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arXiv:1707.00728 · Physical Review Letters

Wave Function and Emergent SU(2) Symmetry in the νT=1 Quantum Hall Bilayer

Biao Lian, Shoucheng Zhang

Published Feb 16, 2018• 17 citations
arXivHTMLPDF

Abstract

We propose a trial wave function for the quantum Hall bilayer system of total filling factor ${\ensuremath{\nu}}_{T}=1$ at a layer distance $d$ to magnetic length $\ensuremath{\ell}$ ratio $d/\ensuremath{\ell}={\ensuremath{\kappa}}_{c1}\ensuremath{\approx}1.1$, where the lowest charged excitation is known to have a level crossing. The wave function has two-particle correlations, which fit well with those in previous numerical studies, and can be viewed as a Bose-Einstein condensate of free excitons formed by composite bosons and anticomposite bosons in different layers. We show the free nature of these excitons indicating an emergent SU(2) symmetry for the composite bosons at $d/\ensuremath{\ell}={\ensuremath{\kappa}}_{c1}$, which leads to the level crossing in low-lying charged excitations. We further show the overlap between the trial wave function, and the ground state of a small size exact diagonalization is peaked near $d/\ensuremath{\ell}={\ensuremath{\kappa}}_{c1}$, which supports our theory.

DOI
10.1103/physrevlett.120.077601

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We propose a trial wave function for the quantum Hall bilayer system of total filling factor \ensuremathνT=1{\ensuremath{\nu}}_{T}=1\ensuremathνT​=1 at a layer distance ddd to magnetic length \ensuremathℓ\ensuremath{\ell}\ensuremathℓ ratio d/\ensuremathℓ=\ensuremathκc1\ensuremath≈1.1d/\ensuremath{\ell}={\ensuremath{\kappa}}_{c1}\ensuremath{\approx}1.1d/\ensuremathℓ=\ensuremathκc1​\ensuremath≈1.1, where the lowest charged excitation is known to have a level crossing. The wave function has two-particle correlations, which fit well with those in previous numerical studies, and can be viewed as a Bose-Einstein condensate of free excitons formed by composite bosons and anticomposite bosons in different layers. We show the free nature of these excitons indicating an emergent SU(2) symmetry for the composite bosons at d/\ensuremathℓ=\ensuremathκc1d/\ensuremath{\ell}={\ensuremath{\kappa}}_{c1}d/\ensuremathℓ=\ensuremathκc1​, which leads to the level crossing in low-lying charged excitations. We further show the overlap between the trial wave function, and the ground state of a small size exact diagonalization is peaked near d/\ensuremathℓ=\ensuremathκc1d/\ensuremath{\ell}={\ensuremath{\kappa}}_{c1}d/\ensuremathℓ=\ensuremathκc1​, which supports our theory.
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sss-wave paired composite-fermion electron-hole trial state for quantum Hall bilayers with ν=1\nu=1ν=1
We introduce a new variational wavefunction for a quantum Hall bilayer at total filling ν=1\nu = 1ν=1, which is based on sss-wave BCS pairing between composite-fermion electrons in one layer and composite-fermion holes in the other. We compute the overlap of the optimized trial function with the ground state from exact diagonalization calculations of up to 14 electrons in a spherical geometry, and we find excellent agreement over the entire range of values of the ratio between the layer separation and the magnetic length. The trial wavefunction naturally allows for charge imbalance between the layers and provides important insights into how the physics at large interlayer separations crosses over to that at small separations in a fashion analogous to the BEC-BCS crossover.
We study the phase diagram of quantum Hall bilayer systems with total filing ν_{T}=1/2+1/2 of the lowest Landau level as a function of layer distances d. Based on numerical exact diagonalization calculations, we obtain three distinct phases, including an exciton superfluid phase with spontaneous interlayer coherence at small d, a composite Fermi liquid at large d, and an intermediate phase for 1.1<d/l_{B}<1.8 (l_{B} is the magnetic length). The transition from the exciton superfluid to the intermediate phase is identified by (i) a dramatic change in the Berry curvature of the ground state under twisted boundary conditions on the two layers and (ii) an energy level crossing of the first excited state. The transition from the intermediate phase to the composite Fermi liquid is identified by the vanishing of the exciton superfluid stiffness. Furthermore, from our finite-size study, the energy cost of transferring one electron between the layers shows an even-odd effect and possibly extrapolates to a finite value in the thermodynamic limit, indicating the enhanced intralayer correlation. Our identification of an intermediate phase and its distinctive features shed new light on the theoretical understanding of the quantum Hall bilayer system at total filling ν_{T}=1.
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.
Quantum Hall bilayers have proven to be an inexhaustible platform for emerging exotic states and novel phase transitions in the strongly correlated arena. Here, the authors study coupled quantum Hall bilayers at half-filled first excited Landau levels by varying the layer distance. The authors find a continuous transition between the exciton superfluid phase and two copies of the Moore-Read state. This transition breaks symmetry and changes topology simultaneously, providing an exotic example of a quantum phase transition beyond the Landau paradigm.