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: 4 He on Vycor Cold atoms on optical lattice Small capacitance Josephson Junction arrays Granular Superconductors

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

4. 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:

5. Exponent  central contrast 0.5 00.10.20.3 0.4 0.3 high Tlow T Z. Hadzibabic et. al., Observation of the BKT transition in 2D bosons, Nature (2006) Vortex proliferation Fraction of images showing at least one dislocation: 0 10% 20% 30% central contrast 0 0.1 0.20.30.4 high T low T Jump in the correlation function exponent is related to the jump in the SF stiffness: Jump in the correlation function exponent  is related to the jump in the SF stiffness: see A.P., E. Altman, E. Demler, PNAS (2006)

6. No off-diagonal disorder: Real Space RG Eliminate the largest coupling: Large charging energy Large Josephson coupling 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.

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

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

9. f 0 and g 0 obey flow equations: These equations describe Kosterlits-Thouless transition (independently confirmed by Monte-Carlo study K. G. Balabanyan, N. Prokof'ev, and B. Svistunov, PRL, 2005) Incomressible Mott Glass Superfluid f 0 ~ U Hamiltonian on the fixed line: Simple perturbative argument: weak interactions are relevant for g 0 1

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

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

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

13. Number of spin ½ sites is irrelevant near the critical point! Random singlet insulator Superfluid f 0 ~ U The transition is governed by the same non-interacting critical point as in the integer case. 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. 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.

14. General story for arbitrary diagonal disorder. 1.The Sf-IN transition is governed by the non-interacting fixed point and it always belongs to KT universality class. 2.Disorder in chemical potential is dangerously irrelevant and does not affect critical properties of the transition as well as the SF phase. g0g0g0g0 f0f0f0f0

15. 3.Insulating phase strongly depends on the type of disorder. a)Integer filling – incompressible Mott glass b)½ - integer filling – random singlet insulator with diverging compressibility c)Generic case – Bose glass with finite compressibility 4.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, 5.In 1D the system restores dynamical symmetry z=1.  g 0 ~1/Log(1/J) Mottglass Boseglass Random-singletinsulator

16. This talk in a nutshell. Coarse-grain the system Coarse-grain the system Effective U decreases: Remaining J decrease, distribution of becomes wide Two possible scenarios: 1.U flows to zero faster than J: superfluid phase, does not matter 2.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!