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REVIEW 3 major objections 4 minor 57 references

Emergent heavy fermion and superconductivity near Mott transition in twisted bilayer graphene

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Twisted bilayer graphene near its Mott transition is governed by an emergent heavy-fermion theory in which an orthogonal fermion and local moments are Kondo-coupled, producing heavy semimetal and superconducting phases.

desk verdict A serious and mostly transparent derivation of the emergent Kondo coupling in TBG's mixed-valence Mott state, but the phase diagram rests on a dilute-limit approximation the paper's own numbers violate. read the letter →

arxiv 2608.12319 v1 pith:ZOMWFYR3 submitted 2026-08-12 cond-mat.str-el

Ya-Hui Zhang This is my paper · ORCID
classification cond-mat.str-el PACS 71.27.+a74.20.-z71.30.+h
keywords twistedbilayergrapheneheavyfermionKondoscreeningMotttransitionorthogonalmixedvalencesuperconductivityancillatheory
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

Near the bandwidth-tuned Mott transition in twisted bilayer graphene (TBG) at integer fillings, the paper claims that the metal dies through an emergent heavy-fermion mechanism rather than a simple one-band Brinkman-Rice collapse. In the projective active-band limit, the charge sector is formed by the active-band electron $c(\mathbf{k})$ hybridizing with an emergent orthogonal fermion $\psi(\mathbf{k})$ (a linear combination of doublon and holon excitations), and this fermion is Kondo-coupled to local moments $\psi'$ with $J_K = \gamma^2 U/(\gamma^2 + U^2/4)$. Increasing the bandwidth by moving away from the magic angle drives a Kondo screening transition: at charge neutrality $\nu=0$ a heavy semimetal with vanishing quasiparticle residue $Z$ develops below $T_K$, and adding an anti-Hund's coupling $J_A$ produces a superconducting dome near the Mott boundary. The significance is that the framework describes both itinerant carriers and local moments from the same $f$ orbital, and it appears to unify the Mott semimetal, heavy-fermion, and resonating-valence-bond pictures of TBG without invoking remote bands.

What carries the argument

The load-bearing objects are the orthogonal fermion $\psi_i = (d_i - \tilde{h}_i)/\sqrt{2}$ (the sign-alternating linear combination of the doublon $d$ and holon $\tilde{h}$ that has no overlap with the microscopic electron $f$) and the emergent Kondo coupling $J_K = \gamma^2 U/(\gamma^2 + U^2/4)$ between $\psi$ and the local-moment spinon $\psi'$. These two objects convert a one-band Hubbard-like model into a two-layer ancilla theory with density constraints $n_\psi = 4-\nu$ and $n_{\psi'} = 4+\nu$, which guarantees the correct Luttinger volume once the moments are screened. The Kondo vertex is attractive in the SU($N_f$) singlet channel, so its mean-field decoupling yields a momentum-selective hybridization $b(\mathbf{k}) = J_K B F(\mathbf{k})$ that generates the heavy band.

What would settle it

A concrete check: recompute the ground state of the THFM charge sector at $\gamma=3U$, $U=30$ meV, enforcing $n_d + n_h \le 1$; if the doublon density differs by order one from the Gaussian value (Fig. 4), the heavy-fermion mean field is unreliable. Experimentally, at $\nu=0$ and $\theta=1.15^\circ$ the paper predicts a heavy semimetal with a vanishing quasiparticle peak and a coherence scale $\sim 26$ K; observing a sharp Dirac cone with large $Z$ at that angle would falsify the Kondo-screening scenario.

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Extended reading notes

Core claim

The central assertion is that the mixed-valence Mott state of TBG at integer filling is exactly an ancilla theory: the physical electron $c(\mathbf{k})$ hybridizes with the orthogonal fermion $\psi \propto (d - \tilde{h})$ through a saturating hybridization $\Phi(\mathbf{k}) = (U/2) k/\sqrt{k^2+k_*^2}$, while $\psi$ is Kondo-coupled to the local-moment spinon $\psi'$ with $J_K = \gamma^2 U/(\gamma^2+U^2/4)$. The theory claims that this Kondo coupling, rather than the bare Hubbard $U$ alone, sets the low-energy scale: $T_K \sim U \exp[-U/(2N_f s^2 |M|)]$, and once $|M|/U$ becomes appreciable the moments are screened, turning the quadratic band-touching semimetal into a heavy semimetal with vanishing $Z$ at $\nu=0$, and producing an intermediate quadratic band-touching semimetal or heavy Fermi liquid at $\nu=\pm1,\pm2$. If the spinons pair through anti-Hund's coupling $J_A$, the Kondo condensate transfers that pairing to the charge sector and gives an s-wave fully gapped or nematic nodal superconductor even at $\nu=0$.

