Non-Fermi liquid induced by U(1) gauge field interactions: a functional renormalization group analysis

Chris A. Hooley; Thomas P. Sheerin

REVIEW 4 major objections 4 minor 98 references

Non-Fermi liquid induced by U(1) gauge field interactions: a functional renormalization group analysis

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A functional renormalization group calculation with gauge-symmetry constraints finds that 2D fermions coupled to a U(1) gauge field form a non-Fermi liquid with dynamical exponent $z=2$ and self-energy…

desk verdict The z=2 claim for the 2D U(1)-gauge-field NFL is cleanly derived and honestly caveated, but it rests on excluding the nonanalytic |Ω|/|q| Landau damping from the boson ansatz, so it deserves a serious referee rather than quick acceptance. read the letter →

arxiv 2411.18245 v1 pith:HO4HC5QN submitted 2024-11-27 cond-mat.str-el hep-th

Thomas P. Sheerin , Chris A. Hooley This is my paper · ORCID

classification cond-mat.str-elhep-th

keywords non-FermiliquidfunctionalrenormalizationgroupU(1)gaugefieldmodifiedWard-TakahashiidentitiesdynamicalcriticalexponentLandaudampingfermionself-energyquantumcriticality

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks what low-energy state a two-dimensional Fermi gas falls into when it interacts with a U(1) gauge field, and it answers with a functional renormalization group that respects the theory's gauge symmetry. It claims the critical point is a non-Fermi liquid with boson dynamical exponent $z=2$ and fermion self-energy scaling as $\omega^{1/2}$, replacing the long-standing $z=3$, $\omega^{2/3}$ picture. The calculation also shows that gauge-symmetry constraints, expressed as modified Ward-Takahashi identities, make the would-be gauge-boson mass irrelevant at the fixed point, which is what one expects because a gauge symmetry cannot break spontaneously. The reason to care is that a canonical model of non-Fermi liquid behavior would then have different universal exponents than commonly believed, and symmetry constraints would change the topology of the renormalization group flow.

What carries the argument

The machinery is the functional renormalization group with a soft frequency regulator $\chi(\omega,\Lambda)=\omega^2/(\omega^2+\Lambda^2)$ for the fermions, which keeps Landau damping from entering as a non-analytic $|\Omega|/|q|$ term at intermediate scales. The gauge symmetry of the regularized theory is encoded in modified Ward-Takahashi identities (mWTIs), one-loop corrected relations between the flowing Yukawa couplings, masses, and four-boson coupling; enforcing them projects the RG flow onto the gauge-symmetric submanifold. The flow equations and mWTIs are solved for seven dimensionless couplings, and the large-$N$ limit $N=k_F/k_{UV}$ is taken to restore the emergent loop-U(1) symmetry.

What would settle it

An unbiased sign-problem-free quantum Monte Carlo simulation of two-dimensional fermions at finite density coupled to a non-compact U(1) gauge field, measuring the fermion self-energy at the Fermi surface and the gauge-field spectral function, could distinguish $\Sigma(\omega)\sim\omega^{1/2}$ with $z=2$ from the standard $\Sigma(\omega)\sim\omega^{2/3}$ with $z=3$; the paper itself names quantum Monte Carlo as the decisive check.

Watch

Extended reading notes

Core claim

The central claim is that, in the large-$N$ limit with $N=k_F/k_{UV}$, the U(1)-gauge-field-induced non-Fermi liquid in $d=2$ has a fixed point with gauge-boson dynamical exponent $z_A=2$ and fermion self-energy at the Fermi surface $\Sigma(\omega,k_F)\sim\omega^{1/2}$. This is obtained with an fRG scheme using a soft frequency cutoff that lets Landau damping develop gradually and keeps the effective average action local at intermediate scales. Enforcing the modified Ward-Takahashi identities does not change most fixed-point couplings, but it does change the gauge-boson mass: without the identities the mass is a relevant operator that must be tuned, while with them it is irrelevant, correctly indicating that the ordered phase is not a spontaneous breaking of gauge symmetry.

Load-bearing premise

The calculation pins the four-fermion interaction to exactly zero throughout the flow to prevent pairing instabilities, and it truncates the effective action to a finite set of vertices; if that interaction or any omitted vertex is actually relevant at the fixed point, the true ground state would be a superconductor or a different non-Fermi liquid rather than the one described here.

