Solutions of the boundary Yang-Baxter equation for ADE models

32 Pages·0.28 MB·English

by  Roger E. Behrend

7 Solutions of the Boundary Yang-Baxter Equation for
9
9 A D E
– – Models
1

n
a
J Roger E. Behrend1 and Paul A. Pearce2

5
Department of Mathematics, University of Melbourne,
1
Parkville, Victoria 3052, Australia

]
h
c
Abstract
e
m
We present the general diagonal and, in some cases, non-diagonal solutions of the
-
t boundaryYang-Baxter equation for anumberof related interaction-round-a-face mod-
a
st els, including the standard and dilute AL, DL and E6,7,8 models.
.
t
a
m 1 . Introduction
-
d
n A two-dimensional lattice spin model in statistical mechanics can be considered as solvable
o
with periodic boundary conditions if its bulk Boltzmann weights satisfy the Yang-Baxter
c
[ equation [1], and as additionally solvable with certain non-periodic boundary conditions if

it admits boundary weights which satisfy the boundary Yang-Baxter equation [2].
2
v Many such models are now known. Restricting our attention to interaction-round-a-
3 (1)
face models, these are the A or eight-vertex solid-on-solid model [3], the A or cyclic
1 ∞ L
2 solid-on-solid models [4], the A or Andrews-Baxter-Forrester models [5, 6, 7], the dilute
L
1
A models [8], and certain higher-rank models associated with A(1), B(1), C(1), D(1) and
1 L n n n n
6 A(2) [9]. Here, we present generalformsoftheboundaryweights forsomeofthese previously-
n
9
considered models and for some additional, related models.
/
t We begin, in Section 2, by outlining the standard relations, including the Yang-Baxter
a
m equation and the boundary Yang-Baxter equation, which may be satisfied by the bulk
- and boundary weights of an interaction-round-a-face model, and we define two important
d
n types of boundary weight, diagonal and non-diagonal. In Section 3, we consider certain
o
intertwining properties which may be satisfied by the bulk and boundary weights of two
c
: appropriately-related interaction-round-a-face models. In Sections 4–9, we obtain boundary
v
i weights, mostly of the diagonal type, which represent general solutions of the boundary
X
Yang-Baxter equation for various models. These models, some of their relationships and the
r
a sections in which they are considered are indicated in Figure 1. We conclude, in Section 10,
with a discussion of general techniques for solving the boundary Yang-Baxter equation.
1E-mail: [email protected]
2E-mail: [email protected]
1
4 A Model restriction- 5 A Models 6 D Models
∞ L L
§ § §
criticality criticality criticality
? ? ?
Critical A Model rest’n- Critical A Models Critical D Models E Models
∞ L L 6,7,8
6 6 6
graph =AL graph =DL graph =E6,7,8
7 Temperley-Lieb Models
§
dilution
?
8 Dilute A Models
L
§
criticality
?
Critical Dilute A Models Dilute D Models Dilute E Models
L L 6,7,8
6 6 6
graph =AL graph =DL graph =E6,7,8
9 Dilute Temperley-Lieb Models
§
Figure 1. Models considered in this paper
2 . Interaction-Round-a-Face Models
An interaction-round-a-face model is generally associated with an adjacency graph whose
G
nodes correspond to the model’s spin values and whose bonds specify pairs of spin values
which are allowed on neighbouring lattice sites.
The adjacency matrix A associated with is defined by
G
A = number of bonds of which connect a to b
ab
G
for any spin values a and b. Here, we consider cases in which contains only bidirectional,
G
single bonds, implying that A is symmetric and that each of its entries is either 0 or 1.
However the formalism can be generalised straightforwardly to accommodate directional or
multiple bonds.
For interaction-round-a-face models with non-periodic boundary conditions, we asso-
ciate a bulk weight W with each set of spin values a, b, c, d satisfying A A A A = 1
ab cb dc da
and a boundary weight B with each set of spin values a, b, c satisfying A A = 1 [10, 6].
ba bc
These weights are assumed to depend on a complex spectral parameter u and are denoted
d c
d c
W u = u (2.1)
a b ! ց
a b
2
and
a
(cid:0)
a
B b u = b(cid:0)u . (2.2)
c @
(cid:12) !
(cid:12) @
(cid:12) c
(cid:12)
Relations which may be satisfied by the(cid:12)se weights are:
The Yang-Baxter equation,
•
f g g d f e
W u v W u W v =
g a b − ! b c ! g d !
X
AfgAgbAgd=1
a g f e e d
W v W u W u v (2.3)
g b c ! a g ! g c − !
