Annales Henri Poincaré最新文献

Given a globally hyperbolic spacetime (M={mathbb {R}}times Sigma ) of dimension four and regularity (C^tau ), we estimate the Sobolev wavefront set of the causal propagator (K_G) of the Klein–Gordon operator. In the smooth case, the propagator satisfies (WF'(K_G)=C), where (Csubset T^*(Mtimes M)) consists of those points ((tilde{x},tilde{xi },tilde{y},tilde{eta })) such that (tilde{xi },tilde{eta }) are cotangent to a null geodesic (gamma ) at (tilde{x}) resp. (tilde{y}) and parallel transports of each other along (gamma ). We show that for (tau >2),

for every ({epsilon }>0). Furthermore, in regularity (C^{tau +2}) with (tau >2),

holds for (0<epsilon <tau +frac{1}{2}). In the ultrastatic case with (Sigma ) compact, we show (WF'^{-frac{3}{2}+tau -epsilon }(K_G)subset C) for (epsilon >0) and (tau >2) and (WF'^{-frac{3}{2}+tau -epsilon }(K_G)= C) for (tau >3) and (epsilon <tau -3). Moreover, we show that the global regularity of the propagator (K_G) is (H^{-frac{1}{2}-epsilon }_{loc}(Mtimes M)) as in the smooth case.

It is well established that the spectral analysis of canonically quantized four-dimensional Seiberg–Witten curves can be systematically studied via the Nekrasov–Shatashvili functions. In this paper, we explore another aspect of the relation between ({mathcal {N}}=2) supersymmetric gauge theories in four dimensions and operator theory. Specifically, we study an example of an integral operator associated with Painlevé equations and whose spectral traces are related to correlation functions of the 2d Ising model. This operator does not correspond to a canonically quantized Seiberg–Witten curve, but its kernel can nevertheless be interpreted as the density matrix of an ideal Fermi gas. Adopting the approach of Tracy and Widom, we provide an explicit expression for its eigenfunctions via an ({{,mathrm{O(2)},}}) matrix model. We then show that these eigenfunctions are computed by surface defects in ({{,mathrm{SU(2)},}}) super Yang–Mills in the self-dual phase of the (Omega )-background. Our result also yields a strong coupling expression for such defects which resums the instanton expansion. Even though we focus on one concrete example, we expect these results to hold for a larger class of operators arising in the context of isomonodromic deformation equations.

In this paper, we show that the higher currents of the sine-Gordon model are super-renormalizable by power counting in the framework of pAQFT. First we obtain closed recursive formulas for the higher currents in the classical theory and introduce a suitable notion of degree for their components. We then move to the pAQFT setting, and by means of some technical results, we compute explicit formulas for the unrenormalized interacting currents. Finally, we perform what we call the piecewise renormalization of the interacting higher currents, showing that the renormalization process involves a number of steps which is bounded by the degree of the classical conserved currents.

We consider the quantum dynamics of a many-fermion system in ({{mathbb {R}}}^d) with an ultraviolet regularized pair interaction as previously studied in Gebert et al. (Ann Henri Poincaré 21(11):3609–3637, 2020). We provide a Lieb–Robinson bound under substantially relaxed assumptions on the potentials. We also improve the associated one-body Lieb–Robinson bound on (L^2)-overlaps to an almost ballistic one (i.e., an almost linear light cone) under the same relaxed assumptions. Applications include the existence of the infinite-volume dynamics and clustering of ground states in the presence of a spectral gap. We also develop a fermionic continuum notion of conditional expectation and use it to approximate time-evolved fermionic observables by local ones, which opens the door to other applications of the Lieb–Robinson bounds.

Building upon previous 2D studies, this research focuses on describing 3D tensor renormalisation group (RG) flows for lattice spin systems, such as the Ising model. We present a novel RG map, which operates on tensors with infinite-dimensional legs and does not involve truncations, in contrast to numerical tensor RG maps. To construct this map, we developed new techniques for analysing tensor networks. Our analysis shows that the constructed RG map contracts the region around the tensor (A_*), corresponding to the high-temperature phase of the 3D Ising model. This leads to the iterated RG map convergence in the Hilbert–Schmidt norm to (A_*) when initialised in the vicinity of (A_*). This work provides the first steps towards the rigorous understanding of tensor RG maps in 3D.

