Momentum-resolved test of small-$q$ Coulomb-energy saving at the superconducting transition in cuprates (the midinfrared scenario)

Turlakov and Leggett (Phys. Rev. B 67, 094517 (2003)) rederived the third-moment frequency sum rule of the density response in the presence of umklapp processes, derived upper and lower bounds on the electron-electron Coulomb energy in layered media, and argued that in cuprates a substantial fraction of the small-wavevector ($1/\xi_0\lesssim q\lesssim q_{TF}$) midinfrared Coulomb energy is in principle available for saving at the superconducting transition — a quantitative version of Leggett's midinfrared scenario. They stressed that a definitive test requires quantitative transmission electron energy-loss spectroscopy (EELS) across $T_c$ in the midinfrared regime over that momentum window (both $qd\ll1$ and $qd\gg1$), or a validated theory of the finite-$q$ dielectric response of the superconducting state, since optical ($q\to0$) data cannot be extrapolated without it. Subsequent work: temperature-dependent ellipsometry across $T_c$ (Levallois et al., PRX 6, 031027 (2016), with Leggett) finds a doping-dependent Coulomb-energy change comparable to the condensation energy but explicitly assumes rather than measures its momentum dependence; momentum-resolved EELS (Mitrano et al. 2018; Husain et al. 2019; de Vries et al. 2026) has so far accessed only the normal state. The problem — to determine, by momentum- and frequency-resolved spectroscopy across $T_c$ (or by a validated theory of $\epsilon(q,\omega)$ below $T_c$), whether, where in $(q,\omega)$ space, and by how much the small-$q$ Coulomb energy is saved in the superconducting state of cuprates — remains open.

Background

High-temperature (cuprate) superconductors are layered copper oxides whose superconductivity emerges from doped CuO$_2$ planes; the microscopic origin of their high transition temperature remains unsettled. A sharp way to pose the mechanism question is energetic. The superconducting condensation energy — the free-energy difference between the normal and superconducting states, of order $1$–$2$ kelvin ($k_B$ per CuO$_2$ unit) in the cuprates — must be paid by some term of the electronic Hamiltonian, and one may ask which. In a first-principles description the Hamiltonian contains only kinetic energy plus Coulomb interactions (electron-electron, electron-ion, ion-ion), and the virial theorem then guarantees that the total Coulomb energy decreases in any transition to a lower-energy state; but in the intermediate-level effective descriptions actually used to discuss mechanisms — band (single-particle) energy plus effective electron-electron interactions — it is not a priori clear whether the saving occurs in the single-particle/band term or in the electron-electron Coulomb term, nor in which range of wavevector $\vec{q}$ and frequency $\omega$.

The relevant experimental object is the density-density response function $\chi(q,\omega)$, which describes the linear response of the electron density to a weak scalar potential; its imaginary part is what inelastic probes measure. In transmission electron energy-loss spectroscopy (EELS) a beam electron deposits energy $\omega$ and in-plane momentum $q$ into the sample; neglecting direct scattering off ionic cores, the cross-section is proportional to $V_q^2\,\mathrm{Im}\,\chi(q,\omega)$, where $V_q$ is the Fourier transform of the Coulomb interaction — equivalently, in a three-dimensional medium, to the loss function $\mathrm{Im}[-1/\epsilon_\parallel(q,\omega)]$ of the longitudinal dielectric function. Optical ellipsometry instead measures the transverse dielectric constant in the limit $\vec{q}\to0$. In the normal state the two must coincide at $\vec{q}\to0$; below $T_c$ this is not guaranteed, and the difference matters (see below).

