We analyse the Hessian of the Thouless-Anderson-Palmer (TAP) free energy for
the Sherrington-Kirkpatrick model, below the de Almeida-Thouless line,
evaluated in Bolthausen's approximate solutions of the TAP equations. We show
that while its empirical spectrum weakly converges to a measure with negative
support, positive outlier eigenvalues occur for some (β,h) below the AT
line. In this sense, TAP free energy may lose concavity in the order parameter
of the theory, i.e. the random spin-magnetisations, even below the AT line.
Possible interpretations of these findings within Plefka's expansion of the
Gibbs potential are not definitive and include the following: i) either higher
order terms shall not be neglected even if Plefka's first convergence criterion
(yielding, in infinite volume, the AT line) is satisfied, ii) Plefka's first
convergence criterion (hence the AT line) is necessary yet hardly sufficient,
or iii) Bolthausen's magnetizations do not approximate the TAP solutions
sufficiently well up to the AT line.Comment: 29 pages, 1 figur