Spin systems on hypercubic Bethe lattices: a Bethe–Peierls approach

Exact equations for the thermodynamic quantities of lattices made of d-dimensional hypercubes are obtainable with the Bethe-Peierls approach.

Journal of Physics A 48, 255001 (2015)

A. Mozeika, A. Coolen

Spin systems on hypercubic Bethe lattices: a Bethe–Peierls approach

We study spin systems on Bethe lattices constructed from d-dimensional hypercubes. Although these lattices are not tree-like, and therefore closer to real cubic lattices than Bethe lattices or regular random graphs, one can still use the Bethe–Peierls method to derive exact equations for the magnetization and other thermodynamic quantities. We compute phase diagrams for ferro- magnetic Ising models on hypercubic Bethe lattices with dimension d = 2, 3, and 4. Our results are in good agreement with the results of the same models on d-dimensional cubic lattices, for low and high temperatures, and offer an improvement over the conventional Bethe lattice with connectivity k = 2d.

More in Spectre of hypercubes

  • Forum of Mathematics, Sigma

    Erdős-Ko-Rado theorem analogue

    A random analogue of the Erdős-Ko-Rado theorem sheds light on its stability in an area of parameter space which has not yet been explored.

  • European Journal of Combinatorics

    Hypercube eigenvalues

    Hamming balls, subgraphs of the hypercube, maximise the graph’s largest eigenvalue exactly when the dimension of the cube is large enough.

  • Discrete Mathematics

    Eigenvalues of neutral networks

    The principal eigenvalue of small neutral networks determines their robustness, and is bounded by the logarithm of the number of vertices.