Finite size Emptiness Formation Probability for the XXZ spin chain at = 1/2

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1 Finite size Emptiness Formation Probability for the XXZ spin chain at = 1/2 Luigi Cantini luigi.cantini@u-cergy.fr LPTM, Université de Cergy-Pontoise (France) CFT AND INTEGRABLE MODELS Bologna, September 12-15, 2011 Based on arxiv:1109.???? Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

2 XXZ spin chain at = 1 2 Hamiltonian of the XXZ spin chain H N ( ) = 1 2 N i=1 σxσ i x i+1 + σyσ i y i+1 + σzσ i z i+1 = q + q 1 2 [Razumov, Stroganov 00] = 1 2 g.s. energy E = 3N 4 N odd (N = 2n + 1) Periodic boundary conditions: σ α N+1 = σα 1. Two-fold degenerate ground state Ψ ± 2n+1, rational-valued components. N even (N = 2n) Twisted periodic boundary conditions: σ z N+1 = σz 1, while σ ± N+1 = e±i 2π 3 σ ± N+1, where σ± = σ x + ±iσ y. A single ground state Ψ e 2n, complex valued components. Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

3 Emptiness Formation Probability (EFP) The definition of the EFP ( (Ψ µ E µ N (k) = N ), ) k i=1 p+ i Ψ µ N (Ψ µ N, Ψµ N ), p + i = σz i µ = ±, e Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

4 Emptiness Formation Probability (EFP) The definition of the EFP ( (Ψ µ E µ N (k) = N ), ) k i=1 p+ i Ψ µ N (Ψ µ N, Ψµ N ), p + i = σz i µ = ±, e we define also a Pseudo EFP ( E ẽ2n(k) Ψ e 2n, ) k i=1 p+ i Ψ e 2n = (Ψ e 2n, Ψe 2n ). Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

5 Emptiness Formation Probability (EFP) The definition of the EFP ( (Ψ µ E µ N (k) = N ), ) k i=1 p+ i Ψ µ N (Ψ µ N, Ψµ N ), p + i = σz i µ = ±, e we define also a Pseudo EFP ( E ẽ2n(k) Ψ e 2n, ) k i=1 p+ i Ψ e 2n = (Ψ e 2n, Ψe 2n ). They have an analytical, factorized form! Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

6 Conjectures [Razumov, Stroganov 00] E 2n+1 (k 1) E 2n+1 (k) = E + 2n+1 (k 1) E + 2n+1 (k) = (2k 2)!(2k 1)!(2n + k)!(n k)! (k 1)!(3k 2)!(2n k + 1)!(n + k 1)! (2k 2)!(2k 1)!(2n + k)!(n k + 1)! (k 1)!(3k 2)!(2n k + 1)!(n + k)! E e 2n(k 1) E e 2n (k) = (2k 2)!(2k 1)!(2n + k 1)!(n k)! (k 1)!(3k 2)!(2n k)!(n + k 1)! E ẽ2n(k 1) = E ẽ2n (k) (2k 2)!(2k 1)!(2n + k 1)!(n k)! q (k 1)!(3k 3)!(3k 1)(2n k)!(n + k 1)! Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

7 Some partial results Asymptotics N [Maillet et al. 02] ( ) 3k 2 3 k lim E Γ(j 1/3)Γ(j + 1/3) N(k) = N 2 Γ(j 1/2)Γ(j + 1/2). j=1 Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

8 Some partial results Asymptotics N [Maillet et al. 02] ( ) 3k 2 3 k lim E Γ(j 1/3)Γ(j + 1/3) N(k) = N 2 Γ(j 1/2)Γ(j + 1/2). j=1 Norm [Di Francesco et al. 06] E ± 2n+1 (n) = A HT (2n + 1) 1 = n j=1 (2j 1)! 2 (2j)! 2 (j 1)!j!(3j 1)!(3j)! E ẽ2n(n) = ( q) n A 2 n = ( q) n CSSC(2n) 1 where: A n = # of n n Alternating Sign Matrices A HT (2n + 1) = # of (2n + 1) (2n + 1) Half-Turn Symmetric Alternating Sign Matrices Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

