The dilute A$_4$ model, the E$_7$ mass spectrum and the tricritical Ising model

24 Pages·0.22 MB·English

by  K. A. Seaton

The dilute A model, the E mass spectrum and
4 7
the tricritical Ising model
∗
2
0
K. A. Seaton†
0
2 C. N. Yang Institute for Theoretical Physics,

n State University of New York, Stony Brook, NY 11794-3840, USA
a
J M. T. Batchelor

9 Department of Mathematics, School of Mathematical Sciences,
2
Australian National University, Canberra ACT 0200, Australia

2
v
1
2 Abstract
0
0 The exact perturbation approach is used to derive the (seven) ele-
1 mentarycorrelationlengthsandrelatedmassgapsofthetwo-dimensional
1 diluteA latticemodelinregime2−fromtheBetheAnsatzsolution. This
4
0 model provides a realisation of the integrable φ perturbation of the
(1,2)
h/ c= 170 conformal field theory, which is known to describe the off-critical
p thermalbehaviourofthetricriticalIsingmodel. TheE7 massespredicted
- from purely elastic scattering theory follow in theapproach tocriticality.
h
Universal amplitudes for thetricritical Ising model are calculated.
t
a
m
:
v
i
X
r
a
∗ExpandedversionofatalkpresentedattheInternational WorkshoponExactlySolvable
ModelsofStatisticalMechanicsandMathematicalPhysics,Asia-PacificCenterforTheoretical
Physics(Seoul, Korea,June26-29,2000).
†Permanentaddress: SchoolofMathematicalandStatisticalSciences,LaTrobeUniversity,
Victoria3086,Australia.
1
I INTRODUCTION
The deep relationship between conformal field theory and criticality has pro-
vided a wealth of detailed information on phase transitions and critical phe-
nomena. Moreover, perturbed conformal field theory provides a description of
the approach to criticality in certain models [1]. One of the most striking ex-
amples is the φ perturbation of the minimal unitary conformal field theory
(1,2)
which is known to describe the scaling limit of the two-dimensional Ising
3,4
M
model at T = T in a magnetic field. In particular, Zamolodchikov’s construc-
c
tionofnontriviallocalintegralsofmotionandthus anintegrablequantumfield
theory led to the remarkable prediction of eight fundamental mass ratios for
the magnetic Ising model [2]. The masses coincide with the components of the
Perron-Frobenius vector of the Cartan matrix of the Lie algebra E .
8
In another development, the exactly solvable dilute A lattice model was
3
discovered [3] and (in regime 2 of its four regimes) seen to be in the same
universalityclass as the magnetic Ising model. Most importantly the dilute A
L
model [3, 4] admits an off-critical extension in which the Boltzmann weights
are parametrisedin terms of elliptic theta functions [3]. In the dilute A model
3
the elliptic nome plays the role of magnetic field. Its hidden E structure has
8
been revealed by a number of studies [5]-[13]. The masses, obtained from the
eigenspectrum, may be summarized by the formula [11, 13]
aπ
m sin , (1)
j
∼ (cid:18) g (cid:19)
Xa
where index j labels the eight particles, g = 30 is the Coxeter number for E
8
and the set of allowed a values is given in Table I.
InadditiontothecorrespondencebetweenthediluteA modelandE ,there
3 8
aresimilarcorrespondencesbetweenthediluteA modelandE ,andthedilute
4 7
A model and E . In regime 2 these models are lattice realizations of the φ
6 6 (1,2)
perturbation of the and minimal unitary conformal field theories
4,5 6,7
M M
respectively, known to have connection to the other exceptional Lie algebras
[14]. Some E-type structures have been observed for these dilute A models
[15, 16].
Based on the results for the eigenspectrum of the dilute A model [13] and
3
general inversion relations, we proposed [17, 18] that, in the thermodynamic
limit and in the appropriate regime, the row transfer matrix eigenvalue excita-
tions
Λ (w)
j
r (w)= lim (2)
j
N→∞Λ0(w)
ofthediluteA ,A andA modelsaregivenbythefollowinggeneralexpression.
