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Strongly interacting cold atoms
Subir SachdevTalks online at http://sachdev.physics.harvard.edu
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Outline
1. Quantum liquids near unitarity: from few-body to many-body physics(a) Tonks gas in one dimension(b) Paired fermions across a Feshbach resonance
2. Optical lattices(a) Superfluid-insulator transition(b) Quantum-critical hydrodynamics via mapping to
quantum theory of black holes.(c) Entanglement of valence bonds
Strongly interacting cold atoms
-
Outline
1. Quantum liquids near unitarity: from few-body to many-body physics(a) Tonks gas in one dimension(b) Paired fermions across a Feshbach resonance
2. Optical lattices(a) Superfluid-insulator transition(b) Quantum-critical hydrodynamics via mapping to
quantum theory of black holes.(c) Entanglement of valence bonds
Strongly interacting cold atoms
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Fermions with repulsive interactions
( ) †
+ short-range repulsive interactions of strength
k k kk
H c c
u
σ σε μ= −∑
0μ < 0μ >
Density
μ
-
Fermions with repulsive interactions
Characteristics of this ‘trivial’ quantum critical point:
• Zero density critical point allows an elegant connection between few body and many body physics.
• No “order parameter”. “Topological” characterization in the existence of the Fermi surface in one state.
• No transition at T > 0.
• Characteristic crossovers at T > 0, between quantum criticality, and low T regimes.
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Fermions with repulsive interactions
Characteristics of this ‘trivial’ quantum critical point:
T
μ
Quantum critical:Particle spacing ~ de Broglie wavelength
Classical Boltzmann gas
Fermi liquid
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Fermions with repulsive interactions
Characteristics of this ‘trivial’ quantum critical point:
2
RG flow characterizing quantum criti
cal
poin
(2 )
:
2
tdu ud udl
= − −d < 2
d > 2
• d > 2 – interactions are irrelevant. Critical theory is the spinful free Fermi gas.
• d < 2 – universal fixed point interactions. In d=1critical theory is the spinless free Fermi gas (Tonks gas).
u
uTonks gas
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Bosons with repulsive interactions2
(2 )2
du ud udl
= − −d < 2
d > 2
• Critical theory in d =1 is also the spinless free Fermi gas (Tonks gas).
• The dilute Bose gas in d >2 is controlled by the zero-coupling fixed point. Interactions are “dangerously irrelevant” and the density above onset depends upon bare interaction strength
(Yang-Lee theory).
Density
u
u
μ
Tonks gas
-
Fermions with attractive interactions2
(2 )2
du ud udl
= − −d > 2
• Universal fixed-point is accessed by fine-tuning to a Feshbachresonance.
• Density onset transition is described by free fermions for weak-coupling, and by (nearly) free bosons for strong coupling. The quantum-critical point between these behaviors is the Feshbachresonance.
Weak-coupling BCS theory
BEC of paired bound state
P. Nikolic and S. Sachdev, Phys. Rev. A 75, 033608 (2007).
-uFeshbachresonance
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Fermions with attractive interactions
ν
μ
Superfluid
Vacuum
BCS
BECdetuning
P. Nikolic and S. Sachdev, Phys. Rev. A 75, 033608 (2007).
1Detuning scattering length
ν = −
-
Fermions with attractive interactions
ν
μ
Superfluid
Vacuum
BCS
BECdetuning
Free fermions
P. Nikolic and S. Sachdev, Phys. Rev. A 75, 033608 (2007).
1Detuning scattering length
ν = −
-
Fermions with attractive interactions
detuning
Free fermions
P. Nikolic and S. Sachdev, Phys. Rev. A 75, 033608 (2007).
ν
μ
Superfluid
Vacuum
BCS
BEC
“Free” bosons
1Detuning scattering length
ν = −
-
Fermions with attractive interactions
detuning
P. Nikolic and S. Sachdev, Phys. Rev. A 75, 033608 (2007).
ν
μ
Superfluid
Vacuum
BCS
BEC
Universal theory of gapless bosons and fermions, with decay of boson into 2 fermions relevant for d < 4
