A master solution of the Yang-Baxter equation and classical discrete - PowerPoint PPT Presentation

A master solution of the Yang-Baxter equation and classical discrete integrable equations. Vladimir Bazhanov (in collaboration with Sergey Sergeev & Vladimir Mangazeev) Australian National University Mathematical Statistical Mechanics, Kyoto University, 1 August 2013 V. Bazhanov (ANU) Master solution of YBE 1 August 2013 1 / 39

Outline Lattice models of statistical mechanics and field theory, Quantum Yang-Baxter equation. Star-triangle relation. low-temperature (quasi-classical) limit and its relation to classical mechanics. New “master” solution to the star-triangle relation (STR) contains all previously known solutions to STR Ising & Kashiwara-Miwa models Fateev-Zamolodchikov & chiral Potts models elliptic gamma-functions & Spiridonov’s elliptic beta integral Low-temperature (quasi-classical) limit of the “master solution”. relation to the Adler-Bobenko-Suris classical non-linear integrable equations on quadrilateral graphs, new integrable models of statistical mechanics where the Boltzmann weights are determined by classical integrable equations ( Q 4 ). V. Bazhanov (ANU) Master solution of YBE 1 August 2013 2 / 39

Space of solutions to the Yang-Baxter equation YBE is an overdetermined system of algebraic equations. Its general solution is unknown even in the simplest cases. Known solutions, various methods: Onsager, McGuire, Yang, Baxter, . . . (over 65 different authors; native languages: Russian 26, Japanese 15, English 9, German 4, French 4, . . . , Norwegian 1.) Algorithmic recipes (Drinfeld,Jimbo) Universal R -matrix for quantized (affine) Lie algebras, or quantum groups. 3D-generalization: tetrahedron equation, Zamolodchikov (1980) followed by Baxter, Bazhanov, Kashaev, Korepanov, Mangazeev, Maillet-Nijhoff, Sergeev, Stroganov,. . . New result (VB-Mangazeev-Sergeev): 3D integrable model with POSITIVE Boltzmann weights V. Bazhanov (ANU) Master solution of YBE 1 August 2013 3 / 39

Yang-Baxter equation in statistical mechanics Local “spins”: σ i ∈ ( set of values), σ i ∈ R � e − E ( σ ) /T , Z = { spins } � E ( { σ } ) = ǫ ( σ i , σ j ) , ( ij ) ∈ edges Boltzmann weights W ( σ i , σ j ) = e − ǫ ( σ i ,σ j ) /T � � Z = W ( σ i , σ j ) . { spins } ( ij ) ∈ edges The problem: calculate partition function when number of edges is infinite, √ log Z = − Nf/T + O ( N ) , N → ∞ Solvable analytically if the Boltzmann weights satisfy the Yang-Baxter equation V. Bazhanov (ANU) Master solution of YBE 1 August 2013 4 / 39

Two types of Boltzmann y weights, depending on the arrangement of rapidity line x y wrt the edge p q p x q W p − q ( x, y ) and W p − q ( x, y ) . W p − q ( x, y ) W p − q ( x, y ) Simplest form of the Yang-Baxter equation: the star-triangle relation � W p − q ( σ, b ) W p − r ( c, σ ) W q − r ( a, σ ) = W p − q ( c, a ) W p − r ( a, b ) W q − r ( c, b ) . σ V. Bazhanov (ANU) Master solution of YBE 1 August 2013 5 / 39

General structure of Boltzmann weights In general, weights W are related to W via � W p − q ( x, y ) = S ( x ) S ( y ) W η − p + q ( x, y ) , where S ( x ) are one-“spin” weights and η is the non-zero crossing parameter (value of an open angle). In most cases the Boltzmann weights W are symmetric, W p − q ( x, y ) = W p − q ( y, x ) . Let for shortness p − q = α 1 , q − r = α 3 . The star-triangle relation takes the form (assume continuous spins) � dx 0 S ( x 0 ) W η − α 1 ( x 1 , x 0 ) W α 1 + α 3 ( x 2 , x 0 ) W η − α 3 ( x 3 , x 0 ) = W α 1 ( x 2 , x 3 ) W η − α 1 − α 3 ( x 1 , x 3 ) W α 3 ( x 1 , x 2 ) V. Bazhanov (ANU) Master solution of YBE 1 August 2013 6 / 39

