Spectral statistics for the Dirac operator on graphs

Bölte, Jens; Harrison, J. M.

doi:10.1088/0305-4470/36/11/307

Cited by 68 publications

(90 citation statements)

References 21 publications

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“…The boundary conditions which define all self-adjoint extensions of the bulk Hamiltonian iγ t γ x ∂ x are [62], [63] …”

Section: Preliminariesmentioning

confidence: 99%

Non-equilibrium steady states of quantum systems on star graphs

Mintchev¹

2011

J. Phys. A: Math. Theor.

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Non-equilibrium steady states of quantum fields on star graphs are explicitly constructed. These states are parametrized by the temperature and the chemical potential, associated with each edge of the graph. Time reversal invariance is spontaneously broken. We study in this general framework the transport properties of the Schrödinger and the Dirac systems on a star graph, modeling a quantum wire junction. The interaction, which drives the system away from equilibrium, is localized in the vertex of the graph. All point-like vertex interactions, giving rise to self-adjoint Hamiltonians possibly involving the minimal coupling to a static electromagnetic field in the ambient space, are considered. In this context we compute the exact electric steady current and the nonequilibrium charge density. We investigate also the heat transport and derive the Casimir energy density away from equilibrium. The appearance of Friedel type oscillations of the charge and energy densities along the edges of the graph is established. We focus finally on the noise power and discuss the non-trivial impact of the point-like interactions on the noise.

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“…The boundary conditions which define all self-adjoint extensions of the bulk Hamiltonian iγ t γ x ∂ x are [62], [63] …”

Section: Preliminariesmentioning

confidence: 99%

Non-equilibrium steady states of quantum systems on star graphs

Mintchev¹

2011

J. Phys. A: Math. Theor.

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“…The semiclassical theory of spinning particles is discussed in [33]; off-diagonal terms of the form factor were considered in a preliminary version in [21], and for quantum graphs in [22].…”

Section: Spinning Particles and The Symplectic Symmetry Classmentioning

confidence: 99%

Periodic-orbit theory of universality in quantum chaos

et al. 2005

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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(τ ) as power series in the time τ . Each term τ 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 non-trivial properties of permutations. We show our series to be equivalent to perturbative implementations of the non-linear sigma models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of orbit pairs are one-to-one with Feynman diagrams known from the sigma model.

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“…The trace of S n may be expanded as a sum over the set P n of periodic orbits of length n, 5) where the periodic orbit p consists of a series of transitions (b 1 , b 2 , . .…”

Section: The Form Factormentioning

confidence: 99%