SciPost Submission Page
On the size of the space spanned by a nonequilibrium state in a quantum spin lattice system
by Maurizio Fagotti
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Maurizio Fagotti 
Submission information  

Preprint Link:  https://arxiv.org/abs/1901.10797v4 (pdf) 
Date accepted:  20190509 
Date submitted:  20190430 02:00 
Submitted by:  Fagotti, Maurizio 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

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