Two-dimensional critical Ising model

=== Space of states 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 space of states 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) & \frac12 & \epsilon & \text{Energy}

\\

\hline

\end{array}

The decomposition of the space of states 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}_\frac12 \otimes \bar{\mathcal{R}}_\frac12

where $\backslash mathcal\{R\}\_\backslash Delta$ is the irreducible highest-weight representation of the Virasoro algebra with the conformal dimension $\backslash Delta$.

In particular, the Ising model is diagonal and unitary.

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

:$$

\begin{align}

\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_{\frac12}(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{align}

where $\backslash eta(q)$ is the Dedekind eta function, and $\backslash theta\_i(0|q)$ are theta functions of the nome $q=e^\{2\backslash pi\; i\backslash tau\}$, for example $\backslash theta\_3(0|q)=\backslash sum\_\{n\backslash in\backslash mathbb\{Z\}\}\; q^\{\backslash frac\{n^2\}\{2\}\}$.

The modular S-matrix, i.e. the matrix $\backslash mathcal\{S\}$ such that $\backslash chi\_i(-\backslash tfrac\{1\}\{\backslash tau\})\; =\; \backslash sum\_j\; \backslash mathcal\{S\}\_\{ij\}\backslash chi\_j(\backslash tau)$, is

:$$

\mathcal{S} = \frac12 \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 $1,\backslash epsilon,\; \backslash sigma$.

The modular invariant partition function is

:$$

Z(q) = \left|\chi_0(q)\right|^2 + \left|\chi_{\frac{1}{16}}(q)\right|^2

+ \left|\chi_\frac12(q)\right|^2 = \frac

\theta_2(0|q)|+ |\theta_3(0|q)|+|\theta_4(0|q)

{2|\eta(q)|}

The fusion rules of the model are

:$$

\begin{align}

\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{align}

The fusion rules are invariant under the $\backslash mathbb\{Z\}\_2$ symmetry $\backslash sigma\; \backslash to\; -\backslash sigma$.

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\sigma\epsilon} = \frac12

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

:$$

\begin{align}

\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|^{-\frac14} \Big(\mathbf{1}(0) + O(z)\Big) +\frac12 |z|^\frac34 \Big(\epsilon(0) + O(z)\Big)

\end{align}

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

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}|^{-\frac14} \ , \ \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}|^{-\frac14}

\ , \ \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 = \frac12 |z_{12}|^{\frac34} |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 $\backslash langle\; \backslash sigma^4\backslash rangle,\; \backslash langle\; \backslash sigma^2\backslash epsilon^2\backslash rangle,\; \backslash langle\; \backslash epsilon^4\backslash rangle$. For a four-point function $\backslash left\backslash langle\backslash prod\_\{i=1\}^4\; V\_i(z\_i)\backslash right\backslash rangle$, let $\backslash mathcal\{F\}^\{(s)\}\_j$ and $\backslash mathcal\{F\}^\{(t)\}\_j$ be the s- and t-channel Virasoro conformal blocks, which respectively correspond to the contributions of $V\_j(z\_2)$ (and its descendants) in the operator product expansion $V\_1(z\_1)V\_2(z\_2)$, and of $V\_j(z\_4)$ (and its descendants) in the operator product expansion $V\_1(z\_1)V\_4(z\_4)$. Let $x=\backslash frac\{z\_\{12\}z\_\{34\}\}\{z\_\{13\}z\_\{24\}\}$ be the cross-ratio.

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

:$$

\begin{align}

& \langle \epsilon^4\rangle = \left|\mathcal{F}^{(s)}_\textbf{1}\right|^2 = \left|\mathcal{F}^{(t)}_\textbf{1}\right|^2

\\

& \mathcal{F}^{(s)}_\textbf{1}

= \mathcal{F}^{(t)}_\textbf{1}

= \left[\prod_{1\leq i

\ \underset{(z_i)=(x, 0,\infty, 1)}{=}\ \frac{1}{x(1-x)} -1

\end{align}

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

:$$

\begin{align}

& \langle \sigma^2\epsilon^2\rangle = \left|\mathcal{F}^{(s)}_\textbf{1}\right|^2 = C_{\sigma\sigma\epsilon}^2\left|\mathcal{F}^{(t)}_\sigma\right|^2 = \frac14\left|\mathcal{F}^{(t)}_\sigma\right|^2

\\

& \mathcal{F}^{(s)}_\textbf{1}

= \frac12 \mathcal{F}^{(t)}_\sigma

=\left[z_{12}^\frac14 z_{34}^{-\frac58}\left(z_{13}z_{24}z_{14}z_{23}\right)^{-\frac{3}{16}} \right]\frac{1-\frac{x}{2}}{x^\frac38(1-x)^\frac{5}{16}}

\ \underset{(z_i)=(x, 0,\infty, 1)}{=}\

\frac{1-\frac{x}{2}}{x^\frac18(1-x)^\frac12}

\end{align}

In the case of $\backslash langle\; \backslash sigma^4\backslash rangle$, fusion rules allow two primary fields in all channels: the identity field and the energy field. In this case we write the conformal blocks in the case $(z\_1,z\_2,z\_3,z\_4)=(x,0,\backslash infty,1)$ only: the general case is obtained by inserting the prefactor

:$$

\begin{align}

\langle \sigma^4\rangle &=

\left|\mathcal{F}_\textbf{1}^{(s)}\right|^2 + \frac14 \left|\mathcal{F}_{\epsilon}^{(s)}\right|^2

