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Subtitles generated by robot

00:09

okay welcome back
feel a little bit frazzled today I spent
the morning reading to my daughter I
read is your mama a llama to her 40
times in a row this is really great
literature if you're if you happen to be
four months old
highly recommended anyways whatever it
kept her from crying and that was what
kept her from crying so is your mama a
llama it is okay sorry that digression
decide today we're going to study

00:41

Ginsberg Landau theory for people who've
studied lambda theory of phase
transitions and Ginsburg lambda theory
before this will look a little bit
familiar but then we will extend onward
to see the relationship to the Andersen
Higgs mechanism application to
superconductors and we'll use Ginsburg
Lando theory to derive some interesting
results about superconductors which
maybe even those people who study phase
transitions might not have seen before
so to start our exploration into
Ginsberg Landau Fieri we're going to
start with the simple case of a neutral

01:14

superfluid neutral superfluid and we'll
write down the Ginsberg Landau free
energy GL free energy which is exactly
the form energy which is exactly the
form of the gross-pitaevskii equation
that we derived earlier so it's an
integral over space
there's a spatial derivative term bring
into sy squared and then there will be

01:46

some expansion in powers of absolute
value si and there may be even further
terms
oops myself out of the way there like
this so this term is side of the fourth
mr. Murray side squared this what we
derived earlier a sign in the fourth
term and this combination the terms
without gradients is often known as V
the potential as a function on the

02:19

absolute value
of sigh now a couple things about about
this potential function first of all
it's it's usually sufficient to to
truncate the series at 4th order or at
least it will be for us sufficient to
truncate the series at 4th order so we
can throw away all terms of higher order
we do want V to be positive for values
of want this to be want positive and

02:52

large for large large sigh and for that
to be true we need alpha for to be
greater than 0 if alpha for o is less
than 0 then V would get very negative
for large sigh and the lowest energy
solution would be for our side to just
blow up and we can't have that so for
stability we want alpha 4 to be greater
than 0 and furthermore we're going to
discover this curve this coefficient

03:26

alpha 2 is going to be related to the
critical temperature it will be a times
t minus critical temperature with a gate
greater than zero so we'll be able to
show that in the in in a moment finally
we're going to start by considering a
uniform system with a system with no
gradients a homogeneous uniform system
the lowest energy homogeneous uniform
system gonna have is if there are no
gradients inside there's that just
greatness inside just raises the energy

03:59

so let's just drop this term drop the in
radiant term drop this to begin with
okay alright so this is our free energy
we want to work with and we'll consider
the case T greater than TC by which I
mean here in the case of alpha 2 greater
than zero so let's plot what the
potential looks like okay so this is sy
here and this is V of sy here and we

04:32

find
it grows what starts Michael and
quadratically at small sigh and then it
then it at larger side goes up that
faster as the fourth power
and the minimum of the free energy min
is a vo psy is at psy equal zero and
that tells us that there is no con
incent that's I equals zero no order

05:00

parameter no condensate okay that's what
we had hoped for T greater than TC you
would have expected no condensate on the
other hand let's consider the case of T
less than TC by which I mean alpha 2 is
less than zero
considering our relationship of alpha to

05:34

2 TC when T is less than T C alpha 2
becomes negative in which case let's
plot the plot the V in this case
okay oops this
No and here the function is going to
start out negative quadratic it's alpha
2 is less than zero but then it will
turn around and go back up do that after

06:14

the fourth term okay so this is absolute
sy and this is the upside ok do the
alpha to the fourth term the function
goes back up four large sigh and the
alpha squared term makes it come down
four small sigh well it's sort of sloppy
looking party curve but you know more or
less what I mean it's a metric okay
now the minimum of this function min

06:44

here it's easy enough to show that if
you're if the function we're minimizing
maybe I'll just write this out here if V
of absolute sy is alpha 2 sine squared
plus alpha 4 SI for the min without for
two negative alpha 2 less than 0 the
minimum of V of sy occurs at help since
I is absolute alpha 2 over 2 alpha 2 for

07:17

alpha 4 so that's a exercise in finding
the minimum of a of a function and so
we're going to define that point to be
called that minimum point we'll call it
sy not I guess we call it sy not and
we'll declare it to be real sign off at
this point here we'll call minus I
naught times minus I naught so sy not is
then this quantity and that is then

07:50

considering our definition of the order
parameter earlier this is what we're
going to define to be the square root of
the super fluid density okay so this is
standard Ginsburg Landau theory and
Ginsburg Landau Theory above TC you have
a something that starts out quadratic
it's always positive and the minimum is
at
si equals zero below TC you break a
symmetry and the minimum is going to be

08:23

at some finite value and that's the the
symmetry broken phase which in our case
is then the superfluid phase okay now I
drew this in this potential V in one
dimension but really sighs complex size
complex so I should have drawn it in in
three dimensions let's see if I can do
this a little bit better in three
dimensions so something looks like this
and then we have a third dimension like

08:57

that okay and that's supposed to be
intersect zero and let's see do this so
the potential the function we're trying
to plot looks kind of like this it's
like this like this
then it turns around on it comes up like
this okay so it's what is typically

09:35

called a Mexican Mexican hat potential
draw us a little better okay so it's
it's radially symmetric it's basically
taking this curve here this curve here
and just rotating it around the y axis
so the axes here are V of sy on the
vertical you have sine the vertical and

10:08

then real upside on this axis and
imaginary of sy on this axis okay so the
V of sy v is only a function of the
absolute value of sy so it's it's it's
rotationally symmetric or if I draw it
correctly it would have been
rotationally symmetric around the
vertical axis okay
this is a terrible Mexican half
potential but but I'm not going to try
to draw it again because my drawing
ability isn't that good make it a little

10:41

clearer let's go straight down like this
right in the middle okay
okay so when we have a Mexican hat
potential like this we've broken the
symmetry we've chosen a particular
direction for the phase of size so if
phase of psy must be chosen in the
symmetry broken phase phase of size
arbitrary but you have to choose some
particular phase when you break when you

11:12

break the symmetry when you go below TC
psy take some non-zero value so you have
to choose some things but doesn't matter
what phase that you choose now when you
have such a situation where a continuous
symmetry has been broken you will always
have a Goldstone mode which is a low
energy expect Goldstone mode which is a

11:43

low which is which is the mode
associated mode associated Low Energy
mode low energy mode mode associated
with reaching the direction that we've
where we break the in the direction
which we break the symmetry so for a

12:12

ferromagnet for example when you go
below TC
ah so that if you have a Heisenberg
ferromagnet where the spins can point in
any any direction when you go below the
critical temperature the spins have to
point and they all point in the same
direction but they have to choose a
direction to point in let's arbitrarily
say that they they point up but there
will be a low energy mode a spin wave
mode which comes down to zero energy
associated with changing that choice of
direction from up to slightly off up so

12:44

it can wiggle in its in its direction
and that's a low energy spin wave
similarly here there should be a mode
associated with changing the complex
phase of this of its order parameter V
choosing
this direction where we've broken the
symmetry break symmetry so in other
words you know we could have chosen any

13:13

point around the rim of this Mexican hat
around the rim of this minimum any point
around around here could have been
chosen as the ground state and then once
we choose that direction on the ground
state say we choose this direction
there's going to be a low energy mode
associated with allowing this direction
to wiggle it's like a spin wave mode so
this will be a sound oops this will be a
sound mode acoustic mode sometimes an

13:43

acoustic mode or a Goldstone boson mode
and I think there is a homework
assignment to try to actually calculate
the the spectrum when this goes and
Goldstone boson okay in order to figure
out the the real properties of the
superfluid we want to put the spatial
dependence back into the system and we
do that by basically deriving what we
call the Ginsberg Landau equation from
the free energy and we did this before

14:18

we take the free energy and we
functionally differentiate it with
respect to say size star L R and we get
putting back in the spatial dependence
we get minus del squared
sigh plus alpha-2 side plus 2 alpha for
size squared
sy so it's setting this equal to zero
this gives us the nonlinear Schrodinger