Load-bearing premise

The construction assumes the doublon and holon operators can be treated as ordinary fermions after averaging over an unpolarized local-moment ensemble, dropping the hard-core constraint $n_d + n_h \le 1$; this dilute-limit assumption is questionable because the paper's own regime of interest (e.g., $\gamma=3U$) gives doublon densities larger than one.

Editorial extensions

If this is right

  • At $\nu=0$, increasing the twist angle away from the magic angle produces a Kondo-screened heavy semimetal with vanishing $Z$; the paper's mean field gives $T_K \simeq 26\,\mathrm{K}$ already at $\theta = 1.15^\circ$.
  • At $\nu = \pm1,\pm2$, the same bandwidth increase first closes the small Mott gap into an intermediate quadratic band-touching semimetal and then enters a heavy Fermi liquid with large Fermi surfaces.
  • With anti-Hund's pairing $J_A$ of the spinons, the Kondo condensate transfers pairing to the charge sector, giving either an $s$-wave fully gapped or a nematic, nodally gapped superconductor near the Mott boundary, including at $\nu=0$.
  • The $O(J_K^2)$ Kondo scattering rate reproduces the recent Hubbard-III/diagrammatic lifetime results for the THFM, identifying those lifetimes as Kondo scattering between the orthogonal fermion and the local moments, and explaining why perturbation theory must break down at low temperature.
  • The ancilla density constraints ensure that the heavy Fermi liquid has the correct Luttinger volume, unifying the itinerant and local degrees of freedom within a single band model.

Reading between the lines

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

  • If the emergent heavy-fermion picture is correct, the superconducting dome in TBG should be centered where $T_K(\theta)$ is largest rather than where the bare bandwidth is smallest; a dome with a nodal (V-shaped) gap at $\nu=0$ is a distinctive prediction.
  • The paper's Gaussian treatment of doublons and holons as canonical fermions is internally strained at $\gamma = 3U$ because the computed doublon density exceeds one; using the exact link-weight theory of its Appendix B inside the Kondo mean field could shift the numerical values of $T_K$ and the dome boundary.
  • The 'dark' zero mode $\psi_0 \propto (\delta n + 1/2)^{-1} f$ at $\mathbf{k}=0$ implies that the low-energy quasiparticle at neutrality is not the bare electron; angle-resolved photoemission or scanning tunneling spectroscopy near $\Gamma_M$ should see a short-lived, low-residue state rather than a coherent Dirac cone.
  • The same ancilla construction, applied to other moiré systems with a Mott transition, predicts the appearance of an orthogonal fermion and a Kondo scale; observing a heavy band with the predicted $T_K$ would indicate that the mechanism is generic.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper claims to derive an emergent heavy-fermion description of twisted bilayer graphene at integer fillings starting from the topological heavy-fermion model. The author restricts the f-orbital valence to n̄−1, n̄, n̄+1, rewrites doublon and holon excitations as canonical fermions after averaging over a free-moment ensemble, and diagonalizes the resulting Gaussian charge sector. This yields an orthogonal fermion ψ that hybridizes with the active conduction band c, producing a quadratic band-touching semimetal at neutrality. The author then derives a Kondo coupling J_K = γ²U/(γ²+U²/4) between ψ and local moments represented by ψ′, argues that the full theory is exactly the ancilla theory, reproduces the O(J_K²) scattering rates of Refs. [33–36], and performs a Kondo mean-field calculation predicting a heavy semimetal below T_K at ν=0, intermediate quadratic band-touching semimetals at other integer fillings, and a superconducting dome driven by an anti-Hund's coupling J_A. The paper emphasizes that the itinerant carrier of the mixed-valence Mott state is the orthogonal fermion rather than the bare electron.

