Phase Diagram of One-Dimensional Bosons in Disordered Potential Anatoli Polkovnikov, Boston University Collaboration: Ehud Altman-Weizmann Yariv Kafri.

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!