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

### 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

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Published as SciPost Phys. 6, 059 (2019)

In this version of the manuscript (v4), the upper bound on the size of the space, presented in the original submitted version (v2), is shown to be saturated. It is also proved that the assumption that the energy cumulants are extensive is almost always fulfilled, with important exceptions when the initial state has power-law decaying correlations.

### 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.

### Submission & Refereeing History

Resubmission 1901.10797v4 on 30 April 2019
Submission 1901.10797v2 on 5 February 2019