Conformal field theory of the 2D Ising model critical point
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
=
1
2
{\displaystyle c={\tfrac {1}{2}}}
.
Correlation functions of the spin and energy operators are described by the
(
4
,
3
)
{\displaystyle (4,3)}
minimal model . While the minimal model has been exactly solved, see also, e.g., the article on Ising critical exponents , the solution does not cover other observables such as connectivities of clusters.
The Kac table of the
(
4
,
3
)
{\displaystyle (4,3)}
minimal model is:
2
1
2
1
16
0
1
0
1
16
1
2
1
2
3
{\displaystyle {\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:[ 1]
Kac table indices
Dimension
Primary field
Name
(
1
,
1
)
or
(
3
,
2
)
0
1
Identity
(
2
,
1
)
or
(
2
,
2
)
1
16
σ
Spin
(
1
,
2
)
or
(
3
,
1
)
1
2
ϵ
Energy
{\displaystyle {\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 space of states into irreducible representations of the product of the left- and right-moving Virasoro algebras is
S
=
R
0
⊗
R
¯
0
⊕
R
1
16
⊗
R
¯
1
16
⊕
R
1
2
⊗
R
¯
1
2
{\displaystyle {\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
R
Δ
{\displaystyle {\mathcal {R}}_{\Delta }}
is the irreducible highest-weight representation of the Virasoro algebra with the conformal dimension
Δ
{\displaystyle \Delta }
.
In particular, the Ising model is diagonal and unitary.
Characters and partition function [ edit ]
The characters of the three representations of the Virasoro algebra that appear in the space of states are[ 1]
χ
0
(
q
)
=
1
η
(
q
)
∑
k
∈
Z
(
q
(
24
k
+
1
)
2
48
−
q
(
24
k
+
7
)
2
48
)
=
1
2
η
(
q
)
(
θ
3
(
0
|
q
)
+
θ
4
(
0
|
q
)
)
χ
1
16
(
q
)
=
1
η
(
q
)
∑
k
∈
Z
(
q
(
24
k
+
2
)
2
48
−
q
(
24
k
+
10
)
2
48
)
=
1
2
η
(
q
)
(
θ
3
(
0
|
q
)
−
θ
4
(
0
|
q
)
)
χ
1
2
(
q
)
=
1
η
(
q
)
∑
k
∈
Z
(
q
(
24
k
+
5
)
2
48
−
q
(
24
k
+
11
)
2
48
)
=
1
2
η
(
q
)
θ
2
(
0
|
q
)
{\displaystyle {\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
η
(
q
)
{\displaystyle \eta (q)}
is the Dedekind eta function , and
θ
i
(
0
|
q
)
{\displaystyle \theta _{i}(0|q)}
are theta functions of the nome
q
=
e
2
π
i
τ
{\displaystyle q=e^{2\pi i\tau }}
, for example
θ
3
(
0
|
q
)
=
∑
n
∈
Z
q
n
2
2
{\displaystyle \theta _{3}(0|q)=\sum _{n\in \mathbb {Z} }q^{\frac {n^{2}}{2}}}
.
The modular S-matrix , i.e. the matrix
S
{\displaystyle {\mathcal {S}}}
such that
χ
i
(
−
1
τ
)
=
∑
j
S
i
j
χ
j
(
τ
)
{\displaystyle \chi _{i}(-{\tfrac {1}{\tau }})=\sum _{j}{\mathcal {S}}_{ij}\chi _{j}(\tau )}
, is[ 1]
S
=
1
2
(
1
1
2
1
1
−
2
2
−
2
0
)
{\displaystyle {\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
,
ϵ
,
σ
{\displaystyle 1,\epsilon ,\sigma }
.
