Complex nonlinear sigma model

TL;DR

This work investigates nonlinear sigma models with complexified couplings as a framework for nonunitary critical phenomena in open quantum systems. Using perturbative renormalization group methods across the tenfold symmetry classes, it reveals that complex fixed points with genuinely complex scaling dimensions arise generically, producing spiral RG flows and a rich global phase structure with both continuous and discontinuous transitions. The findings establish complex universality in field theory and offer a predictive picture for critical behavior in complexified open-system settings, with qualitative robustness beyond finite-order perturbation theory. The results suggest avenues for nonperturbative analyses and potential experimental realizations in measurement-induced or non-Hermitian critical phenomena.

Abstract

Motivated by the recent interest in the criticality of open quantum many-body systems, we study nonlinear sigma models with complexified couplings as a general framework for nonunitary field theory. Applying the perturbative renormalization-group analysis to the tenfold symmetric spaces, we demonstrate that fixed points with complex scaling dimensions and critical exponents arise generically, without counterparts in conventional nonlinear sigma models with real couplings. We further clarify the global phase diagrams in the complex-coupling plane and identify both continuous and discontinuous phase transitions. Our work elucidates universal aspects of critical phenomena in complexified field theory.

Figures (10)

• Figure 1: Perturbative renormalization-group flow of the $\mathrm{O} \left( N \right)$ nonlinear sigma model in the complex-coupling plane. Fixed points (red points) occur either on the real axis or in complex-conjugate pairs, and their scaling dimensions are also shown. The top, middle, and bottom panels correspond to one, two, and three dimensions, respectively. The first and second columns show the results truncated at the cubic and quartic order in perturbation theory for the rescaled coupling $x \coloneqq \left( N-2 \right) t$. The third, fourth, and fifth columns present the quintic-order results for $N=1$, $N=2$, and $N=3$, respectively. In one dimension, all complex fixed points are stable. In two and three dimensions, unstable complex fixed points appear beyond the quartic order.
• Figure 2: Perturbative renormalization-group flow of the $\mathrm{U} \left( N \right)$ nonlinear sigma model in the complex-coupling plane at the quintic order in perturbation theory. Fixed points (red points) occur either on the real axis or in complex-conjugate pairs, and their scaling dimensions are also shown. The top, middle, and bottom panels correspond to one, two, and three dimensions, respectively. The first and second columns show the results for $N=1$ and $N=2$, respectively. The third column presents the results for $N=3$ with a wide-range view shown in the fourth column.
• Figure 3: Perturbative renormalization-group flow of the $\mathrm{Sp} \left( N \right)$ nonlinear sigma model in the complex-coupling plane at the quintic order in perturbation theory. Fixed points (red points) occur either on the real axis or in complex-conjugate pairs, and their scaling dimensions are also shown. The top, middle, and bottom panels correspond to one, two, and three dimensions, respectively. The first and second columns show the results for $N=1$ and $N=2$, respectively.
• Figure 4: Perturbative renormalization-group flow of the $\mathrm{U}\left( N \right)/\mathrm{O}\left( N \right)$ nonlinear sigma model in the complex-coupling plane at the quintic order in perturbation theory. Fixed points (red points) occur either on the real axis or in complex-conjugate pairs, and their scaling dimensions are also shown. The top, middle, and bottom panels correspond to one, two, and three dimensions, respectively. The first and second columns show the results for $N=1$ and $N=2$, respectively.
• Figure 5: Perturbative renormalization-group flow of the $\mathrm{U}\left( 2N \right)/\mathrm{Sp}\left( N \right)$ nonlinear sigma model in the complex-coupling plane at the quintic order in perturbation theory. Fixed points (red points) occur either on the real axis or in complex-conjugate pairs, and their scaling dimensions are also shown. The top, middle, and bottom panels correspond to one, two, and three dimensions, respectively. The first and second columns show the results for $N=1$ and $N=2$, respectively. In two dimensions, the complex fixed points with the spiral renormalization-group flow are stable for $N=1$ but unstable for $N=2$.