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Register a new userLifshitz-point critical behaviour to ${boldsymbol{O(epsilon^2)}}$
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We comment on a recent letter by L. C. de Albuquerque and M. M. Leite (J. Phys. A: Math. Gen. 34 (2001) L327-L332), in which results to second order in $epsilon=4-d+frac{m}{2}$ were presented for the critical exponents $ u_{{mathrm{L}}2}$, $eta_{{mathrm{L}}2}$ and $gamma_{{mathrm{L}}2}$ of d-dimensional systems at m-axial Lifshitz points. We point out that their results are at variance with ours. The discrepancy is due to their incorrect computation of momentum-space integrals. Their speculation that the field-theoretic renormalization group approach, if performed in position space, might give results different from when it is performed in momentum space is refuted.
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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].
The multi-critical fixed points of $O(N)$ symmetric models cease to exist in the $Ntoinfty$ limit, but the mechanism regulating their annihilation still presents several enigmatic aspects. Here, we explore the evolution of high-order multi-critical points in the $(d,N)$ plane and uncover a complex mosaics for their asymptotic behaviour at large $N$. This picture is confirmed by various RG approaches and constitutes a fundamental step towards the full comprehension of critical behaviour in $O(N)$ field theories.
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)].
The critical behaviour of d-dimensional n-vector models at m-axial Lifshitz points is considered for general values of m in the large-n limit. It is proven that the recently obtained large-N expansions [J. Phys.: Condens. Matter 17, S1947 (2005)] of the correlation exponents eta_{L2}, eta_{L4} and the related anisotropy exponent theta are fully consistent with the dimensionality expansions to second order in epsilon=4+m/2-d [Phys. Rev. B 62, 12338 (2000); Nucl. Phys. B 612, 340 (2001)] inasmuch as both expansions yield the same contributions of order epsilon^2/n.
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Yuri M. Pismak n Department of Theoretical Physics State University Saint-Petersburg Russia
2008
The critical behaviour of the O(n)-symmetric model with two n-vector fields is studied within the field-theoretical renormalization group approach in a D=4-2 epsilon expansion. Depending on the coupling constants the beta-functions, fixed points and critical exponents are calculated up to the one- and two-loop order, resp. (eta in two- and three-loop order). Continuous lines of fixed points and O(n)*O(2) invariant discrete solutions were found. Apart from already known fixed points two new ones were found. One agrees in one-loop order with a known fixed point, but differs from it in two-loop order.
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