We study self-propelled dynamics of a droplet due to a Marangoni effect and chemical reactions in a binary fluid with a dilute third component of chemical product which affects the interfacial energy of a droplet. The equation for the migration velocity of the center of mass of a droplet is derived in the limit of an infinitesimally thin interface. We found that there is a bifurcation from a motionless state to a propagating state of droplet by changing the strength of the Marangoni effect.

A functional renormalization group approach to d-dimensional, N -component, non-collinear magnets is performed using various truncations of the effective action relevant to study their long distance behavior. With help of these truncations we study the existence of a stable fixed point for dimensions between d = 2.8 and d = 4 for various values of N focusing on the critical value, Nc(d), that, for a given dimension d, separates a first order region for N < Nc(d) from a second order region for N > Nc(d). Our approach concludes to the absence of stable fixed point in the physical -N = 2, 3 and d = 3 -cases, in agreement with ǫ = 4 − d-expansion and in contradiction with previous perturbative approaches performed at fixed dimension and with recent approaches based on conformal bootstrap program.

We find that the multicritical fixed point structure of the O(N ) models is much more complicated than widely believed. In particular, we find new nonperturbative fixed points in three dimensions (d = 3) as well as at N = ∞. These fixed points come together with an intricate double-valued structure when they are considered as functions of d and N . Many features found for the O(N ) models are shared by the O(N )⊗O(2) models relevant to frustrated magnetic systems.The O(N )-symmetric and Ising statistical models have played an extremely important role in our understanding of second order phase transitions both because many experimental systems show this symmetry and because they have been the playground on which almost all the theoretical formalisms aiming at describing these phase transitions have been developed and tested: Integrability 3 term becomes relevant in d = 3−ǫ and a nontrivial 2-unstable FP emerges from G that becomes 3-unstable. This scenario repeats in each critical dimension d n = 2 + 2/n below which a new n-unstable multicritical FP appears that we call T n . The FP T 2 is tricritical because it lies in the coupling constant space on the boarder separating the domain of second order and first order phase transitions. The common wisdom is that all the T n FPs can be followed by continuity in d down to d = 2 for all values of N . This is corroborated by the fact that in the Ising case (N = 1), it has been rigorously proven that indeed all the T n exist in d = 2 and are nontrivial [9]. Because of Mermin-Wagner theorem, the situation is physically different for N ≥ 2 but at least T 2 can be followed smoothly from d = 3 − ǫ down to d = 2 for N = 2, 3 and 4 [10]. Notice that the N = 2, d = 2 case is peculiar because topological defects can trigger in this case a finite-temperature phase transition.At N = ∞, exact results can be derived such as a closed and exact RG flow equation for the Gibbs effective potential [11]. The common wisdom is that at N = ∞ and in generic dimensions 2 < d < 4, the only nontrivial and nonsingular FP is WF which is simple to obtain after an appropriate rescaling by a factor N [12]. Its nonsingular character means that it is a regular function of the field. The limit N = ∞ is in fact peculiar because in all the d n with n ≥ 2, and only in these dimensions, there also exists a line of FPs. In d = 3, this line corresponds to tricritical FPs sharing all the same (trivial) critical exponents. This line starts at G and terminates at the Bardeen-Moshe-Bander (BMB) FP whose effective potential is nonanalytic at vanishing field, see .It is surprising that this common wisdom about the O(N ) models raises a simple paradox that, to the best of our knowledge, has remained unnoticed up to now.

The large N expansion plays a fundamental role in quantum and statistical field theory. We show on the example of the O(N ) model that at N = ∞, its standard implementation misses some fixed points of the renormalization group in all dimensions smaller than four. These new fixed points show singularities under the form of cusps at N = ∞ in their effective potential that become a boundary layer at finite N . We show that they have a physical impact on the multicritical physics of the O(N ) model at finite N . We also show that the mechanism at play holds also for the O(N )⊗O(2) model and is thus probably generic.The 1/N expansion is one of the most important tools in field theory. It has played a prominent role in QCD [1] as well as in statistical mechanics and condensed matter physics [2,3]. One of its key features is that it can yield reliable results even in strongly coupled models because it is nonperturbative in the coupling constant(s). It also has the enormous advantage of not being linked to a particular dimension, contrary to the usual perturbative expansions. This latter feature has often allowed us to make a bridge between the perturbative expansions performed around the upper and the lower critical dimensions of a model. For instance, at leading order in the 1/N expansion both the Mermin-Wagner theorem in two dimensions and the mean-field behavior at criticality in dimensions d ≥ 4 are retrieved, which is out of reach of both perturbative expansions in = 4 − d [4] and = d − 2 [5].The success of the large N analysis relies on (i) the possibility to extend the original model to arbitrary values of N and (ii) the fact that the model is soluble at N = ∞. This is the case not only for the O(N ) and the gauge SU(N ) models but also for a large class of statistical field theories.We show in this Letter that, surprisingly, even for the O(N ) model, which is the textbook example for the 1/N expansion, the situation is not as simple as it is widely believed. More precisely, we show on the examples of the O(N ) and O(N )⊗O(2) models that at N = ∞ some fixed points (FPs) that play an important role even at a qualitative level were missed by the usual large N approach [6,7]. The presence of these fixed points changes the finite N (multicritical) physics of these models.Over the years, the importance of renormalization group (RG) FPs showing cusps has been recognized. This occurs for FP functions such as thermodynamics potentials that are singular for a certain value of their argument. This is the case of the celebrated random field Ising model and is responsible for the breakdown of supersymmetry and dimensional reduction [8,9]. This is also the case out of equilibrium for some reaction-diffusion problems [10]. To the best of our knowledge, the occurrence of FPs with a cusp is known only in replica theory applied to disordered systems and in field theories describing out of equilibrium statistical models (see, however, Ref. [11]). We prove below that they also play an important role in simple field theories such as...

Antagonistic salts are composed of hydrophilic and hydrophobic ions. In a mixture solvent (wateroil) such ion pairs are preferentially attracted to water or oil, giving rise to a coupling between the charge density and the composition. First, they form a large electric double layer at a water-oil interface, reducing the surface tension and producing mesophases. Here, the cations and anions are loosely bound by the Coulomb attraction across the interface on the scale of the Debye screening length. Second, on solid surfaces, hydrophilic (hydrophobic) ions are trapped in a water-rich (oilrich) adsorption layer, while those of the other species are expelled from the layer. This yields a solvation mechanism of local charge separation near a solid. In particular, near the solvent criticality, disturbances around solid surfaces can become oscillatory in space. In mesophases, we calculate periodic structures, which resemble those in experiments.