Planar graph G , where L is the medial graph V. Bazhanov (ANU) Master solution of YBE 1 August 2013 7 / 39

Low-temperature limit Partition function � p − q, � � � 1 st -type Z = W α ij ( x i , x j ) S ( x m ) dx m , α ij = 2 nd -type η − p + q, m ( ij ) Assume, there is a temperature-like parameter ε , such for ε → 0 W α ( x, y ) = e − Λ α ( x,y ) /ε + O (1) , S ( x ) = ε − 1 / 2 e − C ( x ) /ε + O (1) � � log Z = − 1 ε E ( x ( cl ) ) + O (1) , E ( x ) = Λ α ij ( x i , x j ) + C ( x m ) m ( ij ) and the variables x ( cl ) = { x ( cl ) , x ( cl ) , . . . } solve the variational equations 1 2 � ∂ E ( x ) � � x = x ( cl ) = 0 ∂x j Can one obtain in this way the Q4 system of Adler-Bobenko-Suris, 2003? � x − y � x + y dξ log ϑ 4 (( ξ − i α ) | τ ) dξ log ϑ 4 ( ξ − i α | τ ) Λ α ( x, y ) = − i ϑ 4 ( ξ + i α | τ ) − i ϑ 4 ( ξ + i α | τ ) 0 π/ 2 | x | − π � 2 | x | < π � C ( x ) = 2 . (1) 2 2 V. Bazhanov (ANU) Master solution of YBE 1 August 2013 8 / 39

Z-invariance (Baxter 1979) Partition function depends only on the boundary data (i.e., on values of boundary spins and values of rapidities) but not on details of the lattice inside. Baxter’s factorization theorem (1979) � √ log Z = − 1 f ( α ij ) + O ( N ) T <ij> V. Bazhanov (ANU) Master solution of YBE 1 August 2013 9 / 39

Low-temperature limit of the star-triangle relation � � � � � − E ⋆ ( x 0 ) − E △ ε − 1 / 2 dx 0 exp + O (1) ε + O (1) = exp ε where E ⋆ = Λ η − α 1 ( x 0 , x 1 ) + Λ α 1 + α 3 ( x 0 , x 2 ) + Λ η − α 3 ( x 0 , x 3 ) + C ( x 0 ) , E △ = Λ α 1 ( x 2 , x 3 ) + Λ η − α 1 − α 3 ( x 1 , x 2 ) + Λ α 3 ( x 1 , x 2 ) the STR implies E ⋆ = E △ at the stationary point ∂ E ⋆ = 0 ∂x 0 Any solution of STR, admitting low-temperature expansion, leads to classical discrete integrable system, whose action is invariant under star-triangle moves V. Bazhanov (ANU) Master solution of YBE 1 August 2013 10 / 39

Chiral Potts and Kashiwara-Miwa models N -state chiral Potts model (Albertini, McCoy et al’87, Baxter-Perk-AuYang’87) � µ p � ( a − b ) a − b � y q − ω k x p W pq ( a, b ) = y p − ω k x q µ q k =1 ω N = 1 , and ( x p , y p , µ p ) is a point on genus ≥ 1 algebraic curve Positive Boltzmann weights. Reduces to Ising model for N = 2 . Contains Z N model (Fateev-Zamolodchikov’82) R-matrix R cd ab = W pq ( a, b ) W pq ( b, c ) W pq ( d, c ) W pq ( a, d ) intertwines two cyclic representations of U q ( � sl (2)) (VB-Stroganov’90) V. Bazhanov (ANU) Master solution of YBE 1 August 2013 11 / 39

Chiral Potts and Kashiwara-Miwa models N-state model with broken Z N symmetry (Kashiwara-Miwa’86) W θ ( a, b ) = r θ ( a − b ) t θ ( a + b ) n n � ϑ 1 ( π N ( k − 1 θ � ϑ 4 ( π N ( k − 1 θ 2 ) − 2 N ) 2 ) − 2 N ) r θ ( n ) = 2 N ) , t θ ( n ) = 2 N ) , ϑ 1 ( π N ( k − 1 θ ϑ 4 ( π N ( k − 1 θ 2 ) + 2 ) + k =1 k =1 Reduces to Ising model for N = 2 . In the trig. case reduces to Z N model (Fateev-Zamolodchikov’82) The correponding R-matrix intertwines two (special) cyclic representations of Sklyanin algebra (Hasegawa-Yamada’90) V. Bazhanov (ANU) Master solution of YBE 1 August 2013 12 / 39

Chiral Potts and Kashiwara-Miwa models Is there a generalised KM-model corresponding to the most general cyclic representations of the Sklyanin algebra? (VB-Stroganov,90 unpublished) W θ ( a, b ) = r θ ( a − b, α − β ) t θ ( a + b, α + β ) � N ( θ + φ ) � n/N n � ϑ 1 ( π N ( k − 1 1 2 ) − 2 N ( θ − φ )) r θ ( n, φ ) = ϑ 1 ( π N ( k − 1 1 N ( θ − φ ) 2 ) + 2 N ( θ + φ )) k =1 What is the meaning of the additional parameters? V. Bazhanov (ANU) Master solution of YBE 1 August 2013 13 / 39

Master solution to the star-triangle relation Elliptic gamma-function Γ( x + 1) Γ trig ( x + δ ) Γ ell ( x + δ ) ∼ sinh ( x ) , ∼ ϑ 1 ( x | τ ) = x , Γ( x ) Γ trig ( x ) Γ ell ( x ) V. Bazhanov (ANU) Master solution of YBE 1 August 2013 14 / 39

Elliptic gamma-function Let q , p be the temperature-like parameters (elliptic nomes) q = e i πτ ′ , p = e i πτ Im ( τ, τ ′ ) > 0 . The crossing parameter η > 0 is given by i η = 1 e − 2 η = pq , 2 π ( τ + τ ′ ) . In what follows, we consider the primary physical regimes p ∗ = q . p , q ∈ R η > 0 , or The elliptic gamma-function is defined by     � ∞ � 1 − e 2 i z q 2 j +1 p 2 k +1 e − 2 i zn Φ( z ) = 1 − e − 2 i z q 2 j +1 p 2 k +1 = exp  . k ( q n − q − n )( p n − p − n )  j,k =0 n � =0 V. Bazhanov (ANU) Master solution of YBE 1 August 2013 15 / 39

Properties of Φ : Φ( z ) is π -periodic, Φ( z + π ) = Φ( z ) , log Φ is odd, Φ( z )Φ( − z ) = 1 , Zeros and poles: Zeros of Φ( z ) = {− i η − jπτ − kπτ ′ j, k ≥ 0 } , mod π , Poles of Φ( z ) = { + i η + jπτ + kπτ ′ mod π , j, k ≥ 0 } , Exponential formula for Φ( z ) is valid in the strip − η < Im ( z ) < η . Diference property: ∞ Φ( z − πτ ′ � 2 ) (1 − e 2 i z p 2 n +1 )(1 − e − 2 i z p 2 n +1 ) ∼ ϑ 4 ( z | τ ) , 2 ) = Φ( z + πτ ′ n =0 and similarly with τ ⇆ τ ′ . V. Bazhanov (ANU) Master solution of YBE 1 August 2013 16 / 39

Boltzmann weights V. Bazhanov (ANU) Master solution of YBE 1 August 2013 17 / 39