Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures

@article{Altland1997NonstandardSC,
  title={Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures},
  author={Alexander Altland and Martin R. Zirnbauer},
  journal={Physical Review B},
  year={1997},
  volume={55},
  pages={1142-1161}
}
Normal-conducting mesoscopic systems in contact with a superconductor are classified by the symmetry operations of time reversal and rotation of the electron’s spin. Four symmetry classes are identified, which correspond to Cartan’s symmetric spaces of type C, CI, D, and DIII. A detailed study is made of the systems where the phase shift due to Andreev reflection averages to zero along a typical semiclassical single-electron trajectory. Such systems are particularly interesting because they do… 

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References

Differential Geometry, Lie Groups, and Symmetric Spaces
Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric