WEBVTT 00:00:00.090 --> 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 --> 00:00:13.340 To make a donation or view additional materials 00:00:13.340 --> 00:00:17.217 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.217 --> 00:00:17.842 at ocw.mit.edu. 00:00:21.830 --> 00:00:25.180 PROFESSOR: OK, let's start. 00:00:25.180 --> 00:00:31.820 So we've been looking at the xy model in two dimensions. 00:00:34.910 --> 00:00:40.060 It's a collection of units spins located 00:00:40.060 --> 00:00:44.890 on a site of potentially a lattice in two dimensions. 00:00:44.890 --> 00:00:48.345 Since they are unit vectors, each one of them 00:00:48.345 --> 00:00:52.550 is characterized by an angle theta i. 00:00:52.550 --> 00:00:56.190 And the partition function would be 00:00:56.190 --> 00:01:00.450 obtained by integrating over all of the angles. 00:01:03.490 --> 00:01:09.790 And the weight that we said had the form k e cosine of theta 00:01:09.790 --> 00:01:12.252 i minus theta j. 00:01:12.252 --> 00:01:15.380 So there's a coupling that corresponds 00:01:15.380 --> 00:01:18.480 to the dot per dot of neighboring spins, which 00:01:18.480 --> 00:01:22.500 can be written as this cosine form. 00:01:22.500 --> 00:01:33.290 If we go to low temperatures where k is large, 00:01:33.290 --> 00:01:36.610 then this was roughly an integral 00:01:36.610 --> 00:01:40.150 over all configurations of the continuous field, theta 00:01:40.150 --> 00:01:48.800 of x, with a weight that is the appropriate choice 00:01:48.800 --> 00:01:54.015 of the lattice spacing, the same k integral of gradient of theta 00:01:54.015 --> 00:01:54.515 squared. 00:02:00.860 --> 00:02:07.200 Now if we look at this weight and ask 00:02:07.200 --> 00:02:10.990 what is happening as a function of changing 00:02:10.990 --> 00:02:17.270 temperature or the inverse of this parameter 00:02:17.270 --> 00:02:22.670 so that we are close to zero temperature, 00:02:22.670 --> 00:02:26.430 if we just work with this Gaussian, 00:02:26.430 --> 00:02:28.240 the conclusion would be that if we 00:02:28.240 --> 00:02:33.330 look at the correlation between two spins that are located 00:02:33.330 --> 00:02:42.540 at a distance r from each other, just from this Gaussian weight, 00:02:42.540 --> 00:02:49.000 we found that these correlations decay as something like 1/r, 00:02:49.000 --> 00:02:53.250 or maybe we put some kind for a lattice spacing, 00:02:53.250 --> 00:02:57.270 to the power of an exponent theta that 00:02:57.270 --> 00:02:59.064 was related to this k. 00:03:03.330 --> 00:03:08.380 So the conclusion was that if that is the appropriate weight, 00:03:08.380 --> 00:03:13.460 we have power-law decay correlations. 00:03:13.460 --> 00:03:20.510 There is no long range order for correlations decay very weakly. 00:03:20.510 --> 00:03:24.220 On the other hand, if we start with the full cosine 00:03:24.220 --> 00:03:28.580 and don't do this low temperature expansion, 00:03:28.580 --> 00:03:33.700 just go and do the typical high temperature expansion, 00:03:33.700 --> 00:03:35.530 from the high temperature expansion 00:03:35.530 --> 00:03:37.690 we conclude that these collections decay 00:03:37.690 --> 00:03:46.160 exponentially, which are two totally different forms. 00:03:46.160 --> 00:03:48.520 And presumably, there should be some kind 00:03:48.520 --> 00:03:52.370 of a critical value of k or temperature 00:03:52.370 --> 00:03:56.169 that separates a low temperature formed with power-law decay 00:03:56.169 --> 00:03:58.335 and a high temperature formed with exponential decay 00:03:58.335 --> 00:03:59.147 of correlations. 00:04:01.950 --> 00:04:07.080 Now there was no sign of such a kc when 00:04:07.080 --> 00:04:12.710 we tried to do a low temperature expansion 00:04:12.710 --> 00:04:16.890 like the non-linear sigma model in this particular case 00:04:16.890 --> 00:04:21.480 of the xy model that corresponds to n equals to 2. 00:04:21.480 --> 00:04:23.980 Again, we said that the reason for that 00:04:23.980 --> 00:04:28.120 is that the only high order terms that I can write down 00:04:28.120 --> 00:04:30.990 in this theory close to low temperature gradient of theta 00:04:30.990 --> 00:04:35.680 to the fourth, sixth, et cetera are explicitly irrelevant. 00:04:35.680 --> 00:04:38.055 And unlike n equals 3, et cetera, 00:04:38.055 --> 00:04:40.860 it cannot cause any change. 00:04:40.860 --> 00:04:46.590 So as far as that theory was concerned, all of these 00:04:46.590 --> 00:04:48.640 corresponded to being fixed points. 00:04:52.450 --> 00:04:58.650 Then we said that there is a twist that is not 00:04:58.650 --> 00:05:02.490 taken into account when I make that transformation 00:05:02.490 --> 00:05:06.610 in the first line, as pointed out by Kosterlitz and Thouless. 00:05:13.140 --> 00:05:16.820 What you have are topological defects that are left out. 00:05:24.360 --> 00:05:28.346 And an example of such a defect would 00:05:28.346 --> 00:05:35.620 be a configuration of spins that are kind of radiating out 00:05:35.620 --> 00:05:41.160 from some particular point, such that when I complete 00:05:41.160 --> 00:05:44.920 a circuit going around, the value 00:05:44.920 --> 00:05:55.260 of spin changes by 2 pi for the gradient at the distance r, 00:05:55.260 --> 00:06:00.210 gradient of the change in angle should fall off as 1 over r. 00:06:00.210 --> 00:06:06.050 So if we put one of these defects 00:06:06.050 --> 00:06:11.680 and calculate what the partition function is for one defect, 00:06:11.680 --> 00:06:15.080 z of one defect, we would say, OK, 00:06:15.080 --> 00:06:20.130 what I have to do is to calculate the energy 00:06:20.130 --> 00:06:22.520 cost of this distortion. 00:06:22.520 --> 00:06:26.830 If I use that theory that I have over there, I have k/2. 00:06:26.830 --> 00:06:34.580 I have integral over all space which, because of the symmetry 00:06:34.580 --> 00:06:38.330 here, I can write as 2 pi r to the r. 00:06:41.160 --> 00:06:44.540 And then I have the gradient squared. 00:06:44.540 --> 00:06:48.650 And as I said, the gradient for a topological defect 00:06:48.650 --> 00:06:51.025 such as this goes as 1/r. 00:06:51.025 --> 00:06:55.540 It's square goes like 1 over r squared. 00:06:55.540 --> 00:07:01.940 So this is an integral that is logarithmically divergent. 00:07:01.940 --> 00:07:04.320 It's a an integral of 1/r. 00:07:04.320 --> 00:07:08.610 It's logarithmically divergent both at large distances-- 00:07:08.610 --> 00:07:12.500 let's say of the order of the size of the system-- 00:07:12.500 --> 00:07:15.250 and has difficulty at short distances. 00:07:15.250 --> 00:07:18.680 But at short distances, we know that there is some lattice 00:07:18.680 --> 00:07:23.200 structure, and this approximation will break down 00:07:23.200 --> 00:07:27.540 when I get to some order of some multiple lattice spacing-- so 00:07:27.540 --> 00:07:31.780 let's call that a-- and that the additional energy that 00:07:31.780 --> 00:07:36.260 comes from all of the interactions that are smaller 00:07:36.260 --> 00:07:40.970 than this core value of a I have to separately calculate, 00:07:40.970 --> 00:07:45.820 and I'm going to call them beta epsilon that 00:07:45.820 --> 00:07:50.120 depends on the distance a that I choose for the core. 00:07:53.330 --> 00:07:58.510 This is just a energy term that goes into the Boltzmann factor. 00:07:58.510 --> 00:08:02.670 But this defect can be placed any place 00:08:02.670 --> 00:08:10.070 in a lattice that has size L up to this factor of 1/a 00:08:10.070 --> 00:08:12.300 that I call the size of the core. 00:08:12.300 --> 00:08:17.450 So the number of places goes like L/a squared, the area 00:08:17.450 --> 00:08:19.870 that I'm looking at. 00:08:19.870 --> 00:08:23.940 And so we see that this expression actually 00:08:23.940 --> 00:08:26.820 goes like L/a. 00:08:26.820 --> 00:08:29.620 I have the power 2 here. 00:08:29.620 --> 00:08:31.920 And from this logarithmic interaction, 00:08:31.920 --> 00:08:37.059 I will get here power of 2 pi k log L/a, 00:08:37.059 --> 00:08:39.119 which I can absorb into this. 00:08:42.030 --> 00:08:46.840 And I define the exponential of the core energy 00:08:46.840 --> 00:08:50.220 to be some parameter y. 00:08:50.220 --> 00:08:55.805 That clearly depends on the choice of what I call my core. 00:08:59.260 --> 00:09:03.650 So if I look at just that expression by itself, 00:09:03.650 --> 00:09:08.030 I would say that there is the chance 00:09:08.030 --> 00:09:14.500 to see one defect or no defects as L/a becomes 00:09:14.500 --> 00:09:17.460 large in the thermodynamic limit depending 00:09:17.460 --> 00:09:21.790 on whether this exponent is positive or negative. 00:09:21.790 --> 00:09:27.590 So we are kind of-- oops, I forgot the factor of 2 here. 00:09:27.590 --> 00:09:30.960 It's 1 over pi k. 00:09:30.960 --> 00:09:35.760 We are led from this expression that something interesting 00:09:35.760 --> 00:09:40.920 should happen at some kc for one defect, which 00:09:40.920 --> 00:09:44.450 is let's call it the inverse of the kc. 00:09:44.450 --> 00:09:49.090 That is like a temperature, which is pi/2. 00:09:49.090 --> 00:09:52.870 So maybe around pi/2 here there should 00:09:52.870 --> 00:09:56.050 be something interesting that happens. 00:09:56.050 --> 00:10:01.750 In fact, if the theory was independent vertices, 00:10:01.750 --> 00:10:06.280 I would predict there would be no vertices up to here, 00:10:06.280 --> 00:10:09.100 and then there would be whole bunches vertices 00:10:09.100 --> 00:10:12.940 that would be appearing later on. 00:10:12.940 --> 00:10:16.710 But the point is that these are not 00:10:16.710 --> 00:10:20.000 vertices that don't interact with each other. 00:10:20.000 --> 00:10:23.150 There are interactions between them. 00:10:23.150 --> 00:10:30.440 And if I look at a situation with many vertices, what 00:10:30.440 --> 00:10:36.680 I need to do is to calculate the partition function 00:10:36.680 --> 00:10:40.100 for the defects, for the vertices that 00:10:40.100 --> 00:10:45.700 has a resemblance to the Coulomb system, so we call it z sub q. 00:10:45.700 --> 00:10:50.620 And this z sub q is obtained by summing over 00:10:50.620 --> 00:10:56.650 all numbers of these defects that can appear in the system. 00:10:56.650 --> 00:10:59.370 So let's do a sum over n, starting 00:10:59.370 --> 00:11:02.170 from 0 to however many. 00:11:02.170 --> 00:11:04.102 And if I have a situation in which there 00:11:04.102 --> 00:11:08.030 are n vertices in the system, I clearly 00:11:08.030 --> 00:11:15.290 have to pay a cost of y per core of each one of them. 00:11:15.290 --> 00:11:20.060 So there's a factor of y raised to the power of n. 00:11:20.060 --> 00:11:24.040 These vertices then can be placed anywhere on the lattice, 00:11:24.040 --> 00:11:27.660 in the same way that I had this factor of L/a 00:11:27.660 --> 00:11:29.880 for a single vertex. 00:11:29.880 --> 00:11:36.860 I will have ability to put each one of these vertices 00:11:36.860 --> 00:11:38.150 at some point on the lattice. 00:11:41.080 --> 00:11:47.070 And then we found that when you do the calculation, 00:11:47.070 --> 00:11:50.870 the distortions that are caused by the independent vertices 00:11:50.870 --> 00:11:53.520 clearly add up on top of each other. 00:11:53.520 --> 00:11:56.260 And when we add it up and superimpose 00:11:56.260 --> 00:11:58.460 the gradient of thetas that correspond 00:11:58.460 --> 00:12:02.190 to the different vertices and calculated 00:12:02.190 --> 00:12:05.800 the integral of gradient of theta squared, what we found 00:12:05.800 --> 00:12:09.530 was that there was an interaction between them 00:12:09.530 --> 00:12:14.800 that I could write as 4 pi squared 00:12:14.800 --> 00:12:23.760 k sum over distinct pairs, i less than j, qi qj. 00:12:23.760 --> 00:12:27.150 The Coulomb interaction [? it fits inside ?] xi and xj. 00:12:31.820 --> 00:12:38.490 And actually, the form of this came about as follows. 00:12:38.490 --> 00:12:45.670 Those That basically the charges of these topological defects 00:12:45.670 --> 00:12:50.530 are multiples of 2 pi, but these qi's I have written 00:12:50.530 --> 00:12:52.790 are minus plus 1. 00:12:52.790 --> 00:12:56.480 So the two pis are absorbed in the charges. 00:12:56.480 --> 00:13:02.310 The actual charges being 2 pi is what makes this 4 pi squared. 00:13:02.310 --> 00:13:05.190 k is the [INAUDIBLE] of interaction. 00:13:05.190 --> 00:13:08.320 Furthermore, we have to require the system 00:13:08.320 --> 00:13:15.910 to be overall neutral because, otherwise, there 00:13:15.910 --> 00:13:18.740 would be a large energy for creating 00:13:18.740 --> 00:13:22.860 the monopole in a large system. 00:13:22.860 --> 00:13:26.590 And again, just as a matter of notation, 00:13:26.590 --> 00:13:32.380 our z of x has a 1 over 2 pi itself 00:13:32.380 --> 00:13:37.330 log of the displacement in units of this a 00:13:37.330 --> 00:13:39.920 because we can't allow these things to come 00:13:39.920 --> 00:13:41.340 very close to each other. 00:13:44.070 --> 00:13:53.650 So our task was to calculate properties encoded 00:13:53.650 --> 00:13:58.890 in this partition function, which is, in some sense, 00:13:58.890 --> 00:14:01.965 a grand canonical system of charges 00:14:01.965 --> 00:14:04.570 that can appear and disappear. 00:14:04.570 --> 00:14:11.620 And our expectation is that at the low temperatures, 00:14:11.620 --> 00:14:19.930 essentially all I have are a few dipoles that are kind of small. 00:14:19.930 --> 00:14:23.600 As I go to higher temperature, the two monopoles 00:14:23.600 --> 00:14:28.710 making the dipole can fluctuate and go further from each other. 00:14:28.710 --> 00:14:32.670 And eventually at some point, they will be all mixed 00:14:32.670 --> 00:14:34.810 up together and the picture should 00:14:34.810 --> 00:14:38.617 be regarded as a mixture of plus and minus charges 00:14:38.617 --> 00:14:39.200 in the plasma. 00:14:39.200 --> 00:14:40.156 Yes? 00:14:40.156 --> 00:14:43.396 AUDIENCE: [INAUDIBLE] if we have an external field, 00:14:43.396 --> 00:14:48.760 would this also be fixed [INAUDIBLE], like 00:14:48.760 --> 00:14:51.230 and edge or something external? 00:14:56.190 --> 00:14:58.130 PROFESSOR: It is very hard to imagine 00:14:58.130 --> 00:15:01.930 what that external field has to be in the language of the xy 00:15:01.930 --> 00:15:04.370 model, because what you can do is 00:15:04.370 --> 00:15:06.360 you can put a field that, let's say, 00:15:06.360 --> 00:15:11.410 rotates the spins on one side-- let's say to point down-- 00:15:11.410 --> 00:15:14.260 spins on the other side to point up. 00:15:14.260 --> 00:15:17.080 But then what happens is that the angles 00:15:17.080 --> 00:15:20.750 would adjust themselves so that at 0 temperature, 00:15:20.750 --> 00:15:22.580 you would have a configuration that 00:15:22.580 --> 00:15:27.510 would go from plus to minus, and all topological charges would 00:15:27.510 --> 00:15:31.170 be on top of that base configuration. 00:15:31.170 --> 00:15:35.830 So that kind of field certainly does not have any effect 00:15:35.830 --> 00:15:38.420 that I can ascribe over here. 00:15:38.420 --> 00:15:40.035 If I change my picture completely 00:15:40.035 --> 00:15:42.860 and say forget about the xy model, 00:15:42.860 --> 00:15:46.270 think about this as a system of point charges, 00:15:46.270 --> 00:15:49.290 then I can certainly, like I did last time, 00:15:49.290 --> 00:15:53.682 put an electric field on the system and see what happens. 00:15:53.682 --> 00:15:54.650 Yes? 00:15:54.650 --> 00:15:56.206 AUDIENCE: [INAUDIBLE] saying that you 00:15:56.206 --> 00:15:58.550 can create even bigger defects, where 00:15:58.550 --> 00:16:00.410 q would be [INAUDIBLE] plus minus 1, 00:16:00.410 --> 00:16:02.120 plus minus bigger integer? 00:16:02.120 --> 00:16:05.166 But that's [? discounted ?] as high [INAUDIBLE] effect. 00:16:05.166 --> 00:16:05.790 PROFESSOR: Yes. 00:16:05.790 --> 00:16:08.710 So intrinsically, we could go beyond that. 00:16:08.710 --> 00:16:12.930 We would have a fugacity for a creation of cores 00:16:12.930 --> 00:16:15.560 of singular charge, another for cores 00:16:15.560 --> 00:16:17.770 of double charge, et cetera. 00:16:17.770 --> 00:16:20.710 You expect those y's or double charges 00:16:20.710 --> 00:16:23.475 to be much larger because the configuration is going 00:16:23.475 --> 00:16:25.015 to be more difficult at the core. 00:16:27.870 --> 00:16:30.230 And in some sense, you can imagine 00:16:30.230 --> 00:16:33.550 that we are including something similar to that 00:16:33.550 --> 00:16:37.700 because we can create two single charges that 00:16:37.700 --> 00:16:40.820 are closing off to each other. 00:16:40.820 --> 00:16:41.320 Yes? 00:16:41.320 --> 00:16:44.470 AUDIENCE: So why is a Coulomb state [INAUDIBLE]? 00:16:47.490 --> 00:16:48.308 Is that high order? 00:16:51.000 --> 00:16:52.800 PROFESSOR: Well, you would expect 00:16:52.800 --> 00:16:56.800 that if y is a small parameter, you 00:16:56.800 --> 00:17:00.120 would like to create as few things as possible. 00:17:00.120 --> 00:17:05.310 The reason you create any is because you have entropy gain. 00:17:05.310 --> 00:17:07.800 So I would say energetically, even 00:17:07.800 --> 00:17:10.329 creating a pair is unfavorable. 00:17:10.329 --> 00:17:13.180 But the pair has lots of places that it can go, 00:17:13.180 --> 00:17:15.230 so because of the gain in entropy, 00:17:15.230 --> 00:17:17.109 it is willing to accept that. 00:17:17.109 --> 00:17:19.400 If I create a quadrupole, you say, 00:17:19.400 --> 00:17:21.940 well, I break the quadrupole into two dipoles 00:17:21.940 --> 00:17:23.819 and then I have much more entropy. 00:17:23.819 --> 00:17:28.119 So that's why it's not-- we can have that term, 00:17:28.119 --> 00:17:30.772 but it is going to have much less weight. 00:17:41.220 --> 00:17:45.310 So I won't repeat the calculation, 00:17:45.310 --> 00:17:50.980 but last time we indeed asked what happens if we have, 00:17:50.980 --> 00:17:55.500 let's say, some kind of an electric field. 00:17:55.500 --> 00:17:58.240 And because of the presence of the electric field, 00:17:58.240 --> 00:18:02.280 dipoles are going to be aligned. 00:18:02.280 --> 00:18:08.560 And the effect of that is to reduce the effective strength 00:18:08.560 --> 00:18:11.550 of all kinds of Coulomb interactions. 00:18:11.550 --> 00:18:18.190 We found that the effective strength was reduced by from k 00:18:18.190 --> 00:18:25.150 by an amount that was related to the likelihood of creating 00:18:25.150 --> 00:18:28.430 the dipole of size r. 00:18:28.430 --> 00:18:32.660 And that was clearly proportional to y squared 00:18:32.660 --> 00:18:44.710 into the minus from there, 4 pi squared q log of r/a. 00:18:44.710 --> 00:18:48.553 That's the co-ability we create a dipole of this size. 00:18:51.480 --> 00:18:57.580 And then I have to, in principle, 00:18:57.580 --> 00:19:00.260 integrate over all dipole sizes. 