Quantum graphs: Applications to quantum chaos and universal spectral statistics

Abstract

During the last few years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wavefunction 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. In particular, we summarize 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.

Acknowledgments

We are grateful to Alexander Altland, Gregory Berkolaiko, Peter Braun, Fritz Haake, Jonathan Keating, Tsampikos Kottos, Marek Kuś, Felix von Oppen, Holger Schanz, Michail Solomyak, and Karol Życzkowski for discussions and help through the years. Both authors would like to thank the School of Mathematics at Bristol University for the hospitality extended during the period this manuscript was prepared. Support is acknowledged from the Minerva foundation, the Einstein (Minerva) Center and The Minerva Center for non-linear physics (Weizmann Institute), the Israel Science Foundation, the German-Israeli Foundation, the EPSRC grant GR/T06872/01 and the Institute of Advance Studies, Bristol University.

Notes

¶Current address: School of Mathematical Sciences, University of Nottingham, United Kingdom.

†This limit has to take into account that for k = 0 the general solution of the one-dimensional Schrödinger equation (without magnetic field) is a linear function a+bx. The rest is straightforward. In the following we will always imply a full knowledge of the quantum map when referring to the condition (41) such that the oddities at k = 0 can always be removed by the correct limit. We will only come back to this point if it has some non-trivial consequences.

†In general unistochastic matrices are a subset of bistochastic matrices. A unistochastic matrix defines a Markov process which, can be quantized (though not uniquely) 169–171.

†Star graphs are an example of bipartite graphs for which the directed bonds can be split into two non-empty sets and the transition probabilities within each of the two sets vanish. For star graphs due to its topology there is no transition from any outgoing (incoming) bond to another outgoing (incoming) bond. One may easily construct other examples which rely on the dynamical connectivity as well. Dynamically connected star graphs are generally mixing if the classical dynamics is defined on bonds instead of directed bonds as in sections 8 and 9.

†The definition of reducible and irreducible periodic orbits depends on the code which is used to define a periodic orbit unambiguously. For a simple graph one may use the the sequence of vertices as a code (symbolic dynamics) which uniquely defines a periodic orbit. In the directed bond code the same orbit is given by . Irreducible periodic orbits with respect to the vertex code ( vertex-irreducible periodic orbits ), do not intersect at any vertex such that all i_j are different. Our definition which is based on the directed bond code is not equivalent to the one based on the vertex code. In general, there are bond-irreducible orbits of period n>V and the latter cannot be vertex-irreducible. Both codes have their advantages and disadvantages. The vertex code has the advantage of a smaller number of symbols and a grammar which is readily encoded in the connectivity matrix. In the present context the grammar is less relevant since a code which does not correspond to a periodic orbit has zero weight. The advantage of using the bond code to define reducible orbits, is that weights of reducible orbits are products of weights of non-irreducible orbits.

‡In the non-mixing case a dynamically connected graph is ergodic. With an additional time average over an interval Δ n ≪ n one then gets the sum rule

†One may prove that each finite product over factors 1 − t_p equals 1 − t_a  − t_b +R with if all contributions from periodic orbits that are composed of less than n irreducible periodic orbits appear in the finite product. It is a signature of the bad convergence of the infinite product that the modulus of the remainder |R| for real k in generally does not go to zero in the limit of large n.

‡The final result can also be obtained with Newton's formulas which express the coefficients a_n of a secular determinant through the first N traces tr Aⁿ in a recursive way.

†The k-average destroys any dependence on the value φ which can thus be set to φ = 0 without loss of generality.

†If not all bond lengths of the quantum graphs are rationally independent the number of variables has to be reduced to some number smaller than B. The generalization of the following argument is straightforward. However, we will keep a notation that implies incommensurability of all bond lengths.

†This implies a shift without changing the time difference between the last intersection. Thus, .

†Transpositions of an element with its neighbour (formally n ₁=-1) or next-neighbour (n ₁=0) leads to diagrams which look a little bit different but have the same value.

†In the literature on Andreev billiards it has become a convention to call a system chaotic or integrable according to the dynamics of the normal system where electrons and holes are not coupled. We will always refer to the combined electron–hole dynamics in presence of the superconductor.

‡Another option to restore chaos is to introduce many point scatterers (disorder). Both symmetry classes can also be realized in a completely different context like two coupled spins where chaos in the semiclassical limit (large spins) is not prevented by Andreev reflections.

†At first sight this does not seem to be in accordance with the Bogoliubov–de Gennes equation. Indeed an electron (a hole) propagating along a bond with energy E would acquire a phase () according to the Bogoliubov–de Gennes equation where electron (hole) momentum is () – the sign in the phase of the hole is different since it propagates in opposite direction to its momentum. Since we are interested in the limit E_F ≫E one may expand the momenta . Adding the hole and electron phases (k_e -k_h )L_b the leading part cancels. Keeping fixed in the limit E_F →∞ only 2 k L_b remains.

†One should be aware that there are different conventions for the definition of a supertrace which might differ by an overall sign and consequently for the superdeterminant which may be defined via sdet A= ^estr ln A. There are also different conventions for the integration over anticommuting numbers which differ by an overall constant.

†Note that for s≠ 0 or j _±≠ 0 the second saddle-point equation is changed which leaves only two saddle points in the reduced action (260). These can serve as starting points for perturbative treatments of the correlator. The loop expansion in the diagrammatic periodic-orbit treatment of the form-factor in section 8 is essentially equivalent to an expansion around the saddle-point . Keeping the full saddle-point manifold of S ₀ we are able to go beyond such perturbative expansions.

†For the integral over anticommuting numbers this can easily seen from the action (269) where the shifts decouple the anticommuting parts of Z _ℓ from the anticommuting part of . For the commuting entries one can write down the integral over all real and imaginary parts of the entries and – shifting the entries and changing the contours of integration leads to a similar decoupling such that the action at s=j _±=0 is of the form where and and the terms at s≠0 and j _±≠ 0 only involve Z _{ℓ even}.

Additional information

Notes on contributors

Sven Gnutzmann∥

¶Current address: School of Mathematical Sciences, University of Nottingham, United Kingdom.