1Detuning scattering length
ν = −
-
Fermions with attractive interactions
detuning
P. Nikolic and S. Sachdev, Phys. Rev. A 75, 033608 (2007).
ν
μ
Superfluid
Vacuum
BCS
BEC
Quantum critical point at μ=0, ν=0, forms the basis of the theory of the BEC-BCS crossover, including the transitions to FFLO and normal states with unbalanced densities
1Detuning scattering length
ν = −
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D. E. Sheehy and L. Radzihovsky, Phys. Rev. Lett. 95, 130401 (2005)
Fermions with attractive interactions
Universal phase diagram
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Fermions with attractive interactions
Universal phase diagram
P. Nikolic and S. Sachdev, Phys. Rev. A 75, 033608 (2007).
h – Zeeman field
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D. E. Sheehy and L. Radzihovsky, Phys. Rev. Lett. 95, 130401 (2005)
Fermions with attractive interactions
Universal phase diagram
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D. E. Sheehy and L. Radzihovsky, Phys. Rev. Lett. 95, 130401 (2005)
Fermions with attractive interactions
Universal phase diagram
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( )
( )
3/ 23/ 2
3/ 2 5/ 25/ 2
ln 0.39478
ln 0.02462 16
F
F
εε ε εε
μ ε ε ε εε
Δ= + −
= + −
0.1630.6864
0.3120.5906
F
F
N
N
εμε
Δ= −
= −
Expansion in ε=4-dY. Nishida and D.T. Son, Phys. Rev. Lett. 97, 050403 (2006)
0.54
0.44
F
F
εμε
Δ=
=
Quantum Monte CarloJ. Carlon, S.-Y. Chang, V.R. Pandharipande, and K.E. Schmidt, Phys. Rev. Lett. 91, 050401 (2003).
Fermions with attractive interactionsGround state properties at unitarity and balanced density
Expansion in 1/N with Sp(2N) symmetryM. Y. Veillette, D. E. Sheehy, and L. RadzihovskyPhys. Rev. A 75, 043614 (2007)
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Expansion in 1/Nwith Sp(2N) symmetryM. Y. Veillette, D. E. Sheehy, and L. RadzihovskyPhys. Rev. A 75, 043614 (2007)
Fermions with attractive interactionsGround state properties near unitarity and balanced density
Quantum Monte CarloJ. Carlon, S.-Y. Chang, V.R. Pandharipande, and K.E. Schmidt, Phys. Rev. Lett. 91, 050401 (2003).
-
( )3/ 2 5/ 2
5.3172.104
2.7851.504
/ 0.4050.1322
F
c
c
c
T N
T NP N
Nm T
ε
μ
= +
= +
= +
E. Burovski, N. Prokof’ev, B. Svistunov, and M. Troyer, New J. Phys. 8, 153 (2006)
( )3/ 2 5/ 2
6.579
3.247
/ 0.7762
F
c
c
c
T
TP N
m T
ε
μ
=
=
=
Fermions with attractive interactionsFinite temperature properties at unitarity and balanced density
Expansion in 1/N with Sp(2N) symmetryM. Y. Veillette, D. E. Sheehy, and L. RadzihovskyPhys. Rev. A 75, 043614 (2007)P. Nikolic and S. Sachdev, Phys. Rev. A 75, 033608 (2007).
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V. Gurarie, L. Radzihovsky, and A.V. Andreev, Phys. Rev. Lett. 94, 230403 (2005) C.-H. Cheng and S.-K. Yip, Phys. Rev. Lett. 95, 070404 (2005)
Fermions with attractive interactions in p-wave channel
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Outline
1. Quantum liquids near unitarity: from few-body to many-body physics(a) Tonks gas in one dimension(b) Paired fermions across a Feshbach resonance
2. Optical lattices(a) Superfluid-insulator transition(b) Quantum-critical hydrodynamics via mapping to
quantum theory of black holes.(c) Entanglement of valence bonds
Strongly interacting cold atoms
-
Outline
1. Quantum liquids near unitarity: from few-body to many-body physics(a) Tonks gas in one dimension(b) Paired fermions across a Feshbach resonance
2. Optical lattices(a) Superfluid-insulator transition(b) Quantum-critical hydrodynamics via mapping to
quantum theory of black holes.(c) Entanglement of valence bonds
Strongly interacting cold atoms
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M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
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M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Velocity distribution of 87Rb atoms
Superfliud
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M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Velocity distribution of 87Rb atoms
Insulator
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)()()()()( 212121 GkkknknknknG
−−∝− ∑δ
• 1st order coherence disappears in the Mott-insulating state.
)(kn
• Noise correlation function oscillates at reciprocal lattice vectors; bunching effect of bosons.
Noise correlation (time of flight) in Mott-insulators
Folling et al., Nature 434, 481 (2005); Altman et al., PRA 70, 13603 (2004).
1k
2k
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Two dimensional superfluid-Mott insulator transition
I. B. Spielman et al., cond-mat/0606216.
12/ =REV 20/ =REV 21/ =REV
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Fermionic atoms in optical lattices• Observation of Fermi surface.
