Phase Diagram of OneDimensional Bosons in Disordered Potential

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Phase Diagram of One-Dimensional Bosons in
Disordered Potential
Anatoli Polkovnikov, Boston University
Collaboration
Ehud Altman - Weizmann Yariv Kafri -
Technion Gil Refael - CalTech
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Dirty Bosons
Bosonic atoms on disordered substrate
4He on Vycor Cold atoms on optical lattice Small
capacitance Josephson Junction arrays Granular
Superconductors
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O(2) quantum rotor model
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One dimension Clean limit
Mapped to classical XY model in 11 dimensions
Superfluid
Insulator
Kosterlitz-Thouless transition
y
Universal jump in stifness
K-1
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Z. Hadzibabic et. al., Observation of the BKT
transition in 2D bosons, Nature (2006)
Jump in the correlation function exponent a is
related to the jump in the SF stiffness
see A.P., E. Altman, E. Demler, PNAS (2006)
Vortex proliferation
Fraction of images showing at least one
dislocation
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No off-diagonal disorder
E. Altman, Y. Kafri, A.P., G. Refael, PRL (2004)
Real Space RG
( Spin chains Dasgupta Ma PRB 80, Fisher PRB
94, 95 )
Eliminate the largest coupling
Follow evolution of the distribution functions.
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Possible phases
Superfluid
Clusters grow to size of chain with repeated
decimation
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These equations describe Kosterlits-Thouless
transition (independently confirmed by
Monte-Carlo study K. G. Balabanyan, N. Prokof'ev,
and B. Svistunov, PRL, 2005)
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Diagonal disorder is relevant!!!
Next step in our approach. Consider.
This is a closed subspace under the RG
transformation rules. This constraint still
preserves particle hole symmetry.
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New decimation rule for half-integer sites
Create effective spin ½ site
UW
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Four coupled RG equations f(?), g(b), ,
is an attractive fixed point (corresponding to
relevance of diagonal disorder)
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Number of spin ½ sites is irrelevant near the
critical point!

• The transition is governed by the same
  non-interacting critical point as in the integer
  case.
• Spin ½ sites are (dangerously) irrelevant at the
  critical point.
• Insulating phase is the random singlet insulator
  with infinite compressibility.

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General story for arbitrary diagonal disorder.

• The Sf-IN transition is governed by the
  non-interacting fixed point and it always belongs
  to KT universality class.
• Disorder in chemical potential is dangerously
  irrelevant and does not affect critical
  properties of the transition as well as the SF
  phase.

f0
g0
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• Insulating phase strongly depends on the type of
  disorder.
• Integer filling incompressible Mott glass
• ½ - integer filling random singlet insulator
  with diverging compressibility
• Generic case Bose glass with finite
  compressibility
• We confirm earlier findings (Fisher et. al. 1989,
  Giamarchi and Schulz 1988) that there is a direct
  KT transition from SF to Bose glass in 1D, in
  particular,
• In 1D the system restores dynamical symmetry z1.

G
Random-singlet insulator
g01/Log(1/J)
Bose glass
Mott glass
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This talk in a nutshell.