3 4 6
Proposition. The excitation spectrum of the dilute A , A and A models in
3 4 6
2
regime 2 is given by
E( x6sga/w,x12s)E( x6s(gg−a)/w,x12s)
r (w)= w − − . (3)
j Ya E( x6sgaw,x12s)E( x6s(gg−a)w,x12s)
− −
Here the elliptic nome is p = e−ǫ, w = e−2πu/ǫ, and x = e−π2/rǫ. Regime 2
is specified by the range of the spectral parameter: 0 < u < 3λ, and the value
of the crossing parameter: λ = πs/r where s = L+2 and r = 4(L+1). For
the dilute A model the E Coxeter number is g = 18, while for the A model
4 7 6
the E Coxeter number is g = 12. The standard (conjugate modulus) elliptic
6
function is defined by
∞
E(z,q)= (1 qn−1z)(1 qn/z)(1 qn).
− − −
nY=1
The numbers a appearing in (3) are given in Tables I, II and III. The integers
in these tables have appeared in other contexts in relation to the E-algebras
[19, 20].
In this paper we explicitly derive the elementary excitation spectrum of the
dilute A model, thereby confirming our Proposition in this case. The result
4
(3) leads to the inverse correlation lengths and mass gaps. Our input to these
calculations are the string solutions to the Bethe equations found by Grimm
and Nienhuis [9, 10, 21]. As discussed later in IV, our results are applicable to
the tricriticalIsing model which is in the same universality class. In particular,
the elliptic nome appearing in the dilute A weights in regime 2 corresponds
4
to the leading thermal off-critical perturbation in the tricritical Ising model.
This perturbation is identified with φ [22] and has been shown to exhibit
(1,2)
E structures [14, 23]. We are able to obtain exact results for some universal
7
amplitudes of the tricritical Ising model. These results are in agreement with
those found recently by other means [24, 25].
Theoutlineofthepaperisasfollows. ThediluteA latticemodelisdefined
L
along with the corresponding Bethe equations in II. The bulk free energy and
the eigenvalue expressions in regime 2 for L = 4 associated with the seven E
7
massesare derivedvia the exactperturbationapproachin III (continued in the
Appendix). ThepaperconcludesinIVwithadiscussionoftheresultsandtheir
relevance to universal behaviour in the tricritical Ising model.
II THE DILUTE A MODEL
4
We here give a short summary of facts about the dilute A models [26, 13]
L
which are pertinent to our calculations.
The dilute A model is an exactly solvable, L-state restricted solid-on-solid
L
model defined on the square lattice. Its adjacency diagram is the Dynkin dia-
gramofA withtheadditionalpossibilitythatastatemaybeadjacenttoitself
L
onthelattice. Themodelissolvableinfouroff-criticalregimes,withtheelliptic
3
nome p of its Boltzmann weights taking the model off-critical. At criticality,
the dilute A model can be constructed [3, 4] from the dilute O(n) loop model
L
[27, 28]. In regime 2 of the model the central charge is
6
c=1 .
− L(L+1)
In the majority of exactly solved models the elliptic nome plays the role of
temperature[29]. Inthe diluteA modelthe interpretationofthe ellipticnome
L
differsaccordingtowhetherLisevenorodd. ForLoddtheelliptic nomeplays
the role of a magnetic field [3], and p > 0 and p < 0 are related by simple
labelreversalofthe heights. ForL eventhe nome playsa thermalrole,andthe
behaviourofthe modeldepends onwhetherp>0(regime2+)orp<0 (regime
2−). Morespecifically,itwasshown[26]thatinregime2the nome corresponds
to perturbation of the minimal unitary conformalfield theories by the
L,L+1
M
operator φ .
