Exact solution of the classical dimer model on a triangular lattice Pavel Bleher Indiana University-Purdue University Indianapolis, USA Joint work with Estelle Basor Painlev´ e Equations and Applications: A Workshop in Memory of A. A. Kapaev University of Michigan Ann Arbor, August 26, 2017 Pavel Bleher Dimer model
Dimer Model Dimer Model We consider the classical dimer model on a triangular lattice. It is convenient to view the triangular lattice as a square lattice with diagonals: 2 r 1 q 0 1 2 n Pavel Bleher Dimer model
Main Goal with the weights w h = w v = 1 , w d = t > 0 . Our main goal is to calculate an asymptotic behavior as n → ∞ of the monomer-monomer correlation function K 2 ( n ) between two vertices q and r that are n spaces apart in adjacent rows, in the thermodynamic limit (infinite volume). When t = 1, the dimer model is symmetric, and when t = 0, it reduces to the dimer model on the square lattice, hence changing t from 0 to 1 gives a deformation of the dimer model on the square lattice to the symmetric dimer model on the triangular lattice. Pavel Bleher Dimer model
Block Toeplitz determinant Monomer-monomer correlation function as a block Toeplitz determinant Our starting point is a determinantal formula for K 2 ( n ): K 2 ( n ) = 1 � det T n ( φ ) , 2 where T n ( φ ) is the finite block Toeplitz matrix, T n ( φ ) = ( φ j − k ) , 0 ≤ j , k ≤ n − 1 , where � 2 π φ k = 1 φ ( e ix ) e − ikx dx . 2 π 0 Pavel Bleher Dimer model
Block symbol φ ( e ix ) The 2 × 2 matrix symbol φ ( e ix ) is � p ( e ix ) q ( e ix ) � φ ( e ix ) = σ ( e ix ) , q ( e − ix ) p ( e − ix ) with 1 σ ( e ix ) = t 2 + sin 2 x + sin 4 x � (1 − 2 t cos x + t 2 ) and p ( e ix ) = ( t cos x + sin 2 x )( t − e ix ) , q ( e ix ) = sin x (1 − 2 t cos x + t 2 ) . Pavel Bleher Dimer model
The exact solution of the dimer model The exact solution of the dimer model by Kasteleyn The exact solution of the dimer model begins with the works of Kasteleyn in the earlier ’60s. Kasteleyn finds an expression for the partition function Z = Z MN of the dimer model on the square lattice on the rectangle M × N with free boundary conditions as a Pfaffian of the Kasteleyn matrix A K of the size MN × MN , Z = Pf A K . Pavel Bleher Dimer model
Diagonalization of the Kasteleyn matrix Diagonalization of the Kasteleyn matrix On the square lattice the Kasteleyn matrix A K can be explicitly block-diagonalized with 2 × 2 blocks along diagonal, and this gives a formula for the free energy, as a double integral of the logarithm of the spectral function. The spectral function is an analytic periodic function which vanishes at some points, and this is a manifestation of the fact that the dimer model on a square lattice is critical . Pavel Bleher Dimer model
Periodic boundary conditions The exact solution of the dimer model with periodic boundary conditions Kasteleyn shows that a Pfaffian formula for the partition function is valid for the dimer model on any planar graph , and also he shows that the partition function of the dimer model with periodic boundary conditions is equal to the algebraic sum of four Pfaffians: Z = 1 − Pf A K 1 + Pf A K 2 + Pf A K 3 + Pf A K � � . 4 2 Pavel Bleher Dimer model
The work of Fisher and Stephenson The work of Fisher and Stephenson Fisher and Stephenson in 1963 derive a brilliant formula for the monomer-monomer correlation function of the dimer model on the square lattice K 2 ( n ) along a coordinate axis or a diagonal, in terms of a Toeplitz determinant with the symbol − i cot − 1 ( τ cos θ ) � � a ( θ ) = sgn { cos θ } exp , 2 and β = 1 with jumps at ± π 2 . Pavel Bleher Dimer model
The work of Fisher and Stephenson The work of Fisher and Stephenson Fisher and Stephenson apply then a heuristic argument to show that � � K 2 ( n ) = B 1 + o (1) n → ∞ . , 1 n 2 A rigorous proof of this asymptotics, with an explicit constant B > 0, follows from a general theorem of Deift, Its, and Krasovsky on the Toeplitz determinants of the Fisher–Hartwig type (see also the earlier paper of Ehrhardt ). The polynomial decay of the correlation function indicates that the dimer model on the square lattice exhibits a critical behavior . Pavel Bleher Dimer model
The work of Fendley, Moessner, and Sondhi The work of Fendley, Moessner, and Sondhi In 2002 Fendley, Moessner, and Sondhi use the method of Fisher and Stephenson to derive a determinantal formula for the monomer-monomer correlation function on the triangular lattice, but are unable to analyze its asymptotics. So they ask Basor how to find the asymptotics of their determinant. Pavel Bleher Dimer model
The work of Basor and Ehrhardt The work of Basor and Ehrhardt Basor , together with Ehrhardt , first rewrite the determinantal formula of Fendley, Moessner, and Sondhi as a block Toeplitz determinant, and then they find a nice explicit formula for the order parameter , by using Widom’s extension of the Szeg˝ o theorem to block Toeplitz determinants. Let us describe the result of Basor and Ehrhardt in terms of the block Toeplitz generalization of the Borodin–Okounkov–Case–Geronimo formula. Pavel Bleher Dimer model
BOCG formula To evaluate the asymptotics of det T n ( φ ) as n → ∞ we use a Borodin–Okounkov–Case–Geronimo (BOCG) type formula for block Toeplitz determinants. For any matrix-valued 2 π -periodic matrix-valued function ϕ ( e ix ) consider the corresponding semi-infinite matrices, Toeplitz and Hankel, T ( ϕ ) = ( ϕ j − k ) ∞ H ( ϕ ) = ( ϕ j + k +1 ) ∞ j , k =0 ; j , k =0 , where � 2 π ϕ k = 1 ϕ ( e ix ) e − ikx dx 2 π 0 Pavel Bleher Dimer model
BOCG formula Let ψ ( e ix ) = φ − 1 ( e ix ) , where the matrix symbol φ ( e ix ) was introduced before, and the inverse is the matrix inverse. Then the following BOCG type formula holds: det T n ( φ ) = E ( ψ ) G ( ψ ) n det ( I − Φ) , where det ( I − Φ) is the Fredholm determinant with e − inx ψ ( e ix ) T − 1 � ψ ( e − ix ) e − inx ψ ( e − ix ) T − 1 � ψ ( e ix ) � � � � � � Φ = H H . In our case G ( ψ ) = 1 and t √ E ( ψ ) = 2 t (2 + t 2 ) + (1 + 2 t 2 ) 2 + t 2 (the Basor–Ehrhardt formula). Pavel Bleher Dimer model
Order parameter The Basor–Ehrhardt formula implies that the order parameter is equal to n →∞ K 2 ( n ) = 1 � K 2 ( ∞ ) := lim E ( ψ ) 2 � = 1 t √ 2 + t 2 . 2 2 t (2 + t 2 ) + (1 + 2 t 2 ) Our goal is to evaluate an asymptotic behavior of K 2 ( n ) as n → ∞ . The problem reduces to evaluating an asymptotic behavior of the Fredholm determinant det ( I − Φ), because � K 2 ( n ) = K 2 ( ∞ ) det ( I − Φ) . Pavel Bleher Dimer model
The Wiener–Hopf factorization of φ ( z ) To evaluate det ( I − Φ) we need to invert the semi-infinite Toeplitz matrices T − 1 � ψ ( e ix ) and to do so we use the Wiener–Hopf factorization of the symbol φ . Let z = e ix . Denote � p ( z ) � q ( z ) π ( z ) = , q ( z − 1 ) p ( z − 1 ) so that φ ( z ) = σ ( z ) π ( z ) , where 1 σ ( z ) = t 2 + sin 2 x + sin 4 x � (1 − 2 t cos x + t 2 ) is a scalar function. Pavel Bleher Dimer model
The Wiener–Hopf factorization The Wiener–Hopf factorization Our goal is to factor the matrix-valued symbol φ ( z ) as φ ( z ) = φ + ( z ) φ − ( z ), where φ + ( z ) and φ − ( z − 1 ) are analytic invertible matrix valued functions on the disk D = { z | | z | ≤ 1 } . Denote τ = 1 t . We start with an explicit factorization of the function t 2 + sin 2 x + sin 4 x . Pavel Bleher Dimer model
Factorization of t 2 + sin 2 x + sin 4 x and numbers η 1 , 2 We have that 1 z − 2 − η 2 z − 2 − η 2 z 2 − η 2 z 2 − η 2 t 2 +sin 2 x +sin 4 x = � � � � � � � � , 1 2 1 2 16 η 2 1 η 2 2 where 1 � 1 − 4 t 2 . η 1 , 2 = , µ = � 1 − t 2 ± µ � 2 ± µ − 2 The numbers η 1 , 2 are positive for 0 ≤ t ≤ 1 2 and complex conjugate for t > 1 2 . Pavel Bleher Dimer model
Graphs of η 1 , η 2 The graphs of | η 1 ( t ) | (dashed line), | η 2 ( t ) | (solid line), the upper graphs, and arg η 1 ( t ) (dashed line), arg η 2 ( t ) (solid line), the lower graphs Pavel Bleher Dimer model
Wiener–Hopf factorization Theorem 1. We have the Wiener–Hopf factorization: φ ( z ) = φ + ( z ) φ − ( z ) , where φ − ( z ) = Ψ − 1 ( z − 1 ) , φ + ( z ) = A ( z )Ψ( z ) , with τ A ( z ) = z − τ , and 1 Ψ( z ) = D 0 ( z ) P 1 D 1 ( z ) P 2 D 2 ( z ) P 3 D 3 ( z ) P 4 D 4 ( z ) P 5 , � f ( z ) with Pavel Bleher Dimer model
Wiener–Hopf factorization f ( z ) = ( z 2 − η 2 1 )( z 2 − η 2 2 ) 4 η 1 η 2 and � 1 0 � D 0 ( z ) = , 0 z − τ � z − η 1 0 � � z + η 1 0 � D 1 ( z ) = D 2 ( z ) = , , 0 1 0 1 � z − η 2 � � 1 � 0 0 D 3 ( z ) = D 4 ( z ) = , , 0 1 0 z + η 2 and � 1 � 1 � � p j 0 P j = j = 1 , 2 , 3 , 5; P 4 = , . 0 1 p 4 1 Pavel Bleher Dimer model
Wiener–Hopf factorization Here 1 − 1) 2 − 2 η 1 ( η 2 � τ ( η 2 � p 2 = − i ( η 2 p 1 = i 1 + 1) 1 + 1) , , 2( η 2 η 2 1 − 1) 1 − 1 p 3 = i τ ( η 1 + 1) p 4 = − 2 i η 1 η 2 p 5 = − i τ , , . 2 η 1 τ 2 η 1 Pavel Bleher Dimer model
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