4,898 research outputs found
Thermodynamic Bethe ansatz for the AII sigma-models
We derive thermodynamic Bethe ansatz equations describing the vacuum energy
of the SU(2N)/Sp(N) nonlinear sigma model on a cylinder geometry. The starting
points are the recently-proposed amplitudes for the scattering among the
physical, massive excitations of the theory. The analysis fully confirms the
correctness of the S-matrix. We also derive closed sets of functional relations
for the pseudoenergies (Y-systems). These relations are shown to be the
k-->infinity limit of the Sp(k+1)-related systems studied some years ago by
Kuniba and Nakanishi in the framework of lattice models.Comment: 11 pages, 1 figure, Latex 2e, uses amssymb, graphicx. v2: typos
correcte
Hamiltonian formulation of nonequilibrium quantum dynamics: geometric structure of the BBGKY hierarchy
Time-resolved measurement techniques are opening a window on nonequilibrium
quantum phenomena that is radically different from the traditional picture in
the frequency domain. The simulation and interpretation of nonequilibrium
dynamics is a conspicuous challenge for theory. This paper presents a novel
approach to quantum many-body dynamics that is based on a Hamiltonian
formulation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of
equations of motion for reduced density matrices. These equations have an
underlying symplectic structure, and we write them in the form of the classical
Hamilton equations for canonically conjugate variables. Applying canonical
perturbation theory or the Krylov-Bogoliubov averaging method to the resulting
equations yields a systematic approximation scheme. The possibility of using
memory-dependent functional approximations to close the Hamilton equations at a
particular level of the hierarchy is discussed. The geometric structure of the
equations gives rise to reduced geometric phases that are observable even for
noncyclic evolutions of the many-body state. The formalism is applied to a
finite Hubbard chain which undergoes a quench in on-site interaction energy U.
Canonical perturbation theory, carried out to second order, fully captures the
nontrivial real-time dynamics of the model, including resonance phenomena and
the coupling of fast and slow variables.Comment: 17 pages, revise
Multi-Lagrangians for Integrable Systems
We propose a general scheme to construct multiple Lagrangians for completely
integrable non-linear evolution equations that admit multi- Hamiltonian
structure. The recursion operator plays a fundamental role in this
construction. We use a conserved quantity higher/lower than the Hamiltonian in
the potential part of the new Lagrangian and determine the corresponding
kinetic terms by generating the appropriate momentum map. This leads to some
remarkable new developments. We show that nonlinear evolutionary systems that
admit -fold first order local Hamiltonian structure can be cast into
variational form with Lagrangians which will be local functionals of
Clebsch potentials. This number increases to when the Miura
transformation is invertible. Furthermore we construct a new Lagrangian for
polytropic gas dynamics in dimensions which is a {\it local} functional
of the physical field variables, namely density and velocity, thus dispensing
with the necessity of introducing Clebsch potentials entirely. This is a
consequence of bi-Hamiltonian structure with a compatible pair of first and
third order Hamiltonian operators derived from Sheftel's recursion operator.Comment: typos corrected and a reference adde
Quasi-Exactly Solvable Potentials on the Line and Orthogonal Polynomials
In this paper we show that a quasi-exactly solvable (normalizable or
periodic) one-dimensional Hamiltonian satisfying very mild conditions defines a
family of weakly orthogonal polynomials which obey a three-term recursion
relation. In particular, we prove that (normalizable) exactly-solvable
one-dimensional systems are characterized by the fact that their associated
polynomials satisfy a two-term recursion relation. We study the properties of
the family of weakly orthogonal polynomials defined by an arbitrary
one-dimensional quasi-exactly solvable Hamiltonian, showing in particular that
its associated Stieltjes measure is supported on a finite set. From this we
deduce that the corresponding moment problem is determined, and that the -th
moment grows like the -th power of a constant as tends to infinity. We
also show that the moments satisfy a constant coefficient linear difference
equation, and that this property actually characterizes weakly orthogonal
polynomial systems.Comment: 22 pages, plain TeX. Please typeset only the file orth.te
Are ghost surfaces quadratic-flux-minimizing?
Two candidates for "almost-invariant" toroidal surfaces passing through
magnetic islands, namely quadratic-flux-minimizing (QFMin) surfaces and ghost
surfaces, use families of periodic pseudo-orbits (i.e. paths for which the
action is not exactly extremal). QFMin pseudo-orbits, which are
coordinate-dependent, are field lines obtained from a modified magnetic field,
and ghost-surface pseudo-orbits are obtained by displacing closed field lines
in the direction of steepest descent of magnetic action, . A generalized Hamiltonian definition of ghost
surfaces is given and specialized to the usual Lagrangian definition. A
modified Hamilton's Principle is introduced that allows the use of Lagrangian
integration for calculation of the QFMin pseudo-orbits. Numerical calculations
show QFMin and Lagrangian ghost surfaces give very similar results for a
chaotic magnetic field perturbed from an integrable case, and this is explained
using a perturbative construction of an auxiliary poloidal angle for which
QFMin and Lagrangian ghost surfaces are the same up to second order. While
presented in the context of 3-dimensional magnetic field line systems, the
concepts are applicable to defining almost-invariant tori in other
degree-of-freedom nonintegrable Lagrangian/Hamiltonian systems.Comment: 8 pages, 3 figures. Revised version includes post-publication
corrections in text, as described in Appendix C Erratu
From Interacting Particles to Equilibrium Statistical Ensembles
We argue that a particle language provides a conceptually simple framework
for the description of anomalous equilibration in isolated quantum systems. We
address this paradigm in the context of integrable models, which are those with
particles that are stable against decay. In particular, we demonstrate that a
complete description of equilibrium ensembles for interacting integrable models
requires a formulation built from the mode occupation numbers of the underlying
particle content, mirroring the case of non-interacting particles. This yields
an intuitive physical interpretation of generalized Gibbs ensembles, and
reconciles them with the microcanonical ensemble. We explain how previous
attempts to identify an appropriate ensemble overlooked an essential piece of
information, and provide explicit examples in the context of quantum quenches.Comment: 4 pages + appendice
Inability of spatial transformations of CNN feature maps to support invariant recognition
A large number of deep learning architectures use spatial transformations of
CNN feature maps or filters to better deal with variability in object
appearance caused by natural image transformations. In this paper, we prove
that spatial transformations of CNN feature maps cannot align the feature maps
of a transformed image to match those of its original, for general affine
transformations, unless the extracted features are themselves invariant. Our
proof is based on elementary analysis for both the single- and multi-layer
network case. The results imply that methods based on spatial transformations
of CNN feature maps or filters cannot replace image alignment of the input and
cannot enable invariant recognition for general affine transformations,
specifically not for scaling transformations or shear transformations. For
rotations and reflections, spatially transforming feature maps or filters can
enable invariance but only for networks with learnt or hardcoded rotation- or
reflection-invariant featuresComment: 22 pages, 3 figure
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