# Archive ouverte HAL – On the size of the space spanned by a nonequilibrium state in a quantum spin lattice system

### Maurizio Fagotti 1, 2

#### Maurizio Fagotti. On the size of the space spanned by a nonequilibrium state in a quantum spin lattice system. SciPost Physics, SciPost Foundation, 2019, 6 (5), ⟨10.21468/SciPostPhys.6.5.059⟩. ⟨hal-02292090⟩

We consider the time evolution of a state in an isolated quantum spin lattice system with energy cumulants proportional to the number of the sites $L^d$. We compute the distribution of the eigenvalues of the time averaged state over a time window $[t_0,t_0+t]$ in the limit of large $L$. This allows us to infer the size of a subspace that captures time evolution in $[t_0,t_0+t]$ with an accuracy $1-\epsilon$. We estimate the size to be $\frac{\sqrt{2\mathfrak{e}_2}}{\pi}\mathrm{erf}^{-1}(1-\epsilon) L^{\frac{d}{2}}t$, where $\mathfrak{e}_2$ is the energy variance per site, and $\mathrm{erf}^{-1}$ is the inverse error function.

• 1. LPENS (UMR_8023) - Laboratoire de physique de l'ENS - ENS Paris
• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques