Charge and pair density waves in a spin and valley-polarized system at a Van-Hove singularity

Avigail Gil; Erez Berg

REVIEW 2 major objections 4 minor 1 cited by

Charge and pair density waves in a spin and valley-polarized system at a Van-Hove singularity

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper shows that a fully spin- and valley-polarized electron gas at a Van-Hove singularity develops pair density wave order when its effective interactions are attractive, while repulsive interactions leave it a stable metal.

desk verdict A promising weak-coupling PDW mechanism that is currently undermined by a sign error in the printed one-loop beta function—likely a typo, but load-bearing. read the letter →

arxiv 2504.19321 v1 pith:6MKDPFQA submitted 2025-04-27 cond-mat.str-el

Avigail Gil , Erez Berg This is my paper · ORCID

classification cond-mat.str-el

keywords pairdensitywavechargeVanHovesingularityspinandvalleypolarizationrenormalizationgrouprhombohedralmultilayergrapheneC3vsymmetryfractionalvortices

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

The paper studies a two-dimensional electron gas that is fully spin- and valley-polarized and tuned so that the Fermi level sits at a Van-Hove singularity, where the density of states diverges logarithmically. Its central claim is that the fate of the system is governed by a single effective interaction between symmetry-related saddle points, together with the mass anisotropy of the dispersion. For repulsive interpatch interactions the system remains a stable metal even at the singularity. For attractive interactions it develops either a charge density wave or a pair density wave, a superconducting state whose Cooper pairs carry finite momentum. The three-wavevector pair density wave supports vortices carrying one-sixth of the superconducting flux quantum, a concrete signature that experiments could look for.

What carries the argument

The central object is the Van-Hove patch model: a low-energy theory keeping only fermion fields near the three or six symmetry-related saddle points, where the density of states diverges logarithmically. The heavy lifting is done by a one-loop frequency-shell renormalization group, in which both the BCS (particle-particle) and ZS' (particle-hole) diagrams contribute to a flow coefficient $a(\eta)$ or $A(\eta,\phi)$ that is positive everywhere and diverges at $\eta=1/3$. The mass-anisotropy parameter $\eta$ and the principal-axis angle $\phi$ enter only through these flow coefficients, and the test-vertex flows then determine whether CDW or PDW is the leading instability.

What would settle it

Measure the spin and valley polarization of the normal state in rhombohedral multilayer graphene at the Van-Hove density, for example by quantum oscillation or compressibility probes: seeing Fermi surfaces from both valleys or both spin species would invalidate the single-flavor model's phase diagram. Alternatively, in a candidate 3q PDW state, magnetometry that counts only $h/2e$ vortices per applied flux quantum would contradict the predicted $h/6e$ vortices.

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Extended reading notes

Core claim

The paper's central claim is that in a fully spin- and valley-polarized $C_{3v}$-symmetric two-dimensional metal at a Van-Hove singularity, the weak-coupling physics is governed by the sign and geometry of the interpatch interaction. In the three-VH-point model the one-loop flow of the single coupling is $\dot g=-a(\eta)g^2$ with $a(\eta)>0$, so repulsive $g$ flows to zero and the metal is stable, while attractive $g$ diverges at the energy scale $E_c=\Lambda_0\exp(-1/t_c)$. Tracking test vertices gives susceptibility exponents $\alpha_{\text{CDW}}=-a_{ZS'}(\eta)/a(\eta)$ and $\alpha_{\text{PDW}}=-2a_{BCS}(\eta)/a(\eta)$, producing a crossover at $\eta_c\approx 0.157$ between PDW-dominated ($\eta<\eta_c$) and CDW-dominated ($\eta>\eta_c$) regions. The six-VH-point model adds a second geometric parameter, the angle $\phi$ between principal axes of mirror-related saddle points, and yields a phase diagram in $(\eta,\phi)$ in which CDW fills most of the plane and PDW appears at large mass anisotropy and small angles around $\phi=n\pi/3$. A three-component pair density wave, selected near the transition, breaks translation symmetry and supports $h/6e$ vortices bound to lattice dislocations.

Load-bearing premise

The whole calculation assumes that a fully spin- and valley-polarized normal state already exists; if the actual normal state is not fully polarized, the single-flavor patch model and its predicted phase boundaries do not describe the system.

Editorial extensions

If this is right

A fully polarized normal state with net attractive inter-VH interactions will not remain a Fermi liquid; it develops an ordered state at a parametrically low energy scale $E_c$.
Which order wins is controlled by saddle-point geometry: the mass-anisotropy ratio $\eta$, and for six VH points also the angle $\phi$, select between CDW and PDW, so tuning a displacement field or strain could switch the ground state.
The $3q$ PDW state is a translation-symmetry-breaking superconductor with an accompanying $3q$ CDW, and its elementary defects carry flux $h/6e$ rather than $h/2e$, giving a sharp experimental fingerprint.
In the weak-coupling limit, PDW and CDW orders leave gapless fermionic excitations; PDW states exhibit Bogoliubov Fermi surfaces, so the low-energy spectrum is not fully gapped.
Repulsive interactions are marginally irrelevant even at the diverging density of states, so no Stoner-type or nematic instability appears in this single-flavor model.

Reading between the lines

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

If the polarized normal state is confirmed in rhombohedral multilayer graphene, this mechanism suggests the observed superconductivity is a PDW selected by the specific $\eta$ and $\phi$ of the band structure; tuning the displacement field across the VH point should move the system along the CDW/PDW boundary.
Because the RG treats particle-particle and particle-hole channels on equal footing here, the same single-flavor VH setup could serve as a controlled theoretical laboratory for competing PDW and CDW order in other fully valley-polarized moiré systems, not only rhombohedral graphene.
The predicted $h/6e$ vortices provide a direct test: counting vortices versus applied magnetic field in a SQUID scan, or searching for zero-field dislocation-bound fractional vortices, could distinguish a $3q$ PDW from a conventional $h/2e$ superconductor.
The RPA estimate that an effective interpatch attraction can emerge from repulsive Coulomb interactions depends on wavefunction overlap factors near the VH points; a more quantitative calculation of that overlap would sharpen the experimental regime where the predicted instability applies.

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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

Colleague,

The stress-test note holds up. The one-loop coefficients in Eqs. (B1) and (B2) as printed give a(η) < 0 for 0<η<1/3 and non-real values for η>1/3. That is the opposite of what the text needs for the central claim, and it reverses the stability/instability dichotomy in Eq. (5). This is almost certainly a sign typo—one stray minus sign in each coefficient—but it is load-bearing. The paper as submitted cannot be accepted without the algebra being redone and the conclusion re-derived.

That said, the idea itself is worth taking seriously. The authors isolate a single-flavor (spin and valley polarized) C3v VH system, where both particle-particle and particle-hole susceptibilities diverge only logarithmically, so the weak-coupling RG is on better footing than the spinful VH studies that have to contend with log-squared terms. The three-patch and six-patch analyses are clear, the GL phase diagrams (though qualitative) organize the CDW/PDW competition nicely, and the RPA estimate for an attractive inter-patch interaction in R4G is a concrete attempt to connect to the recent experiments. The h/6e vortex discussion is a clean re-derivation of known PDW physics, correctly attributed.

The other soft spots are secondary. The GL free energy is unbounded at quartic order, so Fig. 4 relies on a phenomenological sixth-order term with undetermined ζ and assumed T(η) profiles—they are honest about this, but it limits the phase diagram's predictive power. And the fully polarized normal state is assumed, not derived. They say so in Sec. IV B, but it means the model applies only if the experiment truly is in a fully isospin-polarized regime. For R4G/R5G that is plausible but not established.

Who gets value from this? People working on rhombohedral graphene superconductivity and on weak-coupling mechanisms for PDW. The paper deserves a serious referee—not a desk reject—because the core model is novel and the error looks fixable. I would not cite the central result until the sign is confirmed. If the authors confirm the flip, the qualitative conclusions likely survive.

Recommendation: send it to review, with instructions to verify the beta function algebra and the sign of a(η) before judging the physics.

