Authors:Po-Ning Chen; Pei-Ken Hung; Mu-Tao Wang; Shing-Tung Yau Pages: 1493 - 1518 Abstract: Abstract We study the space of Killing fields on the four dimensional AdS spacetime \(AdS^{3,1}\) . Two subsets \({\mathcal {S}}\) and \({\mathcal {O}}\) are identified: \({\mathcal {S}}\) (the spinor Killing fields) is constructed from imaginary Killing spinors, and \({\mathcal {O}}\) (the observer Killing fields) consists of all hypersurface orthogonal, future timelike unit Killing fields. When the cosmology constant vanishes, or in the Minkowski spacetime case, these two subsets have the same convex hull in the space of Killing fields. In presence of the cosmology constant, the convex hull of \( {\mathcal {O}}\) is properly contained in that of \({\mathcal {S}}\) . This leads to two different notions of energy for an asymptotically AdS spacetime, the spinor energy and the observer energy. Chruściel et al. (J High Energy Phys 2006(11):084, 2006) proved the positivity of the spinor energy and derived important consequences among the related conserved quantities. We show that the positivity of the observer energy follows from the positivity of the spinor energy. A new notion called the “rest mass” of an asymptotically AdS spacetime is then defined by minimizing the observer energy and is shown to be evaluated in terms of the adjoint representation of the Lie algebra of Killing fields. It is proved that the rest mass has the desirable rigidity property that characterizes the AdS spacetime. PubDate: 2017-05-01 DOI: 10.1007/s00023-017-0555-4 Issue No:Vol. 18, No. 5 (2017)

Authors:Edgar Gasperín; Juan A. Valiente Kroon Pages: 1519 - 1591 Abstract: Abstract The conformal structure of the Schwarzschild–de Sitter spacetime is analysed using the extended conformal Einstein field equations. To this end, initial data for an asymptotic initial value problem for the Schwarzschild–de Sitter spacetime are obtained. This initial data allow to understand the singular behaviour of the conformal structure at the asymptotic points where the horizons of the Schwarzschild–de Sitter spacetime meet the conformal boundary. Using the insights gained from the analysis of the Schwarzschild–de Sitter spacetime in a conformal Gaussian gauge, we consider nonlinear perturbations close to the Schwarzschild–de Sitter spacetime in the asymptotic region. We show that small enough perturbations of asymptotic initial data for the Schwarzschild–de Sitter spacetime give rise to a solution to the Einstein field equations which exists to the future and has an asymptotic structure similar to that of the Schwarzschild–de Sitter spacetime. PubDate: 2017-05-01 DOI: 10.1007/s00023-016-0544-z Issue No:Vol. 18, No. 5 (2017)

Authors:Yannis Angelopoulos; Stefanos Aretakis; Dejan Gajic Pages: 1593 - 1633 Abstract: Abstract We show that degenerate horizons exhibit a new trapping effect. Specifically, we obtain a non-degenerate Morawetz estimate for the wave equation in the domain of outer communications of extremal Reissner–Nordström up to and including the future event horizon. We show that such an estimate requires (1) a higher degree of regularity for the initial data, reminiscent of the regularity loss in the high-frequency trapping estimates on the photon sphere, and (2) the vanishing of an explicit quantity that depends on the restriction of the initial data on the horizon. The latter condition demonstrates that degenerate horizons exhibit a new \(L^{2}\) concentration phenomenon (namely, a global trapping effect, in the sense that this effect is not due to individual underlying null geodesics as in the case of the photon sphere). We moreover uncover a new stable higher-order trapping effect; we show that higher-order estimates do not hold regardless of the degree of regularity and the support of the initial data. We connect our findings to the spectrum of the stability operator in the theory of marginally outer trapped surfaces. Our methods and results play a crucial role in our upcoming works on linear and nonlinear wave equations on extremal black hole backgrounds. PubDate: 2017-05-01 DOI: 10.1007/s00023-016-0545-y Issue No:Vol. 18, No. 5 (2017)

