Topics on abelian spin models and related problems

29 Pages·2011·0.34 MB·English

by  Julien Dubédat

Probability Surveys
Vol.8(2011)374–402
ISSN:1549-5787
DOI:10.1214/11-PS187
Topics on abelian spin models
and related problems
Julien Dub´edat
∗
Columbia University,Department of Mathematics
2990 Broadway, NewYork, NY 10027
e-mail: [email protected]
Abstract: Inthesenotes,wediscussaselectionoftopicsonseveralmodels
of planar statistical mechanics. We consider the Ising, Potts, and more
generallyabelianspinmodels;thediscreteGaussianfreefield;therandom
clustermodel;andthesix-vertexmodel.Emphasisisputonduality,order,
disorderandspinorvariables,andonmappingsbetween thesemodels.
AMS 2000 subject classifications:Primary60G15,82B20.
ReceivedNovember 2011.
1. Introduction
Inthetypeofstatisticalmodelsconsideredhere,afixedunderlyinggraphcarries
a random realization of a structure, consisting of values (boolean, in a finite
alphabet or scalar) carried by bonds (edges) or sites (vertices), and usually
interacting via local rules. We will focus on the Ising and Potts models (and
moregenerallycyclicandabelianspinmodels),whereeachsitecarriesarandom
“spin”;thediscreteGaussianfreefield,whereascalarheightfunctionisdefined
on sites; the random cluster model, where realizations are random subgraphs;
and the six-vertex model, where each edge has a random orientation.
Thekeyquestionistounderstandthelargescalerandomstructuresgenerated
bytheselocalinteractions.Forinstance,onemaystudythecorrelationsbetween
localconfigurationsobservedatlargedistanceontheunderlyinggraph.Wewill
generically refer to (functions of) these local inputs as order variables.
Historically, a fundamental tool, introduced by Kramers and Wannier in
the context of the Ising model, is duality, which maps the model defined on
a (weighted) graph to the same model on another (weighted) graph. Of partic-
ular interest are the “fixed points” of these duality transformations; they often
coincide with a criticalpoint ofthe model, at whichone canobservelargescale
random structures. This type of duality is observed in a variety of models, and
forabelianmodelscanbe phrasedintermsofthe Fourier-Pontryagintransform
and the Poisson summation formula.
AgaininthecontextoftheIsingmodel,KadanoffandCevashowedthatunder
duality, order variables are mapped to “disorder” variables, which represent a
modificationofthestate spaceby theintroductionoflocal“defects”;moreover,
∗PartiallysupportedbyNSFgrantDMS-1005749andtheAlfredP.SloanFoundation.
374
Topicson abelian spin models and related problems 375
the field-theoretic concept of fermion finds a combinatorial incarnation as the
combination of an order and a disorder variable at microscopic distance. Such
combinations are referred to as parafermionic or spinor variables.
In these notes, we will describe duality, order, disorder, and spinor variables
forabelianspinmodelsandthediscretefreefield.Wethendiscusscombinatorial
mappings between the Potts model, the random cluster model, the six-vertex
model, and the dimer model.
For focus and (relative) self-containedness, we do not discuss here a number
ofmajorcontiguoussubjects.Theseinclude: phasetransition,transfermatrices
and Bethe ansatz [2, 38], Yang-Baxter solvability [2], infinite volume measures
[15, 18] and limit shapes [3, 27, 38], scaling limits and Schramm-Loewner Evo-
lution [47, 43].
2. Spin models
2.1. Ising model
Definition
In the Ising model [32, 36, 19], a configuration on an underlying finite graph
Γ = (V,E) consists of an assignment of a spin value σ(v) 1, 1 to each
∈ { − }
vertex v V. The weight of a configuration is:
∈
w(σ)= exp(βJ σ(v)σ(v ))
e ′
e=(vYv′)∈E
where J = 0 is a coupling constant attached to the edge e E and β > 0 is
e
6 ∈
the inverse temperature. Plainly the model (up to normalization) depends only
on the edge weights:
w(e)=exp( 2βJ )
e
−
It is occasionally convenient to allow w(e) = 0, i.e. J = + . Note that these
e
∞
weights are invariant under spin flip: w( σ) = w(σ). One may also consider
−
boundary conditions, that is, fixing the values of spins to 1 on some pre-
± ±
scribedsubsetsV ofV (theconfigurationspacethenconsistsofspincollections
±
(σ(v)) s.t. σ(v) = 1 for v V ). One may also consider some segments
v V ±
∈ ± ∈
on the boundary to be wired, viz. connected by edges with zero weight, so that
the only contributing configurations are constant on these segments (a wired
arc may be equivalently represented as an extended vertex).
The partition function is
= (Γ,(J ) )= w(σ)
e e E
Z Z ∈
σ:VX→{±1}
WedenotetheexpectationundertheprobabilitymeasuregivenbyP(σ =σ )=
0
w(σ )/ by .
0
Z h·i
376 J. Dub´edat
Fig 1. Left:a portion of a planar graph Γ. Right: itsdual Γ†.
A classical object of interest is the spin correlation
σ(v ) σ(v )
1 n
h ··· i
for v ,...,v graphvertices (in particularwhen they are atlarge distance)and
1 n
the spin variables σ(v)’s are fundamental examples of order variables.
The spin correlations depend on the boundary conditions. A wired arc may
be seen as a boundary arc along which couplings are infinite (vanishing edge
weights); in turn, boundary arcs may be represented in terms of wired arcs
±
and spin variables. For example, if γ+,γ are two disjoint boundary arcs, we
−
have
X 1+σ(x) 1−σ(y)
X = · 2 · 2 w,w
h i+,− (cid:10) 1+σ(x) 1−σ(y) (cid:11)
2 · 2 w,w
where X is a generic random vari(cid:10)able, the expec(cid:11)tation on the LHS is for
±
boundary conditions on γ , and the expectations on the RHS are for γ+, γ
± −
(separately) wired arcs. This shows that, allowing for general couplings, it is
enough to treat free boundary conditions.
From now on we will consider the planar case; Γ = (V ,E ) denotes the
† † †
planar dual of Γ, so that faces of Γ correspond to vertices of Γ , and vice-versa
†
(Figure 1).
More precisely, it is rather convenient to think of vertices on a wired arc
of the boundary as a single extended vertex; and to connect vertices on a free
boundary or extended vertices to a vertex at infinity by edges with weight 1
(i.e. with no interaction). If e E, we denote e the corresponding edge of Γ ;
† †
∈
for oriented edges, e crosses e from right to left.
†
Graphical expansions
In the low-temperature regime (β 1), disagreements between neighboring
≫
spins are severely penalized. The low temperature expansion of the Ising model
consists in mapping a spin configuration (σ ) to a subgraph P = (V ,E )
v v V † P
∈
Topicson abelian spin models and related problems 377
of Γ , where E = e = (vv ) E : σ(v)σ(v ) = 1 . At low temperature
† P ′ † † ′
{ ∈ − }
this graph is typically sparse. The spins (σ ) may be recovered from P up to
v
anoverallspinflip(inthe caseoffree boundaryconditions).Itiseasilychecked
that P is an even degree subgraph of P; such subgraphs are called polygons.
The (unnormalized) Ising measure on spins (σ ) projects to a measure on
v
polygons:
w(P)=1 w(e )
P admissible †
{ }
e∈YEP
omitting a multiplicative factor 2 (for overall spin flip).
Letusturntothehigh temperatureexpansion.Sinceσ(v)σ(v )= 1,wemay
′
±
write
exp(βJ σ(v)σ(v ))=cosh(βJ )(1+tanh(βJ )σ(v)σ(v ))
e ′ e e ′
Let us define a dual edge weight:
w′(e)=tanh(βJe)
Inthe hightemperature regime(β 1),these dualedge weightsare closeto 1.
