Title: Phase Diagram of OneDimensional Bosons in Disordered Potential
1Phase Diagram of One-Dimensional Bosons in
Disordered Potential
Anatoli Polkovnikov, Boston University
Collaboration
Ehud Altman - Weizmann Yariv Kafri -
Technion Gil Refael - CalTech
2Dirty Bosons
Bosonic atoms on disordered substrate
4He on Vycor Cold atoms on optical lattice Small
capacitance Josephson Junction arrays Granular
Superconductors
3O(2) quantum rotor model
4One dimension Clean limit
Mapped to classical XY model in 11 dimensions
Superfluid
Insulator
Kosterlitz-Thouless transition
y
Universal jump in stifness
K-1
5Z. 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
6No 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.
7Possible phases
Superfluid
Clusters grow to size of chain with repeated
decimation
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9These equations describe Kosterlits-Thouless
transition (independently confirmed by
Monte-Carlo study K. G. Balabanyan, N. Prokof'ev,
and B. Svistunov, PRL, 2005)
10Diagonal 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.
11New decimation rule for half-integer sites
Create effective spin ½ site
UW
12Four coupled RG equations f(?), g(b), ,
is an attractive fixed point (corresponding to
relevance of diagonal disorder)
13Number 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.
14General 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
15- 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
16This talk in a nutshell.