Transcript Document

Simulations of strongly correlated
electron systems using cold atoms
Eugene Demler
Harvard University
Main collaborators:
Anatoli Polkovnikov
Ehud Altman
Daw-Wei Wang
Vladimir Gritsev
Adilet Imambekov
Ryan Barnett
Mikhail Lukin
Harvard/Boston University
Harvard/Weizmann
Harvard/Tsing-Hua University
Harvard
Harvard
Harvard/Caltech
Harvard
Strongly correlated electron systems
“Conventional” solid state materials
Bloch theorem for non-interacting
electrons in a periodic potential
Consequences of the Bloch theorem
B
VH
d
Metals
I
EF
EF
Insulators
and
Semiconductors
First semiconductor transistor
“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
Ag
Ag
Ag
Non Fermi liquid behavior in novel quantum materials
UCu3.5Pd1.5
Andraka, Stewart,
PRB 47:3208 (93)
CeCu2Si2. Steglich et al.,
Z. Phys. B 103:235 (1997)
Violation of the
Wiedemann-Franz law
in high Tc superconductors
Hill et al., Nature 414:711 (2001)
Puzzles of high temperature superconductors
Unusual “normal” state
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
Maple, JMMM 177:18 (1998)
Applications of quantum materials:
High Tc superconductors
Applications of quantum materials:
Ferroelectric RAM
+ + + + + + + +
V
_ _ _ _ _ _ _ _
FeRAM in Smart Cards
Non-Volatile Memory
High Speed Processing
Modeling strongly correlated
systems using cold atoms
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
New Era in Cold Atoms Research
Focus on Systems with Strong Interactions
• Feshbach resonances
• Rotating systems
• Low dimensional systems
• Atoms in optical lattices
• Systems with long range dipolar interactions
Feshbach resonance and fermionic condensates
Greiner et al., Nature 426:537 (2003); Ketterle et al., PRL 91:250401 (2003)
Ketterle et al.,
Nature 435, 1047-1051 (2005)
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
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 …
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 …
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)
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
Atoms in optical lattices.
Bose Hubbard model
Bose Hubbard model
U
t
tunneling of atoms between neighboring wells
repulsion of atoms sitting in the same well
Bose Hubbard model. Mean-field phase diagram
 U
M.P.A. Fisher et al.,
PRB40:546 (1989)
N=3 Mott
n 1
4
0 N=2
2
N=1
Mott
Superfluid
Mott
0
Superfluid phase
Weak interactions
Mott insulator phase
Strong interactions
Bose Hubbard model
Set
.
Hamiltonian eigenstates are Fock states
2
4
 U
Bose Hubbard Model. Mean-field phase diagram
 U
N=3 Mott
n 1
4
N=2
Mott
Superfluid
2
N=1
Mott
0
Mott insulator phase
Particle-hole excitation
Tips of the Mott lobes
Gutzwiller variational wavefunction
Normalization
Interaction energy
Kinetic energy
z – number of nearest neighbors
Phase diagram of the 1D Bose Hubbard model.
Quantum Monte-Carlo study
Batrouni and Scaletter, PRB 46:9051 (1992)
Optical lattice and parabolic potential
 U
N=3
n 1
4
N=2 MI
2
N=1
MI
0
Jaksch et al.,
PRL 81:3108 (1998)
SF
Superfluid to Insulator transition
Greiner et al., Nature 415:39 (2002)

