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arXiv:0710.5905 · Physical Review B

Plasma picture of the fractional quantum Hall effect with internal SU(K) symmetries

R. de Gail, Nicolas Regnault, M. O. Goerbig

Published Apr 7, 2008• 48 citations
arXivHTMLPDF

Abstract

We consider trial wave functions exhibiting $\text{SU}(K)$ symmetry, which may be well suited to grasp the physics of the fractional quantum Hall effect with internal degrees of freedom. Systems of relevance may be either spin-unpolarized states $(K=2)$, semiconductor bilayers $(K=2,4)$, or graphene $(K=4)$. We find that some introduced states are unstable, undergoing phase separation or phase transition. This allows us to strongly reduce the set of candidate wave functions eligible for a particular filling factor. The stability criteria are obtained with the help of Laughlin's plasma analogy, which we systematically generalize to the multicomponent $\text{SU}(K)$ case. The validity of these criteria is corroborated by exact-diagonalization studies for SU(2) and SU(4). Furthermore, we study the pair-correlation functions of the ground state and elementary charged excitations within the multicomponent plasma picture.

DOI
10.1103/physrevb.77.165310

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We consider trial wave functions exhibiting SU(K)\text{SU}(K)SU(K) symmetry, which may be well suited to grasp the physics of the fractional quantum Hall effect with internal degrees of freedom. Systems of relevance may be either spin-unpolarized states (K=2)(K=2)(K=2), semiconductor bilayers (K=2,4)(K=2,4)(K=2,4), or graphene (K=4)(K=4)(K=4). We find that some introduced states are unstable, undergoing phase separation or phase transition. This allows us to strongly reduce the set of candidate wave functions eligible for a particular filling factor. The stability criteria are obtained with the help of Laughlin's plasma analogy, which we systematically generalize to the multicomponent SU(K)\text{SU}(K)SU(K) case. The validity of these criteria is corroborated by exact-diagonalization studies for SU(2) and SU(4). Furthermore, we study the pair-correlation functions of the ground state and elementary charged excitations within the multicomponent plasma picture.
We investigate even-denominator fractional quantum Hall (EDFQH) states in monolayer graphene, focusing on the experimentally observed filling fractions ν=1/2ν= 1/2ν=1/2 and ν=1/4ν= 1/4ν=1/4. Owing to the approximate SU(4) symmetry arising from spin and valley degrees of freedom, graphene hosts a rich variety of multi-component fractional quantum Hall states with different polarization configurations. Using the Chern-Simons composite fermion framework for SU(4) systems, we construct candidate variational wave functions corresponding to distinct spin, valley, and mixed polarization states. The ground-state energies of these states are evaluated numerically using the Coulomb interaction in spherical geometry. By comparing energies across different polarization sectors, we identify the energetically favored configurations at ν=1/2ν= 1/2ν=1/2 and ν=1/4ν= 1/4ν=1/4. Our variational energy comparison suggests that the energetically favored candidate states correspond to multi-component correlated quantum Hall liquids. Within the restricted family of trial states considered here, these states differ from a simple composite-fermion Fermi-sea description. These findings provide insight into the role of valley polarization and SU(4) symmetry in stabilizing even-denominator fractional quantum Hall states in graphene.
Single-layer and Bilayer of graphene are new classes of two-dimensional electron systems with unconventional band structures and valley degrees of freedom. The ground states and excitations in the integer and fractional quantum Hall regimes are investigated on torus and spherical geometries with the use of the density matrix renormalization group (DMRG) method. At nonzero Landau level indices, the ground states at effective filling factors 1, 1/3, 2/3 and 2/5 are valley polarized both in single-layer and bilayer graphenes. We examine the elementary charge excitations which could couple with the valley degrees of freedom (so called valley skyrmions). The excitation gaps are calculated and extrapolated to the thermodynamic limit. The largest excitation gap at effective filling 1/3 is obtained in bilayer graphene, which is a good candidate for experimental observation of fractional quantum Hall effect.
We report on an exact diagonalization study of fractional quantum Hall states at a filling factor of ν=2/3 in a system with a fourfold degenerate n=0 Landau level and SU(4) symmetric Coulomb interactions. Our investigation reveals previously unidentified SU(3) and SU(4) singlet ground states which appear at a flux quantum shift 2 when a spherical geometry is employed and lie outside the established composite-fermion or multicomponent Halperin state patterns. We evaluate the two-particle correlation functions of these states and discuss quantum phase transitions in graphene between singlet states with a different number of components as the magnetic field strength is increased.
Trial wave functions which, in the thermodynamic limit, describe states relevant to the fractional quantum Hall effect are explicitly constructed for up to eight electrons. Estimates are given for the ground-state energies and pair-correlation functions and for the low-lying excitation energies for states associated with the filling factors \ensuremath{\nu}=(1/3), (2/7), and (2/5). The results help explain why the filling factors \ensuremath{\nu}=${\ensuremath{\nu}}_{m}$=m/(2m+1), m=1,2,..., and \ensuremath{\nu}=1-${\ensuremath{\nu}}_{m}$ show strong quantum Hall effects.
We study the anisotropic effect of the Coulomb interaction on a 1/3-filling fractional quantum Hall system by using an exact diagonalization method on small systems in torus geometry. For weak anisotropy the system remains to be an incompressible quantum liquid, although anisotropy manifests itself in density correlation functions and excitation spectra. When the strength of anisotropy increases, we find the system develops a Hall-smectic-like phase with a one-dimensional charge density wave order and is unstable towards the one-dimensional crystal in the strong anisotropy limit. In all three phases of the Laughlin liquid, Hall-smectic-like, and crystal phases the ground state of the anisotropic Coulomb system can be well described by a family of model wave functions generated by an anisotropic projection Hamiltonian. We discuss the relevance of the results to the geometrical description of fractional quantum Hall states proposed by Haldane [Phys. Rev. Lett. 107, 116801 (2011)].
Quantum Hall bilayer systems at filling fractions near $\ensuremath{\nu}=\frac{1}{2}+\frac{1}{2}$ undergo a transition from a compressible phase with strong intralayer correlation to an incompressible phase with strong interlayer correlations as the layer separation ddd is reduced below some critical value. Deep in the intralayer phase (large separation) the system can be interpreted as a fluid of composite fermions (CFs), whereas deep in the interlayer phase (small separation) the system can be interpreted as a fluid of composite bosons (CBs). The focus of this paper is to understand the states that occur for intermediate layer separation by using trial variational wave functions. We consider two main classes of wave functions. In the first class, previously introduced in M\"oller et al. [Phys. Rev. Lett. 101, 176803 (2008)], we consider interlayer BCS pairing of two independent CF liquids. We find that these wave functions are exceedingly good for $d\ensuremath{\gtrsim}{\ensuremath{\ell}}_{0}$ with ${\ensuremath{\ell}}_{0}$ as the magnetic length. The second class of wave functions naturally follows the reasoning of Simon et al. [Phys. Rev. Lett. 91, 046803 (2003)] and generalizes the idea of pairing wave functions by allowing the CFs also to be replaced continuously by CBs. This generalization allows us to construct exceedingly good wave functions for interlayer spacings of $d\ensuremath{\lesssim}{\ensuremath{\ell}}_{0}$ as well. The accuracy of the wave functions discussed in this work, compared with exact diagonalization, approaches that of the celebrated Laughlin wave function.