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arXiv:1507.02231 · Computer Physics Communications

Using zeros of the canonical partition function map to detect signatures of a Berezinskii–Kosterlitz–Thouless transition

J. C. S. Rocha, L. A. S. Mόl, B. V. Costa

Published Sep 12, 2016• 15 citations
arXivHTMLPDF
DOI
10.1016/j.cpc.2016.08.016

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In this work we show how one can use the zeros of the canonical partition function, the Fisher zeros, to unambiguously characterize a transition as being in the Berezinskii-Kosterlitz-Thouless (BKTBKTBKT) class of universality. By studying the zeros map for the 2D XY-model, we found that its internal border coalesces into the real positive axis in a finite region corresponding to temperatures smaller than the BKTBKTBKT transition temperature. This behavior is consistent with the predicted existence of a line of critical points below the transition temperature, allowing one to distinguish the BKTBKTBKT class of universality from other possibilities.
The locality of field theories strongly constrains the possible behaviors of symmetry-twisted partition functions, and thus they serve as order parameters to detect low-energy realizations of global symmetries, such as spontaneous symmetry breaking (SSB). We demonstrate that the tensor renormalization group (TRG) offers an efficient framework to compute the symmetry-twisted partition functions, which enables us to detect the symmetry-breaking transition and also to study associated critical phenomena. As concrete examples of SSB, we investigate the two-dimensional (2D) classical Ising model and the three-dimensional (3D) classical O ( 2 ) nonlinear sigma model, and we identify their critical points solely from the twisted partition function. By employing the finite-size scaling argument, we find the critical temperature T c = 2.2017 ( 2 ) with the critical exponent ν = 0.663 ( 33 ) for the 3D O ( 2 ) model. In addition, we also study the Berezinskii–Kosterlitz–Thouless (BKT) criticality of the 2D classical O ( 2 ) model by extracting the helicity modulus from the twisted partition functions, and we obtain the BKT transition temperature, T BKT = 0.8928 ( 2 ) .