Renormalization of Fluctuating Tilted-Hexatic Membranes

We consider the tilted-hexatic Hamiltonian on the fluctuating membranes. A renormalization-group analysis leads us to find three critical regions; two correspond to the strong coupling regimes of the gradient cross coupling where we find the (anti-)locked tilted-hexatic to liquid phase transition, the other to the weak coupling regime where we find four phases; the unlocked tiltedhexatic phase, the hexatic phase, the tilted phase, and liquid phase. The crinkled-to-crumpled transition of the fluctuating tilted-hexatic membranes is also described. PACS numbers: 05.70.Jk, 68.10.-m, 87.22.Bt Typeset using REVTEX 1 Lyotropic liquid-crystal systems show a variety of phases with different types of inplane two dimensional order. Among the most interesting are the tilted-hexatic phases, which have quasi-long-range order in two order parameters (the orientation of the local bond and the direction of the local molecular tilt), but only short-range translational order. Recently, there has been considerable progress in understanding tilted-hexatic phases on the rigid layered liquid-crystals. Selinger and Nelson [1] have presented a Landau theory for transitions among tilted-hexatic phases. They consider the tilt-bond interaction potential and find several different hexatic phases differing from each other in the relation between the local bond orientation and the local tilt direction depending on the interaction potential parameters. However, they consider only phase transitions between low temperature phases (tilted-hexatic phases) on the rigid 2-dimensional plane, in which disclinations in the bond orientational angle field θ6(u) and vortices in the tilt-angle field θ1(u) can be neglected. In this Letter, we present a Landau theory, without the tilt-bond interaction potential, for transitions from tilted-hexatic phase to disordered liquid phase on the fluctuating membranes. We consider the fluctuating membranes with the tilt and the hexatic in-plane orders described by the order parameters ψ1 = e iθ1 and ψ6 = e i6θ6 , respectively. The tilt and the hexatic orders are coupled to each other via a gradient cross coupling introduced by Nelson and Halperin [2]. Depending on this gradient cross coupling parameter, we find three critical regions in the phase space of the tilt stiffness K1, the hexatic stiffness K6, and the gradient cross coupling K16; two correspond to the strong coupling and the other the weak coupling. We also show that without the tilt-bond interaction potential there exist a couple of different tilted-hexatic phases. Finally, we discuss the crumpling transition of the fluctuating membranes with the tilt and the hextic in-plane orders. We parametrize the membrane by its position vector as a function of standard Cartesian coordinates x = (x, y); R(x) = (x, h(x)), (1) where h(x) measures the deviation from the flat surface. This is called a Monge gauge. 2 Associated with R(x) is a metric tensor gαβ(x) = ∂αR(x) · ∂βR(x) and a curvature tensor Kαβ(x) defined via Kαβ(x) = N(x) · ∂α∂βR(x), where N(x) is the local unit normal to the surface. From the curvature tensor Kαβ, the mean curvature, H , and the Gaussian curvature, K, are defined as follows: H = 1 2 gKβα, K = det g Kλβ , (2) where g is the inverse tensor of gαβ satisfying g gλβ = δ α β . In the continuum elastic theory, the long-wavelength properties of a fluctuating membrane are described by the HelfrichCanham Hamiltonian [3] HHC = 1 2 ∫