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)
distance to transition
kinetic energy part
0
potential energy part
Mermin-Wagner theorem
• Phase fluctuations in different dimensions
• Energetics of long wavelength fluctuations
no LRDO
??
yes LRDO
• phase fluctuations vs. amplitude fluctuation driven transitions
• 2D – no long range order, but can have algebraically decaying
correlations
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
physical space
– XY vs. Heisenberg
XY model order
parameter space
Sketch of transition: free energy of vortex pairs
• Interaction between a vortex and anti-vortex
bound vortex
anti-vortex pairs
• free energy:
bound
free
transition
free vortices
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
part of ground state
(2nd order
perturbation theory)
Mott Insulator
Superfluid
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
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
Elementary Particles (CMS @ LHC)
Cold Atoms (Munich)
Sherson et. al. Nature 2010
Massless gauge fields (W and Z) acquire mass
Anderson-Higgs mode in 2D ?
Podolsky, Auerbach, Arovas, arXiv:1108.5207
• Danger from scattering on phase modes
f
Higgs
Higgs
f
• In 2D: infrared divergence
(branch cut in susceptibility)
• Different susceptibility has no divergence
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
Outline
• Experimental data
– Setup
– Lattice modulation spectra
• Theoretical modeling
– Gutzwiller
– CMF
• Conclusions
Experimental sequence
(theory)
Mott
Critical
Important features:
(1) close to unit filling in center
(2) gentle modulation drive
(3) number oscillations fixed
(4) high resolution imaging
Superfluid
frequency
absorption
Superfluid
Large Mass
absorption
Mode Softening
QCP
Zero Mass
absorption
frequency
frequency
What about the Trap?
a
b
c
1
4
2
3
5
a
1
4
b
2
5
c
3
6
6
frequency
absorption
Superfluid
Large Mass
absorption
Mode Softening in Trap
QCP
Zero Mass
absorption
frequency
frequency
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
•
lattice modulation spectroscopy
J
U
trap
2D phase diagram
Gutzwiller evolution
What is good?
–
–
–
captures both Higgs and phase modes
effects of trap
non-linearities
What is bad?
–
–
quantitative issues
qp interactions
How to get the eigenmodes?
• step 1: find the ground state. Use the variation wave function
to minimize
• step 2: expand in small fluctuations
How to get eigenmodes ?
• step 3: apply minimum action principle:
• step 4: linearize
• step 5: normalize
Higgs Drum – lattice modulation
spectroscopy in trap
0.1% drive
Breathing
Modes
Higgs
Modes
• 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)
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
Comparison of CMF & Experiment
8Er
9.5Er
9Er
10Er
• Theory: average over particle #, uncertainty in V0
– good: on set, width, absorption amount (no fitting parameters)
– bad: fine structure (due to variational wave function?)
Summary
Experiment
– “gap” disappears at QCP
– wide band
– band spreads out deep in SF
1x1 Clusters (Gutzwiller)
– captures gap
– does not capture width
– {0,1,2,3,4}
2x2 Clusters
– 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)