Master class with Charles Kane
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Formal Metadata
Title 
Master class with Charles Kane

Subtitle 
Topological band theory of insulators and superconductors

Title of Series  
Author 

License 
CC Attribution  NonCommercial  NoDerivatives 3.0 Germany:
You are free to use, copy, distribute and transmit the work or content in unchanged form for any legal and noncommercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
Identifiers 

Publisher 
CityTV.nl, Foundation for Fundamental Research on Matter (FOM)

Release Date 
2012

Language 
English

Producer 

Content Metadata
Subject Area  
Abstract 
We will give a pedagogical introduction to the topological classification of insulators and superconductors with an emphasis on the correspondence between bulk topological invariants and protected gapless boundary modes. We will begin with a discussion of band topology in one dimension followed by the theory of the integer quantum Hall effect, time reversal invariant Z2 topological insulators in two and three dimensions and topological superconductors. We will discuss the general classification of gapped free fermion Hamiltonians, along with its generalization to include topological defects. We will conclude with a discussion of Majorana fermion states at the boundaries of topological superconductors and in superconductor  topological insulator heterostructure devices.

Keywords 
insulators
superconductors

00:00
Effects unit
Order and disorder (physics)
Electronic band structure
Cyclotron
Energy level
Hall effect
Group delay and phase delay
Band gap
Dumpy level
Excited state
Videotape
Quantum Hall effect
Insulator (electricity)
Superconductivity
Ship breaking
Bandstruktur
Stock (firearms)
Superconductivity
Electronics
Lead
Light
Excited state
Potenzialausgleich
Fermion
Plane (tool)
Insulator (electricity)
Caliber
Band gap
Quantum
Superconductivity
02:40
Quality (business)
Bandstruktur
Electronic component
Crystal structure
Energy level
Crystallization
Hall effect
Band gap
Direct current
Potenzialausgleich
Excited state
Insulator (electricity)
Insulator (electricity)
Quantum
Homogeneous isotropic turbulence
Analemma
Magnetism
Quantum
04:45
Ruler
Gang
Ballpoint pen
Measuring cup
Knife
Geokorona
Gas
Particle physics
Northrop Grumman B2 Spirit
Source (album)
Black hole
GALS
Direct current
Solid
Gemstone
Glasfluss
Standard cell
Book design
Measurement
Pager
Book design
Molding (process)
Diamond
Hood (vehicle)
07:06
Prozessleittechnik
Willo'thewisp
Visible spectrum
Electronic band structure
Crystal structure
Order and disorder (physics)
Band gap
Inertial navigation system
Cylinder block
Magnetic moment
Typesetting
Bandstruktur
Valence band
Crystal structure
Map
Solid
Crystallization
Excited state
Direct current
Series and parallel circuits
Satellite
Brillouin zone
Miner
Band gap
Adiabatic process
09:26
Matrix (printing)
Bomb
Schubvektorsteuerung
Day
Hull (watercraft)
Ink
Order and disorder (physics)
Summer (George Winston album)
Minute
Multiplizität
Stromschiene
Angle of attack
Phase (matter)
Separation process
Apparent magnitude
Turning
Microwave
Mechanical watch
Dumpy level
Ship class
Electrical conductor
Spare part
Steckverbinder
Transformer
Membrane potential
Array data structure
Typesetting
Combined cycle
Ruler
Magnetspule
Source (album)
Specific weight
Excited state
Spin (physics)
Direct current
Phase (matter)
Cartridge (firearms)
Magnetic resonance imaging
Membrane potential
Rolling (metalworking)
Quantum
Model building
14:17
Weapon
Brillouin zone
Effects unit
Geokorona
Interstellar medium
Order and disorder (physics)
AMHerculisStern
Phase (matter)
Transformer
Polarization (waves)
Quantum
Charge density
Mechanic
Gauge block
Electronics
Monday
Density
Phase (matter)
Cartridge (firearms)
Halbwellendipol
Measurement
Brillouin zone
Electric charge
Barricade
Hot working
Schubvektorsteuerung
Ink
Year
Electric arc
Electronic band structure
Bulk modulus
Elektrisches Dipolmoment
Dipole
Game
Potentiometer
Alcohol proof
Steckverbinder
Orbital period
Electric charge
Hose coupling
Pressure vessel
Bandstruktur
Ballpoint pen
Magnetspule
Angle of attack
Spin (physics)
Electricity
Color charge
Quantum
Magnetic moment
19:50
Ruler
Stock (firearms)
Transmission line
Bomb
Order and disorder (physics)
Electronics
Foot (unit)
Relative articulation
AMHerculisStern
Phase (matter)
Game
Transformer
Electric charge
22:11
Excited state
Phase (matter)
Hot working
Schubvektorsteuerung
Radioactive decay
Transformer
Order and disorder (physics)
Orbit
AMHerculisStern
Model building
FACTS (newspaper)
Orbital period
25:19
Matrix (printing)
Fur clothing
Order and disorder (physics)
Magnet
AMHerculisStern
Gedeckter Güterwagen
Phase (matter)
Bird vocalization
Pattern (sewing)
Constraint (mathematics)
Pager
Model building
Mat
Field emission microscopy
Ultra
Camshaft
Electronics
Crystal structure
Leyden jar
Ammunition
Excited state
Laserfusion
Direct current
Phase (matter)
Cartridge (firearms)
Plane (tool)
Hot working
Reamer
Bomb
Schubvektorsteuerung
Optical amplifier
Electronic band structure
Scissors
Basis (linear algebra)
Band gap
Apochromat
Ship class
Penumbra: Overture
Bandstruktur
Sigma
Hue
Couch
Sundial
Angle of attack
Game
Spin (physics)
Potenzialausgleich
Band gap
Model building
32:18
Matrix (printing)
Continental drift
Year
Order and disorder (physics)
Crystal structure
Energy level
Domäne <Kristallographie>
Phase (matter)
Bird vocalization
Pulp (paper)
Band gap
Roll forming
Excited state
Mass
Mode of transport
Hose coupling
home office
Mail (armour)
Measuring cup
Car seat
File (tool)
Domäne <Kristallographie>
FACTS (newspaper)
Excited state
Interface (chemistry)
Chiralität <Elementarteilchenphysik>
Gemstone
Continuum mechanics
Fahrgeschwindigkeit
Lowpressure area
Mass
Mode of transport
Model building
Gun
38:28
Brillouin zone
Effects unit
Transmission line
Hail
AMHerculisStern
Energy level
Hall effect
Cylinder (geometry)
Phase (matter)
Separation process
Orbital period
Steckverbinder
Month
Polarization (waves)
Quantum Hall effect
Water vapor
Magnetism
Quantum
Bandstruktur
Hohlzylinder
Gas
Electronics
Crystal structure
Excited state
Melting
Wood
Cartridge (firearms)
Membrane potential
Morning
Band gap
Magnetism
Homogeneous isotropic turbulence
43:15
Bandstruktur
Hohlzylinder
Hail
Order and disorder (physics)
Energy level
Hall effect
Cylinder (geometry)
Focus (optics)
Variable star
Forging
Orbital period
Membrane potential
Measurement
Polarization (waves)
Trade wind
Magnetism
Band gap
Superconductivity
Homogeneous isotropic turbulence
Magnetic moment
45:58
Weapon
Matrix (printing)
Inversionsschicht
Punt (boat)
Submarine
Effects unit
Schubvektorsteuerung
Ink
Year
Crystal structure
Order and disorder (physics)
Minute
Night
Variable star
Band gap
Alcohol proof
Vortex
Hose coupling
Diamond
Model building
Typesetting
Bandstruktur
Sensor
Inversionsschicht
Electronic component
FACTS (newspaper)
Finger protocol
Forging
Direct current
Phase (matter)
Cartridge (firearms)
Vortex
Electronic media
Rotation
Plane (tool)
Measurement
Synthesizer
Band gap
Model building
RWE Dea AG
Last
50:45
Matrix (printing)
Submarine
Geokorona
Ink
Year
Hall effect
Roll forming
Band gap
Cardinal direction
Excited state
Mining
Quantum Hall effect
Model building
Ship breaking
Sensor
Ballpoint pen
Ground station
Electronics
Dipole
Map
Interface (chemistry)
Excited state
Angle of attack
Phase (matter)
Exosphere
Cartridge (firearms)
Mass
Quantum
Model building
54:26
Brillouin zone
Mental disorder
Thursday
Stromschiene
Gunn diode
Turning
Fermion
Stardust (novel)
Hour
Perturbation theory
Quantum
Mat
Coating
Electronics
Weapon
Excited state
Interface (chemistry)
Direct current
Phase (matter)
Swimming (sport)
Mass
Book design
Switch
Engine control unit
Flight information region
Quantum Hall effect
Summer (George Winston album)
Vakuumphysik
Microphone
Map
Cougar
Saw
Band gap
Excited state
European RemoteSensing Satellite
Mining
Extravehicular activity
Burr (edge)
Ramjet
Bandstruktur
Ballpoint pen
Sonic boom
Cogeneration
Forging
Electric power transmission
Mars
Halo (optical phenomenon)
Band gap
Model building
59:37
Ship breaking
Regentropfen
Bandstruktur
Refractive index
Bulk modulus
Energy level
Excited state
Conduction band
Striptease
Band gap
Forging
Automobile
Orbital period
Mode of transport
Book design
Valence band
1:02:14
Inversionsschicht
Order and disorder (physics)
Bulk modulus
Lead
Vakuumphysik
Watercraft
Electronics
Hall effect
Spin (physics)
Excited state
Orbital period
Spare part
Electrical conductor
Quantum Hall effect
Measurement
Quantum
Interplanetary magnetic field
Ship breaking
Buick Century
Effects unit
Tin can
Electronics
Gas
Flight
Excited state
Forging
Electrical conductor
Thrust reversal
Electric current
Insulator (electricity)
Band gap
Quantum
Model building
1:04:55
Scanning acoustic microscope
Roll forming
Excited state
Electrical conductor
Grey
Rail transport operations
Antiparticle
Artillery battery
Superconductivity
Suspension (vehicle)
Counter (furniture)
Bulk modulus
Backscatter
Motor glider
Singing
Ammunition
Excited state
Fiat Panda
Motion capture
Forging
Perturbation theory
Hochseeschlepper
Lawn mower
Cartridge (firearms)
Ionization
1:07:40
Speckle imaging
Rearview mirror
Year
Bulk modulus
Cable
Excited state
Forging
Band gap
Direct current
Ship class
Cartridge (firearms)
Hour
Insulator (electricity)
Book design
Artillery battery
Station wagon
Constraint (mathematics)
1:10:23
Matrix (printing)
Flavour (particle physics)
Hot working
Schubvektorsteuerung
Bulk modulus
Fountain pen
FACTS (newspaper)
Excited state
Rootstype supercharger
Forging
Verdrillung <Elektrotechnik>
Cartridge (firearms)
Halo (optical phenomenon)
Electrical conductor
Orbital period
Reciprocal lattice
Flugbahn
Insulator (electricity)
Book design
Antiparticle
Rail transport operations
Propeller (marine)
Cylinder block
Constraint (mathematics)
1:14:07
Ship breaking
Duty cycle
Speckle imaging
Bandstruktur
Kilogram
Inversionsschicht
Year
Effects unit
Crystal structure
Hall effect
Cogeneration
Excited state
Spin (physics)
Turning
Spin (physics)
Ship class
Gun
Gamma ray
Wind farm
Quantum
Band gap
Flugbahn
1:16:20
Mental disorder
Ferry
Kilogram
Energy level
Metal
Stardust (novel)
Electrical conductor
Orbital period
June
Insulator (electricity)
Hose coupling
Artillery battery
Ballpoint pen
Enclosure (electrical)
Ice
Source (album)
Cogeneration
Excited state
Direct current
Forging
Cartridge (firearms)
Bow (ship)
Insulator (electricity)
Caliber
Book design
Miner
Week
1:19:55
Inversionsschicht
Bandstruktur
Combined cycle
Transmission line
Refractive index
Schubvektorsteuerung
Multiplizität
Crystallization
Excited state
Roll forming
Direct current
Gedeckter Güterwagen
Cartridge (firearms)
Plane (tool)
Flugbahn
Book design
Brillouin zone
Insulator (electricity)
Week
Array data structure
1:22:27
Charge density
Mental disorder
Effects unit
Elevator
Thin film
Hall effect
Particle
Phase (matter)
Zweidimensionales Elektronengas
Fermion
Continuous track
Schwache Lokalisation
Artillery battery
Quantum
Superconductivity
Ship breaking
Car tuning
Hall effect
Electronics
FACTS (newspaper)
Focus (optics)
Excited state
Interface (chemistry)
Direct current
Phase (matter)
Series and parallel circuits
Cartridge (firearms)
Electric charge
Surface energy
Quantum Hall effect
Local Interconnect Network
Scanning acoustic microscope
Anomaly (physics)
Thunderstorm
Energy level
Domäne <Kristallographie>
Fieldeffect transistor
Spin (physics)
Band gap
Automobile
Excited state
Steckverbinder
June
Insulator (electricity)
Quantum Hall effect
Electric current
Semifinished casting products
Bending (metalworking)
Effects unit
Superconductivity
Domäne <Kristallographie>
Snow
Magnet
Cogeneration
Chiralität <Elementarteilchenphysik>
Forging
Schwache Lokalisation
Fermion
Insulator (electricity)
Miner
Band gap
Quantum
Model building
1:28:32
Matrix (printing)
Austauschwechselwirkung
Transmission line
Schubvektorsteuerung
Superconductivity
Crystal structure
Scanning acoustic microscope
Particle
Band gap
Locher
Excited state
Spare part
Material
Magnetism
Superconductivity
Model building
Bandstruktur
Sunday
Minute
Effects unit
Particle
Superconductivity
Magnet
Crystallization
FACTS (newspaper)
Interferometry
Gemstone
Ground effect vehicle
Fermion
BCS theory
Maxwellsche Theorie
Klassifikationsgesellschaft
Insulator (electricity)
Universe
Book design
Quantum
Superconductivity
1:32:20
Bomb
Crystal structure
Global warming
Suitcase
Particle
Game
Locher
Constraint (mathematics)
Spare part
Gamma ray
Rail transport operations
Atmosphere of Earth
Fundamental frequency
Dolch
Combined cycle
Particle
Superconductivity
Quasiparticle
Excited state
Gemstone
Klassifikationsgesellschaft
Book design
Semiconductor
Model building
Superconductivity
1:36:01
Scale (map)
Sunrise
Summer (George Winston album)
Multiplexed Analogue Components
Particle
Band gap
Roll forming
Ship class
Wednesday
Excited state
Constraint (mathematics)
Spare part
Mode of transport
Hose coupling
Rail transport operations
Antiparticle
Electron energy loss spectroscopy
Bulk modulus
Mitsubishi A6M Zero
Particle
Yarn
Superconductivity
Quasiparticle
Source (album)
Cogeneration
Excited state
Cartridge (firearms)
Fermion
Mode of transport
Switch
Model building
Superconductivity
1:42:59
Matrix (printing)
Vehicle
Schubvektorsteuerung
Ink
Amplitude
Superconductivity
Printing press
Kette <Zugmittel>
Particle
Phase (matter)
Microwave
Jack (device)
Measuring instrument
Spare part
Constraint (mathematics)
Model building
Key (engineering)
Superconductivity
Power (physics)
Electronics
Chemical substance
Hovercraft
Hulk (ship)
Sizing
Phase (matter)
Cartridge (firearms)
Membrane potential
Radioactive decay
Model building
Microwave
1:46:03
Weapon
Combined cycle
Green politics
Kette <Zugmittel>
Phase (matter)
Excited state
Pattern (sewing)
Band gap
Fermion
Potenzialausgleich
Chain
Cartridge (firearms)
Fermion
Empennage
Gamma ray
Rail transport operations
Hose coupling
Model building
1:49:26
Effects unit
Geokorona
Schubvektorsteuerung
Superconductivity
Bulk modulus
Particle
Surfboard
Phase (matter)
Fieldeffect transistor
Continuum mechanics
Turning
Automobile
Ground effect vehicle
Excited state
Negation
Week
Measurement
Cylinder block
Model building
Lathe
Sensor
Protection
Green politics
Champ (cryptozoology)
Superconductivity
Cogeneration
Excited state
Forging
Phase (matter)
Cartridge (firearms)
Fermion
Vortex
Book design
Model building
Psyche (psychology)
Week
1:53:21
Hovercraft
Energy level
Particle
Phase (matter)
Bird vocalization
Band gap
Orbital period
Vortex
Mode of transport
Quantum
Magnetism
Water vapor
Hall effect
Superconductivity
Excited state
Black hole
Forging
Mask
Remote control
Vortex
Mode of transport
Universe
Jig (tool)
Bicycle
Analemma
1:56:06
Effects unit
Gravel pit
Hue
Electronics
Light
Hovercraft
Particle
Pattern (sewing)
Spin (physics)
Ship class
Spare part
Insulator (electricity)
Antiparticle
Superconductivity
Orbital period
1:58:59
Weapon
Crystal habit
Hue
Superconductivity
Answering machine
Particle
Excited state
Ship class
Cartridge (firearms)
Automobile
Spare part
Insulator (electricity)
Klassifikationsgesellschaft
Insulator (electricity)
Superconductivity
Orbital period
2:01:08
Ruler
Measurement
Order and disorder (physics)
Manipulator
FACTS (newspaper)
Excited state
Rail transport operations
Fermion
Bahnelement
Fermion
Excited state
Mode of transport
Hose coupling
Quantum
Coherence (signal processing)
Quantum information science
Quantum
Drill bit
Quantum computer
Engine control unit
2:03:20
Ruler
Quasiparticle
Heat exchanger
Cyclotron
Source (album)
Excited state
Rail transport operations
Fermion
Excited state
Mode of transport
Rail transport operations
Quantum
Coherence (signal processing)
Quantum information science
Quantum
Quantum computer
Superconductivity
2:05:35
Quality (business)
Superconductivity
ProximityEffekt
Hall effect
Particle
Resonance
Atom
Sunrise
Excited state
Near field communication
Insulator (electricity)
Membrane potential
Station wagon
Quantum
Magnetism
Wire
Semiconductor
Effects unit
Superconductivity
Single (music)
Quasiparticle
Kopfstütze
Excited state
Metal
Direct current
Mass
