Cavity Control of Strongly Correlated Electrons Beyond Resonant Coupling

Angel Rubio; Dante M. Kennes; Emil Vi\~nas Bostr\"om; Lukas Grunwald; Marios H. Michael; Michael Ruggenthaler; Xinle Cheng

REVIEW 2 major objections 4 minor 1 cited by

Cavity Control of Strongly Correlated Electrons Beyond Resonant Coupling

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Off-resonant cavity control of magnetism is governed by the frequency-integrated photonic density of states, not by any single resonance.

desk verdict Workhorse formalism and a sound central criterion, but the abstract promises a t-J section that isn't there, and the gold-substrate number lacks a damping-sensitivity check. read the letter →

arxiv 2603.18933 v2 pith:GA4MCUO3 submitted 2026-03-19 quant-ph cond-mat.mes-hallcond-mat.str-el

Lukas Grunwald , Xinle Cheng , Emil Viñas Boström , Michael Ruggenthaler , Marios H. Michael , Dante M. Kennes , Angel Rubio This is my paper

classification quant-phcond-mat.mes-hallcond-mat.str-el

keywords cavityquantumelectrodynamicsstronglycorrelatedelectronsmagneticexchangegeneralizedPurcellfactorphotonicdensityofstatessurfacepolaritonsHopfieldquantizationHubbardmodel

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 establishes a design principle for using vacuum fluctuations to control strongly correlated electron systems: off-resonant cavity modifications of the magnetic exchange J are set not by the photon density of states at any single resonance, but by its frequency integral relative to free space—a generalized Purcell factor. Working non-perturbatively in all cavity modes for a half-filled Hubbard model, the authors show that Fabry-Pérot resonators redistribute spectral weight in a way that nearly cancels under integration, leaving negligible effects, while polaritonic surface cavities concentrate spectral weight into a narrow surface-mode peak that survives. A consistent Coulomb-gauge treatment reveals that static dielectric screening and dynamical vector-potential dressing compete, with opposite effects on J; for a gold substrate the net result is a few-percent enhancement at nanometer separations, measurable via two-magnon Raman spectroscopy. The abstract further claims that at weak doping a low-frequency surface cavity can reverse the nodal-antinodal dichotomy of the t-J polaron, observable in ARPES.

What carries the argument

The central object is the generalized Purcell factor, defined as the frequency-integrated photonic density of states relative to free space, which enters through a single cavity-modification function M(x). The paper derives this via a Coulomb-gauge quantization scheme that diagonalizes the polaritonic modes and then removes the scalar potential by a unitary transformation, yielding a screened Coulomb interaction (satisfying the Poisson equation with the static dielectric function) and a vector-potential coupling. The magnetic exchange is obtained by a strong-coupling canonical transformation and a Laplace decoupling of the all-mode sum, reducing an exponentially hard occupation sum to a one-

What would settle it

Measure the two-magnon Raman peak of a cuprate parent compound at 1–10 nm above a gold surface; a shift smaller than predicted (or of the opposite sign) would indicate the lossless Lorentzian model misrepresents the balance. Alternatively, re-run the calculation with a damped Drude dielectric (Q ~ 100): a sign reversal of ΔJ would falsify the prediction.

Watch

Extended reading notes

Core claim

The paper shows that off-resonant vacuum modifications of the magnetic exchange J in a half-filled Hubbard model are controlled by a generalized Purcell factor: the frequency-integrated photonic density of states relative to free space. It further shows that in a surface polaritonic cavity the static dielectric screening of the Coulomb interaction must be included alongside the dynamical vector-potential dressing; for a gold substrate the two compete with opposite signs and a few-percent net enhancement of J remains at nanometer separations, observable in two-magnon Raman spectroscopy.

Load-bearing premise

The quantitative predictions assume a lossless, local Lorentzian (Drude-limit) dielectric for gold down to nanometer separations; if Ohmic damping or nonlocality shifts the balance between screening and dressing, the net effect could change sign or magnitude.

Editorial extensions

If this is right

Fabry-Pérot cavities are ineffective for off-resonant control of correlated electrons, because their periodic spectral-weight redistribution cancels upon frequency integration.
Polaritonic surface cavities are promising platforms, and their strongly peaked photonic density of states justifies single-mode approximations with first-principles coupling constants.
Including static screening is qualitatively essential: neglecting it reverses the sign of the predicted change in J.
A few-percent change in J appears as a 4ΔJ shift in the two-magnon Raman peak, resolvable with existing Raman resolution (~0.5 meV).
The abstract states that a low-frequency surface cavity can reverse the nodal-antinodal dichotomy of the weakly doped t-J polaron, observable in ARPES (no derivation appears in the body).

Reading between the lines

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

Because the only cavity input is the relative photonic density of states, the same integral criterion could be used to screen any cavity geometry (e.g., via macroscopic QED) before expensive many-body calculations.
The predicted balance between screening and dressing is sensitive to the substrate model; a damped Drude description or nonlocal corrections could change the sign or magnitude of ΔJ at the nanometer separations where the effect is largest.
If the t-J polaron reversal holds under a full multi-mode treatment, vacuum fluctuations would provide a mechanism for Fermi-surface reconstruction in correlated metals, beyond mean-field descriptions.

Share X Bluesky LinkedIn Reddit HN

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

I'll cut to the chase. The multi-mode resummation and the Coulomb-gauge Hopfield scheme with static screening are the real goods. I traced Eqs. (17)-(22), including the Laplace decoupling and the PDOS rewriting, and the algebra is consistent. The derivation of the screened interaction W from electrostatics (Appendix B) is clean, and Appendix F2 is an unusually honest treatment of UV regularization. The conclusion that Fabry-Pérot cavities integrate to a near-zero relative PDOS while surface modes contribute a finite integrated weight is physically sensible and well argued. The moment-based validation of the single-mode limit (Appendix G3) is a nice piece of work.

