Post on 01-Jan-2016
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)
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)
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
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
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.
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
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
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
Electric field
correlation
Phase transitionCollective motion
Why it is difficult
Two non-perturbative effectsTwo non-perturbative effects
CurrentNon-equilibrium distribution
we will see..
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
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
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
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
energy gap
The energy spectrum of the Hubbard model with a fixed flux
Metal Insulator
Adiabatic many-body energy levels
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
insulatormetal
insulatormetal
same as above
non-adiabatic tunneling and dielectric breakdown
F < Fth
F > Fth
metal
non-adiabatic tunneling and dielectric breakdown
F < Fth
insulator
F > Fth
same as above
metal
non-adiabatic tunneling and dielectric breakdown
F < Fth
insulator
F > Fth
same as above
p
metal
tunneling rate
1-p
non-adiabatic tunneling and dielectric breakdown
F < Fth
insulator
F > Fth
same as above
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))
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
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 …
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
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
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))
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
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
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
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
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)
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