Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTwist operator correlation functions in O(n) loop models
146
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Using conformal field theoretic methods we calculate correlation functions of geometric observables in the loop representation of the O(n) model at the critical point. We focus on correlation functions containing twist operators, combining these with anchored loops, boundaries with SLE processes and with double SLE processes. We focus further upon n=0, representing self-avoiding loops, which corresponds to a logarithmic conformal field theory (LCFT) with c=0. In this limit the twist operator plays the role of a zero weight indicator operator, which we verify by comparison with known examples. Using the additional conditions imposed by the twist operator null-states, we derive a new explicit result for the probabilities that an SLE_{8/3} wind in various ways about two points in the upper half plane, e.g. that the SLE passes to the left of both points. The collection of c=0 logarithmic CFT operators that we use deriving the winding probabilities is novel, highlighting a potential incompatibility caused by the presence of two distinct logarithmic partners to the stress tensor within the theory. We provide evidence that both partners do appear in the theory, one in the bulk and one on the boundary and that the incompatibility is resolved by restrictive bulk-boundary fusion rules.
rate research
Read More
The O(n) spin model in two dimensions may equivalently be formulated as a loop model, and then mapped to a height model which is conjectured to flow under the renormalization group to a conformal field theory (CFT). At the critical point, the order n terms in the partition function and correlation functions describe single self-avoiding loops. We investigate the ensemble of these self-avoiding loops using twist operators, which count loops which wind non-trivially around them with a factor -1. These turn out to have level two null states and hence their correlators satisfy a set of partial differential equations. We show that partly-connected parts of the four point function count the expected number of loops which separate one pair of points from the other pair, and find an explicit expression for this. We argue that the differential equation satisfied by these expectation values should have an interpretation in terms of a stochastic(Schramm)-Loewner evolution (SLE_kappa) process with kappa=6. The two point function in a simply connected domain satisfies a closely related set of equations. We solve these and hence calculate the expected number of single loops which separate both points from the boundary.
We study mean-field classical $N$-vector models, for integers $Nge 2$. We use the theory of large deviations and Steins method to study the total spin and its typical behavior, specifically obtaining non-normal limit theorems at the critical temperatures and central limit theorems away from criticality. Important special cases of these models are the XY ($N=2$) model of superconductors, the Heisenberg ($N=3$) model (previously studied in cite{KM} but with a correction to the critical distribution here), and the Toy ($N=4$) model of the Higgs sector in particle physics.
In this note we consider non-equilibrium steady states of one-dimensional models of heat conduction (wealth exchange) which are coupled to some reservoirs creating currents. In particular we will give sufficient and necessary conditions which will depend only on the first two moments of the reservoir measures and the redistribution parameter under which the two-point functions are multilinear. This presents the first example of multilinear two-point functions in the absence of product stationary measures.
We compute analytically and in closed form the four-point correlation function in the plane, and the two-point correlation function in the upper half-plane, of layering vertex operators in the two dimensional conformally invariant system known as the Brownian Loop Soup. These correlation functions depend on multiple continuous parameters: the insertion points of the operators, the intensity of the soup, and the charges of the operators. In the case of the four-point function there is non-trivial dependence on five continuous parameters: the cross-ratio, the intensity, and three real charges. The four-point function is crossing symmetric. We analyze its conformal block expansion and discover a previously unknown set of new conformal primary operators.
We consider the $n$-component $|varphi|^4$ lattice spin model ($n ge 1$) and the weakly self-avoiding walk ($n=0$) on $mathbb{Z}^d$, in dimensions $d=1,2,3$. We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance $r$ as $r^{-(d+alpha)}$ with $alpha in (0,2)$. The upper critical dimension is $d_c=2alpha$. For $epsilon >0$, and $alpha = frac 12 (d+epsilon)$, the dimension $d=d_c-epsilon$ is below the upper critical dimension. For small $epsilon$, weak coupling, and all integers $n ge 0$, we prove that the two-point function at the critical point decays with distance as $r^{-(d-alpha)}$. This sticking of the critical exponent at its mean-field value was first predicted in the physics literature in 1972. Our proof is based on a rigorous renormalisation group method. The treatment of observables differs from that used in recent work on the nearest-neighbour 4-dimensional case, via our use of a cluster expansion.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
