The Quantum Hall Effects and Gravity:
A Copernican View
C. S. Unnikrishnan
Gravitation Lab,
Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400005, India
E-mail address: unni@tifr.res.in
Website: www.tifr.res.in/~filab
ISCQI 17 Feb. 2016
Plan
1)
2)
3)
4)
The Quantum Hall Effect and its explanation
The Fractional Quantum Hall Effect and attempts for its explanation
Key Experimental results relevant for this talk
Composite Fermions, a mysterious gauge field and FQHE
a)
b)
c)
d)
The Cosmic Relativity Paradigm
Some established results
Spin-gravity interactions
The cosmic gravitational potentials and a real ‘gauge’ field
FQHE as a QM phenomena in the combined vector potentials
of electrodynamics and Cosmic gravity
ISCQI 17 Feb. 2016
Hall Effects
Larmour frequency:
W = eB / m
r = N / A, n = B / f0 = eB / h, n º r / n
f0
e2
RH = 2 = ; s xy = n
h
ne ne
h
ISCQI 17 Feb. 2016
One electron per flux quantum fills the LL
(Single free particle description)
ISCQI 17 Feb. 2016
ISCQI 17 Feb. 2016
Fractional Quantum Hall Effect
j
n=
2 pj ± 1
Understood partially as a
complex phenomena in a
strongly interacting and
strongly correlated system!
Coulomb interaction
dominate, collective effects
are invoked, multi-particle
relative phases are
crucial…, and iteartive and
heirarchical application of
these effects are required.
Also, fractional charges...
f0
RH = 2 =
ne ne
h
ISCQI 17 Feb. 2016
ISCQI 17 Feb. 2016
Quantum Hall Effect
The entire phenomena can
be measured in a single
sample by varying the
magnetic field or carrier
density (hence varying
filling factor smoothly)
The QHE is understood as a
singe particle effect and
interactions can be
ignored…Then why does one
need a complex description
with interactions and
multiparticle effects for
FQHE?
Need a Copernican view!
ISCQI 17 Feb. 2016
ISCQI 17 Feb. 2016
Nilakantha Somayaji 1444-1545
(Tantrasangraha)
ISCQI 17 Feb. 2016
Tycho Brahe (1546-1601)
300 Million
Light years
(up to Coma)
10 Billion
Light years
U 

All Galaxies
ISCQI 17 Feb. 2016
G  (4 R 2dR) / R  2 G R 2 H
ISCQI 17 Feb. 2016
Description in terms of a ‘composite fermion”
(Jainendra Jain)
CF=electron+ even number of flux quanta/vortices of a
mysterious gauge field
The FQHE is the IQHE of CF!
ISCQI 17 Feb. 2016
The ‘flux’ associated with the electron cancels
part of the applied magnetic field and reduced
the degeneracy to the integer QHE level.
ISCQI 17 Feb. 2016
D* =
AB
f0
*
= D ∓ 2 pN
j
n=
2 pj ± 1
N
n
1
2
n =
=
® ( « 1, « 2...)
D ∓ 2 pN 1∓ 2 pn
3
5
*
ISCQI 17 Feb. 2016
ISCQI 17 Feb. 2016
Physical understanding missing from this successful, but ‘epicyclic’ model.
We need to get to the physical fields and interactions that makes electrons
behave AS IF they are attached to an even number of flux lines of a
mysterious but physically essential ‘gauge field’.
ISCQI 17 Feb. 2016
Universe with all its matter content and gravity existed ages before we
started making our theories…
All our fundamental theories of the physical world were completed well before we
acquired ANY significant knowledge about the physical universe, its content and its
long term evolution.
In particular, the theories of relativity and dynamics (including QM) as well as the
theory of gravity were developed assuming an EMPTY universe.
However, the gravitational potentials of the matter in the universe is a billion (109)
times larger than our local potentials, and if these have any say in dynamics, then we
have completely missed that out in our theories. All our experimental tests, in contrast,
are in the unavoidable presence of cosmic gravity.