Authors:Ivan P. Costa e Silva; José Luis Flores Pages: 1635 - 1670 Abstract: Abstract We study the interplay between the global causal and geometric structures of a spacetime (M, g) and the features of a given smooth \({\mathbb {R}}\) -action \(\rho \) on M whose orbits are all causal curves, building on classic results about Lie group actions on manifolds described by Palais (Ann Math 73:295–323, 1961). Although the dynamics of such an action can be very hard to describe in general, simple restrictions on the causal structure of (M, g) can simplify this dynamics dramatically. In the first part of this paper, we prove that \(\rho \) is free and proper (so that M splits topologically) provided that (M, g) is strongly causal and \(\rho \) does not have what we call weakly ancestral pairs, a notion which admits a natural interpretation in terms of “cosmic censorship.” Accordingly, such condition holds automatically if (M, g) is globally hyperbolic. We also prove that M splits topologically if (M, g) is strongly causal and \(\rho \) is the flow of a complete conformal Killing null vector field. In the second part, we investigate the class of Brinkmann spacetimes, which can be regarded as null analogues of stationary spacetimes in which \(\rho \) is the flow of a complete parallel null vector field. Inspired by the geometric characterization of stationary spacetimes in terms of standard stationary ones (Javaloyes and Sánchez in Class Quantum Gravity 25:168001, 2008), we obtain an analogous geometric characterization of when a Brinkmann spacetime is isometric to a standard Brinkmann spacetime. This result naturally leads us to discuss a conjectural null analogue for Ricci-flat four-dimensional Brinkmann spacetimes of a celebrated rigidity theorem by Anderson (Ann Henri Poincaré 1:977–994, 2000) and highlight its relation with a long-standing conjecture by Ehlers and Kundt (Gravitation: an introduction to current research. Wiley, New York, pp 49–101, 1962). PubDate: 2017-05-01 DOI: 10.1007/s00023-017-0551-8 Issue No:Vol. 18, No. 5 (2017)

Authors:Felix Finster; Moritz Reintjes Pages: 1671 - 1701 Abstract: Abstract We give a nonperturbative construction of a distinguished state for the quantized Dirac field in Minkowski space in the presence of a time-dependent external field of the form of a plane electromagnetic wave. By explicit computation of the fermionic signature operator, it is shown that the Dirac operator has the strong mass oscillation property. We prove that the resulting fermionic projector state is a Hadamard state. PubDate: 2017-05-01 DOI: 10.1007/s00023-017-0557-2 Issue No:Vol. 18, No. 5 (2017)

Authors:Aghil Alaee; Marcus Khuri; Hari Kunduri Pages: 1703 - 1753 Abstract: Abstract We prove a mass-angular momentum-charge inequality for a broad class of maximal, asymptotically flat, bi-axisymmetric initial data within the context of five-dimensional minimal supergravity. We further show that the charged Myers–Perry black hole initial data are the unique minimizers. Also, we establish a rigidity statement for the relevant BPS bound, and give a variational characterization of BMPV black holes. PubDate: 2017-05-01 DOI: 10.1007/s00023-016-0542-1 Issue No:Vol. 18, No. 5 (2017)

Authors:Sven Bachmann; Martin Fraas; Gian Michele Graf Pages: 1755 - 1776 Abstract: Abstract We study the evolution of a driven harmonic oscillator with a time-dependent frequency \(\omega _t \propto t \) . At time \(t=0\) , the Hamiltonian undergoes a point of infinite spectral degeneracy. If the system is initialized in the instantaneous vacuum in the distant past, then the asymptotic future state is a squeezed state, whose parameters are explicitly determined. We show that the squeezing is independent on the sweeping rate. This manifests the failure of the adiabatic approximation at points, where infinitely many eigenvalues collide. We extend our analysis to the situation, where the gap at \(t=0\) remains finite. We also discuss the natural geometry of the manifold of squeezed states. We show that it is realized by the Poincaré disk model viewed as a Kähler manifold. PubDate: 2017-05-01 DOI: 10.1007/s00023-016-0539-9 Issue No:Vol. 18, No. 5 (2017)

Authors:Alexander Müller-Hermes; David Reeb Pages: 1777 - 1788 Abstract: Abstract We prove that the quantum relative entropy decreases monotonically under the application of any positive trace-preserving linear map, for underlying separable Hilbert spaces. This answers in the affirmative a natural question that has been open for a long time, as monotonicity had previously only been shown to hold under additional assumptions, such as complete positivity or Schwarz-positivity of the adjoint map. The first step in our proof is to show monotonicity of the sandwiched Renyi divergences under positive trace-preserving maps, extending a proof of the data processing inequality by Beigi (J Math Phys 54:122202, 2013) that is based on complex interpolation techniques. Our result calls into question several measures of non-Markovianity that have been proposed, as these would assess all positive trace-preserving time evolutions as Markovian. PubDate: 2017-05-01 DOI: 10.1007/s00023-017-0550-9 Issue No:Vol. 18, No. 5 (2017)

Authors:Giuseppe De Nittis; Max Lein Pages: 1789 - 1831 Abstract: Abstract In this work, we present a novel approach to the ray optics limit: we rewrite the dynamical Maxwell equations in Schrödinger form and prove Egorov-type theorems, a robust semiclassical technique. We implement this scheme for periodic light conductors, photonic crystals, thereby making the quantum-light analogy between semiclassics for the Bloch electron and ray optics in photonic crystals rigorous. One major conceptual difference between the two theories, though, is that electromagnetic fields are real, and hence, we need to add one step in the derivation to reduce it to a single-band problem. Our main results, Theorem 3.7 and Corollary 3.9, give a ray optics limit for quadratic observables and, among others, apply to local averages of energy density, the Poynting vector and the Maxwell stress tensor. Ours is the first rigorous derivation of ray optics equations which include all subleading-order terms, some of which are also new to the physics literature. The ray optics limit we prove applies to photonic crystals of any topological class. PubDate: 2017-05-01 DOI: 10.1007/s00023-017-0552-7 Issue No:Vol. 18, No. 5 (2017)

Authors:Chris Bourne; Johannes Kellendonk; Adam Rennie Pages: 1833 - 1866 Abstract: Abstract We study the application of Kasparov theory to topological insulator systems and the bulk–edge correspondence. We consider observable algebras as modelled by crossed products, where bulk and edge systems may be linked by a short exact sequence. We construct unbounded Kasparov modules encoding the dynamics of the crossed product. We then link bulk and edge Kasparov modules using the Kasparov product. Because of the anti-linear symmetries that occur in topological insulator models, real \(C^*\) -algebras and KKO-theory must be used. PubDate: 2017-05-01 DOI: 10.1007/s00023-016-0541-2 Issue No:Vol. 18, No. 5 (2017)

Authors:Johannes Kellendonk Abstract: Abstract The notion of a topological phase of an insulator is based on the concept of homotopy between Hamiltonians. It therefore depends on the choice of a topological space to which the Hamiltonians belong. We advocate that this space should be the \(C^*\) -algebra of observables. We relate the symmetries of insulators to graded real structures on the observable algebra and classify the topological phases using van Daele’s formulation of K-theory. This is related but not identical to Thiang’s recent approach to classify topological phases by K-groups in Karoubi’s formulation. PubDate: 2017-04-12 DOI: 10.1007/s00023-017-0583-0

Authors:Abel Klein; Son T. Nguyen; Constanza Rojas-Molina Abstract: Abstract We extend to the two-particle Anderson model the characterization of the metal–insulator transport transition obtained in the one-particle setting by Germinet and Klein. We show that, for any fixed number of particles, the slow spreading of wave packets in time implies the initial estimate of a modified version of the bootstrap multiscale analysis. In this new version, operators are restricted to boxes defined with respect to the pseudo-distance in which we have the slow spreading. At the bottom of the spectrum, within the regime of one-particle dynamical localization, we show that this modified multiscale analysis yields dynamical localization for the two-particle Anderson model, allowing us to obtain a characterization of the metal–insulator transport transition for the two-particle Anderson model at the bottom of the spectrum. PubDate: 2017-04-11 DOI: 10.1007/s00023-017-0578-x

Authors:Athanasios Chatzistavrakidis; Andreas Deser; Larisa Jonke; Thomas Strobl Abstract: Abstract We study the propagation of bosonic strings in singular target space-times. For describing this, we assume this target space to be the quotient of a smooth manifold M by a singular foliation \({{\mathcal {F}}}\) on it. Using the technical tool of a gauge theory, we propose a smooth functional for this scenario, such that the propagation is assured to lie in the singular target on-shell, i.e., only after taking into account the gauge-invariant content of the theory. One of the main new aspects of our approach is that we do not limit \({{\mathcal {F}}}\) to be generated by a group action. We will show that, whenever it exists, the above gauging is effectuated by a single geometrical and universal gauge theory, whose target space is the generalized tangent bundle \(TM\oplus T^*M\) . PubDate: 2017-04-11 DOI: 10.1007/s00023-017-0580-3

Authors:Alex Clark; Lorenzo Sadun Abstract: Abstract Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam, and Skau conjectured that all minimal, free \(\mathbb {Z}^d\) actions on Cantor sets admit “small cocycles.” These represent classes in \(H^1\) that are mapped to small vectors in \(\mathbb {R}^d\) by the Ruelle–Sullivan (RS) map. We show that there exist \(\mathbb {Z}^2\) actions where no such small cocycles exist, and where the image of \(H^1\) under RS is \(\mathbb {Z}^2\) . Our methods involve tiling spaces and shape deformations, and along the way we prove a relation between the image of RS and the set of “virtual eigenvalues,” i.e., elements of \(\mathbb {R}^d\) that become topological eigenvalues of the tiling flow after an arbitrarily small change in the shapes and sizes of the tiles. PubDate: 2017-04-10 DOI: 10.1007/s00023-017-0579-9

