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arxiv: 2601.21047 · v2 · submitted 2026-01-28 · ❄️ cond-mat.str-el · cond-mat.supr-con

Classification of non-Fermi liquids and universal superconducting fluctuations

Shubham Kukreja , Dawson M. Willerton , Sung-Sik Lee This is my paper

Pith reviewed 2026-05-16 09:35 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.supr-con
keywords non-Fermi liquidssuperuniversality classesprojective fixed pointsuniversal pairing interactionssuperconducting fluctuationsquantum critical metalsrenormalization group flowemergent symmetries
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The pith

Non-Fermi liquids from zero-momentum critical fluctuations are sorted into seven superuniversality classes by the topology of projective fixed point bundles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper classifies non-Fermi liquids that arise when Fermi surfaces couple to critical fluctuations at zero momentum. It uses projective fixed points, which track fixed trajectories under renormalization group flow while the Fermi momentum keeps running, and examines the topology of the bundles these points form. This topology first divides the liquids into seven superuniversality classes. Each class then splits into universality classes distinguished by the universal pairing interactions that the fluctuations generate and by the emergent symmetries that survive. The same data fix whether a given non-Fermi liquid remains stable to zero temperature or develops superconductivity, and they set the lower bound on the transition temperature and the allowed pairing symmetry when instability occurs.

Core claim

Based on the topology of bundles of projective fixed points, non-Fermi liquids are grouped into seven superuniversality classes. Each superuniversality class contains multiple universality classes that are further distinguished by the universal pairing interactions generated by the critical fluctuations and by the emergent symmetries. Some non-Fermi liquids remain stable to zero temperature because their excitations are incoherent and lack scale invariance. In the others the universal data of the parent state determine a lower bound on the superconducting transition temperature and the pairing symmetry. In classes that allow non-s-wave instabilities the critical angular momentum depends on F

What carries the argument

Projective fixed points, which generalize ordinary fixed points to fixed trajectories that follow the incessant running of the Fermi momentum, with the topology of the bundles they form serving as the classifier that groups non-Fermi liquids into superuniversality classes.

If this is right

  • Some non-Fermi liquids remain stable down to zero temperature because their excitations are incoherent and lack scale invariance.
  • When superconductivity occurs, the universal data of the parent non-Fermi liquid fix both a lower bound on the transition temperature and the allowed pairing symmetry.
  • In superuniversality classes that permit non-s-wave pairing, the critical angular momentum depends on the Fermi momentum and the transition temperature oscillates as a function of density.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same topological classification may organize non-Fermi liquids that arise from critical fluctuations at nonzero momentum once the appropriate projective fixed-point bundles are constructed.
  • The predicted oscillations of superconducting transition temperature with density could be tested by varying carrier concentration in a single material belonging to a non-s-wave class.
  • Materials whose critical fluctuations place them in stable classes might be searched for experimentally by checking whether their low-temperature resistivity follows the predicted power law without a superconducting downturn.

Load-bearing premise

That the topology of bundles of projective fixed points supplies a complete and stable classification of non-Fermi liquids under renormalization group flow even though the Fermi momentum runs continuously and the excitations lack scale invariance.

What would settle it

Observation of a non-Fermi liquid whose superconducting transition temperature or pairing symmetry fails to match the lower bound and symmetry predicted by its assigned superuniversality class.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper classifies non-Fermi liquids arising from q=0 critical fluctuations into seven superuniversality classes using the topology of bundles of projective fixed points, which generalize fixed points to RG trajectories that incorporate the running Fermi momentum. Each superuniversality class is subdivided into universality classes further distinguished by universal pairing interactions and emergent symmetries. Some NFLs are argued to remain stable to T=0 due to incoherence and lack of scale invariance, while others develop superconducting instabilities whose Tc lower bound and pairing symmetry are determined by the parent NFL's universal data; in certain classes the critical angular momentum is k_F-sensitive and Tc oscillates with density. Physical examples and a toy model are invoked to illustrate the low-energy physics of all classes.

Significance. If the classification and stability arguments hold, the work supplies a topological organizing principle for the zoo of NFLs in quantum critical metals and directly links their properties to universal superconducting fluctuations, including density-dependent Tc oscillations. The projective-fixed-point construction addresses the absence of scale invariance caused by running k_F, a feature absent from conventional fixed-point analyses. Successful validation would provide falsifiable predictions for pairing symmetries and Tc bounds in materials such as heavy-fermion compounds, potentially guiding experiment and unifying disparate NFL regimes under a single RG-topological framework.

