Investigations into Quantum Compass Models in Two Dimensions [HBNI Th225]

Abstract

Quantum many-body systems with competing interactions are found to be potential candidates to host unconventionally ordered phases and fractionalized excitations. The exactly soluble Kitaev’s 2d spin model has incited intense activity in this context. In this thesis, we investigate various quantum compass models (QCMs) on square and honeycomb lattices. The motivation for studying QCM stems from its unusual symmetries and applications in a wide range of physical platforms, from real materials to various engineered platforms. On the square lattice, we develop a novel mean-field-like approach that respects the stringent constraints set by the "gauge-like" symmetries and self-duality. We find excellent agreement with ab-initio numerical studies (PCUT, PEPS), showing a first-order quantum phase transition (QPT) separating two dual, Ising nematic phases. A qualitative discussion of our results in the context of Kugel-Khomskii spin-orbital physics and other dual models is then made. Next, we discuss the QCM on a honeycomb lattice, where various duality relations uncover an 3d Ising universality as well as a QPT between a higher-order, topological superfluid and a Mott insulator having topological order. A closely related fermionic compass-Hubbard model is then studied using two complementary methods: (a) a two-particle self-consistent approach and (b) strong coupling perturbation theory to study its various weak and strong coupling phases. Finally, we discuss the implications of an experimentally realizable (in cold-atom setups) perturbation on the above strong-coupling phase with the possibility of realizing a Lifshitz type criticality in a qualitative manner. Advances in the synthesis of novel spin-orbit coupled insulators and cold atom technologies may offer the hope of realizing these phases of matter in the near future.