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arXiv:1612.07799 · Physical Review Letters

Particle-Hole Duality in the Lowest Landau Level

Dung Xuan Nguyen, Tankut Can, Andrey Gromov

Published May 19, 2017• 31 citations
arXivHTMLPDF

Abstract

We derive a number of exact relations between response functions of holomorphic, chiral fractional quantum Hall states and their particle-hole (PH) conjugates. These exact relations allow one to calculate the Hall conductivity, Hall viscosity, various Berry phases, and the static structure factor of PH conjugate states from the corresponding properties of the original states. These relations establish a precise duality between chiral quantum Hall states and their PH conjugates. The key ingredient in the proof of the relations is a generalization of Girvin's construction of PH-conjugate states to inhomogeneous magnetic field and curvature. Finally, we make several nontrivial checks of the relations, including for the Jain states and their PH conjugates.

DOI
10.1103/physrevlett.118.206602

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We derive a number of exact relations between response functions of holomorphic, chiral fractional quantum Hall states and their particle-hole (PH) conjugates. These exact relations allow one to calculate the Hall conductivity, Hall viscosity, various Berry phases, and the static structure factor of PH conjugate states from the corresponding properties of the original states. These relations establish a precise duality between chiral quantum Hall states and their PH conjugates. The key ingredient in the proof of the relations is a generalization of Girvin's construction of PH-conjugate states to inhomogeneous magnetic field and curvature. Finally, we make several nontrivial checks of the relations, including for the Jain states and their PH conjugates.
We study fractional quantum Hall states at filling fractions in the Jain sequences using the framework of composite Dirac fermions. Synthesizing previous work, we write an effective field theory consistent with all symmetry requirements, including Galilean invariance and particle-hole symmetry. Employing a Fermi-liquid description, we demonstrate the appearance of the Girvin-Macdonald-Platzman algebra and compute the dispersion relation of neutral excitations and various response functions. Our results satisfy requirements of particle-hole symmetry. We show that while the dispersion relation obtained from the modified random-phase approximation (MRPA) of the Halperin-Lee-Read (HLR) theory is particle-hole symmetric, correlation functions obtained from this scheme are not. The results of the Dirac theory are shown to be consistent with the Haldane bound on the projected structure factor, while those of the MPRA of the HLR theory violate it.
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The particle-hole (PH) symmetry of electrons is an exact symmetry of the electronic Hamiltonian confined to a specific Landau level, and its interplay with the formation of composite fermions has attracted much attention of late. We investigate an emergent symmetry in the fractional quantum Hall effect, namely, the PH symmetry of composite fermions, which relates states at composite fermion filling factors ${\ensuremath{\nu}}^{*}=n+\overline{\ensuremath{\nu}}$ and ${\ensuremath{\nu}}^{*}=n+1\ensuremath{-}\overline{\ensuremath{\nu}}$, where the integer nnn is the $\mathrm{\ensuremath{\Lambda}}$-level index and $0\ensuremath{\le}\overline{\ensuremath{\nu}}\ensuremath{\le}1$. Detailed calculations using the microscopic theory of composite fermions demonstrate the following for low-lying $\mathrm{\ensuremath{\Lambda}}$ levels (small nnn): (i) The two-body interaction between composite-fermion particles is very similar, apart from a constant additive term and an overall scale factor, to that between composite-fermion holes in the same $\mathrm{\ensuremath{\Lambda}}$ level; and (ii) the three-body interaction for composite fermions is an order of magnitude smaller than the two-body interaction. Taken together, these results imply an approximate PH symmetry for composite fermions in low $\mathrm{\ensuremath{\Lambda}}$ levels, which is also supported by exact-diagonalization studies and available