The dynamical structure of topologically massive gravity in the context of the Faddeev-Jackiw symplectic approach is studied. It is shown that this method allows us to avoid some ambiguities arising in the study of the gauge structure via the Dirac formalism. In particular, the complete set of constraints and the generators of the gauge symmetry of the theory are obtained straightforwardly via the zero modes of the symplectic matrix. In order to obtain the generalized Faddeev-Jackiw brackets and calculate the local physical degrees of freedom of this model, an appropriate gauge-fixing procedure is introduced. Finally, the similarities and relative advantages between the Faddeev-Jackiw method and Dirac's formalism are briefly discussed.

We consider two semi-infinite magnetoelecteric media separated by a planar interface whose electromagnetic response is described by axion electrodynamics. The time-dependent Green's function characterizing this geometry is obtained by a method that can be directly generalized to cylindrical and spherical configurations of two magnetoelectrics separated by an interface. We establish the far-field approximation of the Green's function and apply these results to the case of a charged particle moving from one medium to the other at a high constant velocity perpendicular to the interface. From the resulting angular distribution of the radiated energy per unit frequency we provide theoretical evidence for the emergence of reversed Vavilov-Čerenkov radiation in naturally existing magnetoelectric media. In the case where one of the magnetoelectrics is a 3D topological insulator, TlBiSe2, for example, located in front of a regular insulator, we estimate that an average forward Vavilov-Čerenkov radiation with frequency ∼ 2.5 eV (∼ 500 nm) will produce a highly suppressed reversed Vavilov-Čerenkov radiation which can be characterized by an effective frequency in the range of ∼ (4 × 10 −3 − 0.5) meV. However, this value compares favorably with recent measurements in left-handed metamaterials yielding reversed Vavilov-Čerenkov radiation with frequencies of the order of (1.2 − 3.9) × 10 −2 meV. * Electronic address: francamentesantiago@ciencias.unam.mx † Electronic address: urrutia@nucleares.unam.mx ‡ Electronic address: omar.rodriguez@correo.nucleares.unam.mx

When time-reversal symmetry is broken on its surface, topological insulators exhibit a magnetoelectric response which is described by axion electrodynamics. A direct consequence of this theory is the appearance of a magnetic field that resembles the one produced by a magnetic image monopole when a point-like electric charge is located near the surface of the material. In this paper we investigate the more realistic problem when the point-like charge is replaced by a finite size sphere at constant potential. We calculate the electromagnetic fields using the potential formulation in a particular bispherical coordinate system. We find that the electromagnetic fields can be interpreted in terms of point electric and image magnetic charges as if the medium were the vacuum. As a manifestation of the magnetoelectric effect, we highlight the resulting magnetic field, which we analyze in detail along the symmetry axis, since such estimates could be useful in evaluating the experimental possibility of its measurement via sensible magnetometers. Our numerical estimates show that the proposed setup provides a magnetic field strength in the range of 10-100 mG, which is attainable with present day sensitivities in NV center-diamond magnetometers, for example. I. INTRODUCTIONGeneral magnetoelectric (ME) media are characterized by additional relations between the magnetic (electric) field and the polarization (magnetization), aside from the standard connection found in conventional dielectrics [1][2][3][4]. A linear ME material is described by the ME term θ ij E i B j in the free enthalpy of the system, where θ ij is the ME tensor which can be either symmetric or antisymmetric [5]. In the simplest case, when θ ij = θδ ij , we recover the ME coupling θ E · B, which we recognize as that of axion electrodynamics, with θ being the axion field [6]. In the context of particle physics, the axion is an additional pseudoscalar degree of freedom which gives a solution to the strong CP problem [7]. Linear magnetoelectrics can be realized in topological materials, such as topological insulators (TIs) [8][9][10][11] and Weyl semimetals (WSMs) [12].Topological phases are an emerging class of materials which have attracted much attention in condensed matter physics. Among them, the most studied are the TIs, which are time-reversal-symmetric materials characterized by a fully insulating bulk with protected conducting surface states [9,10]. This behavior was first predicted in two-dimensional HgTe/CdTe quantum wells [13][14][15] and less than one year after, it was experimentally observed [16]. The generalization to three-dimensional compounds came shortly afterwards [17][18][19]. In particular, Fu and Kane [20] predicted that the alloy Bi 1−x Sb x would be a three-dimensional (3D) TI in a special range of x, and it was experimentally confirmed one year later [21]. A second generation of 3D TIs was predicted to occur in the stoichiometric crystals Bi 2 Se 3 , Bi 2 Te 3 , and Sb 2 Te 3 [22], and they were experimentally discovered in 2009 [22,23]...

A pure Dirac's framework for 3D Palatini's theory with cosmological constant is performed. By considering the complete phase space, we find out the full structure of the constraints, and their corresponding algebra is computed explicitly. We report that in order to obtain a well defined algebra among the constraints, the internal group corresponds to SO(2, 1). In addition, we obtain the extended action, the extended Hamiltonian, the gauge symmetry, and the Dirac brackets of the theory. Finally, we compare our results with those reported in the literature.

By applying the Faddeev-Jackiw symplectic approach we systematically show that both the local gauge symmetry and the constraint structure of topologically massive gravity with a cosmological constant , elegantly encoded in the zero-modes of the symplectic matrix, can be identified. Thereafter, via a suitable partial gauge-fixing procedure, the time gauge, we calculate the quantization bracket structure (generalized Faddeev-Jackiw brackets) for the dynamic variables and confirm that the number of physical degrees of freedom is one. This approach provides an alternative to explore the dynamical content of massive gravity models.