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CORRELATION FUNCTIONS IN QUANTUM INTEGRABLE MODELS FROM Q-DEFORMATION OF CANONICAL NORMALISED SECOND KIND DIFFERENTIAL. Fedor Smirnov

. – p.1/21

We consider the XXZ spin chain with the Hamiltonian H = 1

2 ∞

• k=−∞
• σ1

kσ1 k+1 + σ2 kσ2 k+1 + ∆σ3 kσ3 k+1

• ,

∆ = 1

2(q + q−1) .

To avoid technicalities, let us accept H as acting on HS =

∞

• j=−∞

C2, but to be more rigourous we have to start with H = lim

N→∞ HN,

HN = 1

2 N

• k=−N+1
• σ1

kσ1 k+1 + σ2 kσ2 k+1 + ∆σ3 kσ3 k+1

• ,

σa

N+1 = σa −N+1 .

. – p.2/21

We studied the vacuum expectation values q2αS(0)OXXZ = vac|q2αS(0)O|vac vac|q2αS(0)|vac , |vac is the ground state, S(k) = 1

2

k

j=−∞ σ3 j , and O is a local operator.

An important generalisation: q2αS(0)OXXZ, β,h = TrS

• e−βH+hSq2αS(0)O
• TrS
• e−βH+hSq2αS(0)

, where TrS stands for the trace on HS. For β → ∞ and h = 0, we return to the previous case. Question: why not to put other local integrals of motion in the exponent?

. – p.3/21

Most general case. Consider the universal L-operator Lj(ζ) = q

1 2

• ζ2q

H+1 2

− q− H+1

2

(q − q−1)ζFq

H−1 2

(q − q−1)ζq− H−1

2 E

ζ2q− H−1

2

− q

H−1 2

• j

. Take 2s + 1 dimensional representation of H, F, E, and a parameter τ denoting corresponding L-operator by Lj,a(ζ/τ). Let Lj,a = Lj,a(1/τ). It is well-known that [ HN, Tra TS,a(1/τ)] = 0, TS,a(1/τ) = lim

N→∞ L−N+1,a(1/τ) · · · LN,a(1/τ)

Along with the space HS, consider the Matsubara space HM = C2s1+1 ⊗ · · · ⊗ C2sn+1 , with an arbitrary spin sm and a spectral parameter τm at- tached to each component.

. – p.4/21

We define Zκ q2αS(0)O

• =

TrSTrM

• TS,Mq2κS+2αS(0)O
• TrSTrM
• TS,Mq2κS+2αS(0)

. Here TS,M is the monodromy matrix associated with HS ⊗ HM: TS,M = TS,n(1/τn) · · · TS,1(1/τ1) . Another way of writing the same thing is TS,M = lim

N→∞ T−N+1,M · · · Tj,M · · · TN,M ,

where Tj,M = Tj,M(1) , Tj,M(ζ) = Lj,n(ζ/τn)Lj,n−1(ζ/τn−1) · · · Lj,1(ζ/τ1) .

. – p.5/21

It is convenient to present the functional Zκ by the following picture: Space

(α+κ) σ 3

= q =q

i j

a r a b u s t a M

κ σ 3

= L

. – p.6/21

Beauty of Matsubara approach. Consider a twisted transfer matrix TM(ζ, λ) = Trj

• Tj,M(ζ)qλσ3

j

. We are interested in λ = κ, λ = α + κ. We assume that TM(ζ, κ), TM(ζ, α + κ) have single eigenvectors |κ, |κ + α such that κ + α|κ = 0 If we denote the eigenvalues by T(ζ, κ), T(ζ, α + κ) are then T(1, κ), T(1, α + κ) are of maximal absolute value. Then the Matrubara transfer-matrices to the right and to the left of O can be replaced by one-dimensional projectors |κT(1, κ)κ| and |κ + αT(1, κ + α)κ + α| .

. – p.7/21

Proper description of the space direction. X = q2αS(0)O is a quasi-local operator with tail α. S(·) = [S, ·] where S = S(∞) is the total spi. We denote by Wα the space of quasi-local operators with tail α, and by Wα,s its subspace of operators of spin s ∈ Z. Consider the space W(α) =

∞

• s=−∞

Wα−s,s . The operator q2αS(0) is called primary field. We were able to define action of creation (t∗(ζ), b∗(ζ),c∗(ζ)) and annihila- tion (b(ζ), c(ζ)) operators on W(α).

. – p.8/21

These are one-parameter families: t∗(ζ) =

∞

• p=1

(ζ2 − 1)p−1t∗

p ,

b∗(ζ) = ζα+2

∞

• p=1

(ζ2 − 1)p−1b∗

p , c∗(ζ) = ζ−α−2 ∞

• p=1

(ζ2 − 1)p−1c∗

p ,

b(ζ) = ζ−α

∞

• p=0

(ζ2 − 1)−pbp , c(ζ) = ζα

∞

• p=0

(ζ2 − 1)−pcp . The operator t∗(ζ) is in the center: [t∗(ζ1), t∗(ζ2)] = [t∗(ζ1), c∗(ζ2)] = [t∗(ζ1), b∗(ζ2)] = 0, [t∗(ζ1), c(ζ2)] = [t∗(ζ1), b(ζ2)] = 0 .