Plane (tool)
Semiconductor
Quantum
2:08:27
Ship breaking
Punt (boat)
Superconductivity
Effects unit
Superconductivity
Electronics
Gas
Desertion
Excited state
Sizing
Roll forming
Phase (matter)
Volumetric flow rate
Fermion
Excited state
Insulator (electricity)
Insulator (electricity)
Hose coupling
Ground (electricity)
Superconductivity
2:10:17
Transmission line
Ink
Year
Order and disorder (physics)
Summer (George Winston album)
Magnet
Domäne <Kristallographie>
Bird vocalization
Band gap
Society of Manufacturing Engineers
Interface (chemistry)
Excited state
Orbital period
Vortex
Insulator (electricity)
Superconductivity
Ship breaking
Effects unit
Superconductivity
Cogeneration
Excited state
Mode of transport
Insulator (electricity)
Caliber
Muon
Superconductivity
Microwave
2:12:44
Weapon
Order and disorder (physics)
Magnet
Watch
Quantum wire
Fermion
Pager
JosephsonKontakt
Semiconductor
Magnetism
Superconductivity
Field emission microscopy
Gang
Ferry
Mental disorder
Semiconductor
Crystal structure
Control valves
Excited state
Interface (chemistry)
Density
Phase (matter)
Mass
Mode of transport
Hot working
Bomb
Day
Superconductivity
Crystal structure
Thor
Map
Cougar
Band gap
Automobile
Excited state
Orbital period
Quantum Hall effect
Insulator (electricity)
Engine
Mode of transport
Hose coupling
Material
Monorail
Superconductivity
Domäne <Kristallographie>
Magnet
Chiralität <Elementarteilchenphysik>
Forging
Potenzialausgleich
Fermion
Halo (optical phenomenon)
Insulator (electricity)
Caliber
Semiconductor
Model building
Superconductivity
2:17:18
Effects unit
Order and disorder (physics)
Interferometry
Biasing
Magnet
Domäne <Kristallographie>
Phase (matter)
Fermion
Locher
Grey
Insulator (electricity)
Electron
Beam splitter
Magnetism
Weather
Superconductivity
Domäne <Kristallographie>
Magnet
Interferometry
Interface (chemistry)
Chiralität <Elementarteilchenphysik>
Phase (matter)
Fermion
Electric charge
Magnetism
2:19:27
Theory of relativity
Channeling
Ground state
Effects unit
Bending (metalworking)
Superconductivity
Order and disorder (physics)
Electronics
Audio frequency
Cogeneration
Flight
Black hole
Excited state
Phase (matter)
Potentiometer
Fermion
Cartridge (firearms)
Wear
Orbital period
Josephson effect
Electric current
JosephsonKontakt
Rail transport operations
Station wagon
2:21:48
Chiralität <Elementarteilchenphysik>
Phase (matter)
Effects unit
Analog signal
Fermion
Excited state
Insulator (electricity)
Automobile platform
Quantum
Crystal structure
Cosmic distance ladder
Interferometry
00:23
so I still undisclosed stock at address so I'm not on Charlie Cain and so it's a it's a real pleasure to be here so why so I have sort of 2 lectures over the first one will be before the break in the 2nd 1 will be after the break and hopefully I will see how it goes up so this is a small enough group that and you know you shouldn't be shy about stopping me in asking questions and so well that'll certainly make it more fun for everybody and if you work if you can do that can't and so am what I wanna do is I tell you about a topological insulators and topological superconductors and in a more general context topological banter and so what China start off with some sort of general things about topological and and there is no lead into the sea to topological like insulator and then after the break and we'll talk about a topological superconductivity said so just as a way to do in order to get started on I'd like to oppose you in the question of which is to think about the insulators and to think about the integer quantum Hall effect so the simplest kind of insulator that you can think of is just a collection of American adults OK and and electrons the sort of tightly bound to each other on and so it's sort of a trivial insulator and has a you know energy gap just because of it's because of the atomic energy levels of the band structure and so you can imagine on you know thinking of of the energy bands as a function of momentum but you know in the Bruins owner and come back and if it's an atomic Slater those bonds will be completely flat Hungarian and the occupied states the empty states will be suffered separated by an energy gap but but on the other hand the think about the injure quartermile effect which of course occurs when you have electrons combined to a twodimensional plane a perpendicular magnetic field around so they're the electrons go around in circles in the
02:41
magnetic field and quantum mechanics similarly quantized as those levels as well and so what you get on or you get it's OK and you have a situation where the lender levels on the occupied the empty when the
02:55
levels are separated from energy gap from going fun and so on I want to ask you a question well and what's the difference between these 2 OK on we know the different because in the quantum Allstate if you applied electric field then you get a whole current that flows perpendicular so so so so it's not an insulator that it sort of looks like an insulated because it has no energy gap and on you know it's so so 1 thing you might say is that well actually when you have a magnetic field on by you can't talk about the on the 2 components of the momentum anymore because the momentum and in the presence of a magnetic field I you know the X and Y components of the momentum don't with each other OK so that's true but so that you can define a unit cell even in the in the you know twodimensional plane and if that unitcell In closes 1 quantum of magnetic flux then the lattice translations in the X wide and directions duking it with each other and so it's just like having a crystal
04:05
you have blocked the mn have energy bonds and so so really this is a band structure just like this Hungarian so the question is what's the difference OK so that is where topology ,comma meets physics is the difference is a matter of topology and so on and so what's topology so far quality of it of course is a branch of mathematics when and on what apologists are interested in is the geometrical properties of of of of objects which are insensitive to smooth defamation so so for example surfaces into
04:46
Indian in 3 dimensions twodimensional surfaces in 3 dimensions on fire can be classified topologically
04:54
and so is sphere where the surface of a ball on their topologically the same because you can imagine switching squishing the on the sphere down and turning it into this likewise the doughnut the source of the doughnut on the surface of the coffee cup ,comma are topologically equivalent because you can imagine stretching and molding the I don't know what is in such a way that turned into the coffee cups of the hole in the doughnut becomes the handle of the coffee cup pumpkin but this standard the doughnut are different you can't continuously Turner spirit into a doughnut without you know taking a knife and poking a hole in a campaign that's against the rules OK so am I and and so these services can be classified topologically by the genus which is basically the number of holes I'm in and out so those fears you 0 lowers a diamond has has 1 now there's a beautiful theorem in mathematics what I call the gals beneath there and what the but Gasconade there and says that if you if you know the Gallician curvature which is basically the product of the 2 radius of recreate curvature of the 2 perpendicular directions then and then by integrating the gassing curvature over the surface on that gives you a quantized of value so for instance 1st fear that gas encourages just 1 over our squared Siew integrated and gives you for OK on any gives you a value which is quantized in units of 4 parts and that quantized value on tells you what the genus gets can't so so this is 1 of topology it's a very you know I have fun branch of mathematics to learn about and if you're interested in learning more of the mathematics is a good math book that sort of written for physicists on by not Hartley's This is the book where I learned most of the most of the topology that I know OK so this is what apologies a branch of mathematics and what we want to apply it to is the theory of solids and so
07:07
let me remind you on about the band theory of solids so so I'm in a periodic crystal 1 has lattice translations symmetry and so you can choose your on heightened states of the Hambletonian to be simultaneously Eigen states of the lattice translation and that defines for for you From Crystal momentum k and on the crystal momentum is like the ordinary momentum except for the fact that on its own and defined periodically because momentum miners and momentum plus apply both give the same translation Eikenberry look at it so the crystal momentum in 2 dimensions is defined not on an infinite plane rather it's final Torres cake is periodic in both directions program and so on now but but when you have on Hambletonian then you can you can think of the Hambletonian as being a function of this crystal momentum and you get a spectrum of ideas states aren't as a function of Crystal Moment of it's about 4 band structure it so it's a mapping if you
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will for on the crystal momentum to be our block Hamilton which 1 is a function of the moment can work well only a mapping from the crystal momentum to the set of energy I can values and the satellite and energy I came back now so that we're going to mean by and so so so so but in order to think about topology 1 has to have some notion of some of the defamation and so on and so on we can sort of imagine what the topological quit equivalence is going to be it it's basically going to be 1 can use smoothly change the Hambletonian in such a way that the energy gap stays .period it's the energy gap staying finite which is crucial because that allows you to define it AI in 88 batik process on high in which found in which the event structure changes OK so that's that that's that's going to be our notion of topological equivalents in the beauty of this series is that band structures fall into on a
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topological classes that we can understand the OK now in order to I I get so that you know or understand the meaning of the
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topological of classes that we're going to find on the severity of important concept upon which is that of the on the very day it's OK and so on and so the very faces something comes up whatever you have on the Hambletonian if you will that depends on some set of parameters so in the present case and then theory the Hambletonian depends continuously on the Crystal Moment Cafe and on and so the the reason that you get a very faces because for every value of case you have a Hambletonian you get a set of Icahnbacked the allied values and surviving vector but nobody tells you what the phase of the item conductors with the face of the item but should be OK so there is an inherent ambiguity and in the face you get the same state if you multiply on the state this the state vector by a phase and that this could be a function of K going now that but when you have a at on so it's sort of like doing that multiple
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interfaces like like doing gage transformation and when you have a gage transformation that is a function of a of a parameter then that introduces a fundamental quantity which is called the connection on which is like a vector potential OK see you know when you do that when you do a gage transformation electromagnetism and you change the face of the wave function and you change the electromagnetic and that's compensated by the change in the Electromagnetic Vector potential and so this I'm not the way this on so when you do this gage transformation on then the vector potential is modified on like this so it behaves exactly the same way that the on Electromagnetic Vector potential behaves when you do a spatially dependent upon a gage transformation and aren't so so physical quantities are going to be the quantities which are invariant with respect to these are gage transformations program but and so 1 important quantity which is invariant with phase transformation is basically the the change in the phase that is acquired when you go through a closed cycle on him in your in your program and on so in that case on the change in the face is basically integral part of the vector potential around that will and that is on gage invariant it's analogous to the 1 sort of you know of magnetic flux on in the electromagnetic case get and so this summer this on phases called the Barry's and on incidents at the Louvre integral part of the of actor potential 1 can write it and a manifestly gage invariant way by introducing the Barry curvature which is the curl of the back of the Patel of the vector potential which is analogous to the magnetic field OK and so on the on an integral part of the vector potential around this loop is going to be the surface and a roll of this son of this varies flocks now that throughout this talk and you may be on 5 talking about sort of specific models which are very useful for a having a concrete calculation you can do in order so you know you're doing look at and on this a very famous simple model for understanding the very phrase which is on just that of a twolevel Hamilton OK and if you have a twolevel Hambletonian you write your Hambletonian is a 2 by 2 matrix and on the side for an overall constant peas which isn't important but 2 by 2 matrix can be written as a on combination of Palin matrices on like this again so you understand that you know sources like a spin in a magnetic field OK so you understand what the on I states are going to do so on and in particular and that if fired on the the IDB energies are going to depend on the magnitude of this vector on but the idea that the the idea that depends on the on the direction and so if you imagine a bomb a a cycle where you take now this on this vector which
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characterizes my Hambletonian around in a closed loop OK or I can imagine I might think of a closed loop in momentum space if if you will then on then the Berry phase on has a very simple geometrical meaning OK it's basically just onehalf of the solid angle against swept out by this unit vector that and so a particular example of this is on and what I like to call my favorite minus sign and Kwan mechanics so you know when you take a spin and you wrote standby 360 degrees and you know that the wave function picks up a minus sign effects and so that minus sign is precisely the very place where in this case the solid angle is half of the sphere were to pot delivery phase of pot OK so and so this is the barricades that said so on the up the question that you should ask is well so we can define this Berry phase What is it me what is the physical meaning of the Berry phase in the context of a band OK and found so on this very on important indeed connection to physics here on which is through the through the on
