WEBVTT 00:00:00.060 --> 00:00:01.780 The following content is provided 00:00:01.780 --> 00:00:04.019 under a Creative Commons license. 00:00:04.019 --> 00:00:06.870 Your support will help MIT OpenCourseWare continue 00:00:06.870 --> 00:00:10.730 to offer high-quality educational resources for free. 00:00:10.730 --> 00:00:13.330 To make a donation, or view additional materials 00:00:13.330 --> 00:00:17.217 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.217 --> 00:00:17.842 at ocw.mit.edu. 00:00:23.560 --> 00:00:26.470 PROFESSOR: OK, let's start. 00:00:26.470 --> 00:00:30.856 So last lecture we started with the strategy of using 00:00:30.856 --> 00:00:40.040 perturbation theory to study our statistical themes. 00:00:40.040 --> 00:00:45.030 For example, we need to evaluate a partition function 00:00:45.030 --> 00:00:50.220 by integrating over all configurations of a field. 00:00:50.220 --> 00:00:54.120 Let's say n components in d dimensions 00:00:54.120 --> 00:01:00.100 with some kind of a weight that we can write as e to the minus 00:01:00.100 --> 00:01:05.740 beta H. And the strategy of perturbation 00:01:05.740 --> 00:01:10.270 was to find part of this Hamiltonian 00:01:10.270 --> 00:01:13.480 that we can calculate exactly. 00:01:13.480 --> 00:01:18.080 And the rest of it, hopefully treating as a small quantity 00:01:18.080 --> 00:01:21.980 and doing perturbative calculations. 00:01:21.980 --> 00:01:25.540 Now, in the context of the Landau-Ginzburg theory 00:01:25.540 --> 00:01:32.050 that we wrote down, this beta H0 was 00:01:32.050 --> 00:01:34.890 bets described in terms of Fourier modes. 00:01:34.890 --> 00:01:38.398 So basically, we could make a change of variables 00:01:38.398 --> 00:01:41.672 to integrate over all configurations of Fourier 00:01:41.672 --> 00:01:43.380 modes. 00:01:43.380 --> 00:01:48.701 And the same breakdown of the weight 00:01:48.701 --> 00:01:52.090 in the language of Fourier modes. 00:01:52.090 --> 00:01:55.360 Since the underlying theories that we were writing 00:01:55.360 --> 00:01:59.740 had translational symmetry, every point in space 00:01:59.740 --> 00:02:04.240 was the same as any other, the composition in to modes 00:02:04.240 --> 00:02:06.262 was immediately accomplished by going 00:02:06.262 --> 00:02:09.199 to Fourier representation. 00:02:09.199 --> 00:02:12.030 And each component of each q-value 00:02:12.030 --> 00:02:15.550 would correspond to essentially an independent weight 00:02:15.550 --> 00:02:19.450 that we could expand in some power series 00:02:19.450 --> 00:02:23.590 in this parameter q, which is an inverse in the wave length. 00:02:23.590 --> 00:02:26.260 And the lowest order terms are what 00:02:26.260 --> 00:02:30.440 determines the longer and longer wavelengths. 00:02:30.440 --> 00:02:38.230 So there is some m of q squared characterizing 00:02:38.230 --> 00:02:41.180 this part of the Hamiltonian. 00:02:41.180 --> 00:02:45.800 And since is real space we had emphasized 00:02:45.800 --> 00:02:53.020 some form of locality, the interaction part in real space 00:02:53.020 --> 00:02:58.660 could be written simply in terms of a power series, let's say, 00:02:58.660 --> 00:02:59.650 in m. 00:03:03.120 --> 00:03:08.580 Which means that if we were to then go to Fourier space, 00:03:08.580 --> 00:03:11.160 things that are local in real space 00:03:11.160 --> 00:03:13.580 become non-local in Fourier space. 00:03:13.580 --> 00:03:17.346 And the first of those terms that we treated 00:03:17.346 --> 00:03:19.980 as a perturbation would involve an integral 00:03:19.980 --> 00:03:25.985 over four factors of m tilde. 00:03:28.800 --> 00:03:33.800 Again, translational invariance forces the four q's 00:03:33.800 --> 00:03:40.700 that appear in the multiplication to add up to 0. 00:03:40.700 --> 00:03:48.410 So I would have m of q1 dot for dot with m of q2, 00:03:48.410 --> 00:03:54.460 m of q3 dot for dot with m of q4, 00:03:54.460 --> 00:03:59.210 which is minus q1 minus q2 minus q3. 00:03:59.210 --> 00:04:03.280 And I can go on and do higher order. 00:04:07.620 --> 00:04:10.260 So once we did this, we could calculate 00:04:10.260 --> 00:04:13.670 various-- let's say, two-point correlation functions, 00:04:13.670 --> 00:04:16.440 et cetera, in perturbation theory. 00:04:16.440 --> 00:04:20.000 And in particular, the two-point correlation function 00:04:20.000 --> 00:04:23.260 was related to the susceptibility. 00:04:23.260 --> 00:04:27.020 And setting q to 0, we found an expression 00:04:27.020 --> 00:04:33.010 for the inverse susceptibility where the 0 order just 00:04:33.010 --> 00:04:36.270 comes from the t that we have over here. 00:04:36.270 --> 00:04:42.580 And because of this perturbation calculated to order of u, 00:04:42.580 --> 00:04:51.240 we had 4u n plus 2 and integral over modes 00:04:51.240 --> 00:04:54.770 of just the variance of the modes if y like. 00:05:03.100 --> 00:05:06.000 Now, first thing that we noted was 00:05:06.000 --> 00:05:09.992 that the location of the point at which the susceptibility 00:05:09.992 --> 00:05:13.110 vanishes, or susceptibility diverges 00:05:13.110 --> 00:05:18.810 or its inverse vanishes is no longer at t equals to 0. 00:05:18.810 --> 00:05:22.675 But we can see just setting this expression to 0 that we have 00:05:22.675 --> 00:05:35.250 a tc, which is minus 4u n plus 2 integral d dk 2 pi to the d 1 00:05:35.250 --> 00:05:40.210 over-- let's put the k here, k squared, potentially 00:05:40.210 --> 00:05:41.366 higher-order terms. 00:05:45.350 --> 00:05:51.760 Now, this is an integral that in dimensions above 2-- 00:05:51.760 --> 00:05:56.210 let's for time being focus on dimensions above 2-- there 00:05:56.210 --> 00:06:00.090 is no singularity as k goes to 0. 00:06:00.090 --> 00:06:03.850 k goes to 0, which is long wavelength, is well-behaved. 00:06:03.850 --> 00:06:06.830 The integral could potentially be singular 00:06:06.830 --> 00:06:10.060 if I were allowed to go all the way to infinity, 00:06:10.060 --> 00:06:12.910 but I don't go all the infinity. 00:06:12.910 --> 00:06:18.520 All of my theories have an underlying short wavelength. 00:06:18.520 --> 00:06:22.240 And hence, there is a maximum in the Fourier modes 00:06:22.240 --> 00:06:26.504 which would render this completely well-behaved 00:06:26.504 --> 00:06:28.746 integral. 00:06:28.746 --> 00:06:31.870 In fact, if I forget higher-order term, 00:06:31.870 --> 00:06:33.150 I could put them. 00:06:33.150 --> 00:06:35.375 But if I forget them, I can evaluate 00:06:35.375 --> 00:06:38.080 what this correction tc is. 00:06:38.080 --> 00:06:43.190 It is minus 4u n plus 2 over k. 00:06:43.190 --> 00:06:49.800 This-- I've been writing-- symmetry at surface area 00:06:49.800 --> 00:06:57.660 of a d dimensional unit sphere divided by 2 pi to the d. 00:06:57.660 --> 00:07:03.240 And then I have the integral of k to the d minus 3, 00:07:03.240 --> 00:07:07.660 which integrates to lambda to the d minus 2 00:07:07.660 --> 00:07:11.780 divided by d minus 2. 00:07:11.780 --> 00:07:15.270 I wrote that explicitly because we 00:07:15.270 --> 00:07:20.880 are going to encounter this combination a lot of times. 00:07:20.880 --> 00:07:24.470 And so we will give it a name k sub d. 00:07:24.470 --> 00:07:29.710 So it's just the solid angle in d dimensions divided by 2 pi 00:07:29.710 --> 00:07:31.030 to the d. 00:07:31.030 --> 00:07:36.030 OK, so essentially, in dimensions greater than 2, 00:07:36.030 --> 00:07:37.410 nothing much happens. 00:07:37.410 --> 00:07:42.120 There is a shift in the location of the singularity compared 00:07:42.120 --> 00:07:43.080 to the Gaussian. 00:07:43.080 --> 00:07:45.800 Because you are no longer at the Gaussian, 00:07:45.800 --> 00:07:49.530 you are at a theory that has additional stabilizing 00:07:49.530 --> 00:07:51.810 terms such as m to the fourth, et cetera. 00:07:51.810 --> 00:07:57.190 So there is no problem now for p going to negative values. 00:07:57.190 --> 00:08:01.170 The thing that was more interesting 00:08:01.170 --> 00:08:04.700 was that when we looked at what happens 00:08:04.700 --> 00:08:11.170 in the vicinity of this new tc, and to lowest order 00:08:11.170 --> 00:08:14.910 we got this form of a divergence. 00:08:14.910 --> 00:08:21.550 And then at the next order, I had a correction. 00:08:21.550 --> 00:08:27.470 Again, coming from this form, 4u n plus 2, an integral. 00:08:30.560 --> 00:08:34.650 And actually, this was obtained by taking the difference of two 00:08:34.650 --> 00:08:38.080 of these factors evaluated at p and tc. 00:08:38.080 --> 00:08:41.410 That's what gave me the t minus tc outside. 00:08:41.410 --> 00:08:51.450 And then I had an integral that involved two of these factors. 00:08:51.450 --> 00:08:54.690 Presumably to be consistent to lowest order, 00:08:54.690 --> 00:08:59.220 I have to evaluate them as small as I can. 00:08:59.220 --> 00:09:02.660 And so I would have two factors of k squared or k squared 00:09:02.660 --> 00:09:03.570 plus something. 00:09:08.500 --> 00:09:09.918 Presumably, higher-order terms. 00:09:14.310 --> 00:09:19.350 The thing about these integrals as opposed to the previous one 00:09:19.350 --> 00:09:21.800 is that, again, I can try to look 00:09:21.800 --> 00:09:25.330 at the behavior at large k and small k. 00:09:25.330 --> 00:09:28.330 At large k, no matter how many terms 00:09:28.330 --> 00:09:31.070 I add to the series, ultimately, I 00:09:31.070 --> 00:09:36.180 will be concerned by cutting it off by lambda. 00:09:36.180 --> 00:09:38.390 Whereas, if I have something that I 00:09:38.390 --> 00:09:43.030 have set t equals to 0 in both of these denominator factors, 00:09:43.030 --> 00:09:45.920 I now have a singularity at k goes 00:09:45.920 --> 00:09:49.200 to 0 in dimensions less than 4. 00:09:49.200 --> 00:09:52.600 The integral would blow up in dimensions less than 4 00:09:52.600 --> 00:09:56.830 if I am allowed to go all the way to 0, which is arbitrarily 00:09:56.830 --> 00:09:59.480 long wavelengths. 00:09:59.480 --> 00:10:03.430 Now in principle, if I am not exactly at tc 00:10:03.430 --> 00:10:07.390 and I'm looking at singularity as being away from tc, 00:10:07.390 --> 00:10:11.476 I expect on physical grounds that fluctuations 00:10:11.476 --> 00:10:16.530 will persist up to some correlation length. 00:10:16.530 --> 00:10:21.320 So the shortest value of k that I should really physically 00:10:21.320 --> 00:10:27.570 be able to go to, irrespective of how careful or careless 00:10:27.570 --> 00:10:31.640 I am with the factors of t and t minus tc that I put here, 00:10:31.640 --> 00:10:34.830 is of the order of the physical correlation length. 00:10:34.830 --> 00:10:37.870 And as we saw, this means that there 00:10:37.870 --> 00:10:42.080 is a correction that is of the form of u k 00:10:42.080 --> 00:10:47.560 to the 4-- k squared psi to the power of 4 minus d. 00:10:47.560 --> 00:10:53.620 And I emphasized that the dimensionless combination 00:10:53.620 --> 00:10:56.320 of the parameter u that potentially 00:10:56.320 --> 00:10:59.880 can be added as a correction to a number of order of 1 00:10:59.880 --> 00:11:04.650 is u k squared divided by some-- multiplied by some length scale 00:11:04.650 --> 00:11:06.775 to the power of 4 minus d. 00:11:06.775 --> 00:11:09.210 Above four dimensions, the integral 00:11:09.210 --> 00:11:12.720 is convergent at small values. 00:11:12.720 --> 00:11:14.630 And the integral will be dominated 00:11:14.630 --> 00:11:17.440 and the length scale that would appear here 00:11:17.440 --> 00:11:21.440 would be some kind of a short distance cutoff, 00:11:21.440 --> 00:11:23.170 like the averaging length. 00:11:23.170 --> 00:11:29.140 Whereas in four dimensions with divergence of the correlation 00:11:29.140 --> 00:11:33.085 length is the thing that will lead this perturbation 00:11:33.085 --> 00:11:35.771 theory to be kind of difficult and [INAUDIBLE]. 00:11:40.490 --> 00:11:48.690 So this is an example of a divergent perturbation theory. 00:11:48.690 --> 00:11:50.930 So what we are going to do in order 00:11:50.930 --> 00:11:53.930 to be able to make sense out of it, 00:11:53.930 --> 00:11:58.600 and see how this divergence here can be translated 00:11:58.600 --> 00:12:01.940 to a change in exponent, which is what we are physically 00:12:01.940 --> 00:12:06.330 expecting to occur, we reorganize this perturbation 00:12:06.330 --> 00:12:10.190 theory in a conceptual way that is 00:12:10.190 --> 00:12:17.258 helped by this perturbative renormalization group approach. 00:12:17.258 --> 00:12:21.490 So we keep the perturbation theory, 00:12:21.490 --> 00:12:25.200 but change the way that we look at perturbation theory 00:12:25.200 --> 00:12:26.910 by appealing to renormalization. 00:12:31.900 --> 00:12:36.626 So you can see that throughout doing this perturbation theory, 00:12:36.626 --> 00:12:44.890 I end up having to do integrals over modes that are defined 00:12:44.890 --> 00:12:49.016 in the space of the Fourier parameter q. 00:12:49.016 --> 00:12:54.280 And a nice way to implement this coarse graining that 00:12:54.280 --> 00:12:58.310 has led to this field theory is to imagine 00:12:58.310 --> 00:13:03.700 that this integration is over some sphere where 00:13:03.700 --> 00:13:09.410 the maximum inverse wavelength that is allowed, 00:13:09.410 --> 00:13:14.090 or q number that is allowed is some lambda. 00:13:14.090 --> 00:13:18.050 And so the task that I have on the first line 00:13:18.050 --> 00:13:24.430 is to integrate over all modes that live in this. 00:13:24.430 --> 00:13:26.950 And just on physical grounds, we don't 00:13:26.950 --> 00:13:31.235 expect to get any singularities from the modes that 00:13:31.235 --> 00:13:33.410 are at the edge. 00:13:33.410 --> 00:13:37.890 We expect to get singularities by considering what's 00:13:37.890 --> 00:13:43.200 going on at long wavelengths or 0q. 00:13:43.200 --> 00:13:47.980 So the idea of renormalization group 00:13:47.980 --> 00:13:50.650 was to follow three steps. 00:13:50.650 --> 00:13:58.960 The first step was to do coarse graining, which 00:13:58.960 --> 00:14:05.140 was to take whatever your shortest wavelength was, 00:14:05.140 --> 00:14:10.350 make it b times larger and average. 00:14:10.350 --> 00:14:11.700 That's in real space. 00:14:11.700 --> 00:14:13.570 In Fourier space, what that amounts 00:14:13.570 --> 00:14:23.640 to is to get rid of all of the variations and frequencies that 00:14:23.640 --> 00:14:26.753 are up to lambda over b. 00:14:29.980 --> 00:14:37.680 So what I can do is to say that I had a whole bunch of modes 00:14:37.680 --> 00:14:38.962 m of q. 00:14:41.620 --> 00:14:47.120 I am going to subdivide them into two classes. 00:14:47.120 --> 00:14:51.500 I will have the modes sigma of q. 00:14:51.500 --> 00:14:54.950 Maybe I will write it in different color, sigma of q. 00:14:58.180 --> 00:15:03.610 That are the ones that sitting here. 00:15:03.610 --> 00:15:06.390 So these are the sigmas. 00:15:06.390 --> 00:15:09.520 And these correspond to wave numbers 00:15:09.520 --> 00:15:15.240 that are between lambda over b and lambda. 00:15:15.240 --> 00:15:20.340 And I will have a bunch of other variables 00:15:20.340 --> 00:15:22.387 that I will call m tilde. 00:15:27.750 --> 00:15:35.080 And this wave close to the singularity, 00:15:35.080 --> 00:15:40.930 but now removed by an amount lambda over b. 00:15:40.930 --> 00:15:45.260 So essentially, getting rid of the picture 00:15:45.260 --> 00:15:49.150 where it had fluctuations at short length scale 00:15:49.150 --> 00:15:54.370 amounts to integrating over Fourier modes that 00:15:54.370 --> 00:15:58.352 represent your field that lie in this integral. 00:16:02.220 --> 00:16:06.585 So I want to do that as an operation that is performed, 00:16:06.585 --> 00:16:10.580 let's say, at the level of the partition function. 00:16:10.580 --> 00:16:16.190 So I can say that my original integration can be broken up 00:16:16.190 --> 00:16:22.840 into integration over this m tilde 00:16:22.840 --> 00:16:25.822 and the integration over sigma. 00:16:25.822 --> 00:16:33.860 So that's just rewriting that rightmost integral up there. 00:16:33.860 --> 00:16:40.760 And then I have a weight exponential. 00:16:40.760 --> 00:16:44.420 OK, let's write it out explicitly. 00:16:44.420 --> 00:16:52.220 So the weight is composed of beta H0 and the u. 00:16:52.220 --> 00:16:58.160 Now, we note that the beta H0 part, just as we did already 00:16:58.160 --> 00:17:00.670 for the case of the Gaussian, does not 00:17:00.670 --> 00:17:04.869 mix up these two classes of modes. 00:17:04.869 --> 00:17:08.460 So I can write that part as an integral from 0 00:17:08.460 --> 00:17:16.269 to lambda over b dd 2 pi over d. 00:17:16.269 --> 00:17:19.800 And these are things that are really inside, 00:17:19.800 --> 00:17:23.289 so I could also label them by q lesser. 00:17:23.289 --> 00:17:27.955 So I have m tilde of q lesser squared. 00:17:30.530 --> 00:17:35.940 And this multiplies t plus k q lesser 00:17:35.940 --> 00:17:38.869 squared and so forth over 2. 00:17:43.560 --> 00:17:49.650 I have a similar term, which is the modes that 00:17:49.650 --> 00:17:54.330 are between lambda over b and lambda. 00:17:54.330 --> 00:18:01.652 So I just simply changed or broke my overall integration 00:18:01.652 --> 00:18:06.380 in beta H0 into two parts. 00:18:06.380 --> 00:18:11.340 Now I have the higher q numbers. 00:18:11.340 --> 00:18:18.570 And these are sigma of q larger squared. 00:18:18.570 --> 00:18:23.820 Again, same weight, t plus k q larger 00:18:23.820 --> 00:18:26.447 squared and so forth over 2. 00:18:30.279 --> 00:18:34.460 Make sure this minus is in line with this. 00:18:34.460 --> 00:18:36.700 And then I have, of course, the u. 00:18:36.700 --> 00:18:42.060 So then I have a minus u. 00:18:42.060 --> 00:18:43.870 Now, I won't write this explicitly. 00:18:43.870 --> 00:18:47.510 I will write it explicitly on the next board. 00:18:47.510 --> 00:18:57.510 But clearly, implicitly it involves both m tilde and sigma 00:18:57.510 --> 00:18:58.780 mixed up into each other. 00:19:07.930 --> 00:19:11.580 So I have just rewritten my partition function 00:19:11.580 --> 00:19:16.170 after subdividing it into these two classes of modes 00:19:16.170 --> 00:19:19.150 and just hiding all of the complexity 00:19:19.150 --> 00:19:21.454 in this function that mixes the two. 00:19:25.570 --> 00:19:29.690 So let's rewrite this. 00:19:29.690 --> 00:19:34.830 I have integral over the modes that I would like to keep. 00:19:34.830 --> 00:19:37.320 The m tilde I would like to keep. 00:19:40.010 --> 00:19:43.370 And there is a weight associated with them 00:19:43.370 --> 00:19:46.095 that will, therefore, not be integrated. 00:19:46.095 --> 00:19:55.870 This is the integral from 0 to lambda over b dd k lesser 2 00:19:55.870 --> 00:20:02.720 pi to the d t plus k, k lesser q lesser squared, et cetera. 00:20:02.720 --> 00:20:10.642 And then I have tilde of q lesser squared. 00:20:15.720 --> 00:20:22.260 Now, if I didn't have this there, the u, 00:20:22.260 --> 00:20:27.680 I could immediately perform the Gaussian integrals over sigmas. 