Zero-Field Composite Fermi Liquids

• Zero-field composite Fermi liquids are compressible metallic phases where composite fermions form emergent Fermi surfaces in the absence of an external magnetic field, achieved via flux attachment or strong interactions.
• They bridge even-denominator quantum Hall physics and Chern band phenomena, exhibiting signatures like suppressed 2kF backscattering and distinct transport responses.
• These states serve as parent phases for novel incompressible states and exotic metallic regimes, offering insights into pairing instabilities and non-Fermi-liquid behavior through numerical and spectroscopic diagnostics.

Zero-field composite Fermi liquids are compressible metallic phases in which composite fermions form an emergent Fermi surface under conditions of vanishing net field. In one lineage, familiar from even-denominator quantum Hall physics, flux attachment cancels the applied magnetic field on average so that the composite fermions experience zero effective magnetic field. In the newer zero-external-field lineage, strong interactions in partially filled $C=1$ Chern bands generate analogous metallic states without any applied magnetic field, most notably the anomalous composite Fermi liquid (ACFL) proposed for moiré systems such as twisted $\mathrm{MoTe}_2$. Across both settings, the central theme is that a non-Fermi-liquid metal with an emergent gauge field can organize nearby incompressible states, pairing instabilities, and unconventional transport (Wang et al., 2016, Goldman et al., 2023).

1. From zero effective field in Landau levels to zero external field in Chern bands

The canonical starting point is the half-filled Landau level. In the Halperin–Lee–Read picture, electrons bind two flux quanta, and at $\nu=1/2$ the average statistical field cancels the external field so that composite fermions see zero net magnetic field and can form a Fermi sea. Infinite-cylinder DMRG gave numerical evidence for an emergent Fermi surface, a particle-hole-symmetric ground state, and suppression of $2k_F$ backscattering in particle-hole-even observables, consistent with Son’s massless Dirac composite fermion description rather than the original particle-hole-asymmetric HLR construction (Geraedts et al., 2015).

A distinct but related lowest-Landau-level formulation replaces flux attachment by a neutral-vortex picture. In this framework, the composite fermion is a charge-neutral particle carrying vorticity, the effective gauge theory has no Chern-Simons term for the emergent gauge field, and the composite-fermion Fermi surface encloses a Berry phase fixed by filling,

$\phi_B=-2\pi \nu .$

This formulation was developed for generic $\nu=1/n$, with explicit discussion of fermions at $\nu=1/2q$ and bosons at $\nu=1/(2q+1)$, and it implies transport responses that differ from HLR-RPA. The same work argued that the bosonic $\nu=1$ composite Fermi liquid has an emergent particle-hole symmetry relating $\nu$ to $\mathrm{MoTe}_2$0 (Wang et al., 2016).

Microscopic wave-function constructions sharpened the particle-hole-symmetric viewpoint. On the sphere, exact particle-hole symmetry at half filling requires

$\mathrm{MoTe}_2$1

with the extra shift interpreted as a half-integer orbital spin carried by the composite fermion. Composite-electron and composite-hole wave functions can then be organized as the two components of a massless Dirac spinor (Yang, 2017). Independent microscopic calculations of pair-correlation oscillations found that the Fermi wave vector is the same at $\mathrm{MoTe}_2$2 and $\mathrm{MoTe}_2$3, while the area of the composite-fermion Fermi sea may slightly violate the naive Luttinger area rule (Balram et al., 2015).

2. Lowest-Landau-level bosons and the non-commutative composite Fermi liquid

For bosons at filling $\mathrm{MoTe}_2$4, the lowest-Landau-level problem admits a particularly explicit zero-field composite-fermion formulation. Building on the Pasquier–Haldane–Read representation, the physical and auxiliary densities $\mathrm{MoTe}_2$5 and $\mathrm{MoTe}_2$6 obey the Girvin–MacDonald–Platzman algebra associated with non-commuting guiding-center coordinates,

$\mathrm{MoTe}_2$7

The resulting low-energy theory is naturally a non-commutative field theory on a plane with

$\mathrm{MoTe}_2$8

and Lagrangian

$\mathrm{MoTe}_2$9

Here $\nu=1/2$0 is the composite-fermion field, the star product encodes non-commutativity, $\nu=1/2$1, and the density is fixed by the background field as $\nu=1/2$2. A defining feature is the absence of any Chern-Simons term for the emergent gauge field $\nu=1/2$3.

At long wavelength and small amplitude, a Seiberg–Witten map yields an approximate commutative description. The mapped theory takes the HLR form,

$\nu=1/2$4

supplemented by subleading higher-derivative corrections $\nu=1/2$5. In this regime the familiar HLR theory re-emerges, but with the effective mass set by interaction energetics rather than the bare mass; for contact interactions the paper reports $\nu=1/2$6. The same analysis states that universal long-wavelength properties such as compressibility and response to weak disorder are captured by the commutative HLR-like action, whereas shorter-wavelength effects, including $\nu=1/2$7 singularities in density correlations, require the full non-commutative theory. Hartree–Fock energetics reveal a weak pairing instability, and extensions are given to spin-$\nu=1/2$8 bosons at total filling $\nu=1/2$9 and to nearby Jain states (Dong et al., 2020).

