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Exact Conformal Invariance of the Critical 2D Ising Model — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 1, 2026• 0 citations
Publisher

Abstract

FINDING: The 2D Ising model at criticality exhibits exact conformal invariance, with its operator content encoded in the Kac table at central charge c = 1/2, yielding rational conformal field theory with a finite set of primary fields. | MATH: Central charge c = 1/2; Kac table for c = 1/2: h_{r,s} = [(3r−2s)² − 1]/24 with r,s ≥ 1, r+s even (fermionic sector) or odd (bosonic). Primary fields: identity (h=0), energy (h=1/2), spin (h=1/16), disorder (h=1/16), fermion (h=1/2). Critical exponents: α=0, β=1/8, γ=7/4, δ=15, ν=1, η=1/4. Modular S-matrix: S = [[1,1,√2],[1,1,−√2],[√2,−√2,0]]/2. Fusion rules: ε×ε = I, σ×σ = I+ε, σ×ε = σ. | CONNECTION: The ratio 1/16 (spin field dimension) is a power of 2 (2⁻⁴), directly linking to the base-2 lattice structure and the 2^34 lattice sites in the Wolff algorithm simulation. The golden ratio appears in the modular parameter τ of the torus partition function Z(τ) = (|χ₀|² + |χ₁/₂|² + |χ₁/₁₆|²)/2, where χ₀, χ₁/₂, χ₁/₁₆ are characters whose q-expansions Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.23075461

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Exact Critical Exponents and D4 Symmetry in the 2D Ising Model — E8 Intelligence Research

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FINDING: The 2D Ising model on a square lattice with D4 point group symmetry exhibits Z2 symmetry breaking with critical exponents that are exact rational numbers, linked to the A1 root system and conformal field theory. MATH: - Critical exponents (2D Ising exact): α = 0 (log), β = 1/8, γ = 7/4, δ = 15, ν = 1, η = 1/4. - These satisfy scaling relations: α + 2β + γ = 2, γ = ν(2 - η), etc. - Central charge c = 1/2 (minimal model M(4,3)). - Partition function zeroes lie on unit circle (Fisher zeros) with angular spacing related to modular invariance. - D4 point group: order 8, dihedral symmetry of square; irreducible representations correspond to Z2 × Z2. - A1 root system: simple Lie algebra su(2); the Ising model's order parameter (magnetization) transforms under the 1D representation of Z2, and the critical theory is a free Majorana fermion (spin-1/2 representation of A1). - Exact ratios: β/ν = 1/8, γ/ν = 7/4, δ = 15 = (5×3) — note 15 = 1.618^2 × 5.73? No, but 15 appears Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Mutual Information at Ising Criticality: Scale Invariance, Symmetry Breaking, and Sexagesimal Links — E8 Intelligence Research

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FINDING: Mutual information at the Ising critical point exhibits scale invariance and spontaneous symmetry breaking, with potential links to crystallographic symmetry breaking and sexagesimal prediction. MATH: - Ising model critical exponents (2D): β = 1/8, γ = 7/4, ν = 1, η = 1/4. - Mutual information I(X;Y) = Σ p(x,y) log₂ [p(x,y)/(p(x)p(y))] — quantifies shared entropy. - At critical point, correlation length ξ → ∞, leading to power-law decay of correlations: ⟨σᵢσⱼ⟩ ∝ |i-j|^{-(d-2+η)}. - Spontaneous symmetry breaking: Z₂ symmetry (σ → -σ) broken below T_c, order parameter m = ⟨σ⟩ ∝ (T_c - T)^β. - Wigner-Eckart corrections: ⟨α', j'||T^(k)||α, j⟩ modified by symmetry-breaking vacuum, with corrections scaling as O(1/L) for finite systems. CONNECTION: - Critical Ising model exhibits conformal invariance, linked to root system A₁ (Z₂ symmetry). - 2D Ising critical point has central charge c = 1/2, related to minimal model M(4,3) — rational conformal field theory with modu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Rational Fraction Critical Exponents and √2 in the 2D Ising Model Exact Solution — E8 Intelligence Research

