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: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: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:Vincent Beaud; Simone Warzel Pages: 3143 - 3166 Abstract: 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: 2017-10-01 DOI: 10.1007/s00023-017-0591-0 Issue No:Vol. 18, No. 10 (2017)

Authors:Hugo Bringuier Pages: 3167 - 3192 Abstract: Abstract Open quantum walks (OQWs), originally introduced in Attal et al. (J Stat Phys 147(4):832–852, 2012), are quantum generalizations of classical Markov chains. Recently, natural continuous time models of OQW have been developed in Pellegrini (J Stat Phys 154(3):838–865, 2014). These models, called continuous time open quantum walks (CTOQWs), appear as natural continuous time limits of discrete time OQWs. In particular, they are quantum extensions of continuous time Markov chains. This article is devoted to the study of homogeneous CTOQW on \(\mathbb {Z}^d\) . We focus namely on their associated quantum trajectories which allow us to prove a central limit theorem for the “position” of the walker as well as a large deviation principle. PubDate: 2017-10-01 DOI: 10.1007/s00023-017-0597-7 Issue No:Vol. 18, No. 10 (2017)

Authors:Raphaël Belliard; Bertrand Eynard; Olivier Marchal Pages: 3193 - 3248 Abstract: 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: 2017-10-01 DOI: 10.1007/s00023-017-0595-9 Issue No:Vol. 18, No. 10 (2017)

Authors:Jan Dereziński Pages: 3249 - 3268 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: 2017-10-01 DOI: 10.1007/s00023-017-0585-y Issue No:Vol. 18, No. 10 (2017)

Authors:Ram Band; Guillaume Lévy Pages: 3269 - 3323 Abstract: Abstract A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of the spectral gap (the first positive Laplacian eigenvalue) on the choice of edge lengths. In particular, starting from a certain discrete graph, we seek the quantum graph for which an optimal (either maximal or minimal) spectral gap is obtained. We fully solve the minimization problem for all graphs. We develop tools for investigating the maximization problem and solve it for some families of graphs. PubDate: 2017-10-01 DOI: 10.1007/s00023-017-0601-2 Issue No:Vol. 18, No. 10 (2017)

Authors:Marco Benini; Matteo Capoferri; Claudio Dappiaggi Pages: 3325 - 3370 Abstract: 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: 2017-10-01 DOI: 10.1007/s00023-017-0593-y Issue No:Vol. 18, No. 10 (2017)

Authors:João L. Costa; Anne T. Franzen Pages: 3371 - 3398 Abstract: 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: 2017-10-01 DOI: 10.1007/s00023-017-0592-z Issue No:Vol. 18, No. 10 (2017)

Authors:Gregory J. Galloway; Carlos Vega Pages: 3399 - 3426 Abstract: 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: 2017-10-01 DOI: 10.1007/s00023-017-0594-x Issue No:Vol. 18, No. 10 (2017)

Authors:Gregory J. Galloway; Eric Ling Pages: 3427 - 3447 Abstract: Abstract The existence, established over the past number of years and supporting earlier work of Ori (Phys Rev Lett 68(14):2117–2120, 1992), of physically relevant black hole spacetimes that admit \(C^0\) metric extensions beyond the future Cauchy horizon, while being \(C^2\) -inextendible, has focused attention on fundamental issues concerning the strong cosmic censorship conjecture. These issues were recently discussed in the work of Sbierski (The \({C}^0\) -inextendibility of the Schwarzschild spacetime and the spacelike diameter in Lorentzian geometry. arXiv:1507.00601v2, (to appear in J. Diff. Geom.), 2015), in which he established the (nonobvious) fact that the Schwarzschild solution in global Kruskal–Szekeres coordinates is \(C^0\) -inextendible. In this paper, we review aspects of Sbierski’s methodology in a general context and use similar techniques, along with some new observations, to consider the \(C^0\) -inextendibility of open FLRW cosmological models. We find that a certain special class of open FLRW spacetimes, which we have dubbed ‘Milne-like,’ actually admits \(C^0\) extensions through the big bang. For spacetimes that are not Milne-like, we prove some inextendibility results within the class of spherically symmetric spacetimes. PubDate: 2017-10-01 DOI: 10.1007/s00023-017-0602-1 Issue No:Vol. 18, No. 10 (2017)

