Potts antiferromagnetic model on a family of fractal lattices: Exact results for an unusual phase

Critical properties of short-range Ising spin glasses on a Wheatstone-bridge hierarchical lattice

An Ising spin-glass model with nearest-neighbor interactions, following a symmetric probability distribution, is investigated on a hierarchical lattice of the Wheatstone-bridge family characterized by a fractal dimension D≈3.58. The interaction distribution considered is a stretched exponential, which has been shown recently to be very close to the fixed-point coupling distribution, and such a model has been considered lately as a good approach for Ising spin glasses on a cubic lattice. An exact recursion procedure is implemented for calculating site magnetizations, mi=〈Si〉T, as well as correlations between pairs of nearest-neighbor spins, 〈SiSj〉T (〈〉T denote thermal averages), for a given set of interaction couplings on this lattice. From these local magnetizations and correlations, one can compute important physical quantities, such as the Edwards-Anderson order parameter, the internal energy, and the specific heat. Considering extrapolations to the thermodynamic limit for the order parameter, such as a finite-size scaling approach, it is possible to obtain directly the critical temperature and critical exponents. The transition between the spin-glass and paramagnetic phases is analyzed, and the associated critical exponents β and ν are estimated as β=0.82(5) and ν=2.50(4), which are in good agreement with the most recent results from extensive numerical simulations on a cubic lattice. Since these critical exponents were obtained from a fixed-point distribution, they are universal, i.e., valid for any coupling distribution considered.

Odd q-state clock spin-glass models in three dimensions, asymmetric phase diagrams, and multiple algebraically ordered phases

Distinctive orderings and phase diagram structures are found, from renormalization-group theory, for odd q-state clock spin-glass models in d=3 dimensions. These models exhibit asymmetric phase diagrams, as is also the case for quantum Heisenberg spin-glass models. No finite-temperature spin-glass phase occurs. For all odd q≥5, algebraically ordered antiferromagnetic phases occur. One such phase is dominant and occurs for all q≥5. Other such phases occupy small low-temperature portions of the phase diagrams and occur for 5≤q≤15. All algebraically ordered phases have the same structure, determined by an attractive finite-temperature sink fixed point where a dominant and a subdominant pair states have the only nonzero Boltzmann weights. The phase transition critical exponents quickly saturate to the high q value.

Ground-state entropies of the Potts antiferromagnet on diamond hierarchical lattices

The ground-state degeneracies of the q-state Potts antiferromagnet on general diamond hierarchical lattices are computed, for q> or =3, by means of two distinct methods. The first method, denominated the recursive approach, is based on exact recursion relations for the total number of ground states, leading to the exact ground-state entropy in the thermodynamic limit. The second method, called the factorization approach, consists in a simple approximation, where the total number of ground states is factorized as a product of the number of ground states at each hierarchy level. The factorization approach appears to be a poor approximation for small values of q, but its accuracy improves substantially as q increases, and it becomes exact in the limit q--> infinity. In spite of the fact that such a model presents no frustration, a residual entropy at zero temperature is found for all q> or =3. Similarly to what happens on Bravais lattices, the residual entropy approaches its maximum allowed value, ln q, as q increases.

Calculation of ground-state entropies of highly frustrated systems on fractal lattices

The extensive ground-state entropy of frustrated systems on fractal lattices is investigated. Two methods of calculation are proposed, namely, recursive and factorization approaches. In the recursive approach the calculation is based on exact recursion relations for the total number of ground states. The latter procedure, which is in principle an approximation, is proposed as an alternative for dealing with complicated systems (for cases where the recursive approach may become impracticable), such as randomly frustrated models; it consists of factorizing the total number of ground states in terms of the number of ground states at each hierarchy level. Some examples of antiferromagnetic Ising models on different fractal lattices are considered, for which both procedures are applied. It is shown that the factorization approach may lead, in some cases, to the exact ground-state entropy, whereas in other cases it yields very accurate (although slightly lower) estimates.

Analytical verification of scaling laws for the Ising model with external field in fractal lattices

We use an exact recursion procedure to verify analytically, without any intermediary numerical calculation, the validity of the hyperscaling (Josephson) law extended to fractals, the Rushbrooke and Griffiths scaling laws for the Ising ferromagnet with external magnetic field in the whole family of Migdal-Kadanoff-like hierarchical lattices.