] - Wannier, G.H. Antiferromagnetism. The Triangular Ising Net. Phys. Rev.
**1950**, 79, 357–364. [Google Scholar] [CrossRef][Green Version] - Potts, R.B. Combinatorial Solution of the Triangular Ising Lattice. Proc. Phys. Soc. A
**1955**, 68, 145–148. [Google Scholar] [CrossRef] - Naya, S. On the Spontaneous Magnetizations of Honeycomb and Kagomé Ising Lattices. Prog. Theor. Phys.
**1954**, 11, 53–62. [Google Scholar] [CrossRef] - Potts, R.B. Spontaneous Magnetization of a Triangular Ising Lattice. Phys. Rev.
**1952**, 88, 352. [Google Scholar] [CrossRef] - Lin, K.Y.; Ma, W.J. Two-dimensional Ising model on a ruby lattice. J. Phys. A Math. Gen.
**1983**, 16, 3895–3898. [Google Scholar] [CrossRef] - Zhang, Z.D. Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices. Phil. Mag.
**2007**, 87, 5309–5419. [Google Scholar] [CrossRef][Green Version] - Wu, F.; McCoy, B.; Fisher, M.; Chayes, L. Comment on a recent conjectured solution of the three-dimensional Ising model. Phil. Mag.
**2008**, 88, 3093–3095. [Google Scholar] [CrossRef] - Perk, J.H. Comment on ‘Conjectures on exact solution of three-dimensional (3D) simple orthorhombic Ising lattices’. Phil. Mag.
**2009**, 89, 761–764. [Google Scholar] [CrossRef][Green Version] - Zhang, Z. Response to the Comment on ‘Conjectures on exact solution of three-dimensional (3D) simple orthorhombic Ising lattices’. Phil. Mag.
**2009**, 89, 765–768. [Google Scholar] [CrossRef][Green Version] - Perk, J.H.H. Erroneous solution of three-dimensional (3D) simple orthorhombic Ising lattices. Bull. Société Sci. Lettres LÓDZ
**2012**, 62, 45–59. [Google Scholar] - Zhang, Z.D. Mathematical structure of the three-dimensional (3D) Ising model. Chin. Phys. B
**2013**, 22, 030513. [Google Scholar] [CrossRef] - Perk, J.H.H. Comment on ‘Mathematical structure of the three-dimensional (3D) Ising model’. Chin. Phys. B
**2013**, 22, 080508. [Google Scholar] [CrossRef][Green Version] - Zhang, Z.; Suzuki, O.; March, N.H. Clifford algebra approach of 3D Ising model. Adv. Appl. Clifford Algebras
**2019**, 29, 12. [Google Scholar] [CrossRef][Green Version] - Suzuki, O.; Zhang, Z. A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure. Mathematics
**2021**, 9, 776. [Google Scholar] [CrossRef] - Zhang, Z.; Suzuki, O. A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases. Mathematics
**2021**, 9, 2936. [Google Scholar] [CrossRef] - Zhang, Z. Topological Quantum Statistical Mechanics and Topological Quantum Field Theories. Symmetry
**2022**, 14, 323. [Google Scholar] [CrossRef] - Viswanathan, G.M.; Portillo, M.A.G.; Raposo, E.P.; da Luz, M.G.E. What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution. Entropy
**2022**, 24, 1665. [Google Scholar] [CrossRef] - Domb, C. On the theory of cooperative phenomena in crystals. Adv. Phys.
**1960**, 9, 245–361. [Google Scholar] [CrossRef] - Wilson, K.G. Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture. Phys. Rev. B
**1971**, 4, 3174–3183. [Google Scholar] [CrossRef][Green Version] - Wilson, K.G. Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior. Phys. Rev. B
**1971**, 4, 3184–3205. [Google Scholar] [CrossRef][Green Version] - Wilson, K.G. The renormalization group and critical phenomena. Rev. Mod. Phys.
**1983**, 55, 583–600. [Google Scholar] [CrossRef] - Goldenfeld, N. Lectures on Phase Transitions and the Renormalization Group; CRC Press: Boca Raton, FL, USA, 1992. [Google Scholar]
- Butera, P.; Comi, M. Critical universality and hyperscaling revisited for Ising models of general spin using extended high-temperature series. Phys. Rev. B
**2002**, 65, 144431. [Google Scholar] [CrossRef][Green Version] - Salman, Z.; Adler, J. High and low temperature series estimates for the critical temperature of the 3D Ising model. Int. J. Mod. Phys. C
**1998**, 9, 195–209. [Google Scholar] [CrossRef] - Jasch, F.; Kleinert, H. Fast-convergent resummation algorithm and critical exponents of ϕ
^{4}-theory in three dimensions. J. Math. Phys.**2001**, 42, 52–73. [Google Scholar] [CrossRef][Green Version] - El-Showk, S.; Paulos, M.F.; Poland, D.; Rychkov, S.; Simmons-Duffin, D.; Vichi, A. Solving the 3D Ising model with the conformal bootstrap. Phys. Rev. D
**2012**, 86, 025022. [Google Scholar] [CrossRef][Green Version] - El-Showk, S.; Paulos, M.F.; Poland, D.; Rychkov, S.; Simmons-Duffin, D.; Vichi, A. Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents. J. Stat. Phys.
**2014**, 157, 869–914. [Google Scholar] [CrossRef][Green Version] - Landau, D.P.; Binder, K. A Guide to Monte Carlo Simulations in Statistical Physics, 4th ed.; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. Equation of State Calculations by Fast Computing Machines. J. Chem. Phys.
**1953**, 21, 1087–1092. [Google Scholar] [CrossRef][Green Version] - Swendsen, R.H.; Wang, J.S. Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett.
**1987**, 58, 86–88. [Google Scholar] [CrossRef] - Wolff, U. Collective Monte Carlo Updating for Spin Systems. Phys. Rev. Lett.
**1989**, 62, 361–364. [Google Scholar] [CrossRef] - Hasenbusch, M. Monte Carlo studies of the three-dimensional Ising model in equilibrium. Int. J. Mod. Phys. C
**2001**, 12, 911–1009. [Google Scholar] [CrossRef] - Blöte, H.W.J.; Heringa, J.R.; Hoogland, A.; Meyer, E.W.; Smit, T.S. Monte Carlo Renormalization of the 3D Ising Model: Analyticity and Convergence. Phys. Rev. Lett.
**1996**, 76, 2613–2616. [Google Scholar] [CrossRef] [PubMed] - Gupta, R.; Tamayo, P. Critical Exponents of the 3-D Ising Model. Int. J. Mod. Phys. C
**1996**, 7, 305–319. [Google Scholar] [CrossRef][Green Version] - Murase, Y.; Ito, N. Dynamic critical exponents of three-dimensional Ising models and two-dimensional three-states Potts models. J. Phys. Soc. Jpn.
**2007**, 77, 014002. [Google Scholar] [CrossRef] - Lundow, P.H.; Markström, K.; Rosengren, A. The Ising model for the bcc, fcc and diamond lattices: A comparison. Phil. Mag.
**2009**, 89, 2009–2042. [Google Scholar] [CrossRef] - Yu, U. Critical temperature of the Ising ferromagnet on the fcc, hcp, and dhcp lattices. Phys. A Stat. Mech. Appl.
**2015**, 419, 75–79. [Google Scholar] [CrossRef] - Ferrenberg, A.M.; Xu, J.; Landau, D.P. Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model. Phys. Rev. E
**2018**, 97, 043301. [Google Scholar] [CrossRef][Green Version] - Netz, R.R.; Berker, A.N. Monte Carlo mean-field theory and frustrated systems in two and three dimensions. Phys. Rev. Lett.
**1991**, 66, 377–380. [Google Scholar] [CrossRef] - Wang, L. Discovering phase transitions with unsupervised learning. Phys. Rev. B
**2016**, 94, 195105. [Google Scholar] [CrossRef][Green Version] - Torlai, G.; Melko, R.G. Learning thermodynamics with Boltzmann machines. Phys. Rev. B
**2016**, 94, 165134. [Google Scholar] [CrossRef][Green Version] - Carrasquilla, J.; Melko, R.G. Machine learning phases of matter. Nat. Phys.
**2017**, 13, 431–434. [Google Scholar] [CrossRef][Green Version] - Hu, W.; Singh, R.R.P.; Scalettar, R.T. Discovering phases, phase transitions, and crossovers through unsupervised machine learning: A critical examination. Phys. Rev. E
**2017**, 95, 062122. [Google Scholar] [CrossRef] - Chung, J.H.; Kao, Y.J. Neural Monte Carlo renormalization group. Phys. Rev. Res.
**2021**, 3, 023230. [Google Scholar] [CrossRef] - Carleo, G.; Cirac, I.; Cranmer, K.; Daudet, L.; Schuld, M.; Tishby, N.; Vogt-Maranto, L.; Zdeborová, L. Machine learning and the physical sciences. Rev. Mod. Phys.
**2019**, 91, 045002. [Google Scholar] [CrossRef][Green Version] - Hu, C.K. Historical review on analytic, Monte Carlo, and renormalization group approaches to critical phenomena of some lattice Models. Chin. J. Phys.
**2014**, 52, 1–76. [Google Scholar] [CrossRef] - Strecka, J.; Jaščur, M. A brief account of the Ising and Ising-like models: Mean-field, effective-field and exact results. Acta Phys. Slovaca
**2015**, 65, 235–367. [Google Scholar] - McCoy, B. Advanced Statistical Mechanics; International Series of Monographs on Physics; Oxford University Press: New York, NY, USA, 2009. [Google Scholar]
- Kardar, M. Statistical Physics of Fields; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- McCoy, B.M.; Wu, T.T. The two-Dimensional Ising Model; Courier Corporation: New York, NY, USA, 2014. [Google Scholar]
- Baxter, R.J. Exactly Solved Models in Statistical Mechanics; Academic Press: London, UK, 1982. [Google Scholar]
- Pelissetto, A.; Vicari, E. Critical phenomena and renormalization-group theory. Phys. Rep.
**2002**, 368, 549–727. [Google Scholar] [CrossRef][Green Version] - Kramers, H.A.; Wannier, G.H. Statistics of the Two-Dimensional Ferromagnet. Part I. Phys. Rev.
**1941**, 60, 252–262. [Google Scholar] [CrossRef] - Fisher, M.E. Transformations of Ising Models. Phys. Rev.
**1959**, 113, 969–981. [Google Scholar] [CrossRef] - Kaufman, B. Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis. Phys. Rev.
**1949**, 76, 1232–1243. [Google Scholar] [CrossRef] - Kaufman, B.; Onsager, L. Crystal Statistics. III. Short-Range Order in a Binary Ising Lattice. Phys. Rev.
**1949**, 76, 1244–1252. [Google Scholar] [CrossRef] - Perk, J. Quadratic identities for Ising model correlations. Phys. Lett. A
**1980**, 79, 3–5. [Google Scholar] [CrossRef] - Callen, H. A note on Green functions and the Ising model. Phys. Lett.
**1963**, 4, 161. [Google Scholar] [CrossRef] - Yang, C.N. Journey through Statistical Mechanics. Int. J. Mod. Phys. B
**1988**, 2, 1325–1329. [Google Scholar] [CrossRef] - Kac, M.; Ward, J.C. A Combinatorial Solution of the Two-Dimensional Ising Model. Phys. Rev.
**1952**, 88, 1332–1337. [Google Scholar] [CrossRef] - Hurst, C.A.; Green, H.S. New Solution of the Ising Problem for a Rectangular Lattice. J. Chem. Phys.
