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Nonequilibrium quantum dynamics in condensed matter:
excitons, chaos, and quantum walk
Takashi Oka (U-Tokyo)
previous talk
“Meson” and their “turbulent higher mode condensation”
Hashimoto, Kinoshita, Murata, TO 2014
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Strong field physics in Condensed matter and Nuclear physics
hole density
T
superconductor
phase diagram of hadron (Fukushima-Hatsuda)
phase diagram of Hi Tc
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Strong field physics in Condensed matter and Nuclear physics
ion collision pump probe exp.
Hirori, Tanaka et al. Nat. Com. 2011
THz laser pulses (now stronger than the Schwinger limit) strong E and B fields
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Difference Movie
Wang et al. … N. Gedik Phys. Rev. Lett. 109, 127401 (2012)
I(E, kx, ky, t<0) data from N. Gedik (MIT)
Pump-probe technique
Time resolved ARPES (angle resolved photo emission spectroscopy)
Gedik@MIT group
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Part I: Condensation of Excitons
Question: How do you obtain
from quantum mechanics?
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Part I: Condensation of Excitons
Proposed phase diagram of the extended Hubbard model at half filling
Jeckelmann PRB 67 (2003)
Mott insulator
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J: Hopping between lattice sites
extended Hubbard model
ij: nearest neighbor site
U: On-site Coulomb interaction V: long-range Coulomb interaction (although it is just neighbor site)
+U +2V
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competition between U and V
Mott insulator (spin density wave)
Mott insulator (spin density wave)
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Excitations from the Mott insulator
Mott insulator (spin density wave) doublon (- charge) hole (+ charge)
Exciton = doublon-hole boundstate
Exciton string (~QCD string?)
CDW droplet with a fractural structure (in higher dim.)
: arbitrary complex number
higher ``exciton” condensate ~
?
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Lu, Sota, Matsueda, Bonča, Tohyama PRL 2012
U=10
Pulse Laser induced CDW (exciton condensation)
short pulse laser
charge-charge correlation
Maybe related to the higher meson condensation
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Question: How do you obtain
from quantum dynamics?
Part II: chaos and quantum walk
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Part II: chaos and quantum walk
Time independent Hamiltonian
fixed
How are cn determined?
Time dependent Hamiltonian
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Spec H(B)
Question: Which are quantum chaotic?
Nakamura Thomas PRL61 ‘88
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Spec H(B)
Question: Which are quantum chaotic?
Chaotic non-chaotic
Nakamura Thomas PRL61 ‘88
cf) classical chaos is distinguished by the Lyapunov exponent
Level repulsion (Wigner distribution)
Level crossing (Poisson distribution)
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Classic case
Quench problem
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Classic case
Quench problem
fluctuation growth
Production rate of B
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Quantum case
slow Quench problem
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Quantum case
Quench problem
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Quantum case
Quench problem
distribution ~ “thermal state”?
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If the system were non-chaotic …
Quench problem
pure state
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②
③
vacuum decay rate/ Euler-Heisenberg Lagrangian (fidelity, Loschmidt echo)
② Oka, Aoki PRL 2005 Hashimoto, Oka 2013
Hashimoto, Kinoshita, Murata, Oka 2014
① Full dynamics
Oka, Arita, Aoki PRL 2003
(Gauge/gravity)
①
③
Oka, Aoki 2010, Oka 2012
production rate
φ two level approximation
Semenoff, Zarembo 2011 Sato, Yoshida 2013,..
Schwinger mechanism
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②
③ Hashimoto, Kinoshita, Murata, Oka 2014
① Full dynamics
Oka, Arita, Aoki PRL 2003
(Gauge/gravity)
①
2. Quantum walk gives a rough sketch of the level dynamics
1. Excited states are multiple doublon-hole pairs ~ exciton string (~ higher meson state)
In the Hubbard model,
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Long time behavior!
creation
creation
annihilation
The dynamics can be decomposed into 2✖2 unitary evolution
The tunneling probability is given by
Schwinger mechanism
Oka, Konno, Aoki PRL 2005
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1D Quantum walk with a reflecting boundary!
introductory reviews; J. Kempe, Contemporary Physics 44, 307 (2003). Nayak et.al quant-ph/0010117. 今野紀雄,「数理科学」 2004 年 6 月号
"量子ウォークの極限定理" in Japanese
[evolution rule]
two state (up, down) at each site n and time τ.
at the boundary
unitary matrix
evolution matrices
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Properties of the distribution!
classical stochastic system!
quantum walk!
quantum interference …. leads to Anderson localization
main difference from classical system
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mapping to a quantum walk!
evolution matrix
p is the Landau-Zener tunneling rate
quantum walk
creation
creation
annihilation
dielectric breakdown of Mott insulator
similar
Oka, Konno, Aoki PRL 2005
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Path integral in energy space!
contribution from each path = product of P,Q matrices!
initial vector generally,!
transition amplitude (2×2 matrix)
Oka, Konno, Aoki PRL 2005
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PQRS method and recurrence formula !recurrence formula
multiplication rule
+
generating function of the wave function
Oka, Konno, Aoki PRL 2005
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time evolution of the wave function!expand in z
asymptotic distribution
ground state
phase interference from various paths
Anderson localization in energy space =dynamical localization
Oka, Konno, Aoki PRL 2005
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localization-delocalization transition !
p=0.01 p=0.2 p=0.4
electric field
δ関数 core
adiabatic evolution!(δfunction)
delocalized state!localized state!
Oka, Konno, Aoki PRL 2005
Higher meson condensation??!
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Conclusion “Meson” and their “higher mode condensation”
Exciton string
1. Similarity with the exciton string/ CDW droplets
2. Localization-delocalization transition in the quantum walk may explain the dynamical higher mode condensation