I summarize recent progress on obtaining rigorous upper bounds on superconducting transition temperature [Formula: see text] in two dimensions independent of pairing mechanism or interaction strength. These results are derived by finding a general upper bound for the superfluid stiffness for a multi-band system with arbitrary interactions, with the only assumption that the external vector potential couples to the kinetic energy and not to the interactions. This bound is then combined with the universal relation between the superfluid stiffness and the Berezinskii–Kosterlitz–Thouless [Formula: see text] in 2D. For parabolic dispersion, one obtains the simple result that [Formula: see text], which has been tested in recent experiments. More generally, the bounds are expressed in terms of the optical spectral weight and lead to stringent constraints for the [Formula: see text] of low-density, strongly correlated superconductors. Results for [Formula: see text] bounds for models of flat-band superconductors, where the kinetic energy vanishes and the vector potential must couple to interactions, are briefly summarized. Upper bounds on [Formula: see text] in 3D remains an open problem, and I describe how questions of universality underlie the challenges in 3D.
The optical conductivity of the 2D
t−J model and the origin of electron incoherence in the high-T
c cuprate superconductors: a variational study
Understanding the origin of electron incoherence is the first step toward a theoretical description of the non-Fermi liquid behavior of the high-T
c cuprate superconductors. Such electron incoherence manifests itself most evidently in the non-Drude behavior of the optical response of the system and the anomalous density fluctuation behavior in the long wave length limit. The spectral weight transfer related to such dissipative response, which is absent in conventional Fermi liquid metal, has direct consequence on the dc transport property of the system in the normal state and the superfluid stiffness in the superconducting state. It is found that such electron incoherence remains significant even in the clean limit and at low temperature and thus must be attributed to the strong electron correlation effect in the cuprate superconductors. Here we study such an intrinsic effect in the 2D
t−J model through the variational calculation of its optical conductivity
σ(ω). We assume a resonating valence bond ground state as our starting point and find that a significant portion of the total optical spectral weight remains incoherent throughout the phase diagram. The optical absorption is found to extend all the way to an energy of the order of the bare band width. We find that both the total optical weight
Kˉ and the integrated incoherent optical weight
I increase monotonically with doping, with their ratio
Rincoh=I/Kˉ decreasing monotonically with doping. Our results indicate that the majority part of electron incoherence in the 2D
t−J model can be attributed to the electron fractionalization mechanism assumed in such a treatment. We also find that the Drude weight deduced from
D=Kˉ−I scales linearly with hole doping, without any sign of a non-monotonic behavior in the overdoped regime.
We construct and analyze a lattice generalization of the Yukawa-Sachdev-Ye-Kitaev model, where spinful fermions experience onsite, 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.