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Phase transitions are interesting phenomena, which are found in many field of research of modern physics. Their study is crucial for understanding a large variety of phenomena. Landau was the first to propose a general framework able to provide a semi-quantitative explanation of the critical phenomena. His theory is based on the assumption that a phase transition is always associated with the growth of a certain quantity called order parameter, a measure of the system transition towards the low-temperature ordered phase. According to Landau, once an order parameter is identi- fied, the symmetry of the system is sufficient to provide an effective theory allowing to obtain a semi-quantitative interpretation of the critical phenomenon. The modern approach to phase transitions is still based on this theory, even though it has been modified and completed over time. Nowadays Landau-Ginsburg-Wilson (LGW) theory represents a powerful technique able to accurately classify a large variety of phase transitions. As the Landau theory, the LGW approach requires the identification of an order parameter. Then, one proceeds constructing an effective field theory whose action is obtained considering the most general fourth-order polynomial in the order parame- ter consistent with the original symmetry of the system. Hence, the universal critical behavior is inferred studying the renormalization group flow of such a theory.
In systems characterized by a gauge symmetry, the LGW approach starts by considering a gauge invariant order parameter implicitly assuming that gauge degrees of freedom do not couple with critical modes. This choice is supported by the fact that, in any spontaneous symmetry breaking phenomenon, gauge symmetry is preserved. However, there are some cases in which this assumption is found lacking. Recent works suggested the hypothesis that, in systems characterized by a gauge symmetry, critical modes might not always be entirely represented by a gauge invariant order parameter.
In order to investigate the correctness of the LGW theory predictions in models featuring a local gauge symmetry, in this thesis, we considered the RP^(N−1) model. It is a lattice model characterized by a global O(N) symmetry, a local Z_2 gauge symmetry. We performed an analysis, based on LGW approach, of the antiferromagnetic RP^(N−1) model. We constructed our LGW Hamiltonian with a staggered gauge-invariant order parameter. According to this analysis, RP^(N−1) models undergo a continuous phase transition only for N ≤ 3. In particular, we verified that for N = 2 the system undergoes a continuous transition that belongs to the O(2) vector model universality class. In the same way, for N = 3 it undergoes a continuous phase transition that belongs to the O(5) universality class with a dynamical enlargement of the symmetry at the critical point. Monte Carlo simulations confirm this result. Instead, for N ≥ 4, we did not find any stable fixed point. This suggests that any phase transition for N ≥ 4 should be of first order. In order to test this result, we performed a Monte Carlo analysis of the RP^3 model. We found a phase transition at the inverse tem- perature β ≈ 6.77. Using a finite-size scaling analysis, we obtained an estimate for the critical exponent of the correlation length ν ≈ 0.66. This value is significantly distant from the limit one ν = 1/3 that would have indicated a first order phase transition. In order to reinforce this statement, we performed an analysis of the energy distribution near the critical point and found no trace of double peaks. Therefore, we are led to the conclusion that the antiferromagnetic RP^3 model undergoes a continuous phase transition.
This result is in contrast with our field theory analysis based on the LGW theory. This proves that a gauge invariant order parameter might not always capture all the relevant critical modes questioning, as a consequence, the validity of the traditional LGW approach when gauge symmetries are present.