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Orthogonally Invariant Spin Glass

An orthogonally invariant spin glass is an Ising spin model whose interaction matrix has a law invariant under orthogonal conjugation. For a diagonal matrix Lambda_N and a Haar-distributed orthogonal matrix U_N, put J_N=U_N^TLambda_NU_N. With temperature absorbed into Lambda_N, its thermodynamic pressure using uniform spin probability is

P_N=1/Nln(2^(-N)sum_(sigma in {-1,1}^N)e^(sigma^TJ_Nsigma/2)).

For example, adding cI to J_N increases this thermodynamic pressure by c/2.

See also

Haar Measure, Orthogonal Matrix, Replica Symmetry Breaking, Spin Glass, Thermodynamic Pressure

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References

Fan, Z. and Wu, Y. "The Replica-Symmetric Free Energy for Ising Spin Glasses with Orthogonally Invariant Couplings." Probab. Th. Rel. Fields 190, 1-77, 2024. https://doi.org/10.1007/s00440-024-01309-w.

Cite this as:

Weisstein, Eric W. "Orthogonally Invariant Spin Glass." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OrthogonallyInvariantSpinGlass.html

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