Priv. Doz. Francesco Parisen Toldin

Research associate (Wissenschaftlicher Mitarbeiter)

About

Privat Dozent and Research associate at the RWTH AAchen University

Research interests: critical phenomena in classical and quantum models, quantum entanglement, strongly correlated electron systems, high-precision Monte Carlo simulations

Available Thesis

Are you looking for a Bachelor or Master Thesis in Theoretical Condensed Matter Physics? Please contact me at the email address above.

Résumé

Summary

Francesco Parisen Toldin

  • Habilitation (Lehrbefähigung) in Theoretical Physics (12/2021)
  • PhD in Physics (11/2007)
  • Email: parisentoldin@physik.rwth-aachen.de
  • Current position: Research associate (Wissenschaftlicher Mitarbeiter) at the Institute for Theoretical Solid State Physics, RWTH Aachen University, Germany

Employement

Research associate

2/2023 - present

Institute for Theoretical Solid State Physics, RWTH Aachen University (Germany)

  • 2023-2025: Position funded by an individual grant (Sachbeihilfe) No. 414456783 of the German Research Foundation (Deutsche Forschungsgemeinschaft)

Research associate

11/2012 - 6/2022

Institut für Theoretische Physik und Astrophysik, University of Würzburg (Germany); group of Prof. F. Assaad

  • 2019-present: Position funded by an individual grant (Sachbeihilfe) No. 414456783 of the German Research Foundation (Deutsche Forschungsgemeinschaft)
  • 2016-2019: Position funded by the DFG Research Group FOR1807
  • Supervision: 3 Bachelor Thesis, 1 Master Thesis

Postdoctoral researcher

10/2010 - 10/2012

Max-Planck Institute for the Physics of Complex Systems (Dresden, Germany); group “Collective Phenomena in Solid State and Materials Physics” of Dr. S. Kirchner

  • Research topic: “Interplay between disorder and interactions in strongly correlated electron systems”

Postdoctoral researcher

11/2007 - 10/2010

Max-Planck Institute for Metal Research (Stuttgart, Germany); group of Prof. S. Dietrich

The institute has been renamed Max Planck Institute for Intelligent Systems in 2011

  • Research topic: “Critical Casimir effect”

Scientific activities

Referee

I am referee (last 5 years) for Eur. Phys. B, J. Phys. A, JSTAT, Phys. Rev. B, Phys. Rev. E, Phys. Rev. Lett., SciPost Physics

Conferences organization

  • Focus Session "Recent Progresses in Criticality in the Presence of Boundaries and Defects" at the DPG Spring meeting 2024, 20 March 2024, Berlin, Germany. Program here and here.
  • bbc2019: Boundary and Bulk Criticality 2019, 1-4 October 2019, Würzburg, Germany
  • bbc2022: Boundary and Bulk Criticality 2022, 21-25 February 2022, online

Education

PhD studies in Physics

2004 - 2007

Scuola Normale Superiore (Pisa, Italy)

16/11/2007: PhD in Physics (Perfezionamento) defense with maximum grade 70/70 cum laude

Title of PhD thesis: "Critical behaviour of magnetic systems in the presence of quenched random disorder". Advisor: prof. E. Vicari (University of Pisa, Department of Physics, Italy)

Undergraduate studies in Physics

1998 - 2003

University of Pisa (Italy) and Scuola Normale Superiore (Pisa, Italy)

20/12/2003: Diploma at Scuola Normale Superiore with maximum grade 70/70 cum laude

28/3/2003: Master degree in Physics (“Laurea specialistica”) with maximum grade 110/110 cum laude. Advisor: prof. E. Vicari (University of Pisa, Department of Physics, Italy)

13/1/2003: Bachelor degree in Physics (“Laurea”) with maximum grade 110/110 cum laude. Advisor: prof. E. Vicari (University of Pisa, Department of Physics, Italy)

9/1998: Successful admission at Scuola Normale Superiore (Pisa, Italy) for the Physics sector

School

6/1998: Diploma at scientific high school “Liceo G. B. Quadri” (Vicenza, Italy) with maximum grade 60/60