Authors:Raphael Ducatez; François Huveneers Abstract: Abstract We study the persistence of localization for a strongly disordered tight-binding Anderson model on the lattice \({\mathbb Z}^d\) , periodically driven on each site. Under two different sets of conditions on the driving, we show that Anderson localization survives if the driving frequency is higher than some threshold value. We discuss the implication of our results for recent development in condensed matter physics, we compare them with the predictions issuing from adiabatic theory, and we comment on the connection with Mott’s law, derived within the linear response formalism. PubDate: 2017-04-10 DOI: 10.1007/s00023-017-0574-1

Authors:Tadahiro Miyao Abstract: Abstract Nagaoka’s theorem on ferromagnetism in the Hubbard model is extended to the Holstein–Hubbard model. This shows that Nagaoka’s ferromagnetism is stable even if the electron–phonon interaction is taken into account. We also prove that Nagaoka’s ferromagnetism is stable under the influence of the quantized radiation field. PubDate: 2017-04-10 DOI: 10.1007/s00023-017-0584-z

Authors:Costanza Benassi; Jürg Fröhlich; Daniel Ueltschi Abstract: Abstract We study a large class of models of two-dimensional quantum lattice systems with continuous symmetries, and we prove a general McBryan–Spencer–Koma–Tasaki theorem concerning algebraic decay of correlations. We present applications of our main result to the Heisenberg, Hubbard, and t-J models, and to certain models of random loops. PubDate: 2017-04-08 DOI: 10.1007/s00023-017-0571-4

Authors:Margherita Disertori; Mareike Lager Abstract: Abstract We consider a two-dimensional random band matrix ensemble, in the limit of infinite volume and fixed but large band width W. For this model, we rigorously prove smoothness of the averaged density of states. We also prove that the resulting expression coincides with Wigner’s semicircle law with a precision \(W^{-2+\delta },\) where \(\delta \rightarrow 0\) when \(W\rightarrow \infty .\) The proof uses the supersymmetric approach and extends results by Disertori et al. (Commun Math Phys 232(1):83–124, 2002) from three to two dimensions. PubDate: 2017-04-08 DOI: 10.1007/s00023-017-0572-3

Authors:Marcus Khuri; Naqing Xie Abstract: Abstract We establish inequalities relating the size of a material body to its mass, angular momentum, and charge, within the context of axisymmetric initial data sets for the Einstein equations. These inequalities hold in general without the assumption of the maximal condition and use a notion of size which is easily computable. Moreover, these results give rise to black hole existence criteria which are meaningful even in the time-symmetric case, and also include certain boundary effects. PubDate: 2017-04-07 DOI: 10.1007/s00023-017-0582-1

Authors:Kohei Iwaki; Olivier Marchal Abstract: Abstract The goal of this article is to prove that the determinantal formulas of the Painlevé 2 system identify with the correlation functions computed from the topological recursion on their spectral curve for an arbitrary nonzero monodromy parameter. The result is established for a WKB expansion of two different Lax pairs associated with the Painlevé 2 system, namely the Jimbo–Miwa Lax pair and the Harnad–Tracy–Widom Lax pair, where a small parameter \(\hbar \) is introduced by a proper rescaling. The proof is based on showing that these systems satisfy the topological type property introduced in Bergère et al. (Ann Henri Poincaré 16:2713, 2015), Bergère and Eynard (arxiv:0901.3273, 2009). In the process, we explain why the insertion operator method traditionally used to prove the topological type property is currently incomplete and we propose new methods to bypass the issue. Our work generalizes similar results obtained from random matrix theory in the special case of vanishing monodromies (Borot and Eynard in arXiv:1011.1418, 2010; arXiv:1012.2752, 2010). Explicit computations up to \(g=3\) are provided along the paper as an illustration of the results. Eventually, taking the time parameter t to infinity we observe that the symplectic invariants \(F^{(g)}\) of the Jimbo–Miwa and Harnad–Tracy–Widom spectral curves converge to the Euler characteristic of moduli space of genus g Riemann surfaces. PubDate: 2017-04-07 DOI: 10.1007/s00023-017-0576-z