A distinguishing feature of fractional quantum Hall (FQH) states is a singular behavior of equilibrium densities at boundaries. In contrast to states at integer filling fraction, such quantum liquids posses an additional dipole moment localized near edges. It enters observable quantities such as universal dispersion of edge states and Lorentz shear stress. For a Laughlin state, this behavior is seen as a peak, or overshoot, in the single-particle density near the edge, reflecting a general tendency of electrons in FQH states to cluster near edges. We compute the singular edge behavior of the one-particle density by a perturbative expansion carried out around a completely filled Landau level. This correction is shown to fully capture the dipole moment and the major features of the overshoot observed numerically. Furthermore, it exhibits the Stokes phenomenon with the Stokes line at the boundary of the droplet, decaying like a Gaussian inside and outside the liquid with different decay lengths. In the limit of vanishing magnetic length, the shape of the overshoot is a singular double layer with a capacity that is a universal function of the filling fraction. Finally, we derive the edge dipole moment of Pfaffian FQH states. The result suggests an explicit connection between the magnitude of the dipole moment and the bulk odd viscosity.
A distinguishing feature of fractional quantum Hall (FQH) states is a singular behavior of equilibrium densities at boundaries. In contrast to states at integer filling fraction, such quantum liquids posses an additional dipole moment localized near edges. It enters observable quantities such as universal dispersion of edge states and Lorentz shear stress. For a Laughlin state, this behavior is seen as a peak, or overshoot, in the single-particle density near the edge, reflecting a general tendency of electrons in FQH states to cluster near edges. We compute the singular edge behavior of the one-particle density by a perturbative expansion carried out around a completely filled Landau level. This correction is shown to fully capture the dipole moment and the major features of the overshoot observed numerically. Furthermore, it exhibits the Stokes phenomenon with the Stokes line at the boundary of the droplet, decaying like a Gaussian inside and outside the liquid with different decay lengths. In the limit of vanishing magnetic length, the shape of the overshoot is a singular double layer with a capacity that is a universal function of the filling fraction. Finally, we derive the edge dipole moment of Pfaffian FQH states. The result suggests an explicit connection between the magnitude of the dipole moment and the bulk odd viscosity.
The discontinuity of guiding-center Hall viscosity (a bulk property) at the edges of incompressible quantum Hall fluids is associated with the presence of an intrinsic electric dipole moment on the edge. If there is a gradient of drift velocity due to a nonuniform electric field, the discontinuity in the induced stress is exactly balanced by the electric force on the dipole. The total Hall viscosity has two distinct contributions: a ``trivial'' contribution associated with the geometry of the Landau orbits and a nontrivial contribution associated with guiding-center correlations. We describe a relation between the guiding-center edge-dipole moment and ``momentum polarization,'' which relates the guiding-center part of the bulk Hall viscosity to the ``orbital entanglement spectrum (OES).'' We observe that using the computationally more-onerous ``real-space entanglement spectrum'' just adds the trivial Landau-orbit contribution to the guiding-center part. This shows that all the nontrivial information is completely contained in the OES, which also exposes a fundamental topological quantity, $\ensuremath{\gamma}$ = $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}\ensuremath{-}\ensuremath{\nu}$, the difference between the ``chiral stress-energy anomaly'' (or signed conformal anomaly) and the chiral charge anomaly. This quantity characterizes correlated fractional quantum Hall fluids and vanishes in uncorrelated integer quantum Hall fluids.
