Walks, Polymers, and Other Tricritical Systems in the Presence of Walls or Surfaces

Semi-infinite O(n) systems are considered at a bulk tricritical point. A continuum model with a |ϕ|6 bulk nonlinearity and a |ϕ|4 surface nonlinearity is used to analyse the critical behaviour at the special transition in d = 3 and d = 3−ε dimensions. For n = 0 the model describes the statistics of polymers at the θ-point, in the presence of a wall. It is shown that the surface critical exponents have a systematic expansion in powers of ε1/2. For d = 3, logarithmic corrections of an unusual form are found for surface quantities. In the special case of vanishing bulk nonlinearity the surface exponents can be expressed in terms of known bulk exponents for systems with long-range interactions.