1. Higgs boson in a 2D superfluid To be, or not to be in d=2 What’s the drama? N. Prokof’ev ICTP, Trieste, July 18, 2012

2. WIBG: : collective excitations are overdamped (classical criticality) In a Galilean system phase and density are canonical variables and the spectrum is exhausted by Bogoliubov quasiparticles Strongly interacting superlfuids: At we have and the amplitude mode energy is comparable to  overlap with other modes. Suppressing by interactions: is the necessary condition for emergence of the new soft mode (Higgs), but … Liquid-Solid first order transition may happen instead Why not to be in a generic superfluid?

3. Bose Hubbard model: Particle-hole symmetric Lorentz-invariant QCP Capogrosso-Sansone, Soyler et al. ‘08

4. To be or not to be in d=3,2 ? Asymptotically exact mean-field Higgs mode is well-defined. Overdamped due to strong decay into two Goldstone modes. No Higgs resonance at low energy in any correlation function in close vicinity to the QCP Chubukov, Sachdev, Ye ’93 Altman, Auerbach ’02 Zwerger ‘04 Podolsky, Auerbach, Arovas ’11 d=3d=2 Does it help to move away from QCP towards Galilean system? [Yes --- mean-field/variational, 1/N, RPA] Huber, Buchler, Theiler, Altman, Blatter ’08, ’07 Menotti, Trivedi ’08 ??? Chubukov, Sachdev, Ye ’93 Podolsky, Auerbach, Arovas ’11 Look at the right response function! Scalar susceptibility is a better candidate

5. Not to be in d=2: 1/N predictions for scalar susceptibility Altman, Auerbach ’02 Polkovnikov, Altman, Demler, Halperin, Lukin ‘05 Podolsky, Auerbach, Arovas (2011) Peak width INCREASES as Peak maximum > non-universal scale, no Higgs resonance in the relativistic limit.

6. Universal scaling predictions Chubukov, Sachdev, Ye ’93 Sachdev ’99 A B Podolsky et al. MISSING SPECTRAL DENSITY

7. Scalar response through lattice modulation Linear response for small Energy dissipation rate : Total energy absorbed: :

8. Recent experiment @ Munich: The onset of quantum critical continuum. Resonance can not be seen due to inhomogeneous broadening. Onset frequency

9. TIME TO CALL WORMS!

10. Quantum Monte Calro: BH model in path integral representation + WA No systematic errors but (ii) finite system size L=20: + explicit checks of no size dependence (i) finite simulation time: for lowest frequencies (ii) imaginary time (Matsubara frequencies)  analytic continuation Ill-posed problem: MaxEnt=“most likely” “ ” “most featureless”

11. space Lattice path-integral = expansion of in hopping transitions, or kinks Kinetic energy = sum of all hopping transitions kink-kink correlation function Results are person, continent, and CPU indendent, and agree with accuracy for the lowest frequencies

12. There is a resonance at low frequency which - emerges at - softens as - gets more narrow as - preserves its amplitude (roughly)

13. Side-by-side comparison

14. Higgs resonance is present only in close vicinity of QCP. Barely seen at U=14, impossible to disentangle from other modes at U=12

16. Higgs resonance in the MI phase – where is the Mexican hat potential?

19. Power-point attempt to compare signals (amplitude adjusted) One (small ?) problem for direct comparison: experiment =

20. Most recent 1/N calculation by Podolsky & Sachdev [arXiv:1205.2700]arXiv:1205.2700 Universal part of the scalar response has an oscillating component ! Pade approximants

21. Conclusions: Universal part QMC simulation Higgs resonance Possible to extract experimentally in traps and at finite temperature.