Isotropic Lifshitz Point Research Articles

Isotropic Lifshitz scaling in four dimensions

The presence of isotropic Lifshitz points for a [Formula: see text]-symmetric scalar theory is investigated with the help of the Functional Renormalization Group. In particular, at the supposed lower critical dimension [Formula: see text], evidence for a continuous line of fixed points is found for the [Formula: see text] theory, and the observed structure presents clear similarities with the properties observed in the two-dimensional Berezinskii–Kosterlitz–Thouless phase.

Beyond the isotropic Lifshitz endpoint

The puzzle of the disappearance of isotropic Lifshitz points in condensed matter physics is explained from the point of view of the Wilsonian renormalization group. In analogy to the commensurate ideas of metamagnetic phase transitions, we describe the physics of thermodynamic states beyond an isotropic Lifshitz endpoint. Such phenomenon may be understood in terms of a statistically isotropic environment of coexisting multiple helicoidal and non-helicoidal states. In addition to the magnetic and condensed matter discussions, we consider also an interesting example in the context of cosmic evolution.

Indications of isotropic Lifshitz points in four dimensions

The presence of isotropic Lifshitz points for a U(1) symmetric scalar theory is investigated with the help of the Functional Renormalization Group at the conjectured lower critical dimension d=4. To this aim, a suitable truncation in the expansion of the effective action in powers of the field is considered and, consequently, the Renormalization Group flow is reduced to a set of ordinary differential equations for the parameters that define the effective action. Within this approximation, indications of a line of Lifshitz points are found, that present evident similarities with the properties shown by the line of fixed points observed in the two dimensional Berezinsky-Kosterlitz-Thouless phase. In particular, this line of Lifshitz points exhibits the vanishing of the expectation value of the field, together with a finite stiffness and, for specific combinations of the parameters that define the effective action, also the algebraic decay at large distance of the order parameter correlations.

Higher-order singletons, partially massless fields, and their boundary values in the ambient approach

Using ambient space we develop a fully gauge and o(d,2)-covariant approach to boundary values of AdSd+1 gauge fields. It is applied to the study of (partially) massless fields in the bulk and (higher-order) conformal scalars, i.e. singletons, as well as (higher-depth) conformal gauge fields on the boundary. In particular, we identify the corresponding generalized Fradkin–Tseytlin equations as obstructions to the extension of the off-shell boundary value to the bulk, generalizing the usual considerations for the holographic anomalies to the partially massless fields. We also relate the background fields for the higher-order singleton to the boundary values of partially massless fields and prove the appropriate generalization of the Flato–Fronsdal theorem, which is in agreement with the known structure of symmetries for the higher-order wave operator. All these facts support the following generalization of the higher-spin holographic duality: the O(N) model at a multicritical isotropic Lifshitz point should be dual to the theory of partially massless symmetric tensor fields described by the Vasiliev equations based on the higher-order singleton symmetry algebra.

Lifshitz-point correlation length exponents from the large-n expansion

The large-n expansion is applied to the calculation of thermal critical exponents describing the critical behavior of spatially anisotropic d-dimensional systems at m-axial Lifshitz points. We derive the leading non-trivial 1/n correction for the perpendicular correlation-length exponent νL2 and hence several related thermal exponents to order O(1/n). The results are consistent with known large-n expansions for d-dimensional critical points and isotropic Lifshitz points, as well as with the second-order epsilon expansion about the upper critical dimension d⁎=4+m/2 for generic m∈[0,d]. Analytical results are given for the special case d=4, m=1. For uniaxial Lifshitz points in three dimensions, 1/n coefficients are calculated numerically. The estimates of critical exponents at d=3, m=1 and n=3 are discussed.

