1. The half-filled Landau level: The case for Dirac composite fermions
by Scott D. Geraedts, Michael P. Zaletel, Roger S. K. Mong, Max A. Metlitski, Ashvin Vishwanath, and Olexei I. Motrunich
Science
Volume 352(6282):
April 8, 2016
Copyright © 2016, American Association for the Advancement of Science

2. Fig. 1 Mapping the Fermi surface by means of density-density correlations.
Mapping the Fermi surface by means of density-density correlations. (A) Boundary conditions on the composite fermions (CFs). Our states can be described by a number of “wires”—slices through the 2D Fermi sea—at various fixed values of ky. The number of wires Nw is dictated by Ly and the boundary condition of the CFs. An odd Nw corresponds to the periodic boundary condition (PBC); an even Nw corresponds to the antiperiodic boundary condition (APBC). (B) Mapping the Fermi surface via the structure factor. The lower panel shows the “guiding center” density-density correlations, exp(q2/2)D(q) = exp(q2/2)〈:δρqδρ–q:〉, measured on a cylinder with Ly = 13. The singularities arise from CF scattering processes across the Fermi surface. The observed scatterings are illustrated in the inset, with colors corresponding to qy. The upper panel shows the derivatives of the correlator, which aid in determining the location of the singularities. (C) Testing Luttinger’s theorem, which states that the area enclosed by the Fermi surface is related to the particle density. On a cylinder, the “area” is given by the sum of the length of each wire in momentum space, which we determined from singularities in plots such as (B). We define Qm to be the length of the Fermi sea slice at ky = (2π/Ly)m for integer m and plot the resulting sums for various circumferences against the Luttinger prediction. Note that we use the relation Q–m = Qm, a consequence of the 180° rotational symmetry. There is excellent agreement between our data and the Luttinger count. Error bars are estimated on the basis of the finite width of the observed singularities, which is due to our finite correlation length.
Scott D. Geraedts et al. Science 2016;352:
Copyright © 2016, American Association for the Advancement of Science

3. Fig. 2 Two-dimensional momentum space view of density-density correlations.
Two-dimensional momentum space view of density-density correlations. D(q) = 〈:δρqδρ–q:〉 is plotted as a function of 2D momenta q = (qx, qy). The cylinder circumference is Ly = 24, where we see eight slices through the Fermi sea. Descendants of a singular 2kF circle in D(q) appear in cuts at qy between –7(2π/Ly) and 7(2π/Ly). At this large circumference, the correlations approach those of a 2D system.
Scott D. Geraedts et al. Science 2016;352:
Copyright © 2016, American Association for the Advancement of Science

4. Fig. 3 Entanglement entropy versus correlation length.
Entanglement entropy versus correlation length. Data are taken at a variety of circumferences Ly; different data points at the same size correspond to different bond dimensions χ = 600 to 12,000. Both S and ξ increase as χ is increased. For a quasi-1D critical system, we expect a linear relationship between S and log ξ with the slope proportional to the central charge c (see Eq. 3). The dashed lines (from bottom to top) correspond to c = 3, 4, 5, and 7.
Scott D. Geraedts et al. Science 2016;352:
Copyright © 2016, American Association for the Advancement of Science

5. Fig. 4 Contrasting orbital entanglement spectra of the n = 0 and n = 1 Landau levels at half-filling.
Contrasting orbital entanglement spectra of the n = 0 and n = 1 Landau levels at half-filling. These results correspond to experiments at v = 1/2 and v = 5/2, respectively. Data are at Ly = 19; n = 0 is the gapless CFL phase, and n = 1 is consistent with the gapped Pfaffian phase. Eent corresponds to the eigenvalues of the reduced density matrix for the left half of the cylinder. Each eigenvalue is associated with a momentum around the cylinder Ky, just like an energy spectrum. PH acts as a reflection Ky ↔ –Ky. The characteristic chiral “dispersion” at n = 1 clearly breaks PH, whereas n = 0 is PH-symmetric.
Scott D. Geraedts et al. Science 2016;352:
Copyright © 2016, American Association for the Advancement of Science

6. Fig. 5 The extinction of 2kF backscattering off PH-symmetric impurities.
The extinction of 2kF backscattering off PH-symmetric impurities. A QH bilayer with chemical potential imbalance μ allows us to continuously tune from a PH-symmetric model (μ = ∞) to a PH-broken one (μ finite). We compute the correlation function of a PH-even operator 〈PqP–q〉 for qy = 2π/Ly at Ly = 13 and plot its derivative with respect to qx to bring out singularities. For the PH-symmetric case, there are many singularities, but noticeably absent is any kink at |q| = 2kF (marked by a dashed line). This demonstrates the Dirac nature of the CFs: P is even under PH, whereas scattering a CF across the Fermi surface to its antipode is PH-odd in the Dirac theory. At finite chemical potential μ, the bilayer setup explicitly breaks PH symmetry, and a kink at 2kF continuously reappears.
Scott D. Geraedts et al. Science 2016;352:
Copyright © 2016, American Association for the Advancement of Science