The notion ofs-holomorphicity will turn out to be important for the convergence theorem (Theorem 6.20). Actually, the semi-discrete holomorphicity provides us only with half of Cauchy-Riemann equations and fortunately, the rest of the information can be “recovered”
by s-holomorphicity.
Roughly speaking, in Section6.2.3, we will define a primitive ofImf2wheref is s-holo-morphic and the convergence of this primitive provides us with additional information for the convergence off. We will discuss this in more details in Sections6.2and6.5.
We define the notion of s-holomorphicity on the mid-edge lattice (Definition5.32) and on the medial lattice (Definition5.33). And they can be proved to be equivalent in Proposi-tion5.34.
Definition 5.32. Letf :Ω[δ→Cbe a function defined on the mid-edge semi-discrete lattice.
It is said to bes-holomorphicif it satisfies the two following properties.
1. Parallelism: fore∈Ω[δ, we havef(e)// τ(e)whereτ(e) = [i(we−ue)]−1/2,ue andwe denote respectively the primal and the dual extremities of the mid-edge e. In other words,
• f(e)∈νRifpe+is a dual vertex, and
• f(e)∈iνRifp+e is a primal vertex, whereν= exp(−iπ/4).
2. Holomorphicity: for all vertexeon the mid-edge latticeΩ[δ, we haveD(δ)f(e) = 0. Definition 5.33. Letg:Ωδ→Cbe a function defined on the medial semi-discrete domain.
It is said to bes-holomorphicif it satisfies the two following properties.
1. Projection: for everye= [pe−p+e]∈Ω[δ, we have
Proj[g(p−e), τ(e)] = Proj[g(p+e), τ(e)] (5.26) whereProj(X, τ)denotes the projection ofXin the direction ofτ:
Proj[X, τ] =1 2
X+τ τ ·X
.
2. Holomorphicity: for all vertexeon medial latticeΩδ, we haveD(δ)f(e) = 0.
5.7. S-holomorphicity These definitions may be linked to their counterparts in the setting of isoradial graphs.
To be more precise, one can use the definitions provided in [CS11] on the latticeGε, which is illustrated in Figure1.7, and recover the above definitions by takingε→0.
We have a correspondence between s-holomorphic functions onΩδ and those onΩ[δ. Proposition 5.34. Given an s-holomorphic functionf :Ω[δ→C, one can defineg:Ωδ→C by:
g(p) =f(ep−) +f(e+p), p∈Ωδ. Then, the new functiongis still s-holomorphic.
Conversely, given an s-holomorphic functiong:Ωδ→C, one can definef :Ω[δ→Cby:
f(e) = Proj[g(pe−), τ(e)] = Proj[g(pe+), τ(e)], e∈Ω[δ. Then, the new functionf is still s-holomorphic.
Proof. Assume thatf :Ω[δ →Cis s-holomorphic. Let us show thatg as defined above is s-holomorphic onΩδ. The projection property is satisfied from the parallelism off and so is the holomorphicity.
Assume that g : Ωδ → C is s-holomorphic. Let us show that f as defined above is s-holomorphic onΩ[δ. The parallelism is clearly satisfied by the definition. We just need to check the holomorphicity off. Lete∈Ω[δ. We can assume thatp+e is primal and pe−dual such thatτ(e)// eiπ/4. We want to calculate∂yf(e).
∂yf(e) =∂yProj[g(p), τ(e)]
=1
2∂y[g(p) + ig(p)]
=1 2
i
δ(g(p+)−g(p−)) + i
−i δ
(g(p+)−g(p−))
= i
δ[f(e+)−f(e−)]
where we useτ(e)/τ(e) = iandτ(e+)/τ(e+) =τ(e−)/τ(e−) =−i.
Chapter 6
Scaling limit of the quantum Ising model
6.1 The quantum random-cluster model
We remind that the semi-discrete lattice and semi-discrete domains were defined in Sec-tion1.2.1. Here, we talk about the semi-discretization of domains inR2, on which we define the quantum Ising model.
6.1.1 Semi-discretization of a continuous domain
A set inR2, orC2, is called adomainif it is open, bounded and simply connected. ADobrushin domainis a domain with two distinct marked pointsa, bon the boundary. It is often denoted by a triplet(Ω, a, b)whereΩis a domain anda, b∈∂Ω.
Here, we explain how tosemi-discretizesuch a domain to get its semi-discrete counter-part, on which the FK-representation of the quantum Ising model will be defined. Consider a Dobrushin domain(Ω, a, b)inCorR2andδ >0. Let us denote by[awδabδ]and[bbδbwδ]two mid-edges withabδ, bbδ ∈Lδ,awδ, bwδ ∈L?
δ and mid-pointsa[δ andb[δ given by minimizing the distances betweenaanda[δ and betweenbandb[δover all possible such mid-edge segments contained inΩ. Once we get these two distinguished edges[awδabδ]and[bbδbwδ], we complete the semi-discrete domain by making approximation with primal horizontal and vertical seg-ment of the arc (aδbδ) then with dual horizontal and vertical segments of the arc (bδaδ). Moreover, we ask that these segments to be insideΩ. This semi-discrete domain lies inΩ and is denoted by(Ωδ, aδ, bδ). See Figure6.1for an example.
We write∂Ωδfor the boundary of this Dobrushin domain. This consists of four compo-nents:
[awδabδ] an horizontal edge
(aδbδ) := (abδbδb) the arc going fromabδtobbδ [bδbawδ] an horizontal edge
(bδaδ) := (bδwawδ) the arc going frombwδ toawδ .
They are ordered counterclockwise in Figure 6.1. Then, the quantum Ising model can be defined on such discretized domains and we also obtain the loop representation of the model
6. Scaling limit of the qantum Ising model
a
b bδ
aδ
Figure 6.1 – An example of approximation of a continuous Dobrushin domain by a semi-dis-crete one with mesh sizeδ.
at the criticality as mentioned in Section1.2.5, adapted to the caseq= 2, dP
QI
λ,µ(D, B)∝
√
2l(D,B)dPρ,ρ(D, B) (6.1)
wherel(D, B)denotes the number of loops in a given configuration(D, B)andρ=p λµ. Note that semi-discrete domains (Ωδ, aδ, bδ)defined above converge to the continuous domain(Ω, a, b)in the Carathéodory topology.
6.1.2 Main result
The result that we will show in this artical concerns the conformal invariance of the interface of the loop representation of the quantum Ising model. Since the conformal invariance also implies the rotational invariance, the parametersλ andµ should be chosen such that the model is isotropic. The good choice of such parameters are given byλ=2δ1 andµ=1δ (thus, ρ= √1
2ρ
). In Section6.2, or more precisely, Equation (6.4), we will see that it is actually a necessary and sufficient condition to get an observable with nice properties.
We are ready to give a formal statement of the main theorem.
Theorem 6.1. Let(Ω, a, b)a Dobrushin domain. Forδ >0, semi-discretize the domain to get a semi-discrete Dobrushin domain(Ωδ, aδ, bδ)and consider the FK-representation of the critical quantum Ising model with parametersλ=2δ1 andµ=1δ on it. Denote byγδthe interface going fromaδtobδin(Ωδ, aδ, bδ), which separates the primal cluster connected to the wired boundary (aδbδ)and the dual cluster connected to the free boundary(bδaδ). Then, the law ofγδconverges weakly to the chordal Schramm-Löwner Evolution SLE16/3running fromatobinΩ.