. – p.9/21

The rest of the operators b, c, b∗, c∗ are fermionic. The only non-vanishing anti-commutators are [b(ζ1), b∗(ζ2)]+ = −ψ(ζ2/ζ1, α) , [c(ζ1), c∗(ζ2)]+ = ψ(ζ1/ζ2, α) , where ψ(ζ, α) = ζα

ζ2+1 2(ζ2−1). Each Fourier mode has the block structure

t∗

p : Wα−s,s → Wα−s,s

b∗

p, cp : Wα−s+1,s−1 → Wα−s,s ,

c∗

p, bp : Wα−s−1,s+1 → Wα−s,s .

Among them, τ = t∗

1/2 plays a special role of the right shift by one site

along the chain. Annihilation operators kill the primary field: b(ζ)

• q2αS(0)

= 0, c(ζ)

• q2αS(0)

= 0 .

. – p.10/21

The set of operators τ mt∗

p1 · · · t∗ pjb∗ q1 · · · b∗ qkc∗ r1 · · · c∗ rk

• q2αS(0)

, where m ∈ Z, j, k ∈ Z≥0, p1 ≥ · · · ≥ pj ≥ 2, q1 > · · · > qk ≥ 1 and r1 > · · · > rk ≥ 1. constitutes a basis of Wα,0. The operators t∗(ζ) are composed of adjoint action of local integrals of motion. Question for classical case. KdV. The primary field eaϕ, Wα corresponds to the space F(u, ux, uxx, · · · )eaϕ, t∗(ζ) ≃ {∂2n−1}, what are the fermions com- muting with them?

. – p.11/21

The main theorem. The space and Matsubara directions are related by: Zκ t∗(ζ)(X)

• = 2ρ(ζ)Zκ{X} ,

Zκ b∗(ζ)(X)

• =

1 2πi

• Γ

ω(ζ, ξ)Zκ c(ξ)(X) dξ2 ξ2 , Zκ c∗(ζ)(X)

• = − 1

2πi

• Γ

ω(ξ, ζ)Zκ b(ξ)(X) dξ2 ξ2 , where Γ goes around ξ2 = 1. This implies Zκ t∗(ζ0

1) · · · t∗(ζ0 k)b∗(ζ+ 1 ) · · · b∗(ζ+ l )c∗(ζ− l ) · · · c∗(ζ− 1 )

• q2αS(0)

=

k

• p=1

2ρ(ζ0

p) × det

• ω(ζ+

i , ζ− j )

• i,j=1,··· ,l .

. – p.12/21

The functions ρ(ζ) and ω(ζ, ξ). These two functions are defined by the properties of the Matsubara direction only. Recall that we had two eigenvectors |κ, |κ + α with corresponding eigenvalues for twisted Matrubara transfer-matrices T(ζ, κ), T(ζ, α + κ). The function ρ(ζ) is simple: ρ(ζ) = T(ζ, α + κ) T(ζ, κ) . The function ω(ζ, ξ) is more interesting. First, it must satisfy the symmetry condition ω(ζ, ξ|κ, α) = ω(ξ, ζ| − κ, −α) . This function is q-deformation of canonical second kind differential. In what follows I explain the meaning of these words.

. – p.13/21

Consider Riemann surface Σ. On Σ × Σ there is a canonical differential satisfying Singulatity ω(x, y) =

• 1

(x − y)2 + O(1)

• dxdy .

Normalisation

• aj

ω(x, y) = 0 . As a corollary of the Riemann bilinear relations this integral is symmetric ω(x, y) = ω(y, x) Convenient form of the Riemann bilinear relations is

• i
• c1

ωi

• c2

˜ ωi −

• c2

ωi

• c1

˜ ωi = 2πi(c1 ◦ c2), res ωid−1˜ ωj = δi,j .

. – p.14/21

Baxter equations. Without going into details we write directly for the eigenvalues: T(ζ, λ)Q±(ζ, λ) = d(ζ)Q±(ζq, λ) + a(ζ)Q±(ζq−1, λ) . These two solutions are as functions of ζ: Q±(ζ, λ) = ζ±λP ±(ζ2). The functions a(ζ), d(ζ) are a(ζ) =

n

• m=1
• q2sm+1ζ2/τ 2

m − 1

• ,

d(ζ) =

n

• m=1
• q−2sm+1ζ2/τ 2

m − 1

• .

Introduce ηf(ζ) = f(ζq), and rewrite the Baxter equation as

• a(ζ)η−1 + d(ζ)η − T(ζ, λ)
• Q±(ζ, λ) = 0 .

This is the quantum version of the classical spectral curve.

. – p.15/21

Symmettries. T(ζ, λ) = T(ζ, −λ), Q+(ζ, λ) = Q−(ζ, −λ) . Deformed Abelian integrals. Introduce the function ϕ(ζ) which satisfies the equation a(ζq)ϕ(ζq) = d(ζ)ϕ(ζ) . This function is elementary, ϕ(ζ) =

n

• m=1

2sm

• k=0
• ζ2q−2sm+2k+1 − 1

−1 . There are two kinds of deformed Abelian integrals,

• Γm

f ±(ζ)Q∓(ζ, κ + α)Q±(ζ, κ)ϕ(ζ)dζ2 ζ2 , where ζ∓αf ±(ζ) is a polynomial in ζ2.

. – p.16/21

In the following, we use the q-difference operator ∆ζf(ζ) = f(ζq) − f(ζq−1) . Deformed Riemann bilinear relations. Consider the following function in two variables r(ζ, ξ) = r+(ζ, ξ) − r−(ξ, ζ) , where r+(ζ, ξ) = r+(ζ, ξ|κ, α), r−(ξ, ζ) = r+(ξ, ζ| − κ, −α), and

. – p.17/21

r+(ζ, ξ|κ, α) = T(ζ, κ)∆−1

ζ

(ψ(ζ/ξ, α)(T(ζ, κ) − T(ξ, κ))) + T(ζ, κ + α)∆−1

ζ

(ψ(ζ/ξ, α)(T(ζ, κ + α) − T(ξ, κ + α))) − T(ζ, κ)∆−1

ζ

(ψ(qζ/ξ, α)(T(ζq, κ + α) − T(ξ, κ + α))) − T(ζ, κ + α)∆−1

ζ

• ψ(q−1ζ/ξ, α)(T(ζq−1, κ) − T(ξ, κ))
• +
• a(ζq) − a(ξ)
• d(ζ)ψ(qζ/ξ, α) −
• d(ζq−1

− d(ξ))a(ζ)ψ(q−1ζ/ξ, α) . Then

• Γi
• Γj

r(ζ, ξ)Q−(ζ, κ + α)Q+(ζ, κ)Q+(ξ, κ + α)Q−(ξ, κ)ϕ(ζ)ϕ(ξ)dζ2 ζ2 dξ2 ξ2 = 0 .

. – p.18/21

Clearly ξαr+(ζ, ξ) is a polynomial in ξ2 and ζ−αr−(ξ, ζ) is a polynomial in ζ2, both of degree n. This allows us to define the polynomials p±

m by

r+(ζ, ξ) =

n

• m=0

ζαp+

m(ζ2)ξ−α+2m ,

r−(ξ, ζ) =

n

• m=0

ξ−αp−

m(ξ2)ζα+2m .

Introduce the (n + 1) × (n + 1) matrices A±

i,j =

• Γi

ζ±α+2jQ∓(ζ, κ + α)Q±(ζ, κ)ϕ(ζ)dζ2 ζ2 , B±

i,j =

• Γi

ζ±αp±

j (ζ2)Q∓(ζ, κ + α)Q±(ζ, κ)ϕ(ζ)dζ2

ζ2 . Then the deformed Riemann bilinear relations read as B+(A−)t = A+(B−)t .

. – p.19/21

Defining properties of ω(ζ, ξ). Singular part ζ−αT(ζ, κ)

• ω(ζ, ξ) − ωsing(ζ, ξ)
• is a polynomial in ζ2 of degree n .

where (recall ψ(ζ) = ζα(ζ2 + 1)/2(ζ2 − 1)) ωsing(ζ, ξ) = −∆ζψ(ζ/ξ, α) + 4 T(ζ, κ)T(ξ, κ)

• a(ξ)d(q−1ξ)ψ(qζ/ξ, α) − a(qξ)d(ξ)ψ(q−1ζ/ξ, α)
• .
• 2. Normalisation. For m = 0, · · · , n
• Γm

T(ζ, κ)

• ω(ζ, ξ) + DζDξ∆−1

ζ ψ(ζ/ξ, α)

• Q−(ζ, κ + α)Q+(ζ, κ)ϕ(ζ)dζ2

ζ2 = 0 , DζF(ζ) = F(qζ) + F(q−1ζ) − 2ρ(ζ)F(ζ) .

. – p.20/21

We prove that det A± = 0. Then we define ω(ζ, ξ|κ, α) = 4 T(ζ, κ)T(ξ, κ)v+(ζ)t(A+)−1B+v−(ξ) + ωsym(ζ, ξ|κ, α) , where v±(ζ) are vectors with components v±(ζ)j = ζ±α+2j, and ωsym(ζ, ξ|κ, α) = 1 T(ζ, κ)T(ξ, κ)

• (4a(ξ)d(ζ) − T(ζ, κ)T(ξ, κ)) ψ(qζ/ξ, α)

− (4a(ζ)d(ξ) − T(ζ, κ)T(ξ, κ)) ψ(q−1ζ/ξ, α) −2ψ(ζ/ξ, α)

• T(ζ, κ)T(ξ, κ + α) − T(ξ, κ)T(ζ, κ + α)
• .

Notice that ωsym(ζ, ξ|κ, α) = ωsym(ξ, ζ| − κ, −α), so the symmetry follows from the Riemann bilinear relations which imply v+(ζ)t(A+)−1B+v−(ξ) = v−(ξ)t(A−)−1B−v+(ζ) .

. – p.21/21