Free field approach to

the 19 Vertex Model at critical regime |q| = 1

可積分研究の現状と展望（２００１年２月３日〜４日、京大会館）予稿小島武夫（こじまたけお）日本大学理工学部数学教室

1 Problem

以下の３つの関係式を満たす関数を構成してやろうというのが本公演の主題です。この関数は、 The 19 Vertex Model at critical regime |q| = 1 の相関関数になることが期待されます。

R-matrix Symmetry.

G_{2N}(⋯, β_{j+1}, β_{j}, ⋯)_{···ϵ_{j+1}ϵ_{j}···}
= ∑_{ϵ′_{j}ϵ′_{j+1}=0,1,2} R(β_{j} − β_{j+1})_{ϵ_{j}ϵ_{j+1}}^{ϵ′_{j}ϵ′_{j+1}}G_{2N}(⋯, β_{j}, β_{j+1}, ⋯)_{···ϵ′_{j}ϵ′_{j+1}···}.  (1)

Cyclicity Condition.

G_{2N}(β_{1} ⋯, β_{2N−1}, β_{2N} + iλ)_{ϵ_{1}···ϵ_{2N}} = G_{2N}(β_{2N}, β_{1} ⋯, β_{2N−1})_{ϵ_{2N} ϵ_{1}···ϵ_{2N−1}}.  (2)

Recursion Relation.

G_{2N}(β_{1}, ⋯, β_{N−1}, β, β + πi, β_{N+2}, ⋯, β_{2N})_{ϵ_{1}···ϵ_{N−1},ϵ,2−ϵ,ϵ_{N+2},···ϵ_{2N}}
= G_{2N−2}(β_{1}, ⋯, β_{N−1}, β_{N+2}, ⋯, β_{2N})_{ϵ_{1}···ϵ_{N−1},ϵ_{N+2},··· ,ϵ_{2N}} .  (3)

ただし、関数の組、 G_{2N}(β_{1}, ⋯, β_{2N})_{ϵ_{1}···ϵ_{2N}} , (ϵ_{1}, ⋯, ϵ_{2N} = 0, 1, 2), は複素数値とします。関数 R(β)_{ϵ_{1}ϵ_{2}}^{ϵ′_{1}ϵ′_{2}} は付録１を見てください。

2 Vertex Operators

前章の３つの関係式をみたす関数の構成法を説明します。

仮に次の３つの性質をみたす Operator Φ_{j}(β), D (以下、それぞれ Vertex Operators と Degree Operatr という) があったとしましょう。 R-matrix Symmetry.

Φ_{j_{2}}(β_{2})Φ_{j_{1}}(β_{1}) = ∑_{k_{1},k_{2}=0,1,2} R(β_{1} − β_{2})_{j_{1}j_{2}}^{k_{1}k_{2}}Φ_{k_{1}}(β_{1})Φ_{k_{2}}(β_{2}).  (1)

Homogeneity Condition.

e^{−λD}Φ_{j}(β)e^{λD} = Φ_{j}(β + iλ), (j = 0, 1, 2),  (2)

Inversion Relation.

Φ_{j}(β)Φ_{2−j}(β + πi) = id, (j = 0, 1, 2).  (3)

すると、これらの Operators の作用する空間 H の上でトレースをとって定める関数 G_{2N} は、１章の３つの条件を満たすことはただちにわかります。

G_{2N}(β_{1}, ⋯, β_{2N})_{j_{1}···j_{2N}} = \begin{matrix}tr & H & (e^{−λD}Φ_{j_{1}}(β_{1}) ⋯ Φ_{j_{2N}} (β_{2N})) \\ tr & H & (e^{−λD}) \end{matrix}.  (4)

そこで、実際ボゾンとフェルミオンの空間に Vertex Opertaors Φ_{j}(β) を構成しました。付録２を見てください。

3 Integral Representation

２章の結果を用いると、 N 点相関関数の積分表示が得られます。ちなみに、 1 点相関関数は次のようになります（λ = 2π）。

G_{2}(β_{1}, β_{2})_{2,0} = e^{− \frac{2}{ξ} β_{2}} \begin{matrix}ch & (\frac{1}{2}(β_{1} − β_{2})) \\ sh & (\frac{1}{ξ}(β_{1} − β_{2} − πi)) \end{matrix}