(1,2)
Usingtheconjugatevariablesintroducedafter(3),andsettingwj =e−2πuj/ǫ,
the eigenvalues of the row transfer matrix of the dilute A models (for a peri-
odic strip of width N where for convenience N has been taken as even) can be
written [5]
E(x4s/w,x2r)E(x6s/w,x2r) N N E(x2sw/w ,x2r)
Λ(w)=ω w1−2s/r j
(cid:20) E(x4s,x2r)E(x6s,x2r) (cid:21) j E(x2sw /w,x2r)
jY=1 j
x2sE(w,x2r)E(x6s/w,x2r) N
+
(cid:20) w E(x4s,x2r)E(x6s,x2r) (cid:21)
N E(w/w ,x2r)E(x6sw /w,x2r)
j j
w
× j E(x2sw /w,x2r)E(x4sw /w,x2r)
jY=1 j j
E(w,x2r)E(x2s/w,x2r) N N E(x8sw /w,x2r)
+ ω−1 x2s w2s/r j
(cid:20) E(x4s,x2r)E(x6s,x2r) (cid:21) j E(x4sw /w,x2r)
jY=1 j
(4)
where ω = exp(iπℓ/(L+1)) for ℓ = 1,... ,L, and s = L+2 and r = 4(L+1)
in regime 2. The Bethe equations which give the N roots u have the form
j
E(x2s/w ,x2r) N
j
ω w
(cid:20) j E(x2sw ,x2r) (cid:21)
j
N E(x2sw /w ,x2r)E(x4sw /w ,x2r)
2s/r j k k j
= w . (5)
− k E(x2sw /w ,x2r)E(x4sw /w ,x2r)
kY=1 k j j k
In the limit p 1 with u/ǫ fixed, or equivalently x 0, the excitations in
| |→ →
theeigenspectrumr (w),definedin(2),breakupintoanumberofdistinctbands
j
labelled by integer powers of w. Numerical investigations of the eigenspectrum
4
[5, 9, 10, 21, 17] have revealed eight and seven thermodynamically significant
excitations for L = 3 and L = 4 respectively, and provided the data in Table
IV.
We previously [12, 13] applied the exact perturbation approachinitiated by
Baxter [30] to calculate the excitations in the eigenspectrum for L = 3. This
involved perturbing away from the strong magnetic field limit at p 1; for
→
L=4 this limit corresponds to moving far away from the critical temperature.
The calculations follow.
III MASS SPECTRUM
A Preliminaries
To apply the perturbation technique [30] to find the form of the excitations
(4), the string structure of the Bethe ansatz roots (5) is required input. The
groundstate roots all have uj pure imaginary, so that wj = e−2πuj/ǫ = aj for
j = 1,... ,N with a = 1; in this sense they all live on a unit circle. For
j
| |
each excitation i, certain roots acquire a real part mπ/20, as shown in Table
IV. (If there are n such roots, one says there is an n -string associated with
i i
the excitation.) For these roots w = b xm, so that the string entries can be
j j
thoughtofaslivingoncirclesofradiusxm withphaseb ,whiletheotherN n
j i
−
roots again lie on the unit circle.
The process of finding the excitations involves using the Bethe equations
(5) to set up recurrence relations for auxiliary functions of the unknown roots
a . As the roots only enter the eigenvalue expression (4) through the auxiliary
j
functions, it just remains to solve the recurrence relations by iteration and to
simplify the resulting expressions. The largest eigenvalue Λ , relative to which
0
excitations are measured, was calculated previously in this way [13] for all L.
The relationship between the excitations (2), the correlation lengths ξ and
j
the mass spectrum m of the associated field theory is
j
ξ−1 = logr =m , (6)
j − j j
where we take the isotropic value u=3λ/2.
It is convenient to use the notation for products:
∞
(z;p ,... ,p ) = (1 pn1 pnkz)
1 k ∞ − 1 ··· k
n1,.Y..nk=0
m
(z ,...z ;p ,... ,p ) = (z ;p ,... ,p )
1 m 1 k ∞ j 1 k ∞
jY=1
5
which satisfy many identities, the ones used repeatedly in what follows being:
(z;p)
∞
=(1 z)
(zp;p) −
∞
(z;p,q)
∞
=(z;q)
∞
(zp;p,q)
∞
(zq/p;p,q) (zq/p;q)
∞ ∞
= .
(z;p,q) (z;p)
∞ ∞
The standard elliptic function is thus re-written as
∞
E(z,q)= (1 qn−1z)(1 qn/z)(1 qn)=(z,q/z,q;q) . (7)
∞
− − −
nY=1
It also proves convenient to use the shorthand notation m a =A .
j=1 j m
Foreachmi,ifthe associatedstringofexcitedrootshQaslengthni,wedefine
the required auxiliary functions of the as-yet-unknownroots to be
N−ni
F (w)= (w/a ;x2r) ,
i j ∞
jY=1
N−ni
G (1/w)= (x2ra /w;x2r) . (8)
i j ∞
jY=1
In fact, we actually solve for combinations of these:
(w)=F (w)/F (x16w)=F (w)/F (x2r−4sw),
i i i i i
F
(1/w)=G (1/w)/G (1/x16w)=G (1/w)/G (1/x2r−4sw), (9)
i i i i i
G
for i = 2,4,5,6,7 (but for i = 1,3 slightly different definitions are convenient
and are given as required).
So far as possible, we write factors and powers which are common to all
eigenvalues (or indeed to all the eigenvalues for other A models) in terms
L
of the generic r and s to distinguish them from the particular integers which
arise from the input strings. Of course, r = 20 and s = 6 throughout. Once
the particular string form for the roots has been applied, the calculations are
straightforward for all masses except m and m . For this reason, we sketch
1 3
belowthe detailsforthefirstthreemasses. Theothercasesfollowsimilarpaths
tom orindeedtomostofthemassesforthediluteA model[13],sowerelegate
2 3
them to the appendix. We make some comments concerning m , m and m
1 3 6
later on.
B Mass m
1
We begin the perturbation argument with the structure w = a for j =
j j
1,... ,N 3 with w = b x−4, w = b x4 and w = b x20, so that
N−1 1 N−2 2 N 3
−
6
the string length is n = 3. From the Bethe equations (5) for j = N 2,
1
−
j = N 1 and j = N in the limit x 0 we can show that b = b = b = b.
1 2 3
− →
The Bethe equation for the other roots a =a is then
k
E(x2s/a) N a2
ω a =(A b3)3/5
− (cid:20) E(x2sa) (cid:21) N−3 b2
E(x4b/a)E(x24b/a)E(x28b/a)N−3E(x2sa/a )E(x4sa /a)
j j
. (10)
× E(x4a/b)E(x24a/b)E(x28a/b) E(x2sa /a)E(x4sa/a )
jY=1 j j
In the x 0 limit this gives the equation
→
1
aN−2+ (A b3)3/5/b2 =0, (11)
N−3
ω
whichisanequationoforder(N 2),sothatthereisamissingrootontheunit
−
circle, a ‘hole’, which we call a . Since this is an equation for the roots, its
N−2
left hand side must be equivalent to N−2(a a ), and equating the constant
j=1 − j
terms from these two expressions weQobtain
1
(A b3)3/5 =A b2 =A a b2, (12)
N−3 N−2 N−3 N−2
ω
(which we later apply to prefactors in Λ ). The Bethe equations for b taken
1
together in this limit, and combined with (12) give
3
1
(A b3)3/5 = b6(A )2 A (a )3 = 1.
N−3 N−3 N−3 N−2
(cid:20)ω (cid:21) − ⇒ −
We use this, together with the fact that each root a , including the hole, must
j
satisfy (11), to show
(a )N−2 = A a (a )N =1. (13)
N−2 N−3 N−2 N−2
− ⇒
We define the following auxiliary functions of the roots (see (8)):
F (w) (x4w/b;x2r)
1 ∞
(w)= ,
F1 F (x16w)(x12w/b;x2r)
1 ∞
G (1/w) (x36b/w;x2r)
1 ∞
(1/w)= .
G1 G (1/x16w)(x24b/w;x2r)
1 ∞
They must satisfy recurrence relations arising from (10)
(x2sa;x2r) N (x24a/a ,x28a/a ;x2r) (x2sa)
∞ N−2 N−2 ∞ 1
(a)= F ,
F1 (cid:20)(x2r−2sa;x2r) (cid:21) (x12a/a ,x16a/a ;x2r) (x4sa)
∞ N−2 N−2 ∞ 1
F
(x2r+2s/a;x2r) N (x36a /a,x40a /a;x2r) (x2s/a)
∞ N−2 N−2 ∞ 1
(1/a)= G .
