Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Six vertex model:
Exact results and open problems
Giuliano Ribeiro and Vladimir Korepin
C.N. Yang Institute for Theoretical Physics
Stony Brook University
2014
V. E. Korepin
Six vertex model with DWBC
1/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model
Phases
Critical exponents
Problem
In statistical physics people believe that in thermodynamic limit the
bulk free energy and correlations should not depend on boundary
conditions. This is often true, but there are counterexamples.
V. E. Korepin
Six vertex model with DWBC
2/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model
Phases
Critical exponents
Six vertex model
Statistical mechanics (classical) model on a square lattice.
Ice model has atoms on vertices and hydrogen bonds on edges
[Pauling J Am Chem Soc 1935, Slater J Chem Phys 1941].
Allowed configurations follow arrow conservation (ice rule).
V. E. Korepin
Six vertex model with DWBC
3/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model
Phases
Critical exponents
phase
Boltzmann weights (zero-field model)
a = exp(‒ε1/T)
b = exp(‒ε2/T)
c = exp(‒ε3/T)
FE
a = sinh(t‒γ)
b = sinh(t+γ)
c = sinh(2γ)
DIS
a = sin(μ‒ω)
b = sin(μ+ω)
c = sin(2μ)
AFE
a = sinh(λ‒υ)
b = sinh(λ+υ)
c = sinh(2λ)
V. E. Korepin
Six vertex model with DWBC
4/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model
Phases
Critical exponents
Periodic boundary conditions (torus)
We consider here an N × N lattice.
V. E. Korepin
Six vertex model with DWBC
5/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model
Phases
Critical exponents
Partition function
Z=
X
Y
W,
W ∈ {a, b, c}
arrow vertices
config.
2
Bulk free energy f = −T ln Z 1/N , as N → ∞.
Calculated by Lieb [PRL, 1967], Sutherland [PRL, 1967].
Baxter [Exactly solved models in statistical mechanics, 1982].
V. E. Korepin
Six vertex model with DWBC
6/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model
Phases
Critical exponents
Phases
Control parameter is ∆ =
a2 + b2 − c2
= − cos 2µ = − cosh 2λ.
2ab
Free energy has different analytic forms when
∆ > 1 (ferroelectric).
−1 < ∆ < 1 (disordered).
∆ < −1 (anti-ferroelectric).
V. E. Korepin
Six vertex model with DWBC
7/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model
Phases
Critical exponents
Phase diagram
b/c
II
III
1
I
IV
1
a/c
Phase I (ferroelectric).
Phase II (ferroelectric).
Phase III (disordered).
Phase IV (anti-ferroelectric).
V. E. Korepin
Six vertex model with DWBC
8/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model
Phases
Critical exponents
Phases I and II (ferroelectric)
∆ > 1.
Phase I:
Phase II:
f = ‒T ln a = ε1
f = ‒T ln b = ε2
No entropy in the ground state. Correlations are classical.
V. E. Korepin
Six vertex model with DWBC
9/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model
Phases
Critical exponents
Phase III (disordered)
a, b, c < 12 (a + b + c) or −1 < ∆ < 1.
f = ε1 − T
Z
∞
−∞
sinh[2(µ + ω)x] sinh[(π − 2µ)x]
dx.
2x sinh(πx) cosh(2µx)
Includes infinite temperature case (a = b = c = 1) where
2
Z 1/N = (4/3)3/2 ,
N → ∞ [Lieb PRL, 1967].
Entropy of ground state.
Correlation decay as power law. Conformal field theory can be used
to describe asymptotic of correlations.
V. E. Korepin
Six vertex model with DWBC
10/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model
Phases
Critical exponents
Phase IV (anti-ferroelectric)
c > a + b or ∆ < −1.
∞
X
e−2mλ sinh[2m(λ + v)]
f = ε1 − T (λ + v) +
.
m cosh(2mλ)
m=1
V. E. Korepin
Six vertex model with DWBC
11/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model
Phases
Critical exponents
Ferroelectric I and disordered phase III
Critical temperature Tc occurs at b + c − a = 0.
With (b + c − a)/a ∝ T − Tc near critical line:
Phase transitions and critical exponents are described in detail in
Baxter’s book.
V. E. Korepin
Six vertex model with DWBC
12/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Other boundary conditions
Free boundaries.
[Owczarek and Baxter JPA, 1989]
Antiperiodic boundaries.