U
Mott insulator
Superfluid
n 1
t/U
Time of flight experiments
Quantum noise interferometry of atoms in an optical lattice
Second order coherence
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)
Hanburry-Brown-Twiss stellar interferometer
Hanburry-Brown-Twiss interferometer
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
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)
Effect of parabolic potential on the second order coherence
Experiment: Spielman, Porto, et al.,
Theory: Scarola, Das Sarma, Demler, cond-mat/0602319
Width of the correlation peak changes across the
transition, reflecting the evolution of Mott domains
Width of the noise peaks
Interference of an array of independent condensates
Hadzibabic et al., PRL 93:180403 (2004)
Smooth structure is a result of finite experimental resolution (filtering)
3
1.4
2.5
1.2
2
1
1.5
0.8
1
0.6
0.5
0.4
0
-0.5
0.2
-1
0
-1.5
0
200
400
600
800
1000
1200
-0.2
0
200
400
600
800
1000
1200
Extended Hubbard Model
- on site repulsion
- nearest neighbor repulsion
Checkerboard phase:
Crystal phase of bosons.
Breaks translational symmetry
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
Extended Hubbard model.
Quantum Monte Carlo study
Hebert et al., PRB 65:14513 (2002)
Sengupta et al., PRL 94:207202 (2005)
Dipolar bosons in optical lattices
Goral et al., PRL88:170406 (2002)
How to detect a checkerboard phase
Correlation Function Measurements
Magnetism in condensed matter systems
Ferromagnetism
Magnetic needle in a compass
Magnetic memory in hard drives.
Storage density of hundreds of
billions bits per square inch.
Stoner model of ferromagnetism
Spontaneous spin polarization
decreases interaction energy
but increases kinetic energy of
electrons
Mean-field criterion
I N(0) = 1
I – interaction strength
N(0) – density of states at the Fermi level
Antiferromagnetism
Maple, JMMM 177:18 (1998)
High temperature superconductivity in cuprates is always found
near an antiferromagnetic insulating state
Antiferromagnetism
Antiferromagnetic Heisenberg model
AF
=
S
=
t
=
AF
=
(
(
(
-
)
)
+
S
+
t
)
Antiferromagnetic state breaks spin symmetry.
It does not have a well defined spin
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
?
Spin liquid behavior in systems
with geometric frustration
Kagome lattice
SrCr9-xGa3+xO19
Ramirez et al. PRL (90)
Broholm et al. PRL (90)
Uemura et al. PRL (94)
Pyrochlore lattice
ZnCr2O4
A2Ti2O7
Ramirez et al. PRL (02)
Engineering magnetic systems
using cold atoms in an optical lattice
Spin interactions using controlled collisions
Experiment: Mandel et al., Nature 425:937(2003)
Theory: Jaksch et al., PRL 82:1975 (1999)
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)
Two component Bose mixture in optical lattice
Example:
. Mandel et al., Nature 425:937 (2003)
t
t
Two component Bose Hubbard model
Quantum magnetism of bosons in optical lattices
Kuklov and Svistunov, PRL (2003)
Duan et al., PRL (2003)
• Ferromagnetic
• Antiferromagnetic
Exchange Interactions in Solids
antibonding
bonding
Kinetic energy dominates: antiferromagnetic state
Coulomb energy dominates: ferromagnetic state
Two component Bose mixture in optical lattice.
Mean field theory + Quantum fluctuations
Altman et al., NJP 5:113 (2003)
Hysteresis
1st order
Coherent spin dynamics in optical lattices
Widera et al., cond-mat/0505492
atoms in the F=2 state
How to observe antiferromagnetic order of
cold atoms in an optical lattice?
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 92:200401 (2004)
Experiment: Folling et al., Nature 434:481 (2005)
See also Hadzibabic et al., PRL 93:180403 (2004)
Probing spin order of bosons
Correlation Function Measurements
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
H  Jx

