Authors:Séverin Charbonnier; Bertrand Eynard; François David Pages: 1611 - 1645 Abstract: 2D quantum gravity is the idea that a set of discretized surfaces (called map, a graph on a surface), equipped with a graph measure, converges in the large size limit (large number of faces) to a conformal field theory (CFT), and in the simplest case to the simplest CFT known as pure gravity, also known as the gravity dressed (3,2) minimal model. Here, we consider the set of planar Strebel graphs (planar trivalent metric graphs) with fixed perimeter faces, with the measure product of Lebesgue measure of all edge lengths, submitted to the perimeter constraints. We prove that expectation values of a large class of observables indeed converge toward the CFT amplitudes of the (3,2) minimal model. PubDate: 2018-06-01 DOI: 10.1007/s00023-018-0662-x Issue No:Vol. 19, No. 6 (2018)
Authors:Thierry Bodineau; Isabelle Gallagher; Laure Saint-Raymond Pages: 1647 - 1709 Abstract: We consider the statistical motion of a convex rigid body in a gas of N smaller (spherical) atoms close to thermodynamic equilibrium. Because the rigid body is much bigger and heavier, it undergoes a lot of collisions leading to small deflections. We prove that its velocity is described, in a suitable limit, by an Ornstein–Uhlenbeck process. The strategy of proof relies on Lanford’s arguments (Lecture notes in physics, vol 38, Springer, New York, pp 1–111, 1975) together with the pruning procedure from Bodineau et al. (Invent Math 203(2):493–553, 2016) to reach diffusive times, much larger than the mean free time. Furthermore, we need to introduce a modified dynamics to avoid pathological collisions of atoms with the rigid body: these collisions, due to the geometry of the rigid body, require developing a new type of trajectory analysis. PubDate: 2018-06-01 DOI: 10.1007/s00023-018-0674-6 Issue No:Vol. 19, No. 6 (2018)
Authors:Rocco Duvenhage; Machiel Snyman Pages: 1747 - 1786 Abstract: The concept of balance between two state-preserving quantum Markov semigroups on von Neumann algebras is introduced and studied as an extension of conditions appearing in the theory of quantum detailed balance. This is partly motivated by the theory of joinings. Balance is defined in terms of certain correlated states (couplings), with entangled states as a specific case. Basic properties of balance are derived, and the connection to correspondences in the sense of Connes is discussed. Some applications and possible applications, including to non-equilibrium statistical mechanics, are briefly explored. PubDate: 2018-06-01 DOI: 10.1007/s00023-018-0664-8 Issue No:Vol. 19, No. 6 (2018)
Authors:Aurelian Gheondea Pages: 1787 - 1816 Abstract: We unify recent Noether-type theorems on the equivalence of symmetries with conservation laws for dynamical systems of Markov processes, of quantum operations, and of quantum stochastic maps, by means of some abstract results on propagation of fixed points for completely positive maps on \(C^*\) -algebras. We extend most of the existing results with characterisations in terms of dual infinitesimal generators of the corresponding strongly continuous one-parameter semigroups. By means of an ergodic theorem for dynamical systems of completely positive maps on von Neumann algebras, we show the consistency of the condition on the standard deviation for dynamical systems of quantum operations, and hence of quantum stochastic maps as well, in case the underlying Hilbert space is infinite dimensional. PubDate: 2018-06-01 DOI: 10.1007/s00023-018-0666-6 Issue No:Vol. 19, No. 6 (2018)
Authors:Yul Otani; Yoh Tanimoto Pages: 1817 - 1842 Abstract: We consider the entanglement entropy for a spacetime region and its spacelike complement in the framework of algebraic quantum field theory. For a Möbius covariant local net (a chiral component of a two-dimensional conformal field theory) satisfying either a certain nuclearity property or the split property, we consider the von Neumann entropy for type I factors between local algebras and introduce an entropic quantity. Then we implement a cutoff on this quantity with respect to the conformal Hamiltonian and show that it remains finite as the distance of two intervals tends to zero. We compare our definition to others in the literature. PubDate: 2018-06-01 DOI: 10.1007/s00023-018-0671-9 Issue No:Vol. 19, No. 6 (2018)