Significance. If the central derivation is valid, the paper is significant: it gives a microscopic route to an emergent heavy-fermion framework for TBG, derives J_K from model parameters rather than fitting it, reproduces quantitative scattering rates obtained by independent diagrammatic calculations, and makes concrete predictions for T_K(θ), a heavy quasiparticle band, and the gap structure of superconductivity. These are falsifiable and experimentally relevant. The strengths include the explicit comparison with Refs. [33–36], the identification of the scattering mechanism as Kondo scattering, and the construction of an ancilla framework with well-defined density constraints. The significance is however conditional: the main quantitative claims rest on treating doublons and holons as canonical fermions in a dilute limit, while the paper's own results show a strongly mixed-valence state in the regime where the heavy semimetal and superconducting phases are predicted.

major comments (3)
  1. [Appendix A (Eq. (A16)) and Appendix B] The derivation of the Gaussian charge sector treats d and h as canonical fermions only after averaging over the free-moment ensemble and explicitly drops the hard-core constraint n_d+n_h≤1 'in the dilute limit with nd,nh≪1.' The paper's own results violate this precondition: Appendix B reports a self-consistent valence distribution (p0,p1,p2)=(0.556,0.197,0.032), so 44% of sites carry a defect, and Appendix C (γ=3U) shows doublon and holon occupations that are not small near k=0. Therefore the statement 'The full theory is therefore exactly the ancilla theory' is not established; the mapping is at best an uncontrolled approximation in exactly the mixed-valence regime where the heavy semimetal and superconducting phases are predicted.
  2. [Appendix A.2 (Eq. (A29))] The derivation of the Kondo coupling explicitly 'ignore[s] the corrections from the finite defect density with non-zero nd and nh,' and Appendix A.1 states that the Gaussian occupation equals the physical doublon probability only to leading order, with O(1) corrections in the mixed-valence regime. Since J_K = γ²U/(γ²+U²/4) is extracted from the frozen-moment self-energy in this truncated treatment, the magnitude of J_K, and hence the Kondo scale T_K ∼ U exp(−1/g), could receive O(1) corrections from the hard-core constraint and defect-defect interactions. The manuscript needs either a controlled estimate of these corrections or a demonstration that J_K is robust beyond the dilute limit before the quantitative comparison with experiment is credible.
  3. [Appendix E and Fig. 2] The mean-field phase diagram uses the ancilla density constraints n_ψ′=4+ν and n_ψ=4−ν together with the dilute-limit J_K, but the physical density of local moments is less than one in the mixed-valence state, and the paper itself notes in Appendix A.2 that n_ψ′=4+ν 'does not necessarily mean that the physical local moment density is n̄.' Consequently the T_K(θ) curve and the b versus Δ coexistence region in Fig. 2(d) are not quantitatively controlled in the regime where they predict a heavy semimetal and superconductivity. The l≥1 link tower of Appendix B provides a possible route to quantify these corrections, but it is not used in the phase diagram, leaving the central quantitative claims unsupported.
minor comments (4)
  1. [Abstract and Introduction] The phrase 'exactly the ancilla theory' overstates the result because Appendix A derives canonical fermions only in the free-moment ensemble average and explicitly neglects the hard-core constraint; I recommend replacing 'exactly' with 'at the Gaussian level' or 'approximately' throughout.
  2. [Appendix A, Eq. (A20)] The parameter κ interpolating between κ=0 and κ=1 is central to the nonzero-filling construction, but its physical value is not determined; Fig. 4 shows agreement between the two schemes only near κ≈1, so the choice κ=1 should be justified from the microscopic THFM parameters.
  3. [Fig. 1(d) and Fig. 6] The color scale and normalization of n_d(k) and n_h(k) in Fig. 1(d) and Fig. 6 are not defined in the captions; please specify whether the plotted quantities are per flavor or summed over flavors and state the total density normalization explicitly.
  4. [References [33–36]] The comparison with Refs. [33–36] is a strength of the paper, but these are arXiv preprints; the quoted equations, especially Eq. (17) of Ref. [35] and the loop result of Ref. [33], should be cited with version numbers and equation numbers in the final version.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the ancilla/Kondo derivation is explicit and self-contained, with only minor non-load-bearing self-citations.

