Navegando por Autor "Cariglia, Marco"
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Item Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets.(2011) Cariglia, Marco; Krtous, Pavel; Kubiznak, DavidIn this paper we derive the most general first-order symmetry operator commuting with the Dirac operator in all dimensions and signatures. Such an operator splits into Clifford even and Clifford odd parts which are given in terms of odd Killing-Yano and even closed conformal Killing-Yano inhomogeneous forms, respectively. We study commutators of these symmetry operators and give necessary and sufficient conditions under which they remain of the first-order. In this specific setting we can introduce a Killing-Yano bracket, a bilinear operation acting on odd Killing-Yano and even closed conformal Killing-Yano forms, and demonstrate that it is closely related to the Schouten-Nijenhuis bracket. An important nontrivial example of vanishing Killing-Yano brackets is given by Dirac symmetry operators generated from the principal conformal Killing-Yano tensor [hep-th/0612029]. We show that among these operators one can find a complete subset of mutually commuting operators. These operators underlie separability of the Dirac equation in Kerr-NUT-(A)dS spacetimes in all dimensions [arXiv:0711.0078].Item Conformal Killing tensors and covariant Hamiltonian dynamics.(2014) Cariglia, Marco; Gibbons, G. W.; Holten, J. W. van; Horvathy, P. A.; Zhang, P. M.A covariant algorithm for deriving the conserved quantities for natural Hamiltonian systems is combined with the non-relativistic framework of Eisenhart, and of Duval, in which the classical trajectories arise as geodesics in a higher dimensional space-time, realized by Brinkmann manifolds. Conserved quantities which are polynomial in the momenta can be built using time-dependent conformal Killing tensors with flux. The latter are associated with terms proportional to the Hamiltonian in the lower dimensional theory and with spectrum generating algebras for higher dimensional quantities of order 1 and 2 in the momenta. Illustrations of the general theory include the Runge-Lenz vector for planetary motion with a time-dependent gravitational constant G(t), motion in a time-dependent electromagnetic field of a certain form, quantum dots, the H´enon-Heiles and Holt systems, respectively, providing us with Killing tensors of rank that ranges from one to six.Item Conformal triality of the Kepler problem.(2016) Cariglia, MarcoWe show that the Kepler problem is projectively equivalent to null geodesic motion on the conformal compactification of Minkowski-4 space. This space realises the conformal triality of Minkowski, dS and AdS spaces.Item Cosmological aspects of the Eisenhart–Duval lift.(2018) Cariglia, Marco; Galajinsky, Anton; Gibbons, G. W.; Horvathy, PeterA cosmological extension of the Eisenhart–Duval metric is constructed by incorporating a cosmic scale factor and the energy-momentum tensor into the scheme. The dynamics of the spacetime is governed by the Ermakov–Milne–Pinney equation. Killing isometries include spatial translations and rotations, Newton–Hooke boosts and translation in the null direction. Geodesic motion in Ermakov–Milne–Pinney cosmoi is analyzed. The derivation of the Ermakov–Lewis invariant, the Friedmann equations and the Dmitriev–Zel’dovich equations within the Eisenhart–Duval framework is presented.Item Curvature-tuned electronic properties of bilayer graphene in an effective four-dimensional spacetime.(2017) Cariglia, Marco; Giambò, Roberto; Perali, AndreaWe show that in AB stacked bilayer graphene low-energy excitations around the semimetallic points are described by massless, four-dimensional Dirac fermions. There is an effective reconstruction of the four-dimensional spacetime, including in particular the dimension perpendicular to the sheet, that arises dynamically from the physical graphene sheet and the interactions experienced by the carriers. The effective spacetime is the Eisenhart-Duval lift of the dynamics experienced by Galilei invariant Lévy-Leblond spin- 1 2 particles near the Dirac points.We find that changing the intrinsic curvature of the bilayer sheet induces a change in the energy level of the electronic bands, switching from a conducting regime for negative curvature to an insulating one when curvature is positive. In particular, curving graphene bilayers allow opening or closing the energy gap between conduction and valence bands, a key effect for electronic devices. Thus, using curvature as a tunable parameter opens the way for the beginning of curvatronics in bilayer graphene.Item Dirac equation in Kerr-NUT-(A)dS spacetimes : intrinsic characterization of separability in all dimensions.(2011) Cariglia, Marco; Krtous, Pavel; Kubiznak, DavidWe intrinsically characterize separability of the Dirac equation in Kerr-NUT-(A)dS spacetimes in all dimensions. Namely, we explicitly demonstrate that, in such spacetimes, there exists a complete set of first-order mutually commuting operators, one of which is the Dirac operator, that allows for common eigenfunctions which can be found in a separated form and correspond precisely to the general solution of the Dirac equation found by Oota and Yasui [Phys. Lett. B 659, 688 (2008)]. Since all the operators in the set can be generated from the principal conformal Killing-Yano tensor, this establishes the (up-to-now) missing link among the existence of hidden symmetry, presence of a complete set of commuting operators, and separability of the Dirac equation in these spacetimes.Item Electron in higher-dimensional weakly charged rotating black hole spacetimes.