1. Eugene Demler Harvard University
Simulations of strongly correlated electron systems using cold atoms
Eugene Demler Harvard University
Main collaborators:
Anatoli Polkovnikov Harvard/Boston University
Ehud Altman Harvard/Weizmann
Daw-Wei Wang Harvard/Tsing-Hua University
Vladimir Gritsev Harvard
Adilet Imambekov Harvard
Ryan Barnett Harvard/Caltech
Mikhail Lukin Harvard

2. Strongly correlated electron systems

3. “Conventional” solid state materials
Bloch theorem for non-interacting
electrons in a periodic potential

4. Consequences of the Bloch theoremVH EF d
Metals I
Insulators
and
Semiconductors EF
First semiconductor transistor

5. “Conventional” solid state materials
Electron-phonon and electron-electron interactions
are irrelevant at low temperatures ky kx
Landau Fermi liquid theory: when frequency and temperature are smaller than EF electron systems
are equivalent to systems of non-interacting fermions kF
Discuss Widemann-Franz law. It works for … Ag Ag Ag

6. Non Fermi liquid behavior in novel quantum materials
Violation of the
Wiedemann-Franz law
in high Tc superconductors
Hill et al., Nature 414:711 (2001)
UCu3.5Pd1.5
Andraka, Stewart,
PRB 47:3208 (93)
CeCu2Si2. Steglich et al.,
Z. Phys. B 103:235 (1997)

7. Puzzles of high temperature superconductors
Unusual “normal” state
Maple, JMMM 177:18 (1998)
Resistivity, opical conductivity,
Lack of sharply defined quasiparticles,
Nernst effect
Mechanism of Superconductivity
High transition temperature,
retardation effect, isotope effect,
role of elecron-electron
and electron-phonon interactions
Competing orders
Role of magnetsim, stripes,
possible fractionalization

8. Applications of quantum materials: High Tc superconductors

9. Applications of quantum materials: Ferroelectric RAM V
_ _ _ _ _ _ _ _
FeRAM in Smart Cards
Non-Volatile Memory
High Speed Processing

10. Modeling strongly correlated
systems using cold atoms

11. Bose-Einstein condensation
Cornell et al., Science 269, 198 (1995)
Ultralow density condensed matter system
Interactions are weak and can be described theoretically from first principles

12. New Era in Cold Atoms Research
Focus on Systems with Strong Interactions
Atoms in optical lattices
Feshbach resonances
Low dimensional systems
Systems with long range dipolar interactions
Rotating systems

13. Feshbach resonance and fermionic condensates
Greiner et al., Nature 426:537 (2003); Ketterle et al., PRL 91: (2003)
Ketterle et al.,
Nature 435, (2005)

14. One dimensional systems
1D confinement in optical potential
Weiss et al., Science (05);
Bloch et al.,
Esslinger et al.,
One dimensional systems in microtraps.
Thywissen et al., Eur. J. Phys. D. (99);
Hansel et al., Nature (01);
Folman et al., Adv. At. Mol. Opt. Phys. (02)
Strongly interacting
regime can be reached
for low densities

15. Atoms in optical lattices
Theory: Jaksch et al. PRL (1998)
Experiment: Kasevich et al., Science (2001);
Greiner et al., Nature (2001);
Phillips et al., J. Physics B (2002)
Esslinger et al., PRL (2004);
and many more …

16. Strongly correlated systems
Electrons in Solids
Atoms in optical lattices
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 …

17. New Era in Cold Atoms Research
Focus on Systems with Strong Interactions
Goals
Resolve long standing questions in condensed matter physics
(e.g. origin of high temperature superconductivity)
Resolve matter of principle questions
(e.g. existence of spin liquids in two and three dimensions)
Study new phenomena in strongly correlated systems
(e.g. coherent far from equilibrium dynamics)

18. Outline
Introduction. Cold atoms in optical lattices. Bose Hubbard model
Two component Bose mixtures
Quantum magnetism. Competing orders. Fractionalized phases
Fermions in optical lattices
Pairing in systems with repulsive interactions. High Tc mechanism
Boson-Fermion mixtures
Polarons. Competing orders
Interference experiments with fluctuating BEC
Analysis of correlations beyond mean-field
Moving condensates in optical lattices
Non equilibrium dynamics of interacting many-body systems
Emphasis: detection and characterzation of many-body states

19. Atoms in optical lattices. Bose Hubbard model

20. Bose Hubbard model U t tunneling of atoms between neighboring wells
repulsion of atoms sitting in the same well

21. Bose Hubbard model. Mean-field phase diagram
M.P.A. Fisher et al.,
PRB40:546 (1989)
N=3
Mott 4
Superfluid
N=2
Mott 2
N=1
Mott
Superfluid phase
Weak interactions
Mott insulator phase
Strong interactions

22. Bose Hubbard model
Set
Hamiltonian eigenstates are Fock states 2 4

23. Bose Hubbard Model. Mean-field phase diagram
Mott 4
Superfluid
N=2
Mott 2
N=1
Mott
Mott insulator phase
Particle-hole excitation
Tips of the Mott lobes

24. Gutzwiller variational wavefunction
Normalization
Interaction energy
Kinetic energy
z – number of nearest neighbors

25. Phase diagram of the 1D Bose Hubbard model. Quantum Monte-Carlo study
Batrouni and Scaletter, PRB 46:9051 (1992)

26. Optical lattice and parabolic potential4
N=2 MI SF 2
N=1 MI
Jaksch et al.,
PRL 81:3108 (1998)

27. Superfluid to Insulator transition
Greiner et al., Nature 415:39 (2002)
t/U
Superfluid
Mott insulator
What if the ground state cannot be described by single particle matter waves? Current experiments are reaching such regimes. A nice example is the experiment by Markus and collaborators in Munich where cold bosons on an optical lattice were tuned across the SF-Mott transition.
Explain experiment.
SF well described by a wavefunction in which the zero momentum state is macroscopically occupied.
Mott state is rather well described by definite RS occupations.

28. Time of flight experiments
Quantum noise interferometry of atoms in an optical lattice
Second order coherence

29. Second order coherence in the insulating state of bosons
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)

30. Hanburry-Brown-Twiss stellar interferometer

31. Hanburry-Brown-Twiss interferometer

32. 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

33. Second order coherence in the insulating state of bosons
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)

34. Effect of parabolic potential on the second order coherence
Experiment: Spielman, Porto, et al.,
Theory: Scarola, Das Sarma, Demler, cond-mat/
Width of the correlation peak changes across the
transition, reflecting the evolution of Mott domains

35. Width of the noise peaks

36. Interference of an array of independent condensates
Hadzibabic et al., PRL 93: (2004)
Smooth structure is a result of finite experimental resolution (filtering)

37. Extended Hubbard Model
- nearest neighbor repulsion
- on site repulsion
Checkerboard phase:
Crystal phase of bosons.
Breaks translational symmetry

38. Extended Hubbard model. Mean field phase diagram
van Otterlo et al., PRB 52:16176 (1995)
Hard core bosons.
Supersolid – superfluid phase with broken translational symmetry

39. Extended Hubbard model. Quantum Monte Carlo study
Hebert et al., PRB 65:14513 (2002)
Sengupta et al., PRL 94: (2005)

40. Dipolar bosons in optical lattices
Goral et al., PRL88: (2002)

41. How to detect a checkerboard phase
Correlation Function Measurements

42. Magnetism in condensed matter systems

43. Ferromagnetism Magnetic needle in a compass
Ferromagnetism is known since the time of the ancient greeks. Magnetic materials find numerous applications
Starting from magnetic needle in a compass and including hard drives which can carry terabytes of information.
Record holder for commercial products: Hitachi hard drives with density of 230 billion bits per sq. inch. This allows
Computer drives capable of storing a trillion bytes (terabyte)
Magnetic memory in hard drives.
Storage density of hundreds of
billions bits per square inch.

44. Stoner model of ferromagnetism
Spontaneous spin polarization
decreases interaction energy
but increases kinetic energy of
electrons
Our current knowledge does not go beyond the original simple argument by Stoner.
Hubbard model for fermions was originally proposed to explain ferromagnetism. After
Many years of numerical studies, it is now believed that the Hubbard model does
Not have ferromagnetism. And Hubbard model is really the simplest example.
When we take systems with more realistic long range interactions, we do not
Really know when such mean-field argument applies. For example, there
Is currently a lot of controversy and debates regarding possible ferromagnetism
in 2d electron gas at low density.
I N(0) = 1
Mean-field criterion
I – interaction strength
N(0) – density of states at the Fermi level

45. Antiferromagnetism
Maple, JMMM 177:18 (1998)
High temperature superconductivity in cuprates is always found
near an antiferromagnetic insulating state

46. ( - ) ( + ) ( + ) Antiferromagnetism t t
Antiferromagnetic Heisenberg model AF =
( ) S =
( ) t =
( ) AF = S t
Antiferromagnetic state breaks spin symmetry.
It does not have a well defined spin

47. Spin liquid states
Alternative to classical antiferromagnetic state: spin liquid states
Properties of spin liquid states:
fractionalized excitations
topological order
gauge theory description
Systems with geometric frustration ?

48. Spin liquid behavior in systems with geometric frustration
Kagome lattice
Pyrochlore lattice
SrCr9-xGa3+xO19
ZnCr2O4
A2Ti2O7
Ramirez et al. PRL (90)
Broholm et al. PRL (90)
Uemura et al. PRL (94)
Ramirez et al. PRL (02)

49. Engineering magnetic systems using cold atoms in an optical lattice

50. Spin interactions using controlled collisions
Experiment: Mandel et al., Nature 425:937(2003)
Theory: Jaksch et al., PRL 82:1975 (1999)

51. Effective spin interaction from the orbital motion.
Cold atoms in Kagome lattices
Santos et al., PRL 93:30601 (2004)
Damski et al., PRL 95:60403 (2005)

52. Two component Bose mixture in optical lattice
Example: Mandel et al., Nature 425:937 (2003) t t
Two component Bose Hubbard model

53. Quantum magnetism of bosons in optical lattices
Kuklov and Svistunov, PRL (2003)
Duan et al., PRL (2003)
Ferromagnetic
Antiferromagnetic

54. Exchange Interactions in Solids
antibonding
bonding
Kinetic energy dominates: antiferromagnetic state
Coulomb energy dominates: ferromagnetic state

55. Two component Bose mixture in optical lattice
Two component Bose mixture in optical lattice. Mean field theory + Quantum fluctuations
Altman et al., NJP 5:113 (2003)
Hysteresis
1st order
2nd order line

56. Coherent spin dynamics in optical lattices
Widera et al., cond-mat/
atoms in the F=2 state

57. How to observe antiferromagnetic order of cold atoms in an optical lattice?

58. Second order coherence in the insulating state of bosons
Second order coherence in the insulating state of bosons. Hanburry-Brown-Twiss experiment
Theory: Altman et al., PRA 70:13603 (2004)
See also Bach, Rzazewski, PRL 92: (2004)
Experiment: Folling et al., Nature 434:481 (2005)
See also Hadzibabic et al., PRL 93: (2004)

59. Probing spin order of bosons
Correlation Function Measurements

60. Engineering exotic phases
Optical lattice in 2 or 3 dimensions: polarizations & frequencies
of standing waves can be different for different directions YY ZZ
Example: exactly solvable model
Kitaev (2002), honeycomb lattice with
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

61. Fermionic atoms in optical lattices
Pairing in systems with repulsive interactions.
Unconventional pairing. High Tc mechanism

62. Fermionic atoms in a three dimensional optical lattice
Kohl et al., PRL 94:80403 (2005)

63. Fermions with attractive interaction
Hofstetter et al., PRL 89: (2002) U t t
Highest transition temperature for
Compare to the exponential suppresion of Tc w/o a lattice

64. Reaching BCS superfluidity in a lattice
Turning on the lattice reduces the effective atomic temperature
K in NdYAG lattice
40K
Li in CO2 lattice
6Li
Superfluidity can be achived even with a modest scattering length

65. Fermions with repulsive interactions
Possible d-wave pairing of fermions

66. High temperature superconductors
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/
After many years of work we still do not understand
the fermionic Hubbard model

67. Positive U Hubbard model
Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)
Antiferromagnetic insulator
D-wave superconductor

68. Second order correlations in the BCS superfluid
n(k)
n(r)
n(r’) k F k
BCS
BEC

69. Momentum correlations in paired fermions
Greiner et al., PRL 94: (2005)

70. Fermion pairing in an optical lattice
Second Order Interference
In the TOF images
Normal State
Superfluid State
measures the Cooper pair wavefunction
One can identify unconventional pairing