Turlakov and Leggett (2003) analyzed the energetics with moment sum rules. The familiar $f$-sum rule fixes the first frequency moment of $\mathrm{Im}\,\chi$. The third-moment (Mihara–Puff-type) sum rule, when rederived in the presence of umklapp processes — momentum-changing scattering events in which the electron's momentum changes by a reciprocal-lattice vector, i.e. the lattice takes up crystal momentum — reads, schematically, $J_3=(n^2q^4/m^2)V_q+(q^2/m^2)\langle\hat{A}\rangle$, where the expectation value $\langle\hat{A}\rangle$ of a lattice (band-structure) operator measures the strength of umklapp scattering. From this and related sum rules they derived upper and lower bounds on the electron-electron Coulomb energy in two and three dimensions, showed that umklapp processes dominate the small-$q$ density-fluctuation spectrum of a layered (nearly two-dimensional) metal, and concluded that in a quasi-two-dimensional system whose optical loss function has a broad, strong midinfrared (MIR) peak — as the cuprates' does — a substantial fraction of the normal-state Coulomb energy locked up at small in-plane wavevectors, $q\lesssim q_{TF}$ (the Thomas–Fermi screening wavevector, the inverse length scale beyond which the Coulomb interaction is screened in the doped planes), is in principle available for saving at the superconducting transition. This quantified the 'midinfrared scenario' (Leggett, PNAS 1999), in which high-$T_c$ pairing is stabilized by Coulomb-energy saving rather than by kinetic-energy lowering.

The decisive momentum window lies between $1/\xi_0$ and $q_{TF}$, where $\xi_0$ is the Cooper-pair radius (coherence length, roughly $15$–$30$ Å in optimally doped cuprates), and the layered structure (interplane spacing $d$) separates the regimes $qd\ll1$ (three-dimensional response) and $qd\gg1$ (independent-plane response, in which both the per-plane Coulomb energy and the EELS cross-section are proportional to $-\mathrm{Im}[1/(1+qK(q,\omega)/\epsilon_{sc})]$, with $K$ the per-plane susceptibility and $\epsilon_{sc}$ the screening constant of the interplane medium). Optical measurements across $T_c$ do show striking superconductivity-induced spectral-weight transfer (e.g. Molegraaf et al., Science 2002, on the Ferrell–Glover–Tinkham sum rule, which states that optical spectral weight lost below the gap energy reappears in the zero-frequency $\delta$-function of the superfluid condensate — its violation would signal a kinetic-energy change). But three complications prevent optical data from settling the Coulomb question in the superconducting state: (i) the Cooper-pair radius $\xi_0$ (or the mean free path) may set a crossover scale below which the optically measured $\vec{q}\to0$ behavior no longer represents the finite-$q$ regime; (ii) the finite-frequency longitudinal and transverse dielectric constants need not be equal below $T_c$; (iii) ellipsometry at non-normal incidence measures a mixture of tensor components. Indeed, when the existing optical data were extrapolated into the relevant regime, they suggested the opposite sign — the MIR Coulomb energy increasing below $T_c$ — but exactly this extrapolation is uncontrolled. Whether the small-$q$ Coulomb energy is actually saved in the superconducting state, and where in $(q,\omega)$ space, therefore requires quantitative transmission EELS across $T_c$, or a validated theory of $\epsilon(q,\omega)$ in the superconducting state at finite $q$. A faithful test must also control the frequency cutoff and core-electron contributions that enter the evaluation of $\langle\hat{A}\rangle$, and the assumptions (static lattice potential, core screening absorbed into a constant $\epsilon_\infty$, neglected disorder and $q$-dependent screening) under which the sum-rule bounds were derived. Modern probes — meV-resolution momentum-resolved EELS (M-EELS) and resonant inelastic x-ray scattering (RIXS, which measures charge and spin excitations at finite $q$ but with a cross-section not reducible to a single Green's function) — have so far been applied to the normal state only.

The sum-rule framework of Turlakov and Leggett (2003) is restricted to homogeneous phases with its stated conventions for $\chi(q,\omega)$.

