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Phase Diagram
of interacting Bose gases
in one
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)
Interplay between disorder and
interactions: …a long-standing problem
Disorder can induce a metal-insulator
transition and in 1d free particles are
always localized
 l
loc
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
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
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 …
Superfluid to Mott-insulator
A spectacular demonstration
(Greiner et al., 2002),
(Bloch et al., 2007).
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 nonsuperfluid) (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
Non-interacting-limit
X. Deng, R.C., A.M. EPJ B, 68, p.435(2009), Exp. Roati et al 2009
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.
Commensurate filling
Incommensurate filling
AG
BG
BG
SF
SF
Uc=3.3
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.
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
Diffraction of
ground-state wf
U=2t, N=50
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
 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
Phase diagram for uniform disorder
X. Deng et al. New Jour. Phys., 15 (2013) 045023
 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)
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
X. Deng et al. Phys. Rev. B 87, 195101 (2013),
New Jour. Phys., 15 (2013) 045023
 Main features: ICDW adiabatically connected to HI; persistence of a
density wave phase
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.
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
you!!
The entanglement spectrum:
behavior of largest eigenvalues
X. Deng et al. New Jour. Phys., 15 (2013) 045023
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 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