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The Complexity of the Thouless-Anderson-Palmer (TAP) solutions of the Ising$p$-spin is investigated in the temperature regime where the equilibrium phaseis one step Replica Symmetry  Breaking. Two solutions of the resulting saddle point equations are found.One is supersymmetric (SUSY) and includes the equilibrium value of the freeenergy while the other is non-SUSY. The two solutions cross exactly at a valueof the free energy where the replicon eigenvalue is zero; at low free energythe complexity is described by the SUSY solution while at high free energy itis described by the non-SUSY solution. In particular the non-SUSY solutiondescribes the total number of solutions, like in the  Sherrington-Kirkpatrick (SK) model. The relevant TAP solutions correspondingto the non-SUSY solution share the same feature of the corresponding solutionsin the SK model, in particular their Hessian has a vanishing isolatedeigenvalue. The TAP solutions corresponding to the SUSY solution, instead, arewell separated minima.Comment: 13 pages, 9 figure

physical review. b, condensed matter and materials physics

Complexity in mean-field spin-glass models: Ising<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>p</mml:mi></mml:math>-spin

Complexity in mean-field spin-glass models: Isingp-spin