Floquet Theory of lattice electrons coupled to an off-resonant cavity

Gian Marcello Andolina; Jules Sueiro; Marco Schir\`o

REVIEW 3 major objections 6 minor 76 references

Floquet Theory of lattice electrons coupled to an off-resonant cavity

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A quantum cavity mode, even in vacuum, mediates electron-electron interactions that are invisible to classical-light treatments, and these interactions become the dominant correction to the SSH topological phase boundary at strong coupling.

desk verdict Careful HFE machinery for off-resonant cavity electrons, but the strong-coupling SSH conclusions outrun the truncation and need an independent many-body benchmark. read the letter →

arxiv 2507.22715 v2 pith:FGDD4NM5 submitted 2025-07-30 cond-mat.str-el cond-mat.quant-gasquant-ph

Jules Sueiro , Gian Marcello Andolina , Marco Schirò This is my paper

classification cond-mat.str-elcond-mat.quant-gasquant-ph

keywords Floquettheoryhigh-frequencyexpansioncavityquantumelectrodynamicscavity-mediatedinteractionsSu-Schrieffer-Heegermodeltopologicalphasediagramlight-matterentanglementoff-resonant

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that when lattice electrons are coupled to a single off-resonant cavity mode, the quantum nature of the cavity field cannot be reduced to a renormalized hopping. Using Floquet theory and the high-frequency expansion, it derives an effective electron-only Hamiltonian in which vacuum fluctuations generate both long-range hopping and cavity-mediated electron-electron interactions, the latter encoded in a first-order commutator term that vanishes for a classical field. Applied to a Su-Schrieffer-Heeger chain, the first-order interactions become the dominant correction at sufficiently strong light-matter coupling for fixed cavity frequency, shifting the topological phase boundary and producing light-matter entanglement that a photon mean-field ansatz misses. The paper also extends the same construction to a cavity driven at its resonance frequency, where the effective dynamics acquire a squeezing term coupled to an electronic current.

What carries the argument

The load-bearing object is the Floquet high-frequency expansion in the rotating frame: the Hamiltonian becomes time-periodic, and its Fourier coefficients are dressed hopping operators built from the operators $\hat\gamma_m(g)$, the $m$-photon components of the Peierls-phase exponential. The first-order correction $[\hat H_m,\hat H_{-m}]/(m\omega_c)$ is evaluated using the tensor-product commutator identity, which produces both a hopping channel and an interaction channel; after normal ordering the interaction channel is a quartic fermion term whose coefficient is a commutator of photon operators, so it exists only for quantum light. Because the projected effective Hamiltonian conserves photon number, projection onto a fixed photon sector is exact, and the Van Vleck generating unitary, the analogue of the Floquet micromotion, is needed to recover lab-frame observables such as photon squeezing and light-matter entanglement. For the SSH chain, the machinery is closed by a Hartree-Fock decoupling of the induced interactions that maps the system back to an effective SSH model with self-consistent hoppings $V_{\rm eff}$ and $W_{\rm eff}$.

What would settle it

Diagonalize the coupled light-matter Hamiltonian exactly for a small SSH chain (for example, $L=4$ to $6$ cells with photon number truncated at $N_{\rm ph}\sim 10$ to $20$) and compute the topological phase boundary as a function of $g$ at fixed $\omega_c$; if the boundary follows the zeroth-order or mean-field curve rather than the first-order HFE curve, or if including the second-order HFE correction changes the boundary by as much as the first-order correction does, the central claim would be falsified.

Watch

Extended reading notes

Core claim

The central claim is that the first-order high-frequency expansion for a single-mode off-resonant cavity, $\hat H_F = \hat H_{m=0} + \sum_{m>0} [\hat H_m,\hat H_{-m}]/(m\omega_c) + O(1/\omega_c^2)$, splits the $1/\omega_c$ correction into two physically distinct pieces: cavity-induced hopping (Eq. 18) and a genuine cavity-mediated electron-electron interaction (Eq. 19), built from commutators of photon operators and quartic fermion terms. Because those commutators vanish identically when the vector potential is a classical number, the interaction term is a direct signature of quantized light. Projected onto the vacuum photon sector, the effective model contains hopping-hopping and current-current interactions whose coefficients decay as power laws at large $g$ while the zeroth-order hopping decays exponentially, so at fixed cavity frequency the first-order terms can dominate at strong coupling. In the SSH application, solving the resulting interacting model by Hartree-Fock self-consistency yields a topological phase boundary that first agrees with previous mean-field results at weak coupling and then deviates strongly at large $g$, a deviation the paper attributes to light-matter entanglement captured through the Floquet micromotion and its Van Vleck unitary equivalent.

Load-bearing premise

The argument rests on assuming that stopping the expansion at first order in the ratio of hopping to cavity frequency is safe in exactly the strong-coupling regime where those first-order terms grow, and that a self-consistent mean-field treatment of the induced interactions is accurate; no estimate is given for the neglected higher-order or beyond-mean-field contributions.

