Creswick, 25 January 2018

Yang-Baxter Solution of Dimers

as a Free-Fermion Six-Vertex Model

Paul A. Pearce & Alessandra Vittorini-Orgeas

School of Mathematics and Statistics

University of Melbourne

”Yang-Baxter solution of dimers as a free-fermion 6-vertex model”, J Phys A: Math and Theor, Vol 50, n 43(2017).

0-1

Outline

• Some History

• Dimers as a Free-Fermion Six Vertex Model

• Counting Dimers on a Periodic M ×N Square Lattice

• Solution on a Cylinder and Torus

• Bulk Conformal Field Theory of Dimers

• Solution on a Strip with Vacuum Boundary Condition

0-2

History: early studies

1937 Fowler, Rushbrooke: Dimer model for diatomic molecules

How many ways are there to fill an

8× 8 lattice with vertical and horizontal

dimers?

ZFT8×8 = 12,988,816.

1961 Kasteleyn: Dimers on a square lattice with free and toroidal boundaries

1961 Temperley, Fisher: Independent solution on the square lattice with free boundaries

1967 Lieb: A non-Yang-Baxter transfer matrix approach

�

�

�

�

1988 Rokhsar, Kivelson: Phase transition properties of a quantum hard-core

dimer gas on a square lattice

1996 Cohn, Elkies, Propp: Local statistics for random domino tilings of the Aztec diamond

0-3

History: Dimers as Critical System

2000 Korepin, Zinn-Justin: Dependence of the bulk free energy on boundary conditions

2003 Izmailian, Oganesyan, Hu: Exact finite-size corrections of the free energy

for the square lattice dimer model under different boundary conditions

2005 Izmailian, Priezzhev, Ruelle, Hu: Logarithmic conformal field theory and

boundary effects in the dimer model

2007 Izmailian, Priezzhev, Ruelle: Non-local finite-size effects in the dimer model

2012 Rasmussen, Ruelle: Refined analysis of conformal spectra in the dimer model

2015 Nigro: Finite size corrections for dimers

2015 Morin-Duchesne, Rasmussen, Ruelle: Dimer representations of the

Temperley-Lieb algebra

2016 Morin-Duchesne, Rasmussen, Ruelle: Integrability and conformal data

of the dimer model

2017 Pearce, Vittorini-Orgeas: Yang-Baxter solution of dimers as a free-fermion 6-vertex model

0-4

The Big Question

Gaussian free theory or Logarithmic Conformal Field Theory?

Strategy:

• Enumerate degrees of freedom (map to λ = π/2 six-vertex model).

• Introduce a spectral parameter (spatial anisotropy).

• Establish Yang-Baxter integrability (rotate faces by 45 degrees).

• Gain control to construct (r, s) type integrable/conformal boundary conditions on the strip.

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Six-Vertex, Particle and Dimer Representations

• Equivalent tiles: Vertex, particle and dimer (Korepin&Zinn-Justin 2000) representations:

or

︸ ︷︷ ︸

a(u)︸ ︷︷ ︸

b(u)︸ ︷︷ ︸

c1(u)︸ ︷︷ ︸

c2(u)

At free-fermion point: λ = π2

a(u) = ρsin(λ− u)

sinλ= ρ cosu

b(u) = ρsinu

sinλ= ρ sinu

c1(u) = ρ g, c2(u) =ρ

g, ρ ∈ R

Counting isotropic dimers:

ρ = g =√2, u = λ

2 = π4

c1(u) = 2, a(u) = b(u) = c2(u) = 1

• The free fermion condition is satisfied at the free-fermion point λ = π2

a(u)2 + b(u)2 = c1(u)c2(u)

• Particle lines are drawn if arrows point down or left.

• Tiles corresponding to a source of horizontal arrows (apricot) have a double degeneracy.

Locally, the mapping is one-to-two for these faces. Sources and sinks of horizontal arrows

appear in pairs so g is a gauge which we fix to g = eiu.

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Lattice Configurations

• A typical periodic vertex configuration on a 6× 4 square lattice: vertex, particle and (one

of the 23 = 8) possible dimer configurations:

• The boundary conditions are periodic such that the left/right edges and top/bottom edges

are identified.

• Sources and sinks of horizontal arrows appear in pairs

• The excess of up arrows over down arrows along a row (2 in this case) is conserved.

• Particles are conserved and move up and to the right around the torus but do not cross.

• An M ×N rectangular lattice is covered by MN dimers.

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Fermionic Algebra

• The face operators of the free-fermion six-vertex model decompose into a sum of

contributions from six elementary tiles

Xj(u) = uj j+1

= a(u)

(

+

)

+ b(u)

(

+

)

+ c1(u) + c2(u)

• In the particle representation, the elementary tiles, as operators, act on an upper row

particle configuration to produce a lower row particle configuration

n00j , n11

j , n10j , n01

j , f†j fj+1, f

†j+1fj

� n00j , n11

j , n10j , n01

j are (diagonal) orthogonal projection operators

n00j + n11

j + n10j + n01

j = I nabj = na

jnbj+1, na

jnbj = δab n

aj , a, b = 0,1

� n1j and n0

j are (diagonal) number operators counting the single site occupancies and

vacancies respectively at position j.

n0j + n1

j = I,

� fj and f†j are (non-diagonal) single-site particle annihilation and creation operators

respectively

{fj, fk} = {f†j , f†k} = 0, {fj, f†k} = δjk

0-8

• All of the elementary tile operators can be written explicitly in terms of the fermion

operators fj and f†j

n1j = f

†j fj n0

j = fjf†j = 1− f

†j fj

• The tiles are expressed in terms of fermi operators as

= n00j = (1− f

†j fj)(1− f

†j+1fj+1) = n01

j = (1− f†j fj)f

†j+1fj+1

= n11j ,= f

†j fjf

†j+1fj+1 = n10

j = f†j fj(1− f

†j+1fj+1)

= f†j fj+1 = f

†j+1fj

• Multiplication of tiles in the fermionic algebra is given diagrammatically

= = = =

= = = =

0-9

From Fermi Algebra to Temperley-Lieb Algebra

• The Temperley-Lieb (TL) algebra has generators I and ej and is defined by

e2j = βej ejej±1ej = ej eiej = ejei, |i− j| ≥ 2

• The T-L algebra admits a planar diagrammatic representation consisting of “monoids”

= I = ej = β = 2cosλ = loop fugacity

• The monoids satisfy

=β =

• The free-fermion algebra gives a representation of the planar T-L algebra

I = + + + , ej = + + x + x−1

where x = eiλ = i and x+ x−1 = 2cosλ = β = 0.

• With these definitions, the diagrammatic relations between fermionic tiles imply the

defining relations of the TL algebra.

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YBE and Inversion Relation

• In terms of the generators of the TL algebra, the face transfer operators of the free-fermion

six vertex model take the form

Xj(u) = u

j j+1

= cosu I + sinu ej

• This form of the face transfer operator is sufficient (Baxter 1982) to guarantee that Xj(u)

satisfies the Yang-Baxter Equation and Inversion Relation

Xj(u)Xj+1(u+ v)Xj(v) = Xj+1(v)Xj(u+ v)Xj+1(u) Xj(u)Xj(−u) = ρ(u) ρ(−u) I

a b

v−u

ef

u

c

d

v =

de

v−u

b c

ua

f

v

cd

−u

a b

u

= ρ2 cos2 u δ(a, d)δ(b, c)

subject to the initial condition Xj(0) = I.

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Commuting Periodic Row Transfer Matrices

YBE + Inversion ⇒ [T (u),T (v)] = 0 ⇒ Integrable

T (u)T (v) =

u u u u u

v v v v v• • • • • v − u• • •u− v

= • •v − uv v v v v

u u u u u• • • • • •• •u− v

=

v v v v v

u u u u u• • • • • •• •• •u− v • •v − u

= T (v)T (u)

• Commuting transfer matrices share a common set of u-independent eigenvectors.

• Since T (u)T = T (λ−u), the row transfer matrices are normal and therefore simultaneously

diagonalizable.

• The eigenvalues spectra can be found by solving functional equations satisfied by T (u).

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Hamiltonian, Free Energy and Residual Entropy

• The logarithmic derivative of the transfer matrix gives the Hermitian free-fermion

Hamiltonian

H =d

dulogT (u)

∣∣∣∣u=0

= −N∑

j=1

ej = −N∑

j=1

(f†j fj+1 + f

†j+1fj)

• The bulk partition function per site,

limM,N→∞

(ZM×N)1

MN = ρ κ(u) = ρ exp(−fbulk(u))

can be obtained by solving the inversion relation κ(u)κ(−u) = cos2 u (Baxter 1982) or by

using the Euler-Maclaurin formula. This gives the bulk free energy

fbulk(u) = −∫ ∞

−∞sinhut sinh(π2 − u)t

t sinhπt cosh πt2

dt = 12 log 2− 1

π

∫ π/2

0log(cosec t+ sin2u)dt

• The residual entropy S has not changed by rotating the orientation of the dimers. Indeed,

it agrees with the known result (Fisher 1961).

W = eS =√2exp(−fbulk(

π4)) = exp(2Gπ ) = 1.791622812 . . . , S = 2G

π = .583121808 . . .

where W is the molecular freedom and G is the Catalan’s constant.

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Periodic Transfer Matrices

Periodic Row Transfer Matrix Partition Function on a Torus

T (u) = u u u u u u

a1 a2 · · · aN

b1 b2 · · · bN

Z = TrT (u)M = T

• The number of particles (down/up arrows) d =∑N

j=1 aj, is conserved under the action of

the transfer matrix.

• Also the total magnetization is conserved

Sz =N∑

j=1

σj = −N,−N +2, . . . , N − 2, N

• Sz is a good quantum number separating the spectrum into sectors labelled by ℓ=|Sz|:

ℓ = |Sz| =

0,2,4, . . . , N, N even

1,3,5, . . . , N, N odd

Z4: N odd, ℓ odd, Ramond:N even, ℓ2 even, Neveu-Schwarz:N even, ℓ

2 odd

• The transfer matrix and the vector space of states thus decompose as

T (u) =N⊕

d=0

T d(u) dimV(N) =N∑

d=0

dimV(N)d =

N∑

d=0

(N

d

)

= 2N

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Inversion Identities

• The periodic free-fermion single row transfer matrix satisfies (Felderhof 73)

T (u)T (u+ λ) =(

cos2N u− sin2N u)

I, N odd

T d(u)T d(u+ λ) = (cosN u+ (−1)d sinN u)2I, N even

• The eigenvalues T(u) of the transfer matrices in a given sector are determined,

up to an overall constant ρ, by the positions uj of their zeros in the analyticity strip

−π/4 ≤ Reu < 3π/4. They are indeed Laurent polynomials in z = eiu

T(u) = ρN∏

j=1

sin(u− uj)

• We solve the inversion identities sector by sector. For example, the factorization of the

right side of the inversion identity for the Z4 sector yields

cos2N u− sin2N u =e−2Niu

22N−1

N∏

j=1

(

e2iu + iǫj tan(2j − 1π)

4N

)(

e2iu − iǫj tan(2j − 1π)

4N

)

for all ǫj=±1.

0-15

• Sharing out the zeros between T(u) and T(u+ λ) gives 2N eigenvalues.

T(u) = ǫ(−i)N/2e−Niu

2N−1/2

N∏

j=1

(

e2iu + iǫj tan(2j − 1)π

4N

)

, Z4: N, ℓ odd

T(u) =ǫR(−i)

N2 e−Niu

2N−1

N∏

j=1

(

e2iu + iǫj tan(2j−1)π

2N

)

, R: N, ℓ/2 even

T(u) =ǫNS(−i)

N2Ne−Niu

2N−1

N∏

j=1

j 6=N/2

(

e2iu + iǫj tanjπ

N

)

, NS: N even, ℓ/2 odd

• The overall sign ǫ = ±1 of each eigenvalue is not fixed by the inversion relation. These

sign factors ǫ are

ǫ = (−1)N−|Sz|

4 , ǫR = ǫNS = (−1)⌊|Sz|+2

4 ⌋,

• Up to the overall choice of sign ǫ, there are either 2N or 2N−2 (NS sector) possible

eigenvalues allowing for all excitations. However, they are not all physical and only[

NN+s

2

]

of

these solutions actually occur as eigenvalues. These are determined by the selection rules.

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Pattern of Zeros of the Transfer Matrix

N, ℓ odd,{

Z4 Sectors N, ℓ even,

Ramond (ℓ/2 even)

Neveu-Schwarz (ℓ/2 odd)ℓ = |Sz|

−π4

π4

π2

3π4

y5y4

y3

y2

y1

−y5−y4−y3

−y2

−y1

N, ℓ odd

−π4

π4

π2

3π4

y5y4

y3

y2

y1

−y5−y4−y3

−y2

−y1

N, ℓ even

• The y-ordinates of 1-strings uj and 1-string energies Ej are

yj = −12 log tan

Ejπ

N, Ej =

12(j −

12), j = 1,2, . . . , N ; Z4

j − 12, j = 1,2, . . . , N/2; Ramond

j, j = 1,2, . . . , N/2− 1; Neveu-Schwarz

• The number of 1-strings mj plus the number of 2-strings nj at any given position is

mj + nj =

1, Z4

2, R, NS

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Physical Combinatorics: Ramond Sectors

