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What are the simplest quantum 1D spin chain models which aren't integrable? Are there any generic criteria for telling whether or not a given quantum 1D spin chain model is integrable?

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    $\begingroup$ Take a Heisenberg spin chain (nearest-neighbor coupling) and add next-nearest-neighbor coupling. I think this breaks the integrability. $\endgroup$
    – Meng Cheng
    Commented Feb 21, 2015 at 18:24
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    $\begingroup$ Not necessarily, for example, the Inozemzev chain, which is a long-range chain is integrable arxiv.org/pdf/1012.3985v3.pdf $\endgroup$
    – Yuri
    Commented Mar 5, 2015 at 1:21

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