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electric polarization OK so let's imagine let's imagine that we're in 1 dimension OK and so on and so of course you know you learn about a letter polarization in your 1st year electromagnetism costs and so the electric polarization is basically the dipole moment per unit length OK and on so if you have a on a bunch of a density of dipole of of course there's a charge at the end which is uncompensated so there's a plus charge q at the end of mine charged on fire at the at the other end and so on in that you know what that in charge is is related to the island to the to the polarization of course if there is a divergent said the bullet if the polarization varies with space and of course the than charge densities that evidence of peace now so what it's actually a nontrivial problem to formulate the theory of the electric polarization in there the reason it is nontrivial is because band theory you're you're always thinking of a system with periodic boundary conditions that and so on but you know what you think about band structure you put on you put on periodic boundary conditions you don't have this end charge OK so on what I would like to convince you of that is that in the context of a band theory the quantum polarization Bond is nothing other than on the other very phase of the arm of this very connection on going around the entire on loop that goes around the entire prolongs the remember 1 dimension prolonged zone is a circle uncle aunt and on and so there is a very phase defined going around that and so on and in order to convince you of this Italia Texel their work to prove it's OK but it's true that in order to allow I show you that it makes sense that on Monday give you a couple of arguments and so the 1st argument 1 is that both of the electric polarization which again is a property of the bowl system it's not a property the hands of the property of the ball both the polarization and the Berry phase are ambiguous OK and on and on in any way mean by that is that both of them are only defined up to an integer OK you can add an integer and nothing changes so in the case of the army Of the polarization on and you know if you think about the end charge so polarize the medium so I introduced articles and in end that builds up charge at the hand but I can always just add electron on yet OK and that that doesn't have anything to do with the ball and I add electron on the end and that's no different than than moving the on charges over on you know in such a way that there's there's an extra electronically and so so so sense you can always just add electron at the end without doing anything to the ball then this unelected polarization is illdefined modulo an integer OK so really on the only thing which makes physical sense is keep in mind OK so that's for the polarization the Berry phase has a similar ambiguity OK now I told him that the Berry phase is on is is invariance OK but there is a slight but I I of a problem with that which is that the on the other the loop where we go around the brawl one's own is a funny kind of because that loop is not the boundaries of an interior and for that reason I I I could do gage transformations on if I do a small gage transformation where I just changed fired by a little bit then of course
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on the very phrase is used on his doesn't change on that but if I do engage transformation where when I go around the loop the is far advances by 2 pine for multiple to then I II III so if if if found if it advances by 2 and then the Berry phase you know bomb increases by the time set OK so so basically on so so the Berry phase and the electric polarization both share the same intrinsic ambiguities and now I II III the polarization by itself is an illdefined up to an integer guide if I am if I start with 1 polarization and then I change something in the book so that I smoothly change the polarization the the change in the polarization is completely well before then you know and I you know as I do in the change I have a rule that I don't add any extra electrons in the Antoine on doing that from then on and so this change in the bowl and likewise the change in the polarization on feet is completely gage invariant because now on when I on I talk about the change in the polarization is going to be the integral of the very phase and on this on on this line minus the integral in this line OK and so now this is the boundary of interior of something on the interior and so on we can now use on stocks there and to rewrite this side that line integral as a surface in the role of the very curvature OK and so on so the polarization by itself is not welldefined up to an integer changes in the polarization are completely welldefined OK so and so this is 1 of the 1st sought a piece of evidence
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that these 2 quantities smell the same thing Grant now and that in order to
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show that they're the same whether sort of a kind of a flippant argument you can give which is to say Well you know the polarization is the expectation value of all are basically and the Berry phase is sort of like the expectation value of ID by decay and you know are as sort of like ID by can so maybe maybe you're happy with this argument on the you really shouldn't be on the problem with that is of course is that these are black states or extended states to go out to infinity so this expectation value of R is really not welldefined so you should be too he shouldn't that be too happy with this argument on a better version of this argument the hope is to define something which is localized OK and so that's something you can do so by constructing on 1 orbitals which are on basically and Fourier transforms of the some of the bloc states OK now on and so these on
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state survive for a transform the bloc states then a in a state associated with every lattice vector all are and it'll be a assuming things they're sort of defined smoothly and it will be you know it will be a localized and a stick now the downside of 1 states is that they are also ambiguous but it's not just the location that's ambiguous actually lead the way function itself is ambiguous the reason is because you on I can change you buy a K dependent gage transformations and if I do that will completely change with these 1 EU states are so you have to keep that in mind that there is nothing you know these won states our our on you know gage depend can't on but nonetheless you could define his life estates in there on on and they are you know they can be reasonably well localized and so you can use them then To compute that I you know the on the expectation value of art which is like the polarization OK and on on and so there's a little calculation you have to do you basically plugging this into here and I haven't done that for so there's actually a little bit of work you have to do in order to show that comes I'm a little bit little work to show this but but if the fact that this polarization and if you do this calculation gives you precisely on this very phase ,comma formula for the you can't alright so the polarization in 1 dimension is very fair any questions about them OK yeah all right so well at the I said that models or work so let me introduce
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the 1st model that is a model where we can see this polarization physics on work in 1 dimension again so this is the smallest call free model it's a famous model that was introduced as a model for on for Polly settling Polly's settling is a conducting polymer that is basically and you know you can think of is the backbone of carbon atoms and this 1 elect 1 electron for every young for every carbon atoms that and so what happens at Holly acetylene is that but you have a halffilled band on and on if the if the polymer Diana rises so that so that alternating bonanzas are short longshore long then on then that opens up an energy gap just like the nearly free electron model it opens up an energy gap at the Fermi energy and since all of the occupied states go down and energy on its energetically favorable to want to introduce this dimerization on and on so I have on my hands and so this side this instability is called the pirates instability so and so on but I interesting thing is that tho is that there are and I know there is a choice that you have to make when you diner as you dying arise with a pattern like there's going you know should short short long or a pattern like this going long short long short on and on and on and so these 2 0 1 different diarization patterns leads to some interesting physics so on so we can write down a simple are tight binding model from for this which is to write down on you know let's forget about spin for the moment on and and then write down a tight binding model which is just having hopping thing which is a little bit stronger on on these bonds and hopping which is little bit weaker on the Xbox game and so on and so what we could do that as we get right this then and in terms of some momentum OK so we 48 writers and for a space is a function of the the seek and then the Hambletonian will have a former on so so there to sublet a scissors and a sublet as an abuse of lettuce and so the Hambletonian will be at 2 by 2 matrix which will be a function of karaoke and so I've written down for for you what the on terms and that matrix will be OK so there will be a a Chernobyl flies Sigma accented turned the multiplies our Civil War In case of this this Hambletonian is the same as those Hambletonian and on so on it's actually interesting to think about what this would decide the vector does so there's 1 thing which is important about this Hambletonian is that there is an extra cemetery that this Hambletonian has on which it is that the 2 Adams on the basis of letters the basis of lettuce are equivalent to each of those things OK and so that means that this on this onedimensional lattice has an inversion symmetry so you can convert about a point at the midpoint of a bomb and that inversion symmetry tells you that is that it that you can have on a term multiplying Sigma's the customers they would be plus on 1 of the items minus on the other but says the equivalent you can't have that OK so so this deal actor due to symmetry is on fire constraints to lie in the plane then and because of the cemetery on there are topological classes to this band structure will on and particularly if you think about this the vector you can go buy you can classify it since it's in the plane you can ask how many times this is the vector wraparound hung around the origin OK and on these are topologically to stay because in order to get from this to this you have to have it go through 0 0 and if it goes to 0 when the D actor is equal to 0 that means the energy gap has gone to 0 OK so these are topologically distinct provided we impose this constraint that Diaz nonzero so now we can as well but what is the polarization OK so remember the polarization on is the very things associated with this again In this case on you know that the vector as I go around it sort of goes like this OK so the Berry phase for that is equal to 0 OK it doesn't it doesn't close any solid angle at all going in going in something like this whereas on in this case here on the Berry phase gets my favor minus OK so in this case on the various phases pie which corresponds to a polarization of you over to program and so it makes sense that on when you go from this I do this you sort of moved every electron over by by half a state OK and so that's the polarization of the overture so on so you can see this polarization physics at work in the sushi for Higa model and and this is the 1st example of a topological structure in a band theory now it was crucial that that that we had this extra constraint if you don't have this extra constraint then the the vector canceled out of the plane OK there's an extra direction here and then this and this would be topologically equivalent you could just go around the 0 camp I can't so so in this onedimensional case the cemetery is crucial OK so and so this is the fishery Vernier model on so I sold 1 dimension if there's no symmetry then there's really no topological classes every onedimensional band structures the same topologically as every other 1 Compaq into dimension is more interested in Kent on all before I get to 2 dimensions and there's 1 more thing I have to talk about sorry about that so which is so
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if we do have this constraint then on
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an interesting thing happens if you have an interface between the 2 different topological regions OK and so this is a famous old problems and so on and so on if you have a domain wall that separated this days on from this phase then will you inevitably have Ari is so there's a gap on the left and on the right inevitably there is a 0 energy state on the boundary camp and on so in order to what describe this what it's useful to do is to on is too well short of that I developed a lowenergy of model focusing on the state's sort of close by on the the energy gap and if you and so this is like Lenny arising for 4 K close to home pie over a can expand this and what you get if you do that is you get a theory which again at 2 2 by 2 matrix but now on why it has the form of a on Dirac Hambletonian compared and so on so it has on you know a master and looks like this the cup is UVA velocity trend the couple's 2 Sigma acts and a master in the couple's signal y and so if this is the masses constant were over here where the masses constant then you just have this this relativistic looking dispersion which which is what this looks like here I'm going so long so the interesting thing is is what happens when this mass changes signs across this interface cares about what's happening this mass is his you know changes sign when you go go from 1 state to the other and so on on and on and again we have this on extra symmetry which is that on you don't have any seats was the terms and so that means that Sigma the actually and he commutes with this Hamilton compliance so I and what this cemetery tells you it's said missing and he commutes with the Hambletonian it tells you that if you operate said museum and Biden state then you're guaranteed to have and I state with negative with exactly the opposite energy and going so that tells you having this Cairo's cemetery which this Hambletonian has tells you that the United States come in pairs with plus and minus at the energy can now and that but what can happen is that you can get you can have a state that's exactly it 0 energy and this is what happens on this domain was so this is a famous problem that has been discovered and rediscovered many many times over the years that he was 1st discovered by the field there is tricky and ready to go on and on and so they solve this problem and and so this is a problem you can solve OK in fact I think I ask solving your homework so well and I see you can write down on you can you can you can solve the shoddy equation and see that there is a state exactly 0 energy and this is this is the This is the way
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function compared on and on and on and so this is a very general the kind of phenomenon that when you have an interface between 2 topologically different regions in Europe that you know you have a gap on on the 2 different sides of the interface but down to the interface and you have some interesting lowenergy degrees of freedom comparing so this is sort of the most important consequence of all this topological stuff OK so that now and so I told you that in 1 dimension we don't have any on topology in 2 dimensions we do and I am not yes this is the only 1 it's the only solution at 0 energy so so there could be other band states which are on file at finite energy look at the of course you know that if there is 1 at energy E there'll be also be wanted energy minuses so there could be more this might not be the only bounced states but it's the only downstate at 0 energy is the only downstate that isn't the doesn't have another state that is it is its partner you OK and so on so all of the state's come in pairs except the 1 equals 0 many of its it's not degenerate is so see so so this tells you that on certain was the acting on the it is sigh he is and I can state with energy minuses so if he is nonzero that it necessarily has a different I can state but if he is equal to 0 these 2 could both be be the same state and that's what happens on in this in this in this situation misstated equals 0 on its said is either status that with energy plus 1 we would like buy plus 1 of and so the important thing here is that you always get this 0 mode and it doesn't matter what the precise form of this of this song masses of function is all that matters is that changes sign now you can have the other ones but they're going to depend on what this precise structure OK so the thing which doesn't change is the existence of this 0 models the 0 mode that is guaranteed for topological reasons everything else does depends on the details thank you the question the question again yes ,comma need financing yes that's right yes yeah look OK so long OK so in 2 dimensions and we can and
