30 research outputs found
Replica field theory and renormalization group for the Ising spin glass in an external magnetic field
We use the generic replica symmetric cubic field-theory to study the
transition of short range Ising spin glasses in a magnetic field around the
upper critical dimension, d=6. A novel fixed-point is found, in addition to the
well-known zero magnetic field fixed-point, from the application of the
renormalization group. In the spin glass limit, n going to 0, this fixed-point
governs the critical behaviour of a class of systems characterised by a single
cubic interaction parameter. For this universality class, the spin glass
susceptibility diverges at criticality, whereas the longitudinal mode remains
massive. The third mode, the so-called anomalous one, however, behaves
unusually, having a jump at criticality. The physical consequences of this
unusual behaviour are discussed, and a comparison with the conventional de
Almeida-Thouless scenario presented.Comment: 5 pages written in revtex4. Accepted for publication in Phys. Rev.
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Is the droplet theory for the Ising spin glass inconsistent with replica field theory?
Symmetry arguments are used to derive a set of exact identities between
irreducible vertex functions for the replica symmetric field theory of the
Ising spin glass in zero magnetic field. Their range of applicability spans
from mean field to short ranged systems in physical dimensions. The replica
symmetric theory is unstable for d>8, just like in mean field theory. For 6<d<8
and d<6 the resummation of an infinite number of terms is necessary to settle
the problem. When d<8, these Ward-like identities must be used to distinguish
an Almeida-Thouless line from the replica symmetric droplet phase.Comment: 4 pages. Accepted for publication in J.Phys.A. This is the accepted
version with the following minor changes: one extra sentence in the abstract;
footnote 2 slightly extended; last paragraph somewhat reformulate
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Parisi Phase in a Neuron
Pattern storage by a single neuron is revisited. Generalizing Parisi's
framework for spin glasses we obtain a variational free energy functional for
the neuron. The solution is demonstrated at high temperature and large relative
number of examples, where several phases are identified by thermodynamical
stability analysis, two of them exhibiting spontaneous full replica symmetry
breaking. We give analytically the curved segments of the order parameter
function and in representative cases compute the free energy, the storage
error, and the entropy.Comment: 4 pages in prl twocolumn format + 3 Postscript figures. Submitted to
Physical Review Letter
The influence of critical behavior on the spin glass phase
We have argued in recent papers that Monte Carlo results for the equilibrium
properties of the Edwards-Anderson spin glass in three dimensions, which had
been interpreted earlier as providing evidence for replica symmetry breaking,
can be explained quite simply within the droplet model once finite size effects
and proximity to the critical point are taken into account. In this paper, we
show that similar considerations are sufficient to explain the Monte Carlo data
in four dimensions. In particular, we study the Parisi overlap and the link
overlap for the four-dimensional Ising spin glass in the Migdal-Kadanoff
approximation. Similar to what is seen in three dimensions, we find that
temperatures well below those studied in Monte Carlo simulations have to be
reached before the droplet model predictions become apparent. We also show that
the double-peak structure of the link overlap distribution function is related
to the difference between domain-wall excitations that cross the entire system
and droplet excitations that are confined to a smaller region.Comment: 8 pages, 8 figure
Large random correlations in individual mean field spin glass samples
We argue that complex systems must possess long range correlations and
illustrate this idea on the example of the mean field spin glass model. Defined
on the complete graph, this model has no genuine concept of distance, but the
long range character of correlations is translated into a broad distribution of
the spin-spin correlation coefficients for almost all realizations of the
random couplings. When we sample the whole phase space we find that this
distribution is so broad indeed that at low temperatures it essentially becomes
uniform, with all possible correlation values appearing with the same
probability. The distribution of correlations inside a single phase space
valley is also studied and found to be much narrower.Comment: Added a few references and a comment phras