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Aside: the BKT phase transiton

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.

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Aside: the BKT phase transiton

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  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) distance to transition kinetic energy part potential energy part 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 XY model order parameter space physical space

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

  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 j i part of ground state (2nd order perturbation theory) Mott Insulator Superfluid

  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 action relativistic Gross-Pitaevskii action phase (Imd) Higgs (Re d)

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

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

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

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

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

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

  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 Large Mass absorption Superfluid frequency absorption frequency absorption QCP frequency Zero Mass

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

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

  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? 0.1% drive Breathing Modes Higgs Modes plots, four lowest Higgs modes in trap (after ~100 modulations)

  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. Comparison of CMF & Experiment • Theory: average over particle #, uncertainty in V0 • good: on set, width, absorption amount (no fitting parameters) • bad: fine structure (due to variational wave function?) 8Er 9.5Er 9Er 10Er

  27. Summary Experiment 1x1 Clusters (Gutzwiller) 2x2 Clusters • “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 action relativistic Gross-Pitaevskii action phase (Imd) Higgs (Re d)

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