Thermodynamic limit of the six-vertex model with reflecting end - PowerPoint PPT Presentation

Thermodynamic limit of the six-vertex model with reflecting end G.A.P. Ribeiro Federal University of S˜ ao Carlos, Brazil May 14, 2015 Collaboration with: V.E. Korepin; T.S. Tavares

Outline ◮ Problem ◮ Six-vertex model ◮ DWBC ◮ Reflecting ◮ determinant formula ◮ Homegeneous limit ◮ Special solution ◮ Entropy

Problem ◮ In statistical physics people believe that in thermodynamic limit the bulk free energy and correlations should not depend on boundary conditions. This is often true, but there are counterexamples. ◮ One of the most proeminent one is the six-vertex model: PBC � = DWBC. ◮ We would like to compute the free-energy and entropy of the six-vertex model with boundaries different boundaries: reflecting end.

Water molecule x ice: six-vertex model ◮ Water molecule: O-H distance (0.95 ˚ A ); angle between O-H: 104 ◦ ◮ Ice: X-ray data (1930s) indicates that O form a hexagonal wurtzite structure (tetraedral): O-O distance (2.76 ˚ A )

Square-ice model: six-vertex model ⇒ Effective model: square ice-model. a a b b c c ❄ ❄ ❄ ✲ ✲ ✻ ✛ ✛ ✲ ✲ ✛ ✛ ✻ ✲ ✻ ✛ ✛✲ ❄ ❄ ❄ ✻ ✻ ✻ ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ω i = e − βε i

Entropy S = Nk log W ⇒ ε 1 = ε 2 = · · · = ε 6 = 0 (or a = b = c = 1). ◮ Pauling (1935) - estimated the entropy of the hexagonal phase of ice (ordinary ice): W = 2 2 (6 / 16) = 3 2 ◮ Lieb (1967) - computed exactly the entropy for the square-ice: 3 ) 3 / 2 = 1 . 5396007 . . . . W = ( 4

Phases: six-vertex model Control parameter is ∆ = a 2 + b 2 − c 2 . 2 ab Free energy has different analytic forms when ◮ ∆ > 1 (ferroelectric). ◮ − 1 < ∆ < 1 (disordered). ◮ ∆ < − 1 (anti-ferroelectric).

Phase diagram b / c II III 1 I IV 1 a / c ◮ Phase I and Phase II (ferroelectric). ◮ Phase III (disordered). ◮ Phase IV (anti-ferroelectric). This phase diagram describes the six-vertex model with PBC (Lieb 1967,Sutherland 1967, Baxter 1982) and with DWBC (V Korepin, P Zinn-Justin 2000, P Zinn-Justin 2000). Rigourous proofs are due P Bleher et al 2006,2009,2010...

Recent realization of vertex models

DWBC In the computation of scalar product os Bethe states, | ψ � N = B ( λ N ) · · · B ( λ 2 ) B ( λ 1 ) |⇑� , appears the (Korepin 1982) Z DWBC ( { λ } , { µ } ) = �⇓| B ( λ N ) · · · B ( λ 2 ) B ( λ 1 ) |⇑� . N µ 1 µ 2 µ 3 µ 4 µ 5 ✻ ✻ ✻ ✻ ✻ ✲ ✛ λ 1 ✲ ✛ λ 2 ✲ ✛ λ 3 ✲ ✛ λ 4 ✲ ✛ λ 5 ❄ ❄ ❄ ❄ ❄

Tsuchiya partition function In the case of open spin chains, the scalar product | φ � N = B ( λ N ) · · · B ( λ 2 ) B ( λ 1 ) |⇑� . leads to another partition function for the six-vertex model Z N ( { λ } , { µ } ) = �⇓| B ( λ N ) · · · B ( λ 2 ) B ( λ 1 ) |⇑� . µ 1 µ 2 µ 3 ✏ ✻ ✻ ✻ ✲ − λ 1 ✑ ✲ λ 1 ✏ ✲ − λ 2 ✑ ✲ λ 2 ✏ ✲ − λ 3 ✑ ✲ λ 3 ❄ ❄ ❄

Reflecting ends The diagonal K -matrix plays the role of the reflecting end, � k 11 ( λ ) � 0 K ( λ ) = . 0 k 22 ( λ ) ✘ ✘ ✛ ✲ k 11 k 22 ✙ ✙ ✲ ✛

Boltzmann weights One can define the Boltzmann weights to the case − 1 < ∆ < 1. In this case, we have a ( λ ) = sin( γ − λ ) , b ( λ ) = sin( γ + λ ) , c ( λ ) = sin(2 γ ) , where 0 < γ < π/ 2 and ∆ = − cos(2 γ ). k 11 ( λ ) = sin( ξ + λ + γ ) k 22 ( λ ) = sin( ξ − λ − γ ) , , sin( ξ ) sin( ξ ) where ξ is the boundary parameter.