Referee Report

2 major / 4 minor

Summary. The manuscript studies a single-flavor (spin- and valley-polarized) two-dimensional Fermi liquid with C3v symmetry tuned to a Van-Hove singularity, using a frequency-cutoff patch renormalization group. For three VH points it derives a one-loop beta function for the single inter-patch coupling and claims that repulsive interactions are marginally irrelevant, while attractive interactions produce either CDW or PDW order depending on mass anisotropy and patch orientation, with a 3q PDW supporting h/6e vortices. A six-VH-point extension and an RPA estimate of the initial coupling for rhombohedral tetralayer graphene are also presented. The paper is clearly written and the RG setup is well motivated, but the central one-loop coefficients as printed have the wrong sign in 0<η<1/3 and are non-real for η>1/3, so the main stability/instability claim is not supported by the calculation as written.

Significance. If the RG calculation is corrected and the claimed sign of a(η) holds, the paper would provide a rare weak-coupling mechanism for PDW and CDW order in a spin/valley-polarized system, directly relevant to recent experiments on rhombohedral multilayer graphene. The manuscript contains explicit derivations of the beta functions, susceptibility exponents, GL coefficients, and an RPA estimate of the initial g(0), which are valuable for reproducibility. The h/6e vortex result is a standard re-derivation and not new. At present, however, the sign/domain inconsistency in the central beta function prevents the main physical conclusion from being accepted.

major comments (2)

[Eq. (5), Appendix B, and Appendix E] The central claim that weak repulsive interactions are marginally irrelevant and weak attractive interactions drive CDW/PDW order rests entirely on a(η)>0 in Eq. (5). The printed coefficients in Eqs. (B1)–(B2) do not satisfy this. For 0<η<1/3, the logarithm in Eq. (B1) is positive (its argument exceeds 1) and the arctangent in Eq. (B2) is positive, while both prefactors carry an overall minus sign, so a_ZS'(η), a_BCS(η), and hence a(η) are negative. For 1/3<η<1, the logarithm in Eq. (B1) has a negative argument and the square root in Eq. (B2) is imaginary, so the beta function is not real. With a(η)<0, Eq. (5) makes positive g grow and negative g flow to zero, which is the opposite of the claimed behavior. Since Eqs. (8) and (9) inherit these signs, the predicted stability/instability dichotomy is not established by the manuscript as written. A similar domain problem appears in Eq. (E1), where the argument of arcsin can exceed unity for allowed values of η and φ, so the six-VH coefficient A(η,φ) is not real over the full domain of Fig. 6. The authors should recompute or correct these coefficients and verify the sign of a(η) on the entire interval 0<η<1.
[Sec. II.D and Appendix D] The selection between single-q and multi-q order is not derived from the microscopic model. The quartic GL functional in Eq. (10) is unbounded below (Appendix D), and the authors stabilize it by adding a phenomenological sixth-order term w[Σ(|Δ|²+ζ|ρ|²)]³ with arbitrary positive coefficients. The 3q PDW region in Fig. 4 therefore depends both on this uncalculated sixth-order term and on the assumed T_PDW(η) and T_CDW(η) profiles stated in the caption; the computed GL coefficients are evaluated only at η=ηc. The one-loop RG instability calculation is not affected, but the 'multiple wavevector' part of the central claim is illustrative rather than controlled. I recommend either computing the sixth-order coefficients from the patch model or explicitly presenting the multi-q phase selection as model-dependent.

minor comments (4)

[Throughout] There are several typos: 'This a consequence' in the Introduction, 'the the CDW phase' in the Introduction, and repeated misspellings of 'anisotropy' as 'anistropy'.
[Eq. (D7)] The lower-left block of the 6x6 matrix in Eq. (D7) appears to be written incorrectly: rows 4–6 begin with cP, bP, bP rather than the corresponding u-type coefficients, so the matrix does not have the expected block structure. Please check this expression.
[Fig. 4 caption] The figure caption states that the phase diagram is independent of w as long as w>0, but the boundary between 3q CDW and 1q PDW depends on the phenomenological parameter ζ; this dependence should be stated in the main text as well.
[Reference [48]] Reference [48] is formatted as 'Jiang-Xiazi et al. Lin'; the author name should be cleaned up to a standard format.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the central RG result is a self-contained one-loop calculation, with a separate non-circular sign/domain issue in Appendix B.

