14634229
Quantum graphs with spin Hamiltonians
The article surveys quantization schemes for metric graphs with spin. Typically quantum graphs are defined with the Laplace or Schrodinger operator which describe particles whose intrinsic angular momentum (spin) is zero. However, in many applications, for example modeling an electron (which has spin-1/2) on a network of thin wires, it is necessary to consider operators which allow spin-orbit interaction. The article presents a review of quantization schemes for graphs with three such Hamiltonian operators, the Dirac, Pauli and Rashba Hamiltonians. Comparing results for the trace formula, spectral statistics and spin-orbit localization on quantum graphs with spin Hamiltonians.
670843
Convergence of spectra of graph-like thin manifolds
Abstract We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace–Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices.
16371459
Spectra of graph neighborhoods and scattering
Let (G e ) e>0 be a family of ‘e-thin’ Riemannian manifolds modeled on a finite metric graph G, for example, the e-neighborhood of an embedding of G in some Euclidean space with straight edges. We study the asymptotic behavior of the spectrum of the Laplace-Beltrami operator on G e , as e→0, for various boundary conditions. We obtain complete asymptotic expansions for the kth eigenvalue and the eigenfunctions, uniformly for k≤Ce −1 , in terms of scattering data on a non-compact limit space. We then use this to determine the quantum graph which is to be regarded as the limit object, in a spectral sense, of the family (G e ). Our method is a direct construction of approximate eigenfunctions from the scattering and graph data, and the use of a priori estimates to show that all eigenfunctions are obtained in this way.
121234849
Convergence of Spectra of Mesoscopic Systems Collapsing onto a Graph
Abstract Let M be a finite graph in the plane and let Me be a domain that looks like the e-fattened graph M (exact conditions on the domain are given). It is shown that the spectrum of the Neumann Laplacian on Me converges when e → 0 to the spectrum of an ODE problem on M. The presence of an electromagnetic field is also allowed. Considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry. The results of the paper extend the ones previously obtained by J. Rubinstein and M. Schatzman.
16874849
Quantum graphs: I. Some basic structures
A quantum graph is a graph equipped with a self-adjoint differential or pseudo-differential Hamiltonian. Such graphs have been studied recently in relation to some problems of mathematics, physics and chemistry. The paper has a survey nature and is devoted to the description of some basic notions concerning quantum graphs, including the boundary conditions, self-adjointness, quadratic forms, and relations between quantum and combinatorial graph models.
119424306
Quantum graphs: Applications to quantum chaos and universal spectral statistics
During the last few years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wavefunction statistics. In the first part of this review we give a detailed introduction to the spectral theory of quantum graphs and discuss exact trace formulae for the spectrum and the quantum-to-classical correspondence. The second part of this review is devoted to the spectral statistics of quantum graphs as an application to quantum chaos. In particular, we summarize recent developments on the spectral statistics of generic large quantum graphs based on two approaches: the periodic-orbit approach and the supersymmetry approach. The latter provides a condition and a proof for universal spectral statistics as predicted by random-matrix theory.
18435430
The spectral form factor for quantum graphs with spin-orbit coupling
We consider quantum graphs with spin-orbit couplings at the vertices. Timereversal invariance implies that the bond S-matrix is in the orthogonal or symplectic symmetry class, depending on spin quantum number s being integer or half-integer, respectively. The periodic-orbit expansion of the spectral form factor is shown to acquire additional weights from spin rotations along orbits. We determine the spin contribution to the coefficients in an expansion of the form factor from properties of the representation of the group of spin transformations on the graph. Consistency with the Circular Orthogonal and Circular Symplectic Ensemble, respectively, of random matrices is obtained. E-mail address: jens.bolte@uni-ulm.de E-mail address: jon@math.tamu.edu
7039408
LETTER TO THE EDITOR: The spin contribution to the form factor of quantum graphs
Following the quantization of a graph with the Dirac operator (spin-1/2), we explain how additional weights in the spectral form factor K(τ) due to spin propagation around orbits produce higher order terms in the small-τ asymptotics in agreement with symplectic random matrix ensembles. We determine conditions on the group of spin rotations sufficient to generate CSE statistics.
