Onsager's Ising Solution, and the Missing ln 2

The two-dimensional Ising model is the most famous exact result in statistical mechanics: Onsager's 1944 free energy, through the phase transition. This video goes through that formula, its two limits, and a simple way to write it down wrong. The same answer appears in several equivalent forms, and the ln 2 moves between them. Take the constant from one correct form and the integrand from another, and you get a formula that is off by ½ ln 2 at every temperature. It still has the right critical temperature, exponents and specific heat, because a constant drops out of every derivative. One number catches it: at infinite temperature the entropy should be ln 2 per spin (two states), and at zero temperature it should be zero, not negative. The wrong formula in this video is a constructed example, labelled as such on screen; it is not a quote from any paper. Then the critical point: Kramers–Wannier duality and sinh 2K_c = 1, the log-divergent specific heat, Yang's magnetization with β = 1/8, the c = ½ conformal field theory, a central charge measured from strip transfer matrices (c = 0.4998), and where c = ½ sits in holography (the fully quantum end, with one proposed gravity dual). Claims in the video are tagged [established], [debated] or [our own work]. The only [our own work] item is the constructed wrong formula. Every on-screen formula and number from the exact solution and the c = ½ theory was checked numerically (4×4 exact enumeration, Kaufman's finite-lattice formula, strip transfer matrices up to width 16, Monte Carlo); mean-field values are textbook, and the gravity-dual statement is from the paper's abstract. Corrections go in the pinned comment. Narration is AI text-to-speech. The ideas are mine; I use AI to help elaborate and draft, and every number on screen is checked with code. Chapters 00:00 Cold open: two formulas, one log two apart 00:42 The model, and why exact matters 02:20 The free energy, and its two limits 03:35 Where the log two goes missing 05:29 The critical point 07:21 The link to holography 09:20 The checklist, and what's next Sources Exact solution (journal papers, no arXiv) • Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev. 65 (1944) 117: https://doi.org/10.1103/PhysRev.65.117 • Kramers & Wannier, Statistics of the Two-Dimensional Ferromagnet. Part I, Phys. Rev. 60 (1941) 252: https://doi.org/10.1103/PhysRev.60.252 • Kaufman, Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis, Phys. Rev. 76 (1949) 1232: https://doi.org/10.1103/PhysRev.76.1232 • Yang, The Spontaneous Magnetization of a Two-Dimensional Ising Model, Phys. Rev. 85 (1952) 808: https://doi.org/10.1103/PhysRev.85.808 • Schultz, Mattis & Lieb, Two-Dimensional Ising Model as a Soluble Problem of Many Fermions, Rev. Mod. Phys. 36 (1964) 856: https://doi.org/10.1103/RevModPhys.36.856 Conformal field theory and finite size • Belavin, Polyakov & Zamolodchikov, Nucl. Phys. B 241 (1984) 333: https://doi.org/10.1016/0550-3213(84)90052-X • Friedan, Qiu & Shenker, Conformal Invariance, Unitarity, and Critical Exponents in Two Dimensions, Phys. Rev. Lett. 52 (1984) 1575: https://doi.org/10.1103/PhysRevLett.52.1575 • Blöte, Cardy & Nightingale, Phys. Rev. Lett. 56 (1986) 742: https://doi.org/10.1103/PhysRevLett.56.742 • Affleck, Phys. Rev. Lett. 56 (1986) 746: https://doi.org/10.1103/PhysRevLett.56.746 Gravity side (arXiv) • Brown & Henneaux, Commun. Math. Phys. 104 (1986) 207: https://doi.org/10.1007/BF01211590 • Strominger, Black Hole Entropy from Near-Horizon Microstates: https://arxiv.org/abs/hep-th/9712251 • Castro, Gaberdiel, Hartman, Maloney & Volpato, The Gravity Dual of the Ising Model: https://arxiv.org/abs/1111.1987 History and 3D • Baxter, Onsager and Kaufman's calculation of the spontaneous magnetization of the Ising model: https://arxiv.org/abs/1103.3347 • El-Showk, Paulos, Poland, Rychkov, Simmons-Duffin & Vichi, Solving the 3D Ising Model with the Conformal Bootstrap: https://arxiv.org/abs/1203.6064 Further reading • Ginsparg, Applied Conformal Field Theory: https://arxiv.org/abs/hep-th/9108028 • Cardy, Conformal Field Theory and Statistical Mechanics: https://arxiv.org/abs/0807.3472 • Witten, Three-Dimensional Gravity Revisited: https://arxiv.org/abs/0706.3359 • Maloney & Witten, Quantum Gravity Partition Functions in Three Dimensions: https://arxiv.org/abs/0712.0155 Series 1. What If Black Holes Have No Horizon? 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