Unconventional exciton evolution from the pseudogap to superconducting phases in cuprates

Above the superconducting transition temperature Tc, cuprate superconductors display an enigmatic pseudogap in various physical quantities, prominently in single-particle spectra from ARPES, especially near the antinodal region. The mechanisms underlying both the pseudogap formation and high-Tc superconductivity remain unresolved. To clarify the origin of the pseudogap, believed to be rooted in strong correlations, two-particle dynamics must be resolved in momentum and energy. Resonant inelastic x-ray scattering (RIXS) enables access to particle–hole (excitonic) charge excitations in correlated cuprates. This work uses RIXS to probe exciton dynamics across the pseudogap and superconducting phases. The conventional Fermi-liquid quasiparticle picture likely breaks down in cuprates. Electron fractionalization—well established in phenomena such as the fractional quantum Hall effect, solitons in polymers, and spin–charge separation—has been proposed for cuprates. A two-component fermion model (TCFM) embodies a dual character: coherent itinerant quasiparticles (c fermions) and localized, hole-bound dark fermions (d fermions). The TCFM can account for the pseudogap without invoking symmetry breaking, consistent with a fractionalization/RVB-like scenario. ARPES, however, probes only single-particle dynamics; thus, combining with two-particle spectroscopy such as RIXS is essential. Prior theoretical work predicted that, if fractionalization operates, RIXS intensity for certain excitons should be enhanced in the superconducting phase relative to the pseudogap phase. The present study directly tests this prediction.

• Numerous studies have questioned the applicability of Landau Fermi-liquid theory to cuprates and documented the pseudogap via ARPES, which shows antinodal spectral weight suppression above Tc.
• Electron fractionalization is established in several contexts (fractional quantum Hall effect with anyons, solitons in polymers, spin–charge separation in 1D). In correlated Mott systems, the Hubbard-band splitting can be viewed as a type of fractionalization. The duality between itinerant quasiparticles and localized electrons underpins a two-component picture in metals.
• The two-component fermion model (TCFM) introduces a coherent c fermion and a localized dark d fermion whose hybridization produces bonding/antibonding bands associated with the coherent LHB and the in-gap band (IGB), offering a symmetry-unbroken origin of the pseudogap and supported by ARPES analyses.
• RIXS provides access to two-particle charge excitations (excitons) and has been developed extensively for cuprates. Prior work suggested an enhancement of certain RIXS excitonic features in the superconducting state if fractionalization is present. Previous exciton measurements (RIXS/optical) generally show minimal temperature dependence (<~1%), emphasizing that any substantial change across Tc would be unusual.
• Optical studies reported small spectral-weight transfers and, in some cases, enhancements in dielectric response at a few eV in underdoped cuprates, potentially consistent with fractionalization-related excitons involving LHB–UHB transitions.

Experimental samples and preparation:

• Single crystals: pristine Bi2.1Sr1.9CaCu2O8+δ (Bi2212) and Pb-doped Bi1.6Pb0.4Sr2CaCu2O8+δ (Pb-Bi2212) grown by traveling-solvent floating-zone method. Hole doping via Tallon’s relation; annealed under O2 partial pressures of 100 Pa at 600 °C and 2 Pa at 600 °C to obtain optimally doped (OP) samples with Tc = 89 K (Bi2212) and 93 K (Pb-Bi2212), p = 0.16 ± 0.005. An overdoped (OD) Pb-Bi2212 crystal was prepared with Tc = 65 K, p = 0.22 ± 0.005.

RIXS/XAS measurements:

• Beamline and instrument: AGM-AGS spectrometer at beamline 41A, Taiwan Photon Source.
• Edges: Cu L3-edge and O K-edge RIXS; XAS recorded at normal incidence in total-electron-yield mode.
• Polarization: Incident x-ray polarization perpendicular to the scattering plane; scattered x-rays detected without polarization analysis.
• Momentum transfers: In-plane Q along antinodal direction, Q⊥ = (π, 0) and (π/2, 0). Scattering geometry defined with Q = k − k′.
• Incident-energy scans: Tuned across the Cu L3 absorption peak (L3) and across O K-edge ZRSB feature (A ≈ 528.5 eV). For Cu L3-edge, L3 approximately corresponds to EF (from fluorescence threshold analysis). For O K-edge, A is taken as EF reference.
• Data treatment: Spectra above/below Tc normalized by integrating energy loss over high-energy windows (e.g., 1.7–13 eV for OP Bi2212; 2–13 eV for OD Pb-Bi2212). Self-absorption not corrected (temperature-induced differences are unaffected). Elastic-tail variations (notably at Q⊥ = (π/2, 0)) attributed to charge-density-wave tails.