Tsuchiya determinant formula - (Tsuchiya 1998) N sin( ξ − µ i ) (sin(2 γ )) N � Z N ( { λ } , { µ } ) = sin(2( λ i + γ )) sin( ξ ) i =1 N � sin( γ − ( λ i − µ j )) sin( γ + λ i − µ j ) sin( γ − ( λ i + µ j )) sin( γ + λ i + µ j ) i , j =1 × N � − sin( λ j − λ i ) sin( µ i − µ j ) sin( λ j + λ i ) sin( µ i + µ j ) i , j =1 i < j × det M , where M is a N × N matrix, whose matrix elements are M ij = φ ( λ i , µ j ) with 1 φ ( λ, µ ) = . sin( γ − ( λ − µ )) sin( γ + λ − µ ) sin( γ − ( λ + µ )) sin( γ + λ + µ )

Homegeneous limit ◮ Taking λ i → λ and µ j → µ . � N � sin( ξ − µ ) Z N ( λ, µ ) = sin(2 γ ) sin(2( λ + γ )) sin( ξ ) [sin( γ − ( λ − µ )) sin( γ + λ − µ ) sin( γ − ( λ + µ )) sin( γ + λ + µ )] N 2 × N ( N − 1) C N [ − sin(2 λ ) sin(2 µ )] 2 × τ N ( λ, µ ) , � 2 . The determinant is given by �� N − 1 where C N = k =1 k ! τ N ( λ, µ ) = det( H ) , where the H -matrix elements are H i , j = ( − ∂ µ ) j − 1 ∂ i − 1 φ ( λ, µ ). λ

Bidimensional Toda equation (Ma 2011, Sylvester 1962) − τ N ∂ 2 µλ τ N + ( ∂ µ τ N )( ∂ λ τ N ) = τ N +1 τ N − 1 , and can be conveniently written as τ N +1 τ N − 1 − ∂ 2 µλ [log( τ N )] = , N ≥ 1 , τ 2 N which is supplemented by the initial data τ 0 = 1 and τ 1 = φ ( λ, µ ).

Special solutions The partition function can be cast directly in simple expressions for some special points. � N � π sin( ξ ∓ µ ) N ( N +1) N ( N − 1) Z N ( λ, µ ; γ = ) = (cos(2 λ )) 2 (sin(2 µ )) 2 . 4 sin( ξ ) For the cases where µ = ± ( λ + γ ) and µ = ± ( λ − γ ), � N � N ( N − 1) N ( N +1) sin( ξ ∓ ( λ + γ )) (sin(2 γ )) N 2 Z N ( λ, ± λ ± γ )) = ( − sin(2 λ )) 2 (sin(2( λ + γ ))) 2 , sin( ξ ) � N � N ( N − 1) sin( ξ ∓ ( λ − γ )) sin(2( γ + λ )) (sin(2 γ )) N 2 Z N ( λ, ± λ ∓ γ ) = (sin(2 λ ) sin(2( γ − λ )) 2 . sin( ξ ) log( ZN ) The thermodynamic limit is trivial in these cases. The free energy F = − lim N →∞ (we set temperature 2 N 2 to 1) is given respectively by � e − 2 F ( λ,µ ; γ = π/ 4) = cos(2 λ ) cos(2 µ ) , � e − 2 F ( λ, ± ( λ + γ )) = sin(2 γ ) − sin(2 λ ) sin(2( λ + γ )) , � e − 2 F ( λ, ± ( λ − γ )) = sin(2 γ ) sin(2 λ ) sinh(2( γ − λ )) .

