Functional Integration in many-body systems: application to ultracold gases

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Functional Integration in many-body systems
application to ultracold gases

• Klaus Ziegler, Institut für Physik, Universität
  Augsburg
• in collaboration with
• Oleksandr Fialko (Augsburg)
• Cenap Ates (MPIPKS Dresden)

Path Integrals, Dresden, September 2007
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Outline

• Mixtures of atoms with elastic scattering
• Competition between thermal and quantum
  fluctuations
• a) asymmetric fermion/bosonic Hubbard model
• b) distribution of heavy fermions
• c) density of states of light fermions
• d) diffusion and localization of light
  fermions
• 2) Mixtures of atoms with inelastic scattering
• a) Holstein-Hubbard model
• b) phases Neel state, density wave,
  alternating dimer state

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Functional Integralsfor many-body systems
Partition function
S
b

Greens function
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Calculational Methods

• Integral (Hubbard-Stratonovich) transform
• Grassmann fields vs. complex fields
• Generalization (e.g. to N components)
• Approximations
• Saddle-point integration
• Gaussian fluctuations
• Monte-Carlo simulation

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Ultracold gases in an optical lattice
counterpropagating Laser fields
Computer for 1D, 2D or 3D many-body
systems Tunable tunneling rate via amplitude of
the Laser field Tunable interaction via magnetic
field (near Feshbach resonance) Tunable density
of particles Free choice of particle statistics
(bosonic or fermionic atoms) Free choice of
spin Free choice of lattice type
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Key experiment BEC-Mott transition
M.Greiner et al. 2002
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Key experiment BEC-Mott transition
M.Greiner et al. 2002
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Bose gas in an optical lattice
Competition between kinetic and repulsive energy
Bose-Einstein Condensate kinetic energy wins
tunneling
Phase coherence but fluctuating particle density
Mott insulator repulsion wins
NO phase coherence but constant local particle
density (n1)
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Phase diagram for T0
Mott insulator
Bose-Einstein Condensate
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Mixture of two atomic species in an optical
lattice

• Two types of atoms with different mass
• a minimal model
• consider two types of fermions (e.g. 6Li and 40K)
• - both species are subject to thermal
  fluctuations
• only light atoms can tunnel
• magnetic trap atoms are spin polarized (Pauli
  exclusion!)
• different species are subject to a local
  repulsive interaction

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Mixture of Heavy and Light Particles
time
Adding heavy particles
heavy particles are thermally distributed
Light particles follow random walks tunneling
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Mixtures in an optical latticeelastic scattering
Asymmetric (Fermion or Boson) Model
light atoms
heavy atoms
Local repulsive interaction
asymmetric Hubbard Model
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Single-site approximation Mott state
Fermionic Mixtures
Bosonic Mixtures
Fermionic-Bosonic Mixtures
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Ising representation of heavy atoms
Greens function of light atoms quenched
average(!)
with respect to the distribution
self-organized disorder
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Schematic phase diagrams
distribution of heavy atoms P(n)
fully occupied
empty
mU/2
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Distribution of heavy atoms

• Monte-Carlo simulation for decreasing temperature
  T

Phase separation
T1/14
T1/7
T1/3
near phase transition
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Phase separation
density of light atoms in a harmonic trap
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Density of States for Light Atoms
mU/2 increasing U
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Propagation vs.localization
propagation
localization
bgtbc
bltbc
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Localization transition
Finite size scaling
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Mixtures in an optical lattice Inelastic
Scattering
Heavy atoms in a Mott state
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Locally oscillating heavy particles
heavy particles can not tunnel but are local
harmonic oscillators
Adding oscillating particles
heavy particles in Mott-insulating state
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Single-Site Approximation
Lang-Firsov transformation
effective fermionic interaction
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Ground states

• Neel and density wave

Ueffgt0
Uefflt0
First order phase transition
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Alternating dimer states
two types of dimers
Degeneracy under p rotation of dimers! dimer
liquid on frustrated lattices?
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Conclusions
Results New quantum states due to competing
atomic species Degeneracy can lead to complex
states Heavy atoms represent correlated random
potential for light atoms heavy fermions -gtIsing
spins correlation effect opening of a
gap disorder effect localization of atoms Open
Questions Can exotic states (e.g. spin liquids)
be realized experimentally? Is there a glass
phase (frustration)?