Chiral Phase Transition In Interacting Systems Of Dirac Fermions

High temperature superconductor (HTSC) and graphene are two strongly cor-related electron systems those have attracted great research interests in recent years. They share one common feature that the low-energy elementary excitations are mass-less Dirac fermions, arising from the special band structures. These Dirac fermions satisfy relativistic Dirac wave equation and can be described by three-dimensional quantum field theory. The dynamical chiral symmetry breaking caused by gauge field and Coulomb interaction, which can describe the low-energy physics of high tempera-ture superconductor and graphene respectively. The theory describing the low-energy physics of of HTSC is (2+1)-dimension QED(QED3), while that is Coulomb inter-action for the Dirac fermions in graphene. The dynamic chiral symmetry breaking (DCSB) in QED3 has been investigated intensively for more than twenty years and about decade for that in graphene, On the one hand, these investigations may help to gain deeper understanding of DCSB in QCD. On the other hand, this non-perturbative phenomenon can be widely used to understand many important physical phenomena in HTSC and graphene.In this thesis, the chiral quantum phase transition and the consequent effects on low-energy physics of system are studied by discussing on the DCSB driven by gauge field and Coulomb interaction in HTSC and graphene, respectively. The DCSB is realized by forming fermion-anti-fermion pairs mediated by strong gauge (Coulomb) interaction. Research shows that when the interaction between Dirac fermions is strong enough, the Dirac fermions will acquire finite dynamic mass by vacuum condensations and break the original chiral symmetry contemporarily. Through the theoretical anal-ysis and calculation of the Dyson-Schwinger (DS) equation, we have identified the conditions of the occurrence of DCSB in the HTSC and the graphene, pointed out that the long-range nature of gauge (Coulomb) interaction play a key role in DCSB. The results are also used to discuss the physical effects induced by DCSB. After the specific calculations, we found that temperature, chemical potential, impurity scattering even the mass of gauge field will make the interaction become short-ranged and prevent the generation of dynamic mass, then reduce the fermion critical flavor number Nc and dynamical mass m.Furthermore, in order to get a better understanding of the physical effects induced by chiral phase transition in graphene, we calculate the specific heat and susceptibil-ity of the system and show that they exhibit distinct behaviors in the semimetal and insulator phases. In the semimetal phase, the Coulomb interaction will lead to non-Fermi liquid behavior of specific heat and susceptibility; while in the insulating phase, the specific heat and susceptibility are strongly suppressed because of finite gap of fermion excitations. Apparently, both specific heat and susceptibility manifest quite different behaviors in these two phases and can distinguish them clearly, which pro-vides important information on experiment research on chiral phase transition. These quantities can be compared with experimental results and hence may help to understand the physical consequence of chiral phase transition.