Two-dimensional critical Ising model

The two-dimensional critical Ising model is the critical limit of the Ising model in two dimensions. It is a two-dimensional conformal field theory whose symmetry algebra is the Virasoro algebra with the central charge $c={\tfrac {1}{2}}$. Correlation functions of the spin and energy operators are described by the $(4,3)$ minimal model. While the minimal model has been exactly solved, the solution does not cover other observables such as connectivities of clusters.

The minimal model

Spectrum and conformal dimensions

The Kac table of the $(4,3)$ minimal model is:

${\begin{array}{c|ccc}2&{\frac {1}{2}}&{\frac {1}{16}}&0\\1&0&{\frac {1}{16}}&{\frac {1}{2}}\\\hline &1&2&3\end{array}}$

This means that the spectrum is generated by three primary states, which correspond to three primary fields or operators:

${\begin{array}{cccc}\hline {\text{Kac table indices}}&{\text{Dimension}}&{\text{Primary field}}&{\text{Name}}\\\hline (1,1){\text{ or }}(3,2)&0&\mathbf {1} &{\text{Identity}}\\(2,1){\text{ or }}(2,2)&{\frac {1}{16}}&\sigma &{\text{Spin}}\\(1,2){\text{ or }}(3,1)&{\frac {1}{2}}&\epsilon &{\text{Energy}}\\\hline \end{array}}$

The decomposition of the spectrum into irreducible representations of the product of the left- and right-moving Virasoro algebras is

${\mathcal {S}}={\mathcal {R}}_{0}\otimes {\bar {\mathcal {R}}}_{0}\oplus {\mathcal {R}}_{\frac {1}{16}}\otimes {\bar {\mathcal {R}}}_{\frac {1}{16}}\oplus {\mathcal {R}}_{\frac {1}{2}}\otimes {\bar {\mathcal {R}}}_{\frac {1}{2}}$

where ${\mathcal {R}}_{\Delta }$ is the irreducible highest-weight representation of the Virasoro algebra with the conformal dimension $\Delta$. In particular, the Ising model is diagonal and unitary.

Characters and partition function

The characters of the three representations of the Virasoro algebra that appear in the spectrum are

${\begin{aligned}\chi _{0}(q)&={\frac {1}{\eta (q)}}\sum _{k\in \mathbb {Z} }\left(q^{\frac {(24k+1)^{2}}{48}}-q^{\frac {(24k+7)^{2}}{48}}\right)={\frac {1}{2{\sqrt {\eta (q)}}}}\left({\sqrt {\theta _{3}(0|q)}}+{\sqrt {\theta _{4}(0|q)}}\right)\\\chi _{\frac {1}{16}}(q)&={\frac {1}{\eta (q)}}\sum _{k\in \mathbb {Z} }\left(q^{\frac {(24k+2)^{2}}{48}}-q^{\frac {(24k+10)^{2}}{48}}\right)={\frac {1}{2{\sqrt {\eta (q)}}}}\left({\sqrt {\theta _{3}(0|q)}}-{\sqrt {\theta _{4}(0|q)}}\right)\\\chi _{\frac {1}{2}}(q)&={\frac {1}{\eta (q)}}\sum _{k\in \mathbb {Z} }\left(q^{\frac {(24k+5)^{2}}{48}}-q^{\frac {(24k+11)^{2}}{48}}\right)={\frac {1}{\sqrt {2\eta (q)}}}{\sqrt {\theta _{2}(0|q)}}\end{aligned}}$

where $\eta (q)$ is the Dedekind eta function, and $\theta _{i}(0|q)$ are theta functions of the nome $q=e^{2\pi i\tau }$, for example $\theta _{3}(0|q)=\sum _{n\in \mathbb {Z} }q^{\frac {n^{2}}{2}}$. The modular S-matrix, i.e. the matrix ${\mathcal {S}}$ such that $\chi _{i}(-{\tfrac {1}{\tau }})=\sum _{j}{\mathcal {S}}_{ij}\chi _{j}(\tau )$, is

${\mathcal {S}}={\frac {1}{2}}\left({\begin{array}{ccc}1&1&{\sqrt {2}}\\1&1&-{\sqrt {2}}\\{\sqrt {2}}&-{\sqrt {2}}&0\end{array}}\right)$

where the fields are ordered as $1,\sigma ,\epsilon$. The modular invariant partition function is