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we also have blocked states and a you know we can define this vector potential but is that affects potential is used on hundreds of lives in 2 dimensions and so on right on what 1 can do then is to find the Berry phase that sort of opposite sides of the Umbro OK now cost you might say well on this line here it is really the same line as this line here OK and that's true OK ,comma but but it's also true that the Berry phase on this line is illdefined up to an integer I can't and so the way I wanted to find this difference is I want a sort of defined as 1 in such a way that if I go continuously across and you know do it smoothly I want house where why have when I get to the end OK and so what you know is that since it's the same line you have to give its different it has to be different by OK but there's no reason that manager has to be 0 can and so the difference of these on is fine if you use as you know stops there and is going to be the eye integral over the entire world 1 zone of this very curvature OK and and this integral on is guaranteed to be an integer object on you know because because we know that these 2 lines are really the same 1 and get in this here is exactly the same kind of thing that happened in the gas beneath their right in that this is like a curvature on which is sort of like the gas curvature on that we had in the gas beneath the considers this topological invariant which is called the called the charter number registered in Raichur number here anywhere but this is this is the true number on the Internet and that's what it is and so it's an integer topological invariant to characterizes any twodimensional band structure so what is this guy topological invariant mean well it's the integer in the integer quantum Hall effect so this is the thing that distinguished the atlanta levels from the atomic orbitals the ladder levels had you know I'm having on 0 China now 1 way you can think about this understand this connection on you know we sort of understand the Berry phase in terms of a polarization and so there is a famous old argument due Laughlin on for thinking about a quartermile effect which is you wrap the quantum Hall effect into a cylinder cylinder is onedimensional and so the cylinder can be characterized by a onedimensional of polarization and then Laughlin as well what happens if you threaten magnetic flux through Saunders In case you wrap up the magnetic flux on through the cylinder on up to the magnetic flux 1 and so what happens when you when you throw the magnetic flux then you know what I fear is law gives you an electric field going around the cylinder and said That's why then gives you a current flowing down the cylinder and so when you wrap 1 flux quantum then you transfer exactly on an electrons from 1 end to the other or others stated another way on the polarization of this onedimensional cylinder changes bye going on now and so this change in the polarization is exactly what you think about it in the woods from a picture like that if you think about the flocks as being like a walk and if you really a few rapid twodimensional thing into a cylinder then I it's like periodic boundary conditions with a really small and you know size and cancel the flux really is like a water again so you can think of these 2 pictures and I'm really evaluating the difference of the polarisations before and after I inserted the flux of began only the in the
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morning and we will continue in this right in saying several months yesterday New York is and saying was was 1 of the most for any of issues you really need this the findings EMU would define KY made 100 a so is he the
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variance yes independence finally the taking of periodic letters the thing in Europe yesterday but ways so so so what I'm so what you have to do so we said so so if you have a onedimensional on the system then on Cassini to variables in order to find the term OK so we have kayaks which is 1 and so on what I'm doing is I'm trading KY for the flux so that means that I have to do not I have to define the cylinder for every value of the flux and so I have a family of onedimensional Hambletonian parameterize by the flux and so that gives me a Hambletonian which is a function of kayaks and the flux which then has a has has hesitant so I'm not sure I answered the question you ask is that is that will this the main question is also full get here in Europe with more flexible Europe during the and yes is argues that a lot of things in it just means you're using this username right if in it has to get the invariant you really need a letter on other but that number but that even if I don't have a lattice I can define a lattice so so let's suppose there's no periodic potential focus on 2 so then of course you know constantly don't you with each other but if I artificially define a unit itself is looking at the potential is 0 but I do find that unitcell that I still have gone the X and Y Crystal moment OK so so you still have you still have a band structure even further than the No Land level problem OK so you can define the turnover in that way and and it's and even for even for the you know the No at all levels the young you evaluate the Chernomyrdin according to the K and procedure you'll get the right look at that and questions OK so on good OK so this is
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the on the the tournament focus now on so I not you know the thing which is interesting is what happens at the edge of the area and so OK so before I get to
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the edge let's talk about a grab on talk about a mile OK so again I told you and I always want to think about done in writing and simple models for all the things we can do so and so this is so there is a beautiful simple model can write for the quorum all effect on that was introduced by holding about 20 years ago on which on is based on graphics you all know about graphemes right on so graphic his famous it was the you know gunmen notice of of that the Nobel Prize last year for this and actually I just read that on that diamond and O'Sullivan our on renowned nights of the queen knighted them so that we have to call them server pondering that now and so on but in any case on my side if you know grabbing is defined on a honeycomb of lattice and and so again and hazardous to solve lattice structure which means that we can describe the band structure sort of in the same same way In terms of the 2 by 2 but matrix again but if the 2 sub lattices are equivalent to the area in the as they are in graphing from then on you can't have any Sigma sea charts again so again this stand and so the absence of the set actually done in order to make sure you don't have anything Aziz you really need to and imposed to different cemeteries you need to impose inversion symmetry which makes the area the piece of lettuce equivalent and you also need to impose time reversal symmetry and if you do that then on this is the component of the arm of the D vector I'm has to be equal 0 so that it it lies in the plan From can't and so now on that and say you know you can you can work out what is the vector is for the for the photographing problem McCain if you've never done that before and that's a good exercise ,comma to do on but 1 thing at a couple of general things that you can cannot understand without even without doing that calculation is that on the same side of the deal actor is is is he is a unit vector now if it's defined on a circle given this year the time soul and the inversion symmetry is defined in a circle as a function of 2 variables that means that it can have Morrissey compared that of more attacks is a place the you know a point in which the and direction of the sort of wraps around on of course at the point of the vortex media vector is is is is is undefined which means DEA has to go to 0 we can't and so that's what happens at the direct points ,comma followed in graphic gay and so on so the fact that the direct ones are allowed men can sort of understand from from this side from this topological type of argument to get in and near these direct points on unaware that the vector goes to 0 you can expand on I 4 small Q around the the vortex .period and and you know you'll get a linear dispersion which describes this on this side of this direct type dispersed from Kent and so on so 1 of the important things that you I notice from this is that when you go around the director .period the the but this ID vector of rotates by 360 degrees in the plane and so there's a barrier there is associated with the direct look itself so this is on but graphene with inversion and time reversal symmetry and actually importantly were not thinking about spending at this point OK there will be a new twist on which announced that so and so on and it's interesting to think of what happens on the when and we come relax these symmetries I want to open up an energy can compare and up and so on so if we do that if we break either the inversion symmetry or the time reversal symmetry then we will open up on energy gaps that the 2 OMB director .period so so I should have said that there are 2 of these direct points which are a plus K and minus
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Kent and so on and so we will open up our energy gaps which mate which make the dispersion near these have sort of on the
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form of a massive Dirac equation with this massive relativistic dispersion and so on so what we can then ask so we have an energy gap in that case and so then we can imagine asking Well what's the what's the I what's the turnover and so on the so Remember the Alamo I so the on the chair numbers the integral the very curvature and bury curvature has to do with the solid angle they get swept out by the diva and so on so basically the charter number on it as a very simple interpretation in this on a tour to buy 2 matrix of model it's just the number of times that this diva actor on wraps around the sphere go on and the on so if we break the peace symmetry then on then I which is what would happen if we made the a and the B sub lettuces have different kinds of Adams on them because of that will open up an energy gap because you're putting a Missouri terms by and so in that case we had a each the masses at the plus K .period in the Mindscape escape wanted to be equal to each other come on and if that's the case then we can sort of think about what is the diva actor do as a function of position will almost everywhere on land it's living on the equator OK because because I when I didn't have this mastermind it was always living on the equator but but but but near the idea points when I put the mastermind go up to the North Pole or the South Pole depending on the sign of the maps I'm going then so if the 2 masses are equal splits a positive then and and then both at the K the indicate primed .period the plus and the minus K . on the detector goes up to the North Pole OK and so you can see from this that on that there's no net solid angle swept out by the defective because it's both knew they cancel each other out and so the number in this case is equal to 0 but on the other hand on if instead of breaking PU break tea organs so this was a Duncan holdings in sick inside 20 years ago and then on the at the plus K . it he has the opposite side of the mass of the mine escape .period Tom and if this is the case then on spot you know again your most the place with most of the time you're living on the equator but close to the case .period goes up to the North Pole and close to the mine escape when he goes to the South Pole congress and the conceded this wraps the sphere wants and so this gives you the "quotation mark Wallstreet so OK so I so now and I now that we have this model we can think about what happens if we have an interface now
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between the quorum Allstate and St. trivial insulin OK and so interesting thing happens when you have interfaces on between topologically distinct regions Tom and so on and so I think about it states now course that states are a familiar things here in the quantum Hall effect the simple view of them is that you know in the ball the electrons are going around in
54:26
circles and that if you have a wall the electrons can sort of bounce off the wall and when they bouncing off the walls then they skip along but they always go in the same direction Grant and so that suggests that on you know there's an energy gap on the ball but there should be a sort of extended propagating states on living on the boundary so so there should be sort of gap states here compared and so on and so on we can solve for these other gapless states if we think in terms of this summer but think in terms of this model this direct model where on 1 side the 2 masses have the same side and on the other side the 2 masses have on opposite sides so let's imagine would think about what happens if we on change this line of minus the on going across the surface well so and so we have so this is what we have on if so near the mine escape .period we have a on a Dirac equation that I looks like this were now the on the master on varies as a function of X OK so on and so on so we can still you know I write down plane waves in the wider actions like and I can have on K Y and Y action and for every value of k we just have the same ready problem that I talked about what I was talking about ,comma about and Polly settling it's really it's exactly the same problem and has exactly for every value KY has exactly the same from but can't and so on so says we have these are about for all KY it's interesting to think about the dispersion so this version is going to be the energies is going to be linear KY OK and so on so you notice that the I'm so there is there's going to be a band of states that crosses through the family energy but the advent of states and has a positive group velocities so those states at the Fermi energy are only going 1 way OK just like just like these are only going 1 way to these are really the same states and cities 1 way states and we call Cairo old direct fermions of Cairo means 1 way in this context To get on and so these are really special things these on side Carroll Dirac fermions states which live on the edge of a quantum Allstate on and on and they're really special because on you know on a oneway street and you can't turn around you have no choice but to go forward OK and that so that means that the on electrical transmission in these states is perfect compare there's no way you can scare them on and on and on and that means that even if you mess it up and make it dirty and you still have no choice but to go forward bookings of these data insensitive disorder the impossible localized on someone so that makes them that's another thing about these on the use of choral director for eons is that's on the onedimensional states but they can't exist purely in 1 dimension became so well you can't have a onedimensional system that has a band structure that looks like this OK and so there's a a nogo theorem called the fermion doubling their on which says that this is impossible and in 1 dimension that that the nogo theorem is really kind of trivial on 1 where had described to me once is on its face with a statement what goes up must come down OK and I so if you think of a band structure on you know the energy has to be a continuous function of K on the brawl 1 zone which means they can if it goes up it has to come back OK so on and so onedimensional coral direct fermions are impossible what makes it possible here is that the choral director Indians living on an interface then and you know that is going to be another interface down here on gained so far 1 of you this is the 1 that the quantum all system as a onedimensional system that I have I have to think about both the edge on the top
