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

The cuprates' phase diagram presents a complex interplay between the pseudogap metal (violating Luttinger's theorem) and *d*-wave superconductivity. Understanding the transition between these phases, especially concerning the role of small Fermi surfaces or arcs, remains a significant challenge. This work investigates the experimental signatures of this transition in both hole- and electron-doped cuprates. While extensively studied in hole-doped materials, recent photoemission experiments on electron-doped cuprates show a reconstructed Fermi surface without long-range antiferromagnetism, and the pairing is also believed to be *d*-wave. The violation of Luttinger's theorem in the pseudogap metal is modeled by associating it with zeros of the electron Green's functions, interpreted as signaling an additional sector of neutral spinon excitations. A complete treatment of the spinon sector is crucial to understanding the emergence of nodal Bogoliubov quasiparticles in the *d*-wave superconductor. The authors employ a theory of the pseudogap metal with fermionic spinons coupled to an SU(2) gauge field in a π-flux background. Fermionic spinons are coupled to physical electrons via a charge *e* boson transforming under the same SU(2) gauge symmetry. In the hole-doped case, the normal state electron Fermi surface exhibits pockets associated with hole density *p*, not the free electron hole-density value 1-*p*. Superconductivity and charge order are treated as competing instabilities of a fractionalized Fermi liquid (FL) pseudogap phase. The π-flux spin liquid is preferred over the U(1) staggered flux spin liquid due to its stability and the additional possibility of charge order. The transition from the pseudogap phase with electron and/or hole pockets and a π-flux spin liquid to a conventional *d*-wave superconductor is analyzed using the Ancilla model, offering a microscopic model for the complete fermion dispersion in the Brillouin zone, arising from an approximation of the Hubbard model. This model maps the low-energy physics to free electrons coupled to a bilayer square lattice antiferromagnet, with the latter serving as ancilla qubits to obtain a wavefunction with non-trivial entanglement on the *c* layer. The ancilla layers are quantum, ensuring compliance with Luttinger-Oshikawa constraints.

The authors review previous theoretical and experimental work on the cuprates, highlighting the pseudogap metal and *d*-wave superconductivity. They discuss models that address the violation of Luttinger's theorem in the pseudogap phase, emphasizing the role of spinon excitations. Several theoretical models of the pseudogap metal are mentioned, focusing on those that associate the violation of Luttinger's theorem with zeros of the electron Green's functions. The authors mention previous work on the condensation of a charge *e* boson from an incoherent normal state and the use of a U(1) staggered flux spin liquid. They contrast this with their use of a π-flux spin liquid, highlighting the advantages of the latter in terms of stability and the possibility of charge order. The Ancilla model is presented as a more complete approach compared to previous phenomenological models, as it provides a microscopic description of the fermion dispersion throughout the Brillouin zone.

The study utilizes the Ancilla model, a theoretical framework that maps the low-energy physics of a single-band Hubbard-like model to a system of free electrons coupled to a bilayer square lattice antiferromagnet. The antiferromagnet layers act as ancilla qubits, introducing non-trivial entanglement. This mapping allows for the canonical transformation of the Hubbard *U* term, avoiding its direct treatment. The Hamiltonian includes terms for electron hopping, Kondo coupling between electrons and the first spin layer (S1), antiferromagnetic coupling between S1 and a second spin layer (S2), and Heisenberg exchange interactions within S1 and S2. The model is solved using a mean-field decoupling approach, where the chemical potentials are adjusted to satisfy filling constraints for electrons and spinons. The π-flux spin liquid is represented in the mean-field theory. The condensation of a two-component, complex boson *B* representing the charge *e* Higgs field is introduced to model the onset of superconductivity. Different condensation patterns of *B* lead to different symmetry-breaking orders. The study focuses on the *d*-wave superconducting order parameter. The electronic spectral density and band structure are calculated using the mean-field Hamiltonian to examine the evolution of the electronic spectrum as *B* condenses. Computations are performed on an 80x80 momentum space grid, and next-nearest neighbor hopping terms are included for both electrons and spinons. Specific hopping parameters are chosen based on fitting to photoemission data in the pseudogap regime for the hole-doped case. The number of nodes, Fermi velocities (*v_c* and *v_o*), and quasiparticle residue (*Z_k*) are analyzed. The spectral density is calculated using A(ω,k) = -π⁻¹ Im[G(ω,k)], with a lifetime parameter of 0.005eV. Quasiparticle weight Zk is calculated by diagonalizing the inverse Green's function and obtaining the eigenvector corresponding to the excitation energy.

The study finds that in both hole- and electron-doped cuprates, the onset of *d*-wave superconductivity leads to the emergence of nodal Bogoliubov quasiparticles. In the hole-doped case, the number of nodes depends on the parameters. For small values of *b* (related to the superconducting pairing strength), 12 nodes appear initially, but for larger *b*, these nodes annihilate in pairs to yield the experimentally observed 4 nodes. The Fermi velocities show a complex dependence on *b* and Φ (representing the strength of the spinon-electron coupling), with *v_c* increasing monotonically and *v_o* exhibiting a non-monotonic behavior. In the electron-doped case, even when the normal state has only electron pockets in the anti-nodal region and a gap in the nodal region, the condensation of *B* results in the re-emergence of 4 nodal points along the zone diagonals. These nodes are associated with the Dirac points of the π-flux spin liquid, hybridizing with the c and f₁ bands. The authors analyze the evolution of quasiparticle excitation energy and residue for both hole-doped and electron-doped scenarios, showing how these quantities are affected by the superconducting transition. The study suggests that nodal Bogoliubov quasiparticles are remnants of Dirac spinons revealed by superconductivity onset. The authors considered three possible routes to a fully gapped *d*-wave superconductor: pairing electron pockets from an FL metal, the onset of strong antiferromagnetic order, and a very large pairing interaction.

The findings suggest that the emergence of nodal quasiparticles in *d*-wave superconductors originates from the underlying π-flux spin liquid, regardless of whether the normal state exhibits hole or electron pockets. The Ancilla model provides a detailed microscopic mechanism explaining this emergence. The transition from 12 to 4 nodes in the hole-doped case highlights the role of parameters in determining the observed nodal structure. The presence of 4 nodes in electron-doped superconductors, even when the normal state is gapped along the diagonal, underscores the importance of the spin liquid's fractionalized excitations. The work's implications are significant for understanding the relationship between the pseudogap phase, the spin liquid, and high-Tc superconductivity. Further research could investigate the effects of coexisting orders like charge order on the superconducting state. While the calculated quantities may not strongly distinguish between different Dirac spin liquids, exploring other experimental probes might differentiate different FL normal states.

This study provides testable predictions for the signatures of conventional *d*-wave superconductivity originating from a pseudogap phase with fractionalized degrees of freedom. The emergence of 4 nodal Bogoliubov quasiparticles, regardless of the normal state's Fermi surface characteristics, is a key finding. Future research could extend this approach to incorporate other phases in the underdoped cuprates, particularly charge order, and explore experimental distinctions between different FL normal states.

The study relies on a mean-field approach to the Ancilla model, neglecting fluctuations which might influence the results. The specific parameter choices for the hopping terms influence the precise numerical results but not the qualitative features of the transition. The study focuses on systems without long-range antiferromagnetic order; however, the effect of such order on the nodal structure remains an open question.