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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,

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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

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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 (Q4).

  • V. Bazhanov

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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

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Yang-Baxter equation in statistical mechanics Local “spins”: σi ∈ (set of values), σi ∈ R Z =

  • {spins}

e−E(σ)/T , E({σ}) =

  • (ij)∈edges

ǫ(σi, σj), Boltzmann weights W(σi, σj) = e−ǫ(σi,σj)/T Z =

  • {spins}
  • (ij)∈edges

W(σi, σj). 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

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Two types of Boltzmann weights, depending on the arrangement of rapidity line wrt the edge Wp−q(x, y) and W p−q(x, y).

p q x y Wp−q(x, y) p q x y W p−q(x, y)

Simplest form of the Yang-Baxter equation: the star-triangle relation

  • σ

W p−q (σ, b) Wp−r (c, σ) W q−r (a, σ) = Wp−q (c, a)W p−r (a, b) Wq−r (c, b) .

  • V. Bazhanov

(ANU) Master solution of YBE 1 August 2013 5 / 39

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General structure of Boltzmann weights In general, weights W are related to W via W p−q(x, y) =

  • S(x)S(y)Wη−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, Wp−q(x, y) = Wp−q(y, x) . Let for shortness p − q = α1 , q − r = α3 . The star-triangle relation takes the form (assume continuous spins)

  • dx0 S(x0) Wη−α1(x1, x0)Wα1+α3(x2, x0)Wη−α3(x3, x0)

= Wα1(x2, x3)Wη−α1−α3(x1, x3)Wα3(x1, x2)

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(ANU) Master solution of YBE 1 August 2013 6 / 39

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Planar graph G, where L is the medial graph

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Low-temperature limit Partition function Z =

(ij)

Wαij(xi, xj)

  • m

S(xm) dxm, αij = p − q, 1st-type η − p + q, 2nd-type Assume, there is a temperature-like parameter ε, such for ε → 0 Wα(x, y) = e−Λα(x,y)/ε+O(1) , S(x) = ε−1/2e−C(x)/ε+O(1) log Z = −1 ε E(x(cl)) + O(1), E(x) =

  • (ij)

Λαij(xi, xj) +

  • m

C(xm) and the variables x(cl) = {x(cl)

1

, x(cl)

2

, . . .} solve the variational equations ∂E(x) ∂xj

  • x=x(cl) = 0

Can one obtain in this way the Q4 system of Adler-Bobenko-Suris, 2003? Λα(x, y) = −i x−y dξ log ϑ4((ξ − iα) | τ) ϑ4(ξ + iα | τ) − i x+y

π/2

dξ log ϑ4(ξ − iα | τ) ϑ4(ξ + iα | τ) C(x) = 2

  • |x| − π

2 2 . |x| < π 2 (1)

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(ANU) Master solution of YBE 1 August 2013 8 / 39

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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 T

  • <ij>

f(αij) + O( √ N)

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Low-temperature limit of the star-triangle relation

  • ε−1/2dx0 exp
  • − E⋆(x0)

ε + O(1)

  • = exp
  • − E△

ε + O(1)

  • where

E⋆ = Λη−α1(x0, x1) + Λα1+α3(x0, x2) + Λη−α3(x0, x3) + C(x0) , E△ = Λα1(x2, x3) + Λη−α1−α3(x1, x2) + Λα3(x1, x2) the STR implies E⋆ = E△ at the stationary point ∂E⋆ ∂x0 = 0 Any solution of STR, admitting low-temperature expansion, leads to classical discrete integrable system, whose action is invariant under star-triangle moves

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Chiral Potts and Kashiwara-Miwa models

N-state chiral Potts model (Albertini, McCoy et al’87, Baxter-Perk-AuYang’87) Wpq(a, b) = µp µq (a−b) a−b

  • k=1

yq − ωkxp yp − ωkxq ωN = 1, and (xp, yp, µp) is a point on genus ≥ 1 algebraic curve Positive Boltzmann weights. Reduces to Ising model for N = 2. Contains ZN model (Fateev-Zamolodchikov’82) R-matrix Rcd

ab = Wpq(a, b)W pq(b, c)Wpq(d, c)W pq(a, d)

intertwines two cyclic representations of Uq( sl(2)) (VB-Stroganov’90)