00:19:03.470 --> 00:19:08.160 But this writing this as an orientationally independent 00:19:08.160 --> 00:19:11.730 result is not correct because in the presence 00:19:11.730 --> 00:19:14.250 of an electric field, you have more likelihood 00:19:14.250 --> 00:19:16.760 to be oriented in one direction as opposed 00:19:16.760 --> 00:19:18.450 to the other direction. 00:19:18.450 --> 00:19:21.340 So this factor of e to the cosine theta, 00:19:21.340 --> 00:19:24.940 et cetera, that we expanded, first of all, 00:19:24.940 --> 00:19:27.820 gave us an average of cosine of theta squared. 00:19:27.820 --> 00:19:30.300 So there was a factor of 1/2 here. 00:19:30.300 --> 00:19:33.660 Rather than full rotation, I was doing an average 00:19:33.660 --> 00:19:38.320 of cosine squared to be that factor. 00:19:38.320 --> 00:19:42.910 Expanding that actually gave me a factor of 4 pi 00:19:42.910 --> 00:19:48.220 squared k because of the Coulomb term that I had up there. 00:19:48.220 --> 00:19:54.320 And then I had essentially the polarizability of one 00:19:54.320 --> 00:19:59.410 of these objects that goes like r squared. 00:19:59.410 --> 00:20:03.320 Again, coming from expanding this factor 00:20:03.320 --> 00:20:05.800 that we had in the exponents. 00:20:05.800 --> 00:20:12.006 Actually, I calculated everything in units of A, 00:20:12.006 --> 00:20:14.250 so I should really do this. 00:20:14.250 --> 00:20:17.960 And this was correct to order of y squared. 00:20:17.960 --> 00:20:20.820 And in principle, one can imagine 00:20:20.820 --> 00:20:24.120 that there are configurations of four charges, quadrupole, 00:20:24.120 --> 00:20:28.730 like things, et cetera, that further modify this. 00:20:28.730 --> 00:20:33.830 And that this is a result that I have to-- oops, 00:20:33.830 --> 00:20:37.120 this was 1 minus. 00:20:37.120 --> 00:20:38.990 It was an overall factor of k. 00:20:38.990 --> 00:20:42.740 This was the correction term that we calculated. 00:20:42.740 --> 00:20:45.990 And then the size of these dipoles, 00:20:45.990 --> 00:20:50.430 we have to integrate from A to the size of the system 00:20:50.430 --> 00:20:52.340 or, if you like, infinity. 00:20:52.340 --> 00:20:58.100 And although we were attempting to make an expansion in powers 00:20:58.100 --> 00:21:02.875 of y, what we see is that because this is giving me 00:21:02.875 --> 00:21:07.760 a factor of k/r, the r has to be integrated 00:21:07.760 --> 00:21:11.000 against these three factors of rdr. 00:21:11.000 --> 00:21:15.770 Whether or not this integral is dominated by its upper cut-off, 00:21:15.770 --> 00:21:19.350 and hence divergent, depends on value of k 00:21:19.350 --> 00:21:22.560 that is related to the same divergence 00:21:22.560 --> 00:21:25.010 that we have for a single vertex. 00:21:25.010 --> 00:21:31.370 So this perturbation theory is, in principle, not valid 00:21:31.370 --> 00:21:35.580 no matter how much y I try to make small as one 00:21:35.580 --> 00:21:40.710 long as my k inverse is greater than pi/2. 00:21:40.710 --> 00:21:47.130 So what we decided to do was not to do this entire integration 00:21:47.130 --> 00:21:52.150 that gives us infinity, but rather to recast this 00:21:52.150 --> 00:22:02.160 as a re-normalization group in which core size is 00:22:02.160 --> 00:22:08.044 changed from a to beta. 00:22:13.890 --> 00:22:17.310 Now one way to see the effect of it-- last time, 00:22:17.310 --> 00:22:19.400 I did this slightly differently-- 00:22:19.400 --> 00:22:26.240 is to ensure that the result for the partition function of one 00:22:26.240 --> 00:22:29.430 charge is unmodified. 00:22:29.430 --> 00:22:32.590 If I simply do this change, the weight 00:22:32.590 --> 00:22:34.820 should not change for one defect. 00:22:34.820 --> 00:22:37.060 And so clearly, you can see that there's 00:22:37.060 --> 00:22:39.090 a change in power of beta that I would 00:22:39.090 --> 00:22:45.200 that I need to compensate by changing 00:22:45.200 --> 00:22:52.012 the core energy by a factor of b to the power of 2 minus pi k. 00:22:56.930 --> 00:23:01.310 So the statement that I have for z 1, in order for z 1 00:23:01.310 --> 00:23:06.010 to be left invariant, I have to rescale the core energies 00:23:06.010 --> 00:23:08.360 by this factor. 00:23:08.360 --> 00:23:13.270 And then over here, I essentially 00:23:13.270 --> 00:23:21.480 just integrate up to a factor of da. 00:23:21.480 --> 00:23:24.930 Just get rid of those interactions. 00:23:24.930 --> 00:23:28.990 And so this becomes minus 4 pi q k 00:23:28.990 --> 00:23:34.130 squared y squared-- as we're looking 00:23:34.130 --> 00:23:43.455 at dipole contribution-- integral from a to ba of dr 00:23:43.455 --> 00:23:47.820 r cubed divided a to the fourth. 00:23:47.820 --> 00:23:54.060 a/r to the power of 2 pi k. 00:23:59.428 --> 00:24:03.243 It means that I probably made a mistake somewhere. 00:24:08.180 --> 00:24:11.165 Yeah, this has a 2 pi. 00:24:11.165 --> 00:24:16.076 I forgot the 2 pi from the definition of the log. 00:24:25.870 --> 00:24:30.070 So these are the recursion relations. 00:24:30.070 --> 00:24:38.290 So basically, this same results at large scale for the Coulomb 00:24:38.290 --> 00:24:41.600 gas can be obtained either by the theory 00:24:41.600 --> 00:24:46.620 that is parametrized by y and original k, 00:24:46.620 --> 00:24:51.110 or after going through this removal of short distance 00:24:51.110 --> 00:24:54.420 degrees of freedom by theory in which y is modified 00:24:54.420 --> 00:24:56.946 by this factor and k is modified by this factor. 00:25:00.520 --> 00:25:07.110 And as usual, we can change these recursion relations 00:25:07.110 --> 00:25:13.530 into flow equations by choosing the value of b 00:25:13.530 --> 00:25:18.190 that is very close to 1, and then essentially converting 00:25:18.190 --> 00:25:26.990 these things to y evaluated at a slightly larger than 1, 00:25:26.990 --> 00:25:30.146 and from that, constructing the y by dl. 00:25:32.920 --> 00:25:40.730 And y by dl simply becomes 2 minus pi k times y. 00:25:40.730 --> 00:25:42.860 And I can do the same thing here. 00:25:42.860 --> 00:25:45.912 This is k plus dk dl. 00:25:45.912 --> 00:25:48.690 The k part cancels. 00:25:48.690 --> 00:25:54.120 And what I will get is that dk by dl 00:25:54.120 --> 00:26:01.430 is minus 4 pi cubed k squared y squared. 00:26:01.430 --> 00:26:06.250 And actually, all I need to do is evaluate this on the shell 00:26:06.250 --> 00:26:08.470 where r equals to a, and you can see 00:26:08.470 --> 00:26:11.920 that the integral essentially gives me 1. 00:26:11.920 --> 00:26:16.090 It gives you a delta l, basically goes over here. 00:26:16.090 --> 00:26:17.950 So these are order of y squared. 00:26:23.130 --> 00:26:30.820 Actually, it is kind of better to cast results 00:26:30.820 --> 00:26:35.340 rather in terms of k, in terms of k inverse, which 00:26:35.340 --> 00:26:39.270 is kind of like a temperature variable. 00:26:39.270 --> 00:26:45.990 And then what we get is that d by dl of k inverse 00:26:45.990 --> 00:26:48.790 essentially is going to be minus 1 00:26:48.790 --> 00:26:52.330 over k squared by dl of k squared. 00:26:52.330 --> 00:26:55.770 So the minus k squared cancels, and it simply 00:26:55.770 --> 00:27:04.760 becomes 4 pi q y squared, order of y to the fourth. 00:27:04.760 --> 00:27:13.491 And divide by dl, it's actually 2 minus pi k y plus O of y. 00:27:21.290 --> 00:27:26.890 So these are the equations that describe the changing 00:27:26.890 --> 00:27:32.150 parameters under rescaling for this Coulomb gas. 00:27:32.150 --> 00:27:34.490 And so we can plot them. 00:27:34.490 --> 00:27:41.440 Essentially, we have two paramters-- y, 00:27:41.440 --> 00:27:46.170 and we have k inverse. 00:27:46.170 --> 00:27:53.650 And what we see is that k inverse, 00:27:53.650 --> 00:27:57.790 its change is always positive. 00:27:57.790 --> 00:28:00.740 So the flow should always be to the right. 00:28:03.950 --> 00:28:10.100 y, whether y decreases or decreases depends on 00:28:10.100 --> 00:28:16.970 whether I am above or below this critical value of 2/pi 00:28:16.970 --> 00:28:20.110 that we keep encountering. 00:28:20.110 --> 00:28:23.552 And in particular, what we will find 00:28:23.552 --> 00:28:31.920 is that there is a trajectory that goes into this point. 00:28:31.920 --> 00:28:37.410 And if you are to the left of that trajectory, 00:28:37.410 --> 00:28:44.140 y is getting smaller, k inverse is getting larger. 00:28:44.140 --> 00:28:46.230 And so you go like this. 00:28:46.230 --> 00:28:52.855 Eventually, you land on a point down here where 00:28:52.855 --> 00:28:54.360 y has gone to 0. 00:28:54.360 --> 00:28:58.370 And if y has gone to 0, then k inverse does not change. 00:28:58.370 --> 00:29:06.150 So you have a structure where you have a line of fixed points 00:29:06.150 --> 00:29:12.120 so any point over here is a fixed point, 00:29:12.120 --> 00:29:14.210 but it is also a stable fixed point. 00:29:14.210 --> 00:29:18.275 It is true that points that are over here, 00:29:18.275 --> 00:29:23.170 if you are exactly at y equals to 0 are fixed points. 00:29:23.170 --> 00:29:26.330 But as soon as you have a little bit of y, 00:29:26.330 --> 00:29:28.054 then they start flowing away. 00:29:31.930 --> 00:29:35.536 And essentially, the general pattern of flow 00:29:35.536 --> 00:29:36.880 is something like this. 00:29:43.400 --> 00:29:51.170 So I go back to my original xy model, 00:29:51.170 --> 00:29:55.530 and I'm at some value at low temperatures-- means that I'm 00:29:55.530 --> 00:29:57.790 down here-- but presumably, there's 00:29:57.790 --> 00:30:04.040 a finite cost for creating the core, so I may be over here. 00:30:04.040 --> 00:30:08.120 And when I go to slightly higher temperature, 00:30:08.120 --> 00:30:10.920 k inverse becomes larger. 