相关论文: Lectures on the Spin and Loop $O(n)$ Models

The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been conjectured that both the spin and the…

A relation between O(n) lattice spin models and Ising models defined on the same lattice was recently put forward [L. Casetti, C. Nardini, and R. Nerattini, Phys. Rev. Lett. 106, 057208 (2011)]. Such a relation, inspired by an energy…

We investigate the O($n$) nonintersecting loop model on the square lattice under the constraint that the loops consist of ninety-degree bends only. The model is governed by the loop weight $n$, a weight $x$ for each vertex of the lattice…

A relation between O$(n)$ models and Ising models has been recently conjectured [L. Casetti, C. Nardini, and R. Nerattini, Phys. Rev. Lett. 106, 057208 (2011)]. Such a relation, inspired by an energy landscape analysis, implies that the…

A family of models for fluctuating loops in a two dimensional random background is analyzed. The models are formulated as O(n) spin models with quenched inhomogeneous interactions. Using the replica method, the models are mapped to the…

We show that the loop $O(n)$ model on the hexagonal lattice exhibits exponential decay of loop sizes whenever $n> 1$ and $x<\tfrac{1}{\sqrt{3}}+\varepsilon(n)$, for some suitable choice of $\varepsilon(n)>0$. It is expected that, for $n…

The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been predicted by Nienhuis that for $0\le…

We study a class of loop models, parameterized by a continuously varying loop fugacity n, on the hydrogen-peroxide lattice, which is a three-dimensional cubic lattice of coordination number 3. For integer n > 0, these loop models provide…

The phase diagram of the O(n) model, in particular the special case $n=0$, is studied by means of transfer-matrix calculations on the loop representation of the O(n) model. The model is defined on the square lattice; the loops are allowed…

We study the loop $O(n)$ model on the honeycomb lattice. By means of local non-planar deformations of the lattice, we construct a discrete stress-energy tensor. For $n\in [0,2]$, it gives a new observable satisfying a part of Cauchy-Riemann…

We explore the phase diagram of the O($n$) loop model on the square lattice in the $(x,n)$ plane, where $x$ is the weight of a lattice edge covered by a loop. These results are based on transfer-matrix calculations and finite-size scaling.…

We consider a gauged O(n) spin model, n >= 2, in one dimension which contains both the pure O(n) and RP(n-1) models and which interpolates between them. We show that this model is equivalent to the non-interacting sum of the O(n) and Ising…

The partition function of the O(n) loop model on the honeycomb lattice is mapped to that of the O(n) loop model on the 3-12 lattice. Both models share the same operator content and thus critical exponents. The critical points are related…

The loop $O(n)$ model is a family of probability measures on collections of non-intersecting loops on the hexagonal lattice, parameterized by a loop-weight $n$ and an edge-weight $x$. Nienhuis predicts that, for $0 \leq n \leq 2$, the model…

High temperature expansions for the free energy, the susceptibility and the second correlation moment of the classical N-vector model [also known as the O(N) symmetric classical spin Heisenberg model or as the lattice O(N) nonlinear sigma…

We show that coarse graining arguments invented for the analysis of multi-spin systems on a randomly triangulated surface apply also to the O(n) model on a random lattice. These arguments imply that if the model has a critical point with…

We reformulate the O(N) sigma model as a loop model whose configurations are the all-order strong coupling graphs of the original model. The loop configurations are represented by a pointer list in the computer and a Monte Carlo update…

A relation between a class of stationary points of the energy landscape of continuous spin models on a lattice and the configurations of a Ising model defined on the same lattice suggests an approximate expression for the microcanonical…

For $n\in [-2,2]$ the $O(n)$ model on a random lattice has critical points to which a scaling behaviour characteristic of 2D gravity interacting with conformal matter fields with $c\in [-\infty,1]$ can be associated. Previously we have…

A critical dilute O($n$) model on the kagome lattice is investigated analytically and numerically. We employ a number of exact equivalences which, in a few steps, link the critical O($n$) spin model on the kagome lattice to the exactly…