Green's functions in condensed matter physics: basics

hi guys and welcome to this virtual lecture course on quantum condensed metaphysics i'm dr andrew mitchell and in this lecture we'll be discussing the theory of green's functions in the context of quantum many body systems many of you may know green's functions from other areas of mathematics or physics where they're very handy in solving differential equations in the context of quantum many body physics the differential equation we'll be solving using these green's functions is of course the schrodinger equation we'll see however that green's functions in condensed metaphysics have a very nice and intuitive interpretation as a single particle propagator it tells us what is the probability that if i put a particle into the system at a particular point at a particular time then i can take it out of the system somewhere else at a different time it tells us about the dynamics of the system how the electrons are moving through the system and what their correlations are in time we'll be discussing this uh theory which is a very rich and powerful theory it's really one of the work horses of quantum condensed metaphysics uh in this lecture we'll start from basics we're going to write down the definitions of these greens functions and we'll see how we can formulate them in terms of the time domain or the frequency depend domain and also in terms of a complex variable z we will see the physical interpretations of these greens functions and how to actually just go ahead and calculate them for a given simple system the system will start with is the simplest possible system of a time independent single quantum orbital with a particular level energy and we'll derive all of the um the greens functions for this system we'll talk about how the greens functions are related to the density of states of the system we'll also talk about the fluctuation dissipation theorem and how the green's functions are related to orbital occupation we'll then have a look at a time-dependent hamiltonian and we'll go through the same process again for this non-equilibrium situation we'll look at the situation of a so-called quantum quench where we suddenly change the hamiltonian at t equals zero and see how the dynamics of the system evolves finally we'll have an outlook towards systems with many quantum orbitals which is of course what we want to look at in this course in the coming lectures we'll return to this and we'll be developing further the theory of green's functions but the foundational material will be contained in this lecture so let's get down to work in this lecture we'll be discussing green's functions in the context of quantum condensed metaphysics but before we get into that let me give some of the mathematical background behind the theory of green's functions suppose that we have a linear differential operator l hat which is a function of the coordinate x let's say that we then wish to solve differential equations of the type l hacked acting on u of x is equal to f of x so here we have on the right hand side a source term f of x and the linear differential operator l hat on the left hand side and we want to solve this equation for this function u of x which u of x when operated on by l hat gives us the source term f of x one way of solving differential equations of this type is to use green's functions we construct a green's function g which is a function of x and s such that when l hat acts upon the green's function we get the delta function the dirac delta function delta of x minus s by multiplying both sides of this equation by f of s and then integrating over s we then easily obtain a solution to u of x so if we know with the green's function g of x and s we can easily find solutions to differential equations of this type some of you may have already encountered green's functions in the context of classical physics in particular it's widely used in the theory of electrodynamics for example let's take the poisson equation where we have a differential operator which is the laplacian del squared acting upon our electrostatic potential phi of r and this is equal on the right hand side to minus one upon epsilon naught times the charge density rho of r in analogy to the above expression involving our lx operator our solution u of x and our source term f of x here we see that the differential operator l hat is the laplacian del squared the source term here is basically the charge density rho of r and the solution that we're after is the electrostatic potential phi of r we can now introduce a green's function g which is a function of two coordinates r and r primed in a translationally invariant system however we can express this in terms of a single variable r minus r prime we define our greens function g of r such that when we operate on it with our laplacian we get the delta distribution delta of r substituting this into the poisson equation we then find an expression for our electrical potential the reason why this construction is useful is that the defining equation for the greens function here that's laplacian acting on the greens function gives the delta function is a much easier equation to solve than the generic poisson equation where the right hand side has some complicated source term in particular we can solve the green's function equation here in this case by means of a simple fourier transformation if we go to k-space then we find very straightforwardly that g of k is just equal to minus 1 upon k squared and i'll leave that as an exercise for you going back to real space we can then substitute our expression for g of r into the equation for the electrical potential this then immediately gives us the well-known expression for the potential created by an arbitrary charge distribution okay so how can we use this green's function theory in the context of quantum condensed metaphysics of course in quantum condensed metaphysics the differential equation that we're interested in solving is the schrodinger equation h psi is equal to e psi or likewise the time-dependent schrodinger equation i d by dt of psi is equal to h psi for a single particle hamiltonian we can then simply define a corresponding green's function which solves the schrodinger equation we can write that e minus h acting on the greens function is equal to the delta function here i've allowed the greens function to have arguments r and r primed but on the right hand side we see that this is equal to the delta function of r minus r primed so in fact r and r primed must enter in the greens function only in the combination r minus r primed in the position representation the greens function is therefore diagonal and we only have a single collective variable r minus r primed we can then naturally identify the inverse green's function g inverse of a variable r as being simply e minus h of r this then implies that g inverse of r multiplied by g of r minus r prime is equal to the delta function r minus r primed as desired this is nice because it provides an alternative way of writing the schrodinger equation g inverse acting on the wave function psi is equal to zero we can now play exactly the same game with the time dependent schrodinger equation which i've written again here we introduce a green's function g which now depends on the position r and the time t as well as r primed and t primed on the right hand side we therefore have dirac delta distributions for position delta of r minus r primed as well as time delta of t minus t prime the same logic applies on the right hand side we see that we have a function only of r minus r primed and t minus t primed so we can define the inverse greens function just as a function of those two collective variables g inverse of r and t is therefore equal to i d by dt minus h of r from these equations it immediately follows that the wave function psi of r and t can be expressed as the integral over all space of the greens function multiplied by the wave function of r primed and t primed this can be very simply verified by taking this expression for the wave function r and t and plugging it in the schrodinger equation here and then using the property that this object in square brackets multiplied by this green's function is equal to these delta functions this expression therefore tells us that the greens function can be understood as a single particle propagator it relates the wave functions at time t primed to a wave function at time t and as we can see here from the integral this involves taking the particle at time t primed at all possible positions r primed and propagating it forwards in time to the specific position r this is rather reminiscent of the five and path integral version of quantum mechanics in which we imagine that the propagation of a particle from time t primed to t involves taking all possible roots from the start point to the end point so in quantum mechanics where the greens functions refer to solutions of the schrodinger equation the green's functions have a physical