1 00:00:00,090 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:21,830 --> 00:00:25,180 PROFESSOR: OK, let's start. 9 00:00:25,180 --> 00:00:31,820 So we've been looking at the xy model in two dimensions. 10 00:00:34,910 --> 00:00:40,060 It's a collection of units spins located 11 00:00:40,060 --> 00:00:44,890 on a site of potentially a lattice in two dimensions. 12 00:00:44,890 --> 00:00:48,345 Since they are unit vectors, each one of them 13 00:00:48,345 --> 00:00:52,550 is characterized by an angle theta i. 14 00:00:52,550 --> 00:00:56,190 And the partition function would be 15 00:00:56,190 --> 00:01:00,450 obtained by integrating over all of the angles. 16 00:01:03,490 --> 00:01:09,790 And the weight that we said had the form k e cosine of theta 17 00:01:09,790 --> 00:01:12,252 i minus theta j. 18 00:01:12,252 --> 00:01:15,380 So there's a coupling that corresponds 19 00:01:15,380 --> 00:01:18,480 to the dot per dot of neighboring spins, which 20 00:01:18,480 --> 00:01:22,500 can be written as this cosine form. 21 00:01:22,500 --> 00:01:33,290 If we go to low temperatures where k is large, 22 00:01:33,290 --> 00:01:36,610 then this was roughly an integral 23 00:01:36,610 --> 00:01:40,150 over all configurations of the continuous field, theta 24 00:01:40,150 --> 00:01:48,800 of x, with a weight that is the appropriate choice 25 00:01:48,800 --> 00:01:54,015 of the lattice spacing, the same k integral of gradient of theta 26 00:01:54,015 --> 00:01:54,515 squared. 27 00:02:00,860 --> 00:02:07,200 Now if we look at this weight and ask 28 00:02:07,200 --> 00:02:10,990 what is happening as a function of changing 29 00:02:10,990 --> 00:02:17,270 temperature or the inverse of this parameter 30 00:02:17,270 --> 00:02:22,670 so that we are close to zero temperature, 31 00:02:22,670 --> 00:02:26,430 if we just work with this Gaussian, 32 00:02:26,430 --> 00:02:28,240 the conclusion would be that if we 33 00:02:28,240 --> 00:02:33,330 look at the correlation between two spins that are located 34 00:02:33,330 --> 00:02:42,540 at a distance r from each other, just from this Gaussian weight, 35 00:02:42,540 --> 00:02:49,000 we found that these correlations decay as something like 1/r, 36 00:02:49,000 --> 00:02:53,250 or maybe we put some kind for a lattice spacing, 37 00:02:53,250 --> 00:02:57,270 to the power of an exponent theta that 38 00:02:57,270 --> 00:02:59,064 was related to this k. 39 00:03:03,330 --> 00:03:08,380 So the conclusion was that if that is the appropriate weight, 40 00:03:08,380 --> 00:03:13,460 we have power-law decay correlations. 41 00:03:13,460 --> 00:03:20,510 There is no long range order for correlations decay very weakly. 42 00:03:20,510 --> 00:03:24,220 On the other hand, if we start with the full cosine 43 00:03:24,220 --> 00:03:28,580 and don't do this low temperature expansion, 44 00:03:28,580 --> 00:03:33,700 just go and do the typical high temperature expansion, 45 00:03:33,700 --> 00:03:35,530 from the high temperature expansion 46 00:03:35,530 --> 00:03:37,690 we conclude that these collections decay 47 00:03:37,690 --> 00:03:46,160 exponentially, which are two totally different forms. 48 00:03:46,160 --> 00:03:48,520 And presumably, there should be some kind 49 00:03:48,520 --> 00:03:52,370 of a critical value of k or temperature 50 00:03:52,370 --> 00:03:56,169 that separates a low temperature formed with power-law decay 51 00:03:56,169 --> 00:03:58,335 and a high temperature formed with exponential decay 52 00:03:58,335 --> 00:03:59,147 of correlations. 53 00:04:01,950 --> 00:04:07,080 Now there was no sign of such a kc when 54 00:04:07,080 --> 00:04:12,710 we tried to do a low temperature expansion 55 00:04:12,710 --> 00:04:16,890 like the non-linear sigma model in this particular case 56 00:04:16,890 --> 00:04:21,480 of the xy model that corresponds to n equals to 2. 57 00:04:21,480 --> 00:04:23,980 Again, we said that the reason for that 58 00:04:23,980 --> 00:04:28,120 is that the only high order terms that I can write down 59 00:04:28,120 --> 00:04:30,990 in this theory close to low temperature gradient of theta 60 00:04:30,990 --> 00:04:35,680 to the fourth, sixth, et cetera are explicitly irrelevant. 61 00:04:35,680 --> 00:04:38,055 And unlike n equals 3, et cetera, 62 00:04:38,055 --> 00:04:40,860 it cannot cause any change. 63 00:04:40,860 --> 00:04:46,590 So as far as that theory was concerned, all of these 64 00:04:46,590 --> 00:04:48,640 corresponded to being fixed points. 65 00:04:52,450 --> 00:04:58,650 Then we said that there is a twist that is not 66 00:04:58,650 --> 00:05:02,490 taken into account when I make that transformation 67 00:05:02,490 --> 00:05:06,610 in the first line, as pointed out by Kosterlitz and Thouless. 68 00:05:13,140 --> 00:05:16,820 What you have are topological defects that are left out. 69 00:05:24,360 --> 00:05:28,346 And an example of such a defect would 70 00:05:28,346 --> 00:05:35,620 be a configuration of spins that are kind of radiating out 71 00:05:35,620 --> 00:05:41,160 from some particular point, such that when I complete 72 00:05:41,160 --> 00:05:44,920 a circuit going around, the value 73 00:05:44,920 --> 00:05:55,260 of spin changes by 2 pi for the gradient at the distance r, 74 00:05:55,260 --> 00:06:00,210 gradient of the change in angle should fall off as 1 over r. 75 00:06:00,210 --> 00:06:06,050 So if we put one of these defects 76 00:06:06,050 --> 00:06:11,680 and calculate what the partition function is for one defect, 77 00:06:11,680 --> 00:06:15,080 z of one defect, we would say, OK, 78 00:06:15,080 --> 00:06:20,130 what I have to do is to calculate the energy 79 00:06:20,130 --> 00:06:22,520 cost of this distortion. 80 00:06:22,520 --> 00:06:26,830 If I use that theory that I have over there, I have k/2. 81 00:06:26,830 --> 00:06:34,580 I have integral over all space which, because of the symmetry 82 00:06:34,580 --> 00:06:38,330 here, I can write as 2 pi r to the r. 83 00:06:41,160 --> 00:06:44,540 And then I have the gradient squared. 84 00:06:44,540 --> 00:06:48,650 And as I said, the gradient for a topological defect 85 00:06:48,650 --> 00:06:51,025 such as this goes as 1/r. 86 00:06:51,025 --> 00:06:55,540 It's square goes like 1 over r squared. 87 00:06:55,540 --> 00:07:01,940 So this is an integral that is logarithmically divergent. 88 00:07:01,940 --> 00:07:04,320 It's a an integral of 1/r. 89 00:07:04,320 --> 00:07:08,610 It's logarithmically divergent both at large distances-- 90 00:07:08,610 --> 00:07:12,500 let's say of the order of the size of the system-- 91 00:07:12,500 --> 00:07:15,250 and has difficulty at short distances. 92 00:07:15,250 --> 00:07:18,680 But at short distances, we know that there is some lattice 93 00:07:18,680 --> 00:07:23,200 structure, and this approximation will break down 94 00:07:23,200 --> 00:07:27,540 when I get to some order of some multiple lattice spacing-- so 95 00:07:27,540 --> 00:07:31,780 let's call that a-- and that the additional energy that 96 00:07:31,780 --> 00:07:36,260 comes from all of the interactions that are smaller 97 00:07:36,260 --> 00:07:40,970 than this core value of a I have to separately calculate, 98 00:07:40,970 --> 00:07:45,820 and I'm going to call them beta epsilon that 99 00:07:45,820 --> 00:07:50,120 depends on the distance a that I choose for the core. 100 00:07:53,330 --> 00:07:58,510 This is just a energy term that goes into the Boltzmann factor. 101 00:07:58,510 --> 00:08:02,670 But this defect can be placed any place 102 00:08:02,670 --> 00:08:10,070 in a lattice that has size L up to this factor of 1/a 103 00:08:10,070 --> 00:08:12,300 that I call the size of the core. 104 00:08:12,300 --> 00:08:17,450 So the number of places goes like L/a squared, the area 105 00:08:17,450 --> 00:08:19,870 that I'm looking at. 106 00:08:19,870 --> 00:08:23,940 And so we see that this expression actually 107 00:08:23,940 --> 00:08:26,820 goes like L/a. 108 00:08:26,820 --> 00:08:29,620 I have the power 2 here. 109 00:08:29,620 --> 00:08:31,920 And from this logarithmic interaction, 110 00:08:31,920 --> 00:08:37,059 I will get here power of 2 pi k log L/a, 111 00:08:37,059 --> 00:08:39,119 which I can absorb into this. 112 00:08:42,030 --> 00:08:46,840 And I define the exponential of the core energy 113 00:08:46,840 --> 00:08:50,220 to be some parameter y. 114 00:08:50,220 --> 00:08:55,805 That clearly depends on the choice of what I call my core. 115 00:08:59,260 --> 00:09:03,650 So if I look at just that expression by itself, 116 00:09:03,650 --> 00:09:08,030 I would say that there is the chance 117 00:09:08,030 --> 00:09:14,500 to see one defect or no defects as L/a becomes 118 00:09:14,500 --> 00:09:17,460 large in the thermodynamic limit depending 119 00:09:17,460 --> 00:09:21,790 on whether this exponent is positive or negative. 120 00:09:21,790 --> 00:09:27,590 So we are kind of-- oops, I forgot the factor of 2 here. 121 00:09:27,590 --> 00:09:30,960 It's 1 over pi k. 122 00:09:30,960 --> 00:09:35,760 We are led from this expression that something interesting 123 00:09:35,760 --> 00:09:40,920 should happen at some kc for one defect, which 124 00:09:40,920 --> 00:09:44,450 is let's call it the inverse of the kc. 125 00:09:44,450 --> 00:09:49,090 That is like a temperature, which is pi/2. 126 00:09:49,090 --> 00:09:52,870 So maybe around pi/2 here there should 127 00:09:52,870 --> 00:09:56,050 be something interesting that happens. 