Dimensional crossover in the non-linear sigma model

Abstract

We consider dimensional crossover for an $O(N)$ model on a $d$-dimensional layered geometry of thickness $L$, in the $\sigma$-model limit, using ``environmentally friendly'' renormalization. We show how to derive critical temperature shifts, giving explicit results to one loop. We also obtain expressions for the effective critical exponents $\delta_{\rm eff}$ and $\beta_{\rm eff}$ that interpolate between their characteristic fixed point values associated with a $d$ and $(d-1)$-dimensional system in the limits $T\rightarrow T_c(L)$, with $L(T-T_c(L))^{\nu}\rightarrow\infty$, and $T\rightarrow T_c(L)$ for $L$ fixed respectively, where $T_c(L)$ is the $L$-dependent critical temperature of the system.