× ∫_{−∞}^{∞} dα\frac{e^{\frac{2}{ξ} α}}{∏_{k=1}^{2} sh (β_{k} − α + i0)}sh (\frac{1}{ξ}(β_{1} − α − πi)) sh (\frac{1}{ξ}(β_{1} − α − 2πi)) .  (1)

G_{2}(β_{1}, β_{2})_{0,2} = e^{− \frac{2}{ξ} β_{1}} \begin{matrix}ch & (\frac{1}{2}(β_{1} − β_{2})) \\ sh & (\frac{1}{ξ}(β_{1} − β_{2} − πi)) \end{matrix}

× ∫_{−∞}^{∞} dα\frac{e^{\frac{2}{ξ} α}}{∏_{k=1}^{2} sh (β_{k} − α + i0)}sh (\frac{1}{ξ}(β_{2} − α)) sh (\frac{1}{ξ}(β_{2} − α + πi)) .  (2)

G_{2}(β_{1}, β_{2})_{1,1} = e^{− \frac{1}{ξ} (β_{1}+β_{2})} \begin{matrix}ch & (\frac{1}{2}(β_{1} − β_{2})) \\ sh & (\frac{1}{ξ}(β_{1} − β_{2} − πi)) \end{matrix}

× ∫_{−∞}^{∞} dα\frac{e^{\frac{2}{ξ} α}}{∏_{k=1}^{2} sh (β_{k} − α + i0)}sh (\frac{1}{ξ}(β_{1} − α − πi)) sh (\frac{1}{ξ}(β_{2} − α)) .  (3)

Reference T.Kojima : The 19 Vertex Model at critical regime |q| = 1, [nlin.SI/005022], to appear in Int.J.Mod.Phys.A(2001).

M.Idzumi : Level 2 Irreducible representations of U_{q}(ŝl_{2}), Vertex Operators and their correlations, Int.J.Mod.Phys.A9, 4449-4482, (1994).

R-matrix

Let us set the R-matrix R(β) ∈ End(ℂ^{3} ⊗ ℂ^{3}) by

R(β) = \frac{1}{κ(β)} ( \begin{matrix}1 \\ & p & & e_{2} \\ & & b & & g_{2} & & c_{2} \\ & e_{1} & & p \\ & & h_{1} & & o & & h_{2} \\ & & & & & p & & e_{2} \\ & & c_{1} & & g_{1} & & b \\ & & & & & e_{1} & & p \\ & & & & & & & & 1 \end{matrix} ) .  (4)

Here the matrix structure of R(β) is relative to the basis v_{ϵ}⊗v_{ϵ′} arranged in the order, (0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2). The matrix elements R(β)_{j_{1},j_{2}}^{k_{1},k_{2}} of the R-matrix represent the Boltzmann weight. The matrix R(β) acts on ℂ^{3} ⊗ℂ^{3} via the natural basis {v_{0}, v_{1}, v_{2}} of ℂ^{3} as following.

R(β)v_{k_{1}} ⊗ v_{k_{2}} = ∑_{j_{1},j_{2}=0,1,2} v_{j_{1}} ⊗ v_{j_{2}}R(β)_{j_{1}j_{2}}^{k_{1}k_{2}}.  (5)

Here we have set the normarized partition as

κ(β) = −\begin{matrix}sh & (\frac{1}{ξ}(β + πi)) \\ sh & (\frac{1}{ξ}(β − πi)) \end{matrix}.  (6)

The nonzero entries are given by

p(β) = −\begin{matrix}sh & (\frac{1}{ξ}β) \\ sh & (\frac{1}{ξ}(β − 2πi)) \end{matrix},  (7)

b(β) = sh (\frac{1}{ξ}β) sh (\frac{1}{ξ}(β + πi)) ; sh (\frac{1}{ξ}(β − πi)) sh (\frac{1}{ξ}(β − 2πi)),  (8)

o(β) = \frac{sh (\frac{πi}{ξ}) sh (\frac{2πi}{ξ}) + sh (\frac{1}{ξ}β) sh (\frac{1}{ξ}(β − πi))}{sh (\frac{1}{ξ}(β − πi)) sh (\frac{1}{ξ}(β − 2πi))},  (9)

