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

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 `$\ln \Lambda$' 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 ${\alpha }_0=1$, the system exhibits a dynamically generated time-dependent mass gap, which at time $t=t_0+\Delta t$ is given by $m(t)=m_0{e}^{-\pi {\alpha }_0\Delta t}$, where $m_0=\Lambda e^{-\pi {\alpha }_0{t}_0}$, 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 ${\alpha }_0$ or for very fast drive rates $\alpha$ such that $\alpha t\gg {\alpha }_0{t}_0$, for any $t<L$, we argue that the system is asymptotically free and approaches the $SU(2{)}_1$ WZNW model, which corresponds to the UV fixed point of the $SU(2)$ GN model.