The central claim—that off-resonant cavity modification of J is governed by the frequency-integrated PDOS relative to free space—appears to be a real result, and the screening-vs-dressing competition is a genuine insight. I did not find a circular step; the cavity enters only through Δρ, and the generalized Purcell factor is an output, not an input.

Now the soft spots.

First, the abstract advertises a variational t-J polaron calculation that reverses the nodal-antinodal dichotomy at weak doping and is observable in ARPES. There is no such section in the body. No methods, no figures, no results. This is either a missing section or an overclaim; either way it cannot stand. The abstract must be reconciled with the content.

Second, the quantitative few-percent prediction for gold depends on modeling the substrate as a lossless, local Drude dielectric down to z≈1 nm. The stress-test note says this is inside the paper's own breakdown region; strictly, the gray region in Fig. 3 is z<1 nm, while the plotted results extend to z=1 nm, which is right at the boundary. So the "inside gray region" phrasing is slightly overblown. The substantive concern is not: a lossless Drude model has no Ohmic damping, and real gold has Q≈100. The surface-mode PDOS peak that drives the effect would broaden and shift, and since the net ΔJ is a cancellation between two large z^{-3} contributions, the sign or magnitude could change. The paper does not quantify this sensitivity. A damped Drude calculation, or even an order-of-magnitude estimate of finite-Q effects on the integrated Δρ, would be needed to support the Raman-observable prediction.

Third, a minor point: Eq. (25) is labeled a generalized Purcell coefficient but is derived only in the small-θ expansion. The full integral (22)-(23) is the more general object; the terminology is fine, but the paper should be clearer that the "coefficient" is the leading-order version.

Overall: the formalism and the design principle are solid, and the paper deserves serious refereeing. The t-J claim must be dealt with, and the quantitative prediction needs a robustness check. I would send it to peer review, not desk-reject it, but with major revision expected.

Referee Report

2 major / 4 minor

Summary. The manuscript develops a Coulomb-gauge Hopfield quantization for a dispersive dielectric substrate coupled to Hubbard electrons, and derives a multi-mode, all-orders expression for the cavity-modified magnetic exchange J in the dark-cavity and strong-coupling (t≪U) limit. The central result is that the modification is controlled by the frequency-integrated relative photonic density of states Δρ(ω)=ρ(ω)−ρ0(ω), leading to a 'generalized Purcell factor' picture. For Fabry–Pérot cavities, spectral-weight redistribution nearly cancels upon integration, giving negligible modifications with a d^-3 scaling; for a lossless Drude gold surface cavity, the surface-mode PDOS dominates and yields a net few-percent enhancement of J at nanometre separations, after strong cancellation between dynamical vector-potential dressing and static dielectric screening. The abstract additionally claims that at weak doping a low-frequency surface cavity reverses the nodal–antinodal dichotomy of the t–J polaron, observable via ARPES.

Significance. If the quantitative claims hold, this is an important conceptual advance: it replaces single-mode phenomenological couplings with a derived, mode-summed figure of merit, provides a controlled route to effective single-mode parameters, and makes falsifiable spectroscopic predictions (two-magnon Raman, RIXS). The algebraic core—Eqs. (17)–(19) via the Laplace representation, the PDOS rewriting Eqs. (20)–(25), and the small-θ closed forms—is internally consistent on tracing; Appendix F2 is unusually candid about the UV-regularization subtleties. The screening-versus-dressing competition is a genuine physical insight and is derived rather than imposed. The main weaknesses are the robustness of the quantitative surface-cavity prediction and an abstract claim that has no corresponding body section.

major comments (2)

[§IV.B, Fig. 3, App. G2 (Eqs. G13, G26)] The percent-level prediction for a gold substrate is computed in a lossless, local, single-Lorentzian Drude model (Eq. G13 with ω_TO→0), with no Ohmic damping and no nonlocal response. This model is applied at the separations where the effect is largest, z≈1–3 nm, and the paper itself marks the macroscopic dielectric description as breaking down in this regime (gray region, Fig. 3). Because the net effect is a near cancellation of two z^-3 contributions of opposite sign, unquantified corrections from damping (gold Q≈100), nonlocal surface-plasmon dispersion, or atomistic dielectric profiles could change the sign or magnitude of ΔJ. The authors should either quantify this sensitivity—for example with a damped Drude Hopfield scheme or a simple nonlocal correction—or explicitly downgrade the quantitative claim. As written, the headline observable prediction is not yet robust.
[Abstract; §III–IV] The abstract states that 'at weak doping, a variational exact diagonalization of the cavity-coupled t–J polaron reveals that a low-frequency surface cavity can reverse the nodal-antinodal dichotomy... and is observable via ARPES.' This result is absent from the body: Sections III and IV concern the half-filled Hubbard exchange and magnon Raman/RIXS signatures, with no t–J polaron model, no variational calculation, and no ARPES analysis anywhere in the manuscript. This is a load-bearing unsupported claim that must either be substantiated by an added section/appendix or removed from the abstract.

minor comments (4)

