Condensed matter systems with flat bands close to the Fermi level generally exhibit, due to their very large density of states, extraordinary high critical ordering temperatures of symmetry breaking orders, such as superconductivity and magnetism. Here we show that the critical temperatures follow one of two universal curves with doping away from a flat band depending on the ordering channel, which completely dictates both the general order competition and the phase diagram. Notably, we find that orders in the particle-particle channel (superconducting orders) survive decisively further than orders in the particle-hole channel (magnetic or charge orders) because the channels have fundamentally different polarizabilities. Thus, even if a magnetic or charge order initially dominates, superconducting domes are still likely to exist on the flanks of flat bands. We apply these general results to both the topological surface flat bands of rhombohedral ABC-stacked graphite and to the van Hove singularity of graphene.c 1 2T − c . We are able to establish the general phase diagram exactly for any interactions in an ideally flat band system. Remarkably, we also show that the results remain valid arXiv:1611.04893v2 [cond-mat.supr-con]

A chiral time-reversal symmetry breaking (d + id)-wave superconducting state is likely to emerge in graphene doped close to the Van Hove singularity. As heavy doping procedures are expected to introduce defects, we here investigate the effects of microscopic defects on the (d + id)-wave superconducting state at the Van Hove singularity. We find that, while the superconducting order is reduced near a defect, the (d + id)-wave state remains intact and recovers in an exponential manner away from the defect. The recovery length is found to be on the order of one lattice constant for weak couplings. This is comparable to the recovery length of a conventional s-wave state, demonstrating that the unconventional (d + id)-wave state is quite resilient to defects. Moreover, we find no significant changes between a single site defect and more extended defects, such as a bivacancy. While the (d + id)-wave state is fully gapped, we also show that defects introduce localized midgap states with non-zero energies, which should be accessible via scanning probe experiments.

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We investigate the induced superconducting pair correlations in junctions between a conventional spin-singlet s-wave superconductor and a disordered normal metal. Decomposing the pair amplitude based on its symmetries in the time domain, we demonstrate that the odd-time, or equivalently oddfrequency, spin-singlet p-wave correlations are both significant in size and entirely robust against random non-magnetic disorder. We find that these odd-frequency correlations can even be generated by disorder. Our results show that anisotropic odd-frequency pairing plays a significant role in disordered superconducting hybrid structures.Superconductivity arises from the formation and condensation of a macroscopic number of electron pairs, or Cooper pairs. The superconducting state is characterized by the Cooper pair amplitude which, due to the fermionic nature of electrons, is antisymmetric under the exchange of all degrees of freedom describing the paired electron states. These symmetries include the positions, spin, and relative time separation of the two electrons.In BCS theory of superconductivity the interparticle interaction is instantaneous and, hence, only static, equal-time, pair amplitudes contribute to the order parameter. Such equal-time pair symmetries are then constrained to be either even in space and odd in spin, as in the conventional spin-singlet s-wave state, or odd in space and even in spin (e.g. spin-triplet p-wave). However, even for BCS superconductors, a growing body of evidence suggests that dynamical pair correlations that are odd in the relative time [1][2][3][4][5] can play a role in the physics of superconducting heterostructures [6][7][8]. Due to their odd time-dependence, these dynamical pair correlations must be either even in space and even in spin or odd in space and odd in spin. Such exotic odd-time pairs are equivalently also referred to as odd-frequency (odd-ω) pairs [1].A well-established host of odd-time/odd-ω pairs are ferromagnet-superconductor (FS) junctions. In these systems, the combination of spatial invariance breaking due to the interface and spin-singlet to spin-triplet conversion due to the ferromagnet, generates finite odd-ω spin-triplet s-wave pairs in the F region even for conventional spin-singlet s-wave BCS superconductors [9,10]. The presence of odd-ω s-wave pairs explains several unconventional features of FS junctions, such as the longrange proximity effect [11-19], zero-bias peaks [20, 21], and paramagnetic Meissner effect [22][23][24].

Twisted bilayer graphene (TBG) develops large moiré patterns at small twist angles with flat energy bands hosting domes of superconductivity. The large system size and intricate band structure have however hampered investigations into the superconducting state. Here, using full-scale atomistic modelling with local electronic interactions, we find at and above experimentally relevant temperatures a highly inhomogeneous superconducting state with nematic ordering on both atomic and moiré length scales. The nematic state has a locally anisotropic real-valued d-wave pairing, with a nematic vector winding throughout the moiré pattern, and is three-fold degenerate. Although d-wave symmetric, the superconducting state has a full energy gap, which we tie to a π-phase interlayer coupling. The superconducting nematicity is further directly detectable in the local density of states. Our results show that atomistic modeling is essential and also that very similar local interactions produce very different superconducting states in TBG and the high-temperature cuprate superconductors.