Editorial extensions

If this is right

Off-resonant cavities can act as mediators of electron-electron interactions even when no real photons are present, so effective low-energy models of cavity-embedded materials must include these terms beyond the usual renormalized hopping.
At fixed cavity frequency, sufficiently strong light-matter coupling makes the first-order interaction corrections dominate over the zeroth-order hopping, so truncating the high-frequency expansion at zeroth order is unreliable in that regime.
The topological phase boundary of the SSH chain coupled to a cavity is not correctly captured at large $g$ by the electron-photon mean-field ansatz of earlier work; the first-order HFE predicts a boundary that changes non-monotonically with $g$ and depends on system size.
Light-matter entanglement, quantified by the Renyi entropy of the reduced electronic state, is controlled mainly by fluctuations of the lowest-order current operator $\hat J_1$, with higher-order Peierls terms contributing only a slowly decaying tail.
For a cavity driven at its resonance frequency, or for a matching laser drive on the electrons, the effective dynamics decompose into photon-number sectors and produce observable consequences such as current-coupled squeezing and $g$-dependent thermalization of edge-state occupations.

Reading between the lines

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

Because the first-order interaction coefficient grows logarithmically in $g^2/L$ while the small parameter is $t/\omega_c$, the same strong-coupling regime in which the paper finds dramatic effects is likely where the second-order $(t/\omega_c)^2$ terms become non-negligible; benchmarking against second-order HFE or exact diagonalization would either confirm or constrain the phase diagram.
The identification of the interaction channel with a commutator of the vector potential suggests a general selection rule: any approximation that linearizes the Peierls phase in the vector potential, keeping only the paramagnetic term, will miss the diamagnetic and density-density interaction channels, so the hierarchy of current versus density interactions can differ between linearized and non-li
The driven-cavity result that a matching laser is equivalent to shifting the cavity into a coherent state implies that the observable heating of the electronic system is controlled by the spread of the coherent state across photon-number sectors; measuring the photon number distribution after driving would directly test this quantum-Rabi picture.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The paper is a serious HFE derivation for electrons coupled to a single off-resonant cavity mode. The first-order effective Hamiltonian gives cavity-mediated hopping and electron-electron interactions in explicit normal-ordered form, and the equivalence to Van Vleck and quantum-Floquet is shown cleanly. The classical limit recovers known Floquet/Peierls results, which is a good sanity check. If you work on cavity-embedded lattice models, this is a useful tool to cite.

What is actually new: the explicit single-mode first-order commutator expansion and the separation into induced hopping and interactions, plus the SSH application with the first-order interactions. The vacuum-projected coefficients in Eq. (24) — the Kodd/Keven current-current and hopping-hopping terms — are new and physically informative.

The soft spot is the strong-coupling claim. The HFE is controlled by t/omega_c, but the g-dependence enters through gamma_m. In the strong-coupling sector, first-order coefficients grow like log x while zeroth-order hopping decays exponentially in x, so first-order dominance happens exactly where the zeroth-order piece is exponentially small. The paper gives no estimate showing that second- and higher-order HFE terms are negligible in that regime. The numerical phase diagram then relies on a self-consistent Hartree-Fock decoupling of the induced interactions. HF is uncontrolled at large g, and the only ED benchmark in the literature (Ref. [38]) is single-particle, which the paper itself correctly says discards all cavity-mediated interactions. So the central conclusion — that mean-field fails and interactions modify the topological boundary — is plausible but not yet established. It needs a small-system ED or MPS calculation at finite g to check.

I want to be clear: the derivation itself looks sound. I checked the HFE algebra and the Van Vleck equivalence; nothing is internally contradictory. The paper is careful and flags many of its own limitations (e.g., App. C). The issue is specifically that the application reaches a physics conclusion in a regime where the method's own expansion parameter is not small and the many-body solver is uncontrolled.

Bottom line: this deserves peer review — the framework is a genuine contribution and the physics question is important. But the referee should push for a benchmark of the strong-coupling phase diagram before claims about mean-field failure are accepted. For my own work, I'd cite the HFE machinery, not the phase diagram.

Referee Report

3 major / 6 minor

Summary. The manuscript develops a Floquet high-frequency expansion (HFE) for electrons coupled to a single-mode off-resonant cavity, treating the Peierls-phase coupling exactly in the bosonic operators. The zeroth-order HFE gives a photon-conserving hopping renormalization, and the first-order HFE produces cavity-mediated hopping and two-body electron-electron interactions (Eq. (18)-(19)). The authors connect the expansion to Van Vleck perturbation theory and to the quantum Floquet picture, and they use the micromotion to compute light-matter entanglement (Eqs. (32)-(34)). The formalism is applied to an SSH chain in a uniform cavity: at zeroth order one obtains exponential hopping renormalization, and at first order a Hartree-Fock decoupling yields a self-consistent SSH Hamiltonian with modified topological phase boundaries (Figs. 5-6). The paper also treats driven-cavity and driven-electron cases. The central physical claim is that first-order cavity-mediated interactions dominate at sufficiently strong coupling for fixed cavity frequency, leading to deviations from the light-matter mean-field result and to sizeable light-matter entanglement.

