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Real critical exponents from the ε-expansion in an interacting U(1) model with non-Hermitian Z4 anisotropy

Eduard Naichuk1,2, Jeroen van den Brink1,3, and Flavio S. Nogueira1

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Phys. Rev. B 113, 115157 – Published 27 March, 2026

DOI: https://doi.org/10.1103/4454-lq3b

Abstract

In quantum optics and condensed matter physics, non-Hermitian phenomena are often studied under the assumption of an open physical system. However, there are many examples of intrinsically non-Hermitian, though often PT-symmetric systems, which do not necessarily need to be characterized as open, where one usually speaks of gain and loss relative to an underlying environment. A particularly intriguing example with experimental consequences in the literature is QCD at finite density. Motivated by the existence of such inherently non-Hermitian systems, here we study the critical behavior of a U(1)-invariant Lagrangian perturbed by a complex, PT-symmetric Z4 anisotropy. We find real critical exponents both in the region of unbroken and broken PT symmetry. In the former the coupling constants for fixed points or lines are real, whereas in the latter they become complex. Importantly, the most stable fixed point corresponds to the flow at large distances towards an effectively Hermitian U(1) symmetric system. This constitutes an example where both the U(1) and the Hermitian character are emergent features of the theory. This tells us about the importance and physical meaning of some non-Hermitian systems beyond interpretations involving gain and loss.

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Phys. Rev. B 113, 115157– Published 27 March, 2026
  • Received 21 October 2025
  • Accepted 10 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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