Two-dimensional spectroscopy of bosonic collective excitations in disordered many-body systems

Albert Liu; Alex G\'omez Salvador; Andrea Cavalleri; Danica Pavicevic; Eugene Demler; Ivan Morera; Marios H. Michael; Pavel E. Dolgirev

REVIEW 3 major objections 6 minor 3 cited by

Two-dimensional spectroscopy of bosonic collective excitations in disordered many-body systems

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A Keldysh-based theory of two-dimensional spectroscopy shows that for disordered bosonic collective excitations the echo peak is governed by the elastic-to-inelastic self-energy ratio, and that interaction-induced quantum fluctuations…

desk verdict Careful extension of the authors' 2DTS formalism to static disorder, with a clean vertex relation (Eq. 39) and an interesting quantum-fluctuation prediction, but the main plots sit outside the stated ladder-validity regime. read the letter →

arxiv 2501.16856 v2 pith:AUPILWF5 submitted 2025-01-28 cond-mat.str-el cond-mat.supr-conquant-ph

Alex Gómez Salvador , Ivan Morera , Marios H. Michael , Pavel E. Dolgirev , Danica Pavicevic , Albert Liu , Andrea Cavalleri , Eugene Demler This is my paper · ORCID

classification cond-mat.str-elcond-mat.supr-conquant-ph

keywords two-dimensionalterahertzspectroscopydisorderedbosonicsystemscollectiveexcitationsKeldyshformalismthird-ordernonlinearresponseechopeakelasticandinelasticscatteringquantumfluctuations

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 develops a theoretical framework for two-dimensional terahertz spectroscopy of bosonic collective excitations in a disordered medium, based on a Keldysh path-integral treatment that sums the infinite series of non-crossing disorder diagrams. In the weak nonlinearity limit it derives a closed formula for the disorder-dressed four-wave-mixing vertex on the echo diagonal, $T_{g-V}(\omega,-\omega)=2ig\,\mathrm{Im}\Sigma_V(\omega)/\mathrm{Im}\Sigma_{\mathrm{bath}}(\omega)$, so the echo peak directly encodes the competition between elastic and inelastic scattering. When interaction-induced quantum fluctuations are included, the same vertex obeys $T_{g-V}=2ig[-\mathrm{Im}\Sigma_V(\omega)+Q(\omega)]/[-\mathrm{Im}\Sigma_{\mathrm{bath}}(\omega)-Q(\omega)]$, and the paper argues that even with no inelastic bath there is no divergence, meaning perfect rephasing cannot be achieved in interacting many-body systems. A reader would care because this gives a concrete experimental diagnostic for separating static disorder from intrinsic many-body dephasing in materials such as cuprates where that separation has been difficult.

What carries the argument

The load-bearing object is the disorder-dressed four-wave-mixing vertex $T_{g-V}(\omega_a,\omega_b)$, obtained by solving a Bethe-Salpeter-type ladder equation for the infinite series of non-crossing impurity lines, $\Gamma_{\mathrm{dis}}(0;\omega_a,\omega_b)=2V_0^2/[1-V_0^2\lambda(0;\omega_a,\omega_b)]$, and then contracting it with the $\phi^4$ interaction. The disorder bubble $\lambda(0;\omega_a,\omega_b)$ is the crucial identity carrier: evaluated at $\omega_a=-\omega_b=\omega$ it equals $V_0^{-2}\mathrm{Im}\Sigma_V(\omega)/\mathrm{Im}\Sigma(\omega)$, which converts the vertex into the self-energy ratio of Eq. (39). The Keldysh contour supplies the causal time-ordering of the two-pulse protocol, and the overall self-consistent construction ensures the self-energy and vertex come from one generating functional, so the approximation is conserving.