00:20:27.680 --> 00:20:30.030 Indeed, we already did this. 00:20:30.030 --> 00:20:32.840 And the answer would be e to the minus-- 00:20:32.840 --> 00:20:35.860 there are n-components to this vector. 00:20:35.860 --> 00:20:40.520 So the answer is going to be multiplied by n. 00:20:40.520 --> 00:20:44.440 1/2 is because the square root that I get from each mode. 00:20:44.440 --> 00:20:47.916 I get a factor of volume integral dd 00:20:47.916 --> 00:20:53.560 q larger 2 pi to the d integrated from lambda over v 00:20:53.560 --> 00:21:01.090 to lambda log of t plus k q greater squared and so forth. 00:21:01.090 --> 00:21:04.190 So if I didn't have the u, this would 00:21:04.190 --> 00:21:08.840 be the answer for doing the Gaussian integration. 00:21:08.840 --> 00:21:12.030 But I have the u, so what should I do? 00:21:12.030 --> 00:21:14.150 The answer is very simple. 00:21:14.150 --> 00:21:22.900 I write it as e to the minus u m tilde m sigma average. 00:21:27.840 --> 00:21:37.270 So what I have done is to say that with this weight, that 00:21:37.270 --> 00:21:41.490 is a Gaussian weight for sigma, I 00:21:41.490 --> 00:21:45.760 average the function e to the minus u. 00:21:45.760 --> 00:21:53.020 If you like, this is a Gaussian sigma. 00:21:53.020 --> 00:21:57.960 So to sort of write it explicitly, what I have stated 00:21:57.960 --> 00:22:03.280 is an average where I integrate out 00:22:03.280 --> 00:22:07.920 the high-frequency short wavelength modes 00:22:07.920 --> 00:22:14.740 is by definition integrate over all configurations of sigma 00:22:14.740 --> 00:22:24.800 with the Gaussian weight whatever object you have, 00:22:24.800 --> 00:22:29.490 and then normalize by the Gaussian. 00:22:35.310 --> 00:22:41.284 Of course, in our case, our O depends both on sigma and m 00:22:41.284 --> 00:22:44.370 tilde, so the result of these averaging 00:22:44.370 --> 00:22:48.810 will be a function of in tilde. 00:22:48.810 --> 00:23:00.590 And indeed, I can write this as an integral over m tilde of q 00:23:00.590 --> 00:23:05.530 with a new weight, e to the minus beta H tilde, which 00:23:05.530 --> 00:23:08.760 only depends on m tilde because I got rid of 00:23:08.760 --> 00:23:10.860 and I integrated over the sigmas. 00:23:13.520 --> 00:23:22.740 And by definition, my beta H tilde 00:23:22.740 --> 00:23:28.810 that depends only on m tilde has a part 00:23:28.810 --> 00:23:40.380 that is the integral from 0 to lambda over b dd q lesser 00:23:40.380 --> 00:23:56.340 2 pi to the d the Gaussian weight over the range of modes 00:23:56.340 --> 00:23:59.530 that are allowed. 00:23:59.530 --> 00:24:03.250 There is a part that is just this constant term when 00:24:03.250 --> 00:24:06.580 I take care-- if I write it in this fashion, 00:24:06.580 --> 00:24:09.040 there is an overall constant. 00:24:09.040 --> 00:24:15.360 Clearly, what this constant is, is the free energy of the modes 00:24:15.360 --> 00:24:19.380 that I have integrated, assuming that they are Gaussian, 00:24:19.380 --> 00:24:21.700 in this interval. 00:24:21.700 --> 00:24:24.670 The answer is proportional to volume. 00:24:24.670 --> 00:24:26.610 But as usual, when we are thinking 00:24:26.610 --> 00:24:30.270 about weights and probabilities, overall constants don't matter. 00:24:30.270 --> 00:24:34.160 But I can certainly continue to write that over here. 00:24:34.160 --> 00:24:39.750 So that part went to here, this part went to here. 00:24:39.750 --> 00:24:43.130 And so the only part that is left 00:24:43.130 --> 00:24:51.510 is minus log of d to the minus u of m tilde and sigma 00:24:51.510 --> 00:24:53.982 after I get rid of the sigmas. 00:25:02.340 --> 00:25:06.955 So far, I have done things that are extremely general. 00:25:09.490 --> 00:25:16.180 But now I note that I am interested in doing 00:25:16.180 --> 00:25:19.100 perturbation. 00:25:19.100 --> 00:25:23.900 So the only place that I haven't really evaluated things 00:25:23.900 --> 00:25:27.860 is where this u is appearing inside the exponential log 00:25:27.860 --> 00:25:30.500 average, et cetera. 00:25:30.500 --> 00:25:34.670 So what I can do is I can perturbatively 00:25:34.670 --> 00:25:38.376 expand this exponential over here. 00:25:38.376 --> 00:25:43.830 So I will get log of 1 minus u, which is approximately minus u. 00:25:43.830 --> 00:25:50.890 So the first term here would be u averaged Gaussian. 00:25:50.890 --> 00:25:57.440 The next term will be minus 1/2 u squared 00:25:57.440 --> 00:26:06.440 average minus u average squared and so forth. 00:26:06.440 --> 00:26:11.850 So you can see that the variance appeared in this stage. 00:26:11.850 --> 00:26:17.120 And generally, the l-th term in the series would be minus 1 00:26:17.120 --> 00:26:22.470 to the l divided by l factorial. 00:26:22.470 --> 00:26:26.660 And we saw this already. 00:26:26.660 --> 00:26:32.490 The log of e to the something is the generator of cumulants. 00:26:32.490 --> 00:26:37.890 So this would be the l-th power of u, 00:26:37.890 --> 00:26:41.500 the cumulant here would appear. 00:26:41.500 --> 00:26:45.205 And again, the cumulant would serve the function 00:26:45.205 --> 00:26:48.770 of cutting off connected pieces as we shall see shortly. 00:26:55.580 --> 00:26:59.270 So that's what we are going to do. 00:26:59.270 --> 00:27:05.290 We are going to insert the u's, all things that 00:27:05.290 --> 00:27:07.920 go beyond the Gaussian-- but initially, 00:27:07.920 --> 00:27:13.005 just the m to the fourth part-- inside this series 00:27:13.005 --> 00:27:18.060 and term by term calculate the corrections to this weight 00:27:18.060 --> 00:27:24.950 that we get after we integrate out the long lambda. 00:27:24.950 --> 00:27:30.291 Or sorry, the short wavelength modes or the long q modes. 00:27:30.291 --> 00:27:30.790 OK? 00:27:33.870 --> 00:27:38.430 So let's focus on this first term. 00:27:38.430 --> 00:27:45.050 So what is this u that depends on both m tilde and sigma? 00:27:48.830 --> 00:27:52.577 And I have the expression for u up there. 00:27:55.560 --> 00:28:07.030 So I can write it as u integral dd q1 dd q2 dd q3 to be 00:28:07.030 --> 00:28:10.686 symmetric in all of the four q's. 00:28:10.686 --> 00:28:15.740 I write an integration over the fourth q, 00:28:15.740 --> 00:28:19.060 but then enforce it by a delta function 00:28:19.060 --> 00:28:23.620 that the sum of the q's should be 0. 00:28:29.260 --> 00:28:34.460 And then I have four factors of m, 00:28:34.460 --> 00:28:40.540 but an m depending on which part of the q space 00:28:40.540 --> 00:28:45.990 I am encountering is either a sigma or an m tilde. 00:28:45.990 --> 00:28:48.320 So without doing anything wrong, I 00:28:48.320 --> 00:28:55.070 can replace each m with an m tilde plus sigma. 00:29:02.790 --> 00:29:07.200 So depending on where my q1 is in the integrations from 0 00:29:07.200 --> 00:29:11.630 to lambda, I will be encountering either this 00:29:11.630 --> 00:29:13.660 or this. 00:29:13.660 --> 00:29:20.230 And then I have the dot product of that with m tilde q2 00:29:20.230 --> 00:29:23.015 plus sigma of q2. 00:29:23.015 --> 00:29:25.230 And then I have the dot product that 00:29:25.230 --> 00:29:33.914 would correspond to m tilde of q3 plus sigma of q3 00:29:33.914 --> 00:29:40.096 with m tilde of q4 plus sigma of q4. 00:29:43.670 --> 00:29:45.550 So that's the structure of my [INAUDIBLE]. 00:29:49.850 --> 00:29:52.900 And again, what I have to do in principle 00:29:52.900 --> 00:29:59.540 is to integrate out the sigmas keeping the m tildes when 00:29:59.540 --> 00:30:02.910 I perform this averaging over here. 00:30:09.470 --> 00:30:12.530 So let's write down, if I were to expand 00:30:12.530 --> 00:30:16.670 this thing before the integration, what 00:30:16.670 --> 00:30:20.580 are the types of terms that I would get? 00:30:20.580 --> 00:30:23.264 And I'll give them names. 00:30:26.050 --> 00:30:29.800 One type of term that is very easy 00:30:29.800 --> 00:30:36.260 is when I have m tilde of q1 dotted with m tilde of Q2, 00:30:36.260 --> 00:30:41.880 m tilde of q3 dotted with m tilde of q4. 00:30:47.460 --> 00:30:50.670 If I expand this so there's 2 terms per bracket 00:30:50.670 --> 00:30:54.540 and there are 4 brackets, so there are 16 terms, only 1 00:30:54.540 --> 00:31:01.210 of these terms is of this variety out of the 16. 00:31:01.210 --> 00:31:04.520 What I will do is also now introduce 00:31:04.520 --> 00:31:08.030 a diagrammatic representation. 00:31:08.030 --> 00:31:15.740 Whenever I see an m tilde, I will include a straight line. 00:31:15.740 --> 00:31:22.140 Whenever I see a sigma, I will include a wavy line. 00:31:22.140 --> 00:31:24.790 So this entity that I have over here 00:31:24.790 --> 00:31:28.980 is composed of four of these straight lines. 00:31:28.980 --> 00:31:40.000 And I will indicate that by this diagram, q1, q2, q3, q4. 00:31:40.000 --> 00:31:43.010 And the reason is, of course-- first of all, 00:31:43.010 --> 00:31:45.136 there are four of these. 00:31:45.136 --> 00:31:49.780 So this is a so-called vertex in a diagrammatic representation 00:31:49.780 --> 00:31:51.940 that has four lines. 00:31:51.940 --> 00:31:57.830 And secondly, the lines are not all totally equivalent because 00:31:57.830 --> 00:32:01.030 of the way that the dot products are arranged. 00:32:01.030 --> 00:32:06.250 Say, q1 and q2 that are dot product together are distinct, 00:32:06.250 --> 00:32:11.020 let's say, from q1 and q3 that are not dot product together. 