128 00:09:56,050 --> 00:10:01,750 In fact, if the theory was independent vertices, 129 00:10:01,750 --> 00:10:06,280 I would predict there would be no vertices up to here, 130 00:10:06,280 --> 00:10:09,100 and then there would be whole bunches vertices 131 00:10:09,100 --> 00:10:12,940 that would be appearing later on. 132 00:10:12,940 --> 00:10:16,710 But the point is that these are not 133 00:10:16,710 --> 00:10:20,000 vertices that don't interact with each other. 134 00:10:20,000 --> 00:10:23,150 There are interactions between them. 135 00:10:23,150 --> 00:10:30,440 And if I look at a situation with many vertices, what 136 00:10:30,440 --> 00:10:36,680 I need to do is to calculate the partition function 137 00:10:36,680 --> 00:10:40,100 for the defects, for the vertices that 138 00:10:40,100 --> 00:10:45,700 has a resemblance to the Coulomb system, so we call it z sub q. 139 00:10:45,700 --> 00:10:50,620 And this z sub q is obtained by summing over 140 00:10:50,620 --> 00:10:56,650 all numbers of these defects that can appear in the system. 141 00:10:56,650 --> 00:10:59,370 So let's do a sum over n, starting 142 00:10:59,370 --> 00:11:02,170 from 0 to however many. 143 00:11:02,170 --> 00:11:04,102 And if I have a situation in which there 144 00:11:04,102 --> 00:11:08,030 are n vertices in the system, I clearly 145 00:11:08,030 --> 00:11:15,290 have to pay a cost of y per core of each one of them. 146 00:11:15,290 --> 00:11:20,060 So there's a factor of y raised to the power of n. 147 00:11:20,060 --> 00:11:24,040 These vertices then can be placed anywhere on the lattice, 148 00:11:24,040 --> 00:11:27,660 in the same way that I had this factor of L/a 149 00:11:27,660 --> 00:11:29,880 for a single vertex. 150 00:11:29,880 --> 00:11:36,860 I will have ability to put each one of these vertices 151 00:11:36,860 --> 00:11:38,150 at some point on the lattice. 152 00:11:41,080 --> 00:11:47,070 And then we found that when you do the calculation, 153 00:11:47,070 --> 00:11:50,870 the distortions that are caused by the independent vertices 154 00:11:50,870 --> 00:11:53,520 clearly add up on top of each other. 155 00:11:53,520 --> 00:11:56,260 And when we add it up and superimpose 156 00:11:56,260 --> 00:11:58,460 the gradient of thetas that correspond 157 00:11:58,460 --> 00:12:02,190 to the different vertices and calculated 158 00:12:02,190 --> 00:12:05,800 the integral of gradient of theta squared, what we found 159 00:12:05,800 --> 00:12:09,530 was that there was an interaction between them 160 00:12:09,530 --> 00:12:14,800 that I could write as 4 pi squared 161 00:12:14,800 --> 00:12:23,760 k sum over distinct pairs, i less than j, qi qj. 162 00:12:23,760 --> 00:12:27,150 The Coulomb interaction [? it fits inside ?] xi and xj. 163 00:12:31,820 --> 00:12:38,490 And actually, the form of this came about as follows. 164 00:12:38,490 --> 00:12:45,670 Those That basically the charges of these topological defects 165 00:12:45,670 --> 00:12:50,530 are multiples of 2 pi, but these qi's I have written 166 00:12:50,530 --> 00:12:52,790 are minus plus 1. 167 00:12:52,790 --> 00:12:56,480 So the two pis are absorbed in the charges. 168 00:12:56,480 --> 00:13:02,310 The actual charges being 2 pi is what makes this 4 pi squared. 169 00:13:02,310 --> 00:13:05,190 k is the [INAUDIBLE] of interaction. 170 00:13:05,190 --> 00:13:08,320 Furthermore, we have to require the system 171 00:13:08,320 --> 00:13:15,910 to be overall neutral because, otherwise, there 172 00:13:15,910 --> 00:13:18,740 would be a large energy for creating 173 00:13:18,740 --> 00:13:22,860 the monopole in a large system. 174 00:13:22,860 --> 00:13:26,590 And again, just as a matter of notation, 175 00:13:26,590 --> 00:13:32,380 our z of x has a 1 over 2 pi itself 176 00:13:32,380 --> 00:13:37,330 log of the displacement in units of this a 177 00:13:37,330 --> 00:13:39,920 because we can't allow these things to come 178 00:13:39,920 --> 00:13:41,340 very close to each other. 179 00:13:44,070 --> 00:13:53,650 So our task was to calculate properties encoded 180 00:13:53,650 --> 00:13:58,890 in this partition function, which is, in some sense, 181 00:13:58,890 --> 00:14:01,965 a grand canonical system of charges 182 00:14:01,965 --> 00:14:04,570 that can appear and disappear. 183 00:14:04,570 --> 00:14:11,620 And our expectation is that at the low temperatures, 184 00:14:11,620 --> 00:14:19,930 essentially all I have are a few dipoles that are kind of small. 185 00:14:19,930 --> 00:14:23,600 As I go to higher temperature, the two monopoles 186 00:14:23,600 --> 00:14:28,710 making the dipole can fluctuate and go further from each other. 187 00:14:28,710 --> 00:14:32,670 And eventually at some point, they will be all mixed 188 00:14:32,670 --> 00:14:34,810 up together and the picture should 189 00:14:34,810 --> 00:14:38,617 be regarded as a mixture of plus and minus charges 190 00:14:38,617 --> 00:14:39,200 in the plasma. 191 00:14:39,200 --> 00:14:40,156 Yes? 192 00:14:40,156 --> 00:14:43,396 AUDIENCE: [INAUDIBLE] if we have an external field, 193 00:14:43,396 --> 00:14:48,760 would this also be fixed [INAUDIBLE], like 194 00:14:48,760 --> 00:14:51,230 and edge or something external? 195 00:14:56,190 --> 00:14:58,130 PROFESSOR: It is very hard to imagine 196 00:14:58,130 --> 00:15:01,930 what that external field has to be in the language of the xy 197 00:15:01,930 --> 00:15:04,370 model, because what you can do is 198 00:15:04,370 --> 00:15:06,360 you can put a field that, let's say, 199 00:15:06,360 --> 00:15:11,410 rotates the spins on one side-- let's say to point down-- 200 00:15:11,410 --> 00:15:14,260 spins on the other side to point up. 201 00:15:14,260 --> 00:15:17,080 But then what happens is that the angles 202 00:15:17,080 --> 00:15:20,750 would adjust themselves so that at 0 temperature, 203 00:15:20,750 --> 00:15:22,580 you would have a configuration that 204 00:15:22,580 --> 00:15:27,510 would go from plus to minus, and all topological charges would 205 00:15:27,510 --> 00:15:31,170 be on top of that base configuration. 206 00:15:31,170 --> 00:15:35,830 So that kind of field certainly does not have any effect 207 00:15:35,830 --> 00:15:38,420 that I can ascribe over here. 208 00:15:38,420 --> 00:15:40,035 If I change my picture completely 209 00:15:40,035 --> 00:15:42,860 and say forget about the xy model, 210 00:15:42,860 --> 00:15:46,270 think about this as a system of point charges, 211 00:15:46,270 --> 00:15:49,290 then I can certainly, like I did last time, 212 00:15:49,290 --> 00:15:53,682 put an electric field on the system and see what happens. 213 00:15:53,682 --> 00:15:54,650 Yes? 214 00:15:54,650 --> 00:15:56,206 AUDIENCE: [INAUDIBLE] saying that you 215 00:15:56,206 --> 00:15:58,550 can create even bigger defects, where 216 00:15:58,550 --> 00:16:00,410 q would be [INAUDIBLE] plus minus 1, 217 00:16:00,410 --> 00:16:02,120 plus minus bigger integer? 218 00:16:02,120 --> 00:16:05,166 But that's [? discounted ?] as high [INAUDIBLE] effect. 219 00:16:05,166 --> 00:16:05,790 PROFESSOR: Yes. 220 00:16:05,790 --> 00:16:08,710 So intrinsically, we could go beyond that. 221 00:16:08,710 --> 00:16:12,930 We would have a fugacity for a creation of cores 222 00:16:12,930 --> 00:16:15,560 of singular charge, another for cores 223 00:16:15,560 --> 00:16:17,770 of double charge, et cetera. 224 00:16:17,770 --> 00:16:20,710 You expect those y's or double charges 225 00:16:20,710 --> 00:16:23,475 to be much larger because the configuration is going 226 00:16:23,475 --> 00:16:25,015 to be more difficult at the core. 227 00:16:27,870 --> 00:16:30,230 And in some sense, you can imagine 228 00:16:30,230 --> 00:16:33,550 that we are including something similar to that 229 00:16:33,550 --> 00:16:37,700 because we can create two single charges that 230 00:16:37,700 --> 00:16:40,820 are closing off to each other. 231 00:16:40,820 --> 00:16:41,320 Yes? 232 00:16:41,320 --> 00:16:44,470 AUDIENCE: So why is a Coulomb state [INAUDIBLE]? 233 00:16:47,490 --> 00:16:48,308 Is that high order? 234 00:16:51,000 --> 00:16:52,800 PROFESSOR: Well, you would expect 235 00:16:52,800 --> 00:16:56,800 that if y is a small parameter, you 236 00:16:56,800 --> 00:17:00,120 would like to create as few things as possible. 237 00:17:00,120 --> 00:17:05,310 The reason you create any is because you have entropy gain. 238 00:17:05,310 --> 00:17:07,800 So I would say energetically, even 239 00:17:07,800 --> 00:17:10,329 creating a pair is unfavorable. 240 00:17:10,329 --> 00:17:13,180 But the pair has lots of places that it can go, 241 00:17:13,180 --> 00:17:15,230 so because of the gain in entropy, 242 00:17:15,230 --> 00:17:17,109 it is willing to accept that. 243 00:17:17,109 --> 00:17:19,400 If I create a quadrupole, you say, 244 00:17:19,400 --> 00:17:21,940 well, I break the quadrupole into two dipoles 245 00:17:21,940 --> 00:17:23,819 and then I have much more entropy. 246 00:17:23,819 --> 00:17:28,119 So that's why it's not-- we can have that term, 247 00:17:28,119 --> 00:17:30,772 but it is going to have much less weight. 248 00:17:41,220 --> 00:17:45,310 So I won't repeat the calculation, 249 00:17:45,310 --> 00:17:50,980 but last time we indeed asked what happens if we have, 250 00:17:50,980 --> 00:17:55,500 let's say, some kind of an electric field. 251 00:17:55,500 --> 00:17:58,240 And because of the presence of the electric field, 252 00:17:58,240 --> 00:18:02,280 dipoles are going to be aligned. 