X
AagAegAgc=1
e d
(cid:0)@ (cid:0)@
f..(cid:0)@ v (cid:0)@@d (cid:0)e(cid:0)@u−v(cid:0)@..c
.. @g→•(cid:0) u @c = f(cid:0) u @→•(cid:0)g ..
.. (cid:0)@ (cid:0) @ (cid:0)@ ..
.(cid:0) @→(cid:0) @→(cid:0) @.
f@u−v(cid:0)b a@ v (cid:0)c
@→(cid:0) @→(cid:0)
a b
for all u and v and all a, b, c, d, e, f satisfying A A A A A A = 1.
ab bc dc ed fe fa
The boundary Yang-Baxter equation,
•
c f g c b a
W u v B f u W u+v B b v = (2.4)
d e − ! e (cid:12) ! f g ! g (cid:12) !
Xfg (cid:12) (cid:12)
AcfAfeAfgAbg=1 (cid:12)(cid:12) (cid:12)(cid:12)
(cid:12) (cid:12)
g c f a c b
B d v W u+v B f u W u v
e (cid:12) ! d g ! g (cid:12) ! f a − !
Xfg (cid:12) (cid:12)
AcfAfaAfgAdg=1 (cid:12)(cid:12) (cid:12)(cid:12)
(cid:12) (cid:12)
a b
(cid:0) (cid:0)@
b(cid:0)v c.(cid:0)u−v@a
(cid:0)@ .@ (cid:0)
c..@(cid:0)u+v@(cid:0)•g = .... (cid:0)@f→•(cid:0)@u
.... @(cid:0)f→•@(cid:0)u c.(cid:0)@u+v@(cid:0)•g
.(cid:0) @ @→(cid:0)
c u−v e v
@ (cid:0) d@
@→(cid:0) @
e
d
for all u and v and all a, b, c, d, e satisfying A A A A = 1.
ba cb cd de
The initial condition,
•
d c
W 0 = δ (2.5)
a b ac
!
3
for all a, b, c, d satisfying A A A A = 1.
ab cb dc da
The boundary initial condition,
•
a
B b 0 = χ δ (2.6)
c a ac
(cid:12) !
(cid:12)
(cid:12)
(cid:12)
for all a, b, c satisfying Aba Abc = 1, where(cid:12)χa are fixed, model-dependent factors.
Invariance of the bulk weights under a symmetry transformation Z of the graph ,
• G
d c Z(d) Z(c)
W u = W u (2.7)
a b Z(a) Z(b)
! !
for all u and all a, b, c, d satisfying A A A A = 1.
ab cb dc da
Invariance of the boundary weights under a symmetry transformation Z of the graph ,
• G
a Z(a)
B b u = B Z(b) u (2.8)
c Z(c)
(cid:12) ! (cid:12) !
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
for all u and all a, b, c satisfying Aba(cid:12)Abc = 1. (cid:12)
Reflection symmetry of the bulk weights,
•
d c b c d a
W u = W u = W u (2.9)
a b a d c b
! ! !
for all u and all a, b, c, d satisfying A A A A = 1.
ab cb dc da
Reflection symmetry of the boundary weights,
•
a c
B b u = B b u (2.10)
c a
(cid:12) ! (cid:12) !
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
for all u and all a, b, c satisfying Aba Ab(cid:12)c = 1. (cid:12)
Crossing symmetry,
•
d c S S 1/2 a b
a c
W u = W λ u (2.11)
a b ! (cid:18)SbSd(cid:19) d c − !
for all u and all a, b, c, d satisfying A A A A = 1, where the crossing parameter λ and
ab cb dc da
crossing factors S are fixed for a particular model.
a
Boundary crossing symmetry,
•
a S2 1/2 b a a
η (u) B b u = d W 2u B d λ u (2.12)
a c (cid:12) ! SaSc! c d ! c (cid:12) − !
(cid:12) Xd (cid:12)
(cid:12)(cid:12) AdaAdc=1 (cid:12)(cid:12)
(cid:12) (cid:12)
a a....... a
(cid:0) S2 1/2 (cid:0)@ (cid:0)
ηa(u) b(cid:0)u = d b(cid:0) 2u @•(cid:0)λ−u
@ SaSc! @ (cid:0)d@
@ @→(cid:0)......@.
c c c
4
for all u and all a, b, c satisfying A A = 1, where λ and S are the same as in (2.11) and
ba bc a
η are fixed, model-dependent functions.
a
The inversion relation,
•
d e d c
W u W u = ρ(u)ρ( u)δ (2.13)
e a b ! e b − ! − ac
AdeXAeb=1
c
(cid:0)@
(cid:0) @
d..@−u (cid:0)..b
.... (cid:0)@e→•(cid:0)@ .... = ρ(u)ρ(−u)δac
.(cid:0) @.
d u b
@ (cid:0)
@→(cid:0)
a
for all u and all a, b, c, d satisfying A A A A = 1, where ρ is a fixed, model-dependent
ab cb dc da
function.