Significance. If the truncation and mean-field steps are valid, the paper provides a transparent, parameter-free framework for deriving effective electron-only Hamiltonians in cavity QED, with explicit analytic coefficients (I0, K0, f), a clean classical limit that recovers known Floquet-Peierls results, and a direct connection between Floquet micromotion and light-matter entanglement. The equivalence proofs with Van Vleck perturbation theory and quantum Floquet theory are a useful consolidation. The SSH application yields falsifiable predictions for the phase boundary as a function of g, omega_c, L, and b0. However, the main strong-coupling conclusion currently rests on an unvalidated first-order HFE truncation and an uncontrolled Hartree-Fock decoupling, and the boundary conditions in the SSH calculation are not specified consistently.

major comments (3)

[Sec. III, Eq. (8); Sec. VI A 2, Fig. 5] The HFE is truncated at first order in t/omega_c, but the regime where the paper concludes that first-order terms dominate is precisely where the hierarchy of the expansion is not established. For large x = g^2 b0^2/L, the zeroth-order hopping in Eq. (51) decays as t e^{-x/2}, while the first-order coefficients f(x) and K_odd from Fig. 1 grow or decay only logarithmically or as 1/x; hence the ratio of first- to zeroth-order terms grows with x. For the parameters of Fig. 5b (b0=0.75, L=100, g up to 10) this ratio reaches order one, so the first-order interaction is not a small perturbation. However, no estimate or numerical check is supplied for the O(1/omega_c^2) terms of the HFE, which involve additional products of the gamma_m operators and can in principle have matrix elements that grow with g. Because the paper's central claim is that first-order effects modify the topological phase boundary, the authors should either compute the next-order HFE for a simplified case or benchmark the first-order result against an exact diagonalization of the original light-matter Hamiltonian at small L with truncated photon number.
[Sec. VI A 2, Eq. (56); App. B 1] The phase diagram in Fig. 5 is obtained by a Hartree-Fock decoupling of the first-order interactions that keeps a single 'global' channel and drops the 'local' channel on the basis of an L versus L^2 counting argument. This counting is not by itself quantitative: the coefficients A_{k,k'} in Eq. (55) carry powers of 1/L, so the number of terms alone does not determine the magnitude of the neglected channel. More importantly, the HF decoupling of a strong interaction is uncontrolled in one dimension. At the parameters of Fig. 5b (omega_c=20, L=100, g around 10) the first-order interaction scale t^2 f(x)/omega_c is comparable to the renormalized hopping t e^{-x/2}, so the self-consistent HF solution need not be close to the exact ground state of Eq. (54). To support the claim that the light-matter mean-field ansatz of Ref. [13] fails, the authors should benchmark the HF treatment against an exact or DMRG solution of Eq. (54) for small L, or against an exact solution of the original light-matter model.
[Sec. VI, Eq. (49) vs Eq. (54)] The manuscript does not specify the boundary conditions used in the SSH application. The Hamiltonian in Eq. (49) has inter-cell hoppings summed from j=1 to L-1, which describes an open chain, whereas the Fourier-transformed first-order Hamiltonian in Eq. (54) and the self-consistent equations in App. B assume translation invariance, which requires periodic boundary conditions. These are inconsistent for a finite chain, and for open boundary conditions the assumption <c^dagger_{k,A} c_{k',B}> proportional to delta_{k,k'} used in the Hartree-Fock decoupling is not valid. The authors should state the boundary conditions, use periodic boundary conditions consistently if the phase diagram is meant to describe the bulk, or explain how the thermodynamic limit is taken.

minor comments (6)

[Sec. III, Eqs. (17)-(20)] The presentation alternates between the normal-ordered form of the first-order Hamiltonian, Eqs. (17)-(19), and the non-normal-ordered form, Eq. (20), and it is not always stated which convention is used in the later projections and in App. B; please clarify.
[Fig. 1] The curves in Fig. 1 are not labeled; add a legend or specify the line styles in the caption, and state the parameter values for which the asymptotic 1/(2g^2) curve is drawn.
[Sec. VI A 2, Eq. (55)] The function f(x) introduced in Eq. (55) is the same as K_0(g,g) from Sec. III A 1; unify the notation by defining f(x) once and referring back to it.
[Sec. VI B 2, Eq. (60)] The coefficients c_n in Eq. (60) are the expansion coefficients of a coherent state; state explicitly that the initial cavity state is taken to be |alpha0> and relate alpha0 to the drive amplitude eta in the text preceding Eq. (60).
[App. B 4] There is a typo in the first sentence of the appendix: 'Hatree-Fock' should be 'Hartree-Fock'.
[Sec. VI A 2, Eq. (54)] The Hamiltonian in Eq. (54) is written with sums over k and k' without explicitly restricting the domain of k; specify that k runs over the Brillouin zone and whether boundary or zero-momentum terms require special care.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the effective Hamiltonian coefficients are parameter-free functions of the model inputs; no fitted quantity is recycled as a prediction.