What would settle it

Measure the echo peak of a disordered bosonic collective mode and extract its cross-diagonal width as a function of the independently determined inelastic linewidth; if Eq. (39) holds, that width should follow the total self-energy while the echo sharpness tracks $V_0^2/\omega\gamma$ in the elastic-dominated regime. Alternatively, compute the disorder-dressed vertex including crossing diagrams at the parameters used in the paper's plots, $(mvd/2/V_0)^2=1/3$; a sizable deviation from $2ig\,\mathrm{Im}\Sigma_V/\mathrm{Im}\Sigma_{\mathrm{bath}}$ would show that the ladder assumption underpinning the central formula fails in that regime.

Watch

Extended reading notes

Core claim

The central claim is that the rephasing (echo) nonlinearity of a disordered bosonic collective mode is controlled by a disorder-dressed vertex whose value on the echo diagonal is, within the paper's conserving ladder approximation, the ratio of the elastic and inelastic parts of the single-particle self-energy: $T_{g-V}(\omega,-\omega)=2ig\,\mathrm{Im}\Sigma_V(\omega)/\mathrm{Im}\Sigma_{\mathrm{bath}}(\omega)$ with $\mathrm{Im}\Sigma_{\mathrm{bath}}=-2\omega\gamma$. This recovers the isolated-two-level-system phenomenology, but with a many-body difference: the almond-shaped peak is asymmetric and elongated toward higher energies, since a zero-momentum collective mode cannot scatter into finite-momentum states below its mass. Including interaction-induced self-energy corrections, the paper obtains $T_{g-V}(\omega,-\omega)=2ig[-\mathrm{Im}\Sigma_V(\omega)+Q(\omega)]/[-\mathrm{Im}\Sigma_{\mathrm{bath}}(\omega)-Q(\omega)]$, where $Q(\omega)$ collects momentum-dependent quantum fluctuations; because $Q(\omega)$ acts like an effective inelastic process, the vertex no longer diverges even when $\Sigma_{\mathrm{bath}}=0$. The paper concludes that perfect rephasing is fundamentally unattainable in interacting many-body systems, and that repulsive interactions suppress while attractive interactions enhance the echo.

Load-bearing premise

The load-bearing assumption is that the ladder (non-crossing) resummation of disorder diagrams captures the rephasing physics, an assumption the paper states holds only for $(mvd/2/V_0)^2 \gg 1$, while the illustrative two-dimensional plots use $V_0^2/m^2=3$, where that combination equals $1/3$.

Editorial extensions

If this is right

The echo peak of a disordered bosonic collective mode carries a direct signature of the ratio $\mathrm{Im}\Sigma_V/\mathrm{Im}\Sigma_{\mathrm{bath}}$: in the elastic-dominated regime the echo sharpens and takes an almond shape, while inelastic scattering broadens it.
When vertex corrections are significant, the diagonal and cross-diagonal cuts of the echo no longer coincide with the linear-response spectral function, so 2DTS provides information beyond linear response.
In the absence of an inelastic bath and of quantum fluctuations, the dressed vertex diverges on the echo diagonal, giving perfect rephasing analogous to a zero-momentum diffuson pole for the bosonic excitation.
Interaction-induced quantum fluctuations, encoded in $Q(\omega)$, add a nonrephasable broadening even when $\Sigma_{\mathrm{bath}}=0$; hence perfect rephasing is fundamentally unattainable in interacting many-body systems.
The sign of the interaction matters: repulsive ($g>0$) interactions suppress the echo, while attractive ($g<0$) interactions enhance it.

Reading between the lines

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

If Eq. (39) holds beyond the specific $\phi^4$ model, experimental 2DTS maps could be used to extract the elastic-to-inelastic scattering ratio directly from the echo peak shape, turning the protocol into a quantitative disorder probe for quantum materials.
The prediction that no perfect rephasing occurs when $Q(\omega)\neq 0$ suggests that the residual cross-diagonal width of an echo, measured with the inelastic bath tuned away, is a direct measure of interaction-induced dephasing in a disordered bosonic system.
A numerical exact-diagonalization study of a disordered Bose-Hubbard model across the ladder-validity boundary $(mvd/2/V_0)^2\sim 1$ could test where crossing diagrams begin to modify the vertex and map the quantitative limits of the central formula.
Applying the same ladder machinery to fifth-order responses could reveal whether bound states of collective excitations rephase sharply or inherit the nonrephasable broadening, a question the paper leaves open.

