Magnetic excitations in quantum rare earth pyrochlores

Rare-earth pyrochlores are materials with chemical formula A₂B₂O₇, where A is the rare-earth ion and B is a transition metal. At low temperature, these systems host various magnetic states such as spin ice, spin liquid state, ferromagnetic ordering, all in-all out, and anti-ferromagnetic ordering. For each rare-earth ion with total angular momentum J, the 2J + 1 fold degeneracy splits into singlets and doublets due to the crystal electric field. However, the crystal electric field ground state for most of the magnetic ions is a doublet that comes into three different varieties, labeled as ┌₃, ┌₄, ┌₅,₆. This work focuses only on systems in which the ground state doublet is well-separated from the first excited state so that we end up with effective two=state systems, referred to as quantum rare-earth pyrochlores. The low temperature excitations of interacting spins have a wave nature and are referred to as spin waves or magnons, where the energy of these waves is quantized. To study these magnons, we apply the Holstein-Primakoff transformation on the effective spin Hamiltonian to construct a bosonic Hamiltonian that describes magnons. In this study, we limit ourselves to the linear spin-wave approximation in which we diagonalize the magnonic Hamiltonian analytically and numerically for various systems of interest. In particular, we study magnons in Nd₂Zr₂O₇ which orders in an all in-all out state near 0.285 K, in Er₂Ti₂O₇ with a antiferromagnetic state below 1.2 K, and finally the Yb₂Ti₂O₇ which orders ferromagnetically near 0.2 K

Many-body perturbation theory algorithm for multiband systems, Floquet Moiré materials and beyond

In the first part of this work we introduce the symbolic determinant method (symDET) for constructing many-body perturbative expansions which is motivated by the Algorithmic Matsubara Integration (AMI) algorithm introduced recently [Taheridehkordi, A., Curnoe, S. H., & LeBlanc, J. P. F. PRB, 99(3), 035120, (2019)]. This algorithm is capable of performing both imaginary and real frequency calculations of physical observables at all coupling parameters, temperatures, etc., making it a promising tool for studying a variety of problems from lattice models to molecular chemistry problems. The current form of our symDET applies to both single and multiband systems with general two-body interactions, but it can be easily extended to beyond two-body interactions by the proper handling of Wick contractions. Although the computational expense increases for multiband problems at higher order perturbation theory, our algorithm is still parallelizable. Furthermore, optimizations still exist. One way could be by following the steps of the connected determinant method (cDET) [ R. Rossi, Phys. Rev. Lett. 119, 045701 (2017)] and the minimal determinant algorithms introduced recently [Šimkovic IV, F., & Ferrero, M. PRB, 105(12), 125104, (2022)]. As an illustration, we applied symDET to a variety of problems such as the hydrogen molecule with 2 and 10 bases, the Hubbard dimer model which is an effective 4 bands system, and the Hubbard model with an effective doubly degenerate band. In the second part of this thesis, we review the Floquet method which is used to study non-equilibrium systems. In particular, we focused on its application to twisted multilayered systems for which the appearance of flat bands at magic angles is a sign of interesting physical states, such as superconductivity which can observed experimentally in those systems. As an illustration, we applied this method to twisted bilayer and trilayer graphene systems. For the first example, we considered the twisted trilayer graphene (TTLG) system with different types of light applied vertically onto layers, mainly circularly polarized light and light from a waveguide, and we focused on the topological maps where we found that for the special case of ABC stacking, those maps are dependent on the handedness of the circularly polarized light. This dependence can be captured via optical conductivity measurements. Secondly, we studied the twisted bilayer graphene (TBLG), with the usual tight binding Hamiltonian together with interlayer hopping interactions, and then on top of that we included the Haldane interaction. The application of circularly polarized and waveguide lights were discussed where we considered the effects of light on the band structure of this model. For the Haldane TBLG we found that the band structure depends on the polarization of the incident light, something that was observed in the TTLG with ABC stacking but never seen in the usual TBLG system, hence we owe that to time reversal symmetry breaking. Lastly, we discuss the possible extensions of symDET to bosonic systems and the possible future application of symDET to twisted multilayer systems that has rich physics with a wide range of applications

Finite-Series Approximation of the Bound States for Two Novel Potentials

We obtain an analytic approximation of the bound states solution of the Schrödinger equation on the semi-infinite real line for two potential models with a rich structure as shown by their spectral phase diagrams. These potentials do not belong to the class of exactly solvable problems. The solutions are finite series (with a small number of terms) of square integrable functions written in terms of Romanovski–Jacobi polynomials

Solutions of the D-Dimensional Schrödinger Equation with the Hyperbolic Pöschl-Teller Potential plus Modified Ring-Shaped Term

We solve the D-dimensional Schrödinger equation with hyperbolic Pöschl-Teller potential plus a generalized ring-shaped potential. After the separation of variable in the hyperspherical coordinate, we used Nikiforov-Uvarov (NU) method to solve the resulting radial equation and obtain explicitly the energy level and the corresponding wave function in closed form. The solutions to the energy eigenvalues and the corresponding wave functions are obtained using the NU method as well