Large-n expansion for m-axial Lifshitz points

The large-n expansion is developed for the study of critical behaviour ofd-dimensionalsystems at m-axial Lifshitz points with an arbitrary numberm of modulation axes. The leading nontrivial contributions toO (1/n) are derived for the two independent correlation exponentsηL2 andηL4, and the relatedanisotropy index θ. Theseries coefficients of these 1/n corrections are given for general values ofm and d with 0≤m≤d and 2+m/2<d<4+m/2 in the form of integrals. For special values ofm andd suchas (m,d) = (1,4), they can be computed analytically, but in general their evaluation requires numerical means. The1/n corrections are shown to reduce in the appropriate limits to those of knownlarge-n expansionsfor the case of d-dimensional isotropic Lifshitz points and critical points, respectively,and to be in conformity with available dimensionality expansions aboutthe upper and lower critical dimensions. Numerical results for the1/n coefficientsof ηL2, ηL4 and θ are presented for the physically interesting case of a uniaxial Lifshitzpoint in three dimensions, as well as for some other choices ofm andd. A universalcoefficient associated with the energy-density pair correlation function is calculated to leading order in1/n for generalvalues of m and d.

Critical, crossover and correction-to-scaling exponents for isotropic Lifshitz points to order (8 $ndash$ 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 ? and ?, the crossover exponent , as well as the (related) wave vector exponent ?q and the correction-to-scaling exponent ? to second order in 8 = 8 ? d. They are compared with the authors' recent -expansion results (2000 Phys. Rev. B 62 12338, 2001 Nucl. Phys. B 612 340) 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 ? 8. This is so despite recent claims to the contrary by de Albuquerque and Leite (2002 J. Phys. A: Math. Gen. 35 1807).

Quantum rotors with regular frustration and the quantum Lifshitz point

We have discussed the zero-temperature quantum phase transition in n-component quantum rotor Hamiltonian in the presence of regular frustration in the interaction. The phase diagram consists of ferromagnetic, helical and quantum paramagnetic phase, where the ferro-para and the helical-para phase boundary meets at a multicritical point called a (d,m) quantum Lifshitz point where (d,m) indicates that the m of the d spatial dimensions incorporate frustration. We have studied the Hamiltonian in the vicinity of the quantum Lifshitz point in the spherical limit and also studied the renormalisation group flow behaviour using standard momentum space renormalisation technique (for finite n). In the spherical limit ( $$(n \to \infty )$$ )one finds that the helical phase does not exist in the presence of any nonvanishing quantum fluctuation for m =d though the quantum Lifshitz point exists for all d > 1+m/2, and the upper critical dimensionality is given by d u = 3 +m/2. The scaling behaviour in the neighbourhood of a quantum Lifshitz point in d dimensions is consistent with the behaviour near the classical Lifshitz point in (d+z) dimensions. The dynamical exponent of the quantum Hamiltonian z is unity in the case of anisotropic Lifshitz point (d>m) whereas z=2 in the case of isotropic Lifshitz point (d=m). We have evaluated all the exponents using the renormalisation flow equations along-with the scaling relations near the quantum Lifshitz point. We have also obtained the exponents in the spherical limit ( $$(n \to \infty )$$ ). It has also been shown that the exponents in the spherical model are all related to those of the corresponding Gaussian model by Fisher renormalisation.

Design of bicontinuous polymeric microemulsions

Recent experiments suggest that thermodynamically stable, bicontinuous microemulsions can be achieved in symmetric ternary blends of two homopolymers and a diblock copolymer by formulating alloys with compositions near mean-field isotropic Lifshitz points. We argue that practical application of this design criterion may require use of homopolymers of unequal molecular weights and block copolymers of different architecture. We demonstrate the existence of, and explicitly locate, mean-field isotropic Lifshitz points in ternary blends with homopolymer molecular weight asymmetry and either AB diblock or ABA triblock copolymer architectures. These calculations considerably expand the parameter space for observing bicontinuous microemulsions and allow for more flexibility in tailoring melt rheological properties and solid-state mechanical properties.

