quantum phase transitions, strongly interacting systems, and cold...
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Quantum Phase Transitions,Strongly Interacting Systems,
and Cold AtomsEugene Demler
Physics Department, Harvard University
Collaborators:
Ehud Altman, Ignacio Cirac, Bert Halperin, Walter Hofstetter, Adilet Imambekov, Ludwig Mathey, Mikhail Lukin, Anatoli Polkovnikov, Anders Sorensen, Charles Wang, Fei Zhou, Peter Zoller
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Classical Phase transitions:Phase Diagram for Water
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Ising Model in Transverse Field1.6
20
4LiHoF
Bitko et al., PRL 77:940 (1996)
HxFerro
Para
H(kOe)
0.4
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Superconductor to Insulator Transition in Thin Films
Marcovic et al., PRL 81:5217 (1998)
Bi films
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High Temperature Superconductors
Maple, JMMM 177:18 (1998)
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Quantum Phase Transition
E
g
E
g
Level crossing at T=0
Avoided level crossing.Second order phase transition
True level crossing.First order phase transition
© Subir Sachdev
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Quantum Critical Region
H
T
Quantum-critical
Quantum critical point controls a wide quantum critical region
Quantum critical region does not have well defined quasiparticles
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Quantum Critical Point in YbRh Si
AF – antiferromagnetic
LFL – Landau Fermi liquid
NFL – non Fermi liquid
2 2
Gegenwart et al., PRL 89:56402(2002)
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Quantum States of Matter.Why are they interesting?
•Understanding fundamental propertiesof complex quantum systems
•Technological applications
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Applications of Quantum Materials: Ferroelectric RAM
Non-Volatile Memory
High Speed Processing
FeRAM in Smart Cards
V+ + + + + + + +
_ _ _ _ _ _ _ _
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Applications of Quantum Materials:High Tc Superconductors
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Bose-Einstein Condensation
Cornell et al., Science 269, 198 (1995)
Ultralow density condensed matter systemInteractions are weak and can be described theoretically from first principles
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New Era in Cold Atoms Research
• Optical lattices• Feshbach resonances• Rotating condensates• One dimensional systems• Systems with long range dipolar
interactions
Focus on systems with strong interactions
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Feshbach Resonance and Fermionic CondensatesGreiner et al., Nature 426:537 (2003)
Zwierlein et al., PRL 91:250401 (2003)See also Jochim et al., Science 302:2101 (2003)
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Atoms in Optical Lattices
Theory: Jaksch et al. PRL 81:3108(1998)
Experiment: Kasevich et al., Science (2001);Greiner et al., Nature (2001);Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004);
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Strongly Correlated SystemsAtoms in optical latticesElectrons in Solids
Simple metals.
Perturbation theory in Coulomb interaction applies. Band structure methods wotk
Strongly Correlated Electron Systems.Band structure methods fail.
Novel phenomena in strongly correlated electron systems:Quantum magnetism, phase separation, unconventional superconductivity,high temperature superconductivity, fractionalization of electrons …
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Cold Atoms with Strong Interactions
• Resolve long standing questions in condensed matter physics (e.g. the origin of high Tc superconductivity)
• Resolve matter of principle questions (e.g. spin liquids in two and three dimensions)
• Find new exciting physics
Goals
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Outline• Cold atoms in optical lattices. Hubbard model• Two component Bose mixture
Quantum magnetism. Competing orders. Fractionalized phases
• Spin one bosonsSpin exchange interactions. Exotic spin order (nematic)
• FermionsPairing in systems with repulsive interactions. Unconventional pairing. High Tc mechanism
• Boson-Fermion mixturesPolarons. Competing orders
• BEC on chipsInterplay of disorder and interactions. Bose glass phase
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Atoms in optical lattice. Bose Hubbard Model
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Bose Hubbard Model
tunneling of atoms between neighboring wells
repulsion of atoms sitting in the same well
U
t
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4
Bose Hubbard Model. Mean-field Phase Diagram.
1+n
Uμ
02
0
M.P.A. Fisher et al.,PRB40:546 (1989)
MottN=1
N=2
N=3
Superfluid
Superfluid phase
Mott insulator phase
Weak interactions
Strong interactions
Mott
Mott
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Set .
Bose Hubbard Model
Hamiltonian eigenstates are Fock states
Uμ
2 4
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Bose Hubbard Model. Mean-field Phase Diagram.
Particle-hole excitation
Mott insulator phase
41+n
Uμ
2
0
MottN=1
N=2
N=3
Superfluid
Mott
Mott
Tips of the Mott lobes
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Gutzwiller variational wavefunction
Normalization
Interaction energy
Kinetic energy
z – number of nearest neighbors
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Phase Diagram of the 1D Bose Hubbard Model. Quantum Monte-Carlo Study
Batrouni and Scaletter, PRB 46:9051 (1992)
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Optical Lattice and Parabolic Potential
41+n
Uμ
2
0
N=1
N=2
N=3
SF
MI
MI
Jaksch et al., PRL 81:3108 (1998)
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Superfluid PhaseOrder parameter
Phase (Bogoliubov) mode. Gapless Goldstone mode.
