Yang's Spontaneous Magnetization via Toeplitz Determinant

Main result

For the isotropic 2D classical Ising model on the square lattice at reduced coupling $K = \beta J$ below the critical coupling $K_c = \tfrac{1}{2}\ln(1 + \sqrt{2})$, the spontaneous magnetization — defined as the long-distance limit of the spin-spin correlation function —

\[M(T) \;\equiv\; \lim_{|\mathbf{r}|\to\infty}\sqrt{\,\bigl\langle \sigma_{\mathbf{0}}\,\sigma_{\mathbf{r}}\bigr\rangle\,}\]

has the exact closed form

\[\boxed{\; M(T) \;=\; \Bigl(\,1 - \sinh^{-4}(2\beta J)\,\Bigr)^{1/8}, \qquad T < T_c. \;}\]

The $(T_c - T)^{1/8}$ vanishing at criticality identifies the order-parameter critical exponent

\[\boxed{\;\beta = \tfrac{1}{8}.\;}\]

The derivation proceeds via three ingredients:

Pfaffian reduction (Kaufman 1949, Montroll–Potts–Ward 1963): the spin-spin correlation function is an $n\times n$ Toeplitz determinant $D_n[a]$ with a known symbol $a(\theta)$.
Strong Szegő limit theorem (Szegő 1915; sharp form Kac 1954): the $n\to\infty$ limit of $D_n[a]$ is $\exp\!\bigl[\sum_{k\ge 1} k\,c_k c_{-k}\bigr]$, with $c_k$ the Fourier coefficients of $\log a(\theta)$.
Wiener–Hopf factorisation of $a(\theta)$ on the symmetric-split form, followed by explicit Fourier-coefficient evaluation.

Substituting in a closed-form relation $\alpha^{2} = \sinh^{-4}(2\beta J)$ between the Wiener–Hopf parameter $\alpha$ and the Ising coupling $K = \beta J$ (Yang 1952 §5) yields the boxed result.

Setup

2D classical Ising model

On an infinite square lattice with reduced coupling $K = \beta J$ and PBC,

\[Z = \sum_{\{s\}} \exp\!\Bigl(K\sum_{\langle i, j\rangle} s_i s_j\Bigr), \qquad s_v \in \{\pm 1\}.\]

For $K > K_c = \tfrac{1}{2}\ln(1 + \sqrt{2})$ (equivalently $\sinh(2K) > 1$, $T < T_c$) the system is in the ordered ferromagnetic phase; the spontaneous magnetization $M(T)$ is the standard order parameter.

Correlation-function definition of $M(T)$

By translational symmetry and cluster decomposition, in the ordered phase

\[\langle\sigma_{\mathbf{0}}\,\sigma_{\mathbf{r}}\rangle \;\xrightarrow[|\mathbf{r}|\to\infty]{}\; \langle\sigma\rangle^{2} \;=\; M(T)^{2},\]

\[M(T) \;=\; \sqrt{\lim_{|\mathbf{r}|\to\infty} \langle\sigma_{\mathbf{0}}\,\sigma_{\mathbf{r}}\rangle}. \tag{1}\]

Goal

Reduce the right-hand side of (1) to a Toeplitz determinant $D_n[a]$ with an explicit symbol, take the $n\to\infty$ limit via Szegő's theorem, evaluate the resulting Fourier-coefficient sum in closed form, and identify $M(T)$ as a function of $\beta J$.

Calculation

Step 1 — Correlation function as a Toeplitz determinant

The 2D Ising correlator along a diagonal direction (Kaufman 1949, MPW 1963) is expressible as the determinant of an $n\times n$ matrix whose $(j, k)$ entry depends only on $j - k$:

\[\bigl\langle\sigma_{0,0}\,\sigma_{n,n}\bigr\rangle \;=\; D_{n}[a] \;:=\; \det\bigl[\,a_{j - k}\,\bigr]_{j, k = 0}^{n - 1}. \tag{2}\]

A matrix of this form is called a Toeplitz matrix, and its determinant $D_n[a]$ is a Toeplitz determinant with symbol $a(\theta) = \sum_{m\in\mathbb{Z}} a_m\,e^{-i m\theta}$.

