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