full rationale

The derivation chain is self-contained and does not reduce to its own inputs. The central reduction (main-text Eq. (2) and Appendix A) starts from the THFM of Refs. [20,21], truncates the f-valence to {nbar-1,nbar,nbar+1}, and rotates the holon/doublon pair into f0,psi; the ancilla form (A18) is obtained by an explicit rotation, not merely imported from the cited ancilla papers. The Kondo coupling JK = gamma^2 U/(gamma^2+U^2/4) is derived in Appendix A by matching the S-dependent part of the exact site self-energy (A28) with a single Kondo insertion and then projecting onto the low-energy band; it is not fitted. The scattering-rate comparison in Appendix D is an external benchmark against Refs. [33-36], which are independent diagrammatic/Hubbard-III calculations, and the TK(theta) mean field in Appendix E uses only THFM parameters from Ref. [38]. The paper's self-citations to prior ancilla work (Refs. [30,31]) identify the resulting structure and the mixed-valence observation, but they do not supply the load-bearing derivation here; Ref. [37] supplies the illustrative RVB pairing mechanism, which is not needed for the heavy-fermion derivation. The substantive weakness is not circularity but control: Appendix A explicitly drops the hard-core constraint 'in the dilute limit with nd,nh << 1' and Appendix A2 states it ignores finite defect-density corrections, while the paper's own self-consistent numbers (p0 ~ 0.556, nd > 1 in some regimes) violate that limit in the mixed-valence window where the heavy semimetal and superconducting dome are predicted. This is an approximation-validity concern, not a circular reduction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 2 invented entities

The central claim rests on the THFM as the microscopic model, on a valence-truncated ancilla representation, and on canonicalization of doublons and holons in a free-moment ensemble. The free inputs are U and J_A (both illustrative), plus κ for the ν≠0 comparison; the invented entities are auxiliary fermions of the ancilla construction.

free parameters (3)
  • U (Hubbard interaction) = 30 meV (illustrative)
    Quoted from the THFM parametrization of Ref. [38] and fixed to 30 meV 'just as an illustration' in the numerical TK(θ) and mean-field plots; quantitative scales depend on this choice.
  • J_A (anti-Hund's coupling) = 0-2 meV (varied)
    Introduced as an illustrative on-site anti-Hund's coupling from optical phonons (Refs. 39-41); it is not derived here, and the superconducting phase appears only for J_A>0.
  • κ (filling-shift parameter at ν≠0) = ≈1
    In Eq. (A20), κ interpolates between bare and filling-tracking Hubbard references; the equivalence of microscopic and ancilla schemes, and hence the validity of the ν≠0 results, holds near κ≈1.
assumptions (4)
  • domain assumption The topological heavy fermion model (THFM) with parameters from Ref. [38] faithfully describes the active bands of twisted bilayer graphene.
    The derivation starts from Eq. (1) and uses γ(θ), M(θ), v*, U from the THFM parametrization; if the THFM is not correct, the emergent heavy-fermion results need not apply to TBG.
  • ad hoc to paper Valence truncation to n_f ∈ {n̄-1, n̄, n̄+1} and restriction to the l=0 doublon-holon pair.
    The main text keeps only the l=0 pair; Appendix B shows higher-link pairs have non-negligible weights (p1≈0.197 at γ=3U), so the truncation is an approximation.
  • domain assumption The local-moment subspace is described by a maximally mixed SU(N_f)-invariant ensemble, making d,h canonical on average (Eq. A16).
    Schur's lemma is invoked for SU(N_f)-invariant ground states; canonicalization and the Gaussian charge sector rely on this ensemble.
  • ad hoc to paper The Kondo coupling derived from a single-insertion frozen-moment self-energy remains valid in the strong-coupling Kondo lattice and mean field.
    J_K is extracted at O(S) via Eq. (A31), then used in the full Kondo mean field with condensate b; this extrapolation is not proven.
invented entities (2)
  • Orthogonal fermion ψ
    purpose: Emergent itinerant charge carrier in the mixed-valence Mott state; hybridizes with the active electron c via Φ(k) to open the Mott gap and forms the heavy quasiparticle after Kondo screening.
    A derived linear combination of doublon and holon operators, ψ=(d-h̃)/√2 at neutrality; it is a composite low-energy field, not a new fundamental particle, and it has no falsifiable handle outside the model.
  • Ancilla spinon ψ'
    purpose: Auxiliary fermion representing the local moment multiplet with density constraint n_ψ'=4+ν; its Kondo condensate with ψ creates the heavy band and its pairing (via J_A) generates superconductivity.
    An Abrikosov representation of the SU(N_f) moment; physical states are obtained only after projecting onto on-site singlets, so it is an auxiliary degree of freedom rather than an observable entity.

