Hidden topological order in one dimensional Bose insulators

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Hidden topological order in one dimensional Bose
insulators
Emanuele Dalla Torre (Weizmann) Erez Berg
(Stanford) Thierry Giamarchi (Geneva)Ehud
Altman (Weizmann)
Dalla-Torre, Berg and E.A. PRL 97, 260401
(2006)Berg, Dalla-Torre, Giamarchi and E. A.
PRB 77, 245119 (2008)
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Polar molecules or atoms in a 1d optical lattice
Polar Molecules Jin, Ye (JILA), Demille (Yale),
Grim (Innsbruck),
Atoms with large magnetic dipole moment (Cr _at_
Stuttgart)Stuhler et. al. PRL 95, 150406
(2005).
E
U
t
polarizing field
We will focus on integer filling
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The conventional phases at unity filling
Note supersolids are expected near half integer
filling, not at integers
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Outline

• A new phase of 1D lattice bosons Haldane
  insulator - Non local string correlations-
  Connection to Haldane phase of spin-1 chains
• Haldane phase as symmetry protected topological
  phase.
• Topological pumping of quantized charge through
  the insulator
• Array of weakly coupled chainsString
  correlations destroyed but Topological order
  remains!

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Numerical investigation with DMRG
Length L256
Excitation gap
Energy gap
Local density
Hidden order ?
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Non local correlations
string order
Parity order
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New insulating phase of bosons characterized by a
highly non local order parameter !
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Understanding the order parameters from spin 1
analogy
Truncate Hilbert space to 3 occupation states
Mott insulator
Pairing of particle and hole fluctuations is
indicated by parity correlations
Haldane insulator
Alternate ordering of particles and holes is
indicated by string correlations
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Related spin-1 chain
Neglect terms breaking p-h symmetry
Numerical phase diagram Chen et. al. PRB (2003)
Disordered phase (MI)
Haldane Gapped phase Haldane (83)
Ising AF (DW)
Heisenberg point
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Continuum limit around the transition MI-HI
(Bosonization)
Schulz, PRB 86 (Spin chains)Berg, Dalla-Torre,
Giamarchi, Altman, 2008 (Bosons)
Parity order parameter
String order parameter
Critical line is a Luttinger liquid with Klt2
decreasing continuosly
Note p-h symmetry breaking terms are irrelevant
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Continuum limit HI and MI phases
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Stability of the Haldane phase (?)
The HI phase is thermodynamically protected
thanks to gap.
But the sharp distinction between HI and MI is
not !
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A relevant perturbation at the HI-MI QCP
Pert. l breaks1) Particle hole symmetry2)
Inversion symmetry
and
Caricature wave-function
Sharp distinction between MI and HI is eliminated
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Another point of view
String
Parity
A local field factors into product of the non
local operators !
The local perturbation effectively couples to non
local order parameter fields !
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Microscopic realization of the perturbation
Use weak secondary optical lattice with half the
wavelength
Adrelini etal. J. Phys. B 2006 (NIST) Foelling
etal. Nature 2007 (Mainz)
Effective hamiltonian within the lowest Bloch
band
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Adiabatic pumping of quantized charge
Cyclic parameter change around the QCP (g0, l0)
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Spontaneous breaking of lattice inversion
symmetry ?
Effective field theory including sub-leading term
Last term becomes relevant on the critical line
when Klt1/2 (provided this occurs before the
transition to the DW phase)
New phase in case s lt 0
Spontaneously broken inversion symmetry !
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Array of coupled tubes
b
a

• Full control of tunneling between chains(by
  tuning intensity of transverse lattice potential)
• Limited control over interchain dipolar
  interaction(e.g. by tuning angle of polarizing
  field)

How do the couplings affect the phase diagram ?
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Weakly coupled chains in the HI
phasequalitative picture

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0
Interchain tunneling creates kinks in the string
order !
Results in exponentially decaying string
correlations(See also Anfuso and Rosch PRB 2007)
Sharp distinction between MI and HI is
eliminated.What is the fate of the direct
transition between these phases?
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Field theory for two chains
Inter-chain Tunneling
Highly relevant at the MI-HI critical point
Direct transition is avoided by emergence of a SF
phase between the MI and HI.
Inter-chain interaction
Irrelevant!Only renormalizes the Luttinger
parameters
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Adiabatic pumping in the coupled tubes
SF
Topological property, quantized charge, remains.
String order is not fundamental to the phase!
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Measuring excitation spectra in the HI phase
1. Periodic modulation of the 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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Measuring excitation spectra
2. Bragg spectroscopy
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Effective low energy theories
MI - HI transition
Sine-Gordon model
Lattice modulation
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Effective low energy theories
HI - DW transition
double sine-Gordon
Mapping to Ising model (near critical point)
Lattice modulation
Bragg spectroscopy
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Absorption spectra
Lattice modulation (q0)
Absorption rate vs. ?
?V-Vc1/(2-K)
?V-Vc
V/t
Bragg spectroscopy (qp)

• particle-hole continua

V/t
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Excitations of the Haldane insulator
Field theory
Numerical results
Lattice modulation (q0)
Particle hole gap Neutral gap
Bragg spectrum at qp
The neutral mode is a phonon
V
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Summary

• Novel Insulating phase of lattice bosons with
  highly non local string correlations.
• Topological order protected by lattice inversion
  symmetry.
• Adiabatic pumping of quantized charge through
  the insulator
• Coupled chainsString correlation unstable to
  interchain tunneling but topological order is!
• Distinct neutral collective mode in the HI phase
  can be detected in Bragg spect.

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Numerical check (t-gt0)
D/t
Gap
Correlation decay exponent
Phase diagram
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Correlations on the MI-HI critical point
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Field theory for two chains
Tunnel coupling is highly relevant !
Direct HI-MI transition is avoided (
intermediate SF phase emerges ).
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Numerical check (V-gt0)
Density wave order
Particle-hole gap
Phase diagram