## SciPost Submission Page

# Statistics of correlations functions in the random Heisenberg chain

### by Luis Colmenarez, Paul A. McClarty, Masudul Haque, David J. Luitz

### Submission summary

As Contributors: | Luis A. Colmenarez · David J. Luitz |

Arxiv Link: | https://arxiv.org/abs/1906.10701v3 |

Date submitted: | 2019-10-08 |

Submitted by: | Colmenarez, Luis A. |

Submitted to: | SciPost Physics |

Domain(s): | Theor. & Comp. |

Subject area: | Quantum Physics |

### Abstract

Ergodic quantum many-body systems satisfy the eigenstate thermalization hypothesis (ETH). However, strong disorder can destroy ergodicity through many-body localization (MBL) -- at least in one dimensional systems -- leading to a clear signal of the MBL transition in the probability distributions of energy eigenstate expectation values of local operators. For a paradigmatic model of MBL, namely the random-field Heisenberg spin chain, we consider the full probability distribution of eigenstate correlation functions across the entire phase diagram. We find gaussian distributions at weak disorder, as predicted by pure ETH. At intermediate disorder -- in the thermal phase -- we find further evidence for anomalous thermalization in the form of heavy tails of the distributions. In the MBL phase, we observe peculiar features of the correlator distributions: a strong asymmetry in $S_i^z S_{i+r}^z$ correlators skewed towards negative values; and a multimodal distribution for spin-flip correlators. A quantitative quasi-degenerate perturbation theory calculation of these correlators yields a surprising agreement of the full distribution with the exact results, revealing, in particular, the origin of the multiple peaks in the spin-flip correlator distribution as arising from the resonant and off-resonant admixture of spin configurations. The distribution of the $S_i^zS_{i+r}^z$ correlator exhibits striking differences between the MBL and Anderson insulator cases.

###### Current status:

### List of changes

1) Panels with $r>1$ are added to this figure, including comments in section III.B .

2) The referee is correct. The kinetic term is the one treated as a perturbation. We thank the referee for pointing out this error and have corrected it in the present version.

3) Distances $r>1$ require higher order perturbation theory because the leading and subleading orders are non-trivial only for $r=1$. Unfortunately, our semianalytical approach becomes numerically very expensive when applied to higher order perturbation theory. A clarification of this point is added at the end of section III.C

4) An inset of the region close to $0$ is added. Comments about the inset are written down in section III.C (paragraph after Eq. 6) now.

5) References suggested by the referee have been included. The duplicated reference has been removed.