14:51

equation that we talked about before
Schrodinger
but Ginsburg Landau equation Schrodinger
Ginsburg Landau equation or was PDF ski
equation if we had derived this from for
interacting bosons um okay incidentally
Ginsburg land up theory was worked out
first in the context of super
conductivity and only later was it
applied to superfluids okay now in a

15:23

uniform system uniform system we know we
already calculated the magnitude of psy
is squared it's gonna be alpha 2 over 2
alpha for the cufflink today do you get
that right no here so I think I this is
magnitude of alpha but I think it's

15:56

magnitude of alpha way back here I think
this is magnitude is I squared should
have been that yeah
that should've been magnitude I was
wrong in my notes sorry about that I
better fix that in that I hope that's
right in the in the type notes I will
check that
so alpha naught squared is alpha
absolute alpha 2 assuming alpha 2 slices
your in other words in the condensed
phase divided by 2 alpha 4 and it's it's
useful this is sort of sets the the
scale of sigh in the in the problem and

16:28

it's useful to rescale rescale sigh bye
sigh not so let's define there we go
let's define F equals sy over absolute
sign on I guess I know it is real she's
a real positive anyway but I'll write it
it's absolutely not and then plug that
in to our nonlinear Schrodinger equation
and what we get is smaller so you see it

17:04

all at the same time
I have to myself out of the way so you
don't see this so then what we have is
minus H bar squared over 2m gradient
square root of F plus alpha 2 F plus 2
alpha 4 then we have sine naught squared
absolute f squared times F equals 0 and
now plugging in the value of of sine

17:40

naught squared into into here we can
this combination ends up being another
factor of about the two which we can
then factor out so the the end result
then becomes right this way sighs
squared del squared F plus F minus
absolute f squared F equals 0 okay so
this is a rather important equation it's

18:13

our nonlinear Schrodinger equation and
it's sort of simplest form where I have
defined here I've defined here
sighs squared X I squared is is this
combination this thing divided by alpha
2 so I'll write that out is well I I
squared is H bar squared over 2m
absolute alpha 2 below TC okay so this
context I is known as the ginsberg Lando

18:45

coherence length and adil coherence
length and it's it's basically a it's
it's a stiffness length of the of the of
the of the superfluid it's it's how it's
the distance scale over which you can
bend psy bend the value of side or bend
the phase of psy
that's the sort of natural length scale
over which psy changes

19:18

obviously terrible and that drawing
these six eye symbols also but
everything that looks like a squiggly is
the same symbol um okay so I'll write
that this is the stiffness length
stiffness length for twisting twisting
sigh and at low temperature for helium 4
for helium 4 at low T sigh can be very
small it can be about one angstrom so
like that so it's a twistable on very

19:58

long small length scales okay now but
note here that alpha 2 alpha 2 is goes
to zero at the critical temperature and
that means as you get close to the
critical temperature as T goes to TC
side divergence explicitly sigh it goes
as t minus TC to the minus 1/2 okay even

20:31

though the coherence length is actually
is very short as you get close to the
critical temperature it will it will
diverge okay at least in this in this it
grows this way at least in Ginsberg
Landau theory okay um so we have this
nonlinear Schrodinger equation let me
actually copy it because we like it I
can use it again copy and bring it down

21:04

here again paste okay this is a the
equation that we want to solve its are
not only a Schrodinger equation and
despite I mean typically nonlinear
equations are very hard to solve but
there are some things we can say about
it their various exact solutions
solutions we can write down one F equals
1 everywhere that is just the constant
uniform solution where you just have
uniform superfluid everywhere and in

21:36

space all the way up to infinity that's
a perfectly good solution isn't it maybe
not that interesting but it's a
perfectly good solution of a uniform
superfluid a more interesting solution
is a solution of the following form that
F equals
hyperbolic tangent of x over square root
of two times X I so what does that look
like let me see if I can draw that so

22:06

this looks like okay again join my axes
here's X here's zero and here is f and
let's put this is F equals one here at
this line F equals one left on this line
and then we have a tan CH which grows
starts by growing linearly and then
converges to this line as a tent and the