00:32:11.020 --> 00:32:14.240 And to indicate that, I make sure that there 00:32:14.240 --> 00:32:19.040 is this dotted line in the vertex that separates 00:32:19.040 --> 00:32:22.445 and indicates which two are dot product to each other. 00:32:27.470 --> 00:32:32.200 Now, the second class of diagram comes 00:32:32.200 --> 00:32:36.410 when I replace one of the m tildes with a sigma. 00:32:36.410 --> 00:32:41.240 So I have sigma of q1 dotted with m tilde 00:32:41.240 --> 00:32:51.700 of q2, m tilde of q3 dotted with m tilde of q4. 00:32:51.700 --> 00:32:55.100 Now clearly, in this case, I had a choice 00:32:55.100 --> 00:33:00.420 of four factors of m tilde to replace with this. 00:33:00.420 --> 00:33:04.780 So of the 16 terms in this expansion, 4 of them 00:33:04.780 --> 00:33:06.880 belong to this class. 00:33:06.880 --> 00:33:10.040 Which if I were to represent diagrammatically, 00:33:10.040 --> 00:33:12.480 I would have one of the legs replaced 00:33:12.480 --> 00:33:16.430 with a wavy line and all the other legs 00:33:16.430 --> 00:33:18.820 staying as solid lines. 00:33:25.530 --> 00:33:30.200 The third class of terms correspond 00:33:30.200 --> 00:33:34.630 to replacing two of the m tildes with sigmas. 00:33:34.630 --> 00:33:38.250 Now here again, I have a choice whether the second one 00:33:38.250 --> 00:33:40.170 is a partner of the first one that 00:33:40.170 --> 00:33:44.260 became sigma, such as this one, sigma of q1 00:33:44.260 --> 00:33:51.240 dotted with sigma of q2, m tilde of q3 dotted 00:33:51.240 --> 00:33:54.140 with m tilde of q4. 00:33:54.140 --> 00:33:56.360 And then clearly, I could have chosen 00:33:56.360 --> 00:34:01.040 one pair or the other pair to change into sigmas. 00:34:01.040 --> 00:34:04.830 So there are two terms that are like this. 00:34:04.830 --> 00:34:09.050 And diagrammatically, the wavy lines 00:34:09.050 --> 00:34:13.400 belong to the same branch of this object. 00:34:20.409 --> 00:34:21.239 OK, next. 00:34:21.239 --> 00:34:23.270 Keep going. 00:34:23.270 --> 00:34:25.429 Actually, I have another thing when 00:34:25.429 --> 00:34:28.840 I replace two of the m tildes with sigma, 00:34:28.840 --> 00:34:32.389 but now belonging to two different elements of this dot 00:34:32.389 --> 00:34:33.173 product. 00:34:33.173 --> 00:34:39.659 So I could have sigma of q1 dotted with m tilde of q2. 00:34:39.659 --> 00:34:44.699 And then I have sigma of q3 dotted with m tilde of q4. 00:34:48.830 --> 00:34:54.179 In which case, in each of the pairs I had a choice of two 00:34:54.179 --> 00:34:56.590 for replacing m tilde with sigma. 00:34:56.590 --> 00:34:58.390 So that's 2 times 2. 00:34:58.390 --> 00:35:01.990 There are four terms that have this character. 00:35:01.990 --> 00:35:05.560 And if I were to represent them diagrammatically, 00:35:05.560 --> 00:35:08.070 I would need to put two wavy lines 00:35:08.070 --> 00:35:09.189 on two different branches. 00:35:14.460 --> 00:35:20.410 And then I have the possibility of three things replaced. 00:35:20.410 --> 00:35:32.450 So I have sigma of q1 sigma of q2 sigma of q3 m tilde of q4. 00:35:32.450 --> 00:35:35.800 And again, now it's the other way around. 00:35:35.800 --> 00:35:39.120 One term is left out of 4 to be m tilde. 00:35:39.120 --> 00:35:43.990 So this is, again, a degeneracy of 4. 00:35:43.990 --> 00:35:50.790 And diagrammatically, I have three lines that are wavy 00:35:50.790 --> 00:35:54.060 and one line that is solid. 00:35:54.060 --> 00:35:58.040 And at the end of story, 6, I will 00:35:58.040 --> 00:36:10.840 have one diagram which is all sigmas, which 00:36:10.840 --> 00:36:14.113 can be represented essentially by all wavy lines. 00:36:21.160 --> 00:36:25.190 And to check that I didn't make any mistake in my calculation, 00:36:25.190 --> 00:36:28.330 the sum of these numbers better be 16. 00:36:28.330 --> 00:36:31.728 So that's 5, 7, 11, 15, 16. 00:36:34.440 --> 00:36:34.940 All right? 00:36:38.220 --> 00:36:44.320 Now, the next step of the story is to do these averages. 00:36:44.320 --> 00:36:46.144 So I have to do the average. 00:37:02.060 --> 00:37:07.230 Now, the first term doesn't involve any sigmas. 00:37:07.230 --> 00:37:09.610 All of my averages here are obtained 00:37:09.610 --> 00:37:12.250 by integrating over sigmas. 00:37:12.250 --> 00:37:14.985 If there is no sigmas to integrate, 00:37:14.985 --> 00:37:18.120 after I do the averaging here I essentially 00:37:18.120 --> 00:37:19.680 get the same thing back. 00:37:19.680 --> 00:37:27.670 So I will get this same expression. 00:37:27.670 --> 00:37:30.640 And clearly, that would be a term 00:37:30.640 --> 00:37:34.710 that would contribute to my beta H tilde, which 00:37:34.710 --> 00:37:37.460 is identical to what I had originally. 00:37:37.460 --> 00:37:40.300 It is, again, m to the fourth. 00:37:40.300 --> 00:37:43.630 So that we understand. 00:37:43.630 --> 00:37:48.920 Now, the second term here, what is 00:37:48.920 --> 00:37:50.620 the average that I have to do here? 00:37:54.140 --> 00:37:58.270 I have one factor of sigma with which I can average. 00:37:58.270 --> 00:38:03.290 But the weight that I have is even in sigma. 00:38:03.290 --> 00:38:08.640 So the average of sigma, which is Gaussian-distributed, is 0. 00:38:08.640 --> 00:38:10.413 So this will give me 0. 00:38:13.180 --> 00:38:17.420 And clearly, here also there is a term 00:38:17.420 --> 00:38:19.885 that involves three factors of sigma. 00:38:19.885 --> 00:38:22.022 Again, by symmetry this will average out to 0. 00:38:31.850 --> 00:38:40.060 Now, there is a way of indicating what happens here. 00:38:40.060 --> 00:38:43.370 See, what happens here is that I will 00:38:43.370 --> 00:38:47.145 have to do an average of this thing. 00:38:47.145 --> 00:38:50.080 The m tildes are not part of the averaging. 00:38:50.080 --> 00:38:51.980 They just go out. 00:38:51.980 --> 00:38:54.550 The average moves all the way over here. 00:38:57.540 --> 00:39:02.460 And the average of sigma of q1, sigma of q2, I know what it is. 00:39:02.460 --> 00:39:06.620 It is going to be-- I could have just written it over here. 00:39:06.620 --> 00:39:13.740 It's 2 pi to the d delta function q1 plus q2 00:39:13.740 --> 00:39:16.560 divided by k q squared. 00:39:16.560 --> 00:39:18.590 Maybe I'll explicitly write it over here. 00:39:18.590 --> 00:39:21.500 So what we have here is that the average of sigma 00:39:21.500 --> 00:39:31.086 of q1 with some index sigma of q2 with some other index 00:39:31.086 --> 00:39:33.377 is-- first of all, the two indices have to be the same. 00:39:36.000 --> 00:39:39.332 I have a delta function q1 plus q2. 00:39:39.332 --> 00:39:46.084 And then I have t plus k q1 squared and so forth. 00:39:46.084 --> 00:39:48.850 So it's my usual Gaussian. 00:39:48.850 --> 00:39:53.370 So essentially, you can see that one immediate consequence 00:39:53.370 --> 00:39:57.520 of this averaging is that previously these things had 00:39:57.520 --> 00:39:59.770 two different momenta and potentially 00:39:59.770 --> 00:40:01.750 two different indices. 00:40:01.750 --> 00:40:05.670 They get to be the same thing. 00:40:05.670 --> 00:40:10.690 And the fact that the labels that 00:40:10.690 --> 00:40:14.690 were assigned to this, the q and the index alpha, 00:40:14.690 --> 00:40:21.890 are forced to be the same, we can diagrammatically 00:40:21.890 --> 00:40:25.570 indicate by making this a closed line. 00:40:25.570 --> 00:40:30.230 So we are going to represent the result of that averaging 00:40:30.230 --> 00:40:36.250 with essentially taking-- these two lines are unchanged. 00:40:36.250 --> 00:40:37.880 They can be whatever they were. 00:40:37.880 --> 00:40:42.640 These two lines really are joined together 00:40:42.640 --> 00:40:44.840 through this process. 00:40:44.840 --> 00:40:48.850 So we indicate them that way. 00:40:48.850 --> 00:40:53.510 And similarly, when I do the same thing over here, 00:40:53.510 --> 00:40:59.120 I do the averaging of this and the answer 00:40:59.120 --> 00:41:05.470 I can indicate by leaving these two lines by themselves 00:41:05.470 --> 00:41:08.845 and joining these two wavy lines together in this fashion. 00:41:16.770 --> 00:41:19.680 Now, when you do-- this one we said is 0. 00:41:19.680 --> 00:41:23.480 So there's essentially one that is left, which is number 6. 00:41:23.480 --> 00:41:25.395 For number 6, we do our averaging. 00:41:27.920 --> 00:41:33.090 And for that we have to use- for average or a product of four 00:41:33.090 --> 00:41:36.050 sigmas that are Gaussian-distributed Wick's 00:41:36.050 --> 00:41:37.550 theorem. 00:41:37.550 --> 00:41:41.840 So one possibility is that sigma 1 and sigma 2 are joined, 00:41:41.840 --> 00:41:45.520 and then sigma 4 and sigma 3 have to be joined. 00:41:45.520 --> 00:41:48.760 So basically, I took sigma 1 and sigma 2 00:41:48.760 --> 00:41:53.670 and joined them, sigma 3 and sigma 4 that I joined them. 00:41:53.670 --> 00:41:56.210 But another possibility is I can take 00:41:56.210 --> 00:41:58.980 sigma 1 with sigma 3 or sigma 4. 00:41:58.980 --> 00:42:01.650 So there are really two choices. 00:42:01.650 --> 00:42:05.098 And then I will have a diagram that is like this. 00:42:19.050 --> 00:42:24.010 Now, each one of these operations and diagrams 00:42:24.010 --> 00:42:28.430 really stands for some integration and result. 00:42:28.430 --> 00:42:32.730 And let's for example, pick our number 3. 00:42:38.670 --> 00:42:42.890 For number 3, what we are supposed to do 00:42:42.890 --> 00:42:46.090 is to do the integration. 00:42:46.090 --> 00:42:47.030 Sorry. 00:42:47.030 --> 00:42:52.750 First of all, number 3 has a numerical factor of 2. 