Renormalizovaná poruchová teorie silně korelovaných elektronů: kritické jevy v nízkorozměrných systémech

• Thesis title in Czech: Renormalizovaná poruchová teorie silně korelovaných elektronů: kritické jevy v nízkorozměrných systémech
• Thesis title in English: Renormalized perturbation theory of strongly correlated electrons: Critical phenomena in low-dimensional systems
• Key words: Nerelativistická kvantová mechanika|statistická fyzika|interagující fermiony|mnohočásticová poruchová teorie|Greenovy funkce|Feynmanovy diagramy|dynamická renormalizace hmoty - selfenergie|dynamická renormalizace náboje - vrcholové funkce|efektivní interakce|Schwingerova-Dysonova rovnice|Betheho-Salpeterovy rovnice|Wardovy identity|parketové rovnice|jednočásticová selfkonsistnce|dvoučásticová selfkonsistence|dalekodosahové uspořádání|kritické chování|itinerantní magnetismus|supravodivost
• English key words: Nonrelativistic quantum mechanics|statistical physics|interacting fermions|many-body perturbation theory|Green functions|Feynman diagrams|dynamical mass renormalization - self-energy|dynamical charge renormalization - vertex function|effective interaction|Schwinger-Dyson equation|Bethe-Salpeter equations|Ward identities|parquet equations|one-particle self-consistency|two-particle self-consistency|long-range order|critical behavior|itinerant magnetism|superconductivity
• Academic year of topic announcement: 2022/2023
• Thesis type: dissertation
• Thesis language: čeština
• Department: Fyzikální ústav AV ČR, v.v.i. (32-FZUAV)
• Supervisor: prof. RNDr. Václav Janiš, DrSc.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 02.08.2022
• Date of assignment: 02.08.2022
• Confirmed by Study dept. on: 04.10.2022

Guidelines

The student will learn during the Ph.D. program modern analytic and numerical techniques to solve problems in the theory of strongly correlated electrons in solids. The basis of the program is the quantum many-body perturbation theory with Green functions and Feynman diagrams. Infinite series of Feynman diagrams must be summed to describe reliably collective phenomena leading to critical points and phase transitions. This will be achieved via self-consistent renormalizations of the perturbation expansion. The first part of the program consists of learning and mastering advanced techniques to build up conserving self-consistent approximations systematically. Later, the construction recently developed by the supervisor and based on the two-particle perturbation theory will be used to study low-temperature, quantum critical phenomena in strong-coupling regimes. The general theory will be reduced to a mean-field-like approximation with a two-particle self-consistency determining the two-particle vertex, serving as an effective interaction. It is the generating functional of this theoretical construction from which one-particle self-energies and all physical quantities will be determined. The approximate scheme will be first applied to the local quantum critical behavior connected with the Kondo effect. Later, critical spatial fluctuations will be added to study quantum critical behavior in extended lattice systems, including phase transitions to states with long-range order. Various levels of self-consistency will be tested to reach reliable results. The ultimate objective is to develop a fully self-consistent scheme for a semi-analytic description of quantum critical behavior in low spatial dimensions.

Prerequisites: Nonrelativistic quantum mechanics, statistical physics, analysis of functions of a complex variable, programming skills

References

[1] G. Mahan: Many-Particle Physics, Kluwer Academic/Plenum Publishers, New York 1990 (2nd edition)
[2] G. Rickayzen: Green's Functions and Condensed Matter, Academic Press, London 1980
[3] H. Bruus and K. Flensferg: Many-Body Quantum Theory in Condensed Matter Physics, Oxford University Press, Oxford (UK) 2004
[4] A.M. Zagoskin: Quantum Theory of Many-Body Systems: Techniques and Applications (Graduate Texts in Contemporary Physics), Springer-Verlag, New York 1998
[5] G. Baym and L. P. Kadanoff: Conservation Laws and Correlation Functions, Phys. Rev. 124, 287 (1961)
[6] G. Baym: Self-Consistent Approximations in Many-Body Systems, Phys. Rev. 127, 1391 (1962)
[7] V. Janiš: Green Functions in the Renormalized Many-Body Perturbation Theory, Chap 7 in Simulating Correlations with Computers, (E. Pavarini and E. Koch eds), Schriften des Forschiuúngszentrums Jülich, Jülich 2021
[8] V. Janiš, P. Zalom, V. Pokorný, and A. Klíč: Strongly correlated electrons: Analytic mean-field theories with two-particle self-consistency, Phys. Rev. B 100, 195114 (2019)
[9] V. Janiš, A. Klíč, and J. Yan: Antiferromagnetic fluctuations in the one-dimensional Hubbard model, AIP Advances 10, 125127 (2020)