The reduction (2) from the original Ising spin variables to a fermion-determinant form goes through:

(i) Jordan–Wigner transformation of the Kaufman transfer matrix, producing a free-fermion quadratic form. The symbol of the resulting covariance matrix is $a(\theta) = \cos\phi(\theta)$ with $\phi(\theta) = \phi(\theta; K_1, K_2)$ the Bogoliubov angle of the $k$-th mode.

(ii) Wick contraction of the spin–spin correlator at sites $(0, 0)$ and $(n, n)$ using the free-fermion 2-point function. The result is a Pfaffian, which for a suitable ordering of Majorana indices collapses to a determinant of Toeplitz form (Montroll–Potts–Ward 1963 Appendix B).

The full reduction is a tour-de-force spread across Kaufman (1949), Yang (1952), and Montroll–Potts–Ward (1963), totalling some 80 journal pages. For the purpose of this note we take (2) as the starting point and focus on the Szegő evaluation, which is where the critical exponent $\beta = 1/8$ ultimately emerges. The reader consulting (e.g.) McCoy–Wu (1973), Ch. VII, finds the reduction worked out end-to-end.

Step 2 — The symbol $a(\theta)$ for $T < T_c$

After the Wiener–Hopf factorisation of the raw Bogoliubov-angle symbol, the effective Toeplitz symbol for the diagonal correlator on the isotropic lattice at $T < T_c$ has the remarkably clean form (McCoy–Wu 1973, eq. VII.2.36a)

\[\boxed{\; a(\theta) \;=\; \left(\frac{1 - \alpha\,e^{+i\theta}} {1 - \alpha\,e^{-i\theta}}\right)^{1/2}, \qquad 0 \le \alpha < 1, \;} \tag{3}\]

with the Wiener–Hopf parameter

\[\boxed{\; \alpha^{2} \;=\; \sinh^{-4}(2\beta J), \qquad \alpha \in [0, 1)\text{ for }T < T_c. \;} \tag{4}\]

Note that $\alpha \to 0$ as $T \to 0$ ($\sinh(2\beta J) \to \infty$) and $\alpha \to 1^{-}$ as $T \to T_c^{-}$ ($\sinh(2\beta_c J) = 1$). The symbol is unimodular ($|a(\theta)| = 1$ for all real $\theta$), which implies $G \equiv \exp(c_0) = 1$ in the Szegő theorem below.

Step 3 — Strong Szegő limit theorem

Theorem (Szegő 1915; sharp form Kac 1954). Let $a(\theta)$ be a non-vanishing function on the unit circle with smooth logarithm, and let

\[\log a(\theta) \;=\; \sum_{k\in\mathbb{Z}} c_k\,e^{i k\theta}, \qquad c_k = \frac{1}{2\pi}\int_{-\pi}^{\pi} e^{-i k\theta}\,\log a(\theta)\,d\theta.\]

Then as $n\to\infty$,

\[\boxed{\; D_n[a] \;=\; G^{n}\,E\,\bigl(1 + o(1)\bigr),\qquad G = e^{c_0},\qquad E = \exp\!\Bigl[\sum_{k=1}^{\infty} k\,c_k\,c_{-k}\Bigr]. \;} \tag{5}\]

The sum $\sum k\,c_k c_{-k}$ converges absolutely whenever $\sum_{k} |k|\,|c_k|^{2} < \infty$ — guaranteed here by the smooth rational form (3), which has $c_k$ decaying geometrically as $\alpha^{|k|}/|k|$ (Step 4).