VSASM We can also fix both spectral parameters and anisotropy parameter γ , such as N − 1 π (6 k + 3)!(2 k + 1)! ) = A VSASM � Z N (0 , 0; = (3 k + 2) = 1 , 3 , 26 , 646 , . . . 1 3 (4 k + 2)!(4 k + 3)! k =0 which is a combinatorial point connected to the number of vertically symmetric alternating sign matrices (VSASM) due to (Kuperberg 2002) Othe special cases are π = 2 N 2 ) = 2 N A VSASM Z N (0 , 0; , 2 4 and 3 N ( N − 3) / 2 N π ( k − 1)!(3 k )! ) / 3 N = A VSASM � Z N (0 , 0; = k ((2 k − 1)!) 2 = 1 , 5 , 126 , . . . , 3 2 N 6 k =1 where A VSASM are the x -enumeration of the vertically symmetric alternating sign matrices (Kuperberg 2002). x

Thermodynamic limit Z N ( λ, µ ) = e − 2 N 2 F ( λ,µ )+ O ( N ) , where F ( λ, µ ) is the bulk free energy and unit temperature. We suppose the following ansatz for the large size behaviour of the determinant τ N ( λ, µ ), τ N ( λ, µ ) = C N e 2 N 2 f ( λ,µ )+ O ( N ) , where sin( γ − ( λ − µ )) sin( γ + λ − µ ) sin( γ − ( λ + µ )) sin( γ + λ + µ ) e − 2 F ( λ,µ ) = e 2 f ( λ,µ ) , � − sin(2 λ ) sin(2 µ )

Liouville equation Substituting the ansatz in the Toda equation (1), we obtain − 2 ∂ 2 µλ f ( λ, µ ) = e 4 f ( λ,µ ) , which is the Liouville equation, whose general solution has the form of � − u ′ ( λ ) v ′ ( µ ) e 2 f ( λ,µ ) = , ( u ( λ ) + v ( µ )) for arbitrary C 2 functions u ( λ ) , v ( µ ).

Solution Our strategy is to chose e 2 f ( λ,µ ) to match with the solution at γ = π/ 4. This leave us a γ dependent parameter to be determined. However the λ, µ dependence was already determined. � � α − sin( αλ ) sin( αµ ) α − sin( αλ ) sin( αµ ) e 2 f ( λ,µ ) = = (1) 2 cos( α 2 ( λ − µ )) cos( α cos( αλ ) + cos( αµ ) 2 ( λ + µ )) where the parameter α = α ( γ ) and α ( π/ 4) = 4.

Solution (GAPR, VE Korepin, 2015) We must use the boundary condition given by µ = ± ( λ + γ ) to determine α parameter. In doing so we see the only possible choice for the parameter is α ( γ ) = π/γ . � γ ) sin( πµ − sin( πλ γ ) π sin( γ − λ + µ ) sin( γ + λ − µ ) sin( γ − λ − µ ) sin( γ + λ + µ ) e − 2 F ( λ,µ ) = . � cos( π ( λ − µ ) ) cos( π ( λ + µ ) 2 γ − sin(2 λ ) sin(2 µ ) ) 2 γ 2 γ (2) ◮ The other points µ = ± ( λ − γ ) are naturally fulfilled. ◮ As an independent check, the solution obtained also reproduces the special points γ = π/ 3 , π/ 4 , π/ 6.

Ferrolectric phase: ∆ > 1 In the case ∆ > 1, one can obtain the expression for the free energy looking at the leading order state. The expression for the free energy can be written as e − 2 F ( λ,µ ) = sinh( λ − | µ | + | γ | ) � sinh( λ + | µ | − γ ) sinh( λ + | µ | + γ ) . However due to the lack of additional boundary condition, we are unable to fix the suitable solution of Liouville equation. µ µ µ µ µ µ ✏ ✏ ✻ ✻ ✻ ✻ ✻ ✻ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ λ λ ✑ ✑ ✻ ✻ ✻ ✻ ✻ ✻ ✲ ✛ ✛ ✛ ✲ ✲ ✲ ✛ λ λ ❄ ❄ ✏ ✏ ✻ ✻ ✻ ✻ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ λ λ ❄ ❄ ✑ ✑ ✻ ✻ ✻ ✻ ✲ ✲ ✛ ✛ ✲ ✲ ✛ ✛ λ λ ❄ ❄ ❄ ❄ ✏ ✏ ✻ ✻ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ λ λ ❄ ❄ ❄ ❄ ✑ ✑ ✻ ✻ ✲ ✲ ✲ ✛ ✲ ✛ ✛ ✛ λ λ ❄ ❄ ❄ ❄ ❄ ❄ γ > 0 γ < 0