≪
Starting from the partition function, we obtain
(Γ,(J ) )= exp( 2βJ 1 )
e e e σ(v)σ(v′)= 1
Z − { − }
σ:V 1 e=(vv′) E
X→{± } Y∈
= cosh(βJe) (1+w′(e)σ(v)σ(v′))
eY∈E σ:VX→{±1}e=(vYv′)∈E
= cosh(βJ ) w (e)σ(v)σ(v )
e ′ ′
eY∈E σ:VX→{±1}EX0⊂EeY∈E0
by fulling expanding the product. Then exchanging summations leads to
(Γ,(w ) )=2V cosh(βJ ) w (e)
e e | | e ′
Z
eY∈E P pXolygone∈YEP
Then comparing high and low temperature expansions, we have the Kramers-
Wannier duality [29, 30] for partition functions:
Z(Γ,(Je)e)=2|V| cosh(βJe)!Z(Γ†,(Je†))
e E
Y∈
where
e−2βJe† =tanh(βJe)
or w′(e) = 11−+ww((ee)). Note that w 7→ 11+−ww is involutive. It exchanges 0 and 1,
so that wired boundary components are exchanged with free boundary compo-
nents. To retain positive weights after the duality transformation, we need to
start from ferromagnetic couplings, i.e. J 0.
e
≥
If Γ is (a portion of) the square lattice, Γ is identical to Γ, up to boundary
†
modifications.Forinstance,onecanconsiderΓtobearectanglewithhalf-wired
378 J. Dub´edat
and half-free boundary conditions, in which case Γ is exactly isomorphic (as a
†
weighted graph) to Γ. Thus for the square lattice the fixed point w =√2 1
sd
−
of the duality mapping w 1 w is self-dual, in the sense that it is fixed for
7→ 1+−w
Kramers-Wannierduality (at least “in the bulk”). The low-temperature regime
w <w and the high-temperature regime w >w are exchanged by duality.
sd sd
Disorder variables
Let us now consider the effect of Kramers-Wannier on spin correlations σ(v )
1
h
σ(v ) .Inthehightemperatureexpansion(andperiodicboundaryconditions
n
··· i
for simplicity), we can write:
(Γ,(J ) ) σ(v ) σ(v ) = σ(v ) σ(v ) eβJeσ(v)σ(v′)
e e 1 n 1 n
Z h ··· i ···
σ∈{±X1}V\V± e=(vYv′)∈E
and then
(Γ,(J ) )
e e
Z cosh(βJ )hσv1···σvni= σv1···σvn w′(e)σvσv′
e∈E e σ∈{±X1}V\V±EX0⊂E eY∈E0
Q =2|V| w′(e)
PX⊂Γe∈YEP
wherethesumbearsonsubgraphsP Γwhichhaveodddegreeat v ,...,v
1 n
⊂ { }
and even degree elsewhere (by σ σ invariance, the correlator is zero if n
↔ −
is odd, in which case the sum is empty). For periodic boundary conditions
(i.e. when Γ is embedded on a torus), one requires additionally that any non-
contractible (with respect to the torus) path on Γ crosses an even number of
†
edges of P.
Such a polygon with defects at v ,...,v does not correspond to a spin
1 n
{ }
configuration on Γ . Following Kadanoff and Ceva [23], this motivates the in-
†
troductionofdisorder variables, whichencodeamodificationofthestatespace.
Thedataof(σ ) isequivalent,uptoglobalspinflip,tothedata(dσ(e)) ,
v v V e E
where dσ(vv )=σ(v∈)σ(v) 1. If ω :E 1 , we define: ∈
′ ′ −
→{± }
dω(f)= ω(e)
e ∂f
Y∈
wheref isafaceofΓand∂f isits(counterclockwiseoriented)boundary.Upto
spin flip, spin configurations (σ ) correspond to closed currents: (ω ) :
v v V e e E
∈ { ∈
dω 1 . (In the periodic case, there is an additional condition: ω(e)=1
≡ } e γ
if γ is a non-contractible path). Plainly the Gibbs weights can be∈written as
Q
functions of these currents.
