arXiv subjects

Sven Gnutzmann

Publications and source records attributed to Sven Gnutzmann.

At least 19 recordsLinked to original sources

arXiv2026

Characterization of Chaotic Evolution in Quantum Systems Induced by Random Hermitian Matrices

In a recent paper, a semiclassical Lyapunov exponent associated with a quantum Hamiltonian represented by a finite-dimensional Hermitian matrix was defined and placed on a mathematical foundation. The Lyapunov exponent characterizes the early stages of the evolution toward the ergodic state, while the late stages are characterized by the spectral gap of the corresponding Markov matrix. Here, we apply this formalism to five random-matrix ensembles. For each ensemble, we derive the mean Lyapunov exponent, its variance, and the spectral gap as functions of energy. We also present the corresponding thermal averages. Extensive numerical data are compared with the theoretical predictions.

nlin.CD↗

arXiv2026

Quantum graph models of quantum chaos: an introduction and some recent applications

Quantum graphs are a paradigmatic model for quantum chaos as well as for spectral theory. We give a concise didactical introduction to quantum graphs, or Schr\"odinger Hamiltonians on metric graphs, with a focus on results related to quantum chaos, periodic orbit theory and spectral theory. We summarise related seminal results, and give an overview over a few more recent developments.

quant-ph↗

arXiv2026

Symmetry Breaking and Phase Transitions in Random Non-Commutative Geometries and Related Random-Matrix Ensembles

Ensembles of random fuzzy non-commutative geometries may be described in terms of finite ($N^2$-dimensional) Dirac operators and a probability measure. Dirac operators of type $(p,q)$ are defined in terms of commutators and anti-commutators of $2^{p+q-1}$ hermitian matrices $H_k$ and tensor products with a representation of a Clifford algebra. Ensembles based on this idea have recently been used as a toy model for quantum gravity, and they are interesting random-matrix ensembles in their own right. We provide a complete theoretical picture of crossovers, phase transitions, and symmetry breaking in the $N \to \infty $ limit of 1-parameter families of quartic Barrett-Glaser ensembles in the one-matrix cases $(1,0)$ and $(0,1)$ that depend on one coupling constant $g$. Our theoretical results are in full agreement with previous and new Monte-Carlo simulations.

math-ph↗

arXiv2024

Information scrambling and chaos induced by a Hermitian Matrix

Given an arbitrary $V \times V$ Hermitian matrix, considered as a finite discrete quantum Hamiltonian, we use methods from graph and ergodic theories to construct a \textit{quantum Poincaré map} at energy $E$ and a corresponding stochastic \textit{classical Poincaré-Markov map} at the same energy on an appropriate discrete \textit{phase space}. This phase space consists of the directed edges of a graph with $V$ vertices that are in one-to-one correspondence with the non-vanishing off-diagonal elements of $H$. The correspondence between quantum Poincaré map and classical Poincaré-Markov map is an alternative to the standard quantum-classical correspondence based on a classical limit $\hbar \to 0$. Most importantly it can be constructed where no such limit exists. Using standard methods from ergodic theory we then proceed to define an expression for the \textit{Lyapunov exponent} $Λ(E)$ of the classical map. It measures the rate of loss of classical information in the dynamics and relates it to the separation of stochastic \textit{classical trajectories} in the phase space. We suggest that loss of information in the underlying classical dynamics is an indicator for quantum information scrambling.

quant-ph↗

arXiv2023

Closed form expressions for the Green's function of a quantum graph -- a scattering approach

In this work we present a three step procedure for generating a closed form expression of the Green's function on both closed and open finite quantum graphs with general self-adjoint matching conditions. We first generalize and simplify the approach by Barra and Gaspard [Barra F and Gaspard P 2001, Phys. Rev. E {\bf 65}, 016205] and then discuss the validity of the explicit expressions. For compact graphs, we show that the explicit expression is equivalent to the spectral decomposition as a sum over poles at the discrete energy eigenvalues with residues that contain projector kernel onto the corresponding eigenstate. The derivation of the Green's function is based on the scattering approach, in which stationary solutions are constructed by treating each vertex or subgraph as a scattering site described by a scattering matrix. The latter can then be given in a simple closed form from which the Green's function is derived. The relevant scattering matrices contain inverse operators which are not well defined for wave numbers at which bound states in the continuum exists. It is shown that the singularities in the scattering matrix related to these bound states or perfect scars can be regularised. Green's functions or scattering matrices can then be expressed as a sum of a regular and a singular part where the singular part contains the projection kernel onto the perfect scar.

quant-ph↗

arXiv2022

A trace formula for metric graphs with piecewise constant potentials and multi-mode graphs

