R. Citro. In collaboration with: A. Minguzzi (LPMMC, Grenoble, France) E. Orignac (ENS, Lyon, France), X. Deng & L. Santos (MP, Hannover, Germany)

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1 Phase Diagram of interacting Bose gases in one-dimensional disordered optical lattices R. Citro In collaboration with: A. Minguzzi (LPMMC, Grenoble, France) E. Orignac (ENS, Lyon, France), X. Deng & L. Santos (MP, Hannover, Germany)

2 Interplay between disorder and interactions: a long-standing problem Disorder can induce a metal-insulator transition and in 1d free particles are always localized loc l Exact solution (Berezinskii, 1974; Abrikosov, 1978) In reduced dimensions strong reinforcement of disorder due to quantum fluctuations! Anderson localization (Anderson, 1958) But when short range repulsive interactions are present competition of Anderson localization vs delocalization! New phases possible! E.g. Bose glass phase for bosons

3 Outline Bosons in quasiperiod 1D optical lattices The noninteracting limit (Aubry-Andrè model) and character of the localization transition Effect of the interactions on the localization transition: The Bose-glass phase The physical quantities of the transition: the superfluid fraction, compressibility and the momentum distribution Bosons in disordered optical lattices Bosons with longer range interaction: polar bosons Effect of the interactions on the insulating phases Phase diagram by entanglement spectrum The analysis: Combined DMRG and bosonization, RG

4 Atoms in optical lattices Ultracold atoms in optical lattices represent an extremely powerful tool for engineering simple quantum systems, thus serving as quantum simulators (Feynman,1982) to reproduce the physics of different systems Theory: Jaksch et al. PRL (1998) Experiment: Kasevich et al., Science (2001); Greiner et al., Nature (2001); Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004); and many more

5 Superfluid to Mott-insulator A spectacular demonstration (Greiner et al., 2002), (Bloch et al., 2007).

6 Disordered Optical Potential: a natural extension Important parameters: b=k 2 /k 1 D=s 2 /s 1 Experiments on the localization in 1d speckle potential: J. Billy et al., Nature (2008), S.S. Kondov, Science (2011)

7 Interacting bosons in disordered potentials For a system defined on a lattice one can derive a zero T model in which particles occupy the fundamental vibrational state:the Bose-Hubbard Fisher 1989, Jaksch 1998 =s2/s1

8 Phase diagram of disordered bosons A new quantum phase appears: The Bose-Glass phase (compressible but nonsuperfluid) (Giamarchi &Shulz, 1988, Fisher, 1989)

9 Direct SF-MI phase transition? One of the possible Fisher scenarios? Superfluid line transition M.A. Fisher, PRB 1989

10 D/t=2

11 X. Deng, R.C., A.M. EPJ B, 68, p.435(2009), Exp. Roati et al 2009 Non-interacting-limit

12 Interplay of disorder and interaction Finite D, finite U/t : Bose-glass is a compressible, but nonsuperfluid phase [Giamarchi&Shulz, 1989, Fisher et al. 1989] Infinite U/t : Anderson localization of the mapped Fermi gas (Tonks-gas), [Graham et al., 2005] L=50,N=25, t=0.5,=2,u=5 for the Bose glass, U=0 for the Anderson local.

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14 Incommensurate filling Commensurate filling AG BG BG Uc=3.3 SF SF MI a )N/Nsites=0.5, with N=10, Nsites=20. b) The SF fraction (main figure) and compressibility gap (inset) in the case of integer filling with N=Nsites=20.

15 Mott lobes in the grand-canonical ensemble Possibility of a direct MI-SF transition: one of the Fisher scenarios NO for true disorder, but YES for a quasi-periodic potential: DMRG calculation (X. Deng, R.C., A. Minguzzi, E. Orignac, PRA 2008) Phase Diagram: Commensurate case <n>=1

16 Diffraction of ground-state wf U=2t, N=50

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18 Interaction and disorder effects for polar bosons Bosons with long-range interaction: dipolar atoms, A. Griesmanier, PRL (2005); polar molecules S. Ospelkaus, Science (2010); M. Baranov, Phys. Rep. (2010)

19 The lattice Hamiltonian: The extended Bose-Hubbard randomly distributed within the interval [-D,D] for the quasi-periodic potential Some questions: The effect of disorder on the insulating phase How does a Bose glass appear? More easily answered for strong interaction: Mapping to a spin-one chain (Holstein-Primakoff tranformation) +bosonization approach

20 Phase diagram of the polar bosons Superfluid phase SF: algebraic decay of the correlation function Incompressible MI phase with hidden parity order Haldane insulator with hidden string order Density wave-phase

21 Phase diagram for uniform disorder X. Deng et al. New Jour. Phys., 15 (2013) Observables: One-particle correlator along the MI-HI Disorder: Random magnetic field along the z-axis Renormalization group theory: Disorder relevant for K=2K a <3/2 Instability of the MI-HI and Bose glass (V-shape phase diagram)

22 The entanglement spectrum (X. Deng-L. Santos, PRB 2011) It is defined as the spectrum of of the effective Hamiltonian obtained by partitioning the Hamiltonian into parts A and B and tracing over A The eigenvalues l i (L A ) and their degeneracy differ significantly in the various phases. X. Deng et al. New Journ. Phys., 15 (2013)

23 Phase diagram for quasi-periodic potential X. Deng et al. Phys. Rev. B 87, (2013), New Jour. Phys., 15 (2013) Main features: ICDW adiabatically connected to HI; persistence of a density wave phase

24 Main messages We showed evidence for a rich phase diagram for a one-dimensional Bose gas in a disordered lattice: emergence of a Bose glass We provided prediction for a direct MI-SF transition and on the behavior of the momentum distribution The phase diagram is strongly modified in the presence of intersite interaction: string hidden order and density waves compete with BG The MI-HI transition line unstable against a Bose-glass: V-shape vs Y-shape phase diagram The rich phase diagram (SF, MI, HI, DW, ICDW) easily recognized by the entanglement spectrum.

25 Outlook Experimental probes: e.g. transport properties and evidence of Bose-glass behavior Temperature effects and Bose-glass collapse Effect of dissipation and particle losses for systems beyond cold atoms Thank you!

26 The entanglement spectrum: behavior of largest eigenvalues X. Deng et al. New Jour. Phys., 15 (2013)

27 DMRG for the quasiperiodic system We consider a system with periodic boundary conditions and use the infinite-size algorithm to build the Hamiltonian up to the length L the Hilbert space of bosons is infinite; to keep a finite Hilbert space in the calculation, we choose the maximal number of boson states approximately of the order 5n, varying nmax between nmax=6 and 15, except close to the Anderson localization phase where we choose the maximal boson states nmax=n. The number of eigenstates of the reduced density matrix are chosen in the range To test the accuracy of our DMRG method, in the case U=0 or for finite U and small chain, we have compared the DMRG numerical results with the exact solution obtained by direct diagonalization The calculations are performed in the canonical ensemble at a fixed number of particles N.

28 DMRG for the quasiperiodic system