Introducingdisordervariablesµ(f ),...,...µ(f )(f ,...,f facesofΓ)con-
1 n 1 n
sists in modifying the state space to:
1
(ωe)e E :dω =( 1) {f1,...,fn}
{ ∈ − }
Topicson abelian spin models and related problems 379
with weights
w(ω)= w(e)
e∈EY:ωe=−1
Weassumethatniseven(otherwisethestatespaceisempty,as dω(f)=
f Γ†
1 for all ω). If v,v V, one can define σ(v )σ(v)= ω(e) w∈here γ
′ ∈ ′ e∈γv→v′ Q v→v′
is a path from v to v on Γ. Notice however that this definition depends on
′ Q
the choice of γ. Indeed, if γ is another such path, ω(e) = ω(e)
′ e γ ± e γ
depending on the parity of the number of f ’s enclosed b∈y the loop γ γ ∈1.
i Q Q′ −
Onecantrivializethisdatabychoosing n “defectlines”(pathsonΓ )joining
2 †
the f ’s pairwise. Then one can find σ : V 1 such that ω(e) = dσ(e) if
i
→ {± }
e does not cross one the defect lines. In terms of this trivialization, we have an
Ising model with antiferromagnetic(negative)couplings J for e E crossing
e
− ∈
a defect line.
We can now define mixed order-disordercorrelators:
σ(v ) µ(f ) (Γ)=2 ω(e) w(e)
i j
h iZ
Yi Yj dωω:=EX→( {1±)11}F eY∈γ e∈E:ωY(e)=−1
−
where v ,...,v Γ, F = f ,...,f Γ , γ is a union of m paths on Γ
1 2m 1 2n †
{ }∈ { }⊂
with endpoints v ,...,v . The sign of the correlator depends on this implicit
1 2m
choiceofpaths(oralternativelyofdefectlines,inwhichcaseonesimplyrequires
that γ does not intersect defect lines).
In this case the Kramers-Wannier duality [29, 30, 23] yields:
σ(v ) µ(f ) = µ(v ) σ(f )
h i j ià h i j iÆ
i j i j
Y Y Y Y
where the couplings on Γ are in duality with those on Γ.
†
KadanoffandCeva[23]identifiedthecombinationofanordervariableσ and
a disorder variable µ at microscopic distance as a discrete version of the field-
theoretic notion of fermion. In particular one denotes ψ =σ µ where v V
vf v f
∈
is on the boundary of the face f.
2.2. Abelian spin models
The Ising model may be generalized in several directions; the first we consider
here is that of abelian spin models, in which the spin variable at each vertex
takesvaluesinafixedfiniteabeliangroupG.InthecaseG= 1 ,onerecovers
{± }
the Ising model.
Given a graph Γ = (V,E) and a finite abelian group G, a configuration is a
mappingσ :V G.Foreache E,wehaveaweightfunctionw :G [0, ),
e
→ ∈ → ∞
whichissymmetric,i.e.w (g 1)=w (g)(relaxingthis conditionleadstochiral
e − e
models and requires the underlying graph Γ to be oriented). Then
w(σ)= w(σ(v )σ(v) 1)
′ −
e=(vYv′)∈E
380 J. Dub´edat
is the weight of a general configuration. For free boundary conditions, we have
w(ρσ)=w(σ) for any ρ G.
IfG=Z/qZ,wehave∈aclockmodel.Iffurthermorew =a+b1 ,wegetthe
e 1
{ }
q-state Potts model. If q = 4 and w is a general symmetric weight, we obtain
e
the Ashkin-Teller model ([2], as phrased by Fan).
