1. Superfluid-Insulator Transition of
One-Dimensional Bosons with Strong Disorder
Ehud Altman (Weizmann)
Yariv Kafri (Technion)
Anatoli Polkovnikov (Boston U)
Gil Refael (Caltech)
PRL 93, (2004),
PRL 100, (2008)

2. Outline
1. Introduction: superfluid-insulator transition with the Bose-Hubbard model.
2. Fully and strongly disordered Bose-Hubbard chain.
3. RG flow of the disordered model: superfluid-insulator transition.
4. Three insulating phases: low energy properties.
Luttinger parameter at the transition.

3. Bose-Hubbard model in the rotor representation
Hopping:
Charging
energy:

4. Screening-charges and local spectrum
integer
Generic
Gap:
Half-integer

5. Bose-Hubbard model – phase diagram
Large U - Mott insulator (no charge fluctuations)
Large J - Superfluid (intense charge fluctuations no phase fluctuations) 1
Superfluid
Mott-insulator
M.I. 2
2.5
1.5
0.5

6. Bose-Hubbard model – phase diagram1
Superfluid
Mott-insulator
M.I. 2
2.5
1.5
0.5
Finite density of:
Half-integer
Bose
Glass
Gapless
Compressible
Fisher, Weichman, Grinstein, Fisher (1989).

7. One-dimension weak disorder: Kosterlitz Thouless SF-BG transition
Giamarchi, Schulz (1987).
Insulator
superfluid
- SF stiffness ~ J
- compressibility ~ 1/U 2
M.I.
Bose SF 1
K=3/2: Luttinger parameter
where phase-slips become relevant MI
Glass
M.I.
G+S: True for all disorder!

8. Random Bose-Hubbard model with strong disorder
Off-diagonal disorder: random hopping and charging. X
Three types of diagonal disorder
(1) no charging disorder
“Commensurate”
(2) general offset-charge allowed
“Generic”
(3) only half integer offset charges present
“particle-hole symmetric incommensurate”
(or Cooper-pair disorder)
Fully disordered B-H model: Three types of insulator, Same superfluid.

9. Eliminate high energy scales iteratively.
Strong Disorder – Strategy X X X X X X X X
Can not use perturbative methods around translational invariance.
New approach: strong randomness RG. Ma, Dasgupta, Hu (1979), DS Fisher (1994)
Eliminate high energy scales iteratively.
Find the largest hopping, J, or charge gap, Δ. Diagonalize local Hamiltonian.
Obtain new H that has same form with: One less site Lower maximum energy ( ).
Repeat until eliminating interactions.

10. Strong disorder renormalization – strong bondX X X X X X X X
Pick strongest term
Case 1 -
Strongest bond:
Set: X X

11. X Large charging energy X X X X X X X X Case 2 -
Strongest charging gap:
Ground state:
Perturbation:

12. Phases of the disordered chain
Superfluid X
Insulator X X X X X

13. RG steps summary X X Bond decimation: Site decimation:
Decimation rules
Flow equations for J, U, distributions

14. RG steps summary X X Bond decimation: Site decimation:
Decimation rules
Flow equations for J, U, distributions
What are the distributions of

15. Universal solution of flow equations +
New strong disorder SF-INS fixed point
For now: Ignore offset charges
The flow equations admit the solution:
Insulator
superfluid
Plugging into the flow equations:
RG flow
Parameter:
Superfluid fixed line
Critical fixed point

16. QMC: Balabanyan, Prokof’ev, Svistunov (2005),
Insulator 1: Commensurate disordered chain
Insulator – gapless and incompressible:
Mott-glass
rare, large grains suppress the charge- gap locally. X
Gap
Compressibility
AKPR, (2004).
QMC: Balabanyan, Prokof’ev, Svistunov (2005),
Sengupta, Haas (2007).
Mott-glass was previously predicted In a related model -
Luttinger Liquid with commensurate potential and random backscattering.
(Orignac, Giamarchi, Le Doussal, 1999 )

17. Case 2: Generic disorder
Universal critical properties – same as before!
Insulator
(??)
Put graphs here
superfluid
1/2 1
Superfluid - maximal chemical potential disorder:
Insulator marginal distribution peaked near the degeneracy point.

18. Low energy mapping to a spin model
Late in the RG all site have:
But:
Eliminate all but lowest doublet:
Like spin-1/2:
Spin operators:

19. X Insulator 2: generic strong disorder Bose Glass! X
At low energies (late in the RG flow, ): X X
z-field:
Distribution of fields:
Bose Glass!

20. X Insulator 3: particle-hole symmetric incommensurate disorder x
At low energies (late in the RG flow, ):
All surviving sites are doubly-degenerate, with X X x
No z-field!
Strongest coupling:
Singlets form in a random fashion over long scales –
Bosons are delocalized over long scales.
Ground state is a singlet:
D.S. Fisher (1994).
From perturbation theory:
Compressibility
and SF susceptibility:
Random-singlet Glass!

