Thermodynamic limit of the six-vertex model with reflecting end - - PowerPoint PPT Presentation

Thermodynamic limit of the six-vertex model with reflecting end

G.A.P. Ribeiro

Federal University of S˜ ao Carlos, Brazil

May 14, 2015

Collaboration with: V.E. Korepin; T.S. Tavares

Outline

◮ Problem ◮ Six-vertex model ◮ DWBC ◮ Reflecting ◮ determinant formula ◮ Homegeneous limit ◮ Special solution ◮ Entropy

Problem

◮ In statistical physics people believe that in thermodynamic

limit the bulk free energy and correlations should not depend

• n boundary conditions. This is often true, but there are

counterexamples.

◮ One of the most proeminent one is the six-vertex model: PBC

= DWBC.

◮ We would like to compute the free-energy and entropy of the

six-vertex model with boundaries different boundaries: reflecting end.

Water molecule x ice: six-vertex model

◮ Water molecule: O-H distance (0.95 ˚

A); angle between O-H: 104◦

◮ Ice: X-ray data (1930s) indicates that O form a hexagonal wurtzite

structure (tetraedral): O-O distance (2.76 ˚ A)

Square-ice model: six-vertex model

⇒ Effective model: square ice-model. a ✻ ✻ ✲ ✲ ω1 a ❄ ❄ ✛ ✛ ω2 b ❄ ❄ ✲ ✲ ω3 b ✻ ✻ ✛ ✛ ω4 c ✻ ❄ ✲ ✛ ω5 c ❄ ✻ ✛✲ ω6 ωi = e−βεi

Entropy

S = Nk log W ⇒ ε1 = ε2 = · · · = ε6 = 0 (or a = b = c = 1).

◮ Pauling (1935) - estimated the entropy of the hexagonal phase of

ice (ordinary ice): W = 22 (6/16) = 3

2

◮ Lieb (1967) - computed exactly the entropy for the square-ice:

W = ( 4

3)3/2 = 1.5396007 . . . .

Phases: six-vertex model

Control parameter is ∆ = a2 + b2 − c2 2ab . Free energy has different analytic forms when

◮ ∆ > 1 (ferroelectric). ◮ −1 < ∆ < 1 (disordered). ◮ ∆ < −1 (anti-ferroelectric).

Phase diagram

b/c a/c

1 1

I II III IV

◮ Phase I and Phase II (ferroelectric). ◮ Phase III (disordered). ◮ Phase IV (anti-ferroelectric).

This phase diagram describes the six-vertex model with PBC (Lieb 1967,Sutherland 1967, Baxter 1982) and with DWBC (V Korepin, P Zinn-Justin 2000, P Zinn-Justin 2000). Rigourous proofs are due P Bleher et al 2006,2009,2010...

Recent realization of vertex models

DWBC

In the computation of scalar product os Bethe states, |ψN = B(λN) · · · B(λ2)B(λ1) |⇑ , appears the (Korepin 1982) Z DWBC

N

({λ}, {µ}) = ⇓| B(λN) · · · B(λ2)B(λ1) |⇑ . ✲ ✲ ✲ ✲ ✲ ✛ ✛ ✛ ✛ ✛ ❄ ❄ ❄ ❄ ❄ ✻ ✻ ✻ ✻ ✻ λ5 λ4 λ3 λ2 λ1 µ1 µ2 µ3 µ4 µ5

Tsuchiya partition function

In the case of open spin chains, the scalar product |φN = B(λN) · · · B(λ2)B(λ1) |⇑ . leads to another partition function for the six-vertex model ZN({λ}, {µ}) = ⇓| B(λN) · · · B(λ2)B(λ1) |⇑ .

✏ ✑ ✏ ✑ ✏ ✑ ✲ ✲ ✲ ✲ ✲ ✲ ❄ ❄ ❄ ✻ ✻ ✻

λ3 −λ3 λ2 −λ2 λ1 −λ1 µ1 µ2 µ3

Reflecting ends

The diagonal K-matrix plays the role of the reflecting end, K(λ) = k11(λ) k22(λ)

• .

k11

✲ ✛ ✘ ✙

k22

✛ ✲ ✘ ✙

Boltzmann weights

One can define the Boltzmann weights to the case −1 < ∆ < 1. In this case, we have a(λ) = sin(γ − λ), b(λ) = sin(γ + λ), c(λ) = sin(2γ), where 0 < γ < π/2 and ∆ = − cos(2γ). k11(λ) = sin(ξ + λ + γ) sin(ξ) , k22(λ) = sin(ξ − λ − γ) sin(ξ) , where ξ is the boundary parameter.

