1,348 research outputs found
Generalized isotropic Lipkin-Meshkov-Glick models: ground state entanglement and quantum entropies
We introduce a new class of generalized isotropic Lipkin-Meshkov-Glick models
with su spin and long-range non-constant interactions, whose
non-degenerate ground state is a Dicke state of su type. We evaluate in
closed form the reduced density matrix of a block of spins when the whole
system is in its ground state, and study the corresponding von Neumann and
R\'enyi entanglement entropies in the thermodynamic limit. We show that both of
these entropies scale as when tends to infinity, where the
coefficient is equal to in the ground state phase with
vanishing su magnon densities. In particular, our results show that none
of these generalized Lipkin-Meshkov-Glick models are critical, since when
their R\'enyi entropy becomes independent of the parameter
. We have also computed the Tsallis entanglement entropy of the ground state
of these generalized su Lipkin-Meshkov-Glick models, finding that it can
be made extensive by an appropriate choice of its parameter only when
. Finally, in the su case we construct in detail the phase
diagram of the ground state in parameter space, showing that it is determined
in a simple way by the weights of the fundamental representation of su.
This is also true in the su case; for instance, we prove that the region
for which all the magnon densities are non-vanishing is an -simplex in
whose vertices are the weights of the fundamental representation
of su.Comment: Typeset with LaTeX, 32 pages, 3 figures. Final version with
corrections and additional reference
Critical behavior of su(1|1) supersymmetric spin chains with long-range interactions
We introduce a general class of su supersymmetric spin chains with
long-range interactions which includes as particular cases the su
Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We
show that this class of models can be fermionized with the help of the
algebraic properties of the su permutation operator, and take advantage
of this fact to analyze their quantum criticality when a chemical potential
term is present in the Hamiltonian. We first study the low energy excitations
and the low temperature behavior of the free energy, which coincides with that
of a -dimensional conformal field theory (CFT) with central charge
when the chemical potential lies in the critical interval , being the dispersion relation. We also analyze the
von Neumann and R\'enyi ground state entanglement entropies, showing that they
exhibit the logarithmic scaling with the size of the block of spins
characteristic of a one-boson -dimensional CFT. Our results thus show
that the models under study are quantum critical when the chemical potential
belongs to the critical interval, with central charge . From the analysis
of the fermion density at zero temperature, we also conclude that there is a
quantum phase transition at both ends of the critical interval. This is further
confirmed by the behavior of the fermion density at finite temperature, which
is studied analytically (at low temperature), as well as numerically for the
su elliptic chain.Comment: 13 pages, 6 figures, typeset in REVTe
Families of classical subgroup separable superintegrable systems
We describe a method for determining a complete set of integrals for a
classical Hamiltonian that separates in orthogonal subgroup coordinates. As
examples, we use it to determine complete sets of integrals, polynomial in the
momenta, for some families of generalized oscillator and Kepler-Coulomb
systems, hence demonstrating their superintegrability. The latter generalizes
recent results of Verrier and Evans, and Rodriguez, Tempesta and Winternitz.
Another example is given of a superintegrable system on a non-conformally flat
space.Comment: 9 page
Exact Solvability of Superintegrable Systems
It is shown that all four superintegrable quantum systems on the Euclidean
plane possess the same underlying hidden algebra . The gauge-rotated
Hamiltonians, as well as their integrals of motion, once rewritten in
appropriate coordinates, preserve a flag of polynomials. This flag corresponds
to highest-weight finite-dimensional representations of the -algebra,
realized by first order differential operators.Comment: 14 pages, AMS LaTe
Quantum models related to fouled Hamiltonians of the harmonic oscillator
We study a pair of canonoid (fouled) Hamiltonians of the harmonic oscillator
which provide, at the classical level, the same equation of motion as the
conventional Hamiltonian. These Hamiltonians, say and , result
to be explicitly time-dependent and can be expressed as a formal rotation of
two cubic polynomial functions, and , of the canonical variables
(q,p).
We investigate the role of these fouled Hamiltonians at the quantum level.
Adopting a canonical quantization procedure, we construct some quantum models
and analyze the related eigenvalue equations. One of these models is described
by a Hamiltonian admitting infinite self-adjoint extensions, each of them has a
discrete spectrum on the real line. A self-adjoint extension is fixed by
choosing the spectral parameter of the associated eigenvalue
equation equal to zero. The spectral problem is discussed in the context of
three different representations. For , the eigenvalue equation is
exactly solved in all these representations, in which square-integrable
solutions are explicity found. A set of constants of motion corresponding to
these quantum models is also obtained. Furthermore, the algebraic structure
underlying the quantum models is explored. This turns out to be a nonlinear
(quadratic) algebra, which could be applied for the determination of
approximate solutions to the eigenvalue equations.Comment: 24 pages, no figures, accepted for publication on JM
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