So, empirical evidence already include all cosmic gravitational effects, if any, whereas
fundamental theories, as constructed, do not – A reconsideration becomes essential.
Gravitational potential “here”
GM E
-9
Earth:
<
10
c 2 RE
Distant masses
dominate!
 gU  1017 m2 / s 2
 gE  108 m2 / s2
300 Million
Light years
(up to Coma)
2
v
1  v2 2
1 c
U
11
12
1
10
2
3
9
8
4
7
 gU  c !
2
6
5
10 Billion
Light years
The necessary paradigm change
ISCQI 17 Feb. 2016
ISCQI 17 Feb. 2016
The universe is filled with gravitational charge at critical density. Just as
an electrically charged matter density in the universe would have
completely changed electrodynamics and Maxwell’s equations for moving
particles (in comparison with an electrically neutral universe),
Gravitationally Charged Universe hugely affects dynamics, relativity and
all of physics, including the the theoretical description of gravity itself.
ISCQI 17 Feb. 2016
If the distant universe were electrically charged, with homogeneity
and isotropy, Maxwell’s equations will still be similar, without any
experimental trace for uniformly moving charges…
But wil give drastically different eperiment experimental results for
accelerating and rotating charges.
Ñ × E = r / e0
Ñ× E = - r / e 0 + r0 / e 0 ® f (r) + F0
‘Maxwell’ would have invoked some electro-inertial pseudoforce (depends on e/m) to keep his equations and some Mach
would have objected and asked look for a material source…
ISCQI 17 Feb. 2016
Moving through the Universe and its CMB marker
Absolute velocity
Absolute (universal) time
Same as average T
(1/105)
Every observer can decide whether he is moving or not, and all clocks
in the universe can be synchronized to this temperature.
ISCQI 17 Feb. 2016
Universe with matter and radiation
There is ONE special frame in which V=0
In all other frames,
SPACE is anisotropic in the frame of a moving observer.
There is a large current of matter (the charge of gravity)
ISCQI 17 Feb. 2016
é g = -1 g = 0
0
0
00
01
ê
0
0
ê g10 = 0 g11 = 1
ê
0
g 21 = 0 g 22 = 1
0
ê
ê
0
0
0
g33 = 1
ë
ù
ú
ú
ú + L(vr : x,t) ®
ú
ú
û
é g = -1 g = 0
0
0
00
01
ê
0
0
ê g10 = 0 g11 = 1
ê
0
g 21 = 0 g 22 = 1
0
ê
ê
0
0
0
g33 = 1
ë
ù
ú
ú
ú
ú
ú
û
Isotropic space remains isotropic after Lorentz transformations on coordinates
Clear conflict with real anisotropic space from matter current
A physical and logically consistent boost transformation should have
returned an anisotropic homogenous metric, reflecting the symmetry of
the transformed space.
ISCQI 17 Feb. 2016
ds = -dt + a (t){dx + dy + dz }
2
2
2
2
2
2
In a frame moving through this matter filled universe, there is
a large matter-current and space is ANISOTROPIC
x' = x -Vt, t ' = t ®
é g ' = -(1- v 2 / c 2 ) g ' = v / c
0
0
00
01
ê
'
'
g
=
v
/
c
g
=1
0
0
ê
10
11
ê
'
'
0
g
=
0
g
=1
0
21
22
ê
ê
0
0
0
g33' = 1
ë
ù
ú
ú
ú
ú
ú
û
Galilean boost gives the physically consistent metric – flat and anisotropic
ISCQI 17 Feb. 2016
Dynamics and Relativity from Cosmic Gravity (Cosmic Relativity)
Principle of Relativity as unobservability of this ‘gauge’ potential.
Same as statement of Lorentz/Galileo Invariance or invariance under boosts.