00:30:10.920 --> 00:30:14.170 But the core energy typically becomes smaller also 00:30:14.170 --> 00:30:16.180 at lower temperatures because everything 00:30:16.180 --> 00:30:19.160 is scaled by scale by 1/kt. 00:30:19.160 --> 00:30:21.850 So as I go to higher and higher temperatures, 00:30:21.850 --> 00:30:28.960 my xy model presumably goes through some trajectory. 00:30:28.960 --> 00:30:32.780 The trajectory of changing the xy model as temperature 00:30:32.780 --> 00:30:36.460 is modified has nothing to do with [? rg ?]. 00:30:36.460 --> 00:30:48.180 So basically is xy model on increasing T. 00:30:48.180 --> 00:30:50.770 And what is happening in the xy model of increasing 00:30:50.770 --> 00:30:55.900 T is that at low temperatures, I'm at some point 00:30:55.900 --> 00:30:59.510 here, which if I look at larger and larger scales, 00:30:59.510 --> 00:31:03.320 I find that eventually I go to a place 00:31:03.320 --> 00:31:07.940 where the effective whole cost for creating vertices 00:31:07.940 --> 00:31:11.960 is so large that they are not created at all. 00:31:11.960 --> 00:31:16.950 So then I'm back to that theory that has no vertices and simply 00:31:16.950 --> 00:31:20.770 gradient squared, and I expect that correlations 00:31:20.770 --> 00:31:25.630 will be given by this power law type of form. 00:31:25.630 --> 00:31:30.180 However, at some point, I am in this region. 00:31:30.180 --> 00:31:34.606 And when I'm in this region, I find that maybe even initially 00:31:34.606 --> 00:31:40.190 the core energy goes down or y goes down. 00:31:40.190 --> 00:31:44.580 But eventually, I end up going to a regime where 00:31:44.580 --> 00:31:49.150 both y is large and effective temperature-- the inverse-- are 00:31:49.150 --> 00:31:49.980 large. 00:31:49.980 --> 00:31:55.940 So essentially anywhere here eventually at large scales, 00:31:55.940 --> 00:32:00.490 I will see that I will be creating vertices pretty much 00:32:00.490 --> 00:32:04.910 at ease and at sufficiently long, large scale. 00:32:04.910 --> 00:32:07.235 My picture should be that of a plasma 00:32:07.235 --> 00:32:09.200 in which the [? plas ?] plus and minus 00:32:09.200 --> 00:32:12.440 charges are moving around. 00:32:12.440 --> 00:32:15.510 And so then there is this transition line 00:32:15.510 --> 00:32:17.150 that separates the two regimes. 00:32:19.800 --> 00:32:23.300 So let's find the behavior of that. 00:32:23.300 --> 00:32:26.650 And clearly, what I need to do is 00:32:26.650 --> 00:32:31.180 to focus in the vicinity of this fixed point that 00:32:31.180 --> 00:32:33.602 controls the transition. 00:32:33.602 --> 00:32:36.610 That is, anything that undergoes the transition 00:32:36.610 --> 00:32:41.810 eventually comes and flows to the vicinity of this point. 00:32:41.810 --> 00:32:47.550 So what we can do is we can construct, if you like, 00:32:47.550 --> 00:32:51.910 a two dimensional blow-up of that. 00:32:51.910 --> 00:32:55.780 And what I'm going to do is to introduce 00:32:55.780 --> 00:33:04.810 a variable of x, which is k inverse minus 2/pi. 00:33:04.810 --> 00:33:09.430 Essentially, how far I have gone from this in this direction-- 00:33:09.430 --> 00:33:12.200 y, I can use as y [INAUDIBLE]. 00:33:12.200 --> 00:33:18.770 And so what we see is that my k inverse 00:33:18.770 --> 00:33:23.386 is 2/pi, my critical value. 00:33:23.386 --> 00:33:25.052 AUDIENCE: I think that should be a pi/2. 00:33:28.370 --> 00:33:31.415 PROFESSOR: k inverse this is pi/2. 00:33:31.415 --> 00:33:32.259 Thank you. 00:33:38.746 --> 00:33:44.894 Which means that this has to be pi/2. 00:33:44.894 --> 00:33:46.040 This has to be pi/2. 00:33:48.930 --> 00:33:54.195 And this I can write as pi/2, 1 plus 2x/pi. 00:33:58.000 --> 00:34:04.750 So that to lowest order in x, k is 2/pi. 00:34:04.750 --> 00:34:07.511 The inverse of this factor, which 00:34:07.511 --> 00:34:13.290 is 1 minus 2x/pi plus order of x squared. 00:34:13.290 --> 00:34:16.350 I'm expanding for small x. 00:34:16.350 --> 00:34:22.940 I put that value in here and I find that my dy by dl 00:34:22.940 --> 00:34:30.170 is now 2 minus pi times what I have over there. 00:34:30.170 --> 00:34:40.845 So it is pi times 2/pi-- so it becomes 2-- plus 4/pi x. 00:34:40.845 --> 00:34:43.730 So essentially, I have minus 4/pi squared. 00:34:43.730 --> 00:34:45.355 I multiple by pi. 00:34:45.355 --> 00:34:48.366 It becomes plus 4/pi x. 00:34:48.366 --> 00:34:50.630 Multiply by y. 00:34:50.630 --> 00:34:56.480 So this is simply a 4/pi xy. 00:34:56.480 --> 00:35:00.040 Now the point is that typically we 00:35:00.040 --> 00:35:05.000 are used to expanding in the vicinity 00:35:05.000 --> 00:35:08.160 of that important fixed point. 00:35:08.160 --> 00:35:10.750 And all the cases that we had seen so far, 00:35:10.750 --> 00:35:14.210 once we did that expression, we ended up 00:35:14.210 --> 00:35:19.510 with a linear behavior-- divide by dl plus something times y. 00:35:19.510 --> 00:35:23.040 Here we see that the vicinity of this point 00:35:23.040 --> 00:35:27.350 is clearly a quadratic type of behavior. 00:35:27.350 --> 00:35:30.810 And this quadratic behavior leads 00:35:30.810 --> 00:35:34.240 to some unusual and interesting critical behavior 00:35:34.240 --> 00:35:37.578 that we are going to explore. 00:35:37.578 --> 00:35:40.790 So let's stick with this a little bit longer. 00:35:40.790 --> 00:35:45.666 We can see that if I look at d by dl of y squared, 00:35:45.666 --> 00:35:49.700 it is going to be 2y divided by dl, 00:35:49.700 --> 00:35:52.720 so I have to multiply this by 2y, 00:35:52.720 --> 00:35:56.620 so I will get 8/pi pi xy squared. 00:35:59.510 --> 00:36:01.290 Why did I do that? 00:36:01.290 --> 00:36:12.570 It's because you can recognize his xy squared shortly. 00:36:15.580 --> 00:36:19.278 Let's go and do the x by dl. 00:36:23.400 --> 00:36:29.440 The x by dl is simply dk inverse by dl, 00:36:29.440 --> 00:36:35.170 so that is 4 pi cubed y squared. 00:36:35.170 --> 00:36:42.150 And now you can see that if I do d by dl of x squared, 00:36:42.150 --> 00:36:51.460 I will have 2x dx by dl, so I will have 8 pi cubed xy 00:36:51.460 --> 00:36:52.040 squared. 00:36:52.040 --> 00:36:58.220 So now we can recognize that these two quantities 00:36:58.220 --> 00:37:01.340 up to some factor of pi to the fourth 00:37:01.340 --> 00:37:03.960 are really the same thing. 00:37:03.960 --> 00:37:10.540 So from here, we conclude that d by dl 00:37:10.540 --> 00:37:21.070 of x squared minus pi to the fourth y squared. 00:37:21.070 --> 00:37:24.330 Essentially once I do that, I will get 0. 00:37:31.750 --> 00:37:37.120 So as I go along these trajectories, 00:37:37.120 --> 00:37:41.890 x and y are changing, but the combination 00:37:41.890 --> 00:37:46.900 x squared minus pi to the fourth y squared is not changing. 00:37:46.900 --> 00:37:50.030 So all of the trajectories that I have drawn-- at least 00:37:50.030 --> 00:37:55.690 sufficiently close at this point around which I am expanding-- 00:37:55.690 --> 00:37:58.340 correspond to lines that are x squared 00:37:58.340 --> 00:38:03.570 minus pi to the fourth y squared is some constant I'll call c. 00:38:08.190 --> 00:38:11.640 And that constant must meet whatever you started with. 00:38:11.640 --> 00:38:15.460 So if I call the trajectory here to be the combination 00:38:15.460 --> 00:38:20.150 x0 by x0-- your original values-- 00:38:20.150 --> 00:38:24.270 I can figure out what my x0 to the fourth minus pi 00:38:24.270 --> 00:38:27.030 the fourth x0 to the fourth is. 00:38:27.030 --> 00:38:30.189 And that's going to be staying constant 00:38:30.189 --> 00:38:31.355 along the entire trajectory. 00:38:33.970 --> 00:38:41.980 So these trajectories are, in fact, portions of a hyperbole. 00:38:41.980 --> 00:38:44.420 And this is the equation that you 00:38:44.420 --> 00:38:46.610 would have for a hyperbola in xy. 00:38:49.260 --> 00:38:53.340 Now clearly there are two types of hyperbole-- 00:38:53.340 --> 00:38:58.520 the ones that go like this and the ones that go like that. 00:38:58.520 --> 00:39:02.880 In fact, this one and this one are pretty much the same thing. 00:39:02.880 --> 00:39:09.400 And what distinguishes this pattern versus that pattern 00:39:09.400 --> 00:39:14.080 is whether this constant c is positive or negative 00:39:14.080 --> 00:39:18.960 because you can see that out here, 00:39:18.960 --> 00:39:24.910 ultimately you end up at the point where y has gone to 0. 00:39:24.910 --> 00:39:28.880 So depending on x positive or negative, it doesn't matter. 00:39:28.880 --> 00:39:31.680 This combination will be positive. 00:39:31.680 --> 00:39:36.277 So throughout here, what I have is that c is positive. 00:39:39.140 --> 00:39:44.060 Whereas what I have up here is c that is negative. 00:39:44.060 --> 00:39:46.335 Presumably, there is other trajectory here 00:39:46.335 --> 00:39:48.520 and down here, c is again positive. 00:39:54.170 --> 00:39:58.660 And again, if you want to ensure y over here c is negative, 00:39:58.660 --> 00:40:03.550 because over here you can see you crossed the line where 00:40:03.550 --> 00:40:06.110 x is 0, but you have some value of y. 00:40:13.890 --> 00:40:21.935 So if I were to blow up that region both as a function of x 00:40:21.935 --> 00:40:27.030 and y, well, first of all, I will 00:40:27.030 --> 00:40:31.320 have a particular set of trajectories-- the ones 00:40:31.320 --> 00:40:36.350 that end up at this important fix point, which 00:40:36.350 --> 00:40:40.340 correspond clearly to c cos to 0. 