full rationale

The paper's central derivation is self-contained. The high-frequency expansion formula in Eq. (8) is taken from the standard result of Ref. [41], and the first-order coefficients I_0(g) and K_0(g,g') in Eqs. (22)-(23) are explicit integrals over the microscopic parameters (t_ij, g, omega_c), with no adjustable parameters and no target observable used as an input. The SSH phase diagram is obtained by solving the self-consistent Hartree-Fock equations in Appendix B for V_eff and W_eff; this is a mean-field closure, not a fit of the phase boundary. The comparison with the earlier mean-field treatment of Ref. [13] is used as a benchmark, not as a premise proving the HFE result; and the exact-diagonalization comparison of Ref. [38] is explicitly identified in Appendix C as restricted to the single-particle sector, so it does not constrain the many-body interaction claim. The main substantive weakness, namely that the first-order HFE truncation is not benchmarked against second- and higher-order terms in the strong-coupling regime where first-order corrections dominate, is a validity and robustness concern rather than circularity: the claimed dominance follows from comparing the computed zeroth-order and first-order terms, not from defining the first-order terms to be dominant. Self-citations appear (Refs. [13], [24], [27], [60]) but none is load-bearing for the derivation; the gauge-invariance caveat and the flow-equation suggestion are auxiliary. Hence no step reduces by construction to its own input.

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

The paper introduces no fitted constants and postulates no new physical entities. It relies on standard HFE and Van Vleck perturbation theory, the Peierls-phase light-matter coupling model, and the assumptions that the cavity frequency is the dominant scale and that the cavity occupies the zero-photon sector after block diagonalization. The numerical application adds a Hartree-Fock decoupling of the induced interactions, which is an uncontrolled approximation.

assumptions (5)

standard math The high-frequency expansion formula H_F = H_0 + sum_{m>0} [H_m, H_-m]/(m omega_c) is taken from Ref. [41] without proof.
The paper relies on this standard Floquet result to build the effective Hamiltonian; it is cited, not derived.
domain assumption The light-matter coupling enters through the full Peierls phase exp(-i(g a^dagger + g* a)), with no truncation to linear order in g.
This defines the model of Eq. (3), and the nonperturbative treatment of g inside each hopping term is essential for the g^4 diamagnetic terms in the interactions.
domain assumption The cavity frequency omega_c is the dominant energy scale, so the expansion parameter delta = t/omega_c is small and truncation at order 1/omega_c is valid.
This off-resonant condition is stated in Section II; the application later enters a regime where first-order terms dominate over zeroth-order terms, which strains this assumption.
domain assumption In equilibrium the physical cavity state lies in the zero-photon sector after the Van Vleck block diagonalization, so projecting the photon-conserving effective Hamiltonian onto n=0 is exact.
The projection in Section III A 1 relies on photon-number conservation of H_F and the choice of the vacuum sector; the physical photon state is recovered only through the unitary transformation.
ad hoc to paper The Hartree-Fock decoupling of the first-order interactions, including the neglect of the local channel, correctly captures the ground state of the effective Hamiltonian.
This approximation is introduced in Section VI A to make the many-body problem tractable; no exact benchmark is provided for the strong-coupling regime.

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Cite this review

Pith. "Pith review of Floquet Theory of lattice electrons coupled to an off-resonant cavity." pith.science (2026). https://pith.science/paper/FGDD4NM5

@misc{pith2026250722715,
  author       = {Pith},
  title        = {Pith review of: Floquet Theory of lattice electrons coupled to an off-resonant cavity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FGDD4NM5}},
  note         = {Machine review of arXiv:2507.22715}
}

read the original abstract

We use Floquet theory and the High-Frequency expansion to derive an effective Hamiltonian for electrons coupled to an off resonant cavity mode, either in its vacuum or driven by classical light. For vacuum fields, we show that long-range hopping and cavity-mediated interactions arise as a direct consequence of quantum fluctuations. As an application, this method is applied to the Su-Schrieffer-Heeger (SSH) model. At high light-matter coupling, our results reveal significant deviations from mean-field predictions, with our framework capturing light-matter entanglement through the Floquet micromotion. Furthermore, the cavity-mediated interactions appearing at first order are shown to be crucial to the description of the system at sufficiently strong light-matter coupling for a fixed cavity frequency. Finally, a drive resonant with the cavity is added with the SSH chain displaying dynamical behavior dependent on the cavity parameters.

Figures

Figures reproduced from arXiv: 2507.22715 by the authors.

**Figure 2.** FIG. 2: Sketch of the SSH chain embedded into a single [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 4.** FIG. 4: Magnitude of the different terms of the mean [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5: Topological phase diagram of the SSH chain cou [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: FIG. 6: Topological phase boundary for the SSH chain [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 8.** Figure 8: FIG. 8: Dynamics of the probability to switch edge state [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

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Reference graph

Works this paper leans on

76 extracted references · 60 canonical work pages

[38]

G. M. Andolina, F. M. D. Pellegrino, V. Giovan- netti, A. H. MacDonald, and M. Polini, Phys. Rev. B102, 125137 (2020), arXiv:2005.09088 [cond-mat, physics:quant-ph]

work page Pith review arXiv 2020
[13]

57 The effective hoppingsV ef f andW ef f appearing in the First order effective Hamiltonian within mean-field of Eq

Self-consistent equation for the effective hoppings of Eq. 57 The effective hoppingsV ef f andW ef f appearing in the First order effective Hamiltonian within mean-field of Eq. 57 are defined self-consistently through : 19 Vef f=v ef f+ 2v2 ef f ωc f( g2b2 0 L ) D ˆTi.c. E +f(− g2b2 0 L ) D ˆT † i.c. E − 2vef fwef f ωc f g2 L b0(1−b 0) D ˆT † e.c. E +f − ...

work page
[1]

In all generality, the indexiregroups the lattice site ⃗Ri, the spinσand if needed sub-lattice indexκ

The tight-binding model coupled through Peierls phase The electrons on a lattice are described by a tight- binding model characterized by the hopping integrals from siteito sitejdenoted byt i,j. In all generality, the indexiregroups the lattice site ⃗Ri, the spinσand if needed sub-lattice indexκ. Formally, the indexiis a triplet :i= ( ⃗Ri, σ, κ). Sums wil...

work page
[2]

On thenphoton sector, the Floquet Hamiltonian of 0 1 2 3 4 5 6 g 0.0 0.5 1.0 Kodd 0 (g, g) Keven 0 (g, g) I0(g) 1/2g2 FIG

Vacuum cavity Since the Floquet Hamiltonian is photon conserving, the projection on a fixed photon number sector is ex- act. On thenphoton sector, the Floquet Hamiltonian of 0 1 2 3 4 5 6 g 0.0 0.5 1.0 Kodd 0 (g, g) Keven 0 (g, g) I0(g) 1/2g2 FIG. 1: Plot of the coefficients appearing in the vac- uum cavity effective Hamiltonian.I 0(g) controls the zero o...

work page
[3]

In this regime the light-matter coupling typically scales asg i,j ∼1/ √ V

Vacuum cavity in the thermodynamic limit It is interesting to discuss the effect of cavity mediated processes in the thermodynamic limit, where the system sizeV− → ∞. In this regime the light-matter coupling typically scales asg i,j ∼1/ √ V. In this limit, the be- havior of the cavity-mediated interaction detailed above show that the leading term is given...

work page
[4]

Classical limit We now consider the classical limit of a cavity in a co- herent state with many photons, as discussed in Ref. [36]. We expect that in this regime the cavity-mediated inter- actions discussed in the previous section would vanish, since it is known that for Floquet driven non-interacting electrons a classical drive can only mediate long-rang...

work page
[5]

Driving with a matching laser mode In the case where the cavity mode and the laser mode coincide, e.g. uniform, so that one can write : gi,jα0 =−η i,j.(42) In other words, the cavity vector potentialA cav(r) and the laser oneA las(r) have to verify : α0 × Z Rj Ri Acav(r)·dr=− Z Rj Ri Alas(r)·dr,(43) so that within the long wavelength approximation, the co...

work page
[6]

The hamiltonian of Eq

Driving with space-dependent light Alernatively, one can imagine the opposite situation where the laser light varies differently that the cavity mode, with its amplitude and phase being modulated in space. The hamiltonian of Eq. 40 can be treated in a manner similar to what is introduced in Ref. [40]. The driving term in Hamiltonian Eq. 41 is written as t...

work page

Show all 76 references

[7]

≡ LX i=1 c† i,Aci,B,(50a) ˆTe.c

Zeroth order Lets us define the following intra- and extra unit-cell hopping global operators : ˆTi.c. ≡ LX i=1 c† i,Aci,B,(50a) ˆTe.c. ≡ L−1X i=1 c† i+1,Aci,B.(50b) So that, at zeroth order in the HFE, the effective Hamil- tonian for the electrons in then-photon sector is giv...
[8]

local” channel of the Hartree-Fock decoupling with respect to the other “global

First order As detailed by Eq. (21), the effective electron-electron interactions appearing in the first order Floquet Hamil- tonian are constructed from terms which couple two hop- ping term initially present in the tight-binding model of Eq.3 with a coefficient controlled by...
[9]

Driving the cavity at its second harmonic This section aims to illustrate the idea developed in Sect. V. We consider the SSH chain embedded in a cavity as described by Eq. (49). The cavity is coherently driven at 2ω c with an amplitudeη. In the notations of Sect. V we can writ...
[10]

V A 1 is applied to the SSH chain

Driving the electrons In this section the formalism developed in Sect. V A 1 is applied to the SSH chain. Specifically, we consider a SSH chain coupled to a single-mode off-resonant cavity just like in Sect. VI. The assumption is made that the coefficients of the modelv,w,ω c,...

2020
[11]

The perturbation will be controlled by a small dimensionless parameterδ≪1 such that ˆV∼δω c

Derivation of the effective Hamiltonian through V an Vleck perturbation theory Very generally, the light-matter Hamiltonian is written as : ˆH= ˆV+1⊗ˆnω c,(A1) withω c being the dominant energy scale of the Hamil- tonian. The perturbation will be controlled by a small dimensio...
[12]

To do so, we are going to link ˆHm defined in 7 with the Quan- tum Floquet Hamiltonian ˆHn,m

Details of the equivalence to the Quantum Floquet picture Let us show that, for a single mode cavity, this formula is equivalent to the Floquet / Van Vleck one. To do so, we are going to link ˆHm defined in 7 with the Quan- tum Floquet Hamiltonian ˆHn,m. First, let us insert t...
[14]

