Emergence of nodal Bogoliubov quasiparticles across the transition from the pseudogap metal to the *d*-wave superconductor

The cuprate phase diagram features a pseudogap metal and a low-temperature d-wave superconducting state. The pseudogap exhibits a carrier density and Fermi surface topology that violate Luttinger’s theorem for a conventional Fermi liquid and is often associated with small Fermi pockets or Fermi arcs. A central open issue is how conventional d-wave superconductivity emerges from this pseudogap parent state and what signatures appear across the transition. While the pseudogap physics has been most extensively studied in hole-doped cuprates, recent photoemission on electron-doped cuprates also indicates reconstructed Fermi surfaces without long-range antiferromagnetic order, and pairing is believed to be d-wave. Motivated by these observations, this work examines both hole- and electron-doped cases. Prior approaches linked Luttinger’s theorem violation to zeros of the electron Green’s function; here, such zeros signal a neutral spinon sector required by non-perturbative Luttinger extensions. Accounting for the spinon sector is essential to explain the emergence of nodal Bogoliubov quasiparticles in the d-wave superconductor. The authors employ a theory with fermionic spinons coupled to an SU(2) gauge field in a π-flux spin liquid background. Physical electrons couple to these spinons via a charge-e boson transforming under the same gauge symmetry. In the hole-doped case, the resulting normal-state electron Fermi surface comprises hole pockets of area p rather than 1−p. Upon condensation of the boson, the gauge symmetry is fully broken and d-wave superconductivity (and other orders) can be inherited by the electrons. Using the Ancilla model, which provides a microscopic dispersion across the Brillouin zone, the paper computes electronic observables through the pseudogap-to-superconductor transition and identifies experimental signatures, including the appearance and evolution of nodal quasiparticles and their anisotropic velocities.

The study builds on extensive experimental and theoretical work on cuprates, including ARPES observations of Fermi arcs and reconstructed Fermi surfaces in both hole- and electron-doped systems, and prior interpretations of pseudogap phenomenology via zeros of the Green’s function that imply fractionalized excitations. Non-perturbative extensions of Luttinger’s theorem (Oshikawa; Senthil et al.) motivate incorporating neutral spinon sectors. The Ancilla framework, previously proposed to connect pseudogap metals (FL*) and large Fermi-surface Fermi liquids within a single-band model using auxiliary layers and emergent gauge fields, allows controlled modeling of dispersions across the Brillouin zone. Earlier bosonic-spinon formulations generated superconducting states with 8 nodes; here, fermionic spinons in a π-flux state are emphasized as essential to produce 4 nodal points in the d-wave superconductor, consistent with experiments. Recent electron-doped studies report reconstructed Fermi surfaces without long-range AF order and d-wave pairing. Numerical studies on electron-doped t-J models suggested possibly fully gapped d-wave states; the present work discusses conditions required for a fully gapped d-wave and argues that, generically, 4 nodal points persist unless additional mechanisms intervene.

