From non-ergodic eigenvectors to local resolvent statistics and back: A random matrix perspective

Davide Facoetti; Pierpaolo Vivo; Giulio Biroli

doi:10.1209/0295-5075/115/47003

From non-ergodic eigenvectors to local resolvent statistics and back: A random matrix perspective

Davide Facoetti, Pierpaolo Vivo, Giulio Biroli

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Abstract

We study the statistics of the local resolvent and non-ergodic properties of eigenvectors for a generalised Rosenzweig-Porter N × N random matrix model, undergoing two transitions separated by a delocalised non-ergodic phase. Interpreting the model as the combination of onsite random energies {a_i} and a structurally disordered hopping, we found that each eigenstate is delocalised over N^2-γ sites close in energy |a_j - a_i| ≤ N^1-γ in agreement with Kravtsov et al. (New J. Phys., 17 (2015) 122002). Our other main result, obtained combining a recurrence relation for the resolvent matrix with insights from Dyson's Brownian motion, is to show that the properties of the non-ergodic delocalised phase can be probed studying the statistics of the local resolvent in a non-standard scaling limit.

Original language	English
Article number	47003
Number of pages	8
Journal	EUROPHYSICS LETTERS
Volume	115
Issue number	4
Early online date	27 Sept 2016
DOIs	https://doi.org/10.1209/0295-5075/115/47003
Publication status	E-pub ahead of print - 27 Sept 2016

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10.1209/0295-5075/115/47003

From non-ergodic eigenvectors_FACOETTI_Publishedonline27September2016_GREEN AAM
This is the peer-reviewed accepted manuscript version of an article published in EPL (Europhysics Letters), v. 115, no. 4 (27 September 2016, © EPLA). The final published version can be viewed online through IOP Science at https://doi.org/10.1209/0295-5075/115/47003
Accepted author manuscript, 457 KBLicence: Other

https://arxiv.org/abs/1607.05942

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title = "From non-ergodic eigenvectors to local resolvent statistics and back: A random matrix perspective",

abstract = "We study the statistics of the local resolvent and non-ergodic properties of eigenvectors for a generalised Rosenzweig-Porter N × N random matrix model, undergoing two transitions separated by a delocalised non-ergodic phase. Interpreting the model as the combination of onsite random energies {ai} and a structurally disordered hopping, we found that each eigenstate is delocalised over N2-γ sites close in energy |aj - ai| ≤ N1-γ in agreement with Kravtsov et al. (New J. Phys., 17 (2015) 122002). Our other main result, obtained combining a recurrence relation for the resolvent matrix with insights from Dyson's Brownian motion, is to show that the properties of the non-ergodic delocalised phase can be probed studying the statistics of the local resolvent in a non-standard scaling limit.",

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T2 - A random matrix perspective

AU - Facoetti, Davide

AU - Vivo, Pierpaolo

AU - Biroli, Giulio

PY - 2016/9/27

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N2 - We study the statistics of the local resolvent and non-ergodic properties of eigenvectors for a generalised Rosenzweig-Porter N × N random matrix model, undergoing two transitions separated by a delocalised non-ergodic phase. Interpreting the model as the combination of onsite random energies {ai} and a structurally disordered hopping, we found that each eigenstate is delocalised over N2-γ sites close in energy |aj - ai| ≤ N1-γ in agreement with Kravtsov et al. (New J. Phys., 17 (2015) 122002). Our other main result, obtained combining a recurrence relation for the resolvent matrix with insights from Dyson's Brownian motion, is to show that the properties of the non-ergodic delocalised phase can be probed studying the statistics of the local resolvent in a non-standard scaling limit.

AB - We study the statistics of the local resolvent and non-ergodic properties of eigenvectors for a generalised Rosenzweig-Porter N × N random matrix model, undergoing two transitions separated by a delocalised non-ergodic phase. Interpreting the model as the combination of onsite random energies {ai} and a structurally disordered hopping, we found that each eigenstate is delocalised over N2-γ sites close in energy |aj - ai| ≤ N1-γ in agreement with Kravtsov et al. (New J. Phys., 17 (2015) 122002). Our other main result, obtained combining a recurrence relation for the resolvent matrix with insights from Dyson's Brownian motion, is to show that the properties of the non-ergodic delocalised phase can be probed studying the statistics of the local resolvent in a non-standard scaling limit.

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From non-ergodic eigenvectors to local resolvent statistics and back: A random matrix perspective

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