Unitary unfoldings of a Bose–Hubbard exceptional point with and without particle number conservation

The conventional non-Hermitian but P T -symmetric three-parametric Bose–Hubbard Hamiltonian H ( γ , v , c ) represents a quantum system of N bosons, unitary only for parameters γ , v and c in a domain D . Its boundary ∂ D contains an exceptional point of order K (EPK; K = N + 1) at c = 0 and γ = v , but even at the smallest non-vanishing parameter c ≠ ~0 the spectrum of H ( v , v , c ) ceases to be real, i.e. the system ceases to be observable. In this paper, the question is inverted: all of the stable, unitary and observable Bose–Hubbard quantum systems are sought which would lie close to the phenomenologically most interesting EPK-related dynamical regime. Two different families of such systems are found. Both of them are characterized by the perturbed Hamiltonians H ( λ ) = H ( v , v , 0 ) + λ V for which the unitarity and stability of the system is guaranteed. In the first family the number N of bosons is assumed conserved while in the second family such an assumption is relaxed. Attention is paid mainly to an anisotropy of the physical Hilbert space near the EPK extreme. We show that it is reflected by a specific, operationally realizable structure of perturbations λ V which can be considered small.

Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)

Tensor product state (TPS) based methods are powerful tools to efficiently simulate quantum many-body systems in and out of equilibrium. In particular, the one-dimensional matrix-product (MPS) formalism is by now an established tool in condensed matter theory and quantum chemistry. In these lecture notes, we combine a compact review of basic TPS concepts with the introduction of a versatile tensor library for Python (TeNPy) [1]. As concrete examples, we consider the MPS based time-evolving block decimation and the density matrix renormalization group algorithm. Moreover, we provide a practical guide on how to implement abelian symmetries (e.g., a particle number conservation) to accelerate tensor operations.

Particle-number conservation in quasiparticle representation in the isovector neutron–proton pairing case

Until now, the Sharp-Bardeen–Cooper–Schrieffer (SBCS) particle-number projection method, in the isovector neutron–proton pairing case, has been developed in the particle representation. However, this formalism is sometimes complicated and cumbersome. In this work, the formalism is developed in the quasiparticle representation. An expression of the projected ground state wave function is proposed. Expressions of the energy as well as the expectation values of the total particle-number operator and its square are deduced. It is shown that these expressions are formally similar to their homologues in the pairing between like-particles case. They are easier to handle than the ones obtained using the particle representation and are more adapted to numerical calculations. The method is then numerically tested within the schematic one-level model, which allows comparisons with exact results, as well as in the case of even–even nuclei within the Woods–Saxon model. In each case, it is shown that the particle-number fluctuations that are inherent to the BCS method are completely eliminated by the projection. In the framework of the one-level model, the values of the projected energy are clearly closer to the exact values than the BCS ones. In realistic cases, the relative discrepancies between projected and unprojected values of the energy are small. However, the absolute deviations may reach several MeV.

Particle-number conservation in odd mass proton-rich nuclei in the isovector pairing case

An expression of a wave function which describes odd–even systems in the isovector pairing case is proposed within the BCS approach. It is shown that it correctly generalizes the one used in the pairing between like-particles case. It is then projected on the good proton and neutron numbers using the Sharp-BCS (SBCS) method. The expressions of the expectation values of the particle-number operator and its square, as well as the energy, are deduced in both approaches. The formalism is applied to study the isovector pairing effect and the number projection one on the ground state energy of odd mass N ≈ Z nuclei using the single-particle energies of a deformed Woods–Saxon mean-field. It is shown that both effects on energy do not exceed 2%, however, the absolute deviations may reach several MeV. Moreover, the np pairing effect rapidly diminishes as a function of (N - Z). The deformation effect is also studied. It is shown that the np pairing effect, either before or after the projection, as well as the projection effect, when including or not the isovector pairing, depends upon the deformation. However, it seems that the predicted ground state deformation will remain the same in the four approaches.