(References: the original Szegő 1915 paper proves $D_n[a] \sim G^n$; the $E$-constant $\exp(\sum k c_k c_{-k})$ is due to Szegő 1952 and in its sharp form Kac 1954. Modern pedagogical treatments: Böttcher–Silbermann 1999, Analysis of Toeplitz Operators, Ch. 1; Simon 2005, Orthogonal Polynomials on the Unit Circle, Ch. 6.)

For the Ising symbol (3), $a$ is unimodular so $c_0 = 0$ and $G = 1$. Therefore $D_n[a] \to E$ as $n\to\infty$.

Step 4 — Fourier coefficients of $\log a(\theta)$

From (3),

\[\log a(\theta) \;=\; \tfrac{1}{2}\log(1 - \alpha e^{+i\theta}) \;-\; \tfrac{1}{2}\log(1 - \alpha e^{-i\theta}).\]

Each logarithm is expanded via $\log(1 - x) = -\sum_{k\ge 1} x^{k}/k$ (convergent for $|\alpha| < 1$):

\[\log(1 - \alpha e^{+i\theta}) \;=\; -\sum_{k=1}^{\infty}\frac{\alpha^{k}}{k}\,e^{+i k\theta},\]

\[\log(1 - \alpha e^{-i\theta}) \;=\; -\sum_{k=1}^{\infty}\frac{\alpha^{k}}{k}\,e^{-i k\theta}.\]

Therefore

\[\log a(\theta) \;=\; -\tfrac{1}{2}\sum_{k=1}^{\infty}\frac{\alpha^{k}}{k}\,e^{+i k\theta} \;+\; \tfrac{1}{2}\sum_{k=1}^{\infty}\frac{\alpha^{k}}{k}\,e^{-i k\theta},\]

reading off the Fourier coefficients:

\[\boxed{\; c_{+k} = -\frac{\alpha^{k}}{2 k},\qquad c_{-k} = +\frac{\alpha^{k}}{2 k}\qquad (k \ge 1),\qquad c_{0} = 0. \;} \tag{6}\]

The zero-th coefficient $c_0 = 0$ confirms $G = 1$ (geometric mean of a unimodular symbol equals unity). Fourier coefficients decay as $\alpha^{|k|}/(2|k|)$; since $|\alpha| < 1$ the series $\sum k |c_k|^{2} \sim \sum \alpha^{2k}/k$ converges.

Step 5 — Evaluate the Szegő constant $E$

Substitute (6) into the constant $E$ of (5):

\[\sum_{k=1}^{\infty} k\,c_{k}\,c_{-k} \;=\; \sum_{k=1}^{\infty} k\,\Bigl(-\frac{\alpha^{k}}{2 k}\Bigr)\Bigl(+\frac{\alpha^{k}}{2 k}\Bigr) \;=\; -\,\frac{1}{4}\sum_{k=1}^{\infty}\frac{\alpha^{2k}}{k}.\]

The inner sum is the standard Taylor series for $-\log(1 - x)$ at $x = \alpha^{2}$:

\[\sum_{k=1}^{\infty}\frac{\alpha^{2k}}{k} \;=\; -\log(1 - \alpha^{2}).\]

Therefore

\[\sum_{k=1}^{\infty} k\,c_{k}\,c_{-k} \;=\; -\,\frac{1}{4}\cdot\bigl(-\log(1 - \alpha^{2})\bigr) \;=\; \tfrac{1}{4}\log(1 - \alpha^{2}),\]

and the Szegő constant is

\[\boxed{\; E \;=\; \exp\!\Bigl[\,\tfrac{1}{4}\log(1 - \alpha^{2})\,\Bigr] \;=\; (1 - \alpha^{2})^{1/4}. \;} \tag{7}\]

Step 6 — Assemble $M(T)$

From Step 3 and (7),

\[\lim_{n\to\infty} D_n[a] \;=\; E \;=\; (1 - \alpha^{2})^{1/4}.\]

The defining relation (1) expresses the magnetization as the square root of this limit:

\[M(T)^{2} \;=\; \lim_{|\mathbf{r}|\to\infty} \bigl\langle\sigma_{\mathbf{0}}\,\sigma_{\mathbf{r}}\bigr\rangle \;=\; (1 - \alpha^{2})^{1/4}.\]

Substituting (4),

\[M(T)^{2} \;=\; \bigl(1 - \sinh^{-4}(2\beta J)\bigr)^{1/4}.\]

Taking the positive square root (magnetization is non-negative for $T < T_c$),

\[M(T) \;=\; \bigl(1 - \sinh^{-4}(2\beta J)\bigr)^{1/8}. \tag{8}\]

This is the Main-result formula.

