To discuss the properties of metamaterials on physical grounds and to consider them in applications, effective material parameters are usually introduced and assigned to a given metamaterial. In most cases, only weak spatial dispersion is considered. It allows to assign local material properties, i.e. a permittivity and a permeability. However, this turned out to be insufficient. To solve this problem, we study here the effective properties of metamaterials with constitutive relations beyond a local response and take strong spatial dispersion into account. The isofrequency surfaces of the dispersion relation are investigated and compared to those of an actual metamaterial. The significant improvement provides evidence for the necessity to use nonlocal material laws in the effective description of metamaterials. The general formulation we choose here renders our approach applicable to a wide class of metamaterials.

Studying basic physical effects sustained in metamaterials characterized by specific constitutive relation is a research topic with a long standing tradition. Besides intellectual curiosity, it derives its importance from the ability to predict observable phenomena that are, if found with an actual metamaterial, a clear indication on its properties. Here, we consider a nonlocal (strong spatial dispersion), lossy, and isotropic metamaterial and study the impact of the nonlocality on the dispersion relation of surface plasmon polaritons sustained at an interface between vacuum and such metamaterial. For that, Fresnel coefficients are calculated and appropriate surface plasmon polaritons existence conditions are being proposed. Predictions regarding the experimentally observable reflection from a frustrated internal reflection geometry are being made. A different behavior for TE and TM polarization is observed. Our work unlocks novel opportunities to seek for traces of the nonlocality in experiments made with nowadays metamaterials.

The parameter retrieval is a procedure in which effective material properties are assigned to a given metamaterial. A widely used technique bases on the inversion of reflection and transmission from a metamaterial slab. Thus far, local constitutive relations have been frequently considered in this retrieval procedure to describe the metamaterial at the effective level. This, however, is insufficient. The retrieved local material properties frequently fail to predict reliably the optical response from the slab in situations that deviate from those that have been considered in the retrieval, e.g. when illuminating the slab at a different incidence angle. To significantly improve the situation, we describe here a parameter retrieval, also based on the inversion of reflection and transmission from a slab, that describes the metamaterial at the effective level with nonlocal constitutive relations. We retrieve the effective material parameters at the example of a basic metamaterial, namely dielectric spheres on a cubic lattice but also on a more advanced, anisotropic metamaterial of current interest, i.e., the fishnet metamaterial. We demonstrate that the nonlocal constitutive relation can describe the optical response much better than local constitutive relation would do. Our approach is widely applicable to a large class of metamaterials. PACS numbers: 41.20. Jb,78.20.Bh,78.20.Ci,78.67.Pt c is the free space wavenumber and ω and c are the frequency of the considered time-harmonic field and the speed of light in vacuum, respectively. We refer to the description of a metamaterial with these two parameters only, as the Weak Spatial Dispersion (WSD) or local approximation. It is a local constitutive relation since the electric displacement D(r, k 0 ) and the magnetic induction B(r, k 0 ) depend only locally on the electric field E(r, k 0 ) and the magnetic field H(r, k 0 ), respectively.However, metamaterials are usually made from building blocks, also called meta-atoms, that have a size in the order of several tens or even hundreds of nanometers, while being designed to operate at optical or nearinfrared wavelengths. This is in stark contrast to natural materials that have critical dimensions of merely a fraction of one nanometer. The disparate length scales between critical feature size and operational wavelength for natural materials justifies their treatment with local constitutive relations. Indeed, it is quite a challenge to trace signatures of a nonlocal character with natural materials 12,13 . The assumption of a local medium, however, ceases to be applicable for optical metamaterials when their critical length scale is no longer much smaller than the wavelength but only smaller. Then, nonlocal effects can no longer be neglected.But how can we judge, which effective description is appropriate? Well, first of all and with the purpose to treat the material as effectively homogeneous, we require it to be sub-wavelength under all circumstances. A fre-arXiv:1808.00748v3 [physics.optics]

When the period of unit-cells constituting metamaterials is no longer much smaller than the wavelength but only smaller, local material laws fail to describe the propagation of light in such composite media when considered at the effective level. Instead, nonlocal material laws are required. They have to be derived by approximating a general response function of the electric field in the metamaterial at the effective level that is accurate but cannot be handled practically. But how to perform this approximation is not obvious at all. Indeed many approximations can be perceived and one should be able to decide as quick as possible which of these possible material laws are mathematically and physically meaningful at all. Here, at the example of a second order Padé approximation of the general response function of the electric field, we present a checklist each possible constitutive relation has to pass in order to be physically and mathematically liable. As will be shown, only one out of these nine Padé approximations passes the checklist. The work is meant to be a guideline applicable to decide which constitutive relation makes actually sense at all. It is an essential ingredient for future research on composite media as any possible constitutive relation to be discussed should pass it.

In this paper we study the critical behavior of an N -component φ 4 -model in hyperbolic space, which serves as a model of uniform frustration. We find that this model exhibits a second-order phase transition with an unusual magnetization texture that results from the lack of global parallelism in hyperbolic space. Angular defects occur on length scales comparable to the radius of curvature. This phase transition is governed by a new strong curvature fixed point that obeys scaling below the upper critical dimension duc = 4. The exponents of this fixed point are given by the leading order terms of the 1/N expansion. In distinction to flat space no order 1/N corrections occur. We conclude that the description of many-particle systems in hyperbolic space is a promising avenue to investigate uniform frustration and non-trivial critical behavior within one theoretical approach. arXiv:1507.02909v1 [cond-mat.stat-mech] 10 Jul 2015