Bulk singularities at critical end points: a field-theory analysis

A class of continuum models with a critical end point is considered whose Hamiltonian ${\mathcal{H}}[\phi,\psi]$ involves two densities: a primary order-parameter field, $\phi$, and a secondary (noncritical) one, $\psi$. Field-theoretic methods (renormalization group results in conjunction with functional methods) are used to give a systematic derivation of singularities occurring at critical end points. Specifically, the thermal singularity $\sim|{t}|^{2-\alpha}$ of the first-order line on which the disordered or ordered phase coexists with the noncritical spectator phase, and the coexistence singularity $\sim |{t}|^{1-\alpha}$ or $\sim|{t}|^{\beta}$ of the secondary density $<\psi>$ are derived. It is clarified how the renormalization group (RG) scenario found in position-space RG calculations, in which the critical end point and the critical line are mapped onto two separate fixed points ${\mathcal P}_{\mathrm{CEP}}^*$ and ${\mathcal P}_{\lambda}^*$ translates into field theory. The critical RG eigenexponents of ${\mathcal P}_{\mathrm{CEP}}^*$ and ${\mathcal P}_{\lambda}^*$ are shown to match. ${\mathcal P}_{\mathrm{CEP}}^*$ is demonstrated to have a discontinuity eigenperturbation (with eigenvalue $y=d$), tangent to the unstable trajectory that emanates from ${\mathcal P}_{\mathrm{CEP}}^*$ and leads to ${\mathcal P}_{\lambda}^*$. The nature and origin of this eigenperturbation as well as the role redundant operators play are elucidated. The results validate that the critical behavior at the end point is the same as on the critical line.