Step 7 — Critical exponent $\beta = 1/8$

Near criticality $T \lesssim T_c$, expand $\alpha^{2} = \sinh^{-4}(2K)$ in small $\delta K \equiv K - K_c > 0$ (recall $K \propto \beta \propto 1/T$, so $\delta K > 0 \Leftrightarrow T < T_c$). Using

\[\sinh(2K) \;=\; \sinh(2 K_c + 2\,\delta K) \;=\; \sinh(2 K_c)\cosh(2\,\delta K) + \cosh(2 K_c)\sinh(2\,\delta K),\]

and the critical-point identities $\sinh(2 K_c) = 1$, $\cosh^{2}(2 K_c) = 1 + \sinh^{2}(2 K_c) = 2$, so $\cosh(2 K_c) = \sqrt{2}$:

\[\sinh(2K) \;=\; 1\cdot\bigl(1 + 2(\delta K)^{2} + \dots\bigr) + \sqrt{2}\cdot\bigl(2\,\delta K + O((\delta K)^{3})\bigr) \;=\; 1 + 2\sqrt{2}\,\delta K + O((\delta K)^{2}).\]

Therefore

\[\sinh^{-4}(2K) \;=\; \bigl(1 + 2\sqrt{2}\,\delta K + \dots\bigr)^{-4} \;=\; 1 - 8\sqrt{2}\,\delta K + O((\delta K)^{2}),\]

and

\[1 - \alpha^{2} \;=\; 1 - \sinh^{-4}(2 K) \;=\; 8\sqrt{2}\,\delta K + O((\delta K)^{2}).\]

Converting to temperature: $K = J/(k_B T)$, so $\delta K = -J\,\delta T/(k_B T^{2}) \approx -J(T - T_c)/(k_B T_c^{2})$, and $\delta K > 0 \Leftrightarrow T < T_c$. Hence

\[1 - \alpha^{2} \;=\; \frac{8\sqrt{2}\,J}{k_B T_c^{2}}(T_c - T) + O((T_c - T)^{2}).\]

From (8),

\[M(T) \;\sim\; (T_c - T)^{1/8} \qquad (T \to T_c^{-}),\]

identifying the order-parameter critical exponent as $\beta = 1/8$.

Step 8 — Limiting-case checks

(i) $T = 0$ saturation. At $T \to 0$, $K \to \infty$ and $\sinh(2 K) \to \infty$, so $\alpha^{2} = \sinh^{-4}(2K) \to 0$ and $M \to 1^{-}$. In fact $M(0) = 1$ is the saturated ferromagnetic limit — all spins aligned in the ground state, $\langle\sigma\rangle = \pm 1$.

(ii) $T = T_c$ criticality. At $K = K_c$, $\sinh(2 K_c) = 1$, so $\alpha^{2} = 1$ and $M(T_c) = (1 - 1)^{1/8} = 0$. Continuity of $M$ at $T_c$ from below confirms the second-order nature of the phase transition.

(iii) Universality-class cross-check. The 2D Ising CFT has primary-operator scaling dimensions $\Delta_{\sigma} = 1/8$ and $\Delta_{\varepsilon} = 1$, and a short-distance OPE $\sigma(\mathbf{r})\,\sigma(\mathbf{0}) \sim C/|\mathbf{r}|^{2\Delta_{\sigma}}$ at the critical point. The exponent $\eta = 2\Delta_{\sigma} = 1/4$ (Fisher exponent for the critical correlator) matches our $M \propto (T_c - T)^{1/8}$ through the scaling relation

\[\beta = \frac{\nu}{2}(d - 2 + \eta)\]

with $\nu = 1$ (correlation-length exponent of 2D Ising), $d = 2$, $\eta = 1/4$: $\beta = (1/2)(0 + 1/4) = 1/8$. ✓

Cross-check with Ising-scaling relations derived in ising-scaling-relations: Widom $\gamma = \beta(\delta - 1)$ with $\delta = 15$ gives $\gamma = 1/8 \cdot 14 = 7/4$, matching the Onsager-susceptibility result.

Remark on why $\beta = 1/8$

The exponent $\beta = 1/8$ traces back to the $(1/2) \times (1/2) × (1/2) = 1/8$ structure of Step 6: the 1/2 from taking the square root in (1) compounds with the 1/4 from the Szegő exponent (7), which is itself 1/2 × 1/2 from the Wiener–Hopf square-root factorisation of the symbol (3). In the closely related $c = 1/2$ Ising minimal-model CFT interpretation, the same factor $2\Delta_{\sigma} = 1/4$ appears as the scaling dimension of the two-point function $\langle\sigma\sigma\rangle$ — see ising-cft-primary-operators and ising-scaling-relations for the CFT-based derivation.

References

C. N. Yang, The spontaneous magnetization of a two-dimensional Ising model, Phys. Rev. 85, 808 (1952). Original derivation. Sections III–V give the Toeplitz reduction; §VI evaluates the limit via what is now called the strong Szegő theorem.
L. Onsager, Discussion remarks, Nuovo Cimento Supp. 6, 261 (1949). The announcement of the result $M = (1 - \sinh^{-4}(2K))^{1/8}$ predates Yang's derivation; Onsager never published his proof.
B. Kaufman, Crystal statistics II: Partition function evaluated by spinor analysis, Phys. Rev. 76, 1232 (1949). Transfer- matrix diagonalisation via JW / Clifford algebra; underlies the Pfaffian reduction of (2).
E. W. Montroll, R. B. Potts, J. C. Ward, Correlations and spontaneous magnetization of the two-dimensional Ising model, J. Math. Phys. 4, 308 (1963). Systematic derivation of the Toeplitz form (2) starting from Kaufman's result.
B. M. McCoy, T. T. Wu, The Two-Dimensional Ising Model, Harvard University Press (1973). Chapter VII is a textbook treatment of the Toeplitz reduction and the Wiener–Hopf factorisation. Equation (VII.2.36a) is our (3), and eq. (5.13) is our (4).
G. Szegő, Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion, Math. Ann. 76, 490 (1915). Original weak Szegő theorem (leading $G^n$).
G. Szegő, On certain Hermitian forms associated with the Fourier series of a positive function, Comm. Sém. Math. Univ. Lund, tome supplémentaire (1952), 228. The sharp form including the constant $E = \exp(\sum k c_k c_{-k})$.
M. Kac, Toeplitz matrices, translation kernels and a related problem in probability theory, Duke Math. J. 21, 501 (1954). Independent rigorous proof of the sharp Szegő theorem.
A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, 2nd ed. (Springer, 2006), Ch. 1–3. Modern pedagogical treatment.

Used by

IsingSquare model page — fetch(IsingSquare(; J, Lx, Ly), SpontaneousMagnetization(); β) returns exactly (8), with β = 1/8 as the order-parameter exponent.
Ising universality class — the exponent $\beta = 1/8$ is one of the four independent 2D Ising critical exponents. The other three ($\alpha = 0\text{ (log)}$, $\gamma = 7/4$, $\nu = 1$) follow by scaling relations (ising-scaling-relations).
Kramers–Wannier duality note — $T_c$ location at the self-dual point; the exponent calculation here confirms the critical point is a second-order transition.