800 research outputs found
Super Rogers-Szeg\"o polynomials associated with type of Polychronakos spin chains
As is well known, multivariate Rogers-Szeg\"o polynomials are closely
connected with the partition functions of the type of Polychronakos
spin chains having long-range interactions. Applying the `freezing trick', here
we derive the partition functions for a class of type of Polychronakos
spin chains containing supersymmetric analogues of polarized spin reversal
operators and subsequently use those partition functions to obtain novel
multivariate super Rogers-Szeg\"o (SRS) polynomials depending on four types of
variables. We construct the generating functions for such SRS polynomials and
show that these polynomials can be written as some bilinear combinations of the
type of SRS polynomials. We also use the above mentioned generating
functions to derive a set of recursion relations for the partition functions of
the type of Polychronakos spin chains involving different numbers of
lattice sites and internal degrees of freedom.Comment: 33 pages, minor typos corrected, journal reference give
Exact solution of type quantum Calogero model through a mapping to free harmonic oscillators
We solve the eigenvalue problem of the type of Calogero model by
mapping it to a set of decoupled quantum harmonic oscillators through a
similarity transformation. In particular, we construct the eigenfunctions of
this Calogero model from those of bosonic harmonic oscillators having either
all even parity or all odd parity. It turns out that the eigenfunctions of this
model are orthogonal with respect to a nontrivial inner product, which can be
derived from the quasi-Hermiticity property of the corresponding conserved
quantities.Comment: 16 page
Supersymmetric analogue of BC_N type rational integrable models with polarized spin reversal operators
We derive the exact spectra as well as partition functions for a class of
type of spin Calogero models, whose Hamiltonians are constructed by
using supersymmetric analogues of polarized spin reversal operators (SAPSRO).
The strong coupling limit of these spin Calogero models yields type of
Polychronakos-Frahm (PF) spin chains with SAPSRO. By applying the freezing
trick, we obtain an exact expression for the partition functions of such PF
spin chains. We also derive a formula which expresses the partition function of
any type of PF spin chain with SAPSRO in terms of partition functions of
several type of supersymmetric PF spin chains, where .
Subsequently we show that an extended boson-fermion duality relation is obeyed
by the partition functions of the type of PF chains with SAPSRO. Some
spectral properties of these spin chains, like level density distribution and
nearest neighbour spacing distribution, are also studied.Comment: 36 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1402.275
Rational quantum integrable systems of D_N type with polarized spin reversal operators
We study the spin Calogero model of D_N type with polarized spin reversal
operators, as well as its associated spin chain of Haldane-Shastry type, both
in the antiferromagnetic and ferromagnetic cases. We compute the spectrum and
the partition function of the former model in closed form, from which we derive
an exact formula for the chain's partition function in terms of products of
partition functions of Polychronakos-Frahm spin chains of type A. Using a
recursion relation for the latter partition functions that we derive in the
paper, we are able to numerically evaluate the partition function, and thus the
spectrum, of the D_N-type spin chain for relatively high values of the number
of spins N. We analyze several global properties of the chain's spectrum, such
as the asymptotic level density, the distribution of consecutive spacings of
the unfolded spectrum, and the average degeneracy. In particular, our results
suggest that this chain is invariant under a suitable Yangian group, and that
its spectrum coincides with that of a Yangian-invariant vertex model with
linear energy function and dispersion relation.Comment: 26 pages, 5 figures, typeset in LaTe
Multi-parameter deformed and nonstandard Yangian symmetry in integrable variants of Haldane-Shastry spin chain
By using `anyon like' representations of permutation algebra, which pick up
nontrivial phase factors while interchanging the spins of two lattice sites, we
construct some integrable variants of Haldane-Shastry (HS) spin chain. Lax
equations for these spin chains allow us to find out the related conserved
quantities. However, it turns out that such spin chains also possess a few
additional conserved quantities which are apparently not derivable from the Lax
equations. Identifying these additional conserved quantities, and the usual
ones related to Lax equations, with different modes of a monodromy matrix, it
is shown that the above mentioned HS like spin chains exhibit multi-parameter
deformed and `nonstandard' variants of Yangian symmetry.Comment: 18 pages, latex, no figure
Pseudo-hermitian interaction between an oscillator and a spin half particle in the external magnetic field
We consider a spin half particle in the external magnetic field which couples
to a harmonic oscillator through some pseudo-hermitian interaction. We find
that the energy eigenvalues for this system are real even though the
interaction is not PT invariant.Comment: Latex, no figs, 8 pages. (To appear in Mod. Phys. Lett. A
noise and integrable systems
An innovative test for detecting quantum chaos based on the analysis of the
spectral fluctuations regarded as a time series has been recently proposed.
According to this test, the fluctuations of a fully chaotic system should
exhibit 1/f noise, whereas for an integrable system this noise should obey the
1/f^2 power law. In this letter, we show that there is a family of well-known
integrable systems, namely spin chains of Haldane-Shastry type, whose spectral
fluctuations decay instead as 1/f^4. We present a simple theoretical
justification of this fact, and propose an alternative characterization of
quantum chaos versus integrability formulated directly in terms of the power
spectrum of the spacings of the unfolded spectrum.Comment: 5 pages, 3 figures, RevTe
Exact spectrum and partition function of SU(m|n) supersymmetric Polychronakos model
By using the fact that Polychronakos-like models can be obtained through the
`freezing limit' of related spin Calogero models, we calculate the exact
spectrum as well as partition function of SU(m|n) supersymmetric Polychronakos
(SP) model. It turns out that, similar to the non-supersymmetric case, the
spectrum of SU(m|n) SP model is also equally spaced. However, the degeneracy
factors of corresponding energy levels crucially depend on the values of
bosonic degrees of freedom (m) and fermionic degrees of freedom (n). As a
result, the partition functions of SP models are expressed through some novel
q-polynomials. Finally, by interchanging the bosonic and fermionic degrees of
freedom, we obtain a duality relation among the partition functions of SP
models.Comment: Latex, 20 pages, no figures, minor typos correcte
Bessel Process and Conformal Quantum Mechanics
Different aspects of the connection between the Bessel process and the
conformal quantum mechanics (CQM) are discussed. The meaning of the possible
generalizations of both models is investigated with respect to the other model,
including self adjoint extension of the CQM. Some other generalizations such as
the Bessel process in the wide sense and radial Ornstein- Uhlenbeck process are
discussed with respect to the underlying conformal group structure.Comment: 28 Page
Inequivalent quantization of the rational Calogero model with a Coulomb type interaction
We consider the inequivalent quantizations of a -body rational Calogero
model with a Coulomb type interaction. It is shown that for certain range of
the coupling constants, this system admits a one-parameter family of
self-adjoint extensions. We analyze both the bound and scattering state sectors
and find novel solutions of this model. We also find the ladder operators for
this system, with which the previously known solutions can be constructed.Comment: 15 pages, 3 figures, revtex4, typos corrected, to appear in EPJ
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