Tsuchiya determinant formula - (Tsuchiya 1998)

ZN({λ}, {µ}) = (sin(2γ))N

N

• i=1

sin(2(λi + γ)) sin(ξ − µi ) sin(ξ) ×

N

• i,j=1

sin(γ − (λi − µj )) sin(γ + λi − µj ) sin(γ − (λi + µj )) sin(γ + λi + µj )

N

• i,j=1

i<j

− sin(λj − λi ) sin(µi − µj ) sin(λj + λi ) sin(µi + µj ) × det M, where M is a N × N matrix, whose matrix elements are Mij = φ(λi , µj ) with φ(λ, µ) = 1 sin(γ − (λ − µ)) sin(γ + λ − µ) sin(γ − (λ + µ)) sin(γ + λ + µ) .

Homegeneous limit

◮

Taking λi → λ and µj → µ. ZN(λ, µ) =

• sin(2γ) sin(2(λ + γ))

sin(ξ − µ) sin(ξ) N × [sin(γ − (λ − µ)) sin(γ + λ − µ) sin(γ − (λ + µ)) sin(γ + λ + µ)]N2 CN [− sin(2λ) sin(2µ)]

N(N−1) 2

× τN(λ, µ), where CN = N−1

k=1 k!

• 2. The determinant is given by

τN(λ, µ) = det(H), where the H-matrix elements are Hi,j = (−∂µ)j−1∂i−1

λ

φ(λ, µ).

Bidimensional Toda equation (Ma 2011, Sylvester 1962)

−τN∂2

µλτN + (∂µτN)(∂λτN) = τN+1τN−1,

and can be conveniently written as −∂2

µλ [log(τN)] =

τN+1τN−1 τ2

N

, N ≥ 1, which is supplemented by the initial data τ0 = 1 and τ1 = φ(λ, µ).

Special solutions

The partition function can be cast directly in simple expressions for some special points. ZN (λ, µ; γ = π 4 ) =

• sin(ξ ∓ µ)

sin(ξ) N (cos(2λ))

N(N+1) 2

(sin(2µ))

N(N−1) 2

. For the cases where µ = ±(λ + γ) and µ = ±(λ − γ), ZN(λ, ±λ ± γ)) =

• sin(ξ ∓ (λ + γ))

sin(ξ) N (sin(2γ))N2 (− sin(2λ))

N(N−1) 2

(sin(2(λ + γ)))

N(N+1) 2

, ZN (λ, ±λ ∓ γ) =

• sin(ξ ∓ (λ − γ)) sin(2(γ + λ))

sin(ξ) N (sin(2γ))N2 (sin(2λ) sin(2(γ − λ))

N(N−1) 2

. The thermodynamic limit is trivial in these cases. The free energy F = − limN→∞

log(ZN ) 2N2

(we set temperature to 1) is given respectively by e−2F(λ,µ;γ=π/4) =

• cos(2λ) cos(2µ),

e−2F(λ,±(λ+γ)) = sin(2γ)

• − sin(2λ) sin(2(λ + γ)),

e−2F(λ,±(λ−γ)) = sin(2γ)

• sin(2λ) sinh(2(γ − λ)).

VSASM

We can also fix both spectral parameters and anisotropy parameter γ, such as ZN(0, 0; π 3 ) = AVSASM

1

=

N−1

• k=0

(3k + 2) (6k + 3)!(2k + 1)! (4k + 2)!(4k + 3)! = 1, 3, 26, 646, . . . which is a combinatorial point connected to the number of vertically symmetric alternating sign matrices (VSASM) due to (Kuperberg 2002) Othe special cases are ZN(0, 0; π 4 ) = 2N AVSASM

2

= 2N2 , and ZN(0, 0; π 6 )/3N = AVSASM

3

= 3N(N−3)/2 2N

N

• k=1

(k − 1)!(3k)! k((2k − 1)!)2 = 1, 5, 126, . . . , where AVSASM

x

are the x-enumeration of the vertically symmetric alternating sign matrices (Kuperberg 2002).

Thermodynamic limit

ZN (λ, µ) = e−2N2F(λ,µ)+O(N), where F(λ, µ) is the bulk free energy and unit temperature. We suppose the following ansatz for the large size behaviour of the determinant τN(λ, µ), τN(λ, µ) = CN e2N2f (λ,µ)+O(N), where e−2F(λ,µ) = sin(γ − (λ − µ)) sin(γ + λ − µ) sin(γ − (λ + µ)) sin(γ + λ + µ)

• − sin(2λ) sin(2µ)

e2f (λ,µ),

Liouville equation

Substituting the ansatz in the Toda equation (1), we obtain −2∂2

µλf (λ, µ) = e4f (λ,µ),

which is the Liouville equation, whose general solution has the form of e2f (λ,µ) =

• −u′ (λ)v′ (µ)

(u(λ) + v(µ)) , for arbitrary C2 functions u(λ), v(µ).