interpretation as the probability amplitude of taking a particle from position r primed at time t primed and ending up at position r at time t if we now define the wave function at position r and at time t as the projection of the wave function psi of t onto the position vector r and given that we know the solution to the time dependent schrodinger equation for a single particle which is written here psi of t primed acted on by the time evolution operator e to the minus i h of t minus t prime propagates the wave function forward to psi of t then we can simply write the following where here i've simply taken this equation and now pre-multiplied by bra r and then interpreted these brackets in terms of psi of r and t on the left hand side and psi of r and t primed on the right hand side i can now insert a complete set of states inside this expression by simply multiplying by the identity operator 1 hat by resolving the identity operator as an integral over position we then obtain the following the wave function psi of r and t is therefore equal to an integral over d r prime of psi of r primed and t primed multiplied by this matrix element that connects r to r primed under the time evolution operator but using our green's function formulation we have a very similar kind of equation comparing these two allows us to get an explicit expression in this case for the greens function we can write that the green's function g as a function r and t and r prime and t primed is equal to minus i theta of t minus t prime times this matrix element this theta function here simply tells us that the time t must be greater than the time t primed that is the order of the events of the propagation that we defined in these equations theta of t minus t primed here is simply the heavyside step function we start with a wavefunction at time t primed and then we imagine propagating forward in time to psi of t this defines the so-called greens function where t is greater than t primed sometimes you will see this denoted with the superscript r in the green's function to emphasize this is the greens function a different but somewhat equivalent solution to the differential equations are the so-called advanced greens functions where here we have plus i theta of t prime minus t and these advanced greens functions are often denoted by the superscript a in this case we assume that the wave function on the left hand side here psi of r and t occurs at a time before the wave function psi of r primed and t primed we can express these in the following way in terms of these brackets we see therefore that the green's function is basically associated to the probability amplitude of finding a particle in a state r and t given that it started at r primed and t primed the greens function is the one where t is greater than t primed and the advanced means function is the one where t primed is greater than t so for the function we start off in the state r primed and t primed and we moved move forward in time to r and t whereas for the advanced greens functions we start in the state r and t and moves forward in time to r primed and t primed now i want to emphasize here that so far we've been considering the dynamics of a system with a single particle however in condensed metaphysics we're usually interested in systems of many particles in fact in the thermodynamic limit we'd have a very large number of particles 10 to the 23 or so so we need to be able to generalize this theory of green's functions to systems involving many quantum particles for the remainder of this lecture and also in the coming lectures we'll now focus on the single particle green's functions but for many particle systems in many particle physics we will adopt the same green's function philosophy that we discussed on the previous slides but we will need to generalize some of the concepts green's functions will describe the dynamics of electrons in many body quantum systems not on the individual level of single electrons but on the statistical level of expectation values we will see that these greens functions are really the workhorse of many body quantum mechanics and they're very very powerful indeed we will see that single particle green's functions only contain part of the full information contained in the wave function however the information that they do contain tells us a lot about the system and in fact are closely related to various experimental observables of interest it turns out that there are various different types of single particle green's functions but the one we'll consider in this lecture course will be the single particle green's functions for fermions i will define it g a b of t and t primed t and t primed here are the time labels and a and b are the uh orbital or site labels we're going to propagate an electron from site b to site a to emphasize that this is a green's function one can add this superscript r here however often i will just drop this and assume that whenever you see a green's function like this i'll be talking about the green's function unless otherwise stated the greens function is defined on the right hand side here it's minus i theta of t minus t primed this of course means that t has to be greater than t prime multiplied by the thermal expectation value of some operators the operators inside here are the anti-commutator of ca and cb dagger evaluated at time t and t primed respectively this might seem like a rather complicated and abstract object but we'll be discussing the physical implications of this in the coming slides writing out this anti-commutator here i see that we have the sum of two terms the first involves the thermal expectation value of ca evaluated at time t and cb dagger evaluated at t primed whereas the second expectation value has the order of these operators switched we have cb dagger evaluated at time t primed and ca evaluated at time t and of course in quantum mechanics the order of these operators matters even more so here because they have different time labels we will denote each of these two terms as so-called keldish green's functions we call the first term g greater and the second one minus g lesser these are denoted by the greater than sign or the less than sign in the superscript here specifically these keldish greens functions are defined as g greater is minus i times the thermal expectation value of c a of t times c b dagger of c t prime whereas g lesser is plus i times the thermal expectation value of cb dagger evaluated at time t primed times ca of t notice that in these definitions we don't see the time ordering there is no theta function that is pulled outside of the bracket here in the definition of the green's function apart from this decorative factor of minus i and plus i here we can easily see what these greens functions mean the physical interpretation of this g grater is the probability that if i put an electron into the many particle system at position or orbital b at time t primed then i can take the electron out at position or orbital a at time t notice that this does not have to be the same electron this is simply the overall thermal expectation value for a such a process likewise for the g lesser function it's the probability that i can take out an electron at orbital a at time t and put one in in orbital b at time t primed and again here there is no time ordering t can be after t primed or t can be before t primed the order of t and t primed is not specified in these keldish greens functions let's now take a look at the special case of systems with time translation invariants this means that we have a steady state situation the hamiltonian does not have a time dependence obviously this is a special case but it's rather a common case and it's something that we're going to be studying a lot in this lecture course of course one can have a non-equilibrium steady state but more commonly we'll be discussing systems at thermal equilibrium this means that the thermal expectation value of some operator evaluated at time t is static it doesn't depend on the time at which you evaluate the expectation value however it's a little more subtle when we're talking about the greens functions because we have two time variables this tells us that the greens function gab of times t and t primed must depend only on the time difference between t and t primed so from now on in this case we'll write g a b of t minus t primed this is because the greens function corresponds to a correlation function involving operators acting at times t and t primed because of the property of time translation and variance in the system i can slide t and t primed simultaneously backwards and forwards in time and still get the same answer however the correlation will depend on the time difference between t and t primed there will very naturally be a difference in the correlation between events separated by a short amount of time and events separated by a large amount of time indeed since we can slide t and t primed backwards and forwards in time and still get the same