59:07
and the edge of the box and the 2 of them together then restore this sort of you know this this summer for me and going there alright so that's the on the the edge states and so on and so so really be that states are something which is very especially if you have then there's no way you can get rid of don't and I so well so really the number of its states you have on is is is a
59:39
topological invariant OK now of course on by soso for instance if you solve the this honeycomb lattice small in on a strip geometry you find that died that there'll be disbanded states on which connects the conduction band the valence band that lives and the states are localized
1:00:02
at 1 of the edges Of course there will be another band states on the other end to attend didn't drop OK around but on I'm so there's no way you can get rid of it's OK now what you could do is you could you could imagine making more states that cut through the Fermi energy by sort of bending it like this but on but if I count the difference between the number of right moving states and the number of left moving states that's always going to be equal to 1 OK and so that 1 is really a topological invariant is not there's no way you can smoothly change that without closing this Balkan energy going and so on and so so now we have 2 topological invariants OK so we have a we have the number a topological invariant that characterizes the Bork twodimensional band structure OK which doesn't say anything about the edges right it's defined with periodic boundary conditions can find so we have a juror number characterizing the book and we have this count of the number of car rolled Furnia modes which
1:01:11
characterizes the boundary and so on and so on it's a deep mathematical there on that these are really the same 1 and the same thing going so so what are you mentioned a TeliaSonera index there and this is another example of the city a index and so a very deep on the idea in mathematics which shot which 1 can understand on many levels which go way beyond what I understand that so out again so this is what I like to call the book boundary that correspondence OK so what now so mall affects me In requires a magnetic field which requires the breaking of time reversal symmetry and carry on you know the the khaled states have to know which way to go OK and so 1 has to have some some breaking of time
1:02:15
reversal symmetry in order for that to be possible on them so for a long time and it was believed that on the only nontrivial topological states you can have on the quantum all states in which require broken time versus summitry on that and there is a sense in which that's true still but there is an escape clause which is and we already socks this .period
1:02:45
on that occurs in graphic kept on hand I explained to you that before there is really inversions metric inversion symmetry in time reversal symmetry that the presence of those that protect the united Europe .period but there is a flight on an argument which is that that I gave you was forced stainless electrons if you have spent the entire reversal symmetry is a little bit different program in particular on if you have spent then on there is a lot of Master and they can write down the doesn't break any of the symmetries of what is master and does is it takes the upstairs and puts them in all the main state with 1 side of the magnetic field in the downstairs put a hold on state with the opposite side of the magnetic field you can now buy this Hambletonian does not break time reversal symmetry because some other time reversal symmetry the you know and this Hambletonian goes to this 1 but also the UPS goes to the downstairs OK and so this on the model home and does not predict that on 1st century to get them what is is it's a model where the upstairs on a quantum Allstate with a with 1 side in the downstairs a formal state with the opposite side of the vessel recalled the quantum spin off so he applied electric field you don't get any of net current flowing but the Austrians go up in the downstairs good answers and that's been part of the floods and so as in the quantum Hall effect on my lives if you have this then you have to have that states and stated kind of interesting because the obstinate states go to the right and the dance it's going away OK and so on the edge 1 hazardous services dispersion that looks on something like that and so they states really form a unique on how onedimensional electronic conductor Kent and in a sense it's like half of an ordinary onedimensional electric gas and the key point is that it's protected by time versus Chancellor may explain a little bit about
1:04:58
time reversal symmetry time reversal symmetry is is is when you have spent on is on a kind of a funny thing and so on because it does 2 things it does it take the complex conjugate of the wave function but also flip suspense and says it takes the accomplice conjugated called onto unitary from operator and it has the important property that forced them to have if you do it twice its squares to minus 1 so this minus 1 is another of my favorite favorite minus signs actually is the same 1 it's the same answer but on in any case on this minus 1 gives us Kramer's there which tells us that every so that 1st spent a half on every Oregon State is at least 242 and so this as an important consequence for the that states because you know and I showed you that you know we have is that states and across here OK on sea
1:06:01
might worry that there could you can add some perturbation which would open up an energy capture OK but if you have time reversal symmetry you can't do that OK because these 2 it states the crossing here on forming Kramer's pair OK and so on so this crossing is protected by time versus symmetry it's even better than that on you can imagine you make this side the edge during then you can imagine that dumb you might have backed steps back scattering which would reduce the conductance but the elastic back scattering is on is forbidden also by time versus symmetry so so on so when you have time reversal symmetry these counter propagating it states form another kind of unique onedimensional conductor that that can't be localized OK so usually we have a onedimensional conducted to make a dirty gets localized pandas compared on these and cannot be about OK so and so so now we see that if you have these on these these states these costed states you can't get rid of OK so it's sort of like we have a sort of a boundary topological invariant and so for that reason will we expect if we believe in the bulk battery correspondents than what we expect is that there should be a balked topological invariant that tells you whether or not you have these on kind of edge states on boundary can and so on and so so what I'd
1:07:43
like to convince you of is that it's on the really too To win only 2 a topological classes some pay and so again on an appeal to this book battery correspondents and so so so I can show you is that there are 2 and only 2 and so so I redrawn this picture of the ad states and so on but I've only drawn half of the Bruins on going from cable Zurich pie the other half is is just a mirror image of you have time reversal symmetry so quiet hours there and tells us that if there are any state Ahmed states bound in the on in the inside the gap down to the edge than they have to be Kramer's Dujarric equals 0 because of these 2 former :colon pair if I go away from cable 0 this guy's partners at minus can so seduced these 2 don't have to be to shatter anymore more so in general
1:08:35
they will slip if you have a spinoff of action cables pies the sailors cables minus pie so these 2 have to be degenerative well so I can see why the 2 and only 2 classes because on easily connect up like this they connect up in pairs OK on in which case this is sort of a trivial case you can imagine getting rid of compact on because you can police and get rid of the estate's completely so this is the trivial case but if they switch partners of the Kramer's chairs here in the grandest Spears years switch partners that you can move these up and down all you want and I and you'll never get away OK so so so this is what happens in the on in the In the sea to a topological interrogated to z to topple up only this is what happens in the the 2 and topological insulin notice that done in this case the Fermi energy intersects the ad states an odd number of times OK whereas in this case intersects an even number of time not even number might be 0 get the odd number cannot be certain so yes thank you because the chair number is equal to 0 if you have time reversal symmetry so so so remember that the charter number tells you the number of Cairo led states that you have but if you have Khaled states you done 2 times worse than the currency going the opposite direction and so that tells you that the charter number is odd undertone person so if you have time reversal symmetry than the chair number has to be equal to 0 and so on but these
1:10:25
soap so if the charter number is all I have these 2 would be topologically equivalent OK but this has got to be something else besides the charter number that distinguishes them so cake and so so this argument here was appealing to the edge so there has to be something a book a topological invariant
1:10:43
and so the mathematics of this is a little bit more on involved so we can write
1:10:48
down 80 to time invariant and for the Balkans so let me just and give you a little bit of a flavor as to how how to on how to write down that a topological invariant so again so so long you start from the wave functions OK on and so all the information is indeed is is is is and these are my conductors and so on and constructively to injury was useful to do is to but for every value of k to define a a matrix which is the overlap of the block factors that K with the time reverse of the block vectors minus 10 now says you have time reversal symmetry the on the you know that this the time reverse overstates the case are related to the time to the states of mind kept and so that means that there is overlap hostage but they might not be in the same gage OK so there's overlap has to be on a unitary a matrix I'm going on in this unitary matrix has some special properties in particular but since found the time reversal operators squares to negative 1 and then on that means that this you know using the anti unitary Cantina charity of the time Ursal operator that says that WAK has to be minus the transpose of W Mindscape just just using just from this and so that means that the special points on in the brawl one's own work a inline skater the same .period working my care related by reciprocal lattice vector so that these 4 points in 2 dimensions at the special points then this W has to be an antisymmetric matrix so it's an antisymmetric unitary matrix now an antisymmetric matrix has a magical property which is it's determinants is a perfect square OK so so far too much to antisymmetric matrix of thought it's right on that the determinant of this disease squared off but this happens for a general and by an antisymmetric matrix it's determinant is the square of something and that something is called the fast case of the fact innovators does get on and so that allows us at each of these points to define the fact the end of the soda fountain is defined it before at that that these 4 points compared to that the determinant is defined everywhere OK and so on now the vaccine will be equal to the square root of the determinant given a choice of sign for the square roots would you be plus or minus 1 go on but since the and determinant is globally defined everywhere I can fix that sign for the square root wants and then on this ratio Of the fat in the square root of the determinant then has me OK and so basically on so we can define this on this ratio the I'm for any of
1:14:08
these 4 points and the Standard & and the product of the 4 of them defines the uneasy to topological against this the statement Our might think anything that will it's very odd odd for an art images spent half a half interest and yet yeah not so spend it's no because there were breaks down is here OK so this is this is this was crucial yeah absolute good OK so I'm OK so this is 1 of
1:14:45
the 2 invariant OK it turns out if it you have on if you have extra symmetry it's easier to compute on Zito invariant on particular year farm if spin is conserved on there and then you could just computer turn numbers for the upstanding then the down separately from that of course they have to be negative each other because of time reversal symmetry but they but you know you don't have it as he is concerned so I can talk about the expense of the dance in separately then the upstairs will have a turnover and so busy to invariant is just whether that your numbers even or odd another cemetery which turned out to be useful on is on inversion symmetry and if you have inversion symmetry it turns out that you can determine the easy to and invariant just by knowing the parity of the occupied of states OK and basically at the special points work a minor scare the same the black states will be I states parity with Nikon value plus or minus 1 and so basically you just multiply all those eigenvalues together and that tells you the
1:15:55
this to OK all so that and so so what we've learned it is that In 2 dimensions Top Gun by band structures fall into 2 your time reversal of Arabian searches fall into 2 topological classes the topological and Slater and this the trivial and so so you can ask the question What have what happens in 3 dimensions and the
1:16:21
story is similar but not exactly the same and so on so the way we can think about it His again to appeal ball battery cost miners were now the boundary is a twodimensional surface Constanzo twodimensional surface will have on a twodimensional role one's and there will be for these special points were carrying miners carry on more of the same and so if you think about the surface states on in the vicinity of these points they have to be Kramer's too generous on exactly the point of I go away in any direction then they will split and so it looks like a director .period then on now so that so that there are lots of direct points among the surface of the twodimensional and on the surface of a threedimensional insulated the interesting thing always happens direct points connected to each other you know they could look like this on in which case you can imagine getting rid of them what they look like this and this is on between any difference 1 of these pairs they continue their have this picture with this picture OK and so if you have it looks like there then you can't get on that can get rid of and so so they're more possibilities than without it turns out that on that there are 4 or topological invariants that that characterize a threedimensional a topological and so on and so on and off the ice the from sort of on a simple it's kind of a topological insulator is what would happen if you just took a on twodimensional topological a bunch of twodimensional topological and slithers and stacked them on top of each other and kept him so if you did that then and then what you have is on you would have the edge states sort of running along the side and you have an isotropic conductor on the side of so so you'd have you know I you know a surface that instructional books like this with the 1st surfaced from service which is nearly flat then you turn on coupling between the layers and native elbow bow out a little bit kept on and now these are surface states and are not as strong as strongly protected is the pure twodimensional surface states would be in particular if you made it sufficiently dirty you could
