CORRELATION FUNCTIONS FOR LATTICE EXACTLY SOLVABLE MODELS.

Fedor Smirnov

H. Boos, M. Jimbo, T. Miwa, F. Smirnov, Y. Takeyama A recursion formula for the correlation functions of an inhomogeneous XXX model. (hep-th/0405044) Reduced qKZ equation and correlation functions of the XXZ model (hep-th/0412191) Traces on the Sklyanin algebra and correlation functions of the eight-vertex model (hep-th/0504072 ) Density matrix of a finite sub-chain of the Heisenberg anti-ferromagnet (hep-th/0506171)

The XXX model. I shall consider the Heisenberg antiferromagnet with the Hamiltonian

H = ∑_{i=−∞}^{∞} (σ_{i}^{1}σ_{i+1}^{1} + σ_{i}^{2}σ_{i+1}^{2} + σ_{i}^{3}σ_{i+1}^{3})

Integrability is due to existence of R-matrix:

R(λ) = \frac{ρ(λ)}{λ + 1} (λ + P) ∈ End (ℂ^{2} ⊗ ℂ^{2})

where

ρ(λ) = −\frac{Γ (\frac{λ}{2}) Γ (\frac{1}{2} − \frac{λ}{2})}{Γ (−\frac{λ}{2}) Γ (\frac{1}{2} + \frac{λ}{2})} .

Considering tensor product of several spaces V_{i} ≃ ℂ^{2} we write R_{i,j}(λ) for R-matrix acting non-trivially in V_{i} ⊗ V_{j}.

Yang-Baxter equation (we always imply λ_{i,j} = λ_{i} − λ_{j}):

R_{1,2}(λ_{1,2})R_{1,3}(λ_{1,3})R_{2,3}(λ_{2,3}) = R_{2,3}(λ_{2,3})R_{1,3}(λ_{1,3})R_{1,2}(λ_{1,2})

Transfer-matrix.

t_{N}(λ) = tr_{V_{a}} (R_{a,−N−1}(λ)R_{a,−N}(λ) ⋯ R_{a,N−1}(λ)R_{a,N}(λ))

Due to Yang-Baxter

[t(λ_{1}), t(λ_{2})] = 0

Moreover t_{N}(0) = U_{N}, U_{N} σ_{j}^{a} U_{N}^{−1} = σ_{j+1}^{a}

\frac{d}{dλ} log t_{N}(λ) = ∑_{n=1}^{∞} λ^{n}I_{N,n} , I_{N,1} = H_{N}

Transfer-matrix can be diagonalised by Bethe ansatz, then one takes limit N → ∞. Ground state is denoted by |vac⟩.

Correlation functions. Consider an operator

𝒪 = σ_{i_{1}}^{a_{1}}σ_{i_{2}}^{a_{2}} ⋯ σ_{i_{k}}^{a_{k}}
i_{1} < i_{2} < ⋯ < i_{k}

The length of operator l(𝒪) = i_{k} − i_{1} + 1. The problem is to find

⟨vac| 𝒪 |vac⟩ L-operator. Let {S_{a}}_{a=1}^{3} be a basis of 𝔰𝔩_{2} satisfying [S_{a}, S_{b}] = 2iϵ_{abc}S_{c}. Define the L-operator which belongs to U(𝔰𝔩_{2}) ⊗ End (ℂ^{2}):

L(λ) = \frac{ρ(λ, d)}{λ + \frac{d}{2}}L^{(0)}(λ),

L^{(0)}(λ) = λ + \frac{1}{2} + \frac{1}{2} ∑_{a=1}^{3} S_{a} ⊗ σ^{a},

where d is related to the Casimir operator as

C = ∑_{a=1}^{3} S_{a}S^{a} = d^{2} − 1

and

ρ(λ, d) = −\frac{Γ (\frac{1}{2} − \frac{d}{4} + \frac{λ}{2}) Γ (1 − \frac{d}{4} − \frac{λ}{2})}{Γ (\frac{1}{2} − \frac{d}{4} − \frac{λ}{2}) Γ (1 − \frac{d}{4} + \frac{λ}{2})} .

In this normalisation we have the unitarity and crossing symmetry in the form

L(λ)L(−λ) = 1, σ^{2} (L(λ))^{t} σ^{2} = −L(−λ − 1).

Yang-Baxter equation:

R_{1,2}(λ_{1,2})L_{1}(λ_{1})L_{2}(λ_{2}) = L_{2}(λ_{2})L_{1}(λ_{1})R_{1,2}(λ_{1,2})

Monodromy matrix.

T (λ) = ⋯ L_{1}(λ)L_{2}(λ)L_{3}(λ)L_{4}(λ) ⋯ ∈ End((ℂ^{2})^{⊗∞}) ⊗ U(𝔰𝔩_{2})

Trace functional. The functional T_{d} (d is ”dimension) is defined by the following sequence:

U(𝔰𝔩_{2}) → U(𝔰𝔩_{2})/[U(𝔰𝔩_{2}), U(𝔰𝔩_{2})] ≃ ℂ[C] · 1 → ℂ[d^{2}] · d

where the last arrow is such that C = d^{2} − 1, T_{d}(1) = d. We shall also use

Tr_{d_{1},··· ,d_{k}} : U(𝔰𝔩_{2})^{⊗k} → ℂ[d_{1}, ⋯, d_{k}]

defined simply as

Tr_{d_{1},··· ,d_{k}}(A_{1} ⊗ ⋯ ⊗ A_{k}) = ∏_{j=1}^{k} Tr_{d_{j}}(A_{j})

.

Function ω(λ). Define

ω(λ) = \frac{d}{dλ} log (ρ(λ) (\frac{λ − 1}{λ + 1})^{\frac{1}{4}})

= \frac{λ}{λ^{2} − 1} (∑_{k=1}^{∞}(−1)^{k} \frac{2k}{λ^{2} − k^{2}} + \frac{1}{2(λ^{2} − 1)})

Taylor expansion of ω(λ):

ω(λ) = ∑_{l=0}^{∞} λ^{2l+1}ζ_{a}(2l + 1),

where the alternating ζ-function is

ζ_{a}(s) = ∑_{k=1}^{∞}(−1)^{k−1}\frac{1}{k^{s}} = (1 − 2^{−s+1}) ζ(s)

Ajoint action of monodromy matrix on operators:

𝒯(λ)(𝒪) = T (λ) · 𝒪 · T (λ)^{−1}

The monodromy matrix is infinite, but in the final formula only finite piece of length l(𝒪) is relevant. Theorem 1.

⟨vac|𝒪|vac⟩ = \frac{1}{2^{∞}} tr_{(ℂ^{2})}^{⊗∞} (ρ(𝒪)) ,

where

ρ = e^{Ω}

with

Ω = \frac{1}{2} ∫∫ \frac{dμ_{1}}{2πi} \frac{dμ_{2}}{2πi} ω(μ_{1,2})Tr_{2,2,μ_{1,2}} (A(μ_{1,2})(𝒯(μ_{1}) ⊗ 𝒯(μ_{2}) ⊗ 𝒯(\frac{μ_{1} + μ_{2}}{2}))) ,

and

A(μ) = \frac{μ}{μ^{2} − 1} ∑_{a=1}^{3} S^{a} ⊗ S^{a} ⊗ I

integrals go around μ_{1}, μ_{2} = 0.

Theorem 2. The action of Ω reduces as follows:

Ω(𝒪) = Ω_{l(𝒪)}(𝒪)

So, we can write alternatively,

⟨vac|𝒪|vac⟩ = \frac{1}{2^{l(𝒪)}} tr_{(ℂ^{2})_{⊗l(𝒪)}} (ρ_{l(𝒪)}(𝒪))

The singularities are as follows \frac{1}{(μ_{1}μ_{2})^{l(𝒪)}} from 𝒯(\frac{μ_{1}+μ_{2}}{2}), \frac{1}{(μ_{1}^{2}−1)^{l(𝒪)}} from 𝒯(μ_{1}) \frac{1}{(μ_{2}^{2}−1)^{l(𝒪)}} from 𝒯(μ_{2})

Structure of the result. Theorem 3. Let

V (μ_{1}, μ_{2}) = Tr_{2,2,μ_{1,2}} (A(μ_{1,2})(𝒯(μ_{1}) ⊗ 𝒯(μ_{2}) ⊗ 𝒯(\frac{μ_{1} + μ_{2}}{2})))

then

[V (μ_{1}, μ_{2}), V (μ_{3}, μ_{4})] ≃ 0

Then the operators

U(α_{1}, α_{2}) = \frac{1}{2} ∫∫ \frac{dμ_{1}}{2πi} \frac{dμ_{2}}{2πi} \frac{V (μ_{1}, μ_{2})}{(α_{1} − μ_{1})(α_{2} − μ_{2})}

(where α_{i} are outside of the contour of integration) define a commutative family: [U(μ_{1}, μ_{2}), U(μ_{3}, μ_{4})] = 0 Further

Ω = ∫ \frac{dα_{1}}{2πi} ∫ \frac{dα_{1}}{2πi} ω(α_{1,2}) U(α_{1}, α_{2}) = ∑_{l=0}^{∞} ζ_{a}(2l + 1)Ω_{2l+1}

with

[Ω_{2l+1}, Ω_{2m+1}] = 0

Reducing to a particular operator 𝒪 one finds only l(𝒪)-order poles, so, the series terminate:

Ω_{l(𝒪)} = ∑_{l=0}^{l(𝒪)−2} ζ_{a}(2l + 1)Ω_{2l+1}

We have additional property of nilpotency: Theorem 4.

U_{l(𝒪)}(α_{1}, α_{2})U_{l(𝒪)}(α_{3}, α_{4}) ⋯ U_{l(𝒪)}(α _{2[\frac{l(𝒪)}{2}]+1}, α _{2[\frac{l(𝒪)}{2}]+2}) = 0

All together it implies:

⟨vac|𝒪|vac⟩ = ∑_{k_{i}≥0} ∑ ; ∑ k_{i}≤[\frac{l(𝒪)}{2}] r_{k_{0},··· ,k_{l(𝒪)−2}} ∏_{i=0}^{l(𝒪)−2} ζ_{a}(2i + 1)^{k_{i}}

with r_{k_{0},··· ,k_{l(𝒪)−2}} ∈ ℚ.

The most general XYZ case. XYZ spin chain is described by the Hamiltonian

H_{XY Z} = ∑_{j=−∞}^{∞} (I^{1}σ_{j}^{1}σ_{j+1}^{1} + I^{2}σ_{j}^{2}σ_{j+1}^{2} + I^{3}σ_{j}^{3}σ_{j+1}^{3})

where

I^{α} = \frac{θ_{α+1}(2η)}{θ_{α+1}}

so, we have two parameters η and τ.

R-matrix. We use t related to λ as follows

λ = ηt

[t] = \frac{θ_{1}(2t)}{θ_{1}(2η)}.

The R matrix is given by

R(t) = ρ(t)\frac{r(t)}{[t + η]} ∈ End(V ⊗ V),

r(t) = \frac{1}{2} ∑_{α=0}^{3} \frac{θ_{α+1}(2t + η)}{θ_{α+1}(η)} σ^{α} ⊗ σ^{α},

ρ(t) solves the equations:

ρ(t)ρ(−t) = 1, ρ(t)ρ(t − η) = \frac{[t]}{[η − t]},

Sklyanin algebra. 𝒜 is generated by S_{0}, S_{1}, S_{2}, S_{3}

[S_{0}, S_{a}] = iJ_{bc}(S_{b}S_{c} + S_{c}S_{b}),
[S_{b}, S_{c}] = i(S_{0}S_{a} + S_{a}S_{0}),

where

J_{bc} = −\frac{J_{b} − J_{c}}{J_{a}} = ε_{a}\frac{θ_{1}(η)^{2}θ_{a+1}(η)^{2}}{θ_{b+1}(η)^{2}θ_{c+1}(η)^{2}},

J_{a} = \frac{θ_{a+1}(2η)θ_{a+1}}{θ_{a+1}(η)^{2}},

with

ε_{2} = −1, ε_{α} = 1 (α ≠ 2).

Casimir elements:

K_{0} = ∑_{α=0}^{3} S_{α}^{2}, K_{2} = ∑_{a=1}^{3} J_{a}S_{a}^{2}.

The Sklyanin algebra is graded: 𝒜 = ⨁_{k=0}^{∞} 𝒜_{k}. Trace functional Tr_{d} is defined by the sequence:

𝒜_{k} → 𝒜_{k}/[𝒜, 𝒜]_{k}

≃ ℂ[K_{0}, K_{2}] · 1 ⊕ ⨁_{α=0}^{3} ℂ[K_{0}, K_{2}] · S_{α} ⊕ ℂ[K_{0}, K_{2}] · S_{0}^{2} ⊕ ℂ[K_{0}, K_{2}] · S_{3}^{2}

→ Θ_{k}(d) + Θ_{k}(d) · d + Θ_{k}(d) · \frac{∂}{∂d} log θ_{1}(ηd)

where the last arrow corresponds to analytical continuation for Sklyanin’s finite-dimensional representations:

K_{0}(d) = 4\frac{θ_{1}(dη)^{2}}{θ_{1}(2η)^{2}}, K_{2}(d) = 4\frac{θ_{1}(dη + η)θ_{1}(dη − η)}{θ_{1}(2η)^{2}}.