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Pith. "Pith review of Emergent heavy fermion and superconductivity near Mott transition in twisted bilayer graphene." pith.science (2026). https://pith.science/paper/ZOMWFYR3

@misc{pith2026260812319,
  author       = {Pith},
  title        = {Pith review of: Emergent heavy fermion and superconductivity near Mott transition in twisted bilayer graphene},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZOMWFYR3}},
  note         = {Machine review of arXiv:2608.12319}
}
abstract

Near a bandwidth-tuned Mott transition, the Fermi velocity $v_F$ and quasiparticle residue $Z$ of a metal often vanish. Here, we show that analogous phenomena emerge in twisted bilayer graphene (TBG) at integer fillings and can be captured by an emergent heavy-fermion framework within a projective active-band limit. Unlike models incorporating remote bands, our effective heavy-fermion description arises from \textit{mixed-valence Mott} physics via decoupled charge and local-moment sectors. In the charge sector, active bands $c(\mathbf{k})$ hybridize with an emergent \emph{orthogonal fermion} $\psi(\mathbf{k})$ to open a large Mott gap at $|\mathbf{k}| > k_*$ ($k_*$ sets the momentum-patch size) and a quadratic band-touching semimetal near $\mathbf{k}=0$ at neutrality ($\nu=0$). The orthogonal fermion is a linear combination of the doublon and holon excitations and may be written as $\psi_i \sim (\delta n^f_i+\frac{1}{2})^{-1} f_i$. An emergent Kondo coupling $J_K \sim U$ ($U$ is the local Hubbard interaction) between $\psi$ and local moments $\psi'$ frames the Mott transition as a Kondo screening transition, tuned by the twist angle $\theta$. Away from the magic angle, a Kondo-screened heavy semimetal develops below $T_K$ (the Kondo temperature) with vanishing $Z$. Introducing anti-Hund's coupling $J_A$ generates an s-wave fully gapped or nematic, nodally gapped superconducting dome near the Mott boundary even at $\nu=0$. At other integer fillings $\nu = \pm 1, \pm 2$, increasing bandwidth first drives the small-gap Mott state into an intermediate quadratic band-touching semimetal before entering a heavy Fermi liquid with large Fermi surfaces. Our results establish a unified framework for emergent heavy fermion physics with both itinerant carriers and local moments from the $f$ orbital.

Figures

Figures reproduced from arXiv: 2608.12319 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Free fermion bands at [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Momentum-integrated [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) The [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Occupations versus [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: shows the f spectral function Af (ω, k) at γ = 5U at the twist angle θ = 1.05◦ , M = 0 for the charge neutrality, including l ∈ {0, 1}. From the self-consistent calculation with the exact metric Eq. (B7) we find weights (wd0 = wh0 = 0.401, wd1 = wh1 = 0.098), with vale…
Figure 6
Figure 6. Figure 6: FIG. 6. Mixed valence of the Mott state from the Gaussian theory Eq. [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

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    As the only interaction in Eq

    Charge sector with holon and doublon operator We derive the ancilla-fermion representation used in the main text directly from the restricted-valence description of the f orbital in the THFM. As the only interaction in Eq. (1) of the main text is the HubbardU on thef orbital, ...

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    ) −∆ψ ∑ iα ( ni;f0−ni;ψ ) + ¯U 2 ∑ iα ( f† 0iαψiα + H.c

    The conduction electron therefore couples to the restricted physical electron and to nothing else, while the whole ofUis carried by thef0–ψsector, H=H c1c2 +γ ∑ kα ( f† 0kαc1kα + H.c. ) −∆ψ ∑ iα ( ni;f0−ni;ψ ) + ¯U 2 ∑ iα ( f† 0iαψiα + H.c. ) ,(A18) with the two filling-depend...

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    The key assumption is the maximally mixed density matrixρlm =Pi;¯n/M of Eq.(A15)

    Effective Kondo coupling In the previous subsection, we provided an effective theory for the charge sector while assuming the local moment is freely fluctuating. The key assumption is the maximally mixed density matrixρlm =Pi;¯n/M of Eq.(A15). Strictly speaking, this is true e...