253 00:18:02,280 --> 00:18:08,560 And the effect of that is to reduce the effective strength 254 00:18:08,560 --> 00:18:11,550 of all kinds of Coulomb interactions. 255 00:18:11,550 --> 00:18:18,190 We found that the effective strength was reduced by from k 256 00:18:18,190 --> 00:18:25,150 by an amount that was related to the likelihood of creating 257 00:18:25,150 --> 00:18:28,430 the dipole of size r. 258 00:18:28,430 --> 00:18:32,660 And that was clearly proportional to y squared 259 00:18:32,660 --> 00:18:44,710 into the minus from there, 4 pi squared q log of r/a. 260 00:18:44,710 --> 00:18:48,553 That's the co-ability we create a dipole of this size. 261 00:18:51,480 --> 00:18:57,580 And then I have to, in principle, 262 00:18:57,580 --> 00:19:00,260 integrate over all dipole sizes. 263 00:19:03,470 --> 00:19:08,160 But this writing this as an orientationally independent 264 00:19:08,160 --> 00:19:11,730 result is not correct because in the presence 265 00:19:11,730 --> 00:19:14,250 of an electric field, you have more likelihood 266 00:19:14,250 --> 00:19:16,760 to be oriented in one direction as opposed 267 00:19:16,760 --> 00:19:18,450 to the other direction. 268 00:19:18,450 --> 00:19:21,340 So this factor of e to the cosine theta, 269 00:19:21,340 --> 00:19:24,940 et cetera, that we expanded, first of all, 270 00:19:24,940 --> 00:19:27,820 gave us an average of cosine of theta squared. 271 00:19:27,820 --> 00:19:30,300 So there was a factor of 1/2 here. 272 00:19:30,300 --> 00:19:33,660 Rather than full rotation, I was doing an average 273 00:19:33,660 --> 00:19:38,320 of cosine squared to be that factor. 274 00:19:38,320 --> 00:19:42,910 Expanding that actually gave me a factor of 4 pi 275 00:19:42,910 --> 00:19:48,220 squared k because of the Coulomb term that I had up there. 276 00:19:48,220 --> 00:19:54,320 And then I had essentially the polarizability of one 277 00:19:54,320 --> 00:19:59,410 of these objects that goes like r squared. 278 00:19:59,410 --> 00:20:03,320 Again, coming from expanding this factor 279 00:20:03,320 --> 00:20:05,800 that we had in the exponents. 280 00:20:05,800 --> 00:20:12,006 Actually, I calculated everything in units of A, 281 00:20:12,006 --> 00:20:14,250 so I should really do this. 282 00:20:14,250 --> 00:20:17,960 And this was correct to order of y squared. 283 00:20:17,960 --> 00:20:20,820 And in principle, one can imagine 284 00:20:20,820 --> 00:20:24,120 that there are configurations of four charges, quadrupole, 285 00:20:24,120 --> 00:20:28,730 like things, et cetera, that further modify this. 286 00:20:28,730 --> 00:20:33,830 And that this is a result that I have to-- oops, 287 00:20:33,830 --> 00:20:37,120 this was 1 minus. 288 00:20:37,120 --> 00:20:38,990 It was an overall factor of k. 289 00:20:38,990 --> 00:20:42,740 This was the correction term that we calculated. 290 00:20:42,740 --> 00:20:45,990 And then the size of these dipoles, 291 00:20:45,990 --> 00:20:50,430 we have to integrate from A to the size of the system 292 00:20:50,430 --> 00:20:52,340 or, if you like, infinity. 293 00:20:52,340 --> 00:20:58,100 And although we were attempting to make an expansion in powers 294 00:20:58,100 --> 00:21:02,875 of y, what we see is that because this is giving me 295 00:21:02,875 --> 00:21:07,760 a factor of k/r, the r has to be integrated 296 00:21:07,760 --> 00:21:11,000 against these three factors of rdr. 297 00:21:11,000 --> 00:21:15,770 Whether or not this integral is dominated by its upper cut-off, 298 00:21:15,770 --> 00:21:19,350 and hence divergent, depends on value of k 299 00:21:19,350 --> 00:21:22,560 that is related to the same divergence 300 00:21:22,560 --> 00:21:25,010 that we have for a single vertex. 301 00:21:25,010 --> 00:21:31,370 So this perturbation theory is, in principle, not valid 302 00:21:31,370 --> 00:21:35,580 no matter how much y I try to make small as one 303 00:21:35,580 --> 00:21:40,710 long as my k inverse is greater than pi/2. 304 00:21:40,710 --> 00:21:47,130 So what we decided to do was not to do this entire integration 305 00:21:47,130 --> 00:21:52,150 that gives us infinity, but rather to recast this 306 00:21:52,150 --> 00:22:02,160 as a re-normalization group in which core size is 307 00:22:02,160 --> 00:22:08,044 changed from a to beta. 308 00:22:13,890 --> 00:22:17,310 Now one way to see the effect of it-- last time, 309 00:22:17,310 --> 00:22:19,400 I did this slightly differently-- 310 00:22:19,400 --> 00:22:26,240 is to ensure that the result for the partition function of one 311 00:22:26,240 --> 00:22:29,430 charge is unmodified. 312 00:22:29,430 --> 00:22:32,590 If I simply do this change, the weight 313 00:22:32,590 --> 00:22:34,820 should not change for one defect. 314 00:22:34,820 --> 00:22:37,060 And so clearly, you can see that there's 315 00:22:37,060 --> 00:22:39,090 a change in power of beta that I would 316 00:22:39,090 --> 00:22:45,200 that I need to compensate by changing 317 00:22:45,200 --> 00:22:52,012 the core energy by a factor of b to the power of 2 minus pi k. 318 00:22:56,930 --> 00:23:01,310 So the statement that I have for z 1, in order for z 1 319 00:23:01,310 --> 00:23:06,010 to be left invariant, I have to rescale the core energies 320 00:23:06,010 --> 00:23:08,360 by this factor. 321 00:23:08,360 --> 00:23:13,270 And then over here, I essentially 322 00:23:13,270 --> 00:23:21,480 just integrate up to a factor of da. 323 00:23:21,480 --> 00:23:24,930 Just get rid of those interactions. 324 00:23:24,930 --> 00:23:28,990 And so this becomes minus 4 pi q k 325 00:23:28,990 --> 00:23:34,130 squared y squared-- as we're looking 326 00:23:34,130 --> 00:23:43,455 at dipole contribution-- integral from a to ba of dr 327 00:23:43,455 --> 00:23:47,820 r cubed divided a to the fourth. 328 00:23:47,820 --> 00:23:54,060 a/r to the power of 2 pi k. 329 00:23:59,428 --> 00:24:03,243 It means that I probably made a mistake somewhere. 330 00:24:08,180 --> 00:24:11,165 Yeah, this has a 2 pi. 331 00:24:11,165 --> 00:24:16,076 I forgot the 2 pi from the definition of the log. 332 00:24:25,870 --> 00:24:30,070 So these are the recursion relations. 333 00:24:30,070 --> 00:24:38,290 So basically, this same results at large scale for the Coulomb 334 00:24:38,290 --> 00:24:41,600 gas can be obtained either by the theory 335 00:24:41,600 --> 00:24:46,620 that is parametrized by y and original k, 336 00:24:46,620 --> 00:24:51,110 or after going through this removal of short distance 337 00:24:51,110 --> 00:24:54,420 degrees of freedom by theory in which y is modified 338 00:24:54,420 --> 00:24:56,946 by this factor and k is modified by this factor. 339 00:25:00,520 --> 00:25:07,110 And as usual, we can change these recursion relations 340 00:25:07,110 --> 00:25:13,530 into flow equations by choosing the value of b 341 00:25:13,530 --> 00:25:18,190 that is very close to 1, and then essentially converting 342 00:25:18,190 --> 00:25:26,990 these things to y evaluated at a slightly larger than 1, 343 00:25:26,990 --> 00:25:30,146 and from that, constructing the y by dl. 344 00:25:32,920 --> 00:25:40,730 And y by dl simply becomes 2 minus pi k times y. 345 00:25:40,730 --> 00:25:42,860 And I can do the same thing here. 346 00:25:42,860 --> 00:25:45,912 This is k plus dk dl. 347 00:25:45,912 --> 00:25:48,690 The k part cancels. 348 00:25:48,690 --> 00:25:54,120 And what I will get is that dk by dl 349 00:25:54,120 --> 00:26:01,430 is minus 4 pi cubed k squared y squared. 350 00:26:01,430 --> 00:26:06,250 And actually, all I need to do is evaluate this on the shell 351 00:26:06,250 --> 00:26:08,470 where r equals to a, and you can see 352 00:26:08,470 --> 00:26:11,920 that the integral essentially gives me 1. 353 00:26:11,920 --> 00:26:16,090 It gives you a delta l, basically goes over here. 354 00:26:16,090 --> 00:26:17,950 So these are order of y squared. 355 00:26:23,130 --> 00:26:30,820 Actually, it is kind of better to cast results 356 00:26:30,820 --> 00:26:35,340 rather in terms of k, in terms of k inverse, which 357 00:26:35,340 --> 00:26:39,270 is kind of like a temperature variable. 358 00:26:39,270 --> 00:26:45,990 And then what we get is that d by dl of k inverse 359 00:26:45,990 --> 00:26:48,790 essentially is going to be minus 1 360 00:26:48,790 --> 00:26:52,330 over k squared by dl of k squared. 361 00:26:52,330 --> 00:26:55,770 So the minus k squared cancels, and it simply 362 00:26:55,770 --> 00:27:04,760 becomes 4 pi q y squared, order of y to the fourth. 363 00:27:04,760 --> 00:27:13,491 And divide by dl, it's actually 2 minus pi k y plus O of y. 364 00:27:21,290 --> 00:27:26,890 So these are the equations that describe the changing 365 00:27:26,890 --> 00:27:32,150 parameters under rescaling for this Coulomb gas. 366 00:27:32,150 --> 00:27:34,490 And so we can plot them. 367 00:27:34,490 --> 00:27:41,440 Essentially, we have two paramters-- y, 368 00:27:41,440 --> 00:27:46,170 and we have k inverse. 369 00:27:46,170 --> 00:27:53,650 And what we see is that k inverse, 370 00:27:53,650 --> 00:27:57,790 its change is always positive. 371 00:27:57,790 --> 00:28:00,740 So the flow should always be to the right. 372 00:28:03,950 --> 00:28:10,100 y, whether y decreases or decreases depends on 373 00:28:10,100 --> 00:28:16,970 whether I am above or below this critical value of 2/pi 374 00:28:16,970 --> 00:28:20,110 that we keep encountering. 375 00:28:20,110 --> 00:28:23,552 And in particular, what we will find 376 00:28:23,552 --> 00:28:31,920 is that there is a trajectory that goes into this point. 