Jean-Sébastien Caux

Student Seminar Theoretical Physics

(February-May 2018)

Three Facets of One Dimension:

An introduction to Bosonization, Bethe Integrability and Conformal Field Theory

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.

Subject coverage

The following subjects will be treated:


Your tasks will be to:


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

2018-02-08Week 1Bosonization [notes on Bosonization and CFT given out] SP C1.112
2018-02-15Week 2 self-study [no class, Master Avond clash]
2018-02-22Week 3[Cancelled] self-study [J-S away] SP C1.112
2018-03-01Week 4Integrability I [notes on Integrability given out] SP C1.112
2018-03-08Week 5Integrability II [eboard notes I&II] SP C1.112
2018-03-15Week 6 CFT [Neil; J-S in Brussels] Notes on CFT (Neil) SP C1.112
2018-03-22Week 7 CFT [Neil; J-S in Leiden] SP C1.112
2018-03-29Week 8Work session SP C1.112
2018-04-05Week 9Work session SP G4.15
2018-04-12Week 10Work session SP G4.15
2018-04-19Week 11Work session SP G4.15
2018-04-26Week 12Work session SP G4.15 [J-S in Canada]
2018-05-03Week 13Work session
Deliver digest of mini-project
SP G4.15 [J-S in Canada]
2018-05-10Week 14 [no class, Hemelvaart]
2018-05-17Week 15Presentations SP G4.15
2018-05-24Week 16Presentations SP G4.15
2018-05-31Week 17Presentations SP G4.15

Mini-project suggestions


See Giamarchi's book, Chapter 2.3.

Possible extensions/directions:
  • Duality with classical XY model. Interpretation of BKT transition in spin language
  • Explore integrability of the sine-Gordon model
Link for integrability:
  • 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)
XXZ to sine-Gordon and Massive Thirring
  • 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)
Nested Bethe Ansatz
  • 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)
Initial paper (RG analysis + bosonization):
Half-filling (RG analysis + bosonization/refermionization)
  • 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).
Integrability of the SO(8) theory and using it to compute stuff:
  • 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
Integrability, Quench Action
Experimental observations: spin chains
  • 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)
Experimental observations: quantum gases
  • 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)