00:40:40.340 --> 00:40:44.710 So that c cos to 0 will give me two straight lines. 00:40:44.710 --> 00:40:50.030 So presumably there is this straight line, 00:40:50.030 --> 00:40:56.670 and then there is another straight line goes out there. 00:40:56.670 --> 00:41:01.320 And then I have this bunch of trajectories 00:41:01.320 --> 00:41:05.920 that are these hyperbole that end up over here. 00:41:05.920 --> 00:41:10.440 I can have hyperboles that will be going out. 00:41:10.440 --> 00:41:14.540 And then I will have hyperboles that are like this. 00:41:17.240 --> 00:41:19.230 This are all in the high temperature phase, 00:41:19.230 --> 00:41:23.038 so let's [INAUDIBLE] like this. 00:41:28.290 --> 00:41:32.500 So one thing that you immediately see 00:41:32.500 --> 00:41:36.540 is that the location of the transition that 00:41:36.540 --> 00:41:43.320 is given by this critical line when c equals to 0. 00:41:43.320 --> 00:41:47.465 So statement number one that we can get 00:41:47.465 --> 00:41:55.900 is that the transition line corresponds to c equals to 0. 00:41:55.900 --> 00:41:59.790 So solving for x as a function of y, 00:41:59.790 --> 00:42:05.580 I will get that x critical is either minus or plus pi squared 00:42:05.580 --> 00:42:06.750 y. 00:42:06.750 --> 00:42:11.060 Clearly from the figure, the solution that I want 00:42:11.060 --> 00:42:13.488 is the one that corresponds to minus. 00:42:16.260 --> 00:42:18.820 My x was k inverse minus pi/2. 00:42:21.390 --> 00:42:25.920 This is kc inverse minus pi/2. 00:42:25.920 --> 00:42:31.130 And so what I see is that kc inverse-- 00:42:31.130 --> 00:42:35.360 the correct transition temperature-- is, 00:42:35.360 --> 00:42:42.630 in fact, lower than the value of pi/2 that we had deduced, 00:42:42.630 --> 00:42:45.500 assuming that there is only a single vertex 00:42:45.500 --> 00:42:52.290 in the entire system by an amount that to lowest order 00:42:52.290 --> 00:42:56.860 is related to the core energy or core fugacity. 00:42:56.860 --> 00:42:59.680 And presumably, there are higher order terms 00:42:59.680 --> 00:43:03.420 that I haven't calculated. 00:43:03.420 --> 00:43:09.430 So this number that we have calculated 00:43:09.430 --> 00:43:11.510 by looking at a single defect, we 00:43:11.510 --> 00:43:14.140 can see that in the presence of multiple defects, 00:43:14.140 --> 00:43:17.680 starts to get lower. 00:43:17.680 --> 00:43:21.350 And this is precisely correct in the limit 00:43:21.350 --> 00:43:23.644 where y is a small quantity. 00:43:28.860 --> 00:43:32.320 Now that's a transition line. 00:43:32.320 --> 00:43:35.120 We can look to the left or to the right. 00:43:35.120 --> 00:43:37.395 Let us just look at the low temperature phase. 00:43:44.470 --> 00:43:48.820 So for the low T phase, we expect c to be negative. 00:43:53.360 --> 00:44:02.850 And I can, for example, make that explicit by writing 00:44:02.850 --> 00:44:11.950 it as-- OK, so c was x0 squared minus pi to the fourth y0 00:44:11.950 --> 00:44:14.620 squared-- what the starting parameters of the system 00:44:14.620 --> 00:44:16.180 dictate. 00:44:16.180 --> 00:44:20.585 If you are under low temperature phase such that c is negative, 00:44:20.585 --> 00:44:24.040 it means that you are at temperatures 00:44:24.040 --> 00:44:27.350 that are smaller than Tc. 00:44:27.350 --> 00:44:31.840 So let's see, T minus Tc, let's write it 00:44:31.840 --> 00:44:36.570 as Tc minus T-- would be positive. 00:44:36.570 --> 00:44:39.150 But this has to be negative, so let me just 00:44:39.150 --> 00:44:41.340 introduce some parameter b. 00:44:41.340 --> 00:44:43.010 It's not the same b as here. 00:44:43.010 --> 00:44:47.030 Just some coefficient that has to be squared. 00:44:47.030 --> 00:44:54.420 So I know that as I hit Tc, this c goes to 0. 00:44:54.420 --> 00:44:58.820 If I'm slightly away from Tc along the trajectory 00:44:58.820 --> 00:45:04.290 that I have indicated over here, right here I'm 0, 00:45:04.290 --> 00:45:07.200 so right I'm slightly negative. 00:45:07.200 --> 00:45:11.230 And there's no reason why the value that I calculated 00:45:11.230 --> 00:45:15.330 from x0 squared minus pi to the fourth y0 squared 00:45:15.330 --> 00:45:18.010 should not be an analytical function. 00:45:18.010 --> 00:45:21.130 So I have expanded that analytical function, 00:45:21.130 --> 00:45:24.640 knowing that that Tc is equal to 0. 00:45:24.640 --> 00:45:26.680 There will be higher order terms for sure, 00:45:26.680 --> 00:45:28.910 but this is the lowest order term 00:45:28.910 --> 00:45:30.430 that I would have in that expansion. 00:45:33.100 --> 00:45:37.030 And as we said, this is preserved 00:45:37.030 --> 00:45:40.460 all along the trajectory that ends on this point. 00:45:40.460 --> 00:45:43.390 So along that trajectory, this is the same 00:45:43.390 --> 00:45:49.660 as x squared minus pi to the fourth y squared, 00:45:49.660 --> 00:45:58.330 which means that I can write y squared to be 1 over pi 00:45:58.330 --> 00:46:01.130 to the fourth. 00:46:01.130 --> 00:46:10.260 x squared plus b squared Tc minus T. So 00:46:10.260 --> 00:46:14.350 if I want to solve for this curve, that's 00:46:14.350 --> 00:46:18.520 what I will have for some value of this quantity. 00:46:18.520 --> 00:46:24.940 And what I do is I look at what that implies for the xy dl. 00:46:24.940 --> 00:46:31.700 The xy dl is 4 pi cubed y squared. 00:46:31.700 --> 00:46:35.456 I substitute the y squared that I have over there. 00:46:35.456 --> 00:46:47.760 I will get 4/pi times x squared plus b squared Tc minus T. 00:46:47.760 --> 00:46:53.920 So under rescaling, this tells me what is happening to x. 00:46:53.920 --> 00:46:59.280 And in particular, what I can do is to integrate this equation. 00:46:59.280 --> 00:47:06.730 I have dx divided by x squared plus b squared Tc minus T is 00:47:06.730 --> 00:47:12.950 4/pi dl-- just rearranging this differential equation. 00:47:12.950 --> 00:47:18.890 And this I can certainly integrate out to l. 00:47:18.890 --> 00:47:23.810 This you should recognize as the differential form 00:47:23.810 --> 00:47:32.890 of the inverse tangent up to a factor of 1 00:47:32.890 --> 00:47:37.650 over b square root of Tc minus T. 00:47:37.650 --> 00:47:39.140 So I integrate this. 00:47:39.140 --> 00:47:42.549 And on the other side, I have 4/pi l. 00:47:47.510 --> 00:48:04.970 So eventually, I know that-- that's what I wanted to do? 00:48:04.970 --> 00:48:10.060 I needed to do this later on, but we'll use it later on. 00:48:10.060 --> 00:48:16.090 What I needed to get is what is the eventual fate 00:48:16.090 --> 00:48:19.170 of this differential equation. 00:48:19.170 --> 00:48:22.530 Eventually, we see that this differential equation arrives 00:48:22.530 --> 00:48:25.715 at the point that I will call x infinity. 00:48:28.400 --> 00:48:33.380 When it arrives at x infinity, this is 0 and y is 0, 00:48:33.380 --> 00:48:35.880 so I immediately know that x infinity-- 00:48:35.880 --> 00:48:40.520 I didn't need to do any of that calculation-- this expression 00:48:40.520 --> 00:48:50.000 has to be 0, is minus the square root of Tc minus T. So let 00:48:50.000 --> 00:48:54.347 me figure out what I did with the signs that is incorrect. 00:48:54.347 --> 00:48:56.255 AUDIENCE: [INAUDIBLE] temperature [INAUDIBLE] 00:48:56.255 --> 00:48:57.209 positive [INAUDIBLE]. 00:49:00.444 --> 00:49:02.110 PROFESSOR: In the low temperature phase, 00:49:02.110 --> 00:49:07.340 I have indeed stated that c has to be positive, 00:49:07.340 --> 00:49:11.600 which means that this coefficient better be positive, 00:49:11.600 --> 00:49:16.100 which means that I would have a minus sign here. 00:49:16.100 --> 00:49:26.790 And then x would be b times Tc minus T. 00:49:26.790 --> 00:49:30.220 Right, so this would be plus or minus. 00:49:30.220 --> 00:49:34.460 The plus solution is somewhere out here, 00:49:34.460 --> 00:49:36.460 which I'm not interested. 00:49:36.460 --> 00:49:40.690 The solution that I'm interested corresponds to this value. 00:49:48.260 --> 00:49:51.230 You say, well, what is important about that? 00:49:56.545 --> 00:50:04.460 You see that various properties of this low temperature phase 00:50:04.460 --> 00:50:08.580 are characterized by this power law as opposed 00:50:08.580 --> 00:50:12.380 to exponential behavior. 00:50:12.380 --> 00:50:17.250 The power law is determined by the value of k 00:50:17.250 --> 00:50:20.070 where the description in terms of this gradient 00:50:20.070 --> 00:50:23.340 squared theory is correct. 00:50:23.340 --> 00:50:26.560 Now out here, the description is not correct 00:50:26.560 --> 00:50:30.190 because I still have the topological difference. 00:50:30.190 --> 00:50:33.360 But if I look at sufficiently large distances, 00:50:33.360 --> 00:50:37.090 I see that the topology defects have disappeared. 00:50:37.090 --> 00:50:40.420 But by the time the topological defects have disappeared, 00:50:40.420 --> 00:50:43.180 I don't have the original value of k. 00:50:43.180 --> 00:50:46.950 I have a slightly different value of k. 00:50:46.950 --> 00:50:51.340 So presumably, the properties are 00:50:51.340 --> 00:50:56.615 going to be described by what value of this k inverse 00:50:56.615 --> 00:50:59.420 is of large behaviors. 