• The building blocks of the spectra in the upper half-plane consist of the q-binomials

[nm

]

q =[

n⌊n/2⌋−σ

]

q= q−

12σ

2 ∑

double−columnsfor fixed σ

q∑

j mjEj ,

σ = ⌊n/2⌋ −m = #right−#left = σmin

Ej = j − 12

[62

]

q = + + + + + + + +1 q 2q2 2q3 3q4 2q5 2q6 q7 q8 (σ = 1)

j = 1j = 2j = 3

• [nm

]

q =[

nn−m

]

qbut different σ so they have different combinatorial interpretations.

• Excitations are generated either by inserting a left-right pair of 1-strings at position j = 1

or incrementing the position j of a 1-string by 1 unit. The selection rules are

σ + σ̄ = ℓ/2, 12(σ − σ̄) ∈ Z

• In a given ℓ sector, the quantum numbers of the groundstate satisfy

σ = σ̄ = ℓ/4, ℓ = 0,4,8, . . . E(σ) + E(σ̄) =ℓ2

16

0-18

Physical Combinatorics: Neveu-Schwarz Sectors

• The building blocks of the in the upper half-plane spectra consist of the q-binomials

[nm

]

q =[

n⌊n/2⌋−σ

]

q= q−

12σ(σ+1)

∑

double−columnsfor fixed σ

q∑

j mjEj ,

σ = ⌊n/2⌋ −m, #right−#left

σmin =

σ, σ ≥ 0

σ +1, σ < 0

Ej = j

[72

]

q = + + + + + + + + + +1 q 2q2 2q3 3q4 3q5 3q6 2q7 2q8 q9 q10 (σ = 1)

j = 1j = 2j = 3

• [nm

]

q =[

nn−m

]

qbut different σ so they have different combinatorial interpretations

• Excitations are generated either by inserting a right or left 1-string at position j = 1 or

incrementing the position j of a 1-string by 1 unit. The selection rules are

σ + σ̄ = (ℓ− 2)/2, 12(σ − σ̄) ∈ Z

• In a given ℓ sector, the quantum numbers of the groundstate satisfy

σ = σ̄ = (ℓ− 2)/4, ℓ = 2,6,10, . . .

(

E(σ) + E(σ̄) =ℓ2 − 4

16

)

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Finitized Modular Invariant Partition Function

• In the R and NS sectors with N even

Z(N)ℓ (q) =

(qq̄)−c/24∑

k∈Zq∆2k+ℓ/2

[2⌊N+2

4⌋

⌊N+2−ℓ

4⌋−k

]

qq̄∆2k−ℓ/2

[2⌊N

4⌋

⌊N−ℓ

4⌋+k

]

q̄, R: ℓ/2 even

(qq̄)−c/24∑

k∈Zq∆2k+ℓ/2

[2⌊N

4⌋+1

⌊N+2−ℓ

4⌋−k

]

qq̄∆2k−ℓ/2

[2⌊N+2

4⌋−1

⌊N−ℓ

4⌋+k

]

q̄, NS: ℓ/2 odd

ZN(q) = Z(N)0 +2

ℓ≤N∑

ℓ∈4NZ(N)ℓ (q) + 2

ℓ≤N∑

ℓ∈4N−2

Z(N)ℓ (q)

= 12(qq̄)

− c24−

18

[ ⌊N+24 ⌋∏

n=1

(1 + qn−12)2

⌊N4 ⌋∏

n=1

(1 + q̄n−12)2 +

⌊N+24 ⌋∏

n=1

(1− qn−12)2

⌊N4 ⌋∏

n=1

(1− q̄n−12)2

]

+2(qq̄)−c24

⌊N4 ⌋∏

n=1

(1 + qn)2⌊N−2

4 ⌋∏

n=1

(1 + q̄n)2

(1a)

The spectra of dimers agrees sector-by-sector with the spectra of critical

dense polymers!