full rationale

The central derivation is a conventional one-loop patch RG starting from the stated model Hamiltonian (1) and dispersions (2)–(3). The beta function (5) and the vertex flows (8) are computed from the printed integrals in Appendices B and C; no parameter is fitted to the target CDW/PDW phases, and the crossover at ηc ≈ 0.157 is obtained by evaluating the resulting integrals, not by imposing the phase diagram. The attractive initial coupling g(0) < 0 is not reverse-engineered from the desired instability: Appendix A estimates it independently via RPA using R4G parameters taken from Ref. [28]. The fractional h/6e vortex claim is explicitly presented as a re-derivation of a known result (Ref. [49]) and is not used to justify the RG flow. The self-citations to Berg and co-workers provide contextual theory, band-structure inputs, or standard formulas (e.g., the RPA polarization expression) and are not load-bearing reductions of the paper's own prediction to its input. One non-circular internal consistency issue should be flagged: as printed, Eqs. (B1)–(B2) give aZS'(η) and aBCS(η) negative for 0 < η < 1/3 and non-real for η > 1/3, contradicting the claim after Eq. (5) that a(η) > 0 for all η. Taken literally, those printed formulas would flip the sign in Eq. (5) and invert the advertised stability/instability dichotomy. This is an arithmetic or typographical correctness problem, not an input–output circularity, so it does not raise the circularity score.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The model rests on several unproven premises: the polarized normal state, the patch approximation with only inter-patch interactions, the sufficiency of one-loop RG, the RPA estimate of the interaction sign, and a phenomenological sixth-order GL term. These are listed as axioms and free parameters.

free parameters (3)

zeta = >0, undetermined
Phenomenological weight of the CDW amplitude in the sixth-order term w[Sum(|Delta|^2 + zeta|rho|^2)]^3; the boundary between 3q CDW and 1q PDW depends on zeta (Sec II D, Fig. 4).
w = 0.1
Coefficient of the phenomenological sixth-order term added to bound the GL free energy from below; authors state the phase diagram is independent of w as long as w > 0.
TPDW/TCDW temperature profiles = TPDW = Tx[1 - (eta - eta_c)/(2 eta_c)], TCDW = Tx[1 + (eta - eta_c)/(2 eta_c)]
Assumed linear temperature dependence used to draw the qualitative phase diagram in Fig. 4; Tx is an overall scale.

assumptions (5)

domain assumption A fully spin and valley polarized normal state exists at the densities of interest
Sec IV B: 'we have assumed that such a state exists, and studied the effects of residual interactions when a VH singularity is approached.' The entire patch model describes only one spin and valley flavor.
domain assumption The patch approximation with a single inter-patch coupling g and no intra-patch interactions captures the low-energy theory
Sec II A: 'there is no momentum-independent intra-patch interaction due to the full spin polarization. There is a single inter-patch interaction parameter, g... Intra and inter-patch momentum dependent interactions are irrelevant under RG, and will be neglected here.'
domain assumption One-loop RG is sufficient; higher-loop corrections do not change the sign of the beta function or the leading instabilities
The analysis is performed to one-loop order (Sec II B, Appendices B and E); no estimate of higher-loop corrections is given.
ad hoc to paper The RPA provides a valid estimate of the effective inter-patch interaction g(0)
Sec IV C and Appendix A use RPA to estimate g(0) < 0 for R4G; the paper admits 'we do not know of a fully controlled way to carry out the first step of this procedure.'
ad hoc to paper The GL free energy can be made bounded by adding a sixth-order term with arbitrary positive coefficients
Sec II D: the quartic GL free energy is unbounded from below; the authors add w[Sum(|Delta|^2 + zeta|rho|^2)]^3 with w = 0.1, zeta > 0 to obtain a phase diagram. This is not derived from the microscopic model.

