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. 2017 Jun 8;8:15791. doi: 10.1038/ncomms15791

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Copyright © 2017, The Author(s)

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

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(a) 3D phase diagram of a PT-symmetric many-body system in the parameter space (K, g_r, g_i). Here K and g_r (g_i) characterize the strength of the inter-particle interaction and the depth of the real (imaginary) part of a complex potential, respectively. The MI and TLL phases are separated by the surface of the BKT transition for K>2 and that of the PT transition for K<2. An example of the BKT (PT) transition is illustrated by the blue (red) curve with the transition point indicated by the filled (open) circle. The MI (TLL) phase corresponds to the 3D region containing the blue (red) shaded plane at g_i=0. On the critical line with K=2 lies a SSCP (black open circle). Dashed lines indicate the phase boundaries on the plane with fixed g_r for comparison with numerical results in Fig. 2a. (b) Hyperbolic RG flows in a PT-unbroken region (g_i<g_r), which reproduce the conventional flow diagram in the sine-Gordon model. (c) RG flows on the two phase boundaries separated by an unconventional fixed line (thick black line). (d) Unconventional semicircular RG flows in a PT-broken region (g_i>g_r). Along each flow, the TLL parameter K monotonically increases, indicating the anomalous enhancement of the superfluid correlation.