Low density: metal high density: band insulator
27
29
29
2940 ,: =FmK
Esslinger et al., PRL 94:80403 (2005)
Fermions with near-unitary interactions in the presence of a periodic potential
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Fermions with near-unitary interactions in the presence of a periodic potential
( )
In the presence of a potential
2 2 2 = cos cos cos
there is a universal phase diagram determined by the ratio of 3 energy scales: , the che
L L L
x y zV r Va a a
V
π π π⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦
2
2
mical potential , and the
recoil energy = 4r L
Ema
μ
E.G. Moon, P. Nikolic, and S. Sachdev, to appear
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Universal phase diagram of fermions with near-unitary interactions in the presence of a periodic potential
E.G. Moon, P. Nikolic, and S. Sachdev, to appear
Expansion in 1/Nwith Sp(2N) symmetry
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E.G. Moon, P. Nikolic, and S. Sachdev, to appear
Boundaries to insulating phases for different values of νaLwhere ν is the detuning from the resonance
Universal phase diagram of fermions with near-unitary interactions in the presence of a periodic potential
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E.G. Moon, P. Nikolic, and S. Sachdev, to appear
Boundaries to insulating phases for different values of νaLwhere ν is the detuning from the resonance
Insulators have multiple band-
occupancy, and are intermediate between
band insulators of fermions and Mott
insulators of bosonicfermion pairs
Universal phase diagram of fermions with near-unitary interactions in the presence of a periodic potential
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Artificial graphene in optical lattices
{
}].)ˆ()([].)ˆ()([
.].)ˆ()([
333
212
111//
cherprp
cherprp
cherprptHAr
t
+++
+++
++=
+
+
∈
+∑
• Band Hamiltonian (σ-bonding) for spin-polarized fermions.
1p2p
3p
yx ppp 21
23
1 +=
yx ppp 21
23
2 −=
ypp −=3
1̂e2ê
3êA
B B
B
Congjun Wu et al
-
Flat bands in the entire Brillouin zone
• Flat band + Dirac cone. • localized eigenstates.
Many correlated phases possible
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Outline
1. Quantum liquids near unitarity: from few-body to many-body physics(a) Tonks gas in one dimension(b) Paired fermions across a Feshbach resonance
2. Optical lattices(a) Superfluid-insulator transition(b) Quantum-critical hydrodynamics via mapping to
quantum theory of black holes.(c) Entanglement of valence bonds
Strongly interacting cold atoms
-
Outline
1. Quantum liquids near unitarity: from few-body to many-body physics(a) Tonks gas in one dimension(b) Paired fermions across a Feshbach resonance
2. Optical lattices(a) Superfluid-insulator transition(b) Quantum-critical hydrodynamics via mapping to
quantum theory of black holes.(c) Entanglement of valence bonds
Strongly interacting cold atoms
-
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
-
Superfluid Insulator
Non-zero temperature phase diagram
Depth of periodic potential
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Superfluid Insulator
Non-zero temperature phase diagram
Depth of periodic potential
Dynamics of the classical Gross-Pitaevski equation
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Superfluid Insulator
Non-zero temperature phase diagram
Depth of periodic potential
Dilute Boltzmann gas of particle and holes
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Superfluid Insulator
Non-zero temperature phase diagram
Depth of periodic potential
No wave or quasiparticledescription
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D. B. Haviland, Y. Liu, and A. M. Goldman, Phys. Rev. Lett. 62, 2180 (1989)
Resistivity of Bi films
( )( )
( )
Superconductor
Insulator
2
Quantum critical point
0
Conductivity
0 0
40
T
T
eTh
σ
σ
σσ → = ∞
→ =
→ ≈
M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990)
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Superfluid Insulator
Non-zero temperature phase diagram
Depth of periodic potential
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Superfluid Insulator
Non-zero temperature phase diagram
Depth of periodic potential
Collisionless-to hydrodynamic crossover of a conformal field theory (CFT)
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
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Superfluid Insulator
Non-zero temperature phase diagram
Depth of periodic potential
Collisionless-to hydrodynamic crossover of a conformal field theory (CFT)
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
Needed: Cold atom experiments in this regime
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Hydrodynamics of a conformal field theory (CFT)
Maldacena’s AdS/CFT correspondence relates the hydrodynamics of CFTs to the quantum gravity theory of the horizon of a black hole in Anti-de Sitter space.
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Maldacena’s AdS/CFT correspondence relates the hydrodynamics of CFTs to the quantum gravity theory of the horizon of a black hole in Anti-de Sitter space.
Holographic representation of black hole physics in a 2+1 dimensional CFT at a temperature equal to the Hawking temperature of the black hole.
Black hole
3+1 dimensional AdS space
Hydrodynamics of a conformal field theory (CFT)
-
Hydrodynamics of a conformal field theory (CFT)
Hydrodynamics of a CFT
Waves of gauge fields in a curved
background
-
Hydrodynamics of a conformal field theory (CFT)
The scattering cross-section of the thermal excitations is universal and so transport co-efficients are universally determined by kBT
2
2
=
4
cB
cDk T
eh
σ
Θ
=Θ
Charge diffusion constant
Conductivity
K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).