G1 (cid:20) (x6s/a;x2r) (cid:21) (x48a /a,x52a /a;x2r) (x4s/a)
∞ N−2 N−2 ∞ 1
G
(14)
7
Solving these we obtain
(x40a/a ;x2r) (x36a/a ,x48a/a ;x12s)
N−2 ∞ N−2 N−2 ∞
(a)= (a) ,
F1 F0 (x16a/a ;x2r) (x12a/a ,x72a/a ;x12s)
N−2 ∞ N−2 N−2 ∞
(x40a /a;x2r) (x36a /a,x96a /a;x12s)
N−2 ∞ N−2 N−2 ∞
(1/a)= (1/a) . (15)
G1 G0 (x64a /a;x2r) (x60a /a,x72a /a;x12s)
N−2 ∞ N−2 N−2 ∞
Here and arise from the square bracketed factors in (14) and give rise
0 0
F G
to the square bracketed factor in (16). They are related to the groundstate
eigenvalue Λ , they are common to the calculation of each mass and we will
0
suppress these factors for m ,... ,m . We now write the eigenvalue expression
2 7
in terms of the auxiliary functions, the first term being
Λ w (x2r−6sw,x2r−4sw,x4s/w,x6s/w;x2r) N
1 ∞
=
3 −a (cid:20) (x2r−6s,x2r−4s,x4s,x6s;x2r) (cid:21)
N−2 ∞
(x28w/a ,x12a /w;x2r)
N−2 N−2 ∞ (x2sw) (1/x2sw). (16)
× (x12w/a ,x28a /w;x2r) F1 G1
N−2 N−2 ∞
Substitutingthesolutions(15)givesanexpressionfortheexcitationr (w)which
1
may be written in elliptic functions (7) as
Λ E( x12/w,x12s)E( x48w,x12s)
1
=w − − , (17)
Λ E( x12w,x12s)E( x48/w,x12s)
0
− −
where we have set a = 1. (The other two terms in the eigenvalue always
N−2
−
give identical elliptic function expressions to the first, upon simplification.)
The Bethe equations involving b and the ‘hole’ equation, which is (10) with
a=a ,canalsobeexpressedintermsoftheauxiliaryfunctions. Application
N−2
of identities and simplification gives:
E(x12b/a ,x2r−4s)=E(x12a /b,x2r−4s)
N−2 N−2
E(x12a ,x12s)E(x48/a ,x12s) N
N−2 N−2 =(a )N.
(cid:20)E(x12/a ,x12s)E(x48a ,x12s)(cid:21) N−2
N−2 N−2
Clearly a = b = 1 (identified initially from numerical studies) satisfy
N−2
−
these conditions; the second reduces to (13) in the x 0 limit, and note the
→
similarities with (17).
C Mass m
2
We begin the perturbation argument with the structure w = a for j =
j j
1,... ,N 2 with w = b x−14 and w = b x14, so that n = 2. From
N−1 1 N 2 2
−
the Bethe equations for j = N 1 and j = N we can show that b = b = b.
1 2
−
8
The Bethe equation for the other roots a =a is then
k
E(x2s/a) N a2
ω a =(A b2)3/5
− (cid:20) E(x2sa) (cid:21) N−2 b2
E(x10b/a)E(x14b/a)N−2E(x2sa/a )E(x4sa /a)
j j
. (18)
× E(x10a/b)E(x14a/b) E(x2sa /a)E(x4sa/a )
jY=1 j j
In the x 0 limit this gives the equation
→
1
aN−2+ (A b2)3/5/b2 =0,
N−2
ω
whichhasthesameorderasthenumberofunknownroots(N 2)sothatthere
−
is no hole. Equating this with N−2(a a ) we obtain
j=1 − j
Q
1
(A b2)3/5 =A b2
N−2 N−2
ω
(which we later apply to prefactors in Λ ). From the other Bethe equations in
2
this limit,
1 2 (A b2)2
(A b2)3/5 = N−2 b2N =1. (19)
(cid:20)ω N−2 (cid:21) b2N ⇒
Treating the Bethe equation (18) as before gives, in terms of the functions
defined in (8) and (9), the recurrences
(x26a/b,x30a/b;x2r) (x2sa)
∞ 2
(a)= F ,
F2 (x10a/b,x14a/b;x2r) (x4sa)
∞ 2
F
(x38b/a,x34b/a;x2r) (x2s/a)
∞ 2
(1/a)= G .
