Bosonic Analogue of Dirac Composite Fermi Liquid

We introduce a particle-hole-symmetric metallic state of bosons in a magnetic field at odd-integer filling. This state hosts composite fermions whose energy dispersion features a quadratic band touching and corresponding $2\pi$ Berry flux protected by particle-hole and discrete rotation symmetries. We also construct an alternative particle-hole symmetric state—distinct in the presence of inversion symmetry—without Berry flux. As in the Dirac composite Fermi liquid introduced by Son [Phys. Rev. X 5, 031027 (2015)], breaking particle-hole symmetry recovers the familiar Chern-Simons theory. We discuss realizations of this phase both in 2D and on bosonic topological insulator surfaces, as well as signatures in experiments and simulations.

Figure 1

Composite-fermion band structures for ${\mathrm{CFL}}_{2\pi }$ and ${\mathrm{CFL}}_0$ states with PH symmetry and varying spatial symmetries. Horizontal planes indicate the chemical potentials of interest. (a) With fourfold rotation symmetry, ${\mathrm{CFL}}_{2\pi }$ exhibits a protected quadratic band touching, unlike ${\mathrm{CFL}}_0$. (b) Breaking rotation down to inversion symmetry allows the band touching to split into two protected Dirac cones but preserves the $2\pi$ Berry flux. (c) Lifting inversion symmetry generically splits the bands, resulting in nonzero Berry curvature indicated by color (blue and red for opposite signs). The sharp distinction between ${\mathrm{CFL}}_{2\pi }$ and ${\mathrm{CFL}}_0$ then disappears. Away from degeneracy points, PH and inversion symmetries, respectively, constrain the Berry curvature $\mathcal{B}$ at momentum $\mathbf{k}$ as $\mathcal{B}(\mathbf{k})=-\mathcal{B}(-\mathbf{k})$ and $\mathcal{B}(\mathbf{k})=\mathcal{B}(-\mathbf{k})$; the presence of both symmetries [i.e., cases (a) and (b)] thus implies $\mathcal{B}=0$.

Figure 2

(a) Alternating strips of bosons at $\nu =2$ and $\nu =0$ yield an average filling $\nu =1$. Edges contain a chiral charge mode and a counterpropagating neutral mode, described by the $K$ matrix ${\sigma }^x$. (b) The same edge-state network arises on the surface of a 3D bosonic SPT with $U(1)\times \mathcal{C}$ symmetry when the antiunitary PH symmetry $\mathcal{C}$ is broken oppositely for neighboring strips.

Figure 3

(a) Massless Dirac band structure for “pure” $N_f=2$ ${\mathrm{QED}}_3$ that is dual to the bosonic models studied here (shown with different velocities to emphasize $N_f=2$). The magnetic field for the bosons maps to a nonzero composite-fermion density that yields two Fermi surfaces. (b) PH and rotation-symmetric perturbations with $|g|>|\mathrm{\Delta }|$ yield ${\mathrm{CFL}}_{2\pi }$ with a quadratic band touching and a single Fermi surface. (c) The transition between ${\mathrm{CFL}}_{2\pi }$ and ${\mathrm{CFL}}_0$ occurs at $|g|=|\mathrm{\Delta }|$ where three bands meet at $k=0$. (d) For $|g|<|\mathrm{\Delta }|$, one obtains ${\mathrm{CFL}}_0$ featuring separated upper and lower bands and a single Fermi surface at an appropriate density. Note that the nature of the partially filled positive-energy band changes qualitatively, with the Berry flux jumping from $2\pi$ to 0.

Figure 4

Both ${\mathrm{CFL}}_{2\pi }$ and ${\mathrm{CFL}}_0$ exhibit vanishing composite-fermion Hall conductance ${\tilde{\sigma }}_{xy}=0$ due to PH symmetry $\mathcal{C}$. Upon breaking $\mathcal{C}$ by the term $m{\mathrm{\Psi }}_2^{†}{\sigma }^z{\mathrm{\Psi }}_2$, ${\tilde{\sigma }}_{xy}=0$ crosses over to the HLR value ${\tilde{\sigma }}_{xy}=1$ (center). For $\mathrm{\Delta }/g<1$, the ${\mathrm{CFL}}_{2\pi }$ band touching is lifted by $g$, resulting in bands with opposite Berry curvature (left). For $\mathrm{\Delta }/g>1$, the partially filled ${\mathrm{CFL}}_0$ band develops Berry curvature as a function of $m$ and changes sign between small and large momenta (right).