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When Can a Cavity Move a Mott Transition? A Spectral-Density Criterion within Gutzwiller Theory

T0 review · 2 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A cavity moves a Mott boundary only if its vacuum field carries bond-scale spectral weight.

desk verdict A clean variational answer to when a cavity can move a Mott transition — single modes can't, bond-varying continua can — with the honest caveat that the quantitative kernel is proven only inside Gutzwiller theory. read the letter →

arxiv 2607.22283 v1 pith:NSYVLOX4 submitted 2026-07-24 cond-mat.str-el quant-ph

Nikhil Vamsodharakan Seshadri , Yu Zhang This is my paper
classification cond-mat.str-elquant-ph
keywords MotttransitioncavityquantumelectrodynamicsGutzwillervariationalmethodBrinkman–RicePauli–FierzspectraldensitysurfacephononpolaritonsmodeextensivityMonteCarlo
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

Can empty-space quantum fluctuations move a bulk metal–insulator transition? This paper answers with a precise condition: a cavity environment shifts a Mott boundary only when its vacuum field carries finite thermodynamic spectral weight that varies across the electronic bond. Built inside the Gutzwiller–Brinkman–Rice variational description, the argument produces a joint frequency–spatial Pauli–Fierz spectral density whose first moment controls the shift. A single normalized bright mode changes the energy density only as O(1/N) and cannot move a regular phase boundary, no matter how large its Rabi splitting; an extensive mode continuum or lossy environment with bond-scale field variation can. The criterion is evaluated for a surface-phonon-polariton geometry, giving an experimentally testable d^-3-to-d^-5 crossover, and supported by finite-coordination variational Monte Carlo checks of the coefficient and M/N scaling.

What carries the argument

The central objects are the joint Pauli–Fierz spectral density JPF(ω,r) and the bounded response kernel K(ω,r)=4A0r/(ω+4A0r). JPF collects each mode's thermodynamic weight per site, coupling strength, frequency, and bond-gradient factor r, while K applies the two physical filters: modes slow enough to leave a residual self-polarization penalty, and fields that actually vary across the electronic bond. The product integrated against JPF gives the leading shift of Uc. The companion normal-state mode-extensivity theorem supplies the counting: M normalized modes change an extensive energy density by O(M/N), with exceptions only for divergent susceptibilities, superradiant order, or superextensiv

What would settle it

Measure or compute with an unbiased many-body method the critical interaction of a correlated layer at several spacer thicknesses d above a surface-phonon-polariton medium: if the shift does not show the predicted d^-3-to-d^-5 crossover with the stated coefficient, or if one properly normalized mode moves the phase boundary by an O(1) amount, the criterion fails.

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Extended reading notes

Core claim

Within the infinite-coordination Gutzwiller variational manifold, the paper separates collective spectroscopic hybridization from thermodynamic phase control. It defines a joint Pauli–Fierz density JPF(ω,r)=Σα ηα λα² δ(ω−ωα)δ(r−rα), where ηα is the mode's thermodynamic weight per site and rα measures the field's variation across a nearest-neighbor bond. The leading shift of the critical interaction is ΔUc = −∫ dω dr JPF(ω,r) 4A0r/(ω+4A0r) + O(JPF²): the bounded kernel 4A0r/(ω+4A0r) suppresses modes that are too fast and fields that are uniform across the bond. A conventionally normalized single mode has ηα=1/N, so its thermodynamic effect vanishes; an environment with finite local spectral w

Load-bearing premise

The load-bearing premise is that the infinite-coordination Gutzwiller–Brinkman–Rice variational description is a faithful proxy for the actual Hubbard-model Mott boundary; the paper itself notes this description omits Hubbard bands and long-range magnetic order.

Editorial extensions

If this is right

  • A single bright cavity mode, even with a large collective Rabi splitting, cannot move a normal-state Mott boundary in the thermodynamic limit; its energy-density effect scales as 1/N.
  • An extensive set of modes or a lossy continuum with finite local Pauli–Fierz spectral weight and bond-scale field variation shifts Uc downward, making localization easier at fixed bare interaction.
  • The shift follows the mode-density law M/N, directly verified in finite-coordination variational Monte Carlo within roughly 5–7% of the infinite-coordination coefficient.
  • For a planar surface-phonon-polariton environment, the shift crosses from d^-3 to d^-5 with increasing spacer thickness, giving a measurable signature by varying the gap.
  • Cavity design should target the bond-projected spectral density, equivalently the first inverse-frequency moment, rather than Rabi splitting or the local density of optical states alone.

