1. Dung-Hai Lee U.C. Berkeley Quantum state that never condenses Condense = develop some kind of order

2. As a solid develops order, some symmetry is broken. Spin rotational symmetry is broken !

3. Ice crystalSuperfluid Neutron starExpanding universe

4. Examples of order

5. Metals are characterized by the Fermi surface Metals do break any symmetry, but they are not stable at zero temperature. Metals always turn into some ordered states with symmetry breaking as T  0.

6. Different types of Fermi surface instability lead to different order. Cooper instability  superconductivity Fermi surface nesting instability  spin density wave, or charge density wave

7. Metal Superconductivity Charge density wave Spin density wave Landau’s paradigm Ordered state is characterized by the symmetry that is broken. All ordered states originate from the metallic state due to Fermi surface instability.

8. Is it possible for a solid not to develop any order at zero temperature ?

9. Insulators with integer filling factor are good candidates Fermion band insulator Boson Mott insulator

10. Fermion Mott insulator Mott insulator Boson Mott insulator Insulating due to repulsion between particles.

11. Examples of electron band insulator C, Si, Ge, GaAs, …

12. YBa 2 Cu 3 O 6 – the parent compound of high temperature superconductor CuO 2 sheet An example of electron Mott insulator

13. An example of boson Mott insulator: optical lattice of neutral atoms Greiner et al, Nature 02

14. Why are we interested in insulators ? Doping make them very useful ! Most of the time, doping make the particle mobile, hence can conduct.

15. Doped band insulator A Silicon chip

16. Doping Mott insulators has produced many materials with interesting properties. High T c superconductors Colossal magneto-resistive materials Doped YBa 2 Cu 3 O 6 Doped LaMnO 3 Doped Mott insulators

17. Is it possible that a solid remains insulating after doping ? Yes

18. An interesting fact: all insulators with fractional filling factor break some kind of symmetry hence exhibit some kind of order. AntiferromagnetDimmerization fermion boson

19. Oshikawa’s theorem If the system is insulating, and if the filling factor = p/q,  the ground state is q-fold degenerate. Usually the required degeneracy is achieved by long range order. Why is uncondensed insulator so rare at fractional filling ? Can a fractional filled insulator exist without symmetry breaking ? Oshikawa PRL 2000

20. It is generally believed that featureless insulators will have very unusual properties. Such as fractional-charge excitations …

21. Anderson’s spin liquid idea Spin liquid is a featureless insulator (at half filling) with no long range order ! It has S=1/2 excitations (spinons). ++... It exists in the parent state of high-temperature superconductors. Resonating singlet patterns Anderson, Science 1987

22. Condensed matter physicists have searched for such insulators for 20 years. The usual search guide line is “frustration”. ?

23. Melts crystal order but never changes the C-M position  preserve 3- fold degeneracy. A new idea: symmetry protected uncondensed quantum state Filling factor =1/3

24. The Quantum Hall effect R xx = V L /I; R xy = V H /I The fractional quantum Hall effect

25. Lee & Leinaas, PRL 2004 One example of this type of state is the fractional quantum Hall liquid

26. Another example is the quantum dimer liquid Moessner & Sondhi, PRL, 2001

27. All existing models in the literature that exhibit uncondensed quantum state conserve the center-of-mass position and momentum.