The fractional quantum Hall effect (FQHE) is the archetype of the strongly correlated systems and the topologically ordered phases. Unlike the integer quantum Hall effect (IQHE) which can be explained by single-particle physics, FQHE exhibits many emergent properties that are due to the strong correlation among many electrons. In this Thesis, among those emergent properties of FQHE, we focus on the guiding-center metric, the guiding-center Hall viscosity, the guiding-center spin, the intrinsic electric dipole moment and the orbital entanglement spectrum. Specifically, we show that the discontinuity of guiding-center Hall viscosity (a bulk property) at edges of incompressible quantum Hall fluids is associated with the presence of an intrinsic electric dipole moment on the edge. If there is a gradient of drift velocity due to a non-uniform electric field, the discontinuity in the induced stress is exactly balanced by the electric force on the dipole. We show that the total Hall viscosity has two distinct contributions: a "trivial'' contribution associated with the geometry of the Landau orbits, and a non-trivial contribution associated with guiding-center correlations. We describe a relation between the intrinsic dipole moment and "momentum polarization'', which relates the guiding-center Hall viscosity to the "orbital entanglement spectrum(OES)''. We observe that using the computationally-more-onerous "real-space entanglement spectrum (RES)'' in the momentum polarization calculation just adds the trivial Landau-orbit contribution to the guiding-center part. This shows that all the non-trivial information is completely contained in the OES, which also exposes a fundamental topological quantity γ = c˜ − ν, the difference between the "chiral stress-energy anomaly'' (or signed conformal anomaly) and the chiral charge anomaly. This quantity characterizes correlated fractional quantum Hall fluids, and vanishes in integer quantum Hall fluids which are uncorrelated.
In this Letter, we investigate the dissipative dynamics at the edge of Laughlin fractional quantum Hall (FQH) states starting from the hydrodynamic framework of the composite boson theory recently developed by G. M. Monteiro et al. [Phys. Rev. B 109, 174525 (2024)]. Critical to this description is the choice of boundary conditions, which ultimately stems from the choice of hydrodynamic variables in condensate degrees of freedom. Given the gapped nature of bulk, one would expect dissipation effects to play an important role only near the FQH edge. Thus, one envisions a scenario where the bulk hydrodynamic equations remain unmodified, while the dissipation effects are introduced at the edge via boundary conditions. We have recently shown that the anomaly requirements fix the boundary conditions of the FQH fluid to be no-penetration and no-stress boundary conditions. In this work, we introduce energy dissipation in the no-stress boundary condition leading to charge diffusion at the boundary. The resulting dissipative edge dynamics is quite rigid from a hydrodynamic perspective, as it has to preserve the edge charge continuity and the anomaly structure. We show that the diffusive edge dynamics with fluctuation-dissipation relations within a power counting scheme belong to the Kardar-Parisi-Zhang universality class.
Abstract Advancing a microscopic framework that rigorously unveils the underlying topological hallmarks of fractional quantum Hall (FQH) fluids is a prerequisite for making progress in the classification of strongly-coupled topological matter. We present a second-quantization framework that reveals an exact fusion mechanism for particle fractionalization in FQH fluids, and uncovers the fundamental structure behind the condensation of non-local operators characterizing topological order in the lowest-Landau-level. We show the first exact analytic computation of the quasielectron Berry connections leading to its fractional charge and exchange statistics, and perform Monte Carlo simulations that numerically confirm the fusion mechanism for quasiparticles. We express the sequence of (bosonic and fermionic) Laughlin second-quantized states, highlighting the lack of local condensation, and present a rigorous constructive subspace bosonization dictionary for the bulk fluid. Finally, we establish universal long-distance behavior of edge excitations by formulating a conjecture based on the DNA, or root state, of the FQH fluid.
The edge of the electronic fractional quantum Hall (FQH) system obeys the law of the chiral Luttinger liquid theory due to its intrinsic topological properties and the relation of bulk-edge correspondence. However, in a realistic experimental system, such as the usual Hall bar setup, the softening of the background confinement potential can induce the reconstruction of the edge spectrum, which breaks the chirality and universality of the FQH edge. The entanglement spectrum (ES) of the FQH ground state has the same counting structure as that of the energy spectrum indicating the topological characters of the quantum state. In this work, we report that the ES can also have an edge reconstruction while sweeping the area of the subsystem in real space cut. Moreover, we found that the critical area of the subsystem matches accurately with the intrinsic building block of the FQH liquids, namely the correlation hole of the FQH liquids. The above results seem to be universal after our study of a series of typical FQH states, such as two Laughlin states at $\ensuremath{\nu}=1/3$ and 1/5, and the Moore-Read state for $\ensuremath{\nu}=5/2$.
The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer.The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE.An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions.One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality.Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus.Having explicit torus wave functions allows us to use the methods developed by Read and Read \& Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information.We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit.