Ising CFT: Primary Operators and Scaling Dimensions

Main result

The 2D classical Ising model at the self-dual critical point $\sinh(2\beta_c J) = 1$ is described at long distances by the simplest non-trivial unitary conformal field theory — the Virasoro minimal model $\mathcal{M}(3, 4)$ of central charge $c = 1/2$. This CFT contains exactly three primary operators, whose left-moving conformal weights $h$ (equal to the right-moving $\bar h$ since the theory is diagonal) and full scaling dimensions $\Delta = h + \bar h = 2h$ are

\[\boxed{\; \begin{array}{c|c|c|c} \text{Operator} & h = \bar h & \Delta & \text{Lattice correspondence}\\ \hline \mathbb{I} \text{ (identity)} & 0 & 0 & \text{1} \\ \sigma \text{ (spin)} & 1/16 & 1/8 & \lim_{a\to 0}\, \sigma^{z}(a\,r) \\ \varepsilon \text{ (energy)} & 1/2 & 1 & \lim_{a\to 0}\,\sigma^{z}_{i}\sigma^{z}_{i+1} - \langle \sigma^{z}\sigma^{z}\rangle_{c}\\ \end{array} \;}\]

The three scaling dimensions $(0, 1/8, 1)$ are the microscopic origin of every 2D Ising critical exponent (via standard CFT scaling relations):

\[\boxed{\;\eta = 2\Delta_{\sigma} = \tfrac{1}{4},\qquad \nu = \tfrac{1}{d - \Delta_{\varepsilon}} = 1,\qquad \beta = \nu\,\Delta_{\sigma} = \tfrac{1}{8},\qquad \alpha = 2 - \nu\,d = 0\;(\log),\qquad \delta = \tfrac{d + 2 - \eta}{d - 2 + \eta} = 15.\;}\]

In particular $\beta = 1/8$ is exactly the Yang-magnetisation exponent derived independently from the Toeplitz determinant in yang-magnetization-toeplitz, and the $\eta = 1/4$ correlation-function exponent matches $2\Delta_{\sigma}$ via Fisher's scaling relation.

The $(\sigma, \varepsilon)$ pair of non-trivial primaries is responsible for the two distinct massive integrable perturbations of the Ising CFT:

• \[\varepsilon\]

  perturbation → free massive Majorana fermion (thermal perturbation, $T \ne T_{c}$);

• \[\sigma\]

  perturbation → interacting integrable QFT with $E_{8}$ mass spectrum (magnetic perturbation, see e8-mass-spectrum-derivation and ising-cft-magnetic-perturbation).

Setup

Virasoro algebra

A 2D CFT with a unique vacuum is characterised by two copies of the Virasoro algebra (holomorphic + antiholomorphic),

\[[L_{m}, L_{n}] \;=\; (m - n)\,L_{m + n} \;+\; \frac{c}{12}\,m(m^{2} - 1)\,\delta_{m + n, 0},\]

with the same relation for $\bar L_{n}$ and $[L_{m}, \bar L_{n}] = 0$. The central charge $c$ parametrises the anomaly in the trace of the stress tensor under conformal rescalings; for the 2D Ising model $c = 1/2$ as we confirm in Step 4.

Primary operators are operators $\phi(z, \bar z)$ on which $L_{n}$ and $\bar L_{n}$ for $n > 0$ annihilate the state $|\phi\rangle = \phi(0)|0\rangle$:

\[L_{n}|\phi\rangle = 0,\qquad \bar L_{n}|\phi\rangle = 0\qquad (n > 0),\]

and $L_{0}|\phi\rangle = h\,|\phi\rangle$, $\bar L_{0}|\phi\rangle = \bar h\,|\phi\rangle$. The eigenvalues $(h, \bar h)$ are the conformal weights; the scaling dimension is $\Delta = h + \bar h$ and the spin is $s = h - \bar h$. All scaling operators in a CFT are either primaries or descendants (acted on by $L_{-n}$ with $n > 0$).