=

\left|\mathcal{F}_\textbf{1}^{(t)}\right|^2 + \frac14 \left|\mathcal{F}_{\epsilon}^{(t)}\right|^2

\\

&=

\frac

1+\sqrt{x}|+|1-\sqrt{x}

{2|x|^\frac14 |1-x|^\frac14}

\ \underset{x\in (0, 1)}{=}\ \frac{1}

x|^\frac14 |1-x|^\frac14} \end{align} In the case of $\backslash langle\; \backslash sigma^4\backslash rangle$, the conformal blocks are: :$$ \begin{align} & \mathcal{F}_\textbf{1}^{(s)} = \frac{\sqrt{\frac{1+\sqrt{1-x}}{2}}}{x^\frac18(1-x)^\frac18} \ ,\;\; \mathcal{F}_{\epsilon}^{(s)} = \frac{\sqrt{2-2\sqrt{1-x}}}{x^\frac18(1-x)^\frac18} \\ & \mathcal{F}_\textbf{1}^{(t)} = \frac{\mathcal{F}^{(s)}_\textbf{1}}{\sqrt{2}} + \frac{\mathcal{F}^{(s)}_\epsilon}{2\sqrt{2}} = \frac{\sqrt{\frac{1+\sqrt{x}}{2}}}{x^\frac18(1-x)^\frac18} \ ,\;\; \mathcal{F}_{\epsilon}^{(t)} = \sqrt{2}\mathcal{F}^{(s)}_\textbf{1} - \frac{\mathcal{F}^{(s)}_\epsilon}{\sqrt{2}} = \frac{\sqrt{2-2\sqrt{x}}}{x^\frac18(1-x)^\frac18} \end{align} From the representation of the model in terms of Dirac fermions, it is possible to compute correlation functions of any number of spin or energy operators: :$$ \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 :$$ \left\langle \prod_{i=1}^{2n} \sigma(z_i)\right\rangle^2 = \frac{1}{2^n}\sum_{\begin{array}{c}\epsilon_i=\pm 1 \\ \sum_{i=1}^{2n}\epsilon_i=0\end{array}} \prod_{1\leq i These formulas have generalizations to correlation functions on the torus, which involve theta functions. == Other observables == === Disorder operator === The two-dimensional Ising model is mapped to itself by a high-low temperature duality. The image of the spin operator $\backslash sigma$ under this duality is a disorder operator $\backslash mu$, which has the same left and right conformal dimensions $(\backslash Delta\_\backslash mu,\backslash bar\backslash Delta\_\backslash mu)\; =\; (\backslash Delta\_\backslash sigma,\backslash bar\; \backslash Delta\_\backslash sigma)=(\backslash tfrac\{1\}\{16\},\backslash tfrac\{1\}\{16\})$. Although the disorder operator does not belong to the minimal model, correlation functions involving the disorder operator can be computed exactly, for example :$$ \left\langle \sigma(z_1)\mu(z_2)\sigma(z_3)\mu(z_4)\right\rangle^2 = \frac12 \sqrt{\frac{|z_{13}z_{24}

z_{12}z_{34}z_{23}z_{14}

}

\Big( |x|+|1-x|-1 \Big)

whereas

:$$

\left\langle \prod_{i=1}^4\mu(z_i)\right\rangle^2

=

\left\langle \prod_{i=1}^4\sigma(z_i)\right\rangle^2

= \frac12 \sqrt{\frac

z_{13}z_{24}

z_{12}z_{34}z_{23}z_{14}

}

\Big( |x|+|1-x|+1 \Big)

The Ising model has a description as a random cluster model due to Fortuin and Kasteleyn. In this description, the natural observables are connectivities of clusters, i.e. probabilities that a number of points belong to the same cluster.

The Ising model can then be viewed as the case $q=2$ of the $q$-state Potts model, whose parameter $q$ can vary continuously, and is related to the central charge of the Virasoro algebra.

In the critical limit, connectivities of clusters have the same behaviour under conformal transformations as correlation functions of the spin operator. Nevertheless, connectivities do not coincide with spin correlation functions: for example, the three-point connectivity does not vanish, while $\backslash langle\backslash sigma\backslash sigma\backslash sigma\backslash rangle=0$. There are four independent four-point connectivities, and their sum coincides with $\backslash langle\backslash sigma\backslash sigma\backslash sigma\backslash sigma\backslash rangle$. Other combinations of four-point connectivities are not known analytically. In particular they are not related to correlation functions of the minimal model, although they are related to the $q\backslash to\; 2$ limit of spin correlators in the $q$-state Potts model.

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{{cite journal | last1=Delfino | first1=Gesualdo | last2=Viti | first2=Jacopo | title=Potts q-color field theory and scaling random cluster model | journal=Nuclear Physics B | date=2011-04-21 | volume=852 | issue=1 | pages=149–173 | doi=10.1016/j.nuclphysb.2011.06.012 | arxiv=1104.4323v2 | bibcode=2011NuPhB.852..149D | s2cid=119183802 }}

{{cite arXiv | last1=Cheng | first1=Miranda C. N. | last2=Gannon | first2=Terry | last3=Lockhart | first3=Guglielmo | title=Modular Exercises for Four-Point Blocks -- I | date=2020-02-25 | class=hep-th | eprint=2002.11125v1 }}

P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997, {{ISBN|0-387-94785-X}}

}}

Category:Exactly solvable models

Category:Conformal field theory

Category:Lattice models

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Category:Statistical mechanics