Breaks U(1) symmetry
Gapped amplitude mode.
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Mott Insulating Phase
Ground state
Particle excitation (gapped)
Hole excitation (gapped)
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2
Excitations of the Bose Hubbard Model
Mott Superfluid
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Superfluid to Insulator TransitionGreiner et al., Nature 415 (02)
Uμ
1−n
t/U
SuperfluidMott insulator
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Excitations of Bosons in the Optical LatticeSchori et al., PRL 93:240402 (2004)
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Time of Flight Experiments.
Quantum Noise Interferometry of Atoms in Optical Lattices.
Second order coherence
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Second Order Coherence in the Insulating State of Bosons.Hanburry-Brown-Twiss experiment
Theory: Altman et al., PRA 70:13603 (2004)
Experiment: Folling et al., Nature 434:481 (2005)
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Hanburry-Brown-Twiss Stellar Interferometer
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Hanburry-Brown-Twiss Interferometer
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Second Order Coherence in the Insulating State of Bosons
Bosons at quasimomentum expand as plane waves
with wavevectors
First order coherence:
Oscillations in density disappear after summing over
Second order coherence:
Correlation function acquires oscillations at reciprocal lattice vectors
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Second Order Coherence in the Insulating State of Bosons.Hanburry-Brown-Twiss experiment
Theory: Altman et al., PRA 70:13603 (2004)
Experiment: Folling et al., Nature 434:481 (2005)
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0 200 400 600 800 1000 1200-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Interference of an Array of Independent Condensates
Hadzibabic et al., PRL 93:180403 (2004)
Smooth structure is a result of finite experimental resolution (filtering)
0 200 400 600 800 1000 1200-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
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Extended Hubbard Model
- on site repulsion - nearest neighbor repulsion
Checkerboard Phase:
Crystal phase of bosons.Breaks translational symmetry
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Extended Hubbard Model. Mean Field Phase diagramvan Otterlo et al., PRB 52:16176 (1995)
Hard core bosons.
Supersolid – superfluid phase with broken translational symmetry
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Extended Hubbard Model. Quantum Monte Carlo study
Sengupta et al., PRL 94:207202 (2005)Hebert et al., PRB 65:14513 (2002)
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Dipolar Bosons in Optical Lattices
Goral et al., PRL88:170406 (2002)
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How to Detect a Checkerboard Phase?
Correlation Function Measurements
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Two Component Bose Mixture in Optical Lattice.
Quantum magnetism. Competing orders. Fractionalized phases
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t
t
Two Component Bose Mixture in Optical LatticeExample: . Mandel et al., Nature 425:937 (2003)
Two component Bose Hubbard Model
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Two Component Bose Mixture in Optical Lattice.Magnetic order in an insulating phase
Insulating phases with N=1 atom per site. Average densities
Easy plane ferromagnet
Easy axis antiferromagnet
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Quantum Magnetism of Bosons in Optical Lattices
Duan et al., PRL (2003)
• Ferromagnetic• Antiferromagnetic
Kuklov and Svistunov, PRL (2003)
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Exchange Interactions in Solidsantibonding
bonding
Kinetic energy dominates: antiferromagnetic state
Coulomb energy dominates: ferromagnetic state
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Two Component Bose Mixture in Optical Lattice.Mean Field Theory + Quantum Fluctuations
2 ndorder line
Hysteresis
1st order
Altman et al., NJP 5:113 (2003)
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Probing Spin Order of Bosons
Correlation Function Measurements
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Engineering exotic phases
• Optical lattice in 2 or 3 dimensions: polarizations & frequenciesof standing waves can be different for different directions
ZZYY
• Example: exactly solvable modelKitaev (2002), honeycomb lattice with
H = Jx< i, j>∈x∑ σ i
xσ jx + Jy
< i, j>∈y∑ σ i
yσ jy + Jz
< i, j>∈z∑ σ i
zσ jz
• Can be created with 3 sets of standing wave light beams !• Non-trivial topological order, “spin liquid” + non-abelian anyons
…those has not been seen in controlled experiments
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Spin F=1 Bosons in Optical Lattices
Spin exchange interactions. Exotic spin order (nematic)
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Spinor Condensates in Optical TrapsSpin symmetric interaction of F=1 atoms
Antiferromagnetic Interactions for
Ferromagnetic Interactions for
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Antiferromagnetic F=1 Condensates
Mean Field
Three species of atoms
Ho, PRL 81:742 (1998) Ohmi, Machida, JPSJ 67:1822 (1998)
Beyond Mean Field. Spin Singlet Ground State
Law et al., PRL 81:5257 (1998); Ho, Yip, PRL 84:4031 (2000)
Experiments: Review in Ketterle’s Les Houches notes
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Antiferromagnetic Spin F=1 Atoms in Optical Lattices
Hubbard Hamiltonian
Symmetry constraints
Demler, Zhou, PRL (2003)
Nematic Mott Insulator
Spin Singlet Mott Insulator
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Nematic Insulating Phase for N=1
Effective S=1 spin model Imambekov et al., PRA 68:63602 (2003)
When the ground state is nematic in d=2,3.
