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.
Lecturers
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:
- 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
Objectives
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]
Workload
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.
Evaluation
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.
Schedule
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 |
Mini-project suggestions
Bosonization:
Integrability:
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
Fundamentals:
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)
References:
- 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)
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
CFT:
Integrability, Quench Action
Theory:
- 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)
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)