Phase Diagram of One-Dimensional Bosons in Disordered Potential

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

2. Dirty Bosons Bosonic atoms on disordered substrate: 4He on Vycor Cold atoms on optical lattice Small capacitance Josephson Junction arrays Granular Superconductors

3. Provided: In continuum systems quantum rotor model is valid after coarse-graining. O(2) quantum rotor model

4. One dimension Clean limit Mapped to classical XY model in 1+1 dimensions: Superfluid Insulator Kosterlitz-Thouless transition y Universal jump in stifness: K-1

5. Exponent a high T low T central contrast 30% 0.5 20% 0.4 10% low T high T 0.3 0 0.2 0.3 0.4 0 0.1 central contrast 0 0.1 0.2 0.3 Z. Hadzibabic et. al., Observation of the BKT transition in 2D bosons, Nature (2006) Jump in the correlation function exponent a is related to the jump in the SF stiffness: see A.P., E. Altman, E. Demler, PNAS (2006) Vortex proliferation Fraction of images showing at least one dislocation:

6. Large Josephson coupling Large charging energy No off-diagonal disorder: E. Altman, Y. Kafri, A.P., G. Refael, PRL (2004) Real Space RG ( Spin chains:Dasgupta & Ma PRB 80, Fisher PRB 94, 95 ) Eliminate the largest coupling: Follow evolution of the distribution functions.

7. Insulator Disconnected clusters Possible phases Superfluid Clusters grow to size of chain with repeated decimation

8. Use parametrization Recursion relations: Assuming typical these equations are solved by simple ansatz

9. Incomressible Mott Glass f0 and g0 obey flow equations: f0 ~ U Superfluid Hamiltonian on the fixed line: Simple perturbative argument: weak interactions are relevant for g0<1 and irrelevant for g0>1 These equations describe Kosterlits-Thouless transition(independently confirmed by Monte-Carlo study K. G. Balabanyan, N. Prokof'ev, and B. Svistunov, PRL, 2005)

10. Transformation rule for : Diagonal disorder is relevant!!! Next step in our approach. Consider. This is a closed subspace under the RG transformation rules. This constraint still preserves particle – hole symmetry.

11. Other decimation rules: New decimation rule for half-integer sites: Create effective spin ½ site U=W

12. N N N N = Remaining three equations are solved by an exponential ansatz Fixed points: Four coupled RG equations: f(), g(b), , is an attractive fixed point (corresponding to relevance of diagonal disorder)

13. Random singlet insulator f0 ~ U Superfluid Number of spin ½ sites is irrelevant near the critical point! • The transition is governed by the same non-interacting critical point as in the integer case. • Spin ½ sites are (dangerously) irrelevant at the critical point. • Insulating phase is the random singlet insulator with infinite compressibility.

14. General story for arbitrary diagonal disorder. • The Sf-IN transition is governed by the non-interacting fixed point and it always belongs to KT universality class. • Disorder in chemical potential is dangerously irrelevant and does not affect critical properties of the transition as well as the SF phase. f0 g0

15. Insulating phase strongly depends on the type of disorder. • Integer filling – incompressible Mott glass • ½ - integer filling – random singlet insulator with diverging compressibility • Generic case – Bose glass with finite compressibility • We confirm earlier findings (Fisher et. al. 1989, Giamarchi and Schulz 1988) that there is a direct KT transition from SF to Bose glass in 1D, in particular, • In 1D the system restores dynamical symmetry z=1. G Random-singlet insulator g0~1/Log(1/J) Bose glass Mott glass

16. Coarse-grain the system Effective U decreases: Remaining J decrease, distribution of becomes wide • Two possible scenarios: • U flows to zero faster than J: superfluid phase, does not matter • J flows to zero faster than U: insulating phase, distribution of determines the properties of the insulating phase Critical properties are the same for all possible filling factors! This talk in a nutshell.