186,169 research outputs found
Spin Glass Field Theory with Replica Fourier Transforms
We develop a field theory for spin glasses using Replica Fourier Transforms
(RFT). We present the formalism for the case of replica symmetry and the case
of replica symmetry breaking on an ultrametric tree, with the number of
replicas and the number of replica symmetry breaking steps generic
integers. We show how the RFT applied to the two-replica fields allows to
construct a new basis which block-diagonalizes the four-replica mass-matrix,
into the replicon, anomalous and longitudinal modes. The eigenvalues are given
in terms of the mass RFT and the propagators in the RFT space are obtained by
inversion of the block-diagonal matrix. The formalism allows to express any
-replica vertex in the new RFT basis and hence enables to perform a standard
perturbation expansion. We apply the formalism to calculate the contribution of
the Gaussian fluctuations around the Parisi solution for the free-energy of an
Ising spin glass.Comment: 39 pages, 3 figure
Replica symmetry breaking related to a general ultrametric space II: RSB solutions and the n\to0 limit
Replica symmetry breaking solutions for the new replica anzats, related to
general ultrametric spaces, are investigated. A variant of analysis on trees is
developed and applied to the computation of the n\to0 limit in the new replica
anzats.Comment: 22 page
Replica symmetry breaking related to a general ultrametric space III: the case of general measure
Family of replica matrices, related to general ultrametric spaces with
general measures, is introduced. These matrices generalize the known Parisi
matrices. Some functionals of replica approach are computed. Replica symmetry
breaking solution is found.Comment: 21 page
Finite size corrections to the Parisi overlap function in the GREM
We investigate the effects of finite size corrections on the overlap
probabilities in the Generalized Random Energy Model (GREM) in two situations
where replica symmetry is broken in the thermodynamic limit. Our calculations
do not use replicas, but shed some light on what the replica method should give
for finite size corrections. In the gradual freezing situation, which is known
to exhibit full replica symmetry breaking, we show that the finite size
corrections lead to a modification of the simple relations between the sample
averages of the overlaps between configurations predicted by
replica theory. This can be interpreted as fluctuations in the replica block
size with a \emph{negative} variance. The mechanism is similar to the one we
found recently in the random energy model [1]. We also consider a simultaneous
freezing situation, which is known to exhibit one step replica symmetry
breaking. We show that finite size corrections lead to full replica symmetry
breaking and give a more complete derivation of the results presented in [2]
for the directed polymer on a tree.Comment: 21 pages, 2 figure
Spontaneous versus explicit replica symmetry breaking in the theory of disordered systems
We investigate the relation between spontaneous and explicit replica symmetry
breaking in the theory of disordered systems. On general ground, we prove the
equivalence between the replicon operator associated with the stability of the
replica symmetric solution in the standard replica scheme and the operator
signaling a breakdown of the solution with analytic field dependence in a
scheme in which replica symmetry is explicitly broken by applied sources. This
opens the possibility to study, via the recently developed functional
renormalization group, unresolved questions related to spontaneous replica
symmetry breaking and spin-glass behavior in finite-dimensional disordered
systems.Comment: 16 page
Thermodynamic Construction of an One-Step Replica-Symmetry-Breaking Solution in Finite Connectivity Spin Glasses
An one-step replica-symmetry-breaking solution for finite connectivity
spin-glass models with K body interaction is constructed at finite temperature
using the replica method and thermodynamic constraints. In the absence of
external fields, this construction provides a general extension of replica
symmetric solution at finite replica number to one-step
replica-symmetry-breaking solution. It is found that this result is formally
equivalent to that of the one-step replica-symmetry-breaking cavity method. To
confirm the validity of the obtained solution, Monte Carlo simulations are
performed for K = 2 and 3. The thermodynamic quantities of the Monte Carlo
results extrapolated to a large-size limit are consistent with those estimated
by our solution for K = 2 at all simulated temperatures and for K = 3 except
near the transition temperature.Comment: 11pages, 19 figures. Added content and references. Accepted to Phys.
Rev.
- …