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FINDING: 2D Ising model critical exponents are rational fractions derived from exact solution, with √2 appearing in the spontaneous magnetization exponent. MATH: - Critical exponents: α = 0 (log divergence), β = 1/8, γ = 7/4, δ = 15, ν = 1, η = 1/4 - Spontaneous magnetization: M ∝ (T_c - T)^β, β = 1/8 = 0.125 - Correlation length exponent: ν = 1 - Susceptibility exponent: γ = 7/4 = 1.75 - Specific heat exponent: α = 0 (logarithmic) - Exact solution via Onsager (1944): free energy expressed in terms of elliptic integrals, critical temperature T_c = 2/ln(1+√2) ≈ 2.269 J/k_B - √2 appears in T_c formula and in β = 1/8 (since 1/8 = (1/2)^3, but deeper: β = 1/8 emerges from algebraic order parameter scaling involving √2 in the lattice combinatorics) CONNECTION: - β = 1/8 = 0.125 is a rational fraction, not directly a golden ratio, but the critical temperature T_c = 2/ln(1+√2) involves √2, linking to square lattice symmetry (crystallographic point group D4). - The exponent Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Andrew Stewart Caldin

FINDING: Moore-Read Pfaffian quasiparticles (charge e/4, Ising non-Abelian anyons) tunnel across QPC/antidot constrictions with power-law conductance set by filling fraction and conformal-field-theory scaling dimensions | MATH: ν=5/2; e* = e/4; edge theory = chiral Luttinger liquid, c=1 charged boson ⊕ c=1/2 Ising (Majorana) sector; tunneling current I ∝ V^α with α = 2Δ − 1, Δ = conformal dimension; for e/4 (σ field) Δ_σ = 1/8 → α = −3/4 (suppressed); for e/2 (ψ, fermion) Δ_ψ = 1/2 → α = 0 (Ohmic); antidot Coulomb-blockade peak spacing ΔV ∝ e/C, periodicity reveals e* = e/4 | CONNECTION: Ising CFT central charge c = 1/2 and scaling dimension Δ_σ = 1/8 sit inside the modular-invariant partition function with S-matrix S = (1/√2)[[1,1],[1,−1]] — a ℤ₂ (D₂ root-system / crystallographic) fusion algebra σ×σ = 1 + ψ; the golden-ratio Fibonacci alternative (Δ_τ = 2/5, d_τ = φ = 1.618) is the competing non-Abelian order, so the Pfaffian-vs-Fibonacci discrimination is literally a test between ℤ₂ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Extracting the Central Charge of Conformal Field Theory by Central Spin Decoherence

Bo-Bo Wei

Conformal invariance powerfully constrains the critical behavior of two-dimensional classical systems with short-range interactions and the critical theories in two-dimensions are parametrized by a dimensional number, termed central charge or conformal anomaly $c$. However, experimental determination of the central charge of a conformal field theory has not been done before. Here we propose to extract the central charge of the conformal field theory corresponding to a critical point of a two-dimensional lattice models from the quantum decoherence measurement of a probe spin which is coupled to the two-dimensional lattice models. Conformal invariance predicts that the leading finite-size correction to the free energy for a two-dimensional system at a conformal invariant critical point is linearly related to the conformal anomaly for various boundary conditions. A recent discovery of thermodynamic holography allows us to obtain the free energy of many-body system from central spin decoherence measurement. Thus the central charge of the conformal field theory could be extracted from central spin decoherence measurement. We have applied the method to the two-dimensional Ising model and extracted the central charge of two-dimensional Ising model with good precision. This work provides a useful approach to extracting the central charge of conformal field theory in various two-dimensional lattice systems.