x
i
i, jx
x
j
 Jy

y
i
i, jy
y
j
 Jz

z
i
z
j
i, jz
• 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
Fermionic atoms in optical lattices
Pairing in systems with repulsive interactions.
Unconventional pairing. High Tc mechanism
Fermionic atoms in a three dimensional optical lattice
Kohl et al., PRL 94:80403 (2005)
Fermions with attractive interaction
Hofstetter et al., PRL 89:220407 (2002)
U
t
t
Highest transition temperature for
Compare to the exponential suppresion of Tc w/o a lattice
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
Fermions with repulsive interactions
U
t
t
Possible d-wave pairing of fermions
High temperature superconductors
Picture courtesy of UBC
Superconductivity group
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
Positive U Hubbard model
Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)
Antiferromagnetic insulator
D-wave superconductor
Second order correlations in the BCS superfluid
n(k)
n(r’)
kF
k
n(r)
BCS
BEC
n(r, r' )  n(r)  n(r' )
n(r,r) BCS  0
Momentum correlations in paired fermions
Greiner et al., PRL 94:110401 (2005)
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
Boson Fermion mixtures
Fermions interacting with phonons.
Polarons. Competing orders
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, …
Boson Fermion mixtures
“Phonons” :
Bogoliubov (phase) mode
Effective fermion-”phonon” interaction
Fermion-”phonon” vertex
Similar to electron-phonon systems
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)
BF mixtures in 2d optical lattices
Wang, Lukin, Demler, PRA (1972)
40K -- 87Rb
(a)
(b)
40K -- 23Na
=1060nm
=765.5nm
=1060 nm
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
Interference experiments with
fluctuating BEC
Analysis of high order correlation
functions in low dimensional systems
Interference of two independent condensates
Andrews et al., Science 275:637 (1997)
Interference of two independent condensates
r’
r
1
r+d
d
2
Clouds 1 and 2 do not have a well defined phase difference.
However each individual measurement shows an interference pattern
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
x
y
Interference amplitude and correlations
Polkovnikov, Altman, Demler, PNAS (2006)
L
For identical condensates
Instantaneous correlation function
Interference between Luttinger liquids
Luttinger liquid at T=0
K – Luttinger parameter
L
For non-interacting bosons
For impenetrable bosons
and
and
Luttinger liquid at finite temperature
Analysis of
can be used for thermometry
Rotated probe beam experiment
For large imaging angle,
q
,
Luttinger parameter K may be
extracted from the angular
dependence of
Interference between two-dimensional
BECs at finite temperature.
Kosteritz-Thouless transition
Interference of two dimensional condensates
Experiments: Stock, Hadzibabic, Dalibard, et al., cond-mat/0506559
Gati, Oberthaler, et al., cond-mat/0601392
Ly
Lx
Lx
Probe beam parallel to the plane of the condensates
Interference of two dimensional condensates.
Quasi long range order and the KT transition
Ly
Lx
Above KT transition
Theory: Polkovnikov, Altman, Demler,
PNAS (2006)
Below KT transition
Experiments with 2D Bose gas
z
Haddzibabic et al., Nature (2006)
Time of
flight
x
Typical interference patterns
low temperature
higher temperature
Experiments with 2D Bose gas
Hadzibabic et al., Nature (2006)
x
integration
over x axis z
z
Contrast after
integration
0.4
low T
integration
middle T
0.2
over x axis
z
high T
integration
over x axis
Dx
0
z
0
10
20
30
integration distance Dx
(pixels)
Experiments with 2D Bose gas
Integrated contrast
Hadzibabic et al., Nature (2006)
0.4
fit by:
C2 ~
low T
1
Dx
 1 

 Dx 
Dx
2


g
(
0
,
x
)
dx ~ 
1

middle T
0.2
Exponent a
high T
0
0
10
20
30
integration distance Dx
if g1(r) decays exponentially
with
:
0.5
0.4
0.3
high T
0
if g1(r) decays algebraically or
exponentially with a large
:
0.1
low T
0.2
0.3
central contrast
“Sudden” jump!?
2a
Experiments with 2D Bose gas
Hadzibabic et al., Nature (2006)
Exponent a
c.f. Bishop and Reppy
0.4
1.0
0
0.5
1.0
1.1
T (K)
1.2
0.3
high T
0
0.1
low T
0.2
0.3
central contrast
He experiments:
universal jump in
the superfluid density
Ultracold atoms experiments:
jump in the correlation function.
KT theory predicts a=1/4
just below the transition
Experiments with 2D Bose gas. Proliferation of
thermal vortices
Haddzibabic et al., Nature (2006)
30%
Fraction of images showing
at least one dislocation
Exponent a
20%
0.5
10%
0.4
low T
high T
0
0
0.1
0.2
0.3
central contrast
The onset of proliferation
coincides with a shifting to 0.5!
Z. Hadzibabic et al., Nature (2006)
0.4
0.3
0
0.1
0.2
central contrast
0.3
Interference between two interacting
one dimensional Bose liquids
Full distribution function
of the interference
amplitude
Gritsev, Altman, Demler, Polkovnikov, cond-mat/0602475
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
?
L
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)
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
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
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
Evolution of the distribution function
Probability P(x)
K=1
K=1.5
K=3
K=5
Narrow distribution
for
.
Approaches Gumble
distribution. Width
Wide Poissonian
distribution for
0
1
x
2
3
4
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, nonintersecting loop model on 2D lattice, growth of random
fractal stochastic interface, high energy limit of multicolor
QCD, …
2D quantum gravity,
non-intersecting loops on 2D lattice
Yang-Lee singularity
Moving condensates in optical lattices
Non equilibrium coherent dynamics
of interacting many-body systems
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), …
Equilibrium superfluid to insulator transition