Polymeric Bicontinuous Microemulsions

High molecular weight block copolymers can be viewed as macromolecular surfactants when blended with thermodynamically incompatible homopolymers. This Letter describes the formation of polymeric bicontinuous microemulsions in mixtures containing a model diblock copolymer and two homopolymers. Although we attribute development of this equilibrium morphology to the effects of fluctuations, mean-field theory provides a quantitative strategy for preparing the bicontinuous state at blend compositions near an isotropic Lifshitz point.

Isotropic Lifshitz behavior in block copolymer-homopolymer blends.

A series of mixtures composed of a symmetric $A\ensuremath{-}B$ diblock copolymer and a symmetric blend of $A$ and $B$ homopolymers was investigated by small-angle neutron scattering. Mean-field theory predicts that a line of lamellar-disorder transitions with wave-vector instability ${q}^{*}&gt;0$ will meet a line of critical points with ${q}^{*}=0$ in the three-component mixture at an isotropic Lifshitz point. Mean-field Lifshitz behavior ($\ensuremath{\gamma}=1$ and $\ensuremath{\nu}=\frac{1}{4}$) was observed in the disordered state at the anticipated composition to within 1 K of the phase transition.

"Soliton" solutions in a field theory of microemulsion.

We discuss a field theory of microemulsion that has been previously studied by a number of techniques. We examine the localized solutions of this theory, with a view to understanding the kinetics of phase growth. However, it is also suggested that, in the region of what was formerly considered to be an isotropic Lifshitz point of the phase diagram, there may be a phase of microscopic multilamellar spherical objects. This new phase may displace the swollen lyotropic lamellar phase.

Multicritical phenomena and microphase ordering in random block copolymers melts

A mean-field analysis of microphase separation and phase behavior is desribed for melts of linear A-B multiblocks copolymers, each chain consisting of a stochastic sequence of flexible segments (each with M monomers), A M and B M , for a total of Q>>1 segments per chain. A phase diagram similar to that for statistical A-B copolymers (M=1) but with an isotropic Lifshitz point that should be accessible for a wide variety of materials was found. Due to the broad distribution of sequence lengths of like monomers, the microphases are pedicted to lack oong-range order and to exhibit unusual temperature sensitivity at the order-disorder transition

Thermodynamics of random copolymer melts.

We report on a mean-field thermodynamic analysis of AB random copolymer melts with (backbone) composition correlations that reflect realistic polymerization conditions. Depending on the strength and sign of these correlations, multiple homogeneous phases, periodic mesophases, or disordered mesophases are possible. We identify an isotropic Lifshitz point and the critical lines that emerge from it. Our results should facilitate the design of new copolymeric materials.

Phase equilibria in copolymer/homopolymer ternary blends: Molecular weight effects

The phase behavior of ternary blends of an A–B random copolymer with two homopolymers (A and B) is investigated within the Flory–Huggins lattice theory. We restrict consideration to the formation of (isotropic) liquid phases. For compositionally symmetric systems in which the two homopolymers have equal molecular weights, two different topologies are found for the phase diagrams according to the length of the copolymer relative to the homopolymers. Namely, upon lowering the temperature, a three-liquid-phase region emerges either continuously via a tricritical point if the copolymer is long enough, or discontinuously otherwise. This change in phase behavior, an entropy of mixing effect, occurs when the copolymer length is a fraction 2/5 of the homopolymer molecular weight. The properties of the (symmetric) tricritical points are discussed, as well as the phase behavior of systems in which the copolymer is not symmetrical in composition and/or the two homopolymers differ in size. This Flory–Huggins approach should be valid for blends containing random copolymers, but also, at high enough temperatures, for blends that contain block copolymers, the temperature range of validity being broader for smaller block copolymers. The critical behavior of systems containing block copolymers is described by the simple Flory–Huggins theory over a range delineated by isotropic Lifshitz points. These Lifshitz points are located within the random phase approximation.

Fluctuation-induced transitions in an isotropic spatially frustrated lattice model.

