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

In this paper we consider a class of (2+1)D schematic models with four-fermion interactions that are effectively used in studying condensed-matter systems with planar crystal structure, and especially graphene. Symmetry breaking in these models occurs due to a possible appearance of condensates. Special attention is paid to the symmetry properties of the appearing condensates in the framework of discrete chiral and $\mathcal C$, $\mathcal P$ and $\mathcal T$ transformations. Moreover, boundary conditions corresponding to carbon nanotubes are considered and their relations with the effect of an applied external magnetic field are studied. To this end we calculated the effective potential for the nanotube model including effects of finite temperature, density and an external magnetic field. As an illustration we made numerical calculations of the chiral symmetry properties in a simpler Gross--Neveu model with only one condensate taken into account. We also investigated the phase structure of the nanotube model under the influence of the Aharonov--Bohm effect and demonstrated that there is a nontrivial relation between the magnitude of the Aharonov--Bohm phase, compactification of the spatial dimension and thermal restoration of the originally broken chiral symmetry.