The authors explore here the interesting ``chemistry'' of anyons -- collective excitations emerging from a large amount of correlated electrons -- in strongly interacting fractional quantum Hall fluids. Anyons can form clusters (``molecules'') because of the effective anyon-anyon interaction induced by tunable electron-electron interactions. Laughlin anyons form hydrogen molecule like bound states with short-range electron interactions, potentially explaining the mystery of the observed 2$e$/3 effective charge at extremely low temperatures. In the non-Abelian Moore--Read phase, fusion channel dependent effective interactions lift the ``1/$\ensuremath{\psi}$'' degeneracy, enabling the possible energetic control of non-Abelian anyons. The authors also propose STM as a useful experimental tool for detecting the fingerprint of various configurations of anyon clusters.
The authors explore here the interesting chemistry'' of anyons -- collective excitations emerging from a large amount of correlated electrons -- in strongly interacting fractional quantum Hall fluids. Anyons can form clusters (molecules'') because of the effective anyon-anyon interaction induced by tunable electron-electron interactions. Laughlin anyons form hydrogen molecule like bound states with short-range electron interactions, potentially explaining the mystery of the observed 2e/3 effective charge at extremely low temperatures. In the non-Abelian Moore--Read phase, fusion channel dependent effective interactions lift the ``1/\ensuremathψ'' degeneracy, enabling the possible energetic control of non-Abelian anyons. The authors also propose STM as a useful experimental tool for detecting the fingerprint of various configurations of anyon clusters.
This thesis investigates interaction-driven phenomena of two-dimensional electrons in strong magnetic fields, with a focus on how the electron-electron (e-e) interaction projected in one Landau level affects the physical properties of the topological bands. The Landau-level-projected interaction favours qualitatively different ground states in different situations, such as topologically ordered fractional quantum Hall liquids that host anyonic quasiparticles, and symmetry-broken electron solids or charge-ordered phases. In the fractional quantum Hall liquid, we study quasiholes as interacting particles rather than featureless point excitations in a dilute and ideal anyon gas. Going beyond model Hamiltonians, we show that realistic e–e interactions generate nontrivial effective anyon–anyon interactions that can endow few-anyon clusters with internal structure and binding, producing “anyon molecules”. Because the same effective interaction also controls how anyons couple to local probes, we compute local density-of-states signatures and show that local tunneling spectra can encode both universal and non-universal information. The analysis is carried out for the Abelian Laughlin states and for the non-Abelian Moore–Read state, the latter one showing characteristic odd–even features tied to its fusion structure. In the high-Landau-level regime, the same interaction-driven physics favors symmetry-broken charge order. We construct a microscopic-to-continuum framework that connects Landau-level-projected Hartree--Fock energetics to elastic constants and, through Kosterlitz--Thouless--Halperin--Nelson--Young defect physics and renormalization-group flow, to a quantitative finite-temperature melting scale. This approach highlights the role of dislocation proliferation-induced thermal renormalization in quantitatively understanding the 2D electron solids' melting, and enables direct comparison with transport signatures of re-entrant insulating phases. Taken together, we provide an interaction-based perspective on both topological quantum Hall liquids and quantum Hall electron solids in Landau levels. This thesis closes by outlining several directions that follow naturally from this work.
The quantum Hall effect has been the prototypical example for many emerging fields of Physics, including the study of topological phases of matter, topological order, and topological insulators. The objects of interest in this thesis are anyons, which are elementary excitations in fractional quantum Hall (FQH) fluids. Anyons are quasiparticles with fractional charge and fractional statistics. The latter prop- erty makes it a subject of immense interest due to application in fault-tolerant quantum computing. Here, we employ a perspective that anyons are the appro- priate degree of freedom to describe the relevant physics of FQH systems. This is because when a topological phase is assumed and the energy gap goes to infinity, anyons (“quasiholes”) are the only possible elementary excitations, since they are not energetically punished by the correspondng model Hamiltonian. With respect to a realistic Hamiltonian, however, this choice of degree of freedom depends on the exact effective interaction in the system. For example, while the Laughlin state is the model state at filling factor n + 1/3 (where n is some integer), when the effective electron-electron interaction is more long-ranged, the quasiholes of a Gaffnian state may be excited and become the better degree of freedom to describe physics in the same filling factor. This perspective results in a surprising observa- tion that a Laughlin quasihole may fractionalize under certain conditions, leading to a quasiparticle of charge e/6 that can potentially be detected in experiments. Another important aspect of anyons is their intrinsic spin and the corresponding spin-statistics relation, which must account for the fact that FQH anyons are not point particles, but objects with finite sizes and intricate internal structures. Our investigation leads to a rigorous derivation of the general spin-statistics relation for anyons in Abelian FQH phases. Furthermore, we show it is possible to fully capture the important features of the statistics by a conformal Hilbert space (CHS) description. Our analysis provides the necessary tools to study such geometric effects, which we use to investigate how disorder in a realistic system may affect different experimental schemes for measuring anyon statistics.
