Multifractal and critical properties of the Ising model
Abstract
Multifractal properties of the two-dimensional Ising model are studied by means of the f(α) spectra of singularities of probability measures supported by energy spectra. These measures are determined through calculations of energy degeneracies for finite-size systems up to 12×12 spins in cases of square and triangular lattices and up to 22×22 spins in the case of the hexagonal lattice. The calculations are performed in an exact manner using the transfer-matrix method. It is argued that, in the thermodynamic limit, the scaling exponent αmax associated with the most probable energy of the system takes at the critical temperature a minimum value and, consequently, it is argued that a given system reveals a phase transition at a finite temperature if αmax possesses a minimum at the finite temperature. It is also shown that, in the thermodynamic limit, the spectrum f(α) shrinks when the critical temperature is approached.
- Publication:
-
Physical Review B
- Pub Date:
- May 1998
- DOI:
- 10.1103/PhysRevB.57.10240
- Bibcode:
- 1998PhRvB..5710240J
- Keywords:
-
- 05.50.+q;
- 05.70.Fh;
- 64.10.+h;
- 68.35.Rh;
- Lattice theory and statistics;
- Phase transitions: general studies;
- General theory of equations of state and phase equilibria;
- Phase transitions and critical phenomena