Publications

Research papers

  1. F. Parisen Toldin, F. F. Assaad, M. A. Metlitski, Extraordinary transition at the edge of a correlated topological insulator, arXiv:2508.00999
  2. D. Przetakiewicz, S. Wessel, F. Parisen Toldin, Boundary operator product expansion coefficients of the three-dimensional Ising universality class, Phys. Rev. Research 7, L032051 (2025), arXiv:2502.14965
  3. F. Parisen Toldin, A. Krishnan, M. A. Metlitski, Universal finite-size scaling in the extraordinary-log boundary phase of 3d O(N) model, Phys. Rev. Research 7, 023052 (2025), arXiv:2411.05089
  4. R. Bärwolf, A. Sushchyev, F. Parisen Toldin, S. Wessel, Phase transitions in the spin-1/2 Heisenberg antiferromagnet on the dimerized diamond lattice, Phys. Rev. B 111, 085136 (2025), arXiv:2410.13706
  5. J. Schwab, F. Parisen Toldin, F. F. Assaad, "Phase diagram of the spin S, SU(N) antiferromagnet on a square lattice", Phys. Rev. B 108, 115151 (2023), arXiv:2304.07329

    Editors' Suggestion

  6. F. Parisen Toldin, The ordinary surface universality class of the 3D O(N) model, Phys. Rev. B 108, L020404 (2023), arXiv:2303.16683
  7. F. Parisen Toldin, M. A. Metlitski, Boundary Criticality of the 3D O(N) Model: From Normal to Extraordinary, Phys. Rev. Lett. 128, 215701 (2022), arXiv:2111.03613
  8. F. Parisen Toldin, Finite-Size Scaling at fixed Renormalization-Group invariant, Phys. Rev. E 105, 034137 (2022), arXiv:2112.00392
  9. ALF Collaboration: F. F. Assaad, M. Bercx, F. Goth, A. Götz, J. S. Hofmann, E. Huffman, Z. Liu, F. Parisen Toldin, J. S. E. Portela, J. Schwab, The ALF (Algorithms for Lattice Fermions) project release 2.0. Documentation for the auxiliary-field quantum Monte Carlo code, SciPost Physics Codebase 1-r2.0 (2022), arXiv:2012.11914
  10. F. Parisen Toldin, Boundary critical behavior of the three-dimensional Heisenberg universality class, Phys. Rev. Lett. 126, 135701 (2021), arXiv:2012.00039
  11. M. Weber, F. Parisen Toldin, M. Hohenadler, Competing Orders and Unconventional Criticality in the Su-Schrieffer-Heeger Model, Phys. Rev. Research 2, 023013 (2020), arXiv:1905.05218
  12. P. Ghosh, T. Müller, F. Parisen Toldin, J. Richter, R. Narayanan, R. Thomale, J. Reuther, Y. Iqbal, Quantum paramagnetism and helimagnetic orders in the Heisenberg model on the body centered cubic lattice, Phys. Rev. B 100, 014420 (2019), arXiv:1902.01179

    Editors' Suggestion

  13. F. Parisen Toldin, T. Sato, F. F. Assaad, Mutual information in heavy-fermion systems, Phys. Rev. B 99, 155158 (2019), arXiv:1811.11194
  14. L. Weber, F. Parisen Toldin, S. Wessel, Nonordinary edge criticaliy of two-dimensional quantum critical magnets, Phys. Rev. B 98, 140403(R) (2018), arXiv:1804.06820
  15. F. Parisen Toldin, F. F. Assaad, Entanglement Hamiltonian of interacting fermionic models, Phys. Rev. Lett. 121, 200602 (2018), arXiv:1804.03163
  16. Z. Wang, F. F. Assaad, F. Parisen Toldin, Finite-size effects in canonical and grand-canonical Quantum Monte Carlo simulations for fermions, Phys. Rev. E 96, 042131 (2017), arXiv:1706.01874
  17. F. Parisen Toldin, F. F. Assaad, S. Wessel, Critical behavior in the presence of an order-parameter pinning field, Phys. Rev. B 95, 014401 (2017), arXiv:1607.04270

    Editors' Suggestion

  18. Y. Iqbal, R. Thomale, F. Parisen Toldin, S. Rachel, J. Reuther, Functional Renormalization Group for three-dimensional Quantum Magnetism, Phys. Rev. B 94, 140408(R) (2016), arXiv:1604.03438
  19. P.-J. Hsu, J. Kügel, J. Kemmer, F. Parisen Toldin, T. Mauerer, M. Vogt, F. Assaad, M. Bode, Coexistence of Charge- and Ferromagnetic-Order in fcc Fe, Nat. Commun. 7:10949 (2016), arXiv:1603.09100
  20. F. Parisen Toldin, M. Hohenadler, F. F. Assaad, I. F. Herbut, Fermionic quantum criticality in honeycomb and $\pi$-flux Hubbard models: Finite-size scaling of renormalization-group-invariant observables from quantum Monte Carlo, Phys. Rev. B 91, 165108 (2015), arXiv:1411.2502
  21. F. Parisen Toldin, M. Tröndle, S. Dietrich, Line contribution to the critical Casimir force between a homogeneous and a chemically stepped surface, J. Phys.: Condens. Matter 27, 214010 (2015), arXiv:1409.5536

