arXiv subjects
Publications and source records attributed to Phillip S. Isaac.
We study two extended Bose-Hubbard-type Hamiltonians representing bosonic networks restricted to the graph of a cube. For both Hamiltonians, we demonstrate that Bethe ansatz methods of solution can be employed after applying a canonical transformation of operators. We provide the resulting Bethe ansatz equations, and corresponding formulae for states and energies of both Hamiltonians.
Let $B$ denote the weighted adjacency matrix of a balanced, symmetric, bipartite graph. We define a class of bosonic networks given by Hamiltonians whose hopping terms are determined by $B$. We show that each quantum Hamiltonian is Yang-Baxter integrable, admitting a set of mutually commuting operators derived through a solution of the classical Yang-Baxter equation. We discuss some applications and consequences of this result.
We study the Dirac oscillator in one, two and three spatial dimensions, showing that the corresponding ladder operators realise the $ \mathbb{Z}_2\times\mathbb{Z}_2 $-graded Lie superalgebras $ \mathfrak{pso}(3|2) $, $ \mathfrak{pso}(3|4) $ and $ \mathfrak{osp}_{01}(1|2) \oplus \mathfrak{sl}_{10}(1|1)$. These algebraic structures are related to parastatistics and their Fock spaces. We demonstrate that these colour algebras and Fock spaces are useful for analysing the Dirac oscillator and its eigenspaces, particularly in $ (1+3) $-dimensions. Apart from this current work, to our knowledge, the recent article by Ito and Nago (arXiv:2501.07311) is the only other such work that makes use of $ \mathbb{Z}_2\times \mathbb{Z}_2 $ graded colour Lie superalgebras in a relativistic setting.
We investigate the structure and representation theory of finite-dimensional $\mathbb{Z}$-graded Lie algebras, including the corresponding root systems and Verma, irreducible, and Harish-Chandra modules. This extends the familiar theory for finite-dimensional semisimple Lie algebras to a much wider class of Lie algebras, and opens up for advances and applications in areas relying on ad-hoc approaches. Physically relevant examples are afforded by the Heisenberg and conformal Galilei algebras, including the Schrödinger algebras, whose $\mathbb{Z}$-graded structures are yet to be fully exploited.
We develop explicit formulae for the eigenvalues of various invariants for highest weight irreducible representations of the quantum supergroup $U_q[gl(m|n)]$. The techniques employed make use of modified characteristic identity methods and allow for the evaluation of generator matrix elements and reduced Wigner coefficients.
We investigate the ground-state energy of a Richardson-Gaudin integrable BCS model, generalizing the closed and open p+ip models. The Hamiltonian supports a family of mutually commuting conserved operators satisfying quadratic relations. From the eigenvalues of the conserved operators we derive, in the continuum limit, an integral equation for which a solution corresponding to the ground state is established. The energy expression from this solution agrees with the BCS mean-field result.
We present a novel realisation of the $\mathbb{Z}_2\times\mathbb{Z}_2$-graded Lie superalgebra $\mathfrak{gl}(m_1,m_2|n_1,n_2)$ inside an algebraic extension of the enveloping algebra of the $\mathbb{Z}_2$-graded Lie superalgebra $\mathfrak{gl}(m|n)$, with $m=m_1+m_2$ and $n=n_1+n_2$. A consequence of this realisation is that the representations of $\mathfrak{gl}(m|n)$ "lift up" to representations of $\mathfrak{gl}(m_1,m_2|n_1,n_2)$, with matrix elements differing only by a sign, which we are able to characterise concisely.
In a previous paper the generator matrix elements and (dual) vector reduced Wigner coefficients (RWCs) were evaluated via the polynomial identities satisfied by a certain matrix constructed from the $R$-matrix $R$ and its twisted counterpart $R^T=T\circ R$. Here we provide an alternative evaluation utilising the $R$-matrix $\tilde{R} = (R^T)^{-1}$. This provides a new direct derivation of the vector RWCs obtained indirectly in earlier work via a symmetry relation. This approach has the advantage that it generalises to the Lie superalgebra case, which will be investigated elsewhere.
In previous work, we introduced an algorithm that utilises differential operator realisations to find polynomial Casimir operators of Lie algebras. In this article we build on this work by applying the algorithm to several classes of finite dimensional conformal Galilei algebras with central extension. In these cases we highlight the utility of an algebra anti-automorphism, and give relevant details through key examples.
Using the exact Bethe Ansatz solution, we investigate methods for calculating the ground-state energy for the $p + ip$-pairing Hamiltonian. We first consider the Hamiltonian isolated from its environment (closed model) through two forms of Bethe Ansatz solutions, which generally have complex-valued Bethe roots. A continuum limit approximation, leading to an integral equation, is applied to compute the ground-state energy. We discuss the evolution of the root distribution curve with respect to a range of parameters, and the limitations of this method. We then consider an alternative approach that transforms the Bethe Ansatz equations to an equivalent form, but in terms of the real-valued conserved operator eigenvalues. An integral equation is established for the transformed solution. This equation is shown to admit an exact solution associated with the ground state. Next we discuss results for a recently derived Bethe Ansatz solution of the open model. With the aforementioned alternative approach based on real-valued roots, combined with mean-field analysis, we are able to establish an integral equation with an exact solution that corresponds to the ground-state for this case.
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.
We introduce a search algorithm that utilises differential operator realisations to find polynomial Casimir operators of Lie algebras. To demonstrate the algorithm, we look at two classes of examples: (1) the model filiform Lie algebras and (2) the Schrödinger Lie algebras. We find that an abstract form of dimensional analysis assists us in our algorithm, and greatly reduces the complexity of the problem.
In this paper fundamental Wigner coefficients are determined algebraically by considering the eigenvalues of certain generalized Casimir invariants. Here this method is applied in the context of both type 1 and type 2 unitary representations of the Lie superalgebra gl(mjn). Extensions to the non-unitary case are investigated. A symmetry relation between two classes of Wigner coefficients is given in terms of a ratio of dimensions.
We generalise the construction of integrals of motion for quantum superintegrable models and the deformed oscillator algebra approach. This is presented in the context of 1D systems admitting ladder operators satisfying a parabosonic algebra involving reflection operators and more generally $c_λ$ extended oscillator algebras with grading. We apply the construction on two-dimensional $c_λ$ oscillators. We also introduce two new superintegrable Hamiltonians that are the anisotropic Dunkl and the singular Dunkl oscillators. We construct the integrals and using this extended approach of the Daskaloyannis method with grading and we present an algebraic derivation of the energy spectrum of the two models from the finite dimensional unitary representations and show how their spectrum divides into different sectors and relates to the physical spectrum.
We consider a generalisation of the p+ip pairing Hamiltonian with external interaction terms. These terms allow for the exchange of particles between the system and its environment. As a result the u(1) symmetry associated with conservation of particle number, present in the p+ip Hamiltonian, is broken. Nonetheless the generalised model is integrable. We establish integrability using the Boundary Quantum Inverse Scattering Method, with one of the reflection matrices chosen to be non-diagonal. We also derive the corresponding Bethe Ansatz Equations, the roots of which parametrise the exact solution for the energy spectrum.
The characteristic identity formalism discussed in our recent articles is further utilized to derive matrix elements of type 2 unitary irreducible $gl(m|n)$ modules. In particular, we give matrix element formulae for all gl(m|n) generators, including the non-elementary generators, together with their phases on finite dimensional type 2 unitary irreducible representations. Remarkably, we find that the type 2 unitary matrix element equations coincide with the type 1 unitary matrix element equations for non-vanishing matrix elements up to a phase.
We present an overview of characteristic identities for Lie algebras and superalgebras. We outline methods that employ these characteristic identities to deduce matrix elements of finite dimensional representations. To demonstrate the theory, we look at the examples of the general linear Lie algebras and Lie superalgebras.
We utilise characteristic identities to construct eigenvalue formulae for invariants and reduced matrix elements corresponding to irreducible representations of osp(m|n). In presenting these results, we further develop our programme of constructive representation theory via characteristic identities.