Experimental focus:

• Identify excitonic features resulting from transitions within/between coherent LHB and IGB, distinct from local dd excitations (~2–3 eV) and magnetic excitations (<0.5 eV). Map incident-energy dependence to track fluorescence-like excitations and exciton dispersions.
• Compare temperature evolution across Tc and across the pseudogap regime in OP samples; perform control on OD sample lacking clear pseudogap.

Theory and calculations:

• Qualitative guidance from a doped square-lattice Hubbard model (t = 0.5 eV, U = 4 eV) to locate coherent/incoherent LHB, IGB, and UHB features and visualize probable RIXS transitions.
• Two-component fermion model (TCFM) Hamiltonian with c (coherent quasiparticle) and d (dark fermion) bands hybridized via A(k); simple square-lattice dispersions with nearest and next-nearest hoppings and chemical potentials. d-wave superconducting gaps Δc(k) and Δd(k) in mean-field.
• Parameters fitted to ARPES/STM: tc1 = 0.1953, tc2 = −0.0762, td1 = 0.0100, td2 = −0.0036, μc = 0.2875, μd = 0.0105, Δc0 = 0.02 eV, Δd0 = 0.067 eV, A0 = 0.0658 eV, A1 = −0.014 eV.
• RIXS intensity computed using Kramers–Heisenberg-type formula with core-hole lifetime width Γ and exciton broadening η. For Cu L3-edge calculations, Γ = 0.3 eV and η = 0.1 eV; for O K-edge, Γ ≈ 0.07 eV and η ≈ 0.1 eV (as specified in figure captions). Incident energy ω0 measured from EF; simulations for Q = (π, 0) and (π/2, 0). Focus on interband transitions forming excitons between IGB electrons and coherent LHB holes.
• Comparison of calculated incident-energy dependence and superconducting vs pseudogap spectral-weight changes to experiment.

• Discovery of a pronounced enhancement of excitonic RIXS spectral weight in the superconducting (SC) phase of optimally doped (OP) Bi2212 relative to the pseudogap (PG) phase, despite exciton energies far above the SC gap (∼1 eV and higher).
• Cu L3-edge, Q⊥ = (π, 0): An exciton feature at energy loss ΔE ≈ 0.8 eV is enhanced below Tc when the incident energy is ∼1 eV above L3 (transition from (0, π) to (π, π)). The ∼0.2 eV difference between ΔE and ω0 is attributed to final-state exciton binding energy.
• Cu L3-edge, Q⊥ = (π/2, 0): A similar enhancement is observed at smaller ω0 (≈0.8 eV above L3) with smaller ΔE (≈0.6 eV).
• O K-edge, Q⊥ = (π/2, 0): Exciton enhancements observed at energy loss ≈1.3 eV and also between 2.5–3 eV with incident energy A + 1 eV, consistent with charge excitations (single-spin-flip forbidden at O K-edge).
• Temperature evolution (OP): From 250 K to 150 K, exciton intensity decreases as the pseudogap develops; upon further cooling below the superconducting fluctuation temperature Tsfer ≈ 110 K, the exciton intensity sharply increases, saturating in the SC phase below Tc = 89–93 K.
• Overdoped control: In OD Pb-Bi2212 (p = 0.22) no SC-induced enhancement of exciton spectral weight is detected across Tc = 65 K, indicating the enhancement is tied to the presence of the pseudogap in OP samples.
• Chemical potential shift: The exciton energy in OD vs OP shifts by ∼0.5 eV, consistent with expected chemical potential changes upon overdoping.
• Magnitude of enhancement: The observed SC-phase enhancement in exciton spectral weight is ∼7–10% relative to the PG region (∼110–150 K), far exceeding typical <1% temperature dependencies reported in prior exciton studies.
• Theory agreement: TCFM calculations reproduce (i) ωpeak ≈ ω0 with widths ∼0.5–1.0 eV and (ii) an enhancement of ∼7.6% at Cu L3-edge for ω0 = 1.2 eV (ωpeak = 1.23 eV), consistent with experiment without fine-tuning beyond ARPES-based parameters.
• Additional high-energy exciton enhancement (~2–3 eV) overlapping dd tail is observed, attributed to a core-hole-induced shake-up that promotes IGB electrons to the UHB, forming higher-energy excitons with coherent LHB holes.
• Edge-dependent energy differences: Different exciton energies at O K (~1.3 eV) vs Cu L3 (~0.8 eV) for ω0 ≈ 1 eV are explained by differing core-hole interactions (repulsive for O 1s with distant bound hole vs attractive for Cu 2p with local excited electron).

The results demonstrate a dramatic restructuring of electronic states across the pseudogap-to-superconducting transition revealed via two-particle (exciton) dynamics. In the TCFM/fractionalization picture, the pseudogap antinodal states carry substantial dark-fermion weight, suppressing transitions to core holes in RIXS (as only quasiparticle components couple). Superconductivity, formed by Cooper pairing of conventional electrons, suppresses fractionalization and restores quasiparticle weight near the antinodes, thereby enhancing excitonic transition probabilities and RIXS intensity at energies well above the SC gap. The observed enhancement magnitude (∼7–10%) across Tc, consistent at both Cu L3 and O K edges and unusual compared to typical <1% changes, supports this mechanism. Differences between Cu and O edges can be rationalized by core-hole potentials affecting the exciton’s final-state energy loss. The additional enhancement near 2–3 eV suggests a shake-up from IGB to UHB in the RIXS intermediate state, consistent with a fractionalization-driven mechanism since the hole remains in the coherent LHB. These findings impose strong constraints on pseudogap theories: simple mean-field symmetry breaking without fractionalization does not naturally produce the observed recovery of quasiparticle weight and associated exciton enhancement in the SC phase. While TCFM captures key trends without parameter adjustment beyond ARPES fits, contributions from dd excitations and other backgrounds complicate line shapes, and alternative interpretations should be explored. Integrating RIXS, ARPES (including unoccupied states), and other spectroscopies will further clarify the role of fractionalization in cuprates.

Cu L3- and O K-edge RIXS measurements on optimally doped Bi2212 reveal a robust enhancement of excitonic spectral weight in the superconducting phase relative to the pseudogap phase at energy losses around 0.6–1.3 eV and also at 2–3 eV. Overdoped samples lacking a clear pseudogap show no such enhancement. The energy dependence and enhancement magnitude align with predictions of the two-component fermion model in which electron fractionalization generates a pseudogap and is suppressed upon superconductivity, restoring quasiparticle weight and increasing exciton intensity. These results provide compelling two-particle evidence linking high-energy excitons to superconductivity and set stringent constraints on pseudogap mechanisms that do not involve fractionalization. Future work should extend models to incorporate the UHB and interactions/damping of fractionalized particles, and pursue combined ARPES–RIXS analyses to quantify unoccupied-state dynamics and exciton lifetimes, potentially illuminating dark-fermion physics and its link to Planckian behavior.

• The excitonic enhancement near ~1 eV at Cu L3-edge is not isolated from overlapping spectral features (dd excitations, paramagnons), complicating precise quantification; background contributions are simpler at O K-edge but still require subtraction.
• The TCFM used is a simplified, low-energy effective model that does not explicitly include the UHB; extensions are needed to fully describe high-energy (2–3 eV) excitons and shake-up processes.
• Model parameters are fitted to ARPES, which lacks unoccupied-state information, introducing uncertainties in parameters governing unoccupied bands/excitons.
• Self-absorption corrections were not applied (argued to be temperature independent for relative comparisons), and scattered x-ray polarization was not analyzed; elastic scattering tails (e.g., from charge-density-wave fluctuations) can vary with momentum and temperature.
• Alternative (non-fractionalization) explanations have not been ruled out; quantitative, stringent tests against other mechanisms remain for future studies.