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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

Here's the short version. This is a real contribution, not a repackaging: it takes the authors' earlier disorder-free 2DTS formalism and properly adds static disorder within the Keldysh/Baym-Kadanoff framework, producing one clean analytical result (Eq. 39) and a physically interesting prediction (Eq. 40) that interaction-induced quantum fluctuations make perfect rephasing impossible. The algebra behind Eq. (39) is transparent and, given the self-energy structure, exact. The comparison with the experimental nonrephasing peak from their earlier paper has no fitting parameters.

The main soft spot is the one the stress-test flags. The ladder (non-crossing) resummation is justified only when (mvd/2/V0)^2 >> 1, which for v=1, d=2 means V0^2/m^2 << 1. The main 2D maps in Figs. 8 and 9 use V0^2/m^2 = 1.5 and 3, so the ratio is 2/3 or 1/3, not >> 1. In that regime crossing diagrams and weak-localization corrections, which the paper itself consigns to future work, enter at order one. So Eq. (39) and the nonrephasing broadening in Eq. (40) are exact within the ladder approximation but are not quantitatively controlled at the plotted parameters. The qualitative almond shape and its asymmetry may well survive—the physics is not obviously wrong—but the paper's claim that the echo peak slices quantitatively map to ImSigma_V/ImSigma_bath is overstated for those figures.

A second, milder issue is the scope of the 'no perfect rephasing' conclusion. It follows from their specific Luttinger-Ward truncation and the definition of Q(omega) within that truncation. The statement in the abstract that perfect rephasing is 'fundamentally unattainable in interacting many-body systems' reaches further than the derivation sustains.

On the plus side: the derivation is careful, the paper is honest about what is neglected (crossing diagrams, localization, fifth-order terms), and the connection to ITLS intuition is genuinely illuminating. Self-citation to Refs. [90] and [62] is fine; they are extending their own prior work, and the new vertex relation is not in those papers. No code or data is shipped, which is a minor omission for a mostly formal paper.

Who gets value: theorists and experimentalists working on 2D THz spectroscopy of collective modes, especially in cuprates. It deserves a serious referee. I'd recommend acceptance after major revision: either soften the claims to match the regime actually plotted, or add a discussion of why crossing diagrams are suppressed at those parameters. Send it out.

Referee Report

3 major / 6 minor

Summary. The manuscript develops a Keldysh field-theoretic framework for computing the third-order nonlinear response, and hence two-dimensional terahertz spectroscopy (2DTS) maps, of bosonic collective excitations in the presence of static disorder. The authors use a Luttinger-Ward functional combined with a ladder (non-crossing) resummation of disorder lines to derive the disorder-dressed four-point vertex, and they specialize to the weak-nonlinearity regime where the echo diagonal vertex takes the closed form T_g-V(ω,-ω) = 2ig ImΣ_V(ω)/ImΣ_bath(ω) (Eq. (39)). They further include one class of interaction-disorder self-energy corrections to obtain Eq. (40), showing that interaction-induced quantum fluctuations produce a nonrephasable broadening that prevents perfect echo rephasing. Numerical 2D maps illustrate the almond-like echo peak, its evolution with disorder strength, and its suppression or enhancement depending on the sign of the interaction.