[Batchelor, Baxter, O’Rourke, Yung JPA, 1995]
In thermodynamic limit the bulk free energy and local correlations
are identical to periodic case.
Can the bulk free energy and correlations depend on boundary
conditions in thermodynamics limit?
V. E. Korepin
Six vertex model with DWBC
13/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Ferroelectric boundary conditions
Ferroelectric-a boundary conditions.
V. E. Korepin
Six vertex model with DWBC
14/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Ferroelectric boundary conditions
Only one arrow configuration in the bulk is compatible with these
boundary conditions.
One can prove this by induction in the lattice size.
f = −T ln a = ε1
V. E. Korepin
Six vertex model with DWBC
15/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Ferroelectric boundary conditions
b/c
FE
a/c
There are no phase transitions. Entropy is zero.
One can tune Boltzmann weights into disordered phase.
Correlations in the center of the lattice are pure classical. No
power law, no conformal filed theory. Unlike periodic
boundary conditions.
This proves that the bulk free energy can depend on boundary
conditions in thermodynamic limit.
V. E. Korepin
Six vertex model with DWBC
16/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Domain wall boundary conditions
Korepin CMP, 1982; Korepin and Zinn-Justin JPA, 2000
[cond-mat/0004250]; Zinn-Justin PRE, 2000.
V. E. Korepin
Six vertex model with DWBC
17/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Partition function (ferroelectric phase)
Derivation by Bethe ansatz.
Izergin-Korepin formula
2
ZN (t) =
[sinh(t + γ) sinh(t − γ)]N
τN (t),
QN −1 2
n!
n=0
(N × N lattice).
Hankel determinant
τN (t) = det
d
dt
i+k−2
φ(t) ,
V. E. Korepin
φ(t) =
sinh(2γ)
sinh(t + γ) sinh(t − γ)
Six vertex model with DWBC
18/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Partition function
Toda (Hirota) equation
′′
′ 2
τN τN
− (τN
) = τN +1 τN −1 ,
∀N ≥ 1,
τ0 = 1, τ1 = φ(t).
In the thermodynamic limit N → ∞, use the ansatz:
T ln ZN (t) = −N 2 f (t) + O(N ).
Bulk free energy is f (t).
V. E. Korepin
Six vertex model with DWBC
19/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Free energy (ferroelectric phase)
Define f (t)/T ≡ −g(t) − ln[sinh(t + γ) sinh(t − γ)].
Equation and solution for g(t)
g ′′ = e2g ,
eg(t) =
k
.
sinh[k(t − t0 )]
with solution parameters k and t0 .
V. E. Korepin
Six vertex model with DWBC
20/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Phase diagram
b/c
II
III
1
I
IV
1
a/c
Phase boundaries identical for PBC and DWBC.
Phases I & II (ferroelectric), III (disordered), and IV
(anti-ferroelectric).
Control parameter ∆ =
a2 + b 2 − c2
= − cos 2µ = − cosh 2λ.
2ab
V. E. Korepin
Six vertex model with DWBC
21/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Ferroelectric phases I and II
(∆ = cosh 2γ) > 1.
Bulk free energy
e−f /T = sinh(t + |γ|) = max(a, b).
Same free energy as ferroelectric phase with PBC.
This was later rigorously proven by Bleher and Liechty,
Commun. Math Phys 2009
2
ZN = (1 − e−4γ ) [sinh (t + |γ|)]N eN (γ−1) (1 + O(e−N
V. E. Korepin
Six vertex model with DWBC
1−ǫ
)), ∀ǫ > 0
22/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Disordered phase III
−1 < (∆ = − cos 2µ) < 1.
Bulk free energy
e−f /T = sin(µ − ω) sin(µ + ω)
1
π
.
2µ cos(πω/2µ)
Free energy with DWBC always greater than PBC case.
ZN
This was later rigorously proven by Bleher and Fokin,
Commun. Math Phys 2006
N 2
κ π sin(µ − ω) sin(µ + ω)
= C(µ) (N cos (πω/2µ))
(1+O(N −ǫ ))
2µ cos(πω/2µ)
where κ =
1
12
−
2µ2
3π(π−2µ)
and C(µ) > 0 is unknown.
V. E. Korepin
Six vertex model with DWBC
23/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Entropy in disordered phase
Consider infinite temperature a = b = c = 1.
Entropy with DWBC
SDWBC = 21 N 2 ln(27/16) ≈ 0.26N 2 .
Entropy with PBC
SPBC = 23 N 2 ln(4/3) ≈ 0.43N 2 .
Entropy with FE
SFE = 0
Entropy of other boundary conditions is bounded by PBC and FE:
SF E ≤ Sother ≤ SPBC
V. E. Korepin
Six vertex model with DWBC
24/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Anti-ferroelectric phase IV with domain wall boundary
conditions
(∆ = − cosh 2λ) < −1.
Bulk free energy
e−f /T = sinh(λ − v) sinh(λ + v)
π
ϑ′1 (0)
.
2λ ϑ2 (πv/2λ)
ϑn (z) are Jacobi theta functions with elliptic modulus q = e−π
2 /2λ
Free energy with DWBC different from PBC case.
Derived by Paul Zinn-Justin.
This was again rigorously proven by Bleher and Liechty,
Commun Pure Appl Math 2010
2
ZN = C(λ)[e−f /T ]N ϑ4 (N (λ + v)π/2λ)(1 + O(N −1 )),
V. E. Korepin
Six vertex model with DWBC
25/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Ferroelectric and disordered phase III
Phase transitions are in the same place
[in the space of Boltzman weights] as for periodic boundary
conditions.
Critical exponents are different.
V. E. Korepin
Six vertex model with DWBC
26/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Correlations for domain wall boundary conditions
Boundary correlation function (determinant formula, multiple
integral):
one-point: Bogoliubov,Pronko,Zvonarev, J PHYS A, 2002
two-point: Colomo, Pronko, JSTAT 2005
e.g one-point boundary correlation:
(r)
HN
−1
= ZN h⇓| B(λN ) · · · B(λr+1 )P↓ B(λr )P↑ B(λr−1 ) · · · B(λ1 ) |⇑i ,
(r)
HN
=
(N − 1)! sinh (2η)
τ
e(t)
[sinh (t + η)]r [sinh (t − η)]N −r+1 τ (t)
,
f ). M
f differs from
where τ (t) = det (Mik ) and τ
e(t) = det (M
ik
ik
i+k−2
d
φ(t) only in the first column:
dt
Mik =
fi,1 =
M
V. E. Korepin
d
dǫ
i−1 (
[sinh ǫ]N −r [sinh (ǫ − 2η)]r−1
[sinh (ǫ + t − η)]N −1
Six vertex model with DWBC
)
ǫ=0
.
27/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Correlations for domain wall boundary conditions
Emptiness formation probability EFP : Colomo, Pronko, NPB
2008
By means of orthogonal polynomials techniques, the EFP can be written as
(r,s)
FN
r
=
(−1)s(s+1)/2 Zs
s!(2πi)s as(s−1) cs
×
s
Y
j,k=1
j6=k
I
···
C0
I
s
Y
[(t2 − 2∆t)zj + 1]s−1
zjr (zj − 1)s
C0 j=1
Y
1
t2 zj zk − 2∆tzj + 1 1≤j<k≤s
(zk − zj )
2
×hN,s (z1 , . . . , zs )hs,s (u1 , . . . , us ) dz1 · · · dzs ,
s
where uj = −
zj −1
and
(t2 −2∆t)zj +1
hN,s (z1 , . . . , zs ) =
s
Y
1
z − zi
i,j=1 j
i<j
(r)
hN (z): generating function of HN
V. E. Korepin
s−i
i−1
det zk (zk − 1)
hN −s+i (zk ) ,
(hN (z) =
Six vertex model with DWBC
PN
(r) r−1
).
r=1 HN z
28/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Artic curve: spatial phase separation
This curve separates the ferroelectric and the disordered phases.
∆ = 0: Artic circle (Elkies, Kuperberg, Larsen, Propp, J Algebraic
Combin, 1992)
(2x − 1)2 + (2y − 1)2 = 1
∆ → −∞: rectangle (Zinn-Justin PRE 2000)
∆ = 12 : ellipse describes the limit shape of alternating sign matrix
(Colomo, Pronko SIAM J Discrete Math, 2010)
(2x − 1)2 + (2y − 1)2 − 4xy = 1
∆ = − 21 : algebraic equation of sixth order
6
6
5
5
4 2
2 4
3 3
5
5
324(x + y ) + 1620(x y + xy ) + 3429(x y + x y ) + 4254x y − 972(x + y )
4
4
3 2
2 3
4
4
3
3
2 2
− 1458(x y + xy ) − 2970(x y + x y ) − 6147(x + y ) − 9150(x y + xy ) − 17462x y
3
3
2
2
2
2
+ 13914(x + y ) + 24086(x y + xy ) − 11511(x + y ) − 17258xy + 4392(x + y) − 648 = 0.
V. E. Korepin
Six vertex model with DWBC
29/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Artic curve: spatial phase separation
Taking the scaling limit of EFP (N, r, s → ∞, such that x = r/N
and y = s/N ) in the saddle-point approximation: Colomo,Pronko,
J STAT PHYS 2010, Colomo,Pronko, Zinn-Justin, JSTAT 2010
−1 ≤ ∆ < 1 (disordered)
x=
y=
ϕ(ζ) =
ϕ′ (ζ + η)Ψ (ζ) − ϕ(ζ + η)Ψ ′ (ζ)
ϕ(ζ + λ)ϕ′ (ζ + η) − ϕ(ζ + η)ϕ′ (ζ + λ)
,
ϕ(ζ + λ)Ψ ′ (ζ) − ϕ′ (ζ + λ)Ψ (ζ)
ϕ(ζ + λ)ϕ′ (ζ + η) − ϕ(ζ + η)ϕ′ (ζ + λ)
sin 2η
sin (λ − η) sin (λ + η)
.
,
Ψ (ξ) = cot ξ − cot(ξ + λ + η) − α cot(αξ) + α cot α(ξ + λ − η).
where α = π/(π − 2η) and ζ ∈ [0, π − λ − η].
V. E. Korepin
Six vertex model with DWBC
30/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Artic curve: spatial phase separation
Taking the scaling limit of EFP (N, r, s → ∞, such that x = r/N
and y = s/N ) in the saddle-point approximation: Colomo,Pronko,
J STAT PHYS 2010, Colomo,Pronko, Zinn-Justin, JSTAT 2010
−∞ < ∆ < −1 (antiferroelectric)
x=
ϕ′ (ζ − η)Ψ (ζ) − ϕ(ζ − η)Ψ ′ (ζ)
ϕ(ζ + λ)ϕ′ (ζ − η) − ϕ(ζ − η)ϕ′ (ζ + λ)
′
y=
ϕ(ζ) =
,
′
ϕ(ζ + λ)Ψ (ζ) − ϕ (ζ + λ)Ψ (ζ)
ϕ(ζ + λ)ϕ′ (ζ − η) − ϕ(ζ − η)ϕ′ (ζ + λ)
sinh 2η
sinh (η − λ) sinh (η + λ)
.
,
′
′
ϑ (αζ)
ϑ (α(ζ + λ + η))
Ψ (ζ) = coth ζ − coth(ζ + λ − η) − α 1
+α 1
ϑ1 (αζ)
ϑ1 (α(ζ + λ + η))
where α = π/2η and ζ ∈ [0, η − λ].
Comment: algebraic curve in roots of unit cases; transcendent elsewhere.
V. E. Korepin
Six vertex model with DWBC
31/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Open problems
Classification of boundary conditions into universality classes:
corresponding to given entropy [in thermodynamic limit].
Prove that the bulk free energy is the same for b.c. in one
universality class.
Prove that for majority of b. c. the phase boundaries are the
same as for periodic case [in the space of Boltzman weights].
Prove that evaluation of the partition function for majority of
boundary conditions are NP hard.
Find new boundary conditions for which the model is exactly
solvable.
Describe behavior of correlations across the arctic circle. How
quantum corrections turn into classical.
V. E. Korepin
Six vertex model with DWBC
32/34
Periodic boundary conditions
Other boundary conditions
Domain wall boundary conditions
Model and partition function
Phases
Summary
Six vertex model was discovered in 1935. We learned a lot
about the model, but there open problems.
V. E. Korepin
Six vertex model with DWBC
33/34
References
1
V Korepin, CMP, vol 86, page 361 (1982).
2
V Korepin and P Zinn-Justin JPA 33, 7053 (2000);
cond-mat/0004250.
3
P Zinn-Justin PRE 62, 3411 (2000).
4
R J Baxter Exactly solved models in statistical mechanics
(AP, London, 1982).
5
V E Korepin, N M Bogoliubov, and A G Izergin Quantum
inverse scattering method and correlation functions (CUP,
Cambridge, 1993).
6
F Colomo and AG Pronko, J STAT PHYS, 138, 662 (2010).
7
F Colomo, AG Pronko and P Zinn-Justin J STAT MECH,
L03002 (2010).
V. E. Korepin
Six vertex model with DWBC
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