Minimal models

BPZ 1984 proved that unitary CFTs with $c < 1$ are discrete: the only allowed values of the central charge are

\[c_{p, p'} \;=\; 1 \;-\; \frac{6\,(p' - p)^{2}}{p\,p'}, \qquad p, p' \in \mathbb{Z}_{\ge 2},\ \gcd(p, p') = 1, \tag{1}\]

with the pair $(p, p')$ labelling the model $\mathcal{M}(p, p')$. Each $\mathcal{M}(p, p')$ has finitely many primary operators, listed by the Kac table (Step 3).

Goal

Derive:

1. \[\mathcal{M}(3, 4)\]

   has central charge $c = 1/2$.
2. The Kac table of $\mathcal{M}(3, 4)$ has exactly three inequivalent primary operators.
3. Their conformal weights are $h \in \{0, 1/16, 1/2\}$.
4. They identify with $\{\mathbb{I}, \sigma, \varepsilon\}$ of the lattice Ising model, reproducing the known critical exponents.

Calculation

Step 1 — Kac formula for conformal weights

For any minimal model $\mathcal{M}(p, p')$, the conformal weights of primary operators are given by the Kac formula

\[\boxed{\; h_{r, s} \;=\; \frac{(p'\,r - p\,s)^{2} - (p' - p)^{2}}{4\,p\,p'}, \qquad 1 \le r \le p - 1,\ \ 1 \le s \le p' - 1. \;} \tag{2}\]

This is the statement that a primary labelled by $(r, s)$ has a null descendant at level $r\,s$ in its Verma module, which constrains its four-point correlator to satisfy a linear ODE and fixes $h_{r, s}$ uniquely. The derivation requires the Kac determinant formula (Kac 1979) for the Gram matrix of a Verma module; see Di Francesco–Mathieu–Sénéchal 1997 §7.3 for the full treatment.

The Kac table has the reflection symmetry

\[h_{r, s} \;=\; h_{p - r,\, p' - s},\]

which follows from (2) by substituting $r \to p - r$, s \to p'

• s:

\[(p'(p - r) - p(p' - s))^{2} = (p'\,p - p'\,r - p\,p' + p\,s)^{2} = (p\,s - p'\,r)^{2} = (p'\,r - p\,s)^{2},\]

so the numerator of (2) is invariant and $h_{r,s} = h_{p-r, p'-s}$. This symmetry is the reason the Kac table has roughly half as many distinct entries as the $(p-1)(p'-1)$ rectangle.

Step 2 — $\mathcal{M}(3, 4)$: the Ising CFT

Substitute $p = 3$, $p' = 4$ into (1):

\[c_{3, 4} \;=\; 1 - \frac{6(4 - 3)^{2}}{3\cdot 4} \;=\; 1 - \frac{6}{12} \;=\; \frac{1}{2}. \tag{3}\]

The central charge $c = 1/2$ is the smallest positive value in the minimal-model series (the $c_{2, 3}$ model is trivial with only the identity operator; the next nontrivial case is $\mathcal{M}(3, 4)$). It is also the central charge of a single free Majorana fermion — not a coincidence, as we discuss at the end.

The Kac rectangle for $\mathcal{M}(3, 4)$ has $(p - 1)(p' - 1) = 2 \cdot 3 = 6$ entries labelled by $(r, s)$ with $r \in \{1, 2\}$ and $s \in \{1, 2, 3\}$.

Step 3 — Kac table for $\mathcal{M}(3, 4)$

Compute $h_{r, s}$ for all six $(r, s)$ pairs using (2) with $p = 3$, $p' = 4$, $p' - p = 1$, $4 p p' = 48$:

\[h_{r, s} \;=\; \frac{(4 r - 3 s)^{2} - 1}{48}.\]

Enumerate:

Apply the Kac-symmetry $h_{r, s} = h_{p - r, p' - s}$:

• \[(1, 1) \leftrightarrow (2, 3)\]

  : both $h = 0$. ✓

• \[(1, 2) \leftrightarrow (2, 2)\]