[App. G2, Eq. (G13)] 'Lorenzian' should be 'Lorentzian'; similarly 'functoin' in §G3.
[App. H1, Eq. (H2)] The incomplete-gamma expression is typeset ambiguously; the arguments of Γ(θ^{-1}) and Γ(θ^{-1},−ḡ²) and the prefactor θḡ^{1/θ} should be clearly defined.
[Fig. 4(c)] The arrow and line darkness should be described explicitly in the caption; the current phrase 'increasing in the direction of the arrow' is vague.
[Eq. (22)] The status of J0 in Eq. (22) should be stated more explicitly: it is the free-space physical exchange after absorbing the free-space vacuum contribution (as in App. F2), not a bare parameter of the lattice model. The main text does explain this, but a parenthetical at Eq. (22) would avoid misreading.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the cavity-dressed exchange J follows from an explicit unitary transformation and resummation with unfitted external parameters; the score reflects minor same-group technical self-citations and an unsupported abstract claim, not a circular chain.

full rationale

The central derivation is self-contained: the screened Coulomb interaction W is obtained by a unitary transformation (Eqs. 5-9) and proven in Appendix B to equal the electrostatic kernel of ε(r,0); the cavity-dressed J is then derived by a Laplace-decoupled Schrieffer-Wolff resummation (Eqs. 16-19), yielding Eq. (22) with M defined by Eq. (23). The generalized Purcell factor (Eq. 25) is a perturbative consequence of that derivation, not an assumed relation. No parameter is fitted to the predicted ΔJ: the surface-cavity calculation uses external inputs (a_ij = 6 Å, U = 5 eV, gold plasma frequency 9.45 eV) and the full PDOS; the single-mode weight in Section G3 is explicitly observable-dependent and is not the basis of the quantitative claim. The paper does rely on same-group technical works (refs 19, 38, 47, 49, 50) for the Schrieffer-Wolff operator and surface-mode numerics, but the relevant formulas are stated in the paper and are standard, so these are minor self-citations rather than load-bearing circularity. Non-circular caveats should be weighed separately: the quantitative gold-substrate prediction is made at z ≈ 1-3 nm, inside the gray region the paper itself marks as the breakdown of the macroscopic dielectric description (Section IV.B, Fig. 3), the lossless local Drude dielectric has no damping or nonlocal corrections, Appendix F2 flags UV-regularization sensitivity, and the abstract's t-J polaron nodal-antinodal reversal claim has no corresponding body section. These are robustness and support issues, not evidence that the derivation reduces to its inputs.

Assumptions & free parameters 6 free parameters · 8 assumptions · 0 invented entities

No new particles, forces, dimensions, or conserved quantities are postulated. The 'dynamical Weyl gauge' is a name for the derived Hamiltonian representation (Eq. 6), and the 'generalized Purcell factor' is a derived functional of the PDOS, not an ad hoc entity. The load-bearing inputs are material parameters (U, a_ij, ω_p, J_0), the UV subtraction prescription, and the moment-matched single-mode weight; the conceptual axioms are the dark-cavity strong-coupling limit, the dipole approximation, and the lossless local dielectric model.

free parameters (6)

Hubbard interaction U = 5 eV (chosen; representative of cuprate parents)
Set by hand in Section IV (Fig. 2b/3b captions: 'a_ij = 6 Å and U = 5 eV'); enters the kernel e^{-ωx/U} and theta = ω_∞/U; the percent-level prediction depends on it.
Bond length a_ij = 6 Å
Chosen by hand; sets the overall coupling scale P_0 = (e a_ij)^2/(2ℏε_0) in Eqs. (20)-(23) and hence the magnitude of delta-J.
Gold plasma frequency ℏω_p = 9.45 eV (literature input)
Input from prior literature for Au entering Eq. (G13); sets the surface-mode limit frequency ω_∞ = ω_p/√2 ≈ 6.7 eV and hence theta ≈ 1.3 for U = 5 eV.
Bare exchange J_0 = 100 meV (illustrative cuprate value)
Used in Section IV.C and Fig. 4 for the Raman/RIXS observability estimate ('changes down to 0.1% assuming a bare exchange of J0 ≈ 100 meV').
UV regularization cutoff η / subtraction prescription = η → 0 in main text; η = (20 eV)^{-1} in App. F2 model scheme
Eq. (19) diverges without regularization; the ab-initio scheme subtracts the free-space PDOS (ρ → ρ - ρ_0). Appendix F2 shows perturbative equivalence to a finite cutoff (corrections ∝ α/(ηU), α ≈ 1e-7), but the subtraction prescription itself is a renormalization choice on which the finite result depends.
Single-mode weight K̄_surf(ω_∞, z) = fixed by moment-matching, Eq. (G28)
Introduced in App. G3 to define the δ-function single-mode approximation and the z^{-3} scaling; fixed by requiring the δ-model to reproduce the full-PDOS observable, so it is calibrated to the theory, not to data.

assumptions (8)

