Ferminoic Basis in Integrable Models: Profile and Prospect Michio - PowerPoint PPT Presentation

Ferminoic Basis in Integrable Models: Profile and Prospect Ferminoic Basis in Integrable Models: Profile and Prospect Michio Jimbo (Rikkyo University, Japan) Workshop “Mathematical Statistical Physics” Yukawa Institute, August 1 2013

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction In this talk we are concerned with an old topic from integrable quantum field theory in two dimensions: to describe the space of local fields and their vacuum expectation values (VEVs) , or one point functions.

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction VEV The significance of VEV has been underlined in the work of Al.Zamolodchikov 1991 on integrable perturbation of conformal field theory.

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction VEV The significance of VEV has been underlined in the work of Al.Zamolodchikov 1991 on integrable perturbation of conformal field theory. To study the correlator of some field Φ ( x ) at short distances, one can apply the operator product expansion , � C i Φ ( x ) Φ (0) = Φ Φ ( x ) A i (0) , i where { A i (0) } is a complete set of local fields in the theory.

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction VEV The significance of VEV has been underlined in the work of Al.Zamolodchikov 1991 on integrable perturbation of conformal field theory. To study the correlator of some field Φ ( x ) at short distances, one can apply the operator product expansion , � C i Φ ( x ) Φ (0) = Φ Φ ( x ) A i (0) , i where { A i (0) } is a complete set of local fields in the theory. The coe ffi cients C i Φ Φ ( x ) are local data accessible by perturbation theory. In contrast, the VEVs � A i (0) � are global data which encode all non-perturbative information. For the characterization of correlation functions, it is necessary to know all of them .

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction Our main example is the sine-Gordon (sG) model µ 2 1 16 π ( ∂ µ ϕ ) 2 − sin πβ 2 ( e − i βϕ + e i βϕ ) . L s G = It is a perturbation of a CFT of massless bosons. In CFT, the space of fields is a Verma module spanned by a e ia ϕ primary field and their descendants, ∂ m 1 ϕ · · · ∂ m K ϕ ¯ ∂ n 1 ϕ · · · ¯ ∂ n L ϕ · e ia ϕ . In the sG model we consider local fields of this form. Among them, VEV has been known for the primary field and for the first non-trivial descendant.

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction Known results about VEV Primary field (LZ 1997) να � e ia ϕ � = [ Γ ( ν ) µ ] 2(1 − ν ) � ∞ sinh 2 ( να t ) ν 2 α 2 � 2(1 − ν ) e − 2 t � dt � � × exp 2 sinh(1 − ν ) t sinh t cosh ν t − . t 0 First non-trivial descendant (FFLZZ 1998) � L − 2 ¯ γ ( − 1 1 1 L − 2 e ia ϕ � = − ( Γ ( ν ) µ ) 4 / ν 2 + α 2 + 2 ν ) γ ( α 2 − 2 ν ) γ ( 1 1 1 � e ia ϕ � (1 − ν ) 2 2 + α 2 − 2 ν ) γ ( α 2 + 2 ν ) where Γ ( x ) ν = 1 − β 2 , γ ( x ) = να = 2 β a . Γ (1 − x ) ,

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction The goal of this talk is to explain that there is a conjectural basis better suited for the systematic description of VEVs.

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction Fermionic basis To create the ‘descendants’, we use two kinds of linear operators acting on the space of local fields.

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction Fermionic basis To create the ‘descendants’, we use two kinds of linear operators acting on the space of local fields. The first is the adjoint action by the local integrals of motion (IM) ¯ i p ( O ) = [¯ i p ( O ) = [ I p , O ] , I p , O ] ( p = 1 , 3 , 5 , · · · )

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction Fermionic basis To create the ‘descendants’, we use two kinds of linear operators acting on the space of local fields. The first is the adjoint action by the local integrals of motion (IM) ¯ i p ( O ) = [¯ i p ( O ) = [ I p , O ] , I p , O ] ( p = 1 , 3 , 5 , · · · ) The second is a set of fermions commuting with IM, p , ¯ β ∗ β ∗ p , γ ∗ γ ∗ p , ¯ ( p = 1 , 3 , 5 , · · · ) . p The basis in question is given by ( ♯ J + = ♯ J − , ♯ ¯ J + = ♯ ¯ i K ¯ J − ¯ β ∗ J − e ia ϕ i ¯ K β ∗ J + γ ∗ J + ¯ γ ∗ J − ) ¯ ¯ where ( J + = { j 1 , · · · , j k } , j 1 < · · · < j k ) , etc.. β ∗ J + = β ∗ j 1 · · · β ∗ j k

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction The VEV of the basis elements are given by β ∗ ⟨ β ∗ J − ¯ J + γ ∗ γ ∗ J − e ia ϕ ⟩ J + ¯ ¯ ¯ = µ 2 ν − 1 ( | J + | + | J − | ) δ J + , ¯ J − δ J − , ¯ J + ⟨ e ia ϕ ⟩ ν cot π i ν cot π i ∏ ∏ × 2 ν ( p + να ) 2 ν ( r − να ) , p ∈ J + r ∈ J − ∑ where | I | = p . p ∈ I The IM do not contribute to VEV: ⟨ [ I p , O ] ⟩ = ⟨ [¯ I p , O ] ⟩ = 0.

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction Existence of such a fermionic basis is our main conjecture.

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction Existence of such a fermionic basis is our main conjecture. The fermions β ∗ Remark. p , etc., are not dynamical variables but rather a member of a symmetry algebra. They are not to be confused with the fermions of the massive Thirring model.

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction Existence of such a fermionic basis is our main conjecture. The fermions β ∗ Remark. p , etc., are not dynamical variables but rather a member of a symmetry algebra. They are not to be confused with the fermions of the massive Thirring model. These fermions seem to appear totally out of the blue, but they appeared already in the literature (Babelon et al. 1997, will comment later.)

Ferminoic Basis in Integrable Models: Profile and Prospect Introduction Plan of the talk 1 6 vertex model and expectation values (Existence theorem) 2 Field theory limit (Conjectures) 3 Relation to previous works: Form factors, Reflection equation 4 Summary Joint work with T.Miwa, F.Smirnov, H.Boos, Y.Takeyama (in part)

Ferminoic Basis in Integrable Models: Profile and Prospect 6 vertex model Lattice regularization: 6 vertex model Consider a six vertex model on an infinite cylinder, (0 < ν < 1 q = e π i ν 2) . On one row, we allow background fields q κσ 3 j (for all j ), q ασ 3 j (for j ≤ 0) and a local dislocation O . One can think of 0 S (0) = 1 � q 2 α S (0) , σ 3 j , 2 j = −∞ as a lattice analog of the primary field e ia ϕ . We take the following set as an analog of the space of local fields. W ( α ) = � W α − s , s , s ∈ Z W α − s , s = { q 2( α − s ) S (0) O | O : local, spin s }

Ferminoic Basis in Integrable Models: Profile and Prospect 6 vertex model Lattice analogs of VEV (more generally, of matrix elements) is the ratio of partition functions with/without dislocation, T [ K , L ] , M (1) q 2 κ S [ K , L ] +2 α S [ K , 0] O { } = ⟨ Φ | Tr [ K , L ] | Ψ ⟩ { } q 2 α S (0) O Z , T [ K , L ] , M (1) q 2 κ S [ K , L ] +2 α S [ K , 0] } { ⟨ Φ | Tr [ K , L ] | Ψ ⟩ where K ≪ 0 ≪ L , ↷ ↶ L n ∏ ∏ T [ K , L ] , M = T j , M (1) , T j , M ( ζ ) = L j , m ( ζ ) j = K m = 1 is the monodromy matrix, and Φ, Ψ are eigen(co)vector of the transfer matrix in the vertical direction [ T j , M ( ζ ) q κσ 3 ] T M ( ζ, κ ) = Tr j . j

Ferminoic Basis in Integrable Models: Profile and Prospect 6 vertex model Fermions on the lattice One can construct fermions acting on W ( α ) (BJMST 2007–2009) b ( ζ ) , c ( ζ ) , b ∗ ( ζ ) , c ∗ ( ζ ) , ∞ ∞ b p ( ζ 2 − 1) − p , p ( ζ 2 − 1) p − 1 , etc . ∑ ∑ b ∗ ( ζ ) = b ∗ b ( ζ ) = p =1 p =1 such that They commute with integrals of motion: t ∗ 1 , t ∗ 2 , · · · The following is a basis of W ( α ) : ( t ∗ 1 ) p t ∗ i 1 · · · t ∗ i r b ∗ j 1 · · · b ∗ j s c ∗ k 1 · · · c ∗ k t ( q 2 α S (0) ) ( i 1 ≥ · · · ≥ i r ≥ 2 , j 1 > · · · > j s ≥ 1 , k 1 > · · · > k t ≥ 1 , p ∈ Z , r , s , t ≥ 0) .

Ferminoic Basis in Integrable Models: Profile and Prospect 6 vertex model The main formula The following is the key result (JMS 2009). { } t ∗ ( η 1 ) · · · t ∗ ( η s ) b ∗ ( ζ 1 ) · · · b ∗ ( ζ r ) c ∗ ( ξ r ) · · · c ∗ ( ξ 1 )( q 2 α S (0) ) Z s ∏ = 2 ρ ( η i ) · det ( ω ( ζ j , ξ k )) , i =1 where ρ ( η ) = T Φ ( η ) T Ψ ( η ) , T M ( η ) | Ψ ⟩ = T Ψ ( η ) | Ψ ⟩ , and ω ( ζ, ξ ) is defined through linear and non-linear integral equations of Thermodynamic Bethe Ansatz type.

Ferminoic Basis in Integrable Models: Profile and Prospect 6 vertex model TBA data auxiliary function characterizing | κ ⟩ K 0 ( ζ/ξ ) log(1 + a ( ξ, κ )) d ξ 2 log a ( ζ, κ ) = − 2 π i νκ + log a ( ζ ) ∫ d ( ζ ) − ξ 2 , γ where γ encircles the Bethe roots clockwise, and a ( ζ ) = (1 − q ζ 2 ) n , d ( ζ ) = (1 − q − 1 ζ 2 ) n , ζ α K α ( ζ ) = ∆ ζ ψ ( ζ, α ) , ψ ( ζ, α ) = ζ 2 − 1 , ∆ ζ f ( ζ ) = f ( q ζ ) − f ( q − 1 ζ ) . resolvent R dress − R dress ⋆ K α = K α , d η 2 ∫ 1 1 f ⋆ g ( ζ, ξ ) = f ( ζ, η ) g ( η, ξ ) η 2 . 1 + a ( η, κ ) ρ ( η ) γ