00:50:59.420 --> 00:51:07.630 And so what I expect is that the effective behavior of this k 00:51:07.630 --> 00:51:22.450 inverse-- actually, the effective behavior of k-- 00:51:22.450 --> 00:51:29.130 as a function of whatever the temperature of the system is. 00:51:29.130 --> 00:51:35.020 We expect that in the original xy model, or any system that 00:51:35.020 --> 00:51:37.950 is described by this behavior, there 00:51:37.950 --> 00:51:43.410 is a critical temperature, Tc, such that at higher 00:51:43.410 --> 00:51:47.880 temperatures, correlations are decaying exponentially. 00:51:47.880 --> 00:51:51.915 So essentially, the effective value of k has gone to 0. 00:51:51.915 --> 00:51:53.730 There is no stiffness parameter. 00:51:53.730 --> 00:51:57.120 So basically at high temperatures, 00:51:57.120 --> 00:51:59.000 you should be over here. 00:52:02.180 --> 00:52:10.450 What I see is that the effective value of k, however, other 00:52:10.450 --> 00:52:15.050 is meaningful all the way to the inverse of 2/pi. 00:52:18.890 --> 00:52:26.220 So there is a value here at 2/pi which 00:52:26.220 --> 00:52:36.330 corresponds to the largest temperature or the smallest 00:52:36.330 --> 00:52:38.930 k that is acceptable. 00:52:38.930 --> 00:52:41.870 Now what I see is that on approaching 00:52:41.870 --> 00:52:52.940 the transition, the value of k-- I have it up there-- is 2/pi. 00:52:52.940 --> 00:52:57.470 This limiting value that we have over here. 00:52:57.470 --> 00:53:05.210 And then there is a correction that is 4 over pi squared x. 00:53:05.210 --> 00:53:09.380 And presumably here, I have to put the x in infinity. 00:53:09.380 --> 00:53:12.270 And what I have for the x infinity 00:53:12.270 --> 00:53:16.720 is something like that, so I will get 2/pi plus 4 00:53:16.720 --> 00:53:23.110 e over pi squared square root of Tc minus T. 00:53:23.110 --> 00:53:32.120 So the prediction is that the effective value of k 00:53:32.120 --> 00:53:35.910 comes to its limiting value of 2/pi 00:53:35.910 --> 00:53:37.620 with a square root singularity. 00:53:42.840 --> 00:53:53.620 So we can replace the theory that 00:53:53.620 --> 00:53:58.130 describes anything that is in this universality class 00:53:58.130 --> 00:54:02.865 in the temperature phase by an effective value of k. 00:54:02.865 --> 00:54:07.680 If we then ask how does that effective value of k change 00:54:07.680 --> 00:54:10.890 as a function of temperature, the prediction 00:54:10.890 --> 00:54:13.500 is that, well, at very low temperature, 00:54:13.500 --> 00:54:17.240 it's presumably inversely related to temperature. 00:54:17.240 --> 00:54:19.780 It will come down, but [INAUDIBLE] 00:54:19.780 --> 00:54:24.280 it will change its behavior, come with a square root 00:54:24.280 --> 00:54:28.880 singularity to a number that is 2/pi, and then jump to 0. 00:54:32.950 --> 00:54:39.390 Now you are justified in saying, well, this is all very obscure. 00:54:39.390 --> 00:54:42.650 Is there any way to see this? 00:54:42.650 --> 00:54:45.730 And the answer is that people have experimentally 00:54:45.730 --> 00:54:50.120 verified this, and I'll tell you how. 00:54:50.120 --> 00:55:00.470 So a system that belongs to this universality class and we've 00:55:00.470 --> 00:55:02.895 mentioned all the way in the class is the superfluid. 00:55:06.670 --> 00:55:11.310 We've said that the superfluid transition is characterized 00:55:11.310 --> 00:55:16.940 by a quantum order parameter that applies a magnitude, 00:55:16.940 --> 00:55:19.880 but then it has a phase theta. 00:55:19.880 --> 00:55:22.350 And roughly, we would say that the phase theta should 00:55:22.350 --> 00:55:27.050 be described by this kind of theory at low temperatures. 00:55:27.050 --> 00:55:31.270 So if we want basically a two dimensional system, what 00:55:31.270 --> 00:55:35.900 we need to do is to look at the superfluid field. 00:55:35.900 --> 00:55:47.250 And this is something that Bishop and Reppy did in 1978 00:55:47.250 --> 00:55:54.760 where they constructed the analog of the Andronikashvili 00:55:54.760 --> 00:56:01.480 experiment that we mentioned in 8333, applied it to the field. 00:56:01.480 --> 00:56:06.770 So let me remind you what the Andronikashvili experiment was. 00:56:06.770 --> 00:56:14.590 Basically, you will have a torsional oscillator. 00:56:14.590 --> 00:56:23.890 This torsional oscillator was connected 00:56:23.890 --> 00:56:28.510 to a vat that had helium in it. 00:56:28.510 --> 00:56:33.880 So basically, this thing was oscillating, 00:56:33.880 --> 00:56:36.780 and the frequency of oscillations 00:56:36.780 --> 00:56:46.090 was related to some kind of a effective torsion of constant k 00:56:46.090 --> 00:56:49.880 divided by some huge mass which is 00:56:49.880 --> 00:56:52.550 contained within the cylinder. 00:56:52.550 --> 00:56:58.700 So basically you can probe classically-- you 00:56:58.700 --> 00:57:00.900 would say there's some kind of a density here. 00:57:00.900 --> 00:57:02.660 You can calculate what the mass is 00:57:02.660 --> 00:57:07.236 if you know what this is, you know what the omega is. 00:57:07.236 --> 00:57:12.630 Now what he noticed was that if this thing was filled 00:57:12.630 --> 00:57:19.330 with liquid helium and you went below Tc of helium, then 00:57:19.330 --> 00:57:22.660 suddenly this frequency changed. 00:57:22.660 --> 00:57:28.010 And the reason was that the mass that was rotating along 00:57:28.010 --> 00:57:32.060 with this whole thing was changed because the part that 00:57:32.060 --> 00:57:35.960 was superfluid was sitting still, 00:57:35.960 --> 00:57:39.800 and the normal part was the part that was oscillating. 00:57:39.800 --> 00:57:44.510 So the mass that was oscillating was reduced, 00:57:44.510 --> 00:57:46.500 frequency would go up. 00:57:46.500 --> 00:57:48.830 And from the change in frequency, 00:57:48.830 --> 00:57:54.070 he could figure out the change in the density of the part that 00:57:54.070 --> 00:57:57.210 was oscillating, and hence calculate 00:57:57.210 --> 00:58:01.350 what the density of the normal part was. 00:58:01.350 --> 00:58:07.430 So what Bishop and Reppy did was to make this two dimensional. 00:58:07.430 --> 00:58:10.220 How did they make it two dimensional? 00:58:10.220 --> 00:58:15.980 Rather than having a container of helium, what they did was 00:58:15.980 --> 00:58:19.250 they made, if you like, some kind of a toilet paper. 00:58:19.250 --> 00:58:21.770 They call it a jelly roll my Mylar. 00:58:21.770 --> 00:58:26.630 So it was Mylar that was wrapped in a cylinder. 00:58:26.630 --> 00:58:32.666 And then the helium was absorbed between the surfaces of Mylar. 00:58:32.666 --> 00:58:36.390 So effectively, it was a two dimensional system 00:58:36.390 --> 00:58:37.520 in this very setup. 00:58:40.860 --> 00:58:43.620 So for that two dimensional system, 00:58:43.620 --> 00:58:46.380 they-- again, with the same thing-- 00:58:46.380 --> 00:58:49.100 they measure the change in frequency. 00:58:49.100 --> 00:58:51.940 They found that if they go to low enough temperatures, 00:58:51.940 --> 00:58:54.730 suddenly there is a change in frequency. 00:58:54.730 --> 00:58:58.080 Of course, the temperature that they were seeing in this case 00:58:58.080 --> 00:59:01.180 was something like 1 degrees Kelvin or a fraction 00:59:01.180 --> 00:59:04.580 of 1 degree Kelvin, whereas when you have the full superfluid, 00:59:04.580 --> 00:59:09.550 it's 2.8 degrees Kelvin clearly because of that dimensionality 00:59:09.550 --> 00:59:11.500 the critical temperature changes. 00:59:11.500 --> 00:59:15.880 But you would say that's not particularly the inverse. 00:59:15.880 --> 00:59:21.760 So they could measure the change in frequency 00:59:21.760 --> 00:59:24.350 and relate the change in frequency 00:59:24.350 --> 00:59:27.845 to the density that became superfluid. 00:59:32.310 --> 00:59:36.880 Now how does the superfluid density 00:59:36.880 --> 00:59:39.750 tell us anything about this curve? 00:59:39.750 --> 00:59:43.130 Well, the answer is that everything 00:59:43.130 --> 00:59:46.300 is going to be weighted by something 00:59:46.300 --> 00:59:51.720 like e the minus beta times some energy. 00:59:51.720 --> 00:59:55.898 The one part of the energy that is associated with oscillations 00:59:55.898 --> 00:59:59.360 is certainly the kinetic energy. 00:59:59.360 --> 01:00:03.384 So let's see what we would write down for beta times 01:00:03.384 --> 01:00:10.340 the kinetic energy of superfluid or superfluid film. 01:00:10.340 --> 01:00:15.790 What I have to do is beta will give me 1/kt. 01:00:18.690 --> 01:00:22.990 The kinetic energy is obtained by integrating 01:00:22.990 --> 01:00:27.060 mass times velocity squared, or density 01:00:27.060 --> 01:00:30.725 integrated against velocity. 01:00:34.170 --> 01:00:36.400 It's a two dimensional film, so we sort of 01:00:36.400 --> 01:00:39.530 integrate as we go along the film. 01:00:39.530 --> 01:00:46.450 The superfluid velocity can be related 01:00:46.450 --> 01:00:55.140 to the mass of helium h bar and the gradient 01:00:55.140 --> 01:00:58.050 of this phase of the superconducting order 01:00:58.050 --> 01:00:59.500 parameter. 01:00:59.500 --> 01:01:04.870 So you can, for example, write your weight function as sidebar 01:01:04.870 --> 01:01:08.810 into the i theta of x, calculate what the current is using 01:01:08.810 --> 01:01:13.430 the usual formula of h bar over m psi star [? grat ?] psi 01:01:13.430 --> 01:01:15.030 [? minus ?] psi [? grat ?] psi star, 01:01:15.030 --> 01:01:19.830 and you would see that effective mass is something like this. 01:01:19.830 --> 01:01:26.753 So this is going to give me rho over kt h bar over n 01:01:26.753 --> 01:01:34.390 helium 4 squared integral gradient of theta 01:01:34.390 --> 01:01:41.580 squared, which you can see is identical to the very 01:01:41.580 --> 01:01:44.570 first line that I wrote down for you. 01:01:44.570 --> 01:01:54.314 And we can see that k can be interpreted as rho-- kt h bar 01:01:54.314 --> 01:01:57.660 over n squared. 