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Scientific Report 2007-2009
Theoretical physics
T12. Statistical mechanics of disordered systems and renormalization
group
The behavior of strongly disordered systems is very
different from the one of pure, homogeneous systems.
Experimentally, the most dramatic effects are observed
in the dynamics. Very slow relaxation and aging, severe
nonequilibrium effects, memory and oblivion, generalizations
of the usual fluctuation-dissipation relations are all
specific features of disordered systems. These dynamic
effects have a static counterpart. For example, in spin
glasses the onset of the slow relaxation is associated with
the divergence of a static quantity, the nonlinear susceptibility.
Unfortunately, our understanding of the static behavior
of strongly disordered systems is rather limited with
the notable exceptions of the Sherrington-Kirkpatrick
model and of Derrida’s random energy model. In most of
the cases, Monte Carlo simulations provide the only tool
to determine the critical behavior and to sort out the
different theories which have been proposed to describe
these systems.
In the last few years we have performed several numerical
studies of the three-dimensional Edwards-Anderson
model. The use of the most advanced numerical techniques
— the random-exchange or parallel-tempering
method, multi-spin coding, cluster algorithms, etc. —
and of very fast computers allowed us to address longstanding
problems and to obtain several new and important
results.
Numerical simulations of random systems are notoriously
very difficult and, in spite of significant algorithmic
progress, numerical simulations are limited to relatively
small system sizes. A significant improvement has
been obtained by developing a new dedicated machine
(JANUS) in collaboration with the University of Ferrara
and several Spanish research groups. JANUS is a
modular, massively parallel, and reconfigurable FPGAbased
computing system. JANUS is tailored for, but
not limited to, the requirements of a class of hard scientific
applications characterized by regular code structure,
unconventional data manipulation instructions, and nottoo-large
database size. In particular, the machine is
well suited for numerical simulations of spin glasses.
On this class of applications JANUS achieves impressive
performances: in some cases one JANUS processing
element outperfoms high-end PCs by a factor of approximately
1000. Several simulations have been performed
on Janus. The critical behavior of the four-state
commutative random-permutation glassy Potts model in
three and four dimensions and of the four-state threedimensional
Potts model have been carefully studied.
More importantly, the use of JANUS allowed us to
study carefully the relaxational dynamics in the threedimensional
Edwards-Anderson model [1], for a time
spanning 11 orders of magnitude, thus approaching the
experimentally relevant scale (i.e., seconds).
One of the most peculiar properties of the mean-field
solution of the Edwards-Anderson model is the so-called
ultrametricity: In the low-temperature phase thermodynamic
states are organized in a hierarchical structure.
One of the long-standing questions is whether such a
structure also holds in the three-dimensional model or
instead is a peculiarity of the mean-field solution as predicted
by the droplet theory. In [2] we studied numerically
the issue and found good evidence for the presence
of an ultrametric structure also in the three-dimensional
case.
Given the difficulty in obtaining clear-cut results for
the three-dimensional spin glass, we also investigated
several spin-glass models which share some of the
properties of the finite-dimension Edwards-Anderson
model, but, at the same time, are significantly simpler
to simulate numerically. For this purpose we introduced
a one-dimensional spin-glass model with long-range interactions.
The interaction between two spins a distance
r apart is either ±1 with a probability that decays with
r as 1/r ρ , or zero. Depending on the exponent ρ, the
model may or may not show mean-field behavior: for
ρ ≤ 4/3 the mean-field approximation is exact, for ρ > 2
no phase transition occurs, while in between the behavior
is nontrivial. Since this model is one-dimensional
and, in spite of the presence of long-range interactions,
each spin only interacts with a finite number of different
(may be far) spins, it is possible to simulate quite large
systems and carefully investigate finite-size effects. In
[3] we studied numerically the model in the absence of
magnetic field for values of p in the intermediate range,
identified the paramagnetic-glassy phase transition, and
characterized the low-temperature phase. We found
both static and dynamic indications in favor of the
so-called replica-symmetry breaking theory. In [4] we
considered the behavior in an external magnetic field
h. The results, obtained by means of a new analysis
method, strongly suggest the presence of a finite-h
transition, as also observed in the mean-field solution of
the Edwards-Anderson model.
References
1. F. Belletti et al., Phys. Rev. Lett. 101, 157201 (2008).
2. P. Contucci et al., Phys. Rev. Lett. 99, 057206 (2007).
3. L. Leuzzi et al., Phys. Rev. Lett. 101, 107203 (2008).
4. L. Leuzzi et al., Phys. Rev. Lett. 103, 267201 (2009).
Authors
G. Parisi, E. Marinari, A. Pelissetto, F. Ricci-Tersenghi, A.
Cavagna, 3 I. Giardina, 3 L. Leuzzi, 3 A. Maiorano, S. Perez
Sapienza Università di Roma 35 Dipartimento di Fisica