0-20

Counting Dimers: Standard Orientation

• The known Pfaffian solution (Kasteleyn1961) for the number of periodic dimer

configurations is

Z̃M×N = 12(Z̃

1/2,1/2M×N + Z̃

0,1/2M×N + Z̃

1/2,0M×N)

Z̃α,βM×N =

N/2−1∏

n=0

M/2−1∏

m=0

4

(

sin22π(n+ α)

N+ sin2

2π(m+ β)

M

)

, M,N = 2,4,6, . . .

• Explicit counting on a M ×N square lattice yields

(Z̃M×N) =

8 36 200 1156 · · ·36 272 3,108 39,952 · · ·200 3,108 90,176 3,113,860 · · ·1,156 39,952 3,113,860 311,853,312 · · ·

... ... ... ... . . .

N,M = 2,4,6, · · ·

Z̃2×2 = 8 ⇒ Z̃8×8 = 311,853,312

0-21

Counting Dimers: Rotated Orientation

• The exact counting of periodic dimer configurations on a finite M × N square lattice, in

the 45 degree rotated orientation, is given by

ZM×N = TrT (N)(π

4

)M=

∑

n≥0

Tn(π

4)M

with ρ =√2 and u = λ

2 = π4.

• The explicit formula is:

ZM×N=

2MN+1N∑

s=−N+2;4

∑

∑Nj=1 ǫj=s

(−1)M(N−s)

4

N∏

j=1

cosM(

ǫjtj − π4

)

, N odd

2MNN∑

s=−Ns = 0 mod 4

∑

∑Nj=1 ǫj=−|s|

(−1)M(2N+s)

4

N∏

j=1

cosM(

ǫjtRj − π

4

)

+ 2MNN∑

s=−Ns = 2 mod 4

∑

∑Nj=1 ǫj=−|s|

ǫMN2

(−1)M(2N+|s|+2)

4

N∏

j=1

cosM(

ǫjtNSj − π

4

)

, N even

• Rotated periodic dimer configurations on an M ×N square lattice in numbers

(ZM×N) =

4 8 16 32 64 . . .8 24 80 288 1,088 . . .16 80 448 2,624 15,616 . . .32 288 2,624 26,752 280,832 . . .64 1,088 15,616 280,832 5,080,064 . . .... ... ... ... ... . . .

, M,N = 1,2,3, . . .

0-22

Z̃2×2 = 8 ⇒ Z̃8×8 = 311,853,312

Z2×2 = 24 ⇒ Z8×8 = 38,735,278,017,380,352

• The asymptotic growth per dimer coincides

(Z̃2M,N)1

MN ∼ (Z̃M,2N)1

MN ∼ (ZM,N)1

MN ∼ exp(2Gπ )

Bulk CFT of Dimers

• The anisotropic partition function is

ZN,M = TrT (u)M =∑

n≥0

Tn(u)M =

∑

n≥0

e−MEn(u)

• Finite-size corrections from conformal invariance

E0 = Nfbulk(u)−πc

6Nsin 2u, En − E0 =

2πi

N

[

(∆+ k)e−2iu − (∆̄ + k̄)e2iu]

• The analytic results using Euler-Maclaurin are

c = −2, ∆j = ∆̄j = j2−18 = −1

8, 0,38, ∆min = −1

8, j = 0,1,2

• For dimers in the standard orientation it is known that (Izmailian et al 2006) in the

scaling limit, the modular invariant conformal partition function is a sesquilinear form in u(1)

characters

Z(q) =∑

∆,∆̄

N∆,∆̄ κ∆(q)κ∆̄(q̄),

where

N∆,∆̄ =

1 0 00 2 00 0 1

, κ∆(q) = q−c/24

∞∑

k=0

d∆(k) q∆+k, q = modular nome

0-23

Modular Invariant Partition Function

• The modular invariant partition function Z(q)(MIPF) of the free-fermion six-vertex model

is given by taking the trace over all Sz sectors with N even.

ZN(q) = Z(N)0 +2

ℓ≤N∑

ℓ∈4NZ(N)ℓ (q) + 2

ℓ≤N∑

ℓ∈4N−2

Z(N)ℓ (q)

Taking the thermodynamic limit N → ∞ gives the conformal modular invariant partition

function

Z(q) = Z0(q)+2∑

ℓ∈2NZℓ(q) =

1

|η(q)|23∑

j=0

|ϑj,2(q)|2 = |κ20(q)|2 +2|κ2

1(q)|2 + |κ22(q)|2

• The u(1) characters are

κnj (q) =

1

η(q)ϑj,n(q), j = 0,1,2

where the Dedekind eta and theta functions are

η(q) = q1/24∞∏

n=1

(1− qn), ϑj,n(q) =∑

k∈Zq(j+2kn)2

4n

• The MIPF Z(q) of dimers agrees with the result for the usual orientation.

