2,260 research outputs found
On infrared divergences in spin glasses
By studying the structure of infrared divergences in a toy propagator in the
replica approach to the Ising spin glass below , we suggest a possible
cancellation mechanism which could decrease the degree of singularity in the
loop expansion.Comment: 13 pages, Latex , revised versio
Dynamics versus replicas in the random field Ising model
In a previous article we have shown, within the replica formalism, that the
conventional picture of the random field Ising model breaks down, by the effect
of singularities in the interactions between fields involving several replicas,
below dimension eight. In the zero-replica limit several coupling constants
have thus to be considered, instead of just one. As a result we had found that
there is no stable fixed point in the vicinity of dimension six. It is natural
to reconsider the problem in a dynamical framework, which does not require
replicas, although the equilibrium properties should be recovered in the large
time limit. Singularities in the zero-replica limit are a priori not visible in
a dynamical picture. In this note we show that in fact new interactions are
also generated in the stochastic approach. Similarly these interactions are
found to be singular below dimension eight. These critical singularities
require the introduction of a time origin at which initial data are
given. The dynamical properties are thus dependent upon the waiting time. It is
shown here that one can indeed find a complete correspondence between the
equilibrium singularities in the limit, and the singularities in the
dynamics when the initial time goes to minus infinity, with replaced
by . There is thus complete coherence between the two
approaches.Comment: 8 pages, latex, no figur
On a Dynamical-Like Replica-Symmetry-Breaking Scheme for the Spin Glass
Considering the unphysical result obtained in the calculation of the
free-energy cost for twisting the boundary conditions in a spin glass, we trace
it to the negative multiplicities associated with the Parisi replica-symmetry
breaking (RSB). We point out that a distinct RSB, that keeps positive
multiplicities, was proposed long ago, in the spirit of an ultra-long time
dynamical approach due to Sompolinsky. For an homogeneous bulk system, both RSB
schemes are known to yield identical free energies and observables. However,
using the dynamical RSB, we have recalculated the twist free energy at the
mean-field level. The free-energy cost of this twist is, as expected, positive
in that scheme, as it should be
The Sherrington-Kirkpatrick model near T_c and near T=0
Some recent results concerning the Sherrington-Kirkpatrick model are
reported. For near the critical temperature , the replica free energy
of the Sherrington-Kirkpatrick model is taken as the starting point of an
expansion in powers of about the
Replica Symmetric solution . The expansion is kept up to 4-th
order in where a Parisi solution emerges, but
only if one remains close enough to .
For near zero we show how to separate contributions from
where the Hessian maintains the standard structure of Parisi Replica Symmetry
Breaking with bands of eigenvalues bounded below by zero modes. For the bands collapse and only two eigenvalues, a null one and a positive
one, are found. In this region the solution stands in what can be called a {\sl
droplet-like} regime.Comment: 11 pages, 3 figures, Published versio
Twist Free Energy
One may impose to a system with spontaneous broken symmetry, boundary
conditions which correspond to different pure states at two ends of a sample.
For a discrete Ising-like broken symmetry, boundary conditions with opposite
spins in two parallel limiting planes, generate an interface and a cost in free
energy per unit area of the interface. For continuum symmetries the order
parameter interpolates smoothly between the end planes carrying two different
directions of the order parameter. The cost in free energy is then proportional
to for a system of characteristic size L. The power of is related
to the lower critical dimension, and the coefficient of this additional free
energy vanishes at the critical temperature. In this note it is shown within a
loop expansion that one does find the expected behavior of this twist free
energy. This is a preamble to the study of situations where the broken
continuum symmetry is believed to be more complex, as in Parisi's ansatz for
the Edwards-Anderson spin glass.Comment: 15 pages, latex, no figur
Twist Free Energy in a Spin Glass
The field theory of a short range spin glass with Gaussian random
interactions, is considered near the upper critical dimension six. In the
glassy phase, replica symmetry breaking is accompanied with massless Goldstone
modes, generated by the breaking of reparametrization invariance of a Parisi
type solution. Twisted boundary conditions are thus imposed at two opposite
ends of the system in order to study the size dependence of the twist free
energy. A loop-expansion is performed to first order around a twisted
background. It is found, as expected but it is non trivial, that the theory
does renormalize around such backgrounds, as well as for the bulk. However two
main differences appear, in comparison with simple ferromagnetic transitions :
(i) the loop expansion yields a (negative) anomaly in the size dependence of
the free energy, thereby lifting the lower critical dimension to a value
greater than two given by (ii) the free energy is lowered
by twisting the boundary conditions. This sign may reflect a spontaneous
spatial non-uniformity of the order parameter.Comment: 15 pages, latex, no figur
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
Scaling and infrared divergences in the replica field theory of the Ising spin glass
Replica field theory for the Ising spin glass in zero magnetic field is
studied around the upper critical dimension d=6. A scaling theory of the spin
glass phase, based on Parisi's ultrametrically organised order parameter, is
proposed. We argue that this infinite step replica symmetry broken (RSB) phase
is nonperturbative in the sense that amplitudes of scaling forms cannot be
expanded in term of the coupling constant w^2. Infrared divergent integrals
inevitably appear when we try to compute amplitudes perturbatively,
nevertheless the \epsilon-expansion of critical exponents seems to be
well-behaved. The origin of these problems can be traced back to the unusual
behaviour of the free propagator having two mass scales, the smaller one being
proportional to the perturbation parameter w^2 and providing a natural infrared
cutoff. Keeping the free propagator unexpanded makes it possible to avoid
producing infrared divergent integrals. The role of Ward-identities and the
problem of the lower critical dimension are also discussed.Comment: 14 page
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