Authors:Mario Berta; Volkher B. Scholz; Marco Tomamichel Pages: 1843 - 1867 Abstract: We show that Araki and Masuda’s weighted non-commutative vector-valued \(L_p\) -spaces (Araki and Masuda in Publ Res Inst Math Sci Kyoto Univ 18:339–411, 1982) correspond to an algebraic generalization of the sandwiched Rényi divergences with parameter \(\alpha = \frac{p}{2}\) . Using complex interpolation theory, we prove various fundamental properties of these divergences in the setup of von Neumann algebras, including a data-processing inequality and monotonicity in \(\alpha \) . We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases \(\alpha \rightarrow \{\frac{1}{2},1,\infty \}\) leading to minus the logarithm of Uhlmann’s fidelity, Umegaki’s relative entropy, and the max-relative entropy, respectively. As a contribution that might be of independent interest, we derive a Riesz–Thorin theorem for Araki–Masuda \(L_p\) -spaces and an Araki–Lieb–Thirring inequality for states on von Neumann algebras. PubDate: 2018-06-01 DOI: 10.1007/s00023-018-0670-x Issue No:Vol. 19, No. 6 (2018)
Authors:Andreas Bluhm; Lukas Rauber; Michael M. Wolf Pages: 1891 - 1937 Abstract: In this work, we investigate the possibility of compressing a quantum system to one of smaller dimension in a way that preserves the measurement statistics of a given set of observables. In this process, we allow for an arbitrary amount of classical side information. We find that the latter can be bounded, which implies that the minimal compression dimension is stable in the sense that it cannot be decreased by allowing for small errors. Various bounds on the minimal compression dimension are proven, and an SDP-based algorithm for its computation is provided. The results are based on two independent approaches: an operator algebraic method using a fixed-point result by Arveson and an algebro-geometric method that relies on irreducible polynomials and Bézout’s theorem. The latter approach allows lifting the results from the single-copy level to the case of multiple copies and from completely positive to merely positive maps. PubDate: 2018-06-01 DOI: 10.1007/s00023-018-0660-z Issue No:Vol. 19, No. 6 (2018)
Authors:Loïc Le Treust; Thomas Ourmières-Bonafos Pages: 1465 - 1487 Abstract: This paper deals with the study of the two-dimensional Dirac operator with infinite mass boundary conditions in sectors. We investigate the question of self-adjointness depending on the aperture of the sector: when the sector is convex it is self-adjoint on a usual Sobolev space, whereas when the sector is non-convex it has a family of self-adjoint extensions parametrized by a complex number of the unit circle. As a by-product of the analysis, we are able to give self-adjointness results on polygonal domains. We also discuss the question of distinguished self-adjoint extensions and study basic spectral properties of the Dirac operator with a mass term in the sector. PubDate: 2018-05-01 DOI: 10.1007/s00023-018-0661-y Issue No:Vol. 19, No. 5 (2018)
Authors:Nikolaos Roidos Pages: 1489 - 1505 Abstract: Let X be a two-dimensional smooth manifold with boundary \(S^{1}\) and \(Y=[1,\infty )\times S^{1}\) . We consider a family of complete surfaces arising by endowing \(X\cup _{S^{1}}Y\) with a parameter-dependent Riemannian metric, such that the restriction of the metric to Y converges to the hyperbolic metric as a limit with respect to the parameter. We describe the associated spectral and scattering theory of the Laplacian for such a surface. We further show that on Y the zero \(S^{1}\) -Fourier coefficient of the generalized eigenfunction of this Laplacian, as a family with respect to the parameter, approximates in a certain sense, for large values of the spectral parameter, the zero \(S^{1}\) -Fourier coefficient of the generalized eigenfunction of the Laplacian for the case of a surface with hyperbolic cusp. PubDate: 2018-05-01 DOI: 10.1007/s00023-018-0669-3 Issue No:Vol. 19, No. 5 (2018)
