The Fractional Quantum Hall Effect 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
QIPA@HRI, 12 Dec. 2015
Core Idea and Plan
All Physical phenomena happens in the gravitational presence of the entire
matter-energy of the universe.
Mass-energy being the charge of gravity and spin its current ALL physical
phenomena that directly involves mass and spin are in fact causally
gravitational, without exception .
In particular, several quantum phenomena of microscopic physics are affected
and often controlled by cosmic gravity. I seek to solve the mysteries of
fractional quantum Hall effect within this paradigm (Cosmic Relativity).
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!
QIPA@HRI, 12 Dec. 2015
Hall Effects
QIPA@HRI, 12 Dec. 2015
Hall Effects (EN LABO)
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
QIPA@HRI, 12 Dec. 2015
One electron per flux quantum fills the LL
(Single free particle description)
QIPA@HRI, 12 Dec. 2015
QIPA@HRI, 12 Dec. 2015
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
QIPA@HRI, 12 Dec. 2015
QIPA@HRI, 12 Dec. 2015
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!
QIPA@HRI, 12 Dec. 2015
The necessary paradigm change
QIPA@HRI, 12 Dec. 2015
300 Million
Light years
(up to Coma)
11
12
1
10
2
3
9
8
4
7
6
5
10 Billion
Light years
U 

All Galaxies
QIPA@HRI, 12 Dec. 2015
G  (4 R 2dR) / R  2 G R 2 H
QIPA@HRI, 12 Dec. 2015
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.
QIPA@HRI, 12 Dec. 2015
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…
QIPA@HRI, 12 Dec. 2015
1. Universe is gravitationally charged…and will hugely affect
dynamics of anything with mass or energy, and spin, which is
the current of the charge of gravity.
2. All physicall effects should be traced to physical interactions of
real charges and currents of one of four fundamental
interactions we have identified. In other words, there cannot be
a real physical effect (change in energy, momentum, angular
momentum etc.) without a real physical interaction.
QIPA@HRI, 12 Dec. 2015
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.
QIPA@HRI, 12 Dec. 2015
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)
QIPA@HRI, 12 Dec. 2015
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.
QIPA@HRI, 12 Dec. 2015
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
QIPA@HRI, 12 Dec. 2015
 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
F gU » c !
2
U 
5
10 Billion
Light years

All Galaxies
QIPA@HRI, 12 Dec. 2015
6
G  (4 R 2dR) / R  2 G R 2 H
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 FROM COSMIC GRAVITY
It is already relativistic and ‘gravito-magnetic’
QIPA@HRI, 12 Dec. 2015
NEWTON’S LAW FROM COSMIC GRAVITY
It is relativistic and ‘gravito-magnetic’
EQUIVALENCE PRINCIPLE FROM COSMIC GRAVITY
If dynamics respects Newton’s law it respects Equivalence principle as well!
C. S. Unnikrishnan, Int. Jl. Mod. Phys. (2014).
QIPA@HRI, 12 Dec. 2015
Cosmic Gravito-magnetic Effects
Universe in rotating frame
Currents of mass generate large vector potential
And its ‘curl’ is a strong gravito-magnetic field
QIPA@HRI, 12 Dec. 2015
QIPA@HRI, 12 Dec. 2015
In the interferometer frame: Mass current with curl  Gravitomagnetic field
QIPA@HRI, 12 Dec. 2015
Atom Interferometer Gyroscope and Cosmic Gravity
QIPA@HRI, 12 Dec. 2015
Current-current interaction (Ampere) in gravity
Fg (i1,i2 ) =
4G
1
c2r
c2r
(mv)
(mv)
«
F
=
1
2
em
2
QIPA@HRI, 12 Dec. 2015
(ev)
(ev)
1
2
2
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.)
QIPA@HRI, 12 Dec. 2015
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
QIPA@HRI, 12 Dec. 2015
An ‘Ampere Experiment’ in Gravity
QIPA@HRI, 12 Dec. 2015
Unnikrishnan, General Theory of Relativity: The Universe between Beauty and Truth
(ResearchGate)
QIPA@HRI, 12 Dec. 2015
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
QIPA@HRI, 12 Dec. 2015
Spin-dependent geometric phase
A. Tomita and R. Y. Chiao, PRL 1986
dj = 2ps (1- cosq )
QIPA@HRI, 12 Dec. 2015
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!
QIPA@HRI, 12 Dec. 2015
Gravitational Vector potential as one moves around…
QIPA@HRI, 12 Dec. 2015
QIPA@HRI, 12 Dec. 2015
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!
QIPA@HRI, 12 Dec. 2015
The ‘flux’ associated with the electron cancels
part of the applied magnetic field and reduced
the degeneracy to the integer QHE level.
QIPA@HRI, 12 Dec. 2015
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
*
QIPA@HRI, 12 Dec. 2015
QIPA@HRI, 12 Dec. 2015
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!
QIPA@HRI, 12 Dec. 2015
QIPA@HRI, 12 Dec. 2015
1473-1543)
QIPA@HRI, 12 Dec. 2015
Nilakantha Somayaji 1444-1545
(Tantrasangraha)
QIPA@HRI, 12 Dec. 2015
Tycho Brahe (1546-1601)
Gravitational Vector potential as one moves around…
QIPA@HRI, 12 Dec. 2015
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!
QIPA@HRI, 12 Dec. 2015
We then need to establish the same coherent picture for the entire sequence
QIPA@HRI, 12 Dec. 2015
Several remaining issues (in all theories of FQHE)…
QIPA@HRI, 12 Dec. 2015
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 and the behaviour of photons and electrons in
helical transport.
The fractional quantum Hall effect has several features that indicate that
cosmic gravity is indeed the extra field required for its single particle
description. 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.
QIPA@HRI, 12 Dec. 2015
QIPA@HRI, 12 Dec. 2015
QIPA@HRI, 12 Dec. 2015
QIPA@HRI, 12 Dec. 2015