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 σ 1 k σ 1 k +1 + σ 2 k σ 2 k +1 + ∆ σ 3 k σ 3 ∆ = 1 2 ( q + q − 1 ) . � � � , k +1 2 k = −∞ ∞ C 2 , but to � To avoid technicalities, let us accept H as acting on H S = j = −∞ be more rigourous we have to start with H = lim N →∞ H N , N H N = 1 � σ 1 k σ 1 k +1 + σ 2 k σ 2 k +1 + ∆ σ 3 k σ 3 � σ a N +1 = σ a � , − N +1 . k +1 2 k = − N +1 . – p.2/21
We studied the vacuum expectation values � q 2 αS (0) O� XXZ = � vac | q 2 αS (0) O| vac � � vac | q 2 αS (0) | vac � , � k | vac � is the ground state, S ( k ) = 1 j = −∞ σ 3 j , and O is a local operator. 2 An important generalisation: � � e − βH + hS q 2 αS (0) O Tr S � q 2 αS (0) O� XXZ, β,h = � , � e − βH + hS q 2 αS (0) Tr S where Tr S stands for the trace on H S . 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 � � H +1 − q − H +1 H − 1 ζ 2 q ( q − q − 1 ) ζFq 2 2 2 1 L j ( ζ ) = q . 2 ( q − q − 1 ) ζq − H − 1 ζ 2 q − H − 1 H − 1 2 E − q 2 2 j Take 2 s + 1 dimensional representation of H, F, E , and a parameter τ denoting corresponding L -operator by L j, a ( ζ/τ ) . Let L j, a = L j, a (1 /τ ) . It is well-known that [ H N , Tr a T S , a (1 /τ )] = 0 , T S , a (1 /τ ) = lim N →∞ L − N +1 , a (1 /τ ) · · · L N, a (1 /τ ) Along with the space H S , consider the Matsubara space H M = C 2 s 1 +1 ⊗ · · · ⊗ C 2 s n +1 , with an arbitrary spin s m and a spectral parameter τ m at- tached to each component. . – p.4/21
We define � � T S , M q 2 κS +2 αS (0) O Tr S Tr M Z κ � � q 2 αS (0) O � . = � T S , M q 2 κS +2 αS (0) Tr S Tr M Here T S, M is the monodromy matrix associated with H S ⊗ H M : T S , M = T S , n (1 /τ n ) · · · T S , 1 (1 /τ 1 ) . Another way of writing the same thing is T S , M = lim N →∞ T − N +1 , M · · · T j, M · · · T N, M , where T j, M = T j, M (1) , T j, M ( ζ ) = L j, n ( ζ/τ n ) L j, n − 1 ( ζ/τ n − 1 ) · · · L j, 1 ( ζ/τ 1 ) . . – p.5/21
It is convenient to present the functional Z κ by the following picture: Space M a t s u b a r a κ σ 3 (α+κ) σ 3 = L = q =q i j . – p.6/21
Beauty of Matsubara approach. Consider a twisted transfer matrix T j, M ( ζ ) q λσ 3 � j � T M ( ζ, λ ) = Tr j . We are interested in λ = κ , λ = α + κ . We assume that T M ( ζ, κ ) , T M ( 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 | κ � T (1 , κ ) � κ | | κ + α � T (1 , κ + α ) � κ + α | . and . – p.7/21
Proper description of the space direction. X = q 2 αS (0) O is a quasi-local operator with tail α . S ( · ) = [ S, · ] where S = S ( ∞ ) is the total spi. We denote by W α the space of quasi-local operators with tail α , and by W α,s its subspace of operators of spin s ∈ Z . Consider the space ∞ W ( α ) = � W α − s,s . s = −∞ The operator q 2 αS (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: ∞ ( ζ 2 − 1) p − 1 t ∗ � t ∗ ( ζ ) = p , p =1 ∞ ∞ ( ζ 2 − 1) p − 1 b ∗ ( ζ 2 − 1) p − 1 c ∗ � � b ∗ ( ζ ) = ζ α +2 p , c ∗ ( ζ ) = ζ − α − 2 p , p =1 p =1 ∞ ∞ ( ζ 2 − 1) − p b p , c ( ζ ) = ζ α ( ζ 2 − 1) − p c p . � � b ( ζ ) = ζ − α p =0 p =0 The operator t ∗ ( ζ ) is in the center: [ t ∗ ( ζ 1 ) , t ∗ ( ζ 2 )] = [ t ∗ ( ζ 1 ) , c ∗ ( ζ 2 )] = [ t ∗ ( ζ 1 ) , b ∗ ( ζ 2 )] = 0 , [ t ∗ ( ζ 1 ) , c ( ζ 2 )] = [ t ∗ ( ζ 1 ) , b ( ζ 2 )] = 0 . . – p.9/21
The rest of the operators b , c , b ∗ , c ∗ are fermionic. The only non-vanishing anti-commutators are [ b ( ζ 1 ) , b ∗ ( ζ 2 )] + = − ψ ( ζ 2 /ζ 1 , α ) , [ c ( ζ 1 ) , c ∗ ( ζ 2 )] + = ψ ( ζ 1 /ζ 2 , α ) , ζ 2 +1 where ψ ( ζ, α ) = ζ α 2( ζ 2 − 1) . Each Fourier mode has the block structure t ∗ p : W α − s,s → W α − s,s b ∗ c ∗ p , c p : W α − s +1 ,s − 1 → W α − s,s , p , b p : W α − s − 1 ,s +1 → W α − 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: q 2 αS (0) � q 2 αS (0) � � � b ( ζ ) = 0 , c ( ζ ) = 0 . . – p.10/21
The set of operators � q 2 αS (0) � τ m t ∗ p 1 · · · t ∗ p j b ∗ q 1 · · · b ∗ q k c ∗ r 1 · · · c ∗ , r k where m ∈ Z , j, k ∈ Z ≥ 0 , p 1 ≥ · · · ≥ p j ≥ 2 , q 1 > · · · > q k ≥ 1 and r 1 > · · · > r k ≥ 1 . constitutes a basis of W α, 0 . The operators t ∗ ( ζ ) are composed of adjoint action of local integrals of motion. Question for classical case. e aϕ , W α KdV. The primary field corresponds to the space 0 F ( u, u x , u xx , · · · ) e aϕ , t ∗ ( ζ ) ≃ { ∂ 2 n − 1 } , what are the fermions com- muting with them? . – p.11/21
The main theorem. The space and Matsubara directions are related by: Z κ � t ∗ ( ζ )( X ) = 2 ρ ( ζ ) Z κ { X } , � � dξ 2 1 � Z κ � b ∗ ( ζ )( X ) ω ( ζ, ξ ) Z κ � � = c ( ξ )( X ) ξ 2 , 2 πi Γ � dξ 2 = − 1 � Z κ � c ∗ ( ζ )( X ) ω ( ξ, ζ ) Z κ � � b ( ξ )( X ) ξ 2 , 2 πi Γ where Γ goes around ξ 2 = 1 . This implies Z κ � q 2 αS (0) �� k ) b ∗ ( ζ + 1 ) · · · b ∗ ( ζ + t ∗ ( ζ 0 1 ) · · · t ∗ ( ζ 0 l ) c ∗ ( ζ − l ) · · · c ∗ ( ζ − � 1 ) k � 2 ρ ( ζ 0 ω ( ζ + i , ζ − � � = p ) × det j ) i,j =1 , ··· ,l . p =1 . – 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 � 1 � ω ( x, y ) = ( x − y ) 2 + O (1) dxdy . Normalisation � ω ( x, y ) = 0 . a j 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 � � � � � res ω i d − 1 ˜ ω i ω i − ˜ ω i ω i = 2 πi ( c 1 ◦ c 2 ) , ˜ ω j = δ i,j . i c 1 c 2 c 2 c 1 . – 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 ± λ P ± ( ζ 2 ) . The functions a ( ζ ) , d ( ζ ) are n n � � q 2 s m +1 ζ 2 /τ 2 q − 2 s m +1 ζ 2 /τ 2 � � � � a ( ζ ) = m − 1 , d ( ζ ) = m − 1 . m = 1 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. Q + ( ζ, λ ) = Q − ( ζ, − λ ) . T ( ζ, λ ) = T ( ζ, − λ ) , Deformed Abelian integrals. Introduce the function ϕ ( ζ ) which satisfies the equation a ( ζq ) ϕ ( ζq ) = d ( ζ ) ϕ ( ζ ) . 2 s m n � − 1 . ζ 2 q − 2 s m +2 k +1 − 1 � � � This function is elementary, ϕ ( ζ ) = m = 1 k =0 There are two kinds of deformed Abelian integrals, f ± ( ζ ) Q ∓ ( ζ, κ + α ) Q ± ( ζ, κ ) ϕ ( ζ ) dζ 2 � ζ 2 , Γ m where ζ ∓ α f ± ( ζ ) is a polynomial in ζ 2 . . – p.16/21
In the following, we use the q -difference operator ∆ ζ f ( 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
Recommend
More recommend
