Symmetry breaking in Pottsφ3field theory. Crossover in random ferromagnets

Abstract

We investigate the effects of quadratic and trilinear symmetry breaking in a ${\ensuremath{\varphi}}^{3}$ field theory for the ($n+1$)-state Potts model by means of renormalized perturbation theory to one-loop order in dimension $d=6\ensuremath{-}\ensuremath{\epsilon}$. For $n+1={2}^{m}$ the break in quadratic symmetry splits the field into $m$ critical and $n\ensuremath{-}m$ noncritical components, and we allow for three trilinear couplings $v$, $w$, $z$ between different field components. In the limit $n=m=0$ the Hamiltonian represents a bond-diluted Ising ferromagnet near the percolation threshold with anisotropy parameter ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{m}}^{2}\ensuremath{\approx}{e}^{\ensuremath{-}\frac{2J}{T}}$ and critical mass $t={p}_{c}(T)\ensuremath{-}p$, where $T$ is the absolute temperature, $p$ the concentration of occupied bonds, and ${p}_{c}(T)$ a point on the critical line. We find that for any nonzero quadratic anisotropy there are only nonsymmetric trilinear fixed-point (FP) couplings ${v}^{*}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\mu}})$, ${w}^{*}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\mu}})$, ${z}^{*}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\mu}})$; $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\mu}}=\frac{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{m}}{\ensuremath{\kappa}}$ and $\ensuremath{\kappa}$ is a scale parameter. The multicritical percolation point ${v}_{\mathrm{II}}^{*}(0)={w}_{\mathrm{II}}^{*}(0)={z}_{\mathrm{II}}^{*}(0)={(\frac{2\ensuremath{\epsilon}}{7})}^{\frac{1}{2}}$ is the only symmetric FP in the parameter space ($v, w, z$), and it is not completely stable under trilinear symmetry breaking even in the absence of quadratic anisotropy, since the stability matrix has a marginal eigenvector. Starting from the percolation point we find that, despite a break in trilinear symmetry induced by quadratic anisotropy, there is a crossover to the critical line with asymptotic mean-field exponents. The flow of the couplings and the effective exponents for the crossover are calculated. A further result is that trilinear symmetry breaking yields a new completely stable, asymmetric FP ${v}_{\mathrm{I}}^{*}={z}_{\mathrm{I}}^{*}=0$, $\frac{{w}_{\mathrm{I}}^{*2}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\mu}})}{(1+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\mu}}}^{2})}=\frac{2\ensuremath{\epsilon}}{7}$, with nonclassical critical exponents ${\ensuremath{\eta}}_{\mathrm{I}}=+\frac{\ensuremath{\epsilon}}{21}$, ${\ensuremath{\nu}}_{\mathrm{I}}^{\ensuremath{-}1}\ensuremath{-}2=\frac{5\ensuremath{\epsilon}}{21}$ for all finite $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\mu}}$.