# How does Haldane conjecture follow from the topological $\Theta$ term

The one dimensional SU(2) Heisenberg quantum spin chain is known to be described by the 1+1d O(3) nonlinear $\sigma$ model with a $\Theta$ term, following the action $$S=\int\mathrm{d}^2x\frac{1}{g}(\partial_\mu n)^2+\frac{i\Theta}{8\pi}\epsilon_{\mu\nu}\epsilon_{abc}n_a\partial_\mu n_b\partial_\nu n_c,$$ where $\Theta=2\pi s$ is related to the spin $s$ on each site. On the other hand, Haldane conjectured that the integer spin chain is gapped, while the half-integer spin chain is gapless (or degenerated?)

My question is how to understand the conjecture based on the topological $\Theta$ term? Or explicitly, how to calculate the ground state degeneracy and the low energy spectrum from the action given above? Is it even possible to write down the ground state wave functions for topological field theory? Is there a systematic way to analyze the low energy spectrum of a topological field theory in general?

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More on Haldane's conjecture: physics.stackexchange.com/q/59986/2451 –  Qmechanic Apr 10 '13 at 9:03
@Qmechanic Thanks for reminding. I am aware of the post which I have answered. But in my answer, I did not provide an argument for Haldane's conjecture, which is what I am asking for here. –  Everett You Apr 10 '13 at 12:55
If the spin chain is gapless , is the ground state degeneracy still well defined ? Thanks. –  K-boy Apr 10 '13 at 17:27
@K-boy For gapless spin chain, the ground state degeneracy is not well defined. I was told that when $\Theta=\pi$ the ground state is either gapless, or gapped but 2-fold degenerated. I have not figured out how to show this explicitly. –  Everett You Apr 10 '13 at 19:07
@K-boy Following Lieb-Schultz-Mattis (LSM), SU(2) invariant tinv. chains with half-integer spin have at least two states with low energy, i.e., they are either gapless or have a degenerate ground state. An example for the former is the spin-1/2 Heisenberg chain, and for the latter the Majumdar-Ghosh model (which breaks tinv.). I would say the degeneracy is always related to local symmetry breaking, since the degeneracy appears also for PBC. Note that LSM has been generalized to 2D (cond-mat/0305505), in which case the degeneracy can be related to topological order (e.g. in topo. spin liquids). –  Norbert Schuch Apr 11 '13 at 10:59
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