Research ArticlePHYSICS

Evidence of pair-density wave in spin-valley locked systems

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Science Advances  29 Mar 2019:
Vol. 5, no. 3, eaat4698
DOI: 10.1126/sciadv.aat4698
  • Fig. 1 Model and Fermi surface.

    (A) The spin-dependent staggered flux pattern for one-spin component with ± Φ flux per plaquette. An opposite flux pattern for the other spin component guarantees time-reversal symmetry. The arrows indicate the direction of positive phase hopping. (B) Our Fermi surface with ti+x^,i;=23ei0.3π and μ = 4.6 in the tight-binding model in Eqs. 1 and 2. Here, the spin-valley locked, circular Fermi pockets are evident.

  • Fig. 2 Evidence of PDW oscillations.

    (A) Arg (Δijsinglet) for all nearest-neighbors with U = +2 for our 3 by 36 lattice with periodic boundary conditions along the short direction and open boundary conditions along the long direction. For visibility, we truncate the plot so that only the third farthest from the edge field is shown. The line thickness is proportional to the pairing amplitude. (B) We plot the real and imaginary components of Δijsinglet and Δijtriplet for i,j along the middle rung of our lattice to present the phase oscillations.

  • Fig. 3 Fourier decomposition of the PDW and bond charge order.

    (A) Fourier transforms of the PDW and charge bond order. Zero momentum, i.e., constant contributions and decay effects have been removed. (B) Depiction of pairing in momentum space. The regions demarcated by dashed lines are the approximate pairing regions.

  • Fig. 4 Lattice and edge field.

    A depiction of our lattice. It is periodic in the short direction with three unit cells and has open boundaries in the long direction. The ellipses on the right signify that multiple lengths are studied: L = 12, 18, 24, 36. The edge field, shown as red lines, is a pair field of the form given in Eq. 4. The nearest-neighbor hopping structure for spin up is also shown with the spin down hopping structure being the complex conjugate of that shown above.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/3/eaat4698/DC1

    Section S1. Phase structure near the pair field

    Section S2. DMRG convergence

    Section S3. Phase structure for attractive interactions

    Section S4. Triplet phase structure for repulsive interactions

    Section S5. Effects of chemical potential on the PDW

    Fig. S1. Singlet phase structure near the edge field.

    Fig. S2. DMRG convergence.

    Fig. S3. Phase structure for attractive interactions.

    Fig. S4. Amplitude for attractive interactions.

    Fig. S5. Triplet phase structure for repulsive interactions.

    Fig. S6. Effect of chemical potential on the PDW phase structure.

    Fig. S7. Effect of chemical potential on the PDW dominant Fourier mode.

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Phase structure near the pair field
    • Section S2. DMRG convergence
    • Section S3. Phase structure for attractive interactions
    • Section S4. Triplet phase structure for repulsive interactions
    • Section S5. Effects of chemical potential on the PDW
    • Fig. S1. Singlet phase structure near the edge field.
    • Fig. S2. DMRG convergence.
    • Fig. S3. Phase structure for attractive interactions.
    • Fig. S4. Amplitude for attractive interactions.
    • Fig. S5. Triplet phase structure for repulsive interactions.
    • Fig. S6. Effect of chemical potential on the PDW phase structure.
    • Fig. S7. Effect of chemical potential on the PDW dominant Fourier mode.

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