We generalize the scattering approach to quantum graphs to quantum graphs with with piecewise constant potentials and multiple excitation modes. The free single-mode case is well-known and leads to the trace formulas of Roth, Kottos and Smilansky. By introducing an effective reduced scattering picture we are able to introduce new exact trace formulas in the more general setting. The latter are derived and discussed in details with some numerical examples for illustration. Our generalization is motivated by both experimental applications and fundamental theoretical considerations. The free single-mode quantum graphs are an extreme idealization of reality that, due to the simplicity of the model allows to understand a large number of generic or universal phenomena. We lift some of this idealization by considering the influence of evanescent modes that only open above threshold energies. How to do this theoretically in a closed model in general is a challenging question of fundamental theoretical interest and we achieve this here for quantum graphs.

quant-ph↗

arXiv2019

Trace formulas for general Hermitian matrices: Unitary scattering approach and periodic orbits on an associated graph

Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix $H$ to a unitary analogue. In the first type the unitary matrix is $e^{i(λ\II - H)}$ where $λ$ is the spectral parameter. The new feature is that the spectral parameter appears in the final form as an argument of Eulerian polynomials -- thus connecting the periodic orbits to combinatorial objects in a novel way. To obtain the second type, one expresses the input in terms of a unitary scattering matrix in a larger Hilbert space. One of the surprising features here is that the locations and radii of the spectral discs of Gershgorin's theorem appear naturally as the pole parameters of the scattering matrix. Both formulas are discussed and possible applications are outlined.

math-ph↗

arXiv2019

On the nodal structure of nonlinear stationary waves on star graphs

We consider stationary waves on nonlinear quantum star graphs, i.e. solutions to the stationary (cubic) nonlinear Schrödinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the star and classify them according to the nodal structure on each edge (i.e. the number of nodal domains or nodal points that the solution has on each edge). We discuss the relevance of these solutions in more applied settings as starting points for numerical calculations of spectral curves and put our results into the wider context of nodal counting such as the classic Sturm oscillation theorem.

math-ph↗

arXiv2018

Extreme values of CUE characteristic polynomials: a numerical study

We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the Circular Unitary Ensemble (CUE) of Random Matrix Theory. In particular, we investigate a range of recent conjectures and theoretical results inspired by analogies with the theory of logarithmically-correlated Gaussian random fields. These include phenomena related to the conjectured freezing transition. Our numerical results are consistent with, and therefore support, the previous conjectures and theory. We also go beyond previous investigations in several directions: we provide the first quantitative evidence in support of a correlation between extreme values of the characteristic polynomials and large gaps in the spectrum, we investigate the rate of convergence to the limiting formulae previously considered, and we extend the previous analysis of the CUE to the C$β$E which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.

cond-mat.stat-mech↗

arXiv2018

Quantum Graphs via Exercises

Studying the spectral theory of Schroedinger operator on metric graphs (also known as quantum graphs) is advantageous on its own as well as to demonstrate key concepts of general spectral theory. There are some excellent references for this study such as a mathematically oriented book by Berkolaiko and Kuchment, a review with applications to theoretical physicsby Gnutzmann and Smilansky, and elementary lecture notes by Berkolaiko. Here, we provide a set of questions and exercises which can accompany the reading of these references or an elementary course on quantum graphs. The exercises are taken from courses on quantum graphs which were taught by the authors.

math.SP↗

arXiv2017

Quantum chaos for nonstandard symmetry classes in the Feingold-Peres model of coupled tops

We consider two coupled quantum tops with angular momentum vectors $\mathbf{L}$ and $\mathbf{M}$. The coupling Hamiltonian defines the Feinberg-Peres model which is a known paradigm of quantum chaos. We show that this model has a nonstandard symmetry with respect to the Altland-Zirnbauer tenfold symmetry classification of quantum systems which extends the well-known threefold way of Wigner and Dyson (referred to as `standard' symmetry classes here). We identify that the nonstandard symmetry classes BD$I_0$ (chiral orthogonal class with no zero modes), BD$I_1$ (chiral orthogonal class with one zero mode) and C$I$ (antichiral orthogonal class) as well as the standard symmetry class A$I$ (orthogonal class). We numerically analyze the specific spectral quantum signatures of chaos related to the nonstandard symmetries. In the microscopic density of states and in the distribution of the lowest positive energy eigenvalue we show that the Feinberg-Peres model follows the predictions of the Gaussian ensembles of random-matrix theory in the appropriate symmetry class if the corresponding classical dynamics is chaotic. In a crossover to mixed and near-integrable classical dynamics we show that these signatures disappear or strongly change.

quant-ph↗

arXiv2017

Stationary waves on nonlinear quantum graphs II: Application of canonical perturbation theory in basic graph structures

