0:00

So it's a great pleasure to

0:02

be giving the talk about

0:04

anions, one of my

0:06

favorite subjects. I have

0:08

this little comment down

0:11

here, respect the young, because

0:13

as you'll see, a lot

0:15

of the great progress in this

0:17

field was made by people who

0:20

are mere spring chickens. So the

0:22

idea of an anionns is really

0:24

trying to answer a question. of

0:27

what happens when you exchange

0:29

two identical particles in quantum

0:31

mechanics. It's an old question.

0:33

It goes back to a

0:35

now famous letter from Satendra

0:37

Bowes to Albert Einstein, written when

0:39

Bowes was 30 years old. The

0:41

letter says, this is about 100 years

0:44

old, 101 years old, now I guess

0:46

this year. It says, respected sir, I

0:48

venture to send you the accompanying article

0:50

for your perusal and opinion. And he

0:52

asked Einstein to help him get it

0:54

published in Zaitre for Friseek, which was

0:57

the leading journal of the time. He

0:59

says, though a complete stranger to you,

1:01

I do not feel any hesitation in

1:03

making this such a request. So what

1:05

Boes had done is he had derived

1:08

using the basic principles of statistical mechanics,

1:10

the distribution function of how photons will fill

1:12

modes in a cavity. Now Einstein read

1:14

this paper he realized that it was

1:16

it was not only correct But it

1:19

could be applied to lots of other

1:21

things to particles that were not photons

1:23

and in and in this way we

1:26

developed Boz Einstein statistics that applies to

1:28

all product particles that are what we

1:30

call Bozons that includes photons pions pions

1:33

fluons phonons exeons and of course the

1:35

famous Higgyz Bozahn the very next year

1:37

When Polly was 25 years old

1:39

he formulated his exclusion principle this

1:41

to remind you is the principle

1:43

that says you can only put

1:45

one Firmion in each orbital It's

1:47

two if you count one spin

1:49

up and one spin down and

1:51

this principle of course is is

1:53

Fundamental to the periodic table and

1:55

and all of chemistry and everything

1:57

else in physics as well realize

2:00

these particles don't obey the same

2:02

kind of statistics as photons. We

2:04

therefore needed another type of distribution,

2:06

another type of particle, and this

2:08

is what we now call fermi-dirac

2:10

statistics, which was derived first by

2:12

neither fermi nor dirac, but it

2:15

was derived by Pascual Jordan, who

2:17

is 23 years old at the

2:19

time. So there's kind of a

2:21

long story about why it is

2:23

that it wasn't named after Jordan.

2:25

Jordan wrote his manuscript. He sent

2:27

it to the journals, actually for

2:29

a physique. The editor was Max

2:31

Bourne, who was a well-intentioned but

2:34

rather forgetful guy. Max put it

2:36

in his suitcase with the best

2:38

of intention to take it out

2:40

and read it, but then he

2:42

forgot about it. And it stayed

2:44

there for the better part of

2:46

a year, during which the same

2:48

result was published by Fermi and

2:50

Iraq. So then we had firmidaract

2:52

statistics which applies to in particular

2:55

electrons, but also to all particles

2:57

that are firmions including nuanced quarks

2:59

and so forth. Now the scientific

3:01

community is usually pretty good about

3:03

correcting errors of attribution. Max Borne

3:05

was very clear that he had

3:07

made a mistake, he was very

3:09

apologetic about it, he told everyone

3:11

that he had made this error,

3:14

he felt guilty about it, he

3:16

felt guilty about it, for the

3:18

rest of his life, having robbed

3:20

Yordan of credit that he rightly

3:22

deserved. And under most conditions, the

3:24

scientific community would have renamed Firmidirect

3:26

statistics into Firmidirect statistics, but this

3:28

didn't happen, and the reason it

3:30

didn't happen is because a few

3:32

years later... Jordan became very prominent

3:35

Nazi and and pretty much no

3:37

one liked him and no one

3:39

felt the need to do him

3:41

any favors. So you know there's

3:43

a moral to this story which

3:45

is don't become a Nazi. This

3:47

is my joke about American politics.

3:49

So anyway by 1930 the of

3:51

quantum mechanics were finished, quantum field

3:54

theory more or less finished by

3:56

1950, and during that time and

3:58

since that time, you might wonder

4:00

if people asked if there are

4:02

other particles out there, there's bosons

4:04

and there's firmions and is there

4:06

something else? And over and over

4:08

people came to the same conclusion,

4:10

which was no. All you have

4:12

is bosons or firmions and nothing

4:15

else. And if you open up

4:17

your favorite quantum mechanics textbook, chances

4:19

are that's what it says. Lots

4:21

and lots of quantum mechanics textbooks

4:23

have that answer in it. And

4:25

they all give the same argument,

4:27

which is very simple, and I'm

4:29

going to give that argument right

4:31

now. It's a pretty easy argument.

4:34

You define an operator called the

4:36

exchange operator, which switches the position

4:38

of two particles. So the exchange

4:40

operator applied to sie of r1,

4:42

gives you sie of r2, r1.

4:44

If you apply this operator twice,

4:46

you get back to where you

4:48

started. exchanging twice the identity, there's

4:50

only two square roots of the

4:52

identity, therefore there's only two possibilities.

4:55

If it's a plus one, you

4:57

call it bosons, if it's a

4:59

minus one, you call it firmions,

5:01

and that's all you're allowed to

5:03

have. This is a great argument,

5:05

it's very simple, it's very clear,

5:07

unfortunately it's also wrong. So this

5:09

was not realized for quite a

5:11

long time, until 1976, with this

5:13

beautiful paper... by the two yons

5:16

in Oslo, Jan Magnolinus and Jan

5:18

Murheim, who are 28 and 30

5:20

years old at the time. Obviously,

5:22

they're a little older in these

5:24

photos. And they pointed out that

5:26

if we lived in a two

5:28

plus one dimensional universe, that's two

5:30

spatial dimensions and one time dimension,

5:32

then you could have other type

5:35

of particles as well. And what

5:37

they envisaged was the idea that

5:39

if you exchange two particles, say

5:41

counterclockwise, the wave function would pick

5:43

up a phase E to a

5:45

phase E to E to the

5:47

E to the I-theta. Theta equals

5:49

zero means no phase, that's bosons.

