1. Nonequilibrium phenomena in 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)

2. 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 Outline Oka & Aoki, to be published in  ``Quantum and Semi-classical Percolation & Breakdown“ (Springer)

3. 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

4. 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

5. Basic rules 1. Hopping between lattice sites Fermi statistics: Pauli principle 2. On-site Coulomb interaction > energy U Hubbard Hamiltonian: minimum model of strongly correlated electron system.

6. 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 super- exchange mechanism

7. Metal-insulator transition due to doping (equilibrium) carrier = hole n =1 n <1 n >1 hole doped metalelectron doped metal carrier = doubly occupied state (doublon) Mott insulator

8. 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

9. Electric field correlation Phase transition Collective motion Why it is difficult Two non-perturbative effects Current Non-equilibrium distribution we will see..

10. 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

11. Dielectric breakdown of Mott insulator Difficulties: In correlated electrons, charge excitation = many-body excitation Q. What is the best quantity to study to understand tunneling in a many-body framework? one body picture is insufficient

12. 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

13. 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

14. energy gap The energy spectrum of the Hubbard model with a fixed flux MetalInsulator Adiabatic many-body energy levels

15. non-adiabatic tunneling and dielectric breakdown F < F th

16. non-adiabatic tunneling and dielectric breakdown F < F th

17. non-adiabatic tunneling and dielectric breakdown F < F th

18. non-adiabatic tunneling and dielectric breakdown F < F th insulator metal

19. insulator metal same as above non-adiabatic tunneling and dielectric breakdown F < F th F > F th

20. metal non-adiabatic tunneling and dielectric breakdown F < F th insulator F > F th same as above

21. metal non-adiabatic tunneling and dielectric breakdown F < F th insulator F > F th same as above

22. p metal tunneling rate 1-p non-adiabatic tunneling and dielectric breakdown F < F th insulator F > F th same as above 

23. 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))

24. 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

25. 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 …

26. 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

27. 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

28. result: localization-delocalization transition p=0.01p=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))

29. 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

30. 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

31. time evolution of the Hubbard model in strong electric fields Dielectric Breakdown of Mott insulators

32. 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

33. Pair creation of electron-hole pairs in the time-independent gauge Quantum tunneling to … charge excitation spin excitation

34. survival probability of the Hubbard model cf)

35. tunneling rate of the Hubbard model fit with dashed line: a is a fitting parameter TO & H. Aoki, PRL 95, 137601 (2005)

36. 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