answer why don't we simply choose t primed to be equal to zero this then gives us that the green's function g a b is a function simply of the time t in terms of the fundamental definition of our greens function we can then write that g a b of time t is equal to minus i theta of t times the thermal expectation value of the anticommutator of c a evaluated at time t and c b dagger evaluated at time zero and just recall here that theta of t is the heavy side step function which is equal to one when t is greater than or equal to zero but theta of t is equal to zero when t is less than zero also note that we can write cb dagger evaluated at time zero simply as cb dagger we can drop the time label if t is equal to zero that was all for the greens function but we can similarly define the keldish components in the same way where we set t primed explicitly to zero in our time translation variance system in most of what follows we're going to be considering these equilibrium situations where the hamiltonian has no explicit time dependence the system will then have a natural time translational invariance and the corresponding greens functions will be functions of a single time variable you should just bear in mind that this time variable t corresponds to the difference in time for the action of our two operators in the greens function later on in this lecture however i will do one example of a system with an explicit time dependence where the green's functions then depend on two time variables but for now let's just stick with the simplest case in this case we can naturally look at the green's functions not in the time domain but in the frequency domain there are actually some subtleties here due to whether we take the fourier transform or the laplace transform and subtleties to do with the convergence of integrals let's define the laplace transformation of the keldish green's function as g a b of the complex variable z is equal to the integral dt from zero to infinity of e to the i z t of the green's function g a b as a function of time i've written this one down explicitly for g greater but this exact same equation also holds for the g lesser here the complex variable z is defined as omega plus i delta where omega and delta are pure real and delta in particular is greater than zero by definition this is needed to ensure the convergence of these integrals as we'll see shortly the laplace transformation of the green's function g a b of t is very similar since the g a b of t is equal to theta of t g greater of t minus g letter of t when we do the laplace transformation this theta of t doesn't matter because we're already going from zero to infinity anyway the time is always positive within this laplace transformation therefore we can simply write that the gab of z is equal to g greater of z minus g lesser of z everything here is a function of the complex variable z is equal to omega plus i delta so how does this relate to green's functions in the frequency domain which would be obtained by doing a fourier transform well the fourier transform is defined as the integral dt from minus infinity to plus infinity of e to the i omega t times the green's function gab in the time domain g of t this would give the green's function in the frequency domain gab of omega however the definition of the greens function in the time domain gab of t as defined in this equation has in the definition of it this theta function theta of t so when we insert that into this expression it will of course kill half of this integral and we'll end up only with the integral dt from zero to plus infinity this means that the greens function in the frequency domain gab of omega is very closely related to that which we obtained by a little laplace transformation g a b of z g a b of z of course is a function of the complex variable omega plus i delta with delta defined as being positive therefore we can make the connection between gab of omega and gab of z by simply taking the limit of the latter as delta goes to zero however since delta is defined as being positive we have to take this limit of delta goes to zero plus that means that delta remains positive but it can become infinitesimal in the end we obtain the fourier transform of the greens function in the frequency domain this might seem rather convoluted and unnecessary however we'll see explicitly shortly that we cannot simply do a regular fourier transform of these greens functions because the integrals don't converge we actually need to include an infinitesimal positive imaginary part for these laplace transformations to be well defined we can then define the fourier transform by simply taking the limit of the imaginary part of z to zero while keeping it infinitesimally positive this is an example of so-called analytic continuation we define g in terms of the complex variable z with the positive imaginary part g is therefore defined everywhere in the upper half complex plane but then we examine the behavior of the green's function g as we approach the real axis and sends delta to 0 while keeping it positive we therefore obtain g of omega which is a function of real frequency omega let's now have a look at an explicit example in fact let's take a look at the simplest possible example of a single non-interacting and time independent quantum orbital the hamiltonian h hat reads simply epsilon c dagger c where epsilon here is the level energy for the single orbital in the occupation number basis we have eigenstates 0 and 1 corresponding to the situation with no electrons or a single electron in our quantum orbital and of course these satisfy the schrodinger equation h psi equals e psi specifically h acting on the zero state gives us an eigenvalue of zero times the zero state back again whereas h acting on the one state gives us epsilon times the one state back again so the energies are simply zero if it's unoccupied and epsilon if it's occupied epsilon can therefore be understood as the single particle energy or local potential energy for this quantum orbital notice here that the hamiltonian does not have an explicit time dependence therefore our green's functions will only involve relative time differences let's take a look at the lesser green's function g letter of t this is defined as before as plus i times the thermal expectation value of the c dagger operator acting at time 0 and the c operator acting at time t so how do we compute such an expectation value well here let's recall that the time evolution of an operator a hat in the heisenberg picture is given by a hat of t is equal to e to the iht times a times e to the minus i h t and here a hat is the time independent version of the operator for and time independent hamiltonian h hat we can therefore write that the g lesser greens function is plus i times the expectation value of a time independent operator c dagger times e to the i h t times a time independent operator c times e to the minus i h t next we need to calculate this thermal expectation value now we should recall that the thermal expectation value of some operator a hat within statistical mechanics is given by 1 over z times the sum over eigenstates j times the boltzmann weights e to the minus beta ej times the matrix element of the operator a sandwiched between the eicon states psi j here we have the schrodinger equation h psi is equal to e psi in particular if i act with the hamiltonian on an eigenstate psi j i get the eigenstate back again times the energy ej this is the energy ej that is featuring in the boltzmann weight here in our statistical sum the factor z is of course the partition function which is simply the sum over all of these boltzmann weights it's basically normalizing this probability finally beta is the so-called inverse temperature it's just one over kbt with kb the usual boltzmann constant so now we have all of the ingredients to simply go ahead and calculate by hand this g lesser greens function so putting all of this together we have the following first we have this factor of i in the original definition divided by the partition function which is the sum of the boltzmann weights that's e to the minus b to epsilon plus e to the minus b to zero because we have two eigen energies zero and epsilon then we have the sum of two terms for the two eigenstates each of these is weighted by its own boltzmann factor the first one involving the occupied state involves the boltzmann factor e to the minus b to epsilon the second one involving the state 0 involves the boltzmann factor e to the minus beta0 the operators inside these matrix elements are the same it's c dagger e to the i h t c and e to the minus i h t we can already make a simplification here because e to the minus b to zero is of course simply one we can also immediately evaluate e to the minus i h t acting on the one state or e to the minus i h t acting on the zero state that's because this exponential operator here involves the hamiltonian and we're working in the eigen bases the first of these terms gives us e to the minus i