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Annales Henri Poincaré    [3 followers]  Follow
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ISSN (Print) 1424-0637 - ISSN (Online) 1424-0661
Published by Springer-Verlag  [2180 journals]   [SJR: 0.977]   [H-I: 26]
• Twisted Equivariant Matter
• Abstract: Abstract We show how general principles of symmetry in quantum mechanics lead to twisted notions of a group representation. This framework generalizes both the classical threefold way of real/complex/ quaternionic representations as well as a corresponding tenfold way which has appeared in condensed matter and nuclear physics. We establish a foundation for discussing continuous families of quantum systems. Having done so, topological phases of quantum systems can be defined as deformation classes of continuous families of gapped Hamiltonians. For free particles, there is an additional algebraic structure on the deformation classes leading naturally to notions of twisted equivariant K-theory. In systems with a lattice of translational symmetries, we show that there is a canonical twisting of the equivariant K-theory of the Brillouin torus. We give precise mathematical definitions of two invariants of the topological phases which have played an important role in the study of topological insulators. Twisted equivariant K-theory provides a finer classification of topological insulators than has been previously available.
PubDate: 2013-12-01

• Renormalization of the Commutative Scalar Theory with Harmonic Term to All Orders
• Abstract: Abstract The noncommutative scalar theory with harmonic term (on the Moyal space) has a vanishing beta function. In this paper, we prove the renormalizability of the commutative scalar field theory with harmonic term to all orders using multiscale analysis in the momentum space. Then, we consider and compute its one-loop beta function, as well as the one on the degenerate Moyal space. We can finally compare both to the vanishing beta function of the theory with harmonic term on the Moyal space.
PubDate: 2013-12-01

• Averaging Fluctuations in Resolvents of Random Band Matrices
• Abstract: Abstract We consider a general class of random matrices whose entries are centred random variables, independent up to a symmetry constraint. We establish precise high-probability bounds on the averages of arbitrary monomials in the resolvent matrix entries. Our results generalize the previous results of Erdős et al. (Ann Probab, arXiv:1103.1919, 2013; Commun Math Phys, arXiv:1103.3869, 2013; J Combin 1(2):15–85, 2011) which constituted a key step in the proof of the local semicircle law with optimal error bound in mean-field random matrix models. Our bounds apply to random band matrices and improve previous estimates from order 2 to order 4 in the cases relevant to applications. In particular, they lead to a proof of the diffusion approximation for the magnitude of the resolvent of random band matrices. This, in turn, implies new delocalization bounds on the eigenvectors. The applications are presented in a separate paper (Erdős et al., arXiv:1205.5669, 2013).
PubDate: 2013-12-01

• Spectral Networks
• Abstract: Abstract We introduce new geometric objects called spectral networks. Spectral networks are networks of trajectories on Riemann surfaces obeying certain local rules. Spectral networks arise naturally in four-dimensional ${\mathcal{N} = 2}$ theories coupled to surface defects, particularly the theories of class S. In these theories, spectral networks provide a useful tool for the computation of BPS degeneracies; the network directly determines the degeneracies of solitons living on the surface defect, which in turn determines the degeneracies for particles living in the 4d bulk. Spectral networks also lead to a new map between flat ${{\rm GL}(K, \mathbb{C})}$ connections on a two-dimensional surface C and flat abelian connections on an appropriate branched cover ${\Sigma}$ of C. This construction produces natural coordinate systems on moduli spaces of flat ${{\rm GL}(K, \mathbb{C})}$ connections on C, which we conjecture are cluster coordinate systems.
PubDate: 2013-11-01

• Convexity of Reduced Energy and Mass Angular Momentum Inequalities
• Abstract: Abstract In this paper, we extend the work in Chruściel and Costa (Class. Quant. Grav. 26:235013, 2009), Chruściel et al. (Ann. Phy. 323:2591–2613, 2008), Costa (J. Math. Theor. 43:285202, 2010), Dain (J. Diff. Geom. 79:33–67, 2008). We weaken the asymptotic conditions on the second fundamental form, and we also give an L 6−norm bound for the difference between general data and Extreme Kerr data or Extreme Kerr–Newman data by proving convexity of the renormalized Dirichlet energy when the target has non-positive curvature. In particular, we give the first proof of the strict mass/angular momentum/charge inequality for axisymmetric Einstein/Maxwell data which is not identical with the extreme Kerr–Newman solution.
PubDate: 2013-11-01

• Eigenfunction Statistics for a Point Scatterer on a Three-Dimensional Torus
• Abstract: Abstract In this paper, we study eigenfunction statistics for a point scatterer (the Laplacian perturbed by a delta-potential) on a three-dimensional flat torus. The eigenfunctions of this operator are the eigenfunctions of the Laplacian which vanish at the scatterer, together with a set of new eigenfunctions (perturbed eigenfunctions). We first show that for a point scatterer on the standard torus all of the perturbed eigenfunctions are uniformly distributed in configuration space. Then we investigate the same problem for a point scatterer on a flat torus with some irrationality conditions, and show uniform distribution in configuration space for almost all of the perturbed eigenfunctions.
PubDate: 2013-11-01

• A Generalized Mass Involving Higher Order Symmetric Functions of the Curvature Tensor
• Abstract: Abstract We define a generalized mass for asymptotically flat manifolds using some higher order symmetric function of the curvature tensor. This mass is non-negative when the manifold is locally conformally flat and the σ k curvature vanishes at infinity. In addition, with the above assumptions, if the mass is zero, then, near infinity, the manifold is isometric to a Euclidean end.
PubDate: 2013-11-01

• Entropic Fluctuations in XY Chains and Reflectionless Jacobi Matrices
• Abstract: Abstract We study entropic functionals/fluctuations of the XY chain with Hamiltonian $$\begin{array}{ll} \frac{1}{2} \sum\limits_{x \in \mathbb{Z}}J_x( \sigma_x^{(1)} \sigma_{x+1}^{(1)} +\sigma_x^{(2)} \sigma_{x+1}^{(2)}) + \lambda_x \sigma_x^{(3)}\end{array}$$ where initially the left (x ≤ 0)/right (x > 0) part of the chain is in thermal equilibrium at inverse temperature β l /β r . The temperature differential results in a non-trivial energy/entropy flux across the chain. The Evans–Searles (ES) entropic functional describes fluctuations of the flux observable with respect to the initial state while the Gallavotti–Cohen (GC) functional describes these fluctuations with respect to the steady state (NESS) the chain reaches in the large time limit. We also consider the full counting statistics (FCS) of the energy/entropy flux associated with a repeated measurement protocol, the variational entropic functional (VAR) that arises as the quantization of the variational characterization of the classical Evans–Searles functional and a natural class of entropic functionals that interpolate between FCS and VAR. We compute these functionals in closed form in terms of the scattering data of the Jacobi matrix hu x  = J x u x+1 + λ x u x + J x−1 u x−1 canonically associated with the XY chain. We show that all these functionals are identical if and only if h is reflectionless (we call this phenomenon entropic identity). If h is not reflectionless, then the ES and GC functionals remain equal but differ from the FCS, VAR and interpolating functionals. Furthermore, in the non-reflectionless case, the ES/GC functional does not vanish at α = 1 (i.e., the Kawasaki identity fails) and does not have the celebrated α ↔ 1 − α symmetry. The FCS, VAR and interpolating functionals always have this symmetry. In the Schrödinger case, where J x  = J for all x, the entropic identity leads to some unexpected open problems in the spectral theory of one-dimensional discrete Schrödinger operators.
PubDate: 2013-11-01

• Anderson’s Orthogonality Catastrophe for One-Dimensional Systems
• Abstract: Abstract The overlap, ${\mathcal{D}_N}$ , between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is ${\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}$ with the so-called Anderson integral ${\mathcal{I}_N}$ . We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit ${\mathcal{I}_N = \gamma\ln N + O(1)}$ as ${N\to\infty}$ . The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on ${\mathcal{D}_N}$ concluding that ${\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}$ with constants C, ${\tilde{C}}$ , and ${\tilde{\gamma}}$ . In particular, ${\mathcal{D}_N\to 0}$ as ${N\to\infty}$ which is known as Anderson’s orthogonality catastrophe.
PubDate: 2013-10-17

• Itsy Bitsy Topological Field Theory
• Abstract: Abstract We construct an elementary, combinatorial kind of topological quantum field theory (TQFT), based on curves, surfaces, and orientations. The construction derives from contact invariants in sutured Floer homology and is essentially an elaboration of a TQFT defined by Honda–Kazez–Matić. This topological field theory stores information in binary format on a surface and has “digital” creation and annihilation operators, giving a toy-model embodiment of “it from bit”.
PubDate: 2013-10-15

• The Reduced Hartree–Fock Model for Short-Range Quantum Crystals with Nonlocal Defects
• Abstract: Abstract In this article, we consider quantum crystals with defects in the reduced Hartree–Fock framework. The nuclei are supposed to be classical particles arranged around a reference periodic configuration. The perturbation is assumed to be small in amplitude, but need not be localized in a specific region of space or have any spatial invariance. Assuming Yukawa interactions, we prove the existence of an electronic ground state, solution of the self-consistent field equation. Next, by studying precisely the decay properties of this solution for local defects, we are able to expand the density of states of the nonlinear Hamiltonian of a system with a random perturbation of Anderson–Bernoulli type, in the limit of low concentration of defects. One important step in the proof of our results is the analysis of the dielectric response of the crystal to an effective charge perturbation.
PubDate: 2013-09-26

• Pseudo-Differential Calculus on Homogeneous Trees
• Abstract: Abstract To study concentration and oscillation properties of eigenfunctions of the discrete Laplacian on regular graphs, we construct in this paper a pseudo-differential calculus on homogeneous trees, their universal covers. We define symbol classes and associated operators. We prove that these operators are bounded on L 2 and give adjoint and product formulas. Finally, we compute the symbol of the commutator of a pseudo-differential operator with the Laplacian.
PubDate: 2013-09-23

• Hypergeometric Type Functions and Their Symmetries
• Abstract: Abstract The paper is devoted to a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F 1 equation (equivalent to the Bessel equation), the Gegenbauer equation and the Hermite equation. In particular, recurrence relations of their solutions, their integral representations and discrete symmetries are discussed.
PubDate: 2013-09-23

• A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian
• Abstract: Abstract We consider the Landau Hamiltonian perturbed by a long-range electric potential V. The spectrum of the perturbed operator consists of eigenvalue clusters which accumulate to the Landau levels. First, we estimate the rate of the shrinking of these clusters to the Landau levels as the number of the cluster tends to infinity. Further, we assume that there exists an appropriate ${\mathbb{V}}$ V , homogeneous of order −ρ with ${\rho \in (0, 1)}$ ρ ∈ ( 0 , 1 ) , such that ${V(x) = \mathbb{V} (x) + O( x ^{-\rho - \varepsilon})}$ V ( x ) = V ( x ) + O ( x - ρ - ε ) , ɛ >  0, as x → ∞, and investigate the asymptotic distribution of the eigenvalues within the qth cluster as q → ∞. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the mean-value transform of ${\mathbb{V}}$ V .
PubDate: 2013-09-21

• Effective Dynamics of an Electron Coupled to an External Potential in Non-relativistic QED
• Abstract: Abstract In the framework of non-relativistic QED, we show that the renormalized mass of the electron (after having taken into account radiative corrections) appears as the kinematic mass in its response to an external potential force. Specifically, we study the dynamics of an electron in a slowly varying external potential and with slowly varying initial conditions and prove that, for a long time, it is accurately described by an associated effective dynamics of a Schrödinger electron in the same external potential and for the same initial data, with a kinetic energy operator determined by the renormalized dispersion law of the translation-invariant QED model.
PubDate: 2013-09-01

