We investigate the critical behavior of three-dimensional antiferromagnetic CP(N-1) (ACP(N-1)) models in cubic lattices, which are characterized by a global U(N) symmetry and a local U(1) gauge symmetry. Assuming that critical fluctuations are associated with a staggered gauge-invariant (Hermitian traceless matrix) order parameter, we determine the corresponding Landau-Ginzburg-Wilson (LGW) model. For N=3 this mapping allows us to conclude that the three-component ACP(2) model undergoes a continuous transition that belongs to the O(8) vector universality class, with an effective enlargement of the symmetry at the critical point. This prediction is confirmed by numerical analyses of the finite-size scaling behaviors of the ACP(2) and the O(8) vector models, which show the same universal features at their transitions. We also present a renormalization-group (RG) analysis of the LGW theories for N≥4. We compute perturbative series in two different renormalization schemes and analyze the corresponding RG flow. We do not find stable fixed points that can be associated with continuous transitions.
We investigate the critical properties of the three-dimensional antiferromagnetic RP^{N-1} model, which is characterized by a global O(N) symmetry and a discrete Z_{2} gauge symmetry. We perform a field-theoretical analysis using the Landau-Ginzburg-Wilson (LGW) approach and a numerical Monte Carlo study. The LGW field-theoretical results are obtained by high-order perturbative analyses of the renormalization-group flow of the most general Φ^{4} theory with the same global symmetry as the model, assuming a gauge-invariant order-parameter field. For N=4 no stable fixed point is found, implying that any transition must necessarily be of first order. This is contradicted by the numerical results that provide strong evidence for a continuous transition. This suggests that gauge modes are not always irrelevant, as assumed by the LGW approach, but they may play an important role to determine the actual critical dynamics at the phase transition of O(N) symmetric models with a discrete Z_{2} gauge symmetry.
Determining whether an additional local symmetry affects the universality class of a sta-tistical model is an important issue in the theory of critical phenomena. A basic example is provided by the RP N −1 model, in which N -component spin variables at each lattice site interact through an Hamiltonian invariant under global O(N ) rotations and local spin re-versals. The local symmetry makes the difference with the usual O(N ) model and amounts to the head-tail symmetry characteristic of liquid crystals. In three dimensions, the weak first order transition observed in numerical simulations of the ferromagnetic model is con-sistent with the mean field scenario. On the other hand, in the two-dimensional case –the one we focus on in this thesis – fluctuations are stronger and minimize the reliability of mean field predictions, as illustrated by the phase transition of the three-state Potts model, which becomes continuous on planar lattices. For the RP N −1 model, the absence of spontaneous breaking of continuous symmetry in two dimensions generically suggests that criticality is limited to zero temperature, and Monte Carlo studies for T → 0 showed a fast growth of the correlation length which made particularly hard to reach the asymp-totic limit and draw conclusions. On the other hand, the possibility of finite temperature topological transitions similar to the Berezinskii-Kosterlitz-Thouless (BKT) one – which should definitely occur for RP 1 ∼ O(2) – and mediated by ”disclination” defects has also been debated in numerical studies. While two-dimensional criticality has allowed for an impressive amount of exact solutions thanks to lattice integrability and conformal field theory, models with local symmetries traditionally remained outside the range of application of these methods. In this thesis, however, we show how the renormalization group fixed points of the RP N −1 model and of its complex generalization, the CP N −1 model, can be accessed in an exact way. This is achieved in the scale invariant scattering framework which, as we review in the introductory part of the thesis, implements in the basis of particle excitations the infinite-dimensional conformal symmetry characteristic of critical points in two dimensions and has provided in the last few years new results for pure and disordered systems. In the last part of the thesis, we will exploit the generality of the scale invariant scattering method to progress with another long standing problem of two-dimensional criticality, namely that of spin clusters in Potts correlated percolation. It has been known for long time that the problem can be addressed considering two coupled Potts models, but the need to consider the number of states of one of these models as a continuous variable has severely limited the analytical or numerical study of the critical points. Also here, the ability of the scattering method to enforce conformal invariance for the internal symmetry characteristic of the universality class will allow us to obtain exact equations for the critical points and to determine their solutions in the relevant limits.