    Special Issue on "Casimir Forces"

  22. M. Hohenadler, F. Parisen Toldin, I. F. Herbut, F. F. Assaad, Phase diagram of the Kane-Mele-Coulomb model, Phys. Rev. B 90, 085146 (2014), arXiv:1407.2708

    Editors' Suggestion

  23. F. F. Assaad, T. C. Lang, F. Parisen Toldin, Entanglement Spectra of Interacting Fermions in Quantum Monte Carlo Simulations, Phys. Rev. B 89, 125121 (2014), arXiv:1311.5851v2

    Editors' Suggestion

  24. F. Parisen Toldin, Critical Casimir force in the presence of random local adsorption preference, Phys. Rev. E 91, 032105 (2015), arXiv:1308.5220
  25. F. Parisen Toldin, M. Tröndle, S. Dietrich, Critical Casimir forces between homogeneous and chemically striped surfaces, Phys. Rev. E 88, 052110 (2013), arXiv:1303.6104v2
  26. F. Parisen Toldin, J. Figgins, S. Kirchner, D. K. Morr, Disorder and quasiparticle interference in heavy-fermion materials, Phys. Rev. B 88, 081101(R) (2013), arXiv:1210.3638v2
  27. F. Parisen Toldin, A. Pelissetto, E. Vicari, Finite-size scaling in two-dimensional Ising spin glass models, Phys. Rev. E 84, 051116 (2011), arXiv:1106.5720v2
  28. F. Parisen Toldin, Improvement of Monte Carlo estimates with finite-size scaling at fixed phenomenological coupling, Phys. Rev. E 84, 025703(R) (2011), arXiv:1104.2500v2
  29. F. Parisen Toldin, S. Dietrich, Critical Casimir forces and adsorption profiles in the presence of a chemically structured substrate, J. Stat. Mech. P11003 (2010), arXiv:1007.3913v2
  30. F. Parisen Toldin, A. Pelissetto, E. Vicari, Universality of the glassy transitions in the two-dimensional ±J Ising model, Phys. Rev. E 82, 021106 (2010), arXiv:1005.4491v1
  31. F. Parisen Toldin, A. Pelissetto, E. Vicari, Strong-disorder paramagnetic-ferromagnetic fixed point in the square-lattice ±J Ising model, J. Stat. Phys. 135, 1039 (2009), arXiv:0811.2101v3

    Special issue dedicated to Edouard Bréezin and Giorgio Parisi

  32. M. Hasenbusch, F. Parisen Toldin, A. Pelissetto, E. Vicari, Universal dependence on disorder of two-dimensional randomly diluted and random-bond ±J Ising models, Phys. Rev. E 78, 011110 (2008), arXiv:0804.2788v2
  33. M. Hasenbusch, F. Parisen Toldin, A. Pelissetto, E. Vicari, Multicritical Nishimori point in the phase diagram of the ±J Ising model on a square lattice, Phys. Rev. E 77, 051115 (2008), arXiv:0803.0444v1
  34. M. Hasenbusch, F. Parisen Toldin, A. Pelissetto, E. Vicari, Magnetic-glassy multicritical behavior of the three-dimensional ±J Ising model, Phys. Rev. B 76, 184202 (2007), arXiv:0707.2866v1
  35. M. Hasenbusch, F. Parisen Toldin, A. Pelissetto, E. Vicari, Critical behavior of the three-dimensional ±J Ising model at the ferromagnetic transition line, Phys. Rev. B 76, 094402 (2007), arXiv:0704.0427v1
  36. M. Hasenbusch, F. Parisen Toldin, A. Pelissetto, E. Vicari, The universality class of 3D site-diluted and bond-diluted Ising systems, J. Stat. Mech. P02016 (2007), arXiv:cond-mat/0611707v2
  37. F. Parisen Toldin, A. Pelissetto and E. Vicari, Critical behaviour of the random-anisotropy model in the strong-anisotropy limit, J. Stat. Mech. P06002 (2006), arXiv:cond-mat/0604124v2
  38. A. Butti and F. Parisen Toldin, The critical equation of state of the three-dimensional O(N) universality class: N > 4, Nucl. Phys. B 704, 527 (2005), arXiv:hep-lat/0406023v2
  39. F. Parisen Toldin, A. Pelissetto and E. Vicari, The scaling equation of state of the 3-D O(4) universality class, JHEP 0307, 029 (2003), arXiv:hep-ph/0305264v2