22:42

the healing little more like a tange
okay and the healing length of this tent
is x i okay so the the properties of
this it's it's nice it goes to the
uniform solution that at large x but it
goes to zero at position x equals zero
so this is the uniform solution out here
and also F equals one I should make me
write this down we call that F equals

23:19

one the uniform solution equals one
implies it's I is this size zero
parameter which is square root of
absolute Delta 2 over 2 alpha four
that's a uniform solution which is it's
the same as f equals f equal but
everywhere in space the same a uniform
solution that we found before that's
when it converges to for large large X
attempts converges to that but for small
exit advantages linearly and this is
basically a particle in a box boundary

23:50

condition oops can't see this can you
again move myself out of the way there
sorry about that so what we have is a
particle particle in a box in
box boundary conditions where the order
parameter goes to 0 at the boundary of
the box so we have a nice tan here
converging to the uniform system the
uniform solution in the bulk and going
to 0 and at the boundary now the one

24:21

thing I haven't haven't shown you is
that this tange
function actually satisfies this
nonlinear Schrodinger equation and
that's actually a a bit of an exercise
in in hyper geometric functions but it's
a it's a fairly easy exercise to do so
maybe do that to entertain yourself a
third solution that we're sort of

24:52

interested in is the vortex solution and
the vortex structure vortex
unfortunately is it's not fully
analytically solvable but it will be
able to say some things about it
so let's write F equals absolute FR
times e to the I theta okay so that the
phase wraps around some central points
so we're imagining here that we have
here's our fluid conventions and there's

25:23

some central point here some central
line and the thing is wraps by 2 pi as
you go around that central line ok now
we know for sure that the magnitude of F
F of our F of R must go to 0 F of our
most good zero at R equals 0 because the
phase is not defined at at R equals 0 so
that will tell us something about about

25:54

the limits of this of this function we
also suspect that F of R will go to 1 at
large R should go to the uniform
solution but we can do actually a little
bit a little bit better we can actually
calculate in in some amount of detail
some
functional form of this F although we
can't calculate it in all of its
entirety but we can get some of the the

26:25

details right by throwing out the
appropriate forms so let me draw roughly
what we expect it to look like so we
expect it to look like something like
this like this okay so this is our this
axis and this is f on this axis again F
should saturate to 1 is 1 along this
line something looks it's like this and

26:57

then it should come she look like
attached also so here it should look
like f goes says well it will have a
phase even I theta 1 minus something and
I'll write down the answer it's actually
the leading term correction is X I
squared over 2 R squared
let's not that come up and then here it
actually starts as VI theta x times R

27:29

and the way we we find these asymptotic
behaviors down here and up here is by
taking appropriate limits of our
equation here's our equation again plus
1/2 minus F squared f0 plugging in the
radial form of the laplacian so del
squared is gonna be 1 over rd by dr r d

28:00

by dr plus I guess 1 over R squared D by
D theta squared that and then just
plugging in the form of F equals the I
theta absolute f
absolutely F and trying to figure out
what the form of absolute Athens I guess
if we you know for a large okay let me
just write out the whole thing then size
squared 1 over R d by dr are given dr

28:35

minus 1 over r squared that comes from
this term here minus 1 over r squared
taking D by D theta squared times an
absolute f plus absolute f minus
absolute f q equals 0 and for for small
are for small radius f is small small R
absolute death is small so you can drop

29:06

the quadratic term drop this term and
for large R the leading term is the d by
dr is going to be small the d by dr term
is oops
no sorry for large R we want to make a
different expansion and we want to write
F equals 1 minus y and expand and RIE
expanded wide okay so I'm not going to

29:38

do those in detail you can you can work
that out for yourself to find these
these limiting forms okay so we know
something about the structure of the
keep saying super conducting vortex this
is super fluid vortex so far where the
entirely neutral superfluids no no no
superconducting yet who might have said
superconductors a couple times and then
I apologize for that
now finally we get to move on to

