Alladi Ramakrishnan Hall

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

Rakesh Dora

IMSc

Two-dimensional electron systems (2DESs) in a strong magnetic field 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 platforms such as graphene heterostructures and ZnO-based 2DES, have further broadened the landscape of quantum Hall physics, unveiling novel correlated phases. This thesis presents a study of various interacting phases in the QH 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), with a few exceptions, such as the recently observed FQHE state at Landau level (LL) filling $4/13$ in graphene. Using the parton framework, which generalizes the CF theory, we propose a ground state wave function for the unconventional FQHE state at $4/13$ and compute its low-lying neutral excitations, unveiling the state’s microscopic structure. We further elucidate the topological properties of the state, including its elementary fractional charges and their Abelian braiding statistics, the number of chiral gapless edge modes, and the degeneracy of the ground state on a genus $g-$surface, among other characteristics.

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 $n/(2n{\pm}1)$ with $n{>}1$, the SMA gap does not provide an accurate description of neutral excitations at any wavelengths, unlike for Laughlin states where it is known to work for small to intermediate wave numbers. In contrast, recent numerical studies for small system sizes indicate that the long-wavelength SMA gap and CFE gap are approximately equal for these states. To resolve this apparent discrepancy, we compute the SMA gap on the sphere semi-analytically for large system sizes and demonstrate that it closely matches the CFE gap for non-Laughlin primary Jain states in the long-wavelength limit. In doing so, we derive a closed algebra of LLL-projected density operators on the sphere, analogous to the Girvin-MacDonald-Platzman algebra in planar geometry. Additionally, we revisit the earlier SMA gap calculations, identify the origin of the long-wavelength discrepancy, and propose 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 LL eigenstate structure. The interaction energy of FQHE liquids and electron solids, such as WC and bubble phases, depends sensitively on the form of LL eigenstates, influencing their stability within a given LL. This motivates us to investigate the stability phase diagram of electron solid phases in several LLs of bilayer and trilayer graphene. Additionally, we study the competition between electron solids and Laughlin FQHE liquids to map out their relative stability in different LLs.