Exact solution of the classical dimer model on a triangular lattice · 2021. 2. 16. · Pavel Bleher Indiana University-Purdue University Indianapolis, USA Joint work with Estelle

Exact solution of the classical dimer model on atriangular lattice

Pavel Bleher

Indiana University-Purdue University Indianapolis, USA

Joint work with Estelle Basor

Painleve 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 isconvenient to view the triangular lattice as a square lattice withdiagonals:

0 1

1

2

2

n

q

r


Main Goal

with the weights

wh = wv = 1, wd = t > 0.

Our main goal is to calculate an asymptotic behavior as n →∞ ofthe monomer-monomer correlation function K2(n) between twovertices q and r that are n spaces apart in adjacent rows, in thethermodynamic limit (infinite volume).When t = 1, the dimer model is symmetric, and when t = 0, itreduces to the dimer model on the square lattice, hence changing tfrom 0 to 1 gives a deformation of the dimer model on the squarelattice to the symmetric dimer model on the triangular lattice.


Block Toeplitz determinant

Monomer-monomer correlation function as a block Toeplitzdeterminant

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

0φ(e ix)e−ikxdx .


Block symbol φ(e ix)

The 2× 2 matrix symbol φ(e ix) is

φ(e ix) = σ(e ix)

(p(e ix) q(e ix)q(e−ix) p(e−ix)

),

with

σ(e ix) =1

(1− 2t cos x + t2)√

t2 + sin2 x + sin4 x

andp(e ix) = (t cos x + sin2 x)(t − e ix),

q(e ix) = sin x(1− 2t cos x + t2).


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 ofKasteleyn in the earlier ’60s. Kasteleyn finds an expression for thepartition function Z = ZMN of the dimer model on the squarelattice on the rectangle M × N with free boundary conditions as aPfaffian of the Kasteleyn matrix AK of the size MN ×MN,

Z = Pf AK .


Diagonalization of the Kasteleyn matrix

Diagonalization of the Kasteleyn matrix

On the square lattice the Kasteleyn matrix AK can be explicitlyblock-diagonalized with 2× 2 blocks along diagonal, and this givesa formula for the free energy, as a double integral of the logarithmof the spectral function.The spectral function is an analytic periodic function whichvanishes at some points, and this is a manifestation of the factthat the dimer model on a square lattice is critical.


Periodic boundary conditions

The exact solution of the dimer model with periodicboundary conditions

Kasteleyn shows that a Pfaffian formula for the partition functionis valid for the dimer model on any planar graph, and also heshows that the partition function of the dimer model with periodicboundary conditions is equal to the algebraic sum of four Pfaffians:

Z =1

2

(−Pf AK

1 + Pf AK2 + Pf AK

3 + Pf AK4

).


The work of Fisher and Stephenson


Fisher and Stephenson in 1963 derive a brilliant formula for themonomer-monomer correlation function of the dimer model on thesquare lattice K2(n) along a coordinate axis or a diagonal, in termsof a Toeplitz determinant with the symbol

a(θ) = sgn {cos θ} exp[−i cot−1(τ cos θ)

],

with jumps at ±π2 and β = 1

2 .




Fisher and Stephenson apply then a heuristic argument to showthat

K2(n) =B

(1 + o(1)

)n

12

, n →∞.

A rigorous proof of this asymptotics, with an explicit constantB > 0, follows from a general theorem of Deift, Its, and Krasovskyon the Toeplitz determinants of the Fisher–Hartwig type (see alsothe earlier paper of Ehrhardt).The polynomial decay of the correlation function indicates that thedimer model on the square lattice exhibits a critical behavior.


The work of Fendley, Moessner, and Sondhi

The work of Fendley, Moessner, and Sondhi

In 2002 Fendley, Moessner, and Sondhi use the method of Fisherand Stephenson to derive a determinantal formula for themonomer-monomer correlation function on the triangular lattice,but are unable to analyze its asymptotics. So they ask Basor howto find the asymptotics of their determinant.


The work of Basor and Ehrhardt

The work of Basor and Ehrhardt

Basor, together with Ehrhardt, first rewrite the determinantalformula of Fendley, Moessner, and Sondhi as a block Toeplitzdeterminant, and then they find a nice explicit formula for theorder parameter, by using Widom’s extension of the Szego theoremto block Toeplitz determinants. Let us describe the result of Basorand Ehrhardt in terms of the block Toeplitz generalization of theBorodin–Okounkov–Case–Geronimo formula.


BOCG formula

To evaluate the asymptotics of det Tn(φ) as n →∞ we use aBorodin–Okounkov–Case–Geronimo (BOCG) type formula forblock Toeplitz determinants. For any matrix-valued 2π-periodicmatrix-valued function ϕ(e ix) consider the correspondingsemi-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

0ϕ(e ix)e−ikxdx


BOCG formula

Let ψ(e ix) = φ−1(e ix), where the matrix symbol φ(e ix) wasintroduced before, and the inverse is the matrix inverse. Then thefollowing BOCG type formula holds:

det Tn(φ) =E (ψ)

G (ψ)ndet (I − Φ) ,

where det (I − Φ) is the Fredholm determinant with

Φ = H(e−inxψ(e ix)

)T−1

(ψ(e−ix)

)H

(e−inxψ(e−ix)

)T−1

(ψ(e ix)

).

In our case G (ψ) = 1 and

E (ψ) =t

2t(2 + t2) + (1 + 2t2)√

2 + t2

(the Basor–Ehrhardt formula).


Order parameter

The Basor–Ehrhardt formula implies that the order parameter isequal to

K2(∞) := limn→∞

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) asn →∞. The problem reduces to evaluating an asymptoticbehavior of the Fredholm determinant det (I − Φ), because

K2(n) = K2(∞)√

det (I − Φ) .


The Wiener–Hopf factorization of φ(z)

To evaluate det (I − Φ) we need to invert the semi-infinite Toeplitzmatrices T−1

(ψ(e ix) and to do so we use the Wiener–Hopf

factorization of the symbol φ. Let z = e ix . Denote

π(z) =

(p(z) q(z)

q(z−1) p(z−1)

),

so thatφ(z) = σ(z)π(z),

where

σ(z) =1

(1− 2t cos x + t2)√

t2 + sin2 x + sin4 x

is a scalar function.


The Wiener–Hopf factorization

The Wiener–Hopf factorization

Our goal is to factor the matrix-valued symbol φ(z) asφ(z) = φ+(z)φ−(z), where φ+(z) and φ−(z−1) are analyticinvertible matrix valued functions on the disk D = {z | |z | ≤ 1}.Denote

τ =1

t.

We start with an explicit factorization of the functiont2 + sin2 x + sin4 x .


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

We have that

t2+sin2 x+sin4 x =1

16η21η

22

(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 ≤ 12 and complex

conjugate for t > 12 .


Graphs of η1, η2

The graphs of |η1(t)| (dashed line), |η2(t)| (solid line), the uppergraphs, and arg η1(t) (dashed line), arg η2(t) (solid line), the lowergraphs


Wiener–Hopf factorization

Theorem 1. We have the Wiener–Hopf factorization:

φ(z) = φ+(z)φ−(z),

whereφ+(z) = A(z)Ψ(z), φ−(z) = Ψ−1(z−1),

withA(z) =

τ

z − τ,

and

Ψ(z) =1√f (z)

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

with



f (z) =(z2 − η2

1)(z2 − η2

2)

4η1η2

and

D0(z) =

(1 00 z − τ

),

D1(z) =

(z − η1 0

0 1

), D2(z) =

(z + η1 0

0 1

),

D3(z) =

(z − η2 0

0 1

), D4(z) =

(1 00 z + η2

),

and

Pj =

(1 pj

0 1

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

(1 0p4 1

).



Here

p1 =i[τ(η2

1 − 1)2 − 2η1(η21 + 1)

]2(η2

1 − 1), p2 = − i(η2

1 + 1)

η21 − 1

,

p3 =iτ(η1 + 1)

2η1, p4 = −2iη1η2

τ, p5 = − iτ

2η1.


Idea of the proof

Idea of the proof

The idea of the proof goes back to the works of McCoy and Wu onthe Ising model, and even before to the works of Hopf andGrothendieck.Let us recall that φ(z) = σ(z)π(z), where σ(z) is a scalarfunction. The difficult part is to factor π(z). To factor π(z) weuse a decreasing power algorithm. In this algorithm at every stepwe make a substitution decreasing the power in z of the matrixentries under consideration.