Tr_{d}1 = d, Tr_{d}S_{α} = 2δ_{α0}\frac{θ_{1}(dη)}{θ_{1}(2η)},

Tr_{d}S_{α}^{2} = ε_{α}\frac{2θ_{α+1}(η)^{2}}{θ′_{1}θ_{1}(2η)^{3}} (\frac{∂}{∂t} − d\frac{∂}{∂η}) (θ_{α+1}(t + η)θ_{α+1}(t − η))

Define the L-operator:

L(t) = \frac{ρ(t, d)}{[t + \frac{d}{2η}]} ∑_{α=0}^{3} \frac{θ_{α+1}(2t + η)}{θ_{α+1}(η)}S_{α} ⊗ σ^{α} ∈ 𝒜 ⊗ End(V).

The operator Ω. Define three transcendental functions:

φ_{1}(t) = \frac{∂}{∂t}ψ(t), φ_{2}(t) = \frac{∂}{∂η}ψ(t), φ_{3}(t) = \frac{∂}{∂τ} ψ(t),

ψ(t) = ρ(t, η, τ) (\frac{[t − η]}{[t + η]})^{\frac{1}{4}}

and three difference operators:

Z_{1}F (u_{1}, u_{2}) = (1 − u_{1,2}Δ_{1}) F (u_{1}, u_{2})

Z_{2}F (u_{1}, u_{2}) = Δ_{1}F (u_{1}, u_{2})
Z_{3}F (u_{1}, u_{2}) = (Δ_{τ} − τΔ_{1}) F (u_{1}, u_{2})

where

Δ_{a}F (u_{1}, u_{2}) = F (u_{1} + a, u_{2}) − F (u_{1}, u_{2})

Define three functions:

F_{1}(u_{1}, u_{2}) = 1, F_{2}(u_{1}, u_{2}) = u_{1,2}, F_{3}(u_{1}, u_{2}) = \frac{∂}{∂u_{1}} log θ_{1}(u_{1,2})

then

Z_{a}F_{b}(u_{1}, u_{2}) = δ_{a,b}

Then Ω is given by

Ω = 2 ∑_{a=1}^{3} ∫∫ \frac{du_{1}}{2πi} \frac{du_{2}}{2πi} ω_{a}(u_{1,2}) Z_{a}Tr_{2,2,u_{1,2}/η} (A(u_{1,2})(𝒯(u_{1}) ⊗ 𝒯(u_{2}) ⊗ 𝒯(\frac{u_{1} + u_{2}}{2}))) ,

where

A(u) = \frac{[2][u]}{2[u + η][u − η]} ∑_{a=1}^{3} \frac{θ_{a+1}(2u)}{θ_{a+1}(2η)} (S_{a} ⊗ S_{a} ⊗ I)

The integrals go around u_{1} = 0, u_{2} = 0,

Z_{a}Tr_{2,2,u_{1,2}/η} (A(u_{1,2})(𝒯(u_{1}) ⊗ 𝒯(u_{2}) ⊗ 𝒯(\frac{u_{1} + u_{2}}{2})))

define three elliptic functions of u_{1}, u_{2}.

Commuting families. Define

U_{a}(v_{1}, v_{2}) = 2 ∫∫ \frac{du_{1}}{2πi} \frac{du_{2}}{2πi} \frac{θ_{0}(v_{1} − u_{1})θ_{0}(v_{2} − u_{2})(θ′_{1})^{2}}{θ_{1}(v_{1} − u_{1})θ_{1}(v_{2} − u_{2})θ_{0}^{2}}

× Z_{a}Tr_{2,2,u_{1,2}/η} (A(u_{1,2})(𝒯(u_{1}) ⊗ 𝒯(u_{2}) ⊗ 𝒯(\frac{u_{1} + u_{2}}{2}))) ,

Then Ω can be rewritten as

Ω = ∑_{a=1}^{3} ∫∫ \frac{du_{1}}{2πi} \frac{du_{2}}{2πi} ω_{a}(u_{1,2}) U_{a}(u_{1}, u_{2}),

and U_{a}(u_{1}, u_{2}) define commutative families:

[U_{a}(u_{1}, u_{2}) , U_{b}(u_{3}, u_{4})] = 0