GSE statistics without spin

Abstract

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...

Spectra and spectral correlations of microwave graphs with symplectic symmetry

“…The scattering matrix S relates the incoming amplitudes a ij with the outgoing ones b ij via b = Sa (5) arXiv:1708.06236v1 [quant-ph] 16 Aug 2017 (Color online) (a) Sketch of the graph proposed in Ref. 13 to study GSE statistics without spin. The four arrows denote bonds along which additional phases are acquired.…”

“…The solutions of the determinant condition (9) generate the spectrum of the graph. To realize graphs with GSE symmetry Joyner et al [13] proposed the graph shown in Fig. 1.…”

Microwave Realization of the Gaussian Symplectic Ensemble

“…1(top) was proposed in Ref. [21]. It contains two geometrically identical subgraphs but with phase shifts by +π/2 and −π/2, respectively, along two corresponding bonds.…”

Microwave Studies of Three Chiral Ensembles in Chains of Coupled Dielectric Resonators

“…where a variance of one for a has been assumed. With (23) the distribution of the non-trivial eigenvalues is given by a χ distribution with f = 2 degrees of freedom. For the chiral unitary case, a and b are complex Gaussian variables and the distribution has f = 4 degrees of freedom.…”

“…3.2 of [20]). For the Gaussian unitary ensemble the number of realizations is still small, the first of them being performed in microwave billiard systems [21,22] and later in microwave networks [23]. The Gaussian symplectic ensemble usually is associated with spin 1/2 systems, but Joyner et al [24] proposed an alternative way to realize the necessary condition T 2 = −1 in a graph with a peculiarly designed geometry.…”