**1960**, 33, 1059–1062. [Google Scholar] [CrossRef] - Montroll, E.W.; Potts, R.B.; Ward, J.C. Correlations and Spontaneous Magnetization of the Two-Dimensional Ising Model. J. Math. Phys.
**1963**, 4, 308–322. [Google Scholar] [CrossRef] - Bornholdt, S.; Wagner, F. Stability of money: Phase transitions in an Ising economy. Phys. A Stat. Mech. Appl.
**2002**, 316, 453–468. [Google Scholar] [CrossRef][Green Version] - Sornette, D.; Zhou, W.X. Importance of positive feedbacks and overconfidence in a self-fulfilling Ising model of financial markets. Phys. A Stat. Mech. Appl.
**2006**, 370, 704–726. [Google Scholar] [CrossRef][Green Version] - Stauffer, D. Social applications of two-dimensional Ising models. Am. J. Phys.
**2008**, 76, 470–473. [Google Scholar] [CrossRef][Green Version] - Weber, M.; Buceta, J. The cellular Ising model: A framework for phase transitions in multicellular environments. J. R. Soc. Interface
**2016**, 13, 20151092. [Google Scholar] [CrossRef] - Matsuda, H. The Ising Model for Population Biology. Prog. Theor. Phys.
**1981**, 66, 1078–1080. [Google Scholar] [CrossRef][Green Version] - Castellano, C.; Fortunato, S.; Loreto, V. Statistical physics of social dynamics. Rev. Mod. Phys.
**2009**, 81, 591–646. [Google Scholar] [CrossRef] - Schneidman, E.; Berry, M.J.; Segev, R.; Bialek, W. Weak pairwise correlations imply strongly correlated network states in a neural population. Nature
**2006**, 440, 1007–1012. [Google Scholar] [CrossRef][Green Version] - Amit, D.J. Modeling Brain Function: The World of Attractor Neural Networks; Cambridge University Press: Cambridge, UK, 1989. [Google Scholar]
- Decelle, A.; Furtlehner, C. Restricted Boltzmann machine: Recent advances and mean-field theory. Chin. Phys. B
**2021**, 30, 040202. [Google Scholar] [CrossRef] - Engel, A.; Van den Broeck, C. Statistical Mechanics of Learning; Cambridge University Press: Cambridge, UK, 2001. [Google Scholar]
- Kaya, T. Relevant spontaneous magnetization relations for the triangular and the cubic lattice Ising model. Chin. J. Phys.
**2022**, 77, 2676–2683. [Google Scholar] [CrossRef] - Kaya, T. Analytic average magnetization expression for the body-centered cubic Ising lattice. Eur. Phys. J. Plus
**2022**, 137, 1130. [Google Scholar] [CrossRef] - Kaya, T. Relevant alternative analytic average magnetization calculation method for the square and the honeycomb Ising lattices. Chin. J. Phys.
**2022**, 77, 747–752. [Google Scholar] [CrossRef] - Suzuki, M. Generalized exact formula for the correlations of the Ising model and other classical systems. Phys. Lett.
**1965**, 19, 267–268. [Google Scholar] [CrossRef] - Suzuki, M. Correlation identities and application. Int. J. Mod. Phys. B
**2002**, 16, 1749–1765. [Google Scholar] [CrossRef] - Baxter, R. Triplet order parameter of the triangular Ising model. J. Phys. A Math. Gen.
**1975**, 8, 1797–1805. [Google Scholar] [CrossRef] - Barry, J.; Múnera, C.; Tanaka, T. Exact solutions for Ising model odd-number correlations on the honeycomb and triangular lattices. Phys. A Stat. Mech. Appl.
**1982**, 113, 367–387. [Google Scholar] [CrossRef] - Pink, D.A. Three-site correlation functions of the two-dimensional Ising model. Can. J. Phys.
**1968**, 46, 2399–2405. [Google Scholar] [CrossRef] - Enting, I.G. Triplet order parameters in triangular and honeycomb Ising models. J. Phys. A Math. Gen.
**1977**, 10, 1737–1743. [Google Scholar] [CrossRef] - Barber, M.N. On the nature of the critical point in the three-spin triangular Ising model. J. Phys. A Math. Gen.
**1976**, 9, L171–L174. [Google Scholar] [CrossRef] - Wood, D.W.; Griffiths, H.P. Triplet order parameters for three-dimensional Ising models. J. Phys. A Math. Gen.
**1976**, 9, 407–411. [Google Scholar] [CrossRef] - Taggart, G.B. Effective field model for Ising ferromagnets: Influence of triplet correlations. J. Appl. Phys.
**1982**, 53, 1907–1909. [Google Scholar] [CrossRef][Green Version] - Baxter, R.J.; Choy, T.C. Local three-spin correlations in the free-fermion and planar Ising models. Proc. R. Soc. Lond. A Math. Phys. Sci.
**1989**, 423, 279–300. [Google Scholar] [CrossRef] - Kaya, T. Exact three spin correlation function relations for the square and the honeycomb Ising lattices. Chin. J. Phys.
**2020**, 66, 415–421. [Google Scholar] [CrossRef] - Lin, K.Y. Three-spin correlation of the Ising model on the generalized checkerboard lattice. J. Stat. Phys.
**1989**, 56, 631–643. [Google Scholar] [CrossRef] - Lin, K.Y.; Chen, B.H. Three-spin correlation of the Ising model on a Kagome lattice. Int. J. Mod. Phys. B
**1990**, 04, 123–130. [Google Scholar] [CrossRef] - Talapov, A.L.; Blöte, H.W.J. The magnetization of the 3D Ising model. J. Phys. A Math. Gen.
**1996**, 29, 5727–5733. [Google Scholar] [CrossRef][Green Version] - Newell, G.F.; Montroll, E.W. On the Theory of the Ising Model of Ferromagnetism. Rev. Mod. Phys.
**1953**, 25, 159. [Google Scholar] [CrossRef] - Binder, K.; Luijten, E. Monte Carlo tests of renormalization-group predictions for critical phenomena in Ising models. Phys. Rep.
**2001**, 344, 179–253. [Google Scholar] [CrossRef]

**Figure 1.**(Color Online) A diagram of FCC lattice. ${\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{12}$ (red spins) denote the neighboring spins whereas ${\sigma}_{0}$ (blue spin) denotes the central spin.

**Figure 2.**(Color Online) The plots of $A\left(K\right)$, $B\left(K\right)$, $C\left(K\right)$, $D\left(K\right)$, $E\left(K\right)$, and $F\left(K\right)$ versus K. The lines loosely-dotted (black), loosely-dash-dotted (magenta), dash-dotted (green), dotted (orange), dashed (red), and solid (blue) correspond to $A\left(K\right)$, $B\left(K\right)$, $C\left(K\right)$, $D\left(K\right)$, $E\left(K\right)$, and $F\left(K\right)$, respectively. The inset zooms in the selected region in the figure.

**Figure 3.**(Color Online) The plot of K versus $\langle \sigma \rangle $. The solid (blue) line and the filled points (red) correspond to Equation (10) and the MC data, respectively. The figure also includes error bars; however, they are invisible because they are smaller than the marker points.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tambaş, B. Relevant Analytic Spontaneous Magnetization Relation for the Face-Centered-Cubic Ising Lattice. *Entropy* **2023**, *25*, 197.
https://doi.org/10.3390/e25020197

**AMA Style**

Tambaş B. Relevant Analytic Spontaneous Magnetization Relation for the Face-Centered-Cubic Ising Lattice. *Entropy*. 2023; 25(2):197.
https://doi.org/10.3390/e25020197

**Chicago/Turabian Style**

Tambaş, Başer. 2023. "Relevant Analytic Spontaneous Magnetization Relation for the Face-Centered-Cubic Ising Lattice" *Entropy* 25, no. 2: 197.
https://doi.org/10.3390/e25020197