Fractional Quantum Hall Liquids: Their Ground States, Neutral Excitations, and Competition with Crystal Phases [HBNI Th263]

Abstract

Two-dimensional electron systems (2DESs) in strong magnetic fields host a rich variety of interacting ground states, including the celebrated fractional quantum Hall effect (FQHE), Wigner crystals (WC), and bubble and stripe phases. Since the discovery of the quantum Hall (QH) effect in GaAs-based 2DES nearly four decades ago, the field has remained an active area of research. Advances in the quality of GaAs heterostructures, along with the emergence of new material platforms such as graphene heterostructures and ZnO-based 2DES, have significantly broadened the landscape of quantum Hall physics, unveiling novel correlated phases. This thesis presents a study of various interacting phases in the quantum Hall regime.

Most of the prominent FQHE states of electrons in the lowest Landau level (LLL) are conveniently described in terms of non-interacting emergent topological particles known as composite fermions (CFs). However, some recently observed states, such as the FQHE state at Landau level (LL) filling factor 4/13 in graphene, deviate from this conventional picture. Using the parton framework—which generalizes the composite fermion theory—this work proposes a ground-state wave function for the unconventional 4/13 FQHE state and computes its low-lying neutral excitations to uncover the microscopic structure of the state. The study further elucidates the topological properties of this state, including its fractional charge excitations, Abelian braiding statistics, chiral edge modes, and ground-state degeneracy on surfaces of different genus.

The energy gap to neutral excitations determines the stability of an FQHE state. Earlier calculations of neutral excitations based on the single-mode approximation (SMA) and the composite fermion exciton (CFE) approach suggested that, for non-Laughlin primary Jain states at fillings 𝑛 / ( 2 𝑛 ± 1 ) n/(2n±1) with 𝑛 > 1 n>1, the SMA gap does not provide an accurate description of neutral excitations at any wavelength—unlike in Laughlin states, where it performs well at small to intermediate wave numbers. However, recent numerical studies for small systems indicate that the long-wavelength SMA and CFE gaps are approximately equal for these states. To resolve this apparent discrepancy, the SMA gap on the sphere is computed semi-analytically for large system sizes, demonstrating that it closely matches the CFE gap for non-Laughlin primary Jain states in the long-wavelength limit. In the process, a closed algebra of LLL-projected density operators on the sphere is derived, analogous to the Girvin–MacDonald–Platzman algebra in planar geometry. The analysis also revisits earlier SMA gap calculations, identifies the source of the long-wavelength discrepancy, and introduces a suitable modification to correct it.

Graphene heterostructures provide a versatile platform for tuning the electronic band structure by varying the number of stacked graphene layers, which in turn modifies the Landau level eigenstate structure. The interaction energy of FQHE liquids and electron solids—such as Wigner crystals and bubble phases—depends sensitively on the form of these eigenstates, influencing their stability within a given Landau level. Motivated by this dependence, this work investigates the stability phase diagram of electron solid phases in several Landau levels of bilayer and trilayer graphene. Furthermore, the competition between electron solids and Laughlin-type FQHE liquids is examined to map their relative stability across different Landau levels.