Universal electrical transport of composite Fermi liquid to Metal transition in Moiré systems

We compute universal electrical transport near continuous transitions between a composite Fermi liquid (CFL) and a metallic phase in moire Chern bands, focusing on fillings $ν=-1/2$ and $ν=-3/4$. The critical theory represents a novel QED-Chern-Simons framework: a charged sector at a bosonic Laughlin-superfluid critical point is coupled, via emergent gauge fields and Chern-Simons mixing, to a neutral spinon Fermi surface. Integrating out matter fields to quadratic order yields an explicit Ioffe-Larkin composition rule for the full resistivity tensor, showing how longitudinal channels add in series while Chern-Simons terms generate Hall response. To obtain the DC limit in the quantum critical fan, we develop a controlled large-N expansion where both fermion flavors and Chern-Simons levels scale with $N$, and solve a quantum Boltzmann equation at leading nontrivial order $1/N$. Gauge-mediated inelastic scattering removes the collisionless Drude singularity and produces a universal scaling function $Σ(ω/T)$ and finite DC conductivities $σ(0) \approx 0.033 e^2/\hbar$ ($ν=-1/2$) and $0.047 e^2/\hbar$ ($ν=-3/4$). We also identify a Chern-Simons “filtering” mechanism that suppresses transmission of Landau damping from the spinon Fermi surface to the critical gauge mode. Our approach provides concrete transport diagnostics for detecting quantum criticality in moire superlattices.

💡 Research Summary

This paper presents a comprehensive theoretical study of the universal electrical transport properties near continuous phase transitions between a composite Fermi liquid (CFL) and a conventional metallic (Fermi‑liquid) phase in moiré Chern‑band systems, focusing on the experimentally relevant fillings ν = −1/2 and ν = −3/4. The authors build on the parton construction in which the physical electron c is decomposed into a charged bosonic sector Φ and a neutral spinon f, coupled by an emergent internal U(1) gauge field aμ. The charged sector undergoes a bosonic Laughlin–superfluid transition at effective filling νΦ = 1/2 (for ν = −1/2) or νΦ = 1/4 (for ν = −3/4). This transition is described by Dirac fermions ψ with a tunable mass M; changing the sign of M switches the induced Chern‑Simons response, thereby interpolating between a Laughlin‑state of Φ (the CFL) and a Higgs‑condensed Φ (the metallic phase). The neutral spinons form a Fermi surface and couple minimally to aμ. An additional internal gauge field bμ couples to aμ and the external electromagnetic field Aμ through a Chern‑Simons (CS) mixing term (b + a + A)∧(b + a + A) − b∧b.

Integrating out the matter fields to quadratic order yields a Gaussian gauge theory whose effective action contains the polarization tensors Πψ (from ψ), Πf (from the spinon Fermi surface), and the CS kernel ΠCS. Solving the resulting equations of motion for the internal gauge fields leads to an Ioffe‑Larkin composition rule for the full electromagnetic response:

(Πphys)⁻¹ = ΠCS⁻¹ + Πf⁻¹ + (Πψ − ΠCS)⁻¹.

In tensor language this means that longitudinal resistivities of the separate sectors add in series, while the antisymmetric Hall component is generated solely by the CS term. This rule reduces the transport problem to the calculation of the individual sector responses.

A crucial subtlety is that the DC conductivity cannot be obtained by a naïve ω→0 limit of the zero‑temperature Kubo formula because the limits ω→0 and T→0 do not commute. At N = ∞ (the large‑N limit with infinite flavor number) the ψ and f excitations are collisionless, producing an unphysical Drude δ(ω) peak. Physical DC transport therefore requires inclusion of inelastic scattering at order 1/N. To achieve a controlled expansion, the authors promote both the number of Dirac flavors and the CS levels to scale with a large parameter N, keeping the CS mixing at the same order as matter‑induced polarizations.

At leading non‑trivial order in 1/N the authors formulate a quantum Boltzmann equation (QBE) for the distribution function of the critical Dirac quasiparticles in the presence of a weak, uniform electric field of frequency ω. The collision integral is evaluated using the equilibrium spectral function of the emergent gauge mode b, which encodes Landau‑damping effects. The resulting transport relaxation rate scales as

1/τtr ∝ (T/N) × F(ω/T),

where F is a universal scaling function. In the ω/T → 0 limit the DC conductivity becomes finite and universal. Explicit solution of the QBE yields

σ(0) ≈ 0.033 e²/ħ for ν = −1/2,

σ(0) ≈ 0.047 e²/ħ for ν = −3/4.

These numbers are independent of microscopic details and constitute universal signatures of the CFL–FL quantum critical point.

An additional key insight is the “Chern‑Simons filtering” mechanism. In ordinary U(1) gauge theories, the spinon Fermi surface generates strong Landau damping of the gauge field, which would in turn damp the critical ψ sector. However, because b couples to a only through a CS term, the CS propagator is purely antisymmetric and proportional to momentum, vanishing in the infrared. Consequently, the damping from the spinon sector cannot be transmitted efficiently to b, and ψ experiences only the intrinsic critical fluctuations. This kinematic suppression protects the universal transport coefficients and leads to a sharp jump in the resistivity tensor at the critical point, as illustrated in the paper’s figures.

The manuscript is organized as follows: Section II reviews the critical theory for ν = −1/2, derives the Ioffe‑Larkin rule, and discusses the expected scaling of the resistivity tensor. Section III introduces the large‑N framework and presents the effective gauge theory after integrating out matter fields. Section IV solves the QBE, extracts the universal scaling function Σ(ω/T), and computes the DC conductivities. Section V repeats the analysis for the more intricate ν = −3/4 case, confirming the same universal structure. The concluding section emphasizes that the predicted universal conductivities and the Chern‑Simons filtering effect provide concrete, experimentally accessible diagnostics for quantum criticality in moiré superlattices, bridging the gap between sophisticated field‑theoretic models and transport measurements.