epsilon t acting on the one state that's because of course the one state is an eigen state of the hamiltonian whereas the second one gives us e to the minus i zero t and again this factor is of course just one in the end if you're curious about the way that these exponential operators act and why i'm allowed to simply replace the operation of the hamiltonian on the eigenstate when it's in the exponent by simply the eigenvalue you can think about just expanding this exponential as a taylor series in powers of the hamiltonian anyway the resulting factor e to the minus i epsilon t or e to the minus i zero t these things are of course just pure numbers and commutes through the rest of these operators so i can collect those exponentials together as common factors then i obtain this expression importantly we see here that this second term is actually equal to zero because we have the action of the annihilation operator c acting on the already empty state so this gives us zero and kills that whole second term likewise in the first term we have the annihilation operator c acting on the occupied state which of course gives us the zero state furthermore we can then act on this zero state by the exponential operator e to the i h t and that will give us e to the i zero t in this case e to the i zero t is of course just a factor one that's a number that can be computed outside of the matrix element which gives us overall the c dagger operator acting on the empty state which of course gives us the occupied state so in the end we're looking at the overlap of the one state with itself and that object is simply equal to one so overall the g lesser green's function as a function of time is simply i e to the minus beta epsilon times e to the minus i epsilon t divided by the partition function one plus e to the minus beta epsilon if i then multiply this expression top and bottom by e to the plus beta epsilon then i get this slightly simplified form furthermore this can be expressed in terms of the fermi dirac distribution f of epsilon defined in the usual way as one over one plus e to the minus b to epsilon as we know the fermi dirac distribution controls the thermal occupation of the orbital and we see it here entering in the g lesser greens function so we see from this explicit expression that the green's function g letter of t is actually something that is periodic in time that is because of this factor e to the minus i epsilon t using euler's relation to write e to the i theta as cos theta plus i sine theta we can then express g lesser of t in this way the imaginary part of g lesser of t as a function of time is therefore simply a cosine the amplitude of the oscillation is simply the fermi function we see it goes between plus f of epsilon 2 minus f of epsilon and the time period is 2 pi upon epsilon we have a periodic structure to the greens function and this goes on forever in time before we discuss the corresponding fourier transforms and the green's functions in frequency domain let me make a comment on the equal time screens function if we consider again now the two times green's function g lesser of t and t primed but now we look at equal times t prime is equal to t then that would have the explicit expression written on the right hand side here due to time and translation in variants of the system given our hamiltonian does not depend on time we can evaluate this green's function any specific time and we can choose in particular t equals zero so in time independent systems the equal times green's function is simply equal to the t equals zero green's function we can now go to our explicit expression for g lesser of t and set t equals to zero to discover that g lesser of zero is simply equal to i times the fermi function f of epsilon we can also see that from the graph here the imaginary part of g lesser is equal to f of epsilon however g letter of 0 has another interpretation plugging t equals 0 into this expression simply gives us the expectation value of c dagger c this is of course the expectation value of the number operator for our single site we can write g lesser of zero therefore simply as i times the expectation value of the occupation of our site so we see that we get the static expectation value of the operators when we look at the zero time or equal time greens function furthermore we recovered the expected result we see here by equating these two expressions that the expectation value of the occupation for our site is given by the fermi function this is precisely the meaning of the fermi function it's the thermal expectation value of the occupation of a quantum orbital so all of this makes good sense okay so what about the green's function in the frequency domain well in the time domain we see that we have an oscillatory function with a well-defined frequency because we have a single frequency to the oscillation we expect in the frequency domain a delta function so let's see how that works out mathematically first of all let's examine the naive fourier transform g latter of omega is equal to the integral dt over all time of e to the i omega t of g letter of t by substituting in this explicit expression for the g lesser greens function in the time domain we obtain the following this looks like a rather simple integral we have i times the fermi function f of epsilon times the integral dt over all time of the fourier factor e to the i omega t times this factor coming from the green's function e to the minus i epsilon t of course we can collect together these two exponentials here as e to the i t multiplied by omega minus epsilon and now we can simply do the integral we obtain this object which is then evaluated at t equals minus infinity and t equals plus infinity and now we have to pause and think a little bit because there's some problems with this expression the first giveaway that there's a problem is that downstairs we have omega minus epsilon what happens when we want to evaluate the green's function g of omega at the point omega equals epsilon this would be undefined because we'd have a divide by zero problem that's the first hint the second hint that something's wrong is that we have to evaluate this integral at t equals minus infinity and plus infinity but as we already saw e to the i t omega minus epsilon is an oscillatory function of time what value do we assign it at t goes to infinity or minus infinity we could write e to the i t omega minus epsilon as cos t omega minus epsilon plus i sine t omega minus epsilon but what value do we assign this at t equals infinity it's oscillating between plus and minus one naively it might appear that this integral is actually undefined however we have to be a bit more careful from the theory of fourier analysis you might recognize the integral dt of e to the i t omega minus epsilon is actually the definition of the delta function so we can actually write that g lesser of omega for pure real frequency omega via the trans fourier transform is given by 2 pi i femi function f epsilon times the delta function the direct delta omega minus epsilon i'll return to this formulation a bit later however first let's look again at the laplace transformation of g lesser of z as before the laplace transformation for g lesser of z is defined as the integral dt from zero to plus infinity this time of e to the i z t times g letter of t here z is a complex variable omega plus i delta where both omega and delta are real and delta is taken to be positive well going through the same steps as before you obtain a very similar expression i f of epsilon times e to the i t z minus epsilon over i z minus epsilon evaluated between t equals zero and infinity the integral is well behaved and convergent first of all let's look downstairs here now we have z minus epsilon on the denominator as long as that we have a finite imaginary part with delta greater than zero here this tells us that the denominator can never actually be equal to zero that's because epsilon is real so there's no divide by zero problem in this equation so let's have a look now at evaluating this definite integral at the two limits t equals zero and t equals infinity okay so this lower limit here is obviously well behaved because we plug in t equals zero there and the top is equal to one what about in the case of t goes to infinity if we substitute explicitly z equals omega plus i delta then we obtain the following in particular notice that the plus i delta factor on the numerator becomes e to the minus delta t and if delta is positive as we've assumed here then this factor will ensure convergence of the integral as t goes to infinity this e to the minus delta t will exponentially suppress anything in this expression here as t becomes large as we've already discussed this first term is oscillatory and is bounded from plus to minus one so this e to the minus delta t will certainly suppress those oscillations at large time giving us a convergent solution to our problem on the bottom we see explicitly that even if omega is equal to epsilon this