5:51

Theta equals pie, EDDI, pie is

5:53

minus one, that's firmions. But they

5:56

pointed out that in fact other

5:58

values of theta, any value of

6:00

theta, is really. also allowed by

6:02

quantum mechanics if you live in

6:04

2 plus 1 dimensions. Now, from

6:06

this paper, there's a number of

6:08

things we can conclude. You might

6:10

be tempted to conclude from this

6:12

paper that everyone in Oslo is

6:15

named Jan. This is, in fact,

6:17

not correct. I assure you. But

6:19

a little bit of a coincidence.

6:21

They both happen to be named

6:23

Jan. But there's other things that

6:25

you should conclude. One thing you

6:27

should conclude that there was something

6:29

wrong with the argument I just

6:31

gave you. And indeed, there is

6:33

something wrong. When you define an

6:36

exchange operator, you need to say

6:38

how you exchange the particles. So

6:40

to make that more clear, in

6:42

2 plus 1, as Shabaji actually

6:44

mentioned this earlier, but I'll give

6:46

the argument again. In two plus

6:48

one dimensions, if you exchange particles

6:50

counterclockwise and you exchange them counterclockwise

6:52

again, if you look at the

6:55

world lines of the particles, the

6:57

paths in space time, you will

6:59

notice that the world lines have

7:01

knotted around each other and it

7:03

becomes more clear if you connect

7:05

up the top to the bottom.

7:07

And now you have two strands

7:09

which are knotted with each other.

7:11

This is not the same as

7:13

having not exchanged the particles at

7:16

all. So two exchanges is not

7:18

equal to the identity. Now the

7:20

reason we got away with saying

7:22

two exchanges is the same as

7:24

the identity is because we usually

7:26

think about three plus one dimensions.

7:28

And in three plus one dimensions,

7:30

two exchanges actually is equal to

7:32

the identity. That comes from a

7:35

topological statement that if you're living

7:37

a space with a total of

7:39

four dimensions, four dimensional space, and

7:41

you have one dimensional strands, you

7:43

cannot make knots in one dimensional

7:45

strands living in four dimensional space.

7:47

If this is not obvious to

7:49

you, ask me at the end.

7:51

We can probably make it obvious.

7:53

But it is a true topological

7:56

statement. OK. So there's some other

7:58

things we can conclude from this

8:00

paper here. One thing that we

8:02

can conclude, which is quite important,

8:04

is that the scientific community isn't

8:06

that good at realizing when something

8:08

important has happened. This paper was

8:10

more or less completely ignored for

8:12

the first. few years of its

8:15

life. It was cited five times

8:17

in the first five years of

8:19

its life. Three of those citations

8:21

are by the young Magnolinas himself.

8:23

So pretty much no one was

8:25

paying attention to this at all.

8:27

But a few years later, this

8:29

person Frank Wilcheck did take notice

8:31

and found it very interesting. Now

8:33

Frank Wilcheck was already very famous.

8:36

for work he did when he

8:38

was 22 years old in 1973,

8:40

which would later win him a

8:42

Nobel Prize, Asymptotic Freedom and QCD,

8:44

which turns out to be very

8:46

important. So people were watching what

8:48

he was doing, and once he

8:50

got interested, then a lot of

8:52

other people got interested in this

8:55

as well. Another thing he did

8:57

is he gave a name to

8:59

these types of particles. He called

9:01

them anions, particles that have any

9:03

statistics besides bosons and firmions. Will

9:05

check is particularly good at coming

9:07

up with cute names. But what

9:09

he was actually concerned with is

9:11

the famous spin statistics theorem. To

9:13

remind you what the spin statistics

9:16

theorem is, it's a statement that

9:18

if you have two identical particles

9:20

and you exchange them, you accumulate

9:22

some phase. Or if you take

9:24

one of those particles and you

9:26

rotate it around its axis by

9:28

two pie, the phases that you

9:30

accumulate in those two processes should

9:32

be the same. For bosons. You

9:34

get no phase for rotating, you

9:37

get no phase for exchanging. For

9:39

firmions, you get a minus one

9:41

for rotating, you get minus one

9:43

for exchanging, and for anyons, does

9:45

the same thing hold up. And

9:47

in fact, it does, and that

9:49

was kind of interesting. He notes

9:51

in his paper, although practical applications

9:53

of these phenomena seem remote, they

9:56

do have considerable methodological interest and

9:58

shed some light on the spin

10:00

statistics connection. So he couldn't imagine

10:02

how you would ever be concerned

10:04

with a two plus one dimensional

10:06

universe, but it's a nice toy

10:08

problem to play with. The same

10:10

year, however, was the discovery of

10:12

the so-called fractional quantum hall effect,

10:14

about which I will say a

10:17

lot more in a moment, but

10:19

it's observed in two-dimensional electrons in

10:21

high magnetic fields and low temperature.

10:23

temperatures, hint, two-dimensional electrons. So how

10:25

do you get two-dimensional electrons? Well,

10:27

in, oh, this was, this discovery

10:29

was made when Horse Stormer was

10:31

33 years old. So to make

10:33

two-dimensional electrons, the way they did

10:36

it was they sandwiched a thin

10:38

layer of gallium arsenide between layers

10:40

of aluminum gallium arsenide and they

10:42

trapped electrons in this thin purple

10:44

layer here. In fact, perhaps the

10:46

more important discovery, even though the

10:48

discovery of fraction quantum hall effect

10:50

was an important discovery, the more

10:52

important discovery was made by a

10:54

horse stormer several years earlier in

10:57

where he figured out... how to

10:59

make such semiconductor structures without introducing

11:01

a lot of disorder into the

11:03

galley marcenide. This is a trick

11:05

known as modulation doping. It's used

11:07

industrially on all sorts of semiconductors.

11:09

It was a very profitable patent

11:11

for Bell Labs, the company was

11:13

working for a long time. The

11:16

patent has now expired, I think.

11:18

Anyway. In the modern era, there's

11:20

other ways to make two-dimensional electrons,

11:22

and a really interesting one is

11:24

the idea of using single atomic

11:26

layers of carbon, what's known as

11:28

graphene. Carbon can make a single

11:30

layer in a little honeycomb pattern

11:32

like this, where each of these

11:34

balls is a carbon atom all

11:37

stuck together. It was discovered that

11:39

you could do that in 2004

11:41

by Kastu Novasilov and Andrew Gaim.

