Universal spectral statistics of Andreev billiards: Semiclassical approach

Gnutzmann, Sven; Seif, Burkhard; Oppen, Felix von; Zirnbauer, Martin R.

doi:10.1103/physreve.67.046225

Cited by 14 publications

(30 citation statements)

References 20 publications

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“…Thus, while the even repetitions contribute with the wrong sign in Eq. (14), these contributions can in principle be balanced (in a class-C system) by correct-sign contributions from other primitive orbits.…”

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confidence: 99%

“…(10) to class C has been considered by Gnutzmann et al [14]. As a prototypical class-C system, they studied a Fermi sea of electrons (with the Fermi surface at E = 0) in a hard-wall billiard in a strong magnetic field (to break time-reversal invariance).…”

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confidence: 99%

“…There is an extra phase factor of −i for each Andreev reflection, in addition to the Maslov phase. In general, the action of a primitive periodic orbit takes the form [14] …”

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confidence: 99%

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Nonclassical Degrees of Freedom in the Riemann Hamiltonian

Srednicki

2011

Phys. Rev. Lett.

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The Hilbert-Polya conjecture states that the imaginary parts of the zeros of the Riemann zeta function are eigenvalues of a quantum hamiltonian. If so, conjectures by Katz and Sarnak put this hamiltonian in Altland and Zirnbauer's universality class C. This implies that the system must have a nonclassical two-valued degree of freedom. In such a system, the dominant primitive periodic orbits contribute to the density of states with a phase factor of -1. This resolves a previously mysterious sign problem with the oscillatory contributions to the density of the Riemann zeros.Comment: 4 pages, no figures; v3-6 have minor corrections to v2, v2 has a more complete solution of the sign problem than v

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Nonclassical Degrees of Freedom in the Riemann Hamiltonian

Srednicki

2011

Phys. Rev. Lett.

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“…The negative spectrum describes hole-like quasiparticle excitations (while the spectrum is negative, the physical energy of such an excitation is positive, of course) and the positive spectrum describes electron-like quasiparticle excitations. It has been known for some time that Andreev reflections reduce the density of states near the Fermi energy and various universality classes have been defined [2,3,4,5,6,7,8,9,10] and related to quantum chaos. Spectral gaps have been found in irregularly shaped Andreev billiards for which the classical dynamics of the normal billiard (with specular reflection at the superconducting interface) is chaotic.…”

Section: Introduction and Physical Backgroundmentioning

confidence: 99%

On the spectral gap in Andreev graphs

Flechsig

Gnutzmann²

2008

Analysis on Graphs and Its Applications

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Abstract. We introduce Andreev scattering (electron-hole conversion at an interface of a normal conductor to a superconductor) at the outer vertices of a quantum star graph and examine its effect on the spectrum. More specifically we show that the density of states in Andreev graphs is suppressed near the Fermi energy where a spectral gap may occur. The size and existence of such a gap depends on one side on the Andreev scattering amplitudes and, on the other side,on the properties of the electron-electron scattering matrix at the central vertex. We also show that the bond length fluctuations have a minor effect on the spectrum near the Fermi energy.

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“…The classical dynamics of the combined SN-system become entirely regular, even and in particular when the normal conducting cavity would feature hard chaos. 18 Unlike chaotic or integrable systems, periodic orbits are no longer isolated but form continuous manifolds that dominate the classical phase space and, in turn, the density of states (DOS). 19,20,21 The BS approach 15,16,22 to the DOS of an Andreev billiard relies on three assumptions: exact retracing of electron-hole trajectories (referred to as assumption A1 in the following), the absence of any quasi-periodic orbits other than the ones caused by Andreev reflection (assumption A2), and the applicability of semiclassical approximations (assumption A3), i.e.…”

Section: Introductionmentioning

confidence: 99%

Non-retracing orbits in Andreev billiards

2006

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The validity of the retracing approximation in the semiclassical quantization of Andreev billiards is investigated. The exact energy spectrum and the eigenstates of normal-conducting, ballistic quantum dots in contact with a superconductor are calculated by solving the Bogoliubov-de Gennes equation and compared with the semiclassical Bohr-Sommerfeld quantization for periodic orbits which result from Andreev reflections. We find deviations that are due to the assumption of exact retracing electron-hole orbits rather than the semiclassical approximation, as a concurrently performed Einstein-Brillouin-Keller quantization demonstrates. We identify three different mechanisms producing non-retracing orbits which are directly identified through differences between electron and hole wave functions.

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Universal spectral statistics of Andreev billiards: Semiclassical approach

Cited by 14 publications

References 20 publications

Nonclassical Degrees of Freedom in the Riemann Hamiltonian

Nonclassical Degrees of Freedom in the Riemann Hamiltonian

On the spectral gap in Andreev graphs

Non-retracing orbits in Andreev billiards

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