Nandini Trivedi: Single- and two-particle energy gaps across the disorder-driven superconductor-insulator transition

Nandini Trivedi presented results of numerical studies of strongly disordered superconductors and superconductor-insulator transition in 2D lattice case. She starts from general discussion of amplitude fluctuations versus phase fluctuations in presence of disorder, and then she compare two basic energy scales which can be defined : single-particle gap Δ in the electron spectrum, and superfluid density ρ_sThe point is that usually in disordered superconductors ρ_s >> Δ (by orders of magnitude) and thus phase fluctuations are irrelevant apart from the region close to T_c Situation changes upon increase of disorder which leads to strong suppression of superfluid density, which eventually becomes comparable to Δ , since its value is much less sensitive to disorder. It is this region there S-I transition occurs.

Going to quantitative discussion, Nandini defines the Hamiltonian to be used, which isattractive Hubbard model with disorder, defined on 2D lattice. The major part of results was obtained for the attraction coupling constant U close in magnitude to the hopping amplitude t (in other terms, U was about ¼ of the full bandwidth). The restriction of U ~ t instead of U << t (which would better correspond to a real superconductor) is due to limited size of a system which can be studied numerically.

Two different methods were used. One of them is a kind of mean-field theory + Hartree-Fock corrections, implemented on the top of exactly determined single-electron wavefunctions. This method disregards phase fluctuations, the results are mainly presented in previous publications (Phys. Rev. Lett. 1998 and Phys.Rev. B 2001). A new recently developed method, which is the main tool of the presented work was called “Detrimental quantum Monte Carlo”, which is “exact” in the sense that no fluctuations are excluded, and also it does not suffer from usual “sign problem”. The outcome of Monte Carlo simulation gives correlation functions defined in imaginary (Matsubara) time. Then one needs to make analytic continuation to real time domain, in order to get observed spectral functions. “Maximum entropy” method was devised for this analytical continuation. Sum rule was controlled individually at each site on the lattice.

It was observed that pairing amplitude Δ(R) developed emergent inhomogenities at the spatial scale much longer than atomic scale where random potential was originally defined. Simultaneously, single-particle spectral gap was determined locally in space. It occurs that local pairing amplitude Δ(R) and local single-particle gap ω₁ anti-correlates in space: the regions with large Δ(R) usually show smaller ω₁.

Distribution of coherence peak heights P(h) was studied at different temperatures. Whereas at T=0.1 Tc maximum of P(h) is at relatively large h, it is shifted to h=0 at T=0.5 Tc where most of the peaks disappear. In presence of sufficiently strong disorder single-particle gap is suppressed already above Tc, this suppression is seen up to higher temperature T* which is associated with “pseudogap”. The value of T* grows with increasing disorder, whereas Tc drops down. At low temperatures hard gap still exists at strong disorder, where coherence peaks are totally gone. Fully developed gap never was seen right at Tc, contrary to the STM experimental data on InOx (reported by Benjamin Sacepe).

Dynamic current-current correlation functions were extracted from Monte-Carlo simulation

and superfluid density ρ_s was obtained at different degrees of disorder. S-I transition was found as a point where ρ_s vanishes.

Dynamic pair susceptibility χ_pair(ω) was also determined from Monte Carlo simulations plus analytic continuation; it was found that has a spectral gap in the insulating state, which vanishes smoothly at the critical disorder. There is a single value of the critical disorder, that separates superconductive state with nonzero Tc from insulating state with nonzero two-particle gap, whereas single-particle gap stays large throughout the whole quantum-critical region. These findings are in qualitative agreement with results of the analytical theory developed previously by M. Feigel’man, L. Ioffe and M. Mezard (Phys. Rev. B 82, 184534 (2010)). This paper is apparently unknown to Nandini, which makes qualitative agreement between the results even more valuable.

Blogged by Mikhail Feigel’man (Landau Institute)