We give several quantum dynamical analogs of the classical Kronecker–Weyl theorem, which says that the trajectory of free motion on the torus along almost every direction tends to equidistribute. As a quantum analog, we study the quantum walk (exp (-textrm{i}t Delta ) psi ) starting from a localized initial state (psi ). Then, the flow will be ergodic if this evolved state becomes equidistributed as time goes on. We prove that this is indeed the case for evolutions on the flat torus, provided we start from a point mass, and we prove discrete analogs of this result for crystal lattices. On some periodic graphs, the mass spreads out non-uniformly, on others it stays localized. Finally, we give examples of quantum evolutions on the sphere which do not equidistribute.

Symmetric tridiagonal matrices appear ubiquitously in mathematical physics, serving as the matrix representation of discrete random Schrödinger operators. In this work, we investigate the top eigenvalue of these matrices in the large deviation regime, assuming the random potentials are on the diagonal with a certain decaying factor (N^{-{alpha }}), and the probability law (mu ) of the potentials satisfies specific decay assumptions. We investigate two different models, one of which has random matrix behavior at the spectral edge but the other does not. Both the light-tailed regime, i.e., when (mu ) has all moments, and the heavy-tailed regime are covered. Precise right tail estimates and a crude left tail estimate are derived. In particular, we show that when the tail (mu ) has a certain decay rate, then the top eigenvalue is distributed as the Fréchet law composed with some deterministic functions. The proof relies on computing one-point perturbations of fixed tridiagonal matrices.

We describe in a simple setting how to extract a braided tensor category from a collection of superselection sectors of a two-dimensional quantum spin system, corresponding to abelian anyons. We extract from this category its fusion ring as well as its F and R-symbols. We then construct the double semion state in infinite volume and extract the braided tensor category describing its semion, anti-semion, and bound state excitations. We verify that this category is equivalent to the representation category of the twisted quantum double (mathcal {D}^{phi }(mathbb {Z}_2)).

We generalize Lévy’s lemma, a concentration-of-measure result for the uniform probability distribution on high-dimensional spheres, to a much more general class of measures, so-called GAP measures. For any given density matrix (rho ) on a separable Hilbert space ({mathcal {H}}), ({textrm{GAP}}(rho )) is the most spread-out probability measure on the unit sphere of ({mathcal {H}}) that has density matrix (rho ) and thus forms the natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue (Vert rho Vert ) of (rho ) is small. We use this fact to generalize and improve well-known and important typicality results of quantum statistical mechanics to GAP measures, namely canonical typicality and dynamical typicality. Canonical typicality is the statement that for “most” pure states (psi ) of a given ensemble, the reduced density matrix of a sufficiently small subsystem is very close to a (psi )-independent matrix. Dynamical typicality is the statement that for any observable and any unitary time evolution, for “most” pure states (psi ) from a given ensemble the (coarse-grained) Born distribution of that observable in the time-evolved state (psi _t) is very close to a (psi )-independent distribution. So far, canonical typicality and dynamical typicality were known for the uniform distribution on finite-dimensional spheres, corresponding to the micro-canonical ensemble, and for rather special mean-value ensembles. Our result shows that these typicality results hold also for ({textrm{GAP}}(rho )), provided the density matrix (rho ) has small eigenvalues. Since certain GAP measures are quantum analogs of the canonical ensemble of classical mechanics, our results can also be regarded as a version of equivalence of ensembles.

We consider the ground state of a Bose gas of N particles on the three-dimensional unit torus in the mean-field regime that is known to exhibit Bose–Einstein condensation. Bounded one-particle operators with law given through the interacting Bose gas’ ground state correspond to dependent random variables due to the bosons’ correlation. We prove that in the limit (N rightarrow infty ) bounded one-particle operators with law given by the ground state satisfy large deviation estimates. We derive a lower and an upper bound on the rate function that match up to second order and that are characterized by quantum fluctuations around the condensate.