17 - 3 - part iii geometric berry phase (difficult advanced material - optional) (2607)

7/28/2019 17 - 3 - Part III Geometric Berry Phase (Difficult Advanced Material - Optional) (2607)

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Finally in this video we're going todiscuss a very interesting class ofquantum phenomenon that appear in thepresence of slow, or so called adiabatictime-dependent perturbations.And such adiabatic perturbations giverise to a very elegant mathematicalstructure.And, in particular, this so calledgeometric, Berry Phase that we're goingto derive.so the this Barry Phase was discoveredtheoretically by Sir Michael Barry.And this paper, published in 1984 in theProceedings of the Royal Society ofLondon.And this is a really very well writtenand elegant paper, and actually eventhough it's a research-level paper whichin all likelihood may lead to a NobelPrize for Sir Michael Berry you arewell-equipped to actually understandeverything in this paper.

So now, both the derivation and the finalresults.because it requires nothing but the basicknowledge of single particle quantummechanics and Schrodinger equation.So we're going to reproduce part of thederivation in today's lecture, in thispart but of course if you want to knowmore details I would like to refer you tothe original paper which again is verywell written.So the problem that Sir Michael Berryconsidered was well, in retrospect a very

simple problem.So basically it was the canonicalSchrdinger equation with sometime-dependent Hamiltonian.And he assumed two things.he looked into adiabatic perturbation.That is to say that So the time-dependentin h of t is, is assumed to be slow.what it means precisely, we're going todefine in the next few slides, but atthis stage that's just leave it at that.So we have some very slowly changingHamiltonian.

And also let's assume that theHamiltonian returns to itself after acertain period let me call it capital T.So basically, we have a periodic in timeslow perturbation.And the question that he asked is whathappens when the wave function As we getto this moment of time, capital T.So what is the wave function at the finalmoment of time?


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So and, it turned out that, as we willsee, so that this wave function has avery interesting contribution sort oftopilogical, or geometric contributions.There are various sort of descriptions ofit that appear in the literature.that was completely counterintuitive, andthat takes some time to digest.even after a derivation.So now let me formulate the problem thathe solved more precisely and proceed tothe actual solution of the problem.And to the formulation of these Theory ofthe Berry phase.So, let's imagine that we have aHamiltonian that depends on a parameterlambda, okay?So this could be a, anything.For example, it can be magnetic field orit can be some parameters of a harmonicoscillator.let's say, the frequency or somethingelse, whatever you want to think about.So let's have with some we have some set

of parameters lambda And we combine thoseparameter in a vector so in principle itcould be more than one parameter.And we just construct a, sort of a vectorin the parameter space which does nothave to be three-dimensional but let mejust for sake of simplicity, let meimagine that they have athree-dimensional parameter space.We will say this is going to be lambda xor lambda one, lambda two, lambda three.Okay.And let me consider the station where

this parameter lambda is changing in, intime.So, for example, again, it can be amagnetic field that is changing.And also, let me consider the situationwhere the after a certain period of timecapital T, so the parameter.Returns to the original point.So if this is my parameter space.So, basically what I'm talking about.I'm talking about a slow evolution ofthis lambda such that it starts at acertain point and returns to the same

point after, time capital T.So, and as I advertised in the previousslide, so what I want to know Is whathappens with the wave function?So, I have some initial condition for thewave function.And,So, the Hamiltonian, after these period.Capital T returns to itself.So, but the wave function, does not


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necessarily have to return to itself.So the question that we're interest in iswhat happens?With this wave function.As a first step in this derivation, letme define precisely what I mean by slow,or, adiabatic perturbation.So what does it actually mean, that theHamiltonian is changing slowly with time?And to provide this definition let meformulate an eigenvalue problem.Instantaneous eigenvalue problem for theHamiltonian.So the Hamiltonian itself is changingwith time so there is no reason for us towrite the stationary Schrodingerequation.But never the less mathematically it iswell defined.So, we can consider at any moment of timet, we can consider an eigenvalue problemfor this h of t.And we can find a set of eigenfunctions.And that the responding eigen values

which I denote as e sub n of t.So, for the sake of simplicity let meactually assume that theSpectrum of the Hamiltonian is discrete.So I have just a set of discrete levels.So n equals 0, n equals 1, et cetera.Also, let me assume that the initialcondition for the wave function, psi at tequals 0, is one of the eigenstates of myHamiltonian at t equals 0.So, for the sake of concreteness, let meactually assume that this is the groundstate.

