Continuous phase transitions provide a natural setting for understanding universality. Near criticality, the correlation length diverges, microscopic details become increasingly irrelevant, and physical observables obey scaling laws characterized by universal critical exponents. Within the Functional Renormalization Group (FRG), this scale dependence is described through the Effective Average Action, which progressively incorporates fluctuations as the momentum scale is lowered. A first part of this thesis is developed within the Local Potential Approximation (LPA), where we study multicritical fixed points of scalar field theories. Their universal properties are compared with results obtained at higher orders in the derivative expansion. Particular attention is devoted to the relation between global fixed-point potentials and their local representations through finite polynomial truncations. We show that the convergence of such expansions is constrained by the analytic structure of the global solution, and that expanding around a suitable nonzero stationary point significantly improves the reconstruction of multicritical branches. Beyond the structure of individual fixed points, a central part of this thesis concerns RG irreversibility. In two dimensions, Zamolodchikov’s c-theorem describes the flow from the ultraviolet to the infrared through a monotonically decreasing c-function. Within the FRG framework, we develop a variational procedure to reconstruct RG trajectories between fixed points and integrate the corresponding c-function along them. The resulting quantity displays monotonic behaviour and becomes increasingly stable as the truncation order is raised, providing a manifestation of RG irreversibility within the approximation considered. Finally, close to the upper critical dimensions, the linearized FRG flow around the Gaussian fixed point is naturally organized in terms of Hermite polynomials. Using the epsilon-expansion, we derive a curvature-based candidate c-function and obtain an analytic explanation for the alternating curvature of successive multicritical fixed points and for their progressive accumulation around the Gaussian fixed point.
A theory of the scaling behavior near the
qc=4 state multicritical point of the two-dimensional Potts lattice gas model is developed. Proceeding from the assumption that a dilution field becomes marginal at the multicritical point while the thermal and ordering fields are relevant, a set of differential renormalization group (RG) equations for these fields are constructed. Keeping terms through second order we find that these equations are characterized by five universal parameters which we evaluate using exact as well as conjectured results. Based upon these RG equations, we investigate the physical properties of the two-dimensional Potts lattice gas for
q near
qc. For the pure Potts model with
q=qc we find logarithmic temperature corrections to the specific heat and the spontaneous magnetization. At
Tc we find
ln(r) corrections to the power law behavior of the spin-spin correlation function. For the dilute Potts model with
q=qc we find that the latent heat, the discontinuity in the magnetization, and the discontinuity in the coexisting densities vanish with an essential singularity as
T approaches the multicritical point from the first-order side. Results for
q>qc and
q<qc are also given.
We discuss the static and dynamic multicritical behavior of three-dimensional systems of
O(n∥)⊕O(n⊥) symmetry as it is explained by the field theoretical renormalization group method. Whereas the static renormalization group functions are currently know within high order expansions, we show that an account of two loop contributions refined by an appropriate resummation technique gives an accurate quantitative description of the multicritical behavior. One of the essential features of the static multicritical behavior obtained already in two loop order for the interesting case of an antiferromagnet in a magnetic field (
n∥=1,
n⊥=2) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points leading to very small transient exponents. We further pursue an analysis of dynamical multicritical behavior choosing different forms of critical dynamics and calculating asymptotic and effective dynamical exponents within the minimal subtraction scheme.