9 CSSC(2n, k 1) E ẽ2n(k 1) CSSC(2n, k 1) E ẽ2n (k) = q CSSC(2n, k) CSSC(2n, k 1) = # of lozange tilings of a regular hexagon of side length 2n, which are invariant under a rotation of π/3 and have a star shaped frozen region of length k, (CSSC(2n) := CSSC(2n, 0) = CSSC(2n, 1)) 2n 3k Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

10 Integrability R-matrix with Twist matrix R i,j (z) = a(z) = a(z) b(z) c 1 (z) 0 0 c 2 (z) b(z) a(z) qz q 1 q q 1 z, b(x) = z 1 q q 1 z, c 1 (z) = (q q 1 )z q q 1 z, c 2(z) = (q q 1 ) q q 1 z. Ω(φ) = ( e iφ 0 0 e iφ ) Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

11 R-Ω commutation [R i,j (x), Ω i (φ) Ω j (φ)] = 0 Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

12 R-Ω commutation Yang-Baxter equation [R i,j (x), Ω i (φ) Ω j (φ)] = 0 R i,j (z i /z j )R i,k (z i /z k )R j,k (z j /z k ) = R j,k (z j /z k )R i,k (z i /z k )R i,j (z i /z j ) Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

13 R-Ω commutation [R i,j (x), Ω i (φ) Ω j (φ)] = 0 Yang-Baxter equation R i,j (z i /z j )R i,k (z i /z k )R j,k (z j /z k ) = R j,k (z j /z k )R i,k (z i /z k )R i,j (z i /z j ) Transfer matrix T N (y z {1,...,N}, φ) = tr 0 [R 0,1 (y/z 1 )R 0,2 (y/z 2 )... R 0,N (y/z N )Ω 0 (φ)] Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

14 R-Ω commutation [R i,j (x), Ω i (φ) Ω j (φ)] = 0 Yang-Baxter equation R i,j (z i /z j )R i,k (z i /z k )R j,k (z j /z k ) = R j,k (z j /z k )R i,k (z i /z k )R i,j (z i /z j ) Transfer matrix T N (y z {1,...,N}, φ) = tr 0 [R 0,1 (y/z 1 )R 0,2 (y/z 2 )... R 0,N (y/z N )Ω 0 (φ)] Crucial idea! Consider the ground state with spectral parameters Ψ µ N Ψµ N (z), T N(y z {1,...,N}, φ)ψ µ N (z) = 1Ψµ N (z) Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

15 quantum Knizhnik Zamolodchikov (qkz) equations [Razumov, Stroganov & Zinn-Justin 07] The ground state satisfies a specialization q = e 2πi/3 of the U q (sl 2 ) qkz equations Ř i,i+1 (z i+1 /z i )Ψ µ N (..., z i, z i+1,... ) = Ψ µ N (..., z i+1, z i,... ) With σψ µ N (z 1, z 2,..., z N 1, z N ) = DΨ µ N (z 2,..., z N 1, z N, sz 1 ). σ(v 1 v 2 v N 1 v N ) = v 2 v N 1 v N v 1 Ř i,i+1 (z) = P i,i+1 R i,i+1 (z) P i,j (e µ i e ν j ) = e ν i e µ j, s = q 6, D = q 3N q 3(sz N +1)/2. Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

16 Inhomogenous EFP Using the solution of the U q (sl 2 ) qkz equations at level 1. we define an inhomogenous version of the EFP E µ N (k; y {1,...,2k}; z {1,...,N k} ) (P N,k (z)(ψ µ N (q 6 y {k+1,...,2k} ;z)), k i=1 p+ i Ψ µ N (y {1,...,k};z)) 1 i<j k (qy i q 1 y j )(qy i+k q 1 y j+k ) with P N,k (z) = N i=k+1 (z ip + i + p i ). and of the Pseudo EFP E ẽn (k; y {1,...,2k}; z {1,...,N k} ) ( Ψ µ N (k;q 6 y 1 {k+1,...,2k} ;z 1 ), k ) i=1 p+ i Ψ µ N (k;y {1,...,k};z) 1 i<j k (qy i q 1 y j )(qy i+k q 1 y j+k ) Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