So n equals 0 at t equals 0.So by the way, it's not necessarilyreally for the main conclusions of thederivation, but for the sake ofcompleteness, let me just be specific.Okay, now let us recall what we discussedactually in the previous lecture, in theprevious part of this lecture where wedealt with the opposite case of fast orsudden perturbations.So there, we know that if we change theHamiltonian very fast, the state, theinitial state, which in this case is the

ground state, would be sort ofredistributed all over the place.so the particle after the quench, orafter a fast sudden change, would existin all possible eigenstates of the newHamiltonian.And sometimes it's natural to define asmall perterbation as the opposite to thesudden perturbation.That is to say that, we will define a


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small perterbation aside that it does notinduce transitions.So, basically what happens here is, let'ssay this is the parameter lambda.It's equal to 0.And we have our initial state, the groundstate, with n equals 0.And, as I change my lambda, so, withtime.So, these energy levels start to move.So, the energy levels really are thisSolutions to this, eigenvalue problem.But the Hamiltonian was changing.And so the, energy levels are moving.And the principal.Also, the explicit, wave functions arechanging.So, and what I can imagine happening isthat.The wave, the particle, which initially,was a.sort of confined to the ground statecanon principal propagate to all otherstates.

And this indeed will, will happen if thechange in Hamiltonian is fast enough.But if this is slow, and this will givethe definition of slow, the the state ofthe system would remain in theinstantaneous ground state of myhamiltonian.And also in this case, if I start formthis point in the parameter space andreturn to the same point in the parameterspace so that at the end of this periodicevolution basically I require then thatthe system, that the system would remain

in the old ground state.And this will be my definition of a slowperturbation.So to mathematically define a precisecondition for this to happen requires alittle bit of work and that provide thisattenuation in the supplementary materialin the notes.And if you go through the attenuation,you will see that the condition for thisto happen for the condition for having notransitions Is that the derivative of theHamiltonian, which is an operator, and

the matrix element of this d H d tbetween two states, let's say this groundstate and for example the first excitedstate, this must be much smaller than thelevel spacing, the distance.between these energy levels divided bythe typical time well, the concentrictime at which the change in theHamiltonian occurs.So this is the definition if you want, if


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you want or, the con, con, constraintthat we have to satisfy in order for theperturbation to be slow.And one thing that we that we notice hereis that, of course, if e n minus e of 0vanishes.That is, if the levels cross, or if theycome very close to one another.So this condition cannot be satisfiedbecause when necessary we'll inducetransitions between these states.But if the gaps don't close.If the level spacing remains large enoughand if the change in the connectonium isslow enough so we can satisfy thiscondition.An do good approximation, we can assumethat the system remains in the groundstate during the revolution.So, now we are at the position toactually derive this geometric berryphase.And what it entails is essentiallysolving the Schrodinger equation, the

time dependent Schrodinger equation underthe assumption of this slow perturbationthat does not induce the transitions.So, this is my Schrodinger equation.I'm just writing it down again.And here I'm basically requiring that psiof t remains proportional to the groundstate.And the only thing I can have here is aphase.Right?So, because, well, the total number ofparticles s conserved, so therefore I

cannot have here any, coefficient whoseabsolute value is larger than 1 becauseit would imply that the probability tofind a the particle in this instantaneousground is larger than 1, and it doesn'tmake any sense.So, and I also require that well, withinmy approximation, there are notransitions, so the particle remains inthe ground state.So there's basically only, only onepossibility for this to happen.if this coefficient relating the wave

function at time t, and the instantaneousground state is a pure phase, either fori phi of t, or if i is a real function.So under these assumptions, what remainsis is to find this one phase.Everything else we know so this is reallythe remaining unknown in our [INAUDIBLE]time-dependent problem.And if we plug in this form of thesolution back into the Schrodinger