18576281
Semiclassical theory of ballistic transport through chaotic cavities with spin-orbit interaction
We investigate the influence of spin-orbit interaction on ballistic transport through chaotic cavities by using semiclassical methods. Our approach is based on the Landauer formalism [IBM J. Res. Dev. 1, 223 (1957); 32, 306 (1988)] and the Fisher-Lee relations [Phys. Rev. B 23, 6851 (1981)], appropriately generalized to spin-orbit interaction, and a semiclassical representation of Green's functions. We calculate conductance coefficients by exploiting ergodicity and mixing of suitably combined classical spin-orbit dynamics, and making use of the Sieber-Richter method [Phys. Scr., T T90, 128 (2001); Phys. Rev. Lett. 89, 206801 (2002)] and its most recent extensions. That way, we obtain weak antilocalization and confirm previous results obtained in the symplectic ensemble of random matrix theory.
5962547
Rashba-effect-induced localization in quantum networks.
We study a quantum network extending in one dimension (chain of square loops connected at one vertex) made up of quantum wires with Rashba spin-orbit coupling. We show that the Rashba effect may give rise to an electron localization phenomenon similar to the one induced by magnetic field. This localization effect can be attributed to the spin precession due to the Rashba effect. We present results both for the spectral properties of the infinite chain and for linear transport through a finite-size chain connected to leads. Furthermore, we study the effect of disorder on the transport properties of this network.
39548001
Rashba effect in quantum networks
We present a formalism to study quantum networks made up by single-channel quantum wires in the presence of Rashba spin-orbit coupling and magnetic field. In particular, linear transport through one-dimensional and two-dimensional finite-size networks is studied by means of the scattering formalism. In some particular quantum networks, the action of the magnetic field or of the Rashba spin-orbit coupling induces localization of the electron wave function. This phenomenon, which relies on both the quantum-mechanical interference and the geometry of the network, is manifested through the suppression of the conductance for specific values of the spin-orbit-coupling strength or of the magnetic field. Furthermore, the interplay of the Aharonov-Bohm phases and of the non-Abelian phases introduced by spin-orbit coupling, is discussed in a number of cases.
119670516
Localization effects in a periodic quantum graph with magnetic field and spin-orbit interaction
A general technique for the study of magnetic Rashba Hamiltonians in quantum graphs is presented. We use this technique to show how manipulating the magnetic and spin parameters can be used to create localized states in a certain periodic graph (T3 lattice).
120207129
The free Dirac operator on compact and noncompact graphs
The free Dirac operator defined on composite one‐dimensional structures consisting of finitely many half‐lines and intervals is investigated. The influence of the connection points between the constituents is modeled by transition conditions for the wave functions or equivalently by different self‐adjoint extensions of the Dirac operator. General relations between the parameters of the extensions and the eigenvalues resp. the scattering coefficients are derived and then applied to the cases of a bundle of half‐lines, a point defect, a branching line, and an eye‐shaped structure.
13963974
Spectral statistics for the Dirac operator on graphs
We determine conditions for the quantization of graphs using the Dirac operator for both two- and four-component spinors. According to the Bohigas–Giannoni–Schmit conjecture for such systems with time-reversal symmetry the energy level statistics are expected, in the semiclassical limit, to correspond to those of random matrices from the Gaussian symplectic ensemble. This is confirmed by numerical investigation. The scattering matrix used to formulate the quantization condition is found to be independent of the type of spinor. We derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation.
17596780
Kirchhoff's rule for quantum wires
We formulate and discuss one-particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with n channels. The corresponding on-shell S-matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy E>0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low-energy behaviour of one theory gives the high-energy behaviour of the transformed theory. Finally, we provide a composition rule by which the on-shell S-matrix of a graph is factorizable in terms of the S-matrices of its subgraphs. All proofs use only known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitian symplectic forms.
15747589
First Order Approach and Index Theorems for Discrete and Metric Graphs
The aim of the present paper is to introduce a unified notion of Laplacians on discrete and metric graphs. In order to cover all self-adjoint vertex conditions for the associated metric graph Laplacian, we develop systematically a new type of discrete graph operators acting on a decorated graph. The decoration at each vertex of degree d is given by a subspace of $${\mathbb{C}^d}$$, generalising the fact that a function on the standard vertex space has only a scalar value. We illustrate the abstract concept by giving classical examples throughout the article. Our approach includes infinite graphs as well. We develop the notion of exterior derivative, differential forms, Dirac and Laplace operators in the discrete and metric case, using a supersymmetric framework. We calculate the (supersymmetric) index of the discrete Dirac operator generalising the standard index formula involving the Euler characteristic of a graph. Finally, we show that for finite graphs, the corresponding index for the metric Dirac operator agrees with the discrete one.
