M2 Quantum Physics -- Academic year 2014-2015 
Broken symmetries and quantum phase transitions 


                                        ORAL EXAM 


Students are invited to give a 20 minute talk on one of the following subjects (any other subject is welcome provided that it fits into the framework of the course). The aim of the oral exam is to take advantage of the techniques (functional integrals, saddle-point approximations, etc.) and concepts (spontaneous broken symmetries, quantum critical phenomena, etc.) taught in the course in order to address a research subject.

It is possible for students to work together on a given subject but the talk (black board only) will be given individually. No more than two students can choose the same subject. 
 

1) Spinor Bose-Einstein condensates: mean-field theory and vortices in a hyperspin F=1 boson gas. 
- T. Ohmi et al., "Bose-Einstein condensation with internal degrees of freedom in alkali atom gases", arXiv:cond-mat/9803160
- T.L. Ho, "Spinor Bose Condensates in Optical Traps", Phys. Rev. Lett. 81, 742–745 (1998)
- M. Ueda et Y. Kawaguchi, arXiv:1001.2072, Secs. 2.1, 2.2.1 à 2.2.3, 3.1, 3.2, 5.1, 6.1, 6.2.
   
2) Superfluid transition near a Feshbach resonance:
- L. Radzihovsky et al., "Superfluid transitions in bosonic atom-molecule mixtures near Feshbach resonance", Phys. Rev. Lett. 92, 160402 (2004).
- L. Radzihovsky et al., "Superfluidity and phase transitions in a resonant Bose gas", Annals of Physics 323, 2376 (2008). 

3) Superfluid--Mott-insulator transition in the Jaynes-Cummings model: 
- J. Koch et K. Lehur, Phys. Rev. A 80, 023811 (2009). 
For the bosonic Mott transition, see 
- D. van Oosten et al., "Quantum phases in optical lattices", Phys. Rev. A 63, 053601 (2001).
- Chapter 10 (Boson Hubbard model) in S. Sachdev, Quantum Phase Transitions (Cambridge University Press, 1999).
For spin coherent states, see references given in subject 9 below. 

4) BCS-BEC crossover in a fermionic superfluid:  
- M. Randeria: "Crossover from BCS theory to Bose-Einstein condensation", in Bose-Einstein Condensation, edited by A. Griffin et al. (Cambridge University Press, 1995).  See also C.A.R. Sa de Melo et al., "Crossover from BCS to Bose superconductivity: Transition temperature and time-dependent Ginzburg-Landau theory", Phys. Rev. Lett. 71, 3202 (1993); J.R. Engelbrecht et al., "BCS to Bose crossover: broken-symmetry state", Phys. Rev. B 55, 15153 (1997). 
- R.B. Diener et al., "Quantum fluctuations in the superfluid state of the BCS-BEC crossover", Phys. Rev. A 77, 023626 (2008). (See in particular section VIII on the unitary limit.)

5) Antiferromagnetic fluctuations in the half-filled Hubbard model: 
- Z.Y. Wen et al., "Path integral approach to the Hubbard model", Phys. Rev. B 43, 3790 (1991). 
- H.J. Schulz, "Functional integrals for correlated electrons", arXiv:cond-mat/9402103v1.
- K. Borejsza & N. Dupuis, "Antiferromagnetism and single-particle properties in the two-dimensional half-filled Hubbard model: A nonlinear sigma model approach" (Secs.I-III), Phys. Rev. B 69, 085119 (2004); "Antiferromagnetism and single-particle properties in the two-dimensional half-filled Hubbard model: Slater vs. Mott-Heisenberg", Europhys. Lett. 63, 722 (2003). 

6) Collective modes in a superconductor.  

7) Fractional quantum Hall effect: 
- S.C. Zhang, Int. J. Mod. Phys. B 6, 25 (1992). 

8) Path integral for a quantum spin; application to quantum spin chains (Haldane's conjecture): 
- A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer Verlag): Secs. 10.1/12.1/12.2/15.1/15.2.
- A. Altland et B. Simons: Secs. 3.3.5  et 9.3.3.
- S. Sachdev, Quantum Phase Transitions (Cambridge University Press): Secs. 13.1/13.3.1/13.3.1.1.

9) The quantum O(N) non-linear sigma model in the large N limit:
- A.V. Chubukov et al, "Theory of two-dimensional quantum Heisenberg antiferromagnets with a nearly critical ground state", Phys. Rev. B 49, 11919 (1994) (in particular the limit N=inf in Sec.III.A).
- Appendix 7.D in http://www.lptl.jussieu.fr/files/chap_rg.pdf

10) Dicke model


For students with some knowledge on the renormalization group: 

11) The Bose gas at the unitary limit:  
- Lee and Lee, "Universality and stability for a dilute Bose gas with a Feshbach resonance", Phys. Rev. A 81, 063613 (2010). 

12) One-dimensional interacting fermions: renormalization group to one-loop order in the g1-g2 model:  
- C. Bourbonnais et al., "Renormalization group technique for quasi-one-dimensional interacting fermion systems at finite temperature" (en particulier Secs.4.1 et 4.2), arXiv:cond-mat/0204163. 
- H.J. Schulz et al., "Fermi liquids and Luttinger liquids" (Sec.3.1), arXiv:cond-mat/9807366
- R. Shankar, "Renormalization-group approach to interacting fermions", Rev. Mod. Phys. 66, 129 (1994) (Sec.IV). 

13) Functional renormalization group: Wilson-Polchinski equation and local potential approximation: 
- C. Bagnuls and C. Bervillier, Phys. Rep. 348, 91 (2001) (Secs. 3.2 et 3.3).
- R.D. Ball et al., "Scheme independence and the exact renormalization group", Phys. Lett. B 347, 80 (1995).
- chapitre 16 in J. Zinn-Justin, Transitions de phase et groupe de renormalisation (EDP Sciences, 2005); voir aussi Appendice 10.1 in J. Zinn-Justin, Quantum field theory and critical phenomena (Clarendon Press, 3ème ou 4ème édition).

15) The non-perturbative renormalization group: 
-B. Delamotte: An Introduction to the Nonperturbative Renormalization Group, arXiv:cond-mat/0702365 

16) The quantum non-linear sigma model in d=2+eps dimensions:
-S. Chakravarty et al., "Two-dimensional quantum Heisenberg antiferromagnet at low temperatures", Phys. Rev. B 39, 2344 (1989). 
-Sec. 7.10.4 in http://www.lptmc.jussieu.fr/files/chap_rg.pdf

17) Super Efimov effect from NPRG (Sergej Moroz).