The phase diagram and surface critical behaviour of the vertex-interacting self-avoiding walk are examined using transfer matrix methods extended using DMRG and coupled with finite-size scaling. Particular attention is paid to the critical exponents at the ordinary and special points along the collapse transition line. The question of the bulk exponents (
ν and
γ) is addressed, and the results found are at variance with previously conjectured exact values.
The phase diagram for the bond-interacting self-avoiding walk is calculated using transfer matrices on finite strips. The model is shown to have a richer phase diagram than the related
Θ-point model. In addition to the standard collapse transition, we conjecture the existence of a line of transitions in the Berezinskii--Kosterlitz--Thouless (BKT) universality class, terminating in a critical end point. Our results are in contradiction with a previous transfer matrix calculation by Machado K D, de Oliveira M J and Stilck J F 2001 {\em Phys Rev E} {\bf 64} 051810.
The critical behaviour of the two-dimensional self-avoiding loop gas model with multiplicity m=1 (singly counted non-intersecting loops) is studied numerically on a square lattice using a recently developed Monte Carlo method. The critical exponents alpha , beta , gamma and delta are evaluated in the 'critical window' between the finite-size rounding and the non-critical ('correction-to-scaling') regime using a recently calculated accurate value of T c . Within error bars, Ising-like critical exponents of the loop gas are obtained.
We study the three-state antiferromagnetic Potts model on the simple-cubic lattice, paying attention to the surface critical behaviors. When the nearest-neighboring interactions of the surface is tuned, we obtain a phase diagram similar to the XY model, owing to the emergent O(2) symmetry of the bulk critical point. For the ordinary transition, we get ${y}_{h1}=0.780(3), {\ensuremath{\eta}}_{\ensuremath{\parallel}}=1.44(1)$, and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=0.736(6)$; for the special transition, we get ${y}_{s}=0.59(1), {y}_{h1}=1.693(2), {\ensuremath{\eta}}_{\ensuremath{\parallel}}=\ensuremath{-}0.391(4)$, and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.179(5)$; in the extraordinary-log phase, the surface correlation function ${C}_{\ensuremath{\parallel}}(r)$ decays logarithmically with decaying exponent
q=0.60(2), however, the correlation ${C}_{\ensuremath{\perp}}(r)$ still decays algebraically with critical exponent ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.442(5)$. If the ferromagnetic next-nearest-neighboring surface interactions are added, we find two transition points, the first one is a special point between the ordinary phase and the extraordinary-log phase, the second one is a transition between the extraordinary-log phase and the
Z6 symmetry-breaking phase, with critical exponent
ys=0.41(2). The scaling behaviors of the second transition is very interesting, the surface spin-correlation function ${C}_{\ensuremath{\parallel}}(r)$, and the surface squared staggered magnetization at this point decays logarithmically with exponent
q=0.37(1); however, the surface structure factor with the smallest wave vector and the correlation function ${C}_{\ensuremath{\perp}}(r)$ satisfy power-law decaying, with critical exponents ${\ensuremath{\eta}}_{\ensuremath{\parallel}}=\ensuremath{-}0.69(1)$ and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.37(1)$, respectively.
We find and discuss the scaling dimensions of the branch 0 manifold of the Nienhuis O(n) loop model on the square lattice, concentrating on the surface dimensions. The results are extracted from a Bethe ansatz calculation of the finite-size corrections to the eigenspectrum of the six-vertex model with free boundary conditions. These results are especially interesting for polymer physics at two values of the crossing parameter lambda . Interacting self-avoiding walks on the Manhattan lattice at the collapse temperature ( lambda = pi /3) and Hamiltonian walks on the Manhattan lattice ( lambda = pi /2) are discussed in detail. Our calculations illustrate the importance of examining both odd and even strip widths when performing finite-size correction calculations to obtain scaling dimensions.
We investigate the finite-size scaling behavior of the tracer surface diffusion coefficient in the vicinity of a second-order phase transition. For this purpose, we use a lattice gas model of repulsively interacting particles on a square lattice. For all lattice sizes L studied, the measured tracer surface diffusion coefficient, D t , is a smooth function, having an inflexion point at critical temperature. Its derivative, ∂ D t /∂(1/ k B T ), exhibits cusp-like maximum which is a) sharply pronounced and b) converged to T c (∞) for large lattice sizes. We have analyzed the finite-size behavior of D t and obtained its critical exponent σ t = 0.665 ± 0.003.