Hidden topological order in one-dimensional Bose Insulators

1
Hidden topological order in one-dimensional Bose
Insulators
Ehud Altman Department of Condensed Matter
Physics The Weizmann Institute of Science
With E. Dalla Torre (Weizmann) E. Berg
(Stanford Weizmann)
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Ultracold atoms in optical lattices. Bose Hubbard
model
Experiment Kasevich et al., Science (2001)
Greiner et al., Nature (2001)
Cataliotti et al., Science
(2001) Phillips et al., J.
Physics B (2002)
Esslinger et al., PRL (2004),
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Superfluid to insulator transition
Theory Fisher et al. PRB (89), Jaksch et al. PRL
(98) Experiment Greiner et al. Nature (01)
U/J
UgtgtJ
UltltJ
Both phases are conceptually very simple. Can we
have something more interesting?
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Polar molecules or atoms in a 1d optical lattice
E
U
t
All three parameters are independently tunable.
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Conventional phases (integer filling)
U/t
MI
CDW
?
?
?
SF
V/t
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Outline

• Phase diagram in 1d (DMRG) - Insulator with
  hidden topological order - Spin analogy
  Haldane gapped state, AKLT
• How to detect hidden order ? - Distinct
  collective mode, quantum criticality - Use MFT
  to calculate absorption spectrum
• Open questions

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First look with DMRG
Length L256
Energy gap
Hidden order ?
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Phase diagram (DMRG)
Length L256
String order
CDW
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Interpretation of the phase Spin 1 analogy
Keep only 3 occupation states
-1
0
1
0
0
Effective spin hamiltonian
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Spin 1 phase diagram
U/J
String correlation
Mott
Neel
Haldane
Den Nijs Rommelse (89) Kennedy Tasaki (92)
V/J
Breaking of hidden Z2 X Z2 symmetry(only Z2 in
Bose system)
X4 (X2) edge degeneracy
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How to detect hidden order ?
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Parametric excitation
Periodic modulation of lattice intensity
Absorption spectrum in linear response
Used to probe excitations in SF and MI
Stoferle et. al., PRL 04 (ETH)
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Absorption spectrum from mean field theory 1
Non local unitary transformation (Kennedy
Tasaki, Oshikawa)
Mean field wave function
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Absorption spectrum from mean field theory 2
Fluctuations
Expand perturbation in terms of the fluctuations
Particle-hole continuum
Neutral mode
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Compare with numerics
U
V
Neutral mode
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Many open questions

• Nature of the quantum phase transitions(effect
  of p.h. symmetry breaking)
• Possibility of observing edge states?
• Extension to quasi 2D(dipole interactions
  between parallel chains)

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Unconventional / conventional 2D phases ? 1D
2D phase transitions tuned by angle of external
field ?
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Conclusions

• Phase diagram of bosons with 1/r3
  repulsionunconventional insulating phase
• Realization and detection with ultra cold dipolar
  bosons in optical lattice.
• Distinct neutral collective mode
• 2D generalizations ?