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Annales Henri Poincaré
Journal Prestige (SJR): 1.097

Citation Impact (citeScore): 2
Number of Followers: 2

Hybrid journal (It can contain Open Access articles)
ISSN (Print) 1424-0637 - ISSN (Online) 1424-0661
Published by Springer-Verlag

[2468 journals]

Spectral and Combinatorial Aspects of Cayley-Crystals
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  Abstract: Abstract Owing to their interesting spectral properties, the synthetic crystals over lattices other than regular Euclidean lattices, such as hyperbolic and fractal ones, have attracted renewed attention, especially from materials and metamaterials research communities. They can be studied under the umbrella of quantum dynamics over Cayley graphs of finitely generated groups. In this work, we investigate numerical aspects related to the quantum dynamics over such Cayley graphs. Using an algebraic formulation of the “periodic boundary condition” due to Lück (Geom Funct Anal 4:455–481, 1994), we devise a practical and converging numerical method that resolves the true bulk spectrum of the Hamiltonians. Exact results on the matrix elements of the resolvent, derived from the combinatorics of the Cayley graphs, give us the means to validate our algorithms and also to obtain new combinatorial statements. Our results open the systematic research of quantum dynamics over Cayley graphs of a very large family of finitely generated groups, which includes the free and Fuchsian groups.
  PubDate: 2024-08-01
  DOI: 10.1007/s00023-023-01373-3
On Unitarity of the Scattering Operator in Non-Hermitian Quantum Mechanics
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  Abstract: Abstract We consider the Schrödinger operator with regular short range complex-valued potential in dimension $d\ge 1$ . We show that, for $d\ge 2$ , the unitarity of scattering operator for this Hamiltonian at high energies implies the reality of the potential (that is Hermiticity of Hamiltonian). In contrast, for $d=1$ , we present complex-valued exponentially localized soliton potentials with unitary scattering operator for all positive energies and with unbroken PT symmetry. We also present examples of complex-valued regular short range potentials with real spectrum for $d=3$ . Some directions for further research are formulated.
  PubDate: 2024-08-01
  DOI: 10.1007/s00023-024-01414-5
Recurrence Relations and General Solution of the Exceptional Hermite
Equation
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  Abstract: Abstract Exceptional orthogonal Hermite polynomials have been linked to the k-step extension of the harmonic oscillator. The exceptional polynomials allow the existence of different supercharges in the Darboux–Crum and Krein–Adler constructions. They also allow the existence of different types of ladder relations and their associated recurrence relations. The existence of such relations is a unique property of these polynomials. Those relations have been used to construct 2D models which are superintegrable and display an interesting spectrum, degeneracies and finite-dimensional unitary representations. In previous works, only the physical or polynomial part of the spectrum was discussed. It is known that the general solutions are associated with other types of recurrence/ladder relations. We discuss in detail the case of the exceptional Hermite polynomials $X_2^{(1)}$ and explicitly present new chains obtained by acting with different types of ladder operators. We exploit a recent result by one of the authors (Chalifour and Grundland in Ann Henri Poincaré 21:3341, 2020), where the general analytic solution was constructed and connected with the non-degenerate confluent Heun equation. The analogue Rodrigues formulas for the general solution are constructed. The set of finite states from which the other states can be obtained algebraically is not unique, but the vanishing arrow and diagonal arrow from the diagram of the 2-chain representations can be used to obtain minimal sets. These Rodrigues formulas are then exploited, not only to construct the states, polynomial and non-polynomial, in a purely algebraic way, but also to obtain coefficients from the action of ladder operators in an algebraic manner based on further commutation relations between monomials in the generators.
  PubDate: 2024-08-01
  DOI: 10.1007/s00023-023-01395-x
Entanglement Entropy of Ground States of the Three-Dimensional Ideal Fermi
Gas in a Magnetic Field
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  Abstract: Abstract We study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in ${{\mathbb {R}}}^3$ subject to a constant magnetic field perpendicular to a plane. As for the case with no magnetic field we find, to leading order $L^2\ln (L)$ , a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region $L\Lambda \subset {{\mathbb {R}}}^3$ as the scaling parameter L tends to infinity. This is in contrast to the two-dimensional case since particles can now move freely in the direction of the magnetic field, which causes the extra $\ln (L)$ factor. The explicit expression for the coefficient of the leading order contains a surface integral similar to the Widom–Sobolev formula in the non-magnetic case. It differs, however, in the sense that the dependence on the boundary, $\partial \Lambda $ , is not solely on its area but on the “surface perpendicular to the direction of the magnetic field”. We utilize a two-term asymptotic expansion by Widom (up to an error term of order one) of certain traces of one-dimensional Wiener–Hopf operators with a discontinuous symbol. This leads to an improved error term of the order $L^2$ of the relevant trace for piecewise $\textsf{C}^{1,\alpha }$ smooth surfaces $\partial \Lambda $ .