• 3D Tensor Field Theory: Renormalization and One-Loop β-Functions
• Abstract: Abstract We prove that the rank 3 analogue of the tensor model defined in Ben Geloun and Rivasseau (Commun Math Phys, arXiv:1111.4997 [hep-th], 2012) is renormalizable at all orders of perturbation. The proof is given in the momentum space. The one-loop γ- and β-functions of the model are also determined. We find that the model with a unique coupling constant for all interactions and a unique wave-function renormalization is asymptotically free in the UV.
PubDate: 2013-09-01

• Inverse Boundary Problems for Systems in Two Dimensions
• Abstract: Abstract We prove identification of coefficients up to gauge equivalence by Cauchy data at the boundary for elliptic systems on oriented compact surfaces with boundary or domains of ${\mathbb{C}}$ . In the geometric setting, we fix a Riemann surface with boundary and consider both a Dirac-type operator plus potential acting on sections of a Clifford module and a connection Laplacian plus potential (i.e. Schrödinger Laplacian with external Yang–Mills field) acting on sections of a Hermitian bundle. In either case we show that the Cauchy data determine both the connection and the potential up to a natural gauge transformation: conjugation by an endomorphism of the bundle which is the identity at the boundary. For domains of ${\mathbb{C}}$ , we recover zeroth order terms up to gauge from Cauchy data at the boundary in first order elliptic systems.
PubDate: 2013-09-01

• Quasilinear Hyperbolic Fuchsian Systems and AVTD Behavior in T 2-Symmetric Vacuum Spacetimes
• Abstract: Abstract We set up the singular initial value problem for quasilinear hyperbolic Fuchsian systems of first order and establish an existence and uniqueness theory for this problem with smooth data and smooth coefficients (and with even lower regularity). We apply this theory in order to show the existence of smooth (generally not analytic) T 2-symmetric solutions to the vacuum Einstein equations, which exhibit asymptotically velocity term dominated behavior in the neighborhood of their singularities and are polarized or half-polarized.
PubDate: 2013-09-01

• Maximal Fermi Charts and Geometry of Inflationary Universes
• Abstract: Abstract A proof is given that the maximal Fermi coordinate chart for any comoving observer in a broad class of Robertson–Walker spacetimes consists of all events within the cosmological event horizon, if there is one, or is otherwise global. Exact formulas for the metric coefficients in Fermi coordinates are derived. Sharp universal upper bounds for the proper radii of leaves of the foliation by Fermi space slices are found, i.e., for the proper radii of the spatial universe at fixed times of the comoving observer. It is proved that the radius at proper time τ diverges to infinity for non inflationary cosmologies as τ → ∞, but not necessarily for cosmologies with periods of inflation. It is shown that any space like geodesic orthogonal to the worldline of a comoving observer has finite proper length and terminates within the cosmological event horizon (if there is one) at the big bang. Geometric properties of inflationary versus non inflationary cosmologies are compared, and opposite inequalities for the inflationary and non inflationary cases, analogous to Hubble’s law, are obtained for the Fermi relative velocities of comoving test particles. It is proved that the Fermi relative velocities of radially moving test particles are necessarily subluminal for inflationary cosmologies in contrast to non inflationary models, where superluminal relative Fermi velocities necessarily exist.
PubDate: 2013-09-01

• Resonant Delocalization on the Bethe Strip
• Abstract: Abstract Recently, Aizenman and Warzel discovered a mechanism for the appearance of absolutely continuous spectrum for random Schrödinger operators on the Bethe lattice through rare resonances (resonant delocalization). We extend their analysis to operators with matrix-valued random potentials drawn from ensembles such as the Gaussian Orthogonal Ensemble. These operators can be viewed as random operators on the Bethe strip, a graph (lattice) with loops.
PubDate: 2013-08-06

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