1. Aside: the BKT phase transiton Spontaneous symmetry breaking Mermin-Wagner: – no continuous symmetry breaking in models with short ranged interactions in dimension less than two Homotopy group Vortex free energy: – origin of Berezinskii-Kosterlitz-Thouless transition

2. Spontaneous symmetry breaking Effective action (d+1 dimensions) potential energy part kinetic energy part distance to transition 0

3. Mermin-Wagner theorem Phase fluctuations in different dimensions Energetics of long wavelength fluctuations phase fluctuations vs. amplitude fluctuation driven transitions 2D – no long range order, but can have algebraically decaying correlations no LRDO yes LRDO ??

4. Ingredients of the BKT transition Important for transition: – phase fluctuations – topological defects (destruction of correlations) What is a topological defect? – a loop in the physical space that maps to a non-trivial element of the fundamental group – XY vs. Heisenberg physical space XY model order parameter space

5. Sketch of transition: free energy of vortex pairs Interaction between a vortex and anti-vortex free energy: bound free transition free vortices bound vortex anti-vortex pairs

6. The Anderson-Higgs mode in a trapped 2D superfluid on a lattice (close to zero temperature) David Pekker, Manuel Endres, Takeshi Fukuhara, Marc Cheneau, Peter Schauss, Christian Gross, Eugene Demler, Immanuel Bloch, Stefan Kuhr (Caltech, Munich, Harvard)

7. Bose Hubbard Model ji Mott Insulator Superfluid part of ground state (2 nd order perturbation theory)

8. What is the Anderson-Higgs mode Motion in a Mexican Hat potential – Superfluid symmetry breaking – Goldstone (easy) mode – Anderson-Higgs (hard) mode Where do these come from – Mott insulator – particle & hole modes – Anti-symmetric combination => phase mode – Symmetric combination => Higgs mode What do these look like – order parameter phase – order parameter amplitude phase mode Higgs mode

9. A note on field-theory MI-SF transition described by Gross-Pitaevskii actionrelativistic Gross-Pitaevskii action phase (Im  ) Higgs (Re  )

10. Anderson-Higgs mode, the Higgs Boson, and the Higgs Mechanism Sherson et. al. Nature 2010 Cold Atoms (Munich)Elementary Particles (CMS @ LHC) Massless gauge fields (W and Z) acquire mass

11. Anderson-Higgs mode in 2D ? Danger from scattering on phase modes In 2D: infrared divergence (branch cut in susceptibility) Different susceptibility has no divergence Higgs   Podolsky, Auerbach, Arovas, arXiv:1108.5207 Higgs

12. Why it is difficult to observe the amplitude mode Stoferle et al., PRL(2004) Peak at U dominates and does not change as the system goes through the SF/Mott transition Bissbort et al., PRL(2010)

13. Outline Experimental data – Setup – Lattice modulation spectra Theoretical modeling – Gutzwiller – CMF Conclusions

14. Experimental sequence Important features: (1)close to unit filling in center (2)gentle modulation drive (3)number oscillations fixed (4)high resolution imaging density Mott Critical Superfluid (theory)

15. Mode Softening QCP Superfluid Zero Mass Large Mass frequency absorption frequency absorption frequency absorption

16. What about the Trap? 4 5 6 4 5 6 a b c a b c 1 2 3 1 2 3

17. Mode Softening in Trap QCP Superfluid Zero Mass Large Mass frequency absorption frequency absorption frequency absorption

18. Higgs mass across the transition Important features: (1)softening at QCP (2)matches mass for uniform system (3)error bars – uncertainty in position of onset (4)dashed bars – width of onset

19. Gutzwiller Theory (in a trap) Bose Hubbard Hamiltonian Gutzwiller wave function Gutzwiller evolution lattice modulation spectroscopytrap U J What is bad? – quantitative issues – qp interactions What is good? – captures both Higgs and phase modes – effects of trap – non-linearities 2D phase diagram

20. How to get the eigenmodes? step 1: find the ground state. Use the variation wave function to minimize step 2: expand in small fluctuations density

21. How to get eigenmodes ? step 3: apply minimum action principle: step 4: linearize step 5: normalize

22. Higgs Drum – lattice modulation spectroscopy in trap Gutzwiller in a trap Gentle drive – sharp peaks – 20 modulations of lattice depth, measure energy – Discrete mode spectrum – Consistent with eigenmodes from linerized theory – Corresponding “drum” modes – Why no sharp peaks in exp. data? plots, four lowest Higgs modes in trap (after ~100 modulations) Higgs Modes Breathing Modes 0.1% drive

23. Character of the eigenmodes Phase modes & out of phase Amplitude modes & in phase Introduce “amplitudeness”

24. Stronger drive Stronger Drive – 0.1%, 1%, 3% lattice depth – Peaks shift to lower freq. & broaden – Spectrum becomes more continuous Features – No fit parameters – OK onset frequency – Breathing mode – Jagged spectrum – Missing weight at high frequencies Averaging over atom # – Spectrum smoothed – Weight still missing

25. CMF – “Better Gutzwiller” Variational wave functions better captures local physics – better describes interactions between quasi-particles Equivalent to MFT on plaquettes

26. 8Er 9Er 9.5Er 10Er Comparison of CMF & Experiment Theory: average over particle #, uncertainty in V 0 – good: on set, width, absorption amount (no fitting parameters) – bad: fine structure (due to variational wave function?)

27. Summary Experiment2x2 Clusters1x1 Clusters (Gutzwiller) – “gap” disappears at QCP – wide band – band spreads out deep in SF – captures gap – does not capture width – {0,1,2,3,4} – captures “gap” – captures most of the width – {0,1,2} Existence & visibility of Higgs mode in a superfluid – softening at transition – consistent with calculations in trap Questions – How do we arrive at GP description deep in SF? where does Higgs mode go? – is it ever possible to see discrete “drum” mode (fine structure of absorption spectrum)

28. Related field-theory consider the GL theory of MI-SF transition Linearize: Gross-Pitaevskii actionrelativistic Gross-Pitaevskii action phase (Im  ) Higgs (Re  )