1:18:41
on you could localized these on the surface states from came on or you could continuously to form the Hambletonian in such a way to want to want to get rid of Bulgarian so the week topological and Slater is not known as robust as on the side of the surface states of the twodimensional a topological insulin notice here that the ferry service on encloses 2 of these time reversal in varying points of the surface level once a so there's another possibility which is that you getting close just single 1 again cancer this is what happens in a strong topological invariant but taught a strong topological an insulator and on in the strong topological insulator on the Fermi surface encloses a single director .period and so this is protected at the same time in the same way that is protected in the twodimensional case so for instance on the states and cannot be localized on by disorder even from source OK so I let me just go on so it's it's time for the break but let me just let me just go on and on a couple more slides and we'll take a break and come back so on and so and so I
1:19:59
told you there for a topological invariants on in 3 D and so the way we can think about those is on the line is on remember I told you that on I can think of I I I can think of the on the C 2 invariant by thinking of this ratio of the fat in the square of the determinant which is defined in each of the time reversal in variant of points and so and in 3 dimensions there are 8 of these some of these of these points came on and so on and so on we can form different combinations of these 8 which make different things so we can forms so free of the topological invariants on on the war a on a factor on which I basically this vector guide describes the Miller indices for some stacking layers on twodimensional states and so on but but of course this that these and Miller
1:21:06
indices are only defined modulo 2 so if I have if I have stacking in 1 direction and stacking in another direction where on the the time Miller indices differ by a multiple of 2 then those 2 states would be topologically equivalent compact and so this gives us all the week topological insulators and then the strong topological invariant on is given by a scalar injury which is just the product overall each of of the fees paid and so so in particular on you if you are on working within the inversion symmetric crystal where these parity Eigen values or something you can look up the book OK so so people do band structure calculations always tell you what these eigenvalues values are actually have a new appreciation of band structure you know there always draw all these you know vans and they labeling with these funny letters so that's funny letters actually have information which is useful get so am I. it's opening a case if you if you know this parity Eigen values and you just multiply them all together and you can get the the strong topological OK so well and so I had the honor but the thing
1:22:27
which is really interesting about topological and insulators from the surface states and on and the thing you know and so that this sort of model system is really just a single but Dirac fermions can and and and it's sort of half of an ordinary twodimensional electron gas because a very circle but each point on the Fermi circle only has a single stated were as ordinarily you have states for span opens for spending and so that makes things are different on in particular it makes it robust disorder on it's on another way of saying that is that on since you get this Highbury phase when you go around the family circle that changes the sign of the weak localization and that's related to the fact that the states are impossible to localize the modem is even more interesting than the snow is what happens if you can open up an energy gap and so you get an exotic state when on when a broken symmetry leads to a surface energy gap
1:23:33
and so there are 2 things I never talk about there's there's 1 thing where you break time reversal symmetry on and give you a quantum warlike state and then the 2nd 1 is the superconducting state and so let me finish up by telling you about the quantum Allstate and then after the break will talk about the superconductor program so so this is the last 1 before the break
1:23:52
so bad that so on and so let's suppose we have surfaced OK then we put on a magnetic field so on and so unique Atlanta levels OK you always get Atlanta levels in a magnetic field but on the Dirac the land little problem for direct from Beyonce is on a little bit different from than the ordinary little problems and in particular on what happens is that there is a land level at exactly 0 energy OK and on and because it is a sort of a particle cemetery here every time you cross a lander level the on the Hall conductivity changes bodies were age but since it's particles symmetric then and now you know it's minus down here and plus appear so that means that it has to be quantized at half integer value so goes from miners and have plus at and so this 1 have is something that's rather special OK is onehalf of course which was 1 of the things that got under a guy and his Nobel Prize on because this is what happened in graphic OK and they saw this in a very unitary clean and beautiful way on grafting tho actually didn't quite see but they didn't see a onehalf they saw 1 have multiplied by 4 can I so there were still measuring the energy an integer quantum Hall effect compliance and now why this really is 1 half which should make you worry a little bit because on if anybody ever tells you that there's such a thing as a fractional integer quantum Hall effect then we tell them that the wrong OK because on you know the deeper reasons that all connectivity it on Indian you know on 4 not interacting electrons is quite which is what we have here is quantized on at integer values and and so this 1 have a kind of a funny thing but I the way around this paradox that always to realize that I know that this is something which occurs on the boundary the system OK so if you imagine you have a slab then you have a boundary on the topic also the boundary on the boss OK and there's a theorem in mathematics and which is that you can't cut the top thunder the top boundary off the bounded bottom battery a boundary can't have a boundary OK so on and so the top and the bottom are always likely to each other OK and so that means that if you measure volcanic was given measuring the tune parallels in the 2 1 has added unit OK and so so the so the integer quantum Hall effect really is always in nature but but but but you can't have a situation where you have gone you know the 1 half on the top and then you have 1 half on the bottom and then those 2 states would share a single at state comparing the United States on goes along and the edge so so it is kind of interesting that interface between onehalf and well if you if you unfolded minus onehalf a gives you a on a lipstick another way you could do that again sought by unfolding at is to on is to deposit in an insulating magnetic material which breaks time reversal symmetry and opens up the gap I'm at the Durant .period and if you do that on 1 side and on the opposite side of the other side so you have a domain wall where the sign of the time reversal symmetry breaking changes then on that's just like unfolding this case and you get a car rolled under domain wall which runs along the such occurrence of this this is 1 of the things which I hope on we get to see experiment was sometime this has not been this event has not been observed yet but I think this is the reason why we can't activist yes you you you you well so that the the wealth so the difference there is a difference in the terms that you write down so so Of course so so so what I put on here was Azeem on track focus on imagining what I put on the magnetic insulator that
1:28:33
the dominant interaction is the sort of exchange interaction on the side the universe generated by the proximity to the magnetic insulin on so this could be an antifur magnet for instance you know something that you know and there could be an exchange interaction with the With the antiterror magnetic thicket thicket that could open up a gap whereas this is coming from the orbital piece so so so on so the difference between these 2 these 2 is is this is coming from the orbital part of the interaction and and and this is coming from moral magnetic field would where this is coming from as a like track and so the 2 places that the magnetic field centers problem question OK so long why we take a break and come back maybe at 9 15 Sunday 15 a OK so
1:29:38
what and so the next 45 minutes so I want to talk about .period topological superconductors my firmly on Sunday and have the related to 1 computing and so on and so are you just sort of start here on so
1:30:01
I superconductivity in mean field theory sort of his news ,comma like a band structure OK so when you do the the the BCS mean field theory then you know be that superconductors are really interacting material so OK you don't get superconductivity without interactions but the BCS mean field theory turns it into a lot of unknown attracting a problem but the twist is that when you do the main field theory you introduce these anomalous terms into the Hamilton kept and so but 1 can still described the on the this mean field theory in terms of a single particle on a description you can on where I so but when you write down Hambletonian now you have to include the fact that you have these on anomalous contracts and so the way you do that is by writing the Hambletonian 2nd Quantas Asian and then you can write it get a on the handle of a 1st quantized Hambletonian on from from the sort of matrix Vogel about and a matrix of structure and so on so this has both the on the sort of normal part of the Hambletonian on the diagonal and it also has the on the parent terms as well OK and so so then you have this on this matrix Hambletonian were now the matrix structure is really coming from the sort of book about Nambu on the structure of the of the of the of the sort of superconducting away function to carry on now but so what you have is that you can solve this you can solve for the No I can vectors and I values of this and in the same way that do an ordinary Kwan context come on line and in particular if you're in a crystal then this is all a function of the crystal momentum carry on so 1 has a band structure that's that some just like in an ordinary band structure and so 1 can think about questions about whether those that those being
1:32:21
structures can be topologically classified now on there is an important step in a twist here tho which
1:32:32
1 is that when you write down the state vector in terms of science I dagger science I dagger or not independent of 1 another OK and so that means that there is a redundancy that is intrinsic redundancy that is built into the system Hamel Hamilton OK and in particular on that so I you know why there is so for instance the Hambletonian that sits here is exactly minus the Hambletonian that sits here on campus and so on but this report is reflected by an intrinsic on particle whole symmetry that this book about Hamilton has to have kept on and so this particles symmetry is on soaring looks a little bit like the time reversal symmetry because it's it's an anti Unitary Symmetry involves the complex conjugation OK and so but died on the part of global symmetries operator which he commutes Armed with the Hambletonian rather than communing with Hamilton look and so on and so 1 has this on this cemetery so so so so you can't just put any the Hambletonian died in here you can only put Hamilton means that have that reflect this on intrinsic of this constraint which reflects the side intrinsic particles can and what the cemetery leads to is a bomb a redundancy so so we know that I if you have this symmetry that if I operate operators operator on on on any I can state that since it and he commutes with the Hambletonian just like with what I was talking about and the young searches here model of it's as any commutes with Tony and it gives you and I can state of some of the Hambletonian with the opposite energy compare so so the United States of of of this hell Tony come in pairs with plus and minus now on the important thing now is that's on the state's plus and minus the are actually fundamentally redundant In the sense that the operator which creates a particle at energy plus the is actually exactly the same operator is the operator that destroys a particle and minus OK so you write these operators out there you know they involved there and that they involve science side Aguirre but remember you know and you know the different commented that this so so sigh dagger the air gamma dagger the Indiana of are both the same combination of science Center compared so they're the same operator which says that this state where this quasiparticles operator is this is quasiparticles is is occupied is the same state is the state where where this 1 empty they is this is different than in ordinary semiconductor were having a particle and having a whole those are 2 different states of we have to keep that in mind keep this this fundamental redundancy in mind on when thinking about on the superconductors so so as I said this particles cemetery kind of seems like
1:36:03
the and it smells like the time reversal symmetry and so you can ask the question whether on the Hambletonian that satisfy you know blocking adultery is that satisfied this particles symmetry constraint and again remember since it's any unitary but that means that is the cemetery takes care to minus scale OK so so what and so can ask whether there these kinds of Hamilton is a satisfy this constraint have topological classes and if they do we'll call them a topological superconductor compare the nontrivial compare so what so that 1st way to think about this is again to think in terms of the bulk boundaries correspondents located
1:36:57
stand on and the simplest version of that actually occurs in 1 dimension compared so I so I so if we had a onedimensional superconductors then let's think about what happens at the end of a onedimensional superconductors so so and so on and so there could be bound states on which I live within the superconducting energy gap but we know that but if there's a bounce state energy than it has to have a partner in energy modesty come the particles symmetry and actually those 2 states far and are related to eat 1 another but that's now another possibility that as in the case was that of the on switch refer here model is 1 could have it's possible for there to be a state exactly and 0 energy a state which is its own partner OK and if we do have this state which is its own partner then added a particle into that state is exactly the same operator as removing a particle from that state which means that the quasiparticles that we are adding is it's only antiparticles OK and so that's what we call a wire on from there so it's an operator of which selfconscious so what Samara manager who were wonderful things and I hope we get to see 1 someday so I sold 1 way 1 can think of it is Elmira firmly on his sort of the real part of a direct from an OK with mention pence or 1 can think of a ordinary direct form firmly on this stairs you know pair of Myron affirming its on and so in a real sense of Myra Fernandez like half of the state in the sense that if I have a pair of my firm announced then that defines a direct firmly on and that simple direct from me on corresponds to 2 quantummechanical states either the firm EON is empty you state is occupied war it's not occupied OK so the to so so to Meyer on affirming islands on defining a single twolevel system compared on so so if I have a I like you know a top a onedimensional topological superconductor which has the 0 modes and it stands on OK and so if the topological superconductor that has the 0 actually 1 other thing that I want to emphasize about this 0 mode if you have a 0 mode then there's no way you can get rid of compare it is topologically protected because think about how could you get rid of it well so it's 0 you could try to push away from 0 but if you did that then the money you pushed away from 0 than it would have to have a partner but it doesn't have a part in case so you can have a state appearing out of nowhere and so on so this the state has no choice but to be it's 0 you can get rid of the only way he could get rid of it is by having the bulk energy gap come no close Grant then so so since these states are unprotected the or arms or topological protected then the bulk boundary correspondence tells us that there must be a you know a class of superconductors that has this and it's Ahmed Attaf complaints about a topological superconductor OK now and I find it should be and there should only be 2 superconductors but that because of I I could either you not have any 0 motorway have 1 0 mode if I have to 0 mode CZ if there were 2 states here than I could computer and so so 2 0 modes is just as good as 0 it's the same Missouri OK so we expect there should only be 2 classes of superconductors the trivial and the topological sources the 2 enduring which you which describes the topological so focus so if we have a topological superconductor has as it has the of 0 modes on each end and you know you have fired you know if if this summer length is finite that actually 0 0 modes will you know will be weakly coupled to each other and so those little bit OK but if this coupling is long that this wedding will be exponentially small but on but these this pair of 0 modes defines a single state where the 2 states this formula is is on using either empty or it is occupied so some of the energy is either 0 yeah so that's are onedimensional a topological superconductor on it so it's nice to have a on solvable this OK and so this business most of the solvable
