Composite Bogoliubov Fermi Liquids (CBFL)

• Composite Bogoliubov Fermi Liquids (CBFL) are gapless topological phases in half-filled Chern bands characterized by neutral Bogoliubov Fermi surfaces arising from superconducting pairing without inversion symmetry.
• They exhibit key experimental signatures such as incompressibility, perfectly quantized Hall response, metallic T-linear specific heat, and a distinct non-analytic q^3 structure factor.
• The theoretical framework employs parton construction and effective field theory to reveal coexisting topological ground-state degeneracy and gapless neutral modes, paving the way for identifying novel quantum Hall daughter states.

The composite Bogoliubov Fermi liquid (CBFL) is a nontrivial gapless topological phase realized in inversion-asymmetric, $C_3$-symmetric half-filled Chern bands. The CBFL emerges from a mechanism where neutral composite fermions undergo superconducting pairing, but, due to the absence of inversion symmetry, this pairing leaves behind neutral, gapless Bogoliubov Fermi surfaces (BFS). The state is distinct from the anomalous composite Fermi liquid (ACFL) and the fully gapped Pfaffian phase: it exhibits incompressibility and perfectly quantized Hall response, yet hosts gapless metallic quasiparticle excitations, non-quantized thermal conductance, Landau-damped density correlations, and a non-analytic $|\mathbf{q}|^3$ structure factor. A notable feature is the coexistence of topological ground-state degeneracy with a gapless neutral sector, signaling a fundamentally new kind of gapless topological order (Shi et al., 14 Jan 2026).

1. Model and Theoretical Framework

CBFL is constructed in a half-filled isolated Chern band, with the Hamiltonian

$H_0 = \sum_k ( \epsilon_k - \mu ) c_k^\dagger c_k$

where $\epsilon_k \neq \epsilon_{-k}$ due to broken inversion, although $C_3$ rotation remains. Projected density–density (Coulomb) interactions are included.

A parton construction is employed: the electron operator is fractionalized as $c = f\,\Phi$, where $f$ is a neutral fermionic "composite fermion" and $\Phi$ is a bosonic charge-$+1$ field. This decomposition introduces an emergent $U(1)$ gauge field $|\mathbf{q}|^3$0 to enforce the constraint. At filling $|\mathbf{q}|^3$1, placing $|\mathbf{q}|^3$2 in a bosonic $|\mathbf{q}|^3$3 Laughlin state and $|\mathbf{q}|^3$4 in a Fermi liquid yields the ACFL effective theory:

$|\mathbf{q}|^3$5

where $|\mathbf{q}|^3$6 is the Chern–Simons field for the bosonic sector. The composite fermions experience zero average internal flux at this filling.

In the presence of attractive interactions in a specific angular momentum channel $|\mathbf{q}|^3$7 (notably $|\mathbf{q}|^3$8 for $|\mathbf{q}|^3$9), composite fermions become susceptible to superconducting pairing:

$H_0 = \sum_k ( \epsilon_k - \mu ) c_k^\dagger c_k$0

Via a Hubbard–Stratonovich decoupling, the system is described by a Bogoliubov–de Gennes Hamiltonian:

$H_0 = \sum_k ( \epsilon_k - \mu ) c_k^\dagger c_k$1

with $H_0 = \sum_k ( \epsilon_k - \mu ) c_k^\dagger c_k$2, $H_0 = \sum_k ( \epsilon_k - \mu ) c_k^\dagger c_k$3. In the absence of inversion, $H_0 = \sum_k ( \epsilon_k - \mu ) c_k^\dagger c_k$4, resulting in a Bogoliubov spectrum

$H_0 = \sum_k ( \epsilon_k - \mu ) c_k^\dagger c_k$5

The condition for BFS emergence is $H_0 = \sum_k ( \epsilon_k - \mu ) c_k^\dagger c_k$6, resulting in closed pockets of gapless neutral modes.

2. Effective Field Theory and Topological Order

The effective low-energy action is derived by integrating out gapped sectors and focusing on phase fluctuations of the pairing order parameter, represented by a dynamical bosonic field $H_0 = \sum_k ( \epsilon_k - \mu ) c_k^\dagger c_k$7:

$H_0 = \sum_k ( \epsilon_k - \mu ) c_k^\dagger c_k$8

where $H_0 = \sum_k ( \epsilon_k - \mu ) c_k^\dagger c_k$9 encodes the paired composite fermion density and current responses. The full CBFL action then incorporates the Chern–Simons field of the bosonic Laughlin sector:

$\epsilon_k \neq \epsilon_{-k}$0

Physical charge response functions are obtained through the Ioffe–Larkin rule. The DC Hall resistivity is

$\epsilon_k \neq \epsilon_{-k}$1

The system is incompressible with

$\epsilon_k \neq \epsilon_{-k}$2

Despite the presence of gapless bulk Bogoliubov quasiparticles, nontrivial topological order persists. The CBFL on a torus exhibits a two-fold ground-state degeneracy, attributable to a self-anomalous $\epsilon_k \neq \epsilon_{-k}$3 1-form symmetry, enforced by the intertwined $\epsilon_k \neq \epsilon_{-k}$4 Chern–Simons topological sector (Shi et al., 14 Jan 2026).