00:42:52.750 --> 00:42:55.180 This is something that is proportional to u 00:42:55.180 --> 00:42:56.730 when we take the average. 00:42:59.590 --> 00:43:11.710 I have in principle to do integration over q1 q2 q3 q4. 00:43:22.895 --> 00:43:23.395 OK. 00:43:26.260 --> 00:43:34.470 The m tilde of q3 and m tilde of q4 in this diagram 00:43:34.470 --> 00:43:37.850 were not averaged over. 00:43:37.850 --> 00:43:39.240 So that term remains. 00:43:42.140 --> 00:43:48.840 I did the averaging over q1 and q2. 00:43:48.840 --> 00:43:51.760 When I did that averaging, I, first of all, 00:43:51.760 --> 00:43:56.720 got a delta alpha alpha because those were two things that 00:43:56.720 --> 00:43:58.780 were dot product to each other, so they 00:43:58.780 --> 00:44:01.690 were carrying the same index to start with. 00:44:01.690 --> 00:44:06.220 I have a 2 pi to the d, a delta function q1 plus q2. 00:44:06.220 --> 00:44:11.485 And I have t plus k q1 squared. 00:44:17.780 --> 00:44:19.790 Now, delta alpha alpha. 00:44:19.790 --> 00:44:25.060 Summing over alpha gives a factor of n. 00:44:25.060 --> 00:44:28.420 And when you look at these diagrams, 00:44:28.420 --> 00:44:32.836 quite generally whenever you see a loop, 00:44:32.836 --> 00:44:37.080 with a loop you would associate the factor of n because 00:44:37.080 --> 00:44:42.080 of the index that runs and gets summed over. 00:44:42.080 --> 00:44:47.840 So this answer is going to be proportional to 2 u n. 00:44:51.660 --> 00:44:57.960 Now, q1 and q2 are said to be 0, the sum. 00:44:57.960 --> 00:44:59.810 So this is 0. 00:44:59.810 --> 00:45:05.140 So q3 and q4 have to add up to 0. 00:45:05.140 --> 00:45:09.290 So the part that involves q3 and q4, the m tilde, 00:45:09.290 --> 00:45:13.720 essentially I will get an integral dd-- 00:45:13.720 --> 00:45:15.980 let's say whatever q3. 00:45:15.980 --> 00:45:19.810 It doesn't matter because it's an index of integration. 00:45:19.810 --> 00:45:26.732 I have m tilde of q3 squared. 00:45:26.732 --> 00:45:31.180 And again, q3, it is something that 00:45:31.180 --> 00:45:33.590 goes with one of these m tildes. 00:45:33.590 --> 00:45:38.952 So this is an integration that I have to do between 0 and lambda 00:45:38.952 --> 00:45:39.944 over b. 00:45:42.920 --> 00:45:45.130 So there is essentially one integration 00:45:45.130 --> 00:45:48.702 left because q1 and q2 are left to be the same. 00:45:48.702 --> 00:45:49.660 So this is an integral. 00:45:49.660 --> 00:45:52.450 Let's call the integration variable 00:45:52.450 --> 00:45:56.240 that was q1-- I could k, it doesn't matter-- 2 pi 00:45:56.240 --> 00:46:02.140 to the d the same integral that we've seen before. 00:46:02.140 --> 00:46:07.640 Except that since this originated from the sigmas, 00:46:07.640 --> 00:46:11.870 the integration here is from lambda over b lambda. 00:46:19.160 --> 00:46:24.340 So this is basically a number that I can take, say, out here 00:46:24.340 --> 00:46:27.640 and regard as a coefficient that multiplies 00:46:27.640 --> 00:46:32.890 a term that is m tilde squared. 00:46:32.890 --> 00:46:37.990 And similarly, 4. 00:46:37.990 --> 00:46:38.490 4. 00:46:38.490 --> 00:46:41.410 We said we have four diagrams of his variety, 00:46:41.410 --> 00:46:46.820 so this would be a contribution that is 4u. 00:46:46.820 --> 00:46:49.620 I can read out the whole-- write out the whole thing. 00:46:49.620 --> 00:46:52.130 Certainly, I have all of this. 00:46:52.130 --> 00:46:54.290 I have all of this. 00:46:54.290 --> 00:47:03.980 In that case, I have m tilde q3 m tilde-- well, let's see. 00:47:03.980 --> 00:47:08.860 I have m tilde of q2 m tilde of q4. 00:47:08.860 --> 00:47:13.730 And they carry different indices because they 00:47:13.730 --> 00:47:18.320 came from two different dot products. 00:47:18.320 --> 00:47:23.725 And then I have to do an average over sigma 1 00:47:23.725 --> 00:47:27.580 and sigma 3 which carry different indices that 00:47:27.580 --> 00:47:29.410 are for beta. 00:47:29.410 --> 00:47:38.654 2 pi to the d delta function q1 plus q3 e plus k q1 squared. 00:47:44.170 --> 00:47:49.490 Again, since q1 plus q3 is 0 and the sum of the four q's is 0, 00:47:49.490 --> 00:47:52.675 these two have to add up to 0. 00:47:52.675 --> 00:47:57.270 So the answer, again, will be written as 4u integral 0 00:47:57.270 --> 00:48:02.955 to lambda over b dd of some q divided by 2 pi 00:48:02.955 --> 00:48:07.890 to the b m tilde of q squared. 00:48:07.890 --> 00:48:09.620 And then actually, the same integration, 00:48:09.620 --> 00:48:18.812 lambda over d lambda dd k 2 pi to the d 1 over 2 00:48:18.812 --> 00:48:20.255 plus k squared. 00:48:28.440 --> 00:48:31.990 So out of the six terms, two are 0. 00:48:31.990 --> 00:48:35.590 Two are explicitly calculated over here. 00:48:35.590 --> 00:48:39.200 One is trivially just m to the fourth. 00:48:39.200 --> 00:48:43.940 The last one is basically summing up all of these things. 00:48:46.750 --> 00:48:52.805 But these explicitly do not depend on m tilde. 00:48:52.805 --> 00:48:58.290 So I'll just call the result of doing all of this sum V delta f 00:48:58.290 --> 00:49:01.260 v at level 1. 00:49:01.260 --> 00:49:08.230 In the same way that integrating the modes sigma that I'm not 00:49:08.230 --> 00:49:10.870 interested and averaging over them 00:49:10.870 --> 00:49:15.380 gave a constant of integration, that constant of integration 00:49:15.380 --> 00:49:20.280 gets corrected to order of u over here. 00:49:20.280 --> 00:49:23.373 I don't need to explicitly take care of it. 00:49:28.650 --> 00:49:32.010 So given all of this information, 00:49:32.010 --> 00:49:36.760 let's write down what our last line from the previous board 00:49:36.760 --> 00:49:37.670 is. 00:49:37.670 --> 00:49:43.000 So our intent was to calculate a weight that 00:49:43.000 --> 00:49:48.050 governed these coarse-grained modes. 00:49:48.050 --> 00:49:50.730 And our answer is that, first of all, 00:49:50.730 --> 00:49:58.977 we will get a bunch of constants delta f v 0 plus delta f v 1 00:49:58.977 --> 00:50:00.060 that we don't really care. 00:50:00.060 --> 00:50:04.040 They're just an overall change that 00:50:04.040 --> 00:50:05.940 doesn't matter for the probabilities. 00:50:05.940 --> 00:50:10.650 It's just contribution to the free energy. 00:50:10.650 --> 00:50:14.180 And then we start to get things. 00:50:14.180 --> 00:50:17.980 And to the lowest order what we had 00:50:17.980 --> 00:50:24.740 was replacing the Gaussian weight, 00:50:24.740 --> 00:50:31.070 but only over this permitted set of wavelengths. 00:50:31.070 --> 00:50:35.460 So I have dd q lesser, let's say, 00:50:35.460 --> 00:50:43.550 2 pi to the d t plus k q lesser squared and so forth. 00:50:43.550 --> 00:50:50.150 Divided by 2 m tilde of q squared. 00:50:54.370 --> 00:50:59.150 Then, term number 1 in the series gave me what? 00:50:59.150 --> 00:51:06.290 It gave me something that was equivalent to my u 00:51:06.290 --> 00:51:09.160 if I were to Fourier transform back to real space, 00:51:09.160 --> 00:51:10.920 m to the fourth. 00:51:10.920 --> 00:51:19.050 Except that my cutoff has been shifted by lambda over b. 00:51:19.050 --> 00:51:22.210 So I don't want to bother to write down 00:51:22.210 --> 00:51:25.660 that full form in terms of Fourier modes. 00:51:25.660 --> 00:51:28.796 Essentially, if I want to write this explicitly, 00:51:28.796 --> 00:51:31.390 it is just like that line that I have, 00:51:31.390 --> 00:51:33.575 except that for the integrations I'll 00:51:33.575 --> 00:51:37.190 have to explicitly indicate 0 to lambda over b. 00:51:39.950 --> 00:51:46.900 So the only terms that we haven't included 00:51:46.900 --> 00:51:48.400 are the ones that are over here. 00:51:51.120 --> 00:51:54.740 Now, you look at those terms and you 00:51:54.740 --> 00:51:59.290 find that the structure of these terms 00:51:59.290 --> 00:52:01.720 is precisely what we have over here. 00:52:07.080 --> 00:52:10.870 Except that there is a modification. 00:52:10.870 --> 00:52:15.500 There is a constant term that is added from this one 00:52:15.500 --> 00:52:18.666 and there's a constant term that is added from that one. 00:52:18.666 --> 00:52:23.000 So the effect of those things I can capture 00:52:23.000 --> 00:52:31.090 by changing the parameters t to something else t tilde. 00:52:31.090 --> 00:52:36.230 So you can see that to order of u squared that I haven't 00:52:36.230 --> 00:52:39.720 calculated, to order of u, the only effect 00:52:39.720 --> 00:52:45.110 of this coarse graining is to modify this one parameter so 00:52:45.110 --> 00:52:48.140 that t goes to t tilde. 00:52:48.140 --> 00:52:52.590 It certainly depends on how much I coarse grain things. 00:52:52.590 --> 00:52:59.330 And this is the original t plus the sum of these things. 00:52:59.330 --> 00:53:12.670 So I will have 2-- n plus 2 u integral dd k 2 00:53:12.670 --> 00:53:17.364 pi to the d 1 over t plus k k squared. 00:53:17.364 --> 00:53:18.780 And presumably, higher-order terms 00:53:18.780 --> 00:53:24.980 are allowed, going from 0 to lambda over here. 00:53:24.980 --> 00:53:26.390 AUDIENCE: Question. 00:53:26.390 --> 00:53:29.080 PROFESSOR: Yes. 00:53:29.080 --> 00:53:29.886 Question? 00:53:29.886 --> 00:53:30.850 AUDIENCE: Yeah. 00:53:30.850 --> 00:53:35.090 When you have an integration over x, if you have previously 00:53:35.090 --> 00:53:41.480 defined lambda to be a cutoff in k space, 00:53:41.480 --> 00:53:45.180 might it be-- is it 1 over b lambda then? 00:53:45.180 --> 00:53:48.135 Or, is it b over lambda? 