17 Inhomogenous EFP Using the solution of the U q (sl 2 ) qkz equations at level 1. we define an inhomogenous version of the EFP E µ N (k; y {1,...,2k}; z {1,...,N k} ) (P N,k (z)(ψ µ N (q 6 y {k+1,...,2k} ;z)), k i=1 p+ i Ψ µ N (y {1,...,k};z)) 1 i<j k (qy i q 1 y j )(qy i+k q 1 y j+k ) with P N,k (z) = N i=k+1 (z ip + i + p i ). and of the Pseudo EFP E ẽn (k; y {1,...,2k}; z {1,...,N k} ) ( Ψ µ N (k;q 6 y 1 {k+1,...,2k} ;z 1 ), k ) i=1 p+ i Ψ µ N (k;y {1,...,k};z) 1 i<j k (qy i q 1 y j )(qy i+k q 1 y j+k ) We shall compute these and then take the specialization z i = y j = 1. Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

18 Properties of E µ N (k; y; z) and of E ẽ2n (k; y; z) 1 Symmetries under z i z j, y i y j. Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

19 Properties of E µ N (k; y; z) and of E ẽ2n (k; y; z) 1 Symmetries under z i z j, y i y j. 2 Specialization z i = 0 or z i lim z 2n 2n+1 E 2n+1 (k; y; z) E z 2n+1 2n(k; e y; z \ z 2n+1 ) E e 2n+2(k; y; z) z2n+2 =0 E + 2n+1 (k; y; z \ z 2n+2) Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

20 Properties of E µ N (k; y; z) and of E ẽ2n (k; y; z) 1 Symmetries under z i z j, y i y j. 2 Specialization z i = 0 or z i lim z 2n 2n+1 E 2n+1 (k; y; z) E z 2n+1 2n(k; e y; z \ z 2n+1 ) E e 2n+2(k; y; z) z2n+2 =0 E + 2n+1 (k; y; z \ z 2n+2) 3 Recursion relation for z j = q ±2 z i z i 2k l=1 qy l q 1 z i q q 1 E µ N (k; y; z) z j =q 2 z i E µ N 2 (k; y; z \ {z i, z j }) 1 l N k l i,j (qz l q 1 z i )(q 3 z i q 1 z l ) (q q 1 ) 2 Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

21 Properties of E µ N (k; y; z) and of E ẽ2n (k; y; z) 1 Symmetries under z i z j, y i y j. 2 Specialization z i = 0 or z i lim z 2n 2n+1 E 2n+1 (k; y; z) E z 2n+1 2n(k; e y; z \ z 2n+1 ) E e 2n+2(k; y; z) z2n+2 =0 E + 2n+1 (k; y; z \ z 2n+2) 3 Recursion relation for z j = q ±2 z i 2k l=1 qy l q 1 z i q q 1 E ẽ2n(k; y; z) zj =q 2 z i E ẽ2n 2 (k; y; z \ {z i, z j }) 1 l 2n k l i,j (qz l q 1 z i )(q 3 z i q 1 z l ) (q q 1 ) 2 Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

22 Properties of E µ N (k; y; z) and of E ẽn (k; y; z) 4 Factorized cases E e/ẽ (qz i q 1 z j )(qz j q 1 z i ) 2k (k; y; z) = (q q 1 ) 2 1 i<j k E + 2k+1 (k + 1; y; z) = E 2k+1 (k; y; z) = 1 i<j k 1 i<j k+1 (qz i q 1 z j )(qz j q 1 z i ) (q q 1 ) 2 k i=1 (qz i q 1 z j )(qz j q 1 z i ) (q q 1 ) 2. z 2 i Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

23 Properties of E µ N (k; y; z) and of E ẽn (k; y; z) 4 Factorized cases E e/ẽ (qz i q 1 z j )(qz j q 1 z i ) 2k (k; y; z) = (q q 1 ) 2 1 i<j k E + 2k+1 (k + 1; y; z) = E 2k+1 (k; y; z) = 1 i<j k 1 i<j k+1 (qz i q 1 z j )(qz j q 1 z i ) (q q 1 ) 2 k i=1 (qz i q 1 z j )(qz j q 1 z i ) (q q 1 ) 2. z 2 i These properties completely determine E µ N (k; y; z) and E ẽ2n(k; y; z)! Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