The boundary inversion relation,
•
d a
B b u B b u = ρˆ (u)δ (2.14)
c (cid:12) ! d (cid:12)− ! a ac
Xd (cid:12) (cid:12)
Abd=1 (cid:12)(cid:12) (cid:12)(cid:12)
(cid:12) (cid:12)
a
(cid:0)
(cid:0)
b.−u
.@
.... @(cid:0)•d = ρˆa(u)δac
.(cid:0)
b u
@
@
c
for all u and all a, b, c satisfying A A = 1, where ρˆ are fixed, model-dependent functions.
ba bc a
These relationsarenotallindependent. Forexample, itcanbeshownthattheboundary
inversion relation is a consequence of the initial condition and the boundary Yang-Baxter
equation, and that boundary crossing symmetry is a consequence of the initial condition,
bulk crossing symmetry and the boundary Yang-Baxter equation.
We note that the boundary Yang-Baxter equation used here differs from the left and
right reflection equations introduced in [6]. However if crossing symmetry is satisfied then
the equations are effectively equivalent.
The bulk and boundary weights can be used to define matrices whose rows and columns
are labelled by elements of (a ,...,a ) A A ... A = 1 , the set of paths of
{ 0 N+1 | a0a1 a1a2 aNaN+1 }
length N+1. The entries of the jth bulk face transfer matrix, for 1 j N, are defined by
≤ ≤
j−1 a b N+1
X (u) = δ W j−1 j u δ (2.15)
j (a0...aN+1),(b0...bN+1) akbk a a akbk
k=0 j j+1 ! k=j+1
Y Y
5
and the entries of the boundary face transfer matrix are defined by
N b
K(u) = δ B a N+1 u . (2.16)
(a0...aN+1),(b0...bN+1) kY=0 akbk N aN+1 (cid:12)(cid:12) !
(cid:12)
(cid:12)
In terms of these matrices, the Yang-Baxter equation is (cid:12)
X (u v) X (u) X (v) = X (v) X (u) X (u v) (2.17)
j j+1 j j+1 j j+1
− −
and the boundary Yang-Baxter equation is
X (u v) K(u) X (u+v) K(v) = K(v) X (u+v) K(u) X (u v) . (2.18)
N − N N N −
In the study of exactly solvable interaction-round-a-face models with non-periodic boundary
conditions, we generally begin with bulk weights which satisfy the Yang-Baxter equation
and then attempt to solve the boundary Yang-Baxter equation for corresponding boundary
weights. Two classes of solution which are of particular interest, since they are needed for
fixed and free boundary conditions respectively, are diagonal solutions, for which
a
B b u = 0 whenever a = c (2.19)
c (cid:12) ! 6
(cid:12)
(cid:12)
(cid:12)
and non-diagonal solutions, for whic(cid:12)h
a
B b u = 0 for all a, b, c . (2.20)
c (cid:12) ! 6
(cid:12)
(cid:12)
(cid:12)
We note that if a diagonal solution s(cid:12)atisfies the boundary initial condition, then χ is given
by
a
χ = B b 0 (2.21)
a a
(cid:12) !
(cid:12)
(cid:12)
for any b satisfying A = 1, and if a diagonal s(cid:12)olution satisfies the boundary inversion
ab (cid:12)
relation, then ρˆ is given by
a a
ρˆ (u) = B b u B b u (2.22)
a a (cid:12)− ! a(cid:12) !
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
for any b satisfying A = 1. (cid:12) (cid:12)
ab
3 . Intertwiners
In this section we outline the formalism of intertwiners [11, 12, 13, 14]. We are considering
two interaction-round-a-face models, with respective finite adjacency graphs and ¯, adja-
cency matrices A and A¯, spin values a,b,... and a¯,¯b,..., crossing factors SGand SG¯ , bulk
a a¯
weights W and W, and boundary weights B and B¯.
6
We now assume that there exists an intertwiner graph ′ each of whose bonds connects
G
a node of to a node of ¯, and whose adjacency matrix C, defined by
G G
C = number of bonds of ′ which connect a to a¯
aa¯
G
for any spin values a and a¯, satisfies the intertwining relation
AC = CA¯ . (3.1)
Here, we consider cases in which ′ contains only single bonds, so that each entry of C is
G
either 0 or 1.