Editorial extensions

If this is right

If the central claim is correct, the transverse gauge field has dynamical exponent $z_A=2$, so its frequency scales as wavevector squared, $\Omega\sim q^2$, rather than $\Omega\sim q^3$.
The fermion self-energy scales as $\omega^{1/2}$, meaning the quasiparticle weight vanishes with a different power than the standard $\omega^{2/3}$, changing predictions for spectral and transport properties of gauge-field-induced strange metals.
With the mWTIs enforced, the gauge-boson mass is irrelevant about the fixed point, so no fine-tuning is required to sit at criticality, and the ordered phase must be described by higher-form symmetry restoration rather than spontaneous gauge symmetry breaking.
The Coulomb field is screened and drops out of the critical physics, leaving the transverse vector potential as the only driver of the non-Fermi liquid behavior.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural test is to apply the same frequency-cutoff fRG scheme to closely related $q=0$ critical boson problems, such as Ising-nematic or conserved-order-parameter transitions, to see whether $z=2$ and $\omega^{1/2}$ appear there as well; the paper's internal logic suggests the soft cutoff, rather than the gauge symmetry itself, may be doing much of the work.
If the $z=2$, $\omega^{1/2}$ exponents survive an unbiased numerical calculation, the widely used Hertz-Millis-style shortcut of inserting the particle-hole bubble into the bare boson propagator would be the main source of the old $z=3$, $\omega^{2/3}$ values, and reanalysis of other gauge-field non-Fermi liquid problems would be warranted.
The constrained-flow result that the mass term is irrelevant implies the RG flow around the critical point is topologically different from a conventional ordering transition; one could look for observable consequences in settings with magnetic fields or disorder that break the relevant higher-form symmetry.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear Colleague,

The thing to know about this paper is that it claims the canonical 2D Fermi surface + U(1) gauge field has z=2 and Σ(ω) ~ ω^{1/2} at the quantum critical point, instead of the standard z=3, ω^{2/3}. It is an fRG calculation with a soft frequency cutoff, and modified Ward-Takahashi identities are imposed on the flow. The result is new relative to the Hertz-Millis literature, and the mWTI handling is careful.

The paper does several things well. The flow equations are derived explicitly, the large-N limits are taken analytically, and the authors are unusually honest: they state plainly that the truncation is uncontrolled and that a QMC study is the right next step. The fact that z_A = 2 appears in both the constrained and unconstrained flows at N → ∞ is reassuring, and the gauge-boson mass term being relevant without mWTI constraints but irrelevant with them is a nice conceptual point, even if it is largely enforced by the identities.

The central exponent claim, however, has a load-bearing soft spot. The boson self-energy ansatz in Eq. (15) contains only local terms (Ω^2 and |q|^2); the nonanalytic |Ω|/|q| Landau damping that produces z=3 in the standard treatment is excluded by hand. The paper argues such terms "ought not to develop at intermediate scales" in a consistent RG, but that is a scheme philosophy, not a proof. The fRG in this truncation cannot generate operators not in the ansatz, so the flow never tests whether the |Ω|/|q| term grows. At Λ=0 the physical particle-hole bubble is recovered, and if that term is relevant, the true exponent is z=3. This is not suppressed by N → ∞, so it directly threatens the headline result. The four-fermion coupling is also pinned to zero to avoid pairing; in the constrained N → ∞ case λ flows to zero anyway, so this is less fatal than the missing Landau damping.

In my view the calculation is internally consistent, the limitations are labelled, and the question is worth asking. But the exponent claim is not yet supported well enough to overturn the conventional picture. A serious referee should push hard on nonlocal operator generation and truncation closure.

Who benefits: people working on non-Fermi liquids, gauge-field criticality, and fRG methodology. It deserves full peer review, not desk rejection. I would bring it to a reading group specifically to argue about the |Ω|/|q| term.

Best, [You]

Referee Report

4 major / 4 minor

Summary. The paper studies a two-dimensional degenerate Fermi gas coupled to a U(1) gauge field using the functional renormalization group with a soft fermionic frequency cutoff. The authors enforce modified Ward-Takahashi identities (mWTIs) arising from the gauge symmetry and compute the RG flow of a truncated set of couplings. Their main findings are a non-Fermi-liquid fixed point with boson dynamical exponent z_A = 2 and fermion self-energy scaling Sigma(omega, k_F) ~ omega^{1/2}, values that differ from the well-known z_A = 3, Sigma ~ omega^{2/3} from Hertz-Millis/RPA treatments. They also report that the gauge-boson mass term is RG-relevant in the unconstrained flow but irrelevant when the mWTIs are enforced.