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Hydrodynamics of a conformal field theory (CFT)
P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, hep-th/0701036
2
3/2
3= 4
3 2
sB
cDk T
N
π
σπ
=
Spin diffusion constant
Spin conductivity
For the (unique) CFT with a SU(N) gauge field and 16 supercharges, we know the exact diffusion
constant associated with a global SO(8) symmetry:
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Outline
1. Quantum liquids near unitarity: from few-body to many-body physics(a) Tonks gas in one dimension(b) Paired fermions across a Feshbach resonance
2. Optical lattices(a) Superfluid-insulator transition(b) Quantum-critical hydrodynamics via mapping to
quantum theory of black holes.(c) Entanglement of valence bonds
Strongly interacting cold atoms
-
Outline
1. Quantum liquids near unitarity: from few-body to many-body physics(a) Tonks gas in one dimension(b) Paired fermions across a Feshbach resonance
2. Optical lattices(a) Superfluid-insulator transition(b) Quantum-critical hydrodynamics via mapping to
quantum theory of black holes.(c) Entanglement of valence bonds
Strongly interacting cold atoms
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H.P. Büchler, M. Hermele, S.D. Huber, M.P.A. Fisher, and P. Zoller, Phys. Rev. Lett. 95, 040402 (2005)
Ring-exchange interactions in an optical lattice using a Raman transition
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Antiferromagnetic (Neel) order in the insulator
; spin operator with =1/2i j iij
H J S S S S= ⇒∑ i
-
Induce formation of valence bonds by e.g. ring-exchange interactions
+ 4-spin exchangei jij
H J S S K= ∑ ∑iA. W. Sandvik, cond-mat/0611343
-
( )1 -2
↑↓ ↓↑
=
-
( )1 -2
↑↓ ↓↑
=
-
( )1 -2
↑↓ ↓↑
=
-
( )1 -2
↑↓ ↓↑
=
-
( )1 -2
↑↓ ↓↑
=
-
( )1 -2
↑↓ ↓↑
=
Entangled liquid of valence bonds (Resonating valence bonds – RVB)
P. Fazekas and P.W. Anderson, Phil Mag 30, 23 (1974).
-
( )1 -2
↑↓ ↓↑
=
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).
Valence bond solid (VBS)
-
( )1 -2
↑↓ ↓↑
=
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001).
Valence bond solid (VBS)
-
( )1 -2
↑↓ ↓↑
=
Excitations of the RVB liquid
-
( )1 -2
↑↓ ↓↑
=
Excitations of the RVB liquid
-
( )1 -2
↑↓ ↓↑
=
Excitations of the RVB liquid
-
( )1 -2
↑↓ ↓↑
=
Excitations of the RVB liquid
-
( )1 -2
↑↓ ↓↑
=
Excitations of the RVB liquid
Electron fractionalization: Excitations carry spin S=1/2 but no charge
-
( )1 -2
↑↓ ↓↑
=
Excitations of the VBS
-
( )1 -2
↑↓ ↓↑
=
Excitations of the VBS
-
( )1 -2
↑↓ ↓↑
=
Excitations of the VBS
-
( )1 -2
↑↓ ↓↑
=
Excitations of the VBS
-
( )1 -2
↑↓ ↓↑
=
Excitations of the VBS
Free spins are unable to move apart: no fractionalization, but confinement
-
Phase diagram of square lattice antiferromagnet
+ 4-spin exchangei jij
H J S S K= ∑ ∑iA. W. Sandvik, cond-mat/0611343
-
+ 4-spin exchangei jij
H J S S K= ∑ ∑i
Phase diagram of square lattice antiferromagnet
VBS orderVBS order
K/J
Neel orderNeel order
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
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+ 4-spin exchangei jij
H J S S K= ∑ ∑i
Phase diagram of square lattice antiferromagnet
VBS orderVBS order
K/J
Neel orderNeel order
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
RVB physics appears at the quantum critical point which has fractionalized excitations: “deconfined criticality”
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Temperature, T
0
Quantum criticality of fractionalized
excitations
K/J
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Phases of nuclear matter
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• Rapid progress in the understanding of quantum liquids near unitarity
• Rich possibilities of exotic quantum phases in optical lattices
• Cold atom studies of the entanglement of large numbers of qubits: insights may be important for quantum cryptography and quantum computing.
• Tabletop “laboratories for the entire universe”: quantum mechanics of black holes, quark-gluon plasma, neutrons stars, and big-bang physics.
Conclusions
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Noise correlation (time of flight) in Mott-insulatorsTwo dimensional superfluid-Mott insulator transitionFermionic atoms in optical lattices Artificial graphene in optical latticesFlat bands in the entire Brillouin zone