11:43

Castillo is 30 years old at

11:45

the time. And this is another

11:47

example of the theorem that the

11:49

scientific community isn't very good at

11:51

understanding when something important has happened.

11:53

In fact, they had a lot

11:56

of trouble getting their work published.

11:58

It took them about a year

12:00

to get it printed anywhere, and

12:02

six years later it already won

12:04

a Nobel Prize. No one realized

12:06

why this was really super interesting,

12:08

but then all of a sudden

12:10

everyone realized, yeah, this is super

12:12

interesting. Anyway. Making single atomic layers

12:14

of carbon is a modern way

12:17

of making two-dimensional electron systems about

12:19

which you'll hear more later. Anyway,

12:21

so in 1982 this effect... fractional

12:23

quantum hall effect was discovered. The

12:25

theory of fractional quantum hall effect

12:27

was laid out its basic parts

12:29

by Bob Laughlin. He was 33

12:31

years old at the time. Actually,

12:33

academically he was even younger than

12:36

33 because he was forced to

12:38

join the military because of the

12:40

draft and he lost quite a

12:42

few of the number of his

12:44

young years and not studying physics,

12:46

which is what you should be

12:48

doing when you're young. Anyway. The

12:50

group of them would win the

12:52

Nobel Prize in 1998. So what

12:54

about Fractional Quantum Hall effect was

12:57

so interesting that it deserves a

12:59

Nobel Prize? Well, the next year,

13:01

two groups managed to show theoretically

13:03

that the low energy particles that

13:05

arise in Fractional Quantum Hall systems

13:07

really are anions. The people involved,

13:09

Bert Halpern, my thesis advisor as

13:11

it turns out, and we'll check

13:13

we've already meant, Rob Shriefer. is,

13:16

well he was 26 years old

13:18

when he did his Nobel Prize

13:20

winning work in 1957, the BCS

13:22

theory of superconductivity, a very important

13:24

major breakthrough in physics, and the

13:26

graduate student who did all the

13:28

work, Dan Arovas, was 23 years

13:30

old at the time. So anyway,

13:32

theoretically, we believe that in these

13:34

fraction quantum hall systems, we do

13:37

have anions running around. So the

13:39

history of the field, just summarizing,

13:41

is by 1920s, we had bosons

13:43

and firmions. The first proposal of

13:45

an anions was in 77. By

13:47

1984, we believe we actually had

13:49

an experimental system where anions exist,

13:51

the theoretical community. accepted this almost

13:53

immediately. It became gospel among quantum

13:56

condensed matter physicists. Everyone learns this

13:58

in graduate school. It's sort of

14:00

fundamental to a lot of our

14:02

understanding of modern condensed matter physics.

14:04

But as Shivaji said, often theory

14:06

outruns experiment. It took a very

14:08

long time before this statement was

14:10

expert. it was confirmed experimentally. Before

14:12

we actually had an experiment where

14:14

we could show that exchanging two

14:17

of these particles would give you

14:19

a phase, which is not plus

14:21

one or minus one. So that's

14:23

what I'm going to talk about.

14:25

So before going on, you might

14:27

ask, why are you interested in

14:29

anions in the first place? Well,

14:31

one reason is because it's a

14:33

fundamental interest. As physicists, we're always

14:35

concerned with what kind of things

14:38

can exist. At least in principle,

14:40

what are its properties, how could

14:42

you use it? So it's just

14:44

fundamentally interesting to begin with. Another

14:46

thing is maybe it's lurking in

14:48

plain sight. Maybe, I mean, there's

14:50

lots of experimental systems where we

14:52

don't actually know what's going on,

14:54

or we think we do, but

14:57

we're not entirely sure. Maybe there's

14:59

anions running around in lots of

15:01

systems, and we just haven't realized

15:03

it yet. There's also a surprisingly

15:05

large number of connections to fields

15:07

like high-energy physics, quantum gravity, pure

15:09

mathematics, and topology, which are also

15:11

interesting in their own right. But

15:13

the field got a huge boost

15:15

in 1997 by this person, Alexei

15:18

Kitayev, who was 33 years old

15:20

at the time, who pointed out

15:22

that if you ever have a

15:24

physical system with anions in it,

15:26

you have a really good way

15:28

to make a quantum memory, which

15:30

would be very useful. for a

15:32

quantum computer, should you ever build

15:34

a quantum computer? This idea took

15:37

hold, then working with Michael Friedman

15:39

shortly thereafter, they proposed the idea

15:41

of a so-called topological quantum computer,

15:43

where all the computations are done

15:45

by moving anions around, an aneons

15:47

of a particular type. Anyway, this

15:49

idea was so important that Microsoft

15:51

invested. I mean, I'm estimating this

15:53

number, but I think the estimate

15:55

is probably fairly accurate. Over a

15:58

billion dollars so far into trying

16:00

to produce a quantum computer that

16:02

runs on this principle. So the

16:04

other person who's involved here is

16:06

Mike Friedman. In 1981, when he

16:08

was 30 years old, he proved

16:10

an important mathematical theorem that won

16:12

him a fields medal. that's like

16:14

the Nobel Prize of Mathematics. The

16:17

very same year he also won

16:19

the American Rock Climbing Championship for

16:21

whatever that's worth. So he's a

16:23

tough guy to keep up with.

16:25

Okay, so what's kind of interesting

16:27

about the experimental confirmation which came

16:29

around 2020, 35, 36 years after

16:31

the proposal that we actually have

16:33

anions, is that it wasn't just

16:35

one experiment that did this, it

16:38

wasn't just one experimental system. a

16:40

number of technologies all matured at

16:42

roughly the same time, and we

16:44

had a bunch of experiments all

16:46

showing the same thing. So the

16:48

first experiment to come out was

16:50

the so-called anion collider experiment done

16:52

by Grendelfevs Group in Paris, done

16:54

with two-dimensional electron gases in Gallium

16:57

arsenide semiconductor heterostructures. Then there was

16:59

the anamine interferometer experiment. done first

17:01

at Purdue by Mike Manford's group

17:03

in galleymarsonite heterostructures and later done

17:05

by the Harvard group and the

17:07

University California Santa Barbara group by

17:09

Philip Kim and Andrea Young, done

17:11

in graphing carbon two-dimensional electron systems.

