In collaboration with:

A. Minguzzi (LPMMC, Grenoble, France)

E. Orignac (ENS, Lyon, France),

X. Deng & L. Santos (MP, Hannover, Germany)

PhasePhase DiagramDiagram

ofof iinteractingnteracting BoseBose gasesgases

in in oneone--dimensionaldimensional disordereddisordered opticaloptical latticeslattices

R. Citro

Interplay between disorder and

interactions: …a long-standing problem

Disorder can induce a metal-insulator

transition and in 1d free particles are

always localized

Exact solution (Berezinskii, 1974; Abrikosov, 1978)

lloc

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

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

The analysis: Combined DMRG and bosonization, RG

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

Atoms in optical lattices

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 …

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

Superfluid to Mott-insulator

(Greiner et al., 2002), (Bloch et al., 2007).

A spectacular demonstration

Disordered Optical Potential: a natural extension

Important parameters: b=k2/k1 D=s2/s1

Experiments on the localization in 1d speckle potential: J. Billy et al., Nature (2008),

S.S. Kondov, Science (2011)

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

Phase diagram of disordered bosons

A new quantum phase appears: The Bose-Glass phase (compressible but non-

superfluid) (Giamarchi &Shulz, 1988, Fisher, 1989)

?

Direct SF-MI phase transition? One of

the possible Fisher scenarios

Superfluid line

transition

M.A. Fisher, PRB 1989

D/t=2

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

Non-interacting-limit

Interplay of disorder and interaction

Finite D, finite U/t : Bose-glass is a compressible, but non-

superfluid 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.

Incommensurate filling Commensurate filling

SF

BG AG

MI

BG

SF

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.

Uc=3.3

Phase Diagram:

Commensurate

case <n>=1

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)

U=2t, N=50

Diffraction of

ground-state wf

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)

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

Phase diagram of the polar bosons

Incompressible MI phase with hidden parity order

Haldane insulator with hidden string order

Superfluid phase SF: algebraic decay of the correlation function

Density wave-phase

Phase diagram for uniform disorder

Observables: One-particle correlator

along the MI-HI

Disorder: Random magnetic field along the z-axis

Renormalization group theory: Disorder relevant for K=2Ka<3/2

Instability of the MI-HI and Bose glass (V-shape phase diagram)

X. Deng et al. New Jour. Phys., 15 (2013) 045023

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 li(LA) and their degeneracy differ significantly in the various

phases.

X. Deng et al. New Journ. Phys., 15 (2013) 045023

Phase diagram for quasi-periodic potential

Main features: ICDW adiabatically connected to HI; persistence of a

density wave phase

X. Deng et al. Phys. Rev. B 87, 195101 (2013),

New Jour. Phys., 15 (2013) 045023

MainMain messagesmessages

We provided prediction for a direct MI-SF transition and on the behavior of

the momentum distribution

We showed evidence for a rich phase diagram for a one-dimensional Bose

gas in a disordered lattice: emergence of a Bose glass

The phase diagram is strongly modified in the presence of intersite

interaction: string hidden order and density waves compete with BG

The rich phase diagram (SF, MI, HI, DW, ICDW) easily recognized by the

entanglement spectrum.

The MI-HI transition line unstable against a Bose-glass: V-shape vs Y-shape

phase diagram

OutlookOutlook

Temperature effects and Bose-glass collapse

Experimental probes: e.g. transport properties and evidence of

Bose-glass behavior

Effect of dissipation and particle losses for systems beyond cold

atoms

ThankThank youyou!!

The entanglement spectrum:

behavior of largest eigenvalues

X. Deng et al. New Jour. Phys., 15 (2013) 045023

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 80–200.

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.

DMRG for the quasiperiodic system

DMRG for the quasiperiodic system