The critical properties of the antiferromagnetic Heisenberg model on the three-dimensional stacked-triangular lattice are studied by means of a large-scale Monte Carlo simulation in order to get insight into the controversial issue of the criticality of the noncollinear magnets with the $O(3)\ifmmode\times\else\texttimes\fi{}O(2)$ symmetry. The maximum size studied is
3843, considerably larger than the sizes studied by the previous numerical works on the model. Availability of such large-size data enables us to examine the detailed critical properties including the effect of corrections to the leading scaling. Strong numerical evidence of the continuous nature of the transition is obtained. Our data indicate the existence of significant corrections to the leading scaling. Careful analysis by taking account of the possible corrections yields critical exponents estimates, $\ensuremath{\alpha}=0.44(3), \ensuremath{\beta}=0.26(2), \ensuremath{\gamma}=1.03(5), \ensuremath{\nu}=0.52(1), \ensuremath{\eta}=0.02(5)$, and the chirality exponents, ${\ensuremath{\beta}}_{\ensuremath{\kappa}}=0.40(3)$ and ${\ensuremath{\gamma}}_{\ensuremath{\kappa}}=0.77(6)$, supporting the existence of the
O(3) chiral [or $O(3)\ifmmode\times\else\texttimes\fi{}O(2)$] universality class governed by a ``chiral'' fixed point. We also obtain an indication that the underlying fixed point is of the focus type, characterized by the complex-valued correction-to-scaling exponent, $\ensuremath{\omega}=0.{1}_{\ensuremath{-}0.05}^{+0.4}+i\phantom{\rule{4pt}{0ex}}0.{7}_{\ensuremath{-}0.4}^{+0.1}$. The focus-like nature of the chiral fixed point accompanied by the spiral-like renormalization-group (RG) flow is likely to be the origin of the apparently complicated critical behavior. The results are compared and discussed in conjunction with the results of other numerical simulations, several distinct types of RG calculations including the higher-order perturbative massive and massless RG calculations and the nonperturbative functional RG calculation, and the conformal-bootstrap program.
We study the phase diagram and critical behaviors of three-dimensional lattice
Z2-gauge
N-vector models, in which an
N-component real field is minimally coupled with
Z2-gauge link variables. These models are invariant under global
O(N) and local
Z2 transformations. They present three phases characterized by the spontaneous breaking of the global
O(N) symmetry and by the different topological properties of the
Z2-gauge correlations. We address the nature of the three transition lines separating the three phases. The theoretical predictions are supported by numerical finite-size scaling analyses of Monte Carlo data for the
N=2 model. In this case, continuous transitions can be observed along both transition lines where the
N-component spins order, in the regimes of small and large inverse gauge coupling
K. Even though these continuous transitions belong to the same
XY universality class, their critical modes turn out to be different. When the gauge variables are disordered (small
K), the relevant order-parameter field is a gauge-invariant bilinear combination of the vector field. On the other hand, when the gauge variables are ordered (large
K), the order-parameter field is the gauge-dependent
N-vector field, whose critical behavior can only be probed by using a stochastic gauge fixing that reduces the gauge freedom.
We investigate the controversial issue of the existence of universality classes describing critical phenomena in three-dimensional systems characterized by a matrix order parameter with symmetry $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(N)$ and symmetry-breaking pattern $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(N)\ensuremath{\rightarrow}\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(N\ensuremath{-}2)$. Physical realizations of these systems are, for example, frustrated spin models with noncollinear order. Starting from the field-theoretical Landau-Ginzburg-Wilson Hamiltonian, we consider the massless critical theory and the minimal-subtraction scheme without
ϵ expansion. The three-dimensional analysis of the corresponding five-loop series shows the existence of a stable fixed point for
N=2 and
N=3, confirming recent field-theoretical results based on a six-loop expansion in the alternative zero-momentum renormalization scheme defined in the massive disordered phase. In addition, we report numerical Monte Carlo simulations of a class of three-dimensional $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(2)$-symmetric lattice models. The results provide further support to the existence of the $\mathrm{O}(2)\ensuremath{\bigotimes}\mathrm{O}(2)$ universality class predicted by the field-theoretical analyses.