and

c_{1}(β) = exp (\frac{2}{ξ}β) sh (\frac{πi}{ξ}) sh (\frac{2πi}{ξ}) ; sh (\frac{1}{ξ}(β − πi)) sh (\frac{1}{ξ}(β − 2πi)),  (10)

e_{1}(β) = −\begin{matrix}exp & (\frac{1}{ξ}β) & sh & (\frac{2πi}{ξ}) \\ sh & (\frac{1}{ξ}(β − 2πi)) \end{matrix},  (11)
c_{2}(β) = exp (−\frac{4}{ξ}β) c_{1}(β), e_{2}(β) = exp (−\frac{2}{ξ}β) ,  (12)
g_{1}(β) = exp (\frac{1}{ξ}(β + πi)) sh (\frac{πi}{ξ}) sh (\frac{1}{ξ}β) ; sh (\frac{1}{ξ}(β − πi)) sh (\frac{1}{ξ}(β − 2πi)),  (13)

h_{1}(β) = \frac{2 exp (\frac{1}{ξ}(β − πi)) sh (\frac{1}{ξ}β) sh (\frac{2πi}{ξ}) ch (\frac{πi}{ξ})}{sh (\frac{1}{ξ}(β − πi)) sh (\frac{1}{ξ}(β − 2πi))},  (14)
g_{2}(β) = exp (−\frac{2}{ξ}β) g_{1}(β), h_{2}(β) = exp (−\frac{2}{ξ}β) h_{1}(β).  (15)

Bosonizations

The purpose of this section is to give the free field realization of the vertex operators. Let us introduce the bose-fields by

[b(t), b(t′)] = −\frac{1}{t} \begin{matrix}sh & (\frac{π}{2} (ξ − 2)t) \\ sh & (\frac{π}{2} ξt) \end{matrix}δ(t + t′).  (16)

Let us set the basic operators by

U_{0}(β) =: exp (∫_{−∞}^{∞} b(t)e^{iβt}dt) :,  (17)
U_{1}(β) =: exp (− ∫_{−∞}^{∞} b(t)e^{iβt}dt) :  (18)

The notation : : is normal ordering. Let us set the fermion-fields by

[ψ(t), ψ(t′)]_{†} = 2ch(πt)δ(t + t′).  (19)

Fourier transformation of the fermion-field is given by

ψ̂(β) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} ψ(t)e^{itβ}dt.  (20)

The free-field realizations of the vertex operators are given by

Φ_{2}(β) = U_{0}(β),  (21)
Φ_{1}(β) = (e^{\frac{πi}{ξ}} + e^{− \frac{πi}{ξ}} ) ∫_{−∞}^{∞} dα \begin{matrix}exp & (\frac{1}{ξ}(α − β)) \\ sh & (\frac{1}{ξ}(α − β + πi)) \end{matrix}

× U_{0}(β)U_{1}(α)ψ̂(α),  (22)
Φ_{0}(β) = e^{\frac{πi}{ξ}} ∫_{−∞}^{∞} dα_{1} ∫_{−∞}^{∞} dα_{2} ∏_{k=1}^{2} \begin{matrix}exp & (\frac{1}{ξ}(α_{k} − β)) \\ sh & (\frac{1}{ξ}(α_{k} − β + πi)) \end{matrix}

× U_{0}(β)U_{1}(α_{1})U_{1}(α_{2})ψ̂(α_{1})ψ̂(α_{2})  (23)

Let us introduce the Fock space H^{b} generated by |vac⟩_{b} which satisfies

b(t)|vac⟩_{b} = 0, if t > 0.  (24)

Let us introduce the Fock space H^{ψ} generated by |vac⟩_{ψ} which satisfies

ψ(t)|vac⟩_{ψ} = 0, if t > 0.  (25)

Set the space H by

H = H^{b} ⊗ H^{ψ}.  (26)

Let us introduce the degree operators D^{b} and D^{ψ} by

D^{b}b(−t)|vac⟩_{b} = tb(−t)|vac⟩_{b}, D^{ψ}ψ(−t)|vac⟩_{ψ} = tψ(−t)|vac⟩_{ψ}, t > 0.  (27)

Set the degree operator D on H by

D = D^{b} ⊗ id + id ⊗ D^{ψ}.  (28)