Nonequilibrium phenomena Nonequilibrium phenomena in strongly correlated in strongly correlated electron systems electron systems Takashi Oka (U-Tokyo) 11/6/2007 The 21COE International Symposium on the Linear Response T in Commemoration of its 50th Anniversary Collaborators: Ryotaro Arita (RIKEN) Norio Konno (Yokohama National U.) Hideo Aoki (U-Tokyo)
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Nonequilibrium phenomena in strongly correlated electron systems Takashi Oka (U-Tokyo) 11/6/2007 The 21COE International Symposium on the Linear Response.
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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.
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