Authors:Jinho Baik; Ji Oon Lee Pages: 1867 - 1917 Abstract: We consider a spherical spin system with pure 2-spin spherical Sherrington–Kirkpatrick Hamiltonian with ferromagnetic Curie–Weiss interaction. The system shows a two-dimensional phase transition with respect to the temperature and the coupling constant. We compute the limiting distributions of the free energy for all parameters away from the critical values. The zero temperature case corresponds to the well-known phase transition of the largest eigenvalue of a rank 1 spiked random symmetric matrix. As an intermediate step, we establish a central limit theorem for the linear statistics of rank 1 spiked random symmetric matrices. PubDate: 2017-06-01 DOI: 10.1007/s00023-017-0562-5 Issue No:Vol. 18, No. 6 (2017)
Authors:Jacob S. Christiansen; Maxim Zinchenko Pages: 1949 - 1976 Abstract: We establish Lieb–Thirring power bounds on discrete eigenvalues of Jacobi operators for Schatten class perturbations under very general assumptions. Our results apply, in particular, to perturbations of reflectionless Jacobi operators with finite gap and Cantor-type essential spectrum. PubDate: 2017-06-01 DOI: 10.1007/s00023-016-0546-x Issue No:Vol. 18, No. 6 (2017)
Authors:Benjamin Landon; Annalisa Panati; Jane Panangaden; Justine Zwicker Pages: 2075 - 2085 Abstract: The dynamic reflection probability of Davies and Simon (Commun Math Phys 63(3):277–301, 1978) and the spectral reflection probability of Gesztesy et al. (Diff Integral Eqs 10(3):521–546, 1997) and Gesztesy and Simon (Helv Phys Acta 70:66–71, 1997) for a one-dimensional Schrödinger operator \(H = - \Delta + V\) are characterized in terms of the scattering theory of the pair \((H, H_\infty )\) where \(H_\infty \) is the operator obtained by decoupling the left and right half-lines \(\mathbb {R}_{\le 0}\) and \(\mathbb {R}_{\ge 0}\) . An immediate consequence is that these reflection probabilities are in fact the same, thus providing a short and transparent proof of the main result of Breuer et al. (Commun Math Phys 295(2):531–550, 2010). This approach is inspired by recent developments in non-equilibrium statistical mechanics of the electronic black-box model and follows a strategy parallel to Jakšić (Commun Math Phys 332:827–838, 2014). PubDate: 2017-06-01 DOI: 10.1007/s00023-016-0543-0 Issue No:Vol. 18, No. 6 (2017)
Authors:Guan Huang Pages: 2087 - 2121 Abstract: Consider a system of periodic pendulum lattice with analytic weak couplings: $$\begin{aligned} \ddot{x}_i+\sin x_i=-\epsilon \sum _{j=i-2}^i\partial _{x_i}\beta _{\alpha }(x_j,x_{j+1},x_{j+2}), \quad x_i=x_{i+N},\quad i\in \mathbb {Z}, \end{aligned}$$ where \(N\geqslant 3\) is an integer, \(\epsilon >0\) is a small parameter, and the function \(\beta _{\alpha }\) is an analytic function of a certain form. It is shown in this paper that for small enough \(\epsilon \) , the system admits motions such that the energy transfers between the pendulums in any predetermined order. PubDate: 2017-06-01 DOI: 10.1007/s00023-017-0561-6 Issue No:Vol. 18, No. 6 (2017)
Authors:Myeongju Chae; Sung-Jin Oh Pages: 2123 - 2198 Abstract: We establish a general small data global existence and decay theorem for Chern–Simons theories with a general gauge group, coupled with a massive relativistic field of spin 0 or 1/2. Our result applies to a wide range of relativistic Chern–Simons theories considered in the literature, including the abelian/non-abelian self-dual Chern–Simons–Higgs equation and the Chern–Simons–Dirac equation. A key idea is to develop and employ a gauge invariant vector field method for relativistic Chern–Simons theories, which allows us to avoid the long-range effect of charge. PubDate: 2017-06-01 DOI: 10.1007/s00023-016-0547-9 Issue No:Vol. 18, No. 6 (2017)
Abstract: We show that recent multivariate generalizations of the Araki–Lieb–Thirring inequality and the Golden–Thompson inequality (Sutter et al. in Commun Math Phys, 2016. doi:10.1007/s00220-016-2778-5) for Schatten norms hold more generally for all unitarily invariant norms and certain variations thereof. The main technical contribution is a generalization of the concept of log-majorization which allows us to treat majorization with regard to logarithmic integral averages of vectors of singular values. PubDate: 2017-07-01
Abstract: We consider a driven open system whose evolution is described by a Lindbladian. The Lindbladian is assumed to be dephasing and its Hamiltonian part to be given by the Landau–Zener Hamiltonian. We derive a formula for the transition probability which, unlike previous results, extends the Landau–Zener formula to open systems. PubDate: 2017-07-01
Abstract: In this paper, we are concerned with the asymptotic behavior of the solution to systems of cubic nonlinear Schrödinger equations in one dimension. It is known that mass transition phenomenon occurs for a system of quadratic nonlinear Schrödinger equations in two dimensions under the mass resonance condition. We show that mass transition phenomenon also occurs for systems with cubic nonlinearity under the corresponding mass resonance conditions. PubDate: 2017-07-01
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: 2017-07-01
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: 2017-07-01
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: 2017-07-01
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: 2017-07-01
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: 2017-07-01
Abstract: Considering different self-adjoint realisations of positively projected massless Coulomb–Dirac operators we find out under which conditions any negative perturbation, however small, leads to emergence of negative spectrum. We also prove some weighted Lieb–Thirring estimates for negative eigenvalues of such operators. In the process we find explicit spectral representations for all self-adjoint realisations of massless Coulomb–Dirac operators on the half-line. PubDate: 2017-07-01