The modular invariant partition function is
Z
(
q
)
=
|
χ
0
(
q
)
|
2
+
|
χ
1
16
(
q
)
|
2
+
|
χ
1
2
(
q
)
|
2
=
|
θ
2
(
0
|
q
)
|
+
|
θ
3
(
0
|
q
)
|
+
|
θ
4
(
0
|
q
)
|
2
|
η
(
q
)
|
{\displaystyle 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 [ edit ]
The fusion rules of the model are
1
×
1
=
1
1
×
σ
=
σ
1
×
ϵ
=
ϵ
σ
×
σ
=
1
+
ϵ
σ
×
ϵ
=
σ
ϵ
×
ϵ
=
1
{\displaystyle {\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
Z
2
{\displaystyle \mathbb {Z} _{2}}
symmetry
σ
→
−
σ
{\displaystyle \sigma \to -\sigma }
.
The three-point structure constants are
C
1
1
1
=
C
1
ϵ
ϵ
=
C
1
σ
σ
=
1
,
C
σ
σ
ϵ
=
1
2
{\displaystyle C_{\mathbf {1} \mathbf {1} \mathbf {1} }=C_{\mathbf {1} \epsilon \epsilon }=C_{\mathbf {1} \sigma \sigma }=1\quad ,\quad C_{\sigma \sigma \epsilon }={\frac {1}{2}}}
Knowing the fusion rules and three-point structure constants, it is possible to write operator product expansions, for example
σ
(
z
)
σ
(
0
)
=
|
z
|
2
Δ
1
−
4
Δ
σ
C
1
σ
σ
(
1
(
0
)
+
O
(
z
)
)
+
|
z
|
2
Δ
ϵ
−
4
Δ
σ
C
σ
σ
ϵ
(
ϵ
(
0
)
+
O
(
z
)
)
=
|
z
|
−
1
4
(
1
(
0
)
+
O
(
z
)
)
+
1
2
|
z
|
3
4
(
ϵ
(
0
)
+
O
(
z
)
)
{\displaystyle {\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
Δ
1
,
Δ
σ
,
Δ
ϵ
{\displaystyle \Delta _{\mathbf {1} },\Delta _{\sigma },\Delta _{\epsilon }}
are the conformal dimensions of the primary fields, and the omitted terms
O
(
z
)
{\displaystyle O(z)}
are contributions of descendant fields .
Correlation functions on the sphere [ edit ]
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.
⟨
1
(
z
1
)
⟩
=
1
,
⟨
σ
(
z
1
)
⟩
=
0
,
⟨
ϵ
(
z
1
)
⟩
=
0
{\displaystyle \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}
⟨
1
(
z
1
)
1
(
z
2
)
⟩
=
1
,
⟨
σ
(
z
1
)
σ
(
z
2
)
⟩
=
|
z
12
|
−
1
4
,
⟨
ϵ
(
z
1
)
ϵ
(
z
2
)
⟩
=
|
z
12
|
−
2
{\displaystyle \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
i
j
=
z
i
−
z
j
{\displaystyle z_{ij}=z_{i}-z_{j}}
.
⟨
1
σ
⟩
=
⟨
1
ϵ
⟩
=
⟨
σ
ϵ
⟩
=
0
{\displaystyle \langle \mathbf {1} \sigma \rangle =\langle \mathbf {1} \epsilon \rangle =\langle \sigma \epsilon \rangle =0}
⟨
1
(
z
1
)
1
(
z
2
)
1
(
z
3
)
⟩
=
1
,
⟨
σ
(
z
1
)
σ
(
z
2
)
1
(
z
3
)
⟩
=
|
z
12
|
−
1
4
,
⟨
ϵ
(
z
1
)
ϵ
(
z
2
)
1
(
z
3
)
⟩
=
|
z
12
|
−
2
{\displaystyle \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}}
⟨
σ
(
z
1
)
σ
(
z
2
)
ϵ
(
z
3
)
⟩
=
1
2
|
z
12
|
3
4
|
z
13
|
−
1
|
z
23
|
−
1
{\displaystyle \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}}
⟨
1
1
σ
⟩
=
⟨
1
1
ϵ
⟩
=
⟨
1
σ
ϵ
⟩
=
⟨
σ
ϵ
ϵ
⟩
