Phase transitions and critical phenomena in chern-simons, $CP^{N-1}$, and four-fermion theories

Abstract

We investigate phase transitions and critical phenomena of the field-theoretical models related to the high-temperature superconductivity. As a Landau-Ginzburg description of high-temperature superconductors, the Maxwell Chern-Simons Higgs theory is considered in the Landau gauge. The critical temperature is determined by the balance between the Higgs effect and the Chern-Simons effect which make the electromagnetic gauge field massive. We examine the effect of finite size (one periodic spatial dimension) on the finite-temperature Neel transition of the large-N $CP^{N-1}$ model. In 1+1 and 2+1 dimensions, the occurrence of the long-range Neel order is valid only for infinite N because of the Mermin-Wagner theorem. The (3+1)-dimensional anisotropic $CP^{N-1}$ model is considered as a model of antiferromagnetism in the high-temperature superconductors where the anisotropic parameter represents the weak interlayer coupling. It can possess two critical temperatures for sufficiently small sizes. Finally we consider critical exponents and scaling relations for the simplest four-fermion theory in $2 < d < 4$. Without introducing temperature, critical exponents satisfy the hyperscaling relations. But thermodynamic critical exponents do not satisfy them. In addition, we obtain the Landau-Ginzburg description of the (2+1)-dimensional Yukawa model by keeping only the relevant terms in the finite-temperature effective action.