Symmetry breaking in Pottsφ3field theory. Crossover in random ferromagnets

We investigate the effects of quadratic and trilinear symmetry breaking in a ${\phi }^3$ field theory for the ($n+1\mathrm{}$)-state Potts model by means of renormalized perturbation theory to one-loop order in dimension $d=6-\epsilon$. For $n+1=2^m$ the break in quadratic symmetry splits the field into $m$ critical and $n-m$ noncritical components, and we allow for three trilinear couplings $v$, $w$, $z$ between different field components. In the limit $n=m=0\mathrm{}$ the Hamiltonian represents a bond-diluted Ising ferromagnet near the percolation threshold with anisotropy parameter ${\tilde{m}}^2\approx e^{-\frac{2J}{T}}$ and critical mass $t=p_c\mathrm{}\left(T\right)-p$, where $T$ is the absolute temperature, $p$ the concentration of occupied bonds, and $p_c\mathrm{}\left(T\right)\mathrm{}$ a point on the critical line. We find that for any nonzero quadratic anisotropy there are only nonsymmetric trilinear fixed-point (FP) couplings $v^{*}\left(\tilde{\mu }\right)$, $w^{*}\left(\tilde{\mu }\right)$, $z^{*}\left(\tilde{\mu }\right)$; $\tilde{\mu }=\frac{\tilde{m}}{\kappa }$ and $\kappa$ is a scale parameter. The multicritical percolation point $v_{\mathrm{II}}^{*}\mathrm{}\left(0\right)=w_{\mathrm{II}}^{*}\mathrm{}\left(0\right)=z_{\mathrm{II}}^{*}\mathrm{}\left(0\right)={\left(\frac{2\epsilon }{7}\right)}^{\frac{1}{2}}$ 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 $v_{\mathrm{I}}^{*}=z_{\mathrm{I}}^{*}=0\mathrm{}$, $\frac{w_{\mathrm{I}}^{*2\mathrm{}}\left(\tilde{\mu }\right)}{(1+{\tilde{\mu }}^2)}=\frac{2\epsilon }{7}$, with nonclassical critical exponents ${\eta }_{\mathrm{I}}=+\frac{\epsilon }{21}$, ${\nu }_{\mathrm{I}}^{-1\mathrm{}}-2=\frac{5\epsilon }{21}$ for all finite $\tilde{\mu }$.