Proceedings with peer review

  1. F. Parisen Toldin, Surface critical behavior of the three-dimensional O(3) model, J. Phys.: Conf. Ser. 2207, 012003 (2022), arXiv:2111.11762

    Proceedings of CCP2021: XXXII IUPAP Conference on Computational Physics

  2. F. Parisen Toldin, F. F. Assaad, Entanglement studies of interacting fermionic models, J. Phys.: Conf. Ser. 1163, 012056 (2019), arXiv:1810.06595

    "Proceedings of CSP 2018: International Conference on Computer Simulation in Physics and beyond, 24-27 September 2018, Moscow, Russia

  3. F. Parisen Toldin, S. Dietrich, Critical Casimir forces involving a chemically structured substrate, Quantum Field Theory Under the Influence of External Conditions (QFEXT09), p. 355, ed. K. A. Milton and M. Bordag, World Scientific, 2010, arXiv:1007.3698v1

    Proceedings of the 9th Conference on Quantum Field Theory Under the Influence of External Conditions (QFEXT09), 21-25 September 2009, Norman, OK, U.S.A.

  4. M. Hasenbusch, F. Parisen Toldin, A. Pelissetto, E. Vicari, Critical and multicritical behavior of the ±J Ising model in two and three dimensions, J. Phys.: Conf. Ser. 145, 012055 (2009), arXiv:0810.0685v1

    Proceedings of HFM2008: Highly Frustrated Magnetism 2008, 7-12 September 2008, Braunschweig, Germany

Other Proceedings

  1. M. Raczkowski, M. Bercx, M. Weber, S. Beyl, J. Hofmann, F. Parisen Toldin, M. Hohenadler, F. F. Assaad, Quantum Monte Carlo simulations of strongly correlated electron systems: the dimensional crossover,Proceedings of the NIC Symposium 2016; 11-12 February 2016, Jülich, Germany, NIC Series vol. 48; ed. K. Binder, M. Müller, M. Kremer, A. Schnurpfeil, p. 241
  2. F. F. Assaad, M. Bercx, F. Goth, M. Hohenadler, F. Parisen Toldin, M. Weber, J. Werner, Correlated topological insulators and semi-metals, Proceedings of the NIC Symposium 2014; 12-13 February 2014, Jülich, Germany, NIC Series vol. 47; ed. K. Binder, G. Münster, M. Kremer, p. 251
  3. A. Butti, F. Parisen Toldin, A. Pelissetto and E. Vicari, The scaling equation of state of the three-dimensional O(N) universality class: N ≥ 4, Nucl. Phys. Proc. Suppl. B 140, 808 (2005), arXiv:hep-lat/0409054v1"

    Talk given at 22nd International Symposium on Lattice Field Theory (Lattice 2004), 21-26 June 2004, Batavia, Illinois, U.S.A."

Teaching

Statistical Physics (Master): Summer Semester 2024

Link to the moddle webpage (requires RWTH account)