  PubDate: 2024-08-01
  DOI: 10.1007/s00023-023-01381-3
Stability of the Spectral Gap and Ground State Indistinguishability for a
Decorated AKLT Model
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  Abstract: Abstract We use cluster expansion methods to establish local the indistiguishability of the finite volume ground states for the AKLT model on decorated hexagonal lattices with decoration parameter at least 5. Our estimates imply that the model satisfies local topological quantum order, and so, the spectral gap above the ground state is stable against local perturbations.
  PubDate: 2024-08-01
  DOI: 10.1007/s00023-023-01398-8
Local Hölder Stability in the Inverse Steklov and Calderón Problems for
Radial Schrödinger Operators and Quantified Resonances
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  Abstract: Abstract We obtain Hölder stability estimates for the inverse Steklov and Calderón problems for Schrödinger operators corresponding to a special class of $L^2$ radial potentials on the unit ball. These results provide an improvement on earlier logarithmic stability estimates obtained in Daudé et al. (J Geom Anal 31(2):1821–1854, 2021) in the case of the Schrödinger operators related to deformations of the closed Euclidean unit ball. The main tools involve: (i) A formula relating the difference of the Steklov spectra of the Schrödinger operators associated to the original and perturbed potential to the Laplace transform of the difference of the corresponding amplitude functions introduced by Simon (Ann Math 150:1029–1057, 1999) in his representation formula for the Weyl-Titchmarsh function, and (ii) a key moment stability estimate due to Still (J Approx Theory 45:26–54, 1985). It is noteworthy that with respect to the original Schrödinger operator, the type of perturbation being considered for the amplitude function amounts to the introduction of a finite number of negative eigenvalues and of a countable set of negative resonances which are quantified explicitly in terms of the eigenvalues of the Laplace-Beltrami operator on the boundary sphere.
  PubDate: 2024-08-01
  DOI: 10.1007/s00023-023-01391-1
Circuit Equation of Grover Walk
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  Abstract: Abstract We consider the Grover walk on the infinite graph in which an internal finite subgraph receives the inflow from the outside with some frequency and also radiates the outflow to the outside. To characterize the stationary state of this system, which is represented by a function on the arcs of the graph, we introduce a kind of discrete gradient operator twisted by the frequency. Then, we obtain a circuit equation which shows that (i) the stationary state is described by the twisted gradient of a potential function which is a function on the vertices; (ii) the potential function satisfies the Poisson equation with respect to a generalized Laplacian matrix. Consequently, we characterize the scattering on the surface of the internal graph and the energy penetrating inside it. Moreover, for the complete graph as the internal graph, we illustrate the relationship of the scattering and the internal energy to the frequency and the number of tails.
  PubDate: 2024-08-01
  DOI: 10.1007/s00023-023-01389-9
Particle Trajectories for Quantum Maps
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  Abstract: Abstract We study the trajectories of a semiclassical quantum particle under repeated indirect measurement by Kraus operators, in the setting of the quantized torus. In between measurements, the system evolves via either Hamiltonian propagators or metaplectic operators. We show in both cases the convergence in total variation of the quantum trajectory to its corresponding classical trajectory, as defined by the propagation of a semiclassical defect measure. This convergence holds up to the Ehrenfest time of the classical system, which is larger when the system is “less chaotic.” In addition, we present numerical simulations of these effects. In proving this result, we provide a characterization of a type of semi-classical defect measure we call uniform defect measures. We also prove derivative estimates of a function composed with a flow on the torus.
  PubDate: 2024-08-01
  DOI: 10.1007/s00023-023-01387-x
Differences Between Robin and Neumann Eigenvalues on Metric Graphs
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  Abstract: Abstract We consider the Laplacian on a metric graph, equipped with Robin ( $\delta $ -type) vertex condition at some of the graph vertices and Neumann–Kirchhoff condition at all others. The corresponding eigenvalues are called Robin eigenvalues, whereas they are called Neumann eigenvalues if the Neumann–Kirchhoff condition is imposed at all vertices. The sequence of differences between these pairs of eigenvalues is called the Robin–Neumann gap. We prove that the limiting mean value of this sequence exists and equals a geometric quantity, analogous to the one obtained for planar domains by Rudnick et al. (Commun Math Phys, 2021. arXiv:2008.07400). Moreover, we show that the sequence is uniformly bounded and provide explicit upper and lower bounds. We also study the possible accumulation points of the sequence and relate those to the associated probability distribution of the gaps. To prove our main results, we prove a local Weyl law, as well as explicit expressions for the second moments of the eigenfunction scattering amplitudes.
  PubDate: 2024-08-01
  DOI: 10.1007/s00023-023-01401-2