• It also precisely coincides with the MIPF of critical dense polymers (MDPR 2013) calculated

by solving the lattice loop model.

• This coincidence is nontrivial because critical dense polymers requires implementation of

a modified (Markov) trace.

0-24

Strip with Vacuum Boundary Conditions

xx−1

0-25

Jordan Cells

• The Hamiltonian for dimers with the (r, s) = (1,1) vacuum boundary condition (no seam)

on the strip coincides with the Uq(sl(2))-invariant XX Hamiltonian

H = −∑N−1j=1 ej = −1

2

∑N−1j=1 (σxj σ

xj+1 + σ

yjσ

yj+1)−

12i(σ

z1 − σzN)

= −∑N−1j=1 (f

†j fj+1 + f

†j+1fj)− i(f

†1f1 − f

†NfN)

where σx,y,zj are Pauli matrices and

fj = σxj − iσyj , f

†j = σxj + iσ

yj .

• This Hamiltonian is manifestly not Hermitian but its eigenvalues are real.

• The Jordan canonical forms for N = 2 and N = 4 are

0⊕(

0 10 0

)

⊕ 0

0⊕(

0 10 0

)

⊕ 0⊕ 0⊕(

0 10 0

)

⊕ 0⊕ (−√2)⊕

(

−√2 1

0 −√2

)

⊕ (−√2)⊕

√2⊕

(√2 1

0√2

)

⊕√2

0-26

The Big Question:

Is the dimer model a Gaussian free theory (c = 1) or a logarithmic CFT(c = −2)?

• A Conformal Field Theory is logarithmic if certain representations of the dilatation

Virasoro generator L0 are non-diagonalizable and exhibit nontrivial Jordan cells.

• In the continuum scaling limit, the Hamiltonian gives the Virasoro dilatation operator L0.

Assuming that the Jordan cells persist in this scaling limit, the representation is reducible

yet indecomposable and so, as a CFT, dimers is logarithmic!

• For dimers with (1, s) boundary conditions the conformal weights are

∆1,s =(2− s)2 − 1

8= 0,−1

8,0,

3

8,1,

15

8, . . . s = 1,2,3,4,5,6, . . .

• Since ∆min is negative and the six-vertex model with λ = π2 on the strip with vacuum

boundary conditions exhibits Jordan cells (e.g. Gainutdinov, Nepomechie et al 2015),

we argue that dimers is nonunitary and logarithmic with central charge c = −2 and

ceff = c− 24∆min = 1.

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Kac Table

• Infinitely extended Kac table

of conformal weights:

∆r,s =(2r − s)2 − 1

8, r, s = 1,2,3, . . .

• Irreducible representations are marked by

......

......

...... . . .

638

358

158

38

−18

38

· · ·

6 3 1 0 0 1 · · ·

358

158

38

−18

38

158

· · ·

3 1 0 0 1 3 · · ·

158

38

−18

38

158

358

· · ·

1 0 0 1 3 6 · · ·

38

−18

38

158

358

638

· · ·

0 0 1 3 6 10 · · ·

−18

38

158

358

638

998

· · ·

0 1 3 6 10 15 · · ·

1 2 3 4 5 6 r

1

2

3

4

5

6

7

8

9

10

s

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Summary and Outlook

• The anisotropic dimer model on the square lattice with 45 degree rotated orientation has

been solved exactly on a torus.

• This is achieved, by viewing the dimer model as a Yang-Baxter integrable free-fermion six-

vertex model and solving the associated inversion identity satisfied by the transfer matrices.

• Explicit formulas are found for the counting of dimer configurations on a finite M × N

lattice.

• The modular invariant partition function is calculated analytically and precisely coincides

with critical dense polymers which is a logarithmic CFT.

• On the strip with vacuum boundary conditions the six-vertex model with λ = π2 exhibits

Jordan cells. Therefore we argue that dimers is a logarithmic, non-unitary theory with central

charge c = −2 and ceff = 1.

• Using Yang-Baxter methods and double row transfer matrices, it is now possible to study

dimers on a strip with many different boundary conditions. Some insight may also be gained

for Aztec diamonds and the six vertex model with domain wall boundary conditions.

Aztec diamond Domain Wall Boundary conditions

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Thank you!

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