1,100 research outputs found
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
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
Quantum annealing with antiferromagnetic fluctuations
We introduce antiferromagnetic quantum fluctuations into quantum annealing in
addition to the conventional transverse-field term. We apply this method to the
infinite-range ferromagnetic p-spin model, for which the conventional quantum
annealing has been shown to have difficulties to find the ground state
efficiently due to a first-order transition. We study the phase diagram of this
system both analytically and numerically. Using the static approximation, we
find that there exists a quantum path to reach the final ground state from the
trivial initial state that avoids first-order transitions for intermediate
values of p. We also study numerically the energy gap between the ground state
and the first excited state and find evidence for intermediate values of p that
the time complexity scales polynomially with the system size at a second-order
transition point along the quantum path that avoids first-order transitions.
These results suggest that quantum annealing would be able to solve this
problem with intermediate values of p efficiently in contrast to the case with
only simple transverse-field fluctuations.Comment: 19 pages, 11 figures; Added references; To be published in Physical
Review
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
Locations of multicritical points for spin glasses on regular lattices
We present an analysis leading to precise locations of the multicritical
points for spin glasses on regular lattices. The conventional technique for
determination of the location of the multicritical point was previously derived
using a hypothesis emerging from duality and the replica method. In the present
study, we propose a systematic technique, by an improved technique, giving more
precise locations of the multicritical points on the square, triangular, and
hexagonal lattices by carefully examining relationship between two partition
functions related with each other by the duality. We can find that the
multicritical points of the Ising model are located at
on the square lattice, where means the probability of ,
at on the triangular lattice, and at on the
hexagonal lattice. These results are in excellent agreement with recent
numerical estimations.Comment: 17pages, this is the published version with some minnor corrections.
Previous title was "Precise locations of multicritical points for spin
glasses on regular lattices
Anomalous behavior of the energy gap in the one-dimensional quantum XY model
We re-examine the well-studied one dimensional spin-1/2 model to reveal
its nontrivial energy spectrum, in particular the energy gap between the ground
state and the first excited state. In the case of the isotropic model --
the model -- the gap behaves very irregularly as a function of the system
size at a second order transition point. This is in stark contrast to the usual
power-law decay of the gap and is reminiscent of the similar behavior at the
first order phase transition in the infinite-range quantum model. The gap
also shows nontrivial oscillatory behavior for the phase transitions in the
anisotropic model in the incommensurate phase. We observe a close relation
between this anomalous behavior of the gap and the correlation functions. These
results, those for the isotropic case in particular, are important from the
viewpoint of quantum annealing where the efficiency of computation is strongly
affected by the size dependence of the energy gap.Comment: 25 pages, 8 figures. arXiv admin note: substantial text overlap with
arXiv:1501.0292
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
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.
High-Temperature Dynamics of Spin Glasses
We develop a systematic expansion method of physical quantities for the SK
model and the finite-dimensional model of spin glasses in
non-equilibrium states. The dynamical probability distribution function is
derived from the master equation using a high temperature expansion. We
calculate the expectation values of physical quantities from the dynamical
probability distribution function. The theoretical curves show satisfactory
agreement with Monte Carlo simulation results in the appropriate temperature
and time regions. A comparison is made with the results of a dynamics theory by
Coolen, Laughton and Sherrington.Comment: 24 pages, figures available on request, LaTeX, uses jpsj.sty, to be
published in J. Phys. Soc. Jpn. 66 No. 7 (1997
Surface terms on the Nishimori line of the Gaussian Edwards-Anderson model
For the Edwards-Anderson model we find an integral representation for some
surface terms on the Nishimori line. Among the results are expressions for the
surface pressure for free and periodic boundary conditions and the adjacency
pressure, i.e., the difference between the pressure of a box and the sum of the
pressures of adjacent sub-boxes in which the box can been decomposed. We show
that all those terms indeed behave proportionally to the surface size and prove
the existence in the thermodynamic limit of the adjacency pressure.Comment: Final version with minor corrections. To appear in Journal of
Statistical Physic
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