G2 (x50b/a,x54b/a;x2r) (x4s/a)
∞ 2
G
Solving these we obtain
(x30a/b,x42a/b;x2r) (x26a/b,x38a/b,x46a/b,x58a/b;x12s)
∞ ∞
(a)= ,
F2 (x14a/b,x26a/b;x2r) (x10a/b,x22a/b,x62a/b,x74a/b;x12s)
∞ ∞
(x38b/a,x50b/a;x2r) (x34b/a,x46b/a,x86b/a,x98b/a;x12s)
∞ ∞
(1/a)= .
G2 (x54b/a,x66b/a;x2r) (x50b/a,x62b/a,x70b/a,x82b/a;x12s)
∞ ∞
We now substitute these into the eigenvalue expression, the first term of which
is
Λ w2(x26w/b,x38w/b,x2b/w,x14b/w;x2r)
2 = ∞ (x2sw) (1/x2sw).
3 b2 (x2w/b,x14w/b,x26b/w,x38b/w;x2r) F2 G2
∞
Thisgivesanexpressionfortheexcitationinelliptic functions(settingb= 1):
−
Λ E( x2/w,x12s)E( x14/w,x12s)E( x38w,x12s)E( x50w,x12s)
2 =w2 − − − − .
Λ E( x2w,x12s)E( x14w,x12s)E( x38/w,x12s)E( x50/w,x12s)
0
− − − −
(20)
9
If the product of the six Bethe equations involving b is expressed in terms
of the auxiliary functions, the equation for b (generalizing b2N = 1 seen in the
x 0 limit in (19)) is clearly satisfied by b= 1:
→ −
E(x2b,x12s)E(x14b,x12s)E(x38/b,x12s)E(x50/b,x12s) N
=b2N.
(cid:20)E(x2/b,x12s)E(x14/b,x12s)E(x38b,x12s)E(x50b,x12s)(cid:21)
Compare the pattern of powers of x in this equation with those in (20); this
equation has a precise analogue for each mass m ,... ,m , which will not be
4 7
given.
D Mass m
3
We begin the perturbation argument with the string structure w = a for
j j
j = 1,... ,N 3 with w = b x−12, w = b x12 and w = b x20. From
N−2 1 N−1 2 N 3
−
theBetheequationsforj =N 2andj =N 1wecanshowthatb =b =α,
1 2
− −
but the Bethe equation for j =N does not link b =b to α in the x 0 limit.
3
→
(This feature was observed also in the L = 3 case, for a string of odd length
[13].) The Bethe equation for the other roots a =a is then
k
E(x2s/a) N a3 E(x4b/a)E(x8b/a)
ω a =(A α2b)3/5
− (cid:20) E(x2sa) (cid:21) N−3 αb2E(x4a/b)E(x8a/b)
E(x12α/a)E(x16α/a)E(x36α/a)N−3E(x2sa/a )E(x4sa /a)
j j
. (21)
× E(x12a/α)E(x16a/α)E(x36a/α) E(x2sa /a)E(x4sa/a )
jY=1 j j
In the x 0 limit this gives the equation
→
1
aN−3 (A α2b)3/5/αb2 =0.
N−3
− ω
Equating this as usual with N−3(a a ), we obtain
j=1 − j
Q
1
((A α2b)3/5 =A αb2
N−3 N−3
ω
(which we later apply to prefactors in Λ ). From the other Bethe equations in
3
this limit,
1 3 (A αb2)3
(A α2b)3/5 = N−3 b2N =1.
(cid:20)ω N−3 (cid:21) b2N ⇒
In this case it is convenient to define
F (w) (x12w/α;x2r)
3
(w)= ,
F3 F (x16w) (x4w/α;x2r)
3
G (1/w) (x28α/w;x2r)
3
(1/w)= , (22)
G3 G (1/x16w)(x36α/w;x2r)
3
10