Reading between the lines

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

  • The same joint spectral-density criterion plausibly extends beyond the single-band Hubbard–Brinkman–Rice case: replacing the Gutzwiller response kernel with a more accurate many-body kernel should preserve the conditions of finite thermodynamic weight and r>0, while changing the numerical coefficient.
  • The d^-3-to-d^-5 crossover is a direct experimental target: if a correlated layer's Uc shifts independently of whether the field resolves the bond, or fails to show the predicted distance law, the criterion is falsified.
  • The criterion reframes nanophotonic engineering for phase control as a classical electromagnetism task: computing the bond-correlated field-difference spectrum, rather than maximizing mode confinement or quality factor.
  • The no-go statement for a single mode is a normal-state result; the paper's own caveats imply that at a critical point with a divergent susceptibility, or with macroscopic photon occupation, the counting can break, so the criterion should not be read as a universal no-go theorem.
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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

2 major / 6 minor

Summary. The paper asks whether vacuum electromagnetic fluctuations can shift a bulk Mott transition, and answers this question within a cavity-extended Gutzwiller/Brinkman-Rice variational description. It defines a thermodynamic mode weight ηα and a bond-gradient factor rα for each normalized electromagnetic mode, proves a normal-state mode-extensivity theorem (SM Theorem 1) stating that a fixed number of normalized modes shifts the energy density only by O(M/N), and derives the central quantitative criterion, Eq. (4): ΔUc = −∫ JPF(ω,r) 4A0r/(ω+4A0r) + O(JPF²), with JPF the joint frequency–spatial Pauli–Fierz spectral density. The criterion is first obtained from a solvable degenerate mode ensemble and then extended to a dilute spectral functional. The paper applies it to a 4H-SiC surface-phonon-polariton environment, obtaining a d⁻³-to-d⁻⁵ crossover for the shift of the Mott boundary and numerical estimates (≈0.37 meV at d=1 nm, ≈5 meV at d=0.5 nm for the stated parameters). Finite-coordination determinant variational Monte Carlo on random regular graphs is used to test the infinite-coordination critical coefficient and the M/N scaling, with agreement at the 5–7% level. The paper is careful to distinguish spectroscopic hybridization from thermodynamic control and repeatedly notes the variational nature of the calculation.

Significance. If the criterion holds, it is a genuinely useful organizing principle: the relevant quantity for thermodynamic cavity control of a correlated phase is not the Rabi splitting or the local density of states, but the bond-projected, frequency-resolved electromagnetic spectral density. The derivation is explicit and self-contained within the stated variational manifold; the mode-extensivity theorem is simple but powerful; the VMC section provides a nontrivial finite-coordination check with graph-resolved error bars; and the connection to the recent QMC result [15] strengthens confidence in the single-mode no-go. The paper is also unusually candid about its limitations, stating in SM Appendix B that the Brinkman–Rice transition is not the exact Hubbard transition and in the VMC section that Hubbard bands and long-range magnetic order are omitted. The main weakness is that the quantitative kernel K(ω,r)=4A0r/(ω+4A0r) is derived from the infinite-coordination Gutzwiller functional, so the numerical predictions for the surface-polariton geometry inherit that approximation; the authors acknowledge this, but the abstract states the criterion without the same qualifiers.

