Following an idea by Joyner et al. [Europhys. Lett. 107, 50004 (2014)], a microwave graph with an antiunitary symmetry T obeying T^{2}=-1 is realized. The Kramers doublets expected for such systems are clearly identified and can be lifted by a perturbation which breaks the antiunitary symmetry. The observed spectral level spacings distribution of the Kramers doublets is in agreement with the predictions from the Gaussian symplectic ensemble expected for chaotic systems with such a symmetry.
We investigate spectral quantities of quantum graphs by expanding them as sums over pseudo orbits, sets of periodic orbits. Only a finite collection of pseudo orbits which are irreducible and where the total number of bonds is less than or equal to the number of bonds of the graph appear, analogous to a cut off at half the Heisenberg time. The calculation simplifies previous approaches to pseudo orbit expansions on graphs. We formulate coefficients of the characteristic polynomial and derive a secular equation in terms of the irreducible pseudo orbits. From the secular equation, whose roots provide the graph spectrum, the zeta function is derived using the argument principle. The spectral zeta function enables quantities, such as the spectral determinant and vacuum energy, to be obtained directly as finite expansions over the set of short irreducible pseudo orbits.
Energy levels statistics following the Gaussian Symplectic Ensemble (GSE) of Random Matrix Theory have been predicted theoretically and observed numerically in numerous quantum chaotic systems. However in all these systems there has been one unifying feature: the combination of halfinteger spin and time-reversal invariance. Here we provide an alternative mechanism for obtaining GSE statistics that is based on geometric symmetries of a quantum system which alleviates the need for spin. As an example, we construct a quantum graph with a particular discrete symmetry given by the quaternion group Q8. GSE statistics is then observed within one of its subspectra.In the 1950s and 1960s Wigner and Dyson pioneered the use of random matrices in modelling the statistical properties of the energy eigenvalues belonging to complicated quantum systems [1,2]. The techniques they developed spawned a new field of mathematics which has since become known as Random Matrix Theory (RMT) and its application has spread far and wide to many areas of Mathematics and Physics [3]. In particular it was later conjectured [4] that the high-lying quantum energy levels of classically chaotic systems are faithful to random matrix averages.One of the cornerstones of RMT is Dyson's three-fold way [2], which groups quantum systems without geometric symmetries into three distinct types. The first occurs if time-reversal invariance is broken, for example by a magnetic field, meaning the quantum Hamiltonian H is inherently complex. The remaining two appear if there is an antiunitary time-reversal operator T which leaves H invariant, i.e. [T , H] = 0. They are then distinguished by either T 2 = 1 or T 2 = −1, in which case H is real symmetric or quaternion-real respectively. For chaotic systems, RMT makes predictions in all three instances by averaging over an ensemble of Hermitian matrices with the appropriate internal structure and Gaussian weighted elements. These are referred to as the Gaussian Unitary, Orthogonal and Symplectic ensembles (GUE, GOE and GSE). We note that the number of symmetry classes can be extended to ten if additional anti-commuting symmetries are present [5,6] but this is beyond the scope of this letter.In systems without geometrical symmetries timereversal invariance with T 2 = −1, and hence GSE statistics, can only arise if the wavefunctions have an even number of components, commonly associated with halfinteger spin. For such systems GSE statistics have been predicted and/or observed numerically in examples such as quantum billiards [7], maps [8] and quantum graphs [9], and explained using periodic-orbit theory [10,11]. However to date there has been no experimental observation.For systems with geometric symmetries the situation becomes more involved. Here the Hilbert space decomposes into subspaces invariant under symmetry transformations, and the spectral statistics inside these subspaces depends both on the system's behaviour under time reversal and on the nature of the subspace. For example 3-fold rotationally invariant...
Abstract. We investigate the eigenvalue statistics of random Bernoulli matrices, where the matrix elements are chosen independently from a binary set with equal probability. This is achieved by initiating a discrete random walk process over the space of matrices and analysing the induced random motion of the eigenvalues -an approach which is similar to Dyson's Brownian motion model but with important modifications. In particular, we show our process is described by a Fokker-Planck equation, up to an error margin which vanishes in the limit of large matrix dimension. The stationary solution of which corresponds to the joint probability density function of certain well-known fixed trace Gaussian ensembles.
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