Non-standard quantum algebras and finite dimensional $\mathcal{PT}$-symmetric systems

In this work, $\mathcal{PT}$-symmetric Hamiltonians defined on quantum $sl(2, \mathbb R)$ algebras are presented. We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard $U_{z}(sl(2, \mathbb R))$ Hopf algebra deformation of $sl(2, \mathbb R)$. By making use of a particular boson representation of the generators of $U_{z}(sl(2, \mathbb R))$, both the co-product and the commutation relations of the quantum algebra are shown to be invariant under the $\mathcal{PT}$-transformation. In terms of these operators, we construct several finite dimensional $\mathcal{PT}$-symmetry Hamiltonians, whose spectrum is analytically obtained for any arbitrary dimension. In particular, we show the appearance of Exceptional Points in the space of model parameters and we discuss the behaviour of the spectrum both in the exact $\mathcal{PT}$-symmetry and the broken $\mathcal{PT}$-symmetry dynamical phases. As an application, we show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits based on asymmetric double quantum dots. Remarkably enough, in this effective model, the deformation parameter $z$ has to be identified with the detuning parameter of the system.