377 00:28:31,920 --> 00:28:37,410 And if you are to the left of that trajectory, 378 00:28:37,410 --> 00:28:44,140 y is getting smaller, k inverse is getting larger. 379 00:28:44,140 --> 00:28:46,230 And so you go like this. 380 00:28:46,230 --> 00:28:52,855 Eventually, you land on a point down here where 381 00:28:52,855 --> 00:28:54,360 y has gone to 0. 382 00:28:54,360 --> 00:28:58,370 And if y has gone to 0, then k inverse does not change. 383 00:28:58,370 --> 00:29:06,150 So you have a structure where you have a line of fixed points 384 00:29:06,150 --> 00:29:12,120 so any point over here is a fixed point, 385 00:29:12,120 --> 00:29:14,210 but it is also a stable fixed point. 386 00:29:14,210 --> 00:29:18,275 It is true that points that are over here, 387 00:29:18,275 --> 00:29:23,170 if you are exactly at y equals to 0 are fixed points. 388 00:29:23,170 --> 00:29:26,330 But as soon as you have a little bit of y, 389 00:29:26,330 --> 00:29:28,054 then they start flowing away. 390 00:29:31,930 --> 00:29:35,536 And essentially, the general pattern of flow 391 00:29:35,536 --> 00:29:36,880 is something like this. 392 00:29:43,400 --> 00:29:51,170 So I go back to my original xy model, 393 00:29:51,170 --> 00:29:55,530 and I'm at some value at low temperatures-- means that I'm 394 00:29:55,530 --> 00:29:57,790 down here-- but presumably, there's 395 00:29:57,790 --> 00:30:04,040 a finite cost for creating the core, so I may be over here. 396 00:30:04,040 --> 00:30:08,120 And when I go to slightly higher temperature, 397 00:30:08,120 --> 00:30:10,920 k inverse becomes larger. 398 00:30:10,920 --> 00:30:14,170 But the core energy typically becomes smaller also 399 00:30:14,170 --> 00:30:16,180 at lower temperatures because everything 400 00:30:16,180 --> 00:30:19,160 is scaled by scale by 1/kt. 401 00:30:19,160 --> 00:30:21,850 So as I go to higher and higher temperatures, 402 00:30:21,850 --> 00:30:28,960 my xy model presumably goes through some trajectory. 403 00:30:28,960 --> 00:30:32,780 The trajectory of changing the xy model as temperature 404 00:30:32,780 --> 00:30:36,460 is modified has nothing to do with [? rg ?]. 405 00:30:36,460 --> 00:30:48,180 So basically is xy model on increasing T. 406 00:30:48,180 --> 00:30:50,770 And what is happening in the xy model of increasing 407 00:30:50,770 --> 00:30:55,900 T is that at low temperatures, I'm at some point 408 00:30:55,900 --> 00:30:59,510 here, which if I look at larger and larger scales, 409 00:30:59,510 --> 00:31:03,320 I find that eventually I go to a place 410 00:31:03,320 --> 00:31:07,940 where the effective whole cost for creating vertices 411 00:31:07,940 --> 00:31:11,960 is so large that they are not created at all. 412 00:31:11,960 --> 00:31:16,950 So then I'm back to that theory that has no vertices and simply 413 00:31:16,950 --> 00:31:20,770 gradient squared, and I expect that correlations 414 00:31:20,770 --> 00:31:25,630 will be given by this power law type of form. 415 00:31:25,630 --> 00:31:30,180 However, at some point, I am in this region. 416 00:31:30,180 --> 00:31:34,606 And when I'm in this region, I find that maybe even initially 417 00:31:34,606 --> 00:31:40,190 the core energy goes down or y goes down. 418 00:31:40,190 --> 00:31:44,580 But eventually, I end up going to a regime where 419 00:31:44,580 --> 00:31:49,150 both y is large and effective temperature-- the inverse-- are 420 00:31:49,150 --> 00:31:49,980 large. 421 00:31:49,980 --> 00:31:55,940 So essentially anywhere here eventually at large scales, 422 00:31:55,940 --> 00:32:00,490 I will see that I will be creating vertices pretty much 423 00:32:00,490 --> 00:32:04,910 at ease and at sufficiently long, large scale. 424 00:32:04,910 --> 00:32:07,235 My picture should be that of a plasma 425 00:32:07,235 --> 00:32:09,200 in which the [? plas ?] plus and minus 426 00:32:09,200 --> 00:32:12,440 charges are moving around. 427 00:32:12,440 --> 00:32:15,510 And so then there is this transition line 428 00:32:15,510 --> 00:32:17,150 that separates the two regimes. 429 00:32:19,800 --> 00:32:23,300 So let's find the behavior of that. 430 00:32:23,300 --> 00:32:26,650 And clearly, what I need to do is 431 00:32:26,650 --> 00:32:31,180 to focus in the vicinity of this fixed point that 432 00:32:31,180 --> 00:32:33,602 controls the transition. 433 00:32:33,602 --> 00:32:36,610 That is, anything that undergoes the transition 434 00:32:36,610 --> 00:32:41,810 eventually comes and flows to the vicinity of this point. 435 00:32:41,810 --> 00:32:47,550 So what we can do is we can construct, if you like, 436 00:32:47,550 --> 00:32:51,910 a two dimensional blow-up of that. 437 00:32:51,910 --> 00:32:55,780 And what I'm going to do is to introduce 438 00:32:55,780 --> 00:33:04,810 a variable of x, which is k inverse minus 2/pi. 439 00:33:04,810 --> 00:33:09,430 Essentially, how far I have gone from this in this direction-- 440 00:33:09,430 --> 00:33:12,200 y, I can use as y [INAUDIBLE]. 441 00:33:12,200 --> 00:33:18,770 And so what we see is that my k inverse 442 00:33:18,770 --> 00:33:23,386 is 2/pi, my critical value. 443 00:33:23,386 --> 00:33:25,052 AUDIENCE: I think that should be a pi/2. 444 00:33:28,370 --> 00:33:31,415 PROFESSOR: k inverse this is pi/2. 445 00:33:31,415 --> 00:33:32,259 Thank you. 446 00:33:38,746 --> 00:33:44,894 Which means that this has to be pi/2. 447 00:33:44,894 --> 00:33:46,040 This has to be pi/2. 448 00:33:48,930 --> 00:33:54,195 And this I can write as pi/2, 1 plus 2x/pi. 449 00:33:58,000 --> 00:34:04,750 So that to lowest order in x, k is 2/pi. 450 00:34:04,750 --> 00:34:07,511 The inverse of this factor, which 451 00:34:07,511 --> 00:34:13,290 is 1 minus 2x/pi plus order of x squared. 452 00:34:13,290 --> 00:34:16,350 I'm expanding for small x. 453 00:34:16,350 --> 00:34:22,940 I put that value in here and I find that my dy by dl 454 00:34:22,940 --> 00:34:30,170 is now 2 minus pi times what I have over there. 455 00:34:30,170 --> 00:34:40,845 So it is pi times 2/pi-- so it becomes 2-- plus 4/pi x. 456 00:34:40,845 --> 00:34:43,730 So essentially, I have minus 4/pi squared. 457 00:34:43,730 --> 00:34:45,355 I multiple by pi. 458 00:34:45,355 --> 00:34:48,366 It becomes plus 4/pi x. 459 00:34:48,366 --> 00:34:50,630 Multiply by y. 460 00:34:50,630 --> 00:34:56,480 So this is simply a 4/pi xy. 461 00:34:56,480 --> 00:35:00,040 Now the point is that typically we 462 00:35:00,040 --> 00:35:05,000 are used to expanding in the vicinity 463 00:35:05,000 --> 00:35:08,160 of that important fixed point. 464 00:35:08,160 --> 00:35:10,750 And all the cases that we had seen so far, 465 00:35:10,750 --> 00:35:14,210 once we did that expression, we ended up 466 00:35:14,210 --> 00:35:19,510 with a linear behavior-- divide by dl plus something times y. 467 00:35:19,510 --> 00:35:23,040 Here we see that the vicinity of this point 468 00:35:23,040 --> 00:35:27,350 is clearly a quadratic type of behavior. 469 00:35:27,350 --> 00:35:30,810 And this quadratic behavior leads 470 00:35:30,810 --> 00:35:34,240 to some unusual and interesting critical behavior 471 00:35:34,240 --> 00:35:37,578 that we are going to explore. 472 00:35:37,578 --> 00:35:40,790 So let's stick with this a little bit longer. 473 00:35:40,790 --> 00:35:45,666 We can see that if I look at d by dl of y squared, 474 00:35:45,666 --> 00:35:49,700 it is going to be 2y divided by dl, 475 00:35:49,700 --> 00:35:52,720 so I have to multiply this by 2y, 476 00:35:52,720 --> 00:35:56,620 so I will get 8/pi pi xy squared. 477 00:35:59,510 --> 00:36:01,290 Why did I do that? 478 00:36:01,290 --> 00:36:12,570 It's because you can recognize his xy squared shortly. 479 00:36:15,580 --> 00:36:19,278 Let's go and do the x by dl. 480 00:36:23,400 --> 00:36:29,440 The x by dl is simply dk inverse by dl, 481 00:36:29,440 --> 00:36:35,170 so that is 4 pi cubed y squared. 482 00:36:35,170 --> 00:36:42,150 And now you can see that if I do d by dl of x squared, 483 00:36:42,150 --> 00:36:51,460 I will have 2x dx by dl, so I will have 8 pi cubed xy 484 00:36:51,460 --> 00:36:52,040 squared. 485 00:36:52,040 --> 00:36:58,220 So now we can recognize that these two quantities 486 00:36:58,220 --> 00:37:01,340 up to some factor of pi to the fourth 487 00:37:01,340 --> 00:37:03,960 are really the same thing. 488 00:37:03,960 --> 00:37:10,540 So from here, we conclude that d by dl 489 00:37:10,540 --> 00:37:21,070 of x squared minus pi to the fourth y squared. 490 00:37:21,070 --> 00:37:24,330 Essentially once I do that, I will get 0. 491 00:37:31,750 --> 00:37:37,120 So as I go along these trajectories, 492 00:37:37,120 --> 00:37:41,890 x and y are changing, but the combination 493 00:37:41,890 --> 00:37:46,900 x squared minus pi to the fourth y squared is not changing. 494 00:37:46,900 --> 00:37:50,030 So all of the trajectories that I have drawn-- at least 495 00:37:50,030 --> 00:37:55,690 sufficiently close at this point around which I am expanding-- 496 00:37:55,690 --> 00:37:58,340 correspond to lines that are x squared 497 00:37:58,340 --> 00:38:03,570 minus pi to the fourth y squared is some constant I'll call c. 498 00:38:08,190 --> 00:38:11,640 And that constant must meet whatever you started with. 499 00:38:11,640 --> 00:38:15,460 So if I call the trajectory here to be the combination 500 00:38:15,460 --> 00:38:20,150 x0 by x0-- your original values-- 501 00:38:20,150 --> 00:38:24,270 I can figure out what my x0 to the fourth minus pi 502 00:38:24,270 --> 00:38:27,030 the fourth x0 to the fourth is. 503 00:38:27,030 --> 00:38:30,189 And that's going to be staying constant 504 00:38:30,189 --> 00:38:31,355 along the entire trajectory. 