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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