Potts model on infinitely ramified Sierpinski-gasket-type fractals and algebraic order at antiferromagnetic phases.

Abstract

We propose families of infinitely ramified fractals, which we call the m-sheet Sierpinski gasket with side b [(mSG${)}_{\mathit{b}}$], on which the q-state Potts model can be exactly solvable through a real-space renormalization-group (RSRG) technique for which there are phase transitions at finite temperatures for m>1. We also propose, within a cell-to-cell RSRG scheme, a criterion for a suitable choice of cells in the study of antiferromagnetic (AF) classical spin models defined on (or approximated by) multirooted hierarchical lattices, and apply it for the AF Potts model on some (mSG${)}_{\mathit{b}}$ fractals. Concerning the Ising model on the (mSG${)}_{2}$ family, we obtain the exact para (P)--ferromagnetic (F) critical temperature as a function of m and verify that, for m=1 and 2, there is no AF order (not even at zero temperature). We calculated the exact P-F and possibly exact P-AF critical frontiers and the corresponding correlation-length critical exponents for q=2-, 3-, and 4-state Potts model on the (mSG${)}_{4}$. The AF q=2 and 4 cases have highly degenerate ground states and each one presents, above a certain critical fractal dimension ${\mathit{D}}_{\mathit{f}}^{\mathit{c}}$(q,m), an unusual low-temperature phase whose attractor occurs at a non-null temperature. For q=2 and ${\mathit{D}}_{\mathit{f}}$(2,m)\ensuremath{\ge}5.1, we prove that the correlations have a power-law decay with distance along this entire phase.