322 research outputs found
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
On Ward-Takahashi identities for the Parisi spin glass
The introduction of ``small permutations'' allows us to derive Ward-Takahashi
identities for the spin-glass, in the Parisi limit of an infinite number of
steps of replica symmetry breaking. The first identities express the emergence
of a band of Goldstone modes. The next identities relate components of (the
Replica Fourier Transformed) 3-point function to overlap derivatives of the
2-point function (inverse propagator). A jump in this last function is
exhibited, when its two overlaps are crossing each other, in the special
simpler case where one of the cross-overlaps is maximal.Comment: this new version includes acknowledgements to funding agencie
Reparametrization invariance: a gauge-like symmetry of ultrametrically organised states
The reparametrization transformation between ultrametrically organised states
of replicated disordered systems is explicitly defined. The invariance of the
longitudinal free energy under this transformation, i.e. reparametrization
invariance, is shown to be a direct consequence of the higher level symmetry of
replica equivalence. The double limit of infinite step replica symmetry
breaking and n=0 is needed to derive this continuous gauge-like symmetry from
the discrete permutation invariance of the n replicas. Goldstone's theorem and
Ward identities can be deduced from the disappearence of the second (and higher
order) variation of the longitudinal free energy. We recall also how these and
other exact statements follow from permutation symmetry after introducing the
concept of "infinitesimal" permutations.Comment: 16 pages, 3 figure
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
Beyond the Sherrington-Kirkpatrick Model
The state of art in spin glass field theory is reviewed.Comment: contribution to the volume "Spin Glasses and Random Fields", ed. P.
Young, World Scientific. Latex file and lprocl.sty (style-file). 41 pages, no
figure
Low temperature spin glass fluctuations: expanding around a spherical approximation
The spin glass behavior near zero temperature is a complicated matter. To get
an easier access to the spin glass order parameter and, at the same
time, keep track of , its matrix aspect, and hence of the Hessian
controlling stability, we investigate an expansion of the replicated free
energy functional around its ``spherical'' approximation. This expansion is
obtained by introducing a constraint-field and a (double) Legendre Transform
expressed in terms of spin correlators and constraint-field correlators. The
spherical approximation has the spin fluctuations treated with a global
constraint and the expansion of the Legendre Transformed functional brings them
closer and closer to the Ising local constraint. In this paper we examine the
first contribution of the systematic corrections to the spherical starting
point.Comment: 16 pages, 2 figure
Random field Ising model: dimensional reduction or spin-glass phase?
The stability of the random field Ising model (RFIM) against spin glass (SG)
fluctuations, as investigated by M\'ezard and Young, is naturally expressed via
Legendre transforms, stability being then associated with the non-negativeness
of eigenvalues of the inverse of a generalized SG susceptibility matrix. It is
found that the signal for the occurrence of the SG transition will manifest
itself in free-energy {\sl fluctuations\/} only, and not in the free energy
itself. Eigenvalues of the inverse SG susceptibility matrix is then approached
by the Rayleigh Ritz method which provides an upper bound. Coming from the
paramagnetic phase {\sl on the Curie line,\/} one is able to use a virial-like
relationship generated by scaling the {\sl single\/} unit length in
higher dimension a new length sets in, the inverse momentum cut off).
Instability towards a SG phase being probed on pairs of {\sl distinct\/}
replicas, it follows that, despite the repulsive coupling of the RFIM the
effective pair coupling is {\sl attractive\/} (at least for small values of the
parameter the coupling and the
effective random field fluctuation). As a result, \lq\lq bound states\rq\rq\
associated with replica pairs (negative eigenvalues) provide the instability
signature. {\sl Away from the Curie line\/}, the attraction is damped out till
the SG transition line is reached and paramagnetism restored. In the
SG transition always precedes the ferromagnetic one, thus the domain in
dimension where standard dimensional reduction would apply (on the Curie line)
shrinks to zero.Comment: te
Stability of the Mezard-Parisi solution for random manifolds
The eigenvalues of the Hessian associated with random manifolds are
constructed for the general case of steps of replica symmetry breaking. For
the Parisi limit (continuum replica symmetry breaking) which is
relevant for the manifold dimension , they are shown to be non negative.Comment: LaTeX, 15 page
Replica Fourier Transforms on Ultrametric Trees, and Block-Diagonalizing Multi-Replica Matrices
The analysis of objects living on ultrametric trees, in particular the
block-diagonalization of 4-replica matrices ,
is shown to be dramatically simplified through the introduction of properly
chosen operations on those objects. These are the Replica Fourier Transforms on
ultrametric trees. Those transformations are defined and used in the present
work.Comment: Latex file, 14 page
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