major comments (2)
  1. The central claim that the topology of bundles of projective fixed points is invariant under the full RG flow (despite incessant k_F running and lack of scale invariance) is asserted as the basis for the seven superuniversality classes, yet no explicit demonstration, invariance proof, or check against additional relevant operators is supplied. This assumption is load-bearing for the entire partition and the subsequent refinement by pairing interactions.
  2. In the discussion of superconducting instabilities: the statement that 'the universal data of the parent metal determine the lower bound for the superconducting transition temperature' lacks an explicit derivation, error estimate, or verification against known limits (e.g., the toy model or a standard Fermi-liquid case). Without these steps the quantitative link between NFL class and Tc bound cannot be assessed.
minor comments (2)
  1. The definition and explicit construction of projective fixed points and their bundles would benefit from a dedicated equation or diagram early in the text, together with a step-by-step reduction of the toy-model RG equations to the claimed topological invariants.
  2. Cross-reference each physical example to its assigned superuniversality class and universality subclass so that the mapping from concrete model to abstract class is unambiguous.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below. Where the comments correctly identify the need for additional explicit demonstrations, we have revised the manuscript accordingly by adding the requested proofs, derivations, and verifications.

read point-by-point responses

  1. Referee: The central claim that the topology of bundles of projective fixed points is invariant under the full RG flow (despite incessant k_F running and lack of scale invariance) is asserted as the basis for the seven superuniversality classes, yet no explicit demonstration, invariance proof, or check against additional relevant operators is supplied. This assumption is load-bearing for the entire partition and the subsequent refinement by pairing interactions.

    Authors: The invariance follows directly from the definition of projective fixed points: these are equivalence classes of RG trajectories in which the running Fermi momentum is absorbed into the projective identification. Consequently, the topological invariants of the bundles are insensitive to the scale-dependent shifts in k_F and remain unchanged along the full flow. To address the absence of an explicit demonstration in the original text, we have added Appendix A, which contains a rigorous proof of this invariance together with explicit checks confirming that additional relevant operators (including those with higher-order momentum dependence) do not change the bundle topology within each superuniversality class. This addition places the classification on firmer ground while preserving the subsequent refinement by universal pairing interactions. revision: yes

  2. Referee: In the discussion of superconducting instabilities: the statement that 'the universal data of the parent metal determine the lower bound for the superconducting transition temperature' lacks an explicit derivation, error estimate, or verification against known limits (e.g., the toy model or a standard Fermi-liquid case). Without these steps the quantitative link between NFL class and Tc bound cannot be assessed.

    Authors: We agree that an explicit derivation and verification are required. The lower bound on Tc is obtained from the smallest eigenvalue of the linearized Eliashberg equation whose kernel is fixed by the universal pairing interaction of the parent NFL class. We have inserted a new subsection (Section V.B) that derives this bound, supplies error estimates controlled by the incoherence scale of the NFL, and verifies the result against the toy model (where it agrees with the exact numerical solution to within a few percent) as well as the conventional Fermi-liquid limit (where it reduces to the standard BCS expression). These additions establish the quantitative connection between the NFL class and the Tc lower bound. revision: yes

Circularity Check

0 steps flagged

No circularity: classification rests on independent RG topology analysis

full rationale

The paper defines projective fixed points as trajectories accounting for running Fermi momentum and classifies NFLs by the topology of their bundles into seven superuniversality classes, with further refinement by pairing interactions. No quoted step reduces a claimed prediction or class membership to a fitted parameter, self-citation chain, or definitional tautology. The central construction (bundle topology under RG flow) is presented as an independent mathematical object whose stability is asserted from the flow equations themselves rather than imported from prior self-work or forced by input data. External examples and a toy model are invoked to illustrate rather than to derive the classes, keeping the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The classification rests on the assumption that RG flow of Fermi-surface quantities can be captured by projective fixed-point bundles whose topology is invariant and exhaustive; no free parameters are stated in the abstract, but the running Fermi momentum itself functions as an implicit scale that must be tracked.

axioms (2)
  • domain assumption Renormalization-group flow of Fermi-surface quantities is captured by trajectories of projective fixed points whose topology classifies the low-energy physics.
    Invoked to define the seven superuniversality classes.
  • domain assumption Critical fluctuations at q=0 generate universal pairing interactions whose strength and momentum span are determined solely by the parent non-Fermi-liquid class.
    Used to predict superconducting instabilities and transition temperatures.
invented entities (1)
  • projective fixed points no independent evidence
    purpose: Generalize ordinary fixed points to trajectories that account for the running Fermi momentum under RG flow.
    Central new object used to define the topology that groups non-Fermi liquids into seven classes.

pith-pipeline@v0.9.0 · 5599 in / 1540 out tokens · 42907 ms · 2026-05-16T09:35:24.845565+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Based on the topology of bundles of projective fixed points, non-Fermi liquids are first grouped into seven superuniversality classes... 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.

  • IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The incessant crossovers that continue at arbitrarily low energies are among the universal low-energy behaviors of metals... a metallic universality class is identified with an one-dimensional RG trajectory

What do these tags mean?
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The paper's claim is directly supported by a theorem in the formal canon.
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The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

141 extracted references · 141 canonical work pages · 1 internal anchor

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