aside: the bkt phase transiton spontaneous symmetry breaking mermin-wagner: – no continuous...
TRANSCRIPT
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)