6,714 research outputs found
Phase transitions in the frustrated Ising model on the square lattice
We consider the thermal phase transition from a paramagnetic to
stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice
Ising model with competing interactions J1<0 (nearest neighbor, ferromagnetic)
and J2 >0 (second neighbor, antiferromagnetic). The striped phase breaks a Z4
symmetry and is stabilized at low temperatures for g=J2/|J1|>1/2. Despite the
simplicity of the model, it has proved difficult to precisely determine the
order and the universality class of the phase transitions. This was done
convincingly only recently by Jin et al. [PRL 108, 045702 (2012)]. Here, we
further elucidate the nature of these transitions and their anomalies by
employing a combination of cluster mean-field theory, Monte Carlo simulations,
and transfer-matrix calculations. The J1-J2 model has a line of very weak
first-order phase transitions in the whole region 1/2<g<g*, where g* = 0.67(1).
Thereafter, the transitions from g above g* are continuous and can be fully
mapped, using universality arguments, to the critical line of the well known
Ashkin-Teller model from its 4-state Potts point to the decoupled Ising limit.
We also comment on the pseudo-first-order behavior at the Potts point and its
neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads
to the appearance of similar effects in the vicinity of the multicritical point
g* in the J1-J2 model. The continuous transitions near g* can therefore be
mistaken to be first-order transitions, and this realization was the key to
understanding the paramagnetic-striped transition for the full range of g>1/2.
Most of our results are based on Monte Carlo calculations, while the cluster
mean-field and transfer-matrix results provide useful methodological
bench-marks for weakly first-order behaviors and Ashkin-Teller criticality.Comment: 13 pages, 13 figure
On the Convergence of (Stochastic) Gradient Descent with Extrapolation for Non-Convex Optimization
Extrapolation is a well-known technique for solving convex optimization and
variational inequalities and recently attracts some attention for non-convex
optimization. Several recent works have empirically shown its success in some
machine learning tasks. However, it has not been analyzed for non-convex
minimization and there still remains a gap between the theory and the practice.
In this paper, we analyze gradient descent and stochastic gradient descent with
extrapolation for finding an approximate first-order stationary point in smooth
non-convex optimization problems. Our convergence upper bounds show that the
algorithms with extrapolation can be accelerated than without extrapolation
The Entropy Function for the Black Holes of Nariai Class
Based on the fact that the near horizon geometry of the extremal
Schwarzschild-de Sitter black holes is Nariai geometry, we define the black
holes of Nariai class as the configuration whose near-horizon geometry is
factorized as two dimensional de Sitter space-time and some compact topology,
that is Nariai geometry. We extend the entropy function formalism to the case
of the black holes of Nariai class. The conventional entropy function (for the
extremal black holes) is defined as Legendre transformation of Lagrangian
density, thus the `Routhian density', over two dimensional anti-de Sitter. As
for the black holes of Nariai class, it is defined as {\em minus} `Routhian
density' over two dimensional de Sitter space-time. We found an exact agreement
of the result with Bekenstein-Hawking entropy. The higher order corrections are
nontrivial only when the space-time dimension is over four, that is, .
There is a subtlety as regards the temperature of the black holes of Nariai
class. We show that in order to be consistent with the near horizon geometry,
the temperature should be non-vanishing despite the extremality of the black
holes.Comment: references added, compatible with the published versio
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