Beyond Integrability Preserving Renormalization-Group Protocol in Non-Hermitian Hamiltonians with Time-Dependent Interaction Strengths

Abstract: It is well established that in time-dependent quantum systems, integrability preserving time-dependent interaction strengths are identical to the renormalization group (RG) trajectories of the corresponding static model when time $t$' in the driven model is identified with the logarithm of the cutoff$\logΛ$' of the static model. We refer to this integrability preserving driving as the RG protocol. In this work we extend the class of time-dependent integrable models to include non-Hermitian quantum models with time-dependent interaction strengths. Using the recently formulated generalized Bethe ansatz framework [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)], we show that the constraints imposed by integrability are more general: The interaction strengths of the static model that flow in the RG follow the respective RG trajectories in the corresponding time-dependent model as described above. In addition, the interaction strengths of the static model that are RG invariant can either be constant or have a specific time-dependence in the corresponding time-dependent model which is constrained by integrability. Thus we establish that in the context of time-dependent non-Hermitian systems, the set of integrability preserving time-dependent strengths is larger than the set corresponding to the RG protocol.

• The paper demonstrates how non-Hermitian Kondo models allow integrability- preserving time-dependent couplings beyond typical RG trajectories
• Integrability-preserving scalable model expansion is applied to time-dependent non-Hermitian systems to derive new dynamical phenomena
• Key observation: General time-dependent driving protocols are identified and may produce new physical implications beyond gravities

Overview

This paper addresses a structural question in the theory of integrable time-dependent quantum systems: is the well-known correspondence between integrability-preserving time-dependent couplings and renormalization group (RG) trajectories fundamental, or merely a special case of a broader set of constraints imposed by integrability? The author answers this question affirmatively for non-Hermitian systems by solving the time-dependent $SU(2)$ non-Hermitian Kondo model with complex, time-dependent exchange coupling $J(t)$ using the generalized Bethe ansatz framework (2608.19519). The central result is that while couplings that flow under RG must follow their RG trajectories when made time-dependent, couplings that are RG invariant in the static model are permitted—by integrability—to acquire specific time dependences beyond simple constancy. Consequently, the set of integrability-preserving driving protocols strictly contains the set corresponding to the "RG protocol."

Background: the integrability–RG connection

Prior work established the connection between time-dependent couplings and RG flows from two directions. Hoare, Levine, and Tseytlin showed that demanding a Lax connection for $\sigma$-models with locally time-dependent couplings forces those couplings to satisfy the one-loop RG equations of the static theory, with time interpreted as logarithmic scale (2608.19519). Pasnoori later reproduced this result within a generalized Bethe ansatz framework for Hermitian models, showing that constraint conditions on time-dependent couplings coincide with RG equations at small coupling; this driving scheme was termed the RG protocol (2608.19519). Related developments include the construction of time-dependent integrable field theories via an extension of four-dimensional Chern-Simons theory (2608.19519). All of these results were obtained for real (Hermitian) couplings.

The static non-Hermitian Kondo model

The model considered is a one-dimensional chiral fermion coupled to a spin-$1/2$ impurity via a contact interaction $J\,\vec{\sigma}\cdot\vec{S}$ at the origin, with $J \in \mathbb{C}$. The static version exhibits three phases controlled by the parameter $f = \frac{1}{2J}(1 - 3J^2/4)$, whose imaginary part $f_i$ is an RG invariant measuring departure from Hermiticity:

Phase	Condition	Impurity physics
Kondo	$f_i < 1/2$	Many-body screening, Kondo scale $T_K = 2\Lambda e^{-\pi f_r}$
$J(t)$0	$J(t)$1	Single-particle bound state screens impurity, $J(t)$2
Local moment	$J(t)$3	Unscreened impurity

The $J(t)$4 phase is notable: unlike conventional Yu-Shiba-Rusinov states, the bulk remains gapless yet supports a stable bound state with finite lifetime ($J(t)$5). A first-order transition occurs at $J(t)$6, where the ground state switches between screened and unscreened configurations. The one-loop RG equation,

$J(t)$7

shows that only $J(t)$8 flows while $J(t)$9 is invariant, generating circular limit cycles centered on the imaginary axis. The author notes explicitly that the relation between $\sigma$0 and $\sigma$1 is non-universal and regularization-dependent, coinciding only in the universal small-coupling regime—an important caveat on the precise functional forms quoted below.

