Critical, crossover, and correction-to-scaling exponents for isotropic Lifshitz points to order $\boldsymbol{(8-d)^2}$

TL;DR

This paper presents a two-loop renormalization group analysis of isotropic Lifshitz points, deriving critical exponents to second order in _8 and clarifying their relation to previous general m-axial results.

Contribution

It provides explicit second-order _8-expansions of critical exponents for isotropic Lifshitz points, confirming their derivation from general m-axial results and addressing recent conflicting claims.

Findings

Derived _8-expansions of critical exponents , ta, , eta_q, and  to second order.

Showed that isotropic Lifshitz point exponents follow from general m-axial results with m=8-_8.

Refuted recent claims disputing the relation between isotropic and general m-axial Lifshitz point expansions.

Abstract

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 $\nu$ 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].