Understanding the nature of quasihole excitations, i.e., anyons that have fractional charge and fractional statistics, has been a challenging problem in condensed matter physics. Our theoretical approach to this problem has been to consider a quantum dot containing a few charged particles, Coulomb coupled to the incompressible fluid. The size of the dot and the separation distance of the dot from the incompressible state strongly influences the energetics of the Laughlin quasiholes. Photoluminescence (PL) spectroscopy studies of this system have previously been able to probe these quasiholes that have confirmed our estimates of the Laughlin quasihole energies. Turning to the Pfaffian state, we now observe that such a theoretical approach is also able to provide valuable information about the Pfaffian quasiholes, viz., the energy dispersion, the charge density distribution and the quasihole energy. The energy dispersion of e/4 quasiholes derived here, clearly reflect the interaction between the quantum dot and the incompressible Pfaffian state. PL spectroscopy experiments on the 5/2 Pfaffian-Laughlin state could perhaps shed light on the energetics we found here of these elusive quasiparticles.
Understanding the nature of quasihole excitations, i.e., anyons that have fractional charge and fractional statistics, has been a challenging problem in condensed matter physics. Our theoretical approach to this problem has been to consider a quantum dot containing a few charged particles, Coulomb coupled to the incompressible fluid. The size of the dot and the separation distance of the dot from the incompressible state strongly influences the energetics of the Laughlin quasiholes. Photoluminescence (PL) spectroscopy studies of this system have previously been able to probe these quasiholes that have confirmed our estimates of the Laughlin quasihole energies. Turning to the Pfaffian state, we now observe that such a theoretical approach is also able to provide valuable information about the Pfaffian quasiholes, viz., the energy dispersion, the charge density distribution and the quasihole energy. The energy dispersion of e/4 quasiholes derived here, clearly reflect the interaction between the quantum dot and the incompressible Pfaffian state. PL spectroscopy experiments on the 5/2 Pfaffian-Laughlin state could perhaps shed light on the energetics we found here of these elusive quasiparticles.
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.
Non-Abelian excitations are an interesting feature of many fractional quantum Hall phases, including those phases described by the Moore-Read (or Pfaffian) wave function. However, the detection of the non-Abelian quasiparticles is challenging. Here, we consider a system described by the Moore-Read wave function and assume that impurity particles bind to its quasiholes. Then, the angular momentum of the impurities, reflected also by the impurity density, provides a useful witness of the physics of the non-Abelian excitations. By demanding that the impurities are constrained to the lowest Landau level, we are able to write down the corresponding many-body wave function describing both the Moore-Read liquid and the impurities. Through Monte Carlo sampling, we determine the impurity angular momentum, and we show that it suggests a quantum-statistical parameter $\ensuremath{\alpha}=a\ensuremath{\nu}\ensuremath{-}b+P/2$ for the quasiholes, where $\ensuremath{\alpha}$ ranges from 0 for bosons to 1 for fermions. A reasonable agreement with the Monte Carlo results is obtained for a=1/4, b=1/8, and P=0,1 depending on the parity of the particle number in the Moore-Read liquid. This parity dependence of the angular momentum serves as an unambiguous demonstration of the non-Abelian nature of the excitations. In addition to the studies of excitations in the Moore-Read liquid, we also apply our scheme to Laughlin liquids, for which we focus on interacting bosonic impurities. With this, the impurities themselves form Laughlin states, which allows for a study of hierarchical fractional quantum Hall states.
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.
The question of anyon interactions and their possible binding plays a key role in the physics of fractional quantum Hall states. Here, we introduce a controlled and scalable approach to study anyon binding by working entirely within the Hilbert space of anyons. The resulting theory is characterized by an effective potential, which captures the electrostatic energy of classical anyon configurations, and a Kähler potential, which simultaneously encodes the anyon Berry phase and the structure of their Hilbert space; both quantities are readily computed using Monte Carlo methods for large systems, enabling reliable extrapolation to the thermodynamic limit. By applying the formalism of geometric quantization on Kähler manifolds, we construct the anyon Hamiltonian, which can be exactly diagonalized in the few-anyon Hilbert space. Applying our approach to the quasiholes of the ν=1/3 Laughlin state with screened Coulomb interaction, we find that Laughlin quasiholes form bound states for screening lengths comparable or smaller than the magnetic length. Remarkably, binding occurs despite both the bare electron-electron interaction and the quasihole electrostatic potential being purely repulsive. The bound-state formation is a Berry phase effect, driven by oscillations in the quasihole density profile on the ℓB scale that are invisible in the quasihole electrostatic potential alone. For multiple quasiholes, we identify a sequence of phases as the screening length is reduced: free e/3 anyons, paired 2e/3 bound states, three-anyon charge-e clusters, and larger composite objects. Finally, we discuss possible signatures in charge imaging experiments on quantum Hall systems and the relevance to the phase diagram of itinerant anyon phases in fractional quantum anomalous Hall materials.
The elementary excitations of a fractional quantum Hall liquid are quasiparticles or quasiholes that are neither bosons nor fermions, but are so-called anyons. Here we study impurity particles immersed in a quantum Hall liquid that bind to the quasiholes via repulsive interactions with the liquid. We show that the angular momentum of an impurity is given by the multiple of a fractional "quantum" of angular momentum, and can directly be observed from the impurity density. In a system with several impurities bound to quasiholes, their total angular momentum interpolates between the values for free fermions and for free bosons. This interpolation is characterized by the fractional statistical parameter of the anyons, which is typically defined via their braiding behavior.
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.