Abstract: Starting from a \(d\times d\) rational Lax pair system of the form \(\hbar \partial _x \Psi = L\Psi \) and \(\hbar \partial _t \Psi =R\Psi \) , we prove that, under certain assumptions (genus 0 spectral curve and additional conditions on R and L), the system satisfies the “topological type property.” A consequence is that the formal \(\hbar \) -WKB expansion of its determinantal correlators satisfies the topological recursion. This applies in particular to all (p, q) minimal models reductions of the KP hierarchy, or to the six Painlevé systems. PubDate: 2017-06-21
Authors:Vincent Beaud; Simone Warzel Abstract: We study a one-dimensional quantum system with an arbitrary number of hard-core particles on the lattice, which are subject to a deterministic attractive interaction as well as a random potential. Our choice of interaction is suggested by the spectral analysis of the XXZ quantum spin chain. The main result concerns a version of high-disorder Fock-space localization expressed here in the configuration space of hard-core particles. The proof relies on an energetically motivated Combes–Thomas estimate and an effective one-particle analysis. As an application, we show the exponential decay of the two-point function in the infinite system uniformly in the particle number. PubDate: 2017-06-20 DOI: 10.1007/s00023-017-0591-0
Authors:João L. Costa; Anne T. Franzen Abstract: Motivated by the strong cosmic censorship conjecture, in the presence of a cosmological constant, we consider solutions of the scalar wave equation \(\Box _g\phi =0\) on fixed subextremal Reissner–Nordström–de Sitter backgrounds \(({\mathcal M}, g)\) , without imposing symmetry assumptions on \(\phi \) . We provide a sufficient condition, in terms of surface gravities and a parameter for an exponential decaying Price Law, for a local energy of the waves to remain bounded up to the Cauchy horizon. The energy we consider controls, in particular, regular transverse derivatives at the Cauchy horizon; this allows us to extend the solutions with bounded energy, to the Cauchy horizon, as functions in \(C^0\cap H^1_\mathrm{loc}\) . Our results correspond to another manifestation of the potential breakdown of strong cosmic censorship in the positive cosmological constant setting. PubDate: 2017-06-15 DOI: 10.1007/s00023-017-0592-z
Authors:Gregory J. Galloway; Carlos Vega Abstract: We begin with a basic exploration of the (point-set topological) notion of Hausdorff closed limits in the spacetime setting. Specifically, we show that this notion of limit is well suited to sequences of achronal sets, and use this to generalize the ‘achronal limits’ introduced by the authors in Galloway and Vega (Ann Henri Poincaré 15(11):2241–2279, 2014). This, in turn, allows for a broad generalization of the notion of Lorentzian horosphere introduced in Galloway and Vega (2014). We prove a new rigidity result for such horospheres, which in a sense encodes various spacetime splitting results, including the basic Lorentzian splitting theorem. We use this to give a partial proof of the Bartnik splitting conjecture (Bartnik in Commun Math Phys 117(4):615–624, 1988), under a new condition involving past and future Cauchy horospheres, which is weaker than those considered in Galloway (Some rigidity results for spatially closed spacetimes. Mathematics of gravitation, part I (Warsaw, 1996), Banach Center Publications, vol 41, Polish Academy of Science, Warsaw, pp 21–34, 1996) and Galloway and Vega (2014). We close with some observations on spacetimes with spacelike causal boundary, including a rigidity result in the positive cosmological constant case. PubDate: 2017-06-15 DOI: 10.1007/s00023-017-0594-x
Authors:Jian Wang Abstract: In symplectic geometry, the action function is a classical object defined on the set of contractible fixed points of the time-one map of a Hamiltonian isotopy. On closed aspherical surfaces, we give a dynamical interpretation of this function, which permits us to generalize it to the case of a diffeomorphism that is isotopic to identity and preserves a Borel finite measure of rotation vector zero. We define a boundedness property on the contractible fixed points set of the time-one map of an identity isotopy. We generalize the classical action function to any Hamiltonian homeomorphism, provided that the proposed boundedness condition is satisfied. We prove that the generalized action function only depends on the time-one map but not on the isotopy. Finally, we define the action spectrum and show that it is invariant under conjugation by an orientation and measure preserving homeomorphism. PubDate: 2017-06-13 DOI: 10.1007/s00023-017-0596-8
Authors:Marco Benini; Matteo Capoferri; Claudio Dappiaggi Abstract: Abelian duality is realized naturally by combining differential cohomology and locally covariant quantum field theory. This leads to a \(\hbox {C}^*\) -algebra of observables, which encompasses the simultaneous discretization of both magnetic and electric fluxes. We discuss the assignment of physically well-behaved states on this algebra and the properties of the associated GNS triple. We show that the algebra of observables factorizes as a suitable tensor product of three \(\hbox {C}^*\) -algebras: the first factor encodes dynamical information, while the other two capture topological data corresponding to electric and magnetic fluxes. On the former factor and in the case of ultra-static globally hyperbolic spacetimes with compact Cauchy surfaces, we exhibit a state whose two-point correlation function has the same singular structure of a Hadamard state. Specifying suitable counterparts also on the topological factors, we obtain a state for the full theory, ultimately implementing Abelian duality transformations as Hilbert space isomorphisms. PubDate: 2017-06-12 DOI: 10.1007/s00023-017-0593-y
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