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Nobel Prize in Physics 2016
Flatland and Topology
Diptiman Sen
Centre for High Energy Physics
Indian Institute of Science
E-mail: [email protected]
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Nobel Laureates
Nobel Prize for "theoretical discoveries of topological phasetransitions and topological phases of matter"
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FlatlandEdwin A. Abbott, 1884
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Peculiarities of low dimensions
Lineland:
In one dimension, particles can only move to the left or to the right,and A will always be on the left of B if crossings are not allowed
A B
Flatland:
Given a closed curve in two dimensions,any point lies either inside it or outside it,and this remains true for all time ifcrossings are not allowed
A B
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What is topology ?Topology is a branch of mathematics where we study those propertiesof a system which remain the same if small changes are made
Example: the genus of a connected, orientable surface is the maximum numberof cuts along non-intersecting closed curves that can be made without makingit fall apart into disconnected pieces
Small deformations of a surface do not change its genus
https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll2.html
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Genus in daily life
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Closed curve in two dimensions
A number which remains the same under small changes is called a topological invariant
Example: the number of times a closed curve in a plane winds around the originin the anticlockwise direction. This integer is called the winding number
http://usf.usfca.edu/vca/PDF/vca-winding.pdf
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Robustness of topological invariants
Since a topological invariant is an integer, it cannot change slowly if some small changesare made in the system
The only way for a topological invariant to change is to become ill-defined at some point
For example, if a closed curve in a plane is gradually deformed, its winding number canonly change if the curve goes through the origin at some time. Exactly at that time, thewinding number is ill-defined
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Closed surface in three dimensions
Another example of a topological invariant is the number of times the surface of one sphere(red) wraps around the surface of another sphere (blue) in three dimensions
http://www3.nd.edu/ mbehren1/presentations/spheres.pdf
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Phase transitions
A phase transition is a qualitative change in the properties of a physical system.It may occur due to a change in temperature (ferromagnetic to paramagneticor solid to liquid) or pressure or the application of a field (superconducting tonormal metal as a magnetic field is applied)
Phase transitions can be first order or continuous, depending on whether thechange is ‘sudden’ (melting of a solid to a liquid where the density changessuddenly) or ‘continuous’ (a ferromagnet changing to a paramagnet wherethe magnetization changes continuously from a non-zero value to zero)
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Continuous phase transitions
A large class of continuous phase transitions involve the breaking of a symmetry
Example: For a magnetic system, there is a critical temperature called Tc
below which the magnetization is non-zero. The magnetization is a vectorand can point in any direction in space in general; the free energy of themagnet is the same for all directions of the vector
However, a given magnet chooses a particular direction for its magnetization.This is called spontaneous breaking of symmetry
(We are talking about a continuous symmetry here. There can also besystems with a discrete symmetry, where the free energy is minimum andequal for two directions of the magnetization which are opposite to each other;the Ising model is an example)
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Order parameter
For a continuous phase transition corresponding to the breaking of a symmetry,we can define an order parameter (such as the magnetization) so that itsvalue is non-zero below Tc and zero above Tc
temperature
orde
r p
aram
eter
orderedphase
disorderedphase
Tc
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Disordered and ordered phases
Disordered phase above Tc Ordered phase below Tc
https://en.wikipedia.org/wiki/Curie_temperature
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Critical exponents
There are various critical exponents associated with a continuous phase transitionin which a symmetry is broken
The order parameter m goes to zero as we approach the critical temperature Tc
from below
m(T ) ∼ (Tc − T )β
There is a correlation length ξ defined through the two-point correlation function of m
〈 m(~r1) m(~r2) 〉 ∼ e−|~r1−~r2|/ξ for T > Tc
〈 m(~r1) m(~r2) 〉 − m2(T ) ∼ e−|~r1−~r2|/ξ for T < Tc
The correlation length diverges as we approach the critical temperature from either side
ξ ∼ |Tc − T |−ν
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Phase transition in two dimensions?
It was shown by Mermin and Wagner (1966) that a continuous symmetry cannot bespontaneously broken at finite temperature for two-dimensional systems
The argument is that at any finite temperature, there will be excitations with sufficientlylow energies that they will be excited in large enough numbers to disorder the system
In a magnetic system, these excitations are spin waves in which the direction ofeach spin is slightly different from that of its neighbors
This suggests that there cannot be a phase transition at finite temperature intwo-dimensional systems with a continuous symmetry
The work of Thouless and Kosterlitz (and Berezinskii) in 1971-74showed that this is not true if the order parameter is two-dimensional
There is a transition between two phases which are both disordered
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Planar spins in a planar system
Consider a ferromagnetic system of spins in two dimensions where the spin vectorsare forced to lie in a plane for some reason
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Planar spins · · ·
There are spin wave excitations which can disorder the system. Hence there isno ordered phase at any finite temperature in agreement with Mermin and Wagner;the magnetization is always zero
However this system has another kind of excitation which is topological in nature
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Vortex and antivortex
Vortices with winding numbers + 1 and − 1 (called an antivortex)
http://scitation.aip.org/content/aip/magazine/physicstoday/article/69/12/10.1063/PT.3.3381
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Disorder due to a vortex
It turns out that a single vortex disorders the spins much more than a spin wave excitation.This is because a vortex changes the spins everywhere in space and by large amounts
https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/advanced.html
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Energy of a vortex
However a single vortex costs a large energy. In a large system of dimensionsL × L, the energy of a vortex is of order ln L
The energy of a planar system of spins is given by
E = − JX
〈ij〉
cos(θi − θj)
where the sum 〈ij〉 is over nearest-neighbor spins, and θi is the angle madeby the i-th spin with respect to the x axis
At a large distance R from the core of a vortex, the relative angle betweentwo spins displaced in the angular direction must be of order 1/R.
Hence cos(θi − θj) ∼ 1/R2 and the total energy of a vortex is
E ∼ J
Z Z
d2~r1
r2∼ πJ
Z L
ardr
1
r2∼ πJ ln
„
L
a
«
where a is a short distance cut-off like the lattice spacing
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Vortex - antivortex pair
While a single vortex costs an energy which grows logarithmically with the systemsize, a vortex - antivortex pair only costs a finite energy and does not changethe spins everywhere
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Entropy of a vortex
Since a vortex - antivortex pair only costs a finite energy, such pairs can appearin the system at low temperatures
At high temperatures, single vortices can also appear. This is becausethere is an entropy S associated with a vortex
In a system with dimensions L × L, a vortex can be centred aroundany one of W ∼ (L/a)2 sites
The entropy of a vortex is therefore of order
S = kB ln W ∼ kB ln
„
L
a
«2
∼ 2kB ln
„
L
a
«
where kB is called the Boltzmann constant
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Entropy
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Effect of temperature
Hence there is a competition between the energy and the entropy of a vortex
The free energy of a single vortex is
F = E − TS ∼ πJ ln
„
L
a
«
− 2kB T ln
„
L
a
«
If the temperature T is smaller than πJ/(2kB), the free energy is minimum ifsingle vortices are absent (but there can be vortex - antivortex pairs)
If the temperature T is larger than πJ/(2kB), the free energy is minimum iflots of single vortices are present
So there is a critical temperature TKT = πJ/(2kB), such that the system is‘more’ disordered above TKT than below TKT
(In real systems, kBTKT /J differs from π/2 for various reasons)
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Two disordered phasesAt all temperatures, the system is disordered and the magnetization m = 〈~Si〉 is zero
However, the amount of disorder is less below TKT than above TKT
Below TKT , the two-point correlation function goes to zero only as a power
〈 ~S(~r1) · ~S(~r2) 〉 ∼ 1
|~r1 − ~r2|η
where η depends on the temperature: η = kBT/(2πJ).
At T = TKT , η = 1/4
Above TKT , the correlation decays exponentially
〈 ~S(~r1) · ~S(~r2) 〉 ∼ e−|~r1−~r2|/ξ
where ξ depends on the temperature: ξ ∼ exp (a/√
T − TKT )
Thus ξ diverges much faster as T approaches TKT compared tothe usual continuous phase transition where ξ ∼ |T − Tc|−ν
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Is the BKT transition continuous?
There is a phase transition at the temperature TKT between two disordered phases
Should we call this transition continuous or not?