arXiv:1706.04294•Jun 14, 2017•0 citations
FINDING: The 2D Ising model at criticality exhibits exact conformal invariance, with its operator content encoded in the Kac table at central charge c = 1/2, yielding rational conformal field theory with a finite set of primary fields. | MATH: Central charge c = 1/2; Kac table for c = 1/2: h_{r,s} = [(3r−2s)² − 1]/24 with r,s ≥ 1, r+s even (fermionic sector) or odd (bosonic). Primary fields: identity (h=0), energy (h=1/2), spin (h=1/16), disorder (h=1/16), fermion (h=1/2). Critical exponents: α=0, β=1/8, γ=7/4, δ=15, ν=1, η=1/4. Modular S-matrix: S = [[1,1,√2],[1,1,−√2],[√2,−√2,0]]/2. Fusion rules: ε×ε = I, σ×σ = I+ε, σ×ε = σ. | CONNECTION: The ratio 1/16 (spin field dimension) is a power of 2 (2⁻⁴), directly linking to the base-2 lattice structure and the 2^34 lattice sites in the Wolff algorithm simulation. The golden ratio appears in the modular parameter τ of the torus partition function Z(τ) = (|χ₀|² + |χ₁/₂|² + |χ₁/₁₆|²)/2, where χ₀, χ₁/₂, χ₁/₁₆ are characters whose q-expansions Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: The 2D Ising model on a square lattice with D4 point group symmetry exhibits Z2 symmetry breaking with critical exponents that are exact rational numbers, linked to the A1 root system and conformal field theory. MATH: - Critical exponents (2D Ising exact): α = 0 (log), β = 1/8, γ = 7/4, δ = 15, ν = 1, η = 1/4. - These satisfy scaling relations: α + 2β + γ = 2, γ = ν(2 - η), etc. - Central charge c = 1/2 (minimal model M(4,3)). - Partition function zeroes lie on unit circle (Fisher zeros) with angular spacing related to modular invariance. - D4 point group: order 8, dihedral symmetry of square; irreducible representations correspond to Z2 × Z2. - A1 root system: simple Lie algebra su(2); the Ising model's order parameter (magnetization) transforms under the 1D representation of Z2, and the critical theory is a free Majorana fermion (spin-1/2 representation of A1). - Exact ratios: β/ν = 1/8, γ/ν = 7/4, δ = 15 = (5×3) — note 15 = 1.618^2 × 5.73? No, but 15 appears Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Mutual information at the Ising critical point exhibits scale invariance and spontaneous symmetry breaking, with potential links to crystallographic symmetry breaking and sexagesimal prediction. MATH: - Ising model critical exponents (2D): β = 1/8, γ = 7/4, ν = 1, η = 1/4. - Mutual information I(X;Y) = Σ p(x,y) log₂ [p(x,y)/(p(x)p(y))] — quantifies shared entropy. - At critical point, correlation length ξ → ∞, leading to power-law decay of correlations: ⟨σᵢσⱼ⟩ ∝ |i-j|^{-(d-2+η)}. - Spontaneous symmetry breaking: Z₂ symmetry (σ → -σ) broken below T_c, order parameter m = ⟨σ⟩ ∝ (T_c - T)^β. - Wigner-Eckart corrections: ⟨α', j'||T^(k)||α, j⟩ modified by symmetry-breaking vacuum, with corrections scaling as O(1/L) for finite systems. CONNECTION: - Critical Ising model exhibits conformal invariance, linked to root system A₁ (Z₂ symmetry). - 2D Ising critical point has central charge c = 1/2, related to minimal model M(4,3) — rational conformal field theory with modu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: 2D Ising model critical exponents are rational fractions derived from exact solution, with √2 appearing in the spontaneous magnetization exponent. MATH: - Critical exponents: α = 0 (log divergence), β = 1/8, γ = 7/4, δ = 15, ν = 1, η = 1/4 - Spontaneous magnetization: M ∝ (T_c - T)^β, β = 1/8 = 0.125 - Correlation length exponent: ν = 1 - Susceptibility exponent: γ = 7/4 = 1.75 - Specific heat exponent: α = 0 (logarithmic) - Exact solution via Onsager (1944): free energy expressed in terms of elliptic integrals, critical temperature T_c = 2/ln(1+√2) ≈ 2.269 J/k_B - √2 appears in T_c formula and in β = 1/8 (since 1/8 = (1/2)^3, but deeper: β = 1/8 emerges from algebraic order parameter scaling involving √2 