Topological Semimetals

• Insulator

  • Integer filling $\nu$

• Metal

  • Fractional number of e- per unit cell per spin
  • Fermi surface of gapless excitations
  • Can be viewed as intermediate phase between two insulators

• Topological semimetals

  • Bulk (semi)metals, which arise at integer filling $\nu$ as intermediate phase between insulators with different electronic structure topology
  • Magnetic Weyl semimetal
    • intermediate phase between ordinary and integer quantum Hall insulators in 3D
  • Non magnetic Weyl semimetal
    • intermediate phase between ordinary and 3D weak TR-invariant TI
  • Type-I Dirac ..

• Topology of Weyl semimetal

  • positive Weyl $k\cdot \sigma$ ($\sigma$ vector on Bloch sphere) $C=+1$
  • negative Weyl $-k\cdot \sigma$, $C=-1$
  • Berry curvature monopole $1/k^2$ decay away from Weyl point
    • $\Omega =$ Area on Bloch sphere/ Area in k space

• Surface Fermi arcs

  • Chern number around closed cylinder (fixed radius around point) around Weyl points

Dirac semimetals

• $N{a}_{3}Bi$

Weyl semimetals

• Separation between Weyl points, Luttinger parameter
• Spontaneous electric polarization $P=\pm V_F/8{\pi }^2$
• SOC at Kramer’s degenerate $k$ points ($k=0,\pi /a$)
• Chiral crystals = no inversion, no mirror, no rotoinversion
  • Allow for Chern number at crossing point

Domain wall network construction

• Single Weyl fermion is anomalous: it may only exist as an edge state of a 4D quantum Hall insulator and can not be localized as long as the insulator is intact
  • This implies localization may only arise from inter-nodal scattering
• Consider only Fourier component of the disorder potential coupling the nodes $\{\mathcal{K}\}$
  • Breaks translational symmetry and opens a gap
• Disorder phase of potential.
  • Vortex lines of the phase are anomalous
  • Gauge invariance fails on vortex line
    • May be cancelled by 1D chiral metal (Callan-Harvey anomaly inflow)
  • Every vortex line carries 1D chiral mode with chirality of the sign of the vortex
  • Restoration of translational symmetry requires vortex lines to percolate through the whole sample, preventing localization
• Consider Weyl semimetal with short-range disorder $<V(r)V(r^{\prime })>=\delta (r-r^{\prime })$
  • impurity self-energy (self-consistent Born approximation)
  • self-consistent equation for scattering rate $1/\tau$ from imaginary part
    • solution for some critical disorder strength, above which there is a diffusive metal (nodes destroyed)
    • Below critical strength, disorder irrelevant

Fermi arcs

• ARPES measurements of surface Fermi arcs
• Cylinder around Weyl point, surface is TI, encloses chiral charge

Electromagnetic response

• Chiral anomaly $E\cdot B$
• magnetoresistance negative for certain magnetic field angles

Current jetting

• Problem: apply magnetic field, it cannot change the total current, but affect the spatial distribution of current density
• If it makes it narrower, the current density will become bigger in the center
• Hard to rule out

Manifestation of Berry curvature

• Anomalous velocity
• Selection rule
  • circular selection rule
  • Opposite Berry curvature
  • optical excitation prefers left handed or right handed
  • Shine circularly polarized light, selective excitation on one side of Weyl point
  • Gives photocurrent
  • Opposite current generated from opposite side excitation
• Pauli Blockade + tilting
  • two sides of Weyl point are not symmetric (tilting)

Quantized photocurrent in topological chiral crystals

$$\frac{d{J}^{CPGE}}{dt}=\frac{e^3}{c{ϵ}_0{\hslash }^2}C{I}_{laser}$$

• depends on Chern number
• Only measures time derivative of current, so use laser pulses

Anomalous Hall effect

• An ideal magnetic Weyl semimetal will have a (semi) quantized AHE.
• Realistic magnetic Weyl semimetals have many Wely points (and trivial Fermi surfaces)
• In nonmagnetic Weyl semimetals, TRS makes total Berry curvature goes to zero
  • Berry curvature dipole
  • Apply E field, Fermi surface tilts and now can get finite Hall current
  • Hall conductivity is proportional to separation in k space of the Fermi surface crossings
  • nonlinear anomalous Hall in bilayer $WT{e}_2$