One dimensional systems are dimerized: Rizzi et al., cond-mat/0506098
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Nematic Insulating Phase for N=1. Two Site Problem
12
0 -2 4
1
Singlet state is favored when
One can not have singlets on neighboring bonds.Nematic state is a compromise. It correspondsto a superposition of andon each bond
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Coherent Spin Dynamics in Optical LatticesWidera et al., cond-mat/0505492
atoms in the F=2 state
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Fermionic Atoms in Optical Lattices
Pairing in systems with repulsive interactions. Unconventional pairing. High Tc mechanism
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Fermionic Atoms in a Three Dimensional Optical Lattice
Kohl et al., PRL 94:80403 (2005)
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Fermions with Attractive Interaction
U
tt
Hofstetter et al., PRL 89:220407 (2002)
Highest transition temperature for
Compare to the exponential suppresion of Tc w/o a lattice
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Reaching BCS Superfluidity in a Lattice
6Li
40K
Li in CO2 lattice
K in NdYAG lattice
Turning on the lattice reduces the effective atomic temperature
Superfluidity can be achived even with a modest scattering length
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Fermions with Repulsive Interactions
t
U
tPossible d-wave pairing of fermions
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Picture courtesy of UBC Superconductivity group
High Temperature Superconductors
Superconducting Tc 93 K
Hubbard model – minimal model for cuprate superconductors
P.W. Anderson, cond-mat/0201429
After many years of work we still do not understand the fermionic Hubbard model
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Positive U Hubbard ModelPossible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)
Antiferromagnetic insulator
D-wave superconductor
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Superfluidity of Fermions in Optical Lattices.Probing excitation spectrum: Bragg scattering
• Pair of non-collinear laser beams create atomic excitation with given frequency and momentum
• Number of excited atoms:
qx = qy = 0.1π , n = 0.6, U / t = −2.5
In superfluid phase:• energy gap• sharp collective mode • broad quasiparticle “continuum”
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Second Order Interference from the BCS Superfluid
)'()()',( rrrr nnn −≡Δ
n(r)
n(r’)
n(k)
k
0),( =Ψ−Δ BCSn rr
BCS
BEC
kF
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Momentum Correlations in Paired FermionsGreiner et al., PRL 94:110401 (2005)
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Fermion Pairing in an Optical Lattice
Second Order InterferenceIn the TOF images
Normal State
Superfluid State
measures the Cooper pair wavefunction
One can identify unconventional pairing
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Boson Fermion MixturesPolarons. Competing orders
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Boson Fermion Mixtures
BEC
Experiments: ENS, Florence, JILA, MIT, Rice, …
Bosons provide cooling for fermionsand mediate interactions. They createnon-local attraction between fermions
Charge Density Wave PhasePeriodic arrangement of atoms
Non-local Fermion PairingP-wave, D-wave, …
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BF Mixtures in 1d Optical Lattices
Cazalila et al., PRL (2003); Mathey et al., PRL (2004)
Spinless fermions Spin ½ fermions
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BF Mixtures in 2d Optical LatticesWang et al., cond-mat/0410492
40K -- 87Rb 40K -- 23Na
=1060 nm(a) =1060nm
(b) =765.5nm
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BEC on microchips.
Interplay of disorder and interactions. Bose glass phase
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Fragmented BEC in magnetic microtraps
Theory: Wang et.al., PRL 92:076802 (2004)
Thywissen et al., EPJD (1999); Kraft et al., JPB (2002);Leanhardt et al., PRL (2002); Fortagh et al., PRA (2002); …
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BEC on atom chips.Esteve et al., PRA 70:43629 (2004)
• Outlook: interplay of interactions and disorder: probing Bose glass phase
SEM image of wire
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Conclusions:
Systems of cold atoms and molecules can be usedto create several types of strongly correlated many-bodysystems. This opens interesting possibilities for
•Simulating fundamental models in CM physics (e.g. Hubbard model)•Understanding quantum magnetism•Studying systems with unconventional fermion pairing•Creating systems with topological order•Understanding the interplay of disorder and interactions•Engineering “quantum states”