17:13

And then in addition, there was

17:15

simulation of anions. on quantum computers,

17:18

on rudimentary quantum computers. And this

17:20

has been done by a huge

17:22

number of groups for people who

17:24

are familiar with it. The Toric

17:26

code is basically an anion model

17:28

or the surface code. This is

17:30

basically the best error-correcting quantum code

17:32

we know of. So more or

17:34

less, every quantum computing effort in

17:37

the world is trying to build

17:39

anion models, more or less. And

17:41

it's been achieved by a number

17:43

of different groups by this time.

17:45

OK. The experiment that I think

17:47

is the nicest and the easiest

17:49

to explain is this one. The

17:51

graphing version has some properties, which

17:53

I like very much, and the

17:55

data is particularly beautiful, so I'm

17:58

going to show this one to

18:00

you. So first, I have to

18:02

explain a little bit about fractional

18:04

quantum hall effect. As I explained,

18:06

you need two-dimensional electrons, minimal amount

18:08

of disorder. You can get rid

18:10

of it all together, that's great.

18:12

We put a magnetic field perpendicular

18:14

to the plane of the sample,

18:17

and you cool it down to

18:19

very, very low temperature. It's one

18:21

10,000th of room temperature is more

18:23

or less where these experiments are

18:25

done. The number you want to

18:27

keep track of is known as

18:29

the filling fraction. It's basically the

18:31

ratio of the density of electrons

18:33

to the magnetic field made dimensionless

18:35

by a flux quantum H-bottom H-bar-Flx

18:38

quantum H-bar-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o- over the charge of

18:40

the electrons. So this is a

18:42

dimensionless ratio. And you can change

18:44

it by changing the magnetic field.

18:46

When this dimensionless ratio, the filling,

18:48

is approximately a ratio of small

18:50

integers. So P over Q could

18:52

be 1 over 3, 2 over

18:54

5, 3 over 7. Then fractional

18:57

quantum hall effect can occur. How

18:59

do you know when you have

19:01

fractional quantum hall effect? Well, you

19:03

measure something. And what you measure.

19:05

is some sort of resistance. So

19:07

you run current through your sample,

19:09

and you measure a voltage in

19:11

the same direction as the current.

19:13

What you measure is zero voltage.

19:15

Now if you remember for a

19:18

second that power dissipated is current

19:20

times voltage in the direction of

19:22

the current. So if the voltage

19:24

in the direction current is zero,

19:26

then you have zero power dissipation,

19:28

which means it's flowing without any

19:30

loss whatsoever. It's like a superconductor

19:32

or super fluid of some sort.

19:34

dissipationless flow. And that's kind of

19:36

interesting. More interesting is what happens

19:39

if you measure the voltage perpendicular

19:41

to the current flow. In this

19:43

case, the ratio of the so-called

19:45

Hall voltage, the voltage perpendicular to

19:47

the current flow, divided by the

19:49

current, this is known as the

19:51

Hall resistance, is 2.5 H bar

19:53

over E squared. E is the

19:55

charged electron times Q over P.

19:58

These two integers down here, exactly.

20:00

to the precision with which it

20:02

can be measured. That's about one

20:04

part in 10 to the 10

20:06

or one part in 10 to

20:08

the 11. That's like measuring the

20:10

distance. from here to California to

20:12

within a centimeter. It's an extraordinary

20:14

amount of precision, considering this is

20:16

a sloppy, messy, solid state experiment.

20:19

So there's disorder in the sample,

20:21

you don't know the shape of

20:23

the sample exactly, you stuck electrodes

20:25

on the sample to measure resistances

20:27

with big, you know, a soldering

20:29

iron, you know, that there's so

20:31

many things about this experiment that

20:33

you don't control precisely, you don't

20:35

control the magnetic field precisely, you

20:38

don't control the temperature precisely, you

20:40

don't control the temperature, you don't

20:42

control the vibrations, the most of

20:44

the things that you don't control,

20:46

and yet the result comes out

20:48

exactly, 2.5 H bar over E

20:50

squared times Q over P. All

20:52

right. So that's kind of cool.

20:54

Here's some real data taken by

20:56

Horstormer. And what you have here

20:59

is the longitudinal resistance down here

21:01

and the hall resistance up here.

21:03

Every time the longitudinal resistance drops

21:05

down to 0, you will see

21:07

that the hall resistance shows a

21:09

flat plateau, exact quantization. And these

21:11

plateaus are labeled by their P

21:13

over Q ratio. So for example,

21:15

this one's two over five, this

21:18

one's one over three, and so

21:20

forth. Each of these is a

21:22

different fractional quantum hall state. The

21:24

one we're going to focus on

21:26

is the simplest, actually the one

21:28

that was first observed in experiment,

21:30

is the so-called new equals one-third

21:32

fractional quantum hall state, which is

21:34

the easiest to understand. In the

21:36

new equals one-third fractional quantum hall

21:39

effect, you start with the ground

21:41

state, then you make some excitations,

21:43

and those excitations are particles that

21:45

surprisingly have fractional charge. You put

21:47

in electrons of charge E, and

21:49

the excitations now have charge E

21:51

over three. You have an emergent

21:53

particle with a fraction of the

21:55

charge of an electron. Now, how

21:58

does that happen? Well, the way

22:00

you should sort of think about

22:02

it is that the electrons form

22:04

a completely uniform soup. of uniform

22:06

density electron. and these particles are

22:08

the defects of that soup. Okay?

22:10

It sort of pushes a fraction

22:12

of a charge of an electron

22:14

away from some region and that

22:16

defect becomes the new low energy

22:19

particle. What's more interesting is that

22:21

these particles are also anions. When

22:23

you exchange them, you pick up

22:25

a phase of the two pie

22:27

I over three. They're neither bosons

22:29

nor firmions. But surely, you must

22:31

say. These particles really live in

22:33

three dimensions. Our universe is three-dimensional.

22:35

How can you know, maybe if

22:38

you have squashed them down so

22:40

they're approximately two-dimensional, are they? Well,

22:42

let's look a little more carefully

22:44

about what we've done. We have

22:46

our sample like this. We've tried

22:48

to squeeze our electrons down into

22:50

this little blue layer here. Let's

22:52

use a little bit of gratuitous

22:54

animation. blow up our system here,

22:56

and the potential felt by the

22:59

electrons is kind of a particle

23:01

in a box, kind of square

23:03

well potential. So the electrons are

23:05

living in here, blow that up,

23:07

look at it more closely. and

23:09

we're putting the electrons in there

23:11

and say, well, it's still living

23:13

in three dimensions, maybe they're sort

23:15

of confined a little bit in

23:18

the well, but they're still really

23:20

living in three dimensions, aren't they?

