Anomalous energy correlations and spectral form factor in the nonergodic phase of the β ensemble

The <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>β</a:mi></a:math> ensemble is a prototypical model of a single-particle system on a one-dimensional disordered lattice with inhomogeneous nearest-neighbor hopping. The corresponding nonergodic phase has an anomalous critical energy scale, <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msub><b:mi>E</b:mi><b:mi>c</b:mi></b:msub></b:math>: Correlations are present above and absent below <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>E</c:mi><c:mi>c</c:mi></c:msub></c:math>, as reflected in the number variance. We study the dynamical properties of the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>β</d:mi></d:math> ensemble where the critical energy controls the characteristic timescales. In particular, the spectral form factor equilibrates at a relaxation time <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msub><e:mi>t</e:mi><e:mi mathvariant="normal">R</e:mi></e:msub><e:mo>≡</e:mo><e:msubsup><e:mi>E</e:mi><e:mi>c</e:mi><e:mrow><e:mo>−</e:mo><e:mn>1</e:mn></e:mrow></e:msubsup></e:mrow></e:math>, which is parametrically smaller than the Heisenberg time, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msub><g:mi>t</g:mi><g:mi mathvariant="normal">H</g:mi></g:msub></g:math>, given by the inverse of the mean level spacing. Incidentally, the dimensionless relaxation time, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:msub><i:mi>τ</i:mi><i:mi mathvariant="normal">R</i:mi></i:msub><i:mo>≡</i:mo><i:msub><i:mi>t</i:mi><i:mi mathvariant="normal">R</i:mi></i:msub><i:mo>/</i:mo><i:msub><i:mi>t</i:mi><i:mi mathvariant="normal">H</i:mi></i:msub><i:mo>≪</i:mo><i:mn>1</i:mn></i:mrow></i:math> is equal to the Dyson index, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>β</m:mi></m:math>. We show that the energy correlations are absent within a temporal window <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:msub><n:mi>t</n:mi><n:mi mathvariant="normal">R</n:mi></n:msub><n:mo><</n:mo><n:mi>t</n:mi><n:mo><</n:mo><n:msub><n:mi>t</n:mi><n:mi mathvariant="normal">H</n:mi></n:msub></n:mrow></n:math>, which we term as the . This is in contrast to the mechanism of equilibration in a typical many-body system. We analytically explain the qualitative behavior of the number variance and the spectral form factor of the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:mi>β</q:mi></q:math> ensemble by a spatially local mapping to the Anderson model.