Transcript Functional integrals in many

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
Outline:
1) 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
Functional Integrals
for many-body systems
Partition function:
=
Green’s function:
S
b
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
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
Key experiment: BEC-Mott transition
M.Greiner et al. 2002
Key experiment: BEC-Mott transition
M.Greiner et al. 2002
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 (n=1)
Phase diagram for T=0
Mott insulator
Bose-Einstein
Condensate
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
Mixture of Heavy and Light Particles
time
Adding heavy particles
Light particles follow random walks
“tunneling”
heavy particles are
thermally distributed
Mixtures in an optical lattice:
elastic scattering
Asymmetric (Fermion or Boson) Model
light atoms:
heavy atoms:
Local repulsive interaction:
asymmetric Hubbard Model
Single-site approximation: Mott state
Fermionic Mixtures
Bosonic Mixtures
Fermionic-Bosonic Mixtures
Ising representation of heavy atoms
Green’s function of light atoms: quenched average(!):
with respect to the distribution
self-organized disorder
Schematic phase diagrams
distribution of heavy atoms P(n)
empty
m=U/2
fully occupied
Distribution of heavy atoms
• Monte-Carlo simulation for decreasing temperature T
Phase separation
T=1/3
T=1/7
near phase transition
T=1/14
Phase separation
density of light atoms in a harmonic trap
Density of States for Light Atoms
m=U/2: increasing U
Propagation
vs.localization
propagation
b>bc
localization
b<bc
Localization transition
Finite size scaling
Mixtures in an optical lattice:
Inelastic Scattering
Heavy atoms in a Mott state
Locally oscillating heavy particles
heavy particles can not tunnel but are local harmonic oscillators
Adding oscillating particles
heavy particles in Mott-insulating state
Single-Site Approximation
Lang-Firsov transformation:
effective fermionic interaction:
Ground states
• Neel and density wave
Ueff>0
Ueff<0
First order phase transition
Alternating dimer states
two types of dimers:
Degeneracy under p rotation of dimers!
dimer liquid on frustrated lattices?
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 ->Ising 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)?