domain assumption Dark-cavity limit: the ground state factorizes as |ψ_e>⊗|0> + O(t/U), so virtual photon occupations are subleading and n=m=0 can be set in Eq. (16) (Section III.C: 'This dark cavity limit is justified within the strong coupling expansion...').
Used to restrict the resummation Eq. (17) to the zero-photon sector; standard for t/U << 1 but asserted rather than controlled at finite coupling.
domain assumption UV regularization: coupling to the free-space electromagnetic field is already included in the bare Hubbard parameters t and U, so the cavity enters only through Δρ = ρ - ρ_0 (Section III.D and App. F2).
Load-bearing: without the subtraction, Eq. (19) diverges; the paper defends it via the ab-initio and model perspectives, but the prescription is a renormalization choice.
domain assumption Long-wavelength (dipole) approximation in the Peierls phase θ_ij = (e/ℏ)(R_j - R_i)·Â (Eqs. 10-11).
Neglects spatial variation of the mode function over the bond; valid for the surface modes that survive at z ~ nm (q ~ 1/z, so q·a << 1), though the PDOS peak is formally at q → ∞ where the approximation is at its limit.
domain assumption Lossless, local single-Lorentzian dielectric description of the substrate with Hopfield normalization (real ε(ω); Eqs. G12-G13), including macroscopic local response down to z ~ 1 nm.
Excludes Ohmic damping, nonlocality, and atomistic structure; the paper's gray region in Fig. 3 concedes z < 1 nm as breakdown, yet the strongest effect is at z ~ 1-3 nm.
domain assumption Two-site (dimer) model for J within the strong-coupling (t/U << 1) Schrieffer-Wolff expansion (Eqs. 13-16 and App. E).
J is extracted from a dimer and the full lattice enters only at the level of the Heisenberg Hamiltonian for the spectra; O((t/U)^3) terms are dropped.
standard math Laplace representation and exchange of summation/integration in Eqs. (17)->(18)->(19).
Standard integral representation; footnote 45 notes regularity is enforced by an explicit Gaussian cutoff sent to zero at the end.
standard math Linear spin-wave theory / Bogoliubov diagonalization for the square-lattice Heisenberg model (App. I).
Standard 1/S expansion used for magnon dispersion, S^{+-}, and two-magnon Raman; the cavity enters only through the renormalized J.
standard math Mode functions and Hopfield normalization for the interface problem (Eqs. G1-G26), with metallic boundary conditions at z = ±L_⊥/2 and the thermodynamic limit.
Established macroscopic-QED mode quantization; the surface-mode PDOS Eq. (G26) is derived from these conventions.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Cavity Control of Strongly Correlated Electrons Beyond Resonant Coupling." pith.science (2026). https://pith.science/paper/GA4MCUO3

@misc{pith2026260318933,
  author       = {Pith},
  title        = {Pith review of: Cavity Control of Strongly Correlated Electrons Beyond Resonant Coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GA4MCUO3}},
  note         = {Machine review of arXiv:2603.18933}
}

abstract

Interfacing materials with electromagnetic cavities offers a route to modify equilibrium properties through structured vacuum fluctuations. The coupling between light and correlated electrons lacks a characteristic energy scale, making vacuum induced ground state modifications of such systems inherently off-resonant and sensitive to the full photon mode structure. We develop a consistent cavity-QED formalism for dispersive electromagnetic environments in the Coulomb gauge, capturing both dynamical dressing via the vector potential and static screening by the dielectric. With this formalism, we perform a non-perturbative study of the cavity-induced modification of magnetism in the Hubbard model close to half-filling, including all cavity modes and with parameters determined from first principles. At half-filling, we show that the modification of the magnetic exchange interaction $J$ is controlled by a generalized Purcell factor, proportional to the frequency integrated photonic density of states relative to free space. This result identifies polaritonic surface cavities as promising platforms to modify correlated systems. For the surface cavity a competition between static screening and dynamical dressing via the vector potential leads to a net enhancement of $J$, directly observable in two-magnon Raman spectroscopy. The inclusion of screening is essential to obtain even qualitatively correct results. At weak doping, a variational exact diagonalization of the cavity-coupled $t$-$J$ polaron reveals that a low-frequency surface cavity can reverse the nodal-antinodal dichotomy of the bare polaron dispersion. This effect lies beyond mean field theory and is observable via ARPES measurements. Our framework establishes a concrete design principle linking cavity geometry to material response in the off-resonant regime, which will guide future experimental and theoretical explorations.

Figures

Figures reproduced from arXiv: 2603.18933 by the authors.

**Figure 1.** Surface cavity interacting with a strongly correlated material. The hybridization of substrate (blue) excitations with the electromagnetic field yields exponentially localized polaritonic modes at the vacuum-dielectric interface. The coupling of longitudinal and transverse electromagnetic field components to a correlated material (red) leads to an intricate interplay that can produce strong modifications of magnetic… view at source ↗

**Figure 2.** Planar cavity dressed magnetic exchange for a Fabry-Perot (FP) cavity with perfect mirrors separated by distance d (inset, top panel) that interacts with a material placed at the midpoint z = d/2. (a) Photonic density of states [Eq. (21)] at the cavity center for a FP resonator (purple) and free space (black) in the thermodynamic limit L∥ → ∞. The oscillatory cavity PDOS nearly averages to zero over each resonance i… view at source ↗

**Figure 4.** Magnon signatures of cavity renormalization. (a) Magnon dispersion for a square lattice antiferromagnet. The light gray dashed line indicates the magnetic Brillouin zone. (b) Transverse dynamical spin structure factor S+−(ω) along the high symmetry lines of the Brillouin zone. The high symmetry points are Γ = (0, 0), M = (π, 0) and X = (π/2, π/2). (c) Two-magnon Raman spectrum I(ω) of a square lattice antiferroma… view at source ↗

Figures from the paper (7 more)

**Figure 5.** Figure 5: Photonic Density of States of a Fabry Perot cavity, measured at the center between the planar mirrors (z = d/2). Shown are the in-plane (blue) and out-of-plane component, relative to their free space counterparts. The former density is relevant for the coupling to embe…

**Figure 6.** Figure 6: Surface Cavity constituted by an interface between a dispersive medium (blue), modeled via ε(ω) in the lower plane z < 0 and vacuum in the upper plane z > 0. The cavity couples to the cavity embedded material (red) with distance z to the substrate. In-plane we use per…

**Figure 7.** Figure 7: Mode Dispersion at an interface. Shown is the bulk dispersion (blue) as well as the surface dispersion (red). The surface mode only exists in a finite energy window ωTO < ω < ω∞ with limit frequency ω∞ [Eq. (G23)]. A common approximation neglects the weak momentum depe…

**Figure 8.** Figure 8: Mode resolved photonic density of states of SrTiO3 as a function of frequency for various distances above the substrate in natural units λ∞ = 2πc/ω∞. Results are shown relative to free space. (a) Bulk mode contribution to the PDOS. As we approach the surface, a peak em…

**Figure 9.** Figure 9: Dimensionless Weight function determined from the n-th moment of the PDOS [Eq. (G28)], relative to the n = 0 weight. While the weights are approximately constant for the surface polariton cavity (red), we observe strong variations for the Fabry Perot (blue). In partic…

**Figure 10.** Figure 10: Bulk and surface resolved magnetic modifications due to coupling to surface (red) or bulk (blue) modes of a SrTiO3 substrate previously introduced [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

**Figure 11.** Figure 11: Gauge contribution resolved scaling of dressed magnetic exchange due to surface cavity coupling as a functions of the substrates plasma frequency ωp for aij = 6 Åand U = 5 eV for various substrate distances z. Shown is the dynamical dressing (reds) and the static scr…

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

When Can a Cavity Move a Mott Transition? A Spectral-Density Criterion within Gutzwiller Theory
cond-mat.str-el 2026-07 conditional novelty 7.0 of 10

A Gutzwiller derivation shows cavity-induced shifts of the Brinkman–Rice Mott boundary are controlled by the bond-gradient-projected Pauli–Fierz spectral density, making single normalized modes thermodynamically inert...

Reference graph

Works this paper leans on

68 extracted references · 2 canonical work pages · cited by 1 Pith paper

[1]

(17) in the thermody- namic limit, which after Laplace decoupling reproduces our result from the main text Eq

Scaling Derivation We provide an alternative expression for the mag- netic exchange interaction Eq. (17) in the thermody- namic limit, which after Laplace decoupling reproduces our result from the main text Eq. (19). To start, no- tice that we can decompose the summation P k into distinct terms containingsvirtual photons, via P k =P∞ s=0 P {P λ kλ=s}. The...
[2]

(19) has to be regularized due to the UV divergence of the PDOSρ(ω)∼ω 2 in the UVω→ ∞

Regularization Our result Eq. (19) has to be regularized due to the UV divergence of the PDOSρ(ω)∼ω 2 in the UVω→ ∞. Here, we provide two viewpoints and show their (pertur- bative) equivalence. Ab-Initio PerspectiveIn the main text, we argued that the coupling of our theory to the free space elec- tromagnetic field leads to a finite electronic mass and em...
[3]

Fabry Perot Cavity Consider an idealized Fabry Perot (FP) cavity with co-planer mirrors atz= 0, d, illustrated in the inset of Fig. 2(a). At these idealized surface, the parallel compo- nentoftheelectricfieldE ∥ = 0[Eq.(G5)]andthenormal component of the magnetic fieldB⊥ = 0[Eq. (G6)] van- ishes. Inamorerealisticsetup, themetallicmirrorscould be modeled as...
[4]

The cavity couples to the cavity embedded material (red) with distancezto the substrate

Surface Polariton Cavity Figure 6.Surface Cavityconstituted by an interface be- tween a dispersive medium (blue), modeled viaε(ω)in the lower planez <0and vacuum in the upper planez >0. The cavity couples to the cavity embedded material (red) with distancezto the substrate. In-plane we use periodic bound- ary conditions with system sizeL∥, while we impose...
[5]

Single mode cavity approximations are hence commonly used in prac- tice

Single Mode Limit The inclusion of many cavity modes, let alone all of them, is practically impossible in many applications be- cause of the exponential complexity scaling. Single mode cavity approximations are hence commonly used in prac- tice. Formally, the contribution of a single cavity mode vanishes in the thermodynamic limit, but the contribu- tiono...
[6]

The delta-distributed PDOS allows an explicit evaluation of the cavity correction function Eq

Evaluation for single mode limit For the surface mode contribution, the single-mode limit [Section G3] provides high-fidelity results. The delta-distributed PDOS allows an explicit evaluation of the cavity correction function Eq. (23), M(x) = ¯g2 e−xθ −1 ,(H1) withθ=ω ∞/Uand effective single-mode coupling¯g2 = P0ρ0ω2 ∞ ¯Ksurf(ω∞, z)— compare with Eq. (25)...
[7]

8(a)] and surface [Fig

Bulk Contribution The vast separation in scale between bulk [Fig. 8(a)] and surface [Fig. 8(b)] PDOS, suggest that bulk modes are irrelevant for the evaluation of the cavity induces magnetic exchange modificationsJ[Eq. (22)]. But while the surface modes are constrained to a small energy win- dow, bulk modes couple on a much larger energy scale. Figure 10....
[8]

Scaling of Surface Cavity The contribution resolved scaling of the surface cav- ity induced magnetic exchange modificationJ−J 0 is shown in Fig. 11. The dynamical dressing contribution shows a saturation behavior with weak linear decrease for small plasma frequenciesℏωp ≲UeV, while it is strongly quenched at largerω p. In this regime the total effect is s...