Let Gˆ = Hom(G,C ) be the dual of G (Gˆ and G are isomorphic, though
∗
not canonically). The basic order variables are of type χ(σ(v)), χ Gˆ, with
∈
associated correlators:
χ (σ(v ))
i i
h i
i
Y
wherev V,χ Gˆ.Forfreeboundaryconditions,thisiszerounless χ =1.
i ∈ i ∈ i i
Kramers-Wannier duality has been extended successively to various models,
Q
see in particular [48, 41]. By Fourier-Pontryaginduality [40], one can write:
1
w = w (χ)χ
e G1/2 e
| | χXGˆ
∈
c
where w is the Fourier-Pontryagintransform
e
1
b we(χ)= G1/2 we(g)χ¯(g)
| | g G
X∈
b
At the level of partition functions, we have the summation:
(Γ,G,(w ) )= w (σ(v )σ(v) 1)
e e e ′ −
Z
σ:XV→Ge=(vYv′)∈E
= G−12|E| we(χe)χe(σ(v′)σ(v)−1)
| |
σ:XV→Ge=(vYv′)∈EχXe∈Gˆ
= G−21|E| we(χe)χce(σ(v′)σ(v)−1)
| |
χσ::XEV→GGˆe=Y(vv′)
→ c
= G|V|−21|E| we(χe)
| |
χ:E→XGˆ:d∗χ≡1e=Y(vv′)σXv∈G
c
where d χ(v) = χ . Let us identify χ : E Gˆ with χ : E Gˆ via
∗ v′ v e → † →
χ(e ) = χ(e) for e E∼. Then d χ 1 is equivalent (in the simply connected
†
case)totheexistenQc∈eofσˆ :V ∗Gˆ s≡uchthatχ(ff )=gˆ(f )gˆ(f) 1.(Implicitly,
† ′ ′ −
→
we fixeda reference orientationfor edgesof Γ ; by symmetry ofthe weight,one
†
can write eachterm w (σ(v )σ(v) 1) so that v (resp. v ) is the left (resp. right)
e ′ − ′
vertex wrt the oriented edge of Γ separating them). We conclude that:
†
(Γ,G,(we)e)= G|V|−12|E|+1 (Γ†,Gˆ,(we)e)
Z | | Z
Disorder variables are indexed by a face f V and an element g G, and
†
∈ c ∈
denoted µ (f). The corresponding modified state space consists of G-valued 1-
g
forms ω : −→E G defined on oriented edges and antisymmetric in the sense
→
Topicson abelian spin models and related problems 381
ttheractloωc(k−x→wyi)se=oωri(e−y→nxt)e−d1b.oWuenddaerfiyneofdωf.(fT)h=en t→−hee∈∂sftaωte(−→esp)a,cwehceorrer∂esfpoisntdhinegcotuona-
Q
disorder µ (f ) is:
i gi i
1
Q ω :dω = g fi
i
(cid:26) i (cid:27)
Y
with weight w(ω) = w (ω(e)) (there is an overall factor G compared
e E e | |
with the earliernormaliza∈tion).Again this may be trivialized by fixing a defect
Q
linegoingthroughallf ’sandmodifyingcouplingsalongthis line.Remarkthat
i
f = 1 (otherwise the state space is empty). For simplicity let us consider a
i i
pairχ(σ(v )),χ 1(σ(v )ofspinvariables,andapairµ (f ),µ (f )ofdisorder
2 − 1 g 2 g−1 1
Q
variables. Let us fix non intersecting paths γ (resp. γ ) on Γ (resp. Γ ) from v
† † 1
to v (resp. from f to f ). We can write a mixed correlator:
2 1 2
χ(σ(v ))χ 1(σ(v ))µ (f )µ (f )
Zh 2 − 1 g 2 g−1 1 i
= χ(σ(v )σ(v) 1) w (σ(v )σ(v) 1)
′ − e′ ′ −
σ:XV→G(vYv′)∈γ (vvY′)∈E
where w = w if e does not cross γ and w (.) = w (g.) otherwise (in which
e′ e † e′ e
case we orient e from right to left of γ ). Repeating the argument above leads
†
to
hχ0(σ(v2))χ−01(σ(v1))µg(f2)µg−1(f1)iΓ =hµχ0(v2)µχ−01(v1)χf2(g)χf1(g−1)iΓ†
wheretheχ ’sarethedualordervariables;weightsonΓ areFouriercoefficients
f †
of those on Γ. This extends to any number of insertions.
