1. Quantum Field Theory and Renormalization Group 2. Gaussian Expectation Values. Steepest Descent Method . 3. Universality and Continuum Limit 4. Classical Statistical Physics: One Dimension 5. Continuum Limit and Path Integral 6. Ferromagnetic Systems. Correlations 7. Phase transitions: Generalities and Examples 8. Quasi-Gaussian Approximation: Universality, Critical Dimension 9. Renormalization Group: General Formulation 10. Perturbative Renormalization Group: Explicit Calculations 11. Renormalization group: N-component fields 12. Statistical Field Theory: Perturbative Expansion 13. The sigma4 Field Theory near Dimension 4 14. The O(N) Symmetric (phi2)2 Field Theory: Large N Limit 15. The Non-Linear sigma-Model 16. Functional Renormalization Group Appendix 1. Quantum Field Theory and Renormalization Group 2. Gaussian Expectation Values. Steepest Descent Method . 3. Universality and Continuum Limit 4. Classical Statistical Physics: One Dimension 5. Continuum Limit and Path Integral 6. Ferromagnetic Systems. Correlations 7. Phase transitions: Generalities and Examples 8. Quasi-Gaussian Approximation: Universality, Critical Dimension 9. Renormalization Group: General Formulation 10. Perturbative Renormalization Group: Explicit Calculations 11. Renormalization group: N-component fields 12. Statistical Field Theory: Perturbative Expansion 13. The sigma4 Field Theory near Dimension 4 14. The O(N) Symmetric (phi2)2 Field Theory: Large N Limit 15. The Non-Linear sigma-Model 16. Functional Renormalization Group Appendix1. Quantum Field Theory and Renormalization Group 2. Gaussian Expectation Values. Steepest Descent Method . 3. Universality and Continuum Limit 4. Classical Statistical Physics: One Dimension 5. Continuum Limit and Path Integral 6. Ferromagnetic Systems. Correlations 7. Phase transitions: Generalities and Examples 8. Quasi-Gaussian Approximation: Universality, Critical Dimension 9. Renormalization Group: General Formulation 10. Perturbative Renormalization Group: Explicit Calculations 11. Renormalization group: N-component fields 12. Statistical Field Theory: Perturbative Expansion 13. The sigma4 Field Theory near Dimension 4 14. The O(N) Symmetric (phi2)2 Field Theory: Large N Limit 15. The Non-Linear sigma-Model 16. Functional Renormalization Group Appendix 1. Quantum Field Theory and Renormalization Group 2. Gaussian Expectation Values. Steepest Descent Method . 3. Universality and Continuum Limit 4. Classical Statistical Physics: One Dimension 5. Continuum Limit and Path Integral 6. Ferromagnetic Systems. Correlations 7. Phase transitions: Generalities and Examples 8. Quasi-Gaussian Approximation: Universality, Critical Dimension 9. Renormalization Group: General Formulation 10. Perturbative Renormalization Group: Explicit Calculations 11. Renormalization group: N-component fields 12. Statistical Field Theory: Perturbative Expansion 13. The sigma4 Field Theory near Dimension 4 14. The O(N) Symmetric (phi2)2 Field Theory: Large N Limit 15. The Non-Linear sigma-Model 16. Functional Renormalization Group Appendix

Oxford University Press eBooksJan 1, 20139 citations

Nonperturbative Renormalization Group

Taiki Haga

Springer thesesJan 1, 20190 citations

The nonperturbative functional renormalization group and its applications

N. Dupuis, Léonie Canet, Astrid Eichhorn +4 more

arXiv:2006.04853Physics ReportsJan 18, 2021644 citations

Functional renormalization group beyond the perturbative regime

Nils Wentzell

This thesis aims at developing new schemes for the treatment of correlation effects in condensed matter systems using quantum field theoretical approaches. In particular, our goal is to extend the description of correlation physics at the two-particle level. This is necessary for an unbiased treatment of condensed matter systems that exhibit electronic correlations and competing ordering tendencies. In this respect, the functional renormalization group (fRG) approaches have surely contributed substantially over the last years, as they account for all scattering channels and their mutual feedback effects in an unbiased way. In spite of its flexibility, the application of the fRG is limited by its inherent perturbative nature. To go beyond the conventional weak-coupling implementations, we discuss the general idea to extend fRG based computational schemes by using an exactly solvable interacting reference problem as starting point for the RG flow. The systematic expansion around this solution accounts for a non-perturbative inclusion of correlations at both, the one-particle (self-energy) and two-particle (vertex functions) level. The full treatment of the two-particle vertex functions, however, poses a huge limitation to the numerical performance, not only in the fRG, but in several forefront many-body algorithms. In this perspective, we provide a detailed diagrammatic analysis of the frequency and momentum structures of the vertex functions, together with their physical interpretation. This constitutes the basis for sophisticated parametrization schemes. We then explain the technical details necessary for cutting-edge numerical implementations, and further benchmark our ideas using refined implementations of both, the fRG and the parquet approximation (PA).This thesis aims at developing new schemes for the treatment of correlation effects in condensed matter systems using quantum field theoretical approaches. In particular, our goal is to extend the description of correlation physics at the two-particle level. This is necessary for an unbiased treatment of condensed matter systems that exhibit electronic correlations and competing ordering tendencies. In this respect, the functional renormalization group (fRG) approaches have surely contributed substantially over the last years, as they account for all scattering channels and their mutual feedback effects in an unbiased way. In spite of its flexibility, the application of the fRG is limited by its inherent perturbative nature. To go beyond the conventional weak-coupling implementations, we discuss the general idea to extend fRG based computational schemes by using an exactly solvable interacting reference problem as starting point for the RG flow. The systematic expansion around this solution accounts for a non-perturbative inclusion of correlations at both, the one-particle (self-energy) and two-particle (vertex functions) level. The full treatment of the two-particle vertex functions, however, poses a huge limitation to the numerical performance, not only in the fRG, but in several forefront many-body algorithms. In this perspective, we provide a detailed diagrammatic analysis of the frequency and momentum structures of the vertex functions, together with their physical interpretation. This constitutes the basis for sophisticated parametrization schemes. We then explain the technical details necessary for cutting-edge numerical implementations, and further benchmark our ideas using refined implementations of both, the fRG and the parquet approximation (PA).

Jul 11, 20170 citations

Functional renormalization group for non-Hermitian and $\mathcal{PT}$ -symmetric systemsFunctional renormalization group for non-Hermitian and $\mathcal{PT}$-symmetric systems

Lukas Grunwald

We generalize the vertex expansion approach of the functional renormalization group to non-Hermitian systems. As certain anomalous expectation values might not vanish, additional terms as compared to the Hermitian case can appear in the flow equations. We investigate the merits and shortcomings of the vertex expansion for non-Hermitian systems by considering an exactly solvable $\mathcal{PT}$ -symmetric non-linear toy-model and reveal, that in this model, the fidelity of the vertex expansion in a perturbatively motivated truncation schema is comparable with that of the Hermitian case. The vertex expansion appears to be a viable method for studying correlation effects in non-Hermitian systems.We generalize the vertex expansion approach of the functional renormalization group to non-Hermitian systems. As certain anomalous expectation values might not vanish, additional terms as compared to the Hermitian case can appear in the flow equations. We investigate the merits and shortcomings of the vertex expansion for non-Hermitian systems by considering an exactly solvable $\mathcal{PT}$-symmetric non-linear toy-model and reveal, that in this model, the fidelity of the vertex expansion in a perturbatively motivated truncation schema is comparable with that of the Hermitian case. The vertex expansion appears to be a viable method for studying correlation effects in non-Hermitian systems.

May 1, 20227 citations

Renormalized Perturbation Theory and Field-Theoretic Renormalization Group

Lothar Schäfer

Springer eBooksJan 1, 19991 citation

Functional renormalization group for non-Hermitian and

\mathcal{PT}

-symmetric systems

\mathcal{PT}

-symmetric non-linear toy-model and reveal, that in this model, the fidelity of the vertex expansion in a perturbatively motivated truncation schema is comparable with that of the Hermitian case. The vertex expansion appears to be a viable method for studying correlation effects in non-Hermitian systems.