505 00:38:33,970 --> 00:38:41,980 So these trajectories are, in fact, portions of a hyperbole. 506 00:38:41,980 --> 00:38:44,420 And this is the equation that you 507 00:38:44,420 --> 00:38:46,610 would have for a hyperbola in xy. 508 00:38:49,260 --> 00:38:53,340 Now clearly there are two types of hyperbole-- 509 00:38:53,340 --> 00:38:58,520 the ones that go like this and the ones that go like that. 510 00:38:58,520 --> 00:39:02,880 In fact, this one and this one are pretty much the same thing. 511 00:39:02,880 --> 00:39:09,400 And what distinguishes this pattern versus that pattern 512 00:39:09,400 --> 00:39:14,080 is whether this constant c is positive or negative 513 00:39:14,080 --> 00:39:18,960 because you can see that out here, 514 00:39:18,960 --> 00:39:24,910 ultimately you end up at the point where y has gone to 0. 515 00:39:24,910 --> 00:39:28,880 So depending on x positive or negative, it doesn't matter. 516 00:39:28,880 --> 00:39:31,680 This combination will be positive. 517 00:39:31,680 --> 00:39:36,277 So throughout here, what I have is that c is positive. 518 00:39:39,140 --> 00:39:44,060 Whereas what I have up here is c that is negative. 519 00:39:44,060 --> 00:39:46,335 Presumably, there is other trajectory here 520 00:39:46,335 --> 00:39:48,520 and down here, c is again positive. 521 00:39:54,170 --> 00:39:58,660 And again, if you want to ensure y over here c is negative, 522 00:39:58,660 --> 00:40:03,550 because over here you can see you crossed the line where 523 00:40:03,550 --> 00:40:06,110 x is 0, but you have some value of y. 524 00:40:13,890 --> 00:40:21,935 So if I were to blow up that region both as a function of x 525 00:40:21,935 --> 00:40:27,030 and y, well, first of all, I will 526 00:40:27,030 --> 00:40:31,320 have a particular set of trajectories-- the ones 527 00:40:31,320 --> 00:40:36,350 that end up at this important fix point, which 528 00:40:36,350 --> 00:40:40,340 correspond clearly to c cos to 0. 529 00:40:40,340 --> 00:40:44,710 So that c cos to 0 will give me two straight lines. 530 00:40:44,710 --> 00:40:50,030 So presumably there is this straight line, 531 00:40:50,030 --> 00:40:56,670 and then there is another straight line goes out there. 532 00:40:56,670 --> 00:41:01,320 And then I have this bunch of trajectories 533 00:41:01,320 --> 00:41:05,920 that are these hyperbole that end up over here. 534 00:41:05,920 --> 00:41:10,440 I can have hyperboles that will be going out. 535 00:41:10,440 --> 00:41:14,540 And then I will have hyperboles that are like this. 536 00:41:17,240 --> 00:41:19,230 This are all in the high temperature phase, 537 00:41:19,230 --> 00:41:23,038 so let's [INAUDIBLE] like this. 538 00:41:28,290 --> 00:41:32,500 So one thing that you immediately see 539 00:41:32,500 --> 00:41:36,540 is that the location of the transition that 540 00:41:36,540 --> 00:41:43,320 is given by this critical line when c equals to 0. 541 00:41:43,320 --> 00:41:47,465 So statement number one that we can get 542 00:41:47,465 --> 00:41:55,900 is that the transition line corresponds to c equals to 0. 543 00:41:55,900 --> 00:41:59,790 So solving for x as a function of y, 544 00:41:59,790 --> 00:42:05,580 I will get that x critical is either minus or plus pi squared 545 00:42:05,580 --> 00:42:06,750 y. 546 00:42:06,750 --> 00:42:11,060 Clearly from the figure, the solution that I want 547 00:42:11,060 --> 00:42:13,488 is the one that corresponds to minus. 548 00:42:16,260 --> 00:42:18,820 My x was k inverse minus pi/2. 549 00:42:21,390 --> 00:42:25,920 This is kc inverse minus pi/2. 550 00:42:25,920 --> 00:42:31,130 And so what I see is that kc inverse-- 551 00:42:31,130 --> 00:42:35,360 the correct transition temperature-- is, 552 00:42:35,360 --> 00:42:42,630 in fact, lower than the value of pi/2 that we had deduced, 553 00:42:42,630 --> 00:42:45,500 assuming that there is only a single vertex 554 00:42:45,500 --> 00:42:52,290 in the entire system by an amount that to lowest order 555 00:42:52,290 --> 00:42:56,860 is related to the core energy or core fugacity. 556 00:42:56,860 --> 00:42:59,680 And presumably, there are higher order terms 557 00:42:59,680 --> 00:43:03,420 that I haven't calculated. 558 00:43:03,420 --> 00:43:09,430 So this number that we have calculated 559 00:43:09,430 --> 00:43:11,510 by looking at a single defect, we 560 00:43:11,510 --> 00:43:14,140 can see that in the presence of multiple defects, 561 00:43:14,140 --> 00:43:17,680 starts to get lower. 562 00:43:17,680 --> 00:43:21,350 And this is precisely correct in the limit 563 00:43:21,350 --> 00:43:23,644 where y is a small quantity. 564 00:43:28,860 --> 00:43:32,320 Now that's a transition line. 565 00:43:32,320 --> 00:43:35,120 We can look to the left or to the right. 566 00:43:35,120 --> 00:43:37,395 Let us just look at the low temperature phase. 567 00:43:44,470 --> 00:43:48,820 So for the low T phase, we expect c to be negative. 568 00:43:53,360 --> 00:44:02,850 And I can, for example, make that explicit by writing 569 00:44:02,850 --> 00:44:11,950 it as-- OK, so c was x0 squared minus pi to the fourth y0 570 00:44:11,950 --> 00:44:14,620 squared-- what the starting parameters of the system 571 00:44:14,620 --> 00:44:16,180 dictate. 572 00:44:16,180 --> 00:44:20,585 If you are under low temperature phase such that c is negative, 573 00:44:20,585 --> 00:44:24,040 it means that you are at temperatures 574 00:44:24,040 --> 00:44:27,350 that are smaller than Tc. 575 00:44:27,350 --> 00:44:31,840 So let's see, T minus Tc, let's write it 576 00:44:31,840 --> 00:44:36,570 as Tc minus T-- would be positive. 577 00:44:36,570 --> 00:44:39,150 But this has to be negative, so let me just 578 00:44:39,150 --> 00:44:41,340 introduce some parameter b. 579 00:44:41,340 --> 00:44:43,010 It's not the same b as here. 580 00:44:43,010 --> 00:44:47,030 Just some coefficient that has to be squared. 581 00:44:47,030 --> 00:44:54,420 So I know that as I hit Tc, this c goes to 0. 582 00:44:54,420 --> 00:44:58,820 If I'm slightly away from Tc along the trajectory 583 00:44:58,820 --> 00:45:04,290 that I have indicated over here, right here I'm 0, 584 00:45:04,290 --> 00:45:07,200 so right I'm slightly negative. 585 00:45:07,200 --> 00:45:11,230 And there's no reason why the value that I calculated 586 00:45:11,230 --> 00:45:15,330 from x0 squared minus pi to the fourth y0 squared 587 00:45:15,330 --> 00:45:18,010 should not be an analytical function. 588 00:45:18,010 --> 00:45:21,130 So I have expanded that analytical function, 589 00:45:21,130 --> 00:45:24,640 knowing that that Tc is equal to 0. 590 00:45:24,640 --> 00:45:26,680 There will be higher order terms for sure, 591 00:45:26,680 --> 00:45:28,910 but this is the lowest order term 592 00:45:28,910 --> 00:45:30,430 that I would have in that expansion. 593 00:45:33,100 --> 00:45:37,030 And as we said, this is preserved 594 00:45:37,030 --> 00:45:40,460 all along the trajectory that ends on this point. 595 00:45:40,460 --> 00:45:43,390 So along that trajectory, this is the same 596 00:45:43,390 --> 00:45:49,660 as x squared minus pi to the fourth y squared, 597 00:45:49,660 --> 00:45:58,330 which means that I can write y squared to be 1 over pi 598 00:45:58,330 --> 00:46:01,130 to the fourth. 599 00:46:01,130 --> 00:46:10,260 x squared plus b squared Tc minus T. So 600 00:46:10,260 --> 00:46:14,350 if I want to solve for this curve, that's 601 00:46:14,350 --> 00:46:18,520 what I will have for some value of this quantity. 602 00:46:18,520 --> 00:46:24,940 And what I do is I look at what that implies for the xy dl. 603 00:46:24,940 --> 00:46:31,700 The xy dl is 4 pi cubed y squared. 604 00:46:31,700 --> 00:46:35,456 I substitute the y squared that I have over there. 605 00:46:35,456 --> 00:46:47,760 I will get 4/pi times x squared plus b squared Tc minus T. 606 00:46:47,760 --> 00:46:53,920 So under rescaling, this tells me what is happening to x. 607 00:46:53,920 --> 00:46:59,280 And in particular, what I can do is to integrate this equation. 608 00:46:59,280 --> 00:47:06,730 I have dx divided by x squared plus b squared Tc minus T is 609 00:47:06,730 --> 00:47:12,950 4/pi dl-- just rearranging this differential equation. 610 00:47:12,950 --> 00:47:18,890 And this I can certainly integrate out to l. 611 00:47:18,890 --> 00:47:23,810 This you should recognize as the differential form 612 00:47:23,810 --> 00:47:32,890 of the inverse tangent up to a factor of 1 613 00:47:32,890 --> 00:47:37,650 over b square root of Tc minus T. 614 00:47:37,650 --> 00:47:39,140 So I integrate this. 615 00:47:39,140 --> 00:47:42,549 And on the other side, I have 4/pi l. 616 00:47:47,510 --> 00:48:04,970 So eventually, I know that-- that's what I wanted to do? 617 00:48:04,970 --> 00:48:10,060 I needed to do this later on, but we'll use it later on. 618 00:48:10,060 --> 00:48:16,090 What I needed to get is what is the eventual fate 619 00:48:16,090 --> 00:48:19,170 of this differential equation. 620 00:48:19,170 --> 00:48:22,530 Eventually, we see that this differential equation arrives 621 00:48:22,530 --> 00:48:25,715 at the point that I will call x infinity. 