23:22

Well, think about that more carefully.

23:24

Remember that they form discrete eigenstates

23:26

in the z direction in that

23:28

well, and they occupy some of

23:30

the different eigenstates at low temperature.

23:32

They all get frozen down into

23:34

the lowest eigenstate, and you remove

23:36

any ability for them to change

23:39

their wavefunction in the z direction.

23:41

frozen, there's no freedom to change

23:43

anything in the Z direction, and

23:45

so they can only move in

23:47

the X and Y direction, they

23:49

become strictly two-dimensional objects. Okay? So,

23:51

people might be thinking something else,

23:53

another objection, but surely these aren't

23:55

fundamental particles, not... like an electron

23:58

is a fundamental particle, is it?

24:00

Well, you know, maybe nothing is.

24:02

You know, that we think of

24:04

an electron as being a fundamental

24:06

particle because on the energy scales

24:08

available to our experiments, we have

24:10

not seen it break up into

24:12

other things. We have not seen

24:14

it emerge from other things. But

24:16

that just means the energy scales

24:19

available to us, it looks fundamental.

24:21

It's the same thing here in

24:23

this low-tem temperature. person living in

24:25

this two-dimensional low-temperature quantum well, you

24:27

would swear that the particles, the

24:29

fundamental particles, are charged E over

24:31

three. And it's only when you

24:33

got yourself out of that two-dimensional

24:35

layer and could go up to

24:37

higher energies that you would notice,

24:40

oh, actually, it's the electrons that

24:42

are running around and the E

24:44

over three is just emergent. We're

24:46

always in the business of describing

24:48

physical systems at the relevant scale

24:50

for the experiments we can do.

24:52

All right. So this is sort

24:54

of the history of the story.

24:56

And the experiment I'm going to

24:59

explain is this one here. So

25:01

to explain this experiment in the

25:03

remaining 15 minutes is I'm going

25:05

to need to tell you a

25:07

couple things more about fractional quantum

25:09

hall effect, but not much. So

25:11

the first thing I have to

25:13

tell you about is quantum hall

25:15

edge states. So here, I've drawn

25:17

the blue region and the white

25:20

region. The blue region is where

25:22

there are electrons, my fractional quantum

25:24

hall effect, and then the white

25:26

region is outside of the sample,

25:28

that's a vacuum. I know that

25:30

there's going to be an electric

25:32

field, well, minus the electric field

25:34

is going to point into this

25:36

sample, so there's electric force holding

25:39

the electrons in. How do I

25:41

know it's there? Well, if it

25:43

wasn't there, the electrons would leak

25:45

out. And they're not leaking out,

25:47

so there's an electric field there.

25:49

So, and then there's a magnetic

25:51

field perpendicular to the sample. And

25:53

I know from basic E&M that

25:55

whenever I have a crossed electric

25:57

and magnetic field... there's a drift

26:00

velocity of any charge. And you

26:02

can calculate the drift velocity just

26:04

by finding the reference frame in

26:06

which the Lorentz force, E plus

26:08

V cross B, turns out to

26:10

be zero. So if I put

26:12

a charge, in particular, one of

26:14

these particles on the edge, it

26:16

will drift along like this. And

26:19

just because of the E-cross-B effect.

26:21

Now, we're going to use that

26:23

to our advantage to transport these

26:25

particles around our system. So

26:27

here's the geometry of the sample

26:29

we're going to use. We're going

26:31

to take, you know, the blue

26:33

region again is quantum hall fluid,

26:36

and we're going to pinch it

26:38

down in some region in what

26:40

experimentals call a quantum point contact.

26:42

It's a point contact, and then

26:44

they put the word quantum because

26:46

they like the word quantum. So

26:48

anyway, so if you send these

26:50

charges in along the edge, and

26:52

they kind of move along the

26:54

edge. bump, bump, bump, bump, bump,

26:56

like that, and most of them

26:58

go through, but occasionally you'll discover

27:00

that one of them comes along

27:02

and then jumps across the narrow

27:04

neck and gets back reflected instead.

27:07

Okay? So we should think of

27:09

this constriction as being a half-silvered

27:11

mirror. to send some through and

27:13

reflect some back. Just as a

27:15

side comment, the first measurement of

27:17

the fractional charge of these particles

27:19

was done with a single point

27:21

contact like this. You measure that

27:23

when the current coming back at

27:25

you, and you measure some total

27:27

current, but you notice that the

27:29

noise in that current is indicating

27:31

that the charges are coming back

27:33

to you in units of E

27:35

over three rather than in units

27:38

of E. And this experiment was

27:40

done in the 90s, by several

27:42

groups, and it's now... not controversial

27:44

that this works as described. All

27:46

right, this is the experiment I'm

27:48

going to describe. It was proposed

27:50

in 97 by our first speaker,

27:52

Shabaji, and his friends, Claudio Shimon,

27:54

Denise Freed, Steve Colston, and Shalgonwen.

27:56

The idea is that we're going

27:58

to have two of the. point

28:00

contact. One of them called T1

28:02

and one of them called T2.

28:04

This is going to act as

28:07

a beam splitter and a mirror

28:09

and if you remember your optics

28:11

this is basically a Fabri-Pro interferometer.

28:13

So the idea is that a

28:15

particle can come along like this.

28:17

It will split into two partial

28:19

waves. One partial wave jumps across

28:21

the other partial wave goes on

28:23

and is reflected around the cavity

28:25

and then they vinterfer and go

28:27

on their way. Now, if you

28:29

count both of those partial waves,

28:31

the wave function of the particle

28:33

coming back at you is the

28:35

sum of the part that went

28:38

across T1 here and the part

28:40

that went across T2 here. But

28:42

the part that went around T2

28:44

picks up an additional phase for

28:46

having gone around the cavity. Then,

28:48

if you want to know the current

28:50

coming back, you have to

28:53

square the wave function. the

28:55

usual way, you know, probabilities

28:57

are squares of amplitudes, gives

28:59

you T1 square plus T2,

29:02

T1, T2, cosine phi, assuming

29:04

T1 and T2 are real

29:06

for simplicity. Okay? So this

29:08

is basically Fabrio interferometry physics,

29:11

and the thing we're going to

29:13

be interested in is this phase

29:15

phi down here. So we're going to

29:17

try to change that phase phi and

29:19

measure the change in the backscattered... current

29:21

and the way we change Phi is

29:23

by actually changing the shape of the

29:25

cavity slightly by pushing on the edge.