Show all 68 references

[9]

Magnon Raman scattering The Raman scattering cross-section for incident radi- ation of polarizationes and scattered radiation of polar- izatione s′ is given by [58, 59] Pss′(ω) = X n |⟨Ψn|HR(s, s′)|Ψ0⟩|2δ(ℏω+E 0 −E n), (I5) whereω=ω in −ω sc is the frequency difference be- twe...
[10]

Spin structure factor We also evaluate the spin structure factor, which is di- rectly related to the magnon dispersion, and measurable both in inelastic neutron scattering or resonant inelastic X-ray scattering experiments. More precisely, the spin structure factor is defined ...
[11]

I.-T. Lu, D. Shin, M. Kamper Svendsen, S. Latini, H. Hübener, M. Ruggenthaler, and A. Rubio, Advances in Optics and Photonics17, 441 (2025)

2025
[12]

Schlawin, D

F. Schlawin, D. M. Kennes, and M. A. Sentef, Applied Physics Reviews9, 011312 (2022), arXiv:2112.15018 [cond-mat, physics:quant-ph]

2022 arXiv
[13]

D. M. Kennes and A. Rubio, inSketches of Physics: The Celebration Collection, edited by R. Citro, M. Lewen- stein, A. Rubio, W. P. Schleich, J. D. Wells, and G. P. Zank (Springer International Publishing, Cham, 2023) pp. 1–39

2023
[14]

F. J. Garcia-Vidal, C. Ciuti, and T. W. Ebbesen, Science 373, eabd0336 (2021)

2021
[15]

G. Kipp, H. M. Bretscher, B. Schulte, D. Herrmann, K. Kusyak, M. W. Day, S. Kesavan, T. Matsuyama, X. Li, S. M. Langner, J. Hagelstein, F. Sturm, A. M. Potts, C. J. Eckhardt, Y. Huang, K. Watanabe, T. Taniguchi, A. Rubio, D. M. Kennes, M. A. Sentef, E. Baudin, G. Meier, M. H. ...

1926
[16]

G. Jarc, S. Y. Mathengattil, A. Montanaro, F. Giusti, E. M. Rigoni, R. Sergo, F. Fassioli, S. Winnerl, S. Dal Zilio, D. Mihailovic, P. Prelovšek, M. Eckstein, and D. Fausti, Nature622, 487 (2023)

2023
[17]

G. L. Paravicini-Bagliani, F. Appugliese, E. Richter, F. Valmorra, J. Keller, M. Beck, N. Bartolo, C. Rössler, T. Ihn, K. Ensslin, C. Ciuti, G. Scalari, and J. Faist, Nature Physics15, 186 (2019)

2019
[18]

Appugliese, J

F. Appugliese, J. Enkner, G. L. Paravicini-Bagliani, M. Beck, C. Reichl, W. Wegscheider, G. Scalari, C. Ciuti, and J. Faist, Science375, 1030 (2022)

2022
[19]

Enkner, L

J. Enkner, L. Graziotto, D. Boriçi, F. Appugliese, C. Re- ichl, G. Scalari, N. Regnault, W. Wegscheider, C. Ciuti, and J. Faist, Nature641, 884 (2025)

2025
[20]

Keren, T

I. Keren, T. A. Webb, S. Zhang, J. Xu, D. Sun, B. S. Y. Kim, D. Shin, S. S. Zhang, J. Zhang, G. Pereira, J. Yao, T. Okugawa, M. H. Michael, E. Viñas Boström, J. H. Edgar, S. Wolf, M. Julian, R. P. Prasankumar, K. Miya- gawa, K. Kanoda, G. Gu, M. Cothrine, D. Mandrus, M. Buzzi,...

2026
[21]

Vacuum-dressed superconductivity in nbn observed in a high-qterahertz cavity,

H. Xu, A. Baydin, Q. Yi, I.-T. Lu, N. Zhu, T. E. Kritzell, J. Doumani, D. Kim, F. Tay, A. Rubio, and J. Kono, “Vacuum-dressed superconductivity in nbn observed in a high-qterahertz cavity,” (2026), arXiv:2601.08191

2026
[22]

W. Gao, X. Li, M. Bamba, and J. Kono, Nature Pho- tonics12, 362–367 (2018)

2018
[23]

Dirnberger, R

F. Dirnberger, R. Bushati, B. Datta, A. Kumar, A. H. MacDonald, E. Baldini, and V. M. Menon, Nature Nan- otechnology17, 1060–1064 (2022)

2022
[24]

Dirnberger, J

F. Dirnberger, J. Quan, R. Bushati, G. M. Diederich, M. Florian, J. Klein, K. Mosina, Z. Sofer, X. Xu, A. Kamra, F. J. García-Vidal, A. Alù, and V. M. Menon, Nature620, 533–537 (2023)

2023
[25]

D. Kim, J. Hou, G. Lee, A. Agrawal, S. Kim, H. Zhang, D. Bao, A. Baydin, W. Wu, F. Tay, S. Huang, E. E. M. Chia, D.-S. Kim, M. Seo, A. D. Mohite, D. Hagen- müller, and J. Kono, Nature Communications16(2025), 10.1038/s41467-025-63810-7

2025 doi
[26]

Schäfer, M

C. Schäfer, M. Ruggenthaler, V. Rokaj, and A. Rubio, ACS Photonics7, 975 (2020)

2020
[27]

to put on, to be covered, to clothe/dress

V. Rokaj, M. Ruggenthaler, F. G. Eich, and A. Rubio, Physical Review Research4, 013012 (2022). [18]endyo(“to put on, to be covered, to clothe/dress”), and is used to denote matter degrees of freedom dressed by vacuum fluctuations. It was coined during the Flag- ship School on ...

2022
[28]

The- ory of Quantum Light-Matter Interaction in Cavities: Extended Systems and the Long Wavelength Approxi- mation,

M. K. Svendsen, M. Ruggenthaler, H. Hübener, C. Schäfer, M. Eckstein, A. Rubio, and S. Latini, “The- ory of Quantum Light-Matter Interaction in Cavities: Extended Systems and the Long Wavelength Approxi- mation,” (2023), arXiv:2312.17374

2023 arXiv
[29]

P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006)

2006
[30]

Purcell, Physical Review69, 681 (1964)

E. Purcell, Physical Review69, 681 (1964)

1964
[31]

Ruggenthaler, D

M. Ruggenthaler, D. Sidler, and A. Rubio, Chemical Reviews123, 11191 (2023)

2023
[32]

In the literature usually denoted as Helmholtz eigenvalue equation [56]
[33]

R. J. Glauber and M. Lewenstein, Physical Review A43, 467 (1991)

1991
[34]

Lenac, Physical Review A68, 063815 (2003)

Z. Lenac, Physical Review A68, 063815 (2003)

2003
[35]

C. R. Gubbin, S. A. Maier, and S. De Liberato, Physical Review B94, 205301 (2016)

2016
[36]

Bechler, Journal of Physics A: Mathematical and Gen- eral39, 13553 (2006)

A. Bechler, Journal of Physics A: Mathematical and Gen- eral39, 13553 (2006)

2006
[37]

B. A. Ferreira, B. Amorim, A. Chaves, and N. Peres, Physical Review A101, 033817 (2020)

2020
[38]

Longitudinal fields can also appeasr for non-dispersive, but spatially varying dielectrics

This is not unique for dispersive media. Longitudinal fields can also appeasr for non-dispersive, but spatially varying dielectrics
[39]

J. M. Luttinger, Physical Review84, 814 (1951)

1951
[40]

Dmytruk and M

O. Dmytruk and M. Schiró, Physical Review B103, 075131 (2021)

2021
[41]

Di Stefano, A

O. Di Stefano, A. Settineri, V. Macrì, L. Garziano, R. Stassi, S. Savasta, and F. Nori, Nature Physics15, 803 (2019)

2019
[42]

O. D. Stefano, A. Settineri, V. Macrì, L. Garziano, R. Stassi, S. Savasta, and F. Nori, Nature Physics20, 379 (2024)

2024
[43]

Garziano, A

L. Garziano, A. Settineri, O. Di Stefano, S. Savasta, and F. Nori, Physical Review A102, 023718 (2020)

2020
[44]

Stokes and A

A. Stokes and A. Nazir, Nature Physics20, 376 (2024)

2024
[45]

Savasta, O

S. Savasta, O. Di Stefano, A. Settineri, D. Zueco, S. Hughes, and F. Nori, Physical Review A103, 053703 (2021)

2021
[46]

J. Li, D. Golez, G. Mazza, A. J. Millis, A. Georges, and M. Eckstein, Physical Review B101, 205140 (2020)

2020
[47]

Cavity Spectroscopy for Strongly Correlated Systems,

L. Grunwald, E. V. Boström, M. K. Svendsen, D. M. Kennes, and A. Rubio, “Cavity Spectroscopy for Strongly Correlated Systems,” (2024), arXiv:2410.21515 [cond-mat]

2024
[48]

Weber, E

L. Weber, E. Viñas Boström, M. Claassen, A. Rubio, and D. M. Kennes, Communications Physics6, 1 (2023)

2023
[49]

M. A. Sentef, J. Li, F. Künzel, and M. Eckstein, Physical Review Research2, 033033 (2020)

2020
[50]

Viñas Boström, A

E. Viñas Boström, A. Sriram, M. Claassen, and A. Ru- bio, npj Computational Materials9, 1 (2023)

2023
[51]

D. M. Welakuh, V. Rokaj, M. Ruggenthaler, and A. Ru- 24 bio, Physical Review Research7, 013093 (2025)

2025
[52]

Li and M

J. Li and M. Eckstein, Phys. Rev. Lett.125, 217402 (2020)

2020
[53]

Formally, this follows from the ob- servation thatf(x 1, x2) =f 1(x1)f2(x2)if and only if ∂x1 ∂x2 log(f(x 1, x2)) = 0

An algebraic factorization of the summations at finite ˜Uis impossible. Formally, this follows from the ob- servation thatf(x 1, x2) =f 1(x1)f2(x2)if and only if ∂x1 ∂x2 log(f(x 1, x2)) = 0. For our function, this relation is not fulfilled, implying that it cannot be factorize...
[54]

Here regularity is enforced by an explicit Gaussian cutoff, which we sent to zero at the end of our calculation
[55]

M. K. Svendsen, K. S. Thygesen, A. Rubio, and J. Flick, Journal of Chemical Theory and Computation20, 926 (2024)

2024
[56]

Surface-mediated ultra-strong cavity coupling of two-dimensional itinerant electrons,

C. J. Eckhardt, A. Grankin, D. M. Kennes, M. Ruggen- thaler, A. Rubio, M. A. Sentef, M. Hafezi, and M. H. Michael, “Surface-mediated ultra-strong cavity coupling of two-dimensional itinerant electrons,” (2024), arXiv:2409.10615 [cond-mat, physics:quant-ph]

2024 arXiv
[57]

Equilibrium non-linear phononics by elec- tric field fluctuations of terahertz cavities,

E. V. Boström, M. H. Michael, C. Eckhardt, and A. Rubio, “Equilibrium non-linear phononics by elec- tric field fluctuations of terahertz cavities,” (2024), arXiv:2409.19063

2024 arXiv
[58]

Eckhardt,Cavitronics in Low-Dimensional Systems, Ph.D

C. Eckhardt,Cavitronics in Low-Dimensional Systems, Ph.D. thesis, RWTH Aachen University (2024)

2024
[59]

Cavity control of multiferroic order in single-layer NiI$_2$,

C. Fan, E. V. Boström, X. Cheng, L. Grunwald, Z. Zhang, D. M. Kennes, D. N. Basov, and A. Ru- bio, “Cavity control of multiferroic order in single-layer NiI$_2$,” (2026), arXiv:2602.09641 [cond-mat]