We consider exact and asymptotic solutions of the stationary cubic nonlinear Schrödinger equation (NLSE) on metric graphs. We focus on some basic example graphs. The asymptotic solutions are obtained using the canonical perturbation formalism developed in our earlier paper \cite{paper1}. For closed example graphs (interval, ring, star graph, tadpole graph) we calculate spectral curves and show how the description of spectra reduces to known characteristic functions of linear quantum graphs in the low intensity limit. Analogously for open examples we show how nonlinear scattering of stationary waves arises and how it reduces to known linear scattering amplitudes at low intensities. In the short-wave length asymptotics we discuss how genuine nonlinear effects such as may be described using the leading order of canonical perturbation theory: bifurcation of spectral curves (and the corresponding solutions) in closed graphs and multistability in open graphs.

nlin.PS↗

arXiv2016

Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory

In this paper we present a general framework for solving the stationary nonlinear Schrödinger equation (NLSE) on a network of one-dimensional wires modelled by a metric graph with suitable matching conditions at the vertices. A formal solution is given that expresses the wave function and its derivative at one end of an edge (wire) nonlinearly in terms of the values at the other end. For the cubic NLSE this nonlinear transfer operation can be expressed explicitly in terms of Jacobi elliptic functions. Its application reduces the problem of solving the corresponding set of coupled ordinary nonlinear differential equations to a finite set of nonlinear algebraic equations. For sufficiently small amplitudes we use canonical perturbation theory which makes it possible to extract the leading nonlinear corrections over large distances.

nlin.PS↗

arXiv2015

Electron-hole coherent states for the Bogoliubov-de Gennes equation

We construct a new set of generalized coherent states, the electron-hole coherent states, for a (quasi-)spin particle on the infinite line. The definition is inspired by applications to the Bogoliubov-de Gennes equations where the quasi-spin refers to electron- and hole-like components of electronic excitations in a superconductor. Electron-hole coherent states generally entangle the space and the quasi-spin degrees of freedom. We show that the electron-hole coherent states allow obtaining a resolution of unity and form minimum uncertainty states for position and velocity where the velocity operator is defined using the Bogoliubov-de Gennes Hamiltonian. The usefulness and the limitations of electron-hole coherent states and the phase space representations built from them are discussed in terms of basic applications to the Bogoliubov-de Gennes equation such as Andreev reflection.

math-ph↗

arXiv2015

Quantum walks and quantum search on graphene lattices

Quantum walks have been very successful in the development of search algorithms in quantum information, in particular in the development of spatial search algorithms. However, the construction of continuous-time quantum search algorithms in two-dimensional lattices has proved difficult, requiring additional degrees of freedom. Here, we demonstrate that continuous-time quantum walk search is possible in two-dimensions by changing the search topology to a graphene lattice, utilising the Dirac point in the energy spectrum. This is made possible by making a change to standard methods of marking a particular site in the lattice. Various ways of marking a site are shown to result in successful search protocols. We further establish that the search can be adapted to transfer probability amplitude across the lattice between specific lattice sites thus establishing a line of communication between these sites.

quant-ph↗

arXiv2015

Microwave Experiments Simulating Quantum Search and Directed Transport in Artificial Graphene

A series of quantum search algorithms have been proposed recently providing an algebraic speedup compared to classical search algorithms from $N$ to $\sqrt{N}$, where $N$ is the number of items in the search space. In particular, devising searches on regular lattices has become popular in extending Grover's original algorithm to spatial searching. Working in a tight-binding setup, it could be demonstrated, theoretically, that a search is possible in the physically relevant dimensions 2 and 3 if the lattice spectrum possesses Dirac points. We present here a proof of principle experiment implementing wave search algorithms and directed wave transport in a graphene lattice arrangement. The idea is based on bringing localized search states into resonance with an extended lattice state in an energy region of low spectral density---namely, at or near the Dirac point. The experiment is implemented using classical waves in a microwave setup containing weakly coupled dielectric resonators placed in a honeycomb arrangement, i.e., artificial graphene. Furthermore, we investigate the scaling behavior experimentally using linear chains.

cond-mat.mes-hall↗

arXiv2014

Sigma models for quantum chaotic dynamics

We review the construction of the supersymmetric sigma model for unitary maps, using the color- flavor transformation. We then illustrate applications by three case studies in quantum chaos. In two of these cases, general Floquet maps and quantum graphs, we show that universal spectral fluctuations arise provided the pertinent classical dynamics are fully chaotic (ergodic and with decay rates sufficiently gapped away from zero). In the third case, the kicked rotor, we show how the existence of arbitrarily long-lived modes of excitation (diffusion) precludes universal fluctuations and entails quantum localization.

nlin.CD↗

arXiv2013

Quantum search on graphene lattices

We present a continuous-time quantum search algorithm on a graphene lattice. This provides the sought-after implementation of an efficient continuous-time quantum search on a two-dimensional lattice. The search uses the linearity of the dispersion relation near the Dirac point and can find a marked site on a graphene lattice faster than the corresponding classical search. The algorithm can also be used for state transfer and communication.

quant-ph↗