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

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

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

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

Critical behavior at m-axial Lifshitz points: field-theory analysis and ϵ\epsilon-expansion results

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 Carneiro's recent field-theory analysis [Phys. Rev. B {\bf 59},13954 (1999)].Comment: Latex file with one figure (eps-file). Latex file uses texdraw to generate figures that are included in the tex

Thermodynamics of the incommensurate state in Rb_2WO_4: on the Lifshitz point in A`A``BX_4 compounds

We consider the evolution of the phase transition from the parent hexagonal phase $P6_{3}/mmc$ to the orthorhombic phase $Pmcn$ that occurs in several compounds of $A'A''BX_{4}$ family as a function of the hcp lattice parameter $c/a$. For compounds of $K_{2}SO_{4}$ type with $c/a$ larger than the threshold value 1.26 the direct first-order transition $Pmcn-P6_{3}/mmc$ is characterized by the large entropy jump $Rln2$. For compounds $Rb_{2}WO_{4}$, $K_{2}MoO_{4}$, $K_{2}WO_{4}$ with $c/a<1.26$ this transition occurs via an intermediate incommensurate $(Inc)$ phase. DSC measurements were performed in $Rb_{2}WO_{4}$ to characterize the thermodynamics of the $Pmcn-Inc-P6_{3}/mmc$ transitions. It was found that both transitions are again of the first order with entropy jumps $0.2Rln2 and$0.3Rln2$. Therefore, at$c/a ~ 1.26$the$A'A''BX_{4}$compounds reveal an unusual Lifshitz point where three first order transition lines meet. We propose the coupling of crystal elasticity with$BX_{4}$ tetrahedra orientation as a possible source of the transitions discontinuity.Comment: 13 pages,1 Postscript figure. Submitted as Brief Report to Phys. Rev. B, this paper reports a new work in Theory and Experiment, directed to Structural Phase Transition

Boundary critical behavior at m-axial Lifshitz points for a boundary plane parallel to the modulation axes

The critical behavior of semi-infinite $d$-dimensional systems with $n$-component order parameter $\bm{\phi}$ and short-range interactions is investigated at an $m$-axial bulk Lifshitz point whose wave-vector instability is isotropic in an $m$-dimensional subspace of $\mathbb{R}^d$. The associated $m$ modulation axes are presumed to be parallel to the surface, where $0\le m\le d-1$. An appropriate semi-infinite $|\bm{\phi}|^4$ model representing the corresponding universality classes of surface critical behavior is introduced. It is shown that the usual O(n) symmetric boundary term $\propto \bm{\phi}^2$ of the Hamiltonian must be supplemented by one of the form $\mathring{\lambda} \sum_{\alpha=1}^m(\partial\bm{\phi}/\partial x_\alpha)^2$ involving a dimensionless (renormalized) coupling constant $\lambda$. The implied boundary conditions are given, and the general form of the field-theoretic renormalization of the model below the upper critical dimension $d^*(m)=4+{m}/{2}$ is clarified. Fixed points describing the ordinary, special, and extraordinary transitions are identified and shown to be located at a nontrivial value $\lambda^*$ if $\epsilon\equiv d^*(m)-d>0$. The surface critical exponents of the ordinary transition are determined to second order in $\epsilon$. Extrapolations of these $\epsilon$ expansions yield values of these exponents for $d=3$ in good agreement with recent Monte Carlo results for the case of a uniaxial ($m=1$) Lifshitz point. The scaling dimension of the surface energy density is shown to be given exactly by $d+m (\theta-1)$, where $\theta=\nu_{l4}/\nu_{l2}$ is the anisotropy exponent.Comment: revtex4, 31 pages with eps-files for figures, uses texdraw to generate some graphs; to appear in PRB; v2: some references and additional remarks added, labeling in figure 1 and some typos correcte

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.

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

Numerical calculations of the phase diagram of cubic blue phases in cholesteric liquid crystals

We study the static properties of cubic blue phases by numerically minimising the three-dimensional, Landau-de Gennes free energy for a cholesteric liquid crystal close to the isotropic-cholesteric phase transition. Thus we are able to refine the powerful but approximate, semi-analytic frameworks that have been used previously. We obtain the equilibrium phase diagram and discuss it in relation to previous results. We find that the value of the chirality above which blue phases appear is shifted by 20% (towards experimentally more accessible regions) with respect to previous estimates. We also find that the region of stability of the O5 structure -- which has not been observed experimentally -- shrinks, while that of BP I (O8-) increases thus giving the correct order of appearance of blue phases at small chirality. We also study the approach to equilibrium starting from the infinite chirality solutions and we find that in some cases the disclination network has to assemble during the equilibration. In these situations disclinations are formed via the merging of isolated aligned defects.Comment: 16 pages, 5 figures. Accepted for publication in Phys. Rev.