will not be a divide by zero problem because we have a finite imaginary part so overall the integral converges we have a well-behaved function it evaluates to minus f of epsilon over z minus epsilon let me come back to the physical interpretation of this expression when we discuss the corresponding green's functions in the frequency domain i will also discuss the relationship between the results for the fourier transform and the result for the laplace transform so far we've been considering the g lesser function now let's turn to the corresponding g greater function g greater of t is defined as being minus i times the thermal expectation value of the operator c of t and c dagger of zero we can now evaluate explicitly this expression in exactly the same way as we did for the g lesson using the previous definitions of the time evolution of the operator c of t and also using the expression for the expectation value from statistical mechanics we obtain the following this is exactly an analogy to the g lesser function that we considered previously as before we can immediately make some simplifications first of all um e to the minus b to zero is of course one in both of these two factors also we have c dagger acting on the occupied state this time and this is equal to zero whereas in the second term we have c dagger acting on the unoccupied state which gives us the one state the expression for g greater of t therefore collapses to this e to the minus i h t acting on the occupied state of course gives us a factor of e to the minus i epsilon t times the occupied state back again e to the minus i epsilon t is of course just a number commutes through these operators and we can collect it to the front of the expression we can therefore write e to the minus i epsilon t times c acting on the one state this of course just gives us the empty state then we act one more time with e to the plus i h t on the empty state which gives us a factor of one and then we simply have the brackets of zero with itself which is one so collecting all this together we have that g greater of t is equal to minus i times e to the minus i epsilon t divided by 1 plus e to the minus beta epsilon as before we can now multiply top and bottom by e to the plus beta epsilon and then i again recognize the fermi function 1 over 1 plus e to the beta epsilon so in the end we see a slightly different expression for g greater we have a minus sign and this extra factor of e to the beta epsilon in fact with a bit more manipulation we can actually eliminate this factor e to the beta epsilon by expressing it in terms of the fermi function i will leave it as an exercise for you to prove that the g greater function can be written in this form as minus i into 1 minus the fermi function times the factor e to the minus i epsilon t now we can interpret the factor of 1 minus f of epsilon here as being the population of holes if f of epsilon would was equal to the expectation value of the number operator that would of course give us the occupation of particles so one minus f is then the occupation of the holes it might seem rather strange to talk about the population of holes but it's simply a complementary way of counting things we can either count the number of particles relative to zero occupation or we can count the number of holes relative to an occupied site so we see that the g lesser function is basically the green's functions for the particles and the g greater greens function is the green's function for the holes the imaginary part of g greater of t therefore again has an oscillatory form this time the amplitude goes between 1 minus f of epsilon and f minus epsilon minus 1. of course we see the same time period of 2 pi over epsilon for the oscillations let's now take a look at the greens function in the frequency domain as before we'll actually apply the laplace transformation and relate g greater over z to g greater of t by integrating over time from zero to infinity of e to the i z t times the time domain g uh greater of t function following exactly the same steps as before we obtain that g greater of z is equal to one minus the fermi function divided by z minus epsilon alternatively we can perform the regular fourier transform and in that case very similar to the case of g lesser we obtain minus 2 pi i one minus f of epsilon times the delta function omega minus epsilon these are two equivalent ways of performing the transformation either to the complex argument z or to the real frequency omega so let me now summarize in the time domain for the simple example of a single non-interacting time independent quantum orbital we have g lesser functions and g greater functions they're related to this factor e to the minus i epsilon t and we have these fermi factors f of epsilon entering in particular we're now in a position to calculate the green's function g of t as we showed earlier this is simply theta of t the heavy side step function multiplied by g greater minus g lesser we can now simply substitute in our expressions for g lesser and greater and find out an explicit expression for g when we do that we find that g is simply minus i theta of t times e to the minus i epsilon t the fermi functions exactly cancel out this is very interesting because it tells us that g has no temperature dependence notice that the temperature dependence in our thermal expectation values entered entirely through these fermi functions and they've cancelled out here g has no temperature dependence for this non-interacting system so our greens functions do not depend on the thermal or statistical occupation of our quantum orbital so what information does it contain well to answer that question let's have a look again at the frequency domain in the frequency domain and here i mean functions of the complex variable z obtained by the laplace transformation we have g lesser of z being minus f of epsilon over z minus epsilon whereas g greater of z is one minus f of epsilon over z minus epsilon the corresponding green's function in the frequency domain g of said is therefore g greater of z minus g lesser of z as before which of course gives us in this case one over z minus epsilon again we see that the fermi functions cancel out they drop out we have no temperature dependence to the greens function in particular we can now also relate the lesser and greater greens functions to the function we find that g lesser is equal to minus the fermi function times g whereas g greater is equal to one minus the fermi function times g these are actually related to the so-called fluctuation dissipation theorem and are quite general in fact they really apply very generally in any equilibrium system as we'll see g greater and g lesser can of course also be directly related and we find that g greater of z is equal to minus g lester of z times the factor e to the beta epsilon note here that there's no simple relationship between g as a function of real frequency omega obtained by the standard fourier transformation and the g lesser and greater functions as a function of real frequency omega obtained by their corresponding fourier transforms and that is because if we look at the definition of g in the time domain given here we see that it contains this extra theta function so the fourier transform of g of t is not simply equal to the difference of the fourier transforms of g greater and g lesser because of this extra factor theta of t but we'll see an alternative formulation for that shortly so let's now focus on the greens function for this single orbital we've just derived that g of z is equal to one over z minus epsilon with a complex z equals omega plus i delta and delta positive we can now look at the real and imaginary parts of g of z and we find the following expressions in particular you'll see from these expressions that g of said is well-behaved and analytic meaning not singular everywhere for delta greater than zero and we say that the delta regularizes these greens functions in particular i can write minus 1 upon pi of the imaginary parts of the green's function g of omega plus i delta in this way i write it as 1 over pi delta all divided by 1 plus omega minus epsilon over delta squared this is the very definition of the lorentzian function centered on omega equals epsilon and of width delta so let's now plot this function as a function of epsilon for a given delta for a fairly large deltan we see the larencian has a rather broad width in fact it is easy to show from this explicit form that the peak is located exactly at omega equals epsilon the peak height is one over pi delta and the half width at half maximum of the peak is exactly delta therefore as we go to a smaller delta we see that the peak gets higher and it gets narrower but it's still centered on omega equals epsilon in fact as delta goes to zero but remains infinitesimally positive we'll see that this will turn into a delta peak meaning that it has infinitesimal width and infinite height in fact the lorentzian is a mathematically well-defined approximation