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Register a new userCritical Behavior at M-Axial Lifshitz Points
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H.W.Diehl
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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.
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We investigate the critical behavior that d-dimensional systems with short-range forces and a n-component order parameter exhibit at Lifshitz points whose wave-vector instability occurs in a m-dimensional isotropic subspace of ${mathbb R}^d$. Utilizing dimensional regularization and minimal subtraction of poles in $d=4+{mover 2}-epsilon$ dimensions, we carry out a two-loop renormalization-group (RG) analysis of the field-theory models representing the corresponding universality classes. This gives the beta function $beta_u(u)$ to third order, and the required renormalization factors as well as the associated RG exponent functions to second order, in u. The coefficients of these series are reduced to m-dependent expressions involving single integrals, which for general (not necessarily integer) values of $min (0,8)$ can be computed numerically, and for special values of m analytically. The $epsilon$ expansions of the critical exponents $eta_{l2}$, $eta_{l4}$, $ u_{l2}$, $ u_{l4}$, the wave-vector exponent $beta_q$, and the correction-to-scaling exponent are obtained to order $epsilon^2$. These are used to estimate their values for d=3. The obtained series expansions are shown to encompass both isotropic limits m=0 and m=d.
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 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 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.
The large-n expansion is developed for the study of critical behaviour of d-dimensional systems at m-axial Lifshitz points with an arbitrary number m of modulation axes. The leading non-trivial contributions of O(1/n) are derived for the two independent correlation exponents eta_{L2} and eta_{L4}, and the related anisotropy index theta. The series coefficients of these 1/n corrections are given for general values of m and d with 0<m<d and 2+m/2<d<4+m/2 in the form of integrals. For special values of m and d such as (m,d)=(1,4), they can be computed analytically, but in general their evaluation requires numerical means. The 1/n corrections are shown to reduce in the appropriate limits to those of known large-n expansions for the case of d-dimensional isotropic Lifshitz points and critical points, respectively, and to be in conformity with available dimensionality expansions about the upper and lower critical dimensions. Numerical results for the 1/n coefficients of eta_{L2}, eta_{L4} and theta are presented for the physically interesting case of a uniaxial Lifshitz point in three dimensions, as well as for some other choices of m and d. A universal coefficient associated with the energy-density pair correlation function is calculated to leading order in 1/n for general values of m and d.
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