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![Page 1: Nonequilibrium phenomena in strongly correlated electron systems Takashi Oka (U-Tokyo) 11/6/2007 The 21COE International Symposium on the Linear Response.](https://reader030.fdocuments.us/reader030/viewer/2022032709/56649ec75503460f94bd36a1/html5/thumbnails/1.jpg)
Nonequilibrium phenomena Nonequilibrium phenomena in strongly correlated electron systemsin strongly correlated electron systems
Takashi Oka (U-Tokyo)
11/6/2007
The 21COE International Symposium on the Linear Response Theory, in Commemoration of its 50th Anniversary
Collaborators:Ryotaro Arita (RIKEN)Norio Konno (Yokohama National U.)Hideo Aoki (U-Tokyo)
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1. Introduction: Strongly Correlated Electron System,
Heisenberg-Euler’s effective Lagrangian2. 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
Outline
Oka & Aoki, to be published in �``Quantum and Semi-classical Percolation & Breakdown“ (Springer)Oka & Aoki, to be published in �``Quantum and Semi-classical Percolation & Breakdown“ (Springer)
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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 ) (1949 Mott)Copper oxides, Vanadium oxides ,
・ Superconductivity (from 1980’s) Copper oxides (Hi-Tc), organic compounds
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Correlated electrons + non-equilibrium
Recent experimental progress:
Attaching electrodes to clean films (crystal) and observe the IV-characteristics which reflects correlation effects.
Non-linear transport:
Non-linear optical response:
Hetero-structure:
Kishida et. al Nature (2000)
Asamitsu et. al Nature (1997), Kumai et. al Science (2000), …
Ohtomo et. al Nature (2004)
Experimental breakthrough have been made recently
excitation in AC fields
fine control of layer-by-layer doping
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Basic rules
1. Hopping between lattice sites
Fermi statistics: Pauli principle2. On-site Coulomb interaction
>energy
UHubbard Hamiltonian: minimum model of strongly correlated electron system.
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Equilibrium phase transitions
Magic filling When the filling takes certain values and , the groundstatetend 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 super-
exchange mechanism
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Metal-insulator transition due to doping (equilibrium)
carrier = hole
n =1
n <1 n >1hole doped metal electron doped metal
carrier = doubly occupied state (doublon)
Mott insulator
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metal-insulator ``transition” in nonequilibrium
We consider production of carriers due to DC electric fields
doublon-hole pairsQuestions:
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
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Electric field
correlation
Phase transitionCollective motion
Why it is difficult
Two non-perturbative effectsTwo non-perturbative effects
CurrentNon-equilibrium distribution
we will see..
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Similar phenomenon: Dielectric breakdown of the vacuum
Schwinger mechanism of electron-hole pair production
tunneling problem of the ``pair wave function”
production rate (Schwinger 1951)
threshold( ) behavior
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Dielectric breakdown of Mott insulator
Difficulties: In correlated electrons, charge excitation = many-body excitation
Q. What is the best quantity to studyto understand tunneling in amany-body framework?
one body picture is insufficient
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Heisenberg-Euler’s effective Lagrangian
In the following, we will calculate this quantity using
Heisenberg-Euler’s effective Lagrangian
Non-adiabatic extension of the Berry phase theory of polarization introduced by Resta, King-Smith Vanderbilt
(Euler-Heisenberg Z.Physik 1936)tunneling rate (per length L) non-linear polarization
TO & H. Aoki, PRL 95, 137601 (2005)
(1) time-dependent gauge (exact diagonalization)(2) quantum random walk(3) time-independent gauge (td-DMRG)
in …
position operator
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L: #sites
Two gauges for electric fields
Time independent gauge
Time dependent gauge
F=eEa, (a=lattice const.)
suited for open boundary condition
suited for periodic boundary condition
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energy gap
The energy spectrum of the Hubbard model with a fixed flux
Metal Insulator
Adiabatic many-body energy levels
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non-adiabatic tunneling and dielectric breakdown
F < Fth
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non-adiabatic tunneling and dielectric breakdown
F < Fth
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non-adiabatic tunneling and dielectric breakdown
F < Fth
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non-adiabatic tunneling and dielectric breakdown
F < Fth
insulatormetal
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insulatormetal
same as above
non-adiabatic tunneling and dielectric breakdown
F < Fth
F > Fth
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metal
non-adiabatic tunneling and dielectric breakdown
F < Fth
insulator
F > Fth
same as above
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metal
non-adiabatic tunneling and dielectric breakdown
F < Fth
insulator
F > Fth
same as above
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p
metal
tunneling rate
1-p
non-adiabatic tunneling and dielectric breakdown
F < Fth
insulator
F > Fth
same as above
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p
1-p
Answer 1: Carriers are produced by many-body LZ transition
F: field, Δ : Mott gap , : const.
Landau-Zener formula gives the creation rate
threshold electric field
field strength: F/2
(TO, R. Arita & H. Aoki, PRL 91, 066406 (2003))
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Question 2:
What is the property of the distribution?
In equilibrium,
and see its long time limit.
but here, we continue our coherent time-evolution based on
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branching of paths
pair productionpair 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 …
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 …
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Diffusion in energy space
The wave function (distribution) is determined by diffusion in energy space
The wave function (distribution) is determined by diffusion in energy space
Quantum (random) walk
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Quantum walk – model for energy space diffusion
Multiple-LZ transition
=
1 dim quantum walk with a boundary
= +
+=
Difference from classical random walk1. Evolution of wave function2. Phase interference between paths
Review: A. Nayak and A. Vishwanath, quant-ph/0010117
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result: localization-delocalization transition
p=0.01 p=0.2 p=0.4
electric field
δ function core
adiabatic evolution( δfunction )
delocalized statelocalized state
phase interference
(TO, N. Konno, R. Arita & H. Aoki, PRL 94, 100602 (2005))
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Test by time dependent density matrix renormalization group
Time dependent DMRG:
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
right Block (m dimension)left Block
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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
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Time-dependent DMRG, N=50, U=4, m=150, Half-Filled Hubbard
time evolution of the Hubbard model in strong electric fields
Dielectric Breakdown of Mott insulators
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time evolution of the Hubbard model in strong electric fields
Numerical experiments
creation > annihilation
Time-dependent DMRG, N=50, U=4, m=150, Half-Filled Hubbard
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Pair creation of electron-hole pairs in the time-independent gauge
Quantum tunneling to …
charge excitation
spin excitation
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survival probability of the Hubbard model
cf)
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tunneling rate of the Hubbard model
fit with
dashed line:
a is a fitting parameter TO & H. Aoki, PRL 95, 137601 (2005)
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ConclusionDielectric 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