Key Observable for Linear Thermalization

For studies on thermalization of an isolated quantum many-body system, the fundamental issue is to determine whether a given system thermalizes or not. However, most studies tested only a small number of observables, and it was unclear whether other observables thermalize. Here, we study whether `linear thermalization' occurs for all additive observables: We consider a quantum many-body system prepared in an equilibrium state and its unitary time evolution induced by a small change $\Delta f$ of a physical parameter $f$ of the Hamiltonian, and examine whether \emph{all} additive observables relax to the equilibrium values in a manner fully consistent with thermodynamics up to the linear order in $\Delta f$. We find that the additive observable conjugate to $f$ is key for linear thermalization in that its linear thermalization guarantees, under physically reasonable conditions, linear thermalization of all additive observables. Such a linear thermalization occurs in the timescale of $\mathcal{O}(|\Delta f|^0)$, and lasts at least for a period of $o(1/\sqrt{|\Delta f|})$. We also consider linear thermalization against the change of other parameters, and find that linear thermalization of the key observable against $\Delta f$ guarantees its linear thermalization against small changes of any other parameters. Furthermore, we discuss the generalized susceptibilities for cross responses and their consistency between quantum mechanics and thermodynamics. We demonstrate our main result by performing numerical calculations for spin models. The present paper offers an efficient way of judging linear thermalization because it guarantees that examination of the single key observable is sufficient.