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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:

1 1 2 2 n

q r

Pavel Bleher Dimer model

Main Goal

with the weights wh = wv = 1, wd = t > 0. Our main goal is to calculate an asymptotic behavior as n → ∞ of the monomer-monomer correlation function K2(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 K2(n): K2(n) = 1 2

• det Tn(φ),

where Tn(φ) is the finite block Toeplitz matrix, Tn(φ) = (φj−k), 0 ≤ j, k ≤ n − 1, where φk = 1 2π 2π φ(eix)e−ikxdx.

Pavel Bleher Dimer model

Block symbol φ(eix)

The 2 × 2 matrix symbol φ(eix) is φ(eix) = σ(eix) p(eix) q(eix) q(e−ix) p(e−ix)

• ,

with σ(eix) = 1 (1 − 2t cos x + t2)

• t2 + sin2 x + sin4 x

and p(eix) = (t cos x + sin2 x)(t − eix), q(eix) = sin x(1 − 2t cos x + t2).

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 = ZMN of the dimer model on the square lattice on the rectangle M × N with free boundary conditions as a Pfaffian of the Kasteleyn matrix AK of the size MN × MN, Z = Pf AK .

Pavel Bleher Dimer model

Diagonalization of the Kasteleyn matrix

Diagonalization of the Kasteleyn matrix On the square lattice the Kasteleyn matrix AK 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

• f 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 2

• −Pf AK

1 + Pf AK 2 + Pf AK 3 + Pf AK 4

• .

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 K2(n) along a coordinate axis or a diagonal, in terms

• f a Toeplitz determinant with the symbol

a(θ) = sgn {cos θ} exp

• −i cot−1(τ cos θ)
• ,

with jumps at ±π

2 and β = 1 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 K2(n) = B

• 1 + o(1)
• n

1 2

, n → ∞. A rigorous proof of this asymptotics, with an explicit constant B > 0, follows from a general theorem of Deift, Its, and Krasovsky

• n 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

• rder parameter, by using Widom’s extension of the Szeg˝
• 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 Tn(φ) as n → ∞ we use a Borodin–Okounkov–Case–Geronimo (BOCG) type formula for block Toeplitz determinants. For any matrix-valued 2π-periodic matrix-valued function ϕ(eix) consider the corresponding semi-infinite matrices, Toeplitz and Hankel, T(ϕ) = (ϕj−k)∞

j,k=0 ;

H(ϕ) = (ϕj+k+1)∞

j,k=0 ,

where ϕk = 1 2π 2π ϕ(eix)e−ikxdx

Pavel Bleher Dimer model

BOCG formula

Let ψ(eix) = φ−1(eix), where the matrix symbol φ(eix) was introduced before, and the inverse is the matrix inverse. Then the following BOCG type formula holds: det Tn(φ) = E(ψ) G(ψ)n det (I − Φ) , where det (I − Φ) is the Fredholm determinant with Φ = H

• e−inxψ(eix)
• T −1

ψ(e−ix)

• H
• e−inxψ(e−ix)
• T −1

ψ(eix)

• .

In our case G(ψ) = 1 and E(ψ) = t 2t(2 + t2) + (1 + 2t2) √ 2 + t2 (the Basor–Ehrhardt formula).

Pavel Bleher Dimer model

Order parameter

The Basor–Ehrhardt formula implies that the order parameter is equal to K2(∞) := lim

n→∞ K2(n) = 1

2

• E(ψ)

= 1 2

• t

2t(2 + t2) + (1 + 2t2) √ 2 + t2 . Our goal is to evaluate an asymptotic behavior of K2(n) as n → ∞. The problem reduces to evaluating an asymptotic behavior of the Fredholm determinant det (I − Φ), because K2(n) = K2(∞)

• 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(eix) and to do so we use the Wiener–Hopf factorization of the symbol φ. Let z = eix. Denote π(z) = p(z) q(z) q(z−1) p(z−1)

• ,

so that φ(z) = σ(z)π(z), where σ(z) = 1 (1 − 2t cos x + t2)

• t2 + sin2 x + sin4 x

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 t2 + sin2 x + sin4 x.

Pavel Bleher Dimer model

Factorization of t2 + sin2 x + sin4 x and numbers η1,2

We have that t2+sin2 x+sin4 x = 1 16η2

1η2 2

• z−2 − η2

1

z−2 − η2

2

z2 − η2

1

z2 − η2

2

• ,

where η1,2 = 1

• 2 ± µ − 2
• 1 − t2 ± µ

, µ =

• 1 − 4t2 .

The numbers η1,2 are positive for 0 ≤ t ≤ 1

2 and complex

conjugate for t > 1

2 .