01:02:00.210 --> 01:02:05.020 So all of these quantities-- h bar m, you know. 01:02:05.020 --> 01:02:08.080 T is the temperature that you're measuring. 01:02:08.080 --> 01:02:13.232 Rho you get through the change in this frequency. 01:02:13.232 --> 01:02:18.360 And so then they can plot what [INAUDIBLE] 01:02:18.360 --> 01:02:23.976 rho is as a function of temperature. 01:02:23.976 --> 01:02:27.490 And we see that it's very much related to k. 01:02:27.490 --> 01:02:33.550 And indeed, they find that the rho that they measure 01:02:33.550 --> 01:02:37.140 has some kind of behavior such as this. 01:02:40.110 --> 01:02:45.000 And then they go and change their Mylar, make the films 01:02:45.000 --> 01:02:46.860 thicker or whatever. 01:02:46.860 --> 01:02:49.100 They find that the transition temperature 01:02:49.100 --> 01:02:55.870 changes so that a different type of film 01:02:55.870 --> 01:02:58.295 would show a behavior such as this. 01:02:58.295 --> 01:03:02.630 And thicker films will have a higher critical temperature. 01:03:02.630 --> 01:03:06.520 They do it for a number of film thicknesses, 01:03:06.520 --> 01:03:11.860 and they got things' behavior such as this, 01:03:11.860 --> 01:03:16.390 and found that this behavior followed this gray line, which 01:03:16.390 --> 01:03:18.940 is exactly what is predicted from here. 01:03:18.940 --> 01:03:23.000 It was predicted that rho c over Tc 01:03:23.000 --> 01:03:31.690 should kb m over h bar squared times what 01:03:31.690 --> 01:03:43.090 the critical value of k is that we've calculated to be 2/pi 01:03:43.090 --> 01:03:47.725 So they could precisely check these 2/pi 01:03:47.725 --> 01:03:49.400 that we've calculated. 01:03:49.400 --> 01:03:53.210 They could more or less see this square root approach 01:03:53.210 --> 01:03:54.840 to this singularity. 01:03:54.840 --> 01:03:57.150 I'm not sure the data at that point were good 01:03:57.150 --> 01:04:00.849 enough so that they could say this exponent is precisely 1/2. 01:04:15.520 --> 01:04:20.090 So this was for the low temperature phase. 01:04:20.090 --> 01:04:22.210 What can I say about the high temperature phase? 01:04:29.830 --> 01:04:35.470 So in the high temperature phase is where my c is negative. 01:04:39.070 --> 01:04:46.480 So there I can write x0 squared minus pi 01:04:46.480 --> 01:04:52.483 to the fourth y0 squared as being a negative number, 01:04:52.483 --> 01:04:58.740 which I will write as minus b squared T minus Tc. 01:04:58.740 --> 01:05:02.710 So I have now T that is greater that Tc, multiply 01:05:02.710 --> 01:05:07.550 with some constant, and I get this. 01:05:07.550 --> 01:05:10.450 And this is the same all along the trajectory. 01:05:13.880 --> 01:05:17.190 So as I go further and x and y change, 01:05:17.190 --> 01:05:19.430 they will change in a manner that 01:05:19.430 --> 01:05:23.300 is consistent with this, which implies 01:05:23.300 --> 01:05:31.890 that as x changes with l and y changes with l, the two of them 01:05:31.890 --> 01:05:45.551 will be related by y squared is being x squared plus b squared 01:05:45.551 --> 01:05:53.190 T minus Tc divided by pi to the fourth. 01:05:53.190 --> 01:05:55.770 So this is where I don't really see 01:05:55.770 --> 01:05:57.730 the endpoint of the trajectory. 01:05:57.730 --> 01:06:01.300 I just want to see how the trajectory is behaving. 01:06:01.300 --> 01:06:06.300 So I go back to this equation, dx by dl 01:06:06.300 --> 01:06:11.700 is 4 pi cube y squared. 01:06:11.700 --> 01:06:17.120 Substitute that y squared, I will get 4/pi 1 01:06:17.120 --> 01:06:21.330 over x squared plus b squared T minus Tc. 01:06:35.040 --> 01:06:38.920 And then I rearrange this in a form 01:06:38.920 --> 01:06:40.850 I can see how to integrate. 01:06:40.850 --> 01:06:50.340 dx x squared plus b squared T minus Tc is 4/pi dl. 01:06:50.340 --> 01:06:52.660 I integrate the left-hand side. 01:06:52.660 --> 01:06:55.620 And as I already jumped ahead, it 01:06:55.620 --> 01:07:02.670 is the inverse tangent of x divided 01:07:02.670 --> 01:07:10.291 by b square root of T minus Tc is 4/pi times l. 01:07:15.190 --> 01:07:20.660 So what do I want to do with this expression? 01:07:23.570 --> 01:07:30.390 So what I want to do is to see the trajectories that 01:07:30.390 --> 01:07:34.150 just cross to the high temperature side. 01:07:34.150 --> 01:07:38.880 So I start with a point that is just slightly 01:07:38.880 --> 01:07:42.690 to the right of this transition line. 01:07:42.690 --> 01:07:46.140 Presumably, what is happening is that I 01:07:46.140 --> 01:07:51.880 will follow the transition trajectory for a long while, 01:07:51.880 --> 01:07:57.300 then I will start to head out, which 01:07:57.300 --> 01:08:02.580 means that for this trajectory, if I look 01:08:02.580 --> 01:08:07.080 at the system over larger and larger scales, initially 01:08:07.080 --> 01:08:10.500 I find that it becomes harder and harder 01:08:10.500 --> 01:08:13.730 to create these topological defects. 01:08:13.730 --> 01:08:16.920 The core energy for them becomes large. 01:08:16.920 --> 01:08:19.729 The fugacity for them becomes small. 01:08:19.729 --> 01:08:27.430 But ultimately I manage to break that, and I go to a regime 01:08:27.430 --> 01:08:30.250 where it becomes easier and easier 01:08:30.250 --> 01:08:33.189 to create these topological defects. 01:08:33.189 --> 01:08:36.969 And presumably at some point out here, everything 01:08:36.969 --> 01:08:40.540 that I have said I have to throw out because I'm making 01:08:40.540 --> 01:08:45.340 an expansion, assuming that y is small, x is small, et cetera. 01:08:45.340 --> 01:08:52.080 So presumably, as I integrate, I come to a point where I say, 01:08:52.080 --> 01:08:55.220 OK, differential equations break down. 01:08:55.220 --> 01:09:00.170 But my intuition tells me that I have reached the regime where 01:09:00.170 --> 01:09:09.189 I can create pretty much plus and minus charges at ease. 01:09:09.189 --> 01:09:14.479 So I would say that once I have reached that region where 01:09:14.479 --> 01:09:18.880 x and y have managed to escape the region where they are 01:09:18.880 --> 01:09:22.920 small, they have become of the order of 1. 01:09:22.920 --> 01:09:26.580 Maybe you can put them 1/3, 1/4-- it doesn't matter. 01:09:26.580 --> 01:09:31.890 Once they have become something that is not infinitesimal, 01:09:31.890 --> 01:09:35.740 then I can create these charges more or less at will. 01:09:35.740 --> 01:09:38.300 I will have a system where I have 01:09:38.300 --> 01:09:41.490 lots of charges that can be created at ease. 01:09:41.490 --> 01:09:45.740 And my intuition tells me that in that system, 01:09:45.740 --> 01:09:52.229 I shall have this kind of decay-- exponential decay. 01:09:52.229 --> 01:09:59.890 So how far did I have to go in order to reach that value of l? 01:09:59.890 --> 01:10:03.090 I have to go to a correlation length 01:10:03.090 --> 01:10:07.270 for a size that is larger than 1 what 01:10:07.270 --> 01:10:15.210 I started with by a factor of e to the l where the value of x 01:10:15.210 --> 01:10:19.240 became something that is of the order of 1. 01:10:19.240 --> 01:10:21.170 And actually, you can see from here 01:10:21.170 --> 01:10:24.000 that if I'm very close to the transition, 01:10:24.000 --> 01:10:26.150 it doesn't matter whether I choose here 01:10:26.150 --> 01:10:31.300 to be 1/10, 1/2, even 1/100. 01:10:31.300 --> 01:10:34.070 As long as I'm close enough to Tc, 01:10:34.070 --> 01:10:38.180 I'm dividing something by something that is close to 0. 01:10:38.180 --> 01:10:43.250 And this is tan inverse of a large number. 01:10:43.250 --> 01:10:45.810 And tan inverse of a large number 01:10:45.810 --> 01:10:48.730 is tan inverse of 90 degrees. 01:10:48.730 --> 01:10:51.590 So essentially, I go to some value 01:10:51.590 --> 01:10:58.230 of x where I can approximate this by pi over 2. 01:10:58.230 --> 01:11:01.660 And you can see that that is really insensitive to what 01:11:01.660 --> 01:11:05.600 I choose to be my x's as long as I'm sufficiently 01:11:05.600 --> 01:11:07.640 close to the critical point. 01:11:07.640 --> 01:11:09.728 So you can see that once I have done 01:11:09.728 --> 01:11:13.840 that, I have figured out what my l star is, if you like. 01:11:13.840 --> 01:11:16.360 And if I substituted that over there, 01:11:16.360 --> 01:11:27.320 I will get a behavior that is about pi/4 times pi/2 times 01:11:27.320 --> 01:11:30.060 1 over the square root of T minus Tc. 01:11:32.700 --> 01:11:39.210 Now these coefficients out front are not that important. 01:11:39.210 --> 01:11:44.080 What you see is that, indeed, we get a correlation length 01:11:44.080 --> 01:11:48.080 that, as we approach Tc, diverges, 01:11:48.080 --> 01:11:50.820 but it is not that at all of any of the forms 01:11:50.820 --> 01:11:52.090 that we had seen before. 01:11:52.090 --> 01:11:55.540 So typically, we wrote that the correlation lens diverges 01:11:55.540 --> 01:11:59.110 T minus Tc to some exponent minus mu. 01:11:59.110 --> 01:12:01.350 This is not that type of divergence. 01:12:01.350 --> 01:12:04.870 It's a very different type of divergence. 01:12:04.870 --> 01:12:09.430 And again, it's root is in the non-linear version 01:12:09.430 --> 01:12:13.300 of the recursion undulations that we have. 01:12:13.300 --> 01:12:16.420 The closest thing to this that we have 01:12:16.420 --> 01:12:19.540 is when we were calculating the correlation length 01:12:19.540 --> 01:12:23.340 for the non-linear sigma model where we had something that 01:12:23.340 --> 01:12:26.980 had a 1 over temperature type of behavior. 01:12:26.980 --> 01:12:30.770 This is even more complicated. 01:12:30.770 --> 01:12:33.848 Now once you know the singular behavior of the correlation 01:12:33.848 --> 01:12:38.660 length, you would say that in the two dimensional system, 01:12:38.660 --> 01:12:40.880 the singular part of the free energy 01:12:40.880 --> 01:12:45.510 should scale like c to the minus 2. 01:12:45.510 --> 01:12:48.900 Essentially, you break your system 01:12:48.900 --> 01:12:52.490 into pieces that are of the size correlation length. 01:12:52.490 --> 01:12:57.090 The number of those pieces is l over xi squared, 01:12:57.090 --> 01:12:58.580 because you are in two dimensions. 01:12:58.580 --> 01:12:59.970 So you would get this. 01:12:59.970 --> 01:13:04.270 So that says that your singularity of the free energy 01:13:04.270 --> 01:13:08.080 is something like, I don't know, pi squared 01:13:08.080 --> 01:13:14.640 8 4b square root of T minus Tc. 01:13:14.640 --> 01:13:17.370 Again, not a popular singularity. 01:13:17.370 --> 01:13:18.620 It's an essential singularity. 01:13:27.052 --> 01:13:32.140 And essential singularity is a kind of singular function 01:13:32.140 --> 01:13:34.860 that no matter how many derivatives 01:13:34.860 --> 01:13:40.100 you take, as T comes to Tc, there is no singularity. 01:13:40.100 --> 01:13:42.940 So for example, if I take two derivatives 01:13:42.940 --> 01:13:49.600 to get the heat capacity, what I would 01:13:49.600 --> 01:13:56.920 plot as a function of T at Tc should have no signatures. 01:13:56.920 --> 01:14:01.040 So basically what you would see because if this 01:14:01.040 --> 01:14:02.730 is that the curve just continues. 01:14:02.730 --> 01:14:06.465 There is no signature of a transition at this heat 01:14:06.465 --> 01:14:08.170 capacity. 01:14:08.170 --> 01:14:10.070 And indeed, people later on, they 01:14:10.070 --> 01:14:12.870 did numerical simulations, et cetera. 01:14:12.870 --> 01:14:15.430 What they find is that the heat capacity actually 01:14:15.430 --> 01:14:20.240 has kind of smooth peak a little bit later than Tc, which 01:14:20.240 --> 01:14:23.580 is the location where there's lots and lots of vortex 01:14:23.580 --> 01:14:25.060 unbinding going, gone. 01:14:25.060 --> 01:14:28.576 But at Tc itself, there is no signature of a singularity. 01:14:36.520 --> 01:14:43.480 As far as I know, there's no experimental case 01:14:43.480 --> 01:14:45.810 with this correlation length as we observed. 01:14:55.950 --> 01:15:06.500 So the lesson that we can take from this particular system 01:15:06.500 --> 01:15:13.300 is that two dimensional system are kind of potentially 01:15:13.300 --> 01:15:16.710 interesting and different. 01:15:16.710 --> 01:15:20.000 We had this Mermin-Wagner Theorem 01:15:20.000 --> 01:15:21.690 that we mentioned in the beginning that 01:15:21.690 --> 01:15:24.010 said that there should be no true long range 01:15:24.010 --> 01:15:25.990 order than two dimensions. 01:15:25.990 --> 01:15:27.780 That is still true. 01:15:27.780 --> 01:15:30.856 But despite that, there could be phase transitions 01:15:30.856 --> 01:15:36.240 with quite observable consequences. 01:15:36.240 --> 01:15:42.670 Add a particular type of transition in two dimension 01:15:42.670 --> 01:15:47.440 that we will pursue next lecture-- so I'll give you 01:15:47.440 --> 01:15:51.428 a preview-- is that of melting. 01:15:56.670 --> 01:16:01.210 So the prototype of a phase transition you may think of 01:16:01.210 --> 01:16:05.130 is either liquid gas or a liquid solid. 01:16:05.130 --> 01:16:09.640 And you can say, well, you have studied phase transitions 01:16:09.640 --> 01:16:11.340 to such a degree. 01:16:11.340 --> 01:16:16.470 Why not go back and talk about the melting transition, 01:16:16.470 --> 01:16:18.800 for example? 01:16:18.800 --> 01:16:24.320 The reason is that the straight melting transition is typically 01:16:24.320 --> 01:16:25.620 first order. 01:16:25.620 --> 01:16:28.670 And we've seen that universality and all of those things 01:16:28.670 --> 01:16:33.290 emerge when you have a diverging correlation length. 01:16:33.290 --> 01:16:38.130 So you want to have a place where 01:16:38.130 --> 01:16:40.420 there is a possible potential for continuous phase 01:16:40.420 --> 01:16:41.530 transition. 01:16:41.530 --> 01:16:46.140 And it turns out that melting in two dimensions provides that. 01:16:46.140 --> 01:16:48.730 So in two dimensions, you could have 01:16:48.730 --> 01:16:53.570 a bunch of points that could, for example, 01:16:53.570 --> 01:16:58.270 in a minimum energy configuration at T close to 0 01:16:58.270 --> 01:17:00.560 form a triangular lattice. 01:17:03.360 --> 01:17:05.540 Now when you go to finite temperature-- 01:17:05.540 --> 01:17:08.750 as we discussed, again, at the very first lecture-- 01:17:08.750 --> 01:17:13.990 you will start to have distortions around this. 01:17:13.990 --> 01:17:15.920 We can describe these distortions 01:17:15.920 --> 01:17:19.630 to effect of u of x and y. 01:17:19.630 --> 01:17:23.380 And then go to that appropriate continuum limit 01:17:23.380 --> 01:17:27.920 that describes the elasticity of these things. 01:17:27.920 --> 01:17:30.180 And it is going to look very much 01:17:30.180 --> 01:17:32.220 like that gradient of theta squared term 01:17:32.220 --> 01:17:35.970 that we wrote at the beginning, except that since this u is 01:17:35.970 --> 01:17:39.780 a vector, as we saw, even for an isotropic material, 01:17:39.780 --> 01:17:41.970 you will have the potential for having 01:17:41.970 --> 01:17:45.180 multiple elastic constants. 01:17:45.180 --> 01:17:49.110 But modeled on that, the conclusion that you would have 01:17:49.110 --> 01:17:53.430 is that as long as it is OK for me 01:17:53.430 --> 01:17:58.800 to make an expansion that is like the elastic theory-- 01:17:58.800 --> 01:18:03.170 some kind of a gradient of u expansion-- 01:18:03.170 --> 01:18:08.170 the conclusion would be that the correlations in u 01:18:08.170 --> 01:18:11.880 will grow logarithmically as a function of size. 01:18:11.880 --> 01:18:16.670 And you will not have too long range of order, 01:18:16.670 --> 01:18:19.640 but you will have some kind of a power of behavior, 01:18:19.640 --> 01:18:23.730 such as the one that we have indicated over there. 01:18:23.730 --> 01:18:26.290 On the other hand, when you go to very high temperature, 01:18:26.290 --> 01:18:28.612 presumably this whole thing melts. 01:18:28.612 --> 01:18:31.290 There is no reason to have correlations 01:18:31.290 --> 01:18:35.520 beyond a few atoms that are close to you. 01:18:35.520 --> 01:18:37.500 Add so typically at high temperature, 01:18:37.500 --> 01:18:41.140 correlations will decay exponentially. 01:18:41.140 --> 01:18:43.280 So this is low temperature expansion 01:18:43.280 --> 01:18:47.200 is elastic theory expansion that we have written down 01:18:47.200 --> 01:18:50.990 has to break down also in this case. 01:18:50.990 --> 01:18:55.220 And a particular mechanism for its breakdown in two dimensions 01:18:55.220 --> 01:18:58.590 is to create these topological defects, which 01:18:58.590 --> 01:19:01.890 in the case of solid will correspond to these location 01:19:01.890 --> 01:19:04.670 lines that, for example, correspond 01:19:04.670 --> 01:19:10.040 to adding an addition row of particles here terminating 01:19:10.040 --> 01:19:11.990 at some point. 01:19:11.990 --> 01:19:15.620 And we can go through exactly the same kind of story 01:19:15.620 --> 01:19:21.570 as we had before and conclude that these dislocations, 01:19:21.570 --> 01:19:24.370 because of the competition between their energy 01:19:24.370 --> 01:19:27.950 cost soaring logarithmically and their entropy gain growing 01:19:27.950 --> 01:19:31.150 logarithmically, we need to unbind 01:19:31.150 --> 01:19:33.740 at the critical temperature. 01:19:33.740 --> 01:19:38.870 And so that provides a mechanism for describing 01:19:38.870 --> 01:19:41.875 the melting of two-dimensional materials 01:19:41.875 --> 01:19:45.690 in a language that is very similar to this, 01:19:45.690 --> 01:19:51.380 except for the complications that have to do with this being 01:19:51.380 --> 01:19:55.110 a vector rather than the scale of quantity. 01:19:55.110 --> 01:19:59.650 And so what we find is that these topological charges 01:19:59.650 --> 01:20:02.920 are different from minus 2 plus 2 pi. 01:20:02.920 --> 01:20:06.930 The interactions between them is a particular version 01:20:06.930 --> 01:20:12.900 of the Coulomb interaction, but that many of the other results 01:20:12.900 --> 01:20:14.135 go through. 01:20:14.135 --> 01:20:22.820 And we will get an idea of what happens when the solid melts 01:20:22.820 --> 01:20:26.520 because of the unbinding of these locations. 01:20:26.520 --> 01:20:28.150 But there is a puzzle that we will 01:20:28.150 --> 01:20:31.530 find if we're not melting to a liquid, 01:20:31.530 --> 01:20:34.090 but into something which is more like a liquid crystal. 01:20:34.090 --> 01:20:37.670 So [INAUDIBLE] did they discover also something 01:20:37.670 --> 01:20:39.675 about liquid crystals.