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Continuum models with critical end points are considered whose Hamiltonian ${mathcal{H}}[phi,psi]$ depends on two densities $phi$ and $psi$. Field-theoretic methods are used to show the equivalence of the critical behavior on the critical line and at the critical end point and to give a systematic derivation of critical-end-point singularities like the thermal singularity $sim|{t}|^{2-alpha}$ of the spectator-phase boundary and the coexistence singularities $sim |{t}|^{1-alpha}$ or $sim|{t}|^{beta}$ of the secondary density $<psi>$. The appearance of a discontinuity eigenexponent associated with the critical end point is confirmed, and the mechanism by which it arises in field theory is clarified.
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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.
An introduction to the theory of critical behavior at Lifshitz points is given, and the recent progress made in applying the field-theoretic renormalization group (RG) approach to $phi^4$ $n$-vector models representing universality classes of $m$-axial Lifshitz points is surveyed. The origins of the difficulties that had hindered a full two-loop RG analysis near the upper critical dimension for more than 20 years and produced long-standing contradictory $epsilon$-expansion results are discussed. It is outlined how to cope with them. The pivotal role the considered class of continuum models might play in a systematic investigation of anisotropic scale invariance within the context of thermal equilibrium systems is emphasized. This could shed light on the question of whether anisotropic scale invariance implies an even larger invariance, as recently claimed in the literature.
The critical behaviour of semi-infinite $d$-dimensional systems with short-range interactions and an O(n) invariant Hamiltonian is investigated at an $m$-axial Lifshitz point with an isotropic wave-vector instability in an $m$-dimensional subspace of $mathbb{R}^d$ parallel to the surface. Continuum $|bphi|^4$ models representing the associated universality classes of surface critical behaviour are constructed. In the boundary parts of their Hamiltonians quadratic derivative terms (involving a dimensionless coupling constant $lambda$) must be included in addition to the familiar ones $proptophi^2$. Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in $d=4+frac{m}{2}-epsilon$ dimensions (with $epsilon>0$) are located at $lambda=lambda^*=Or(epsilon)$. At second order in $epsilon$, the surface critical exponents of both the ordinary and the special transitions start to deviate from their $m=0$ analogues. Results to order $epsilon^2$ are presented for the surface critical exponent $beta_1^{rm ord}$ of the ordinary transition. The scaling dimension of the surface energy density is shown to be given exactly by $d+m (theta-1)$, where $theta= u_{l4}/ u_{l2}$ is the bulk anisotropy exponent.
The critical behavior of d-dimensional systems with an n-component order parameter is reconsidered at (m,d,n)-Lifshitz points, where a wave-vector instability occurs in an m-dimensional subspace of ${mathbb R}^d$. Our aim is to sort out which ones of the previously published partly contradictory $epsilon$-expansion results to second order in $epsilon=4+frac{m}{2}-d$ are correct. To this end, a field-theory calculation is performed directly in the position space of $d=4+frac{m}{2}-epsilon$ dimensions, using dimensional regularization and minimal subtraction of ultraviolet poles. The residua of the dimensionally regularized integrals that are required to determine the series expansions of the correlation exponents $eta_{l2}$ and $eta_{l4}$ and of the wave-vector exponent $beta_q$ to order $epsilon^2$ are reduced to single integrals, which for general m=1,...,d-1 can be computed numerically, and for special values of m, analytically. Our results are at variance with the original predictions for general m. For m=2 and m=6, we confirm the results of Sak and Grest [Phys. Rev. B {bf 17}, 3602 (1978)] and Mergulh{~a}o and Carneiros recent field-theory analysis [Phys. Rev. B {bf 59},13954 (1999)].
A two-loop renormalization group analysis of the critical behaviour at an isotropic Lifshitz point is presented. Using dimensional regularization and minimal subtraction of poles, we obtain the expansions of the critical exponents $ u$ and $eta$, the crossover exponent $phi$, as well as the (related) wave-vector exponent $beta_q$, and the correction-to-scaling exponent $omega$ to second order in $epsilon_8=8-d$. These are compared with the authors recent $epsilon$-expansion results [{it Phys. Rev. B} {bf 62} (2000) 12338; {it Nucl. Phys. B} {bf 612} (2001) 340] for the general case of an $m$-axial Lifshitz point. It is shown that the expansions obtained here by a direct calculation for the isotropic ($m=d$) Lifshitz point all follow from the latter upon setting $m=8-epsilon_8$. This is so despite recent claims to the contrary by de Albuquerque and Leite [{it J. Phys. A} {bf 35} (2002) 1807].
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