Significance. The paper's main strength is that Eq. (39) is a derived, parameter-free relation connecting the echo-diagonal vertex to the elastic-to-inelastic self-energy ratio; this goes beyond a fit and provides a concrete, potentially falsifiable prediction for 2DTS experiments such as those on Josephson plasmons in cuprates. The framework also correctly reproduces the ITLS echo phenomenology in a collective-mode language and identifies a genuinely many-body effect (quantum-fluctuation-induced nonrephasable broadening). If the central results are robust, this is a valuable contribution to the theory of nonlinear spectroscopy of disordered quantum materials. The main caveat is that the quantitative predictions are obtained within a specific non-crossing approximation, and the numerical demonstration is carried out at parameter values that lie outside the stated validity regime of that approximation.

major comments (3)

[Sec. V A; Figs. 8 and 9] The ladder resummation is stated to be valid for (mvd/2/V0)^2 >> 1, which for d = 2 and v = 1 reduces to (m/V0)^2 >> 1. The main echo maps use V0^2/m^2 = 1.5 and 3 in Fig. 8(a)-(b) and V0^2/m^2 = 3 in Fig. 9, corresponding to (m/V0)^2 = 2/3 and 1/3, respectively. These parameters are not in the stated validity regime, so the quantitative echo-peak slices in Fig. 8(c) and the quantum-fluctuation broadening shown in Fig. 9 are not controlled by the non-crossing approximation. The authors should either rerun the main figures at parameters satisfying (m/V0)^2 >> 1, provide a quantitative estimate of the neglected crossing-disorder-diagram corrections at the plotted parameters, or explicitly justify that the ladder result is quantitatively accurate beyond its nominal validity range.
[Sec. VII, Eq. (40)] The conclusion that perfect rephasing is 'fundamentally unattainable in interacting many-body systems' is stronger than what is demonstrated. Equation (40) is derived within a specific Luttinger-Ward functional (Fig. 7) that retains one mixed disorder-interaction self-energy term (Eq. (35)) while neglecting the disorder bubble (Eq. (37)) and the inelastic vertex corrections of Appendix D. The formula is therefore established for this class of non-crossing diagrams, not as a general theorem for all interacting many-body systems. Please either prove that Q(omega) remains nonzero for all conserving approximations that include momentum-dependent self-energy corrections, or explicitly scope the conclusion to the non-crossing approximation used in this work.
[Sec. VI, text after Eq. (39)] The statement that Eq. (39) is 'exact within perturbation theory in g in all spatial dimensions' should be qualified. Within the chosen self-energy structure (Eqs. (25)-(33)) the algebra leading to Eq. (39) is indeed exact, but the vertex Gamma_dis itself comes from the ladder resummation and excludes crossing disorder diagrams. As written, the sentence could be read as a model-independent exactness claim, which is not supported by the derivation.

minor comments (6)

[Abstract and Sec. I] 'non-pertubative' should be 'non-perturbative'.
[Sec. III B, Eq. (15)] The frequency convention for the four-point vertex Gamma(omega3, omega2, omega1) is not specified at the point of introduction; please state explicitly that the arguments follow the shifted-frequency convention of Eq. (12) to avoid confusion.
[Fig. 2 caption] The caption uses V0^2/m^2 v^2 = 2 while the main text sets v = 1; a single dimensionless notation (e.g., V0^2/m^2) should be used consistently.
[Appendix C, first paragraph] 'caclulation' should be 'calculation'.
[Sec. IV B, Eq. (22)] The regulator eta is described as a finite broadening, while later inelastic damping is denoted gamma; please clarify the relationship between eta and gamma to avoid notational confusion.
[References] Reference [99] is listed as 'A. Gomez Salvador, unpublished'; if this is intended as a companion work, it should be replaced by a proper citation or removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central vertex relation and the quantum-fluctuation broadening are derived from the paper's stated self-consistent equations, while the overlapping-author citations are not load-bearing.

