Post on 23-Jan-2016
Phase Diagram of One-Dimensional Bosons in Phase Diagram of One-Dimensional Bosons in Disordered PotentialDisordered Potential
Anatoli Polkovnikov,Anatoli Polkovnikov,Boston UniversityBoston University
Collaboration:
Ehud AltmanEhud Altman -- WeizmannWeizmannYariv Kafri Yariv Kafri - - TechnionTechnionGil Refael Gil Refael - - CalTechCalTech
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 In continuum systems quantum rotor model is valid after coarse-graining.coarse-graining.
One dimension Clean limit
Mapped to classical XY model in 1+1 dimensions:
Superfluid Insulator
K-1
y
Kosterlitz-Thouless transition
Universal jump in stifness:
Exp
onen
t
central contrast
0.5
0 0.1 0.2 0.3
0.4
0.3 high T low T
Z. Hadzibabic et. al., Observation of the BKT transition in 2D bosons, Nature (2006)Z. Hadzibabic et. al., Observation of the BKT transition in 2D bosons, Nature (2006)
Vortex proliferationVortex proliferation
Fraction of images showing at least one dislocation:
0
10%
20%
30%
central contrast0 0.1 0.2 0.3 0.4
high T low T
Jump in the correlation function Jump in the correlation function exponent exponent is related to the is related to the jump in the SF stiffness: jump in the SF stiffness:
see A.P., E. Altman, E. Demler, PNAS (2006)see A.P., E. Altman, E. Demler, PNAS (2006)
No off-diagonal disorder:No off-diagonal disorder:
Real Space RGReal Space RG
Eliminate the largest coupling:
Large charging energy
Large Josephson coupling
E. Altman, Y. Kafri, A.P., G. Refael, PRL (2004)E. Altman, Y. Kafri, A.P., G. Refael, PRL (2004)
( Spin chains: Dasgupta & Ma PRB 80, Fisher PRB 94, 95 )
Follow evolution of the distribution functions.Follow evolution of the distribution functions.
Possible phasesSuperfluid Clusters grow to size of chain with repeated decimation
Insulator Disconnected clusters
Use parametrizationUse parametrization
01 (capacitance), log , logj jj jU J
Recursion relations:
1 1 , 1j j j j
Assuming typical these equations are solved by simple ansatz
f0 and g0 obey flow equations:
These equations describe Kosterlits-Thouless transitionThese equations describe Kosterlits-Thouless transition (independently confirmed by Monte-Carlo study K. G. Balabanyan, N. (independently confirmed by Monte-Carlo study K. G. Balabanyan, N. Prokof'ev, and B. Svistunov, PRL, 2005)Prokof'ev, and B. Svistunov, PRL, 2005)
Incomressible Mott Glass
Superfluid
ff 0 0 ~
U~
UHamiltonian on the fixed line:
Simple perturbative Simple perturbative argument: weak interactions argument: weak interactions are relevant for are relevant for gg00<<1 and 1 and
irrelevant for irrelevant for gg00>>1 1
Diagonal disorder is relevant!!!Diagonal disorder is relevant!!!
Transformation rule for :Transformation rule for :n 1j jn n n
Next step in our approach. Consider. Next step in our approach. Consider. {0, 1 2}jn
This is a closed subspace under the RG transformation rules. This is a closed subspace under the RG transformation rules. This constraint still preserves particle – hole symmetry.This constraint still preserves particle – hole symmetry.
New decimation rule for half-integer sites:New decimation rule for half-integer sites:
U=U=
Create effective Create effective spin ½ sitespin ½ site
0 1
1 2
Other decimation rules:Other decimation rules:
Four coupled RG equations: Four coupled RG equations: f(f(), g), g , ,
is an attractive fixed pointis an attractive fixed point(corresponding to relevance (corresponding to relevance of diagonal disorder)of diagonal disorder)
NN NN
NN NN==
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Fra
ctio
n of
site
s
Log()
integer sites half-integer sites
Remaining three equations are solved by an exponential ansatzRemaining three equations are solved by an exponential ansatz
Fixed points:Fixed points:
0 0
0 0
0, 1, 0 - SF
, 0, 1 - IN
f g s
f g s
Number of spin ½ sites is irrelevant near the critical point!Number of spin ½ sites is irrelevant near the critical point!
Random singlet insulator
Superfluid
ff 0 0 ~
U~
U
0 50 100 150 200
1E-3
0.01
0.1
1
s
f0=0.5, g
0=1.7
f0=0.5, g
0=1.7017
• The transition is governed by the same The transition is governed by the same non-interactingnon-interacting critical point as in the critical point as in the integer case.integer case.
• Spin ½ sites are (dangerously) irrelevant at the critical point.Spin ½ sites are (dangerously) irrelevant at the critical point.
• Insulating phase is the random singlet insulator with infinite compressibility.Insulating phase is the random singlet insulator with infinite compressibility.
General story for arbitrary diagonal disorder.General story for arbitrary diagonal disorder.
1.1. The Sf-IN transition is governed by the non-interacting fixed The Sf-IN transition is governed by the non-interacting fixed point and it always belongs to KT universality class.point and it always belongs to KT universality class.
2.2. Disorder in chemical potential is dangerously irrelevant and Disorder in chemical potential is dangerously irrelevant and does not affect critical properties of the transition as well as does not affect critical properties of the transition as well as the SF phase.the SF phase.
0 1 2 3 40.00
0.05
0.10
0.15
0.20
0.25
n
gg00
ff 00
3.3. Insulating phase strongly depends on the type of disorder.Insulating phase strongly depends on the type of disorder.
a)a) Integer filling – Integer filling – incompressible Mott glassincompressible Mott glass
b)b) ½ - integer filling – random ½ - integer filling – random singlet insulator with singlet insulator with diverging compressibilitydiverging compressibility
c)c) Generic case – Bose glass Generic case – Bose glass with finite compressibilitywith finite compressibility
4.4. We confirm earlier findings (Fisher et. al. 1989, Giamarchi We confirm earlier findings (Fisher et. al. 1989, Giamarchi and Schulz 1988) that there is a direct KT transition from and Schulz 1988) that there is a direct KT transition from SF to Bose glass in 1D, in particular,SF to Bose glass in 1D, in particular,
5.5. In 1D the system restores dynamical symmetry In 1D the system restores dynamical symmetry zz=1.=1.
gg 00~1/
Log
(1/J
)~1
/Log
(1/J
)
Mott Mott glassglass
BoseBoseglassglass
Random-singletRandom-singletinsulatorinsulator
1, ~ exp(1 )L
This talk in a nutshell.This talk in a nutshell.
Coarse-grain the systemCoarse-grain the system
Effective Effective U U decreases:decreases: 22 2 2
2 2 2 2
nn n UU U
Remaining Remaining JJ decrease, distribution of becomes wide decrease, distribution of becomes widen
Two possible scenarios:Two possible scenarios:
1.1. UU flows to zero faster than flows to zero faster than JJ: superfluid phase, does not matter : superfluid phase, does not matter 2.2. JJ flows to zero faster than U: insulating phase, distribution of flows to zero faster than U: insulating phase, distribution of
determines the properties of the insulating phasedetermines the properties of the insulating phase
nn
Critical properties are the same for all possible filling factors!Critical properties are the same for all possible filling factors!