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    The rotor coordinateˆL carriesallof the site’s charge: ˆL=n f−¯n, while the spinonsψ′ carry the spin

    The representation Write thefelectron as a rotor times a spinon, f† iα =e iθiψ′† iα, H U = U 2 ∑ i ˆL2 i, ˆLi = ∑ α ψ′† iαψ′ iα−¯n,(B1) with[ θi, ˆLj] = iδij andeiθ|ℓ⟩ =|ℓ + 1⟩ on the rotor ladderℓ∈Z . The rotor coordinateˆL carriesallof the site’s charge: ˆL=n f−¯n, while the...

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    ˜h†), andℓ = 1,− 2,

    Link operators and the exact algebra Decompose the rotor raising operator into its links,eiθ =∑ ℓ|ℓ + 1⟩⟨ℓ|, and define one composite fermion per link and flavor: D† ℓ,iα≡|ℓ+ 1⟩⟨ℓ| i⊗ψ′† iα, f † iα = ∑ ℓ D† ℓ,iα.(B2) Theℓ = 0link is the doublon operator of the main text (¯n→¯n...

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    The untruncated Hamiltonian therefore is in the form H= ∑ i U 2 ˆL2 i +γ ∑ iα [ c† 1,iα ∑ ℓ Dℓ,iα + H.c

    Canonical link fermions and the untruncated Gaussian theory Summing Eq.(B3) over the links telescopes,{fiα,f† iβ} =δαβ ∑ ℓPℓ =δαβ: the electron is theequal-amplitudesum of all links and is exactly canonical. The untruncated Hamiltonian therefore is in the form H= ∑ i U 2 ˆL2 i...

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    With the dominantl = 0pair alone, f0,iα =√wh0 ˜h0,iα +√wd0d0,iα, ψ iα =√wh0d0,iα−√wd0 ˜h0,iα,(B13) which reproduces Eq

    The orthogonal fermion and the dark zero mode in rotor language The physical electron is one specific combination of link fermions,fiα =∑ l(√wdldl,iα +√whl ˜hl,iα); the orthogonal fermions are themetric-orthogonal complementof this combination in link space. With the dominantl...

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    With more pairs kept there are more dark combinations

    is f0 = ( ˜h +d)/ √ 2, ψ = (d− ˜h)/ √ 2 of the main text.ψ is the sign-alternating combination of links—compactly,ψ∝e −iθ sgn(ˆL− 1 2)ψ′ in the truncated space. With more pairs kept there are more dark combinations. The link space holds2npairs orbitals per flavor. The hybridiz...

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    In the projective limitζ→ 1they reduce to the projected results of Ref

    or on controlled diagrammatic expansions [33, 35, 36], but now we have a clear physical interpretation in terms of a Kondo coupling. In the projective limitζ→ 1they reduce to the projected results of Ref. [34], including theNf + 1 prefactor. The γ4/(γ2 +U2/4)dependence at gene...

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    The projected model Per valley–spin flavor (four of them, each carrying a two-component orbital spinor, soNf = 8in total), H= ∑ k [ c† k [ ha(k)−µ c ] ck + (νU 8 −µψ ) ψ† kψk + ( c† kΦkψk + H.c. ) −µψ′ψ′† kψ′ k ] +JK ∑ kk′ F(k)F(k′) ∑ αβ ψ† kαSαβ k−k′ψk′β.(E1) HereΦ k is the s...

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    The Kondo term is exactly an attraction in the singlet channel Before any approximation, the vertex can be written as a single bilinear-squared. Using Eq.(A33) at reference filling¯n, ∑ αβ ψ† iαSαβ i ψiβ = (1−¯x)nψ i −Q† iQi,Q i≡ ∑ α ψ′† iαψiα,(E6) an operator identity, exact ...

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    The free energy depends on|b| only, so the two signs are one saddle, not two

    Saddle point Writebi =JK⟨Qi⟩, uniform and real: the global U(1) phase ofψ′ is a gauge freedom of the Abrikosov representation, which we fix byb≥ 0. The free energy depends on|b| only, so the two signs are one saddle, not two. With the separable vertex the saddle closes on a si...

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    Onset condition At the transitionb→0. Linearizing,⟨ψ ′†ψk⟩=b(k)K(k)with K(k) = ∑ n ⏐⏐uψn(k) ⏐⏐2 nF (−µψ′)−n F [εn(k)] εn(k) +µψ′ ,(E9) where n runs over the four charge-sector branches at the constrained levels,|uψn|2 is their orthogonal-fermion weight, and all energies are me...

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