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Nonequilibrium phenomena
in strongly correlated electron systems
Takashi Oka (U-Tokyo)
Collaborators:
Ryotaro Arita (RIKEN)
Norio Konno (Yokohama National U.)
Hideo Aoki (U-Tokyo)
The 21COE International Symposium on the Linear Response Theory,
in Commemoration of its 50th Anniversary
11/6/2007
Outline
1. Introduction:
Strongly Correlated Electron System,
Heisenberg-Euler’s effective Lagrangian
2. Dielectric Breakdown of Mott insulators
(TO, R. Arita & H. Aoki, PRL 91, 066406 (2003))
3. Dynamics in energy space, non-equilibrium
distribution
(TO, N. Konno, R. Arita & H. Aoki, PRL 94, 100602 (2005))
4. Time-dependent DMRG
(TO & H. Aoki, PRL 95, 137601 (2005))
5.
Summary
Oka & Aoki, to be published in •
``Quantum and Semi-classical Percolation & Breakdown“ (Springer)
Introduction： Strongly correlated electron system
Coulomb interaction
In some types of materials, the effect of Coulomb interaction
is so strong that it changes the properties of the system a lot.
Strongly correlated electron system
・Metal-insulator transition （Mott transition）
Copper oxides, Vanadium oxides ,
・Superconductivity
(from 1980’s)
Copper oxides (Hi-Tc), organic compounds
(1949 Mott)
Correlated electrons + non-equilibrium
Recent experimental progress:
Experimental breakthrough have been made recently
Non-linear transport:
Asamitsu et. al Nature (1997), Kumai et. al Science (2000), …
Attaching electrodes to clean films (crystal) and observe
the IV-characteristics which reflects correlation effects.
Hetero-structure:
Ohtomo et. al Nature (2004)
fine control of layer-by-layer doping
Non-linear optical response:
excitation in AC fields
Kishida et. al Nature (2000)
Basic rules
1. Hopping between lattice sites
Fermi statistics: Pauli principle
2. On-site Coulomb interaction
energy
>
Hubbard Hamiltonian:
U
minimum model of strongly correlated electron system.
Equilibrium phase transitions
Magic filling
When the filling takes certain values and , the groundstate
tend to show non-trivial orders.
n =1 (half-filling)
Mott Insulator
1. Insulator:
no free carriers
2. Anti-ferromagnetic order:
spin-spin interaction due to superexchange mechanism
Metal-insulator transition due to doping (equilibrium)
n >1
n <1
hole doped metal
electron doped metal
n =1
Mott insulator
carrier = hole
carrier = doubly occupied state
(doublon)
metal-insulator ``transition” in nonequilibrium
We consider production of carriers due to DC electric fields
doublon-hole pairs
Questions:
1. How are the carriers produced?
Many-body Landau-Zener transition
(cf. Schwinger mechanism in QED)
2. What is the distribution of the
non-equilibrium steady state?
Quantum random walk, suppression of tunneling
Why it is difficult
Electric field
correlation
Two non-perturbative effects
Current
Non-equilibrium distribution
we will see..
Phase transition
Collective motion
Similar phenomenon: Dielectric breakdown of the vacuum
Schwinger mechanism of electron-hole pair production
production rate (Schwinger 1951)
threshold(
tunneling problem of the ``pair wave function”
) behavior
Dielectric breakdown of Mott insulator
Difficulties: In correlated electrons,
charge excitation = many-body excitation
one body picture is insufficient
Q. What is the best quantity to study
to understand tunneling in a
many-body framework?
