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Phase Diagram of One-Dimensional Bosons in
Disordered Potential
Anatoli Polkovnikov,
Boston University
Collaboration:
Ehud Altman
Yariv Kafri
Gil Refael
-
Weizmann
Technion
CalTech
Dirty Bosons
Bosonic atoms on disordered substrate:
4He
on Vycor
Cold atoms on optical lattice
Small capacitance Josephson Junction arrays
Granular Superconductors
O(2) quantum rotor model
Provided:
In continuum systems quantum rotor model is valid after
coarse-graining.
One dimension Clean limit
Mapped to classical XY model in 1+1 dimensions:
Superfluid
y
Insulator
Kosterlitz-Thouless
transition
Universal jump in stifness:
K-1
Exponent a
Z. Hadzibabic et. al., Observation of the BKT transition in 2D bosons, Nature (2006)
0.5
Jump in the correlation function
exponent a is related to the jump
in the SF stiffness:
0.4
0.3
high T
low T
see A.P., E. Altman, E. Demler, PNAS (2006)
0
0.1
0.2
0.3
central contrast
Vortex proliferation
Fraction of images showing at least one dislocation:
30%
20%
10%
low T
high T
0
0
0.1
0.2
0.3
0.4
central contrast
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:
Large charging energy

Large Josephson coupling

Follow evolution of the distribution functions.
Possible phases
Superfluid
Insulator
Clusters grow to size of chain with repeated decimation
Disconnected clusters
Use parametrization
0


j 
 1 (capacitance),  j  log ,   log
Uj
Jj

Recursion relations:
   j   j 1 ,    j   j 1  1
Assuming typical
these equations are solved by simple ansatz


Incomressible
Mott Glass
f0 ~ U
f0 and g0 obey flow equations:
These equations describe Kosterlits-Thouless transition
Superfluid
(independently confirmed by Monte-Carlo study K. G. Balabanyan, N.
Prokof'ev, and B. Svistunov, PRL, 2005)
Hamiltonian on the fixed line:
Simple perturbative
argument: weak interactions
are relevant for g0<1 and
irrelevant for g0>1
Diagonal disorder is relevant!!!
Transformation rule for
n:
n  n j  n j 1
Next step in our approach. Consider. n j  {0, 1 2}
This is a closed subspace under the RG transformation rules.
This constraint still preserves particle – hole symmetry.
New decimation rule for half-integer sites:
U=
1
2
Create effective
spin ½ site
0
Other decimation rules:
1
Four coupled RG equations: f(), g(), N , N
N
=
N
is an attractive fixed point
(corresponding to relevance
of diagonal disorder)
Fraction of sites
1.0
integer sites
half-integer sites
0.8
0.6
0.4
0.2
0.0
0
1
2
Log()
3
4
Remaining three equations are solved by an exponential ansatz
Fixed points:
f 0  0, g0  1, s  0 - SF
f 0  , g0  0, s  1 - IN
Number of spin ½ sites is irrelevant near the critical point!
1
Random singlet
insulator
f0=0.5, g0=1.7
f0=0.5, g0=1.7017
f0 ~ U
s
0.1
0.01
1E-3
0
50
 100
150
200
Superfluid
• 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.
General story for arbitrary diagonal disorder.
1. The Sf-IN transition is governed by the non-interacting fixed
point and it always belongs to KT universality class.
2. Disorder in chemical potential is dangerously irrelevant and
does not affect critical properties of the transition as well as the
SF phase.
0.25
0.20
f0
n
0.15
0.10
0.05
0.00
0
1
2

3
4
g0
3. Insulating phase strongly depends on the type of disorder.
g0~1/Log(1/J)
a) Integer filling –
incompressible Mott glass
b) ½ - integer filling – random
singlet insulator with
diverging compressibility
c) Generic case – Bose glass
with finite compressibility

Random-singlet
insulator
Mott
glass
Bose
glass
4. 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,
L, 1 ~ exp(1
)
5. In 1D the system restores dynamical symmetry z=1.
This talk in a nutshell.
Coarse-grain the system
n
n U ( 2n )
Effective U decreases: U
U

2
2 2 2
2
Remaining J decrease,
2
distribution of
2
n becomes wide
Two possible scenarios:
1. U flows to zero faster than J: superfluid phase, n does not matter
2. J flows to zero faster than U: insulating phase, distribution of n
determines the properties of the insulating phase
Critical properties are the same for all possible filling factors!