As the temperature approaches TKT from above, the divergence of ξ is so rapidand appears over such a small temperature range that the singularity of thespecific heat is unobservable experimentally. So the transition seems to bemore continuous than the usual continuous phase transition
On the other hand, the long-distance correlation changes suddenly from a power-lawdecay given by η = 1/4 to an exponential decay when we cross TKT
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BKT transition · · ·
Another quantity that changes suddenly across TKT is the spin stiffness
If we impose a small twist on all the spins given by θ(~r) = θ(~0) + ~k · ~r,
this costs a free energy
F (~k) = F (~0) +1
2A ρ ~k2
where A is the area of the systems and ρ is called the spin stiffness
It turns out that ρ is non-zero below TKT and zero above TKT . When thetemperature crosses TKT , there is a universal jump in ρ given by (2/π)kBTKT
Chaikin and Lubensky, Principles of Condensed Matter Physics
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Experimental observation
The BKT transition occurs in any two-dimensional system where the order parameteris two-dimensional
Examples: superfluids (where the order parameter is the wave function of the atomsforming the Bose-Einstein condensate) and superconductors (where the orderparameter is the wave function of the Cooper pairs of electrons)
The magnitude of the wave function cannot vary much since that costs a lot of energy,but the phase θ of the wave function can vary more easily; it plays the same roleas the orientation of the planar spins in our earlier model
A vortex in a superfluid is a region where the superfluid rotates around some point
The equivalent of the ‘spin stiffness’ in a superfluid is ρ = (~2/m2) ρs whereρs is the superfluid density and m is the mass of the atoms forming the superfluid(4He in the experiments)
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Experimental observation · · ·
When the temperature is changed across TKT , the superfluid density ρs
should jump by (2/π)(m2/~2)kBTKT in all samples, even though thevalue of TKT can be quite different in different samples
Bishop and Reppy, Phys. Rev. Lett. 40, 1727 (1978)
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Quantum Hall effect
Thouless and collaborators gave a topological understanding of the quantizationof the Hall conductance of a quantum Hall system
An interface between two semiconductors GaAs and GaAlAs traps electronsin a two-dimensional layer. In the presence of a strong magnetic field and atvery low temperatures, the conductances show some striking features
If an electric field Ex is applied in the x direction, there can be currentdensities in both the x and y directions, called Jx and Jy
σxx = Jx/Ex and σyx = Jy/Ex are called the longitudinal and Hallconductances respectively
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Quantum Hall effect · · ·
As a function of the magnetic field, σxy = −σyx shows a series of plateaus atvalues equal to an integer times e2/h (integer quantum Hall effect, 1980) ora fraction times e2/h (fractional quantum Hall effect, 1982)
σxx vanishes wherever σxy shows plateaus
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Quantum Hall effect · · ·
The plateaus in the Hall conductance are sample independent and flat to aboutone part in 109 at low temperatures (about 100 mK)
This is because the Hall current is carried entirely by modes near the edgesof the system; on each edge, these modes move in only one direction
So even if there is an impurity at the edge, the modes do not get reflected backbut continue to move in the same direction
Thus a small number of impurities do not change the Hall conductance at all !p.32/50
TKNN invariant
There is a topological way of understanding the quantization of σxy
Brief explanation: consider an electron in a uniform magnetic field and a potentialwhich is periodic in the x and y directions with periods a and b respectively
The wave functions can be labelled by the Bloch momenta kx and ky whichlie in the ranges [−π/a, π/a] and [−π/b, π/b] respectively
Thouless and others showed that σxy is given by (e2/h)(1/2π) timesa line integral around the unit cell in k-space which is related to the changein the phase of the wave function. This must be an integer multiple of 2π
for the wave function to be single-valued. Hence σxy must be an integermultiple of e2/h
Thouless, Kohmoto, Nightingale and den Nijs, Phys. Rev. Lett. 49, 405 (1982)
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From flatland to lineland
Consider spins located at the sites of a lattice in one dimension. The spin valuesdepend on the system; for example, Cu2+ has spin-1/2 while Ni2+ has spin-1
Suppose that spins on neighboring sites have Heisenberg interactions of theantiferromagnetic form. The Hamiltonian is
H = JX
n
~Sn · ~Sn+1
where J is positive. Classically, the ground state will have neighboring spinspointing in opposite directions
↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓
Quantum mechanically, the situation is not so simple; the state shown aboveis not an eigenstate of H
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Spin wave theoryFor a spin-1/2 antiferromagnetic chain, the energy spectrum was found exactly byBethe (1931). The ground state has total spin zero and the excitations are gapless
Spin wave spectrum in the absence of a magnetic field
For the case of a large spin S ≫ 1, a spin wave theory was developedby Anderson (1952). This again showed that the ground state of anantiferromagnetic chain has total spin zero and the excitations are gapless
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What about other spin values?