In this paper we discuss the effects of fluctuations on the phase diagram of an isotropic Ising model with ferromagnetic first-neighbor and antiferromagnetic second- and third-neighbor interactions. We find that, in certain cases, there are substantial deviations from mean-field behavior. Thus the phase transitions between the one-dimensional modulated phases and paramagnetic phase are found to undergo fluctuation-induced first-order phase transitions. We also discuss that point of the phase diagram which, within mean-field theory, was predicted to be an isotropic Lifshitz point. It is argued that the existence of such a point is unlikely, and some alternative scenarios are proposed.

Continuously varying critical exponents to O(1/n).

In the context of a phase transition, the critical singularity is characterized by a set of numbers, the critical exponents. It is widely believed that for transitions in systems with pair interactions on Bravais lattices (as opposed to hierarchical lattices), the critical exponents are determined solely by parameters d and n, representing the dimensionality of space and of the order parameter, respectively, and by \ensuremath{\sigma}, the interaction range. In this paper we consider a certain generalization of the Lifshitz singularity in the n-vector model in which a new parameter \ensuremath{\lambda}, describing a type of anisotropy of the interactions, is introduced. This is described by a model Hamiltonian in k space in which the leading term is of the form [${\mathrm{\ensuremath{\Sigma}}}_{\mathit{i}=1}^{\mathit{d}}$(${\mathit{k}}_{\mathit{i}}^{2}$${)}^{\mathrm{\ensuremath{\sigma}}/(2\ensuremath{\lambda})}$]\ensuremath{\lambda}. It should be noted that, as for the usual Lifshitz case, what would otherwise be the dominant ${\mathit{k}}^{2}$ term is supposed to have already been cancelled. For this model the parameters d, n, and \ensuremath{\sigma} are not sufficient to determine the value of the critical exponents to O(1/n), although they are sufficient for the first order in the \ensuremath{\epsilon} expansion.Calculations exact to O(1/n) show that, for a given set d, n, and \ensuremath{\sigma}, the exponents vary continuously with the parameter \ensuremath{\lambda} and include as a special case (for \ensuremath{\lambda}=1) the critical exponents for an anisotropic Lifshitz point. The critical exponents for this Lifshitz point differ markedly in value from those for the usually studied isotropic Lifshitz point, which is also a special case (for \ensuremath{\lambda}=\ensuremath{\sigma}/2) of our model. For \ensuremath{\sigma}=2, however, the two cases corresponding to \ensuremath{\lambda}=1 and \ensuremath{\sigma}/2 become degenerate. We provide another explicit example of the dependence of the critical exponents on \ensuremath{\lambda} by performing, in addition to the 1/n expansion, an expansion in \ensuremath{\sigma}-2. This helps to shed some light on the approach to the degeneracy at \ensuremath{\sigma}=2. The analysis of the Feynman integrals arising in the O(1/n) calculation provides interesting examples of the applications of hypergeometric functions; these integrals have themselves led to new results (published separately) in the theory of hypergeometric functions.

Isotropic Lifshitz point in2&lt;d&lt;4dimensions

The scaling behavior of the spherical model and other Ornstein-Zernike systems in the region of an isotropic Lifshitz point in $2&lt;d&lt;4$ dimensions is investigated, and a different interpretation of the Lifshitz point than the one previously emphasized is given. Our scaling results are consistent with renormalization-group calculations where they overlap. The scaling relations are modified appropriately for a Lifshitz point occurring at zero temperature. A means of generalizing the results to non-Ornstein-Zernike systems is suggested.

A new type of critical behavior: k-space instability on the γ line

Abstract Critical exponents at a general Lifshitz point are calculated in the spherical model limit and to first order in a double ϵ expansion. In addition, the exponents at an isotropic Lifshitz point are calculated to second order in ϵ and to first order in 1/n.

Critical exponents at a Lifshitz point to O(1/ n)

The critical exponents at a general Lifshitz point are calculated in the spherical model limit, as are those of an isotropic Lifshitz point to O(1/ n). These results are in exact agreement in the overlap region with those obtained using an ϵ-expansion.