30:09

superconductors so we want to view our
superconductors as a charged superfluid
superfluid
so a superconductor is a charged
superfluid ie a superconductor
superconductor and we then want to write
down a appropriate Ginsburg lined out
theory for the for the superconductor
so f of size gonna look extremely
similar to what we had before the only
difference okay there's maybe two

30:50

difference is one is that we are going
to now have minimal coupling to to the
electromagnetic field so I squared like
this okay so the momentum term gets
minimally coupled to the electromagnetic
field we're going to still have alpha 2
so I squared plus alpha 4 sorry for and

31:23

it will still be the case that alpha 2
is going to be a times t minus TC with a
greater than zero and it will still be
the case that alpha 4 is has to be
spending between in the zero okay so
this looks entirely the same as as what
we had for the neutral superfluid except
for the minimal coupling to the
electromagnetic field but to be careful
we should also have an have the free
energy of the electromagnetic field as
well so we'll write that we know how to

31:55

write the energy of an electromagnetic
field it's gonna be V squared over 2 mu
naught and V squared over 2 epsilon
naught in SI units and one should be a
little bit more careful even that the if
you're really careful about it you
realize that this integral is only over
the superconducting sample whereas this
integral is over all of space so
electromagnetic fields that leave the
sample have to have to be included but

32:28

often what we're going to drop the
energy of the electric field but
this is just because it's it's you know
it's not doing anything interesting for
us actually the electric field and the
superconductors zero but so we may may
drop that later on ok so again I
analogous to the case of of the neutral
superfluid below TC we're going to have

32:58

the absolute value of sy is going to be
yes I not which will be square root of
alpha 2 over 2 alpha 4 exactly as we had
before which we'll call the super fluid
density and here I've I've put stars as
we have been doing before on the mass
and the electron charge and and the
superfluid density again to represent
the fact that we don't know how big the
the elementary boson cluster is whether

33:29

it's a charge to boson or charge for
boson or charge eight boson or charge
one boson we don't know that
and only the ratios of a over Tamara are
going to matter here although as I
mentioned once you look at the flux want
and then you can tell how big the the
charge of the boson is so we might
expect
write this down might expect
well Goldstone boson associated with

34:03

associated with changing the phase of
the order parameter around the rim of
the Mexican hat associated with going
around around the rim of the Mexican hat
exactly as we had exactly as we had in
the case of exactly as we had in the

34:36

case of the neutral superfluid however
there is none but there is none birthday
is and this is this rather surprising
result maybe I should say what I mean
but but by there is none superconductors
super cond have no know Low Energy

35:05

no no energy electronic noise so there's
no electronic sound mode tronic loads no
electronic sound modes or no phonon of
the electronic degrees of freedom they
that's what you would expect from a
Goldstone boson a sound mode of the of
the bosons but but in fact
superconductors don't have them and why
don't they this is why it's because of

35:40

the Anderson Higgs mechanism there's
Senator Higgs mechanism mechanism and I
think it's mechanism I think it's fair
to call it anderson eggs because it
anderson explained this in the context
of condensed matter in 1962 and Higgs in
1964 and in fact Higgs even cites
Anderson and and says that in fact what

36:11

Higgs was doing it was just a
generalization of the the Anderson
mechanism also I guess to give fair
credit uhé Nambu also was very involved
in developing the understanding of
spontaneous symmetry breaking with with
a gauge field Phil Anderson the
discoverer of the
Anderson he's Matt the mechanism passed
away just just last week actually um at

36:40

the age of 96 and and it's this
particular course is you know probably
the most prominent scientists that we
mentioned over and over in this course
is is probably landau but it's probably
not a exaggeration to say that within
the condensed matter field anderson
might have been the most influential
condensed matter physicist of the of the
20th century very hard to think of
anyone who's had the same kind of

37:13

influence even even landau maybe
although it was a little bit later okay
okay we're we're going to work with this
free energy in order to describe the the
anderson Higgs mechanism but this is a
little bit of a cheap to use Ginsburg
Landau theory and the reason it's a
cheat is because I mean people usually
use the Lagrangian or a Hamiltonian
description and the reason for that is
because you want to be able to keep

