Universality and Conformal Invariance for the Ising Model in Domains With

Universality and conformal invariance for the Ising model in domains with boundary∗ by R. P. Langlands, Marc-Andre´ Lewis and Yvan Saint-Aubin Abstract. The partition function with boundary conditions for various two•dimensional Ising models is examined and previ• ously unobserved properties of conformal invariance and univer• sality are established numerically. ∗ First appeared in J. Stat. Physics 98, Nos. 1/2, pp. 131–244, 2000 1. Introduction. Although the experiments of this paper, statistical and numerical, were undertaken in pursuit of a goal not widely shared, they may be of general interest since they reveal a number of curious properties of the two•dimensional Ising model that had not been previously observed. The goal is not difficult to state. Although planar lattice models of statistical mechanics are in many respects well understood physically, their mathematical investigation lags far behind. Since these models are purely mathematical, this is regrettable. It seems to us that the problem is not simply to introduce mathematical standards into arguments otherwise well understood; rather the statistical•mechanical consequences of the notion of renormalization remain obscure. Our experiments were undertaken to support the view that the fixed point (or points) of the renormalization procedure can be realized as concrete mathematical objects and that a first step in any attempt to come to terms with renormalization is to understand what they are. We have resorted to numerical studies because a frontal mathematical attack without any clear notion of the possible conclusions has little chance of success. We are dealing with a domain in which the techniques remain to be developed. A fixed point is a point in a space of presumably an infinite number of dimensions; so this point and all other points of the space are defined by an infinite number of coordinates. Some will presumably be superfluous, so that the total space is realized as a submanifold of some larger coordinate space. The total space will be the carrier, in some sense, of the renormalization transformation, but the transformation will not appear explicitly in this paper. The point does! The implicit condition on each quantity serving as a possible coordinate of the fixed point is that, at the fixed point itself, it remains invariant under renormalization and that, at a critical point of any model within the class considered, its value approaches a limit under repeated renormalization because renormalization drives the critical point to the fixed point. Since repeated renormalization is in coarsest terms nothing more than passage to larger and larger blocks or to smaller and smaller mesh, the condition is that the quantity has a meaning as the mesh length goes to zero, the dimensions otherwise remaining the same. For percolation this is a property of crossing probabilities. Our point of view is that any such quantity is a candidate as a possible coordinate in the space of the fixed point. Rather than a single numerical quantity we can consider several at once, which amounts in the customary mathematical way to considering objects lying in some given space, finite•dimensional or infinite•dimensional, for example, a space of probability distributions, and if these objects satisfy this criterion, thus if, for each model at the critical point, they tend to a limit as the mesh goes to zero, then this limit or rather its coordinates in the given space can also serve as coordinates of the fixed point. Such objects are described in the paper. Universality and conformal invariance 2 There are at least two possibilities: one modeled on the considerations for the free boson of [L]; the other on the crossing probabilities for percolation [LPPS, LPS]. The second possibility was suggested to us by Haru Pinson to whom we are grateful. Thus to each form M of the Ising model (taken at the critical temperature) we will attach two points pD(M) and pC (M), each defined by an infinite number of coordinates. Both are, in so far as this can be confirmed by experiments, universal and conformally invariant in the sense of [LPS]. It is unlikely that these two points are independent. One set of coordinates may well be deducible from another, but we have not examined this possibility. Crossing probabilities may or may not be peculiar to a few models. The evidence for their conformal invariance and universality is not difficult to present and appears in Section 4. There is, however, one point to underline. The Ising model is considered in regions that may be bounded or unbounded. Crossing probabilities are defined for crossings within a region that may or may not coincide with the region in which the model is considered. It may be smaller. In contrast to crossing probabilities for percolation, in which there is no interaction, those for the Ising model depend on both the region in which the crossings are allowed to occur and the region of thermalization on which the Ising model is considered. Conformal invariance refers to the simultaneous action of a conformal mapping on the pair of regions. The coordinates modeled on the free boson should, on the other hand, be available for a large class of models. Their definition is, in principle, quite general, but we have confined ourselves to the Ising model. The states σ of the Ising model take values in the set 1 which {± } is contained in the set of all complex numbers z with z =1. This set in turn is covered by the | | line z = exp(2πix). We simply develop the circle on the line. We first assign to one site p0 in the lattice the value h(p )=0 and then choose for all other p the value h(p) = mπ, m Z, 0 ∈ so that exp(ih(p) ih(p )) = σ(p)/σ(p ). Of course, there has to be more method than that. − 0 0 For example, for the square lattice we introduce clusters: maximal collections of lattice sites of the same spin that are connected through bonds joining nearest•neighbor sites. Each cluster is surrounded by a curve constructed from edges of the dual lattice and this curve separates it from all other clusters. To each of these curves an orientation is assigned randomly and, for nearest neighbors p and q, we set h(p) h(q) = π, the sign being determined by the − ± relative orientation of the bond from p to q and the curve it crosses. If it crosses no curve, then σ(p) = σ(q) and we take h(p) = h(q). Thus to every state σ are attached several functions h, but h determines σ up to sign. For a finite lattice the measure on H, the set of all possible h is taken to be such that the measure, mI , on H assigns the same mass to all points lying above a given σ. Their sum is one•half the mass of σ. Fix now a bounded planar region D and consider the Ising model in this region with respect to a square lattice whose mesh a approaches 0. Since the model is to be critical, the Universality and conformal invariance 3 contribution to the Boltzmann weight of a pair of neighboring spins is eJσ1σ2 , sinh(2J)=1, J = .440687. Each h is in effect a function on the whole region if we take the value in the open square of side a about the site p to be h(p). (The ambiguity at the boundary is disregarded here; it has to be confronted in various ways from experiment to experiment.) If C is a (smooth) curve in R, which can lie entirely in the interior or run entirely or partially along the boundary, then we can restrict each function in H to C. This yields a set of points, each carrying a mass, in the set DC of Schwartz distributions on C, and thus a probability measure on DC. Experiments to be described in Sections 1 and 2 suggest strongly that this measure has a limit as the mesh tends to 0 and that the limit is universal and conformally invariant. This is perhaps the most important conclusion of the paper. These measures have surprised the authors more than once. When C is the boundary, the measure has a number of properties, to which we devote considerable attention, that suggest it is gaussian. It is not. There is no reason to restrict ourselves to planar regions and we begin our study with the cylinder, because the ambiguities at the boundary are then absent. A long cylinder (effectively semi•infinite) is, provided we stay close to the end, to be regarded as equivalent to a disk. The simplest conformally invariant distribution on the set of distributions on the boundary of a disk is the gaussian distribution with respect to the quadratic form defined by the Dirichlet form. For a function ϕ this form is obtained by extending ϕ to a harmonic function ϕ˜ in the interior and then taking g ∂ϕ˜ 2 ∂ϕ˜ 2 + dxdy. (1) 4π ∂x ∂y Z ! The first experiments described in Section 2 strongly suggest that the measure given by our construction is in some respects very similar to this gaussian with a constant g = gB that appears to be universal; the final experiments of that section show, however, that it differs in important respects from a gaussian. If C is interior, the measure on DC is in no respect similar to a gaussian. Our construction is different from but not unrelated to familiar constructions relating the Ising model to SOS models. For the Ising model on a triangular lattice our construction is equivalent in many respects to the usual one for the O(1)•model.

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