Abstract: Abstract We consider four-dimensional vacuum spacetimes which admit a nonvanishing spacelike Killing field. The quotient with respect to the Killing action is a three-dimensional quotient spacetime (M, g). We establish several results regarding maximal hypersurfaces (spacelike hypersurfaces of zero mean curvature) in such quotient spacetimes. First, we show that a complete noncompact maximal hypersurface must either be a cylinder \(S^1 \times {\mathbb {R}}\) with flat metric or else conformal to the Euclidean plane \({\mathbb {R}}^2\) . Second, we establish positivity of mass for certain maximal hypersurfaces, referring to a analogue of ADM mass adapted for the quotient setting. Finally, while lapse functions corresponding to the maximal hypersurface gauge are necessarily bounded in the four-dimensional asymptotically Euclidean setting, we show that nontrivial quotient spacetimes admit the maximal hypersurface gauge only with unbounded lapse. PubDate: 2017-10-20

Abstract: Abstract We study the quasi-classical limit of a quantum system composed of finitely many nonrelativistic particles coupled to a quantized field in Nelson-type models. We prove that, as the field becomes classical and the corresponding degrees of freedom are traced out, the effective Hamiltonian of the particles converges in resolvent sense to a self-adjoint Schrödinger operator with an additional potential, depending on the state of the field. Moreover, we explicitly derive the expression of such a potential for a large class of field states and show that, for certain special sequences of states, the effective potential is trapping. In addition, we prove convergence of the ground-state energy of the full system to a suitable effective variational problem involving the classical state of the field. PubDate: 2017-10-17

Authors:Giulia Basti; Alessandro Teta Abstract: Abstract We consider a three-particle quantum system in dimension three composed of two identical fermions of mass one and a different particle of mass m. The particles interact via two-body short range potentials. We assume that the Hamiltonians of all the two-particle subsystems do not have bound states with negative energy and, moreover, that the Hamiltonians of the two subsystems made of a fermion and the different particle have a zero-energy resonance. Under these conditions and for \(m<m_* = (13.607)^{-1}\) , we give a rigorous proof of the occurrence of the Efimov effect, i.e., the existence of infinitely many negative eigenvalues for the three-particle Hamiltonian H. More precisely, we prove that for \(m>m_*\) the number of negative eigenvalues of H is finite and for \(m<m_*\) the number N(z) of negative eigenvalues of H below \(z<0\) has the asymptotic behavior \(N(z) \sim \mathcal {C}(m) \log z \) for \(z \rightarrow 0^-\) . Moreover, we give an upper and a lower bound for the positive constant \(\mathcal {C}(m)\) . PubDate: 2017-10-14 DOI: 10.1007/s00023-017-0608-8

Authors:Sylvain Lacroix; Benoît Vicedo Abstract: Abstract Let \(\mathfrak {g}\) be a semisimple Lie algebra over \(\mathbb {C}\) . Let \(\nu \in \hbox {Aut}\, \mathfrak {g}\) be a diagram automorphism whose order divides \(T \in \mathbb {Z}_{\ge 1}\) . We define cyclotomic \(\mathfrak {g}\) -opers over the Riemann sphere \(\mathbb {P}^1\) as gauge equivalence classes of \(\mathfrak {g}\) -valued connections of a certain form, equivariant under actions of the cyclic group \(\mathbb {Z}/ T\mathbb {Z}\) on \(\mathfrak {g}\) and \(\mathbb {P}^1\) . It reduces to the usual notion of \(\mathfrak {g}\) -opers when \(T = 1\) . We also extend the notion of Miura \(\mathfrak {g}\) -opers to the cyclotomic setting. To any cyclotomic Miura \(\mathfrak {g}\) -oper \(\nabla \) , we associate a corresponding cyclotomic \(\mathfrak {g}\) -oper. Let \(\nabla \) have residue at the origin given by a \(\nu \) -invariant rational dominant coweight \(\check{\lambda }_0\) and be monodromy-free on a cover of \(\mathbb {P}^1\) . We prove that the subset of all cyclotomic Miura \(\mathfrak {g}\) -opers associated with the same cyclotomic \(\mathfrak {g}\) -oper as \(\nabla \) is isomorphic to the \(\vartheta \) -invariant subset of the full flag variety of the adjoint group G of \(\mathfrak {g}\) , where the automorphism \(\vartheta \) depends on \(\nu \) , T and \(\check{\lambda }_0\) . The big cell of the latter is isomorphic to \(N^\vartheta \) , the \(\vartheta \) -invariant subgroup of the unipotent subgroup PubDate: 2017-10-13 DOI: 10.1007/s00023-017-0616-8

Authors:Michele Cirafici Abstract: Abstract A large class of \({\mathcal {N}}=2\) quantum field theories admits a BPS quiver description, and the study of their BPS spectra is then reduced to a representation theory problem. In such theories the coupling to a line defect can be modeled by framed quivers. The associated spectral problem characterizes the line defect completely. Framed BPS states can be thought of as BPS particles bound to the defect. We identify the framed BPS degeneracies with certain enumerative invariants associated with the moduli spaces of stable quiver representations. We develop a formalism based on equivariant localization to compute explicitly such BPS invariants, for a particular choice of stability condition. Our framework gives a purely combinatorial solution to this problem. We detail our formalism with several explicit examples. PubDate: 2017-10-13 DOI: 10.1007/s00023-017-0611-0

Authors:Carlos E. González-Guillén; Cécilia Lancien; Carlos Palazuelos; Ignacio Villanueva Abstract: Abstract It is now a well-known fact that the correlations arising from local dichotomic measurements on an entangled quantum state may exhibit intrinsically non-classical features. In this paper we delve into a comprehensive study of random instances of such bipartite correlations. The main question we are interested in is: given a quantum correlation, taken at random, how likely is it that it is truly non-explainable by a classical model' We show that, under very general assumptions on the considered distribution, a random correlation which lies on the border of the quantum set is with high probability outside the classical set. What is more, we are able to provide the Bell inequality certifying this fact. On the technical side, our results follow from (i) estimating precisely the “quantum norm” of a random matrix and (ii) lower-bounding sharply enough its “classical norm”, hence proving a gap between the two. Along the way, we need a non-trivial upper bound on the \(\infty {\rightarrow }1\) norm of a random orthogonal matrix, which might be of independent interest. PubDate: 2017-10-12 DOI: 10.1007/s00023-017-0615-9

Authors:Horia D. Cornean; Domenico Monaco Abstract: Abstract We investigate the possibility of constructing exponentially localized composite Wannier bases, or equivalently smooth periodic Bloch frames, for three-dimensional time-reversal symmetric topological insulators, both of bosonic and of fermionic type, so that the bases in question are also compatible with time-reversal symmetry. This problem is translated in the study (of independent interest) of homotopy classes of continuous, periodic, and time-reversal symmetric families of unitary matrices. We identify three \(\mathbb {Z}_2\) -valued complete invariants for these homotopy classes. When these invariants vanish, we provide an algorithm which constructs a “multi-step logarithm” that is employed to continuously deform the given family into a constant one, identically equal to the identity matrix. This algorithm leads to a constructive procedure to produce the composite Wannier bases mentioned above. PubDate: 2017-10-10 DOI: 10.1007/s00023-017-0621-y