632 research outputs found
Nature of the phase transition of the three-dimensional isotropic Heisenberg spin glass
Equilibrium properties of the three-dimensional isotropic Heisenberg spin
glass are studied by extensive Monte Carlo simulations, with particular
attention to the nature of its phase transition. A finite-size-scaling analysis
is performed both for the spin-glass (SG) and the chiral-glass (CG) orders. Our
results suggest that the model exhibits the CG long-range order at finite
temperatures without accompanying the conventional SG long-range order, in
contrast to some of the recent works claiming the simultaneous SG and CG
transition. Typical length and time scales which represent a crossover from the
spin-chirality coupling regime at short scales to the spin-chirality decoupling
regime at long scales are introduced and examined in order to observe the true
asymptotic transition behavior. On the basis of these crossover scales,
discussion is given concerning the cause of the discrepancy between our present
result and those of other recent numerical works.Comment: 22 pages, 26 figures, minor change
Higher-order tensor renormalization group approach to lattice glass model
In this study, the higher-order tensor renormalization group (HOTRG) method
is applied to a lattice glass model that has local constraints on the
occupation number of neighboring particles represented by many-body
interactions. This model exhibits first- and second-order transitions depending
on a certain model parameter. The results obtained by using the HOTRG method
for the model were confirmed to be consistent with those obtained by a
Markov-chain Monte Carlo (MCMC) method for systems of relatively small sizes.
The transition points are accurately estimated by the HOTRG calculation for the
systems of large sizes, which is challenging to perform using the MCMC method.
These results demonstrate that the HOTRG method can be an efficient method for
studying systems with many-body interactions.Comment: 8 pages, 15 figure
Statistical-mechanical Analysis of Linear Programming Relaxation for Combinatorial Optimization Problems
Typical behavior of the linear programming (LP) problem is studied as a
relaxation of the minimum vertex cover, a type of integer programming (IP)
problem. A lattice-gas model on the Erd\"os-R\'enyi random graphs of
-uniform hyperedges is proposed to express both the LP and IP problems
of the min-VC in the common statistical-mechanical model with a one-parameter
family. Statistical-mechanical analyses reveal for that the LP
optimal solution is typically equal to that given by the IP below the critical
average degree in the thermodynamic limit. The critical threshold for
good accuracy of the relaxation extends the mathematical result , and
coincides with the replica symmetry-breaking threshold of the IP. The LP
relaxation for the minimum hitting sets with , minimum vertex
covers on -uniform random graphs, is also studied. Analytic and
numerical results strongly suggest that the LP relaxation fails to estimate
optimal values above the critical average degree where the
replica symmetry is broken.Comment: 12 pages, 5 figures; typos are fixe
Numerical Study on Aging Dynamics in the 3D Ising Spin-Glass Model. III. Cumulative Memory and `Chaos' Effects in the Temperature-Shift Protocol
The temperature ()-shift protcol of aging in the 3 dimensional (3D)
Edwards- Anderson (EA) spin-glass (SG) model is studied through the
out-of-phase component of the ac susceptibility simulated by the Monte Carlo
method. For processes with a small magnitude of the -shift, , the
memory imprinted before the -shift is preserved under the -change and the
SG short-range order continuously grows after the -shift, which we call the
cumulative memory scenario. For a negative -shift process with a large
the deviation from the cumulative memory scenario has been observed
for the first time in the numerical simulation. We attribute the phenomenon to
the `chaos effect' which, we argue, is qualitatively different from the
so-called rejuvenation effect observed just after the -shift.Comment: 8 pages, 5 figure
Minimum vertex cover problems on random hypergraphs: replica symmetric solution and a leaf removal algorithm
We study minimum vertex cover problems on random \alpha-uniform hypergraphs
using two different approaches, a replica method in statistical mechanics of
random systems and a leaf removal algorithm. It is found that there exists a
phase transition at the critical average degree e/(\alpha-1). Below the
critical degree, a replica symmetric ansatz in the statistical-mechanical
method holdsand the algorithm estimates a solution of the problem which
coincide with that by the replica method. In contrast, above the critical
degree, the replica symmetric solution becomes unstable and these methods fail
to estimate the exact solution.These results strongly suggest a close relation
between the replica symmetry and the performance of approximation algorithm.Comment: 5 pages, 2 figure
Evidence of one-step replica symmetry breaking in a three-dimensional Potts glass model
We study a 7-state Potts glass model in three dimensions with first, second,
and third neighbor interactions with a bimodal distribution of couplings by
Monte Carlo simulations. Our results show the existence of a spin-glass
transition at a finite temperature T_c, a discontinuous jump of an order
parameter at T_c without latent heat, and a non-trivial structure of the
order-parameter distribution below T_c. They are compatible with a one-step
replica symmetry breaking.Comment: 5 pages, 7 figure
Numerical Detection of the Ergodicity Breaking in a Lattice Glass Model
We directly detect the ergodicity breaking in a lattice glass model by a
numerical simulation. The obtained results nicely agree with those by the
cavity method that the model on a regular random graph exhibits a dynamical
transition with the ergodicity breaking at an occupation density. The present
method invented for a numerical detection of the ergodicity breaking is
applicable to glassy systems in finite dimensions.Comment: 5 pages, 4 figure
Typical behavior of the linear programming method for combinatorial optimization problems: From a statistical-mechanical perspective
Typical behavior of the linear programming problem (LP) is studied as a
relaxation of the minimum vertex cover problem, which is a type of the integer
programming problem (IP). To deal with the LP and IP by statistical mechanics,
a lattice-gas model on the Erd\"os-R\'enyi random graphs is analyzed by a
replica method. It is found that the LP optimal solution is typically equal to
that of the IP below the critical average degree c*=e in the thermodynamic
limit. The critical threshold for LP=IP is beyond a mathematical result, c=1,
and coincides with the replica-symmetry-breaking threshold of the IP.Comment: 5 pages, 3 figure
Eigenvalue analysis of an irreversible random walk with skew detailed balance conditions
An irreversible Markov-chain Monte Carlo (MCMC) algorithm with skew detailed
balance conditions originally proposed by Turitsyn et al. is extended to
general discrete systems on the basis of the Metropolis-Hastings scheme. To
evaluate the efficiency of our proposed method, the relaxation dynamics of the
slowest mode and the asymptotic variance are studied analytically in a random
walk on one dimension. It is found that the performance in irreversible MCMC
methods violating the detailed balance condition is improved by appropriately
choosing parameters in the algorithm.Comment: 14 pages, 6 figure
Phase transitions and ordering structures of a model of chiral helimagnet in three dimensions
Phase transitions in a classical Heisenberg spin model of a chiral helimagnet
with the Dzyaloshinskii--Moriya (DM) interaction in three dimensions are
numerically studied. By using the event-chain Monte Carlo algorithm recently
developed for particle and continuous spin systems, we perform equilibrium
Monte Carlo simulations for large systems up to about spins. Without
magnetic fields, the system undergoes a continuous phase transition with
critical exponents of the three-dimensional \textit{XY} model, and a uniaxial
periodic helical structure emerges in the low temperature region. In the
presence of a magnetic field perpendicular to the axis of the helical
structure, it is found that there exists a critical point on the temperature
and magnetic-field phase diagram and that above the critical point the system
exhibits a phase transition with strong divergence of the specific heat and the
uniform magnetic susceptibility.Comment: 10 pages, 18 figure
- …