Critical Behavior of an Ising Model with Aperiodic Interactions

We write exact renormalization-group recursion relations for a ferromagnetic Ising model on the diamond hierarchical lattice with an aperiodic distribution of exchange interactions according to a class of generalized two-letter Fibonacci sequences. For small geometric fluctuations, the critical behavior is unchanged with respect to the uniform case. For large fluctuations, the uniform fixed point in the parameter space becomes fully unstable. We analyze some limiting cases, and propose a heuristic criterion to check the relevance of the fluctuations.

Authorship

S. T. R. Pinho

UniversidadeUniversidadeUniversidade, Instituto de Física, São Paulo, SP, BrazilUniversidadeUniversidadeUniversidadeBrazilSão Paulo, SP, BrazilUniversidadeUniversidadeUniversidade, Instituto de Física, São Paulo, SP, Brazil

T. A. S. Haddad

S. R. Salinas

SCIMAGO INSTITUTIONS RANKINGS

Figures

Figures (5)

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Figure 1.
Some stages of the construction of a diamond hierarchical lattice with q = 2 branches. Letters A and B indicate the exchange interactions (J_A and J_B).

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Figure 3.
Second iterates and fixed points (black circles) of the recursion relations in the parameter space, x_A = tanh K_A versus x_B = tanhK_B, for an Ising model on a diamond hierarchical lattice with q = 2 and exchange interactions given by the generalized Fibonacci rules, A®AB, and B ®AA (irrelevant fluctuations). We draw the stable trivial fixed points (at zero and infinite temperatures) and the physical saddle point. The dashed lines and the black arrows indicate the flows of the second iterates of the map. The uniform model is recovered along the diagonal, x_A = x_B. The light arrows indicate the stable direction in the neighborhood of the physical fixed point.

Thumbnail

Figure 4.
Second iterates and fixed points for the Ising chain: (a) exchange interactions according to the rules A® AB, and B® AA (irrelevant fluctuations); (b) exchange interactions according to the rules A® ABB, and B® AAA (relevant fluctuations).

Thumbnail

Figure 5.
Second iterates and fixed points (black circles) of the recursion relations in the parameter space, x_A = tanhK_A versus x_B = tanhK_B, for an Ising model on a diamond hierarchical lattice with q = 2 and exchange interactions given by the generalized Fibonacci rules, A ®ABB, and B ®AAA (relevant fluctuations). We draw the stable trivial fixed points (at zero and infinite temperatures) and the unstable node. The uniform model is recovered along the diagonal, x_A = x_B. The dashed lines are indications of the flow associated with the second iterates of the map. The light arrows indicate the directions of the eigenvectors in the neighborhood of the physical fixed point.

Figure 1. Some stages of the construction of a diamond hierarchical lattice with q = 2 branches. Letters A and B indicate the exchange interactions (J_A and J_B).

Figure 3. Second iterates and fixed points (black circles) of the recursion relations in the parameter space, x_A = tanh K_A versus x_B = tanhK_B, for an Ising model on a diamond hierarchical lattice with q = 2 and exchange interactions given by the generalized Fibonacci rules, A®AB, and B ®AA (irrelevant fluctuations). We draw the stable trivial fixed points (at zero and infinite temperatures) and the physical saddle point. The dashed lines and the black arrows indicate the flows of the second iterates of the map. The uniform model is recovered along the diagonal, x_A = x_B. The light arrows indicate the stable direction in the neighborhood of the physical fixed point.

Figure 4. Second iterates and fixed points for the Ising chain: (a) exchange interactions according to the rules A® AB, and B® AA (irrelevant fluctuations); (b) exchange interactions according to the rules A® ABB, and B® AAA (relevant fluctuations).

Figure 5. Second iterates and fixed points (black circles) of the recursion relations in the parameter space, x_A = tanhK_A versus x_B = tanhK_B, for an Ising model on a diamond hierarchical lattice with q = 2 and exchange interactions given by the generalized Fibonacci rules, A ®ABB, and B ®AAA (relevant fluctuations). We draw the stable trivial fixed points (at zero and infinite temperatures) and the unstable node. The uniform model is recovered along the diagonal, x_A = x_B. The dashed lines are indications of the flow associated with the second iterates of the map. The light arrows indicate the directions of the eigenvectors in the neighborhood of the physical fixed point.

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