3,166 research outputs found
Exact results on spin glass models
Exact and/or rigorous results are reviewed for the Ising and XY/Villain spin
glasses in finite dimensions, such as the exact energy, correlation identities
and a functional relation between the distribution functions of ferromagnetic
and spin glass order parameters. This last relation is useful to prove that the
phase space is not complex on a line in the phase diagram. The spin wave theory
neglecting periodicity is shown to give exact results for the Villain model on
a line in the phase diagram. Implications of this fact on the
renormalization-group results for the random-phase XY/Villain model in two
dimensions are discussed.Comment: 9 pages, Proceedings of STATPHYS21, to be published in Physica
Complex and Non-Complex Phase Structures in Models of Spin Glasses and Information Processing
The gauge theory of spin glasses and statistical-mechanical formulation of
error-correcting codes are reviewed with an emphasis on their similarities. For
the gauge theory, we explain the functional identities on dynamical
autocorrelation functions and on the distribution functions of order
parameters. These functional identities restrict the possibilities of slow
dynamics and complex structure of the phase space. An inequality for
error-correcting codes is shown to be interpreted to indicate non-monotonicity
of spin orientation as a function of the temperature in spin glasses.Comment: 13 pages; Proceedings of the International Symposium on Slow Dynamics
in Nature, Seoul, Korea, November 2001; to be published in Physica
Derivatives and inequalities for order parameters in the Ising spin glass
Identities and inequalities are proved for the order parameters, correlation
functions and their derivatives of the Ising spin glass. The results serve as
additional evidence that the ferromagnetic phase is composed of two regions,
one with strong ferromagnetic ordering and the other with the effects of
disorder dominant. The Nishimori line marks a crossover between these two
regions.Comment: 10 pages; 3 figures; new inequalities added, title slightly change
Multicritical point of spin glasses
We present a theoretical framework to accurately calculate the location of
the multicritical point in the phase diagram of spin glasses. The result shows
excellent agreement with numerical estimates. The basic idea is a combination
of the duality relation, the replica method, and the gauge symmetry. An
additional element of the renormalization group, in particular in the context
of hierarchical lattices, leads to impressive improvements of the predictions.Comment: 6 pages, 3 figures, Dedicated to Prof. A. Nihat Berker on the
occasion of his sixtieth birthday. to appear in Physica
Self-Duality and Phase Structure of the 4D Random-Plaquette Z_2 Gauge Model
In the present paper, we shall study the 4-dimensional Z_2 lattice gauge
model with a random gauge coupling; the random-plaquette gauge model(RPGM). The
random gauge coupling at each plaquette takes the value J with the probability
1-p and -J with p. This model exhibits a confinement-Higgs phase transition. We
numerically obtain a phase boundary curve in the (p-T)-plane where T is the
"temperature" measured in unit of J/k_B. This model plays an important role in
estimating the accuracy threshold of a quantum memory of a toric code. In this
paper, we are mainly interested in its "self-duality" aspect, and the
relationship with the random-bond Ising model(RBIM) in 2-dimensions. The
"self-duality" argument can be applied both for RPGM and RBIM, giving the same
duality equations, hence predicting the same phase boundary. The phase boundary
curve obtained by our numerical simulation almost coincides with this predicted
phase boundary at the high-temperature region. The phase transition is of first
order for relatively small values of p < 0.08, but becomes of second order for
larger p. The value of p at the intersection of the phase boundary curve and
the Nishimori line is regarded as the accuracy threshold of errors in a toric
quantum memory. It is estimated as p=0.110\pm0.002, which is very close to the
value conjectured by Takeda and Nishimori through the "self-duality" argument.Comment: 14 pages, 7 figure
Inequalities for the Local Energy of Random Ising Models
We derive a rigorous lower bound on the average local energy for the Ising
model with quenched randomness. The result is that the lower bound is given by
the average local energy calculated in the absence of all interactions other
than the one under consideration. The only condition for this statement to hold
is that the distribution function of the random interaction under consideration
is symmetric. All other interactions can be arbitrarily distributed including
non-random cases. A non-trivial fact is that any introduction of other
interactions to the isolated case always leads to an increase of the average
local energy, which is opposite to ferromagnetic systems where the Griffiths
inequality holds. Another inequality is proved for asymmetrically distributed
interactions. The probability for the thermal average of the local energy to be
lower than that for the isolated case takes a maximum value on the Nishimori
line as a function of the temperature. In this sense the system is most stable
on the Nishimori line.Comment: 10 pages. Submitted to J. Phys. Soc. Jp
Multicritical points for the spin glass models on hierarchical lattices
The locations of multicritical points on many hierarchical lattices are
numerically investigated by the renormalization group analysis. The results are
compared with an analytical conjecture derived by using the duality, the gauge
symmetry and the replica method. We find that the conjecture does not give the
exact answer but leads to locations slightly away from the numerically reliable
data. We propose an improved conjecture to give more precise predictions of the
multicritical points than the conventional one. This improvement is inspired by
a new point of view coming from renormalization group and succeeds in deriving
very consistent answers with many numerical data.Comment: 11 pages, 9 figures, 7 tables This is the published versio
Stability in microcanonical many-body spin glasses
We generalize the de Almeida-Thouless line for the many-body Ising spin glass
to the microcanonical ensemble and show that it coincides with the canonical
one. This enables us to draw a complete microcanonical phase diagram of this
model
Error counting in a quantum error-correcting code and the ground-state energy of a spin glass
Upper and lower bounds are given for the number of equivalence classes of
error patterns in the toric code for quantum memory. The results are used to
derive a lower bound on the ground-state energy of the +/-J Ising spin glass
model on the square lattice with symmetric and asymmetric bond distributions.
This is a highly non-trivial example in which insights from quantum information
lead directly to an explicit result on a physical quantity in the statistical
mechanics of disordered systems.Comment: 15 pages, 7 figures, JPSJ style, latex style file include
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