3. Anomalous composite Fermi liquids at zero magnetic field

The direct zero-external-field realization arises in partially filled Chern bands. Motivated by experimental observations of fractional quantum anomalous Hall states in twisted bilayer $2k_F$0, exact diagonalization of the projected Coulomb Hamiltonian was used to argue for composite Fermi liquids at zero magnetic field at fillings $2k_F$1 and $2k_F$2. These states were termed anomalous composite Fermi liquids. The numerical evidence includes fully spin/valley-polarized and gapless ground states, many-body spectra and momentum structure matching those of the lowest-Landau-level composite Fermi liquid, and uniform momentum occupation inconsistent with a conventional Fermi liquid. At larger twist angles, where the bands are more dispersive, a conventional Fermi liquid appears instead (Goldman et al., 2023).

A complementary exact-diagonalization and DMRG study of twisted $2k_F$3 at half filling likewise identified a zero-field composite Fermi liquid over a large portion of the phase diagram centered around twist angle $2k_F$4. In that work, the composite-Fermi-liquid regime competes with a Fermi liquid and can be tuned by a displacement field. The topological valence band was found to have excellent quantum geometry over a wide range of twist angles and a small bandwidth that is reduced by interactions for a range of angles; these features were identified as key ingredients stabilizing the zero-field quantum Hall-like phases. The same paper proposed an optical signature based on extinguished optical responses in bands with ideal quantum geometry (Dong et al., 2023).

The long-wavelength ACFL theory at half filling is

$2k_F$5

The periodic scalar potential $2k_F$6 is intrinsic to the moiré superlattice. Flux attachment is encoded by

$2k_F$7

so that at $2k_F$8 the emergent field vanishes, $2k_F$9. A central distinction from HLR in a conventional two-dimensional electron gas is that the background “flux” originates in the Chern band and the moiré periodicity rather than in an external magnetic field.

4. Multiband and paired zero-field descendants

Zero-field composite Fermi liquids admit several generalizations beyond the single half-filled $\phi_B=-2\pi \nu .$0 band. The conjugate-composite Fermi liquid arises when two Chern bands with opposite Chern numbers are both half-filled. In the proposed cCFL, each valley or spin sector can independently realize a composite Fermi liquid, described either by Dirac composite fermions or by HLR-like variables, while time-reversal symmetry exchanges the two sectors. When an in-plane spin order develops and Higgses the spin/valley gauge field, the cCFL can enter a quantum bad metal phase with metallic character but zero-temperature longitudinal resistivity satisfying $\phi_B=-2\pi \nu .$1. In that regime the paper derives a new Wiedemann–Franz law in which thermal conductivity is proportional to electrical resistivity rather than conductivity. Inter-valley composite-fermion exciton condensation, $\phi_B=-2\pi \nu .$2, yields proximate superconducting or chiral spin liquid phases (Myerson-Jain et al., 2023).

A different instability is pairing within a single inversion-asymmetric, $\phi_B=-2\pi \nu .$3-symmetric half-filled Chern band. The resulting composite Bogoliubov Fermi liquid is incompressible and has quantized Hall conductance, but its paired composite fermions host a neutral Bogoliubov Fermi surface. The phase therefore combines zero longitudinal resistance and a Hall plateau at $\phi_B=-2\pi \nu .$4 with metallic $\phi_B=-2\pi \nu .$5-linear specific heat, non-quantized thermal conductance, Landau damping of density fluctuations, a non-analytic equal-time structure factor $\phi_B=-2\pi \nu .$6, and two-fold torus ground-state degeneracy. The same work emphasizes the absence of quantum oscillations under doping or applied magnetic field, in contrast to the compressible ACFL (Shi et al., 14 Jan 2026).

State	Setting	Distinguishing properties
ACFL	Half-filled or $\phi_B=-2\pi \nu .$7-filled $\phi_B=-2\pi \nu .$8 moiré Chern band at zero magnetic field	Compressible, gapless, Jain-parent metal, intrinsic commensurability oscillations
cCFL	Two half-filled Chern bands with opposite Chern numbers	Compressible; can become a quantum bad metal; exciton condensate gives SC or CSL
CBFL	Paired composite fermions in an inversion-broken $\phi_B=-2\pi \nu .$9 half-filled Chern band	Incompressible, quantized Hall conductance, neutral Bogoliubov Fermi surface, no quantum oscillations

5. Numerical, spectroscopic, and transport diagnostics

The principal numerical diagnostics are inherited from Landau-level studies but adapted to zero external field. In twisted $\nu=1/n$0, DMRG structure factors were used to identify the hidden composite-fermion Fermi surface, while the electron momentum distribution lacked the jump expected for a conventional Fermi liquid. The same study proposed several experimental probes: a tunneling pseudogap $\nu=1/n$1, $\nu=1/n$2 in the clean limit for the composite Fermi liquid whereas the Fermi liquid has diverging $\nu=1/n$3, strong violation of Wiedemann–Franz, and surface-acoustic-wave response with $\nu=1/n$4 for the composite Fermi liquid versus $\nu=1/n$5 for the Fermi liquid. It also pointed to near-perfect circular dichroism, with extinguished opposite-circular-polarization optical response, as a signature of ideal quantum geometry (Dong et al., 2023).

