We present a multipolar model of surface -lattice resonances (SLRs) in 2d arrays of spheres including the electric dipole, magnetic dipole, and electric quadrupole moments of the spheres. We identify SLRs of dipolar and multipolar character, show the importance of non-resonant multipoles in their description, and discuss the sensitivity of SLRs to illumination conditions. We link SLRs to an excitation of modes supported by the array, and we propose a simplified model of the mode dispersion relations that explains the sensitivity of SLRs and the band gap in mode dispersion found at low frequencies. Finally we discuss the resonant features associated with a direct coupling to a mode which can occur in addition to the diffractive coupling signalled by SLRs.

Electronic nematicity, proposed to exist in a number of transition metal materials, can have different microscopic origins. In particular, the anisotropic resistivity and meta-magnetic jumps observed in Sr 3 Ru 2 O 7 are consistent with an earlier proposal that the isotropic-nematic transition is generically first order and accompanied by meta-magnetism when tuned by a magnetic field. However, additional striking experimental features such as a non-Fermi liquid resistivity and critical thermodynamic behaviour imply the presence of an unidentified quantum critical point (QCP). Here we show that orbital degrees of freedom play an essential role in revealing a nematic QCP, even though it is overshadowed by a nearby meta-nematic transition at low temperature. We further present a finite temperature phase diagram including the entropy landscape and discuss our findings in light of the phenomena observed in Sr 3 Ru 2 O 7 .A variety of transition metal materials such as cuprates [1], Ru-oxides [2] and Fe-pnictides [3] have been proposed to harbour an electronic nematic phase [4]. Electronic nematic phases are broadly characterized by the presence of spontaneously broken rotational symmetry and viewed as the quantum counterpart of nematic classical liquid crystal phases. The theoretical proposal of nematic quantum liquid crystals became more concrete when experiments on ultra-pure bilayer ruthenate (Sr 3 Ru 2 O 7 ) samples subjected to a magnetic field along the c-axis revealed an unusual phase characterized by a pronounced residual resistivity in place of a putative meta-magnetic quantum critical point (QCP) [5]. Interestingly, Sr 3 Ru 2 O 7 was initially viewed as a prototype for the study of quantum phase transitions, exhibiting a striking non-Fermi liquid resistivity thought to originate from the putative magnetic field tuned QCP [6]. The unusual phase found in ultra-pure samples is delimited by two consecutive first order meta-magnetic transitions at low temperature and, remarkably, exhibits a significant inplane magnetoresistive anisotropy when the external field is slightly tilted towards one of the in-plane crystal axes [2]. These observations strongly imply the formation of an anisotropic metallic, i.e. electronic nematic, phase in the bilayer ruthenate compound.Based at first on the two consecutive meta-magnetic transitions, an electronic nematic phase was proposed and generic features of nematic phase formation were theoretically explored early on [5,7,8]. It was found that the transition between the isotropic and nematic phase is generally first order, and that nematic order typically develops near a van Hove singularity (vHS) to avoid a Lifshitz transition. Varying the chemical potential, the nematic phase is bounded at low and high values by two isotropic phases, while the concomitant first order transitions lead to jumps in the electron density. When a magnetic field is applied (and the chemical potential is held fixed at, say, some low value), Zeeman coupling acts as a spin-dependent chemical potential te...

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We present all the periodic Green function dyadics that enter a description of a 2D array of emitters at the level that includes the electric dipole, magnetic dipole and electric quadrupole moment of each emitter. We find a concise analytic form for the radiative contributions to the periodic Green function dyadics that give rise to radiation reaction fields, and so our description of the scattered light explicitly satisfies the optical theorem; we give the non-radiative contributions that do not affect energy balance in the form of rapidly converging series. Finally, we present an approximation scheme for evaluating periodic Green function dyadics at long wavelengths that rigorously respects energy conservation. The scheme extends the range of validity of the usual static approximation by the inclusion of a simple dynamic correction.

We present an electrodynamic approach to the description of the linear response of solids to electromagnetic fields. For time and spatially varying applied fields we solve the dynamical equations satisfied by the gaugeinvariant Green function and find microscopic charge and current densities that result in a form allowing for an easy construction of the multipole expansion of applied fields. Restricting ourselves to static and uniform electric and magnetic fields, we construct microscopic expressions for polarization and magnetization fields associated with each lattice site. The approach is in the spirit of the Power-Zienau-Wooley (PZW) treatment but generalized to account for the motion of the charge between lattice sites. We show that the macroscopic polarization and magnetization can be understood as the spatial average of the generalized PZW microscopic fields.