=
⟨
σ
σ
σ
⟩
=
⟨
ϵ
ϵ
ϵ
⟩
=
0
{\displaystyle \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
⟨
σ
4
⟩
,
⟨
σ
2
ϵ
2
⟩
,
⟨
ϵ
4
⟩
{\displaystyle \langle \sigma ^{4}\rangle ,\langle \sigma ^{2}\epsilon ^{2}\rangle ,\langle \epsilon ^{4}\rangle }
. For a four-point function
⟨
∏
i
=
1
4
V
i
(
z
i
)
⟩
{\displaystyle \left\langle \prod _{i=1}^{4}V_{i}(z_{i})\right\rangle }
, let
F
j
(
s
)
{\displaystyle {\mathcal {F}}_{j}^{(s)}}
and
F
j
(
t
)
{\displaystyle {\mathcal {F}}_{j}^{(t)}}
be the s- and t-channel Virasoro conformal blocks , which respectively correspond to the contributions of
V
j
(
z
2
)
{\displaystyle V_{j}(z_{2})}
(and its descendants) in the operator product expansion
V
1
(
z
1
)
V
2
(
z
2
)
{\displaystyle V_{1}(z_{1})V_{2}(z_{2})}
, and of
V
j
(
z
4
)
{\displaystyle V_{j}(z_{4})}
(and its descendants) in the operator product expansion
V
1
(
z
1
)
V
4
(
z
4
)
{\displaystyle V_{1}(z_{1})V_{4}(z_{4})}
. Let
x
=
z
12
z
34
z
13
z
24
{\displaystyle x={\frac {z_{12}z_{34}}{z_{13}z_{24}}}}
be the cross-ratio.
In the case of
⟨
ϵ
4
⟩
{\displaystyle \langle \epsilon ^{4}\rangle }
, fusion rules allow only one primary field in all channels, namely the identity field.[ 2]
⟨
ϵ
4
⟩
=
|
F
1
(
s
)
|
2
=
|
F
1
(
t
)
|
2
F
1
(
s
)
=
F
1
(
t
)
=
[
∏
1
≤
i
<
j
≤
4
z
i
j
−
1
3
]
1
−
x
+
x
2
x
2
3
(
1
−
x
)
2
3
=
(
z
i
)
=
(
x
,
0
,
∞
,
1
)
1
x
(
1
−
x
)
−
1
{\displaystyle {\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
⟨
σ
2
ϵ
2
⟩
{\displaystyle \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.[ 2]
⟨
σ
2
ϵ
2
⟩
=
|
F
1
(
s
)
|
2
=
C
σ
σ
ϵ
2
|
F
σ
(
t
)
|
2
=
1
4
|
F
σ
(
t
)
|
2
F
1
(
s
)
=
1
2
F
σ
(
t
)
=
[
z
12
1
4
z
34
−
5
8
(
z
13
z
24
z
14
z
23
)
−
3
16
]
1
−
x
2
x
3
8
(
1
−
x
)
5
16
=
(
z
i
)
=
(
x
,
0
,
∞
,
1
)
1
−
x
2
x
1
8
(
1
−
x
)
1
2
{\displaystyle {\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}}}
In the case of
⟨
σ
4
⟩
{\displaystyle \langle \sigma ^{4}\rangle }
, fusion rules allow two primary fields in all channels: the identity field and the energy field.[ 2] In this case we write the conformal blocks in the case
(
z
1
,
z
2
,
z
3
,
z
4
)
=
(
x
,
0
,
∞
,
1
)
{\displaystyle (z_{1},z_{2},z_{3},z_{4})=(x,0,\infty ,1)}
only: the general case is obtained by inserting the prefactor
x
1
24
(
1
−
x
)
1
24
∏
1
≤
i
<
j
≤
4
z
i
j
−
1
24
{\displaystyle x^{\frac {1}{24}}(1-x)^{\frac {1}{24}}\prod _{1\leq i<j\leq 4}z_{ij}^{-{\frac {1}{24}}}}
, and identifying
x
{\displaystyle x}
with the cross-ratio.