17085199
First-order operators and boundary triples
The aim of the present paper is to introduce a first-order approach to the abstract concept of boundary triples for Laplace operators. Our main application is the Laplace operator on a manifold with boundary; a case in which the ordinary concept of boundary triples does not apply directly. In our first-order approach, we show that we can use the usual boundary operators in abstract Green’s formula as well. Another motivation for the first-order approach is to give an intrinsic definition of the Dirichlet-to-Neumann map and intrinsic norms on the corresponding boundary spaces. We also show how the first-order boundary triples can be used to define a usual boundary triple leading to a Dirac operator.
123018066
The Dirac Equation
Ever since its invention in 1929 the Dirac equation has played a fundamental role in various areas of modern physics and mathematics. Its applications are so widespread that a description of all aspects cannot be done with sufficient depth within a single volume. In this book the emphasis is on the role of the Dirac equation in the relativistic quantum mechanics of spin-1/2 particles. We cover the range from the description of a single free particle to the external field problem in quantum electrodynamics. Relativistic quantum mechanics is the historical origin of the Dirac equation and has become a fixed part of the education of theoretical physicists. There are some famous textbooks covering this area. Since the appearance of these standard texts many books (both physical and mathematical) on the non- relativistic Schrodinger equation have been published, but only very few on the Dirac equation. I wrote this book because I felt that a modern, comprehensive presentation of Dirac's electron theory satisfying some basic requirements of mathematical rigor was still missing.
119849268
Quantum signatures of chaos
The distinction between level clustering and level repulsion is one of the quantum analogues of the classical distinction between globally regular and predominantly chaotic motion (see Figs. 1, 2, 3). In order to reveal level repulsion under conditions of global classical chaos special care may be necessary: (i) subspectra referring to different values of the quantum numbers related to symmetries must be dealt with separately and (ii) for systems with quantum localization only levels whose wavefunctions have overlapping support must be admitted. A “level” may either be an energy eigenvalue E in the case of autonomous systems or, for periodically driven systems, a quasi-energy φ, i.e. an eigenphase of the unitary Floquet operator transporting the wavevector from period to period.
17510999
Periodic Orbit Theory and Spectral Statistics for Quantum Graphs
Abstract We quantize graphs (networks) which consist of a finite number of bonds and vertices. We show that the spectral statistics of fully connected graphs is well reproduced by random matrix theory. We also define a classical phase space for the graphs, where the dynamics is mixing and the periodic orbits proliferate exponentially. An exact trace formula for the quantum spectrum is developed in terms of the same periodic orbits, and it is used to investigate the origin of the connection between random matrix theory and the underlying chaotic classical dynamics. Being an exact theory, and due to its relative simplicity, it offers new insights into this problem which is at the forefront of the research in quantum chaos and related fields.
7903735
Spin filtering on a ring with the Rashba Hamiltonian
We consider a quantum graph consisting of a ring with the Rashba Hamiltonian and an arbitrary number of semi-infinite wires attached. We describe the scattering matrix for this system and investigate spin filtering for a three-terminal device.
121385423
Quantum Chaos on Graphs
We quantize graphs (networks) which consist of a finite number of bonds and nodes. We show that their spectral statistics is well reproduced by random matrix theory. We also define a classical phase space for the graph, where the dynamics is mixing and the periodic orbits (loops on the graph) proliferate exponentially. An exact trace formula for the quantum spectrum is developed and used to investigate the origin of the connection between random matrix theory and the underlying chaotic classical dynamics. Being an exact theory, and due to its relative simplicity, it offers new insights into this problem which is at the forefront of the research in quantum chaos and related fields. [S0031-9007(97)04707-8]
15039930
A semiclassical approach to theDirac equation
We derive a semiclassical time evolution kernel and a trace formula for the Dirac equation. The classical trajectories that enter the expressions are determined by the dynamics of relativistic point particles. We carefully investigate the transport of the spin degrees of freedom along the trajectories which can be understood geometrically as parallel transport in a vector bundle with SU(2) holonomy. Furthermore, we give an interpretation in terms of a classical spin vector that is transported along the trajectories and whose dynamics, dictated by the equation of Thomas precession, gives rise to dynamical and geometric phases every orbit is weighted by. We also present an analogous approach to the Pauli equation which we analyse in two diierent limits.