Problem Statement

Determine whether the electron-electron Coulomb energy of the cuprates at small in-plane wavevectors is reduced at the superconducting transition, and quantitatively account for what fraction of the condensation energy it provides and in which region of $(q,\omega)$ space the saving occurs. The small-$q$ midinfrared Coulomb energy in question lies within the in-plane momentum window $1/\xi_0\lesssim q\lesssim q_{TF}$, in both the $qd\ll1$ and $qd\gg1$ geometries; the underlying question is whether that energy is actually saved in the superconducting state of cuprates, as the midinfrared scenario postulates, and by how much relative to the independently known condensation energy.

References

1. M. Turlakov and A. J. Leggett, "Sum rule analysis of umklapp processes and Coulomb energy: Application to cuprate superconductivity," Physical Review B 67, 094517 (2003). DOI: 10.1103/PhysRevB.67.094517, https://doi.org/10.1103/PhysRevB.67.094517 (preprint: arXiv:cond-mat/0207422, https://arxiv.org/abs/cond-mat/0207422)
2. A. J. Leggett, "A 'midinfrared' scenario for cuprate superconductivity," Proceedings of the National Academy of Sciences USA 96, 8365-8372 (1999). DOI: 10.1073/pnas.96.15.8365, https://doi.org/10.1073/pnas.96.15.8365
3. H. J. A. Molegraaf, C. Presura, D. van der Marel, P. H. Kes, and M. Li, "Superconductivity-Induced Transfer of In-Plane Spectral Weight in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$," Science 295, 2239-2241 (2002). DOI: 10.1126/science.1069947, https://doi.org/10.1126/science.1069947
4. M. Turlakov, "Umklapp scattering in cuprates," Journal of Physics A: Mathematical and General 36, 9399-9404 (2003). DOI: 10.1088/0305-4470/36/35/324, https://doi.org/10.1088/0305-4470/36/35/324
5. J. Levallois, M. K. Tran, D. Pouliot, C. N. Presura, L. H. Greene, J. N. Eckstein, J. Uccelli, E. Giannini, G. D. Gu, A. J. Leggett, and D. van der Marel, "Temperature-Dependent Ellipsometry Measurements of Partial Coulomb Energy in Superconducting Cuprates," Physical Review X 6, 031027 (2016). DOI: 10.1103/PhysRevX.6.031027, https://doi.org/10.1103/PhysRevX.6.031027 (preprint: arXiv:1512.00672, https://arxiv.org/abs/1512.00672)
6. M. Mitrano, A. A. Husain, S. Vig, A. Kogar, M. S. Rak, S. I. Rubeck, J. Schmalian, B. Uchoa, J. Schneeloch, R. Zhong, G. D. Gu, and P. Abbamonte, "Anomalous density fluctuations in a strange metal," Proceedings of the National Academy of Sciences USA 115, 5392-5396 (2018). DOI: 10.1073/pnas.1721495115, https://doi.org/10.1073/pnas.1721495115
7. A. A. Husain, M. Mitrano, M. S. Rak, S. Rubeck, B. Uchoa, K. March, C. Dwyer, J. Schneeloch, R. Zhong, G. D. Gu, and P. Abbamonte, "Crossover of Charge Fluctuations across the Strange Metal Phase Diagram," Physical Review X 9, 041062 (2019). DOI: 10.1103/PhysRevX.9.041062, https://doi.org/10.1103/PhysRevX.9.041062 (preprint: arXiv:1903.04038, https://arxiv.org/abs/1903.04038)
8. N. de Vries, E. Hoglund, D. Chaudhuri, S. H. Bae, J. Chen, X. Guo, D. Balut, G. Gu, P. Huang, J. Hachtel, and P. Abbamonte, "Reexamining the strange metal charge response with transmission inelastic electron scattering," arXiv:2602.02348 (2026), https://arxiv.org/abs/2602.02348 (unpublished preprint)
