Relation between the completely packed O(n) loop model andexactly solved coloring models

——————————————————Yougang Wang

Leiden University

Outline:1. Mapping between loop model and coloring model

2. Coloring model and analytical solutions

3. Transfer matrix method and numerical evaluations

Supervisor:

Prof. Henk W.J. Blote (Lorentz Institute, Leiden University)

1) Mapping between loop model and coloring model

I. Completely packed O(n) loop model:

- Every site of the lattice is visited by a loop

- Full coverage of all the edges of the lattice

- Three different types of vertices exist:

x c z

- Partition sum of the system is

Zloop =∑GzNzcNcxNxnNn =

∑GzNz

(c

n

)NcxNxnNl

II. Coloring model:

- Vertex weights are denoted Rλµ(αβ) where λ, µ denote the colors

of the bonds in the −x,+x directions, and α, β apply to the −y,+y

directions respectively.

Rλµ(αβ) = Wdαλδαβδλµ +W r

αβδαλδβµ +W lαβδαµδβλ

with

W rαβ = W r(1− δαβ), W l

αβ = W l(1− δαβ)

and Wdαβ = Wdδαβ +W0(1− δαβ).

- Additional symmetry condition W l = W r

- Partition sum is Zcm =∑C∏vR

λvµv(αvβv)

III. Mapping between the two models:

- The four different ways in which the incoming bonds at a vertex can

be connected by the remaining configuration of the generalized loop

model.

Z1 Z2 Z3 Z4

- In terms of the restricted sums, the partition sum of the coloring

model can be written as

Zcm = [(n2−n)W r +nWd](Z1 +Z2) + [(n2−n)W0 +nWd]Z3 +nWdZ4

and the partition sum of the loop model can be written as

Zloop = [(n2+n)z+nx+nc](Z1+Z2)+n(2z+nx+c)Z3+n(2z+x+c)Z4

- The equivalence of both models follows if the prefactors of Z1 +Z2,

Z3 and Z4 are the same in both partition sums. Therefore,

W0 = x

Wd = 2z + x+ c

W r = z

which is the bridge relation between the two models.

Branch CaseVertex Weights

z x c

1 IIA1 1 0 0

2 IIA2a 1 0 −1 +√n− 1

3 IIA2b 1 0 −1−√n− 1

4 IIB1a 1 2−n4 0

5 IIB1b 1 n−24

2−n2

6 IIB2a 0 1 0

7 IIB2b 0 1 -2

2) Coloring model and analytical solutions

I. Branch 1, subcase IIA1

- No crossing bond, no cubic vertex, only z-type colliding vertices

- Existing mapping between Potts model and the completely packed

loop model with n =√q

- Analytical solution for free energy density (Lieb & Baxter)

n > 2, f = 12θ +

∑∞k=1

exp(−kθ) tanh(kθ)k , cosh θ = n/2

n = 2, f = 2 ln Γ(1/4)2 Γ(3/4)

−2 < n < 2, f = 12

∫∞−∞ dx

tanhµx sinh(π−µ)xx sinhπx , µ = arccos(n/2)

n = −2, f = 0

By continuing Lieb’s parametrization to n < −2, we obtained

n < −2, f = 12θ +

∑∞k=1

[− exp(−θ)]k tanh(kθ)k , θ = θ + πi

- Schultz’s solution for free energy density

n > 2, f = ln u2√n−2

+ 2∑∞k=1 ln 1−u2−8k

1−u−2−8k

where u = 12[(n− 2)

12 + (n+ 2)

12].

II. Branch 2 & 3, subcase IIA2a & IIA2b

- No crossing bond, both colliding (z-type) and cubic vertices (c-type)

exist

- Schultz’s solution for free energy density

n > 2, f = ln{(1±√n− 1) n−1

n−2×

×∏∞k=1

1−n−1±√n−1

1±√n−1

(n−1)−2k

1−n−1±√n−1

1±√n−1

(n−1)−1−2k

×∏∞k=1

1− 1±√n−1

n−1±√n−1

(n−1)1−2k

1− 1±√n−1

n−1±√n−1

(n−1)−2k

}

III. Branch 4, subcase IIB1a

- No cubic vertex, both colliding vertices (z-type) and crossing bond

(x-type) exist

- Rietman’s solution for free energy density

f =(

12 −

n4

) Γ2(

14

)Γ2(

34

) Γ2(β)Γ2(α)

where α = n−64(n−2), β = 3n−10

4(n−2)

- Schultz’s solution for free energy density

f =(

12 −

n4

) Γ2(

14

)Γ2(

34

) Γ2(β)Γ2(α)

ctg(απ)

with one extra factor: ctg(απ)

IV. Branch 5, subcase IIB1b

- All three types of vertices exist

- Schultz’s solution specifies the same free energy density as it does

for branch 4

V. Branch 6 & 7, subcase IIB2a & IIB2b

- No colliding vertex, only crossing bonds exist for branch 6, both

crossing bonds and cubic vertices exist for branch 7.

- Schultz’s solution of free energy density

f = ln

Γ(

1nb

)Γ(

1+(n−2)bnb

)Γ(

1−bnb

)Γ(

1+(n−1)bnb

), with b = W r

Wd = z2z+x+c = 0

suggests f = 0 for both subcases when we take the limit at b→ 0.

3) Transfer matrix method and numerical evaluations

I. The transfer matrix method

- The transfer matrix specifies how an added row contributes to the

partition sum. Z(M+1) = TM · Z(1)

- Free energy and correlation lengths (Λi(L) is the eigenvalue)

f(L) = L−1 ln Λ0(L)

ξ−1k (L) = ln

Λ0

|Λk|

Xk(L) =L

2πξk(L)=

L

2πln

Λ0

|Λk|

Xt(L) =L

2π[ln Λ0(L)− ln |Λ2(L)|)]

Xh(even L) =L

2π

[ln Λ0(L)−

1

2(ln |Λk(L− 1)|+ ln |Λk(L+ 1)|)

]

Xh(odd L) =L

2π

[1

2(ln |Λ0(L− 1)|+ ln |Λ0(L+ 1)|)− ln Λk(L)

]

II. Coding and decoding of the connectivites

- To generate all possible L-point connectivities of the model (CL con-nectivites) and represent the connectivities by means of consecutiveintegers 1, 2, ...- CL is a function of L and determines the size of the transfer matrix

vertex types even/odd Lmax CLmax

zeven 30 9694845

odd 27 20058300

z, ceven 22 8414640

odd 19 6906900

z, xeven 16 2027025

odd 15 2027025

z, c, xeven 14 4373461

odd 13 4373461

III. Numerical results for free energy density

- Branch 1

1. Extrapolated free energy density in good agreement with Baxter’s

solution (Schultz’s solution at n > 2) in range |n| ≤ 2 and |n| ≥ 10.

2. Elsewhere with poor convergence finite-size data

3. Extrapolated conformal anomaly ca agrees with Coulomb gas pre-

dictions.

- Branch 2 & 3

1. For n > 2, good agreement with Schultz’s solution

2. No analytical solution yet for 1 < n < 2

3. Trivial case at n = 1 where two branches are connected and we

have f = 0.

4. For branch 3, finite size data of free energy density behaves like

f(L) ≈ f(∞) + a/L, instead of f(L) ' f + πca6L2.

- Branch 4 & 5

1. For n ≤ 2, good agreement with Rietman’s solution, or Schultz’ssolution divided by ctg(απ).2. Extrapolated free energy density is symmetric with respect to n = 2.3. Inspired by the symmetry, a general expression for free energydensity may be conjectured from Rietman’s solution for n ≤ 2 as

f(n) = ln

(1

2−

2− |n− 2|4

)+ 2 ln Γ

(4 + 3|n− 2|

4|n− 2|

)4. Conformal anomaly ca ≈ n−1 for 1 ≤ n ≤ 2, and ca ≈ n/2 for n ≥ 2.

- Branch 6 & 7:

f(n,L) =a(n)

L

Branch 6 is a trivial case with diagonal transfer matrix whose elementsand eigenvalues all equal to n. a(n) = ln(n)For branch 7:

a(n) =

ln(n), if L is even

ln(n− 2), if L is odd

IV. Numerical results for scaling dimensions

- Branch 1

1. For n ≥ 1, the extrapolated Xt matches Coulomb gas results.

2. For n < 1, the leading thermal scaled gap converges to X1,2 = 4.

2

2.5

3

3.5

4

4.5

5

5.5

6

-1.5 -1 -0.5 0 0.5 1 1.5 2

Xt

n

3. Magetic scaled gap in good agreement with Coulomb gas results.

- Branch 2 & 3

1. For large n, extrapolated Xt tends to vanish.

Xt

0

2

4

6

8

10

cn

-0.3 -0.2 -0.1 0 0.1 0.2

Xh

0

1

2

3

cn

-0.3 -0.2 -0.1 0 0.1 0.2

n = 40

2. Scaling dimension results suggest first-order phase transition forlarge n

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

< σ

>

ρ

i

n = 100, cn = 340

- Branch 4 & 5

1. These two branches always have the same largest leading eigenvaluefor their transfer matrices.2. Different subdominant eigenvalues lead to difference in scaled gapcalculation.