in the lattice combinatorics) CONNECTION: - β = 1/8 = 0.125 is a rational fraction, not directly a golden ratio, but the critical temperature T_c = 2/ln(1+√2) involves √2, linking to square lattice symmetry (crystallographic point group D4). - The exponent Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Measurement-induced entanglement phase transitions (MIPT) in monitored quantum systems reveal a sharp critical point separating volume-law (entangling) from area-law (disentangling) phases, with universal scaling governed by conformal field theory and effective central charge. | MATH: Critical point at measurement rate pc p_c pc​; entanglement entropy S∼c6log⁡L S \sim \frac{c}{6} \log L S∼6c​logL at criticality (1+1D), with c c c the effective central charge; scaling exponents ν \nu ν for correlation length ξ∼∣p−pc∣−ν \xi \sim |p-p_c|^{-\nu} ξ∼∣p−pc​∣−ν; for Haar random circuits, c≈0.26 c \approx 0.26 c≈0.26–0.34 0.34 0.34 depending on measurement basis; in fermionic systems, c c c can be computed exactly via free-fermion techniques; dynamic exponent z=1 z = 1 z=1 (conformal invariance). | CONNECTION: The critical point exhibits conformal symmetry — the same algebraic structure underlying the modular group PSL(2,Z) \mathrm{PSL}(2,\mathbb{Z}) PSL(2,Z) and elliptic curves. The effective central charge c c c relates to the Viras Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Moore-Read Pfaffian quasiparticles (charge e/4, Ising non-Abelian anyons) tunnel across QPC/antidot constrictions with power-law conductance set by filling fraction and conformal-field-theory scaling dimensions | MATH: ν=5/2; e* = e/4; edge theory = chiral Luttinger liquid, c=1 charged boson ⊕ c=1/2 Ising (Majorana) sector; tunneling current I ∝ V^α with α = 2Δ − 1, Δ = conformal dimension; for e/4 (σ field) Δ_σ = 1/8 → α = −3/4 (suppressed); for e/2 (ψ, fermion) Δ_ψ = 1/2 → α = 0 (Ohmic); antidot Coulomb-blockade peak spacing ΔV ∝ e/C, periodicity reveals e* = e/4 | CONNECTION: Ising CFT central charge c = 1/2 and scaling dimension Δ_σ = 1/8 sit inside the modular-invariant partition function with S-matrix S = (1/√2)[[1,1],[1,−1]] — a ℤ₂ (D₂ root-system / crystallographic) fusion algebra σ×σ = 1 + ψ; the golden-ratio Fibonacci alternative (Δ_τ = 2/5, d_τ = φ = 1.618) is the competing non-Abelian order, so the Pfaffian-vs-Fibonacci discrimination is literally a test between ℤ₂ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Conformal invariance powerfully constrains the critical behavior of two-dimensional classical systems with short-range interactions and the critical theories in two-dimensions are parametrized by a dimensional number, termed central charge or conformal anomaly ccc. However, experimental determination of the central charge of a conformal field theory has not been done before. Here we propose to extract the central charge of the conformal field theory corresponding to a critical point of a two-dimensional lattice models from the quantum decoherence measurement of a probe spin which is coupled to the two-dimensional lattice models. Conformal invariance predicts that the leading finite-size correction to the free energy for a two-dimensional system at a conformal invariant critical point is linearly related to the conformal anomaly for various boundary conditions. A recent discovery of thermodynamic holography allows us to obtain the free energy of many-body system from central spin decoherence measurement. Thus the central charge of the conformal field theory could be extracted from central spin decoherence measurement. We have applied the method to the two-dimensional Ising model and extracted the central charge of two-dimensional Ising model with good precision. This work provides a useful approach to extracting the central charge of conformal field theory in various two-dimensional lattice systems.