First-order phase transitions in polymerized phantom membranes

Abstract

The crumpled-to-flat phase transition that occurs in D-dimensional polymerized phantom membranes embedded in a d-dimensional space is investigated nonperturbatively using a field expansion up to order eight in powers of the order parameter. We get the critical dimension dcr(D) that separates a second order region from a first order one everywhere between D = 4 and D = 2. Our approach strongly suggests that the phase transitions that take place in physical membranes are of first order in agreement with most recent numerical simulations.PACS numbers: 87.16. 11.10.Hi, 11.15.Tk Fluctuating or random surfaces are a recurrent concept in physics [1,2]. They occur in soft matter physics or in biology as assemblies of amphiphilic molecules that can form plane or closed structures (vesicles) according to the chemical composition of the membrane itself and its surroundings. Random surfaces also appear in highenergy physics, especially in string theory, as the worldsheet swept out by a string during its spacetime evolution. More recently membranes have received a renewed interest in condensed matter physics where it has been realized that, from the point of view of their mechanical properties, novel materials, like graphene [3], identify with polymerized membranes, providing the first and unique example of genuinely two-dimensional membrane [4,5]. The coexistence of two-dimensional geometry and thermal fluctuations is at the origin of a variety of behaviours depending on the nature of the internal structure of the membrane. Fluid membranes are made of molecules that freely diffuse and re-arrange rapidly when a shear or a stress is performed. This implies that, in absence of an external tension, the dominant energy is the bending energy. It has been shown that, in this case, the height fluctuations are sufficiently strong to prevent the appearance of long-range order; fluid membranes are thus always crumpled [6,7]. Polymerizedor tethered -membranes display a drastically different behaviour. Indeed the existence of an underlying network of linked molecules induces elastic (shearing and stretching) energy contributions that lead to a coupling between height and transverse -phonons -modes. It results from this situation a frustration of the height fluctuations [8] that are strongly reduced at low temperatures giving rise to the appearance of a flat phase with longrange order between the normals [9,10]. The existence of a low-temperature phase accompanied with spontaneous symmetry breaking of rotational invariance is a priori in contradiction with the Mermin-Wagner theorem. However it appears that the effective phonon-mediated interaction between the height fields (more precisely between the Gaussian curvatures) is of long-range kind, allowing to evade the conditions of application of the MerminWagner theorem [9]. Correlatively the low-temperature phase of membranes is characterized by non trivial scaling behaviour in the infrared [11][12][13]:where G hh (q) and G uu (q) are the correlation functions of the out-of-plane and ...

Skyrmions on 2D elastic surfaces with fixed boundary frame

“…These S 1 and S 2 are called Gaussian bond potential and bending energy, respectively, and ℓ ij (= |r j −r i |) is the length of bond ij, and n i and n j in S 2 are the unit normal vectors of the triangles that share the bond ij. The parameter κ is the bending rigidity of the surface [29,30,31,32,33]. We should note that the first term S 1 depends on the size of projected area of lattice because S 1 is the sum of bond length squares of surface which spans the boundary of projected area A p , and for this reason the model becomes dependent on the lattice spacing a.…”

“…Here we should note that "dynamical" variable refers to the variable that is integrated in the partition function in the context of statistical mechanics. Therefore, a fluctuating lattice is considered to be a direct tool for this purpose, where the lattice fluctuation is allowed not only into the in-plane direction but also into the out-of-plane direction [27,28,29,30,31,32,33,34]. On the fluctuating lattices, a certain magnetoelastic coupling is implicitly included in the DM interaction, and for simplicity, only this magnetoelastic coupling is taken into account under the presence of external magnetic field in this paper.…”

“…The DM interaction on the continuous 2D plane depends on both spins and coordinates, and hence, we are able to calculate mechanical properties such as surface tension under the presence of skyrmions if the coordinate, or equivalently the surface, is treated as a dynamical variable [27,28,29,30,31,32,33,34]. Moreover, DM interaction is invariant under an arbitrary rotation around the axis perpendicular to the surface.…”

Mathematical modeling and simulations for large-strain J-shaped diagrams of soft biological tissues

“…We comment on the reason why we use the FG model instead of standard or canonical models, which are defined without the variable σ [30][31][32][33][34]. The problem is whether or not the results of the canonical 2D and 3D models are identical to those of the FG models.…”

Optimized regulator for the quantized anharmonic oscillator

“…Recently, this optimization strategy has often been used. [12][13][14][15][16][17][18] At first glance, it seems to be rather a mathematical condition. Nevertheless, it can be made plausible by simple physical reasoning.…”