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-NCP^{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≥3 would usually be taken as an indication that these models generically undergo first-order 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.
The critical behavior of an MN-component order parameter Ginzburg-Landau model with isotropic and cubic interactions describing antiferromagnetic and structural phase transitions in certain crystals with complicated ordering is studied in the framework of the four-loop renormalization group (RG) approach in $(4-2\ve)$-dimensions. Using dimensional regularization and the minimal subtraction scheme, the perturbative expansions for RG functions are deduced for generic M and N and resummed by the Borel transformation combined with a conformal mapping. Investigation of the global structure of RG flows for the physically significant cases M=2 and N=2, N=3 shows that the model has a three-dimensionally stable fixed point different from the Bose one. The critical dimensionality is proved to be exactly two times smaller than its counterpart in the real cubic model: N_c^C ={1/2} N_c^R. The numerical value
NcC=1.447±0.020 is obtained from resumming the known five-loop $\ve$-series for N_c^R. Since N_c^C < 2, the critical thermodynamics of the model relevant to the phase transitions in real substances should be governed by the complex cubic fixed point with a new set of critical exponents: $\gm=1.404(25)$,
ν=0.715(10), $\et=0.0343(20)$ for N=2 and $\gm=1.390(25)$,
ν=0.702(10), $\et=0.0345(15)$ for N=3.
Scaling, universality, and renormalization are three pillars of modern critical phenomena. According to the hypothesis of universality, continuous phase transitions fall into classes mainly determined by spatial dimensionality and symmetry of the order parameter. The latter is usually reflected by the degeneracy of the ground state of the Hamiltonian. However, for certain systems at criticality, a higher symmetry emerges in the order parameter, and the associated critical behavior may become very rich. Such emergent symmetry has been found in spin ice systems, deconfined quantum critical points, high-
Tc superconductors, and so forth, and are often accompanied by very interesting critical phenomena. New results presented here are based on Monte Carlo simulations and finite-size scaling of a series of
q-state Potts models on the simple cubic lattice with ferromagnetic interactions in one lattice direction and antiferromagnetic interactions in the other two directions. The staggered magnetization appears to display an emergent continuous O(
n) symmetry with
n=
q-1, as illustrated by two-dimensional intersections of the distribution functions. Also the estimated critical exponents are consistent with the O(
n=
q-1) universality classes.
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.
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.