Heisenberg-Euler’s effective Lagrangian
position operator
TO & H. Aoki, PRL 95, 137601 (2005)
Heisenberg-Euler’s effective Lagrangian (Euler-Heisenberg Z.Physik 1936)
tunneling rate (per length L)
non-linear polarization
Non-adiabatic extension of the Berry phase theory of polarization
introduced by Resta, King-Smith Vanderbilt
In the following, we will calculate this quantity using
in …
(1) time-dependent gauge (exact diagonalization)
(2) quantum random walk
(3) time-independent gauge (td-DMRG)
Two gauges for electric fields
Time independent gauge
F=eEa, (a=lattice const.)
suited for open boundary condition
Time dependent gauge
L: #sites
suited for periodic boundary condition
Adiabatic many-body energy levels
The energy spectrum of the Hubbard model with a fixed flux
Metal
Insulator
energy gap
non-adiabatic tunneling and dielectric breakdown
F < Fth
non-adiabatic tunneling and dielectric breakdown
F < Fth
non-adiabatic tunneling and dielectric breakdown
F < Fth
non-adiabatic tunneling and dielectric breakdown
F < Fth
metal
insulator
non-adiabatic tunneling and dielectric breakdown
F < Fth
metal
F > Fth
same as above
insulator
non-adiabatic tunneling and dielectric breakdown
F < Fth
metal
F > Fth
same as above
insulator
non-adiabatic tunneling and dielectric breakdown
F < Fth
metal
F > Fth
same as above
insulator
non-adiabatic tunneling and dielectric breakdown
F < Fth
metal
insulator
F > Fth
tunneling rate
p
same as above
D
1-p
Answer 1:
Carriers are produced by many-body LZ transition
p
1-p
Landau-Zener formula gives the creation rate
F: field, Δ：Mott gap , a: const.
threshold electric field
field strength: F/D2
(TO, R. Arita & H. Aoki, PRL 91, 066406 (2003))
Question 2:
What is the property of the distribution?
In equilibrium,
but here, we continue our coherent time-evolution based on
and see its long time limit.
branching of paths
pair production
pair annihilation
Related physics:
multilevel system: M. Wilkinson and M. A. Morgan (2000)
spin system: H.De Raedt S. Miyashita K. Saito D. Garcia-Pablos and N. Garcia (1997)
destruction of tunneling: P. Hanggi et. al …
Diffusion in energy space
The wave function (distribution) is determined
by diffusion in energy space
Quantum (random) walk
Quantum walk – model for energy space diffusion
Multiple-LZ transition
1 dim quantum walk
with a boundary
＝
＝
＝
＋
＋
Difference from classical random walk
1. Evolution of wave function
2. Phase interference between paths
Review: A. Nayak and A. Vishwanath, quant-ph/0010117
result: localization-delocalization transition
p=0.01
p=0.4
p=0.2
phase interference
δ function
adiabatic evolution
（δfunction）
core
localized state
delocalized state
electric field
(TO, N. Konno, R. Arita & H. Aoki, PRL 94, 100602 (2005))
Test by time dependent density matrix
renormalization group
Time dependent DMRG:
left Block
right Block (m dimension)
M. A. Cazalilla, J. B. Marston (2002)
G.Vidal, S.White (2004), A J Daley, C Kollath, U Schollwöck and G Vidal (2004)
review: Schollwöck RMP
Dielectric Breakdown of Mott insulators
time evolution of the Hubbard model in strong electric fields
Time-dependent DMRG, N=50, U=4, m=150, Half-Filled Hubbard
Dielectric Breakdown of Mott insulators
time evolution of the Hubbard model in strong electric fields
Time-dependent DMRG, N=50, U=4, m=150, Half-Filled Hubbard
Numerical experiments
time evolution of the Hubbard model in strong electric fields
creation
> annihilation
Time-dependent DMRG, N=50, U=4, m=150, Half-Filled Hubbard
Pair creation of electron-hole pairs in the time-independent gauge
Quantum tunneling to …
charge excitation
spin excitation
survival probability of the Hubbard model
cf)
tunneling rate of the Hubbard model
fit with
dashed line:
a is a fitting parameter
TO & H. Aoki, PRL 95, 137601 (2005)
Conclusion
Dielectric breakdown of Mott insulators
Answers to Questions:
1. How are the carriers produced?
Many-body Landau-Zener transition
(cf. Schwinger mechanism in QED)
2. What is the distribution of the
non-equilibrium steady state?
Quantum random walk, suppression of tunneling
interesting relation between physical models