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equation and recall by definition thatthis 0, t is the instantaneous eigenstateof the Hamiltonian.So simply get, so basically combiningthese 2 things will lead to theSchrodinger equation in this form.So it will be I H D over D T of phi of T,in this form is equal to the energy ofteh ground state which is time dependant.And this is because energy is movingaround.si of t.Okay?And well so this is no longer an operatoractually.This is just a function and naively youmay sort of assume there exists a simplesolution to this problem.where if I would be equal to sort of ageneralization of usual quantum phasethat appears even in the time-independentproblems.So we could naively just write it asminus 1 over h bar, an integral from zero

to t, E naught of t prime, d t prime.So again, if the Hamiltonian were not todepend, on time, that is, in this picturewould mean that basically Lambda's apoint which stays there.So I would just get the quantum phasebeing e to the power i minus i over hbar, energy times time, which is theusual quantum phase that appears instationary quantum mechanics.So here, instead, I may just go ahead andintegrate from 0 to t.Well it turns out however that this is

not entirely correct.And there exists an importantcontribution which appears in well, tothis quantum fix, and let me write thiscontribution as gamma.And this addtional phase gamma that wewill derive in a second is exactly whatis called the Berry phase.So this is my, our main quantity ofinterest.While the first part is, is sort of theusual dynamical phase.It's actually called dynamical phase, let

me, you note it as Theta with thesubscript of D, and this is calleddynamical phase.If we plug in now, back, you know,basically, this expression for the phaseback into the Schrodinger equation, we'regoing to get in the left hand side, we'regoing to get the following term.So there's going to be derivative pf e tothe power i theta, e to the power i


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is equal, basically, to this guy, i dover dt 0t.So finally, what we can do here, we cantake advantage of the fact that we havenormalized states.So the number of particles is conserved,and therefore, and hold up, and ourparticles exist in this ground state,therefore, this bracket product of thisinstantaneous ground state with itself isequal to one.And so if we multiply this equation fromthe left by a broad vector of zero t, sowe're going to have simply one in theleft hand side.And we're going to have you know, zero tg over d t zero t in the right hand side.So therefore, what we can do, we canwrite the, the Berry phase, well, thisBerry phase gamma at the time T as anintegral from 0 to capital T.So this is the time that it takes theparticle, well, the, the system, toreturn to the original Hamiltonian.

And under this integral we're going tohave zero T, D over D T, zero T, D T.So it's an integral over time.So we get this interesting term which isthe Berry phase.And and as we shall discuss in the lastslide, so this term has a veryinteresting, geometric rotation, that'swhy it's called a geometric phase.To see this let us recall the definitionof these eigenstates, zero T, so thoseare the instantaneous eigenstates of theHamiltonian.

But, per our assumption, so theHamiltonian depends on time only throughthe parameter lambda, this extraparameter lambda.So instead of writing this guise as beingparametrized by t, we can as wellparametrize them by this parameterlambda, in this parameter space here.And if we do so, we can transform thisintegral over time into an integral overlambda in this parameter space.And this would require a very curious, asI said, geometric interpretation.

So almost the last step here is toperform a change of variables and to gofrom an integral over time, to anintegral over Landa.And this can be done by simply writing.So here, this dd, d over dt can bewritten, so this is the identity.It can be written as d lambda over dt dover d lambda.Okay?


