Non-Hermitian physics in open quantum many-body systems provides novel opportunities for the discovery of exotic quantum phenomena unexpected in Hermitian systems. A previous study of the non-Hermitian Kondo problem in ultracold atoms reports reversion of renormalization group flows which violates the $g$ theorem and produces an unusual quantum phase transition. In this work, we study the effect of electron-electron interactions by considering the non-Hermitian Kondo problem in a Luttinger liquid. By performing a perturbative renormalization group analysis to two-loop order, we find that the interplay between non-Hermitian Kondo couplings and electron-electron interactions can produce a pair of complex fixed points. Complex fixed points have often been discussed in an attempt to understand the extremely long correlation length of Hermitian systems with weakly first-order transitions. Here, we show that complex fixed points arise naturally and can be physically realized in open quantum systems. We discuss consequences of the complex fixed points and future directions.

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• Kondo effect
• Non-Hermitian systems
• Luttinger liquid model
• Renormalization group