37:43

track of the dynamics of the theories as
well as the static energies if you want
to you can think of the free energy as
being the Hamiltonian or negative of the
Lagrangian where you've taken all the
side dot terms all the time dependent
terms and if you've thrown them out or
if you set them equal to zero this is
almost gonna work for us not quite I'll
tell you where we have to wave our hands
a little bit and and pretend we're
putting back into dynamical terms and
try to understand what they would do but
I want to stick with the Ginsburg Landau
free energy because that's what we've

38:15

been we've been doing and and you know
there's plenty of references that will
go through the Higgs mechanism in the so
called abelian Higgs mechanism from the
lagrangian standpoint so i'm going to
try to do it just thinking about three
hundred free energies here now the key
to the the Higgs mechanism here is to
realize that the that the the free
energy here is

38:46

has gauge invariant and so you can make
a gauge transformation on it we already
know that the electromagnetic fields are
gauge invariant you can make a
transformation on the vector potential
and electromagnetic fields stay
unchanged here it doesn't look gauge
invariant but it is gauge invariant
under simultaneous gauge transform on
both the vector potential and the field
sigh so the particular gauge transform
that we can we are allowed to make is

39:16

gauge transform which leaves leaves F or
H or L Lagrangian or the Hamiltonian
invariant is of the following form you
take sy to e to the I alpha times I and
you take simultaneously East are a
vector field vector potential take East

39:49

are a times plus gradient F of alpha
times I guess times H bar so if you look
up here the what's gonna happen here is
that the gradient here is going to hit
the e to the I alpha here but then it's
gonna be corrected for by the change in
in a so the combination is going to
remain gauge invariant if you do both of
these at the same time okay now we're
going to consider the case of the
temperature less than the critical

40:21

temperature and in this case we know the
the magnitude of sy will and its minimum
it's called sine not we calculated
that's absolute alpha absolute alpha 2
over 2 alpha for some positive number
this minimizes the the free energy this
quantity is known as the valve in high
energy physics that stands for vacuum
expectation value

40:53

expectation which tells you that that
even in the vacuum with no excitation at
all
the field psy has a has a finite finite
value so let's see if I can I can draw
that I already discovered them pretty
good at drawing anything but okay here's
the Mexican Hat here just a part of the
Mexican Hat and this minimum here is has
magnitude its I not now there's what we
want to do is we want to describe the

41:27

excitations around some point on the
minimum in terms of excitations up the
hill they'll like this which we'll call
H and excitation z' around the rim which
we'll call Chi okay so there's two
different directions we can go if we're
sitting at the bottom of that of that
hill of the Mexican Hat one is to go up
the hill its H and the other is to go
around the bottom of the Hat that's the
thing we found at energetic and it had
no energy there was a Goldstone mode

41:58

previously in the neutral fluid neutral
superfluid so here we'll write sy is sy
naught plus h both real real fields well
sign on is just a constant H is a real
field here maybe the AI Chi going around
real okay now we then want to rewrite
our our free energy or that's just me
right V of sy in terms of sign on an H
so we can write alpha to sign on plus h

42:32

squared plus alpha 4 sy naught plus h so
forth
and if we expand that out what we know
what it's gonna look like it's gonna be
a bunch of constants which are not
interested in plus some constant C okay
let's call this constant C prime over
here C times H squared plus dot dot dot
it's gonna be quadratic going up and
down this this hill whenever you're at
the minimum you can always expand
quadratically you know it's gonna be
he'll and if you actually do this

43:04

expansion you'll discover the value of
that constant see there is 2 times alpha
- ok so we know that they could need
energy to go up and down that hill now
the whole key to the anderson Higgs
mechanism is is the following that we if
you try to go around this around the rim
by changing Chi you can always gauge
transform in a way that we're allowed to
make this gauge transform and we can

43:36

gauge transform this sigh here to get
rid of any attempt to go around the edge
of this rim so preventing you from going
around the rim all together by just
choosing a different gauge so this is a
rather interesting thing to do we're
going to change make the the gauge
transformation we're always gonna work
and engage where sigh sigh prime is even
the minus I Chi x sigh so sigh prime is
always going to be a real real valued