We consider a non-interacting fermion problem which is diagonalized in the basis|µ⟩with associated creation and annihilation operatorsc † µ andc µ

Evaluation of the fluctuations of a one-body observable on a non-interacting ground state. We consider a non-interacting fermion problem which is diagonalized in the basis|µ⟩with associated creation and annihilation operatorsc † µ andc µ. We consider a one body operator : ˆJ= ...
[15]

Calculations for the entropy To compute the Renyi entenglement entropy in the ground state of the SSH chain coupled to an off-resonant cavity, we use the result obtained in Sect. IV. In the par- ticular case of the SSH model, the ˆJn operator defined in Eq. 32 is given by : ˆJ...
[16]

VI B 2 through a time-dependent mean-field

Details of the time-dependent mean-field This appendix details the procedure used to compute the dynamics of electron observables in Sect. VI B 2 through a time-dependent mean-field. In thenphoton sector, the dynamics of the electrons are controlled by the effective Hamiltonia...
[17]

Mukamel, Annual Review of Physical Chemistry41, 647 (1990)

S. Mukamel, Annual Review of Physical Chemistry41, 647 (1990)

1990
[18]

Giannetti, M

C. Giannetti, M. Capone, D. Fausti, M. Fabrizio, F. Parmigiani, and D. M. and, Advances in Physics65, 58 (2016), https://doi.org/10.1080/00018732.2016.1194044

2016
[19]

Nicoletti and A

D. Nicoletti and A. Cavalleri, Adv. Opt. Photon.8, 401 (2016)

2016
[20]

J. W. McIver, B. Schulte, F.-U. Stein, T. Matsuyama, G. Jotzu, G. Meier, and A. Cavalleri, Nat. Phys.16, 38 (2020), publisher: Nature Publishing Group

2020
[21]

Fausti, R

D. Fausti, R. I. Tobey, N. Dean, S. Kaiser, A. Dienst, M. C. Hoffmann, S. Pyon, T. Takayama, H. Takagi, and A. Cavalleri, Science331, 189 (2011), publisher: Ameri- can Association for the Advancement of Science

2011
[22]

Mitrano, A

M. Mitrano, A. Cantaluppi, D. Nicoletti, S. Kaiser, A. Perucchi, S. Lupi, P. Di Pietro, D. Pontiroli, M. Ricc` o, S. R. Clark, D. Jaksch, and A. Cavalleri, Nature530, 461 (2016)

2016
[23]

Oka and S

T. Oka and S. Kitamura, Annual Review of Con- densed Matter Physics10, 387 (2019), eprint: https://doi.org/10.1146/annurev-conmatphys-031218- 013423

2019 doi
[24]

Brune, F

M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Ha- gley, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 76, 1800 (1996)

1996
[25]

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
[26]

J. B. Curtis, Z. M. Raines, A. A. Allocca, M. Hafezi, and V. M. Galitski, Phys. Rev. Lett.122, 167002 (2019)

2019
[27]

X. Wang, E. Ronca, and M. A. Sentef, Phys. Rev. B99, 235156 (2019), arXiv:1903.00339 [cond-mat]

2019 arXiv
[28]

Ciuti, Phys

C. Ciuti, Phys. Rev. B104, 155307 (2021)

2021
[29]

Dmytruk and M

O. Dmytruk and M. Schir` o, Commun Phys5, 1 (2022), number: 1 Publisher: Nature Publishing Group

2022
[30]

Nguyen, G

D.-P. Nguyen, G. Arwas, and C. Ciuti, Phys. Rev. B 110, 195416 (2024)

2024
[31]

Rokaj, J

V. Rokaj, J. Wang, J. Sous, M. Penz, M. Ruggenthaler, and A. Rubio, Phys. Rev. Lett.131, 196602 (2023)

2023
[32]

Fausti, G

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

2023
[33]

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

2021
[34]

G. L. Paravicini-Bagliani, F. Appugliese, E. Richter, F. Valmorra, J. Keller, M. Beck, N. Bartolo, C. R¨ ossler, T. Ihn, K. Ensslin, C. Ciuti, G. Scalari, and J. Faist, Na- ture Phys15, 186 (2019), arXiv:1805.00846 [cond-mat, physics:quant-ph]

2019 arXiv
[35]

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), publisher: Amer- ican Association for the Advancement of Science

2022
[36]

Enkner, L

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

2025
[37]

Mazza and A

G. Mazza and A. Georges, Phys. Rev. Lett.122, 017401 (2019)

2019
[39]

Guerci, P

D. Guerci, P. Simon, and C. Mora, Phys. Rev. Lett.125, 257604 (2020), arXiv:2005.08994 [cond-mat]

2020 arXiv
[40]

Dmytruk and M

O. Dmytruk and M. Schir´ o, Phys. Rev. B103, 075131 (2021)

2021
[41]

Collective theory for an interacting solid in a single-mode cavity,

K. Lenk, J. Li, P. Werner, and M. Eckstein, “Collective theory for an interacting solid in a single-mode cavity,” (2022), arXiv:2205.05559 [cond-mat.str-el]

2022 arXiv
[42]

