Simulating the Fermi -Hubbard model with ultracold atoms...
59
Experiments in Tilman Esslinger’s group, ETH Zurich Leticia Tarruell Les Houches – 02/07/2015 Simulating the Fermi-Hubbard model with ultracold atoms in optical lattices
Transcript of Simulating the Fermi -Hubbard model with ultracold atoms...
Experiments in Tilman Esslinger’s group, ETH Zurich
Leticia Tarruell
Les Houches – 02/07/2015
Simulating the Fermi-Hubbard model with ultracold atoms in optical lattices
« with a suitable class of quantum machines you could imitate any quantum system »
Quantum simulation
Many-body system
R. P. Feynman, 1981
Quantum simulator
Model
t U ?
Quantum simulation
t U ? Strongly correlated
materials Model
« with a suitable class of quantum machines you could imitate any quantum system »
R. P. Feynman, 1981
Quantum simulator
Quantum simulation
t U ? Strongly correlated
materials Fermi-Hubbard
model
« with a suitable class of quantum machines you could imitate any quantum system »
R. P. Feynman, 1981
Quantum simulator
Quantum simulation
t U ? Strongly correlated
materials Fermi-Hubbard
model
Ultracold fermions in optical lattices
The Fermi-Hubbard model
tunneling interaction
t U
Repulsive phase diagram
Metal Mott insulator
Magnetic order Doping: high-Tc?
Building the Fermi-Hubbard model Metal-Mott insulator transition Short-range magnetic correlations
Outline
Building the Fermi-Hubbard model Metal-Mott insulator transition Short-range magnetic correlations
Outline
100000 trapped fermionic atoms T<0.1TF
TF
(fermionic isotopes of K, Li, Yb, Sr, Dy, Er, Cr)
Creating a periodic potential
Quantum gases in optical lattices
x
y
z
100x100x20 sites
Brillouin zones
(k=2π/λ)
quasi momentum
Realizing tight-binding models
Time-of-flight absorption imaging
CCD
Atoms
Glass cell (ultra-high vacuum)
Lens
Band mapping
Deep lattice Weak lattice Free atoms
lattice switch-off time
Metal – band insulator transition
Metal Band insulator filling
M. Köhl, H. Moritz, T. Stöferle, K. Günter and T. Esslinger, Phys. Rev. Lett. 94, 080403 (2005)
Observing the Fermi surface
Other non-standard lattices: aligned dimers, plaquettes, Kagome, Lieb (NIST, Munich, Bonn, Hamburg, Berkeley, Kyoto)
Many lattice geometries possible
Chequerboard
Triangular
Dimer 1D chains
Square Honeycomb
L. Tarruell et al., Nature 483, 302 (2012)
Studying strongly correlated systems
The Fermi-Hubbard model
t U
Control: tunneling (lattice depth) interaction (Feshbach resonances) filling (atom number) spin (internal states) lattice geometry gauge fields …
Correlated materials vs. cold atoms
x
y
z
High-temperature superconductor Ultra-cold fermions + optical lattice
Lattice spacing: ~ 500 nm Density: ~ 1013 atoms/cm3
Temperature: ~ nK
Lattice spacing: ~ nm Density: ~ 1022 electrons/cm3 Temperature: ~ K
40K
Some specificities of cold atom systems Trap: varying chemical potential over the cloud
U Vtrap
t
Varying filling: several phases in one cloud
Some specificities of cold atom systems
Isolated: total entropy of the system is fixed
Building the Fermi-Hubbard model Metal-Mott insulator transition Short-range magnetic correlations
Outline
The metal – Mott insulator transition
kinetic energy interaction energy
Bosons : superfluid – Mott insulator
Fermions : metal – Mott insulator
M. Greiner et al. Nature 415, 39 (2002)
U/t
Delocalization vs. interactions
R. Jördens et al., Nature 455, 204 (2008); U. Schneider et al., Science 322, 1520 (2008)
Temperature scales
U>>t
ener
gy
T > U: metallic behaviour
T < U: Mott insulator
T
Which observable? Mainz: cloud compressibility (Mott insulator = incompressible)
U. Schneider et al., Science 322, 1520 (2008)
Which observable? Zurich: occupation of lattice sites
Mott insulator: reduced number fluctuations
Metal Mott insulator
In-situ fluorescence imaging
Harvard, Munich, Tokyo (bosons)
Harvard, Glasgow, MIT (fermions)
Which observable?
Double occupancy
Zurich: occupation of lattice sites
Mott insulator: reduced number fluctuations
Metal Mott insulator
4. Expansion and Stern-Gerlach separation
Measuring double occupancy
2. Induce energy shift
1. Suppress tunneling 3. RF transfer
Doubly occupied sites
Measuring double occupancy
mF=-9/2 mF=-5/2 mF=-7/2
Doubly occupied sites
Measure D for values as low as 1%
Qualitative results
Qualitative results
U/6t = 0 Kinetic energy and trap dominated U/6t = 4.8 Interaction energy dominated
R. Jördens, N. Strohmaier, K. Günter, H. Moritz and T. Esslinger, Nature 455, 204 (2008)
U/6t=4.8
U/6t=0 Non interacting
Mott insulator
Quantitative comparison
R. Jördens, L. Tarruell, D. Greif, T. Uehlinger, N. Strohmaier, H. Moritz, T. Esslinger, L. De Leo, C. Kollath, A. Georges, V. Scarola, L. Pollet, E. Burovski, E. Kozik, and M. Troyer
Phys. Rev. Lett. 104, 180401 (2010)
Weakly interacting Mott insulator Intermediate
U/6t
DMFT and high-temperature series expansions
Metal Mott insulator
Next challenge
Quantum magnetism
Building the Fermi-Hubbard model Metal-Mott insulator transition Short-range magnetic correlations
Outline
Magnetism: a temperature challenge
U>>t
ener
gy
T > U: metallic behavior
T < U: Mott insulator
T
T < J: spin ordering
T
R. Jördens et al., Phys. Rev. Lett. 104, 180401 (2010) P. Duarte et al., Phys. Rev. Lett. 114, 070403 (2015)
J=4t2/U
Superexchange J
Approaches to magnetism
S. Trotzky et al., Science 319, 295 (2008) S. Nascimbène et al., Phys. Rev. Lett. 108, 205301 (2012) S. Murmann et al., Phys. Rev. Lett. 114, 080402 (2015)
Isolated double-wells or plaquettes (Munich, Heidelberg)
Approaches to magnetism
J. Simon et al., Nature 472, 307 (2011)
Ising spin chains (Harvard)
J. Struck et al., Science 333, 996 (2011) J. Struck et al., Nature Phys. 9, 738 (2013)
Classical magnetism, Ising XY (Hamburg)
Mappings
Approaches to magnetism
Dipolar interactions (JILA, Paris)
B. Yan et al., Nature 501, 521-525 (2013) A. de Paz et al., Phys. Rev. Lett. 111, 185305 (2013)
D. Greif et al., Science 340, 1307 (2013) R. A. Hart et al., Nature 519, 211 (2015)
Approaches to magnetism
Short-range quantum magnetism in the Fermi-Hubbard model (ETH, Rice)
J < T < Jd,s
T J
Jd
Jd > J
Dimerized lattice
ener
gy
Enhancing magnetic correlations
Magnetic correlations T < J
Js > J
Anisotropic cubic lattice
Jd,s
Magnetic correlations in dimerized lattice
singlet
triplet
Jd
Spin correlations on neighboring sites
T < Jd : NS > NT
Local spin correlations in cubic lattice Nearest-neighbor spin correlations vs. temperature
antiferromagnetic transition
DCA simulation 3D Fermi-Hubbard model
S. Fuchs, E. Gull, L. Pollet, E. Burovski, E. Kozik, T. Pruschke, and M. Troyer, Phys. Rev. Lett. 106, 030401 (2011)
Merging lattice sites
Chequerboard
Dimer
Square
Detecting magnetic correlations
singlet
or
triplet t0
singlet triplet t0
Dimerized lattice
Singlet-Triplet Imbalance
Measuring singlets and triplets
𝑝𝑆
𝑝𝑡𝑡 Sin
glet
s Tr
iple
ts
Merging neighboring sites Singlet-triplet oscillations
Singlet-triplet oscillations: S. Trotzky et al., Phys. Rev. Lett. 105, 265303 (2010)
Theory: second order high-temperature series expansion of coupled dimers
Dependence on dimerization
s=1.7 kB
Jd
T J
isotropic strongly dimerized
D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and T. Esslinger, Science 340, 1307 (2013)
Dependence on entropy
Theory: second order high-temperature series expansion of coupled dimers
U/t = 11.0(8) td/t = 22(2) t/h = 67(3) Hz
Jd T
J
D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and T. Esslinger, Science 340, 1307 (2013)
Anisotropic simple cubic lattice
transverse spin correlator ⟺ population difference
AFM correlations along x
Effective 1D chains
Dependence on anisotropy
isotropic strongly anisotropic
VY,Z = 11.0(3) ER s = 1.8 kB
normalized spin correlator
D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and T. Esslinger, Science 340, 1307 (2013)
Dependence on entropy
tS /t=7.3
D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and T. Esslinger, Science 340, 1307 (2013)
Comparison with theory
Theory: DCA+LDA for anisotropic simple cubic lattice J. Imriška, M. Iazzi, L. Wang, E. Gull, D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, T. Esslinger, and M. Troyer, Phys. Rev. Lett. 112, 115301 (2014)
Correlations over 2 sites
T<t
Analogous results with DMRG: B. Sciolla et al., Phys. Rev. A 88, 063629 (2013)
Conclusion
Metal-Mott insulator transition
Nearest-neighbor magnetic correlations
Building the Fermi-Hubbard model
With a bit more cooling…
High-T phase diagram of cuprates
QCP
Dimers AFM
T/J
J/Jd
Geometry-induced quantum phase transitions
Frustration
Gregor Jotzu Daniel Greif L. T. Thomas Uehlinger Tilman Esslinger
Robert Jördens
Henning Moritz
Thomas Uehlinger
Niels Strohmaier
Daniel Greif
L. T. Tilman Esslinger
Zurich: V. Scarola, L. Pollet, E. Burovski, E. Kozik, J. Imriška, M. Iazzi, L. Wang, E. Gull, M. Troyer Paris: L. De Leo, C. Kollath, A. Georges Geneva/Bonn: B. Sciolla, A. Tokuno, S. Uchino, P. Bartmettler, T. Giamarchi, C. Kollath
THEORY
Ultracold Quantum Gases group @
L. T.
Pierrick Cheiney
César Cabrera
www.qge.icfo.es
Luca Tanzi
Jordi Sastre
Julio Sanz
Manel Bosch (now at Laboratoire Kastler Brossel, Paris) Vincent Lienhard (now student at ENS Cachan) Lisa Saemisch (now at ICFO’s Molecular Nanophotonics group)
