14,690 research outputs found
Non-integrability of measure preserving maps via Lie symmetries
We consider the problem of characterizing, for certain natural number ,
the local -non-integrability near elliptic fixed points of
smooth planar measure preserving maps. Our criterion relates this
non-integrability with the existence of some Lie Symmetries associated to the
maps, together with the study of the finiteness of its periodic points. One of
the steps in the proof uses the regularity of the period function on the whole
period annulus for non-degenerate centers, question that we believe that is
interesting by itself. The obtained criterion can be applied to prove the local
non-integrability of the Cohen map and of several rational maps coming from
second order difference equations.Comment: 25 page
Signatures of integrability in charge and thermal transport in 1D quantum systems
Integrable and non-integrable systems have very different transport
properties. In this work, we highlight these differences for specific one
dimensional models of interacting lattice fermions using numerical exact
diagonalization. We calculate the finite temperature adiabatic stiffness (or
Drude weight) and isothermal stiffness (or ``Meissner'' stiffness) in
electrical and thermal transport and also compute the complete momentum and
frequency dependent dynamical conductivities and
. The Meissner stiffness goes to zero rapidly with system
size for both integrable and non-integrable systems. The Drude weight shows
signs of diffusion in the non-integrable system and ballistic behavior in the
integrable system. The dynamical conductivities are also consistent with
ballistic and diffusive behavior in the integrable and non-integrable systems
respectively.Comment: 4 pages, 4 figure
Integrability and level crossing manifolds in a quantum Hamiltonian system
We consider a two-spin model, represented classically by a nonlinear
autonomous Hamiltonian system with two degrees of freedom and a nontrivial
integrability condition, and quantum mechanically by a real symmetric
Hamiltonian matrix with blocks of dimensionalities K=l(l+1)/2, l=1,2,... In the
six-dimensional (6D) parameter space of this model, classical integrability is
satisfied on a 5D hypersurface, and level crossings occur on 4D manifolds that
are completely embedded in the integrability hypersurface except for some
lower-D sub-manifolds. Under mild assumptions, the classical integrability
condition can be reconstructed from a purely quantum mechanical study of level
degeneracies in finite-dimensional invariant blocks of the Hamiltonian matrix.
Our conclusions are based on rigorous results for K=3 and on numerical results
for K=6,10.Comment: 8 pages, 3 figure
Open problems, questions, and challenges in finite-dimensional integrable systems
The paper surveys open problems and questions related to different aspects
of integrable systems with finitely many degrees of freedom. Many of the open
problems were suggested by the participants of the conference “Finite-dimensional
Integrable Systems, FDIS 2017” held at CRM, Barcelona in July 2017.Postprint (updated version
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