622 00:48:28,400 --> 00:48:33,380 When it arrives at x infinity, this is 0 and y is 0, 623 00:48:33,380 --> 00:48:35,880 so I immediately know that x infinity-- 624 00:48:35,880 --> 00:48:40,520 I didn't need to do any of that calculation-- this expression 625 00:48:40,520 --> 00:48:50,000 has to be 0, is minus the square root of Tc minus T. So let 626 00:48:50,000 --> 00:48:54,347 me figure out what I did with the signs that is incorrect. 627 00:48:54,347 --> 00:48:56,255 AUDIENCE: [INAUDIBLE] temperature [INAUDIBLE] 628 00:48:56,255 --> 00:48:57,209 positive [INAUDIBLE]. 629 00:49:00,444 --> 00:49:02,110 PROFESSOR: In the low temperature phase, 630 00:49:02,110 --> 00:49:07,340 I have indeed stated that c has to be positive, 631 00:49:07,340 --> 00:49:11,600 which means that this coefficient better be positive, 632 00:49:11,600 --> 00:49:16,100 which means that I would have a minus sign here. 633 00:49:16,100 --> 00:49:26,790 And then x would be b times Tc minus T. 634 00:49:26,790 --> 00:49:30,220 Right, so this would be plus or minus. 635 00:49:30,220 --> 00:49:34,460 The plus solution is somewhere out here, 636 00:49:34,460 --> 00:49:36,460 which I'm not interested. 637 00:49:36,460 --> 00:49:40,690 The solution that I'm interested corresponds to this value. 638 00:49:48,260 --> 00:49:51,230 You say, well, what is important about that? 639 00:49:56,545 --> 00:50:04,460 You see that various properties of this low temperature phase 640 00:50:04,460 --> 00:50:08,580 are characterized by this power law as opposed 641 00:50:08,580 --> 00:50:12,380 to exponential behavior. 642 00:50:12,380 --> 00:50:17,250 The power law is determined by the value of k 643 00:50:17,250 --> 00:50:20,070 where the description in terms of this gradient 644 00:50:20,070 --> 00:50:23,340 squared theory is correct. 645 00:50:23,340 --> 00:50:26,560 Now out here, the description is not correct 646 00:50:26,560 --> 00:50:30,190 because I still have the topological difference. 647 00:50:30,190 --> 00:50:33,360 But if I look at sufficiently large distances, 648 00:50:33,360 --> 00:50:37,090 I see that the topology defects have disappeared. 649 00:50:37,090 --> 00:50:40,420 But by the time the topological defects have disappeared, 650 00:50:40,420 --> 00:50:43,180 I don't have the original value of k. 651 00:50:43,180 --> 00:50:46,950 I have a slightly different value of k. 652 00:50:46,950 --> 00:50:51,340 So presumably, the properties are 653 00:50:51,340 --> 00:50:56,615 going to be described by what value of this k inverse 654 00:50:56,615 --> 00:50:59,420 is of large behaviors. 655 00:50:59,420 --> 00:51:07,630 And so what I expect is that the effective behavior of this k 656 00:51:07,630 --> 00:51:22,450 inverse-- actually, the effective behavior of k-- 657 00:51:22,450 --> 00:51:29,130 as a function of whatever the temperature of the system is. 658 00:51:29,130 --> 00:51:35,020 We expect that in the original xy model, or any system that 659 00:51:35,020 --> 00:51:37,950 is described by this behavior, there 660 00:51:37,950 --> 00:51:43,410 is a critical temperature, Tc, such that at higher 661 00:51:43,410 --> 00:51:47,880 temperatures, correlations are decaying exponentially. 662 00:51:47,880 --> 00:51:51,915 So essentially, the effective value of k has gone to 0. 663 00:51:51,915 --> 00:51:53,730 There is no stiffness parameter. 664 00:51:53,730 --> 00:51:57,120 So basically at high temperatures, 665 00:51:57,120 --> 00:51:59,000 you should be over here. 666 00:52:02,180 --> 00:52:10,450 What I see is that the effective value of k, however, other 667 00:52:10,450 --> 00:52:15,050 is meaningful all the way to the inverse of 2/pi. 668 00:52:18,890 --> 00:52:26,220 So there is a value here at 2/pi which 669 00:52:26,220 --> 00:52:36,330 corresponds to the largest temperature or the smallest 670 00:52:36,330 --> 00:52:38,930 k that is acceptable. 671 00:52:38,930 --> 00:52:41,870 Now what I see is that on approaching 672 00:52:41,870 --> 00:52:52,940 the transition, the value of k-- I have it up there-- is 2/pi. 673 00:52:52,940 --> 00:52:57,470 This limiting value that we have over here. 674 00:52:57,470 --> 00:53:05,210 And then there is a correction that is 4 over pi squared x. 675 00:53:05,210 --> 00:53:09,380 And presumably here, I have to put the x in infinity. 676 00:53:09,380 --> 00:53:12,270 And what I have for the x infinity 677 00:53:12,270 --> 00:53:16,720 is something like that, so I will get 2/pi plus 4 678 00:53:16,720 --> 00:53:23,110 e over pi squared square root of Tc minus T. 679 00:53:23,110 --> 00:53:32,120 So the prediction is that the effective value of k 680 00:53:32,120 --> 00:53:35,910 comes to its limiting value of 2/pi 681 00:53:35,910 --> 00:53:37,620 with a square root singularity. 682 00:53:42,840 --> 00:53:53,620 So we can replace the theory that 683 00:53:53,620 --> 00:53:58,130 describes anything that is in this universality class 684 00:53:58,130 --> 00:54:02,865 in the temperature phase by an effective value of k. 685 00:54:02,865 --> 00:54:07,680 If we then ask how does that effective value of k change 686 00:54:07,680 --> 00:54:10,890 as a function of temperature, the prediction 687 00:54:10,890 --> 00:54:13,500 is that, well, at very low temperature, 688 00:54:13,500 --> 00:54:17,240 it's presumably inversely related to temperature. 689 00:54:17,240 --> 00:54:19,780 It will come down, but [INAUDIBLE] 690 00:54:19,780 --> 00:54:24,280 it will change its behavior, come with a square root 691 00:54:24,280 --> 00:54:28,880 singularity to a number that is 2/pi, and then jump to 0. 692 00:54:32,950 --> 00:54:39,390 Now you are justified in saying, well, this is all very obscure. 693 00:54:39,390 --> 00:54:42,650 Is there any way to see this? 694 00:54:42,650 --> 00:54:45,730 And the answer is that people have experimentally 695 00:54:45,730 --> 00:54:50,120 verified this, and I'll tell you how. 696 00:54:50,120 --> 00:55:00,470 So a system that belongs to this universality class and we've 697 00:55:00,470 --> 00:55:02,895 mentioned all the way in the class is the superfluid. 698 00:55:06,670 --> 00:55:11,310 We've said that the superfluid transition is characterized 699 00:55:11,310 --> 00:55:16,940 by a quantum order parameter that applies a magnitude, 700 00:55:16,940 --> 00:55:19,880 but then it has a phase theta. 701 00:55:19,880 --> 00:55:22,350 And roughly, we would say that the phase theta should 702 00:55:22,350 --> 00:55:27,050 be described by this kind of theory at low temperatures. 703 00:55:27,050 --> 00:55:31,270 So if we want basically a two dimensional system, what 704 00:55:31,270 --> 00:55:35,900 we need to do is to look at the superfluid field. 705 00:55:35,900 --> 00:55:47,250 And this is something that Bishop and Reppy did in 1978 706 00:55:47,250 --> 00:55:54,760 where they constructed the analog of the Andronikashvili 707 00:55:54,760 --> 00:56:01,480 experiment that we mentioned in 8333, applied it to the field. 708 00:56:01,480 --> 00:56:06,770 So let me remind you what the Andronikashvili experiment was. 709 00:56:06,770 --> 00:56:14,590 Basically, you will have a torsional oscillator. 710 00:56:14,590 --> 00:56:23,890 This torsional oscillator was connected 711 00:56:23,890 --> 00:56:28,510 to a vat that had helium in it. 712 00:56:28,510 --> 00:56:33,880 So basically, this thing was oscillating, 713 00:56:33,880 --> 00:56:36,780 and the frequency of oscillations 714 00:56:36,780 --> 00:56:46,090 was related to some kind of a effective torsion of constant k 715 00:56:46,090 --> 00:56:49,880 divided by some huge mass which is 716 00:56:49,880 --> 00:56:52,550 contained within the cylinder. 717 00:56:52,550 --> 00:56:58,700 So basically you can probe classically-- you 718 00:56:58,700 --> 00:57:00,900 would say there's some kind of a density here. 719 00:57:00,900 --> 00:57:02,660 You can calculate what the mass is 720 00:57:02,660 --> 00:57:07,236 if you know what this is, you know what the omega is. 721 00:57:07,236 --> 00:57:12,630 Now what he noticed was that if this thing was filled 722 00:57:12,630 --> 00:57:19,330 with liquid helium and you went below Tc of helium, then 723 00:57:19,330 --> 00:57:22,660 suddenly this frequency changed. 724 00:57:22,660 --> 00:57:28,010 And the reason was that the mass that was rotating along 725 00:57:28,010 --> 00:57:32,060 with this whole thing was changed because the part that 726 00:57:32,060 --> 00:57:35,960 was superfluid was sitting still, 727 00:57:35,960 --> 00:57:39,800 and the normal part was the part that was oscillating. 728 00:57:39,800 --> 00:57:44,510 So the mass that was oscillating was reduced, 729 00:57:44,510 --> 00:57:46,500 frequency would go up. 730 00:57:46,500 --> 00:57:48,830 And from the change in frequency, 731 00:57:48,830 --> 00:57:54,070 he could figure out the change in the density of the part that 732 00:57:54,070 --> 00:57:57,210 was oscillating, and hence calculate 733 00:57:57,210 --> 00:58:01,350 what the density of the normal part was. 734 00:58:01,350 --> 00:58:07,430 So what Bishop and Reppy did was to make this two dimensional. 735 00:58:07,430 --> 00:58:10,220 How did they make it two dimensional? 736 00:58:10,220 --> 00:58:15,980 Rather than having a container of helium, what they did was 737 00:58:15,980 --> 00:58:19,250 they made, if you like, some kind of a toilet paper. 738 00:58:19,250 --> 00:58:21,770 They call it a jelly roll my Mylar. 739 00:58:21,770 --> 00:58:26,630 So it was Mylar that was wrapped in a cylinder. 740 00:58:26,630 --> 00:58:32,666 And then the helium was absorbed between the surfaces of Mylar. 