Research ArticleMATERIALS SCIENCE

Entanglement signatures of emergent Dirac fermions: Kagome spin liquid and quantum criticality

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Science Advances  09 Nov 2018:
Vol. 4, no. 11, eaat5535
DOI: 10.1126/sciadv.aat5535
  • Fig. 1 Entanglement of quantum critical Dirac fermions.

    (A) Cylinder with a flux insertion. (B) π-flux square lattice model. The different cylinder types—YCn-0, YCn-1, and YCn-2—correspond to identifying x with a, b, or c, respectively. (C) The energy dispersion at V = 0 shows two Dirac cones at (π/2, π/2). (D and E) Blue (red) lines show allowed momenta for Φ = 0 (π) in the Brillouin zone for an infinite cylinder with circumference Ly = 8. The green crosses show the positions of the Dirac points. (F to I) EE versus flux Φ on a YC8-0 infinite cylinder for V/t = 0, 0.8, and 1.2. (H) EE for the YC8-2 cylinder at V/t = 0.8. In (F) to (I), the lines are the best fits to Eq. 2; insets show the data plotted in terms of the Φ-dependent part of Eq. 2 and the linear fits. (J) Fitting parameter B versus the repulsion strength V/Vc for various cylinder types. The dashed line is the prediction for noninteracting Dirac fermions.

  • Fig. 2 Kagome cylinders and allowed momenta.

    (A) The different types of kagome cylinders—YC8-0, YC8-2, or YC8-4—correspond to identifying site x with site a, b, or c, respectively. The blue (solid) and red (dashed) lines show the allowed momenta for an infinite cylinder of types (B) YC8-0 and (C) YC8-2. The gray rectangle is the magnetic Brillouin zone due to the π-fluxes in the hexagons. The two Dirac points of the QSL are at ±Q = ±(π/2, π/2).

  • Fig. 3 EE for the kagome Heisenberg model.

    DMRG results for the EE versus the flux Φ for the spin-1/2 kagome antiferromagnetic Heisenberg model on an infinite cylinder with eight sites in the periodic direction. (A) to (C) are for the YC8-0 cylinder, and (D) to (F) are for the YC8-2 cylinder. The red lines are best fits to Eq. 2. Insets: EE plotted as a function of the Φ-dependent part of Eq. 2 and the corresponding linear fits.

  • Table 1 Values of the shifts Embedded Image in Eq. 2 for the kagome model.

    On the YC8-0 cylinder, Embedded Image equals the internal gauge flux φ = π.

    Dirac flavorYC8-0YC8-2
    ↑, QEmbedded Imageππ/2
    ↑, − QEmbedded Imageπ− π/2
    ↓, QEmbedded Imageπ− π/2
    ↓, − QEmbedded Imageππ/2

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/11/eaat5535/DC1

    Section S1. EE for free Dirac fermions on the cylinder

    Section S2. EE of interacting Dirac fermions at large N

    Section S3. Quantum critical point of Dirac fermions on the honeycomb lattice

    Section S4. EE in the gapped phase

    Fig. S1. Bipartition of an infinite cylinder, which is threaded by a flux Φ.

    Fig. S2. Entanglement of quantum critical Dirac fermions on the honeycomb lattice.

    Fig. S3. Entanglement in the charge-ordered phase.

    References (3437)

  • Supplementary Materials

    This PDF file includes:

    • Section S1. EE for free Dirac fermions on the cylinder
    • Section S2. EE of interacting Dirac fermions at large N
    • Section S3. Quantum critical point of Dirac fermions on the honeycomb lattice
    • Section S4. EE in the gapped phase
    • Fig. S1. Bipartition of an infinite cylinder, which is threaded by a flux Φ.
    • Fig. S2. Entanglement of quantum critical Dirac fermions on the honeycomb lattice.
    • Fig. S3. Entanglement in the charge-ordered phase.
    • References (3437)

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