(2013) Cariglia, Marco; Frolov, Valeri P.; Krtous, Pavel; Kubiznak, DavidWe demonstrate separability of the Dirac equation in weakly charged rotating black hole spacetimes in all dimensions. The electromagnetic field of the black hole is described by a test field approximation, with the vector potential proportional to the primary Killing vector field. It is shown that the demonstrated separability can be intrinsically characterized by the existence of a complete set of mutually commuting first-order symmetry operators generated from the principal Killing-Yano tensor. The presented results generalize the results on integrability of charged particle motion and separability of charged scalar field studied in V. P. Frolov and P. Krtous [Phys. Rev. D 83, 024016 (2011)].Item Electronic properties of curved few-layers graphene : a geometrical approach.(2018) Cariglia, Marco; Giambò, Roberto; Perali, AndreaWe show the presence of non-relativistic Lévy-Leblond fermions in flat three- and four-layers graphene with AB stacking, extending the results obtained in Cariglia et al. 2017 for bilayer graphene. When the layer is curved we obtain a set of equations for Galilean fermions that are a variation of those of Lévy-Leblond with a well defined combination of pseudospin, and that admit Lévy-Leblond spinors as solutions in an approriate limit. The local energy of such Galilean fermions is sensitive to the intrinsic curvature of the surface. We discuss the relationship between two-dimensional pseudospin, labelling layer degrees of freedom, and the different energy bands. For Lévy-Leblond fermions, an interpretation is given in terms of massless fermions in an effective 4D spacetime, and in this case the pseudospin is related to four dimensional chirality. A non-zero energy band gap between conduction and valence electronic bands is obtained for surfaces with positive curvature.Item General theory of Galilean gravity.(2018) Cariglia, MarcoWe obtain the complete theory of Newton-Cartan gravity in a curved spacetime by considering the large c limit of the vielbein formulation of general relativity. Milne boosts originate from local Lorentzian transformations, and the special cases of torsionless and twistless torsional geometries are explained in the context of the larger locally Lorentzian theory. We write the action for Newton-Cartan fields in the first order Palatini formalism and the large c limit of the Einstein equations. Finally, we obtain the generalized Eisenhart-Duval lift of the metric that plays an important role in nonrelativistic holographyItem Geometry of Lax pairs : particle motion and Killing-Yano tensors.(2013) Cariglia, Marco; Frolov, Valeri P.; Krtous, Pavel; Kubiznak, DavidA geometric formulation of the Lax pair equation on a curved manifold is studied using the phase-space formalism. The corresponding (covariantly conserved) Lax tensor is defined and the method of generation of constants of motion from it is discussed. It is shown that when the Hamilton equations of motion are used, the conservation of the Lax tensor translates directly to the well-known Lax pair equation, with one matrix identified with components of the Lax tensor and the other matrix constructed from the (metric) connection. A generalization to Clifford objects is also discussed. Nontrivial examples of Lax tensors for geodesic and charged particle motion are found in spacetimes admitting a hidden symmetry of Killing-Yano tensors.Item Hidden symmetries of dynamics in classical and quantum physics.(2014) Cariglia, MarcoThis article reviews the role of hidden symmetries of dynamics in the study of physical systems, from the basic concepts of symmetries in phase space to the forefront of current research. Such symmetries emerge naturally in the description of physical systems as varied as nonrelativistic, relativistic, with or without gravity, classical or quantum, and are related to the existence of conserved quantities of the dynamics and integrability. In recent years their study has grown intensively, due to the discovery of nontrivial examples that apply to different types of theories and different numbers of dimensions. Applications encompass the study of integrable systems such as spinning tops, the Calogero model, systems described by the Lax equation, the physics of higher-dimensional black holes, the Dirac equation, and supergravity with and without fluxes, providing a tool to probe the dynamics of nonlinear systems.Item Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux.(2012) Cariglia, MarcoThe Eisenhart-Duval lift allows embedding nonrelativistic theories into a Lorentzian geometrical setting. In this paper we study the lift from the point of view of the Dirac equation and its hidden symmetries. We show that dimensional reduction of the Dirac equation for the Eisenhart-Duval metric in general gives rise to the nonrelativistic Le´vy-Leblond equation in lower dimension. We study in detail in which specific cases the lower dimensional limit is given by the Dirac equation, with scalar and vector flux, and the relation between lift, reduction, and the hidden symmetries of the Dirac equation. While there is a precise correspondence in the case of the lower dimensional massive Dirac equation with no flux, we find that for generic fluxes it is not possible to lift or reduce all solutions and hidden symmetries. As a byproduct of this analysis, we construct new Lorentzian metrics with special tensors by lifting Killing-Yano and closed conformal Killing-Yano tensors and describe the general conformal Killing-Yano tensor of the Eisenhart-Duval lift metrics in terms of lower dimensional forms. Last, we show how, by dimensionally reducing the higher dimensional operators of the massless Dirac equation that are associated with shared hidden symmetries, it is possible to recover hidden symmetry operators for the Dirac equation with flux.Item Integrability of spinning particle motion in higher-dimensional rotating black hole spacetimes.(2012) Kubiznak, David; Cariglia, MarcoWe study the motion of a classical spinning particle (with spin degrees of freedom described by a vector of Grassmann variables) in higher-dimensional general rotating black hole spacetimes with a cosmological constant. In all dimensions n we exhibit n bosonic functionally independent integrals of spinning particle motion, corresponding to explicit and hidden symmetries generated from the principal conformal Killing-Yano tensor. Moreover, we demonstrate that in 4-, 5-, 6-, and 7-dimensional black hole spacetimes such integrals are in involution, proving the bosonic part of the motion integrable. We conjecture that the same conclusion remains valid in all higher dimensions. Our result generalizes the result of Page et al. [Phys. Rev. Lett. 98, 061102 (2007)] on complete integrability of geodesic motion in these spacetimes.Item Ion traps and the memory effect for periodic gravitational waves.(2018) Zhang, P. M.; Cariglia, Marco; Duval, C.; Elbistan, M.; Gibbons, G. W.; Horvathy, P. A.The Eisenhart lift of a Paul trap used to store ions in molecular physics is a linearly polarized periodic gravitational wave. A modified version of Dehmelt’s Penning trap is, in turn, related to circularly polarized periodic gravitational waves, sought in inflationary models. Similar equations also govern the Lagrange points in celestial mechanics. The explanation is provided by anisotropic oscillators.Item Is the pseudogap a topological state?(2015) Paredes, Alfredo Andres Vargas; Cariglia, Marco; Doria, Mauro MelchiadesWe conjecture that the pseudogap is an inhomogeneous condensate above the homogeneous state whose existence is granted by topological stability. We consider the simplest possible order parameter theory that provides this interpretation of the pseudogap and study its angular momentum states. Also we obtain a solution of the Bogomol'nyi self-duality equations and find skyrmions. The normal state gap density, the breaking of the time reversal symmetry and the checker board pattern are naturally explained under this view. The pseudo gap is a lattice of skyrmions and the inner weak local magnetic field falls below the experimental threshold of observation given by NMR/NQR and μSR experiments.Item On integrability of the geodesic deviation equation.(2018) Cariglia, Marco; Houri, Tsuyoshi; Krtous, Pavel; Kubiznak, DavidThe Jacobi equation for geodesic deviation describes finite size effects due to the gravitational tidal forces. In this paper we showhowone can integrate the Jacobi equation in any spacetime admitting completely integrable geodesics. Namely, by linearizing the geodesic equation and its conserved charges, we arrive at the invariant Wronskians for the Jacobi system that are linear in the ‘deviation momenta’ and thus yield a system of first-order differential equations that can be integrated. The procedure is illustrated on an example of a rotating black hole spacetime described by the Kerr geometry and its higher-dimensional generalizations. A number of related topics, including the phase space formulation of the theory and the derivation of the covariant Hamiltonian for the Jacobi system are also discussed.Item Topologically stable gapped state in a layered superconductor.(2014) Cariglia, Marco; Paredes, Alfredo Andres Vargas; Doria, Mauro MelchiadesWe show that a layered superconductor, described by a two-component order parameter, has a gapped state above the ground state, topologically protected from decay, containing flow and counterflow in the absence of an applied magnetic field. This state is made of skyrmions, breaks time reversal symmetry and produces a weak local magnetic field. We estimate the density of carriers that condense into the pseudogap of the cuprate superconductors based on the assumption that the pseudogap is a skyrmion state.Item Zero helicity states in the LaAlO3-SrTiO3 interface : the origin of the mass anisotropy.(2017) Rodrigues, Edinardo Ivison Batista; Doria, Mauro Melchiades; Paredes, Alfredo Andres Vargas; Cariglia, Marco; Perali, AndreaWe consider the transverse magnetic moment and torque observed by Li et al. (Nat. Phys. 7, 762 (2011)) in the LaAlO3/SrTiO3 interface and the theoretical model for it based on the zero helicity states. The transverse magnetic moment is explained in terms of an asymmetry between the two sides of the interface. We show here that there is an intrinsic magnetization which gives rise to a mass anisotropy in each side of the interface.