Universal spectral statistics in Wigner-Dyson, chiral, and Andreev star graphs. I. Construction and numerical results

Gnutzmann, Sven; Seif, Burkhard

doi:10.1103/physreve.69.056219

Cited by 18 publications

(18 citation statements)

References 36 publications

(54 reference statements)

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“…In this paper, however, we consider Fermi energies away from zero, where the normal and the chiral universality classes lead to the same results. 29 Section III on spectral statistics and Sec. IV on transport can both be read independently.…”

Section: B Antiunitary Symmetries Of the Effective Theorymentioning

confidence: 99%

Edge effects in graphene nanostructures: Semiclassical theory of spectral fluctuations and quantum transport

2011

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We investigate the effect of different edge types on the statistical properties of both the energy spectrum of closed graphene billiards and the conductance of open graphene cavities in the semiclassical limit. To this end, we use the semiclassical Green's function for ballistic graphene flakes [seeJ. Wurm, K. Richter, anḋ I. Adagideli, Phys. Rev. B 84, 075468 (2011)]. First, we study the spectral two-point correlation function or, more precisely, its Fourier transform the spectral form factor, starting from the graphene version of Gutzwiller's trace formula for the oscillating part of the density of states. We calculate the two leading-order contributions to the spectral form factor, paying particular attention to the influence of the edge characteristics of the system. Then, we consider transport properties of open graphene cavities. We derive generic analytical expressions for the classical conductance, the weak localization correction, the size of the universal conductance fluctuations, and the shot-noise power of a ballistic graphene cavity. Again, we focus on the effects of the edge structure. For both the conductance and the spectral form factor, we find that edge-induced pseudospin interference affects the results significantly. In particular, intervalley coupling mediated through scattering from armchair edges is the key mechanism that governs the coherent quantum interference effects in ballistic graphene cavities.

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Section: B Antiunitary Symmetries Of the Effective Theorymentioning

confidence: 99%

Edge effects in graphene nanostructures: Semiclassical theory of spectral fluctuations and quantum transport

2011

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“…The DFT star graph is known to show universal spectral statistics [125] -the classical map is simply M DFT⋆ bb ′ = 1/B for which all eigenvalues apart from ν 1 = 1 vanish exactly. Thus the gap condition is fulfilled maximally.…”

Section: The Diagonal Approximationmentioning

confidence: 99%

“…Note also, that the secular equations for a graph with scattering matrix S ⋆ = S 0 D and a graph with the scattering matrix S ′ ⋆ = D 1/2 S 0 D 1/2 are completely equivalent. Time reversal invariant star graphs of symmetry class AII can be constructed by adding a spin freedom to the wave function [125]. This will be implemented in Section 8.4.…”

Section: Spectral Form Factor For Star Graphsmentioning

confidence: 99%

“…Here, we shall introduce spin degrees of freedom which are only effected by the scattering at the vertices. As in the previous section we shall discuss only star graphs [125,126]. The spin degree of freedom is added to a graph by considering wave functions ψ bσ (x b ) on the bond b which have a spin up σ = 1/2 and a spin down σ = −1/2 component.…”

Section: Graphs With Spin and Their Spectral Statisticsmentioning

confidence: 99%

“…Following these pioneering beginnings of Kottos and Smilansky quantum graphs have become a new paradigm of quantum chaos and have been applied to various problems, including also disorder and diffusion [251,76]. All aspects of quantum chaos have been covered: spectral statistics in finite [25,38,44,45,39,41,42,58,59,162,194,195,213,215,220,234,235,236,125,126,127,128] and infinite periodic structures [85], localisation and wavefunction statistics [37,146,214,124,149,216,43,46,251], chaotic scattering [161,163], transport through chaotic devices [217,74], resonances and decay in open chaotic 1. Introduction 3 systems [203,164,165].…”

Section: Introductionmentioning

confidence: 99%

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Quantum graphs: Applications to quantum chaos and universal spectral statistics

Gnutzmann

Smilansky

2006

Advances in Physics

313

450

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During the last years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wave function 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. Especially, we summarise 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. Chapter 1 IntroductionThe general mathematical concept of a graph (network) as a set of elements which are connected by some relation has found applications in many branches of science, engineering, and also social science. A street network of a traffic engineer, the network of neurons studied by a neuroscientist, the structure of databases in computer science can all be described by graphs.Recently the Laplacian on a metric graph has gained a lot of attention in physics and mathematics in terms of the diffusion equation or Schrödinger equation. They have now become known as quantum graphs but different aspects are studied under various names such as quantum networks or quantum wires. They have a long history in mathematics and physics.In physics, the first application has probably been in the context of free electron models for organic molecules about seventy years ago by Pauling [198], an approach which has been further developed in subsequent years [172,173,201,209,77,182,204]. Quantum graphs have also been applied successfully to superconductivity in granular and artificial materials [12], acoustic and electromagnetic waveguide networks [112,181], the Anderson transition in a disordered wire [18,222], quantum Hall systems [71,151,150], fracton excitations in fractal structures [22,190], and mesoscopic quantum systems [142,166,242,243,241]. Quantum graphs have also been simulated experimentally [141].The construction of self-adjoint operators, or wave equations with appropriate boundary conditions on graphs has first been addressed by Ruedenberg and Scherr [209] (see also [204]). They considered graphs as an idealisation of networks of wires or wave guides of finite cross-section in the limit where the diameter of the wire is much smaller than any other length scale. Similar approaches to graphs as networks of thin wires with a finite diameter, or fat quantum graphs as they are now called, have been a topic in mathematical physics recently [211,212,208,167,106,202].Another interesting aproach which has been discussed mainly by Exner and his coworkers is based on leaky graphs [101,100,102,103,105,104 , 108, 109]. Here, a finite attractive potential in the Schrödinger equation is centered on a metric graph. A leaky g...

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