  : both $h = 1/16$. ✓

• \[(1, 3) \leftrightarrow (2, 1)\]

  : both $h = 1/2$. ✓

So the six entries collapse to three inequivalent conformal weights

\[\boxed{\;h \in \{0,\; 1/16,\; 1/2\}.\;} \tag{4}\]

Each of these is a self-conjugate primary (for a diagonal CFT like the Ising model, $(h, \bar h)$ are equal — left = right). Therefore the 2D Ising CFT has exactly three scalar primary operators, with full scaling dimensions

\[\Delta \in \{0,\ 1/8,\ 1\}. \tag{5}\]

Step 4 — Identification with lattice operators

We now identify the three primaries with physical operators on the Ising lattice. The identification uses the scaling dimension of lattice operators in the critical two-point function:

\[\bigl\langle\mathcal{O}(\mathbf{r})\,\mathcal{O}(\mathbf{0})\bigr\rangle_{T_{c}} \;\sim\; |\mathbf{r}|^{-2\Delta_{\mathcal{O}}}. \tag{6}\]

Identity $\mathbb{I}$, $\Delta = 0$: the trivial constant operator. Its correlator is the vacuum expectation, constant in $|\mathbf{r}|$.

Spin $\sigma$, $\Delta = 1/8$: the continuum limit of the Ising spin variable $\sigma^{z}_{i}$. The two-point function decays as

\[\langle\sigma^{z}(\mathbf{r})\,\sigma^{z}(\mathbf{0})\rangle_{T_{c}} \;\sim\; |\mathbf{r}|^{-2\Delta_{\sigma}} \;=\; |\mathbf{r}|^{-1/4}.\]

This matches the $\eta = 1/4$ Fisher exponent of the 2D Ising model (Fisher 1964; Stanley 1971 §12.4), derivable from the Toeplitz-determinant analysis of yang-magnetization-toeplitz.

Energy $\varepsilon$, $\Delta = 1$: the continuum limit of the bond-energy operator $\varepsilon_{i} \propto \sigma^{z}_{i}\sigma^{z}_{i+1} - \langle\sigma^{z}\sigma^{z}\rangle_{c}$ (the connected bond-bond correlator). Its conformal dimension $1$ reflects the thermal operator — perturbing by $\varepsilon$ tunes $T$ away from $T_{c}$, with $T - T_{c} \propto \int d^{2}x\,\varepsilon(x)$. The relation $\nu = 1/(d - \Delta_{\varepsilon}) = 1/(2 - 1) = 1$ reads off the correlation-length exponent directly.

The lattice → CFT identifications are

\[\sigma^{z}_{i} \;\xrightarrow[a \to 0]{}\; a^{\Delta_{\sigma}}\,\sigma(\mathbf{r}_{i}) \;=\; a^{1/8}\,\sigma(\mathbf{r}_{i}),\]

\[\sigma^{z}_{i}\sigma^{z}_{i+1} - \langle\sigma^{z}\sigma^{z}\rangle \;\xrightarrow[a \to 0]{}\; a^{\Delta_{\varepsilon}}\,\varepsilon(\mathbf{r}_{i}) \;=\; a^{1}\,\varepsilon(\mathbf{r}_{i}),\]

where the powers of the lattice spacing $a$ accompany the RG blocking of each operator under a rescaling $\mathbf{r} \to b\mathbf{r}$.

Step 5 — Critical exponents from $(\Delta_{\sigma}, \Delta_{\varepsilon})$

All critical exponents of the 2D Ising universality class follow from the two non-trivial scaling dimensions $\Delta_{\sigma} = 1/8$ and $\Delta_{\varepsilon} = 1$ via standard CFT / scaling relations (Cardy 1996 Ch. 3).