44:06

field sigh not plus h so there's this
size no longer a complex value field
it's no unreal sigh prime here is now a
real valued field because we got rid of
any fluctuations around the bottom of
the rim bye-bye gaze transform if we
plug this back in ok at the same time we
have to engage transform a this goes
along for the ride
H bar H bar is unrelated to aids in my

44:41

bad notation so the free energy here if
we plug in these new new parameters we
can write the free energy is some
constant they'll be integral D R then we
have a leading term 1 over m 2m star the
kinetic term rather H bar gradient -
like a sea star a prime and then it's
going to be sy naught plus H quantity
squared and then the potential term the
the potential going up the hill is 2

45:14

alpha 2
eight squared and then we always have
the electromagnetic terms which are look
the same in if we write them in terms of
B and E they'll always look like 1 over
2 mu naught B squared plus epsilon 1
over 2 e squared but it's probably good
to keep in mind that B and E are written

45:44

in terms of the vector potential a but
they're also gauge invariant quantities
so in particular I guess B is the curl
of a and either gauge and E is minus
gradient of a not a vector I probably
have a minus sign wrong here like this
um in in either gauge that's acceptable
okay

46:15

now writing this expanding this this out
here and canceling some terms we get the
following following expression we then
get the free energy is constants which
we're not interested in plus integral
over space H bar squared over 2m
gradient of H squared that's the
fluctuation up and down the hill plus 2

46:47

alpha 2 H squared plus sy naught squared
e star squared over 2m astara times a
prime squared and then plus the
electromagnetic terms plus well again
maybe I'll write it del cross a prime
squared over nu not oh yeah 2 nu naught
plus I guess the epsilon over 2 minus
gradient a
et a vector squared okay being a

47:22

different integral plus er okay good so
let's look at these terms here this is
now the fluctuation up and down the hill
this is an energetic fluctuation no
matter what you put in here what
whatever normalize function you put in
for H is going to cost you a finite
amount of energy because of this term
which is a which is a so called mass
term now it's a little bit hard to see
that here but if you remember that we we
had dropped all of it in our dynamical
terms in the Lagrangian or the

47:55

Hamiltonian so really there should have
been a you know you think of H here as
being an x-coordinate in in the system
and there's a conjugate key coordinate
that me that we had dropped this is
telling you that basically you have a
harmonic oscillator going up and down
the hill of H and it has a gap to to
create an excitation of that of that
harmonic oscillator the H dr. miss is is
missing from the Hamiltonian because
we're working with a free energy this
gradient term tells me that if the

48:25

harmonic oscillator isn't is in uniform
in space if each if each point in space
is not fluctuating the same that cost
you additional energy at any rate the H
field is massive this is the so-called
Higgs field its massive it has a gap now
more interesting for us is this a prime
term a prime squared term this is a gap
for the photon or mass for the photon so
let me just label this term here this is

48:56

massive photon massive photon so let me
try and explain why it is this term is a
photon is similar to the case of H that
this this term is telling me that any
fluctuation of the electromagnetic field
that I make is going to cost me a finite
amount of energy that is unlike the case
of regular photons where you can go to
larger and larger a larger wavelength
and the energy drops further and further

49:29

further so let's try to understand why
that is if we if we drop this this mass
term for the if we drop this mass term
for the photon then we know that well
okay because both terms here in the
magnetic term and the electric term have
derivatives will have a frequency term
that's up here which is going to be
proportional to the speed of light times
times the times the wave vector if you

50:02

insert and and and the key for a
massless photon if you drop this massive
term here for a massless photon the the
key is if you go to very small wave
vector the frequency drops all the way
down to zero so you have a massless
field because you can get energies
arbitrarily low however if you take any
field any finite field and you plug it
into this a a prime squared term over
here this will give you a finite result

50:32

it will always give you a finite energy
even at long wavelength it will give you
a finite energy when you plug it in to
this a Prime's a term so so this is for
a massless massless and Omega goes as
constant plus dot dot dot plus probably
Q squared or something is massive okay
so what does it mean that the the photon
is massive what does it mean what does