First step

First step

As the first step, we write π(z) as

π(z) = τ−2

(1 00 z − τ

)ρ(z)

(z−1 − τ 0

0 1

).

where

ρ(z) =

(ρ11(z) ρ12(z)ρ21(z) ρ22(z)

)with


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(τ2 + 1)z

2+

i(τ2 + 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) = −τz4

+1

2+

τ

2z+

1

2z2− τ

4z3.

Let us factor ρ(z). Observe that

det ρ(z) =τ2

16η21η

22

(z−2 − η2

1

) (z−2 − η2

2

) (z2 − η2

1

) (z2 − η2

2

).


Second step

Let p1 be a constant. We have that(1 p1

0 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 thatρ11(η1)− p1ρ21(η1) = 0.

Then, since det ρ(η1) = 0 and ρ21(η1) 6= 0, automatically

ρ12(η1)− p1ρ22(η1) = 0,

and (z − η1) can be factored out in the first row.


Second step

As a result we obtain that

ρ(z) =

(1 p1

0 1

) (z − η1 0

0 1

)ρ(1)(z).

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

ρ(1)11 (z) = −τz

2

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).


Proof of Theorem 1

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


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),

whereθ−(z) = σ3φ

T−(z), θ+(z) = φT

+(z)σ3.


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(e ix)

)T−1

(ψ(e−ix)

)H

(e−inxψ(e−ix)

)T−1

(ψ(e ix)

).

This Φ is not very handy for an asymptotic analysis. We haveanother useful representation of det(I − Φ):

det(I − Φ) = det(I − Λ),

whereΛ = H(z−nα)H(z−nβ)

with

α(z) = φ−(z)θ−1+ (z), β(z) = θ−1

− (z−1)φ+(z−1).


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

0α(e ix)e−ikxdx , βk =

1

2π

∫ 2π

0β(e ix)e−ikxdx .

We point out that this representation allows for a more directcomputation of the determinant of interest without the morecomplicated formula involving the operator inverses.


Asymptotics of the coefficients αk , βk

The following theorem gives the asymptotics of the coefficients αk ,βk :

Theorem 2. Assume that 0 < t < 12 . Then as k →∞, αk , βk

admit the asymptotic expansions

αk ∼e−k ln η2

√k

∞∑j=0

a0j + (−1)ka1

j

k j,

βk ∼e−k ln η2

√k

∞∑j=0

b0j + (−1)kb1

j

k j.


Asymptotics of the monomer-monomer correlationfunction for 0 < t < 1

2 .

Asymptotics of the monomer-monomer correlation functionfor 0 < t < 1

2 .

Theorem 3. Let 0 < t < 12 . 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) .


Asymptotics of the monomer-monomer correlationfunction for 1

2 < t < 1 .

Asymptotics of the monomer-monomer correlation functionfor 1

2 < t < 1 .

If t > 12 , then η1, η2 are complex conjugate numbers,

η1 = es−iθ, η2 = es+iθ;

s = ln |η1| = ln |η2| > 0; 0 < θ <π

4.



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 12 < 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 asymptoticexpansions

αk(p, σ) ∼ σke−k ln ηp

√k

∞∑j=0

aj(p, σ)

k j,

βk(p, σ) ∼ σke−k ln ηp

√k

∞∑j=0

bj(p, σ)

k j.



2 < t < 1 .

Theorem 4. Assume that 12 < 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.


Conclusion

ConclusionIn this work we obtain the asymptotics of the monomer-monomercorrelation function for subcritical, 0 < t < 1

2 , and supercritical,12 < 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)

],


Open problems

Open problems

1. Double scaling limit as t → 0 and n →∞. We expect thatthe double scaling asymtpotics of the monomer-monomercorrelation function as t → 0 and n →∞ is expressed interms of a solution to PIII.

2. Asymtpotics of the monomer-monomer correlation function att = 1

2 .

3. Asymtpotics of the monomer-monomer correlation functionfor t > 1. The Wiener–Hopf factorization exhibit indices.

4. Critical asymptotics at t = 1.


Reference

Reference

E. Basor and P. Bleher, Exact solution of the classical dimer modelon a triangular lattice: monomer-monomer correlations. ArXiv:1610.08021 (to appear in Commun. Math. Phys.).


Thank you!

The End

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Documents

Exact solution of the classical dimer model on a triangular lattice · 2021. 2. 16. · Pavel Bleher Indiana University-Purdue University Indianapolis, USA Joint work with Estelle