741 00:58:32,666 --> 00:58:36,390 So effectively, it was a two dimensional system 742 00:58:36,390 --> 00:58:37,520 in this very setup. 743 00:58:40,860 --> 00:58:43,620 So for that two dimensional system, 744 00:58:43,620 --> 00:58:46,380 they-- again, with the same thing-- 745 00:58:46,380 --> 00:58:49,100 they measure the change in frequency. 746 00:58:49,100 --> 00:58:51,940 They found that if they go to low enough temperatures, 747 00:58:51,940 --> 00:58:54,730 suddenly there is a change in frequency. 748 00:58:54,730 --> 00:58:58,080 Of course, the temperature that they were seeing in this case 749 00:58:58,080 --> 00:59:01,180 was something like 1 degrees Kelvin or a fraction 750 00:59:01,180 --> 00:59:04,580 of 1 degree Kelvin, whereas when you have the full superfluid, 751 00:59:04,580 --> 00:59:09,550 it's 2.8 degrees Kelvin clearly because of that dimensionality 752 00:59:09,550 --> 00:59:11,500 the critical temperature changes. 753 00:59:11,500 --> 00:59:15,880 But you would say that's not particularly the inverse. 754 00:59:15,880 --> 00:59:21,760 So they could measure the change in frequency 755 00:59:21,760 --> 00:59:24,350 and relate the change in frequency 756 00:59:24,350 --> 00:59:27,845 to the density that became superfluid. 757 00:59:32,310 --> 00:59:36,880 Now how does the superfluid density 758 00:59:36,880 --> 00:59:39,750 tell us anything about this curve? 759 00:59:39,750 --> 00:59:43,130 Well, the answer is that everything 760 00:59:43,130 --> 00:59:46,300 is going to be weighted by something 761 00:59:46,300 --> 00:59:51,720 like e the minus beta times some energy. 762 00:59:51,720 --> 00:59:55,898 The one part of the energy that is associated with oscillations 763 00:59:55,898 --> 00:59:59,360 is certainly the kinetic energy. 764 00:59:59,360 --> 01:00:03,384 So let's see what we would write down for beta times 765 01:00:03,384 --> 01:00:10,340 the kinetic energy of superfluid or superfluid film. 766 01:00:10,340 --> 01:00:15,790 What I have to do is beta will give me 1/kt. 767 01:00:18,690 --> 01:00:22,990 The kinetic energy is obtained by integrating 768 01:00:22,990 --> 01:00:27,060 mass times velocity squared, or density 769 01:00:27,060 --> 01:00:30,725 integrated against velocity. 770 01:00:34,170 --> 01:00:36,400 It's a two dimensional film, so we sort of 771 01:00:36,400 --> 01:00:39,530 integrate as we go along the film. 772 01:00:39,530 --> 01:00:46,450 The superfluid velocity can be related 773 01:00:46,450 --> 01:00:55,140 to the mass of helium h bar and the gradient 774 01:00:55,140 --> 01:00:58,050 of this phase of the superconducting order 775 01:00:58,050 --> 01:00:59,500 parameter. 776 01:00:59,500 --> 01:01:04,870 So you can, for example, write your weight function as sidebar 777 01:01:04,870 --> 01:01:08,810 into the i theta of x, calculate what the current is using 778 01:01:08,810 --> 01:01:13,430 the usual formula of h bar over m psi star [? grat ?] psi 779 01:01:13,430 --> 01:01:15,030 [? minus ?] psi [? grat ?] psi star, 780 01:01:15,030 --> 01:01:19,830 and you would see that effective mass is something like this. 781 01:01:19,830 --> 01:01:26,753 So this is going to give me rho over kt h bar over n 782 01:01:26,753 --> 01:01:34,390 helium 4 squared integral gradient of theta 783 01:01:34,390 --> 01:01:41,580 squared, which you can see is identical to the very 784 01:01:41,580 --> 01:01:44,570 first line that I wrote down for you. 785 01:01:44,570 --> 01:01:54,314 And we can see that k can be interpreted as rho-- kt h bar 786 01:01:54,314 --> 01:01:57,660 over n squared. 787 01:02:00,210 --> 01:02:05,020 So all of these quantities-- h bar m, you know. 788 01:02:05,020 --> 01:02:08,080 T is the temperature that you're measuring. 789 01:02:08,080 --> 01:02:13,232 Rho you get through the change in this frequency. 790 01:02:13,232 --> 01:02:18,360 And so then they can plot what [INAUDIBLE] 791 01:02:18,360 --> 01:02:23,976 rho is as a function of temperature. 792 01:02:23,976 --> 01:02:27,490 And we see that it's very much related to k. 793 01:02:27,490 --> 01:02:33,550 And indeed, they find that the rho that they measure 794 01:02:33,550 --> 01:02:37,140 has some kind of behavior such as this. 795 01:02:40,110 --> 01:02:45,000 And then they go and change their Mylar, make the films 796 01:02:45,000 --> 01:02:46,860 thicker or whatever. 797 01:02:46,860 --> 01:02:49,100 They find that the transition temperature 798 01:02:49,100 --> 01:02:55,870 changes so that a different type of film 799 01:02:55,870 --> 01:02:58,295 would show a behavior such as this. 800 01:02:58,295 --> 01:03:02,630 And thicker films will have a higher critical temperature. 801 01:03:02,630 --> 01:03:06,520 They do it for a number of film thicknesses, 802 01:03:06,520 --> 01:03:11,860 and they got things' behavior such as this, 803 01:03:11,860 --> 01:03:16,390 and found that this behavior followed this gray line, which 804 01:03:16,390 --> 01:03:18,940 is exactly what is predicted from here. 805 01:03:18,940 --> 01:03:23,000 It was predicted that rho c over Tc 806 01:03:23,000 --> 01:03:31,690 should kb m over h bar squared times what 807 01:03:31,690 --> 01:03:43,090 the critical value of k is that we've calculated to be 2/pi 808 01:03:43,090 --> 01:03:47,725 So they could precisely check these 2/pi 809 01:03:47,725 --> 01:03:49,400 that we've calculated. 810 01:03:49,400 --> 01:03:53,210 They could more or less see this square root approach 811 01:03:53,210 --> 01:03:54,840 to this singularity. 812 01:03:54,840 --> 01:03:57,150 I'm not sure the data at that point were good 813 01:03:57,150 --> 01:04:00,849 enough so that they could say this exponent is precisely 1/2. 814 01:04:15,520 --> 01:04:20,090 So this was for the low temperature phase. 815 01:04:20,090 --> 01:04:22,210 What can I say about the high temperature phase? 816 01:04:29,830 --> 01:04:35,470 So in the high temperature phase is where my c is negative. 817 01:04:39,070 --> 01:04:46,480 So there I can write x0 squared minus pi 818 01:04:46,480 --> 01:04:52,483 to the fourth y0 squared as being a negative number, 819 01:04:52,483 --> 01:04:58,740 which I will write as minus b squared T minus Tc. 820 01:04:58,740 --> 01:05:02,710 So I have now T that is greater that Tc, multiply 821 01:05:02,710 --> 01:05:07,550 with some constant, and I get this. 822 01:05:07,550 --> 01:05:10,450 And this is the same all along the trajectory. 823 01:05:13,880 --> 01:05:17,190 So as I go further and x and y change, 824 01:05:17,190 --> 01:05:19,430 they will change in a manner that 825 01:05:19,430 --> 01:05:23,300 is consistent with this, which implies 826 01:05:23,300 --> 01:05:31,890 that as x changes with l and y changes with l, the two of them 827 01:05:31,890 --> 01:05:45,551 will be related by y squared is being x squared plus b squared 828 01:05:45,551 --> 01:05:53,190 T minus Tc divided by pi to the fourth. 829 01:05:53,190 --> 01:05:55,770 So this is where I don't really see 830 01:05:55,770 --> 01:05:57,730 the endpoint of the trajectory. 831 01:05:57,730 --> 01:06:01,300 I just want to see how the trajectory is behaving. 832 01:06:01,300 --> 01:06:06,300 So I go back to this equation, dx by dl 833 01:06:06,300 --> 01:06:11,700 is 4 pi cube y squared. 834 01:06:11,700 --> 01:06:17,120 Substitute that y squared, I will get 4/pi 1 835 01:06:17,120 --> 01:06:21,330 over x squared plus b squared T minus Tc. 836 01:06:35,040 --> 01:06:38,920 And then I rearrange this in a form 837 01:06:38,920 --> 01:06:40,850 I can see how to integrate. 838 01:06:40,850 --> 01:06:50,340 dx x squared plus b squared T minus Tc is 4/pi dl. 839 01:06:50,340 --> 01:06:52,660 I integrate the left-hand side. 840 01:06:52,660 --> 01:06:55,620 And as I already jumped ahead, it 841 01:06:55,620 --> 01:07:02,670 is the inverse tangent of x divided 842 01:07:02,670 --> 01:07:10,291 by b square root of T minus Tc is 4/pi times l. 843 01:07:15,190 --> 01:07:20,660 So what do I want to do with this expression? 844 01:07:23,570 --> 01:07:30,390 So what I want to do is to see the trajectories that 845 01:07:30,390 --> 01:07:34,150 just cross to the high temperature side. 846 01:07:34,150 --> 01:07:38,880 So I start with a point that is just slightly 847 01:07:38,880 --> 01:07:42,690 to the right of this transition line. 848 01:07:42,690 --> 01:07:46,140 Presumably, what is happening is that I 849 01:07:46,140 --> 01:07:51,880 will follow the transition trajectory for a long while, 850 01:07:51,880 --> 01:07:57,300 then I will start to head out, which 851 01:07:57,300 --> 01:08:02,580 means that for this trajectory, if I look 852 01:08:02,580 --> 01:08:07,080 at the system over larger and larger scales, initially 853 01:08:07,080 --> 01:08:10,500 I find that it becomes harder and harder 854 01:08:10,500 --> 01:08:13,730 to create these topological defects. 855 01:08:13,730 --> 01:08:16,920 The core energy for them becomes large. 856 01:08:16,920 --> 01:08:19,729 The fugacity for them becomes small. 857 01:08:19,729 --> 01:08:27,430 But ultimately I manage to break that, and I go to a regime 858 01:08:27,430 --> 01:08:30,250 where it becomes easier and easier 859 01:08:30,250 --> 01:08:33,189 to create these topological defects. 860 01:08:33,189 --> 01:08:36,969 And presumably at some point out here, everything 861 01:08:36,969 --> 01:08:40,540 that I have said I have to throw out because I'm making 862 01:08:40,540 --> 01:08:45,340 an expansion, assuming that y is small, x is small, et cetera. 