Since antiferromagnetic spin chains with both spin-1/2 and large spin havegapless excitations, it was natural to think that chains with any spin will havegapless excitations
Haldane showed that this is not true in 1981-83, using techniques fromrelativistic quantum field theory
Very few people believed Haldane’s result and his original paper was rejectedin 1981. He posted it last week on the arXiv: https://arxiv.org/abs/1612.00076
A different version of Haldane’s paper was published in 1983
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Haldane’s theoryHaldane showed that antiferromagnetic spin chains with integer spins (S = 1, 2, · · · )have gapped excitations while chains with half-odd-integer spins (S = 1/2, 3/2, · · · )have gapless excitations
Numerical calculations confirmed this later
Spin wave spectrum for a spin-1 chain
The spectrum is shown in units of ∆ ∼ 0.41Jp.37/50
Haldane’s theory · · ·
The theory begins with the order parameter of an antiferromagnet. Unlikea ferromagnet where the order parameter is directly the magnetization,the order parameter of an antiferromagnet is the staggered magnetization
~φ = (−1)n~Sn
S
where S is the value of the spin. ~φ is defined in such a way that it is a unit vector.In a quantum field theory, the properties of a system are governed by an integralover all possible configurations of the order parameter ~φ(x, t)
Z =
Z
D~φ(x, t) eiA
where A is called the action. In terms of ~φ(x, t), the action is usually given by
A =
Z Z
dt dx
2
4
1
2vg
∂~φ
∂t
!2
− v
2g
∂~φ
∂x
!23
5
where v is the velocity of spin waves and g is the strength of the interactionsbetween the spin waves p.38/50
Topological termApart from the usual
A =
Z Z
dt dx
2
4
1
2vg
∂~φ
∂t
!2
− v
2g
∂~φ
∂x
!23
5
Haldane discovered that there is another term in the action given by
Atop =θ
4π
Z Z
dt dx ~φ · ∂~φ
∂t× ∂~φ
∂x
where θ is equal to 0 if the spin is an integer and π if the spin is a half-odd-integer.It turns out that Atop has a topological significance. The space-time (x, t) can,by stereographic projection, be mapped to the surface of a sphere. Also, ~φ is aunit vector and can be directly mapped to the surface of a sphere. The quantity
1
4π
Z Z
dt dx ~φ · ∂~φ
∂t× ∂~φ
∂x
is the wrapping number of the sphere of space-time to the sphere of ~φ and istherefore always an integer
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Topological term · · ·
Picture of a configuration ~φ(x, t) with wrapping number 1, with ~φ pointingin the − z direction at the origin of space-time, (x, t) = (0, 0), and in the+z direction at infinity
Due to the topological term, configurations with wrapping number W contributeto the integral Z with an amplitude eiθW
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Integer versus half-odd-integer spin
The topological term gives rise to an amplitude eiθW for configurations with wrappingnumber W
For half-odd-integer spin, θ = π and configurations of ~φ(x, t) with even wrapping numbercontribute with amplitude + 1 while configurations with odd wrapping number contributewith amplitude − 1. Due to this destructive interference, the excitations are gapless
(Think of a double well potential in quantum mechanics. If the total tunneling betweenthe two ground states was zero due to some destructive interference, the ground stateand first excited state would be degenerate)
For integer spin, θ = 0 so the topological term plays no role. All configurations of~φ(x, t) contribute with the same amplitude and the excitations are gapped
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Experimental observationHaldane: “There’s nothing like experimental confirmation to quiet the critics”
Neutron scattering experiments on the spin-1 chain CsNiCl3 directly showthe existence of a gap
Kenzelmann et al, Phys. Rev. B 66, 024407 (2002)
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Experimental observation
If the excitations have a gap ∆, the magnetic susceptibility will rapidly go to zeroat low temperatures as χ ∼ e−∆/kBT , rather than as a power of T.
