Curvature Induced Phase Transition in a Four-Fermion Theory Using the Weak Curvature Expansion

Curvature induced phase transition is thoroughly investigated in a four- fermion theory with $N$ components of fermions for arbitrary space-time dimensions $(2 \leq D < 4)$. We adopt the $1/N$ expansion method and calculate the effective potential for a composite operator $\bar{\psi}\psi$. The resulting effective potential is expanded asymptotically in terms of the space-time curvature $R$ by using the Riemann normal coordinate. We assume that the space-time curves slowly and keep only terms independent of $R$ and terms linear in $R$. Evaluating the effective potential it is found that the first-order phase transition is caused and the broken chiral symmetry is restored for a large positive curvature. In the space-time with a negative curvature the chiral symmetry is broken down even if the coupling constant of the four-fermion interaction is sufficiently small. We present the behavior of the dynamically generated fermion mass. The critical curvature, $R_{cr}$, which divides the symmetric and asymmetric phases is obtained analytically as a function of the space-time dimension $D$. At the four-dimensional limit our result $R_{cr}$ agrees with the exact results known in de Sitter space and Einstein universe.