Newton’s law of dynamics and the Equivalence principle are gra Cosmic
gravitational consequences of cosmic matter-energy.
They already relativistic and ‘gravito-magnetic’
ISCQI 17 Feb. 2016
C. S. Unnikrishnan, Int. Jl. Mod. Phys. (2014).
Cosmic Gravito-magnetic Effects
Universe in rotating frame
Currents of mass generate large vector potential
And its ‘curl’ is a strong gravito-magnetic field
ISCQI 17 Feb. 2016
In the interferometer frame: Mass current with curl  Gravitomagnetic field
ISCQI 17 Feb. 2016
Atom Interferometer Gyroscope and Cosmic Gravity
ISCQI 17 Feb. 2016
Cosmic Gravity and Spin Physics
SPIN (both classical and quantum) will couple to this because spin and
angular momentum are currents of the charge of gravity – mass currents
So, ALL spin-orbit effects (including second order effects) on neutral
particles are due to gravitational interaction, coupled to the gravitational
mass of the particle – there is no exception.
Unnikrishnan: Relativity and Dynamics in the once-given Universe (in prep.)
ISCQI 17 Feb. 2016
An Ampere Experiment in Electromagnetism
Current-Current Interaction
The flip of the magnetic moment is due to a reversed current-current
interaction or reversed magnetic field, now written as
ISCQI 17 Feb. 2016
An ‘Ampere Experiment’ in Gravity
ISCQI 17 Feb. 2016
Unnikrishnan, General Theory of Relativity: The Universe between Beauty and Truth
(ResearchGate)
ISCQI 17 Feb. 2016
Several applications, insights and solutions…
1. Geometric phase on photon in helical fiber
2. Fine structure of atoms and Thomas precession
3. Spin-orbit energy splitting in chiral motion
4. Spin-transport in helical (bio) molecules
5. The spin-statistics connection
6. Fractional quantum Hall effect
ISCQI 17 Feb. 2016
Spin-dependent geometric phase
A. Tomita and R. Y. Chiao, PRL 1986
dj = 2ps (1- cosq )
ISCQI 17 Feb. 2016
R. Naaman, D. W. Waldeck, Ann. Rev. Phys. Chem. 2015
r ∓ 0.5 nm,v ∓ 5 ´10 m / s ® W ∓ 10 rad / s
5
15
s × Bg ∓ 1eV > kBT
Gravity-controlled spin valve!
ISCQI 17 Feb. 2016
Gravitational Vector potential as one moves around…
ISCQI 17 Feb. 2016
Bg = 2W = 2B(e / m)
Since Aem = rB / 2,
e
Ag = rBg / 2 = - B(e / m) = -2 Aem
m
For LL with index n, Bg = 2nW,
Ag = -2nAem (e / m) ® even number of flux lines!
The extra ‘gauge field’ in FQHE is possibly gravity!
ISCQI 17 Feb. 2016
We then need to establish the same coherent picture for the entire sequence
ISCQI 17 Feb. 2016
Several remaining issues (in all theories of FQHE)…
ISCQI 17 Feb. 2016
Summary:
Cosmic Relativity, in which gravity of the matter-energy in the universe
determines dynamics and relativistic effects, is the essential gravitational
ingredient in classical and quantum dynamics.
Several quantum phases conventionally described as spin-orbit phases
are in fact cosmic gravitational phases (G,m etc.). This is confirmed in
Thomas precession, the behaviour of photons and electrons in helical
transport, and in the case of Sagnac interferometer.
The fractional quantum Hall effect has several features that indicate that
cosmic gravity is indeed the extra field required for its single particle
description. Gravity is an essential player in quantum dynamics. The
pseudo composite-fermion is just a projected view, successful yet
missing the true physics. However, several details need to be checked
and matched and hopefully a more complete picture will emerge soon.
ISCQI 17 Feb. 2016
ISCQI 17 Feb. 2016