We show that the pseudorelativistic physics of graphene near the Fermi level can be extended to three dimensional (3D) materials. Unlike in phase transitions from inversion symmetric topological to normal insulators, we show that particular space groups also allow 3D Dirac points as symmetry protected degeneracies. We provide criteria necessary to identify these groups and, as an example, present ab initio calculations of β-cristobalite BiO(2) which exhibits three Dirac points at the Fermi level. We find that β-cristobalite BiO(2) is metastable, so it can be physically realized as a 3D analog to graphene.

We report on a Dirac-like Fermi surface in three-dimensional bulk materials in a distorted spinel structure on the basis of density functional theory (DFT) as well as tight-binding theory. The four examples we provide in this paper are BiZnSiO4, BiCaSiO4, BiMgSiO4, and BiAlInO4. A necessary characteristic of these structures is that they contain a Bi lattice which forms a hierarchy of chain-like substructures, with consequences for both fundamental understanding and materials design.Following the discovery of topological insulators, there 6 has been considerable interest in studying semimetal- 66As shown in Fig. 1, the cubic unit cell distorts to an or- (Fig. 3). Both the crystal and electronic

We present ab initio and k • p calculations of the spin texture on the Fermi surface of tensile strained HgTe, which is obtained by stretching the zincblende lattice along the (111) axis. Tensile strained HgTe is a semimetal with pointlike accidental degeneracies between a mirror symmetry protected twofold degenerate band and two nondegenerate bands near the Fermi level. The Fermi surface consists of two ellipsoids which contact at the point where the Fermi level crosses the twofold degenerate band along the (111) axis. However, the spin texture of occupied states indicates that neither ellipsoid carries a compensating Chern number. Consequently, the spin texture is locked in the plane perpendicular to the (111) axis, exhibits a nonzero winding number in that plane, and changes winding number from one end of the Fermi ellipsoids to the other. The change in the winding of the spin texture suggests the existence of singular points. An ordered alloy of HgTe with ZnTe has the same effect as stretching the zincblende lattice in the (111) direction. We present ab initio calculations of ordered HgxZn1−xTe that confirm the existence of a spin texture locked in a 2D plane on the Fermi surface with different winding numbers on either end.

We investigate the electromagnetic Casimir interactions of an object contained within an otherwise empty, perfectly conducting spherical shell. For a small object we present analytical calculations of the force, which is directed away from the center of the cavity, and the torque, which tends to align the object opposite to the preferred alignment outside the cavity. For a perfectly conducting sphere as the interior object, we compute the corrections to the proximity force approximation (PFA) numerically. In both cases the results for the interior configuration match smoothly onto those for the corresponding exterior configuration.

We investigate a stable Casimir force configuration consisting of an object contained inside a spherical or spheroidal cavity filled with a dielectric medium. The spring constant for displacements from the center of the cavity and the dependence of the energy on the relative orientations of the inner object and the cavity walls are computed. We find that the stability of the force equilibrium can be predicted based on the sign of the force, but the torque cannot be.The Casimir force between atoms or macroscopic objects arises from quantum fluctuations of the electrodynamic field [1,2]. In all known examples it is found that the force is attractive, as long as the space between the objects is empty, and the magnetic susceptibility of the objects is negligible compared to their electric susceptibilities. But when space is filled with a medium with electric permittivity M intermediate between that of two objects, 1 < M < 2 , the force between the two becomes repulsive [3]. This effect has recently been verified experimentally [4]. But while repulsive forces are nothing new, they become interesting for applications when they produce stable equilibria, which, for example, the Coulomb force cannot.For an infinite cylinder enclosed in another, the Casimir force has recently been shown to have a stable equilibrium in the two directions perpendicular to the cylinder axes, when the material properties are chosen so that the force between two slabs of the same materials would be repulsive [5]. If the inner and outer cylinders have square cross sections, the direction of the torque exerted by one cylinder on the other is found to agree qualitatively with the predictions of the pairwise summation or proximity force approximations (PSA or PFA). Orientation dependence of the Casimir energy has also been studied recently for a small spheroid facing another spheroid or an infinite plate [6], or for a small spheroid located inside a spherical metal cavity [7]. (The smallness of the spheroid refers to keeping only the first term in the series expansion of the Casimir energy in the largest length scale of the spheroid.) The preferred orientation of the small spheroid flips when it is moved between the inside and the outside of the spherical shell; there is no torque if the spheroid faces an infinite metal plate, which is the limit of an infinitely large shell.In this paper we investigate the first configuration, depicted in Fig. 1, in which a compact object levitates stably due to the Casimir force alone. We characterize the Casimir energyby the spring constant k and the coefficients k n in a series expansion in a/R, where a is the magnitude of the displacement from the center of the cavity and R the radius of the (undeformed) spherical cavity. (k has units of energy here.) Unlike the case of infinite cylinders, where one object can be displaced along the cylinder axes without changing the energy, our case exhibits, for appropriately chosen materials, true stability in all directions and applies to realistic situations. For example...

We study the electromagnetic Casimir interaction of a compact object contained inside a closed cavity of another compact object. We express the interaction energy in terms of the objects' scattering matrices and translation matrices that relate the coordinate systems appropriate to each object. When the enclosing object is an otherwise empty metallic spherical shell, much larger than the internal object, and the two are sufficiently separated, the Casimir force can be expressed in terms of the static electric and magnetic multipole polarizabilities of the internal object, which is analogous to the Casimir-Polder result. Although it is not a simple power law, the dependence of the force on the separation of the object from the containing sphere is a universal function of its displacement from the center of the sphere, independent of other details of the object's electromagnetic response. Furthermore, we compute the exact Casimir force between two metallic spheres contained one inside the other at arbitrary separations. Finally, we combine our results with earlier work on the Casimir force between two spheres to obtain data on the leading-order correction to the proximity force approximation for two metallic spheres both outside and within one another. Keywords Casimir force Disciplines Quantum Physics CommentsSuggested Citation: Zaheer, S., Rahi, S.J., Emig, T. and Jaffe, R.L. (2010 We study the electromagnetic Casimir interaction of a compact object contained inside a closed cavity of another compact object. We express the interaction energy in terms of the objects' scattering matrices and translation matrices that relate the coordinate systems appropriate to each object. When the enclosing object is an otherwise empty metallic spherical shell, much larger than the internal object, and the two are sufficiently separated, the Casimir force can be expressed in terms of the static electric and magnetic multipole polarizabilities of the internal object, which is analogous to the Casimir-Polder result. Although it is not a simple power law, the dependence of the force on the separation of the object from the containing sphere is a universal function of its displacement from the center of the sphere, independent of other details of the object's electromagnetic response. Furthermore, we compute the exact Casimir force between two metallic spheres contained one inside the other at arbitrary separations. Finally, we combine our results with earlier work on the Casimir force between two spheres to obtain data on the leading-order correction to the proximity force approximation for two metallic spheres both outside and within one another.

Using the ray-optics approximation, we analyze the Casimir force in a two dimensional domain formed by two metallic blocks adjacent to parallel metallic sidewalls, which are separated from the blocks by a finite distance h. For h > 0, the ray-optics approach is not exact because diffraction effects are neglected. Nevertheless, we show that ray optics is able to qualitatively reproduce a surprising effect recently identified in an exact numerical calculation: the force between the blocks varies non-monotonically with h. In this sense, the ray-optics approach captures an essential part of the physics of multi-body interactions in this system, unlike simpler pairwise-interaction approximations such as PFA. Furthermore, by comparison to the exact numerical results, we are able to quantify the impact of diffraction on Casimir forces in this geometry.

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