The subject of the Thesis is a theoretical description of the critical behavior of non-ideal systems. Their non-ideality manifests itself in the presence of quenched disorder, one or two surfaces, strong spatial anisotropy. These special properties are generally termed as spatial inhomogeneities.
Disordered systems are studied using the Parisi's ``massive'' field theory in fixed space dimensions $d$. We implement the idea of Parisi in non-integer dimensions and construct a smooth dependence of critical exponents on continuously varying parameter $d$. Required Feynman integrals and renormalization group (RG) functions are calculated explicitly for generic values of $d$. The massive field theory is further generalized to the case of the low-temperature phase $T<T_c$. Universal critical amplitude combinations are calculated in the three-loop approximation.
The massive field theory with appropriate normalization conditions is formulated for semi-infinite systems with a flat boundary and different boundary conditions in $2<d\le4$ space dimensions. Numerical estimates of surface critical exponents at the special and ordinary transitions are obtained in $d=3$. They are compared with available experimental, theoretical and Monte Carlo data. The effect of the surface disorder on the special transition and changes of the ordinary-transition exponents in the presence of the bulk disorder are addressed.
A new approach is proposed for the calculation of critical exponents of strongly anisotropic systems at the $m$-axial Lifshitz point (LP). The correct $\varepsilon=4+m/2-d$-expansions for all LP exponents up to $O(\varepsilon^2)$ are derived, valid for arbitrary number of anisotropy axes $m$. In particular, the non-classical anisotropy index $\theta$ is found $\forall m\in]0,d[$ in non-trivial interacting theories. We study the problem of the influence of crystal structure (cubic anisotropy) of the $m$-axial modulation subspace on the critical behavior of LPs. We calculate the corresponding crossover exponent and conclude that this anisotropy can influence the LP critical behavior. We give an RG description of semi-infinite anisotropic systems with a boundary perpendicular to one of the modulation axes.
A thorough check is performed of the local scaling invariance hypothesis by M. Henkel, an attempt to generalize the conformal invariance to anisotropic systems. We found a number of inaccuracies in its formulation and obtained new information regarding the mathematical structure and explicit form of two-point functions at the LP. Explicit expressions for correlation functions $\langle\phi\phi\rangle$ and $\langle\phi^2\phi^2\rangle$ up to $O(\varepsilon^2)$ have more complex mathematical structure as predicted by the hypothesis, which appears to be valid only for interaction-free models.
A $1/n$-expansion (with a large number of the order-parameter components $n$) for generic $m$-axial LPs is formulated, for all admissible $d$ and $0\le m\le d$ between the lower and upper critical dimensions $d_\ell(m)=2+m/2$ and $d^*(m)=4+m/2$. The results are consistent with four limiting cases: isotropic critical exponents to $O(1/n)$ at $m=0$;
isotropic LP ones at $m=d$; $\varepsilon_\ell$- and $\varepsilon$-expansions for small $\varepsilon_\ell=d-d_\ell(m)$ and $\varepsilon=d^*(m)-d$. A number of other special cases are considered, in particular, important instances of uniaxial systems in three and four dimensions. Schematic patterns of correlation exponents $\eta_2(m,d)$ and $\eta_4(m,d)$ as functions of $d$ are suggested. They are confirmed in a work of other authors, by means of exact RG equations.
Finally, we study effects of a geometric restriction of statistical systems by two parallel surfaces with different boundary conditions, where, at criticality, fluctuation-induced forces arise analogous to Casimir forces in quantum electrodynamics; their value is determined by the Casimir amplitude. We show that $\varepsilon$-expansions of Casimir amplitudes for systems with film geometry and periodic and special boundary conditions are nonanalytic. We explicitly find corrections of order $O(\varepsilon^{3/2})$ for these amplitudes. The reason for nonanalyticity is explained, numerical estimates are performed in $d=3$. For possible future progress with higher-order approximations, an alternative calculational scheme of Casimir amplitudes is studied, based on integrals of products of Hurwitz zeta functions. For strongly anisotropic systems, we argue that fluctuation-induced forces depend on the orientation of boundaries with respect to directions of anisotropy axes. For two orientations: parallel and perpendicular with respect to these directions, we perform explicit calculations of Casimir amplitudes and provide their numerical estimates in three dimensions.
