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In this paper we consider a class of (2+1)D schematic models withfour-fermion interactions that are effectively used in studyingcondensed-matter systems with planar crystal structure, and especiallygraphene. Symmetry breaking in these models occurs due to a possible appearanceof condensates. Special attention is paid to the symmetry properties of theappearing condensates in the framework of discrete chiral and $\mathcal C$,$\mathcal P$ and $\mathcal T$ transformations. Moreover, boundary conditionscorresponding to carbon nanotubes are considered and their relations with theeffect of an applied external magnetic field are studied. To this end wecalculated the effective potential for the nanotube model including effects offinite temperature, density and an external magnetic field. As an illustrationwe made numerical calculations of the chiral symmetry properties in a simplerGross--Neveu model with only one condensate taken into account. We alsoinvestigated the phase structure of the nanotube model under the influence ofthe Aharonov--Bohm effect and demonstrated that there is a nontrivial relationbetween the magnitude of the Aharonov--Bohm phase, compactification of thespatial dimension and thermal restoration of the originally broken chiralsymmetry.

Phase transitions in hexagonal, graphene-like lattice sheets and nanotubes under the influence of external conditions