9. R. D. Dawson, X. Shi, K. S. Rabinovich, D. Putzky, Y.-L. Mathis, G. Christiani, G. Logvenov, B. Keimer, and A. V. Boris, "High-precision measurement of the Ferrell-Glover-Tinkham sum rule in a cuprate high-temperature superconductor," Physical Review B 108, 104501 (2023). DOI: 10.1103/PhysRevB.108.104501, https://doi.org/10.1103/PhysRevB.108.104501
10. M. Hepting, L. Chaix, E. W. Huang, R. Fumagalli, Y. Y. Peng, B. Moritz, K. Kummer, N. B. Brookes, W. C. Lee, M. Hashimoto, T. Sarkar, J.-F. He, C. R. Rotundu, Y. S. Lee, R. L. Greene, L. Braicovich, G. Ghiringhelli, Z. X. Shen, T. P. Devereaux, and W. S. Lee, "Three-dimensional collective charge excitations in electron-doped copper oxide superconductors," Nature 563, 374-378 (2018). DOI: 10.1038/s41586-018-0648-3, https://doi.org/10.1038/s41586-018-0648-3
11. A. Nag, M. Zhu, M. Bejas, J. Li, H. C. Robarts, H. Yamase, A. N. Petsch, D. Song, H. Eisaki, A. C. Walters, M. García-Fernández, A. Greco, S. M. Hayden, and K.-J. Zhou, "Detection of Acoustic Plasmons in Hole-Doped Lanthanum and Bismuth Cuprate Superconductors Using Resonant Inelastic X-Ray Scattering," Physical Review Letters 125, 257002 (2020). DOI: 10.1103/PhysRevLett.125.257002, https://doi.org/10.1103/PhysRevLett.125.257002
12. F. Boschini, M. Minola, R. Sutarto, E. Schierle, M. Bluschke, S. Das, Y. Yang, Y. C. Shao, X. Feng, S. Ono, R. D. Zhong, J. A. Schneeloch, G. D. Gu, E. Weschke, F. He, Y. D. Chuang, B. Keimer, A. Damascelli, A. Frano, and E. H. da Silva Neto, "Dynamic electron correlations with charge order wavelength along all directions in the copper oxide plane," Nature Communications 12, 597 (2021). DOI: 10.1038/s41467-020-20824-7, https://doi.org/10.1038/s41467-020-20824-7
13. N. Sellati, J. Fiore, C. Castellani, and L. Benfatto, "Optical Absorption in Tilted Geometries as an Indirect Measurement of Longitudinal Plasma Waves in Layered Cuprates," Nanomaterials 14, 1021 (2024). DOI: 10.3390/nano14121021, https://doi.org/10.3390/nano14121021 (preprint: arXiv:2404.10519, https://arxiv.org/abs/2404.10519)
14. J. Chen, X. Guo, C. Boyd, S. Bettler, C. Kengle, D. Chaudhuri, F. Hoveyda, A. Husain, J. Schneeloch, G. Gu, P. Phillips, B. Uchoa, T.-C. Chiang, and P. Abbamonte, "Consistency between reflection M-EELS and optical spectroscopy measurements of the long-wavelength density response of Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$," arXiv:2306.03681 (2023), https://arxiv.org/abs/2306.03681 (unpublished preprint)
15. C. Setty, B. Padhi, K. Limtragool, P. Abbamonte, A. A. Husain, M. Mitrano, and P. W. Phillips, "Inequivalence of the zero-momentum Limits of Transverse and Longitudinal Dielectric Response in the Cuprates," arXiv:1803.05439 (2018), https://arxiv.org/abs/1803.05439 (unpublished preprint)
16. L. Zinni, H. Yamase, M. Hepting, M. Bejas, and A. Greco, "Strongly correlated model of acousticlike plasmons persisting across the phase diagram of cuprate superconductors," Physical Review B 114, L051103 (2026). DOI: 10.1103/k1m6-dbpk, https://doi.org/10.1103/k1m6-dbpk
17. I. Sung, "Long-wavelength density response and momentum resolution in strange-metal charge spectroscopy," Journal of Physics: Condensed Matter 38, 305601 (2026). DOI: 10.1088/1361-648X/ae8d6c, https://doi.org/10.1088/1361-648X/ae8d6c