Generalized Bethe ansatz solution and consistency conditions

For the time-dependent Hamiltonian with $\sigma$2, the author constructs an exact many-body wavefunction satisfying the time-dependent Schrödinger equation. The particle number is conserved, and the wavefunction is decomposed into amplitudes labeled by orderings of particles relative to each other and to the impurity, related by time- and position-dependent S-matrices. Periodic boundary conditions yield matrix difference equations of the quantum Knizhnik-Zamolodchikov (qKZ) type, with monodromy operators transporting each particle around the system.

Consistency of these transport operators—the requirement that transporting particles $\sigma$3 and $\sigma$4 around the system commute—imposes the condition

$\sigma$5

where $\sigma$6 encodes the interaction strength. In the universal regime, this yields the central result: the most general integrability-preserving time-dependent coupling satisfies

$\sigma$7

The Yang-Baxter equations for the particle-impurity and particle-particle S-matrices hold identically along these trajectories.

Beyond the RG protocol

When $\sigma$8 is real, differentiating the solution recovers exactly the static RG equation under $\sigma$9, reproducing the RG protocol. Here $1/2$0 is constant—a dynamical invariant playing the role of $1/2$1—and the trajectory traces circles analogous to RG limit cycles.

The essential new observation is that $1/2$2 may be complex. In that case both $1/2$3 and $1/2$4 vary linearly with time, and no identification of $1/2$5 with $1/2$6 maps the dynamics onto the static RG flow. Nevertheless, the driven trajectories remain closed circles in the $1/2$7 plane, with centers displaced off the imaginary axis along a line whose slope interpolates continuously between the y-axis (RG protocol) and the x-axis. Two quantities are invariant in time even in the general case: the radius of the circle and its center, located at $1/2$8. These constitute newly identified dynamical invariants, although—as the author states plainly—their physical significance has not yet been explored.

An interesting structural point emerges here: since RG limit cycles are periodic, one might have expected the generalization to require periodic driving, but the linear-in-inverse-coupling form is already periodic as a trajectory. In the adiabatic limit, the system follows instantaneous eigenstates, so a slow drive varying $1/2$9 can carry the system across the three phases of the static model while remaining adiabatic. For arbitrary driving rates, the paper anticipates new dynamical phenomena but does not analyze them.

Limitations and open questions

Several qualifications bear directly on the results. First, the analysis is confined to the universal (weak-coupling) regime; away from it, the relation between $J\,\vec{\sigma}\cdot\vec{S}$0 and $J\,\vec{\sigma}\cdot\vec{S}$1 is regularization-dependent, and the precise allowed functional forms may differ, though the qualitative conclusion should be robust. Second, the physical interpretation of the new dynamical invariants (circle radius and center) is left unaddressed. Third, the dynamics at non-adiabatic driving rates—including possible phase transitions induced dynamically by sweeping across the $J\,\vec{\sigma}\cdot\vec{S}$2 boundaries—is explicitly deferred to future work. Fourth, while the construction is claimed to generalize directly to other models with complex couplings (sine-Gordon, chiral Gross-Neveu), those cases are asserted rather than worked out here. Finally, the experimental realization of time-dependent complex couplings in the dissipative alkaline-earth platform where the static model has been realized raises practical questions the paper does not treat.

Conclusion

By solving the time-dependent non-Hermitian $J\,\vec{\sigma}\cdot\vec{S}$3 Kondo model within the generalized Bethe ansatz, this work demonstrates that the integrability–RG correspondence, previously established in Hermitian settings, is not exhaustive once couplings are allowed to be complex. Integrability constrains couplings flowing under RG to follow their static trajectories, but permits RG-invariant couplings to acquire nontrivial time dependence, enlarging the space of integrable drives beyond the RG protocol. Whether this enlargement persists in other non-Hermitian integrable models, and what dynamical phenomena the new trajectories produce at arbitrary driving rates, remain open questions raised by the analysis (2608.19519).