BCS to incoherent superconductivity crossovers in the Yukawa-SYK model on a lattice

We provide a quantitative and controlled analysis of the phase diagram of the the Yukawa-SYK model on a lattice, in the normal and superconducting states. We analyze the entire crossover from BCS/weak-coupling to Eliashberg/strong coupling superconductivity, as a function of fermion-boson interaction strength and hopping parameter. Cooper pairs of sharp Fermi-liquid quasiparticles at weak coupling evolve into pairing of fully incoherent fermions in the non-Fermi liquid regime. The crossovers leave observable traces in the critical temperature, the zero-temperature and zero-energy gap, the entropy, and the phase stiffness.Comment: 7 pages, 4 figures; companion paper: arXiv:2302.1313

Quantum critical Eliashberg theory, the Sachdev-Ye-Kitaev superconductor and their holographic duals

Superconductivity is abundant near quantum critical points, where fluctuations suppress the formation of Fermi liquid quasiparticles and the BCS theory no longer applies. Two very distinct approaches have been developed to address this issue: quantum-critical Eliashberg theory and holographic superconductivity. The former includes a strongly retarded pairing interaction of ill-defined fermions, the latter is rooted in the duality of quantum field theory and gravity theory. We demonstrate that both are different perspectives of the same theory. We derive holographic superconductivity in form of a gravity theory with emergent space-time from a quantum many-body Hamiltonian—the Yukawa Sachdev-Ye-Kitaev model—where the Eliashberg formalism is exact. Exploiting the power of holography, we then determine the dynamic pairing susceptibility of the model. Our holographic map comes with the potential to use quantum gravity corrections to go beyond the Eliashberg regime

Correlation between phase stiffness and condensation energy across the non-Fermi to Fermi-liquid crossover in the Yukawa-Sachdev-Ye-Kitaev model on a lattice

We construct and analyze a lattice generalization of the Yukawa-Sachdev-Ye-Kitaev model, where spinful fermions experience on-site, random, all-to-all interactions with an Einstein bosonic mode, and random intersite coherent hopping. We obtain the exact self-consistent numerical solution of the model at mean-field level, and analytical approximations, for all values of fermion-boson coupling and hopping, under the spin-singlet ansatz and at particle-hole symmetry, both in the normal and superconducting states, thus tracing the entire phase diagram. In the normal state, the competition between hopping and coupling leads to crossovers between Fermi-liquid and non-Fermi liquid states, as reflected by the fermionic and bosonic spectral functions and the normal-state entropy. We calculate the finite phase stiffness of the superconducting state through the equilibrium electromagnetic response. Furthermore, we study the critical temperature $T_c$, as well as the spectral functions, the quasiparticle weight, the gap, and the condensation energy in the superconducting state. At weak coupling, we retrieve a disordered generalization of Bardeen-Cooper-Schrieffer theory. At strong coupling, asymptotically $T_c$ saturates but the stiffness decreases, which suggests strong superconducting fluctuations. $T_c$ is maximum in the single-dot limit, while the stiffness peaks exactly at the crossover between non-Fermi liquid and Fermi-liquid phases. We discover that the quasiparticle weight, the stiffness, and the condensation energy, are all correlated as a function of coupling, reminiscent of the correlations observed in high-temperature cuprate superconductors.Comment: 53 pages, 22 figures; companion paper: arXiv:2302.1313

Quantum critical scaling and holographic bound for transport coefficients near Lifshitz points

The transport behavior of strongly anisotropic systems is significantly richer compared to isotropic ones. The most dramatic spatial anisotropy at a critical point occurs at a Lifshitz transition. The present study uses scaling arguments and the gauge - gravity duality to investigate universal bounds appearing in strongly-coupled quantum anisotropic systems near a Lifshitz point. Explicit examples are merging Dirac or Weyl points or Lifshitz points near the superconductor-insulator quantum phase transition. Using scaling arguments we propose a generalization of both the shear-viscosity to entropy-density ratio and the charge-diffusivity bounds to the anisotropic case. We demonstrate that the electric conductivity and viscosity of the same material vanish along certain directions yet diverge along others. Thus, at such a quantum Lifshitz point the non-quasi-particle transport in the strong coupling limit is both metallic and insulating, depending on the electric field direction. We investigate the strongly-coupled phase of such systems in a gravitational Einstein-Maxwell-dilaton model with a linear massless scalar. The holographic computation demonstrates that some elements of the viscosity tensor can be related to the ratio of the electric conductivities. From the IR critical geometry, we express the charge diffusion constants in terms of the square butterfly velocities. The proportionality factor turns out to be direction-independent, linear in the inverse temperature, and related to the critical exponents which parametrize the anisotropic scaling of the dual field theory