the kicked rotor: from classical chaos to integer quantum hall … · 2020. 1. 17. · outline •a...
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The kicked rotor: from classical chaos to integer quantum Hall effect
Chushun Tian
Institute of theoretical Physics Chinese Academy of Sciences
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Outline • A review of kicked rotor
• Planck’s quantum-driven IQHE from chaos
1. formulation of problem and summary of main results
2. analytic theory
3. numerical confirmation
4. chaos origin
• Conclusion and outlook
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Outline • A review of kicked rotor
• Planck’s quantum-driven IQHE from chaos
1. formulation of problem and summary of main results
2. analytic theory
3. numerical confirmation
4. chaos origin
• Conclusion and outlook
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What is kicked rotor? What is the problem?
controlled by two parameters:
• Planck’s quantum he
• nonlinear parameter K
a free rotating particle under the influence of sequential time-periodic driving
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he→0 :Chirikov standard map - the birth of classical KR
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Why ”Chirikov standard map” standard?
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classical KR
Liouville integrability: •regular foliation of phase space •action variables = complete sets of invariants of Hamiltonian flow
nnn
nnn
lKll+=+=
+
++
θθθ
1
11 sin
l
θ
K=0
nnn
nn
lll
+==
+
+
θθ 1
1
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classical KR
transition from weak chaoticity (KAM, confined motion in angular momentum l space) to strong chaoticity (deconfined motion in l space)
nnn
nnn
lKll+=+=
+
++
θθθ
1
11 sin
from B. Chirikov and D. Shepelyansky, Scholarpedia 3, 3550 (2008)
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classical KR
transition from “classical insulator” to “classical normal metal”
.)(space-in walk random
on memory lose : general and large
constttE
lK
=
θ
nnn
nnn
lKll+=+=
+
++
θθθ
1
11 sin
from B. Chirikov and D. Shepelyansky, Scholarpedia 3, 3550 (2008)
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he>0 :QKR
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irrational values of he/(4π) quantum suppression of classical chaos: dynamical localization
classical chaos
Casati, G., Chirikov, B. V., Ford, J. and Izrailev, F. M. Lecture Notes in Physics 93, 334-352 (1979).
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•take a cloud of cold atoms, each of which is a two-level system
F. L. Moore, J. C. Robinson, C. F. Bharucha, B. Sundaram, and M. G. Raizen, Phys. Rev. Lett. 75, 4598 (1995).
g
e0ω
atom-optics realization of QKR
mass: M dipole moment:
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•subjected to two counterpropagating laser beams → dipole-electric field coupling
F. L. Moore, J. C. Robinson, C. F. Bharucha, B. Sundaram, and M. G. Raizen, Phys. Rev. Lett. 75, 4598 (1995).
g
e0ω
atom-optics realization of QKR
mass: M dipole moment:
field amplitude: E0 wavenumber: kL wave frequency: L
kL -kL
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F. L. Moore, J. C. Robinson, C. F. Bharucha, B. Sundaram, and M. G. Raizen, Phys. Rev. Lett. 75, 4598 (1995).
g
e0ω
atom-optics realization of QKR
field amplitude: E0 wavenumber: kL wave frequency: L
kL -kL
•Schroedinger equation of ground state wave function
- :detuning
:frequency Rabi
0
0
LL
E
ωωδ
µ
=
=Ω mass: M
dipole moment:
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F. L. Moore, J. C. Robinson, C. F. Bharucha, B. Sundaram, and M. G. Raizen, Phys. Rev. Lett. 75, 4598 (1995).
g
e0ω
atom-optics realization of QKR
field amplitude: E0 wavenumber: kL wave frequency: L
kL -kL
•switch on laser beams at multiple time periods (T)
mass: M dipole moment:
gti
nL
L
e
nTtxkxM
ti
ψψ
ψδδ
ψ
ψ
8/
2
2
22
2
)()2cos(82
Ω=
−Ω
−∂∂
−
=∂∂
∑
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dynamical-Anderson localization analogy
Bloch-Floquet theory mapping to Anderson-like model
energyeigen-quasi:eigenstate:)(
operatorFloquet :ˆ
α
α
ωφ tU
Fishman, S., DR. Grempel and RE. Prange, Phys. Rev. Lett. 49 (1982) 508.
kicking (before)after right )( of valuethe:)( tt αα φφ ±
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dynamical-Anderson localization analogy
Bloch-Floquet theory mapping to Anderson-like model
energyeigen-quasi:eigenstate:)(
operatorFloquet :ˆ
α
α
ωφ tU
Fishman, S., DR. Grempel and RE. Prange, Phys. Rev. Lett. 49 (1982) 508.
kicking (before)after right )( of valuethe:)( tt αα φφ ±
no disorders! highly nonlinear!
???
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Ehrenfest time effects
transition to dynamical localization Ehrenfest time:
−−−= 2/1
2/3)4()4(3
42)(L
EE t
tttttDtE θπ
KtE ln1λ
=
4tE
CT, A. Kamenev, and A. Larkin, Phys. Rev. Lett. 93, 124101 (2004); Phys. Rev. B 72, 045108 (2005).
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Ehrenfest time effects
E. B. Rozenbaum, S. Ganeshan, and V. Galitski, Phys. Rev. Lett. 118, 086801 (2017)
[ ]22 )0(ˆ),(ˆ)( ntnhtC e=out-of-time-ordered correlator:
transition in C(t) •before Ehrenfest time: C(t) grows exponentially with a constant rate .
•after Ehrenfest time: The growth rate decays in t.
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sensitivity to the value of he/(4π): he/(4π) =p/q
small q: nonuniversal
perfect crystal, no dissipation
2~)( ttE
qnnU
+→ ˆˆ:on translatiunder theinvariant ˆ
supermetal
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sensitivity to the value of he/(4π): he/(4π) =p/q
P. Fang, CT, and J. Wang, Phys. Rev. B 92, 235437 (2015).
Rescaled E(t) exhibits a universal metal-supermetal dynamics crossover:
large q: universal
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sensitivity to the value of he/(4π): he/(4π) =p/q
universality class of metal-supermetal dynamics crossover = universality class of RMT
Rescaled E(t) exhibits a universal metal-supermetal dynamics crossover:
large q: universal
P. Fang, CT, and J. Wang, Phys. Rev. B 92, 235437 (2015).
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QKR can simulate:
•1D disordered solids; •1D perfect crystal.
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QKR can simulate:
•1D disordered solids; •1D perfect crystal.
beyond 1D?
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sfrequencie modulation : ~,...,~2 dωω
exploring d-dimensional physics in 1D
ψψδωθωθθψ HktttVnHt
ik
ddˆ)()~,...,~,ˆ()ˆ( 22110 ≡
−+++=
∂∂ ∑
∑∑→
∑→ === ∂
∂∂∂
−∂∂
−d
i ii
d
i ii
d
i ii ttt
eHeHe 222
~~~
ˆˆ , θω
θω
θω
ψψ
ii
kd
d
iii inktVnnH
ti
θψδθθθωψ
∂∂
−=
−++=
∂∂ ∑∑
=
ˆ ,)()ˆ,...,ˆ,ˆ(ˆ~)ˆ( 212
10
1D quasiperiodic QKR driven by (d-1)frequencies = d-D periodic QKR
G. Casati, I. Guarneri, and D. L. Shepelyansky, Phys. Rev. Lett. 62, 345 (1989).
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quasiperiodic QKR: irrational he/(4π)
J. Chabe et al., Phys. Rev. Lett. 101, 255702 (2008); G. Lemarie et al., ibid. 105, 090601 (2010).
Localized
Critic
008.0571.1:)transition(Anderson
ssimulation01.059.1
:(QKR) ssimulation05.063.1
:(QKR) experiment
±=
±=
±=
ν
ν
ν
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quasiperiodic QKR: rational he/(4π)
• Anderson insulator turned into supermetal (E~t2) ;
• Anderson metal-insulator transition turned into metal-supermetal transition
CT, A. Altland, and M. Garst, Phys. Rev. Lett. 107, 074101 (2011) J. Wang, CT, and A. Altland, Phys. Rev. B 89, 195105 (2014)
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rich Planck’s quantum-driven phenomena in spinless kicked rotors;
associated with the restoration (breaking) of translation symmetry in the angular momentum space.
spinful kicked rotor?
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Outline • A review of kicked rotor
• Planck’s quantum-driven IQHE from chaos
1. formulation of problem and summary of main results
2. analytic theory
3. numerical confirmation
4. chaos origin
• Conclusion and outlook
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Outline • A review of kicked rotor
• Planck’s quantum-driven IQHE from chaos
1. formulation of problem and summary of main results
2. analytic theory
3. numerical confirmation
4. chaos origin
• Conclusion and outlook
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θ1
quasiperiodically quantum kicked spin-1/2 rotor
incommensurate with 2π
matrix Pauli :
)~,()~,(3
12121
i
i
ii
V
tVtV
σ
σ
σωθθωθθ
⋅≡
+=+ ∑=
11 / lITl →tTt →/ωω ~~ →T
Y. Chen and CT, Phys. Rev. Lett. 113, 216802 (2014) CT, Y. Chen, and J. Wang, Phys. Rev. B 93, 075403 (2016) (38 pages)
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quasiperiodically quantum kicked spin-1/2 rotor
incommensurate with 2π
matrix Pauli :
)~,()~,(3
12121
i
i
ii
V
tVtV
σ
σ
σωθθωθθ
⋅≡
+=+ ∑=
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θ1
quasiperiodically quantum kicked spin-1/2 rotor
symmetry class A
incommensurate with 2π
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θ1
quasiperiodically quantum kicked spin-1/2 rotor
?
σ
σωθθωθθ
⋅≡
+=+ ∑=
V
tVtV i
ii
3
12121 )~,()~,(
incommensurate with 2π
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θ1
quasiperiodically quantum kicked spin-1/2 rotor
Microscopically, controlled by single parameter – Planck’s quantum he.
σ
σωθθωθθ
⋅≡
+=+ ∑=
V
tVtV i
ii
3
12121 )~,()~,(
incommensurate with 2π
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θ1
quasiperiodically quantum kicked spin-1/2 rotor
Macroscopically, controlled by two phase parameters – energy growth rate (EGR) and (hidden) quantum number namely quantized topological theta angle.
σ
σωθθωθθ
⋅≡
+=+ ∑=
V
tVtV i
ii
3
12121 )~,()~,(
incommensurate with 2π
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Planck’s quantum-driven IQHE (I)
EGR at large t
(Planck’s quantum)
-1
for almost all Planck’s quantum, EGR vanishes at large t limit →insulator
Planck’s quantum dependence of EGR – E(t)/t
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Planck’s quantum-driven IQHE (I)
EGR at large t
(Planck’s quantum)
𝜎𝜎 *
-1
insulator
(quantum) metal
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Planck’s quantum-driven IQHE(II)
(Planck’s quantum)
𝜎𝜎 *
-1
σ* order of unity; independent of the details of V and the critical points
insulator
(quantum) metal
EGR at large t
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Planck’s quantum-driven IQHE (III)
𝜎𝜎 * equally spaced when Planck’s quantum is small
insulator
σ* order of unity; independent of the details of V and the critical points
(Planck’s quantum) -1
EGR at large t
(quantum) metal
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Planck’s quantum-driven IQHE (III)
𝜎𝜎 * equally spaced when Planck’s quantum is small
Is the insulating phase the same as usual Anderson insulator?
σ* order of unity; independent of the details of V and the critical points
(Planck’s quantum) -1
EGR at large t
(quantum metal)
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Planck’s quantum-driven IQHE(IV)
Quantum Number
(Planck’s quantum)
n+4 𝜎𝜎 *
n+3
n+2
n+1
n -1
insulator characterized by an integer
EGR at large t
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Planck’s quantum-driven IQHE(V)
Quantum Number Energy Growth Rate
(Planck’s quantum)
n+4 𝜎𝜎 *
n+3
n+2
n+1
n -1
insulator characterized by an integer
quantum number increases by unity
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Planck’s quantum-driven IQHE(VI)
Quantum Number Energy Growth Rate
(Planck’s quantum)
n+4 𝜎𝜎 *
n+3
n+2
n+1
n -1
insulator characterized by an integer
This quantum number is of topological nature.
quantum number increases by unity
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Planck’s quantum-driven IQHE(VI)
Quantum Number Energy Growth Rate
(Planck’s quantum)
n+4 𝜎𝜎 *
n+3
n+2
n+1
n -1
insulator characterized by an integer
quantized topological theta angle
quantum number increases by unity
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Integer quantum Hall effect
ρxx ρxy
two dimensional electron gas (MOSFET)
strong magnetic field
quantized Hall conductance
Claus von
Klitzing
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phenomenological analogy to conventional IQHE
• energy growth rate → longitudinal conductivity
• quantum number → quantized Hall conductivity
• inverse Planck’s quantum → filling fraction
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fundamental differences from conventional IQHE
• no magnetic field, no electromagnetic response, driven by Planck’s quantum
• strong chaoticity origin
• one-body system → no concept such as integer filling
• one-dimensional, far-from equilibrium system
• no translation symmetry, no adiabatic parameter cycle → TKNN?