00:53:48.135 --> 00:53:48.637 Because-- 00:53:48.637 --> 00:53:49.220 PROFESSOR: OK. 00:53:52.870 --> 00:53:53.460 You're right. 00:53:53.460 --> 00:53:57.460 So previously, maybe the best way to write this 00:53:57.460 --> 00:54:05.120 would have been that there is a shortest length scale a. 00:54:05.120 --> 00:54:08.760 So I should really indicate what's 00:54:08.760 --> 00:54:13.070 happening here as shortest length scale having gone 00:54:13.070 --> 00:54:17.770 to v. And there is always a relationship 00:54:17.770 --> 00:54:23.330 between the a and lambda, which is inverse relation, 00:54:23.330 --> 00:54:26.835 but there are factors of 2 pi and things like that which 00:54:26.835 --> 00:54:28.676 I don't really want to bother. 00:54:28.676 --> 00:54:29.610 It doesn't matter. 00:54:33.980 --> 00:54:34.480 Yes. 00:54:34.480 --> 00:54:36.915 AUDIENCE: Shouldn't it be t tilde is 00:54:36.915 --> 00:54:40.319 equal to 2 plus 4 multiplied by? 00:54:40.319 --> 00:54:41.110 PROFESSOR: Exactly. 00:54:41.110 --> 00:54:41.680 Good. 00:54:41.680 --> 00:54:46.750 Because the coefficients here are divided by 2. 00:54:46.750 --> 00:54:47.975 So that 2 I forgot. 00:54:47.975 --> 00:54:49.175 And I should restore it. 00:54:52.520 --> 00:54:57.800 And if I had gone a little bit further, 00:54:57.800 --> 00:55:01.734 I would have then started comparing this formula 00:55:01.734 --> 00:55:05.320 with this formula. 00:55:05.320 --> 00:55:08.890 And I realized that I should have had the 4. 00:55:08.890 --> 00:55:14.130 Clearly, the two formula are telling me the same thing. 00:55:14.130 --> 00:55:16.330 You can see that they are almost exactly 00:55:16.330 --> 00:55:20.760 the same with the exception of how much integration. 00:55:20.760 --> 00:55:21.657 Yes? 00:55:21.657 --> 00:55:22.740 AUDIENCE: One other thing. 00:55:22.740 --> 00:55:25.710 Are bounds of those integrals for your tb, 00:55:25.710 --> 00:55:29.540 shouldn't they be lambda over d to lambda? 00:55:29.540 --> 00:55:30.165 PROFESSOR: Yes. 00:55:33.630 --> 00:55:36.170 Lambda over d to lambda. 00:55:36.170 --> 00:55:39.520 And it is because of that that I don't really 00:55:39.520 --> 00:55:43.210 have to worry because we saw that when we were doing 00:55:43.210 --> 00:55:46.830 straightforward perturbation theory, 00:55:46.830 --> 00:55:50.000 the reason that perturbation theory was blowing up 00:55:50.000 --> 00:55:54.000 in my face was integrating all the way to the origin. 00:55:54.000 --> 00:56:00.040 And the trick of renormalization group is by averaging of there, 00:56:00.040 --> 00:56:06.190 I don't really yet reach the singularity 00:56:06.190 --> 00:56:09.690 that I would have at k equals to 0. 00:56:09.690 --> 00:56:13.860 This integral by itself is not problematic at k equals to 0, 00:56:13.860 --> 00:56:15.410 but future integrals would. 00:56:19.640 --> 00:56:22.378 Any other mistakes? 00:56:22.378 --> 00:56:24.760 No. 00:56:24.760 --> 00:56:27.450 All right. 00:56:27.450 --> 00:56:29.430 But the other part of this story is 00:56:29.430 --> 00:56:34.400 that the effect of this coarse graining at lowest order mu 00:56:34.400 --> 00:56:38.600 was to modify this parameter t. 00:56:38.600 --> 00:56:41.260 But importantly, to do nothing else. 00:56:41.260 --> 00:56:45.432 That is, we can see that in this Hamiltonian that we 00:56:45.432 --> 00:56:49.340 had written, the k that is rescaled 00:56:49.340 --> 00:56:51.610 is the same as the old k. 00:56:51.610 --> 00:56:57.970 And the u is the same as the old u. 00:56:57.970 --> 00:57:06.330 That is, to the lowest order the only effect of coarse graining 00:57:06.330 --> 00:57:10.955 was to modify the parameter that covers coefficient 00:57:10.955 --> 00:57:11.736 of t squared. 00:57:18.880 --> 00:57:21.940 But we have not completed our task 00:57:21.940 --> 00:57:24.866 of constructing a renormalization group. 00:57:27.715 --> 00:57:31.080 Renormalization group had three steps. 00:57:31.080 --> 00:57:34.820 The most difficult step, which was the coarse graining 00:57:34.820 --> 00:57:40.220 we have completed, but now we have generated a grainy picture 00:57:40.220 --> 00:57:44.660 in which the shortest wavelengths are a factor of b 00:57:44.660 --> 00:57:47.160 larger than what we had before. 00:57:47.160 --> 00:57:50.850 In order to make our pictures look the same, 00:57:50.850 --> 00:57:59.326 we had to do steps 2 and 3 of rg. 00:57:59.326 --> 00:58:03.060 Step 2 was to shrink all of the lengths 00:58:03.060 --> 00:58:10.800 in real space, which amounts to q prime being b times q. 00:58:10.800 --> 00:58:15.650 And that will restore the upper part of the q integration 00:58:15.650 --> 00:58:17.510 to be lambda. 00:58:17.510 --> 00:58:22.120 And there was a rescaling that we 00:58:22.120 --> 00:58:26.540 had to perform for the magnitude of the fluctuations which 00:58:26.540 --> 00:58:34.690 amounted to replacing the m tilde of q with z N prime. 00:58:34.690 --> 00:58:38.290 So this would be, if you like, q lesser 00:58:38.290 --> 00:58:41.200 and this would be m prime. 00:58:51.410 --> 00:58:53.220 Now, the reason these steps are trivial 00:58:53.220 --> 00:58:56.140 is because just whenever I see a q, 00:58:56.140 --> 00:58:59.700 I replace it with b inverse q prime. 00:58:59.700 --> 00:59:05.190 Whenever I see an m tilde, I replace it with z m prime. 00:59:05.190 --> 00:59:14.140 So then I will find the Hamiltonian that characterizes 00:59:14.140 --> 00:59:22.030 the m prime variables, the rg implemented variables, 00:59:22.030 --> 00:59:26.764 which is-- OK, there is a bunch of constants out front. 00:59:26.764 --> 00:59:29.942 There is the fv. 00:59:29.942 --> 00:59:34.860 Actually, it's 0 delta fv1. 00:59:34.860 --> 00:59:40.030 In the sign, I had to put as plus. 00:59:40.030 --> 00:59:42.140 But really, it doesn't matter. 00:59:46.740 --> 00:59:51.660 Then, I go and write down what I have. 00:59:51.660 --> 00:59:58.310 The integration over q prime after the rescaling is 00:59:58.310 --> 01:00:04.225 performed goes back to the same cutoff or the same shortest 01:00:04.225 --> 01:00:05.500 wavelength as before. 01:00:08.060 --> 01:00:12.160 Except that when I do this replacement of q 01:00:12.160 --> 01:00:18.040 lesser with q prime, I will get a factor of b to the minus d 01:00:18.040 --> 01:00:19.716 down here. 01:00:19.716 --> 01:00:21.860 And there are b integrations. 01:00:21.860 --> 01:00:24.650 And then I have t tilde. 01:00:24.650 --> 01:00:32.720 The next term is k tilde, but k tilde is the same thing as k. 01:00:32.720 --> 01:00:34.230 Goes with q squared. 01:00:34.230 --> 01:00:37.440 So this becomes q prime squared, and then I 01:00:37.440 --> 01:00:40.040 will get a b to the minus 2. 01:00:40.040 --> 01:00:43.766 Higher orders will get more factors of b to the minus 2 01:00:43.766 --> 01:00:46.040 over 2. 01:00:46.040 --> 01:00:51.590 And then I have m tilde, which is replaced by z. 01:00:51.590 --> 01:00:53.578 And there are two of them. 01:00:53.578 --> 01:00:57.805 I will get m prime of q prime squared. 01:01:03.395 --> 01:01:05.770 If I were to explicitly now write 01:01:05.770 --> 01:01:12.340 the factors that go in construction of u, 01:01:12.340 --> 01:01:17.410 since u had three integrations over q-- 01:01:17.410 --> 01:01:21.980 left board over there-- I will get three integrations 01:01:21.980 --> 01:01:33.160 over q prime giving me a factor of b to the minus 3d. 01:01:36.780 --> 01:01:44.087 And then I have four factors of m tilde that become m prime. 01:01:56.820 --> 01:02:01.450 Along the way, I will pick up four factors of z. 01:02:01.450 --> 01:02:04.658 And then, of course, order of u squared. 01:02:07.860 --> 01:02:14.050 So under this three steps of rg, what 01:02:14.050 --> 01:02:21.430 happened was that I generated t prime, which was z 01:02:21.430 --> 01:02:28.650 squared b to the minus d t tilde. 01:02:28.650 --> 01:02:34.390 I generated u prime, which was z to the fourth 01:02:34.390 --> 01:02:40.702 b to the minus 3d u, the original u. 01:02:40.702 --> 01:02:44.510 And I generated the k prime, which 01:02:44.510 --> 01:02:50.570 was z squared b to the minus d minus 2 k. 01:02:50.570 --> 01:02:53.667 And I could do the same for various other parameters. 01:02:58.440 --> 01:03:01.655 Now again, we come up with this issue 01:03:01.655 --> 01:03:06.760 of what to choose for zeta. 01:03:06.760 --> 01:03:08.630 Sorry, for z. 01:03:08.630 --> 01:03:11.770 And what I had said previously was 01:03:11.770 --> 01:03:15.070 that we went through all of these exercise of doing 01:03:15.070 --> 01:03:18.530 the Gaussian model via rg in order 01:03:18.530 --> 01:03:21.150 to have an anchoring point. 01:03:21.150 --> 01:03:24.640 And there we saw that the thing that we were interested 01:03:24.640 --> 01:03:28.840 was to look at the point where k prime was the same as k. 01:03:28.840 --> 01:03:32.770 So let's stick with that choice and see 01:03:32.770 --> 01:03:34.720 what the consequences are. 01:03:34.720 --> 01:03:38.885 So choose z such that k prime is the same 01:03:38.885 --> 01:03:43.720 a k, which means the I choose my z to be b to the 1 01:03:43.720 --> 01:03:47.100 plus d over 2 exactly as I had done previously 01:03:47.100 --> 01:03:49.105 for the Gaussian model. 01:03:49.105 --> 01:03:52.900 So now I can substitute these values. 01:03:52.900 --> 01:03:55.660 And therefore, see that following 01:03:55.660 --> 01:04:02.080 a rescaling by a factor of b, the value of my t prime. 01:04:02.080 --> 01:04:03.970 z squared b to the minus d. 01:04:03.970 --> 01:04:08.830 I will get b to the power of 2 plus d minus d. 01:04:08.830 --> 01:04:11.960 So that would give me b squared. 