2026
[60]

Anyonic Chern insulator in graphene induced by surface electromagnon vacuum fluc- tuations,

X. Cheng, E. V. Boström, F. Y. Gao, E. Baldini, D. M. Kennes, and A. Rubio, “Anyonic Chern insulator in graphene induced by surface electromagnon vacuum fluc- tuations,” (2025), arXiv:2511.10836 [cond-mat]

2025
[61]

Scheel and S

S. Scheel and S. Buhmann, Acta Physica Slovaca. Re- views and Tutorials58(2008), 10.2478/v10155-010-0092- x

2008 doi
[62]

J. R. Schrieffer and P. A. Wolff, Physical Review149, 491 (1966)

1966
[63]

Bravyi, D

S. Bravyi, D. P. DiVincenzo, and D. Loss, Annals of Physics326, 2793 (2011)

2011
[64]

Zinn-Justin,Quantum Field Theory and Critical Phe- nomena, 4th ed., International Series of Monographs on Physics No

J. Zinn-Justin,Quantum Field Theory and Critical Phe- nomena, 4th ed., International Series of Monographs on Physics No. 113 (Clarendon Press ; Oxford University Press, Oxford : New York, 2002)

2002
[65]

J. D. Jackson,Classical Electrodynamics, 3rd ed. (Wiley, Hoboken, NY, 2009)

2009
[66]

(23), this is not true for the magnetic modification Eq

Note that while the PDOS enter additively in the cavity modification function Eq. (23), this is not true for the magnetic modification Eq. (22)!
[67]

E. V. n. Boström, T. S. Parvini, J. W. McIver, A. Rubio, S. V. Kusminskiy, and M. A. Sentef, Phys. Rev. B104, L100404 (2021)

2021
[68]

Viñas Boström, T

E. Viñas Boström, T. S. Parvini, J. W. McIver, A. Rubio, S. V. Kusminskiy, and M. A. Sentef, Phys. Rev. Lett. 130, 026701 (2023)

2023

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

(17) in the thermody- namic limit, which after Laplace decoupling reproduces our result from the main text Eq

Scaling Derivation We provide an alternative expression for the mag- netic exchange interaction Eq. (17) in the thermody- namic limit, which after Laplace decoupling reproduces our result from the main text Eq. (19). To start, no- tice that we can decompose the summation P k into distinct terms containingsvirtual photons, via P k =P∞ s=0 P {P λ kλ=s}. The...

[2] [2]

(19) has to be regularized due to the UV divergence of the PDOSρ(ω)∼ω 2 in the UVω→ ∞

Regularization Our result Eq. (19) has to be regularized due to the UV divergence of the PDOSρ(ω)∼ω 2 in the UVω→ ∞. Here, we provide two viewpoints and show their (pertur- bative) equivalence. Ab-Initio PerspectiveIn the main text, we argued that the coupling of our theory to the free space elec- tromagnetic field leads to a finite electronic mass and em...

[3] [3]

Fabry Perot Cavity Consider an idealized Fabry Perot (FP) cavity with co-planer mirrors atz= 0, d, illustrated in the inset of Fig. 2(a). At these idealized surface, the parallel compo- nentoftheelectricfieldE ∥ = 0[Eq.(G5)]andthenormal component of the magnetic fieldB⊥ = 0[Eq. (G6)] van- ishes. Inamorerealisticsetup, themetallicmirrorscould be modeled as...

[4] [4]

The cavity couples to the cavity embedded material (red) with distancezto the substrate

Surface Polariton Cavity Figure 6.Surface Cavityconstituted by an interface be- tween a dispersive medium (blue), modeled viaε(ω)in the lower planez <0and vacuum in the upper planez >0. The cavity couples to the cavity embedded material (red) with distancezto the substrate. In-plane we use periodic bound- ary conditions with system sizeL∥, while we impose...

[5] [5]

Single mode cavity approximations are hence commonly used in prac- tice

Single Mode Limit The inclusion of many cavity modes, let alone all of them, is practically impossible in many applications be- cause of the exponential complexity scaling. Single mode cavity approximations are hence commonly used in prac- tice. Formally, the contribution of a single cavity mode vanishes in the thermodynamic limit, but the contribu- tiono...

[6] [6]

The delta-distributed PDOS allows an explicit evaluation of the cavity correction function Eq

Evaluation for single mode limit For the surface mode contribution, the single-mode limit [Section G3] provides high-fidelity results. The delta-distributed PDOS allows an explicit evaluation of the cavity correction function Eq. (23), M(x) = ¯g2 e−xθ −1 ,(H1) withθ=ω ∞/Uand effective single-mode coupling¯g2 = P0ρ0ω2 ∞ ¯Ksurf(ω∞, z)— compare with Eq. (25)...

[7] [7]

8(a)] and surface [Fig

Bulk Contribution The vast separation in scale between bulk [Fig. 8(a)] and surface [Fig. 8(b)] PDOS, suggest that bulk modes are irrelevant for the evaluation of the cavity induces magnetic exchange modificationsJ[Eq. (22)]. But while the surface modes are constrained to a small energy win- dow, bulk modes couple on a much larger energy scale. Figure 10....

[8] [8]

Scaling of Surface Cavity The contribution resolved scaling of the surface cav- ity induced magnetic exchange modificationJ−J 0 is shown in Fig. 11. The dynamical dressing contribution shows a saturation behavior with weak linear decrease for small plasma frequenciesℏωp ≲UeV, while it is strongly quenched at largerω p. In this regime the total effect is s...