21. Transport at the critical point – Luttinger parameter?
Insulator
superfluid
RSG BG MG 1 MG BG
RSG
Stiffness:
Compressibility:
Insulator
superfluid 1
Critical Luttinger parameter:
Disorder GS
superfluid
Insulator
AKPR
initial disorder

22. (w/ V. Gurarie, J. Chalker)
Superfluid phase: Phonon localization
(w/ V. Gurarie, J. Chalker) x X
Insulator
superfluid 1
(weak disorder: )

23. Summary and conclusions: A tale of three insulators
SF susc.
Comp.
Commensurate disorder
P-H symmetric disorder
Generic disorder
Mott-Glass
Random-singlet
Glass
Bose-Glass
Gap
All superfluid-insulator transition are in the same universality class.
Type of insulator determined by symmetry of the disorder.
Obtained full universal coupling distributions from RG.
Calculated Luttinger parameter at the transition: universality lost at finite disorder?
More work: low-energy spectrum, numerics, comparison to experiments.

24. (near) Future work: random pancakes
(w/ D. Pekker, E. Demler)
Random chain where each site is a 2d condensate with random stiffness X X X X X X
Competition between interlayer tunneling and vortex formation.

25. Case 2: Generic disorder
Universal critical properties – same as before!
Insulator
(??)
Put graphs here
superfluid
1/2 1
Superfluid - maximal chemical potential disorder:
Insulator marginal distribution peaked near the degeneracy point.
U and δn are correlated:
When:

26. Comparison to numerics
Carried out the RG flow numerically with block and gaussian initial distributions:
Put graphs here
Insulator
superfluid
Example of distributions during RG flow:
Universal!

27. Measurable physical implications – Bosons with off diagonal disorder
The strong-randomness RG approach yields (in addition to phase diagram):
Compressibility and Gap.
Spectrum of charge excitations.
Bound on the SF stiffness.
Full distributions → low T thermodynamics.
DOES NOT yield correct plasma-wave spectrum.

28. Summary and conclusions: A tale of three insulators
Superfluid
susc.
Comp.
Gap
Commensurate disorder
Mott-Glass
P-H symmetric disorder
Random-singlet
Glass
Generic disorder
Bose-Glass
All superfluid-insulator transition are in the same universality class.
Type of insulator determined by symmetry of the disorder.
Real-space RG allows us to find the full universal coupling distributions.
More work: find low-energy spectrum, plasmons, numerics.

29. Summary and conclusions
We used the real space RG technique to analyze a bosonic 1D system with off diagonal randomness.
Discovered behavior that is quite different from the pure case. Floppy superfluid, and Mott-glass.
A new critical quantum fixed point with strictly classical physics
dominates low energy physics + Universal distributions.
Next Goals:
Predict tunneling conductance.
Connect with transport measurements.
Generalization to diagonal+off diagonal disorder.
Application to higher dimensions? (optimistic…)

30. Diagonal disorder – Bose glass1 2 3
Superfluid
Mott-insulator
M.I.
Bose
Glass
Commensurate
In-commensurate
Local incommensurability:
Anderson localization V
Fisher, Weichman, Grinstein, Fisher (1989).

31. Weak disorder - ‘Harris’ criterion
Imaginary time action:
With the disorder obeying:
Rescaling:
Weak disorder is irrelevant.
But what about strong disorder?

32. Flow equations II X Define: : Bond decimation: Site decimation:
(charging
energy)
(Josephson)
Bond decimation: X
Site decimation:

33. Gapless, incompressible
Physical properties
superfluid
insulator
Compressibility
Gap
Stiffness
Gapless, incompressible
Mott Glass
‘floppy’ superfluid
Finite disorder
QCP

34. Creates charge fluctuations – delocalizes bosons
Hopping term – number and phase operators
Hopping:
Creates charge fluctuations – delocalizes bosons
Imposes phase coherence.

35. Strong bond X X X X X X X X X X Pick strongest coupling
Strongest bond: X X

36. Universal properties - correlations
Correlation length:
Correlation time :
Put graphs here
Insulator
superfluid
Dynamical scaling exponent :

37. Off-diagonal + particle hole symmetric disorder
Superfluid
w/ probability: p q
n+1
M.I. n
M.I.
Mott-insulator
n-1
- Example: Chain of superconducting grains with even or odd electron number. X X X X X X X X

38. X Generic disorder X X X X What if is completely unrestricted?
Coulomb blockade: X
If: X X 1 3 X
Cluster formation: X
If:

39. What are the distributions of and ?
Flow equations
Bond decimation: X
Site decimation: X X
What are the distributions of and ?

40. Large charging energy – but small gapX X X X X X X X
Strongest charging energy:
In-commensurate
Ground state:
Like spin-1/2
Density of spins: s
Spin operators:

41. x Strong bond between spin sites X 2 3 1 Strongest coupling:x 3 1
Strongest coupling:
Ground state is a singlet:
From perturbation theory: