131 research outputs found
Solution of the classical Yang--Baxter equation with an exotic symmetry, and integrability of a multi-species boson tunneling model
Solutions of the classical Yang-Baxter equation provide a systematic method
to construct integrable quantum systems in an algebraic manner. A Lie algebra
can be associated with any solution of the classical Yang--Baxter equation,
from which commuting transfer matrices may be constructed. This procedure is
reviewed, specifically for solutions without skew-symmetry. A particular
solution with an exotic symmetry is identified, which is not obtained as a
limiting expansion of the usual Yang--Baxter equation. This solution
facilitates the construction of commuting transfer matrices which will be used
to establish the integrability of a multi-species boson tunneling model. The
model generalises the well-known two-site Bose-Hubbard model, to which it
reduces in the one-species limit. Due to the lack of an apparent reference
state, application of the algebraic Bethe Ansatz to solve the model is
prohibitive. Instead, the Bethe Ansatz solution is obtained by the use of
operator identities and tensor product decompositions.Comment: 22 pages, no figure
Generalised Heine-Stieltjes and Van Vleck polynomials associated with degenerate, integrable BCS models
We study the Bethe Ansatz/Ordinary Differential Equation (BA/ODE)
correspondence for Bethe Ansatz equations that belong to a certain class of
coupled, nonlinear, algebraic equations. Through this approach we numerically
obtain the generalised Heine-Stieltjes and Van Vleck polynomials in the
degenerate, two-level limit for four cases of exactly solvable
Bardeen-Cooper-Schrieffer (BCS) pairing models. These are the s-wave pairing
model, the p+ip-wave pairing model, the p+ip pairing model coupled to a bosonic
molecular pair degree of freedom, and a newly introduced extended d+id-wave
pairing model with additional interactions. The zeros of the generalised
Heine-Stieltjes polynomials provide solutions of the corresponding Bethe Ansatz
equations. We compare the roots of the ground states with curves obtained from
the solution of a singular integral equation approximation, which allows for a
characterisation of ground-state phases in these systems. Our techniques also
permit for the computation of the roots of the excited states. These results
illustrate how the BA/ODE correspondence can be used to provide new numerical
methods to study a variety of integrable systems.Comment: 24 pages, 9 figures, 3 table
Ground-State Analysis for an Exactly Solvable Coupled-Spin Hamiltonian
We introduce a Hamiltonian for two interacting spins. We use a
mean-field analysis and exact Bethe ansatz results to investigate the
ground-state properties of the system in the classical limit, defined as the
limit of infinite spin (or highest weight). Complementary insights are provided
through investigation of the energy gap, ground-state fidelity, and
ground-state entanglement, which are numerically computed for particular
parameter values. Despite the simplicity of the model, a rich array of
ground-state features are uncovered. Finally, we discuss how this model may be
seen as an analogue of the exactly solvable pairing Hamiltonian
Bethe Ansatz Solutions of the Bose-Hubbard Dimer
The Bose-Hubbard dimer Hamiltonian is a simple yet effective model for
describing tunneling phenomena of Bose-Einstein condensates. One of the
significant mathematical properties of the model is that it can be exactly
solved by Bethe ansatz methods. Here we review the known exact solutions,
highlighting the contributions of V.B. Kuznetsov to this field. Two of the
exact solutions arise in the context of the Quantum Inverse Scattering Method,
while the third solution uses a differential operator realisation of the su(2)
Lie algebra.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
On the boundaries of quantum integrability for the spin-1/2 Richardson-Gaudin system
We discuss a generalised version of Sklyanin's Boundary Quantum Inverse
Scattering Method applied to the spin-1/2, trigonometric sl(2) case, for which
both the twisted-periodic and boundary constructions are obtained as limiting
cases. We then investigate the quasi-classical limit of this approach leading
to a set of mutually commuting conserved operators which we refer to as the
trigonometric, spin-1/2 Richardson-Gaudin system. We prove that the rational
limit of the set of conserved operators for the trigonometric system is
equivalent, through a change of variables, rescaling, and a basis
transformation, to the original set of trigonometric conserved operators.
Moreover we prove that the twisted-periodic and boundary constructions are
equivalent in the trigonometric case, but not in the rational limit.Comment: 29 page
Exact solution of the p+ip Hamiltonian revisited: duality relations in the hole-pair picture
We study the exact Bethe Ansatz solution of the p+ip Hamiltonian in a form
whereby quantum numbers of states refer to hole-pairs, rather than
particle-pairs used in previous studies. We find an asymmetry between these
approaches. For the attractive system states in the strong pairing regime take
the form of a quasi-condensate involving two distinct hole-pair creation
operators. An analogous feature is not observed in the particle-pair picture.Comment: 19 pages, 2 figures, 2 table
The Yang-Baxter paradox
Consider the statement "Every Yang-Baxter integrable system is defined to be
exactly-solvable". To formalise this statement, definitions and axioms are
introduced. Then, using a specific Yang-Baxter integrable bosonic system, it is
shown that a paradox emerges. A generalisation for completely integrable
bosonic systems is also developed.Comment: 16 pages, revised version correcting typographical error
Energy-level crossings and number-parity effects in a bosonic tunneling model
An exactly solved bosonic tunneling model is studied along a line of the
coupling parameter space, which includes a quantum phase boundary line. The
entire energy spectrum is computed analytically, and found to exhibit multiple
energy level crossings in a region of the coupling parameter space. Several key
properties of the model are discussed, which exhibit a clear dependence on
whether the particle number is even or odd.Comment: 12 pages, 7 figure
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