71. Boson Fermion mixtures
Fermions interacting with phonons.
Polarons. Competing orders

72. Boson Fermion mixtures
Experiments: ENS, Florence, JILA, MIT, Rice, …
BEC
Bosons provide cooling for fermions
and mediate interactions. They create
non-local attraction between fermions
Charge Density Wave Phase
Periodic arrangement of atoms
Non-local Fermion Pairing
P-wave, D-wave, …

73. Boson Fermion mixtures
“Phonons” :
Bogoliubov (phase) mode
Effective fermion-”phonon” interaction
Fermion-”phonon” vertex
Similar to electron-phonon systems

74. Boson Fermion mixtures in 1d optical lattices
Cazalila et al., PRL (2003); Mathey et al., PRL (2004)
Spinless fermions
Spin ½ fermions
Note: Luttinger parameters can be determined using correlation function
measurements in the time of flight experiments. Altman et al. (2005)

75. BF mixtures in 2d optical lattices
Wang, Lukin, Demler, PRA (1972)
40K -- 87Rb
40K -- 23Na
(a)
=1060nm
=1060 nm
(b)
=765.5nm

77. Systems of cold atoms with strong interactions and correlations
Goals
Resolve long standing questions in condensed matter physics
(e.g. origin of high temperature superconductivity)
Resolve matter of principle questions
(e.g. existence of spin liquids in two and three dimensions)
Study new phenomena in strongly correlated systems
Interference experiments with fluctuating BEC
Analysis of high order correlation functions in low dimensional
systems
Moving condensates in optical lattices
Non equilibrium dynamics of interacting many-body systems

78. Interference experiments with fluctuating BEC
Analysis of high order correlation
functions in low dimensional systems

79. Interference of two independent condensates
Andrews et al., Science 275:637 (1997)

80. Interference of two independent condensates1
r+d d 2
Clouds 1 and 2 do not have a well defined phase difference.
However each individual measurement shows an interference pattern

81. Interference of one dimensional condensates
Experiments: Schmiedmayer et al., Nature Physics (2005) d
Amplitude of interference fringes, ,
contains information about phase fluctuations
within individual condensates x1 x2 y x

82. Interference amplitude and correlations
Polkovnikov, Altman, Demler, PNAS (2006) L
For identical condensates
Instantaneous correlation function

83. Interference between Luttinger liquids
Luttinger liquid at T=0 L
K – Luttinger parameter
For non-interacting bosons and
For impenetrable bosons and
Luttinger liquid at finite temperature
Analysis of can be used for thermometry

84. Rotated probe beam experiment
For large imaging angle, ,
Luttinger parameter K may be
extracted from the angular
dependence of q

85. Interference between two-dimensional
BECs at finite temperature.
Kosteritz-Thouless transition

86. Interference of two dimensional condensates
Experiments: Stock, Hadzibabic, Dalibard, et al., cond-mat/
Gati, Oberthaler, et al., cond-mat/ Lx Ly Lx
Probe beam parallel to the plane of the condensates

87. Interference of two dimensional condensates
Interference of two dimensional condensates. Quasi long range order and the KT transition Ly Lx
Theory: Polkovnikov, Altman, Demler,
PNAS (2006)
Above KT transition
Below KT transition

88. Experiments with 2D Bose gas
Haddzibabic et al., Nature (2006) z
Time of
flight x
Typical interference patterns
low temperature
higher temperature

89. Experiments with 2D Bose gas
Hadzibabic et al., Nature (2006) x z
integration
over x axis
integration distance Dx
(pixels)
Contrast after
integration
0.4
0.2 10 20 30
middle T z
low T
high T
integration
over x axis z
integration
over x axis Dx z

90. Experiments with 2D Bose gas
Hadzibabic et al., Nature (2006)
0.4
0.2 10 20 30
low T
middle T
high T
fit by:
Integrated contrast
Exponent a
central contrast
0.5
0.1
0.2
0.3
0.4
high T
low T
“Sudden” jump!?
integration distance Dx
if g1(r) decays exponentially
with :
if g1(r) decays algebraically or
exponentially with a large :