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Chiral Potts and Kashiwara-Miwa models

N-state model with broken ZN symmetry (Kashiwara-Miwa’86) Wθ(a, b) = rθ(a − b) tθ(a + b) rθ(n) =

n

  • k=1

ϑ1( π

N (k − 1 2) − θ 2N )

ϑ1( π

N (k − 1 2) + θ 2N ),

tθ(n) =

n

  • k=1

ϑ4( π

N (k − 1 2) − θ 2N )

ϑ4( π

N (k − 1 2) + θ 2N ) ,

Reduces to Ising model for N = 2. In the trig. case reduces to ZN model (Fateev-Zamolodchikov’82) The correponding R-matrix intertwines two (special) cyclic representations of Sklyanin algebra (Hasegawa-Yamada’90)

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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) Wθ(a, b) = rθ(a − b, α − β) tθ(a + b, α + β) rθ(n, φ) = N(θ+φ) N(θ−φ) n/N

n

  • k=1

ϑ1( π

N (k − 1 2) − 1 2N (θ−φ))

ϑ1( π

N (k − 1 2) + 1 2N (θ+φ))

What is the meaning of the additional parameters?

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Master solution to the star-triangle relation Elliptic gamma-function

Γ(x + 1) Γ(x) = x , Γtrig(x + δ) Γtrig(x) ∼ sinh (x) , Γell(x + δ) Γell(x) ∼ ϑ1(x|τ)

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Elliptic gamma-function Let q, p be the temperature-like parameters (elliptic nomes) q = eiπτ ′ , p = eiπτ Im(τ, τ ′) > 0 . The crossing parameter η > 0 is given by e−2η = pq , iη = 1 2π(τ + τ ′) . In what follows, we consider the primary physical regimes η > 0 , p, q ∈ R

  • r

p∗ = q . The elliptic gamma-function is defined by Φ(z) =

∞

  • j,k=0

1 − e2izq2j+1p2k+1 1 − e−2izq2j+1p2k+1 = exp   

  • n=0

e−2izn k(qn − q−n)(pn − p−n)    .

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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) = {−iη − jπτ − kπτ ′ mod π , j, k ≥ 0} , Poles of Φ(z) = {+iη + 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 )

Φ(z + πτ ′

2 ) = ∞

  • n=0

(1 − e2izp2n+1)(1 − e−2izp2n+1) ∼ ϑ4(z | τ) , and similarly with τ ⇆ τ ′.

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Boltzmann weights

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Weights W and W Define the weights W and W by Wα(x, y) = κ(α)−1 Φ(x − y + iα) Φ(x − y − iα) Φ(x + y + iα) Φ(x + y − iα) and Wα(x, y) =

  • S(x)S(y)Wη−α(x, y) ,

S(x) = eη/2 2π ϑ1(2x | τ)ϑ1(2x | τ ′) . Normalization factor (partition function per edge – exact solution) κ(α) is given by κ(α) = exp   

  • n=0

e4αn n(pn − p−n)(qn − q−n)(pnqn + p−nq−n)    . It satisfies κ(η − α) κ(α) = Φ(iη − 2iα) , κ(α)κ(−α) = 1 .

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Plots Plot of the real π-periodic function Rα(x) = Φ(x + iα) Φ(x − iα) for p = q = 1

2 and

red: α = η

4

blue: α = η

2

black: α = 3η

4

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Plots Plot of the real π-periodic function Rα(x) = Φ(x + iα) Φ(x − iα) for α = η/4 and red: p = q = 0.5 blue: p = q = 0.6 black: p = q = 0.7

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Properties of W and W The weights Wα(x, y) and Wα(x, y) are real positive for x, y ∈ R and 0 < α < η The weights are symmetric and π-periodic, Wα(x, y) = Wα(y, x) = Wα(−x, y) = Wα(x + π, y) = . . . . Difference properties of the weights: Wα(x − πτ ′

2 , y)

Wα(x + πτ ′

2 , y) = ϑ4(x − y + iα | τ)

ϑ4(x − y − iα | τ) ϑ4(x + y + iα | τ) ϑ4(x + y − iα | τ) and similarly with τ ⇆ τ ′.

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Connection with the theory of elliptic hypergeometric functions

As a mathematical identity the star-triangle relation for this solution is equivalent to Spiridonov’s celebrated elliptic beta integral (2001). This identity lies in the basis of the theory of elliptic hypergeometric functions. Its connection with the Yang-Baxter equation (star-triangle relation) was not hitherto known

  • V. Bazhanov

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Particular cases of the master solution “Trigonometric” limit. τ = ib/R , τ ′ = ib−1/R ,

R → ∞

Gamma-function with small argument Φ(π

Rσ) → ϕ(σ) = exp

1 4

  • pv

dw w e−2iσw sinh (bw) sinh (w/b)

  • Gamma-function with big argument

Φ(π

Rσ + const) → 1 ,

const = O(R0) . Two regimes of the star-triangle equation: xj = const + π

Rσj

and xj = π

Rσj .

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Low temperature limit We consider the low-temperature limit outside the primary physical regime: p2 = e2iπτ and q2 = e−T /N 2ω , ω = e2πi/N ,

T → 0 .

Asymptotic of W: the low-T expansion Wα(x, y) = exp

  • −Λα(x, y)

T

  • · Wα(x, y) · (1 + O(T))

where the Lagrangian density Λα(x, y) is

π N periodic in x and y while the

finite part Wα(x, y) is π-periodic. Asymptotic of the partition function: Z =

  • . . .
  • 0≤xm≤π

exp

  • −E({x})

T

+ O(1) + O(T)

m

dxm √

T , T → 0 ,

where E({x}) is an action for a classical discrete integrable system. The ground state of the system is highly degenerate due to π/N periodicity.

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π N -comb structure

Plot of abs Φ(x + iα) Φ(x − iα)

  • with p = 1

2 and

q = 0.99 · eiπ/5 . The peaks are at x = π N (n + 1

2),

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Star-triangle equation in the low temperature limit

Expression for the Lagrangian density: Λα(x, y) = 2iN x−y dξ log ϑ3(N(ξ − iα) | Nτ) ϑ3(N(ξ + iα) | Nτ) + 2iN x+y

π/2N

dξ log ϑ3(N(ξ − iα) | Nτ) ϑ3(N(ξ + iα) | Nτ) Λη−α(x, y) = π2 2 − (Nx)2 − (Ny)2 +2iN x−y dξ log ϑ1

  • N(iα + ξ) | Nτ
  • ϑ1
  • N(iα − ξ) | Nτ

+ 2iN x+y

π/2N

dξ log ϑ1

  • N(iα + ξ) | Nτ
  • ϑ1
  • N(iα − ξ) | Nτ

. (2) C(x) = 2

  • x − π

2 2 . 0 < x < π N (3) Energy for the regular square lattice E(X) =

  • (ij)

Λ(α | xi, xj) +

  • (kl)

Λ(η − α | xk, xl) +

  • m

C(xm) , (4) Variational equations (Adler-Bobenko-Suris Q4 eqns.) ∂E(X) ∂xi = 0, ⇒ Ψ3

  • x, xr
  • Ψ3(x, xℓ) = Ψ1
  • x, xu
  • Ψ1(x, xd) ,

Ψj(x, y) = ϑj

  • N(x − y + iα) | Nτ
  • ϑj
  • N(x + y + iα) | Nτ
  • ϑj
  • N(x − y − iα) | Nτ
  • ϑj
  • N(x + y − iα )| Nτ

, j = 1, 2, 3, 4. The star-triangle relation

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Zeroth order Due to

π N -periodicity of the leading term, we introduce the discrete spin

variables nj, xj = ξj + π N nj , 0 < Re(ξj) < π 2N , nj ∈ ZN where parameter ξ0 is the solution of the variational equation (in general: parameters ξj are solution of classical integrable equations). Canceling then the T −1 term, we come to the most general discrete-spin star-triangle equation:

  • n0∈ZN

W pq(x0, x1)Wpr(x0, x2)W qr(x0, x3) = RpqrWpq(x2, x3)W pr(x1, x3)Wqr(x1, x2) Note: we consider the star-triangle equation in the orders T −1 and T 0, however it is satisfied in all orders of T-expansion.

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Hybrid model

Z =

  • . . .
  • 0≤xm≤π

exp

  • −E({x})

T

+ O(1) + O(T)

m

dxm √

T , T → 0 ,

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General hybrid model

  • I. Rapidity lattice
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General hybrid model

  • II. Bipartite graph, to each site assign a pair (ξj, nj), where ξj are continuous

and nj ∈ ZN.

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General hybrid model

  • III. Fix all boundary variables (ξi, ni).
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General hybrid model

  • IV. Solve classical integrable variational equations for the parameters ξj in

the bulk (Dirichlet problem for the Adler-Bobenko-Suris system)

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General hybrid model

  • V. All discrete-spin Boltzmann weights W and W entering the partition

function are now defined, the lattice statistical mechanics begins.

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General hybrid model Asymptotics of the partition function: log Z = −E({ξ(cl)})

T

+ log Z0 + O(T) , where {ξ(cl)} denote the stationary point of the classical action, ∂E({ξ}) ∂ξm

  • {ξ}={ξ(cl)} = 0 ,

and Z0 = Z0({ξ(cl)}) is the partition function for the discrete-spin system.

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A new solution of the tetrahedron equation

Yang-Baxter equation R12 R13 R23 = R23 R13 R12 (5) Tetrahedron equation R123R145R246R356 = R356R246R145R123 , (6) where R123 acts in a product of three oscillator Fock spaces, n = 0, 1, 2 . . . Rn′

1,n′ 2,n′ 3

n1,n2,n3 = δn1+n2,n′

1+n′ 2 δn2+n3,n′ 2+n′ 3qn2(n2+1)−(n2−n′ 1)(n2−n′ 3)

×φn1

1 φn2 2 φn3 3 φn′

1

4 n2

  • r=0

(q−2n′

1; q2)n3−r

(q2; q2)n3−r (q2+2n3; q2)r (q2; q2)r q−2r(n2+n′

1+1)

For 0 < q < 1 all nonzero matrix elements of R are positive. Layer-to-layer transfer matrix of the size M × N, possesses rank-size duality for Uq( slN) and Uq( slM) T({φ}) = ⊕

µ T

  • slN

M (µ) = ⊕ ν T

  • slM

N

(ν)

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Quasiclassical limit leads to 3D circular nets (Bobenko, Konopelchenko-Schief)

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Summary We presented a new solution to the star-triangle equation expressed in terms of elliptic Gamma-functions This solution involves two temperature-like parameters (elliptic nomes p and q) This solution contains as specials cases all previously known solutions of the star-triangle equation both with discrete and continuous spin variables When one elliptic nome tends to a root of unity, q2 → e2πi/N, we obtain a hybrid of a classical non-linear integrable system and a solvable model

  • f statistical mechanics. In particular, it contains the chiral Potts and

Kashiwara-Miwa models. This is analogous to the background field quantization in Quantum Field Theory. Connection to superconformal indices and electric-magnetic dualities (Dolan-Osborn, Spiridonov-Vartanov)

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THANK YOU

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Few references

Bazhanov, V. V. and Sergeev, S. M. “A master solution of the quantum Yang-Baxter equation and classical discrete integrable equations”, 2010. arXiv:1006.0651. Bazhanov, V. V. and Sergeev, S. M. “Quasi-classical expansion of the Yang-Baxter equation and integrable systems on planar graphs”, 2010. To appear. Bazhanov, V. V., Mangazeev, V. V., and Sergeev, S. M. Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry. Nuclear Physics B 784 [FS] (2007) pp 234-258 Spiridonov, V. P. “On the elliptic beta function”. Успехи Математических Наук 56 (1) (2001) 181–182. Spiridonov, V. P. “Essays on the theory of elliptic hypergeometric functions”. Успехи Математических Наук 63 (2008) 3–72. Adler, V. E. and Suris, Y. B. “Q4: integrable master equation related to an elliptic curve”. Int. Math. Res. Not. (2004) 2523–2553. Bobenko, A. I. and Suris, Y. B. “On the Lagrangian structure of integrable quad-equations”. Lett. Math. Phys. 92 (2010) 17–31.

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