We report a numerical evidence of the discontinuous transition of a tethered membrane model which is defined within a framework of the membrane elasticity of Helfrich. Two kinds of phantom tethered membrane models are studied via the canonical Monte Carlo simulation on triangulated fixed connectivity surfaces of spherical topology. A surface model is defined by the Gaussian term and the bending energy term, and the other, which is tensionless, is defined by the bending energy term and a hard wall potential. The bending energy is defined by using the normal vector at each vertex. Both of the models undergo the first-order phase transition characterized by a gap of the bending energy. The phase structure of the models depends on the choice of discrete bending energy.

It is reported that a surface model of Polyakov strings undergoes a first-order phase transition between smooth and crumpled (or branched polymer) phases. The Hamiltonian of the model contains the Gaussian term and a deficit angle term corresponding to the weight of the integration measure dX in the partition function. PACS. 64.60.-i General studies of phase transitions -68.60.-p Physical properties of thin films, nonelectronic -87.16.Dg Membranes, bilayers, and vesicles

We study a model of elastic surfaces that was first constructed by Baillie et al. for an interpolation between the models of fluid and crystalline membranes. The Hamiltonian of the model is a linear combination of the Gaussian energy and a squared scalar curvature energy. These energy terms are discretized on dynamically triangulated surfaces that are allowed to self-intersect. We confirm that the model has not only crumpled phases but also a branched polymer phase, and find that the model undergoes a first-order phase transition between the branched polymer phase and one of the crumpled phases. We find also that the model undergoes a second-͑or higher-͒ order phase transition between the branched polymer phase and another crumpled phase.

By using the grand canonical Monte Carlo simulations on spherical surfaces with two fixed vertices separated by the distance L, we find that the second-order phase transition changes to the first-order one when L is sufficiently large. We find that string tension σ = 0 in the smooth phase while σ → 0 in the wrinkled phase.

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