$Z(q)=\left|\chi _{0}(q)\right|^{2}+\left|\chi _{\frac {1}{16}}(q)\right|^{2}+\left|\chi _{\frac {1}{2}}(q)\right|^{2}={\frac {|\theta _{2}(0|q)|+|\theta _{3}(0|q)|+|\theta _{4}(0|q)|}{2|\eta (q)|}}$

Fusion rules and operator product expansions

The fusion rules of the model are

${\begin{aligned}\mathbf {1} \times \mathbf {1} &=\mathbf {1} \\\mathbf {1} \times \sigma &=\sigma \\\mathbf {1} \times \epsilon &=\epsilon \\\sigma \times \sigma &=\mathbf {1} +\epsilon \\\sigma \times \epsilon &=\sigma \\\epsilon \times \epsilon &=\mathbf {1} \end{aligned}}$

The fusion rules are invariant under the $\mathbb {Z} _{2}$ symmetry $\sigma \to -\sigma$. The three-point structure constants are

$C_{\mathbf {1} \mathbf {1} \mathbf {1} }=C_{\mathbf {1} \epsilon \epsilon }=C_{\mathbf {1} \sigma \sigma }=1\quad ,\quad C_{\sigma \epsilon \epsilon }={\frac {1}{2}}$

Knowing the fusion rules and three-point structure constants, it is possible to write operator product expansions, for example

${\begin{aligned}\sigma (z)\sigma (0)&=|z|^{2\Delta _{\mathbf {1} }-4\Delta _{\sigma }}C_{\mathbf {1} \sigma \sigma }{\Big (}\mathbf {1} (0)+O(z){\Big )}+|z|^{2\Delta _{\epsilon }-4\Delta _{\sigma }}C_{\sigma \sigma \epsilon }{\Big (}\epsilon (0)+O(z){\Big )}\\&=|z|^{-{\frac {1}{4}}}{\Big (}\mathbf {1} (0)+O(z){\Big )}+{\frac {1}{2}}|z|^{\frac {3}{4}}{\Big (}\epsilon (0)+O(z){\Big )}\end{aligned}}$

where $\Delta _{\mathbf {1} },\Delta _{\sigma },\Delta _{\epsilon }$ are the conformal dimensions of the primary fields, and the omitted terms $O(z)$ are contributions of descendent fields.

Correlation functions on the sphere

Any one-, two- and three-point function of primary fields is determined by conformal symmetry up to a multiplicative constant. This constant is set to be one for one- and two-point functions by a choice of field normalizations. The only non-trivial dynamical quantities are the three-point structure constants, which were given above in the context of operator product expansions.

$\left\langle \mathbf {1} (z_{1})\right\rangle =1\ ,\ \left\langle \sigma (z_{1})\right\rangle =0\ ,\ \left\langle \epsilon (z_{1})\right\rangle =0$

$\left\langle \mathbf {1} (z_{1})\mathbf {1} (z_{2})\right\rangle =1\ ,\ \left\langle \sigma (z_{1})\sigma (z_{2})\right\rangle =|z_{12}|^{-{\frac {1}{4}}}\ ,\ \left\langle \epsilon (z_{1})\epsilon (z_{2})\right\rangle =|z_{12}|^{-2}$

with $z_{ij}=z_{i}-z_{j}$.

$\langle \mathbf {1} \sigma \rangle =\langle \mathbf {1} \epsilon \rangle =\langle \sigma \epsilon \rangle =0$

$\left\langle \mathbf {1} (z_{1})\mathbf {1} (z_{2})\mathbf {1} (z_{3})\right\rangle =1\ ,\ \left\langle \sigma (z_{1})\sigma (z_{2})\mathbf {1} (z_{3})\right\rangle =|z_{12}|^{-{\frac {1}{4}}}\ ,\ \left\langle \epsilon (z_{1})\epsilon (z_{2})\mathbf {1} (z_{3})\right\rangle =|z_{12}|^{-2}$

$\left\langle \sigma (z_{1})\sigma (z_{2})\epsilon (z_{3})\right\rangle ={\frac {1}{2}}|z_{12}|^{\frac {3}{4}}|z_{13}|^{-1}|z_{23}|^{-1}$

$\langle \mathbf {1} \mathbf {1} \sigma \rangle =\langle \mathbf {1} \mathbf {1} \epsilon \rangle =\langle \mathbf {1} \sigma \epsilon \rangle =\langle \sigma \epsilon \epsilon \rangle =\langle \sigma \sigma \sigma \rangle =\langle \epsilon \epsilon \epsilon \rangle =0$