3. Quasiparticle Spectrum and Physical Observables

The neutral Bogoliubov Fermi surfaces are closed contours in momentum space where the mean-field quasiparticle excitation gap vanishes. Their existence is protected by the lack of inversion symmetry and is robust against disorder and weak interactions within the framework discussed.

Specific Heat

Both the BFS and the massive gauge ($\epsilon_k \neq \epsilon_{-k}$5) fluctuations contribute:

• Metallic $\epsilon_k \neq \epsilon_{-k}$6-linear specific heat: $\epsilon_k \neq \epsilon_{-k}$7.
• The BFSs yield

$\epsilon_k \neq \epsilon_{-k}$8

• The $\epsilon_k \neq \epsilon_{-k}$9-field acquires a Higgs mass $C_3$0, so its gauge fluctuations contribute $C_3$1.

Thermal Transport

The gapless BFS quasiparticles are neutral but carry heat; the longitudinal thermal conductivity obeys $C_3$2, non-quantized, in contrast to the gapped Pfaffian’s quantized thermal Hall $C_3$3. The edge thermal Hall effect is non-quantized due to bulk conduction.

Dynamical Density Response

In the London limit ($C_3$4, $C_3$5), Landau damping is present:

$C_3$6

And thus, for the physical density after flux attachment:

$C_3$7

yielding $C_3$8 but with vanishing compressibility.

Equal-Time Structure Factor

CBFL exhibits a non-analytic $C_3$9 component:

$c = f\,\Phi$0

The $c = f\,\Phi$1 term is distinct from the analytic behavior of the Pfaffian and $c = f\,\Phi$2 from the CFL.

Quantum Oscillations

There are no quantum oscillations in $c = f\,\Phi$3 or $c = f\,\Phi$4 as a function of net magnetic field or doping, since effective composite fermion orbits do not close due to the momentum dependence and sign-changing of the effective charge $c = f\,\Phi$5 on the BFS:

$c = f\,\Phi$6

4. Vortices and Fractionalized Descendant States

A $c = f\,\Phi$7 vortex of the composite fermion superconductor carries $c = f\,\Phi$8 flux of the gauge field $c = f\,\Phi$9. Due to the mutual Chern–Simons term, each such vortex binds charge $f$0 of $f$1 and corresponds to a semionic anyon $f$2 in the $f$3 theory. The presence of gapless BFS renders the vortex Magnus mode overdamped.

Away from exact half-filling or with effective net magnetic field, the CBFL accommodates additional emergent quantum Hall states. At small deviations, a vortex lattice forms. Melting this lattice at larger deviations leads to gapped fractional quantum Hall (“daughter”) states. The filling fraction is determined by

$f$4

with $f$5 the paired CF fraction and $f$6, $f$7 labeling unpaired and paired sectors, respectively. The construction interpolates between the Jain states ($f$8) and $f$9 daughters ($\Phi$0).

5. Distinction from Related Phases and Experimental Hallmarks

CBFL is sharply distinguished from the ACFL, Pfaffian, and traditional CFL as follows:

Property	CBFL	ACFL/CFL	Pfaffian
Incompressibility	Yes ($\Phi$1)	No	Yes
Hall resistivity $\Phi$2	$\Phi$3	$\Phi$4	$\Phi$5
Ground state degeneracy (torus)	$\Phi$6	$\Phi$7	$\Phi$8
Bogoliubov Fermi surface	Present (neutral, gapless)	Absent	Absent
$\Phi$9 at $+1$0	$+1$1-linear (metallic)	$+1$2-linear	Exponential suppression
Quantum oscillations	Absent	Present	Absent
Structure factor $+1$3	$+1$4 non-analytic	$+1$5	Analytic
Edge/bulk thermal Hall $+1$6	Non-quantized	Non-quantized	Quantized ($+1$7)

CBFL's combination of incompressibility, perfect Hall quantization, gapless neutral surface, absence of quantum oscillations, and topological ground state degeneracy provides a diagnostic for experimentally distinguishing it from both ACFL and the Pfaffian phase (Shi et al., 14 Jan 2026).

6. Broader Significance

CBFL exemplifies a broader class of gapless topological phases realized through paired composite fermion mechanisms in lattice Chern bands beyond the conventional Landau level setting. The results provide theoretical foundations and concrete predictions for future experimental and numerical exploration of gapless topological order and its unique observable manifestations (Shi et al., 14 Jan 2026).