## SciPost Submission Page

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

### by Maurizio Fagotti

#### - Published as SciPost Phys. 6, 059 (2019)

### Submission summary

As Contributors: | Maurizio Fagotti |

Arxiv Link: | https://arxiv.org/abs/1901.10797v4 |

Date accepted: | 2019-05-09 |

Date submitted: | 2019-04-30 |

Submitted by: | Fagotti, Maurizio |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Quantum Physics |

Approach: | Theoretical |

### Abstract

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.

### Ontology / Topics

See full Ontology or Topics database.Published as SciPost Phys. 6, 059 (2019)

### Author comments upon resubmission

### List of changes

- An appendix (Appendix A) has been added with a proof that the cumulants of a quasilocal Hamiltonian are extensive, provided that the state has finite correlation lengths.

- Section 4 has been improved.

- Section 4.1 now includes a practical application of the main result: it provides a physical criterion to fix the time step of the numerical simulations of the dynamics.

- References to the appendices have been added in the main text.

- Some typos have been fixed.