1. Hidden topological order in one-dimensional Bose Insulators Ehud Altman Department of Condensed Matter Physics The Weizmann Institute of Science With: E. Dalla Torre (Weizmann) E. Berg (Stanford & Weizmann)

2. Ultracold atoms in optical lattices. Bose Hubbard model Experiment: Kasevich et al., Science (2001) Greiner et al., Nature (2001) Cataliotti et al., Science (2001) Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004), …

3. Superfluid to insulator transition U>>J U<<J U/J Theory: Fisher et al. PRB (89), Jaksch et al. PRL (98) Experiment: Greiner et al. Nature (01) Both phases are conceptually very simple. Can we have something more interesting?

4. Polar molecules or atoms in a 1d optical lattice t U E All three parameters are independently tunable.

5. Conventional phases (integer filling) U/t V/t MI CDW SF ? ? ?

6. Outline Phase diagram in 1d (DMRG) - Insulator with hidden topological order - Spin analogy: Haldane gapped state, AKLT How to detect hidden order ? - Distinct collective mode, quantum criticality - Use MFT to calculate absorption spectrum Open questions

7. First look with DMRG MI CDW SF U/t V/t Length: L=256 Energy gap Filling: Hidden order ? Measure the local density:

8. Phase diagram (DMRG) Length: L=256 Filling: Correlation functions SF: CDW: String: String order CDW

9. Interpretation of the phase: Spin 1 analogy Keep only 3 occupation states: 0 0 0 1 Effective spin hamiltonian:

10. Spin 1 – phase diagram V/J U/J Mott Haldane Neel Breaking of hidden Z 2 X Z 2 symmetry (only Z 2 in Bose system) String correlation: Den Nijs & Rommelse (89) Kennedy &Tasaki (92) X4 (X2) edge degeneracy

11. How to detect hidden order ?

12. Parametric excitation Periodic modulation of lattice intensity: Used to probe excitations in SF and MI Absorption spectrum in linear response: Stoferle et. al., PRL 04 (ETH)

13. Absorption spectrum from mean field theory 1 Non local unitary transformation (Kennedy & Tasaki, Oshikawa): Mean field wave function:

14. Absorption spectrum from mean field theory 2 Fluctuations: Expand perturbation in terms of the fluctuations: Neutral mode Particle-hole continuum

15. Compare with numerics U V Neutral mode

16. Many open questions Nature of the quantum phase transitions (effect of p.h. symmetry breaking) Possibility of observing edge states? Extension to quasi 2D (dipole interactions between parallel chains)

17. Unconventional / conventional 2D phases ? 1D – 2D phase transitions tuned by angle of external field ?

18. Conclusions Phase diagram of bosons with 1/r 3 repulsion: unconventional insulating phase Realization and detection with ultra cold dipolar bosons in optical lattice. Distinct neutral collective mode 2D generalizations ?