Log of the lessons:
  • 10/4: Introduction: basic phenomenology, critical exponents, scaling ansatz. Universality in critical phenomena. Basic models: Ising model and applications of it, O(N) models. Exact solution of the Ising model in one dimension.
  • 17/4: Mean-field theory of the Ising model
  • 24/4: Infinite-range Ising model. Landau theory of phase transitions: Ising model and O(N) models. Appearance of Goldstone modes in the low-temperature phase. Landau theory of tricritical point. Limit of applicability of mean field theory: Ginzburg criterion.
  • 8/5: Introduction to the Renormalization Group. Coarse graining and scale transformations. Basic properties of the RG flow, fixed points and their classification. Linearized RG flow, scaling fields. Classification of scaling fields at a fixed point. Critical surface, flow of the RG close to a fixed point.
  • 15/5: Identity operator. Scaling law of the free energy. Relation between the critical exponents and the RG dimensions of the relevant operators. Scaling law for the correlation function. Irrelevant operators and corrections to scaling. Scaling law for the mean field theory.
  • 29/5: Recap of scaling form of the free energy. Irrelevant operators. Crossover behavior. Crossover to long-range behavior. RG flow of the 1D Ising model.
  • 5/6: Finished RG-flow of 1D Ising model. Real space RG for the Ising model on the triangular lattice. Midgal-Kadanoff RG.
  • 12/6: Field-theoretical approach to critical phenomena. Hubbard-Stratonovich transformation, effective Φ4 theory. Gaussian fixed point. Two-point function at the Gaussian fixed point.
  • 19/6: Breaking of hyperscaling: dangerously irrelevant operator at the Gaussian fixed point. Failure of naïve perturbation theory in d < 4. Asymptotic series and Borel summability. Canonical and anomalous dimension: the role of the cutoff. Introduction to functional methods in field theories. Wick's theorem.
  • 26/6: Feynman diagrams. Generator of connected and 1PI correlations. Feynman diagrams in momentum space.
  • 3/7: Loop expansion. Divergences in field theories. Classification of theories in terms of renormalization. Operator insertions.
  • 10/7: Regularization methods: cuf-off/Pauli-Villars, lattice, dimensional regularization. Mean-field theory and renormalization. Renormalization of Φ4 theory in d=4: general setup, calculation of 1PI 2-point function in dimensional regularization. Started computation of 1PI 4-point function.
  • 17/7: Finished computation of 1PI 4-point function. Renormalization prescription: minimal subtraction and prescription consistent with tree-level. Functionals for correlations of renormalized fields. Callan-Symanzik equations. Massless limit of CS equations. Homogeneous CS equations (RG equations) for a massless and a massive theory. Solution to RG equations.

Critical Phenomena

In statistical physics, Critical Phenomena refer to the universal behavior which is found close to a continous phase transition. The theoretical framework which allows to understand Critical Phenomena is the so-called Renormalization Group, which plays an important role in many area of physics. The course provides an introduction to Critical Phenomena and Renormalization Group theory, as well as some applications.

Contents of the course:
  • Basic phenomenology: universality, scaling, critical exponents
  • Mean-field theories
  • Renormalization Group theory and scaling behavior
  • Real space Renormalization Group for spin models
  • Renormalization Group in momentum space
  • Duality and high/low temperature expansion
  • Finite-size Scaling theory
  • Exact results
Literature:
  • H. Nishimori, G. Ortiz, Elements of Phase Transitions and Critical Phenomena, Oxford University Press (2011)
  • I. Herbut, A Modern Approach to Critical Phenomena, Cambridge University Press (2006)
  • J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press (1996)
  • N. Goldenfeld, Lectures on phase transitions and the Renormalization Group, Addison-Wesley (1992)
  • J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford University Press (2002)
Review article:

Critical Phenomena: Winter Semester 2018/19

Contents of the course: see here

Link to the moddle webpage where the exercise sheets can be downloaded

Log of the lessons:
  • Tue 16/10. Introduction: basic phenomenology, critical exponents, scaling ansatz.
  • Wed 17/10. Universality in critical phenomena. Basic models: Ising model and applications of it, O(N) models. Exact solution of the Ising model in one dimension.
  • Tue 23/10. Mean field theory of the Ising model.
  • Wed 24/10. Correlations function in the mean-field Ising model. Correction of the first exercise sheet.
  • Tue 30/10. Correction of the second part of the first exercise sheet. Infinite-range Ising model.
  • Wed 31/10. Landau theory of phase transitions: Ising model and O(N) models. Appearance of Goldstone modes in the low-temperature phase.
  • Tue 6/11. Correction of the second exercise sheet.
  • Wed 7/11. Landau theory of tricritical point. Product probability measure approach to mean field.
  • Tue 13/11. Limit of applicability of mean field theory: Ginzburg criterion, upper critical dimension. Introduction to the Renormalization Group. Coarse graining and scale transformations.
  • Wed 14/11. Correction of the third exercise sheet. Basic properties of the RG flow, fixed points and their classification.
  • Tue 20/11. Linearized RG flow, scaling fields. Classification of the fixed points, and of the scaling fields at a fixed point. Critical surface, flow of the RG close to a fixed point.
  • Wed 21/11. Identity operator. Scaling law of the free energy. Relation between the critical exponents and the RG dimensions of the relevant operators. Scaling law for the correlation function.
  • Tue 27/11. Irrelevant operators and corrections to scaling. Scaling law for the mean field theory. Crossover behavior.
  • Wed 28/11. Finished discussion on crossover behavior. Crossover to long range behavior. Finite-Size Scaling theory.
  • Tue 4/12. RG equations for the 1-dimensional Ising model.
  • Wed 5/12. Real space RG for the Ising model on the triangular lattice.
  • Tue 11/12. Correction of the fourth exercise sheet. Migdal-Kadanoff Renormalization Group.
  • Wed 12/12. Field-theoretical approach to critical phenomena. Hubbard-Stratonovich transformation, effective Φ4 theory. Gaussian fixed point.
  • Tue 18/12. Correlation function at the Gaussian fixed point. Dangerously irrelevant operators.
  • Wed 19/12. Failure of naïve perturbation theory in d < 4. Asymptotic series and Borel summability. Canonical and anomalous dimension: the role of the cutoff. Introduction to the ε-expansion.
  • Tue 8/1. Functional techniques for field theories. Wick's theorem. Momentum-space Renormalization Group: first order in perturbation theory.
  • Wed 9/1. Computation of the correction to the quartic interaction in second-order perturbation theory.
  • Tue 15/1. RG equation in differential form. Fixed points of the RG in ε-expansion: Gaussian and Wilson-Fisher fixed points. Critical exponents in ε-expansion.
  • Wed 16/1. Irrelevant operators. Computation of the RG dimension of the Φ6 operator at the Gaussian and Wilson-Fisher fixed points.
  • Tue 22/1. Critical exponents of the O(N) universality class. Redundant operators.
  • Wed 23/1. High- and low-temperature series expansion of the bidimensional Ising model. Kramers-Wannier duality. Exact value of the critical coupling and critical energy in the Ising model on a square lattice.
  • Tue 29/1. Duality by Fourier transformation. Application to the Ising and Potts model on a square lattice. Exact value of the critical coupling of the Potts model on a square lattice.
  • Wed 30/1. Correction of the fifth exercise sheet.
  • Wed 6/2. Partition function and critical behavior of the O(N) model in the large-N limit.

Critical Phenomena: Winter Semester 2015/16

Contents of the course: see here

Link to the moddle webpage where the exercise sheets can be downloaded

Log of the lessons:

  • Tue 13/10. Introduction: basic phenomenology, critical exponents, scaling ansatz.
  • Wed 14/10. Universality in critical phenomena. Basic models: Ising model and applications of it, O(N) models. Exact solution of the Ising model in one dimension.
  • Tue 20/10. Mean field theory of the Ising model.
  • Wed 21/10. Correlations function in the mean-field Ising model. Correction of the first exercise sheet.
  • Tue 27/10. Correction of the second part of the first exercise sheet. Infinite-range Ising model.
  • Wed 28/10. Landau theory of phase transitions: Ising model and O(N) models. Appearance of Goldstone modes in the low-temperature phase.
  • Tue 3/11. Correction of the second exercise sheet.
  • Wed 4/11. Landau theory of tricritical point. Product probability measure approach to mean field.
  • Tue 10/11. Limit of applicability of mean field theory: Ginzburg criterion, upper critical dimension. Introduction to the Renormalization Group. Coarse graining and scale transformations.
  • Wed 11/11. Correction of the third exercise sheet. Basic properties of the RG flow, fixed points and their classification. Linearized RG flow, scaling fields.
  • Tue 17/11. Classification of the fixed points, and of the scaling fields at a fixed point. Critical surface, flow of the RG close to a fixed point. Identity operator. Scaling law of the free energy. Relation between the critical exponents and the RG dimensions of the relevant operators.
  • Wed 18/11. Scaling law for the correlation function. Irrelevant operators and corrections to scaling.
  • Mon 23/11. Scaling law for the mean field theory. Crossover behavior.
  • Tue 24/11. Crossover to long range behavior. Finite-Size Scaling theory.
  • Wed 25/11. RG equations for the 1-dimensional Ising model.
  • Tue 1/12. Real space RG for the Ising model on the triangular lattice.
  • Wed 2/12. Correction of the fourth exercise sheet. Migdal-Kadanoff Renormalization Group.
  • Mon 7/12. Field-theoretical approach to critical phenomena. Hubbard-Stratonovich transformation, effective Φ4 theory. Gaussian fixed point.
  • Tue 8/12. Correlation function at the Gaussian fixed point. Dangerously irrelevant operators.
  • Wed 9/12. Failure of naïve perturbation theory in d < 4. Asymptotic series and Borel summability. Canonical and anomalous dimension: the role of the cutoff. Introduction to the ε-expansion. Functional techniques for field theories. Wick's theorem.
  • Tue 15/12. Momentum-space Renormalization Group: first order in perturbation theory.
  • Wed 16/12. Computation of the correction to the quartic interaction in second-order perturbation theory.
  • Tue 12/1. RG equation in differential form. Fixed points of the RG in ε-expansion: Gaussian and Wilson-Fisher fixed points. Critical exponents in ε-expansion.
  • Wed 13/1. Irrelevant operators. Computation of the RG dimension of the Φ6 operator at the Gaussian and Wilson-Fisher fixed points.
  • Tue 19/1. High- and low-temperature series expansion of the bidimensional Ising model. Kramers-Wannier duality. Exact value of the critical coupling and critical energy in the Ising model on a square lattice.
  • Mon 25/1. Correction of the fifth exercise sheet. Redundant operators.
  • Tue 26/1. Duality by Fourier transformation. Application to the Ising and Potts model on a square lattice. Exact value of the critical coupling of the Potts model on a square lattice.
  • Wed 27/1. Partition function of the O(N) model in the large-N limit.
  • Tue 2/2. Correction of the sixth exercise sheet.
  • Wed 3/2. Critical behavior of the O(N) model in the large-N limit.