51:05

this mean what does this mean that we
have a massive photon what it means here
is that even at long wavelength the
photon costs you energy with you're
inside a superconductor why does that
make sense well let's imagine you're a
photon say a magnetic field impinging on
your superconductor that you know them
in an EM field is coming in towards the
superconductor when it goes into the
superconductor its energy goes up and

51:37

it's reflected it's like going up a hill
because the energy in the superconductor
is is
even if it's a long wavelength low
energy photon it's then goes up this
hill extender Jesus planet in the
superconductor and then it's reflected
back out because well it has finite
energy and it can't go into the
superconductor because there's a gap
there's a mass gap to getting the photon
into the superconductor what this means
is exactly the Meissner effect Meissner

52:07

effect photons photons are expelled
expelled from superconductor due to
their energy due to their mass okay
and we've also then discovered
if there is no Goldstone boson no alone
energy excitation no Goldstone boson and

52:38

the the catchphrase that people always
use is that the Goldstone boson has been
eaten by the the gauge field a and a the
gauge field gets massive so Goldstone
boson Goldstone is eaten eaten by the
gauge field a and a having eaten gets
massive because when you eat you get

53:10

massive okay so it's worth just for a
second counting the degrees of freedom
that we have in the problem degrees of
freedom to make sure we haven't lost
anything and above TC above TC what do
we have well we have sy which has two
degrees of freedom two degrees of

53:42

freedom because it's a complex field can
oscillate in you know so you can
oscillate you know you're sort of at the
bottom of this of this parabola and you
can oscillate in either the real or
imaginary
a direction it's terrible drawing but
you know I mean you can either go this
way or that way there's two degrees of
freedom in the side field here at the
bottom of this parabolic well the vector
potential is a massless is massless and
a massless photon has two polarizations

54:14

it can oscillate the electromagnetic
field can oscillate in either direction
perpendicular to its vector of
propagation so it's a total of four
degrees of freedom total okay on the
other hand we can look below TC below TC
we have h is a massive field field

54:52

that's the Higgs field but it's real
it's one degree of freedom okay you're
here it's a oscillation in the radial
direction of this Mexican hat okay
a is now massive and a massive field has
three polarizations three degrees of
freedom so like a phonon which can be
massive you can oscillate in the
direction of propagation you've got a

55:22

longitudinal as well as transverse as
well as two transverse so we can a
longitudinal plus two transverse modes
it cannot have a longitudinal
oscillation of a massless field moving
at the at the speed of light so here we
have one degree of freedom in the in the
H field and in the magnitude of the
field and three degrees of freedom in
the in the gauge field so again we get
four two piece of freedom total so we

55:54

haven't lost any degrees of freedom
we've just moved them around the the
gauge field has eaten one of the degrees
of freedom when it becomes mass
so just before we end this lecture I
guess we'll do we'll apply a Ginsberg
LAN dout theory to two superconductors
in the in the next lecture but I should
comment a couple things about about I
mean you see an awful lot about the
Higgs boson in the in the popular media

56:25

so the Higgs boson is the excitation up
and down this this Mexican hat
the massive excitation and you'll often
see written in the popular media like
the Higgs boson gives mass to the
departed calls like um like like
electrons get their mass from the Higgs
boson that's not really true what
actually gives the the photon mass gives

56:55

the the gauge field mass is is the valve
the vacuum expectation value this
quantity here that's not the Higgs boson
the Higgs boson is the boson the
excitation is is this H field they have
stations up and down that the hill the
they've it's what gives the particles
particles in mass there's another thing
that you will often see in the popular
media that all mass in the universe is

57:26

is created by the by the Higgs by the
Higgs field or the I guess is what they
mean and that's not true either it's
only true for leptons for for protons
for example most of the mass is is
contained in gluon physics not in Higgs
physics all right so so next time we
will start up on applying Ginsburg
Landau theory to superconductors and
finding a little bit about their their

57:59

properties okay until until we meet
again

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