118109042
LETTER TO THE EDITOR: The semiclassical origin of the logarithmic singularity in the symplectic form factor
Sieber and Richter achieved a breakthrough towards a proof of the universality of spectral fluctuations of chaotic quantum systems conjectured by Bohigas, Giannoni and Schmidt by calculating semiclassically the first term beyond the diagonal approximation of the orthogonal form factor. In this letter, the semiclassical origin of the logarithmic singularity of the symplectic form factor is deduced perturbatively by combining this result with the contribution that arises due to the spin. This approach stands in contrast to the duality approach introduced by Bogomolny and Keating, which is essentially non-perturbative, and where the structure around the Heisenberg time is related to the structure for very small time which can be deduced using the diagonal approximation.
119766827
Periodic orbits and a correlation function for the semiclassical density of states
A principle of uniform density of periodic orbits in the phase space of a Hamiltonian system with bound classical motion is proposed and used to obtain information about the semiclassical quantum eigenvalue spectrum. It supplies a more refined statistic than the 'one state per Planck cell' rule for the average semiclassical density of states, namely the limiting behaviour of a certain correlation function of the density of states. Unlike the average, this correlation shows markedly different behaviour for systems with integrable and ergodic classical motion.
121529466
Correlations between periodic orbits and their rôle in spectral statistics
We consider off-diagonal contributions to double sums over periodic orbits that arise in semiclassical approximations for spectral statistics of classically chaotic quantum systems. We identify pairs of periodic orbits whose actions are strongly correlated. For a class of systems with uniformly hyperbolic dynamics, we demonstrate that these pairs of orbits give rise to a τ2 contribution to the spectral form factor K(τ) which agrees with random matrix theory. Most interestingly, this contribution has its origin in a next-to-leading-order behaviour of a classical distribution function for long times.
16127665
Leading off-diagonal correction to the form factor of large graphs.
Using periodic-orbit theory beyond the diagonal approximation we investigate the form factor, K(tau), of a generic quantum graph with mixing classical dynamics and time-reversal symmetry. We calculate the contribution from pairs of self-intersecting orbits that differ from each other only in the orientation of a single loop. In the limit of large graphs, these pairs produce a contribution -2tau(2) to the form factor which agrees with random-matrix theory.
5620122
Form factor for a family of quantum graphs: an expansion to third order
For certain types of quantum graphs we show that the random matrix form factor can be recovered to at least third order in the scaled time τ from periodic-orbit theory. We consider the contributions from pairs of periodic orbits represented by diagrams with up to two self-intersections connected by up to four arcs and explain why all other diagrams are expected to give higher-order corrections only. For a large family of graphs with ergodic classical dynamics the diagrams that exist in the absence of time-reversal symmetry sum to zero. The mechanism for this cancellation is rather general which suggests that it also applies at higher orders in the expansion. This expectation is in full agreement with the fact that in this case the linear-τ contribution, the diagonal approximation, already reproduces the random matrix form factor for τ < 1. For systems with time-reversal symmetry there are more diagrams which contribute at third order. We sum these contributions for quantum graphs with uniformly hyperbolic dynamics, obtaining +2τ3, in agreement with random matrix theory. As in the previous calculation of the leading-order correction to the diagonal approximation we find that the third-order contribution can be attributed to exceptional orbits representing the intersection of diagram classes.
29826589
Periodic-orbit theory of universality in quantum chaos.
We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory, that full classical chaos is paralleled by quantum energy spectra with universal spectral statistics, in agreement with random-matrix theory. For dynamics from all three Wigner-Dyson symmetry classes, we calculate the small-time spectral form factor K(tau) as power series in the time tau. Each term tau(n) of that series is provided by specific families of pairs of periodic orbits. The contributing pairs are classified in terms of close self-encounters in phase space. The frequency of occurrence of self-encounters is calculated by invoking ergodicity. Combinatorial rules for building pairs involve nontrivial properties of permutations. We show our series to be equivalent to perturbative implementations of the nonlinear sigma models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of orbit pairs have a one-to-one relationship with Feynman diagrams known from the sigma model.
4496197
Spectral correlations of individual quantum graphs.
We investigate the spectral properties of chaotic quantum graphs. We demonstrate that the energy-average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric nonlinear -model action. This proves that spectral correlations of individual quantum graphs behave according to the predictions of Wigner-Dyson random matrix theory. We explore the stability of the universal random matrix behavior with regard to perturbations, and discuss the crossover between different types of symmetries.