1:43:00
models of are due to curtail on and on and on and so could take that as a very on a nice and simple and beautiful model for a onedimensional on superconductor which is just a on stainless electrons on a onedimensional chain on I with a chemical potential and a key wave on parenting amplitude that's defined on the nearest neighbor pair of electrons so just adds a and the nearest neighbor pair of electrons look at and so this is a on a simple model to write down and solved on and off in particular but so what if we write down the the Bogle about Agenda matrix were now this is the matrix in the Anambra space or call the Palin matrices power in this case on then and I then this is the this is the Hambletonian for the curtail model and again as in the sewers referred here model we can write it on you know as a vector darted into the and the power matrix OK but now this is almost like this usually freedom all but on by it has a slightly different cemeteries of the particles symmetry that we have in the sutra vehicular model is not the same particles symmetry that we have in the in the superconductor model so Sobhraj in particular in this case we have it and we have a constraint that the diva acted the particles cemetery gives us a constraint so so in the US here Molly was DC equals 0 remember that that that was the constraint we had on in this case and what we have is is is a constraint that relates the decay to the admonished press so it's a slightly different it's not exactly the sutra vehicular model of closely related in a particular part this model also has 2 different phases depending on the value of the chemical potential and if the chemical potential is very large then under the vector but doesn't wrap around the origin OK on so that died but I know it looks like this and so this is the topologically trivial cases whereas if the chemical potential is the size smaller than we have a weak parent phase where I am where the director does wrap around the origin and this is the topologically nontrivial case the 1 which has the instruments and a now 1 by there's
1:45:52
even a nicer version of this again due to curtail on which is to take this small which has 3 parameters and on and simplify and in particular
1:46:04
but so what we can and intend to write it in terms of Mark Fuhrman's awarded did is really taking each of these and other sites which has an operator Sea Island on it and we're right CI but in terms of 2 Meyer on a far beyond operators gal 1 I'm going to try to and get and if we do that then then what will have is that each of these I was sort of green and blue ovals describes a given our studio site of the lattice with the the 2 Gamez the live on and so so it's not too hard to translate the model that I had there into into the Gamez it's just a matter of love you know doing during this and what 1 finds is that for a particular the ratio of gamma to T when when area of Delta to T1 Delta is equal to tea then it becomes just a nearest neighbor chain in terms of the my rights so if I if I draw out the Meyer honors from you know in order of like this then I only have nearest neighbor interactions on 1 side in this case and for this particular combination and so now we have on 2 different values of tea and which 1 correspond to the operators sort of within a of you know a lattice site and then the ones between the letter sites compared and arms so now you can see now it looks even more like the Sioux Schriever here modeled on on where it looks like you have an alternating from alternating I bonds and in particular but I know what you can see that they're going to be 2 phases depending on whether you have a dimerization pattern like this in which case of every I Adam has a every mile on has a program that it gets gap with can't or looks like this where the sort of left over Myra fermions at the end go on and on and so
1:48:24
on so this is a very simple and intuitive way of understanding how you get these on Myron and states along in this column in this tale of a my congress OK so on and and again this simple thing you can imagine is that done but you can imagine the Ltd which team won the strong bond is very strong in T 2 those 2 0 0 and in that case you know each of these pairs sort of gets a big tapped so there's a big gap in the ball but that these these wines don't coupled anything and so the of his remarks OK and then done and if I make this tea on the little T to finite then on you know on there'll be an exponentially small coupling between these of these and states OK so that the Meyer but so
1:49:26
I so the next thing to think about is that you know a twodimensional from superconductor and on so and so again but let's 1st think about the book boundary correspondents so if I have a twodimensional Of the bodies and so you know theory what kind of protected states might have OK and so on so what so the inequality in the ordinary case we had the court ruled that states on 5 where you know it didn't look like this now invisible theory remember the states of positive energy are going to be redundant with the states of negative energy from going on and so on so this higher role eds state with this particle redundancy that is called a car Ohlmeyer as opposed to a choral director from and stick together and so on and so this gone and then if we have this then you know again is protected in the same way that the ad states and foremost affect our protect can get rid of AlQaeda and so on and so on and so these will occur at the edge of a twodimensional topological superconductor so on so examples of of of a twodimensional topological superconductor but if you have a PX was psyche wise lists of superconductors gives you on it gives you an equals 1 but if you have a on a coral Triplett pwave superconductor like like most strongly resonate then so that also is a topological supra note with an to so so there's a model looking right down on top of that actually make it through goes back to a model that read green studied when they were looking when they're explaining the new was 5 halves of Cornwall effect on and on and so on and so on and so on can write down a continuum theory or 1 can write down a lattice some version of it that looks something like this again with a deal vector and that's defined as far as a function of you know it's a function of kayaks and KY but it's a threedimensional vector and then the on the interesting so the question is on so this these other Bogle but in Hamilton audience will be classified by the Chernoff OK just like the ordinary on my block Hambletonian as it in 2 dimensions is classified by Cher number so is the on the above about John Hamilton and so so it depends on the turnover that are in this detector model it depends on the number of times the D vector wraps around the sphere so in this case the strong .period face on the tour numbers 0 doesn't wrap around the origin of all whereas on in the week .period face on it wraps around the origin and intuitive ,comma 1 considers this is the topological so the week pairing faces the topological superconductor stronger in phases of trivial look at so and so
1:53:00
that's the twodimensional a topological superconductor that 1 of the interesting things about the on a twodimensional topological superconductors that you can get on 1 of the 0 dimensional Meyer estates and analogous to the end stable onedimensional topological serving after you get those at a vortex unity on
1:53:22
in a twodimensional topological superconductor and wait and think about that is on a matching you on so vortex is of course occurs when a quantum of magnetic flux on by a thread through a hole in the superconductor and and seeking Nagin cutting a hole in superconductor and so then what's going to happen let's make the whole you know kind of big and if we do that then and then there's an edge army and there's going to be in that state running around the hall I'm going on and on so what it's actually down 1 has to think a little bit about what the boundary condition for the wave function of the I want the particles it goes around the whole so turns it it turns out that because the way you go around the Holzer rotates by 2 pine and if there's no vortex there then it picks up a minus sign OK and so on so it picks up a phase which is minus 1 to be lied to the people is 1 where P is the number of each of the universe superconducting a 4th quarter you have thrown in the water to grab and so so what's going to happen then is since this state has a finite length but momentum on his status could be quantized possessed periodic boundary conditions program and on the periodic boundary conditions is such that when the when PC even there and the state's Miss 0 whereas 1 piece on the state there is a state exactly at 0 look at and so if you have a single H over 2 we vortex then on then would you expect is that there should be on a 0 mode on that vortex now in general on you know you have is a remote area lots of other states all these other states of finite energy to and so there is that that the classic on theory of this bike religion and that you're kind of shows that don't you know in the you know what I didn't get a medal honor the this level
1:55:42
spacing and is given by the square of the superconducting energy gap divided by the firm in craft stuff was so abide by the important thing here is that there is is is this song is this is this OK now what
1:56:07
you have to do so so we said we've seen that done you know extra cemeteries time reversal of particles symmetry change the kinds of topological structure that we have invented the and done so it turns out that these are all part of a big beautiful mathematical structure ,comma by which was worked out by the on the lexicon tale and also by undress Woodward and his collaborators and on time and so on and so the question is in and so what would these others were on able to do is to classify all the possible cemeteries on the can have on and I and so it turns out that there are 10 and only 10 symmetry classes which are distinguished by the presence of some of the time reversal symmetry which square plus 1 minus 1 so so if you have on stainless electron spin 0 then it's worse plusone 1 compared worse than half minus 1 you can also have the particles symmetry the squares to plus or minus 1 again the for 1st half electrons and its squares to plus 1 on and then you can also have a cemetery which is like the product of the 2 which gives you sort of a I'm kind of a particle like cemeteries become a unitary coral symmetry that like what happened in the in the in the original and social freedom all that talk about craft and so on and so on and so these different possibilities I give you 10 possible cemetery classes and so on and so on what 1 can show is that the pattern of some of topological structures as a function of symmetry class and dimension of space have this remarkable long pattern to the Hungarian so there's very deep mathematics that
1:58:27
goes into this into this into this pattern and 1 can say words like a theory and but curiously that I don't want to go into what those with this meeting because I only have very dim understanding of them myself so on but what we can do is we can understand on this table on the things that we already know OK so a particular on Class a is on is the class that doesn't have any of these asymmetries program so that for example what happens in the warm all effect we don't have time reversal you don't have a
1:59:00
particle you don't have anything on and I Class AA in 2 dimensions is classified by an integer topological invariant that's the formal connect similarly Class H 2 is what happens when you have time reversal symmetry the squares to minus 1 so that spent I have with time reversal symmetry that's the topological insulator which has a C 2 topological classification in 2 dimensions and easy to topological classification in 3 dimensions of these with the on the 2 and threedimensional topological insulators that we thought that we already learned about interestingly in 4 dimensions of a topological legislators have any integer topological invariant so why and then similarly the on Class indeed is the case that has the particles cemetery squared plus 1 but we saw that we had in 1 dimension we saw that we have as you too a topological invariant because we habit and state or we don't OK that's
2:00:11
Zito whereas into dimensions we have an integer topological invariant because you can have any number of arms of car Ohlmeyer on states so so these also the things we know all fit into this pattern and and and there's a lot more here on and maybe there are other states that we have not discovered yet that can fill out other parts of this diagram so for instance Class D 3 Is is superconductors with time reversal symmetry and in particular on this 1 corresponds to a phase of helium3 and so on and so you know so there are time reversal invariant and so it would be injured and this would be a threedimensional topological superconductor if we could find 1 Tokyo foreign so on so this is the Periodic Table and
2:01:09
so on so let me say a few words about Hawaii we like Myron of fermions so much in 1 of the 5 important ideas again going back to curtail is the idea that if 1 could have these Myra fermion bound states OK then on 1 could use the properties to manipulate quantum information come on and so the basic idea is this this this fact which told you which is that on a pair Of these Meyer affirming unbound states defines a single direct for me on which itself defines from a twolevel system can you see the occupied or it's not occupied so to Cuban OK so it's an element of quantum information and the beauty of it is that is that if the admirer of fermions are separated from each other so if you have to enter these Meyer is that you have and Cuba it's like the beauty is therefore if they're far apart from each other
2:02:19