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And on the other hand, when this guy isgoing to multiply.This differential, so we can simplyreplace this product by d lambda.And so another way to write this thisBarry phase, so let me present it here,so gamma can be written as an integraland now this integral will go actuallyuhhhm around this loop in this bramoshirspace.So it's going to be closed integral solets save this contour, we'll call it C,so it's going to be basically encirclingthe C, and here were going to have zero dover d lambda 0 lambda d lambda.Okay.So this is another way to writeessentially the same quantity, but thetime is completely gone from ourdescription even though we started withtime-dependent problem so here time isnot essential anymore.So, another thing we can do, we canintroduce a new notation, let me call it

A a differentiation A is the function ofLambda, which is by definition is goingto be this guy.And so in this notations we can, we cansee that the,A vary phase is nothing but a circulationof this potential if you want, A oflambda, around this contour.Okay, so another thing that we can dohere.We can take advantage of certainidentities from the vector analysis.Hopefully familiar to some of you who

have studied the theory ofelectromagnetism, namely we can definethe curl of this vector A, so let'sdefine the curl A Curl with this respectto this lambda.And let me call it b of lambda.And if we do so this berry phase gammacan be rewritten as a flux of this I willcall it, fictitious magnetic field,through the area enclosed by this loopthat corresponds to thisquantum evolution, okay.So essentially well it's very complicated

but, so those of you who again who arefamiliar with the theory ofelectromagnetism.Now remember, that to describe magneticfield in real space is often timesconvenient to introduce a vectorpotential such that the curl of thisvector potential is the magnetic field.And the, so here, instead of the realspace have this sort of fictitious


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parameter space, which does not have tobe a real space.It can be something else.Can be just be a set of parameters in ourproblem.And what we found is that the quantumphase that will be required by the wavefunction as a result of this evolution.It's essentially a flux of somefictitious magnetic field that exists inthis model.Okay?So, by the way, notice that even thoughwe started with a time-dependent problem,this b of lambda, and the, correspondingvector potential a can be calculated bythemselves.So if we just have a, Hamiltonian, whichdepends on Lambda, whether or not itdepends on time is a separate matter, wecan just go ahead and calculate theseproperties.And so, what we may think about is that.When we have a quantum system, which is

parameterized by some parameter lambda,so there exists an internal and in somesense fictitious magnetic field in thisparameter space.And so it exists whether or not weperform a quantum evolution, but if we doperform quantum evolution and if we havethis sort of periodic Time depend onperturbation.So the quantum phase that would beacquired by the particle is going to bebasically the flux of this magnetic fieldif you want, through this area enclosed

by this loop.And that's why it's called the geometricphase, sometimes it's also called thepological phase.It's a very non-trivial concept, so Imean, first of all it was not discoveredin the early days of quantum mechanics.It was discovered only in the 80s.And furthermore, only now, and this isbasically the subject of currentresearch, we're starting to understandthe true importance of this Berry phase.And, one of the recent discoveries that

you might have heard about Is so calledtopological insulators.Something I didn't have time to talkabout these quite amazing discoveries,but I just wanted to mention them inpassing to sort of emphasize that this isnot just a pure mathematical construct,which is sort of elegant by itself, butalso does have Important physicalapplications.


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And, this is the subject of currentresearch.So, the final thing I'm going to mentionhere sort of concluding this technicaldiscussion, is that unlike the magneticfield in real space, which does not havesources or sinks.So, there are no monopoles as we know.So, the divergence of the magnetic fieldis 0.So this fictitious magnetic fieldactually it can have sources.And this sources are, are, well some sortof monopoles in this parameter spacecorrespond to the degeneracies in thisspectrum of the problem.So what it actually means is and this isone of the results eh, understood byBerry in his original paper.Is that if we have some, value, somespecial values of parameter lambda, let'ssay, equal lambda star, such that, thereexist, two, there may exist two wavefunctions with the same energy.

So basically, the energy levels cross Sothen these points, let's say this point,serve as a source of the monopoles ofthis magnetic field.So in this parameter is best.This is also a very interesting result.Well, it's a very interesting subject.I would definitely talk more about it.It's a very exciting field.But I feel that I probably should stophere.Because I've been talking for twenty-fiveminutes now on this last part of the last

lecture.And I just would like to thank you verymuch for your attention.I hope that this lecture and in generalthe course were interesting and useful,at least some parts of it.And I wish you all the best, thank youvery much.

17 - 3 - part iii geometric berry phase (difficult advanced material - optional) (2607)

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