Authors:Bas Janssens Pages: 1587 - 1610 Abstract: Generalised spin structures describe spinor fields that are coupled to both general relativity and gauge theory. We classify those generalised spin structures for which the corresponding fields admit an infinitesimal action of the space–time diffeomorphism group. This can be seen as a refinement of the classification of generalised spin structures by Avis and Isham (Commun Math Phys 72:103–118, 1980). PubDate: 2018-05-01 DOI: 10.1007/s00023-018-0667-5 Issue No:Vol. 19, No. 5 (2018)
Authors:Chao Liu; Todd A. Oliynyk Abstract: We establish the existence of 1-parameter families of \(\epsilon \) -dependent solutions to the Einstein–Euler equations with a positive cosmological constant \(\Lambda >0\) and a linear equation of state \(p=\epsilon ^2 K \rho \) , \(0<K\le 1/3\) , for the parameter values \(0<\epsilon < \epsilon _0\) . These solutions exist globally to the future, converge as \(\epsilon \searrow 0\) to solutions of the cosmological Poisson–Euler equations of Newtonian gravity, and are inhomogeneous nonlinear perturbations of FLRW fluid solutions. PubDate: 2018-06-04 DOI: 10.1007/s00023-018-0686-2
Authors:David Krejčiřík; Rafael Tiedra de Aldecoa Abstract: We consider the Dirichlet Laplacian in a two-dimensional strip composed of segments translated along a straight line with respect to a rotation angle with velocity diverging at infinity. We show that this model exhibits a “raise of dimension” at infinity leading to an essential spectrum determined by an asymptotic three-dimensional tube of annular cross section. If the cross section of the asymptotic tube is a disc, we also prove the existence of discrete eigenvalues below the essential spectrum. PubDate: 2018-06-02 DOI: 10.1007/s00023-018-0684-4
Authors:Zhituo Wang Abstract: We study a quartic matrix model with partition function \(Z=\int d\ M\exp \mathrm{Tr}\ (-\Delta M^2-\frac{\lambda }{4}M^4)\) . The integral is over the space of Hermitian \((\varLambda +1)\times (\varLambda +1)\) matrices, the matrix \(\Delta \) , which is not a multiple of the identity matrix, encodes the dynamics and \(\lambda >0\) is a scalar coupling constant. We proved that the logarithm of the partition function is the Borel sum of the perturbation series and hence is a well-defined analytic function of the coupling constant in certain analytic domain of \(\lambda \) , by using the multi-scale loop vertex expansions. All the non-planar graphs generated in the perturbation expansions have been taken care of on the same footing as the planar ones. This model is derived from the self-dual \(\phi ^4\) theory on the 2-dimensional Moyal space also called the 2-dimensional Grosse–Wulkenhaar model. This would also be the first fully constructed matrix model which is non-trivial and not solvable. PubDate: 2018-06-02 DOI: 10.1007/s00023-018-0688-0
Authors:Tadahiro Miyao Abstract: We study the ground state properties of the Holstein–Hubbard model on some bipartite lattices at half-filling; The ground state is proved to exhibit ferrimagnetism whenever the electron–phonon interaction is not so strong. In addition, the antiferromagnetic long-range order is shown to exist in the ground state. In contrast to this, we prove the absence of the long-range charge order. PubDate: 2018-05-31 DOI: 10.1007/s00023-018-0690-6
Authors:Jim Bryan; Zinovy Reichstein; Mark Van Raamsdonk Abstract: We study a question which has natural interpretations both in quantum mechanics and in geometry. Let \(V_{1},\cdots , V_{n}\) be complex vector spaces of dimension \(d_{1},\ldots ,d_{n}\) and let \(G= {\text {SL}}_{d_{1}} \times \cdots \times {\text {SL}}_{d_{n}}\) . Geometrically, we ask: Given \((d_{1},\ldots ,d_{n})\) , when is the geometric invariant theory quotient \(\mathbb {P}(V_{1}\otimes \cdots \otimes V_{n})/\!/G\) non-empty' This is equivalent to the quantum mechanical question of whether the multipart quantum system with Hilbert space \(V_{1}\otimes \cdots \otimes V_{n}\) has a locally maximally entangled state, i.e., a state such that the density matrix for each elementary subsystem is a multiple of the identity. We show that the answer to this question is yes if and only if \(R(d_{1},\cdots ,d_{n})\geqslant 0\) where $$\begin{aligned} R(d_{1},\cdots ,d_{n}) = \prod _{i}d_{i} +\sum _{k=1}^{n} (-1)^{k}\sum _{1\leqslant i_{1}<\cdots <i_{k}\leqslant n} \left( \gcd (d_{i_{1}},\cdots ,d_{i_{k}}) \right) ^{2}. \end{aligned}$$ We also provide a simple recursive algorithm which determines the answer to the question, and we compute the dimension of the resulting quotient in the non-empty cases. PubDate: 2018-05-31 DOI: 10.1007/s00023-018-0682-6