The ACFL long-wavelength theory makes additional predictions specific to moiré bands. Upon doping away from half filling, the composite fermions experience an effective field

$\nu=1/n$6

Because the periodic moiré potential is intrinsic, the theory predicts commensurability oscillations without any externally imposed superlattice. In the proposed oscillation pattern, minima in resistivity and maxima in compressibility occur when

$\nu=1/n$7

The same framework predicts zero-field Hofstadter sub-gaps accessible by density tuning and a large Hall angle $\nu=1/n$8 in the clean limit (Goldman et al., 2023).

Dirac-specific diagnostics remain conceptually important because they distinguish a particle-hole-symmetric composite Fermi liquid from older mean-field descriptions. In the half-filled Landau level, DMRG observed the suppression of $\nu=1/n$9 backscattering in particle-hole-even observables, precisely the type of signature expected from a Dirac composite fermion with a $\nu=1/2q$0 Berry phase around the Fermi surface (Geraedts et al., 2015). This suggests that, where an emergent particle-hole symmetry is operative, backscattering anomalies can carry information about the internal geometry of the composite-fermion metal.

At the level of systematic modeling, composite-fermion density functional theory provides a formalism for inhomogeneous states near even-denominator fillings. In that framework the ground-state energy is a functional of the electron density $\nu=1/2q$1 and the composite-fermion paramagnetic current density $\nu=1/2q$2, and for uniform ground states at $\nu=1/2q$3 one has $\nu=1/2q$4. The construction is designed for strong magnetic fields, but it was proposed as a route to understanding artificial structures and possible zero-field composite-fermion liquids in more general topological settings (Zhang et al., 2019).

6. Descendant phases, transitions, and conceptual boundaries

A recurring role of zero-field composite Fermi liquids is as parent states of incompressible Hall phases. For the ACFL in a moiré Chern band, doping away from half filling leads the composite fermions to fill $\nu=1/2q$5 Landau levels and produces a Jain sequence of fractional quantum anomalous Hall states with

$\nu=1/2q$6

The bosonic non-commutative composite Fermi liquid at $\nu=1/2q$7 furnishes an analogous systematic description of nearby Jain states, with non-commutativity entering through corrections to effective masses and gap estimates (Goldman et al., 2023, Dong et al., 2020).

Zero-field composite-Fermi-liquid ideas also enter descriptions of phase transitions out of metallic states. A critical theory for a continuous transition at $\nu=1/2q$8 between a Fermi liquid and an FCI$\nu=1/2q$9 phase was formulated in terms of a composite Fermi liquid of composite bosons crossing into a superfluid$\nu=1/(2q+1)$0 phase. On the Fermi-liquid side, the electronic wave function contains an additional factor $\nu=1/(2q+1)$1. The finite-temperature transport in this regime was argued to resemble an “anyon gas” atop the FCI, with $\nu=1/(2q+1)$2 close to $\nu=1/(2q+1)$3, while ordinary Fermi-liquid behavior with $\nu=1/(2q+1)$4 is recovered only at very low temperature (Zhang, 29 Jul 2025).

A separate conceptual issue concerns how much of zero-field band structure survives into composite-fermion physics. For conventional $\nu=1/(2q+1)$5 composite fermions in high field, there is generally no straightforward relation between the zero-field electron Fermi contour and the composite-fermion Fermi surface. Rotationally symmetric zero-field dispersions can produce annular or multiply connected electron Fermi seas while leaving the $\nu=1/(2q+1)$6 composite-fermion Fermi surface circular, and higher-order $\nu=1/(2q+1)$7 distortions with $\nu=1/(2q+1)$8 are transferred only weakly, with the response rapidly decreasing as $\nu=1/(2q+1)$9 increases (Ippoliti et al., 2017, Bhatt et al., 2020). This suggests caution when inferring zero-field composite-Fermi-liquid phenomenology directly from noninteracting band Fermiology alone.

Taken together, these developments place zero-field composite Fermi liquids at the center of a broad program linking lowest-Landau-level dualities, topological Chern bands, fractional quantum anomalous Hall hierarchies, bad metals, superconductivity, and gapless topological order. The common structure is not a conventional Landau quasiparticle metal, but a metal or metal-adjacent phase controlled by emergent gauge fields, Berry phases, and the topology and quantum geometry of the underlying band or Landau level.