experiments. This symmetry, which relates states at electron filling factors $\ensuremath{\nu}=\frac{n+\overline{\ensuremath{\nu}}}{2(n+\overline{\ensuremath{\nu}})\ifmmode\pm\else\textpm\fi{}1}$ and $\ensuremath{\nu}=\frac{n+1\ensuremath{-}\overline{\ensuremath{\nu}}}{2(n+1\ensuremath{-}\overline{\ensuremath{\nu}})\ifmmode\pm\else\textpm\fi{}1}$, is not present in the original Hamiltonian and owes its existence entirely to the formation of composite fermions. With increasing $\mathrm{\ensuremath{\Lambda}}$-level index, the two-body and three-body pseudopotentials become comparable, but at the same time they both diminish in magnitude, indicating that the interaction between composite fermions becomes weak as we approach $\ensuremath{\nu}=1/2$.
We propose a derivative operator formed as a function of derivatives of the electron coordinates. When the derivative operator is applied to the Laughlin wave function, two new wave functions in the lowest Landau level at filling factor 1/2 are generated. For systems of 4, 6, and 8 electrons in spherical geometry, it is shown that the first wave function has nearly unity overlap with the particle-hole conjugate of the Moore-Read Pfaffian wave function, therefore together with the Moore-Read Pfaffian state forms a particle-hole conjugate pair. The second wave function has essentially perfect particle-hole symmetry itself, with a positive parity when the number of electron pairs N/2 is an even integer and and a negative parity when N/2 is an odd integer. An equivalent form suggests the first wave function forms a f-wave pairing of composite fermions, and the second wave function forms a p-wave pairing. The corresponding Non-Abelian statistics quasiparticle wave functions are also proposed.
Using methods based on conformal field theory, we construct model wave functions on a torus with arbitrary flat metric for all chiral states in the abelian quantum Hall hierarchy. These functions have no variational parameters, and they transform under the modular group in the same way as the multicomponent generalizations of the Laughlin wave functions. Assuming the absence of Berry phases upon adiabatic variations of the modular parameter τττ, we calculate the quantum Hall viscosity and find it to be in agreement with the formula, given by Read, which relates the viscosity to the average orbital spin of the electrons. For the filling factor ν=2/5ν=2/5ν=2/5 Jain state, which is at the second level in the hierarchy, we compare our model wave function with the numerically obtained ground state of the Coulomb interaction Hamiltonian in the lowest Landau level, and find very good agreement in a large region of the complex τττ-plane. For the same example, we also numerically compute the Hall viscosity and find good agreement with the analytical result for both the model wave function and the numerically obtained Coulomb wave function. We argue that this supports the notion of a generalized plasma analogy that would ensure that wave functions obtained using the conformal field theory methods do not acquire Berry phases upon adiabatic evolution.
The Read-Rezayi (RR) parafermion states form a series of exotic non-Abelian fractional quantum Hall (FQH) states at filling $\ensuremath{\nu}=k/(k+2)$. Computationally, the wave functions of these states are prohibitively expensive to generate for large systems. We introduce a series of parton states, denoted 2‾k1k+1{\overline{2}}^{k}{1}^{k+1}2k1k+1, and show that they lie in the same universality classes as the particle-hole-conjugate RR (``anti-RR'') states. Our analytical results imply that a $[U{(1)}_{k+1}\ifmmode\times\else\texttimes\fi{}U{(2k)}_{\ensuremath{-}1}]/[SU{(k)}_{\ensuremath{-}2}\ifmmode\times\else\texttimes\fi{}U{(1)}_{\ensuremath{-}1}]$ coset conformal field theory describes the edge excitations of the 2‾k1k+1{\overline{2}}^{k}{1}^{k+1}2k1k+1 state, suggesting nontrivial dualities with respect to previously known descriptions. The parton construction allows wave functions in anti-RR phases to be generated for hundreds of particles. We further propose the parton sequence n‾2‾214\overline{n}{\overline{2}}^{2}{1}^{4}n2214, with n=1,2,3n=1,2,3n=1,2,3, to describe the FQH states observed at $\ensuremath{\nu}=2+1/2$, 2+2/52+2/52+2/5, and 2+3/82+3/82+3/8.