Lindblad dynamics of the quantum spherical model

The purely relaxational non-equilibrium dynamics of the quantum spherical model as described through a Lindblad equation is analysed. It is shown that the phenomenological requirements of reproducing the exact quantum equilibrium state as stationary solution and the associated classical Langevin equation in the classical limit $g\to 0$ fix the form of the Lindblad dissipators, up to an overall time-scale. In the semi-classical limit, the models' behaviour become effectively the one of the classical analogue, with a dynamical exponent $z=2$, and an effective temperature $T_{\rm eff}$, renormalised by the quantum coupling $g$. A distinctive behaviour is found for a quantum quench, at zero temperature, deep into the ordered phase $g\ll g_c(d)$, for $d>1$ dimensions. Only for $d=2$ dimensions, a simple scaling behaviour holds true, with a dynamical exponent $z=1$, while for dimensions $d\ne 2$, logarithmic corrections to scaling arise. The spin-spin correlator, the growing length scale and the time-dependent susceptibility show the existence of several logarithmically different length scales.