to the dirac delta distribution we can see that because the normalization of the lorentzian which is the integral d omega from omega is minus infinity to plus infinity of this object minus one upon pi of the imaginary part of g is equal precisely to one and that's independent of the delta this means that if i would integrate this red curve here i would find an area under that curve of one if i integrate the blue curve i would find an area of one and indeed as i send delta to 0 keeping it infinitesimally positive of course i still find that the normalization of that is equal to 1. and that's one of the important properties of the delta distribution so we see that as we take the limit of delta goes to zero plus of minus one upon pi of the imaginary part of g we see exactly the delta function of delta omega minus epsilon this is an important quantity and we give it a name we call it the spectral function the spectral function a of omega is defined as being the limit of delta goes to zero plus of minus one upon pi of the imaginary part of the corresponding greens for a single quantum orbital we derived that the g of z is equal to one over z minus epsilon which gives a spectral function a of omega is simply the delta function delta of omega minus epsilon the spectral function a of omega gives the so-called density of states now for the simple case that we've considered involving a single quantum orbital the spectral function is simply a delta function this is a very trivial example we have a single state in the system located at omega equals epsilon therefore when we're looking at a of omega as a function of omega we have zero until we hit omega equals epsilon and then this delta function basically counts a single state for us notice that the integral over all omega of a of omega is equal to one the total density of states is normalized to one the spectral function a of omega can therefore be thought of as being a probability distribution containing the probability of finding a state of a given energy for a system of just a single orbital we have just a delta function located at the energy of that state we can also reconstruct the green's function with a knowledge of the spectral function a of omega and the real part of the green's function however given that the green's function g is an analytic function of the complex variable z we know that the real and imaginary parts are related through the chromos chronic transformation recall here that an analytic function of a complex variable z is one that has a well-defined and convergent taylor series in powers of z we can use the chromos-chronic relation to obtain the so-called spectral representation of the green's function g of said for any complex sed we can therefore write g in this fashion purely in terms of the spectral function as for the real part of g we can evaluate it simply as one over omega minus epsilon provided that omega is not equal to epsilon this is referred to as the principal value of one over z minus epsilon in this case the real part of g looks something like this with a divergence around epsilon the infinitesimal imaginary part of the frequency omega plus i zero plus here regularizes this function so that there's no real divergence at omega equals epsilon but for all intents and purposes when we're not at omega equals epsilon we simply see a regular function one over omega minus epsilon so this function one over z minus epsilon in the limit of delta goes to zero plus is rather an interesting one the imaginary part is only finite when omega is equal to epsilon when we get a delta function whereas the real part is only finite and regular when omega is not equal to epsilon in which case we get one over omega minus epsilon from our explicit expressions for the g lesser and g greater obtained by the laplace transformation of a complex variable z and then sending the imaginary part to a positive infinitesimal we can relate the lesser and greater greens functions to the spectral function a of omega on the right hand side indeed given that g can be expressed as g greater minus g lesser i can take the imaginary part of both sides of the equation and simply relate the imaginary part of g to the spectral function and of course we recover the definition of the spectral function that minus one upon pi of g is a of omega previously i mentioned that it was not so simple to relate g as a function of pure real frequency omega obtained by the regular fourier transformation in terms of the fourier transforms of g greater and g lesser however the spectral function provides a means to connect these objects the fourier transform to real frequency omega of g lesser of omega and g greater than omega can now be cast in terms of the spectral function a of omega we've derived this explicitly here because g lesser of omega and g greater of omega were expressed in terms of the delta function omega minus epsilon for a site of energy epsilon and we also related the spectral function a of omega to the delta function however it turns out that these expressions are totally general and apply in all cases we can prove that by the so-called layman representation which we'll talk about in a future lecture these important relations are known as the fluctuation dissipation theorem they relate the occupations of a site to the spectral functions we can see this by writing the orbital occupation which is the expectation value of the operator n hat which can also be written of course as the thermal expectation value of c dagger c in terms of our lesser greens function we can write it as minus i g lesser evaluated at t equals zero this is then the equal times lesser green's function which gives us the orbital occupation we can obtain g lesser of t from g lesser of omega using the inverse fourier transform we write 1 upon 2 pi times the integral d omega of e to the minus i omega t times g lesser of omega the zero time lesser greens function is therefore simply obtained by setting t equals to zero on the right hand side which then just means that we integrate g lesser of omega over all omega we can express the orbital occupation which is the expectation value of n hat then simply as minus i upon 2 pi integral d omega of g letter of omega substituting our expression for g lesser of omega into this integral in terms of our spectral function then gives us the following the orbital occupation is given as the integral d omega of the fermi function f of omega times the spectral function a of omega and this is an important result it tells us how the orbital occupation is related to the spectral function the orbital occupation is something that depends on temperature of course the spectral function does not depend on temperature but we see how temperature enters this equation through the fermi functions just a final comment on this this formulation involves green's functions of the real frequency omega and we were able to do the inverse fourier transformation here of course we could have used these equations at the top in terms of our complex variables variable z but then we would have had to have done the inverse laplace transformation which is a bit more cumbersome of course they both give the same answer but this version involving the inverse fourier transformation is certainly easier finally we're in a position to show that the greens function as a function of the real frequency omega as would be obtained by a regular fourier transform can be expressed in terms of the green's function as a function of the complex variable z as we take the imaginary part to be a positive infinitesimal we can also express this in terms of the spectral representation in the following way there are another class of green's functions that i've not yet mentioned called the advanced greens functions contrary to the green's functions however the only difference in the definition is that on the bottom here we have omega minus i zero plus rather than omega plus i zero plus for the greens function and therefore since the only difference is the sine of the imaginary part we can obtain g advanced simply by taking the complex conjugate of g and this is really the only difference sometimes you'll see these advanced greens functions mentioned in books or papers however we won't discuss them further in this course we'll just focus on the green's functions and suffice it to say that if we wanted the advanced functions we can obtain them simply by this relation now i want to do an example of a situation with a time dependent hamiltonian in that case we have green's functions that depend explicitly on two time variables here i've written the green's function in terms of its original definition and one can of course express this in terms of the greater and lesser functions these greater and lesser functions are written down here for the case where we have t1 and t2 being arbitrary times so the example that i want to look at is a so-called quantum quench this is a paradigmatic situation where the total hamiltonian h of t on the left here has