Quantum Glass, Spin Liquids, Fragmentation, Fractional Orbitals, and Flat Band Physics in Classical and Quantum Spin Models

Abstract

The world of many-body physics, encompassing both classical and quantum systems, is a breeding ground for novel phases of matter. Recent attention revolves around exotic phases characterized by fractionalized particles, emergent gauge fields, and long-range entanglement that hold immense potential for applications in quantum computing, information processing, and advanced sensing. Spin liquids (SLs) are a prime example, where the local spins exhibit neither a long-range order nor a complete disorder but rather a unique correlation structure with long-range entanglement. In the quantum SLs, gauge fields emerge at low energies, potentially interacting with the (fractional) particle-like excitations. Despite extensive research, numerous puzzles remain unsolved across quantum and classical SLs (Q/CSLs), such as understanding the dynamics of different gauge degrees of freedom, their influence on the dynamics of particle-like excitations, and their unique statistical properties. The present thesis delves into the study of the QSLs and CSLs in corresponding frustrated systems.

In the first project, I investigated the 2D Kitaev SL model via the Density-Matrix Renormalization Group (DMRG) study. The Kitaev model, one of the very few exactly solvable lattice models in 2D, predicts an interplay between Majorana fermions and Z$_2$ gauge fields. The ground state consists of uniform zero fluxes with gapped or gapless Majorana dispersions depending on parameters. A magnetic field along the [111] direction disrupts this uniformity, and the model becomes not exactly solvable. Within DMRG calculations, I identify five phases with distinct gauge sectors. The phases are uniform zero-flux, $\pi$-flux gas, amorphous $\pi$-flux crystal, a novel quantum glass, and a ferromagnet. Intriguingly, the origin of this glassiness arises from the restricted dynamics of excitations such as Majorana-flux or flux-flux pairs connected by finite-length strings, but not from conventional disorder or conserved quantities.

In the next project, I explore periodic arrangements of $\pi$-fluxes in the Kitaev model and their impact on the low-energy Majorana dispersions. By tuning the $\pi$-flux pair length and the coupling constants, I observe nearly flat-band Majorana dispersions with gaps and gap-less linearly dispersing Dirac points and tunable bandwidth and gaps. These gapped bands possess non-trivial Chern numbers and quantum metrics, meeting the criteria for ideal fractional Chern insulators. In the presence of interactions, these bands give the fractional Chern insulator phase for the Majorana fermions. I study the fractional Chern insulator phase with the mean-field theory involving density-wave orders. I systematically construct a low-energy tight-binding model via the Wannierization technique to describe the Majorana fermions in the presence of $Z_2$ gauge fields. This model introduces a gauge potential through a superexchange-like interaction.

In the third project, I explore CSLs. CSLs are cooperative paramagnets that have extensive degeneracy and finite zero-temperature entropy, in contrast to QSLs with zero entropy. Gauge theories also emerge here, but for different reasons. In recent years, significant interest in CSLs has been spurred by the emergence of higher-rank gauge theories and fragmentation properties. Traditionally, CSLs are studied using models like the spin-ice rule or the Luttinger-Tisza approximation. I developed a group theoretical method based on forming a vector space representation for spins within a plaquette of the lattice. The key to this method involves the decomposition of the spin vector into irreducible representations of point-group symmetry. The on-site unit-length spin constraint and frustration play pivotal roles in the emergence of CSLs. This method has been verified with the experimentally relevant XXZ model with Dzyaloshinskii-Moriya (DM) interaction. Among various interesting findings, I observe AFM-vortex/Anti-AFM-vortex-like structures in the ordered phases, a fragmented phase due to the coexistence of ordered phase and disordered CSLs. The group theoretical method helps unify all these phases into a single picture. Notably, the method allows us to quantify the exact ratio of order to disordered components in the case of classical continuous spins.