29:28

Oh, this is the particle going around

29:30

the cavity. There it goes. So we're interested

29:32

in the phase of the particle going around

29:34

the cavity. That shows up here as

29:36

Phi. And we're going to change... this

29:38

cosine phi by changing the shape of

29:40

the cavity so that the phase accumulated

29:42

by the particle going around the cavity

29:45

changes. So you do that with an

29:47

electrode that sort of pushes the electrons

29:49

out of the way and changes the

29:51

shape of the cavity. So this is

29:53

all changed with an electrode and it

29:55

will change cosine phi and so the

29:57

current you measure backscattered is going to

29:59

oscillate. sinusoidally, it's just like taking a

30:01

Fabri-Prointerferometer, you have two mirror, you have

30:04

a half silver mirror and a solid

30:06

mirror and you just move them back

30:08

and forth and you'll see interference fringes.

30:10

Okay, now the interesting part of this

30:13

experiment is what happens if you add

30:15

an anion to the center of the

30:17

cavity. Well, if one particle has now

30:20

gone around another particle, it picks up

30:22

a braiding phase. 2.5 over 3, braiding

30:24

phase. So without the particle, if

30:26

you see the black curve, with

30:29

the particle, you'll see a shifted

30:31

curve, the blue curve. And if

30:33

I add another particle to the

30:35

middle, the curve shifts again by

30:37

another 2.5 over 3. That's what

30:39

we're going to try to see. OK.

30:42

So it sounds like an easy

30:44

experiment, right? Nope. So it was

30:46

about 15 years of effort trying

30:48

to make this experiment work, and

30:50

people eventually came to the conclusion

30:52

that it's actually a very hard

30:55

experiment. It might even be impossible.

30:57

So the reason it is hard

30:59

is sort of a conflict between

31:01

two issues. For any finite temperature,

31:03

there's a coherence length beyond which

31:06

you don't see any interference. So

31:08

where that comes from is that

31:10

the phase can be stated as

31:12

E to E to the E to

31:14

the E to E to the E

31:16

length. times a wave vector. Now the

31:19

wave vector is a function of energy,

31:21

so you expand this, K-0 plus D-K-D-E,

31:23

times D-E. So depending on the energy

31:25

of the incoming particle, you get a

31:28

different phase. But at any finite temperature,

31:30

say even 30 milli Kelvin,

31:32

D-E is big enough so

31:34

that the changes in this

31:36

term end up scrambling the

31:38

phase completely. And the only

31:40

way you cannot scramble the phase is

31:42

if you make L very small. So this

31:44

is going to force you to do the

31:46

experiment on a very small sample on the

31:49

micron scale, even at 30 millicelvin. I mean,

31:51

if you could go to zero temperature, zero

31:53

temperature, you could do it in a much

31:55

bigger sample. But 30 millicelvin is about the

31:58

limit of what you can do experimentally. However,

32:00

there's a conflict with that, which

32:02

is that adding a single electric

32:04

charge of E over 3 to

32:06

a micron-sized object is a very

32:09

strong perturbation. It changes the position

32:11

of all the edge states, and

32:13

then you're measuring something completely different

32:15

once you add the E over

32:17

3. So you're not seeing the

32:20

change just from the statistics of

32:22

the particle, you're seeing the change

32:24

from the coolum interaction with the

32:26

particles running around. So this is

32:28

problematic. And then on top of

32:31

that, even at 30 mil a

32:33

Kelvin, there's significant thermal noise from

32:35

various sources that you have to

32:37

wrestle with. So all of these

32:39

things were addressed by Mike Manford's

32:41

group in 2020 using Galli Marcinite

32:44

heterostructure with lots and lots of

32:46

tricks to get around these problems.

32:48

And it was done successfully in

32:50

a beautiful tour-to-force experiment, but that's

32:52

not the experiment I'm going to

32:55

describe. I'm going to describe this

32:57

experiment, which was done more recently

32:59

by the Harvard group, Thomas Workmeister,

33:01

a graduate student who's the lead

33:03

author. And the reason I like

33:05

it, is because it invokes some

33:08

of the things that we admit.

33:10

in this earlier paper from 2006.

33:12

So the idea of the experiment

33:14

is we're going to do exactly

33:16

that experiment. We're not going to

33:19

change any edge voltage. We're not

33:21

going to change the shape of

33:23

the interferometer. We're just going to

33:25

wait. So you just sit in

33:27

your experiment and you measure some

33:29

current backscattering and you would think,

33:32

okay, I'm just, I'm not changing

33:34

anything, the current backscattering should be

33:36

exactly the same, it should just

33:38

not change at all. But it

33:40

does change, it sort of jumps

33:43

around, it sort of jumps around,

33:45

after half a minute it jumps

33:47

up to this blue level, and

33:49

then it jumps back down to

33:51

this green level. It's jumping all

33:53

over the place, it looks like

33:56

it's a noisy sample, and typically

33:58

what you do with noisy samples,

34:00

and typically what you do with

34:02

noisy samples, and typically what you

34:04

do with noisy samples, is you

34:07

do with noisy samples, is you

34:09

throw them, is you throw them,

34:11

you do with noisy samples, you

34:13

throw them, and you do, you

34:15

do with noisy samples, But then

34:17

if you look at this for

34:20

a little longer, you realize that

34:22

actually it's only jumping between three

34:24

different levels. The green level, the

34:26

blue level, and the purple level.

34:28

So let's plot those three levels

34:31

over here. And then, once you've

34:33

accumulated data to find what these

34:35

three levels are, then you change

34:37

the shape of the interferon. and

34:39

you trace out three curves, which

34:42

are three sinusortal curves shifted by

34:44

two pie over three. This is

34:46

exactly these three curves here. What

34:48

you're seeing is you're seeing telegraph

34:50

noise as one particle is jumping

34:52

in and out of the interferometer.

