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arXiv:1403.2822 · Physical Review B

Guiding-center Hall viscosity and intrinsic dipole moment along edges of incompressible fractional quantum Hall fluids

YeJe Park, F. D. M. Haldane

Published Jul 18, 2014• 48 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.90.045123

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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, \ensuremathγ\ensuremath{\gamma}\ensuremathγ = c\ifmmode~\else˜\fi\ensuremath−\ensuremathν\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}\ensuremath{-}\ensuremath{\nu}c\ifmmode~\else˜\fi\ensuremath−\ensuremathν, 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.
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 1/51/51/5, and the Moore-Read state for $\ensuremath{\nu}=5/2$.
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.
This paper investigates the bulk and boundary dynamics of Laughlin states, which are modeled using composite boson theory within a fluid dynamics framework. In this work, we adopt an alternative starting point based on a hydrodynamic action with topological terms, which fleshes out the fluid aspects of the Laughlin state manifestly. For a particular choice of the velocity field, the fluid equation for this action is akin to first-order hydrodynamic equations, supplemented with an additional constitutive equation known as the Hall constraint. When a hard wall boundary is present, one of the topological terms in the fluid action triggers anomaly inflow, indicating the presence of gauge anomaly at the edge. The first-order hydrodynamic equations require a second boundary condition, which, in the absence of dissipation, can be either a no-slip or a no-stress condition. We find that the no-slip condition, where the fluid adheres to the wall, is incompatible with the chiral edge dynamics. On the other hand, the no-stress condition, which allows the fluid to move along the wall without friction, is consistent with the expected chiral edge dynamics of the Laughlin state. Furthermore, our work derives this modified no-stress boundary condition within a variational principle. This is accomplished by incorporating a chiral boson action within the boundary action that is nonlinearly coupled to the edge density, thus systematically extending the edge chiral Luttinger liquid theory.
Abstract The description of chiral quantum incompressible fluids by the W∞ symmetry can be extended from the edge, where it encompasses the conformal field theory approach, to the non-conformal bulk. The two regimes are characterized by excitations with different sizes, energies and momenta within the disk geometry. In particular, the bulk quantities have a finite limit for large droplets. We obtain analytic results for the radial shape of excitations, the edge reconstruction phenomenon and the energy spectrum of density fluctuations in Laughlin states.
The quantum Hall effect is due to discontinuities in the chemical potential at certain bulk electron densities. It has been conjectured that the resulting edge states in some fractional Hall gaps can have an intricate composite structure which reflects the correlations in the incompressible bulk fluid. We present direct numerical evidence obtained from exact-diagonalization studies in favor of this conjecture. We show that the states on the \ensuremath{\nu}=2/3 fractional quantum Hall edge constitute a two-component system in which a \ensuremath{\nu}=1/3 droplet of holes is contained within a \ensuremath{\nu}=1 droplet of electrons.