Abstract
In Chap. 1, we briefly discussed how symmetries and phases of matter are intertwined. In fact under Landau–Ginzburg–Wilson paradigm, classification of various phases are understood in terms of broken symmetries. In the last couple of decades, however, another set of symmetries have started playing most important role-time reversal, charge conjugation and sublattice symmetry. Why are there only three intrinsic symmetries? Moreover, these three symmetries seem qualitatively distinct, how is it that they can exhaust the complete classification? Why is this a “ten” fold way, and not more? How does one set up Hamiltonians in generic settings, say on a polymer? How does one then think about these symmetries physically? This chapter answers all these questions.
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Notes
- 1.
This new basis is not unique. In fact, existence of such a basis implies that any other basis related by a real orthogonal matrix will also be an equally valid one.
References
Altland A, Zirnbauer MR (1997) Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys Rev B 55:1142–1161
Heinzner P, Huckleberry A, Zirnbauer M (2005) Symmetry classes of disordered fermions. Commun Math Phys 257(3):725–771
Zirnbauer MR (2010) Symmetry classes. ArXiv e-prints, arXiv:1001:0722
Dyson FJ (1962) The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics. J Math Phys 3(6):1199–1215
Sakurai JJ, Tuan S-F, Commins ED (1995) Modern quantum mechanics, revised edn
Wigner EP (1959) Group theory and its application to the quantum mechanics of atomic spectra. Academic, New York
Parthasarathy KR (1969) Projective unitary antiunitary representations of locally compact groups. Commun Math Phys 15(4):305–328
Ludwig AWW (2016) Topological phases: classification of topological insulators and superconductors of non-interacting fermions, and beyond. Phys Scripta 2016(T168):014001. arXiv:1512.08882
Dreiner HK, Haber HE, Martin SP (2010) Two-component spinor techniques and feynman rules for quantum field theory and supersymmetry. Phys Rep 494(12):1–196
Caselle M, Magnea U (2004) Random matrix theory and symmetric spaces. Phys Rep 394(23):41–156
Gilmore R (1974) Lie groups, Lie algebras, and some of their applications. Wiley, New York
Bernevig BA, Hughes TL (2013) Topological insulators and topological superconductors. Princeton University Press, Princeton
Shen S-Q (2013) Topological insulators: Dirac equation in condensed matters, vol 174. Springer Science & Business Media, Berlin
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Agarwala, A. (2019). Tenfold Way. In: Excursions in Ill-Condensed Quantum Matter. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-21511-8_2
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