12,148 research outputs found
The full replica symmetry breaking in the Ising spin glass on random regular graph
In this paper, we extend the full replica symmetry breaking scheme to the
Ising spin glass on a random regular graph. We propose a new martingale
approach, that overcomes the limits of the Parisi-M\'ezard cavity method,
providing a well-defined formulation of the full replica symmetry breaking
problem in random regular graphs. Finally, we define the order parameters of
the system and get a set of self-consistency equations for the order parameters
and the free energy. We face up the problem only from a technical point of
view: the physical meaning of this approach and the quantitative evaluation of
the solution of the self-consistency equations will be discussed in next works.Comment: 23 page
Michel theory of symmetry breaking and gauge theories
We extend Michel's theorem on the geometry of symmetry breaking [L. Michel,
{\it Comptes Rendus Acad. Sci. Paris} {\bf 272-A} (1971), 433-436] to the case
of pure gauge theories, i.e. of gauge-invariant functionals defined on the
space of connections of a principal fiber bundle. Our proof follows
closely the original one by Michel, using several known results on the geometry
of . The result (and proof) is also extended to the case of gauge
theories with matter fields.Comment: 24 pages. An old paper posted for archival purpose
Two optimization problems in thermal insulation
We consider two optimization problems in thermal insulation: in both cases
the goal is to find a thin layer around the boundary of the thermal body which
gives the best insulation. The total mass of the insulating material is
prescribed.. The first problem deals with the case in which a given heat source
is present, while in the second one there are no heat sources and the goal is
to have the slowest decay of the temperature. In both cases an optimal
distribution of the insulator around the thermal body exists; when the body has
a circular symmetry, in the first case a constant heat source gives a constant
thickness as the optimal solution, while surprisingly this is not the case in
the second problem, where the circular symmetry of the optimal insulating layer
depends on the total quantity of insulator at our disposal. A symmetry breaking
occurs when this total quantity is below a certain threshold. Some numerical
computations are also provided, together with a list of open questions.Comment: 11 pages, 7 figures, published article on Notices Amer. Math. Soc. is
available at
http://www.ams.org/publications/journals/notices/201708/rnoti-p830.pd
Path integrals and symmetry breaking for optimal control theory
This paper considers linear-quadratic control of a non-linear dynamical
system subject to arbitrary cost. I show that for this class of stochastic
control problems the non-linear Hamilton-Jacobi-Bellman equation can be
transformed into a linear equation. The transformation is similar to the
transformation used to relate the classical Hamilton-Jacobi equation to the
Schr\"odinger equation. As a result of the linearity, the usual backward
computation can be replaced by a forward diffusion process, that can be
computed by stochastic integration or by the evaluation of a path integral. It
is shown, how in the deterministic limit the PMP formalism is recovered. The
significance of the path integral approach is that it forms the basis for a
number of efficient computational methods, such as MC sampling, the Laplace
approximation and the variational approximation. We show the effectiveness of
the first two methods in number of examples. Examples are given that show the
qualitative difference between stochastic and deterministic control and the
occurrence of symmetry breaking as a function of the noise.Comment: 21 pages, 6 figures, submitted to JSTA
Variational Principle of Bogoliubov and Generalized Mean Fields in Many-Particle Interacting Systems
The approach to the theory of many-particle interacting systems from a
unified standpoint, based on the variational principle for free energy is
reviewed. A systematic discussion is given of the approximate free energies of
complex statistical systems. The analysis is centered around the variational
principle of N. N. Bogoliubov for free energy in the context of its
applications to various problems of statistical mechanics and condensed matter
physics. The review presents a terse discussion of selected works carried out
over the past few decades on the theory of many-particle interacting systems in
terms of the variational inequalities. It is the purpose of this paper to
discuss some of the general principles which form the mathematical background
to this approach, and to establish a connection of the variational technique
with other methods, such as the method of the mean (or self-consistent) field
in the many-body problem, in which the effect of all the other particles on any
given particle is approximated by a single averaged effect, thus reducing a
many-body problem to a single-body problem. The method is illustrated by
applying it to various systems of many-particle interacting systems, such as
Ising and Heisenberg models, superconducting and superfluid systems, strongly
correlated systems, etc. It seems likely that these technical advances in the
many-body problem will be useful in suggesting new methods for treating and
understanding many-particle interacting systems. This work proposes a new,
general and pedagogical presentation, intended both for those who are
interested in basic aspects, and for those who are interested in concrete
applications.Comment: 60 pages, Refs.25
Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions
It is an open question how well tensor network states in the form of an
infinite projected entangled pair states (iPEPS) tensor network can approximate
gapless quantum states of matter. Here we address this issue for two different
physical scenarios: i) a conformally invariant quantum critical point
in the incarnation of the transverse field Ising model on the square lattice
and ii) spontaneously broken continuous symmetries with gapless Goldstone modes
exemplified by the antiferromagnetic Heisenberg and XY models on the
square lattice. We find that the energetically best wave functions display {\em
finite} correlation lengths and we introduce a powerful finite correlation
length scaling framework for the analysis of such finite- iPEPS states. The
framework is important i) to understand the mild limitations of the finite-
iPEPS manifold in representing Lorentz-invariant, gapless many body quantum
states and ii) to put forward a practical scheme in which the finite
correlation length combined with field theory inspired formulae can be
used to extrapolate the data to infinite correlation length, i.e. to the
thermodynamic limit. The finite correlation length scaling framework opens the
way for further exploration of quantum matter with an (expected)
Lorentz-invariant, massless low-energy description, with many applications
ranging from condensed matter to high-energy physics.Comment: 16 pages, 11 figure
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