Abstract:
The possibility of non-s-wave superconductivity induced by phonons is investigated using a simple model that is inspired by Sr$_2$RuO$_4$. The model assumes a two-dimensional electronic structure, a two-dimensional spin-fluctuation spectrum, and three-dimensional electron-phonon coupling. Taken separately, each interaction favors formation of spin-singlet pairs (of s symmetry for the phonon interaction and d$_{x^2-y^2}$ symmetry for the spin interaction), but in combination, a variety of more unusual singlet and triplet states are found, depending on the interaction parameters. This may have important implications for Sr$_2$RuO$_4$, providing a plausible explanation of how the observed spin fluctuations, which clearly favor d$_{x^2-y^2}$ pairing, may still be instrumental in creating a superconducting state with a different (e.g., p-wave) symmetry. It also suggests an interpretation of the large isotope effect observed in Sr$_2$RuO$_4$. These results indicate that phonons could play a key role in establishing the order-parameter symmetry in Sr$_2$RuO$_4$, and possibly in other unconventional superconductors.

Abstract:
Giant and nil proximity effects and unconventional symmetry of cuprate superconductors are explained as a result of the strong Froehlich interaction of holes with c-axis polarised optical phonons acting together with an anisotropic nonlocal deformation potential.

Abstract:
It has gone almost unquestioned that superexchange in the t-J (or Hubbard) model, and not phonons, is responsible for the unconventional ("d-wave") pairing symmetry of cuprate superconductors. However a number of advanced numerical studies have not found superconductivity in the Hubbard model. On the other hand compelling experimental evidence for a strong electron-phonon interaction (EPI) has currently arrived. Here I briefly review some phonon-mediated unconventional pairing mechanisms. In particular the anisotropy of sound velocity makes the phonon-mediated attraction of electrons non-local in space providing unconventional Cooper pairs with a nonzero orbital momentum already in the framework of the conventional BCS theory with weak EPI. In the opposite limit of strong EPI rotational symmetry breaking appears as a result of a reduced Coulomb repulsion between unconventional bipolarons. Using the variational Monte-Carlo method we have found that a relatively weak finite-range EPI induces a d-wave BCS state also in doped Mott-Hubbard insulators or strongly-correlated metals. These results tell us that poorly screened EPI with conventional phonons is responsible for the unconventional pairing in cuprate superconductors.

Abstract:
Superconductivity without phonons has been proposed for strongly correlated electron materials that are tuned close to a zero-temperature magnetic instability of itinerant charge carriers. Near this boundary, quantum fluctuations of magnetic degrees of freedom assume the role of phonons in conventional superconductors, creating an attractive interaction that glues electrons into superconducting pairs. Here we show that superconductivity can arise from a very different spectrum of fluctuations associated with a local or Kondo-breakdown quantum-critical point that is revealed in isotropic scattering of charge carriers and a sub-linear temperature-dependent electrical resistivity. At this critical point, accessed by applying pressure to the strongly correlated, local-moment antiferromagnet CeRhIn5, magnetic and charge fluctuations coexist and produce electronic scattering that is maximal at the optimal pressure for superconductivity. This previously unanticipated source of pairing glue opens possibilities for understanding and discovering new unconventional forms of superconductivity.

Abstract:
We develop a theory for interlayer pairing of chiral electrons in graphene materials which results in an unconventional superconducting (S) state with s-wave spin-triplet order parameter. In a pure bilayer graphene, this superconductivity exhibits a gapless property with an exotic effect of temperature-induced condensation causing an increase of the pairing amplitude (PA) with increasing temperature. We find that a finite doping opens a gap in the excitation spectrum and weakens this anomalous temperature-dependence. We further explore the possibility of realizing variety of pairing patterns with different topologies of the Fermi surface, by tuning the difference in the doping of the two layers. In trillayer graphene, the interlayer superconductivity is characterized by a two components order parameter which can be used to define two distinct phases in which only one of the components is non vanishing. For ABA stacking the stable state is determined by a competition between these two phases. By varying the relative amplitude of the corresponding coupling strenghes, a first order phase transition can occur between these two phases. For ABC stacking, we find that the two phases coexist with a possibility of a similar phase transition which turns out to be second order.

Abstract:
The formation of electron pairs is a prerequisite of superconductivity. The fermionic nature of electrons yields four classes of superconducting correlations with definite symmetry in spin, space and time. Here, we suggest double quantum dots coupled to conventional s-wave superconductors in the presence of inhomogeneous magnetic fields as a model system exhibiting unconventional pairing. Due to their small number of degrees of freedom, tunable by gate voltages, quantum-dot systems provide an ideal tool to gain fundamental insight in unconventional pairing. We propose two detection schemes for unconventional superconductivity, based on either Josephson or Andreev spectroscopy.

Abstract:
We show that finite angular momentum pairing chiral superconductors on the triangular lattice have point zeroes in the complex gap function. A topological quantum phase transition takes place through a nodal superconducting state at a specific carrier density $x_c$ where the normal state Fermi surface crosses the isolated zeros. For spin singlet pairing, we show that the second nearest neighbor $d+id$-wave pairing can be the dominant pairing channel. The gapless critical state at $x_c\simeq0.25$ has six Dirac points and is topologically nontrivial with a $T^3$ spin relaxation rate below $T_c$. This picture provides a possible explanation for the unconventional superconducting state of Na$_x$CoO$_2\cdot y$H$_2$O. Analyzing a pairing model with strong correlation using the Gutzwiller projection and symmetry arguments, we study these topological phases and phase transitions as a function of Na doping.

Abstract:
We report observation of the nonlinear Meissner effect (NLME) in Nb films by measuring the resonance frequency of a planar superconducting cavity as a function of the magnitude and the orientation of a parallel magnetic field. Use of low power rf probing in films thinner than the London penetration depth, significantly increases the field for the vortex penetration onset and enables NLME detection under true equilibrium conditions. The data agree very well with calculations based on the Usadel equations. We propose to use NLME angular spectroscopy to probe unconventional pairing symmetries in superconductors.

Abstract:
At sufficiently high densities, cold dense three-flavor quark matter is in the color-flavor locked (CFL) phase, in which all nine quarks pair in a particularly symmetric fashion. Once the heaviness of the strange quark (mass $m_s$) and the requirements of electric and color neutrality are taken into account, the CFL pattern of color superconductivity requires the pairing of quarks that would, in the absence of pairing, have Fermi momenta that differ by of order $m_s^2/\mu$, with $\mu$ the quark number chemical potential. This means that at sufficiently small $\mu$, the ``stress'' on the pairing is large enough that the system can lower its energy by breaking pairs, resulting in some unconventional color superconductor which includes gapless excitations, spatial inhomogeneity, counter-propagating currents, or all three. In this paper we ask whether there is some less symmetric but still conventional pattern of pairing that can evade the stress. In other words, is there a pattern of pairing in which, once electric and color neutrality are imposed by suitable chemical potentials, pairing only occurs among those quarks whose Fermi momenta would be equal in the absence of pairing? We use graph-theoretical methods to classify 511 patterns of conventional color superconducting pairing, and show that none of them meet this requirement. All feel a stress, and all can be expected to become unstable to gapless modes at a density comparable to that at which the CFL phase becomes unstable.

Abstract:
The possibility of superconducting pairing of electrons in doped graphene due to in-plane and out-of-plane phonons is studied. Quadratic coupling of electrons with out-of-plane phonons is considered in details, taking into account both deformation potential and bond-stretch contributions. The order parameter of electron-electron pairing can have different structures due to four-component spinor character of electrons wave function. We consider s-wave pairing, diagonal on conduction and valence bands, but having arbitrary structure with respect to valley degree of freedom. The sign and magnitude of contribution of each phonon mode to effective electron-electron interaction turns out to depend on both the symmetry of phonon mode and the structure of the order parameter. Unconventional orbital-spin symmetry of the order parameter is found.