Phase transitions in three-dimensional loop models and theCPn−1sigma model

Abstract

We consider the statistical mechanics of a class of models involving close-packed loops with fugacity n on three-dimensional lattices. The models exhibit phases of two types as a coupling constant is varied: in one, all loops are finite, and in the other, some loops are infinitely extended. We show that the loop models are discretisations of CP n−1 σ models. The finite and infinite loop phases represent, respectively, disordered and ordered phases of the σ model, and we discuss the relationship between loop properties and σ model correlators. On large scales, loops are Brownian in an ordered phase and have a non-trivial fractal dimension at a critical point. We simulate the models, finding continuous transitions between the two phases for n = 1, 2, 3 and first order transitions for n ≥ 4. We also give a renormalisation group treatment of the CP n−1 model that shows how a continuous transition can survive for values of n larger than (but close to) two, despite the presence of a cubic invariant in the Landau-Ginzburg description. The results we obtain are of broader relevance to a variety of problems, including SU (n) quantum magnets in (2+1) dimensions, Anderson localisation in symmetry class C, and the statistics of random curves in three dimensions.

Three-dimensional antiferromagneticCPN−1models

“…Because of the presence of the cubic term, on the basis of meanfield arguments, one expects the system to undergo a first-order transition for any N > 2, unless the Hamiltonian parameters are tuned so that w = 0 in the effective model. This prediction is, however, contradicted by recent numerical studies [2,8,10,11], which find evidence of continuous transitions in models that are expected to be in the same universality class as that of the 3D CP 2 model. In particular, a numerical study of 3D loop models [8] provided the estimate ν = 0.536 (13) for the correlation-length critical exponent.…”

“…This prediction is, however, contradicted by recent numerical studies [2,8,10,11], which find evidence of continuous transitions in models that are expected to be in the same universality class as that of the 3D CP 2 model. In particular, a numerical study of 3D loop models [8] provided the estimate ν = 0.536 (13) for the correlation-length critical exponent. These results imply the existence of a 3D CP 2 universality class characterized by a U(3) global symmetry and U(1) gauge invariance, with a corresponding fixed point (FP) that cannot be determined in perturbation theory at fixed N .…”

“…This approach requires the identification of the order parameter associated with the critical modes. Assuming that the critical behavior is essentially driven by gauge-invariant modes, a plausible choice for ferromagnetic (J < 0) systems is the gauge-invariant site variable [2,8] …”

Universality class of the two-dimensional polymer collapse transition

“…(7) (see Refs. [12,22,39] for more detail). We take a model with N -component complex vectors Z located on the links of the lattice, with fixed length | Z| 2 = N .…”

“…To obtain the above we classify the allowed perturbations of simple models for the θ point that show the DS exponents, making use of mappings to concrete lattice field theories [12,[19][20][21][22]. The lattice field theories for these models have SU(N ) symmetry.…”