Anomalous energy correlations and spectral form factor in the nonergodic phase of the $β$-ensemble
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arXiv
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| Format: | Preprint |
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2025
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| author | Roy, Basudha Das, Adway Kumar Ghosh, Anandamohan Khaymovich, Ivan M. |
| author_facet | Roy, Basudha Das, Adway Kumar Ghosh, Anandamohan Khaymovich, Ivan M. |
| contents | The $β$-ensemble is a prototypical model of a single particle system on a one-dimensional disordered lattice with inhomogeneous nearest neighbor hopping. Corresponding nonergodic phase has an anomalous critical energy scale, $E_c$: correlations are present above and absent below $E_c$ as reflected in the number variance. We study the dynamical properties of the $β$-ensemble where the critical energy controls the characteristic timescales. In particular, the spectral form factor equilibrates at a relaxation time, $t_\mathrm{R} \equiv E_c^{-1}$, which is parametrically smaller than the Heisenberg time, $t_\mathrm{H}$, given by the inverse of the mean level spacing. Incidentally, the dimensionless relaxation time, $τ_\mathrm{R} \equiv t_\mathrm{R}/t_\mathrm{H} \ll 1$ is equal to the Dyson index, $β$. We show that the energy correlations are absent within a temporal window $t_\mathrm{R} < t < t_\mathrm{H}$, which we term as the correlation void. 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 $β$-ensemble by a spatially local mapping to the Anderson model. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_15283 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Anomalous energy correlations and spectral form factor in the nonergodic phase of the $β$-ensemble Roy, Basudha Das, Adway Kumar Ghosh, Anandamohan Khaymovich, Ivan M. Disordered Systems and Neural Networks Other Condensed Matter Quantum Physics The $β$-ensemble is a prototypical model of a single particle system on a one-dimensional disordered lattice with inhomogeneous nearest neighbor hopping. Corresponding nonergodic phase has an anomalous critical energy scale, $E_c$: correlations are present above and absent below $E_c$ as reflected in the number variance. We study the dynamical properties of the $β$-ensemble where the critical energy controls the characteristic timescales. In particular, the spectral form factor equilibrates at a relaxation time, $t_\mathrm{R} \equiv E_c^{-1}$, which is parametrically smaller than the Heisenberg time, $t_\mathrm{H}$, given by the inverse of the mean level spacing. Incidentally, the dimensionless relaxation time, $τ_\mathrm{R} \equiv t_\mathrm{R}/t_\mathrm{H} \ll 1$ is equal to the Dyson index, $β$. We show that the energy correlations are absent within a temporal window $t_\mathrm{R} < t < t_\mathrm{H}$, which we term as the correlation void. 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 $β$-ensemble by a spatially local mapping to the Anderson model. |
| title | Anomalous energy correlations and spectral form factor in the nonergodic phase of the $β$-ensemble |
| topic | Disordered Systems and Neural Networks Other Condensed Matter Quantum Physics |
| url | https://arxiv.org/abs/2506.15283 |