91. Experiments with 2D Bose gas
Hadzibabic et al., Nature (2006)
Exponent a
T (K)
1.0
1.1
1.2
c.f. Bishop and Reppy
0.5
0.1
0.2
0.3
0.4
high T
low T
central contrast
Ultracold atoms experiments:
jump in the correlation function.
KT theory predicts a=1/4
just below the transition
He experiments:
universal jump in
the superfluid density

92. Experiments with 2D Bose gas
Experiments with 2D Bose gas. Proliferation of thermal vortices Haddzibabic et al., Nature (2006)
10%
20%
30%
central contrast
0.1
0.2
0.3
0.4
high T
low T
Fraction of images showing
at least one dislocation
Exponent a
0.5
0.1
0.2
0.3
0.4
central contrast
The onset of proliferation
coincides with a shifting to 0.5!
Z. Hadzibabic et al., Nature (2006)

93. Full distribution function of the interference amplitude
Interference between two interacting
one dimensional Bose liquids
Full distribution function
of the interference
amplitude
Gritsev, Altman, Demler, Polkovnikov, cond-mat/

94. Higher moments of interference amplitude
is a quantum operator. The measured value of
will fluctuate from shot to shot.
Can we predict the distribution function of ?
Higher moments
Changing to periodic boundary conditions (long condensates)
Explicit expressions for are available but cumbersome
Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995)

95. Impurity in a Luttinger liquid
Expansion of the partition function in powers of g
Partition function of the impurity contains correlation functions
taken at the same point and at different times. Moments
of interference experiments come from correlations functions
taken at the same time but in different points. Euclidean invariance
ensures that the two are the same

96. Relation between quantum impurity problem and interference of fluctuating condensates
Normalized amplitude
of interference fringes
Distribution function
of fringe amplitudes
Relation to the impurity partition function
Distribution function can be reconstructed from
using completeness relations for the Bessel functions

97. Bethe ansatz solution for a quantum impurity
can be obtained from the Bethe ansatz following
Zamolodchikov, Phys. Lett. B 253:391 (91); Fendley, et al., J. Stat. Phys. 79:799 (95)
Making analytic continuation is possible but cumbersome
Interference amplitude and spectral determinant
is related to the single particle Schroedinger equation
Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999)
Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001)
Spectral determinant

99. Evolution of the distribution function
Narrow distribution
for
Approaches Gumble
distribution. Width
Wide Poissonian
distribution for

100. From interference amplitudes to conformal field theories
correspond to vacuum eigenvalues of Q operators of CFT
Bazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999
When K>1, is related to Q operators of CFT with c<0. This includes 2D quantum gravity, non-intersecting loop model on 2D lattice, growth of random
fractal stochastic interface, high energy limit of multicolor QCD, …
Yang-Lee singularity
2D quantum gravity,
non-intersecting loops on 2D lattice

101. Moving condensates in optical lattices
Non equilibrium coherent dynamics
of interacting many-body systems

102. Atoms in optical lattices. Bose Hubbard model
Theory: Jaksch et al. PRL 81:3108(1998)
Experiment: Kasevich et al., Science (2001)
Greiner et al., Nature (2001)
Cataliotti et al., Science (2001)
Phillips et al., J. Physics B (2002)
Esslinger et al., PRL (2004), …

103. Equilibrium superfluid to insulator transition
Theory: Fisher et al. PRB (89), Jaksch et al. PRL (98)
Experiment: Greiner et al. Nature (01) U m
Superfluid
Mott insulator
What if the ground state cannot be described by single particle matter waves? Current experiments are reaching such regimes. A nice example is the experiment by Markus and collaborators in Munich where cold bosons on an optical lattice were tuned across the SF-Mott transition.
Explain experiment.
SF well described by a wavefunction in which the zero momentum state is macroscopically occupied.
Mott state is rather well described by definite RS occupations.
t/U

104. Moving condensate in an optical lattice. Dynamical instability
Theory: Niu et al. PRA (01), Smerzi et al. PRL (02)
Experiment: Fallani et al. PRL (04) v
Related experiments by
Eiermann et al, PRL (03)

