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arXiv:2605.05111

Time-Dependent Dynamical Dimensional Transmutation in the $SU(2)$ Gross-Neveu Model with Time-Dependent Interaction Strength

Parameshwar R. Pasnoori

Published May 6, 2026• 0 citations
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Abstract

In this work we study driven $SU(2)$ Gross-Neveu (GN) model whose time-dependent interaction strength is constrained by quantum integrability within the recently developed generalized Bethe ansatz framework. Integrability requires the coupling strengths to evolve according to the same equation as the renormalization-group (RG) flow of the corresponding static theory, where time `$t$' of the time-dependent model is identified with the logarithm of the cutoff `$\ln Λ$' of the static model. We reformulate the exact quantum Knizhnik-Zamolodchikov (qKZ) solution in terms of a Yang-Yang functional, whose saddle-point equations determine the long-time dynamics of the system. In the scaling regime, this analysis reveals the emergence of a characteristic time scale $t_0$. In the case of coupling strength decreasing with time, we show that in the adiabatic regime, which corresponds to $t\sim t_0$ for drive rate $α_0=1$, the system exhibits a dynamically generated time-dependent mass gap, which at time $t=t_0+Δt$ is given by $m(t)=m_0 e^{-πα_0Δt}$, where $m_0=Λe^{-πα_0 t_0}$, which possesses the same functional dependence as the non-perturbatively generated mass scale of the static GN model. We thereby establish a time-dependent analogue of dynamical dimensional transmutation and identify the adiabatic regime of the driven system with the scaling regime of the static theory. In the case of very large time scales $t\gg t_0$ for drive rate $α_0$ or for very fast drive rates $α$ such that $αt \gg α_0t_0$, for any $t

DOI
10.48550/arxiv.2605.05111

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In this paper we consider quantum Hamiltonians with time-dependent interaction strengths, and following the recently formulated generalized Bethe ansatz framework [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)], we show that constraints imposed by integrability take the same form as the renormalization group flow equations corresponding to the respective Hamiltonians with constant interaction strengths. As a concrete example, we consider the anisotropic time-dependent Kondo model characterized by the time-dependent interaction strengths J∥(t)J_{\parallel}(t)J∥​(t) and J⊥(t)J_{\perp}(t)J⊥​(t). We construct an exact solution to the time-dependent Schrodinger equation and by applying appropriate boundary conditions on the fermion fields we obtain a set of matrix difference equations called the quantum Knizhnik-Zamolodchikov (qKZ) equations corresponding to the XXZ R-matrix. The consistency of these equations imposes constraints on the time-dependent interaction strengths J∥(t)J_{\parallel}(t)J∥​(t) and J⊥(t)J_{\perp}(t)J⊥​(t), such that the system is integrable. Remarkably, the resulting temporal trajectories of the couplings are shown to coincide exactly with the RG flow trajectories of the static Kondo model, establishing a direct and universal correspondence between integrability and renormalization-group flow in time-dependent quantum systems.
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 `ttt' in the driven model is identified with the logarithm of the cutoff `log⁡Λ\logΛlogΛ' 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 generalized Bethe ansatz framework formulated in [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)] provides a unified framework to find exact solutions to quantum many-body systems with time-dependent coupling strengths with periodic boundary conditions. In this work we extend this framework to the case of open boundary conditions where in addition to the time-dependent interactions in the bulk, the boundary conditions are explicitly time-dependent. We show that for integrable time-dependent bulk coupling strengths, the generalized Bethe ansatz framework provides the time-dependent boundary conditions compatible with integrability and reduces the time-dependent Schrodinger equation to a set of matrix difference equations called the boundary quantum Knizhnik-Zamolodchikov (BqKZ) equations. The solution to the BqKZ equations provides the explicit form of the exact wavefunction. We further show that the RG invariants of the corresponding static model identify with the dynamical invariants in the time-dependent model.
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In this Letter we consider the time-dependent Kondo model where a magnetic impurity interacts with the electrons through a time-dependent interaction strength J(t)J(t)J(t). We develop an alternative framework based on the Bethe ansatz and construct an exact solution to the time-dependent Schr\"odinger equation. We show that when periodic boundary conditions are applied, the consistency of the solution results in a constraint equation which relates the amplitudes corresponding to a certain ordering of the particles in the configuration space. This constraint equation takes the form of a matrix difference equation, and the associated consistency conditions restrict the interaction strength J(t)J(t)J(t) for the system to be integrable. For a given J(t)J(t)J(t) satisfying these constraints, the solution to the matrix difference equations provides the exact many-body wave function that satisfies the time-dependent Schr\"odinger equation. We provide a concrete example of J(t)J(t)J(t) which satisfies these constraint equations. We show that in this case, the matrix difference equations turn into quantum Knizhnik-Zamolodchikov (qKZ) equations, which are well studied in the literature. The framework developed in this Letter allows one to probe the nonequilibrium physics of the Kondo model, and being general, it also allows one to solve a new class of Hamiltonians with time-dependent interaction strengths which are based on quantum Yang-Baxter algebra.