Quantum Matter Lecture 8

Quantum Matter Lecture 8

SUBTITLE'S INFO:

Language: English

Type: Robot

Number of phrases: 1031

Number of words: 7387

Number of symbols: 31883

DOWNLOAD SUBTITLES:

DOWNLOAD AUDIO AND VIDEO:

SUBTITLES:

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

DOWNLOAD SUBTITLES: