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 Tc, 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 Tc saturates but the stiffness decreases, which suggests
strong superconducting fluctuations. Tc 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