Significance. If correct, the claimed exponents would overturn a long-standing result for a canonical non-Fermi-liquid model and would demonstrate that symmetry constraints plus a frequency-cutoff fRG can access strong-coupling criticality. The manuscript has genuine strengths: the mWTI construction is explicit, the flow equations are derived in detail in the appendices, and the paper is unusually candid about the uncontrolled nature of its truncation. However, the central claim depends on a bosonic ansatz that excludes the nonanalytic Landau-damping term, and on pinning the four-fermion coupling to zero; neither step is controlled. The paper's value as a benchmark for symmetry-constrained fRG is real, but the headline exponents are not established.

major comments (4)

[Sec. IV C, Eq. (15)] The central result z_A = 2 is obtained from the boson propagator ansatz (15), which contains only the local terms (B_{A,Omega}-1)Omega^2 + (B_{A,q}-1)|q|^2. The nonanalytic |Omega|/|q| Landau-damping term, which produces z = 3 in the standard treatment and is generated by the particle-hole bubble, is excluded by hand. The paper's argument that nonanalytic operators should not develop at intermediate scales (Sec. IV C) does not establish that they remain absent at Lambda = 0, where the ordinary WTIs are recovered and the physical particle-hole bubble should reappear. Because the truncation is not closed under the flow, the beta functions (24)-(25) and the extraction of eta_{A,Omega} = 1 via Eq. (27) test only the local subspace; they provide no evidence that a singular |Omega|/|q| term is not generated. A concrete consistency check would be to include that term with its own beta function and determine whether the fixed point moves to z = 3. Until such a check is performed, the claimed exponent is an artifact of the ansatz, not a result about the model.
[Sec. IV A, Eq. (9)] The four-fermion interaction is manually pinned to zero in the effective average action (9). The justification given, that this 'prevents pairing instabilities and allows access to the critical point', is an ad hoc assumption rather than a controlled approximation. If the four-fermion coupling or higher vertices omitted from Eq. (9) are relevant at the purported fixed point, the true critical point could be a pairing instability or a different non-Fermi liquid. The constrained mWTI (23) drives the particular coupling lambda to zero at large N, but this does not control the full frequency- and momentum-dependent four-fermion vertex or the neglected higher vertices. The paper acknowledges that the truncation is 'uncontrolled' and possibly 'ill-motivated' in the Introduction, and this limitation is load-bearing for the central claim.
[Sec. VII A, Eq. (23)] The claim that the gauge-boson mass term is irrelevant when the mWTIs are enforced is largely enforced by construction. The constrained procedure fixes delta through the identities (23), so delta and g are not independent flowing couplings; the constrained flow is projected onto the mWTI surface. The comparison in Fig. 5 between unconstrained and constrained flows therefore does not provide independent dynamical evidence about the relevance of a gauge-invariant mass term; it shows that a constrained trajectory lies on a surface on which delta is tied to g and kappa. This weakens the paper's conclusion that the mWTI constraints 'correctly ensure' irrelevance and that the unconstrained relevance is 'physically inaccurate'.
[App. B and Eq. (27)] The Appendix B derivation of z_A -> 2 uses the same local truncation as the main text and relies on eta_{A,q} = 0 and eta_{A,Omega} = 1, both of which are fixed-point values obtained within the local ansatz. Even if the algebra is internally consistent, it does not address the stability of the fixed point against nonlocal terms. The comparison with the literature in Table II, while helpful, is also not probative: none of the cited z = 2 examples involve a conserved U(1) gauge field with the same coupling structure, and the paper itself notes the order parameter is not conserved in those cases. Thus the 'plausibility' argument does not compensate for the missing consistency check on the |Omega|/|q| term.

minor comments (4)

[Sec. III] There is a typo: 'neccessary' should be 'necessary'.
[Eq. (22)] The definition of N = k_F/k_UV and the claim that the results are most trustworthy at N -> infinity should be more carefully separated from the finite-N flows shown in Figs. 3-5; the figure captions could state explicitly which plots are large-N extrapolations.
[Sec. V, Eq. (19)] The mWTI for M_phi^2 is acknowledged not to vanish at Lambda -> 0, which is an artefact of the frequency cutoff. Since the paper drops the Coulomb field soon afterward, this is not fatal, but the reader should be told whether the artefact affects the constrained fixed point for the A field through the mWTIs beyond the stated 'no effect within our parametrization'.
[Appendix D] The derivation of the mWTIs is sketched rather than fully presented; in particular, the step from Eq. (D22) to the explicit forms used in Sec. V would benefit from a few more intermediate equations, especially regarding the treatment of the regulator insertions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central exponents follow from the paper's own fixed-point equations, and the mass-term statement is a consistency check of gauge symmetry rather than an input renamed as a prediction.

full rationale

The claimed NFL exponents z=2 and Σ(ω,k_F)∼ω^{1/2} are obtained by solving the coupled beta functions (24)-(25) for the fixed point in the N→∞ limit, not by inserting the result into the ansatz. The boson propagator ansatz (15) is local and omits the nonanalytic |Ω|/|q| Landau-damping term; that is a truncation assumption that bears on the validity of the result, but it is not a circularity, because the value z=2 emerges from the computed anomalous dimensions η_{A,Ω}=1, η_{A,q}=0 through Eq. (27) rather than being assumed. Similarly, the statement that the gauge-boson mass δ is irrelevant in the constrained flow is not a tautology: the mWTIs (23) define a submanifold, and the irrelevance is a property of the flow restricted to that submanifold, which must be computed; the paper explicitly finds different flow topology between constrained and unconstrained cases (Fig. 5). The only self-citation, Ref. [29] (Trott & Hooley), is used for the soft-frequency cutoff and for the practice of pinning the four-fermion interaction, but the cutoff scheme is attributed to the independent Ref. [28] (Maier & Strack), and no uniqueness theorem or central claim rests on the authors' prior work. The paper itself flags its main limitations—manual pinning of the four-fermion coupling (Sec. IV A), locality of the effective action (Sec. IV C), the instanton caveat, and N-dependence—but these are assumptions that affect physical correctness, not equations that reduce to their own inputs. Therefore there is no significant circularity.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a set of modeling axioms: truncation of the effective action, zeroed four-fermion coupling, N→∞ limit, instanton irrelevance, validity of the mWTI constraints, and symmetry assumptions. These are flagged in the paper but are load-bearing.

free parameters (1)

N = k_F/k_UV = ∞ (large-N limit taken)
The fixed-point couplings and exponents depend on N; the paper selects N→∞ as the physical limit, so the large-N choice is load-bearing for the claimed universal results.

assumptions (6)

domain assumption The truncation of the effective average action to the vertices in Eq. (9) is sufficient; higher vertices and Γ(0,3) are neglected.
Sec IV A states higher vertices are neglected for analytic tractability and Γ(0,3) is omitted by assuming a second-order transition.
ad hoc to paper The four-fermion interaction remains exactly zero throughout the flow.
Sec IV A: 'We have also manually pinned the four-Fermi interaction to zero; this is common in investigations of NFLs since it prevents pairing instabilities and allows access to the critical point.'
domain assumption Instantons do not change the low-energy fixed point, so the analysis can apply to compact U(1) gauge theory.
Sec IV A and Sec VIII note this is contentious and cite Refs [62,63].
domain assumption The N→∞ limit gives the physical critical behavior and removes cutoff artifacts.
Sec VII argues N→∞ is enforced by loop-U(1) symmetry and eliminates the k_UV dependence of the exponents.
standard math The modified Ward-Takahashi identities (Eq. 23) remain valid under truncation and can be used to fix dependent couplings.
Sec VI uses the standard approach of Refs [68,78] to enforce identities; the mWTIs follow from the master identity (7).
domain assumption Time-reversal or parity symmetry is present, forbidding a Chern-Simons term.
Sec I states this; relaxing it would give a different universality class.

