Critical Behavior of Lines and Surfaces

  • Condensed and soft matter systems often contain thin structures such as (i) domain boundaries, vortex lines, polymers, and filaments, or (ii) interfaces, monolayers, and membranes etc which behave as 1-dimensional lines or 2-dimensional surfaces. These low-dimensional manifolds undergo thermally excited shape fluctuations in order to increase their configurational entropy. These fluctuations are scale-invariant and can be characterized by a corresponding critical exponent. [1] [2] One particularly interesting case are domain boundaries and interfaces in quasi-periodic systems.

  • Because of their molecular architecture, 1-dimensional lines and 2-dimensional surfaces interact via electrostatic and van der Waals forces or other constraints. The shape fluctuations renormalize these latter interactions and lead to unbinding transitions between bound and unbound states of the manifolds. [3] These latter transitions represent wetting [4], adhesion [5], and adsorption transitions [6] for interfaces, membranes, and polymers, respectively.

  • In all cases, one encounters a strong fluctuation regime [7] with universal critical behavior [8] [9] as well as an intermediate fluctuation regime with more complex behavior [10] [11]. In the latter regime, the unbinding transitions are discontinuous and characterized by a kink in the internal energy but the shape fluctuations exhibit unusual properties and are governed by a power law distribution [12].

  • The scaling properties of different systems are found to be related in unexpected ways. The continuous unbinding of two membranes in three dimensions [5], for example, exhibits the same scaling properties as the unbinding of two domain boundaries or "strings" in two dimensions [13]. For 1-dimensional lines, there is even an intimate relationship between the scaling properties of bound and unbound states [14].
    Both bundles of N strings in two dimensions [15] [16] and bunches of N membranes in three dimensions [17] undergo continuous unbinding transitions whereas bundles of N filaments unbind discontinuously in three dimensions [18]. In the continuous case, extensive numerical studies [15] [17] and analytical solutions based on a Bethe Ansatz [16] have led to somewhat different conclusions about the N-dependence of the critical exponents which remains an open issue [19].

  • Recent studies have focused on the critical behavior of filaments and semiflexible polymers:
  • For more information, see publications on lines and surfaces.