Critical Phenomena: Winter Semester 2014/15

Contents of the course: see here

Link to the moddle webpage where the exercise sheets can be downloaded

Log of the lessons:

  • Tue 7/10. Introduction: basic phenomenology, critical exponents, scaling ansatz.
  • Wed 8/10. Universality in critical phenomena. Basic models: Ising model and applications of it, O(N) models. Exact solution of the Ising model in one dimension.
  • Tue 14/10. Mean field theory of the Ising model.
  • Wed 15/10. Correction of the first exercise sheet.
  • Tue 21/10. Infinite-range Ising model. Landau theory of phase transitions: Ising model and O(N) models. Appearance of Goldstone modes in the low-temperature phase.
  • Wed 22/10. Landau theory of tricritical point. Product probability measure approach to mean field. Limit of applicability of mean field theory: Ginzburg criterion, upper critical dimension.
  • Tue 28/10. Correction of the second exercise sheet.
  • Wed 29/10. Correction of the second part of the second exercise sheet. Introduction to the Renormalization Group. Coarse graining and scale transformations.
  • Tue 4/11. Basic properties of the RG flow, fixed points. Linearized RG flow. Classification of the fixed points, and of the scaling fields at a fixed point. Critical surface, flow of the RG close to a fixed point.
  • Wed 5/11. Correction of the third exercise sheet. Identity operator. Scaling law of the free energy. Relation between the critical exponents and the RG dimensions of the relevant operators.
  • Tue 11/11. Scaling law for the correlation function. Irrelevant operators and corrections to scaling.
  • Wed 12/11. Scaling law for the mean field theory. Crossover behavior.
  • Tue 18/11. Crossover to long range behavior. Finite-Size Scaling theory.
  • Wed 19/11. RG equations for the 1-dimensional Ising model.
  • Thu 20/11. Real space RG for the Ising model on the triangular lattice.
  • Tue 25/11. Correction of the fourth exercise sheet. Migdal-Kadanoff Renormalization Group.
  • Wed 26/11. Field-theoretical approach to critical phenomena. Hubbard-Stratonovich transformation, effective Φ4 theory. Gaussian fixed point.
  • Tue 2/12. Correlation function at the Gaussian fixed point. Dangerously irrelevant operators.
  • Wed 3/12. Failure of naïve perturbation theory in d < 4. Asymptotic series and Borel summability. Canonical and anomalous dimension: the role of the cutoff. Introduction to the ε-expansion. Functional techniques for field theories. Wick's theorem.
  • Thu 4/12. Momentum-space Renormalization Group: first order in perturbation theory.
  • Tue 9/12. Computation of the correction to the quartic interaction in second-order perturbation theory. RG equation in differential form.
  • Wed 10/12. Fixed points of the RG in ε-expansion: Gaussian and Wilson-Fisher fixed points. Critical exponents in ε-expansion.
  • Thu 11/12. Irrelevant operators. Computation of the RG dimension of the Φ6 operator at the Gaussian and Wilson-Fisher fixed points. Redundant operators.
  • Tue 16/12. High- and low-temperature series expansion of the bidimensional Ising model. Kramers-Wannier duality. Exact value of the critical coupling and critical energy in the Ising model on a square lattice.
  • Wed 17/12. Duality by Fourier transformation. Application to the Ising and Potts model on a square lattice. Exact value of the critical coupling of the Potts model on a square lattice.
  • Thu 18/12. Correction of the fifth exercise sheet.
  • Wed 21/1. Correction of the sixth exercise sheet.
  • Tue 27/1. Partition function of the O(N) model in the large-N limit.
  • Thu 29/1. Critical behavior of the O(N) model in the large-N limit.

