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

Spontaneous symmetry breaking

• Effective action (d+1 dimensions)

potential energy partkinetic energy part

distance to transition

0

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

??

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 spaceXY model order parameter space

Sketch of transition: free energy of vortex pairs

• Interaction between a vortex and anti-vortex

• free energy:boundfree

transition

free vorticesbound vortex

anti-vortex pairs

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)

Bose Hubbard Model

j i

Mott Insulator Superfluid

part of ground state (2nd order perturbation theory)

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

phasemode

Higgsmode

A note on field-theory• MI-SF transition described by

Gross-Pitaevskii action relativistic Gross-Pitaevskii action

phase (Im d)

Higgs (Re d)

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

Anderson-Higgs mode in 2D ?

• Danger from scattering on phase modes

• In 2D: infrared divergence (branch cut in susceptibility)

• Different susceptibility has no divergence

Higgsf

f

Podolsky, Auerbach, Arovas, arXiv:1108.5207

Higgs

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)

Outline

• Experimental data– Setup– Lattice modulation spectra

• Theoretical modeling– Gutzwiller– CMF

• Conclusions

Experimental sequence

Important features:(1) close to unit filling in center(2) gentle modulation drive(3) number oscillations fixed(4) high resolution imaging

densi

tydensi

tydensi

ty

Mott

Critical

Superfluid

(theory)

Mode Softening

QCP

Supe

rflui

d

Zero Mass

Large Mass

frequency

abso

rptio

n

frequency

abso

rptio

nfrequency

abso

rptio

n

What about the Trap?

4

5

6

4

5

6

abc

a

b

c

123

1

2

3

Mode Softening in Trap

QCP

Supe

rflui

d

Zero Mass

Large Mass

frequency

abso

rptio

n

frequency

abso

rptio

nfrequency

abso

rptio

n

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

Gutzwiller Theory (in a trap)• Bose Hubbard Hamiltonian

• Gutzwiller wave function

• Gutzwiller evolution

lattice modulation spectroscopy trap

UJ

What is bad?– quantitative issues– qp interactions

What is good?– captures both Higgs and phase modes– effects of trap– non-linearities

2D phase diagram

How to get the eigenmodes?• step 1: find the ground state. Use the variation wave function

to minimize

• step 2: expand in small fluctuations

densi

ty

How to get eigenmodes ?

• step 3: apply minimum action principle:

• step 4: linearize

• step 5: normalize

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

Character of the eigenmodes

• Phase modes

& out of phase

• Amplitude modes

& in phase

• Introduce “amplitudeness”

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

CMF – “Better Gutzwiller”

• Variational wave functions better captures local physics– better describes interactions between quasi-particles

• Equivalent to MFT on plaquettes

8Er

9Er

9.5Er

10Er

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?)

SummaryExperiment 2x2 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)

Related field-theory• consider the GL theory of MI-SF transition

• Linearize:

Gross-Pitaevskii action relativistic Gross-Pitaevskii action

phase (Im d)

Higgs (Re d)