¯
We now associate an interwiner cell I with each set of spin values a, b, a¯, b satisfying
A A¯ C C = 1,
ba ¯ba¯ aa¯ b¯b
¯ b ¯b
b b
I = . (3.2)
a a¯! ց
a a¯
Relations which may be satisfied by the intertwiner cells are:
The first intertwiner inversion relation,
•
a c¯
AaXbCbb¯b=1I ab a¯¯b! I ab ¯bc¯! = aa¯....ց...•ց¯b.b......¯b = δa¯c¯ (3.3)
for all a, a¯, ¯b, c¯satisfying A¯ A¯ C C = 1.
a¯¯b c¯¯b aa¯ ac¯
The second intertwiner inversion relation,
•
c¯ ¯b
AbaXCbb¯b=1 SS¯¯¯ba¯SSab I ab ¯cb¯! I ab a¯¯b! = SS¯¯¯ab¯SSab a.տ.....ba.•ց.......¯ab¯ = δa¯c¯ (3.4)
for all a, a¯, ¯b, c¯satisfying A¯ A¯ C C = 1.
¯ba¯ ¯bc¯ aa¯ ac¯
The bulk weight intertwining relation,
•
¯
a d a c¯ d b
W u I I =
¯
b c d b c a¯
! ! !
Xd
AadAdcCd¯b=1
¯ ¯
a c¯ b d c¯ b
I I W u (3.5)
¯ ¯
b d c a¯ d a¯
! ! !
Xd¯
A¯c¯d¯A¯d¯a¯Cbd¯=1
c¯ ¯b a¯ ¯b
(cid:0)@
ր ր (cid:0) @
a.. d(cid:0)•@ ..c = c¯..@ u (cid:0)..a¯
.(cid:0) @. . d¯@→(cid:0) .
a u c c¯ • a¯
@ (cid:0)
@→(cid:0) ր ր
a c
b b
7
for all u and all a, b, c, a¯, ¯b, c¯satisfying A A A¯ A¯ C C = 1.
ab bc c¯¯b ¯ba¯ ac¯ ca¯
The boundary weight intertwining relation,
•
¯ ¯
c a b a b a¯
B a u I = I B¯ ¯b u (3.6)
d c a¯ b c¯ c¯
AacXCcca¯=1 (cid:12)(cid:12)(cid:12)(cid:12) ! ! A¯¯bc¯XCc¯bc¯=1 ! (cid:12)(cid:12)(cid:12)(cid:12) !
(cid:12) (cid:12)
¯b a¯ a¯
(cid:0)
ր (cid:0)
a.. (cid:0)•c = ¯b..@u
.(cid:0) . @
a u ¯b •c¯
@
@ ր
b a b
for all u and all a, b, a¯, ¯b satisfying A A¯ C = 1.
ab ¯ba¯ a¯b
The bulk and boundary weight intertwining relations can be combined with the inter-
twiner inversion relations to give expressions for the weights of one model in terms of those
of the other model. For example, assuming the first intertwiner inversion relation, we find
that the bulk weight intertwining relation is equivalent to
¯
d c¯
W u δ = (3.7)
a¯ ¯b d¯d¯′
!
d d¯ a a¯ c c¯ d d¯′ d c
I I I I W u
¯ ¯
a a¯ b b b b c c¯ a b
! ! ! ! !
Xabc
AabAcbAdcAdaCaa¯Cb¯bCcc¯=1
d¯′ c¯ ¯b
..
.
.
(cid:0)c¯@ d.. (cid:0)ր•@cր•..b ...
d¯(cid:0)@ u (cid:0)@¯b δd¯d¯′ = d...(cid:0)@ u (cid:0)@•...b ....
@→(cid:0) d. @→•(cid:0)a •.b ..
a¯ ց ց .
.
..
d¯ a¯ ¯b
for all u and all d, a¯, ¯b, c¯, d¯, d¯′ satisfying A¯ A¯ A¯ A¯ C C = 1. Similarly, assuming
a¯¯b c¯¯b d¯′c¯ d¯a¯ dd¯ dd¯′
the second intertwiner inversion relation, we find that the bulk weight intertwining relation
is equivalent to
¯
d c¯
W u δ = (3.8)
a¯ ¯b ¯b¯b′
!
S¯¯bSd I d d¯ I a a¯ I c c¯ I d d¯ W d c u
Xacd S¯d¯Sb a a¯! b ¯b! b ¯b′! c c¯! a b !
AabAcbAdcAdaCaa¯Ccc¯Cdd¯=1
8
d¯ c¯ ¯b′
..