major comments (2)
  1. [Abstract and Conclusion] The abstract states without qualification that "a Mott transition shifts only when the electromagnetic environment supplies finite thermodynamic spectral weight with bond-scale variation." This is established only within the normal-state, non-superradiant, infinite-coordination Brinkman–Rice/Gutzwiller manifold. SM Appendix B explicitly says that the Brinkman–Rice transition is not the exact Hubbard-model transition, and the VMC section says the trial state omits Hubbard bands and long-range magnetic order. Since the title correctly carries the qualifier "within Gutzwiller Theory," the abstract and the closing summary should carry the same qualifier, e.g., "within this variational description." Without this, the reader may take the criterion as a proven theorem for the actual Hubbard-model boundary rather than a controlled variational-manifold result.
  2. [Eq. (4) and SM Appendices F,H] The spectral functional is derived to first order in JPF; the O(JPF²) corrections from the self-consistent renormalization of Ac are identified in SM but not estimated. The paper should state the relevant small parameter (e.g., Λloc/A0 or νmλ²/(8A0)) and verify it for the SiC values in Table I. At d=0.5 nm, Λloc≈8.5 meV against A0=0.2 eV, so the ratio is about 4%, but this bound is not given in the main text. Relatedly, Eq. (8) substitutes a continuous spectral density J_surf(Ω,q) into a formula derived for a discrete mode ensemble with independent variational displacements; the additivity of the first-order kernel is plausible, but the conditions under which the continuum limit commutes with the variational minimization should be stated. This would materially strengthen confidence in the quantitative d⁻³-to-d⁻⁵ numbers.
minor comments (6)
  1. [SM Appendix J.5] The sentence "The resulting coefficients are The finite-coordination estimate..." has a grammatical break; the table reference and sentence should be cleaned up.
  2. [References] Reference [56] is a duplicate of reference [5]; consolidate.
  3. [Fig. 3(c)] The legend distinguishes "M=1, varying N" and "N=16, varying M" only by symbols. Please make the fixed parameters explicit in the caption or axis labels, since both data sets are plotted against the same horizontal axis label.
  4. [Abstract] "M/Nscaling" should read "M/N scaling" (missing space).
  5. [Main text, mode-extensivity paragraph] "Theorem1intheSM" appears with missing spaces; similarly, a few other LaTeX artifacts remain (e.g., in the abstract).
  6. [SM Theorem 1 proof] The phrase "the difference between two uncoupled energy densities" is unclear; it should read "the difference between the energy densities of the two phases of the matter Hamiltonian" or similar.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity in the central derivation; Eq. (4) is derived from the variational functional and the O(M/N) no-go is a theorem, not a fitted input.

full rationale

The central result, Eq. (4), is not assumed; it is obtained by expanding the solved degenerate-ensemble critical line (SM Eqs. A8-A27) and then expressed as the joint Pauli-Fierz spectral density. The single-mode O(1/N) result follows from the mode normalization condition sum_i |u_alpha i|^2 = 1, which gives eta_alpha = 1/N, together with the normal-state mode-extensivity theorem (SM Appendix D). The theorem's assumptions are stated explicitly (non-superextensive connected correlation/response kernels), and its exceptional cases are acknowledged; it is not a conclusion smuggled into the assumptions. The surface-polariton d^-3-to-d^-5 crossover is a direct integration of Eq. (4) with independently measured 4H-SiC dielectric data and a chosen lattice cutoff; no parameter fitted to the predicted shift is reused as a prediction. The VMC section tests the coefficient by sampling the finite-coordination Gutzwiller state without imposing the infinite-z form Z(D)=8D(1-2D); this is a consistency check within the same variational family, frankly labeled as such, not an unbiased external solver. Self-citations [37,38] are grouped with external references for Lang-Firsov/Gutzwiller constructions and provide background QED formalism; they are not load-bearing for the central claim, and no uniqueness or no-go theorem is imported from the authors' own prior work. The paper explicitly disclaims its own scope: SM Appendix B says the Brinkman-Rice transition 'is therefore exact within that manifold, but it is not the exact Hubbard-model transition,' and the main text says the VMC 'is not, however, an unbiased finite-dimensional Hubbard solution: it omits Hubbard bands and long-range magnetic order.' These are limitations on external validity of the quantitative kernel, not circularity. No fitted parameter is renamed as a prediction and no definition is constructed from the target result. The score of 1 reflects only the presence of minor, non-load-bearing self-citations; there are no circular steps.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central criterion itself is parameter-free; parameters appear only in the illustrative SiC estimates and are not fitted to the predicted shift. The main axioms are the variational manifold, normal-state conditions, leading-order spectral-weight truncation, and the macroscopic-QED model. No new physical entities are postulated; JPF and r are constructed observables, not new fields or particles.

free parameters (5)
  • A0 (uncorrelated kinetic energy per site) = 0.2 eV (illustrative)
    Chosen for the SiC numerical estimate in Fig. 2; the Mott shift scales linearly with A0. Not fitted to the target shift.
  • p_z (out-of-plane charge-transfer dipole) = 2 eÅ (illustrative)
    Assumed for the correlated layer; ΔUc scales as p_z², so results for other dipoles are rescaled directly.
  • a (nearest-neighbor bond length) = 0.4 nm (illustrative)
    Sets the bond form factor rb(q)=1−J0(qa) and the momentum cutoff qmax=π/a.
  • qmax (high-momentum cutoff) = π/a
    Chosen so the continuum Green tensor is not used beyond the electronic Brillouin-zone scale; physical cutoff supplied by Wannier form factors and nonlocal response.
  • 4H-SiC dielectric parameters (ε∞, ωTO, ωLO, γ) = 6.7, 797 cm⁻¹, 969 cm⁻¹, 4 cm⁻¹
    Measured inputs from infrared ellipsometry [53]; not fit to the Mott shift.
assumptions (6)
  • domain assumption The infinite-coordination Gutzwiller/Brinkman–Rice variational wavefunction faithfully represents the Mott transition.
    All analytical results are derived in the z→∞ Gutzwiller manifold; the paper notes this is not the exact Hubbard-model transition and that DMFT/QMC would modify the response kernel.
  • domain assumption Normal-state/non-superradiant conditions: no macroscopic photon occupation, O(1) connected response kernels, finite frequencies, and couplings that do not grow with N.
    Theorem 1 (SM Appendix D) and the spectral criterion assume these; divergent susceptibility, superradiance, and superextensive coupling are excluded.
  • domain assumption Leading-order truncation O(JPF²)=0 for general multimode environments.
    Eq. (4) is the dilute spectral-weight functional; only the degenerate ensemble is solved nonperturbatively (SM Appendix F).
  • domain assumption In the infinite-coordination normal phase, connected intersite residual self-polarization terms are subleading for the isotropic mode ensemble.
    SM Appendix E.2 states off-diagonal density contractions are subleading; a general finite-range kernel would change the coefficient.
  • domain assumption Nonretarded macroscopic QED with the scattering Green tensor and a single-oscillator dielectric response describes the 4H-SiC surface-phonon-polariton environment.
    SM Appendix I; uses measured SiC parameters and the q≫k₀ approximation.
  • domain assumption The determinant-VMC estimators are unbiased within the chosen variational family on finite random regular graphs.
    SM Appendix J describes the trial state and estimators; results are variational, not exact finite-dimensional Hubbard solutions.

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Pith. "Pith review of When Can a Cavity Move a Mott Transition? A Spectral-Density Criterion within Gutzwiller Theory." pith.science (2026). https://pith.science/paper/NSYVLOX4

@misc{pith2026260722283,
  author       = {Pith},
  title        = {Pith review of: When Can a Cavity Move a Mott Transition? A Spectral-Density Criterion within Gutzwiller Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NSYVLOX4}},
  note         = {Machine review of arXiv:2607.22283}
}
abstract

Can vacuum electromagnetic fluctuations shift a bulk Mott transition? Within the Gutzwiller variational method, we derive a criterion that separates collective spectroscopic hybridization from thermodynamic phase control. We show that a Mott transition shifts only when the electromagnetic environment supplies finite thermodynamic spectral weight with bond-scale variation. A joint frequency--spatial Pauli--Fierz density gives the leading shift. Surface phonon polaritons yield a $d^{-3}$-to-$d^{-5}$ crossover, while finite-coordination variational Monte Carlo supports the predicted critical coefficient and $M/N$ scaling.

Figures

Figures reproduced from arXiv: 2607.22283 by the authors.

Figure 1
Figure 1. FIG. 1. Cavity-induced displacement of the Brinkman–Rice [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Bond-scale field variation controls the surface-polariton shift. (a) Geometry and surface response of 4H-SiC. (b) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Finite-coordination validation of the spectral-density criterion. (a) Weak-coupling cavity energy density versus measured [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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    General electronic and Pauli–Fierz Hamiltonian Partition the electronic system into localized correlated unitsR, which may denote sites, correlated orbitals, molecules, or clusters. We write a Hamiltonian including local correlation and two-site nonlocal correlation [32], ˆHel...

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    Variational displacement and the photon-dressed electronic Hamiltonian We use the polaron-dressed Gutzwiller state |ΨcEGA⟩= exp " −i X α ˆpα ˆFα # Y R ˆPR ! |Ψ0⟩ |χph⟩,(A5) where|Ψ 0⟩is a Slater determinant or Bogoliubov vacuum,ˆPR is a local Gutzwiller correlator, and ˆFα = X...