full rationale

The central analytical claims do not reduce to their inputs by construction. Equation (39) follows by inserting the Bethe-Salpeter disorder vertex of Eq. (29), the self-consistent Born self-energy of Eq. (25), and the identity of Eq. (33) into the vertex definition of Eq. (36); the factor ImSigma_V/ImSigma_bath emerges algebraically from the denominator 1 - V0^2 lambda, and no parameter is fitted to the echo peak. Equation (40) is likewise obtained from the same vertex formula combined with the self-consistent relation Eq. (26) and the definition Q = 4 V0^2 sum_k |D_k|^2 Im Sigma_{g-V}; the statement that a nonzero Q removes the perfect-rephasing divergence is a mathematical consequence of that denominator structure, not an input. The 2D maps are computed from the derived chi^(3) with model parameters (g, V0, gamma, m) that are not adjusted to reproduce the predicted echo line shapes. The manuscript cites prior work by overlapping authors, Refs. [62], [90], and [99], but the mean-field result from Ref. [90] is rederived in Appendix A and Sec. IV A, and Ref. [62] is used as an experimental comparison, so the self-citations are not the load-bearing evidence for the central disorder results. The stated ladder validity condition (m v d / 2 / V0)^2 >> 1 in Sec. V A, together with plots at V0^2/m^2 = 1.5-3 in d = 2, is a legitimate correctness/regime risk for crossing-diagram corrections that the authors acknowledge in Secs. V A and VIII; it does not make the derivation circular because the paper's equations are solved as stated rather than being fitted to the quantities they are said to predict.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The model introduces no new particles or forces. Its free parameters are the interaction strength g, disorder strength V0^2, and bath broadening gamma, plus a regulator eta used for one experimental comparison. The main axioms are Gaussian white-noise disorder, zero-momentum coupling, delta-function pulses, and the ladder approximation, with the ladder validity condition violated in some plotted regimes.

free parameters (4)

g (phi-fourth interaction strength) = g/m^2 = 0.1, 0.25, 0.01, 0.5, -0.4, 1 across figures
Chosen by hand to explore weak and strong nonlinear regimes; the sign of g controls suppression or enhancement of the echo.
V0^2 (disorder strength) = V0^2/m^2 = 1.5, 2, 3
Chosen to access inelastic, intermediate, and elastic regimes; controls the ratio V0^2/m gamma in Eq. (39).
gamma (inelastic bath coefficient) = gamma/m = 0.1, 0.075, 0.05, 0.015, 0.01
Phenomenological bath self-energy -i 2 omega gamma; varied to tune the competition between elastic and inelastic scattering.
eta (regulator in Eq. (22)) = eta/m = 0.1
Used for the analytically computed nonrephasing peak compared with experiment in Fig. 4(b); acts as an effective linewidth.

assumptions (6)

domain assumption Gaussian white-noise disorder with V(r)V(r') = V0^2 delta(r-r')
Introduced in Sec. V, Eq. (23-24); central to the diagrammatic disorder resummation.
domain assumption Ladder (non-crossing) approximation is sufficient for the disorder rephasing physics
Sec. V A; justified only for (mvd/2/V0)^2 >> 1, while several plots use values violating this condition.
domain assumption Bath self-energy has the form -i 2 omega gamma with negligible inelastic vertex corrections
Used in Sec. V C and Eq. (39); Appendix D argues for its validity when the bath is a heavy optical phonon.
domain assumption Light couples only to the zero-momentum component of the field, and the signal is measured at zero momentum
Stated in Sec. III B; standard for optical drive of collective modes but restricts the domain of the claims.
standard math Excitation pulses are perfect Dirac delta functions
Used in Sec. II, Eq. (1-2), to extract the third-order susceptibility from the 2DTS protocol.
domain assumption A self-consistent, momentum-independent total self-energy exists for the disorder problem
Eqs. (25)-(27); needed to derive the exact-looking relations in Eqs. (33) and (39).