Robustness of Flat Bands on the Perturbed Kagome and the Perturbed
Super-Kagome Lattice
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  Abstract: Abstract We study spectral properties of perturbed discrete Laplacians on two-dimensional Archimedean tilings. The perturbation manifests itself in the introduction of non-trivial edge weights. We focus on the two lattices on which the unperturbed Laplacian exhibits flat bands, namely the $(3.6)^2$ Kagome lattice and the $(3.12^2)$ “Super-Kagome” lattice. We characterize all possible choices for edge weights which lead to flat bands. Furthermore, we discuss spectral consequences such as the emergence of new band gaps. Among our main findings is that flat bands are robust under physically reasonable assumptions on the perturbation, and we completely describe the perturbation-spectrum phase diagram. The two flat bands in the Super-Kagome lattice are shown to even exhibit an “all-or-nothing” phenomenon in the sense that there is no perturbation, which can destroy only one flat band while preserving the other.
  PubDate: 2024-08-01
  DOI: 10.1007/s00023-023-01399-7
Algebraic Localization of Wannier Functions Implies Chern Triviality in
Non-periodic Insulators
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  Abstract: Abstract For gapped periodic systems (insulators), it has been established that the insulator is topologically trivial (i.e., its Chern number is equal to 0) if and only if its Fermi projector admits an orthogonal basis with finite second moment (i.e., all basis elements satisfy $\int \varvec{x} ^2 w(\varvec{x}) ^2 \,\text {d}{\varvec{x}} < \infty $ ). In this paper, we extend one direction of this result to non-periodic gapped systems. In particular, we show that the existence of an orthogonal basis with slightly more decay ( $\int \varvec{x} ^{2+\epsilon } w(\varvec{x}) ^2 \,\text {d}{\varvec{x}} < \infty $ for any $\epsilon > 0$ ) is a sufficient condition to conclude that the Chern marker, the natural generalization of the Chern number, vanishes.
  PubDate: 2024-08-01
  DOI: 10.1007/s00023-024-01444-z
The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity
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  Abstract: Abstract 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 $C\subset T^*(M\times M)$ 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 $\tau >2$ , $$\begin{aligned} WF'^{-2+\tau -{\epsilon }}(K_G)\subset C \end{aligned}$$ for every ${\epsilon }>0$ . Furthermore, in regularity $C^{\tau +2}$ with $\tau >2$ , $$\begin{aligned} C\subset WF'^{-\frac{1}{2}}(K_G)\subset WF'^{\tau -\epsilon }(K_G)\subset C \end{aligned}$$ 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 $H^{-\frac{1}{2}-\epsilon }_{loc}(M\times M)$ as in the smooth case.
  PubDate: 2024-07-27
  DOI: 10.1007/s00023-024-01462-x
Correction to: ‘On Multimatrix Models Motivated by Random Noncommutative
Geometry II: A Yang-Mills-Higgs Matrix Model’
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  PubDate: 2024-07-18
  DOI: 10.1007/s00023-024-01456-9
Discrete Symplectic Fermions on Double Dimers and Their Virasoro
Representation
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  Abstract: Abstract A discrete version of the conformal field theory of symplectic fermions is introduced and discussed. Specifically, discrete symplectic fermions are realised as holomorphic observables in the double-dimer model. Using techniques of discrete complex analysis, the space of local fields of discrete symplectic fermions on the square lattice is proven to carry a representation of the Virasoro algebra with central charge $-2$ .
  PubDate: 2024-07-17
  DOI: 10.1007/s00023-024-01455-w
Sharp Semiclassical Spectral Asymptotics for Local Magnetic Schrödinger
Operators on $${\mathbb {R}}^d$$ Without Full Regularity
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  Abstract: Abstract We consider operators acting in $L^2({\mathbb {R}}^d)$ with $d\ge 3$ that locally behave as a magnetic Schrödinger operator. For the magnetic Schrödinger operators, we suppose the magnetic potentials are smooth and the electric potential is five times differentiable and the fifth derivatives are Hölder continuous. Under these assumptions, we establish sharp spectral asymptotics for localised counting functions and Riesz means.
  PubDate: 2024-07-16
  DOI: 10.1007/s00023-024-01471-w
Painlevé Kernels and Surface Defects at Strong Coupling
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  Abstract: Abstract 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.
  PubDate: 2024-07-14
  DOI: 10.1007/s00023-024-01469-4
Renormalization of Higher Currents of the Sine-Gordon Model in pAQFT
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  Abstract: Abstract 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.
  PubDate: 2024-07-12
  DOI: 10.1007/s00023-024-01468-5
On Lieb–Robinson Bounds for a Class of Continuum Fermions
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  Abstract: Abstract 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.
  PubDate: 2024-07-12
  DOI: 10.1007/s00023-024-01453-y
3D Tensor Renormalisation Group at High Temperatures
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  Abstract: Abstract 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.
  PubDate: 2024-07-09
  DOI: 10.1007/s00023-024-01464-9
Ergodic Theorems for Continuous-Time Quantum Walks on Crystal Lattices and
the Torus
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  Abstract: Abstract 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.
  PubDate: 2024-07-08
  DOI: 10.1007/s00023-024-01470-x

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