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Transcript

Nonequilibrium phenomena in strongly correlated electron systems Takashi Oka (U-Tokyo) Collaborators: Ryotaro Arita (RIKEN) Norio Konno (Yokohama National U.) Hideo Aoki (U-Tokyo) The 21COE International Symposium on the Linear Response Theory, in Commemoration of its 50th Anniversary 11/6/2007 Outline 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 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） Copper oxides, Vanadium oxides , ・Superconductivity (from 1980’s) Copper oxides (Hi-Tc), organic compounds (1949 Mott) Correlated electrons + non-equilibrium Recent experimental progress: Experimental breakthrough have been made recently Non-linear transport: Asamitsu et. al Nature (1997), Kumai et. al Science (2000), … Attaching electrodes to clean films (crystal) and observe the IV-characteristics which reflects correlation effects. Hetero-structure: Ohtomo et. al Nature (2004) fine control of layer-by-layer doping Non-linear optical response: excitation in AC fields Kishida et. al Nature (2000) Basic rules 1. Hopping between lattice sites Fermi statistics: Pauli principle 2. On-site Coulomb interaction energy > Hubbard Hamiltonian: U minimum model of strongly correlated electron system. 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 superexchange mechanism Metal-insulator transition due to doping (equilibrium) n >1 n <1 hole doped metal electron doped metal n =1 Mott insulator carrier = hole carrier = doubly occupied state (doublon) 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 Why it is difficult Electric field correlation Two non-perturbative effects Current Non-equilibrium distribution we will see.. Phase transition Collective motion Similar phenomenon: Dielectric breakdown of the vacuum Schwinger mechanism of electron-hole pair production production rate (Schwinger 1951) threshold( tunneling problem of the ``pair wave function” ) behavior Dielectric breakdown of Mott insulator Difficulties: In correlated electrons, charge excitation = many-body excitation one body picture is insufficient Q. What is the best quantity to study to understand tunneling in a many-body framework? Heisenberg-Euler’s effective Lagrangian position operator TO & H. Aoki, PRL 95, 137601 (2005) Heisenberg-Euler’s effective Lagrangian (Euler-Heisenberg Z.Physik 1936) tunneling rate (per length L) non-linear polarization Non-adiabatic extension of the Berry phase theory of polarization introduced by Resta, King-Smith Vanderbilt In the following, we will calculate this quantity using in … (1) time-dependent gauge (exact diagonalization) (2) quantum random walk (3) time-independent gauge (td-DMRG) Two gauges for electric fields Time independent gauge F=eEa, (a=lattice const.) suited for open boundary condition Time dependent gauge L: #sites suited for periodic boundary condition Adiabatic many-body energy levels The energy spectrum of the Hubbard model with a fixed flux Metal Insulator energy gap 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 metal insulator non-adiabatic tunneling and dielectric breakdown F < Fth metal F > Fth same as above insulator non-adiabatic tunneling and dielectric breakdown F < Fth metal F > Fth same as above insulator non-adiabatic tunneling and dielectric breakdown F < Fth metal F > Fth same as above insulator non-adiabatic tunneling and dielectric breakdown F < Fth metal insulator F > Fth tunneling rate p same as above D 1-p Answer 1: Carriers are produced by many-body LZ transition p 1-p Landau-Zener formula gives the creation rate F: field, Δ：Mott gap , a: const. threshold electric field field strength: F/D2 (TO, R. Arita & H. Aoki, PRL 91, 066406 (2003)) Question 2: What is the property of the distribution? In equilibrium, but here, we continue our coherent time-evolution based on and see its long time limit. 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 … 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 walk 1. Evolution of wave function 2. Phase interference between paths Review: A. Nayak and A. Vishwanath, quant-ph/0010117 result: localization-delocalization transition p=0.01 p=0.4 p=0.2 phase interference δ function adiabatic evolution （δfunction） core localized state delocalized state electric field (TO, N. Konno, R. Arita & H. Aoki, PRL 94, 100602 (2005)) Test by time dependent density matrix renormalization group Time dependent DMRG: left Block right Block (m dimension) 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 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 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 Numerical experiments time evolution of the Hubbard model in strong electric fields 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) 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