By analogy with the Ising model, on the square lattice we are particularly
interested in self-dual weights, viz. weights such that w w φ for some iso-
morphism φ : Gˆ G. If we assume weights are nonneg∝ative◦, by Parseval we
→
have w =w φ. b
◦
Cyclicbmodels
Let us specialize to the cyclic case: G = Z/qZ. U = z C : zq = 1 . Let
q
{ ∈ }
ξ =exp(2iπ/q). Identifying G with Gˆ in the usual way, the Fourier transform
0
is written:
q 1
1 −
(Ff)(j)= √q f(i)ξ0−ij
i=0
X
Finding a self-dual weight consists in finding eigenvectors for . This operator
F
is the discrete Fourier transform (DFT), involved in fast Fourier transform al-
gorithm. The question of its diagonalization is classical, if rather delicate. By
Fourier inversion, the Fourier transform : CG CG is of order 4: 4 = Id,
F → F
and consequently the eigenvalues of are 1, i . It is known that the mul-
tiplicity of the eigenvalue 1 is q +1F. {± ± }
⌊4⌋
382 J. Dub´edat
First we look for self-dual Potts model weights. In this case: w 1+aδ .
0
∝
Plainly Fw = √aq +√qδ0. The fixed point equation Fw = w is thus solved
for a = √q. For Ising (q = 2), we recover the self-dual weight function (up to
multiplicative constant).
As soon as q 4, the multiplicity of 1 is 2. Similarly to [20], one can look
≥ ≥
for a nice operator D End(CG) (almost) commuting with . An educated
∈ F
guess gives:
D =(a+bχ−1)R+(c+dχ−1)
where R is the right shift: (Rf)(k) = f(k+1), and χ CG is the character:
∈
χ(k) = ξk. Classically, R = χ (χ acting by pointwise multiplication) and
0 F F
χ 1 =R . Besides Rχ=ξ χR. Then
− 0
F F
D =(aχ+bξ χR+c+dR)
0
F F
Consequently, D =λχD (λ C) iff:
F F ∈
(a,b,c,d)=λ(bξ ,d,a,c)
0
that is: λ4 = ξ0−1, (a,b,c,d) ∝ (1,λ3,λ,λ2). An element w of KerD satisfies a
trivial recursion:
w(k+1)(1+λ3ξ0−k)+w(k)(λ+λ2ξ0−k)=0
To ensure positivity, we choose λ= exp( iπ/2q); then we find
− −
k 1 sin(πk + π)
− q 4q
w(k)=
sin(π(k+1) π)
jY=0 q − 4q
and check that w(q)=w(0)=1. Then KerD is a line which is invariant under
, and thus w = w is an invariant weight. This is the Fateev-Zamolodchikov
Fpoint [11] for the ZF/qZ model (identified in [11] as part of a family of weights
satisfying the Yang-Baxter relations).
Ashkin-Teller model
In the Ashkin-Teller model ([2], as phrased by Fan), each vertex v V carries
∈
a pairofspins (σ ,ρ ) 1 2;the interactionweightonthe edgee=(vv )is:
v v ′
∈{± }
exp(β(Jeσvσv′ +Je′ρvρv′ +Je′′σvσv′ρvρv′))
We then recognize the abelian spin model with G = 1 2 (the simplest non
cyclic finite abelian group). Identifying G with Gˆ we g{e±t (h}ere (ε ,ε ) G)
1 2
∈
1
( f)(ε ,ε )= (f(1,1)+ε f( 1,1)+ε f(1, 1)+ε ε f( 1, 1))
1 2 1 2 1 2
F 2 − − − −
leading to the following simple self-duality condition for the weight w : G R
→
(i.e. satisfying w = w):
F
w(1,1)=w( 1,1)+w( 1,1)+w( 1, 1)
− − − −
Topicson abelian spin models and related problems 383
The basic order variables are σ,ρ,σρ and there are naturally corresponding
disorder variables µ,ν,µν. Remark that in the isotropic case J =J , the model
′
becomes a Z/4Z model.
Parafermions
LetusgobacktothegeneralcaseofanabelianmodelwithspingroupG.Order
(resp.disorder)variablesareindexedbyGˆ (resp. G).Byanalogywiththe Ising
model, one can consider a parafermion (or spinor)
ψχ0,g0 =χ (σ )µ (f)
vf 0 v g0
where(χ ,g ) Gˆ G (and arefixed andomitted fromnow on), v Γ, f Γ
0 0 †
∈ × ∈ ∈
adjacent.Whentrackinganordervariableχ (σ)alongacyclearoundadisorder
0
variable µ (f) (or vice versa), one picks a phase χ (g ) 1. In particular, when
g0 0 0 ±
thespinorrotatesuntoitself(egfixf andfollowv along∂f),itismultipliedby
χ (g ) 1.Thismaybeformalizedasfollows.ConsideragraphΓ withverticesat
0 0 ± ′
midpoints ofsegments [vf], v Γ, f Γ , v onthe boundary off (denote such
†
∈ ∈
a vertex (vf)); edges of Γ are such that (vf) (v f) if v,v are consecutive
′ ′ ′
∼
vertices on ∂f and dually (vf) (vf ) if v ∂f ∂f . The “line bundle”
′ ′
C(vf) is equipped with∼a connection∈(in th∩e terminology of [25] eg),
(vf) Γ′
i.e. a c∈ollection of isomorphisms φ : Cu Cu for adjacent vertices u,u in
L u,u′ → ′ ′
Γ. These are defined by:
′
f v v f
′ ′
φ (z)=exp isarg − z, φ (z)=exp isarg − z
(vf),(v′f) f v (vf),(vf′) v f
(cid:18) − (cid:19) (cid:18) − (cid:19)
wheresisthespin,andargumentsarechosenin(−π,π)(sothatφu,u′ =φ−u′1,u).
Let us consider a general correlator:
n m
F(v ,f )= ψ χ (v ) µ (f )
0 0 h v0f0 i i gj j i
i=1 j=1
Y Y
We regard all insertions as fixed except v,f, which are adjacent. In order for
this correlator to be non-trivial, we need χ χ = 1 , g g = 1 .
0 i 1 i Gˆ 0 j 1 j G
To assign an actual value to the correlator,we al≥so have to fix a de≥fect line (or
Q Q
defecttree)connectingalldisorderoperatorsonΓ .Alternatively,onecanthink
†
of F as being multivalued (with a phase ambiguity) or as a section of the line
bundle described above (with the necessary modifications around v ,f ).
i j
Consider the local situation around an edge e=(vv ) E and its dual edge
′
∈
(ff ) = (vv ) E ; this gives four possible locations for the parafermion. We
′ ′ † †
∈
assumethat the defect line γ chosenfor f is the concatenationof (ff ) and the
′
defect line γ for f (which has f as an endpoint).
′ ′ ′
For g G, let us consider the partial (modified) partition function:
∈
= 1 χ (σ(v )) w (yx 1)
Zg σ(v)=1,σ(v′)=g i i e′ −
σ:XV→G iY≥1 e=(xy)∈YE,e6=(vv′)