| The S=1 Haldane gap phase exhibits edge states (free S=1/2's at the ends), and a nonlocal order measured by the string order parameter. Kennedy and Tasaki demonstrated that these two features are consequences of spontaneous breaking of hidden, discrete Z2*Z2 symmetry. Generalizing to S>1, the same Z2*Z2 symmetry was found as a hidden symmetry for any integer spin. However, there were several puzzling aspects. For example, the Haldane gap phase for even integer spins (S=2,4,6,....) does not break the hidden symmetry. In this talk, I revisit this (old) problem utilizing an analogy with quantum spin hall insulator. I will clarify the physical meaning of the hidden symmetry, and discuss its consequenses on quantum phases. |
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