Solution

Our strategy is to chose e2f (λ,µ) to match with the solution at γ = π/4. This leave us a γ dependent parameter to be determined. However the λ, µ dependence was already determined. e2f (λ,µ) = α

• − sin(αλ) sin(αµ)

cos(αλ) + cos(αµ) = α

• − sin(αλ) sin(αµ)

2 cos( α

2 (λ − µ)) cos( α 2 (λ + µ))

(1) where the parameter α = α(γ) and α(π/4) = 4.

Solution (GAPR, VE Korepin, 2015)

We must use the boundary condition given by µ = ±(λ + γ) to determine α parameter. In doing so we see the

• nly possible choice for the parameter is α(γ) = π/γ.

e−2F(λ,µ) = π sin(γ − λ + µ) sin(γ + λ − µ) sin(γ − λ − µ) sin(γ + λ + µ) 2γ

• − sin(2λ) sin(2µ)
• − sin( πλ

γ ) sin( πµ γ )

cos( π(λ−µ)

2γ

) cos( π(λ+µ)

2γ

) . (2)

◮

The other points µ = ±(λ − γ) are naturally fulfilled.

◮

As an independent check, the solution obtained also reproduces the special points γ = π/3, π/4, π/6.

Ferrolectric phase: ∆ > 1

In the case ∆ > 1, one can obtain the expression for the free energy looking at the leading order state. The expression for the free energy can be written as e−2F(λ,µ) = sinh(λ − |µ| + |γ|)

• sinh(λ + |µ| − γ) sinh(λ + |µ| + γ).

However due to the lack of additional boundary condition, we are unable to fix the suitable solution of Liouville equation. γ > 0

✏ ✑ ✏ ✑ ✏ ✑ ✲ ✲ ✲ ✲ ✲ ✲ ❄ ❄ ❄ ✻ ✻ ✻ ✛ ✛ ✛ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✛ ✲ ✲ ✛ ✛ ✲ ✲ ❄ ❄ ✻ ❄ ❄ ✻ ❄ ✻ ✻ ❄ ✻ ✻ ✻ ✻ ✻

λ λ λ λ λ λ µ µ µ γ < 0

✏ ✑ ✏ ✑ ✏ ✑ ✲ ✲ ✲ ✲ ✲ ✲ ❄ ❄ ❄ ✻ ✻ ✻ ✛ ✛ ✛ ✲ ✲ ✲ ✛ ✛ ✲ ✲ ✲ ✛ ✲ ✲ ✲ ✲ ✲ ✲ ✻ ❄ ❄ ✻ ❄ ❄ ✻ ✻ ❄ ✻ ✻ ❄ ✻ ✻ ✻

λ λ λ λ λ λ µ µ µ

Entropy

The number of alternating sign matrix (ASM) is given by ZDWBC

N

(λ − µ = π 3 ; γ = π 3 ) = AASM

N

=

N−1

• k=0

(3k + 1)! (N + k)! = 1, 2, 7, 42, 429, · · · . (3) Taking the large limit we obtain the entropy of the six-vertex model with domain-wall boundary SDWBC = 1 2 ln

• 33

24

• .

(4) The six-vertex model with reflecting end (Tsuchiya partition function) is related to the number of vertically symmetric alternating sign matrices (VSASM) ZN (0, 0; π 3 ) = AVSASM

1

=

N−1

• k=0

(3k + 2) (6k + 3)!(2k + 1)! (4k + 2)!(4k + 3)! = 1, 3, 26, 646, . . . (5)

◮

Taking the large limit (N → ∞) we again obtain the same value for the entropy, which means STSUCHIYA = SDWBC .

200 400 600 800 1000 N 0.245 0.250 0.255 0.260 Z

Z_Tsuchiya Z_DWBC S12 ln3^32^4

Entropy - other boundary conditions

◮

Ferroelectric boundary (Wu 1973) ZFE = 1, (6)

✲ ✲ ✲ ✲ ✻ ✻ ✻ ✻ ✲ ✲ ✲ ✲ ✻ ✻ ✻ ✻ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻

λ4 λ3 λ2 λ1 µ1 µ2 µ3 µ4 SFE = 0

Entropy - other boundary conditions

◮

DWBC Descendent (TS Tavares, GAPR, VE Korepin 2015)

✲ ✲ ✲ ✛ ✛ ✛ ❄ ❄ ❄ ✻ ✻ ✻

s3 ¯ s3 s1 s1 s4 s4 s2 ¯ s2 λ5 λ4 λ3 λ2 λ1 µ1 µ2 µ3 µ4 µ5 where si =↑, ↓ or →, ← and ¯ si is its reverse. S = SDWBC (likewise the case of reflecting end presented before.)

Entropy - other boundary conditions

◮

Fusion of FE and DWBC (TS Tavares, GAPR, VE Korepin 2015)

✲ ✲ ✲ ✛ ✛ ✛ ✛ ✛ ❄ ❄ ❄ ✻ ✻ ✻ ✻ ✻ ❄

n

✲

n

✛ ✛ ✻ ✻ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ❄ ❄ ❄ ❄ ❄ ❄ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻

λ5 λ4 λ3 λ2 λ1 µ1 µ2 µ3 µ4 µ5 ZfDWBC

N

=

• ij∈ n×n

b(λi − µj ) × ZDWBC

n

. Therefore, we see that the entropy at infinity temperature is given by SfDWBC = lim

N→∞

n N 2 SDWBC . (7) SFE ≤ S ≤ SDWBC

Entropy - other boundary conditions

◮

N´ eel boundary (TS Tavares, GAPR, VE Korepin 2015)

✛ ✛ ✲ ✲ ✛ ✛ ✲ ✲ ❄ ❄ ✻ ✻ ❄ ❄ ✻ ✻

λ4 λ3 λ2 λ1 µ1 µ2 µ3 µ4

0.1 0.2 0.3 0.4 0.5 2 4 6 8 10 12 14 16 18 20 S N NE PBC 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 (SPBC-SNE)/SPBC 1/N

ZNE

N

({λ}, {µ}) = ↑↓ . . . ↑↓| D(λN)A(λN−1) · · · D(λ2)A(λ1) |↑↓ . . . ↑↓ , SNE = SPBC (1 − γ N ), where γ ∼ 2.

Entropy x boundary: six-vertex model

We believe that the entropy varies continuously in the interval, SFE ≤ S ≤ SPBC, but we still need to compute the entropy in the SDWBC < S < SPBC.

Concluding remarks

◮ We determined the free-energy in the disordered phase (|∆| < 1). ◮ The leading ferroelectric state was identified. ◮ The entropy at a = b = c = 1 of the six-vertex model with

reflecting end was found to be the same as DWBC.

◮ What is the free-energy in the antiferroelectric phase? ◮ Is there any limiting shape curve in the case of reflecting end

boundary? Further question:

◮ Are there any limiting shape curves in the case of other fixed

boundaries (FE/DWBC, N´ eel,...)

References

◮

H.J. Brascamp, H. Kunz, F.Y. Wu, J. Math. Phys. 14 (1973) 1927.

◮

R J Baxter Exactly solved models in statistical mechanics (1982).

◮

V E Korepin, N M Bogoliubov, and A G Izergin Quantum inverse scattering method and correlation functions (1993).

◮

O Tsuchiya, J. Math. Phys., 39 (1998), 5946.

◮

AG Izergin, DA Coker and VE Korepin, J. Phys. A: Math. Gen. 25 (1992) 4315.

◮

V Korepin and P Zinn-Justin JPA 33, 7053 (2000).

◮

G Kuperberg, Ann. of Math. 156 (2002) 835.

◮

P Zinn-Justin PRE 62, 3411 (2000).

◮

P.M. Bleher, V.V. Fokin, Comm. Math. Phys., 268 (2006) 223; P.M. Bleher, K. Liechty, Comm. Math.

• Phys. 286 (2009) 777; P.M. Bleher, K. Liechty, J. Stat. Phys. 134 (2009) 463; P.M. Bleher, K. Liechty,
• Comm. on Pure and Appl. Math., 63 (2010) 779.

◮

V Korepin, CMP, vol 86, page 361 (1982).

◮

GAP Ribeiro and VE Korepin, J. Phys. A: Math. Theor. 48 (2015) 045205.

◮

TS Tavares, GAP Ribeiro and VE Korepin, to appear in J.Stat.Mech (arXiv:1501.02818 [cond-mat.stat-mech]). Acknowledgments C.N. Yang Institute for Theoretical Physics at Stony Brook for hospitality and the Brazilian agency FAPESP for financial support.