34:55

And the blue curve will be

34:57

when you have one, four, seven.

34:59

particles in the interferometer, the purple

35:01

will be 258, and the green

35:03

will be 036, and so forth,

35:06

and it's jumping back and forth

35:08

between them, but at any number

35:10

of particles in the interferometer, you're

35:12

on one of these three sinusoidal

35:14

curves. So, how did we address

35:16

these problems? Well, this one, we

35:19

got rid of the thermal noise

35:21

by making lemonate out of lemons,

35:23

I guess. So we used it

35:25

to our advantage. But what about

35:27

the conflict between the size of

35:30

the device and the coolment interaction?

35:32

Well, here, what they did was

35:34

they screened the coolment interaction by

35:36

slapping down a metal plate very

35:38

close to the two-dimensional electron gas

35:40

that you're interested in. That if

35:43

you put a metal right near

35:45

your two-dimensional electron gas, then every

35:47

time you have a charge in

35:49

the two-dimensional electron gas, you have

35:51

a mirror charge in the... in

35:54

the metal plate. So instead of

35:56

having columbic interactions between E over

35:58

3 and E over 3 over

36:00

here, you have dipolar interactions much

36:02

weaker, dipolar interactions between this pair

36:04

and this pair. Now, doing that

36:07

in gallium arsenide was really a

36:09

very difficult trick because, well, the

36:11

gallium arsenide, the quantum wells are

36:13

100 nanometers to begin with, and

36:15

the... the galleymarshanide ball needs a

36:18

cap and then the metal plane

36:20

can only be so close, but

36:22

with graphing it's super easy to

36:24

do because graphing is only an

36:26

atom thick and you can plunk

36:28

it right down on top of

36:31

a piece of metal within a

36:33

couple of anxstrom so you can

36:35

screen the... the Coulomb interaction extremely

36:37

effectively and that's why some of

36:39

these new graphing experiments are so

36:42

so nice. Anyway, that more or

36:44

less ends the story after about

36:46

36 years we can finally put

36:48

a checkmark next to the experimental

36:50

confirmation of... Okay, we can finally

36:53

put a checkmark next to the

36:55

experimental confirmation. of anion statistics and

36:57

I'll thank you for listening just

36:59

in time. So there's a legend

37:01

that when Horse Stormer and Dan

37:03

Suy were taking the first data

37:06

on fractional quantum hall effect, they

37:08

were sort of, the way, you

37:10

scan the magnetic field slowly and

37:12

you see these plateaus form. They

37:14

saw this plateau form at three

37:17

times the other plateau that they'd

37:19

seen. And Dan Sui immediately said,

37:21

oh. Quarks! And it was completely

37:23

a joke, but he realized immediately

37:25

that the quantized one-third would be

37:27

consistent with a one-third particle. It's

37:30

not quarks. You know, the quarks

37:32

are bound with enormous, enormous energy,

37:34

orders of magnitude. higher than anything

37:36

in these experiments. But nonetheless, you

37:38

know, it has that odd similarity.

37:41

But there are other facts from

37:43

quantum hall states where the charged

37:45

particles are, you know, E over

37:47

5 or E over 7 or

37:49

any number like that. So three

37:51

is sort of the minimal odd

37:54

number beyond one. Yeah, it's... It's

37:56

a little, yeah, it is a

37:58

complicated combination of effects. So the

38:00

question is why are the width

38:02

of the plateau is what they

38:05

are? So there's a theorem which

38:07

says that if you had no

38:09

disorder at all, there would be

38:11

no plateaus anymore. So you need

38:13

some amount of disorder. And it

38:15

actually depends on not only the

38:18

amount of disorder, the tendency to

38:20

initially grow wider as you reduce

38:22

the disorder, but it also depends

38:24

on the type of disorder, the

38:26

range of disorder. And when quantum

38:29

hall effect, because of the precision

38:31

of the effect, is used for

38:33

metrology, for setting resistance standards, if

38:35

you want to ask, how do

38:37

you define an ome really accurately?

38:39

You do it this way, use

38:42

quantum hall effect. And they use

38:44

very special samples with a particular

38:46

type of disorder, which is known

38:48

to give a. wide plateaus. So

38:50

it's actually a combination of things

38:53

that goes into the width of

38:55

the plateau. But it has to

38:57

be sufficiently clean, but then the

38:59

details of this order actually matter

39:01

too. Yeah. So the question is,

39:04

why do you need the integer

39:06

ratios to be small? It only

39:08

comes from the statement that as

39:10

you get higher integers. the gaps

39:12

tend to get smaller. And this

39:14

is going to have to be

39:17

the case, because otherwise you're going

39:19

to have a double staircase, where

39:21

there's a different quantum hall effect.

39:23

At each epsilon, you change the

39:25

magnetic field. So as you get

39:28

to cleaner and cleaner samples, lots

39:30

of more fractions do start emerging

39:32

between other old ones. But the

39:34

ones with the lower denominators are

39:36

the ones that emerge first. Yeah,

39:39

so the question is about the

39:41

rationality or irrationality of these of

39:44

these these effects So the so

39:46

I wouldn't say this this is

39:48

this is not a physical constant

39:51

where we're measuring where we're measuring

39:53

a number the I guess I

39:56

would say that we are measuring

39:58

a third of an electron to

40:00

very high precision in some ways.

40:03

Although, to be honest, the experimental

40:05

measurement that tells you directly that

40:07

you're measuring the charge on these

40:10

things is one-third. Unless you are

40:12

saying that the hall resistance itself,

40:14

which is very easy to measure.

40:17

is evidence that the charge is

40:19

fractionalized. And theoretically, you might make

40:21

the connection. But if you want

40:24

a direct measurement of the charge

40:26

on that particle, which you can

40:29

do by noise measurements, or you

40:31

can, these days, they can do

40:33

it actually by using a very

40:36

sensitive electrometer, and you scan over

40:38

the sample, and you say, oh,

40:40

there's a bump, and it's charges

40:43

about E over three. But those

40:45

experiments. are not accurate to a

40:47

part in 10 to the 10.

40:50

Those experiments are accurate to say

40:52

5% something like that. So it's

40:55

consistent, but it's not highly, highly

40:57

accurate the way the Hall resistance

40:59

experiment is. Yeah, okay, it's a

41:02

very good question. So there was

41:04

actually, there was a, in the

41:06

early days of fraction quantum hall

41:09

effect. It was believed to be

41:11

a theorem that all denominators had

41:13

to be odd. And that actually

41:16

comes from the fact that the

41:18

underlying particle electrons that you're putting

41:20

in is a fermion. And so

41:23

it's a little bit of a

41:25

complicated connection, but from the fermionic

41:28

statistics, the statement is that you

41:30

would need to have an odd

41:32

denominator and three is the smallest

41:35

odd denominator higher than one. which

41:37

gives you the integer quantum hall

41:39

effect in which there's no fractionalization.

41:42

That turned out not to be

41:44

true, actually, that people have measured

41:46

even denominator fractions, and that comes

41:49

from a more subtle effect, where

41:51

the electrons pair in... into bosons

41:53

like a superconductor and the mat

41:56

condenses. So you can have even

41:58

denominators too. It could be that

42:01

the first one we measured was

42:03

at one half, but it just

42:05

turns out that the one half

42:08

plateau is weaker and a little

42:10

bit harder to see. So they

42:12

have been seen, but only more

42:15

weakly and more high, you know,

42:17

heat in cleaner samples. So they

42:19

can exist. Even denominators can exist.

42:22

But some things, you know, people

42:24

have observed something like, something like.

42:27

80 or 100 different fractions in

42:29

fractional quantum hall experiments of which

42:31

all but a very few have

42:34

odd denominators. So there's a community

42:36

in the world who wants to

42:38

build quantum computers out of anions.

42:41

Now initially, Microsoft was the home

42:43

of this. Sorry, I should repeat

42:45

the question. The question is. is

42:48

where do you see this being

42:50

applied to in technology and where

42:52

do you see this being applied

42:55

in fundamental physics. So in technology,

42:57

this community that wants to use

43:00

anions to build quantum computers, and

43:02

they initially started exploring fraction quantum

43:04

hall effect very intensively. And that's

43:07

why people did this 15 years

43:09

of experiments, of unsuccessful experiments. Actually

43:11

and they at some point after

43:14

doing this for about eight years

43:16

Microsoft said okay That's not the

43:18

way to do it. We're going

43:21

to do something else. They're still

43:23

thinking about So any on-based quantum

43:26

computers or myrona based quantum computers

43:28

is very similar But they switched

43:30

the platform to using superconductor semiconductor

43:33

structures. It's not quantum hall effect

43:35

anymore So it has a lot

43:37

of similarities, but not exactly the

43:40

same so that's something that they're

43:42

really pushing very hard right now

43:44

and that could be a real

43:47

technology although it's not fractional quantum

43:49

politics. Although there's some people in

43:51

the world, myself included, who love

43:54

to see fractional quantum hall effect

43:56

come back into the quantum computing

43:59

game and still think that that's

44:01

a you know not an insane

44:03

possibility. For fundamental physics, so in

44:06

some ways I have to ask

44:08

maybe what do you mean by

44:10

fundamental to begin with, but I

44:13

would say that this is fundamental

44:15

as anything else you will come

44:17

up with and you know seeing

44:20

this this principle that particles don't

44:22

need to be don't need to

44:24

be bosons or firmions, is a

44:27

really fundamental advance. And that is

44:29

what I would call fundamental physics.

44:32

I probably should have. As I

44:34

went further in, I mean, to

44:36

some extent it's not fair because

44:39

I think a lot of the

44:41

modern work is much more frequently

44:43

done in collaborations than it used

44:46

to be. And so there will

44:48

almost always be a student on

44:50

the paper, a postdoc, and a

44:53

senior. you know, senior faculty member

44:55

or several and so forth. And

44:58

then the question arises, you know,

45:00

whose work was this? You know,

45:02

is it really the graduate student

45:05

who came up with the great

45:07

idea? I mean, sometimes it actually

45:09

is, and the faculty member is

45:12

just the one who raised the

45:14

money to pay the graduate student.

45:16

But other times, it's more of

45:19

a collaboration, so I think it

45:21

becomes harder to say whether the

45:23

ideas are still coming from the

45:26

young people. My guess is that

45:28

in fact a lot of the

45:31

ideas still are coming from the

45:33

young people, but it's harder to

45:35

prove. Yeah. Yeah, exactly. So

45:39

actually, there's a couple things that

45:41

I can say that where it's

45:43

not an irrelevant connection. So the

45:46

underlying theory of anion models are

45:48

so-called topological quantum field theories that

45:50

Shavaji mentioned earlier, where you throw

45:52

out all space and all you

45:54

care about is where the one

45:57

thing went around another. And topological

45:59

quantum field theories were. actually cooked

46:01

up by string theorists in the

46:03

1980s, more or less, and they

46:05

were thinking about theories of quantum

46:08

gravity. If you go down one

46:10

dimension to a two plus one

46:12

dimensional universe instead of a three

46:14

plus one dimensional universe, it is

46:17

known that the gravity is very

46:19

different in two plus one dimensions,

46:21

it becomes completely topological. And a

46:23

lot of the structure goes away,

46:25

and these kind of theories actually

46:28

do describe universes in lower dimensions.

46:30

It's beyond my pay grave to

46:32

say whether any of that survives

46:34

in our... three plus one dimensional

46:36

universe or not. But it's definitely

46:39

interesting to study. Yeah, what are

46:41

the, the question is, what are

46:43

the statistics of anion's analogous to

46:45

Boz Einstein and Fermian statistics? You

46:47

can write down a distribution function

46:50

for anion statistics, which is somewhere

46:52

between Bozon and Fermion as well.

46:54

There's another. description of statistics, it

46:56

also arises, which is interesting, which

46:58

is to ask the question, you

47:01

have some Hilbert space, and you

47:03

ask how big it is, and

47:05

then when you add a particle

47:07

to it, how much smaller did

47:10

it get? How many fewer orbitals

47:12

are allowed for the next particle

47:14

that comes in? So for bosons,

47:16

if I put a particle in,

47:18

the next particle I put in

47:21

has exactly the same many options.

47:23

For firmions, if I put a

47:25

particle in, the next particle has

47:27

fewer options. With any ions, it's

47:29

somewhere in between. That you put

47:32

two particles in, and then you

47:34

reduce the number of options by

47:36

one, for example. So it does,

47:38

I mean, it always seems to

47:40

interpolate between the two possibilities. Thank

48:06

you.