*(February-May 2018)*

This year's Student Seminar Theoretical Physics will guide you through a thorough introduction to the multifaceted world of many-body quantum physics in one dimension.

These three pillars of theory are of general use for all theorists interested in strongly-correlated systems, from condensed matter to high-energy physics.

This course will consist of a mixture of lectures, self-study and student presentations.

The main lecturer will be Jean-Sébastien Caux,

with assistance from Sasha Gamayun, Eoin Quinn, Neil Robinson and Enej Ilievski, who will be happy to offer you assistance for your mini-project.

The following subjects will be treated:

- Bosonization
- The Tomonaga-Luttinger model
- The bosonization identity
- Asymptotics of correlation functions

- Integrability and the Bethe Ansatz
- The notion of quantum integrability
- The Lieb-Liniger model
- Heisenberg chains
- Thermodynamics
- The Algebraic Bethe Ansatz

- Conformal Field Theory
- Conformal invariance
- The operator formalism
- The Virasoro algebra and its representations
- Minimal models

Your tasks will be to:

- read through and develop at least a basic level of proficiency in each of the three core themes [weeks 1-7]
- select/conceive a research mini-project, and perform it [weeks 8-11]
- Take a (set of) research paper(s) using the methods, and reproduce the results
- Teamwork is allowed

- write up a research digest on your mini-project (approx. 8-page (per person) summary) [weeks 12-14]
- Due for delivery in week 14; stack of digests will be made available to all

**What is a good digest?**During your mini-project, you'll hopefully have learned something interesting, rederived some results from the literature, perhaps even tried to work out some of your own ideas further. Your digest should be a document from which other students can understand what you did (and we, the teachers, can clearly see what you did or tried to do). - prepare a 15-minute presentation on your mini-project [weeks 12-14]
- give your presentation (and attend all other presentations!) [weeks 15-17]

This is a 6 EC course, and since each EC equates to 28 hours of work, you should reserve about 10 hours per week for this course, on average, throughout its duration.

There is no grading for this course. 6 ECs will be give to students who have worked through the material, delivered a research digest, and gave a presentation.

Thursdays 15:00-18:00

2018-02-08 | Week 1 | Bosonization [notes on Bosonization and CFT given out] | SP C1.112 |

2018-02-15 | Week 2 | self-study [no class, Master Avond clash] | |

2018-02-22 | Week 3 | [Cancelled] | self-study [J-S away] SP C1.112 |

2018-03-01 | Week 4 | Integrability I [notes on Integrability given out] | SP C1.112 |

2018-03-08 | Week 5 | Integrability II [eboard notes I&II] | SP C1.112 |

2018-03-15 | Week 6 | CFT [Neil; J-S in Brussels] Notes on CFT (Neil) | SP C1.112 |

2018-03-22 | Week 7 | CFT [Neil; J-S in Leiden] | SP C1.112 |

2018-03-29 | Week 8 | Work session | SP C1.112 |

2018-04-05 | Week 9 | Work session | SP G4.15 |

2018-04-12 | Week 10 | Work session | SP G4.15 |

2018-04-19 | Week 11 | Work session | SP G4.15 |

2018-04-26 | Week 12 | Work session | SP G4.15 [J-S in Canada] |

2018-05-03 | Week 13 | Work session Deliver digest of mini-project |
SP G4.15 [J-S in Canada] |

2018-05-10 | Week 14 | [no class, Hemelvaart] | |

2018-05-17 | Week 15 | Presentations | SP G4.15 |

2018-05-24 | Week 16 | Presentations | SP G4.15 |

2018-05-31 | Week 17 | Presentations | SP G4.15 |

- P. Schlottmann, The Kondo problem. I. Transformation of the model and its renormalization, Phys. Rev. B 25, 4815 (1982)
- see also: Phys. Rev. B 25, 4828 (1982) and Phys. Rev. B 25, 4838 (1982)

- N. Andrei, Diagonalization of the Kondo Hamiltonian, Phys. Rev. Lett. 45, 379 (1980)
- P. Wiegmann, Exact solution of s-d exchange model at T=0, JETP Lett. 31, 392 (1980)

See Giamarchi's book, Chapter 2.3.

- Duality with classical XY model. Interpretation of BKT transition in spin language
- Explore integrability of the sine-Gordon model

- A. Imambekov and L. I. Glazman, Universal theory of non-linear Luttinger liquids, Science 323 228 (2009)
- A. Imambekov, T. L. Schmidt, and L. I. Glazman, One-dimensional quantum liquids: Beyond the Luttinger liquid paradigm, Rev. Mod. Phys. 84 1253 (2012)

- R. G. Pereira, J. Sirker, J.-S. Caux, R. Hagemans, J. M. Maillet, S. R. White, I. Affleck, Dynamical structure factor at small q for the XXZ spin-1/2 chain, J. Stat. Mech. P08022 (2007)
- R. G. Pereira, S. R. White, and I. Affleck, Spectral function of spinless fermions on a one-dimensional lattice, Phys. Rev. B 79, 165113 (2009)
- F. H. L. Essler, Threshold singularities in the one-dimensional Hubbard model, Phys. Rev. B 81 205120 (2010)

- Bosonic and fermonic lattice formulations of XXZ
- Continuum limit of fermionic formulation (see e.g. Tsvelik book)
- Bosonization of fermion theory to sine-Gordon (see e.g. Tsvelik or Giamarchi book)
- Heisenberg limit: marginal terms and how to eliminate them with next-neighbor coupling (Affleck and Eggert) and relation to CFT
- Duality between sine-Gordon and Massive Thirring (see Coleman's paper)

- Solution of the Hubbard model via nested Bethe ansatz (see the Hubbard book)
- Extensions: bosonization of the Hubbard model. RG treatment. (Covered in Hubbard book)

- C. K. Lai, J. Math. Phys. 15, 167 (1974)
- B. Sutherland, Phys. Rev. B 12, 3795 (1975)
- P. Schlottmann, Phys. Rev. B 36, 5177 (1987)
- P. A. Bares and G. Blatter, Phys. Rev. Lett. 64, 2567 (1990)
- S. Sarkar, J. Phys. A 23, L409 (1990)
- S. Sarkar J. Phys. A 24 1137 (1991)
- P.A.Bares, G.Blatter, and M.Ogata, Phys. Rev. B 44, 130 (1991)
- S. Sarkar, J. Phys. A 24, 5775 (1991)
- F. H. L. Essler and V. E. Korepin, Phys. Rev. B 46, 9147 (1992)

- L. Balents and M. P. A. Fisher, Weak-coupling phase diagram of the two-chain Hubbard model, Phys. Rev. B 53 12133 (1996).

- H.-H. Lin, L. Balents and M. P. A. Fisher, Exact SO(8) symmetry in the weakly-interacting two-leg ladder, Phys. Rev. B 58 1794 (1998).

- R. M. Konik and A. W. W. Ludwig, Exact zero-temperature correlation functions for two-leg Hubbard ladders and carbon nanotubes, Phys. Rev. B 64, 155112 (2001)
- F. H. L. Essler and R. M. Konik, Applications of Massive Integrable Quantum Field Theories to Problems in Condensed Matter Physics, arXiv:cond-mat/0412421

- P. Calabrese and J. Cardy, Time Dependence of Correlation Functions Following a Quantum Quench, Phys. Rev. Lett. 96, 136801 (2006)
- P. Calabrese and J. Cardy, Quantum quenches in extended systems, J. Stat. Mech. 2007 P06008 (2007)

- J.-S. Caux, The Quench Action, J. Stat. Mech. 064006 (2016)
- J. De Nardis, B. Wouters, M. Brockmann, J.-S. Caux, Solution for an interaction quench in the Lieb-Liniger Bose gas, Phys. Rev. A 89, 033601 (2014)

- J.-S. Caux, J.-M. Maillet, Computation of dynamical correlation functions of Heisenberg chains in a field, Phys.Rev.Lett. 95 077201 (2005)
- J.-S. Caux, R. Hagemans, J.-M. Maillet, Computation of dynamical correlation functions of Heisenberg chains: the gapless anisotropic regime, J. Stat. Mech. P09003 (2005)
- J.-S. Caux, P. Calabrese, Dynamical density-density correlations in the one-dimensional Bose gas, Phys.Rev. A 74 031605 (2006)
- J.-S. Caux, P. Calabrese, N. A. Slavnov, One-particle dynamical correlations in the one-dimensional Bose gas, J. Stat. Mech. P01008 (2007)

- M. Mourigal, M. Enderle, A. Klöpperpieper, J.-S. Caux, A. Stunault, H. M. Rønnow, Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain, Nature Physics 9, 435-441 (2013)
- B. Lake, D. A. Tennant, J.-S. Caux, T. Barthel, U. Schollwöck, S. E. Nagler, C. D. Frost, Multispinon continua at zero and finite temperature in a near-ideal Heisenberg chain, Phys. Rev. Lett. 111, 137205 (2013)

- F. Meinert, M. Panfil, M. J. Mark, K. Lauber, J.-S. Caux, H.-C. Nägerl, Probing the Excitations of a Lieb-Liniger Gas from Weak to Strong Coupling, Phys. Rev. Lett. 115, 085301 (2015)
- N. Fabbri, M. Panfil, D. Clément, L. Fallani, M. Inguscio, C. Fort, J.-S. Caux, Dynamical structure factor of one-dimensional Bose gases: experimental signatures of beyond-Luttinger liquid physics, Phys. Rev. A 91, 043617 (2015)

- A. B. Zamolodchikov, "Irreversibility" of the Flux of the Renormalization Group in a 2-D Field Theory, JETP Lett. 43 730–732 (1986)
- J. Cardy, Is there a c-theorem in four dimensions?, Phys. Lett. B 215: 749 (1988)
- Z. Komargodski, A. Schwimmer, On renormalization group flows in four dimensions, JHEP 2011(12) 99 (2011)

- J. Cardy, Boundary Conformal Field Theory, arXiv:hep-th/0411189

- S. Rychkov, EPFL Lectures on Conformal Field Theory in D>=3 Dimensions, arXiv:1601.05000
- D. Poland and D. Simmons-Duffin, The conformal bootstrap, Nature Phys. 12 535 (2016)

- S. El-Showk et al., Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86 025022 (2012)
- S. El-Showk et al., Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents, J. Stat. Phys. 157 869 (2014)