Fisher exponent $\eta$ (anomalous dimension of the order parameter). Definition: $\langle\sigma\sigma\rangle \sim |\mathbf{r}|^{-(d - 2 + \eta)}$. Compare with (6): $2\Delta_{\sigma} = d - 2 + \eta$, hence

\[\eta \;=\; 2\Delta_{\sigma} - d + 2 \;=\; 2\cdot\tfrac{1}{8} - 2 + 2 \;=\; \tfrac{1}{4}.\]

Correlation-length exponent $\nu$. Definition: $\xi \sim |T - T_{c}|^{-\nu}$. Scaling dimension of the thermal coupling $\delta K = \beta(T - T_{c})/J$ is $d - \Delta_{\varepsilon}$ (since $\int d^{2}x\,\varepsilon$ must be dimensionless). Hence

\[\nu \;=\; \frac{1}{d - \Delta_{\varepsilon}} \;=\; \frac{1}{2 - 1} \;=\; 1.\]

Order-parameter exponent $\beta$. Definition: $M \sim |T_{c} - T|^{\beta}$. Scaling of $M$ as $\xi^{-\Delta_{\sigma}}$ gives $M \sim \xi^{-\Delta_{\sigma}} \sim |T_{c} - T|^{\nu\Delta_{\sigma}}$, hence

\[\beta \;=\; \nu\,\Delta_{\sigma} \;=\; 1\cdot\tfrac{1}{8} \;=\; \tfrac{1}{8}.\]

Cross-check with yang-magnetization-toeplitz Step 7, which obtains $\beta = 1/8$ from the Szegő-theorem evaluation of the Toeplitz determinant: both derivations agree to first principles.

Susceptibility exponent $\gamma$. From Widom scaling $\gamma = \nu(2 - \eta) = 1 \cdot (2 - 1/4) = 7/4$. Or directly: the susceptibility scales as $\int d^{2}x\,\langle\sigma\sigma\rangle$ which diverges as $\xi^{d - 2\Delta_{\sigma}}$, giving $\chi \sim |T - T_{c}|^{-\nu(d - 2\Delta_{\sigma})} = |T - T_{c}|^{-7/4}$.

Specific-heat exponent $\alpha$. Josephson scaling \alpha = 2

• \nu d = 2 - 2 = 0$, i.e. $\alpha = 0$ with a logarithmic

divergence (Onsager 1944). This matches the exact 2D Ising specific heat $C \sim |\ln|T - T_{c}||$.

Critical isotherm $\delta$. Widom relation \delta = (d + 2

• \eta)/(d - 2 + \eta) = (4 - 1/4)/(0 + 1/4) = (15/4)/(1/4) = 15$.

Summary of the five independent 2D Ising exponents — all derived from the Kac table (2) + scaling relations:

Further scaling-relation cross-checks in ising-scaling-relations; the QAtlas stored values are in Ising universality class page.

Step 6 — Why $c = 1/2$? The free-Majorana-fermion

construction

The central charge $c = 1/2$ is also that of a single free Majorana fermion ($c = N/2$ for $N$ real Majoranas; a complex Dirac fermion has $c = 1$). This is not a coincidence: the 2D Ising CFT is a free Majorana fermion, via the Jordan–Wigner transformation of the lattice TFIM at the critical point $h = J$.

From jw-tfim-bdg Step 7(iii), the thermodynamic-limit dispersion of the TFIM at $h = J$ is $\Lambda(k) = 4 J\,|\sin(k/2)|$, vanishing linearly at $k = 0$ (and $k = 2\pi$). Linearising near $k = 0$ gives two chiral modes $\Psi_{R}(x), \Psi_{L}(x)$ forming a single Majorana fermion doublet. The CFT of the long-wavelength theory is therefore a single free Majorana — central charge $c = 1/2$. The three primaries $\mathbb{I}, \sigma, \varepsilon$ correspond to the identity, the disorder operator (non-local in fermion language, local in the $\mathbb{Z}_{2}$-dual Ising formulation), and the fermion bilinear $\bar\psi\psi$ (= $\sigma^{z}\sigma^{z}$ after JW) respectively.

The spin–disorder duality between the $\sigma$ and the dual-theory spin is the CFT avatar of Kramers–Wannier duality; see kramers-wannier-duality.