01:04:11.960 --> 01:04:16.850 And I have t plus 4u n plus 2. 01:04:16.850 --> 01:04:21.490 This integral from lambda over b to lambda 01:04:21.490 --> 01:04:28.560 to dk 2 pi to the d 1 over d plus k, 01:04:28.560 --> 01:04:30.909 k squared, and so forth. 01:04:30.909 --> 01:04:35.750 Presumably, order of u squared. 01:04:35.750 --> 01:04:44.255 And that my factor of u rescaled by b. 01:04:44.255 --> 01:04:49.670 I have to put four factors of z. 01:04:49.670 --> 01:04:52.900 So I will get b to the 4 plus 2d, 01:04:52.900 --> 01:04:57.740 and then 3d gets subtracted, so I will get b to the 4 minus 01:04:57.740 --> 01:05:00.330 d and u again. 01:05:00.330 --> 01:05:02.730 But presumably, order of u squared. 01:05:15.470 --> 01:05:20.130 So the first factors are precisely the things 01:05:20.130 --> 01:05:25.280 that we had done and obtained for the Gaussian model. 01:05:25.280 --> 01:05:29.440 So the only thing that we gained by this exercise so far 01:05:29.440 --> 01:05:31.045 is this correction to t. 01:05:33.720 --> 01:05:37.670 We will see that that correction is not that important. 01:05:37.670 --> 01:05:41.480 And in order to really gain an understanding, 01:05:41.480 --> 01:05:44.030 we have to go to the next order. 01:05:44.030 --> 01:05:48.700 But let's use this opportunity to also make 01:05:48.700 --> 01:05:51.120 some changes of terminology that is useful. 01:05:53.980 --> 01:05:58.450 So clearly, the way that we had constructed this-- 01:05:58.450 --> 01:06:00.220 and in particular, if you are thinking 01:06:00.220 --> 01:06:04.600 about averaging over spins, et cetera, in real space-- 01:06:04.600 --> 01:06:06.500 the natural thing to think about is 01:06:06.500 --> 01:06:10.350 that maybe your b is a factor of 2, or a factor of 3. 01:06:10.350 --> 01:06:14.010 You sort of group things by a factor of twice as much 01:06:14.010 --> 01:06:16.440 and then do the averaging. 01:06:16.440 --> 01:06:19.230 But when you look at things from the perspective 01:06:19.230 --> 01:06:25.620 of this momentum shell, this b can be anything. 01:06:25.620 --> 01:06:30.730 And it turns out to be useful just as a language tool, not as 01:06:30.730 --> 01:06:35.670 a conceptual tool, to make this b to be very close to 1 01:06:35.670 --> 01:06:39.900 so that effectively you are just removing a very, very 01:06:39.900 --> 01:06:45.380 tiny, thin shell around the boundary. 01:06:45.380 --> 01:06:54.620 So essentially what I am saying is choose b that is almost 1, 01:06:54.620 --> 01:06:58.550 but maybe a little bit shifted from 1. 01:07:01.340 --> 01:07:06.060 Then clearly, as b goes to 1, then t prime 01:07:06.060 --> 01:07:11.020 has to go to t-- prime has to go to u. 01:07:11.020 --> 01:07:15.580 So what I can do is I can define t prime scaled 01:07:15.580 --> 01:07:22.850 by a factor of 1 plus delta l to be t plus a small amount dt 01:07:22.850 --> 01:07:25.830 by dl. 01:07:25.830 --> 01:07:27.835 And in order of delta l squared. 01:07:31.930 --> 01:07:39.570 And similarly, I can write u prime to be u plus delta l du 01:07:39.570 --> 01:07:44.680 by dl and higher order. 01:07:44.680 --> 01:07:53.230 And the reason is that if we now look at the parameter space-- 01:07:53.230 --> 01:07:58.060 t, u, whatever-- the effect of this procedure 01:07:58.060 --> 01:08:01.610 rather than being some jump from some point to another point 01:08:01.610 --> 01:08:04.100 because we rescaled by a factor of b 01:08:04.100 --> 01:08:07.850 is to go to a nearby point. 01:08:07.850 --> 01:08:11.980 So these things, dt by dl, du by dl, 01:08:11.980 --> 01:08:15.690 essentially point to the direction 01:08:15.690 --> 01:08:19.279 in which the parameters would change. 01:08:19.279 --> 01:08:23.260 And so they allow you to replace these jumps 01:08:23.260 --> 01:08:28.464 that you have by things that are called flows in this parameter 01:08:28.464 --> 01:08:29.580 space. 01:08:29.580 --> 01:08:34.819 So basically, you have constructed flows and vectors 01:08:34.819 --> 01:08:39.390 that describe the flows in this parameter space. 01:08:39.390 --> 01:08:48.399 Now, if I do that over here, we can also 01:08:48.399 --> 01:08:51.310 see some other things emerging. 01:08:51.310 --> 01:08:57.279 So b squared I can write as 1 plus 2 delta l. 01:09:00.779 --> 01:09:02.920 And then here, I have t. 01:09:06.330 --> 01:09:12.000 Now clearly here, if d was 1, the integral would be 0. 01:09:12.000 --> 01:09:14.955 I am integrating over a tiny shell. 01:09:14.955 --> 01:09:18.430 So the answer here when b goes to 1 01:09:18.430 --> 01:09:24.760 is really just the area of that sphere multiplied 01:09:24.760 --> 01:09:26.720 by the thickness. 01:09:26.720 --> 01:09:28.790 And so what do I get? 01:09:28.790 --> 01:09:35.109 I get 4u n plus 2 is just the overall coefficient. 01:09:35.109 --> 01:09:39.649 Then, what is the surface area? 01:09:39.649 --> 01:09:43.609 I have the solid angle divided by 2 pi to the d. 01:09:43.609 --> 01:09:46.475 OK, you divide solid angle divided by 2 pi 01:09:46.475 --> 01:09:49.960 to the d to be kd. 01:09:49.960 --> 01:09:53.350 What's the value of the integrand on shell? 01:09:53.350 --> 01:09:56.530 On shell, I have to replace k with the lambda. 01:09:56.530 --> 01:10:01.680 So I will get t plus k lambda squared. 01:10:01.680 --> 01:10:09.000 Actually, the surface area is Sd lambda to the d minus 1. 01:10:09.000 --> 01:10:14.000 But then the thickness is lambda delta l. 01:10:21.164 --> 01:10:23.320 The second one is actually very easy. 01:10:23.320 --> 01:10:28.755 It is 1 plus 4 minus b delta l times u. 01:10:32.870 --> 01:10:42.460 So you can see that if I match things to order of delta l, 01:10:42.460 --> 01:10:45.010 I get the following rg flows. 01:10:47.550 --> 01:10:53.650 So the left-hand side is t plus delta l dt by dl. 01:10:53.650 --> 01:10:56.920 The right-hand side-- if I expand, there is a factor of t. 01:10:56.920 --> 01:10:59.020 So that it gets rid of. 01:10:59.020 --> 01:11:03.930 And I'm left with a term that is proportional to delta l, whose 01:11:03.930 --> 01:11:07.370 coefficient on the left-hand side is dt by dl. 01:11:07.370 --> 01:11:11.393 And the right-hand side, I get either a factor of 2t 01:11:11.393 --> 01:11:16.460 from multiplying the 2 delta l with t or from multiplying 01:11:16.460 --> 01:11:20.140 the 1 with the result of this integration. 01:11:20.140 --> 01:11:27.035 I will get a 4u n plus 2 kd lambda 01:11:27.035 --> 01:11:32.170 to the d divided by t plus k lambda squared. 01:11:32.170 --> 01:11:35.820 And I don't need to evaluate any integrals. 01:11:38.410 --> 01:11:44.390 And the second flow equation for the parameter u du by dl 01:11:44.390 --> 01:11:46.650 is 4 minus d u. 01:11:56.540 --> 01:12:00.680 Now, clearly in the language of flows, 01:12:00.680 --> 01:12:06.282 a fixed point is when there is no flow. 01:12:06.282 --> 01:12:12.560 So I could have dt by dl du by dl need to be 0. 01:12:15.210 --> 01:12:18.530 And du by dl being 0 immediately tells ms 01:12:18.530 --> 01:12:23.000 that u star has to be 0. 01:12:23.000 --> 01:12:25.630 And if I set u star equals to 0, I 01:12:25.630 --> 01:12:29.960 will see that t star has to be 0. 01:12:29.960 --> 01:12:33.370 So these equations have one and only 01:12:33.370 --> 01:12:36.480 one fixed point at this order. 01:12:36.480 --> 01:12:42.900 And then looking for relevance and irrelevance of going away 01:12:42.900 --> 01:12:46.855 from the fixed point can be captured by linearizing. 01:12:46.855 --> 01:12:51.660 That is, I write my t to be t star plus delta l. 01:12:51.660 --> 01:12:54.765 Of course, my t star and u star are both 0. 01:12:54.765 --> 01:12:59.560 But in general, if they were not at 0, 01:12:59.560 --> 01:13:07.650 I would linearize my in general, non-linear rg recursions 01:13:07.650 --> 01:13:11.620 by going slightly away from the fixed point. 01:13:11.620 --> 01:13:14.940 And then the linearized form of the equation 01:13:14.940 --> 01:13:21.030 would say that going away be a small amount delta t delta u-- 01:13:21.030 --> 01:13:25.350 and there could be more and more of these operators-- an 01:13:25.350 --> 01:13:32.800 be written in terms of a matrix multiply delta t and delta u. 01:13:32.800 --> 01:13:36.460 And clearly, the matrix for u is very simple. 01:13:36.460 --> 01:13:42.600 It is simply proportional to 4 minus d delta u. 01:13:42.600 --> 01:13:45.720 The matrix for t? 01:13:45.720 --> 01:13:47.170 Well, there's two terms. 01:13:47.170 --> 01:13:50.060 First of all, there is this 2. 01:13:50.060 --> 01:13:53.070 And then if I am taking a derivative, 01:13:53.070 --> 01:13:55.340 there will be a derivative of this expression 01:13:55.340 --> 01:13:57.700 because there is a t-dependence here. 01:13:57.700 --> 01:13:59.930 But ultimately, since I'm evaluating it 01:13:59.930 --> 01:14:06.230 at u star equals to 0, I don't need to include that term here. 01:14:06.230 --> 01:14:08.800 But if I now make a variation in u, 01:14:08.800 --> 01:14:15.870 I will get an off-diagonal term here, 01:14:15.870 --> 01:14:19.086 which is k lambda squared. 01:14:19.086 --> 01:14:22.394 So these two can combine with each other. 01:14:27.180 --> 01:14:32.730 Now, looking for relevance or irrelevance 01:14:32.730 --> 01:14:36.110 is then equivalent-- previously, we were talking about it 01:14:36.110 --> 01:14:42.520 in terms of the full equations with b that could have been 01:14:42.520 --> 01:14:45.190 anything, but now we have gone to this limit 01:14:45.190 --> 01:14:47.210 of infinitesimal b. 01:14:47.210 --> 01:14:52.190 Then, what I have to do is to find the eigenvalues 01:14:52.190 --> 01:14:54.973 of the matrix that I have for these flows. 