Authors:Jinho Baik; Ji Oon Lee Pages: 1867 - 1917 Abstract: 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:Per von Soosten; Simone Warzel Pages: 1919 - 1947 Abstract: Abstract We apply Feshbach–Krein–Schur renormalization techniques in the hierarchical Anderson model to establish a criterion on the single-site distribution which ensures exponential dynamical localization as well as positive inverse participation ratios and Poisson statistics of eigenvalues. Our criterion applies to all cases of exponentially decaying hierarchical hopping strengths and holds even for spectral dimension \( d > 2 \) , which corresponds to the regime of transience of the underlying hierarchical random walk. This challenges recent numerical findings that the spectral dimension is significant as far as the Anderson transition is concerned. PubDate: 2017-06-01 DOI: 10.1007/s00023-016-0549-7 Issue No:Vol. 18, No. 6 (2017)

Authors:Jacob S. Christiansen; Maxim Zinchenko Pages: 1949 - 1976 Abstract: 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:Davide Gabrielli; Fabio Roncari Pages: 1977 - 2006 Abstract: Abstract We call Alphabet model a generalization to N types of particles of the classic ABC model. We have particles of different types stochastically evolving on a one-dimensional lattice with an exchange dynamics. The rates of exchange are local, but under suitable conditions the dynamics is reversible with a Gibbsian-like invariant measure with long-range interactions. We discuss geometrically the conditions of reversibility on a ring that correspond to a gradient condition on the graph of configurations or equivalently to a divergence-free condition on a graph structure associated with the types of particles. We show that much of the information on the interactions between particles can be encoded in associated Tournaments that are a special class of oriented directed graphs. In particular we show that the interactions of reversible models are corresponding to strongly connected tournaments. The possible minimizers of the energies are in correspondence with the Hamiltonian cycles of the tournaments. We can then determine how many and which are the possible minimizers of the energy looking at the structure of the associated tournament. As a by-product we obtain a probabilistic proof of a classic Theorem of Camion (C R Acad Sci Paris 249: 2151–2152, 1959) on the existence of Hamiltonian cycles for strongly connected tournaments. Using these results, we obtain in the case of an equal number of k types of particles new representations of the Hamiltonians in terms of translation invariant k-body long range interactions. We show that when \(k=3,4\) the minimizer of the energy is always unique up to translations. Starting from the case \(k=5\) , it is possible to have more than one minimizer. In particular, it is possible to have minimizers for which particles of the same type are not joined together in single clusters. PubDate: 2017-06-01 DOI: 10.1007/s00023-017-0558-1 Issue No:Vol. 18, No. 6 (2017)

Authors:Benoît Laslier; Fabio Lucio Toninelli Pages: 2007 - 2043 Abstract: Abstract We study a reversible continuous-time Markov dynamics on lozenge tilings of the plane, introduced by Luby et al. (SIAM J Comput 31:167–192, 2001). Single updates consist in concatenations of n elementary lozenge rotations at adjacent vertices. The dynamics can also be seen as a reversible stochastic interface evolution. When the update rate is chosen proportional to 1 / n, the dynamics is known to have special features: a certain Hamming distance between configurations contracts with time on average (Luby et al. in SIAM J Comput 31:167–192, 2001), and the relaxation time of the Markov chain is diffusive (Wilson in Ann Appl Probab 14:274–325, 2004), growing like the square of the diameter of the system. Here, we present another remarkable feature of this dynamics, namely we derive, in the diffusive timescale, a fully explicit hydrodynamic limit equation for the height function (in the form of a nonlinear parabolic PDE). While this equation cannot be written as a gradient flow w.r.t. a surface energy functional, it has nice analytic properties, for instance it contracts the \({\mathbb {L}}^2\) distance between solutions. The mobility coefficient \(\mu \) in the equation has non-trivial but explicit dependence on the interface slope and, interestingly, is directly related to the system’s surface free energy. The derivation of the hydrodynamic limit is not fully rigorous, in that it relies on an unproven assumption of local equilibrium. PubDate: 2017-06-01 DOI: 10.1007/s00023-016-0548-8 Issue No:Vol. 18, No. 6 (2017)

Authors:Tristan Benoist; Clément Pellegrini; Francesco Ticozzi Pages: 2045 - 2074 Abstract: Abstract We study the stability of quantum pure states and, more generally, subspaces for stochastic dynamics that describe continuously monitored systems. We show that the target subspace is almost surely invariant if and only if it is invariant for the average evolution and that the same equivalence holds for the global asymptotic stability. Moreover, we prove that a strict linear Lyapunov function for the average evolution always exists, and latter can be used to derive sharp bounds on the Lyapunov exponents of the associated semigroup. Nonetheless, we also show that taking into account the measurements can lead to an improved bound on stability rate for the stochastic, non-averaged dynamics. We discuss explicit examples where the almost sure stability rate can be made arbitrary large while the average one stays constant. PubDate: 2017-06-01 DOI: 10.1007/s00023-017-0556-3 Issue No:Vol. 18, No. 6 (2017)

Authors:Benjamin Landon; Annalisa Panati; Jane Panangaden; Justine Zwicker Pages: 2075 - 2085 Abstract: 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: 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: 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)

Authors:Piotr Tourkine Pages: 2199 - 2249 Abstract: Abstract In this work, we argue that the \(\alpha '\rightarrow 0\) limit of closed string theory scattering amplitudes is a tropical limit. The motivation is to develop a technology to systematize the extraction of Feynman graphs from string theory amplitudes at higher genus. An important technical input from tropical geometry is the use of tropical theta functions with characteristics to rigorously derive the worldline limit of the worldsheet propagator. This enables us to perform a non-trivial computation at two loops: we derive the tropical form of the integrand of the genus-two four-graviton type II string amplitude, which matches the direct field theory computations. At the mathematical level, this limit is an implementation of the correspondence between the moduli space of Riemann surfaces and the tropical moduli space. PubDate: 2017-06-01 DOI: 10.1007/s00023-017-0560-7 Issue No:Vol. 18, No. 6 (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: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:Asao Arai; Fumio Hiroshima 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: 2017-05-24 DOI: 10.1007/s00023-017-0586-x

Authors:Eduardo Garibaldi; Samuel Petite; Philippe Thieullen 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: 2017-05-19 DOI: 10.1007/s00023-017-0589-7

Authors:Alexander Schenkel; Jochen Zahn 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: 2017-05-19 DOI: 10.1007/s00023-017-0590-1

Authors:Tadeusz Balaban; Joel Feldman; Horst Knörrer; Eugene Trubowitz 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: 2017-05-18 DOI: 10.1007/s00023-017-0587-9

Authors:Leonid Perlov; Michael Bukatin 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: 2017-05-17 DOI: 10.1007/s00023-017-0588-8

Authors:Jan Dereziński Abstract: Abstract A holomorphic family of closed operators with a rank one perturbation given by the function \(x^{\frac{m}{2}}\) is studied. The operators can be used in a toy model of renormalization group. PubDate: 2017-05-16 DOI: 10.1007/s00023-017-0585-y