Composite Bogoliubov Fermi liquid in a half-filled Chern band

TL;DR

This work introduces the composite Bogoliubov Fermi liquid (CBFL), a gapless topological phase at half filling of inversion-asymmetric Chern bands with $C_3$ symmetry, where composite fermions pair in a $p+ip$ channel to form neutral Bogoliubov Fermi surfaces while the charge sector remains gapped. Using a quantum Ginzburg-Landau framework with a Chern-Simons gauge structure, the CBFL exhibits incompressibility and a quantized Hall conductance despite gapless neutral excitations, alongside Landau-damped density fluctuations, a linear-in-$T$ specific heat, a nonanalytic $S(\mathbf{q}) \sim |\mathbf{q}|^3$, and a two-fold torus ground-state degeneracy arising from emergent 1-form symmetry. The paper also develops the topological and dynamical properties of CBFL, including the Ioffe-Larkin composition for transport, the role of a Higgsed gauge field, and the monopole-based electron operator structure, and investigates how doping and external fields produce CF daughter states with Jain-like or $U(1)_8$-type topologies. Overall, the CBFL broadens the landscape of gapless topological phases beyond Landau-level physics and motivates future experimental and theoretical exploration of lattice Chern bands.

Abstract

The composite Fermi liquid (CFL) in the half-filled Landau level is a cornerstone of the quantum Hall phase diagram. Recent experiments and numerics indicate that an anomalous composite Fermi liquid (ACFL) can also arise at half filling of a Chern band without any external magnetic field, opening new possibilities for paired states of composite fermions beyond the fully gapped Pfaffian phase. We argue that in inversion-asymmetric Chern bands with lattice rotational symmetry reduced to $C_3$, as realized in experimental platforms where signatures of the ACFL have been observed, composite fermions can form a superconductor with neutral gapless Bogoliubov Fermi surfaces. We term the resulting electronic state {\it the composite Bogoliubov Fermi liquid (CBFL)}. This phase has a number of remarkable properties that make it distinct from both the ACFL and the fully gapped Pfaffian. For instance, it is incompressible, has quantized Hall conductance, shows no quantum oscillations as a function of magnetic field or doping, and has topological ground state degeneracy on a torus despite the presence of gapless quasiparticles. At the same time, the neutral Bogoliubov Fermi surface yields metallic $T$-linear specific heat, non-quantized thermal conductance, Landau damping of density fluctuations, and a non-analytic $|\mathbf{q}|^3$ contribution to the equal-time structure factor $S(\mathbf{q})$. We also briefly discuss vortex physics and possible fractionalized daughter states induced by doping or external magnetic fields. Our results pave the way for a broader understanding of gapless topological phases arising from paired composite fermions in Chern bands that go beyond the conventional Landau level paradigm.

Figures (4)

• Figure 1: Illustrative solution of the self-consistent mean-field equation for the chiral $p+ip$ SC order parameter with the composite fermion (CF) dispersion $\epsilon_f(\bm{k})=\frac{k^2}{2m}(1+\lambda \cos 3\theta_{\bm{k}})$ and attractive interactions in the $l=1$ channel (see SM Sec. \ref{['subapp:gapequation']} for details). a) The CF Fermi surface in a normal state is shown in black. The resulting Bogoliubov Fermi surface at $T=0$ is shown in red. b) Schematic momentum dependence of the quadratic part of the Ginzburg-Landau action at $T=T_c$. The red cross denotes the global minimum at $\boldsymbol{q}=0$.
• Figure 2: Illustration of the zero-temperature pairing amplitude $|\ev{f_{-\boldsymbol{k}} f_{\boldsymbol{k}}}|$ for $\Delta(\boldsymbol{k}) = 1.525 e^{i\theta_{\boldsymbol{k}}}, \;\lambda = 0.2,\; \mu = 10, \;m = 1$.
• Figure 3: Illustration of various angular-dependent quantities parameterizing the BFS. Sectors indicated by the black dashed line are determined from the condition in Eq. \ref{['eq:cond_1']}. Blue (red) solid lines show the value of $k_{F,+}(\theta)/\sqrt{2m\mu}$ ($k_{F,-}(\theta)/\sqrt{2m\mu}$). From this, we can see that $k_{F,+}(\theta)$ parameterizes the outer segment of the BFS pockets, whereas $k_{F,-}(\theta)$ parameterizes the inner segments. Yellow lines indicate the region allowed by the inequality in Eq. \ref{['eq:cond3']}, whereas the green lines show the region allowed by Eq. \ref{['eq:cond2']}. Parameters used: $\bar{\Delta}=0.1$, $\lambda=0.2$, $\sqrt{2m\mu}=1$.
• Figure 4: Left panel: angular dependence of $\Gamma_{\hat{\bm{q}}}\equiv\Gamma_{\hat{\bm{q}}}(\alpha\rightarrow0^+)$ in units of $m^{3/2}/\mu^{1/2}$, defined in Eq. \ref{['eq:def-Gamma']} (and also given more explicitly for our illustrative model in Eq. \ref{['eq:Gamma_explicit']}). Reft panel: angular dependence of $\Gamma^{TT}_{\hat{\bm{q}}}\equiv\Gamma_{\hat{\bm{q}}}^{TT}(\alpha\rightarrow0^+)$ in units of $\sqrt{m\mu}$, defined in Eq. \ref{['eq:Gamma_IJ']}. Parameters: $\lambda=0.2$, and $\Delta/\mu=0.1525$.