01:14:54.973 --> 01:14:58.460 Now, a matrix that has this structure where there's 01:14:58.460 --> 01:15:01.790 0 on one side of the diagonal, I immediately 01:15:01.790 --> 01:15:07.260 know that the eigenvalues are 2 and 4 minus d. 01:15:07.260 --> 01:15:12.130 So basically, depending on whether I'm 01:15:12.130 --> 01:15:15.060 in dimensions greater than 4 or less than 4, 01:15:15.060 --> 01:15:20.410 I will have either two or one relevant direction. 01:15:20.410 --> 01:15:23.780 In particular, if I look at what is happening 01:15:23.780 --> 01:15:28.730 for d that is greater than 4-- for d greater than 4, 01:15:28.730 --> 01:15:31.970 I will just have one relevant direction. 01:15:31.970 --> 01:15:36.480 And if I look at the behavior and the flows 01:15:36.480 --> 01:15:44.000 that I am allowed to have in the two parameters t and u. 01:15:44.000 --> 01:15:47.530 Now, if my Hamiltonian only has t and u, 01:15:47.530 --> 01:15:49.620 I'm only allowed to look at the case 01:15:49.620 --> 01:15:52.530 where u is positive in order not to have 01:15:52.530 --> 01:15:55.140 weights that are unbounded. 01:15:55.140 --> 01:16:00.142 My fixed point occurs at 0, 0. 01:16:00.142 --> 01:16:10.480 And then one simple thing is that if u is 0, it will stay 0. 01:16:10.480 --> 01:16:13.950 And then dt by dl is 2t. 01:16:13.950 --> 01:16:18.320 So I know that I always have an eigen-direction that 01:16:18.320 --> 01:16:23.810 is along the t and is flowing away with a velocity of 2 01:16:23.810 --> 01:16:24.405 if you like. 01:16:27.660 --> 01:16:32.250 The other eigen-direction is not the axis t equals to 0. 01:16:32.250 --> 01:16:37.060 Because if t is 0 originally but 0 is nonzero, 01:16:37.060 --> 01:16:41.940 this term will generate some positive amount of t for me. 01:16:41.940 --> 01:16:45.085 So if I start somewhere on this axis, 01:16:45.085 --> 01:16:49.210 t will go in the direction of becoming positive. 01:16:49.210 --> 01:16:52.060 Above four dimensions, you will become less. 01:16:52.060 --> 01:16:54.280 So above four dimension, you can see 01:16:54.280 --> 01:16:57.930 that the general trend of flows is 01:16:57.930 --> 01:17:01.092 going to be something like this. 01:17:01.092 --> 01:17:04.700 Indeed, there has to be a second eigen-direction 01:17:04.700 --> 01:17:07.430 because I'm dealing with a 2 by 2 matrix. 01:17:07.430 --> 01:17:09.110 And if you look at it carefully, you'll 01:17:09.110 --> 01:17:14.360 find that the second eigen-direction is down here 01:17:14.360 --> 01:17:17.730 and corresponds to a negative eigenvalue. 01:17:17.730 --> 01:17:22.290 So I basically would be having flows that go towards that. 01:17:22.290 --> 01:17:25.180 And in general, the character of the flows, 01:17:25.180 --> 01:17:30.380 if I have parameters somewhere here, 01:17:30.380 --> 01:17:31.919 they would be flowing there. 01:17:31.919 --> 01:17:33.915 If I have parameters somewhere here, 01:17:33.915 --> 01:17:35.412 they would be flowing there. 01:17:38.410 --> 01:17:43.220 So physically, I have something like iron in five dimension, 01:17:43.220 --> 01:17:43.990 for example. 01:17:43.990 --> 01:17:47.880 And it corresponds to being somewhere here. 01:17:47.880 --> 01:17:51.213 When I change the temperature of iron in five dimension 01:17:51.213 --> 01:17:55.900 and I execute some trajectory such as this, 01:17:55.900 --> 01:17:59.680 all of the points that are on this side of the trajectory 01:17:59.680 --> 01:18:01.980 on their flow will go to a place where 01:18:01.980 --> 01:18:04.780 u becomes small and t becomes positive. 01:18:04.780 --> 01:18:09.130 So I will essentially go to this Gaussian-like fixed point 01:18:09.130 --> 01:18:12.520 that describes independent spins. 01:18:12.520 --> 01:18:14.780 All the things that are down here, 01:18:14.780 --> 01:18:16.950 which previously in the Gaussian model 01:18:16.950 --> 01:18:19.130 we could not describe, because the Gaussian model 01:18:19.130 --> 01:18:21.870 did not allow me to go to t negative. 01:18:21.870 --> 01:18:23.670 Because now I have a positive view, 01:18:23.670 --> 01:18:25.600 I have no problem with that. 01:18:25.600 --> 01:18:29.410 And I find that essentially I go to large negative t 01:18:29.410 --> 01:18:31.270 corresponding to some positive u. 01:18:31.270 --> 01:18:34.621 I can figure out what the amount of magnetization is. 01:18:34.621 --> 01:18:36.900 And so then in between them, there 01:18:36.900 --> 01:18:41.790 is a trajectory that separates paramagnetic and ferromagnetic 01:18:41.790 --> 01:18:42.370 behavior. 01:18:42.370 --> 01:18:45.985 And clearly, that trajectory at long scales 01:18:45.985 --> 01:18:48.420 corresponds to the fixed point that 01:18:48.420 --> 01:18:52.690 is simply the gradient squared. 01:18:52.690 --> 01:18:55.640 Because all the other terms went to 0, so 01:18:55.640 --> 01:18:57.950 we know all of the correlation functions, 01:18:57.950 --> 01:19:02.400 et cetera, that we should see in this system. 01:19:02.400 --> 01:19:12.690 Unfortunately, if I look for d less than 4, what happens 01:19:12.690 --> 01:19:16.520 is that I still have the same fixed point. 01:19:16.520 --> 01:19:19.410 And actually , the eigen-directions don't change 01:19:19.410 --> 01:19:20.760 all that much. 01:19:20.760 --> 01:19:23.500 u equals to 0 is still an eigen-direction 01:19:23.500 --> 01:19:26.390 that has relevance, too. 01:19:26.390 --> 01:19:30.334 But the other direction changes from being 01:19:30.334 --> 01:19:32.190 irrelevant to being relevant. 01:19:32.190 --> 01:19:35.470 So I have something like this. 01:19:35.470 --> 01:19:38.600 And the natural types of flows that I 01:19:38.600 --> 01:19:41.250 get kind of look like this. 01:19:45.820 --> 01:19:49.890 And now if I take my iron in three dimensions 01:19:49.890 --> 01:19:53.950 and change its temperature, I go from behavior 01:19:53.950 --> 01:19:57.120 that is kind of paramagnetic-like 01:19:57.120 --> 01:20:00.240 to ferromagnetic-like. 01:20:00.240 --> 01:20:04.120 And there is a transition point. 01:20:04.120 --> 01:20:06.180 But that transition point I don't 01:20:06.180 --> 01:20:07.983 know what fixed point it goes to. 01:20:07.983 --> 01:20:09.000 I have no idea. 01:20:11.780 --> 01:20:14.550 So the only difference by doing this analysis 01:20:14.550 --> 01:20:18.190 from what we had done just on the basis of scaling 01:20:18.190 --> 01:20:20.420 and Gaussian theory, et cetera, is 01:20:20.420 --> 01:20:23.080 that we have located the shift in t 01:20:23.080 --> 01:20:27.640 with u, which is what we had done before. 01:20:27.640 --> 01:20:30.780 So was it worth it? 01:20:30.780 --> 01:20:34.000 The answer is up to here, no. 01:20:34.000 --> 01:20:37.080 But let's see if you had gone one step further what 01:20:37.080 --> 01:20:39.470 would have happened. 01:20:39.470 --> 01:20:42.670 The series is an alternating series. 01:20:42.670 --> 01:20:46.860 So the next order term that I get here I expect 01:20:46.860 --> 01:20:49.300 will come with some negative u squared. 01:20:49.300 --> 01:20:52.880 So let's say there will be some amount of work 01:20:52.880 --> 01:20:53.750 that I have to do. 01:20:53.750 --> 01:20:57.680 And I calculate something that is minus a u squared. 01:20:57.680 --> 01:21:00.005 I do some amount of work and I calculate 01:21:00.005 --> 01:21:04.410 something that is minus b u squared. 01:21:04.410 --> 01:21:09.340 But then you can see that if I search for the fixed point, 01:21:09.340 --> 01:21:13.570 I will find another one at u star, which 01:21:13.570 --> 01:21:17.340 is 4 minus d over b. 01:21:17.340 --> 01:21:21.890 So I will find another point. 01:21:21.890 --> 01:21:26.310 And indeed, we'll find that things that left here 01:21:26.310 --> 01:21:29.860 will go to that point. 01:21:29.860 --> 01:21:34.650 And so that point has one direction here. 01:21:34.650 --> 01:21:37.310 It was relevant that becomes irrelevant. 01:21:37.310 --> 01:21:39.530 And the other direction is the analog 01:21:39.530 --> 01:21:43.780 of this direction, which still remains relevant, 01:21:43.780 --> 01:21:45.740 but with modified exponents. 01:21:45.740 --> 01:21:49.870 We will figure out what that is. 01:21:49.870 --> 01:21:52.300 So the next step is to find this b, 01:21:52.300 --> 01:21:55.490 and then everything would be resolved. 01:21:55.490 --> 01:22:01.920 The only thing to then realize is, what are we perturbing it? 01:22:01.920 --> 01:22:04.210 Because the whole idea of perturbation 01:22:04.210 --> 01:22:09.190 theory is that you should have a small parameter. 01:22:09.190 --> 01:22:13.905 And if we are perturbing a u and then 01:22:13.905 --> 01:22:17.720 basing our results on what is happening here, 01:22:17.720 --> 01:22:20.330 the location of this fixed point better 01:22:20.330 --> 01:22:23.080 be small-- close to the original one 01:22:23.080 --> 01:22:26.010 around which I am perturbing. 01:22:26.010 --> 01:22:28.590 So what do I have to make small? 01:22:28.590 --> 01:22:30.860 4 minus d. 01:22:30.860 --> 01:22:33.700 So we thought we were making a perturbation in u, 01:22:33.700 --> 01:22:37.840 but in order to have a small quantity the only thing that we 01:22:37.840 --> 01:22:40.760 can do is to stay very close to four dimension 01:22:40.760 --> 01:22:43.630 and make a perturbation expansion around four 01:22:43.630 --> 01:22:45.180 dimension.