Ising Conformal Field Theory

The Ising conformal field theory describes the universal long-distance limit of the two-dimensional Ising model at its continuous transition, and of the critical one-dimensional quantum Ising chain. Its small operator spectrum makes it an exact laboratory for scaling, fusion, and relevant deformations. The CFT is the critical continuum theory, not the lattice model at arbitrary temperature or field.

The critical Ising operator spectrum

Consider the diagonal bosonic minimal model $\mathcal M(3,4)$, with left and right central charges $c=\bar c=1/2$. Its scalar primary fields are

Field	Meaning	Left/right weights	Scaling dimension
$\mathbf1$	Identity	$(0,0)$	$0$
$\sigma$	Spin or order field	$(1/16,1/16)$	$1/8$
$\varepsilon$	Energy field	$(1/2,1/2)$	$1$

The scaling dimension is $\Delta=h+\bar h$. Descendants generated by the conformal algebra complete each primary family. The fusion rules are

$$[\sigma]\times[\sigma]=[\mathbf1]+[\varepsilon], \qquad [\sigma]\times[\varepsilon]=[\sigma], \qquad [\varepsilon]\times[\varepsilon]=[\mathbf1].$$

These are statements about which conformal families can occur, not numerical values of all OPE coefficients. The minimal-model construction appears in Di Francesco, Mathieu, and Sénéchal 1997, §§7.4.2 and 8.4.

From dimensions to measurable scaling

For a unit-normalized scalar primary, scale invariance gives a two-point function proportional to $\lvert x\rvert^{-2\Delta}$. Thus at criticality,

$$\langle\sigma(x)\sigma(0)\rangle\propto\lvert x\rvert^{-1/4}, \qquad \langle\varepsilon(x)\varepsilon(0)\rangle \propto\lvert x\rvert^{-2}.$$

The long-distance exponents are universal; the proportionality constants for a microscopic lattice spin depend on its continuum normalization.

A Euclidean deformation $\int\mathrm d^2x\,[t\varepsilon+h\sigma]$ has coupling dimensions $[t]=1$ and $[h]=15/8$. At zero magnetic field, the thermal coupling therefore gives $\xi\propto\lvert t\rvert^{-1}$ near the transition. The proportionality factor and the identification of the sign of $t$ require a microscopic dictionary.

Majorana fields and global qualifications

A free massless Majorana fermion realizes the same local stress-tensor algebra. The thermal deformation becomes a fermion mass, which makes one departure from criticality especially tractable. However, the bosonic Ising model also contains spin fields and requires an appropriate combination of fermionic sectors. A fermion theory on one fixed spin structure is not automatically the complete bosonic Ising theory on every surface.

The model is useful for exact tests of conformal blocks, finite-size spectra, and relevant perturbations. Its small central charge and exact solvability do not imply that all critical points share its exponents.

Develop the calculation

Minimal Models and Fusion Rules develops the classification. Ising CFT, Majorana Fermions, and Tricritical Extensions explains the fermionic realization. Compare the continuously varying spectrum of the Compact Free Boson.

References

• Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Springer, 1997, §§7.4.2 and 8.4. DOI.