Step 7 — Limiting-case and consistency checks

(i) Identity operator. $\Delta_{\mathbb{I}} = 0$ as required of any CFT: $\langle\mathbb{I}\rangle = 1$ is constant in $\mathbf{r}$.

(ii) Unitarity of $\mathcal{M}(3, 4)$. The FQS theorem (Friedan–Qiu–Shenker 1984) classifies unitary minimal models as those with $p' = p + 1$; checking $\mathcal{M}(3, 4)$ has $p' - p = 1$, so it is unitary. The three primaries have non-negative conformal weights (0, 1/16, 1/2), consistent.

(iii) Numerical verification against finite-size ED. $\sigma$-type operator: the two-point function of $\sigma^{z}$ on the critical TFIM decays as $|\mathbf{r}|^{-1/4}$; the exponent is extracted from QAtlas's closed-form correlator in the "criticality: scaling toward CFT exponent -1/4" testset of test/models/test_TFIM_dynamics.jl, which asserts the effective doubling-ratio slope at $N \in \{80, 160, 240\}$ converges monotonically to $-1/4$ within $0.10$ at the largest $N$. The central charge $c = 1/2$ itself is extracted to $\lesssim 1\%$ from the PBC entanglement entropy in test/verification/test_entanglement_central_charge.jl.

(iv) Yang-magnetisation exponent $\beta = 1/8$. Derived independently from the Toeplitz determinant in yang-magnetization-toeplitz. Both derivations yield $\beta = 1/8$ — the microscopic confirmation of the CFT prediction $\beta = \nu\,\Delta_{\sigma} = 1 \cdot \tfrac{1}{8}$.

(v) Logarithmic specific heat. Onsager 1944 exactly solved the 2D Ising free energy and found $C \sim -\ln|T - T_{c}|$ near criticality. The CFT value $\alpha = 0$ corresponds to this logarithmic (not power-law) divergence; the $\alpha = 0$ value is "degenerate" with a $\ln$ correction, a common feature of 2D CFTs.

References

• A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241, 333 (1984). Original introduction of minimal models; central-charge formula (1) is BPZ eq. (3.24).
• V. G. Kac, Contravariant form for infinite-dimensional Lie algebras and superalgebras, in Group Theoretical Methods in Physics, Springer (1979) 441–445. The Kac determinant formula that determines the null-vector structure and hence (2).
• D. Friedan, Z. Qiu, S. Shenker, Conformal invariance, unitarity and critical exponents in two dimensions, Phys. Rev. Lett. 52, 1575 (1984). FQS classification of unitary minimal models.
• J. L. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press (1996), Ch. 3. Critical exponents from CFT + scaling relations.
• P. Di Francesco, P. Mathieu, D. Sénéchal, Conformal Field Theory, Springer (1997), Ch. 7. Pedagogical treatment of minimal models; the Kac formula derivation is in §7.3, and the Ising-CFT / free-Majorana identification is in §12.
• J. L. Cardy, Conformal field theory and statistical mechanics, Les Houches lecture notes (2008), arXiv:0807.3472. Modern review.

Used by

• Ising universality class — the stored exponents $\beta = 1/8, \gamma = 7/4, \nu = 1, \eta = 1/4, \alpha = 0, \delta = 15$ are read off from the scaling dimensions derived here.
• yang-magnetization-toeplitz — independent microscopic derivation of $\beta = 1/8$ via the Szegő / Toeplitz determinant; agrees with the CFT $\beta = \nu\,\Delta_{\sigma}$ prediction.
• ising-scaling-relations — the scaling relations linking all six exponents are verified in Step 5 above.
• e8-mass-spectrum-derivation — the $\sigma$-perturbation channel produces the $E_{8}$ massive integrable theory.
• ising-cft-magnetic-perturbation — conserved-charges mechanism for the $E_{8}$ perturbation.
• kramers-wannier-duality — the CFT version of KW duality is the spin↔disorder duality between $\sigma$ and its $\mathbb{Z}_{2}$-dual primary.
• TFIM model page — $h = J$ is the lattice realisation of the Ising CFT critical point.