We study the trapping of Abelian anyons (quasiholes and quasiparticles) by a local potential (e.g., induced by an atomic force microscopy tip) in a microscopic model of fractional quantum Hall liquids with long-range Coulomb interaction and edge confining potential. We find, in particular, that at Laughlin filling fraction $\ensuremath{\nu}=1∕3$, both quasihole and quasiparticle states can emerge as the ground state of the system in the presence of the trapping potential. As expected, we find that the presence of an Abelian quasihole has no effect on the edge spectrum of the quantum liquid, unlike in the non-Abelian case [X. Wan et al., Phys. Rev. Lett. 97, 256804 (2006)]. Although quasiholes and quasiparticles can emerge generically in the system, their stability depends on the strength of the confining potential and the strength and the range of the trapping potential. We discuss the relevance of the calculation to the high-accuracy generation and control of individual anyons in potential experiments, in particular, in the context of topological quantum computing.
Anyons are low-dimensional quasiparticles that obey fractional statistics, hence interpolating between bosons and fermions. In two dimensions, they exist as elementary excitations of fractional quantum Hall states and they are believed to enable topological quantum computing. One-dimensional (1D) anyons have been theoretically proposed, but their experimental realization has proven to be difficult. Here, we observe anyonic correlations, which emerge through the phenomenon of spin-charge separation, in a 1D strongly-interacting quantum gas. The required spin degree of freedom is provided by a mobile impurity, whose effective anyonic correlations are associated with an experimentally tunable statistical angle. These anyonic correlations are measured by monitoring the impurity momentum distribution, whose asymmetric feature demonstrates the transmutation of bosons via anyons to fermions. Going beyond equilibrium conditions, we study the dynamical properties of the anyonic correlations via dynamical fermionization of the anyons. Our work opens up the door to the exploration of non-equilibrium anyonic phenomena in a highly controllable setting.
We use segment DMRG on the infinite cylinder to compute energies of charged excitations in gate-screened fractional quantum Hall states. For the ν=1/3 Laughlin, ν=2/5 Jain, and ν=5/2 anti-Pfaffian states, we find screening can bind like-charged anyons into molecules, with a strong dependence on filling-factor, gate distance, and fusion channel. In the Laughlin state, stable ±2e/3 molecules and larger clusters appear over a broad gate-distance window. The Jain state is molecular throughout the range we consider. In the anti-Pfaffian, binding is strongest on the hole side, where the charge-e/2 molecule is fused into the ψ channel over a broad window of gate distances. In all three cases, screening suppresses long-range repulsion and exposes an intermediate-range attraction encoded in the oscillatory density tail of the fundamental anyon. We discuss consequences for addition spectra, interferometry, Wigner crystallization, anyon superconductivity, and entropy measurements.
We show the low-lying excitations at filling factor ν=n+1/3 with realistic interactions can be understood as quantum fluids with "Gaffnian quasiholes" as the proper elementary degrees of freedom. Each Laughlin quasihole can thus be understood as a bound state of two Gaffnian quasiholes, which in the lowest Landau level (LLL) behaves like "partons" with "asymptotic freedom" mediated by neutral excitations acting as "gluons." Near the experimentally observed nematic FQH phase in higher LLs, quasiholes become weakly bound and can fractionalize with rich dynamical properties. By studying the effective interactions between quasiholes, we predict a finite temperature phase transition of the Laughlin quasiholes even when the Laughlin ground state remains incompressible, and derive relevant experimental conditions for its possible observations.