an explicit time dependence it is however a simple example of time dependence because we go from an initial hamiltonian which is some static time independent hamiltonian h i to some final static hamiltonian hf and then we see these theta functions of minus t and theta of t here so basically this says the hamiltonian h of t is equal to h i when time is less than zero and then it instantaneously changes to hf for t greater than zero so at t equals zero we just change from the initial hamiltonian to the final hamiltonian and that's referred to as a quench here we'll consider the simplest possible example of this quantum quench where our initial or final hamiltonian this is h hat alpha where alpha is either i or f for initial or final is a single quantum orbital c dagger c with a level energy epsilon that changes depending on whether or not we're in the initial or final state one can think about this as having a hamiltonian h of t that has an energy level epsilon of t that depends explicitly on time and this is multiplied by the number operator for our site if how to plot epsilon of t as a function of time we would see that it takes a constant value epsilon i for t less than zero and then discontinuously changes to epsilon f for times t greater than zero so this is a very simple example of time dependence of the hamiltonian but we'll still see that it has some interesting physics and we can explore that physics using these time dependent greens functions i should note here that for time t less than 0 we have an equilibrium situation the hamiltonian is constant if we go to the infinitely distant future t goes to plus infinity then we'll also have an equilibrium situation where we've relaxed to our new equilibrium with a different epsilon this is the epsilon f however for finite t greater than zero we'll see some transient dynamics when a system is relaxing from its old equilibrium to its new equilibrium and it's this transient behavior that we'd like to understand here we will see that it's a little more complicated when we discuss the green's functions of two time variables t1 and c2 however because t1 might be before the quench at t less than zero and t2 might be after the quench at t greater than zero or we could have both t1 and t2 less than zero or both of them greater than zero these will correspond to physically quite different regimes so let's just do the calculation and see what we get so let's now suppose that the initial level energy epsilon i and the final energy epsilon f are both less than zero this will mean that the occupied state is the ground state of the system at all times t less than zero as well as t greater than zero for simplicity let's also specialize now to the zero temperature limit this means that we'll only have excitations from this unique ground state this immediately implies that the g greater function is actually equal to zero for all times that's because on the right hand side here i have a c dagger operator but if we're only considering excitations from the occupied ground state in the zero temperature limit then we will not be able to apply this c dagger operator acting with the c dagger operator at any time on this occupied state will annihilate that state by contrast if we look at the g lesser function then the order of the operators is reversed this means that here on the right hand side i have the annihilation operator acting on the occupied state and this will be finite expressing the time dependent operators in the heisenberg form i can now write the lesser greens function in this way here i'm time evolving the operators using e to the iht and e to the minus iht however the hamiltonian has to be evaluated at the particular time of interest for example in this term we originally had c of time t1 and therefore we need to evaluate the hamiltonian at that time t1 as i mentioned earlier we therefore need to break this down into the different regimes first let's look at the case where both t1 and t2 are negative in this case h of t is always the initial hamiltonian h i as i've written here in explicitly now we can evaluate this string of operators as before and we simply obtain i e to the epsilon i into t2 minus t1 we see in this case that g lesser is actually a function only of t2 minus t1 the relative difference between t1 and t2 this is the situation we saw previously for the time independent greens functions the reason why we're getting that again here is that both t1 and t2 are negative and therefore the dynamics of the system is evolving under a static time independent hamiltonian h i exactly the same logic holds when both t1 and t2 are greater than zero here the system is evolving under the final hamiltonian hf which is time independent the greens functions therefore look exactly the same except we simply replaced epsilon i with epsilon f the more interesting case arises when t1 and t2 have different signs for example let's look at the case when t1 is greater than zero that's after the quench and t2 is less than zero before the quench therefore in the string of operators this first term which has evolved at time t1 applies to the final hamiltonian hf whereas this operator is evolved under the initial hamiltonian because we're at time t2 which is less than zero when we evaluate this we find that g lesser is i e to the i epsilon i t 2 minus epsilon f t1 so let's now discuss the laplace transformation to the frequency domain of these green's functions first of all we have to take care of the fact that we have two time arguments t1 and t2 will therefore introduce the label t to mean the average time between t1 and t2 and the label tau to indicate the time difference t1 minus t2 with this change of variables i can write g lesser as a function of t and tau is equal to i e to the minus i epsilon f tau we now perform the laplace transformation on the variable tau this is the time difference between t1 and c2 this will give us a green's function g lesser which is something that depends on time but also on this complex variable z we want to keep this time argument here because we want to know the green's functions as a function of the real time in the system the dynamics of the system in these correlation functions are encoded in the time difference which is tau and this is the variable that will laplace transforming to the complex variable z evaluating this integral explicitly we obtain minus one over z minus epsilon f and we see on the right hand side that this screen's function is not depending explicitly on the time labeled t this is actually as expected because as we argued right at the beginning when both t1 and t2 have the same sign we'll have an equilibrium situation in this case with t1 and t2 both greater than zero the dynamics are controlled by the final hamiltonian and therefore we have epsilon f appearing downstairs here as before we can now define a green's function of the complex variable z in terms of g greater minus g letter it's just that now we carry this additional time label t we also argued that at zero temperature this g greater function was equal to zero and therefore overall we have that the green's function is simply one over z minus epsilon f again this confirms our original assumption that if both t1 and t2 are positive then we should obtain the equilibrium green's function for the final hamiltonian which we know already is one over z minus epsilon f in an exactly analogous fashion if t1 is less than zero and t2 is less than zero then we have an equilibrium green's function for the initial hamiltonian g lesser of t and tau takes this form performing the laplace transformation to the complex variable z from tau we obtain g lesser of t and z and then finally just as before we can obtain the greens function this again takes exactly the form we expected with one over z minus epsilon i now with epsilon i being the level energy for the initial hamiltonian much more interesting is the case when t1 is greater than zero and t2 is less than zero in which case we can define a lesser green's function of our two times t1 and t2 which we worked out earlier as this thing let's now convert this to t and tau we can write g letter of t and tau as i e to the i t into the difference between epsilon i and epsilon f minus a half tau of the sum of epsilon i plus epsilon f when we perform at the laplace transformation of this we obtain g lesser of t and z in the following form the spectral function a of t and omega therefore takes this rather unusual form we see that we have a delta function exactly at the average position of epsilon i and epsilon f but that the amplitude of this has this oscillatory form in time then we also get this piece corresponding to the real part of the greens function times the sine of t epsilon i minus epsilon f and this is finite whenever we're not exactly at the pole position omega equals a half of epsilon i plus epsilon f you'll also notice that this spectral function actually can be negative this is not a feature of equilibrium spectral functions but it's