Superfluid-Insulator Transition of One-Dimensional Bosons with Strong Disorder

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, 150402 (2004), PRL 100, 170402(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. Charging energy: Hopping: Bose-Hubbard model in the rotor representation

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

5. 2.5 2 M.I. Superfluid 1.5 1 Mott-insulator 0.5 M.I. 0 Bose-Hubbard model – phase diagram • Large U - Mott insulator (no charge fluctuations) • Large J - Superfluid (intense charge fluctuations no phase fluctuations)

6. 2.5 • Finite density of: Superfluid 2 M.I. 1.5 Half-integer Bose Gapless Compressible 1 Mott-insulator 0.5 Glass M.I. 0 Bose-Hubbard model – phase diagram Fisher, Weichman, Grinstein, Fisher (1989).

7. - SF stiffness ~ J - compressibility ~ 1/U Insulator superfluid One-dimension weak disorder: Kosterlitz Thouless SF-BG transition Giamarchi, Schulz (1987). 2 M.I. SF Bose K=3/2: Luttinger parameter where phase-slips become relevant 1 MI Glass M.I. 0 G+S: True for all disorder!

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

9. 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. 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.

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

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

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

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

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

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

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

17. Superfluid - maximal chemical potential disorder: • Insulator - marginal distribution peaked near the degeneracy point. Case 2: Generic disorder • Universal critical properties – same as before! Insulator (??) Put graphs here 0 1/2 1 superfluid

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

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

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

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

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

23. SF susc. Comp. Gap Commensurate disorder Mott-Glass Generic disorder Bose-Glass P-H symmetric disorder Random-singlet Glass Summary and conclusions: A tale of three insulators • 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. Superfluid - maximal chemical potential disorder: U and δn are correlated: • Insulator - marginal distribution peaked near the degeneracy point. When: Case 2: Generic disorder • Universal critical properties – same as before! Insulator (??) Put graphs here 0 1/2 1 superfluid

26. Comparison to numerics • Example of distributions during RG flow: • Carried out the RG flow numerically with block and gaussian initial distributions: Put graphs here Insulator superfluid 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. Commensurate M.I. 3 Bose 2 M.I. Superfluid Glass In-commensurate • Anderson localization M.I. 1 V Mott-insulator 0 Diagonal disorder – Bose glass • Local incommensurability: Fisher, Weichman, Grinstein, Fisher (1989).

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

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

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

34. 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 • Pick strongest coupling Strongest bond: X X

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

37. w/ probability: p q Off-diagonal + particle hole symmetric disorder Superfluid 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. What if is completely unrestricted? • Coulomb blockade: X If: X X 1 3 • Cluster formation: X X If: Generic disorder X

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

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

41. X X X x • Ground state is a singlet: • From perturbation theory: Strong bond between spin sites 2 3 0 1 • Strongest coupling: