Critical phenomena on k-booklets.

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Mathematical Physics

Background:

• Standard critical phenomena are well-understood in infinite and semi-infinite lattices (k=1, 2).
• The behavior of statistical systems on more complex geometries, like k-booklets, is less explored.

Purpose of the Study:

• To investigate critical phenomena in self-avoiding walks, Ising model, and percolation on k-booklets for k≥3.
• To compare the behavior on k-booklets with standard lattice geometries.

Main Methods:

• Definition of k-booklets as k semi-infinite planes glued at y=0.
• Analysis of self-avoiding random walks, Ising model, and percolation on these k-booklets.
• Examination of critical transitions and order parameter behavior.

Main Results:

• For k≥3, self-avoiding walks exhibit a first-order transition with no power-law scaling.
• Ising model and percolation display hybrid transitions, with standard scaling laws alongside order parameter discontinuities.
• Ergodicity breaks at the critical temperature (T=T_{c}) for the Ising model on k-booklets.
• Correlations show significant anisotropy for small y values across all studied models.

Conclusions:

• K-booklets (k≥3) present distinct critical phenomena compared to standard lattices.
• Hybrid transitions and anisotropic correlations are key features of these systems.
• The geometry of the lattice significantly influences critical behavior and ergodicity.