Resonant delocalisation in random operators
Theoretical Condensed Matter Physics
Final Report Abstract
The project lies at the interface of the physics of disordered systems and mathematics. It was partially motivated by numerous discussions on many-body localization in the physics literature. They revolve around the question whether the Anderson transition survives the onset of generic interactions. The Anderson transition is a universal feature of quantum systems which is either modelled by a random operator of Schrödinger type or in terms of more effective random band matrix models. In the one-particle random Schrödinger setup, various elaborate methods of spectral theory have been developed for an understanding the localization part of this story. Unfortunately, these techniques do not easily carry over to neither effective banded random matrices nor to the many-particle set-up. During this project, we investigated the persistence of localization in the special case of the integrable XY-spin chain in random field as well as the disordered Tonks-Girardeau gas. In particular, we established the decay of time-dependent correlation functions as well as a proof of the absence of Bose-Einstein condensation and superfluidity. Another huge mathematical challenge in this broader area are proofs of delocalization. Even in the one-particle set-up such proofs are only known in very special tree-graph situations. The major part of this project was devoted to the study of delocalization in effective random matrix models. In this area, delocalization for mean-field random matrix models of GOE or GUE-type has been established as part of Erdös’ and Yau’s random matrix universality programme. In contrast, we studied the hierarchical random matrix ensemble beyond the mean-field regime. In particular, we established delocalization of eigenvectors as well as random matrix universality of its eigenvalue process. The study of this ensemble is partially motived by similar open question for power-law random band matrices. Interestingly, the hierarchical ensemble allows to define an infinite-volume random operator, for which we prove spectral delocalization – adding this operator to the sparse collection of random operators for which this was established. Proofs for the hierarchical ensemble are enabled by a renormalization group analysis. As part of its one-step problem, we used and developed a stochastic differential equation approach towards the spectral analysis of deformed GOE matrices. The latter are also known as Rosenzweig-Porter ensemble. As a side result, we rigorous establish the existence of an intermediate regime for this ensemble, which was recently predicted in the physics literature. In this regime the eigenvectors are not completely delocalized, but the eigenvalue process already follows GOE statistics.
Publications
- Boosted Simon-Wolff Spectral Criterion and Resonant Delocalization. Commun. Pure Appl. Math. 69: 2195-2218 (2016)
M. Aizenman, S. Warzel
(See online at https://doi.org/10.1002/cpa.21625) - Decay of correlations and absence of superfluidity in the disordered Tonks-Girardeau gas. New Journal of Physics 18: 035002 (2016)
R. Seiringer, S. Warzel
(See online at https://doi.org/10.1088/1367-2630/18/3/035002) - Decay of Determinantal and Pfaffian Correlation Functionals in One-Dimensional Lattices. Commun. Math. Phys. 347: 903–931 (2016). Erratum: Commun. Math. Phys. 361 825–826 (2018)
R. Sims, S. Warzel
(See online at https://doi.org/10.1007/s00220-016-2612-0 https://doi.org/10.1007/s00220-018-3171-3) - Renormalization Group Analysis of the Hierarchical Anderson Model, Ann. Henri Poincare, 18: 1919-1947 (2017)
P. von Soosten, S. Warzel
(See online at https://doi.org/10.1007/s00023-016-0549-7) - Delocalization and continuous spectrum for ultrametric random operators. In: Ann. Henri Poincare
P. von Soosten, S. Warzel
(See online at https://doi.org/10.1007/s00023-019-00809-z) - Hierarchical Disordered Quantum Systems. PhD thesis TUM 2018
P. von Soosten
- Singular Spectrum and Recent Results on Hierarchical Operators Mathematical Problems in Quantum Physics, Contemp. Math. 717: 215-225 (2018)
P. von Soosten, S. Warzel
(See online at https://doi.org/10.1090/conm/717) - The Phase Transition in the Ultrametric Ensemble and Local Stability of Dyson Brownian Motion. Electron. J. Probab., 23: 24 pp. (2018)
P. von Soosten, S. Warzel
(See online at https://doi.org/10.1214/18-EJP197) - Non-ergodic delocalization in the Rosenzweig-Porter model. Lett. Math. Phys., 109: 905–922 (2019)
P. von Soosten, S. Warzel
(See online at https://doi.org/10.1007/s11005-018-1131-7)