⟨
σ
4
⟩
=
|
F
1
(
s
)
|
2
+
1
4
|
F
ϵ
(
s
)
|
2
=
|
F
1
(
t
)
|
2
+
1
4
|
F
ϵ
(
t
)
|
2
=
|
1
+
x
|
+
|
1
−
x
|
2
|
x
|
1
4
|
1
−
x
|
1
4
=
x
∈
(
0
,
1
)
1
|
x
|
1
4
|
1
−
x
|
1
4
{\displaystyle {\begin{aligned}\langle \sigma ^{4}\rangle &=\left|{\mathcal {F}}_{\textbf {1}}^{(s)}\right|^{2}+{\frac {1}{4}}\left|{\mathcal {F}}_{\epsilon }^{(s)}\right|^{2}=\left|{\mathcal {F}}_{\textbf {1}}^{(t)}\right|^{2}+{\frac {1}{4}}\left|{\mathcal {F}}_{\epsilon }^{(t)}\right|^{2}\\&={\frac {|1+{\sqrt {x}}|+|1-{\sqrt {x}}|}{2|x|^{\frac {1}{4}}|1-x|^{\frac {1}{4}}}}\ {\underset {x\in (0,1)}{=}}\ {\frac {1}{|x|^{\frac {1}{4}}|1-x|^{\frac {1}{4}}}}\end{aligned}}}
In the case of
⟨
σ
4
⟩
{\displaystyle \langle \sigma ^{4}\rangle }
, the conformal blocks are:
F
1
(
s
)
=
1
+
1
−
x
2
x
1
8
(
1
−
x
)
1
8
,
F
ϵ
(
s
)
=
2
−
2
1
−
x
x
1
8
(
1
−
x
)
1
8
F
1
(
t
)
=
F
1
(
s
)
2
+
F
ϵ
(
s
)
2
2
=
1
+
x
2
x
1
8
(
1
−
x
)
1
8
,
F
ϵ
(
t
)
=
2
F
1
(
s
)
−
F
ϵ
(
s
)
2
=
2
−
2
x
x
1
8
(
1
−
x
)
1
8
{\displaystyle {\begin{aligned}&{\mathcal {F}}_{\textbf {1}}^{(s)}={\frac {\sqrt {\frac {1+{\sqrt {1-x}}}{2}}}{x^{\frac {1}{8}}(1-x)^{\frac {1}{8}}}}\ ,\;\;{\mathcal {F}}_{\epsilon }^{(s)}={\frac {\sqrt {2-2{\sqrt {1-x}}}}{x^{\frac {1}{8}}(1-x)^{\frac {1}{8}}}}\\&{\mathcal {F}}_{\textbf {1}}^{(t)}={\frac {{\mathcal {F}}_{\textbf {1}}^{(s)}}{\sqrt {2}}}+{\frac {{\mathcal {F}}_{\epsilon }^{(s)}}{2{\sqrt {2}}}}={\frac {\sqrt {\frac {1+{\sqrt {x}}}{2}}}{x^{\frac {1}{8}}(1-x)^{\frac {1}{8}}}}\ ,\;\;{\mathcal {F}}_{\epsilon }^{(t)}={\sqrt {2}}{\mathcal {F}}_{\textbf {1}}^{(s)}-{\frac {{\mathcal {F}}_{\epsilon }^{(s)}}{\sqrt {2}}}={\frac {\sqrt {2-2{\sqrt {x}}}}{x^{\frac {1}{8}}(1-x)^{\frac {1}{8}}}}\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:[ 1]
⟨
∏
i
=
1
2
n
ϵ
(
z
i
)
⟩
2
=
|
det
(
1
z
i
j
)
1
≤
i
≠
j
≤
2
n
|
2
{\displaystyle \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}}
⟨
∏
i
=
1
2
n
σ
(
z
i
)
⟩
2
=
1
2
n
∑
ϵ
i
=
±
1
∑
i
=
1
2
n
ϵ
i
=
0
∏
1
≤
i
<
j
≤
2
n
|
z
i
j
|
ϵ
i
ϵ
j
2
{\displaystyle \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<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 .[ 1]
The two-dimensional Ising model is mapped to itself by a high-low temperature duality. The image of the spin operator
σ
{\displaystyle \sigma }
under this duality is a disorder operator
μ
{\displaystyle \mu }
, which has the same left and right conformal dimensions
(
Δ
μ
,
Δ
¯
μ
)
=
(
Δ
σ
,
Δ
¯
σ
)
=
(
1
16
,
1
16
)
{\displaystyle (\Delta _{\mu },{\bar {\Delta }}_{\mu })=(\Delta _{\sigma },{\bar {\Delta }}_{\sigma })=({\tfrac {1}{16}},{\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[ 1]
⟨
σ
(
z
1
)
μ
(
z
2
)
σ
(
z
3
)
μ
(
z
4
)
⟩
2
=
1
2
|
z
13
z
24
|
|
z
12
z
34
z
23
z
14
|
(
|
x
|
+
|
1
−
x
|
−
1
)
{\displaystyle \left\langle \sigma (z_{1})\mu (z_{2})\sigma (z_{3})\mu (z_{4})\right\rangle ^{2}={\frac {1}{2}}{\sqrt {\frac {|z_{13}z_{24}|}{|z_{12}z_{34}z_{23}z_{14}|}}}{\Big (}|x|+|1-x|-1{\Big )}}
whereas
⟨
∏
i
=
1
4
μ
(
z
i
)
⟩
2
=
⟨
∏
i
=
1
4
σ
(
z
i
)
⟩
2
=
1
2
|
z
13
z
24
|
|
z
12
z
34
z
23
z
14
|
(
|
x
|
+
|
1
−
x
|
+
1
)
{\displaystyle \left\langle \prod _{i=1}^{4}\mu (z_{i})\right\rangle ^{2}=\left\langle \prod _{i=1}^{4}\sigma (z_{i})\right\rangle ^{2}={\frac {1}{2}}{\sqrt {\frac {|z_{13}z_{24}|}{|z_{12}z_{34}z_{23}z_{14}|}}}{\Big (}|x|+|1-x|+1{\Big )}}
Connectivities of clusters [ edit ]
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
{\displaystyle q=2}
of the
q
{\displaystyle q}
-state Potts model , whose parameter
q
{\displaystyle 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
⟨
σ
σ
σ
⟩
=
0
{\displaystyle \langle \sigma \sigma \sigma \rangle =0}
. There are four independent four-point connectivities, and their sum coincides with
⟨
σ
σ
σ
σ
⟩
{\displaystyle \langle \sigma \sigma \sigma \sigma \rangle }
.[ 3] Other combinations of four-point connectivities are not known analytically. In particular they are not related to correlation functions of the minimal model,[ 4] although they are related to the
q
→
2
{\displaystyle q\to 2}
limit of spin correlators in the
q
{\displaystyle q}
-state Potts model.[ 3]
^ a b c d e f P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory , 1997, ISBN 0-387-94785-X
^ a b c Cheng, Miranda C. N.; Gannon, Terry; Lockhart, Guglielmo (2020-02-25). "Modular Exercises for Four-Point Blocks -- I". arXiv :2002.11125v1 [hep-th ].
^ a b Delfino, Gesualdo; Viti, Jacopo (2011-04-21). "Potts q-color field theory and scaling random cluster model". Nuclear Physics B . 852 (1): 149–173. arXiv :1104.4323v2 . Bibcode :2011NuPhB.852..149D . doi :10.1016/j.nuclphysb.2011.06.012 . S2CID 119183802 .
^ Delfino, Gesualdo; Viti, Jacopo (2010-09-07). "On three-point connectivity in two-dimensional percolation". Journal of Physics A: Mathematical and Theoretical . 44 (3): 032001. arXiv :1009.1314v1 . doi :10.1088/1751-8113/44/3/032001 . S2CID 119246430 .