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Cite this review

Pith. "Pith review of Two-dimensional spectroscopy of bosonic collective excitations in disordered many-body systems." pith.science (2026). https://pith.science/paper/AUPILWF5

@misc{pith2026250116856,
  author       = {Pith},
  title        = {Pith review of: Two-dimensional spectroscopy of bosonic collective excitations in disordered many-body systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AUPILWF5}},
  note         = {Machine review of arXiv:2501.16856}
}

read the original abstract

We present a novel theoretical approach for computing and analyzing two-dimensional spectroscopy of bosonic collective excitations in disordered many-body systems. Specifically, we employ the Keldysh formalism to derive, within a non-pertubative treatment of disorder effects, the third-order nonlinear response and obtain two-dimensional spectroscopy maps. In the weak nonlinear regime of our formalism, we demonstrate the ability of the echo peak to distinguish between elastic and inelastic scattering processes, in perfect agreement with the intuition developed in isolated two-level systems. Furthermore, we discuss unique many-body effects on the echo peak signature arising from interaction induced quantum fluctuations. In particular, we show that these quantum fluctuations induce a finite nonrephasable broadening and examine how the echo peak is influenced by the attractive or repulsive nature of the collective excitations.

Figures

Figures reproduced from arXiv: 2501.16856 by the authors.

**Figure 2.** FIG. 2. (a)-(b) Cartoon of the effects of disorder on ITLS [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Luttinger-Ward functional approximation to low [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4. (a) Two-sided Feynman diagrams for bosonic excitations. The solid arrows indicate the interaction with the first and [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5. Intuition behind the diagrams for the echo peak. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 7.** Figure 7: FIG. 7. Luttinger-Ward functional employed for the compu [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: FIG. 8. Weak nonlinear regime of the 2D map in the pres [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: FIG. 9. Echo peak in the presence of interaction induced [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: FIG. 10. (a) Echo and nonrephasing peaks in the lowest [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 11.** Figure 11: FIG. 11. (a) Imaginary part of the self-energy due to the [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

Discussion (0). Continue with ORCID to comment.

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We subsequently fix the renormalized bare mass m0 to match the physically ob- served mass m

Self-energy The self-energy calculation results in a constant term (potentially diverging depending on the dimensionality) Σg(ω) = δΦ[D] δDR(ω) = = ig Ld X k Z dΩ 2π DK k (Ω) (17) which renormalizes the mass. We subsequently fix the renormalized bare mass m0 to match the physically ob- served mass m

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3, and it is simply given by Γ(0) = δ2Φ[D] δDK(ω)δDR(ω) = = ig, (18) compare with Eq

Vertex The bare vertex is obtained by cutting open two Green’s function from the Luttinger-Ward functional, see (a) in Fig. 3, and it is simply given by Γ(0) = δ2Φ[D] δDK(ω)δDR(ω) = = ig, (18) compare with Eq. (13)

[5] [3]

Bubble The bare bubble can be computed directly as gB0(ω) = = 2ig Ld X k Z dΩ 2π DR k (ω + Ω)DK k (Ω). (19)

[6] [4]

mean field

Third-order response Collecting all the previously computed ingredients, the vertex correction is simply given by (note that the exter- nal legs have been amputated) iΓRPA(ω3, ω2, ω1) = permutations 2 1 , (20) and we arrive directly at the third-order response χ(3) RPA(ω1 + ω2 + ω3, ω1 + ω2, ω1) = 2g − 3X i=1 g 1 − gB0(ω1 + ω2 + ω3 − ωi) ! | {z } iΓRPA(ω3...

[7] [5]

(17) and (25) respectively

Self-energy The interaction and disorder-only contributions to the self-energy arising from the Luttinger-Ward functional are given by Eqs. (17) and (25) respectively. There are two additional contributions coming from the disorder- interaction part: Σg−V (ω, k) = = 2 × 4ig L2d X k,q Z dΩ 2π DR p (ω)DR p+q−k(Ω)DK q (Ω) × Γdis(q − k; ω, Ω). (35) Furthermor...

[8] [6]

The RPA contribution is given in Eq

Vertices The bare vertices contributing to the nonlinear re- sponse are given by taking a derivative with respect to the Keldysh Green’s function and setting the exter- nal momenta to 0. The RPA contribution is given in Eq. (18) and the interaction-disorder contribution to the Luttinger-Ward results in the following vertices: =ig 4 Ld X k DR k (ωa)DR −k(ω...