Abstract: We study the 11-dimensional supergravity equations which describe a low-energy approximation to string theories and are related to M-theory under the AdS/CFT correspondence. These equations take the form of a nonlinear differential system, on \(\mathbb B^7\times \mathbb S^4\) with the characteristic degeneracy at the boundary of an edge system, associated with the fibration with fiber \(\mathbb S^4.\) We compute the indicial roots of the linearized system from the Hodge decomposition of the 4-sphere following the work of Kantor, and then using the edge calculus and scattering theory, we prove that the moduli space of solutions, near the Freund–Rubin states, is parametrized by three pairs of data on the bounding 6-sphere. PubDate: 2018-05-30 DOI: 10.1007/s00023-018-0689-z
Authors:Marco Benini; Claudio Dappiaggi; Alexander Schenkel Abstract: We analyze quantum field theories on spacetimes M with timelike boundary from a model-independent perspective. We construct an adjunction which describes a universal extension to the whole spacetime M of theories defined only on the interior \(\mathrm {int}M\) . The unit of this adjunction is a natural isomorphism, which implies that our universal extension satisfies Kay’s F-locality property. Our main result is the following characterization theorem: Every quantum field theory on M that is additive from the interior (i.e., generated by observables localized in the interior) admits a presentation by a quantum field theory on the interior \(\mathrm {int}M\) and an ideal of its universal extension that is trivial on the interior. We shall illustrate our constructions by applying them to the free Klein–Gordon field. PubDate: 2018-05-30 DOI: 10.1007/s00023-018-0687-1
Authors:Alexander C. R. Belton; Michał Gnacik; J. Martin Lindsay Abstract: We give a simple and direct treatment of the strong convergence of quantum random walks to quantum stochastic operator cocycles, via the semigroup decomposition of such cocycles. Our approach also delivers convergence of the pointwise product of quantum random walks to the quantum stochastic Trotter product of the respective limit cocycles, thereby revealing the algebraic structure of the limiting procedure. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described. PubDate: 2018-04-21 DOI: 10.1007/s00023-018-0676-4
Authors:Volker Betz; Steffen Dereich; Peter Mörters Abstract: We consider effective models of condensation where the condensation occurs as time t goes to infinity. We provide natural conditions under which the buildup of the condensate occurs on a spatial scale of 1 / t and has the universal form of a Gamma density. The exponential parameter of this density is determined only by the equation and the total mass of the condensate, while the power law parameter may in addition depend on the decay properties of the initial condition near the condensation point. We apply our results to some examples, including simple models of Bose–Einstein condensation. PubDate: 2018-04-17 DOI: 10.1007/s00023-018-0673-7
Authors:P. J. Forrester; J. R. Ipsen; Dang-Zheng Liu Abstract: We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We prove that the eigenvalues form a bi-orthogonal ensemble, which reduces asymptotically to the Hermite Muttalib–Borodin ensemble. Explicit expressions for the bi-orthogonal functions as well as the correlation kernel are provided. Scaling the latter near the origin gives a limiting kernel involving Meijer G-functions, and the functional form of the global density is calculated. As a part of this study, we introduce a new matrix transformation which maps the space of polynomial ensembles onto itself. This matrix transformation is closely related to the so-called hyperbolic Harish-Chandra–Itzykson–Zuber integral. PubDate: 2018-02-24 DOI: 10.1007/s00023-018-0654-x
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