The three non-trivial four-point functions are of the type $\langle \sigma ^{4}\rangle ,\langle \sigma ^{2}\epsilon ^{2}\rangle ,\langle \epsilon ^{4}\rangle$. For a four-point function $\left\langle \prod _{i=1}^{4}V_{i}(z_{i})\right\rangle$, let ${\mathcal {F}}_{j}^{(s)}$ and ${\mathcal {F}}_{j}^{(t)}$ be the s- and t-channel Virasoro conformal blocks, which respectively correspond to the contributions of $V_{j}(z_{2})$ (and its descendents) in the operator product expansion $V_{1}(z_{1})V_{2}(z_{2})$, and of $V_{j}(z_{4})$ (and its descendents) in the operator product expansion $V_{1}(z_{1})V_{4}(z_{4})$. Let $x={\frac {z_{12}z_{34}}{z_{13}z_{24}}}$ be the cross-ratio.

In the case of $\langle \epsilon ^{4}\rangle$, fusion rules allow only one primary field in all channels, namely the identity field.

${\begin{aligned}&\langle \epsilon ^{4}\rangle =\left|{\mathcal {F}}_{\textbf {1}}^{(s)}\right|^{2}=\left|{\mathcal {F}}_{\textbf {1}}^{(t)}\right|^{2}\\&{\mathcal {F}}_{\textbf {1}}^{(s)}={\mathcal {F}}_{\textbf {1}}^{(t)}=\left[\prod _{1\leq i<j\leq 4}z_{ij}^{-{\frac {1}{3}}}\right]{\frac {1-x+x^{2}}{x^{\frac {2}{3}}(1-x)^{\frac {2}{3}}}}\ {\underset {(z_{i})=(x,0,\infty ,1)}{=}}\ {\frac {1}{x(1-x)}}-1\end{aligned}}$

In the case of $\langle \sigma ^{2}\epsilon ^{2}\rangle$, fusion rules allow only the identity field in the s-channel, and the spin field in the t-channel.

${\begin{aligned}&\langle \sigma ^{2}\epsilon ^{2}\rangle =\left|{\mathcal {F}}_{\textbf {1}}^{(s)}\right|^{2}=C_{\sigma \sigma \epsilon }^{2}\left|{\mathcal {F}}_{\sigma }^{(t)}\right|^{2}={\frac {1}{4}}\left|{\mathcal {F}}_{\sigma }^{(t)}\right|^{2}\\&{\mathcal {F}}_{\textbf {1}}^{(s)}={\frac {1}{2}}{\mathcal {F}}_{\sigma }^{(t)}=\left[z_{12}^{\frac {1}{4}}z_{34}^{-{\frac {5}{8}}}\left(z_{13}z_{24}z_{14}z_{23}\right)^{-{\frac {3}{16}}}\right]{\frac {1-{\frac {x}{2}}}{x^{\frac {3}{8}}(1-x)^{\frac {5}{16}}}}\ {\underset {(z_{i})=(x,0,\infty ,1)}{=}}\ {\frac {1-{\frac {x}{2}}}{x^{\frac {1}{8}}(1-x)^{\frac {1}{2}}}}\end{aligned}}$

From the representation of the model in terms of Dirac fermions, it is possible to compute correlation functions of any number of spin or energy operators:

$\left\langle \prod _{i=1}^{2n}\epsilon (z_{i})\right\rangle ^{2}=\left|\det \left({\frac {1}{z_{ij}}}\right)_{1\leq i\neq j\leq 2n}\right|^{2}$

$\left\langle \prod _{i=1}^{2n}\sigma (z_{i})\right\rangle ^{2}=\sum _{\begin{array}{c}\epsilon _{i}=\pm 1\\\sum _{i=1}^{2n}\epsilon _{i}=0\end{array}}\prod _{1\leq i<j\leq 2n}|z_{ij}|^{\frac {\epsilon _{i}\epsilon _{j}}{2}}$

These formulas have generalizations to correlation functions on the torus, which involve theta functions.

Relations with other CFTs

Free fermions? Fermionic Ising model

Better distinguish the minimal model from the CFT. Cite Delfino-Viti? Potts model. Mention connectivities. Other observables outside the minimal model: disorder fields, cf BYB.

3pt connectivity obviously not described by minimal model, whose spin operator has vanishing three-point function.