Classification of non-Fermi liquids and universal superconducting fluctuations

In quantum critical metals, a plethora of different non-Fermi liquids arises depending on the nature of critical fluctuations coupled to Fermi surfaces. In this paper, we classify non-Fermi liquids that arise from q=0 critical fluctuations and characterize their universal superconducting fluctuations. The essential tool is the projective fixed points, which generalizes the notion of fixed points to fixed trajectories that take into account the incessant running of the Fermi momentum under the renormalization group flow. Based on the topology of bundles of projective fixed points, non-Fermi liquids are first grouped into seven superuniversality classes. Each superuniversality class includes multiple universality classes, which are further classified by the universal pairing interactions and emergent symmetries. Despite the pairing interaction generated by critical fluctuations, some non-Fermi liquids remain stable down to zero temperature due to the incoherence of excitations and the lack of scale invariance caused by Fermi momentum. Depending on the strength and span of the universal pairing interaction in momentum space, the emergent symmetry of non-Fermi liquids may or may not be lower than that of Fermi liquids. In non-Fermi liquids that become superconductors at low temperatures, the universal data of the parent metal determine the lower bound for the superconducting transition temperature and the associated pairing symmetry. In superuniversality classes that contain non-Fermi liquids prone to non-s-wave superconducting instabilities, the critical angular momentum above which pairing instability becomes inevitable is sensitive to the Fermi momentum, and the associated superconducting transition temperature oscillates as a function of the density. We use physical examples, as well as a toy model, to elucidate the universal low-energy physics of all superuniversality classes.

💡 Research Summary

This paper tackles the long‑standing problem of classifying the myriad non‑Fermi‑liquid (NFL) metals that emerge near quantum critical points and of understanding how such metals give way to superconductivity. The authors focus on the class of quantum critical metals whose critical fluctuations carry zero momentum (q = 0) and couple to a full Fermi surface. Because the Fermi momentum kF grows without bound relative to the low‑energy cutoff μ under renormalization‑group (RG) flow, conventional scale‑invariant fixed‑point analysis fails. To overcome this, the authors introduce the notion of a “projective fixed point” (PFP): a one‑dimensional RG trajectory along which the ratio kF/μ runs inexorably to ±∞. The two asymptotic limits of a PFP are classified as stable, unstable, or marginal fixed points, and the collection of all PFPs for a given microscopic theory forms a “bundle”.

By studying the topology of these bundles, the authors find that all q = 0 NFLs fall into seven distinct “super‑universality classes”. Each super‑universality class groups together several “universality classes” (individual low‑energy phases) that share the same bundle topology but differ in the details of their pairing interactions and emergent symmetries. The classification proceeds along three axes: (i) the strength and momentum‑space extent of the universal pairing interaction generated by critical bosons, (ii) the symmetry of the leading superconducting instability (s‑wave, d‑wave, f‑wave, etc.), and (iii) any residual symmetry left after the critical fluctuations are taken into account (e.g., U(1), C2).

A central physical insight is that two opposing effects compete in every NFL: (1) incoherence of the fermionic quasiparticles, caused by strong scattering off critical bosons, which tends to suppress Cooper pairing, and (2) the same critical bosons provide an attractive “pairing glue” that promotes superconductivity. The RG analysis shows how both effects scale with kF/μ and with angular momentum ℓ. When incoherence dominates, the NFL can remain stable down to zero temperature provided the bare four‑fermion attraction is not too strong. When the glue dominates, the metal inevitably becomes superconducting, and the transition temperature Tc is set by universal data of the parent NFL rather than by microscopic details.

For non‑s‑wave channels (ℓ > 0) the authors uncover a novel density‑dependent phenomenon: the critical angular momentum ℓc above which pairing becomes unavoidable depends sensitively on kF. As the electron density (and thus kF) is varied, ℓc jumps, leading to an oscillatory dependence of Tc on kF. This predicts that in certain NFLs the superconducting dome will display non‑monotonic wiggles as a function of carrier concentration—a clear experimental signature.

The paper illustrates the classification with four concrete sets of examples. (1) A U(1) gauge field coupled to a Fermi surface yields a metallic PFP that can flow to a stable NFL, a critical NFL, or a charge‑2 superconducting fixed point, depending on the bare four‑fermion coupling. (2) A C2n Ising‑nematic quantum critical metal is analyzed in three regimes (d > dSC, d = dSC, d < dSC), producing stable NFL, s‑wave critical, and non‑s‑wave critical super‑universality classes, respectively. Complex asymptotic fixed points, quasi‑universality, and universal ratios Tc/kF are derived. (3) Hybrid theories that interpolate between the previous cases are discussed, identifying a candidate non‑s‑wave super‑universality class (B) and its proximity to class A. (4) A solvable toy model is constructed with three elementary PFPs (A, B, C). By combining them the authors explicitly realize all seven super‑universality classes (A, AB, AC, ABC, B, BC, C) and compute their RG flows, pairing interactions, and universal Tc/kF ratios. The toy model reproduces the oscillatory Tc(kF) behavior for class B and the universal constant Tc/kF for class C.

Technical details are provided in a functional RG framework that includes boson self‑energy, fermion self‑energy, cubic vertex corrections, and the full set of four‑fermion couplings (λ0, λ1, λ2). The authors demonstrate the absence of a finite Hermitian fixed point in the full metallic theory, reinforcing the necessity of the PFP approach.

In summary, the work establishes a unified language—projective fixed points and their bundles—for describing the low‑energy landscape of quantum‑critical metals. It shows that the fate of an NFL is dictated by two universal ingredients: the incoherence scale and the universal pairing interaction. The resulting classification predicts which NFLs remain metallic, which become superconducting, the symmetry of the emergent order, and quantitative bounds on Tc, including novel density‑dependent oscillations. This framework provides a powerful guide for both theorists constructing models of strange metals and experimentalists probing unconventional superconductors.