Magic Angle Chaotic Precession (Presentation)

“Magic Angle Chaotic Precession”
(Recurrent Holonomies)
Bernd Binder
Quanics.com, Germany, 88679 Salem, P.O. Box 1247
binder@quanics.com
Chaotic Modeling and Simulation International Conference (CHAOS2008)
June 3, 2008, Chania Crete Greece
corrected version from 13.6.2008
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Recurrent Precession
Self-Adjusting Geometry via Spin Holonomy and
Corresponding Recurrent Chaotic Processes
Targets: rather simple spin-orbit coupling systems.
Observables: geometric phases and coupling constants.
Could be relevant to
•
Coriolis effects
•
Complex Vortex Systems
•
Coupled Rotations in Curved Paths
•
Quantum Physics (anomalous moments)
•
Relativity (anomalous coupling)
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Topological and Geometric Phases
Some stages could be recognized in the historical development (*):
1.
2.
3.
4.
Mid-19th century stage: advanced description of rigid body
rotation kinematics by Euler; Gauss-Bonnet, Rodrigues's
formula (noncommutativity property of two finite rotations),
Hamilton's lectures on Quaternions.
First half and mid-20th century: Thomas precession; Osgood‘s
curvature or “bending“, relation to magnetic monopoles.
Ishlinskii was the first who directly approached the holonomy
of gyroscopes; Aharonov-Bohm effect.
After 1970s: Wu and Yang monopole, Anandan, nonholonomic
effect in the framework of Hamiltonian formalism, new
relations to Quantum physics from Berry‘s Phase; then Hannay
angle; Chern-Simons theory.
Last 2 decades: measurements in applied optics, mesoscopic
and nanoscale physics, role in quantum computing,
superconductivity. Is now found in almost all kind of
dynamics, even on the Fermi surface and in the DNA.
(* See, e.g., G. B. Malykin1 et al 2003 Phys.-Usp. 46 957-965
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New: Holonomies and Geometric Phases
dominated by a Recurrent Chaotic Dynamics
Holonomy appears as the rotation of parallel translated tangent
vectors. It is now and then involved in chaotic dynamics (Berry).
Binder (2002):
Geometric phases can iteratively control orbital dynamics (Curvature,
holonomy, coupling strength, …).
Example: cyclic conic rolling and precession, where the cone apex determining
the orbital radius is a function of the velocity and frequency.
See more at
www.quanics.com,
“Magic Angle Precession”
In this talk: Recurrent Holonomy
⇒ Magic angles/phase precession
⇒ Winding numbers
⇒ Chaotic attractor
⇒ Bifurcation instabilities
⇒ Chaotic oscillator (Chua-type)
⇒ Neural spin net
⇒ Coupling constants
⇒ Coupling “anomalies”
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Covariant Geometric Gauss Flux
in Hilbert Space driven by Holonomy
∂ µ Ψ = ∑ ∂ µ ui i , ∂ µ ≡
i
∂
∂g µ
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Covariant derivative of
parameters in Hilbert space.
Aµ = −i Ψ | ∂ µ Ψ
Berry Connection, Vector Potential.
Fµν = ∂ µ Aν − ∂ν Aµ
Berry Curvature, Magnetic flux density,
satisfies Gauss law.
e
= exp i ∫ Aµ dg
iθ Γ
Γ
θS = π (1 − cos θ )
2
ωθ
j
=π
=π
ωψ
N
µ
Berry phase for a closed directed path,
Stokes theorem can be applied.
Berry geometric phase on the sphere S2.
Generally, we have an integral winding
number j, a Chern or Gauss-Bonnet
invariant, a Dirac monopole quantum
number, and other topological
invariants (Hopf,…) on closed paths.
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Observable Angles and Frequencies in a
Rotated Rotator (Gyroscope)
z
φ
ψ
One fixed point + 3 Euler
angles + external coupling
θ
O
ω ∆ = ωψ − ωθ = ω r − ωϕ
ω ∆ ωθ
=
= cos θ
ωψ ωθ
ωϕ = d ϕ / dt
ωψ = dψ / dt
ωθ = d θ / dt
Nutation
ω ∆ = d ∆ / dt
Dynamic Shift
ωr = ωϕ + ω∆
Euler ang. velocity
ωθ = d θ / dt
Geometric Shift,
Holonomy
Spin
Precession
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Dimensionless Numbers
Numbers describing precession N, M, j
cos θ =
(Winding numbers. Integral
j,N signals quantum spin,
integral M signals quantum
precession, where M is a
Dirac monopole quantum
number)
N−j
, ωr = ωψ M
N
Spin-orbit coupling parameter
ω∆ cos θ
1 ωθ
N−j
α=
=
=
−
=
ωr
M
M ωr
NM
Frequency ratios
ω1 =
ωr
MN
=
(generalized fine structure
constant, j/NM is the Berry
Phase component, Binder
2002)
ωr > ωϕ > ωψ > ω∆ > ωθ > ωθ
ωϕ
N ( M − 1) + j
=
ωψ
N
=
ω∆
N−j
=
ωθ
j
=
N ωθ
j( N − j)
(in geometric units)
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Magic Angle Precession (MAP)
θ Berry Phase
on S2
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θ =π /3
Extremum
Order N-j
AB Geom. Phase
Holonomy
θ
2π j
xy-Tshebysheff
Polynomial
Order N
2π j
Orbit is chaotic if both, apex and
rolling path are affected by holonomy:
SO(3) recurrent holonomy (angular defect) is "contracting" not only the
azimuthal (ψ, 0...2π) but also the zenithal range (θ, 0...π) down by a
rational factor (N-j)/N. θ and the azimuthal range ∆ are linearly coupled!
ω∆ ωθ M θ
=
=
= cos θ
ωψ ωθ
N θ
j∆
θ=
= π jα
MN
∆ = π N cos θ = π ( N − j )
M θ t +1 = ± jπ cos θt
1 dθ
= − M θ ± jπ cos θ
ω dt
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Berry Quantum Precession => Monopole
Quantum Precession Number
M = Dirac Monopole Charge
ωθ M θ
=
ωθ
N θ
M = CEIL [π / arcos(1 − 1/ N )]
Holonomy,
Curvature
Hamiltonian,
Spin dynamics
≈π /M
θ
Boundaries,
Winding numbers
2π / N
A rotated rotor “charged” by precession θ linearly
induced by a geometric phase, iteratively searches the
optimum precession angle subject to topological and
geometric boundaries like winding numbers.
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Cosine Map with Bifurcations
Mθ Linear
Spin/Precession
Coupling
Transcendental
Holonomy cosθ
α
Bifurcation
singularity
condition
θ
−θ tan θ = 1,
θ = ±0.8603336...
0
π
Chua oscillator
with
transcendental
nonlinearity
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MAP in a Powerball (M=12)
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(Dynabee, Powerball,… patented by A. L. Mishler 1973)
The function of the Gyrotwister or Powerball is based on a
mechanical gear-type coupling between precession and spin phase: if
the spin axis rolls in the bearing (on the rim of the groove by friction)
and the device has enough spin, the onset of linear spin-precession
coupling enables to control spin by precession. If the conic motion of
the hand is such that the number N of axis rotations for one
precession period is reduced by j, the device auto-tunes recurrently
and follows the external holonomy induced by the hand. Now it can
be powered-up to high spin frequencies with precession frequency
multiplier N/j. Near dynamic equilibrium we get curves of constant
precession (Darboux) in the Frenet description, which are rolling cone
paths showing the dynamics of a precession axis pendulum.
The Powerball or Gyrotwister has
M ≈ 12 with N ≈ 30 rotation/spin
units. For j = 1 we have about θ ≈
180°/12 =15° .
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Chua-type Oscillator Model
Spin as a
linear oscillator is linearly coupled by a monopole field
times precession spin current to a
M
M is a kind of
von Klitzing
conductivity
nonlinear
geometric
boundary
π cos  j (θtot − θ ) 
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MAP Dynamics
part of a modified Chua Circuit: exchange of spin
(current) between rotator and precession of rotator
1 dθ
= M θ tot − θ ± π cos  j (θ tot − θ ) 
ωψ dt
(
)
1 dθtot
= ϕ − M θtot − θ
ωψ dt
(
1 dϕ
ϕ
= −θ tot −
k
ωψ dt
Oscillator
θ dyn =
j∆
= j θ tot − θ = π jα
NM
(
)
)
Geometric
regulation
Coupling
term
k = 0, M = j = 12
Spin current φ
Spiral
term
Precession
Potential
geometric phase =>
gauge potential
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„Charged Pair“: MAP Bifurcation shows
a Weak Partner Singularity
MAP can generate
asymmetric opposite
charge pairs
+
1
0
j/M
_
Mθt +1 = ±π j cos θt
Shows up if j is about 5% below M
j/M = 0.95
1.00
1.05
M=20, k=10
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MAP as a Recurrent Neural Net
with „Charge“ = Number of Precession Units
Extension of one rotated rotor element to a recurrent neural
network of Z “charges” coupling by precession and driven by spin
 Z −1

θ k ,t +1 = π J k cos  ∑ ωikθi ,t 
 i =0

In a symmetric situation all elements could synchronize to a
−1
common precession dynamics with coupling ωik = M
Z −1
θis ,t +1 = ∑ ωikθ k ,t +1 =
k =0
Z
θt +1
M
which behaves as one MAP element with charge Z
θis ,t +1 =
π Ji
M
j = JZ
j = 1, J = 12 , Z = 2
Z −1
∑ cos (θks,t ) = π
k =0
Ji Z
cos (θis ,t )
M
The total winding number is spin times charge
Winding number 1 as the minimum requirement
for one closed unit preferes a pairing of half spin
elements with Z=2. Could be a chaotic model for
superconductivity and Cooper pairs
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Chua-MAP: Symmetry Breaking
without CP and T Symmetry?
Parity violation but
CPT-Symmetry?
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mirror
point
M= j = 20
k=5
For k < ∞
there is a
tilt
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Chua-MAP: Non-Zero Nutation
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Chua-MAP: looks like a „Black Hole“
(dynamics transformed into space or frequency coordinates)
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Neutral Mid Layer for j ~ 3M, 5M,…
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∆ = θ / Z = π : Extremum is Quark-like
Dirac condition 2M = 1/α, Berry Phase = Dynamic Phase
cos ( JZπα ) = M α = ±
θ =2π/3
v
θ =π/3
π/6
c
1
2
∆ = MN θ / j = π N (1 − x ) = π
x = 0.5, N = 2
j = xN = 1
θ = π jα = ±π / 3, ± 2π / 3
α = ±1/ 3, ± 2 / 3
M = ∓1/ 2α = 3 / 2, 3 / 4
Z = 1/ J = ±1/ 3, ± 2 / 3
Extreme values can be assigned to the minimum in N=2 and j=1 with
maximum ∆=π, where the Berry phase equals the dynamic phase. In this
case we have a rational exact solution without chaotic transcendental
part. Spin and charge follow from SU(3) symmetry, where the Dirac
monopole charge condition NM = ± 1/α fits very well to Z = j/J = ±1/3,
±2/3, the quark charge. This spin has some serious support in literature,
see i.e., J. Franklin, “Fermion Quarks of Spin 3/2”, Phys. Rev. 180, 1583 - 1587 (1969).
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M = 137, Geometric Phase Part of
Chaotic Quantum Electrodynamics
cos θ N − j
α=
=
M
NM
Berry‘s phase part of the Sommerfeld Fine
Structure and Rydberg Constant (Binder, 2002)
Mαt +1 = ± cos( jπαt )
Dirac Monopol Number M = 137, j = ZJ = 1, N = 3804, …., 3808
1/α = M + Geometric Phase/π
= 137, 035995… or 137, 035999… or 137, 036005…
2006 CODATA: 137, 035999… + error
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M=137, Instability for Z>115, J=1/2
Network becomes instable near bifurcation at Z=115.05275…
−θ tan θ = 1
θ = ±0.8603336...
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Very Large N, M
“Anomalies” in General Relativity
Gyro-Precession
Spin frequency shift
linear
ar
line
ea
r
θ
no
nl
in
Curv./Acceler.
Relativity,
Holonomy
Orbital boundaries,
path, winding
θ
∆
Could affect precessing satellites, stars, …
see table of candidates on www.quanics.com
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