Bulk and Boundary Critical Behavior at Lifshitz Points

Lifshitz points are multicritical points at which a disordered phase, a homogeneous ordered phase, and a modulated ordered phase meet. Their bulk universality classes are described by natural generalizations of the standard $\phi^4$ model. Analyzing these models systematically via modern field-theoretic renormalization group methods has been a long-standing challenge ever since their introduction in the middle of the 1970s. We survey the recent progress made in this direction, discussing results obtained via dimensionality expansions, how they compare with Monte Carlo results, and open problems. These advances opened the way towards systematic studies of boundary critical behavior at $m$-axial Lifshitz points. The possible boundary critical behavior depends on whether the surface plane is perpendicular to one of the $m$ modulation axes or parallel to all of them. We show that the semi-infinite field theories representing the corresponding surface universality classes in these two cases of perpendicular and parallel surface orientation differ crucially in their Hamiltonian's boundary terms and the implied boundary conditions, and explain recent results along with our current understanding of this matter.Comment: Invited contribution to STATPHYS 22, to be published in the Proceedings of the 22nd International Conference on Statistical Physics (STATPHYS 22) of the International Union of Pure and Applied Physics (IUPAP), 4--9 July 2004, Bangalore, Indi

Specific heat amplitude ratios for anisotropic Lifshitz critical behaviors

We determine the specific heat amplitude ratio near a $m$-axial Lifshitz point and show its universal character. Using a recent renormalization group picture along with new field-theoretical $\epsilon_{L}$-expansion techniques, we established this amplitude ratio at one-loop order. We estimate the numerical value of this amplitude ratio for $m=1$ and $d=3$. The result is in very good agreement with its experimental measurement on the magnetic material $MnP$. It is shown that in the limit $m \to 0$ it trivially reduces to the Ising-like amplitude ratio.Comment: 8 pages, RevTex, accepted as a Brief Report in Physical Review

Statistical mechanics of double-stranded semi-flexible polymers

We study the statistical mechanics of double-stranded semi-flexible polymers using both analytical techniques and simulation. We find a transition at some finite temperature, from a type of short range order to a fundamentally different sort of short range order. In the high temperature regime, the 2-point correlation functions of the object are identical to worm-like chains, while in the low temperature regime they are different due to a twist structure. In the low temperature phase, the polymers develop a kink-rod structure which could clarify some recent puzzling experiments on actin.Comment: 4 pages, 3 figures; final version for publication - slight modifications to text and figure

A new picture of the Lifshitz critical behavior

New field theoretic renormalization group methods are developed to describe in a unified fashion the critical exponents of an m-fold Lifshitz point at the two-loop order in the anisotropic (m not equal to d) and isotropic (m=d close to 8) situations. The general theory is illustrated for the N-vector phi^4 model describing a d-dimensional system. A new regularization and renormalization procedure is presented for both types of Lifshitz behavior. The anisotropic cases are formulated with two independent renormalization group transformations. The description of the isotropic behavior requires only one type of renormalization group transformation. We point out the conceptual advantages implicit in this picture and show how this framework is related to other previous renormalization group treatments for the Lifshitz problem. The Feynman diagrams of arbitrary loop-order can be performed analytically provided these integrals are considered to be homogeneous functions of the external momenta scales. The anisotropic universality class (N,d,m) reduces easily to the Ising-like (N,d) when m=0. We show that the isotropic universality class (N,m) when m is close to 8 cannot be obtained from the anisotropic one in the limit d --> m near 8. The exponents for the uniaxial case d=3, N=m=1 are in good agreement with recent Monte Carlo simulations for the ANNNI model.Comment: 48 pages, no figures, two typos fixe

Compatibility of 1/n and epsilon expansions for critical exponents at m-axial Lifshitz points

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.Comment: 8 pages, submitted to J. Phys.

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

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].Comment: 11 pages, Latex, uses iop stylefiles, some graphs are generated automatically via texdra

Local scale invariance and strongly anisotropic equilibrium critical systems

A new set of infinitesimal transformations generalizing scale invariance for strongly anisotropic critical systems is considered. It is shown that such a generalization is possible if the anisotropy exponent \theta =2/N, with N=1,2,3 ... Differential equations for the two-point function are derived and explicitly solved for all values of N. Known special cases are conformal invariance (N=2) and Schr\"odinger invariance (N=1). For N=4 and N=6, the results contain as special cases the exactly known scaling forms obtained for the spin-spin correlation function in the axial next nearest neighbor spherical (ANNNS) model at its Lifshitz points of first and second order.Comment: 4 pages Revtex, no figures, with file multicol.sty, to appear in PR

Periodic vacuum and particles in two dimensions

Different dynamical symmetry breaking patterns are explored for the two dimensional phi4 model with higher order derivative terms. The one-loop saddle point expansion predicts a rather involved phase structure and a new Gaussian critical line. This vacuum structure is corroborated by the Monte Carlo method, as well. Analogies with the structure of solids, the density wave phases and the effects of the quenched impurities are mentioned. The unitarity of the time evolution operator in real time is established by means of the reflection positivity.Comment: Final version, additional references and the proof of reflection positivity added, 41 pages, 16 figure

Lattice models and Landau theory for type II incommensurate crystals

Ground state properties and phonon dispersion curves of a classical linear chain model describing a crystal with an incommensurate phase are studied. This model is the DIFFOUR (discrete frustrated phi4) model with an extra fourth-order term added to it. The incommensurability in these models may arise if there is frustration between nearest-neighbor and next-nearest-neighbor interactions. We discuss the effect of the additional term on the phonon branches and phase diagram of the DIFFOUR model. We find some features not present in the DIFFOUR model such as the renormalization of the nearest-neighbor coupling. Furthermore the ratio between the slopes of the soft phonon mode in the ferroelectric and paraelectric phase can take on values different from -2. Temperature dependences of the parameters in the model are different above and below the paraelectric transition, in contrast with the assumptions made in Landau theory. In the continuum limit this model reduces to the Landau free energy expansion for type II incommensurate crystals and it can be seen as the lowest-order generalization of the simplest Lifshitz-point model. Part of the numerical calculations have been done by an adaption of the Effective Potential Method, orginally used for models with nearest-neighbor interaction, to models with also next-nearest-neighbor interactions.Comment: 33 pages, 7 figures, RevTex, submitted to Phys. Rev.

General criteria for the stability of uniaxially ordered states of Incommensurate-Commensurate Systems

Reconsidering the variational procedure for uniaxial systems modeled by continuous free energy functionals, we derive new general conditions for thermodynamic extrema. The utility of these conditions is briefly illustrated on the models for the classes I and II of incommensurate-commensurate systems.Comment: 5 pages, to be published in Phys. Rev. Let