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On the size of the space spanned by a nonequilibrium state in a quantum spin lattice system
by Maurizio Fagotti
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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.
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.