.
.
d¯(cid:0)@(cid:0)@→uc¯(cid:0)@(cid:0)@¯b δ¯b¯b′ = SS¯¯¯bd¯SSdb ......... ddd•••......@(cid:0)@(cid:0)ր→u••@(cid:0)ac@(cid:0)ր......bbb
a¯ . ց ց
.
..
d¯ a¯ ¯b
for all u and all b, a¯, ¯b, ¯b′, c¯, d¯satisfying A¯ A¯ A¯ A¯ C C = 1, and that the boundary
a¯¯b c¯¯b′ d¯c¯ d¯a¯ b¯b b¯b′
weight intertwining relation is equivalent to
a¯ S S¯ b ¯b b ¯b a
B¯ ¯b u = b c¯ I I B b u (3.9)
c¯(cid:12)(cid:12) ! Xab ScS¯¯b a a¯! c c¯! c (cid:12)(cid:12) !
(cid:12)(cid:12) AbaAbcCaa¯Cb¯b=1 (cid:12)(cid:12)
(cid:12) (cid:12)
a¯ a¯ a•...•a
¯b@(cid:0)(cid:0)u = SSbSS¯¯c¯ ¯b տ•@(cid:0)b (cid:0)u
@ c ¯b ւ ..@.
c¯ c¯ c c
for all u and all c, a¯, ¯b, c¯ satisfying A¯ A¯ C = 1. We note that the right sides of (3.7),
¯ba¯ ¯bc¯ cc¯
(3.8) and (3.9) must be independent of d, b and c respectively.
The importance of the intertwiner relations (3.3)–(3.6) is that they imply that if the
Yang-Baxter equation and boundary Yang-Baxter equation are satisfied by the weights of
one model, then they are also satisfied by the weights of the other model. For if, on each side
of the Yang-Baxter equation for the weights of one model, we introduce three intertwiner
cells, apply (3.5) three times successively, introduce a further three intertwiner cells, and
apply (3.3) three times, then we obtain the Yang-Baxter equation for the weights of the
other model. Similarly, if, on each side of the boundary Yang-Baxter equation for the
weights of one model, we introduce two intertwiner cells, apply (3.5) twice and (3.6) twice,
introduce a further two intertwiner cells, andapply (3.4) twice, then we obtain the boundary
Yang-Baxter equation for the weights of the other model.
9
A
4 . Model
∞
4.1 Bulk Weights
Throughout this and subsequent sections, we shall use the elliptic theta functions,
∞
ϑ (u,q) = 2q1/4sinu 1 2q2ncos2u+q4n 1 q2n
1
− −
nY=1(cid:16) (cid:17)(cid:16) (cid:17)
∞
ϑ (u,q) = 2q1/4cosu 1+2q2ncos2u+q4n 1 q2n
2
−
nY=1(cid:16) (cid:17)(cid:16) (cid:17) (4.1)
∞
ϑ (u,q) = 1+2q2n−1cos2u+q4n−2 1 q2n
3
−
nY=1(cid:16) (cid:17)(cid:16) (cid:17)
∞
ϑ (u,q) = 1 2q2n−1cos2u+q4n−2 1 q2n .
4
− −
nY=1(cid:16) (cid:17)(cid:16) (cid:17)
Since the nome q is generally fixed, we shall abbreviate
ϑ (u) = ϑ (u,q) . (4.2)
i i
We now consider the A or eight-vertex solid-on-solid model [15]. The spins in this model
∞
take values from the set of all integers and the adjacency graph is
(4.3)
A∞ = • • • • • .
−2 −1 0 1 2
The bulk weights are
a 1 a ϑ (λ u)
1
W ± u = −
a a∓1 ! ϑ1(λ)
a a 1 ϑ ((a 1)λ+w )ϑ ((a+1)λ+w ) 1/2 ϑ (u)
1 0 1 0 1
W ± u = − (4.4)
a∓1 a ! ϑ1(aλ)2 ! ϑ1(λ)
a a 1 ϑ (aλ+w u)
1 0
W ± u = ±
a±1 a ! ϑ1(aλ+w0)
where λ and w are arbitrary.
0
These weights satisfy the initial condition, reflection symmetry, crossing symmetry with
crossing parameter λ and crossing factors
S = ϑ (aλ+w ) , (4.5)
a 1 0
the inversion relation with
ϑ (λ u)
1
ρ(u) = − , (4.6)
ϑ (λ)
1
and the Yang-Baxter equation.
The bulk weights for the critical A model are obtained by taking q 0 so that all of
∞
→
the ϑ functions in (4.4) become sin functions.
1
10