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    Extended-Gutzwiller operator equivalences The electronic state in Eq. A5 is |ΨG⟩= Y R ˆPR |Ψ0⟩.(A13) Introduce quasiparticle fermionsfRa and the one-body density matrix ∆RR′,ab =⟨Ψ 0|f † RafR′b |Ψ0⟩.(A14) The local constraints are ⟨Ψ0| ˆP † R ˆPR |Ψ0⟩= 1,(A15) ⟨Ψ0| ˆP † R ˆPRf...

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    A22 to the residual operators in Eq

    Cavity-extended energy functional and saddle-point equations Application of Eq. A22 to the residual operators in Eq. A11 gives Ecav res = 1 2 X α " X R ⟨ΦR| ˆB2 αR |ΦR⟩+ X R̸=R′ bαRbαR′ −Tr[T B αR∆RR′T B αR′∆R′R] # ,(A23) where bαR = ⟨ΦR| ˆBαR |ΦR⟩. The first intersite contrib...

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    A23 is subleading for the isotropic mode ensemble used in the main text

    Reduction to the density-coupled single-band model For the Hubbard model, choose ˆXα =λ α X i uαiδni, ˆFα = λαξα ωα X i uαiδni.(A27) 11 The transformed hopping is c† iσcjσ →c † iσcjσ exp " i X α λαξα ωα ˆpα(uαi −u αj) # ,(A28) whereas the residual interaction is ˆHres = 1 2 X ...

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    i λξ ω X α ˆpα(uαi −u αj) # .(A6) The Gaussian average is D ei(λξ/ω) P α ˆpα(uαi−uαj ) E χσ = exp

    Photon transformation of hopping Using[δn i, c† iσ] =c † iσ, one obtains ˆU † ξ c† iσcjσ ˆUξ =c † iσcjσ exp " i λξ ω X α ˆpα(uαi −u αj) # .(A6) The Gaussian average is D ei(λξ/ω) P α ˆpα(uαi−uαj ) E χσ = exp " − λ2ξ2σ 4ω X α |uαi −u αj|2 # .(A7) Averaging over bonds and using ...

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    Therefore 1 N X α λ2(1−ξ) 2 2 ⟨X 2 α⟩=ν mλ2(1−ξ) 2D.(A11) We used ⟨(δni)2⟩G = 2D.(A12) For a general finite-range kernel, the extended-Gutzwiller equivalences of Ref

    Residual self-polarization term The displacement shifts ˆU † ξ ˆqα ˆUξ = ˆqα + λξ ω X i uαiδni.(A9) Consequently, ˆU † ξ ˆHα ˆUξ = 1 2 " ˆp2 α +ω 2 ˆqα − λ(1−ξ) ω Xα 2# .(A10) For the isotropic ensemble used in the main text, the diagonal part of the mode kernel isνm and off-d...

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    Squeezing energy For one pure Gaussian mode, 1 2 ⟨p2⟩+ ω2 2 ⟨q2⟩ −ω 2 = ω 4 (σ+σ −1 −2).(A13) There areM=ν mNeffective modes, so the squeezing energy density is Esq =ν m ω 4 (σ+σ −1 −2).(A14) 14

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    Infinite-coordination variational energy functional Combining Eqs. A5, A8, A11, and A14, the variational energy per site is E(D, ξ, σ) =−AZ(D) + [U+νmλ2(1−ξ) 2]D+ν m ω 4 (σ+σ −1 −2).(A15) Define Ueff =U+ν mλ2(1−ξ) 2, u= Ueff 8A .(A16) Stationarity with respect toDgives −8A(1−4...

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    Stationarity conditions for displacement and squeezing Using the envelope theorem, derivatives with respect toξandσmay be taken at fixed optimizedD. Since ∂A ∂ξ =−2ν mr λ2 2ω σξA,(A20) we find νmr λ2 ω σξAZ= 2ν mλ2(1−ξ)D.(A21) UsingZ/D= 4(1 +u)gives ξ= ω ω+ 2rσA(1 +u) .(A22) S...

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    Adiabatic limit Forω≪4rA 0, ξc ≃ ω 4rA0 , A c =A 0 +O(ω),(A4) so that Uc = 8A0 −ν mλ2 +O(ω).(A5)

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    Therefore Uc = 8A0 exp −νmr λ2 2ω +O(ω −2).(A7)

    Antiadiabatic limit Forω≫4rA 0, ξc = 1− 4rA0 ω +O(ω −2),(A6) and the residual term is higher order. Therefore Uc = 8A0 exp −νmr λ2 2ω +O(ω −2).(A7)

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    Dilute-mode expansion of the critical interaction Here we show explicitly that the dilute-mode expansion of the critical equations above produces the bounded kernel used in the main text and in the general spectral functional derived below. We define the cavity-induced shift o...

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    Combining Eqs

    Macroscopic-QED mapping For a local dipolepat positionsR i, the electromagnetic coupling spectral-density matrix is [45, 48] J (g) ij (ω) = ω2 πℏϵ0c2 p·ImG(R i,R j;ω)·p,(A1) 19 whereGis the classical dyadic Green tensor. Combining Eqs. A8 and A1, the Pauli–Fierz self-polarizat...

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    The peak ofImR p lies inside the Reststrahlen band and produces the spectrum shown in Fig

    Dielectric response of 4H-SiC The isotropic single-oscillator approximation is ϵ(ω) =ϵ ∞ ω2 LO −ω 2 −iγω ω2 TO −ω 2 −iγω ,(A9) with ϵ∞ = 6.7, ω TO = 797 cm−1, ω LO = 969 cm−1, γ= 4 cm −1,(A10) based on infrared ellipsometry of 4H-SiC [53]. The peak ofImR p lies inside the Rest...

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    A8 is4A0rb(q)/Ω

    Distance asymptotics Whend≫a, the dominant momenta satisfyq∼d −1 and rb(q) = 1−J 0(qa) = q2a2 4 +O(q 4a4).(A11) If additionally4A 0rb(q)≪Ω, the kernel in Eq. A8 is4A0rb(q)/Ω. Using Z ∞ 0 dqq 4e−2qd = 3 4d5 ,(A12) we obtain |∆Uc| ≃3p2 zA0a2 8π2ϵ0d5 Z ∞ 0 dΩ ImR p(Ω) Ω2 .(A13) T...

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    Surface-polariton contribution to the Mott-boundary shift

    Numerical values TABLE I. Surface-polariton contribution to the Mott-boundary shift. HereΛloc is the total local Pauli–Fierz self-polarization weight before projection onto the electronic bond form factor. d(nm)|∆U c|(meV)Λ loc (meV)|∆U c|/Λloc 0.5 5.04 8.54 0.590 1.0 0.372 1....

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    The reference determinant|Φ0⟩ fills the lowestN/2one-particle orbitals for each spin

    Finite-graph trial state For az-regular graph withN sites, the one-body hopping is−t∗/√z on every edge. The reference determinant|Φ0⟩ fills the lowestN/2one-particle orbitals for each spin. We use |Ψ(g, ξ, σ)⟩=ˆUξg ˆDtot |Φ0⟩ |χσ⟩, ˆDtot = X i ni↑ni↓,(A1) with0 < g≤ 1and the v...

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    Equation A5 is evaluated by ordinary Metropolis sampling of Eq

    Photon-integrated local-energy estimator Let Ci→j,σ denote a configuration obtained by moving a spin-σ electron from occupied sitei to an empty sitej, and define the determinant-Gutzwiller ratio Rijσ (C) = ψg(Ci→j,σ) ψg(C) .(A3) The Gaussian photon overlap associated with this...

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    Localized-mode estimator and weak-coupling slope For the mode-extensivity benchmark we choose orthonormal site-local modes on a selected setS, uαi =δ i,iα , i α ∈ S, M=|S|.(A6) On any regular graph this ensemble hasνm =M/Nandr= 1exactly. For a hop across bondij, define sij =1 ...

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    Direct extraction of the critical-shift coefficient Within the Gutzwiller variational description, the cavity contribution to the critical interaction is the derivative of the cavity energy with respect to the per-site double occupancy at the projected endpoint. Define y(D) = ...

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    3(a,b) of the main text useN = 24random regular graphs with z = 4, 8, 12, 16, 50 independent graph realizations for each coordination,t∗ = 1, andω= 2

    Simulation parameters and statistical analysis The critical-shift data in Fig. 3(a,b) of the main text useN = 24random regular graphs with z = 4, 8, 12, 16, 50 independent graph realizations for each coordination,t∗ = 1, andω= 2. The sampled Gutzwiller parameters are g∈ {0.08,...

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