105. ??? This discussion: How to connect
the dynamical instability (irreversible, classical)
to the superfluid to Mott transition (equilibrium, quantum)
Unstable
??? p
U/J
p/2
Stable SF MI
This discussion
U/t p SF MI
???
Possible experimental
sequence:

106. Superconductor to Insulator transition in thin films
Marcovic et al., PRL 81:5217 (1998)
Bi films d
Superconducting films
of different thickness

107. Dynamical instability
Classical limit of the Hubbard model.
Discreet Gross-Pitaevskii equation
Current carrying states
Linear stability analysis: States with p>p/2 are unstable
Amplification of
density fluctuations
unstable
unstable r

108. Dynamical instability for integer filling
Order parameter for a current carrying state
Current
GP regime
Maximum of the current for
When we include quantum fluctuations, the amplitude of the
order parameter is suppressed
decreases with increasing phase gradient

109. Dynamical instability for integer fillingSF MI p
U/J
p/2
Vicinity of the SF-I quantum phase transition.
Classical description applies for
The amplitude and the phase of the order parameter can only be defined if we coarse grain the system on the scale ksi.
Applying the same argument as before we find the condition that p ksi should be of the order of pi over two.
Near the SF-I transition ksi diverges. So the critical phase gradient goes to zero.
Dynamical instability occurs for

110. Dynamical instability. Gutzwiller approximation
Wavefunction
Time evolution
We look for stability against small fluctuations
Phase diagram. Integer filling

111. Optical lattice and parabolic trap. Gutzwiller approximation
The first instability develops near the edges, where N=1
U=0.01 t
J=1/4
Gutzwiller ansatz simulations (2D)

112. Beyond semiclassical equations. Current decay by tunneling
phase j
phase j
phase j
Current carrying states are metastable.
They can decay by thermal or quantum tunneling
Thermal activation
Quantum tunneling

113. Decay of current by quantum tunneling
phase j
phase
Quantum
phase slip j
Escape from metastable state by quantum tunneling.
WKB approximation
S – classical action corresponding to the motion in an inverted potential.

114. Decay rate from a metastable state. Example

115. For d>1 we have to include transverse directions.
Need to excite many chains to create a phase slip
Longitudinal stiffness
is much smaller than
the transverse.
The transverse size of the phase slip diverges near a phase slip. We can use continuum approximation to treat transverse directions

116. Weakly interacting systems. Gross-Pitaevskii regime.
Decay of current by quantum tunneling SF MI p
U/J
p/2
Fallani et al., PRL (04)
Quantum phase slips are
strongly suppressed
in the GP regime

117. Strongly interacting regime. Vicinity of the SF-Mott transitionMI p
U/J
p/2
Close to a SF-Mott transition
we can use an effective
relativistivc GL theory
(Altman, Auerbach, 2004)
Metastable current carrying state:
This state becomes unstable at corresponding to the maximum of the current:

118. Strongly interacting regime. Vicinity of the SF-Mott transition
Decay of current by quantum tunneling SF MI p
U/J
p/2
Action of a quantum phase slip in d=1,2,3
- correlation length
Strong broadening of the phase transition in d=1 and d=2
is discontinuous at the transition. Phase slips are not important.
Sharp phase transition

119. Decay of current by quantum tunneling

121. Decay of current by thermal activation
phase j
phase
Thermal
phase slip j DE
Escape from metastable state by thermal activation

122. Thermally activated current decay. Weakly interacting regime
phase slip DE
Activation energy in d=1,2,3
Thermal fluctuations lead to rapid decay of currents
Crossover from thermal
to quantum tunneling

123. Decay of current by thermal fluctuations
Phys. Rev. Lett. (2004)

124. Conclusions
We understand well: electron systems in semiconductors and simple metals.
Interaction energy is smaller than the kinetic energy. Perturbation theory works
We do not understand: strongly correlated electron systems in novel materials.
Interaction energy is comparable or larger than the kinetic energy.
Many surprising new phenomena occur, including high temperature
superconductivity, magnetism, fractionalization of excitations
Ultracold atoms have energy scales of 10-6K, compared to 104 K for
electron systems. However, by engineering and studying strongly interacting
systems of cold atoms we should get insights into the mysterious properties
of novel quantum materials
Our big goal is to develop a general framework for understanding strongly
correlated systems. This will be important far beyond AMO and condensed
matter