Two-Dimensional Heisenberg Model with Nonlinear Interactions

We investigate a two-dimensional classical $N$-vector model with a nonlinear interaction (1 + \bsigma_i\cdot \bsigma_j)^p in the large-N limit. As observed for N=3 by Bl\"ote {\em et al.} [Phys. Rev. Lett. {\bf 88}, 047203 (2002)], we find a first-order transition for $p>p_c$ and no finite-temperature phase transitions for $p p_c$, both phases have short-range order, the correlation length showing a finite discontinuity at the transition. For $p=p_c$, there is a peculiar transition, where the spin-spin correlation length is finite while the energy-energy correlation length diverges.Comment: 7 pages, 2 figures in a uufile. Discussion of the theory for p = p_c revised and enlarge

Renormalization-group flow and asymptotic behaviors at the Berezinskii-Kosterlitz-Thouless transitions

We investigate the general features of the renormalization-group flow at the Berezinskii-Kosterlitz-Thouless (BKT) transition, providing a thorough quantitative description of the asymptotc critical behavior, including the multiplicative and subleading logarithmic corrections. For this purpose, we consider the RG flow of the sine-Gordon model around the renormalizable point which describes the BKT transition. We reduce the corresponding beta-functions to a universal canonical form, valid to all perturbative orders. Then, we determine the asymptotic solutions of the RG equations in various critical regimes: the infinite-volume critical behavior in the disordered phase, the finite-size scaling limit for homogeneous systems of finite size, and the trap-size scaling limit occurring in 2D bosonic particle systems trapped by an external space-dependent potential.Comment: 16 pages, refs adde

Three-dimensional ferromagnetic CP(N-1) models

We investigate the critical behavior of three-dimensional ferromagnetic CP(N-1) models, which are characterized by a global U(N) and a local U(1) symmetry. We perform numerical simulations of a lattice model for N=2, 3, and 4. For N=2 we find a critical transition in the Heisenberg O(3) universality class, while for N=3 and 4 the system undergoes a first-order transition. For N=3 the transition is very weak and a clear signature of its discontinuous nature is only observed for sizes L>50. We also determine the critical behavior for a large class of lattice Hamiltonians in the large-N limit. The results confirm the existence of a stable large-N CP(N-1) fixed point. However, this evidence contradicts the standard picture obtained in the Landau-Ginzburg-Wilson (LGW) framework using a gauge-invariant order parameter: the presence of a cubic term in the effective LGW field theory for any N>2 would usually be taken as an indication that these models generically undergo first-order transitions.Comment: 14 page

Critical mass renormalization in renormalized phi4 theories in two and three dimensions

We consider the O(N)-symmetric phi4 theory in two and three dimensions and determine the nonperturbative mass renormalization needed to obtain the phi4 continuum theory. The required nonperturbative information is obtained by resumming high-order perturbative series in the massive renormalization scheme, taking into account their Borel summability and the known large-order behavior of the coefficients. The results are in good agreement with those obtained in lattice calculations.Comment: 4 page

Interacting N-vector order parameters with O(N) symmetry

We consider the critical behavior of the most general system of two N-vector order parameters that is O(N) invariant. We show that it may a have a multicritical transition with enlarged symmetry controlled by the chiral O(2)xO(N) fixed point. For N=2, 3, 4, if the system is also invariant under the exchange of the two order parameters and under independent parity transformations, one may observe a critical transition controlled by a fixed point belonging to the mn model. Also in this case there is a symmetry enlargement at the transition, the symmetry being [SO(N)+SO(N)]xC_2, where C_2 is the symmetry group of the square.Comment: 14 page

Operator Product Expansion on the Lattice: a Numerical Test in the Two-Dimensional Non-Linear Sigma-Model

We consider the short-distance behaviour of the product of the Noether O(N) currents in the lattice nonlinear sigma-model. We compare the numerical results with the predictions of the operator product expansion, using one-loop perturbative renormalization-group improved Wilson coefficients. We find that, even on quite small lattices (m a \approx 1/6), the perturbative operator product expansion describes that data with an error of 5-10% in a large window 2a \ltapprox x \ltapprox m^{-1}. We present a detailed discussion of the possible systematic errors.Comment: 53 pages, 11 figures (26 eps files

Discrete non-Abelian groups and asymptotically free models

We consider a two-dimensional $\sigma$-model with discrete icosahedral/dodecahedral symmetry. Using the perturbative renormalization group, we argue that this model has a different continuum limit with respect to the O(3) $\sigma$ model. Such an argument is confirmed by a high-precision numerical simulation.Comment: 5 pages including 6 postscript figures. Talk given at HEP01 in Budapest, Hungary, in July 200