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Pith. "Pith review of Charge and pair density waves in a spin and valley-polarized system at a Van-Hove singularity." pith.science (2026). https://pith.science/paper/6MKDPFQA

@misc{pith2026250419321,
  author       = {Pith},
  title        = {Pith review of: Charge and pair density waves in a spin and valley-polarized system at a Van-Hove singularity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6MKDPFQA}},
  note         = {Machine review of arXiv:2504.19321}
}

abstract

We study a single component (i.e., single valley, spin-polarized) two-dimensional electron gas with $C_{3v}$ symmetry tuned to a Van-Hove (VH) singularity. Generically, there may be either three or six VH points at the Fermi level, related to each other by symmetry. Using a renormalization group analysis, we show that when the effective interactions between electrons at the VH points are positive, the system is stable. In contrast, if the effective interactions are negative, the system develops an instability toward either pair density wave (PDW) or charge density wave (CDW) orders, depending on the anisotropy of the dispersion at the VH points. The PDW may have either a single wavevector or multiple wavevectors. The PDW phase with three coexisting wavevectors can support fractional $\tfrac{h}{6e}$ vortices. The interplay between the geometry of the Fermi surface and the singularity of the density of states is the key that enables PDW formation.

Figures

Figures reproduced from arXiv: 2504.19321 by the authors.

**Figure 1.** (a) Iso-energetic lines for R4G in the +K valley conduction band with a potential difference of ∆E = 70meV between the top and bottom layers. The model parameters are taken from [28]. Here, a ≈ 0.246nm is the lattice constant of graphene. The three distinct VH points are drawn schematically in orange on the contours and the black lines represent the principal axes of the saddle points. The VH points are reached for … view at source ↗

**Figure 2.** Diagrams contributing to the renormalization of g up to one loop order. The indices α and β label the different patches (VH points). The outer frequencies are set to zero, and the frequency inside the loops lies in a shell of width dΛω being integrated over. For our purpose, the important characteristics of a(η) are that it is positive for all η and it diverges at η = 1 3 . For η = 1 3 , the contribution of the par… view at source ↗

**Figure 3.** The exponents of the PDW and CDW susceptibilities, χi ≈ (t − tc) γi . The inter-patch PDW is shown in blue and the CDW is drawn in orange. For η < ηc, the dominant instability is toward an inter-patch PDW order and for η > ηc the CDW instability is dominant. of the vertices, assuming that close to tc, the order parameter evolves as (t − tc) αi (i = CDW,PDW), we [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Qualitative phase diagrams for the three VH point model. The GL free energy coefficients were set to their values at η = ηc. The coefficient of the phenomenological sixth-order term was set to w = 0.1. Tx is the temperature where TPDW = TCDW cross at η = ηc. We have a…

**Figure 5.** Figure 5: (a) Iso-energetic lines for R3G vs. ⃗k in the +K valley conduction band with a potential difference ∆E = 20meV between the top and bottom layers. The model parameters were taken from Ref. [44] . The six distinct VH points are drawn schematically in orange on the conto…

**Figure 6.** Figure 6: Leading instability of the six VH point model as a function of η and ϕ. The type of order with the most divergent susceptibility is indicated for each (η, ϕ). The various orders correspond to the test vertices given in Eq. (16). The CDW phases dominates most of the pha…

**Figure 7.** Figure 7: The effective interaction potential for R4G using RPA (a) The static polarization calculated along the vector connecting VH points 2 and 3 (see [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: The diagrams that contribute to the renormalization of the various test vertices up to one loop [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: The Ginzburg - Landau coefficients as a function of [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 10.** Figure 10: Examples of the contributing diagrams to the Ginzburg-Landau free energy in the three VH points [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

**Figure 11.** Figure 11: The vectors Q⃗ 1,2,3 of the three components of the PDW. The two vectors Q⃗ 1 − Q⃗ 2 and Q⃗ 2 − Q⃗ 3 are reciprocal lattice vectors of the lattice corresponding to the modulation of the charge in the PDW state. In order to write the free energy, it is convenient to ch…

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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has a divergence at η = 1 3, due to the presence of a susceptibility that diverges as log squared of the IR cutoff. 6 C. Instabilities To probe the tendency to develop each order, we add test vertices to the Hamiltonian and track how they flow. The test vertices and the diagrams that contribute to the renormalization in this model are similar to those in ...

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Thus, near η = 1 3 our description breaks down

For η = 1 3, the contribution of the particle- hole channel has a log-squared divergence, due to the perfect nesting of the Fermi surface. Thus, near η = 1 3 our description breaks down. An analysis of the case of η = 1 3 for spinful fermions can be found in [42]. Solving Eq. (5), We find that for an initial repulsive interaction (g > 0), g flows towards ...

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