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Cite this review

Pith. "Pith review of Non-Fermi liquid induced by U(1) gauge field interactions: a functional renormalization group analysis." pith.science (2026). https://pith.science/paper/HO4HC5QN

@misc{pith2026241118245,
  author       = {Pith},
  title        = {Pith review of: Non-Fermi liquid induced by U(1) gauge field interactions: a functional renormalization group analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HO4HC5QN}},
  note         = {Machine review of arXiv:2411.18245}
}

abstract

We study the non-Fermi-liquid state formed by an isotropic, degenerate Fermi gas in two spatial dimensions interacting with a U(1) gauge field. Our calculation uses the functional renormalization group (fRG) with a soft frequency cutoff for the fermions. The fRG scheme we employ takes account of the gauge symmetry, which imposes relations (modified Ward-Takahashi identities) between the couplings which constrain the RG flow. The critical exponents and couplings we find for the resulting non-Fermi liquid are mostly insensitive to whether or not we enforce the gauge symmetry constraints, which signifies either that the constraints are superfluous or that the frequency-cutoff scheme is particularly robust. The exception is the gauge-boson mass term, which is RG-relevant about the fixed point without the constraints, but is irrelevant when they are enforced. The latter is physically accurate, as a gauge symmetry cannot be spontaneously broken. In addition, we find $z=2$ and $\Sigma(\omega,k_F)\sim\omega^{1/2}$ for the boson dynamical exponent and scaling of the fermion self-energy, respectively. These results differ considerably from those of past works on the model, though we argue for their plausibility.

Figures

Figures reproduced from arXiv: 2411.18245 by the authors.

**Figure 1.** Diagrammatic modified Ward-Takahashi identities for the model. The boxed lines represent the propagator [RΛ f (k + q) − RΛ f (k)]GΛ(k)GΛ(k + q). In the third identity the diagram with the boxed fermionic propagator represents the sum of three diagrams, in each of which the box is placed on a different one of the fermionic propagator lines. 1 ξ q 2q · A(q) − qµ δΓ Λ δAµ(−q) − e Z k ψ¯(k − q) δΓ Λ δψ¯(k) + δΓ Λ δψ(k… view at source ↗

**Figure 2.** Visualization of constraining in coupling [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 4.** The fixed-point value of the four-point coupling as a function of log N (N = kF /kUV), in the unconstrained case. It is identically 0 in the constrained case. quired by restoration of gauge symmetry at the end of the flow), while it only tends to zero as N → ∞, as in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: The functional renormalization group flow (a) [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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5 2 QCP δ g (b) 2 3 4

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There, the endpoint of the constrained trajectory is more likely to ﬁnish close to the true low-energy eﬀective average action; an uncon- strained ﬂow runs the risk of being driven away from the sub-manifold by an RG-relevant coupling. VII. RESULTS In addition to solving the system in the presence of mWTI constraints, we also compute ﬂows and ﬁxed- point ...

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log N ηω ηk

that depends strongly on initial conditions. log N ηω ηk

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25 -0. 25 0 0 2 4 6 Figure 3: The fermion anomalous dimensions as functions of the non-universal parameter log N (N =kF/kUV). The solid lines are from the “constrained” analysis (i.e. with modiﬁed Ward-Takahashi identities enforced); the dashed lines are from the unconstrained analysis. It is diﬃcult to draw strong conclusions from this — the most obvious...

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4 QCP δ g Figure 5: The functional renormalization group ﬂow (a) without and (b) with modiﬁed Ward-Takahashi identities enforced. We have taken N → ∞ in (a), and N = 1 in (b) (this is to facilitate easy plotting; the ﬂow topology is insensitive to N ). The ﬂows in (a) have all the other couplings set at their ﬁxed-point values, while those in (b) have bee...

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The ﬂow equation for BA, Ω is ∂Λ BA, Ω = ˜g2 A ∂2 ∂Ω 2 ⏐ ⏐ ⏐ ⏐ q=0 [ ∂R Λ ∫ k G(k)G(k +q) sin2θ ] , (D5) where θ is the angle between k and q

Flow equations with purely bosonic or fermionic loops The ﬂows for BA, Ω, BA, q,M 2 A and ˜λ fall into this category. The ﬂow equation for BA, Ω is ∂Λ BA, Ω = ˜g2 A ∂2 ∂Ω 2 ⏐ ⏐ ⏐ ⏐ q=0 [ ∂R Λ ∫ k G(k)G(k +q) sin2θ ] , (D5) where θ is the angle between k and q. The ﬂow equation for BA, q is ∂Λ BA, q = − ˜g2 A 2 gij ∂2 ∂q i∂q j ⏐ ⏐ ⏐ ⏐ q=0 [ ∂R Λ ∫ k G(k)G(...