then the quantum information stored in these 2 bits is topologically protected in the sense that In order to measure the state public you that you have to do a nonlocal measurement the couples to both of the 1 of the Myron on if you just do a local national which only measures 1 that there's no way you can measure the state of that on you know that that's encoded in In those those moronic about states and if you can't measure it by a local measurement than that means that the environment can't measure it with a local national and that's the whole problem with making a quantum computer is trying to keep the environment from accidently measuring the state because of the environment measures at the state you know gets destroyed program and so that's the the coherence problem for quantum computing and all of
2:03:21
them so but these Meyer estates and to the extent that they are exponentially decoupled from 1 another From there they are immune from local sources of the coherence there so this is something that's a very important property and that they have found in addition if you want to have a quantum computer would you like to be able to do you like to be able to you would like to build a strong quantum information than you'd like to be able to do to manipulate to do an operation to like to build operate with the unitary operator that you can control find on your quantum state and on so so this is not possible to to a property of these of these are some of the states which I is that they're not a billion statistics in particular on time if you break them around so so there now and then you can now buy you can change the change the state's OK so if you create pairs of these where you create them and their bimbo each pair is in the
2:04:31
0 state then if you do an operation where you exchange 1 of them then you will end up in this Parliament entangled state compare and so on so this allows you bye you know doing breeding operations were you move the quasiparticles or all around each other are you exchange that allows you to do some but not all but on unitary operations that you would need to make universal on computer OK so just by doing riding alone you can't do universal ,comma Kwan computation of his wanted to add something to this nor do that but you can do something you can do a nontrivial operation cancer even saying that I think would be would be an exciting thing program so and so this is that you know you know 1 of the things which sort of motivates us to today keep trying on this and that but it's a hard problem OK on so I see you talking you the topological superconductivity the problem is that topological
2:05:37
superconductors don't come so easily compare and so on and so 1 possible the route which is the 1 people been thinking about it and for the longest time on is the quality of particle expectations in the in the more read the quantum state that is believed to occur at times but I knew it was 5 House OK and so on there actually on a fair number of experiments of people trying to observe the signatures of these Meyer on estates and at this point there's some I it's pretty encouraging on the 0 I don't think it's conclusive commented that at this point I that .period you know that the quasiparticles on indeed have the expected of a dynamic billion statistics another possible route 1 might I go used to look for an unconventional superconductor see most of the conventional superconductors that we know about the moral trivial superconductors OK so it's it's so we are really haven't been handed any topological superconductors yet and so 1 idea is on is this and I was strontium originate a single plane of strontium resonate on there's also some of the ideas that may be what could make these kinds of states with along with cold Adams ,comma and then there's another possibility but there is a massive soap business selling is on you know are best topological insulator on it turns out if you don't with copper it becomes a superconducting metal is not it's not insularity anymore but it's it's a metal on but but but on this is a superconductor OK I'm done by on there's some reason to believe that this might be a topological soup company with don't know at this point we still don't know now I know there are possible on 1 can do and so is of our spend the rest of my time on discussing 1 is to I used to use ordinary superconductors because there are lots of those but to combine them with on other things to give us proximity effect devices OK and so in particular on what I'd like to discuss what happens if you combine superconductors with topological insulators and then there are also on the rise number of ideas too use on a more conventional semiconductors on our combined with some sort of magnetism more magnetic field located in particular are very promising direction is on the site snide quantum
2:08:24
watched OK so on
2:08:28
and so what happens if you combine a topological insulated with a superconductor so and so what can happen is that on the Cooper pairs from the superconductor Candide tunnel into the surface states of the topological insulated and so on and you know and make them superconductor OK and so on and so by coupling nearby by having electrons flow back and forth between the topological insulator and superconductor that effectively makes the you know so this the the surface state of topological is insulated from inherits the superconductivity from Super so it opens up a superconducting on energy at home and so this superconducting state on of the surface states but is a lot like a topological superconductor so particular you know I told you that the service is like half of the ordinary on the electron deserts and 90 generates from electron gas OK there's no spending generously each
2:09:37
state each point on the Fermi circle as 1 state OK and so it's like happen ordinary superconductor on and so it's like a specialists superconductors that also only have 1 state on this form a circle and going on and so on and so particular and so this side this size superconducting state is similar to a people IP superconductor Grant ,comma except a people's IP superconductor necessarily has to break time reversal symmetry France it's time reverses a P minus ii the superconductor compared the on the superconducting state of a topological insulator
2:10:18
and does not break time reversal symmetry OK it's really S waves singling superconductivity the difference between the peoples IP and the topological but was super going on surface superconductor is there's a minus sign you have to get you know you know on the minus side you have a piece state OK but I am and that minus sign you have you have because it's really coming from the Polly principle a people pieces superconductor it you know if you have a specialist superconductor the pairing has to be the way because the because of the of the Kelly the principal focus but that minus sign In the topological insulator as elsewhere superconductor is that my side is provided by the by the Pied bury things that you get when you go around and the firm circles OK so that might decide which which gives you the correct my sign that allows you to have an S waves superconductor on the surface of your topological so on diet died but the important thing tho is that like that it was likely if you have a vortex on in this in this song by surface superconductor then you get a moron about please so someone so this is in principle where you can engineer Mark has
2:11:40
so so so so 1 way to do it is to have a superconductor on top of that of a topological insulator and the threat of war taxfree drawl and then the prediction is that there be about state on at this summer at the same now in order to have this bad state and be well separated by an energy gap but it's important that you not have the Delta square over here on the statesman and so 1 thing 1 can do so on the topological and letters if you can tune the family energy so that it is exactly at the direct .period that equals 0 OK so so that the dead Delta square over a year can be as big as as as as dealt with and so on so you get this or that there is a remote another way you can get a 0 mode is on the line if you have an interface but between a super could you know so so the superconductor on 5 year coupled to a superconductor opened up a gap in the surface
2:12:44
states it's a superconducting searching another way you can open up a gap is by breaking time reversal symmetry and couple into a magnetic to perfection compare that give us the surface quantum Hall effect in the twodimensional in the in this on the surface of the threedimensional topological watch closely so on and so on so the 2 different kinds of mass gaps that you can open up and so something happen interesting happens on the interface between them so in particular this is in the twodimensional and topological and so are the ones that Holland's letter that has the edge states if you I make 1 half of its superconducting and the other half have a magnetic energy gap then on the interface between the 2 it's like you have a it's like you have amassed the changes signed and there's a domain wall as 0 mode Meyer on his arm of that occurs on that on that now infects so 1
2:13:40
can also do this with me and with the on the surface of a threedimensional topological and so you could have an interface between magnetic regions superconducting region and this will have a bomb in a car all the time that I my on affirming that the governments of Canada on the America also but if 1 has a superconductor but I knew Josephson junction on a topological insulator than when the phase difference across this Josephson junction is equal apply that you get on the 9th viral somewhere for me on that lives on this on the structure terms of the various ways that 1 can engineer a fermions using these topological and slid a superconductor devices but
2:14:31
so the other on idea on which is to use it and so so so 1 problem is that the topological insulator materials have their own problems OK so on so the topological ancillary materials that we know the insulators but they're not such good insulators OK they still conducted the VOC content on it and so I you know you so so another idea is to use materials the war highly developed on more ordinary semiconductor materials on care and on but in order to do that and you need to on you need to I have a magnetic material and superconductor and so the idea is that if you haven't of conventional semiconductor with a strong spin orbit interaction then on you get these on the sort of rushed the dispersion of the some of the bonds to get this this and direct .period and the states on that the comes from this Rosh for splitting and so so so this is you know this is a direct .period but it's not good not because you also have this surface here and so the idea is 2 and but turn on a magnetic field which breaks time reversal symmetry on war is a minefield would you know which breaks time reversal symmetry to open up a gap at this direct .period but then you still have this Fermi surface here and then if you do that correctly this Fermi surface here again makes it look like a people aside the most analysts a superconductor program so this is a route toward the engineering from ATI twodimensional a topological superconductor work there is a variant on this where you can on I do this with a onedimensional on semiconductor on for instance these on India Morrison I a onedimensional quantum wire or something and so on is actually on the acquired the effort going on these days too I'm tried to on factories and our Snyder quantum wires and some of this work is going on here in the Netherlands on it by the time you know and and some of the motivation for that is is is this is this is this proposed it's hard the challenges is that you have to have these on semiconductors and we had a very low density OK in order to get the Fermi energy into this sort of energy gap here and if you have semiconductors a low density than it and it has to be very pure or otherwise of the model the localized pain so this is it's a heart problem but done but maybe it's something we can make some OK
2:17:18
and on let's see I think well although this very quickly so so you can ask yourself on let's suppose that you had Mark Fuhrman's what might you do in order to tell you have guys really is you have to do some experiments on that Salemme Justice Robert coupled experiment so so the simplest experiment you could do use you could tumble into it and try to see the 0 bias anomaly complains so that maybe will be the 1st thing to do another thing that 1 can do on his 1 can make a kind of interferometry I and and so the idea here is the following that on I want combine a topological insulator with magnetic materials mom of opposite sides and so on the interface where on on the on the domain wall where the magnetism changes signed there's a choral director firmly on the properties on again and now I want to put a superconductor in the middle of this and that Carroll Dirac fermions will splits into 2 Tyrol Myron affirming its kind and so it's like a beam splitter for the Myron on affirming someplace so 1 of the moron fermions goes around this way the other 1 goes around this way and then on the other side they recombined into a choral director and so this is sort of like
2:18:41
a interferometry for and for the Myron firm a particular now of course I the Myron firmly on the gamble it it can have any phase can only have phase 0 or Pike is Gamez real B plus or minus OK and so what 1 can do so is 1 can imagine on not having a flux through the superconductor or you have a flux disappeared after cost flexes superconductors quantum OK so if you have a single h over to fluxes superconductor than that has the effect of changing signs of 1 of the moron from him so wet weather will water recombined it has the
2:19:28
opposite side and that has the effect of Channel Changing electrons into a hole Hungarian so what would it really expect is that as a function of the flights the current that flows across Europe should go up and down the you're sending electrons in but if you have an odd flux you're getting a whole come out if you haven't even flux you're having electron ,comma OK now costs at the whole comes out then that means the Cooper pair goes into the superconductors OK so this is that this is 1 of a particular on signature of 1 of the moron on states that would have on another thing that
2:20:09
1 might be able to do is to do is to make a kind of justice and device wear on you have my from states which are coupled across Josephson junction and then on 5 white and what happens is that as you advance the phase across this Josephson junction by 2 pot so you normally expect is that when you advance the phase across the Josephson junction by 2 pi or get back to where you started so that on the on the current phase relation has to apply operators that an ordinary justice in this case on when you advance the phase of the Josephson junction by to apply these Meyer on estates change sign OK and so that means that when you started off in the ground state of this despair of ironic that might remind my honor for me and you end up in the excited state OK and so that means that as you advance the phase on you don't get back to where you started on until you get to 4 instead of 2 .period can now cost but fundamentally the Justin of justice effect has to be to Piper OK it's about means there has to be another state as well that that that that looks like this and the reason that these 2 states cannot Cross is that in order to get from 1 state to the other you have to have a Myron fermion on you have to have it in electron basically has 2 tall from the top to the bottom of the samples cancer that can happen if these are separated and by
2:21:49
buying a large distances and so so what this time of predicts that there should be a 4 pie Imperia the which 1 maybe could measure within a siege of some of the effect OK so on so with that I think I may be I'm not I'm finished so
2:22:06
on so I guess this conclusion is just for the 2nd half but does basically there are lots of theoretical challenges you know the 1 can take off from this and I even more experimental challenges so I hope but I hope you got something out of a sow's 1 for me so thank you very much and I am happy to stay