Model: Start from a single-band Hubbard model for physical electrons (c) and introduce two ancilla spin layers (S1, S2) forming a bilayer antiferromagnet. The Hubbard interaction is decoupled via a paramagnon field identified with rung-triplet excitations of the ancilla bilayer. In the effective Ancilla Hamiltonian, c-electrons (carrying doping 1−p) couple via Kondo coupling J_K to S1 spinons (f1), while S1 and S2 are antiferromagnetically coupled and each has intra-layer Heisenberg exchange. S1 is realized as a fermionic spinon π-flux spin liquid (with Dirac nodes) coupled to an emergent SU(2) gauge field; S2 is represented by fermionic spinons (f2). A two-component complex boson B=(B1,B2), a gauge-fundamental chargon (spin singlet), couples the spinon layers; B1 plays the role of a Higgs field. Mean-field treatment: After mean-field decoupling, the effective Hamiltonian contains c-dispersions, f-dispersions, c–f1 hybridization Φ, inter-spinon couplings via B, and adjustable chemical potentials μc, μf, μ1 fixed by constraints on fillings (c at 1−p; f1 and f2 at one fermion per site). The π-flux spin liquid gauge is chosen to yield two band minima; the chargon condensate is expanded about these minima and a d-wave pairing channel is selected by choosing B-components such that Δ=εabBaBb is condensed with ansatz Ba+ = (−b, b), Ba− = (b, b) to induce d-wave pairing inherited by c-electrons when B condenses. Normal states considered: (i) hole-doped FL* with hole pockets of volume p formed by hybridized c–f1 states (Φ condensed, B gapped); (ii) electron-doped states with either only electron pockets at (π,0)/(0,π) or both electron anti-nodal pockets and hole-like nodal pockets, depending on parameters and Φ. Parameter choices: For hole-doped comparisons to ARPES, c-electron hoppings t0=0.22 eV, t1=−0.034 eV, t20=0.036 eV, t11=−0.007 eV; f1 hoppings t0=−0.1 eV, t1=0.03 eV, t2=0.01 eV; π-flux spin liquid hopping t2 (second layer) = 0.14 eV. Fits assumed Φ≈0.09 and p≈0.206. For electron-doped, c hoppings same as hole-doped; f1 hoppings adjusted (e.g., t0=−0.1 eV, next-nearest −0.07 eV, others zero) to realize only electron pockets; alternative values (e.g., t0=−0.06 eV, t02=0.02 eV) yield coexisting electron and hole pockets. Computations: Spectral densities A(ω,k)=−π−1 Im G(ω,k) computed from tight-binding Hamiltonian in momentum space. Chemical potentials determined on an 80×80 k-grid via bisection within filling error 0.01. Spectral functions evaluated on a quarter Brillouin zone with 200×200 grid; lifetime broadening 0.005 eV. Quasiparticle weights Zk obtained by diagonalizing G−1(ω,k)=ω−H(k); Zk is the norm square of the relevant eigenvector component at excitation energy Δk. Nodal velocities vF (tangent to kx=ky) and vΔ (normal to kx=ky) extracted by least-squares linear fits to dispersions near nodes using dense sampling along diagonal cuts (e.g., 80,000 points, fits over 500 nearest points). Analyses focus on how spectra and velocities evolve as B condenses (parameter b) for various Φ in both hole- and electron-doped cases.

- Universal emergence of 4 nodal Bogoliubov quasiparticles: Across both hole- and electron-doped pseudogap metals, the onset of conventional d-wave superconductivity via B-condensation yields gapless Bogoliubov quasiparticles at 4 nodal points along the Brillouin zone diagonals with strong velocity anisotropy, consistent with BCS phenomenology. - Hole-doped case (starting from FL* with hole pockets): As b increases from 0, there exists a very narrow window (0<b<bc) in which 12 nodes (3 per quadrant) appear due to small particle–hole asymmetry of the f1 spinon bands. With further increase of b, pairs of nodes annihilate leaving 4 nodes, matching experimental observations. The specifics of which nodes annihilate first can depend on the sign of certain hopping parameters of the spin liquid. - Electron-doped case with only anti-nodal electron pockets in the normal state: Despite a normal-state gap along the diagonals and Fermi surfaces only near (π,0)/(0,π), superconductivity gaps the anti-nodal pockets and simultaneously produces 4 nodal points along the diagonals. These nodal quasiparticles descend from Dirac points of the π-flux spin liquid hybridized with c and f1 bands and cannot be gapped without additional symmetry breaking (e.g., strong spin-rotation symmetry breaking). - Electron-doped case with both anti-nodal electron pockets and nodal hole pockets: Upon superconductivity, finite spectral weight persists near nodes while all other k-points are gapped by a d-wave gap of varying magnitude. - Nodal velocity evolution (hole-doped illustrative results): vΔ (normal to diagonal) starts at 0 for b=0, becomes finite with B condensation, increases with b when b≲Φ (reflecting growing effective pairing), then decreases at large b as c electrons decouple from spinons; it can approach 0 as b→large. vF (tangent to diagonal) begins at the normal-state value and increases monotonically with b, saturating at the Fermi velocity of the decoupled c-band. For all parameters studied, vΔ/vF remains small, indicating strong nodal anisotropy. - Conditions for a fully gapped d-wave state: The conventional d-wave superconductor with pairing not exceeding the Fermi energy has 4 nodal points. A fully gapped d-wave state requires additional mechanisms: (i) an SC* state where the spin liquid remains active (starting from FL* with electron pockets then pairing), (ii) strong coexisting antiferromagnetism gapping nodes via pairwise annihilation across the magnetic Brillouin zone, or (iii) pairing strength comparable to or exceeding the Fermi energy leading to nodal annihilation at high-symmetry points.

The work addresses how d-wave superconductivity emerges from a pseudogap metal that violates Luttinger’s theorem by invoking fractionalized spinon sectors in a π-flux spin liquid coupled to electrons via an SU(2) gauge structure. Condensation of a charge-e Higgs field (B) both breaks the gauge symmetry and transmits d-wave pairing to the electrons. A key outcome is that nodal Bogoliubov quasiparticles in the superconductor can be understood as descendants of Dirac spinons of the π-flux state, explaining their robust appearance at 4 diagonal points even when the normal state features only anti-nodal electron pockets and a diagonal gap (electron-doped case). In the hole-doped case, the transient 12-node regime for small b highlights how small particle–hole asymmetries in spinon dispersions can affect the immediate post-onset spectra, but the system flows to the experimentally observed 4-node structure as pairing strengthens. The computed evolution of nodal velocities offers quantitative signatures for ARPES: vF increases and saturates, while vΔ is non-monotonic and generally smaller, maintaining strong anisotropy characteristic of cuprates. The framework suggests that fully gapped d-wave states require non-generic additional ingredients (SC* character, strong AF order, or very strong pairing), aligning with the general robustness of nodal quasiparticles in cuprates. These results provide concrete, testable predictions for spectral signatures across the pseudogap-to-superconductor transition in both hole- and electron-doped materials.

Using the Ancilla model of a π-flux spin liquid coupled to physical electrons, the study provides a unified description of the transition from a pseudogap metal with reconstructed Fermi surfaces to a conventional d-wave superconductor. It predicts that, generically, 4 nodal Bogoliubov quasiparticles emerge along zone diagonals with strong velocity anisotropy, even when the normal state lacks nodal Fermi surfaces (electron-doped case). In hole-doped systems, a narrow parameter window with 12 nodes may appear immediately after pairing onset, but the spectrum rapidly evolves to 4 nodes as pairing strengthens. The evolution of nodal velocities with the Higgs condensate b and hybridization Φ yields specific trends for experimental verification. The work delineates the exceptional conditions required to fully gap a d-wave state and argues that conventional scenarios retain nodes. Future directions include extending the framework to incorporate charge order and devising experimental probes to distinguish among possible Dirac spin-liquid parent states and fractionalized Fermi liquid normal states.

- Mean-field treatment neglects dynamical fluctuations of the emergent SU(2) gauge fields; while argued to be benign for low-energy dispersions in the studied phases, this remains an approximation. - Parameter choices (hoppings, Φ, b) are partly fitted to existing ARPES on hole-doped cuprates; electron-doped parameters are less constrained, potentially affecting quantitative predictions. - The π-flux spin liquid is assumed as the parent fractionalized state; while nodal results may be similar for other Dirac spin liquids, stability and specific spectral details can vary. - Long-range antiferromagnetic order is not included in the primary analysis; although the authors argue results persist, coexisting AF could qualitatively modify nodal structure under certain conditions. - Finite lifetime broadening and finite momentum grids may blur fine spectral features; extraction of velocities involves linear fits near nodes and may be sensitive to discretization and parameter choices.