something that can happen out of equilibrium due to these quantum quenches so this was a simple example of a time dependent hamiltonian and time-dependent greens functions that arise for such systems in this particular case we considered the zero temperature quantum quench case which is one of the simplest it quickly gets more complicated with more complicated systems but the basic ideas are the same the final topic of today's lecture is how we calculate green's functions for many particle systems this is actually a big topic that will be developed further in the future lectures however today i just want to touch on a simple example of the quantum gas so far we just considered a single quantum orbital in fact now we're going to go back to the time independent case but we're going to add more orbitals a quantum gas is one that is written in this so-called diagonal representation we have a bunch of quantum orbitals but they don't talk to each other and all of the electrons in the system are independent this is a quadratic hamiltonian meaning that we just have pairs of these fermionic operators and there are no electronic interactions in the system furthermore each of these orbitals is decoupled from all the others the important simplification in the case of a quantum gas is that the exact many particle eigenstates of this hamiltonian are product states in the occupation bases of each orbital we can write an exact eigenstate of the system in this ket notation involving the orbital occupation for site number one two three and so on and we can express this simply as the direct product of these complex vector spaces for orbital one orbital two orbital three and so on where n1 n2 and n3 are the occupation numbers for those orbitals this is an exact eigenstate of the hamiltonian because if we apply h hat to this ket we get the ket back again multiplied by some scalar quantity that's the energy in this case of course which is simply the sum over the orbital occupation for a given site times the site energy and we sum over all sites these nk's being summed over here are not operators they're simply the quantum numbers corresponding to the orbital occupation in this kit so let's now have a look at the corresponding greens functions for this system i can define a lesser greens function for example for orbital number j i will label this g lesser with the subscripts j j since the hamiltonian is time independent i know that this will be a green's function of a single time variable t i express this as i times the thermal expectation value of the operator cj dagger multiplied by cj evaluated at time t and j here labels the orbital this is therefore the local greens function for orbital j using the usual definition of the thermal expectation value in terms of statistical mechanics and also the heisenberg representation of the operator evolved forwards in time to a time t we obtain the following said here is as usual the full partition function these ket x's here are exact eigen states which are obviously going to be of this product state form that i wrote out earlier e x is the eigenenergy which is this object and uh inside here we have this bunch of operators we have the creation and annihilation operators for site j but we also also have these time evolution operators note here that we're summing over all possible eigenstates of our many particle system the important point however is that this cj operator and this cj dagger operator only act on site j and they commute with operators for all the other orbitals in the system that means that even though the eigenstate denoted ket x here involves all of the orbitals 1 2 3 and so on these operators only act on the jth orbital indeed given that we have a ket x here and a bra x on this side the only thing that's changing by this string of operators is what's happening on the jth orbital everything else remains unchanged because of the product state structure of the eigenstates one can then simply show that the lesser green's function for orbital j is equal to i e to the minus i e j t times the fermi function evaluated at epsilon j the form of this green's function equation is exactly the same as that which we obtained for a single quantum orbital so we see that for the quantum gas where we have a bunch of non-interacting quantum orbitals that don't talk to each other the greens function for each of those orbitals is as if we have just a single orbital and the same results also hold for the greater and green's functions i will leave it as an exercise for you to confirm these results indeed when we perform the laplace transformation to our complex variable z we find that the local greens function for orbital j is simply one over z minus epsilon j as you might have expected of course in our many particle system we can also define non-local green's functions for example consider g lesser i j of t where i and j now refer to different orbitals this would be defined as before as i times the thermal expectation value of these operators we have c dagger for orbital i and c of t for orbital j we can write this out in the same way as before using our definition of the time evolution of the operator and our expression for the thermal expectation value from statistical mechanics remember that these kets here are the many particle eigenstates and we're summing over all of those we can now immediately see that this entire matrix element here is proportional to the kronecker delta delta ij that means that this object is precisely equal to zero if i and j are not the same this can be understood by imagining the action of a cj annihilation operator for side j upon some arbitrary ket which i've labeled here in the occupation basis we see that this will give exactly zero if nj is equal to zero that's because we're trying to annihilate something in orbital j and if there's nothing there then this will give zero on the other hand if nj is equal to one then i can annihilate the electron in that orbital and i would obtain this cat let's now consider feeding this ket back in and operating on it with ci dagger the creation operator for orbital i this will again annihilate the ket if n i this time is equal to one that's because of the pauli principle i cannot create an electron in the ith orbital if it's already occupied on the other hand if ni is equal to zero then i will get this ket so the only nonzero contribution when i act with ci dagger cj is a ket where there are no electrons in the ith orbital and one electron in the jth orbital when i act with this operator i then end up with a state with one electron in the ith orbital and no electrons in the jth orbital note also that the occupation numbers of all the other orbitals are exactly the same on the left hand side and on the right hand side the c i dagger cj operator doesn't affect the number of electrons in all of the other orbitals now the important thing is is that the matrix elements appearing in our greens functions involve the same state ket x and bra x is the same x in both cases x of course is just shorthand notation for all of these occupation numbers here and now you can see that if i were to pre-multiply this equation by the bra of this state then i obtain this matrix element and this is equal to zero because on the right hand side i would have the bracket of this state with this state and these are orthogonal so this shows that if the operators i and j are not the same then that means that this matrix element will be equal to zero fundamentally this is because the hamiltonian conserves the individual particle numbers of the different orbitals and this operator ci dagger cj is changing the orbital occupation of the two sides i and j in our green's functions we have additionally these time evolution operators but you can easily put those in yourself and convince yourself that this argument goes through and what this shows us in the end is that our green's functions for the non-interacting electron gas where the orbitals are totally decoupled from each other are exactly local this means that g lesser i j of time t is proportional to the kronecker delta i j if i is equal to j then we get the familiar result that we derived previously for a single orbital if i is not equal to j then the green's function vanishes and this holds also for the greater functions and the functions as well as their fourier transformed counterparts of course this locality condition on the green's functions is a peculiar feature of the non-interacting electron gas where each orbital is decoupled from the others we'll see in the coming lectures how in realistic models of real materials and molecules for example we don't have eigenstates which conserve the occupation number of each orbital in that case we can have finite non-local greens functions which correspond to the electron propagation from one side to the other we'll be discussing those in detail in the coming lectures

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Length: 83min 58sec (5038 seconds)

Published: Tue Apr 13 2021