Pavel Bleher Dimer model

Graphs of η1, η2

The graphs of |η1(t)| (dashed line), |η2(t)| (solid line), the upper graphs, and arg η1(t) (dashed line), arg η2(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) = A(z)Ψ(z), φ−(z) = Ψ−1(z−1), with A(z) = τ z − τ , and Ψ(z) = 1

• f (z)

D0(z)P1D1(z)P2D2(z)P3D3(z)P4D4(z)P5, with

Pavel Bleher Dimer model

Wiener–Hopf factorization

f (z) = (z2 − η2

1)(z2 − η2 2)

4η1η2 and D0(z) = 1 z − τ

• ,

D1(z) = z − η1 1

• ,

D2(z) = z + η1 1

• ,

D3(z) = z − η2 1

• ,

D4(z) = 1 z + η2

• ,

and Pj = 1 pj 1

• ,

j = 1, 2, 3, 5; P4 = 1 p4 1

• .

Pavel Bleher Dimer model

Wiener–Hopf factorization

Here p1 = i

• τ(η2

1 − 1)2 − 2η1(η2 1 + 1)

• 2(η2

1 − 1)

, p2 = −i(η2

1 + 1)

η2

1 − 1

, p3 = iτ(η1 + 1) 2η1 , p4 = −2iη1η2 τ , p5 = − iτ 2η1 .

Pavel Bleher Dimer model

Idea of the proof

Idea of the proof The idea of the proof goes back to the works of McCoy and Wu on the Ising model, and even before to the works of Hopf and Grothendieck. Let us recall that φ(z) = σ(z)π(z), where σ(z) is a scalar

• function. The difficult part is to factor π(z). To factor π(z) we

use a decreasing power algorithm. In this algorithm at every step we make a substitution decreasing the power in z of the matrix entries under consideration.

Pavel Bleher Dimer model

First step

First step As the first step, we write π(z) as π(z) = τ −2 1 z − τ

• ρ(z)

z−1 − τ 1

• .

where ρ(z) = ρ11(z) ρ12(z) ρ21(z) ρ22(z)

• with

Pavel Bleher Dimer model

First step

ρ11(z) = z(cos x + τ sin2 x) = −τz3 4 + z2 2 + τz 2 + 1 2 − τ 4z , ρ12(z) = sin x (z − τ)(z−1 − τ) = iτz2 2 − i(τ 2 + 1)z 2 + i(τ 2 + 1) 2z − iτ 2z2 , ρ21(z) = − sin x = i(z2 − 1) 2z = iz 2 − i 2z , ρ22(z) = z−1(cos x + τ sin2 x) = −τz 4 + 1 2 + τ 2z + 1 2z2 − τ 4z3 . Let us factor ρ(z). Observe that det ρ(z) = τ 2 16η2

1η2 2

• z−2 − η2

1

z−2 − η2

2

z2 − η2

1

z2 − η2

2

• .

Pavel Bleher Dimer model

Second step

Let p1 be a constant. We have that 1 p1 1 −1 ρ(z) = ρ11(z) − p1ρ21(z) ρ12(z) − p1ρ22(z) ρ21(z) ρ22(z)

• .

Let us take p1 = ρ11(η1) ρ21(η1) , so that ρ11(η1) − p1ρ21(η1) = 0. Then, since det ρ(η1) = 0 and ρ21(η1) = 0, automatically ρ12(η1) − p1ρ22(η1) = 0, and (z − η1) can be factored out in the first row.

Pavel Bleher Dimer model

Second step

As a result we obtain that ρ(z) = 1 p1 1 z − η1 1

• ρ(1)(z).

where ρ(1)(z) is a rational matrix valued function with leading terms at infinity ρ(1)

11 (z) = −τz2

4 + O(z), ρ(1)

12 (z)

= iτz 2 + O(1) . Observe that the degrees of the functions ρ(1)

11 (z), ρ(1) 12 (z) in z

decrease by one comparing to ρ11(z), ρ12(z).

Pavel Bleher Dimer model

Proof of Theorem 1

We repeat this factorization procedure several times, to different rows, and, as a result, we obtain the desired explicit Wiener–Hopf factorization of the symbol. This finishes the proof of Theorem 1.

Pavel Bleher Dimer model

Minus-plus factorization of φ(z)

Applying the symmetry relation, φ(z) = σ3φT(z)σ3, to the plus-minus factorization of φ(z), φ(z) = φ+(z)φ−(z), we obtain a minus-plus factorization of φ(z): φ(z) = θ−(z)θ+(z), where θ−(z) = σ3φT

−(z),

θ+(z) = φT

+(z)σ3.

Pavel Bleher Dimer model

A useful formula for the Fredholm determinant det(I − Φ)

Our goal is to evaluate the Fredholm determinant det(I − Φ), with Φ = H

• e−inxψ(eix)
• T −1

ψ(e−ix)

• H
• e−inxψ(e−ix)
• T −1

ψ(eix)

• .

This Φ is not very handy for an asymptotic analysis. We have another useful representation of det(I − Φ): det(I − Φ) = det(I − Λ), where Λ = H(z−nα)H(z−nβ) with α(z) = φ−(z)θ−1

+ (z),

β(z) = θ−1

− (z−1)φ+(z−1).

Pavel Bleher Dimer model

The matrix elements of the matrix Λ

The matrix elements of the matrix Λ are Λjk =

∞

• a=0

αj+n+a+1βk+n+a+1, where αk = 1 2π 2π α(eix)e−ikxdx, βk = 1 2π 2π β(eix)e−ikxdx. We point out that this representation allows for a more direct computation of the determinant of interest without the more complicated formula involving the operator inverses.

Pavel Bleher Dimer model

Asymptotics of the coefficients αk, βk

The following theorem gives the asymptotics of the coefficients αk, βk: Theorem 2. Assume that 0 < t < 1

2 . Then as k → ∞, αk, βk

admit the asymptotic expansions αk ∼ e−k ln η2 √ k

∞

• j=0

a0

j + (−1)ka1 j

kj , βk ∼ e−k ln η2 √ k

∞

• j=0

b0

j + (−1)kb1 j

kj .

Pavel Bleher Dimer model

Asymptotics of the monomer-monomer correlation function for 0 < t < 1

2 .

Asymptotics of the monomer-monomer correlation function for 0 < t < 1

2 .

Theorem 3. Let 0 < t < 1

2 . Then as n → ∞,

K2(n) = K2(∞)

• 1 − e−2n ln η2

2n

• C1 + (−1)n+1C2 + O(n−1)
• ,

with some explicit C1, C2 > 0.

• Corollary. This gives that the correlation length is equal to

ξ = 1 2 ln η2 . As t → 0, ξ = 1 2t + O(1) .

Pavel Bleher Dimer model

Asymptotics of the monomer-monomer correlation function for 1

2 < t < 1 .

Asymptotics of the monomer-monomer correlation function for 1

2 < t < 1 .

If t > 1

2 , then η1, η2 are complex conjugate numbers,

η1 = es−iθ, η2 = es+iθ; s = ln |η1| = ln |η2| > 0; 0 < θ < π 4 .

Pavel Bleher Dimer model

Asymptotics of the monomer-monomer correlation function for 1

2 < t < 1 .

The following theorem gives the asymptotics of the coefficients αk, βk in the supercritical case, 1

2 < t < 1 :

Theorem 4. Assume that 1

2 < t < 1 . Then

αk =

• p=1,2; σ=±1

αk(p, σ), βk =

• p=1,2; σ=±1

βk(p, σ), where as k → ∞, αk(p, σ), βk(p, σ) admit the asymptotic expansions αk(p, σ) ∼ σke−k ln ηp √ k

∞

• j=0

aj(p, σ) kj , βk(p, σ) ∼ σke−k ln ηp √ k

∞

• j=0

bj(p, σ) kj .

Pavel Bleher Dimer model

Asymptotics of the monomer-monomer correlation function for 1

2 < t < 1 .

Theorem 4. Assume that 1

2 < t < 1 . Then as n → ∞,

K2(n) = K2(∞)

• 1 − e−2ns

2n

• C1 cos(2θn + ϕ1)

+ C2(−1)n cos(2θn + ϕ2) + C3 + C4(−1)n + O(n−1)

• ,

with s = ln |η1| = ln |η2|, θ = | arg η1| = | arg η2|, and explicit C1, C2, C3 , C4, ϕ1, ϕ2.

Pavel Bleher Dimer model

Conclusion

Conclusion In this work we obtain the asymptotics of the monomer-monomer correlation function for subcritical, 0 < t < 1

2 , and supercritical, 1 2 < t < 1 , cases:

K2(n) = K2(∞)

• 1 − e−2n ln η2

2n

• C1 + (−1)n+1C2 + O(n−1)
• ,

and K2(n) = K2(∞)

• 1 − e−2ns

2n

• C1 cos(2θn + ϕ1)

+ C2(−1)n cos(2θn + ϕ2) + C3 + C4(−1)n + O(n−1)

• ,

Pavel Bleher Dimer model

Open problems

Open problems

• 1. Double scaling limit as t → 0 and n → ∞. We expect that

the double scaling asymtpotics of the monomer-monomer correlation function as t → 0 and n → ∞ is expressed in terms of a solution to PIII.

• 2. Asymtpotics of the monomer-monomer correlation function at

t = 1

2.

• 3. Asymtpotics of the monomer-monomer correlation function

for t > 1. The Wiener–Hopf factorization exhibit indices.

• 4. Critical asymptotics at t = 1.

Pavel Bleher Dimer model

Reference

Reference

• E. Basor and P. Bleher, Exact solution of the classical dimer model
• n a triangular lattice: monomer-monomer correlations. ArXiv:

1610.08021 (to appear in Commun. Math. Phys.).

Pavel Bleher Dimer model

Thank you!

The End Thank you!

Pavel Bleher Dimer model