Teória self-konzistentných náhodných matíc

• Thesis title in Czech: Teória self-konzistentných náhodných matíc
• Thesis title in English: Self-consistent random matrix ensembles
• English key words: Power-law banded random matrix Self-consistent mean-field: Hartree|Bogoliubov--de-Gennes|slave-boson mean-field Multifractality
• Academic year of topic announcement: 2021/2022
• Thesis type: diploma thesis
• Department: Department of Condensed Matter Physics (32-KFKL)
• Supervisor: Ing. Richard Korytár, Ph.D.

Guidelines

The project can start with a numerical exploration of multifractality in a power-law banded random matrices. These matrices can be thought as single-particle Hamiltonians of one-dimensional quantum wires with a long-range hopping.
The numerical procedure would then be enhanced by the self-consistency rule. The latter could originate from superconducting correlations (Bogoliubov--de-Gennes) or electron-electron interactions.
The phenomenon of multifractality will be revisited with the self-consistency rule.
Similarly, other aspects of random-matrix theory (density of states, level correlations, criticality, Anderson transitions) can be investigated in presence of interactions at the mean-field level.

References

Random matrix theory: Wigner-Dyson statistics and beyond. (Lecture notes of a course given at SISSA (Trieste, Italy)), https://arxiv.org/abs/0911.0639
Anderson transitions, F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008)
Fluctuations of the Inverse Participation Ratio at the Anderson Transition, F. Evers and A. D. Mirlin, Phys. Rev. Lett. 84, 3690 (2000)
Random-matrix theory of quantum transport, C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997)
Random-matrix theory of Majorana fermions and topological superconductors, C. W. J. Beenakker, Rev. Mod. Phys. 87, 1037 (2015)

Preliminary scope of work in English

Theory of random matrices (RMT) describes systems with effectively non-interacting particles in a random potential, such as nuclear matter or electrons in mesoscopic devices. In this project we would extend the random-matrix paradigm by including particle interactions at the mean-field level. Examples of mean-field theories are Hartree (for the electron-electron interaction), Hartree-Fock, Bogoliubov--de-Gennes (for the Bardeen-Cooper-Schrieffer interaction) and slave-boson mean-field (for the Kondo interaction). It is foreseen that entirely new features in the RMT emerge at the interacting level, such as new phases and critical points. The purpose of this project is to undergo a numerical exploration of this rather unknown area.