Communications in Mathematical Physics最新文献

Let B be a spacetime region of width (2R >0), and (varphi ) a vector state localized in B. We show that the vacuum relative entropy of (varphi ), on the local von Neumann algebra of B, is bounded by (2pi R)-times the energy of the state (varphi ) in B. This bound is model-independent and rigorous; it follows solely from first principles in the framework of translation covariant, local Quantum Field Theory on the Minkowski spacetime.

In this paper we first prove that the maximal ideal of the universal affine vertex operator algebra (V^k(sl_n)) for (k=-n+frac{n-1}{q}) is generated by two singular vectors of conformal weight 3q if (n=3), and by one singular vector of conformal weight 2q if (ngeqslant 4). We next determine the associated varieties of the simple vertex operator algebras (L_k(sl_3)) for all the non-admissible levels (k=-3+frac{2}{2m+1}), (mgeqslant 0). The varieties of the associated simple affine W-algebras (W_k(sl_3,f)), for nilpotent elements f of (sl_3), are also determined.

We formulate a family of algebras, twisted Yangians (of split type) in current generators and relations, via a degeneration of the Drinfeld presentation of affine (imath )quantum groups (associated with split Satake diagrams). These new algebras admit PBW type bases and are shown to be a deformation of twisted current algebras; presentations for twisted current algebras are also provided. For type AI, it matches with the Drinfeld presentation of twisted Yangian obtained via Gauss decomposition. We conjecture that our split twisted Yangians are isomorphic to the corresponding ones in RTT presentation.

The goal of this paper is to analyze how the celebrated phase transitions of the XY model are affected by the presence of a non-elliptic quenched disorder. In dimension (d=2), we prove that if one considers an XY model on the infinite cluster of a supercritical percolation configuration, the Berezinskii–Kosterlitz–Thouless (BKT) phase transition still occurs despite the presence of quenched disorder. The proof works for all (p>p_c) (site or edge). We also show that the XY model defined on a planar Poisson-Voronoi graph also undergoes a BKT phase transition. When (dge 3), we show in a similar fashion that the continuous symmetry breaking of the XY model at low enough temperature is not affected by the presence of quenched disorder such as supercritical percolation (in (mathbb {Z}^d)) or Poisson-Voronoi (in (mathbb {R}^d)). Adapting either Fröhlich–Spencer’s proof of existence of a BKT phase transition (Fröhlich and Spencer in Commun Math Phys 81(4):527–602, 1981) or the more recent proofs (Lammers in Probab Theory Relat Fields 182(1–2):531–550, 2022; van Engelenburg and Lis in Commun Math Phys 399(1):85–104, 2023; Aizenman et al. in Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models, 2021. arXiv preprint arXiv:2110.09498; van Engelenburg and Lis in On the duality between height functions and continuous spin models, 2023. arXiv preprint arXiv:2303.08596) to such non-uniformly elliptic disorders appears to be non-trivial. Instead, our proofs rely on Wells’ correlation inequality (Wells in Some Moment Inequalities and a Result on Multivariable Unimodality. PhD thesis, Indiana University, 1977).

We establish the Anderson localization and exponential dynamical localization for a class of quasi-periodic Schrödinger operators on (mathbb Z^d) with bounded or unbounded Lipschitz monotone potentials via multi-scale analysis based on Rellich function analysis in the perturbative regime. We show that at each scale, the resonant Rellich function uniformly inherits the Lipschitz monotonicity property of the potential via a novel Schur complement argument.

When two spatially separated parties make measurements on an unknown entangled quantum state, what correlations can they achieve? How difficult is it to determine whether a given correlation is a quantum correlation? These questions are central to problems in quantum communication and computation. Previous work has shown that the general membership problem for quantum correlations is computationally undecidable. In the current work we show something stronger: there is a family of constant-sized correlations—that is, correlations for which the number of measurements and number of measurement outcomes are fixed—such that solving the quantum membership problem for this family is computationally impossible. Thus, the undecidability that arises in understanding Bell experiments is not dependent on varying the number of measurements in the experiment. This places strong constraints on the types of descriptions that can be given for quantum correlation sets. Our proof is based on a combination of techniques from quantum self-testing and undecidability results for linear system nonlocal games.

We use the theory of (x-y) duality to propose a new definition/construction for the correlation differentials of topological recursion; we call it generalized topological recursion. This new definition coincides with the original topological recursion of Chekhov–Eynard–Orantin in the regular case and allows, in particular, to get meaningful answers in a variety of irregular and degenerate situations.

We study quantum neural networks made by parametric one-qubit gates and fixed two-qubit gates in the limit of infinite width, where the generated function is the expectation value of the sum of single-qubit observables over all the qubits. First, we prove that the probability distribution of the function generated by the untrained network with randomly initialized parameters converges in distribution to a Gaussian process whenever each measured qubit is correlated only with few other measured qubits. Then, we analytically characterize the training of the network via gradient descent with square loss on supervised learning problems. We prove that, as long as the network is not affected by barren plateaus, the trained network can perfectly fit the training set and that the probability distribution of the function generated after training still converges in distribution to a Gaussian process. Finally, we consider the statistical noise of the measurement at the output of the network and prove that a polynomial number of measurements is sufficient for all the previous results to hold and that the network can always be trained in polynomial time.

A rigorous derivation of point vortex systems from kinetic equations has been a challenging open problem, due to singular layers in the inviscid limit, giving a large velocity gradient in the Boltzmann equations. In this paper, we derive the Helmholtz–Kirchhoff point-vortex system from the hydrodynamic limits of the Boltzmann equations. We construct Boltzmann solutions by the Hilbert-type expansion associated to the point vortices solutions of the 2D Navier–Stokes equations. We give a precise pointwise estimate for the solution of the Boltzmann equations with small Strouhal number and Knudsen number.

We study the geometry of integrable systems of hydrodynamic type of the form (w_t=Xcirc w_x) where (circ ) is the product of a regular F-manifold. In the first part of the paper, we present a general construction of a connection compatible with the F-manifold structure starting from a frame of vector fields defining commuting flows of hydrodynamic type. In the second part of the paper, using this construction, we study regular F-manifolds with compatible connection and Euler vector field, ((nabla ,circ ,e,E)), associated with integrable hierarchies obtained from the solutions of the equation (dcdot d_L ,a_0=0) where (L=Ecirc ). In particular, we show that n-dimensional F-manifolds associated to regular operators L are classified by n arbitrary functions of a single variable. Moreover, we show that flat connections (nabla ) correspond to linear solutions (a_0).