Authors:Costanza Benassi; Jürg Fröhlich; Daniel Ueltschi Pages: 2831 - 2847 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: 2017-09-01 DOI: 10.1007/s00023-017-0571-4 Issue No:Vol. 18, No. 9 (2017)
Authors:Tadahiro Miyao Pages: 2849 - 2871 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: 2017-09-01 DOI: 10.1007/s00023-017-0584-z Issue No:Vol. 18, No. 9 (2017)
Authors:Tadeusz Balaban; Joel Feldman; Horst Knörrer; Eugene Trubowitz Pages: 2873 - 2903 Abstract: Abstract This paper is a contribution to a program to see symmetry breaking in a weakly interacting many boson system on a three-dimensional lattice at low temperature. It provides an overview of the analysis, given in Balaban et al. (The small field parabolic flow for bosonic many-body models: part 1—main results and algebra, arXiv:1609.01745, 2016, The small field parabolic flow for bosonic many-body models: part 2—fluctuation integral and renormalization, arXiv:1609.01746, 2016), of the ‘small field’ approximation to the ‘parabolic flow’ which exhibits the formation of a ‘Mexican hat’ potential well. PubDate: 2017-09-01 DOI: 10.1007/s00023-017-0587-9 Issue No:Vol. 18, No. 9 (2017)
Authors:Eduardo Garibaldi; Samuel Petite; Philippe Thieullen Pages: 2905 - 2943 Abstract: Abstract The Frenkel–Kontorova model describes how an infinite chain of atoms minimizes the total energy of the system when the energy takes into account the interaction of nearest neighbors as well as the interaction with an exterior environment. An almost periodic environment leads to consider a family of interaction energies which is stationary with respect to a minimal topological dynamical system. We focus, in this context, on the existence of calibrated configurations (a notion stronger than the standard minimizing condition). In any dimension and for any continuous superlinear interaction energies, we exhibit a set, called projected Mather set, formed of environments that admit calibrated configurations. In the one-dimensional setting, we then give sufficient conditions on the family of interaction energies that guarantee the existence of calibrated configurations for every environment. The main mathematical tools for this study are developed in the frameworks of discrete weak KAM theory, Aubry–Mather theory and spaces of Delone sets. PubDate: 2017-09-01 DOI: 10.1007/s00023-017-0589-7 Issue No:Vol. 18, No. 9 (2017)
Authors:Jian Wang Pages: 2945 - 2993 Abstract: 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: 2017-09-01 DOI: 10.1007/s00023-017-0596-8 Issue No:Vol. 18, No. 9 (2017)
Authors:Asao Arai; Fumio Hiroshima Pages: 2995 - 3033 Abstract: In an abstract framework, a new concept on time operator, ultra-weak time operator, is introduced, which is a concept weaker than that of weak time operator. Theorems on the existence of an ultra-weak time operator are established. As an application of the theorems, it is shown that Schrödinger operators \({H_V}\) with potentials V obeying suitable conditions, including the Hamiltonian of the hydrogen atom, have ultra-weak time operators. Moreover, a class of Borel measurable functions \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) such that \(f({H_V})\) has an ultra-weak time operator is found. PubDate: 2017-09-01 DOI: 10.1007/s00023-017-0586-x Issue No:Vol. 18, No. 9 (2017)
Authors:Leonid Perlov; Michael Bukatin Pages: 3035 - 3048 Abstract: Abstract In this paper, we research all possible finite-dimensional representations and corresponding values of the Barbero–Immirzi parameter contained in EPRL simplicity constraints by using Naimark’s fundamental theorem of the Lorentz group representation theory. It turns out that for each nonzero pure imaginary with rational modulus value of the Barbero–Immirzi parameter \(\gamma = i \frac{p}{q}, p, q \in Z, p, q \ne 0\) , there is a solution of the simplicity constraints, such that the corresponding Lorentz representation is finite-dimensional. The converse is also true—for each finite-dimensional Lorentz representation solution of the simplicity constraints \((n, \rho )\) , the associated Barbero–Immirzi parameter is nonzero pure imaginary with rational modulus, \(\gamma = i \frac{p}{q}, p, q \in Z, p, q \ne 0\) . We solve the simplicity constraints with respect to the Barbero–Immirzi parameter and then use Naimark’s fundamental theorem of the Lorentz group representations to find all finite-dimensional representations contained in the solutions. PubDate: 2017-09-01 DOI: 10.1007/s00023-017-0588-8 Issue No:Vol. 18, No. 9 (2017)
