{ "id": "2411.08095", "version": "v1", "published": "2024-11-12T19:00:00.000Z", "updated": "2024-11-12T19:00:00.000Z", "title": "The $J_1$-$J_2$ spin chain and the $O(3)$ non-linear sigma model at $θ= π$", "authors": [ "Adam J. McRoberts", "Chris Hooley", "A. G. Green" ], "comment": "4 pages, 2 figures", "categories": [ "cond-mat.stat-mech", "cond-mat.str-el", "hep-th" ], "abstract": "There are two competing pictures for how the $O(3)$ non-linear sigma model (NLSM) with a topological theta term renormalises to the $\\mathfrak{\\hat{su}}(2)_1$ Wess-Zumino-Witten model. In the Affleck-Haldane picture, an `extension field' parametrises the correlated fluctuations of the couplings and extends the target space from $S^2$ to $S^3$; recently, however, Zirnbauer has advanced a competing picture where the extension field is frozen out and the infrared theory is more naturally thought of as a compact free boson dual to $\\mathfrak{\\hat{su}}(2)_1$. We show that the $J_1$-$J_2$ spin chain evinces a phase transition between these pictures -- corresponding to whether the dimer field that parametrises local singlet order is part of a joint $O(4)$ N\\'eel-singlet order parameter or appears as a separate gapped mode. For the $O(3)$ NLSM itself, this analysis supports the Zirnbauer picture.", "revisions": [ { "version": "v1", "updated": "2024-11-12T19:00:00.000Z" } ], "analyses": { "keywords": [ "non-linear sigma model", "extension field", "parametrises local singlet order", "compact free boson dual", "neel-singlet order parameter" ], "note": { "typesetting": "TeX", "pages": 4, "language": "en", "license": "arXiv", "status": "editable" } } }