Theory: Fisher et al. PRB (89), Jaksch et al. PRL (98)
Experiment: Greiner et al. Nature (01)
U
Superfluid
Mott
insulator
n 1
t/U
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)
This discussion: How to connect
the dynamical instability (irreversible, classical)
to the superfluid to Mott transition (equilibrium, quantum)
p
p/2
Unstable
Stable
???
SF
This discussion
MI
U/J
p
???
Possible experimental
U/t
sequence:
SF
MI
Superconductor to Insulator
transition in thin films
Bi films
d
Superconducting films
of different thickness
Marcovic et al., PRL 81:5217 (1998)
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
unstable
unstable
Amplification of
density fluctuations
r
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
.
Dynamical instability for integer filling
s
(p)
sin(p)
p
p/2
I(p)
p
0.0
0.1
0.2
0.3
U/J
*
0.4
0.5
Condensate momentum p/
Vicinity of the SF-I quantum phase transition.
Classical description applies for
Dynamical instability occurs for
SF
MI
Dynamical instability. Gutzwiller approximation
Wavefunction
Time evolution
We look for stability against small fluctuations
0.5
unstable
0.4
d=3
Phase diagram. Integer filling
d=2
p/p
0.3
d=1
0.2
stable
0.1
0.0
0.0
0.2
0.4
U/Uc
0.6
0.8
1.0
Center of Mass Momentum
Optical lattice and parabolic trap.
Gutzwiller approximation
0.00
0.17
0.34
0.52
0.69
0.86
N=1.5
N=3
0.2
0.1
The first instability
develops near the edges,
where N=1
0.0
-0.1
U=0.01 t
J=1/4
-0.2
0
100
200
300
Time
400
500
Gutzwiller ansatz simulations (2D)
j
phase
j
phase
phase
Beyond semiclassical equations. Current decay by tunneling
Current carrying states are metastable.
They can decay by thermal or quantum tunneling
Thermal activation
Quantum tunneling
j
phase
phase
Decay of current by quantum tunneling
Quantum
phase slip
j
j
Escape from metastable state by quantum tunneling.
WKB approximation
S – classical action corresponding to the motion in an inverted potential.
Decay rate from a metastable state. Example
S
0
0
 1  dx 2

2
3
d 
  x  bx 


 2m  d 



  ( pc  p )  0
For d>1 we have to include transverse directions.
Need to excite many chains to create a phase slip
J||  J cos p,
J  J
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
Weakly interacting systems. Gross-Pitaevskii regime.
Decay of current by quantum tunneling
p
p/2
U/J
SF
MI
Fallani et al., PRL (04)
Quantum phase slips are
strongly suppressed
in the GP regime
Strongly interacting regime. Vicinity of the SF-Mott transition
p
p/2
Close to a SF-Mott transition
we can use an effective
relativistivc GL theory
(Altman, Auerbach, 2004)
U/J
SF
M
I
2 2 ip x


1

p
 e
Metastable current carrying state:
This state becomes unstable at pc  1 3 corresponding to the
maximum of the current: I  p   p 1  p 2 2  .
2
Strongly interacting regime. Vicinity of the SF-Mott transition
Decay of current by quantum tunneling
p
p/2
U/J
SF
Action of a quantum phase slip in d=1,2,3
MI
- 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
Decay of current by quantum tunneling
0.5
unstable
0.4
d=3
d=2
d=1
p/
0.3
0.2
stable
0.1
0.0
0.0
0.2
0.4
U/Uc
0.6
0.8
1.0
phase
phase
Decay of current by thermal activation
Thermal
phase slip
j
j
E
Escape from metastable state by thermal activation
Thermally activated current decay. Weakly interacting regime
E
Thermal
phase slip
Activation energy in d=1,2,3
Thermal fluctuations lead to rapid decay of currents
Crossover from thermal
to quantum tunneling
Decay of current by thermal fluctuations
Phys. Rev. Lett. (2004)
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