Authors:Michał Eckstein; Tomasz Miller Pages: 3049 - 3096 Abstract: Abstract Drawing from the theory of optimal transport we propose a rigorous notion of a causal relation for Borel probability measures on a given spacetime. To prepare the ground, we explore the borderland between Lorentzian geometry, topology and measure theory. We provide various characterisations of the proposed causal relation, which turn out to be equivalent if the underlying spacetime has a sufficiently robust causal structure. We also present the notion of the ‘Lorentz–Wasserstein distance’ and study its basic properties. Finally, we outline the possible applications of the developed formalism in both classical and quantum physics. PubDate: 2017-09-01 DOI: 10.1007/s00023-017-0566-1 Issue No:Vol. 18, No. 9 (2017)
Authors:Kohei Iwaki; Olivier Marchal Pages: 2581 - 2620 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: 2017-08-01 DOI: 10.1007/s00023-017-0576-z Issue No:Vol. 18, No. 8 (2017)
Authors:Athanasios Chatzistavrakidis; Andreas Deser; Larisa Jonke; Thomas Strobl Pages: 2641 - 2692 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: 2017-08-01 DOI: 10.1007/s00023-017-0580-3 Issue No:Vol. 18, No. 8 (2017)
Authors:Alexander Schenkel; Jochen Zahn Pages: 2693 - 2714 Abstract: Abstract We formulate an algebraic criterion for the presence of global anomalies on globally hyperbolic space-times in the framework of locally covariant field theory. We discuss some consequences and check that it reproduces the well-known global SU(2) anomaly in four space-time dimensions. PubDate: 2017-08-01 DOI: 10.1007/s00023-017-0590-1 Issue No:Vol. 18, No. 8 (2017)
Authors:Christian Gérard; Michał Wrochna Pages: 2715 - 2756 Abstract: Abstract We consider the massive Klein–Gordon equation on a class of asymptotically static spacetimes (in the long-range sense) with Cauchy surface of bounded geometry. We prove the existence and Hadamard property of the in and out states constructed by scattering theory methods. PubDate: 2017-08-01 DOI: 10.1007/s00023-017-0573-2 Issue No:Vol. 18, No. 8 (2017)
Authors:Paul T. Allen; Iva Stavrov Allen Pages: 2789 - 2814 Abstract: Abstract We show that any polyhomogeneous asymptotically hyperbolic constant-mean-curvature solution to the vacuum Einstein constraint equations can be approximated, arbitrarily closely in Hölder norms determined by the physical metric, by shear-free smoothly conformally compact vacuum initial data. PubDate: 2017-08-01 DOI: 10.1007/s00023-017-0565-2 Issue No:Vol. 18, No. 8 (2017)
Authors:Marcus Khuri; Naqing Xie Pages: 2815 - 2830 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: 2017-08-01 DOI: 10.1007/s00023-017-0582-1 Issue No:Vol. 18, No. 8 (2017)
Authors:Alan D. Rendall; Juan J. L. Velázquez Abstract: Abstract This paper continues the investigation of the formation of naked singularities in the collapse of collisionless matter initiated in Rendall and Velázquez (Annales Henri Poincaré 12:919–964, 2011). There the existence of certain classes of non-smooth solutions of the Einstein–Vlasov system was proved. Those solutions are self-similar and hence not asymptotically flat. To obtain solutions which are more physically relevant it makes sense to attempt to cut off these solutions in a suitable way so as to make them asymptotically flat. This task, which turns out to be technically challenging, will be carried out in this paper. PubDate: 2017-08-14 DOI: 10.1007/s00023-017-0607-9
Authors:Ko Sanders Abstract: Abstract For a massless free scalar field in a globally hyperbolic space-time we compare the global temperature \(T=\beta ^{-1}\) , defined for the \(\beta \) -KMS states \(\omega ^{(\beta )}\) , with the local temperature \(T_{\omega }(x)\) introduced by Buchholz and Schlemmer. We prove the following claims: (1) whenever \(T_{\omega ^{(\beta )}}(x)\) is defined, it is a continuous, monotonically decreasing function of \(\beta \) at every point x. (2) \(T_{\omega }(x)\) is defined when M is ultra-static with compact Cauchy surface and non-trivial scalar curvature \(R\ge 0\) , \(\omega \) is stationary, and a few other assumptions are satisfied. Our proof of (2) relies on the positive mass theorem. We discuss the necessity of its assumptions, providing counter-examples in an ultra-static space-time with non-compact Cauchy surface and \(R<0\) somewhere. Our results suggest that under suitable circumstances (in particular in the absence of acceleration, rotation and violations of the weak energy condition in the background space-time) both notions of temperature provide qualitatively similar information, and hence the Wick square can be used as a local thermometer. PubDate: 2017-08-11 DOI: 10.1007/s00023-017-0603-0
Authors:Adam Sawicki; Katarzyna Karnas Abstract: Abstract We consider the problem of deciding if a set of quantum one-qudit gates \(\mathcal {S}=\{g_1,\ldots ,g_n\}\subset G\) is universal, i.e. if \({<}\mathcal {S}{>}\) is dense in G, where G is either the special unitary or the special orthogonal group. To every gate g in \(\mathcal {S}\) we assign the orthogonal matrix \(\mathrm {Ad}_g\) that is image of g under the adjoint representation \(\mathrm {Ad}:G\rightarrow SO(\mathfrak {g})\) and \(\mathfrak {g}\) is the Lie algebra of G. The necessary condition for the universality of \(\mathcal {S}\) is that the only matrices that commute with all \(\mathrm {Ad}_{g_i}\) ’s are proportional to the identity. If in addition there is an element in \({<}\mathcal {S}{>}\) whose Hilbert–Schmidt distance from the centre of G belongs to \(]0,\frac{1}{\sqrt{2}}[\) , then \(\mathcal {S}\) is universal. Using these we provide a simple algorithm that allows deciding the universality of any set of d-dimensional gates in a finite number of steps and formulate a general classification theorem. PubDate: 2017-08-10 DOI: 10.1007/s00023-017-0604-z
Authors:Michael J. Cole; Juan A. Valiente Kroon Abstract: Abstract We describe the construction of a geometric invariant characterising initial data for the Kerr–Newman spacetime. This geometric invariant vanishes if and only if the initial data set corresponds to exact Kerr–Newman initial data, and so characterises this type of data. We first illustrate the characterisation of the Kerr–Newman spacetime in terms of Killing spinors. The space-spinor formalism is then used to obtain a set of four independent conditions on an initial Cauchy hypersurface that guarantee the existence of a Killing spinor on the development of the initial data. Following a similar analysis in the vacuum case, we study the properties of solutions to the approximate Killing spinor equation and use them to construct the geometric invariant. PubDate: 2017-08-07 DOI: 10.1007/s00023-017-0606-x
Authors:Sigmund Selberg; Achenef Tesfahun Abstract: Abstract Lower bound on the rate of decrease in time of the uniform radius of spatial analyticity of solutions to the quartic generalized KdV equation is derived, which improves an earlier result by Bona, Grujić and Kalisch. PubDate: 2017-08-05 DOI: 10.1007/s00023-017-0605-y
Authors:Benjamin Dodson; Nishanth Gudapati Abstract: Abstract We consider the Cauchy problem of \(2+1\) equivariant wave maps coupled to Einstein’s equations of general relativity and prove that two separate (nonlinear) subclasses of the system disperse to their corresponding linearized equations in the large. Global asymptotic behavior of \(2+1\) Einstein-wave map system is relevant because the system occurs naturally in \(3+1\) vacuum Einstein’s equations. PubDate: 2017-07-12 DOI: 10.1007/s00023-017-0599-5
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