Critical Phenomena: Winter Semester 2013/14

Contents of the course: see here

Link to the moddle webpage where the exercise sheets can be downloaded

Log of the lessons:

  • Mon 14/10. Introduction: basic phenomenology, critical exponents, scaling ansatz, universality.
  • Wed 16/10. Basic models: Ising model and applications of it, O(N) models. Exact solution of the Ising model in one dimension.
  • Mon 21/10. Mean field theory of the Ising model.
  • Wed 24/10. Correction of the first exercise sheet.
  • Mon 28/10. Infinite-range Ising model. Landau theory of phase transitions: Ising model and O(N) models. Appearance of Goldstone modes in the low-temperature phase.
  • Wed 30/10. Landau theory of tricritical point. Product probability measure approach to mean field.
  • Mon 4/11 Correction of the second exercise sheet. Limit of applicability of mean field theory: Ginzburg criterion, upper critical dimension.
  • Wed 6/11 Introduction to the Renormalization Group. Coarse graining and scale transformations. Basic properties of the RG flow, fixed points.
  • Mon 11/11 Linearized RG flow. Classification of the fixed points, and of the scaling fields at a fixed point. Critical surface, flow of the RG close to a fixed point. Identity operator.
  • Wed 13/11 Correction of the third exercise sheet. Scaling law of the free energy. Relation between the critical exponents and the RG dimensions of the relevant operators.
  • Mon 18/11 Scaling law for the correlation functions. Irrelevant operators and corrections to scaling.
  • Wed 20/11 RG equations for the 1-dimensional Ising model.
  • Mon 25/11 Scaling law for the mean field theory. Crossover behavior.
  • Tue 26/11 Real space RG for the Ising model on the triangular lattice.
  • Wed 27/11 Correction of the fourth exercise sheet. Migdal-Kadanoff Renormalization Group.
  • Mon 2/12 Field-theoretical approach to critical phenomena. Hubbard-Stratonovich transformation, effective Φ4 theory.
  • Tue 3/12 Gaussian fixed point. Dangerously irrelevant operators.
  • Mon 16/12 Failure of naïve perturbation theory in d < 4. Asymptotic series and Borel summability. Canonical and anomalous dimension: the role of the cutoff. Introduction to the ε-expansion. Functional techniques for field theories. Wick's theorem.
  • Tue 17/12 Momentum-space Renormalization Group: first order in perturbation theory.
  • Wed 18/12 Computation of the correction to the quartic interaction in second-order perturbation theory
  • Mon 13/1 Recap of the Momentum-space Renormalization Group calculation up to second order. RG equation in differential form
  • Tue 14/1 Fixed points of the RG in ε-expansion: Gaussian and Wilson-Fisher fixed points. Critical exponents in ε-expansion.
  • Wed 15/1 Irrelevant operators. Computation of the RG dimension of the Φ6 operator at the Gaussian and Wilson-Fisher fixed points.
  • Mon 20/1 High- and low-temperature series expansion of the bidimensional Ising model. Kramers-Wannier duality. Exact value of the critical coupling and critical energy in the Ising model on a square lattice.
  • Tue 21/1 Duality by Fourier transformation. Application to the Ising and Potts model on a square lattice. Exact value of the critical coupling of the Potts model on a square lattice.
  • Wed 22/1 Correction of the fifth exercise sheet
  • Mon 27/1 Partition function of the O(N) model in the large-N limit
  • Wed 29/1 Correction of the sixth exercise sheet
  • Mon 3/2 Critical behavior of the O(N) model in the large-N limit
  • Wed 5/2 Finite-Size Scaling theory