• semiclassical regime (small Planck’s quantum)
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• no magnetic field, no electromagnetic response, driven by Planck’s quantum
• modulation frequency commensurate with 2π →always insulator
• one-body system → no concept such as integer filling
• one-dimensional, far-from equilibrium system
• no translation symmetry, no adiabatic parameter cycle → TKNN?
• semiclassical regime (small Planck’s quantum)
fundamental differences from conventional IQHE
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• no magnetic field, no electromagnetic response, driven by Planck’s quantum
• modulation frequency commensurate with 2π →always insulator
• one-body system → no concept such as integer filling
• one-dimensional, far-from equilibrium system
• semiclassical regime (small Planck’s quantum) ||||||4
12121 V
VVV
VVddI
⋅
∂×∂−= ∫∫ θθθθ
π
fundamental differences from conventional IQHE
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Outline • A review of kicked rotor
• Planck’s quantum-driven IQHE from chaos
1. formulation of problem and summary of main results
2. analytic theory
3. numerical confirmation
4. chaos origin
• Conclusion and outlook
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Analytic theory (I)
•mapping onto 2D periodic quantum dynamics
two-particle Green function
interference between advanced and retarded quantum amplitudes
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• introduce supervector=(complex/bosonic number, Grassmann/fermionic number): and express Kω in terms a functional integral over ψ
Analytic theory (II)
indexspin :space etry)(supersymm rmionicbosonic/fe :/
space etardedadvanced/r :
sfb
±
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• no Hubbard-Stratonovich transformation, instead, color-flavor transformation (Zirnbauer ‘96)
Analytic theory (III)
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• no Hubbard-Stratonovich transformation, instead, color-flavor transformation (Zirnbauer ‘96)
Analytic theory (III)
( )( ) ( )∫∫×
=FB MM
dQZZd ~,µ
)1()1()1,1(
UUUM B ×
=)1()1(
)2(UU
UM F ×=
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• no Hubbard-Stratonovich transformation, instead, color-flavor transformation (Zirnbauer ‘96)
Supermatrix field Z describes the collective mode of coherent motion.
Analytic theory (III)
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• functional integral expression for Kω
Analytic theory (IV)
( )[ ] ( )( ) ( )( )∫ −+−+
−−−
−+−+
−−= bsNbsNbNsbNsZZS ZZZZZZeZZD
ssNsNsK
'',''1
,1~, ~~1~1)~,(
''',ω
• express Kω in terms a functional integral over ψ and Z, with an action quadratic in ψ
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• In general, ZNN’ is off-diagonal in angular momentum space;
• (N+N’)/2 is center-of-mass coordinate of the coherent motion, while the off-diagonality encodes the information of the angular relaxation;
• Strong chaoticity renders the memory about the angle lost → off-diagonality is eliminated.
Analytic theory (V)
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• In general, ZNN’ is off-diagonal in angular momentum space;
• (N+N’)/2 is center-of-mass coordinate of the coherent motion, while the off-diagonality encodes the information of the angular relaxation;
• Strong chaoticity renders the memory about the angle lost → off-diagonality is eliminated.
Analytic theory (V)
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2nd order hydrodynamic expansion (supersymmetry version of Pruisken’s replica field theory)
topological θ-term: first time seen to emerge from microscopic chaos
bare coupling constants
Analytic theory (VI)
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2nd order hydrodynamic expansion (supersymmetry version of Pruisken’s replica field theory)
bare coupling constants
Analytic theory (VI)
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2nd order hydrodynamic expansion (supersymmetry version of Pruisken’s topological field theory)
bare coupling constants
Analytic theory (VI)
short-time energy growth rate “classical Hall conductivity”
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•renormalization group analysis recipe: background field formalism (Pruisken ‘80)
Analytic theory (VII)
minimal coupling to the effective field theory - “virtual electromagnetic response”
“transport parameters”
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Analytic theory (VII)
minimal coupling to the effective field theory - “virtual electromagnetic response”
“transport parameters”
•renormalization group analysis recipe: background field formalism (Pruisken ‘80)
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•perturbative + nonperturbative instanton corrections
Analytic theory (VII)
renormalization of short-time energy growth rate
renormalization of topological angle
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•two-parameter renormalization group flow
Analytic theory (VIII)
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•two-parameter renormalization group flow
Analytic theory (VIII)
unity of period a with ,~in periodic Hσ
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insulating phase: σ=0, σH=n (quantization of topological theta angle: origin of quantum number)
metallic phase: σ=σ*, σH=n+1/2
critical lines
Analytic theory (VIII)
•two-parameter renormalization group flow
Khmelnitskii’s RG flow for conventional IQHE (‘83)
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insulating phase: σ=0, σH=n (quantization of topological theta angle: origin of quantum number)
metallic phase: σ=σ*, σH=n+1/2
critical lines
Analytic theory (VIII)
•two-parameter renormalization group flow
Khmelnitskii’s RG flow for conventional IQHE (‘83)
decreasing of he
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•estimation of critical exponent
Analytic theory (IX)
•estimation of critical energy growth rate
021~ =
+=nLHσ
β
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Outline • A review of kicked rotor
• Planck’s quantum-driven IQHE from chaos
1. formulation of problem and summary of main results
2. analytic theory
3. numerical confirmation
4. chaos origin
• Conclusion and outlook
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Numerical test (t<102): chaoticity
linear energy growth in short times
blue dots are simulation results for the energy growth rate in short times;
fluctuations of eigen quasi-energies follow Wigner-Dyson statistics of unitary type.
( )( )2121 coscos18.0,sin,sin
,2arctan2
θθθθ
σ
−−=
⋅=
d
dd
dV
red line is the theoretical prediction.
Beenakker et. al. ‘11
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Numerical test (t<6×105): transition between topological insulating phases
Hall plateaux (n=0,1,2,…) critical points (n=1/2,3/2, …)
Analytic results for σH (he) predict three transition points at 1/he =0.73,2.19,3.60 for 0.23< he <1.50.
Simulations indeed show three transition points at 1/he =0.77,2.13,3.45 .
Simulations show that the growth rate at the critical point is universal.
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Numerical test (t<6×105): transition between topological insulating phases
Hall plateaux (n=0,1,2,…) critical points (n=1/2,3/2, …)
Analytic results for σH (he) predict three transition points at 1/he =0.73,2.19,3.60 for 0.23< he <1.50.
Simulations indeed show three transition points at 1/he =0.77,2.13,3.45 .
Simulations show that the growth rate at the critical point is universal.
Simulations show that the transition is robust against the change of H0.
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Universality of critical energy growth rate
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expectation from conventional IQHE: 25.0* =σ
Universality of critical energy growth rate
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Robustness of Planck’s quantum-driven IQHE against the change of kicking potential
( )( )2121 coscos18.0,sin,sin
,2arctan2
θθθθ
σ
−−=
⋅=
d
dd
dV
µ
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Robustness of Planck’s quantum-driven IQHE against the change of free rotation Hamitonian
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Outline • A review of kicked rotor
• Planck’s quantum-driven IQHE from chaos
1. formulation of problem and summary of main results
2. analytic theory
3. numerical confirmation
4. chaos origin
• Conclusion and outlook
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θ1
periodically kicked rotor, with a time period of q!
commensurate with 2π
σ
σωθθωθθ
⋅≡
+=+ ∑=
V
tVtV i
ii
3
12121 )~,()~,(
coprime ),( ,2~ qpqpπω =
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Equivalent 2D system is decomposed into a family of decoupled (quasi) 1D subsystems, each of which is governed by a good quantum number, namely, the Bloch momentum.
Analytic theory (I)
0)1()1(
)2()1()1(
)1,1(1 =
×
×× UU
UUU
Uπ
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Analytic theory (II)
NO topological term!
Eq : energy growth within a single (q) period
• effective field theory
• energy growth at long times
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Numerical confirmation (I)
qp
=πω2
~ rational
no localization-delocalization transitions occurs; the system is always insulating.
scaling behavior of the saturation value confirmed
eh critical2
~ irrationalπω
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Numerical confirmation (II)
the equivalent 2D system exhibits ballistic motion in the virtual (n2) direction.
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• a connection between chaos and IQHE
• Planck’s quantum ↔ magnetic field;
energy growth rate ↔ longitudinal conductivity;
hidden quantum number ↔ quantized Hall conductivity;
• strong chaoticity origin
Conclusions
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Outlook
chaos (classical or quantum)
kicked rotor
condensed matter
theoretical tools: nonlinear dynamics, field theory, random matrix theory, functional analysis, …
experimental tools: cold atoms, molecular systems, …
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Outlook
chaos (classical or quantum)
kicked rotor
condensed matter
theoretical tools: nonlinear dynamics, field theory, random matrix theory, functional analysis, …
experimental tools: cold atoms, molecular systems, …
more surprises to come!