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)

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.

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

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

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

Bose-Hubbard model – phase diagram 1 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).

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!

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.

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.

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

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

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

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

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

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

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 )

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.

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:

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!

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!

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

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

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.

(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.

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:

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!

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.

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.

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…)

Diagonal disorder – Bose glass 1 2 3 Superfluid Mott-insulator M.I. Bose Glass Commensurate In-commensurate Local incommensurability: Anderson localization V Fisher, Weichman, Grinstein, Fisher (1989).

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

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

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

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

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

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

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

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:

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

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

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: