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

In this work we consider the time-dependent $SU(2)$ Gross-Neveu model, which is a quantum field theory of fermions which interact with each other through spin exchange interaction with time-dependent coupling strength $g(t)$. Using the recently formulated generalized Bethe ansatz framework, we show that the system is integrable provided the time-dependent coupling strength is such that its trajectories in time are exactly same as that of the renormalization group (RG) flow equations corresponding to the static model, where time `$t$' of the time-dependent model is identified with the logarithm of the cutoff `$\ln \Lambda$' of the static model. In the scaling regime $\Lambda\rightarrow\infty$, the above relation between time and the logarithm of the cutoff provides a characteristic time scale $t_0$. We analyze the exact time-dependent wavefunction in the case of coupling strength decreasing with time and show that in the adiabatic regime, which corresponds to $t\sim t_0$ for drive rate $\alpha_0=1$, the system exhibits a time-dependent dynamical dimensional transmutation where a time dependent mass gap is generated, which at time $t=t_0+\Delta t$ is given by $m(\Delta t)=m_0 e^{-\pi\alpha_0\Delta t}$, where $m_0=\Lambda e^{-\pi \alpha_0 t_0}$. Comparing this with the mass gap of the static model, we identify the adiabatic regime of the time-dependent model with the scaling regime of the static model. 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 $\alpha t \gg \alpha_0t_0$, for any $t<L$, we argue that the system is asymptotically free and approaches the $SU(2)_1$ Wess-Zumino-Novikov-Witten (WZNW) model, which corresponds to the UV fixed point of the $SU(2)$ Gross-Neveu model. Hence we establish that progression of time in the time-dependent model is equivalent to RG flow in the corresponding static model.