K. Lenk, J. Li, P. Werner, and M. Eckstein, Phys. Rev. B106, 245124 (2022)

2022
[43]

Dmytruk and M

O. Dmytruk and M. Schir` o, Phys. Rev. B110, 075416 (2024), publisher: American Physical Society

2024
[44]

Many-body photon blockade and quantum light gen- eration from cavity quantum materials,

B. Kass, S. Talkington, A. Srivastava, and M. Claassen, “Many-body photon blockade and quantum light gen- eration from cavity quantum materials,” (2024), arXiv:2411.08964 [cond-mat.str-el]

2024 arXiv
[45]

Ruggenthaler, J

M. Ruggenthaler, J. Flick, C. Pellegrini, H. Appel, I. V. Tokatly, and A. Rubio, Phys. Rev. A90, 012508 (2014)

2014
[46]

Flick and P

J. Flick and P. Narang, Phys. Rev. Lett.121, 113002 (2018)

2018
[47]

Bacciconi, G

Z. Bacciconi, G. M. Andolina, T. Chanda, G. Chiriac` o, M. Schir` o, and M. Dalmonte, SciPost Phys.15, 113 (2023)

2023
[48]

Bacciconi, H

Z. Bacciconi, H. B. Xavier, I. Carusotto, T. Chanda, and M. Dalmonte, Phys. Rev. X15, 021027 (2025)

2025
[49]

Shaffer, M

D. Shaffer, M. Claassen, A. Srivastava, and L. H. Santos, Phys. Rev. B109, 155160 (2024)

2024
[50]

Arwas and C

G. Arwas and C. Ciuti, Phys. Rev. B107, 045425 (2023)

2023
[51]

G. M. Andolina, A. De Pasquale, F. M. D. Pellegrino, I. Torre, F. H. L. Koppens, and M. Polini, Phys. Rev. B 109, 104513 (2024), arXiv:2210.10371 [cond-mat]

2024 arXiv
[52]

M. A. Sentef, J. Li, F. K¨ unzel, and M. Eckstein, Phys. Rev. Research2, 033033 (2020)

2020
[53]

J. Li, L. Schamriß, and M. Eckstein, Phys. Rev. B105, 165121 (2022), arXiv:2105.08711 [cond-mat]

2022 arXiv
[54]

Perez-Gonzalez, G

B. Perez-Gonzalez, G. Platero, and A. Gomez-Leon, Quantum9, 1633 (2025), arXiv:2302.12290 [cond-mat]

2025 arXiv
[55]

Fermion parity switches imprinted in the photonic field of cavity embedded kitaev chains,

V. F. Becerra and O. Dmytruk, “Fermion parity switches imprinted in the photonic field of cavity embedded kitaev chains,” (2025), arXiv:2506.06237 [cond-mat.mes-hall]

2025
[56]

Goldman and J

N. Goldman and J. Dalibard, Phys. Rev. X4, 031027 (2014)

2014
[57]

Eckardt and E

A. Eckardt and E. Anisimovas, New J. Phys.17, 093039 (2015), publisher: IOP Publishing

2015
[58]

Bukov, L

M. Bukov, L. D’Alessio, and A. P. and, Advances in Physics64, 139 (2015), https://doi.org/10.1080/00018732.2015.1055918

2015
[59]

Bukov, M

M. Bukov, M. Kolodrubetz, and A. Polkovnikov, Phys. Rev. Lett.116, 125301 (2016), publisher: American 23 Physical Society

2016
[60]

Perez-Gonzalez, A

B. Perez-Gonzalez, A. Gomez-Leon, and G. Platero, Phys. Chem. Chem. Phys.24, 15860 (2022)

2022
[61]

D. Zhao, Z. Wang, L. Yang, Y. Zhong, X. Xi, Z. Zhu, X. Jiao, Q.-a. Tu, Y. Meng, B. Yan, C. Shang, and Z. Gao, Nature Communications16, 5914 (2025)

2025
[62]

Dalibard, N

J. Dalibard, N. Goldman, M. Aidelsburger, and N. R. Cooper, Phys. Rev. A91, 033632 (2015), publisher: American Physical Society

2015
[63]

G. M. Andolina, F. M. D. Pellegrino, V. Giovannetti, A. H. MacDonald, and M. Polini, Phys. Rev. B100, 121109 (2019)

2019
[64]

K. Lenk, J. Li, P. Werner, and M. Eckstein, (2022), 10.48550/arXiv.2205.05559, arXiv:2205.05559 [cond-mat]

2022 doi arXiv
[65]

J. Li, D. Golez, G. Mazza, A. Millis, A. Georges, and M. Eckstein, Phys. Rev. B101, 205140 (2020), arXiv:2001.09726 [cond-mat]

2020 arXiv
[66]

De Bernardis, P

D. De Bernardis, P. Pilar, T. Jaako, S. De Liberato, and P. Rabl, Phys. Rev. A98, 053819 (2018), publisher: American Physical Society

2018
[67]

Di Stefano, A

O. Di Stefano, A. Settineri, V. Macr ` ı, L. Garziano, R. Stassi, S. Savasta, and F. Nori, Nat. Phys.15, 803 (2019), number: 8 Publisher: Nature Publishing Group

