Dirac Fermion Hierarchy of Composite Fermi Liquids

Jie Wang

Phys. Rev. Lett. 122, 257203 – Published 26 June 2019

Abstract

Composite Fermi liquids (CFLs) are compressible states that can occur for 2D interacting fermions confined in the lowest Landau level at certain Landau level fillings. They have been understood as Fermi seas formed by composite fermions which are bound states of electromagnetic fluxes and electrons as reported by Halperin, Lee, and Read [Phys. Rev. B 47, 7312 (1993)]. At half filling, an explicitly particle-hole symmetric theory based on Dirac fermions was proposed by Son [Phys. Rev. X 5, 031027 (2015)] as an alternative low energy description. In this work, we investigate the Berry curvature of CFL model wave functions at a filling fraction of one-quarter, and observe that it is uniformly distributed over the Fermi sea except at the center where an additional $π$ phase was found. Motivated by this, we propose an effective theory which generalizes Son’s half filling theory, by internal gauge flux attachment, to all filling fractions in which fermionic CFLs can occur. The numerical results support the idea of internal gauge flux attachment.

Received 12 September 2018
Revised 13 April 2019

DOI:https://doi.org/10.1103/PhysRevLett.122.257203

Physics Subject Headings (PhySH)

Composite fermions Dirac fermions Quantum Hall effect

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jie Wang ^*

Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

^*jiew@princeton.edu; jiewang.phy@gmail.com

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Vol. 122, Iss. 25 — 28 June 2019

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Images

Figure 1
Berry curvature distribution (right) obtained from $Ψ_{n = 1}^{1 / 4}$ model wave function by a linear regression on a FS (left) consisting of $N = 37$ dipoles. The red dashed line represents the path, which can be interpreted as FS boundary, along which anticlockwisely transporting a single composite fermion has $- 2 π ν$ Berry phase. The area enclosed by the red dashed line contains 46 grids. The Berry curvature has a peak of around $- 0.25 + 0.011 = - 0.239$ (in units of $π$ ), while the rest of the values are around $- (2 / 4 - 1) / 46 = 0.011$ . It suggests an interesting Berry curvature distribution for CFLs at a generic filling fraction in the thermodynamic limit: a $- π$ singularity at center and $- (2 π ν - π)$ uniformly distributed over FS.
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Figure 2
Variational energies (red dots for $Ψ_{n = 1}^{1 / 4}$ , blue dots for $Ψ_{n = 2}^{1 / 4}$ ) and exactly diagonalized Coulomb energies (dashed lines) as a function of many-body momentum [35] $(K_{1}, K_{2})$ for $N = 10$ electrons for the $ν = 1 / 4$ filled LLL on a square torus. Energies are plotted in units of $e^{2} / ε l_{B}$ . For each $K_{1}$ , $K_{2}$ is chosen to match the momentum of the lowest energy state. Because of inversion symmetry, only $K_{1} \in [0, 5]$ are plotted.
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Figure 3
Comparison of the Berry phases $Φ_{Γ}$ associated with various clockwise paths ${Γ} = Γ_{1}, \dots, Γ_{6}$ on a FS of $N = 37$ dipoles (see Fig. 1 for FS) computed from $Ψ_{n = 1}^{1 / 4}$ (red dots) and from the formula $Φ_{Γ} = δ_{Γ} π + (2 π ν - π) A_{Γ} / A_{FS}$ (black lines), where $δ_{Γ}$ is the winding number of $Γ$ relative to the FS center, $A_{Γ}$ and $A_{FS}$ are the $k$ -space areas enclosed by the path $Γ$ and FS area, respectively. See Supplemental Material for details about paths and more examples, including $N = 69$ FS and $ν = 1 / 3$ [36].
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Figure 4
Guiding center structure factor $S (k)$ as a function of $k_{x}$ along $k_{y} = 0$ axes computed from $Ψ_{n = 1}^{1 / 4}$ . The plot is obtained after a finite size scaling for MWFs of $\sqrt{N} \times \sqrt{N}$ square FSs where $N$ is the number of electrons. The red lines are $2 k_{F} l_{B} = \sqrt{2 π ν}$ , a value of twice that of the Fermi wave vector obtained by applying Luttinger theorem on a square FS. The fact that $S (k)$ plots fit into one curve and the numerical singularities match with the analytical value implies that Luttinger theorem is true for CFLs.
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Figure 5
Illustrations of FS, particles, band structure in Son’s half filled DF theory (middle), and theory $L_{ψ_{+}}$ in Eq. (8). PH acts like time reversal, thus flipping the fluxes (arrows) attached to the DFs (black dots). FS sizes of PH conjugate states are the same, fixed by Luttinger theorem. The valance band, represented as dashed straight line, has been integrated out. The band mass is conjectured to be negligible.
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