24 Combinatorial point q = e 2πi/3 Let ρ, σ be strictly increasing infinite sequences of nonnegative integers. Introduce the following matrices z ρ 1 1 z ρ z ρr+s z ρ 1 2 z ρ z ρr+s z ρ 1 r z ρ 2 r... zr ρr+s z σ 1 1 z σ z σr+s 1 M ( ρ, σ) (r, s; y; z) = z σ 1 2 z σ z σr+s z σ 1 r z σ 2 r... zr σr+s y ρ 1 1 y ρ y ρr+s 1 y σ 1 1 y σ y σr+s y ρ 1 2s y ρ 2 2s... y ρr+s 2s y σ 1 2s y σ 2 2s... y σr+s 2s Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

25 Combinatorial point q = e 2πi/3 Define the following polynomials S ( ρ, σ) (r, s; y; z) = det M ( ρ, σ) (r,s;y;z) 1 i<j r (z i z j ) 2 1 i<j 2s (y i y j ). 1 i r (z i y j ) 1 j 2s Using the Laplace expansion along the first r + s columns we can write S ( ρ, σ) (r, s; y; z) as a bilinear in Schur polynomials I {1,...2s} I =s S ( ρ, σ) (r, s; y; z) = ( 1) ɛ(i ) i<j I (y i y j ) i<j I c (y i y j ) 1 i<j 2s (y i y j ) S ρ(r+s) (z, y I )S σ(r+s) (z, y I c), Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

26 Combinatorial point q = e 2πi/3 Now let us introduce the following family of integer sequences λ i (r) = 3i 3 + r, 2 λ(0) = {0, 1, 3, 4, 6, 7,... } λ(1) = {0, 2, 3, 5, 6, 8,... } λ(2) = {1, 2, 4, 5, 7, 8,... } We claim that E 2n+1 (k; y; z) S( λ(0), λ(1)) (2n + 1 k, k; y; z) E + 2n+1 (k; y; z) S( λ(1), λ(2)) (2n + 1 k, k; y; z) E2n(k; e y; z) S ( λ(0), λ(1)) (2n k, k; y; z) E ẽ2n(k; y; z) S ( λ(0), λ(2)) (2n k, k; y; z) Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

27 t specialization Setting and z = {z 1 = t 0, z 2 = t 1,..., z N k = t N k 1 } y = {y 1 = t N k,..., y 2k = t N+k 1 } the functions E µ N (k; y; z) and E ẽ2n(k; y; z), as rational functions of t, have simple factors. Using the usual t-numbers and t-factorial [i] t = ti 1 t 1 and [n] t! = n [i] t. i=1 Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

28 t specialization: result E2n(k e 1; t) E2n e (k; t) [2n + k 1] t![n k] t3![2k 1] t3![2k 2] t3! [2n k] t![n + k 1] t 3![k 1] t 3![3k 2] t! E ẽ2n(k 1; t) ( q)e ẽ2n (k; t) [2n + k 1] t![n k] t3![2k 1] t3![2k 2] t3! [2n k] t![n + k 1] t 3![k 1] t 3![3k 3] t![3k 1] t E 2n+1 (k 1; t) E 2n+1 (k; t) [2n + k] t![n k] t 3![2k 1] t 3![2k 2] t 3! [2n k + 1] t![n + k 1] t 3![k 1] t 3![3k 2] t! E + 2n+1 (k 1; t) E + 2n+1 (k; t) [2n + k] t![n k + 1] t3![2k 1] t3![2k 2] t3! [2n k + 1] t![n + k] t 3![k 1] t 3![3k 2] t! ( ) k 1 The coefficients in front are of the form t β [3]t 3 1. t 1 Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

29 What if q is generic? We have found a multiresidua formula n k {qz i } l=1 E µ 2n ɛ(µ) (k; y; z) dw l w α(µ) 2k j=1 (w l y j ) 2πi 2n k ɛ(µ) i=1 (w l qz i )(w l q 1 z i ) K n k(w) where ɛ(±) = ±1, ɛ(e) = ɛ(ẽ) = 0, α(+) = α(e) = 0, α(ẽ) = 1, α( ) = 2 and K N (w) = (w i w j ) 2 (qw i q 1 w j )(qw j q 1 w i ). 1 i<j N Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

30 Perspectives Consider other correlation functions. Other boundary conditions (e.g. open chain with diagonal K matrices) Higher spins or also U q (sl N ) spin chain. Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21

31 Thank you! Cantini (LPTM Cergy-Pontoise) EFP at = 1/2 Bologna, / 21