2019
[68]

H¨ ubener, U

H. H¨ ubener, U. De Giovannini, C. Sch¨ afer, J. Andberger, M. Ruggenthaler, J. Faist, and A. Rubio, Nat. Mater. 20, 438 (2021), publisher: Nature Publishing Group

2021
[69]

Passetti, C

G. Passetti, C. J. Eckhardt, M. A. Sentef, and D. M. Kennes, Phys. Rev. Lett.131, 023601 (2023), publisher: American Physical Society

2023
[70]

C. J. Eckhardt, G. Passetti, M. Othman, C. Karrasch, F. Cavaliere, M. A. Sentef, and D. M. Kennes, Commun Phys5, 1 (2022), number: 1 Publisher: Nature Publish- ing Group

2022
[71]

C. B. Dag and V. Rokaj, Phys. Rev. B110, L121101 (2024), arXiv:2311.02806 [cond-mat]

2024 arXiv
[72]

R. W. Bomantara, Phys. Rev. B111, 075429 (2025), publisher: American Physical Society

2025
[73]

M. J. Collett and C. W. Gardiner, Phys. Rev. A30, 1386 (1984)

1984
[74]

C. W. Gardiner and M. J. Collett, Phys. Rev. A31, 3761 (1985)

1985
[75]

F. J. Wegner, Physics Reports348, 77 (2001)

2001
[76]

S. J. Thomson, D. Magano, and M. Schir` o, SciPost Phys. 11, 028 (2021)

2021

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [38]

G. M. Andolina, F. M. D. Pellegrino, V. Giovan- netti, A. H. MacDonald, and M. Polini, Phys. Rev. B102, 125137 (2020), arXiv:2005.09088 [cond-mat, physics:quant-ph]

work page Pith review arXiv 2020

[2] [13]

57 The effective hoppingsV ef f andW ef f appearing in the First order effective Hamiltonian within mean-field of Eq

Self-consistent equation for the effective hoppings of Eq. 57 The effective hoppingsV ef f andW ef f appearing in the First order effective Hamiltonian within mean-field of Eq. 57 are defined self-consistently through : 19 Vef f=v ef f+ 2v2 ef f ωc f( g2b2 0 L ) D ˆTi.c. E +f(− g2b2 0 L ) D ˆT † i.c. E − 2vef fwef f ωc f g2 L b0(1−b 0) D ˆT † e.c. E +f − ...

work page

[3] [1]

In all generality, the indexiregroups the lattice site ⃗Ri, the spinσand if needed sub-lattice indexκ

The tight-binding model coupled through Peierls phase The electrons on a lattice are described by a tight- binding model characterized by the hopping integrals from siteito sitejdenoted byt i,j. In all generality, the indexiregroups the lattice site ⃗Ri, the spinσand if needed sub-lattice indexκ. Formally, the indexiis a triplet :i= ( ⃗Ri, σ, κ). Sums wil...

work page

[4] [2]

On thenphoton sector, the Floquet Hamiltonian of 0 1 2 3 4 5 6 g 0.0 0.5 1.0 Kodd 0 (g, g) Keven 0 (g, g) I0(g) 1/2g2 FIG

Vacuum cavity Since the Floquet Hamiltonian is photon conserving, the projection on a fixed photon number sector is ex- act. On thenphoton sector, the Floquet Hamiltonian of 0 1 2 3 4 5 6 g 0.0 0.5 1.0 Kodd 0 (g, g) Keven 0 (g, g) I0(g) 1/2g2 FIG. 1: Plot of the coefficients appearing in the vac- uum cavity effective Hamiltonian.I 0(g) controls the zero o...

work page

[5] [3]

In this regime the light-matter coupling typically scales asg i,j ∼1/ √ V

Vacuum cavity in the thermodynamic limit It is interesting to discuss the effect of cavity mediated processes in the thermodynamic limit, where the system sizeV− → ∞. In this regime the light-matter coupling typically scales asg i,j ∼1/ √ V. In this limit, the be- havior of the cavity-mediated interaction detailed above show that the leading term is given...

work page

[6] [4]

Classical limit We now consider the classical limit of a cavity in a co- herent state with many photons, as discussed in Ref. [36]. We expect that in this regime the cavity-mediated inter- actions discussed in the previous section would vanish, since it is known that for Floquet driven non-interacting electrons a classical drive can only mediate long-rang...

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[7] [5]

Driving with a matching laser mode In the case where the cavity mode and the laser mode coincide, e.g. uniform, so that one can write : gi,jα0 =−η i,j.(42) In other words, the cavity vector potentialA cav(r) and the laser oneA las(r) have to verify : α0 × Z Rj Ri Acav(r)·dr=− Z Rj Ri Alas(r)·dr,(43) so that within the long wavelength approximation, the co...

work page

[8] [6]

The hamiltonian of Eq

Driving with space-dependent light Alernatively, one can imagine the opposite situation where the laser light varies differently that the cavity mode, with its amplitude and phase being modulated in space. The hamiltonian of Eq. 40 can be treated in a manner similar to what is introduced in Ref. [40]. The driving term in Hamiltonian Eq. 41 is written as t...

work page