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Lee, Yuan-Harng; Links, Jon; Zhang, Yao-Zhong (2011)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

The quasi-Gaudin algebra was introduced to construct integrable systems which are only quasi-exactly solvable. Using a suitable representation of the quasi-Gaudin algebra, we obtain a class of bosonic models which exhibit this curious property. These models have the notable feature that they do not preserve U(1) symmetry, which is typically associated to a non-conservation of particle number. An exact solution for the eigenvalues within the quasi-exactly solvable sector is obtained via the al...

Marquette, Ian (2013)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog and we extend Daskaloyannis' construction in obtained in context of quadratic algebras and we obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain th...

Birrell, Andrew; Isaac, Phillip S.; Links, Jon (2013)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

We demonstrate a novel approach that allows the determination of very general classes of exactly solvable Hamiltonians via Bethe ansatz methods. This approach combines aspects of both the co-ordinate Bethe ansatz and algebraic Bethe ansatz. The eigenfunctions are formulated as factorisable operators acting on a suitable reference state. Yet, we require no prior knowledge of transfer matrices or conserved operators. By taking a variational form for the Hamiltonian and eigenstates we obtain gen...

Birrell, A.; Isaac, P. S.; Links, J. (2011)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

We introduce a variational approach for the Quantum Inverse Scattering Method to exactly solve a class of Hamiltonians via Bethe ansatz methods. We undertake this in a manner which does not rely on any prior knowledge of integrability through the existence of a set of conserved operators. The procedure is conducted in the framework of Hamiltonians describing the crossover between the low-temperature phenomena of superconductivity, in the Bardeen-Cooper-Schrieffer (BCS) theory, and Bose-Einste...

Dunning, Clare; Isaac, Phillip S.; Links, Jon; Zhao, Shao-You (2011)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

We analyse a p+ip-wave pairing BCS Hamiltonian, coupled to a single bosonic degree of freedom representing a molecular condensate, and investigate the nature of the BEC-BCS crossover for this system. For a suitable restriction on the coupling parameters, we show that the model is integrable and we derive the exact solution by the algebraic Bethe ansatz. In this manner we also obtain explicit formulae for correlation functions and compute these for several cases. We find that the crossover bet...

Marquette, Ian; Links, Jon (2012)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

We introduce an integrable Hamiltonian which is an extended d+id-wave pairing model. The integrability is deduced from a duality relation with the Richardson-Gaudin (s-wave) pairing model, and associated to this there exists an exact Bethe ansatz solution. We study this system using the continuum limit approach and solve the corresponding singular integral equation obtained from the Bethe ansatz solution. We also conduct a mean-field analysis and show that results from these two approaches co...

Mattei, Eduardo; Links, Jon (2012)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

We introduce a Hamiltonian for two interacting $su(2)$ 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 r...

Links, Jon (2012)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

Since the advent of Drinfel’d’s double construction, Hopf algebraic structures have been a centrepiece for many developments in the theory and analysis of integrable quantum systems. An integrable anyonic pairing Hamiltonian will be shown to admit Hopf algebra symmetries for particular values of its coupling parameters. While the integrable structure of the model relates to the well-known six-vertex solution of the Yang–Baxter equation, the Hopf algebra symmetries are not in terms of the quan...

Marquette, Ian (2012)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

In the case of a one-dimensional nonsingular Hamiltonian $H$ and a singular supersymmetric partner $H_a$, the Darboux and factorization relations of supersymmetric quantum mechanics can be only formal relations. It was shown how we can construct an adequate partner by using infinite barriers placed where are located the singularities on the real axis and recover isospectrality. This method was applied to superpartners of the harmonic oscillator with one singularity. In this paper, we apply th...

Marquette, I.; Quesne, C. (2013)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

Projects: ARC | Topological properties of exactly-solvable, two-dimensional quantum systems (DP110101414)

New ladder operators are constructed for a rational extension of the harmonic oscillator associated with type III Hermite exceptional orthogonal polynomials and characterized by an even integer $m$. The eigenstates of the Hamiltonian separate into $m+1$ infinite-dimensional unitary irreducible representations of the corresponding polynomial Heisenberg algebra. These ladder operators are used to construct a higher-order integral of motion for two superintegrable two-dimensional systems separab...