This is found to be case in the spin-1 chain [Ni(C2H8N2)2NO2](BF4)
Cizmar et al, New J. Phys. 10, 033008 (2008)
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The usefulness of guesswork
We can often get a good understanding of the ground state of a quantum mechanicalsystem by clever ‘guessing’. We make a guess about the form of the ground statewave function and use that to calculate the energy
Sometimes the guessed wave function depends on some parameters which wevary to get the minimum possible energy — called a ‘variational wave function’.This can give extremely accurate estimates of the ground state energy dependingon the number of parameters used
At other times, the guessed wave function has no variable parameters — calleda ‘trial wave function’. This also can give quite accurate estimates of the groundstate energy
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AKLT state
For a spin-1 antiferromagnetic chain, a useful trial wave function is obtained bythinking of each spin-1 as being made out of two spin-1/2 objects (two black dots).Then we can pair up spin-1/2’s on nearest-neighbor sites to form a valence bond
This picture suggests that an open chain will have free spin-1/2’s at the ends eventhough the system only has spin-1 at each site. The spin-1/2’s at the ends can bedetected by their contribution to the magnetic susceptibility at low temperatures.The other spins do not contribute to the susceptibility as they all form valence bondswhich have total spin zero
Affleck, Kennedy, Lieb and Tasaki, Comm. Math. Phys. 115, 477 (1988)
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Last words (almost)
Thouless, Kosterlitz and Haldane introduced new ways of thinking about old problems
By doing so they overturned some conventional wisdoms
(i) phase transitions are not possible at finite temperature in two-dimensional systemswith a continuous symmetry
(ii) Heisenberg antiferromagnetic chains have gapless excitations for any spin
Kosterlitz says that his complete ignorance was an advantage !
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Modern developmentsThe work of the Nobel laureates has inspired a huge amount of research on thetopological aspects of condensed matter systems
Some systems have topological and non-topological phases which are separatedby continuous phase transitions. A topological phase has the following properties
• The bulk of the system is gapped, namely, there is a finite energy gap between theground state and the excited states. Hence the bulk is an insulator at low temperatures
• The band structure of the bulk of the system is characterized by a topologicalinvariant which is a non-zero integer
• There are gapless states at the boundaries of the system; these contribute toelectronic transport
• Bulk-boundary correspondence: The number of boundary states is equal to thetopological invariant; it does not change if the parameters in the Hamiltonian arechanged a bit or if a small amount of disorder is present
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Topological phases of matter
A system in a topological phase has no local order parameter,unlike magnets, superfluids and superconductors
It is impossible to examine a small part of the bulk of an insulator anddiscover whether it is topological or not
The topological invariant is a globally defined quantity (it requires a knowledgeof the band structure for all momenta), and we cannot calculate it withoutknowing about the entire system
As a parameter in the Hamiltonian is changed, a topological insulator can turn intoa non-topological insulator. The transition is continuous. Exactly at the transition,the bulk of the system is gapless and is therefore not an insulator.The topological invariant is ill-defined at that point
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Examples of topological phases
1. Quantum Hall systems: two-dimensional systems of electrons in the presence ofa strong magnetic field. Only the edge states contribute to the Hall conductance σxy
and this is quantized to be an integer or fraction times e2/h. For the integer quantumHall effect, σxy is given by the TKNN invariant
2. Two-dimensional topological insulators: these have states at the edges whichcontribute to charge and spin transport. The number of edge states is given by atopological invariant which is a wrapping number (also called Chern number)
3. Three-dimensional topological insulators: these have states at the surfaces.The number of surface states is either even or odd and is given by atopological invariant which can only take two values, 0 or 1
4. p-wave superconducting wire: there are zero energy states at the ends of along system whose number is a topological invariant which is a winding number.These states behave like ‘half an electron’ and are called Majorana fermions
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References
Nobel Prize and Physics Today
Popular information:
https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/popular.html
http://scitation.aip.org/content/aip/magazine/physicstoday/article/69/12/10.1063/PT.3.3381
Advanced information:
https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/advanced.html
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