Real critical exponents from the $\varepsilon$-expansion in an interacting $U(1)$ model with non-Hermitian $Z_4$ anisotropy

TL;DR

The paper analyzes a $U(1)$-invariant scalar theory perturbed by a complex, $PT$-symmetric $Z_{4}$ anisotropy to explore critical behavior in intrinsically non-Hermitian systems. Using an $\varepsilon$-expansion up to $O(\varepsilon^2)$, the authors derive two-loop RG equations that reveal an RG invariant $k$ and a family of fixed points and fixed lines, including Gaussian, Heisenberg, generalized Ising, and generalized Cubic classes. Remarkably, all critical exponents $\eta$ and $\nu$ are real and $k$-independent in both the $PT$-unbroken and $PT$-broken regions, with the Heisenberg fixed point being the most stable and corresponding to emergent Hermiticity. This work demonstrates that non-Hermitian perturbations can preserve or even yield Hermitian-like universality in the infrared, offering insight into non-Hermitian QFT beyond the standard gain-loss interpretation.

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 examples of intrinsically non-Hermitian, though often $\mathcal{PT}$ (parity-time) symmetric, not necessarily open systems, in which case the concept of gain and loss relative to an underlying environment is not primordial. 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, $\mathcal{PT}$ symmetric $Z_{4}$ anisotropy. We find real critical exponents both in the region of unbroken and broken $\mathcal{PT}$ 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.

Figures (2)

• Figure 1: RG flows for the non-Hermitian model associated to the Lagrangian of Eq. \ref{['Eq:L']}. The RG flows shown here are for $d=4-\varepsilon$. In all panels the red dot represents the fixed point and red line represents the fixed line. Panel (a) show full picture of RG flows. Panels (b), (c) and (d) show separately the $\Tilde{g}_3=0$, $\Tilde{g}_3=0.8\Tilde{g}_2$ and $\Tilde{g}_3=1.5\Tilde{g}_2$ planes appearing in panel (a). For panels (b) and (c), $k<1$ and the $\mathcal{PT}$ symmetry is unbroken. For panel (d), $k>1$, the fixed lines become complex (only two fixed points remain) and the $\mathcal{PT}$ symmetry is broken.
• Figure 2: Behavior of the real and imaginary parts of the coupling constants $\Tilde{g}_1^*$, $\Tilde{g}_2^*$, and $\Tilde{g}_3^*$ for the case of Ising (a, b, c) and Cubic (d, e, f) fixed lines with respect to the parameter $k$. Region $k\in(-1,1)$ corresponds to the region of unbroken $\mathcal{PT}$ symmetry and in this region the imaginary part is always zero. Accordingly, the region $k\in(-\infty,-1)\cup(1,+\infty)$ corresponds to the region of broken $\mathcal{PT}$ symmetry and in it the imaginary part is nonzero. Points $k=\pm 1$ correspond to exceptional points, and at these points the values diverge.