936 research outputs found
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
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
Fluctuation Theorems on Nishimori Line
The distribution of the performed work for spin glasses with gauge symmetry
is considered. With the aid of the gauge symmetry, which leads to the
exact/rigorous results in spin glasses, we find a fascinating relation of the
performed work as the fluctuation theorem. The integral form of the resultant
relation reproduces the Jarzynski-type equation for spin glasses we have
obtained. We show that similar relations can be established not only for the
distribution of the performed work but also that of the free energy of spin
glasses with gauge symmetry, which provides another interpretation of the phase
transition in spin glasses.Comment: 10 pages, and 1 figur
Quantum Annealing in the Transverse Ising Model
We introduce quantum fluctuations into the simulated annealing process of
optimization problems, aiming at faster convergence to the optimal state.
Quantum fluctuations cause transitions between states and thus play the same
role as thermal fluctuations in the conventional approach. The idea is tested
by the transverse Ising model, in which the transverse field is a function of
time similar to the temperature in the conventional method. The goal is to find
the ground state of the diagonal part of the Hamiltonian with high accuracy as
quickly as possible. We have solved the time-dependent Schr\"odinger equation
numerically for small size systems with various exchange interactions.
Comparison with the results of the corresponding classical (thermal) method
reveals that the quantum annealing leads to the ground state with much larger
probability in almost all cases if we use the same annealing schedule.Comment: 15 pages, RevTeX, 8 figure
Location of the Multicritical Point for the Ising Spin Glass on the Triangular and Hexagonal Lattices
A conjecture is given for the exact location of the multicritical point in
the phase diagram of the +/- J Ising model on the triangular lattice. The
result p_c=0.8358058 agrees well with a recent numerical estimate. From this
value, it is possible to derive a comparable conjecture for the exact location
of the multicritical point for the hexagonal lattice, p_c=0.9327041, again in
excellent agreement with a numerical study. The method is a variant of duality
transformation to relate the triangular lattice directly with its dual
triangular lattice without recourse to the hexagonal lattice, in conjunction
with the replica method.Comment: 9 pages, 1 figure; Minor corrections in notatio
Griffiths Inequalities for Ising Spin Glasses on the Nishimori Line
The Griffiths inequalities for Ising spin glasses are proved on the Nishimori
line with various bond randomness which includes Gaussian and bond
randomness. The proof for Ising systems with Gaussian bond randomness has
already been carried out by Morita et al, which uses not only the gauge theory
but also the properties of the Gaussian distribution, so that it cannot be
directly applied to the systems with other bond randomness. The present proof
essentially uses only the gauge theory, so that it does not depend on the
detail properties of the probability distribution of random interactions. Thus,
the results obtained from the inequalities for Ising systems with Gaussian bond
randomness do also hold for those with various bond randomness, especially with
bond randomness.Comment: 13pages. Submitted to J. Phys. Soc. Jp
Symmetry, complexity and multicritical point of the two-dimensional spin glass
We analyze models of spin glasses on the two-dimensional square lattice by
exploiting symmetry arguments. The replicated partition functions of the Ising
and related spin glasses are shown to have many remarkable symmetry properties
as functions of the edge Boltzmann factors. It is shown that the applications
of homogeneous and Hadamard inverses to the edge Boltzmann matrix indicate
reduced complexities when the elements of the matrix satisfy certain
conditions, suggesting that the system has special simplicities under such
conditions. Using these duality and symmetry arguments we present a conjecture
on the exact location of the multicritical point in the phase diagram.Comment: 32 pages, 6 figures; a few typos corrected. To be published in J.
Phys.
Multicritical Points of Potts Spin Glasses on the Triangular Lattice
We predict the locations of several multicritical points of the Potts spin
glass model on the triangular lattice. In particular, continuous multicritical
lines, which consist of multicritical points, are obtained for two types of
two-state Potts (i.e., Ising) spin glasses with two- and three-body
interactions on the triangular lattice. These results provide us with numerous
examples to further verify the validity of the conjecture, which has succeeded
in deriving highly precise locations of multicritical points for several spin
glass models. The technique, called the direct triangular duality, a variant of
the ordinary duality transformation, directly relates the triangular lattice
with its dual triangular lattice in conjunction with the replica method.Comment: 18 pages, 2, figure
Image restoration using the chiral Potts spin-glass
We report on the image reconstruction (IR) problem by making use of the
random chiral q-state Potts model, whose Hamiltonian possesses the same gauge
invariance as the usual Ising spin glass model. We show that the pixel
representation by means of the Potts variables is suitable for the gray-scale
level image which can not be represented by the Ising model. We find that the
IR quality is highly improved by the presence of a glassy term, besides the
usual ferromagnetic term under random external fields, as very recently pointed
out by Nishimori and Wong. We give the exact solution of the infinite range
model with q=3, the three gray-scale level case. In order to check our
analytical result and the efficiency of our model, 2D Monte Carlo simulations
have been carried out on real-world pictures with three and eight gray-scale
levels.Comment: RevTex 13 pages, 10 figure
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