863 01:08:45,340 --> 01:08:52,080 So presumably, as I integrate, I come to a point where I say, 864 01:08:52,080 --> 01:08:55,220 OK, differential equations break down. 865 01:08:55,220 --> 01:09:00,170 But my intuition tells me that I have reached the regime where 866 01:09:00,170 --> 01:09:09,189 I can create pretty much plus and minus charges at ease. 867 01:09:09,189 --> 01:09:14,479 So I would say that once I have reached that region where 868 01:09:14,479 --> 01:09:18,880 x and y have managed to escape the region where they are 869 01:09:18,880 --> 01:09:22,920 small, they have become of the order of 1. 870 01:09:22,920 --> 01:09:26,580 Maybe you can put them 1/3, 1/4-- it doesn't matter. 871 01:09:26,580 --> 01:09:31,890 Once they have become something that is not infinitesimal, 872 01:09:31,890 --> 01:09:35,740 then I can create these charges more or less at will. 873 01:09:35,740 --> 01:09:38,300 I will have a system where I have 874 01:09:38,300 --> 01:09:41,490 lots of charges that can be created at ease. 875 01:09:41,490 --> 01:09:45,740 And my intuition tells me that in that system, 876 01:09:45,740 --> 01:09:52,229 I shall have this kind of decay-- exponential decay. 877 01:09:52,229 --> 01:09:59,890 So how far did I have to go in order to reach that value of l? 878 01:09:59,890 --> 01:10:03,090 I have to go to a correlation length 879 01:10:03,090 --> 01:10:07,270 for a size that is larger than 1 what 880 01:10:07,270 --> 01:10:15,210 I started with by a factor of e to the l where the value of x 881 01:10:15,210 --> 01:10:19,240 became something that is of the order of 1. 882 01:10:19,240 --> 01:10:21,170 And actually, you can see from here 883 01:10:21,170 --> 01:10:24,000 that if I'm very close to the transition, 884 01:10:24,000 --> 01:10:26,150 it doesn't matter whether I choose here 885 01:10:26,150 --> 01:10:31,300 to be 1/10, 1/2, even 1/100. 886 01:10:31,300 --> 01:10:34,070 As long as I'm close enough to Tc, 887 01:10:34,070 --> 01:10:38,180 I'm dividing something by something that is close to 0. 888 01:10:38,180 --> 01:10:43,250 And this is tan inverse of a large number. 889 01:10:43,250 --> 01:10:45,810 And tan inverse of a large number 890 01:10:45,810 --> 01:10:48,730 is tan inverse of 90 degrees. 891 01:10:48,730 --> 01:10:51,590 So essentially, I go to some value 892 01:10:51,590 --> 01:10:58,230 of x where I can approximate this by pi over 2. 893 01:10:58,230 --> 01:11:01,660 And you can see that that is really insensitive to what 894 01:11:01,660 --> 01:11:05,600 I choose to be my x's as long as I'm sufficiently 895 01:11:05,600 --> 01:11:07,640 close to the critical point. 896 01:11:07,640 --> 01:11:09,728 So you can see that once I have done 897 01:11:09,728 --> 01:11:13,840 that, I have figured out what my l star is, if you like. 898 01:11:13,840 --> 01:11:16,360 And if I substituted that over there, 899 01:11:16,360 --> 01:11:27,320 I will get a behavior that is about pi/4 times pi/2 times 900 01:11:27,320 --> 01:11:30,060 1 over the square root of T minus Tc. 901 01:11:32,700 --> 01:11:39,210 Now these coefficients out front are not that important. 902 01:11:39,210 --> 01:11:44,080 What you see is that, indeed, we get a correlation length 903 01:11:44,080 --> 01:11:48,080 that, as we approach Tc, diverges, 904 01:11:48,080 --> 01:11:50,820 but it is not that at all of any of the forms 905 01:11:50,820 --> 01:11:52,090 that we had seen before. 906 01:11:52,090 --> 01:11:55,540 So typically, we wrote that the correlation lens diverges 907 01:11:55,540 --> 01:11:59,110 T minus Tc to some exponent minus mu. 908 01:11:59,110 --> 01:12:01,350 This is not that type of divergence. 909 01:12:01,350 --> 01:12:04,870 It's a very different type of divergence. 910 01:12:04,870 --> 01:12:09,430 And again, it's root is in the non-linear version 911 01:12:09,430 --> 01:12:13,300 of the recursion undulations that we have. 912 01:12:13,300 --> 01:12:16,420 The closest thing to this that we have 913 01:12:16,420 --> 01:12:19,540 is when we were calculating the correlation length 914 01:12:19,540 --> 01:12:23,340 for the non-linear sigma model where we had something that 915 01:12:23,340 --> 01:12:26,980 had a 1 over temperature type of behavior. 916 01:12:26,980 --> 01:12:30,770 This is even more complicated. 917 01:12:30,770 --> 01:12:33,848 Now once you know the singular behavior of the correlation 918 01:12:33,848 --> 01:12:38,660 length, you would say that in the two dimensional system, 919 01:12:38,660 --> 01:12:40,880 the singular part of the free energy 920 01:12:40,880 --> 01:12:45,510 should scale like c to the minus 2. 921 01:12:45,510 --> 01:12:48,900 Essentially, you break your system 922 01:12:48,900 --> 01:12:52,490 into pieces that are of the size correlation length. 923 01:12:52,490 --> 01:12:57,090 The number of those pieces is l over xi squared, 924 01:12:57,090 --> 01:12:58,580 because you are in two dimensions. 925 01:12:58,580 --> 01:12:59,970 So you would get this. 926 01:12:59,970 --> 01:13:04,270 So that says that your singularity of the free energy 927 01:13:04,270 --> 01:13:08,080 is something like, I don't know, pi squared 928 01:13:08,080 --> 01:13:14,640 8 4b square root of T minus Tc. 929 01:13:14,640 --> 01:13:17,370 Again, not a popular singularity. 930 01:13:17,370 --> 01:13:18,620 It's an essential singularity. 931 01:13:27,052 --> 01:13:32,140 And essential singularity is a kind of singular function 932 01:13:32,140 --> 01:13:34,860 that no matter how many derivatives 933 01:13:34,860 --> 01:13:40,100 you take, as T comes to Tc, there is no singularity. 934 01:13:40,100 --> 01:13:42,940 So for example, if I take two derivatives 935 01:13:42,940 --> 01:13:49,600 to get the heat capacity, what I would 936 01:13:49,600 --> 01:13:56,920 plot as a function of T at Tc should have no signatures. 937 01:13:56,920 --> 01:14:01,040 So basically what you would see because if this 938 01:14:01,040 --> 01:14:02,730 is that the curve just continues. 939 01:14:02,730 --> 01:14:06,465 There is no signature of a transition at this heat 940 01:14:06,465 --> 01:14:08,170 capacity. 941 01:14:08,170 --> 01:14:10,070 And indeed, people later on, they 942 01:14:10,070 --> 01:14:12,870 did numerical simulations, et cetera. 943 01:14:12,870 --> 01:14:15,430 What they find is that the heat capacity actually 944 01:14:15,430 --> 01:14:20,240 has kind of smooth peak a little bit later than Tc, which 945 01:14:20,240 --> 01:14:23,580 is the location where there's lots and lots of vortex 946 01:14:23,580 --> 01:14:25,060 unbinding going, gone. 947 01:14:25,060 --> 01:14:28,576 But at Tc itself, there is no signature of a singularity. 948 01:14:36,520 --> 01:14:43,480 As far as I know, there's no experimental case 949 01:14:43,480 --> 01:14:45,810 with this correlation length as we observed. 950 01:14:55,950 --> 01:15:06,500 So the lesson that we can take from this particular system 951 01:15:06,500 --> 01:15:13,300 is that two dimensional system are kind of potentially 952 01:15:13,300 --> 01:15:16,710 interesting and different. 953 01:15:16,710 --> 01:15:20,000 We had this Mermin-Wagner Theorem 954 01:15:20,000 --> 01:15:21,690 that we mentioned in the beginning that 955 01:15:21,690 --> 01:15:24,010 said that there should be no true long range 956 01:15:24,010 --> 01:15:25,990 order than two dimensions. 957 01:15:25,990 --> 01:15:27,780 That is still true. 958 01:15:27,780 --> 01:15:30,856 But despite that, there could be phase transitions 959 01:15:30,856 --> 01:15:36,240 with quite observable consequences. 960 01:15:36,240 --> 01:15:42,670 Add a particular type of transition in two dimension 961 01:15:42,670 --> 01:15:47,440 that we will pursue next lecture-- so I'll give you 962 01:15:47,440 --> 01:15:51,428 a preview-- is that of melting. 963 01:15:56,670 --> 01:16:01,210 So the prototype of a phase transition you may think of 964 01:16:01,210 --> 01:16:05,130 is either liquid gas or a liquid solid. 965 01:16:05,130 --> 01:16:09,640 And you can say, well, you have studied phase transitions 966 01:16:09,640 --> 01:16:11,340 to such a degree. 967 01:16:11,340 --> 01:16:16,470 Why not go back and talk about the melting transition, 968 01:16:16,470 --> 01:16:18,800 for example? 969 01:16:18,800 --> 01:16:24,320 The reason is that the straight melting transition is typically 970 01:16:24,320 --> 01:16:25,620 first order. 971 01:16:25,620 --> 01:16:28,670 And we've seen that universality and all of those things 972 01:16:28,670 --> 01:16:33,290 emerge when you have a diverging correlation length. 973 01:16:33,290 --> 01:16:38,130 So you want to have a place where 974 01:16:38,130 --> 01:16:40,420 there is a possible potential for continuous phase 975 01:16:40,420 --> 01:16:41,530 transition. 976 01:16:41,530 --> 01:16:46,140 And it turns out that melting in two dimensions provides that. 977 01:16:46,140 --> 01:16:48,730 So in two dimensions, you could have 978 01:16:48,730 --> 01:16:53,570 a bunch of points that could, for example, 979 01:16:53,570 --> 01:16:58,270 in a minimum energy configuration at T close to 0 980 01:16:58,270 --> 01:17:00,560 form a triangular lattice. 981 01:17:03,360 --> 01:17:05,540 Now when you go to finite temperature-- 982 01:17:05,540 --> 01:17:08,750 as we discussed, again, at the very first lecture-- 983 01:17:08,750 --> 01:17:13,990 you will start to have distortions around this. 984 01:17:13,990 --> 01:17:15,920 We can describe these distortions 985 01:17:15,920 --> 01:17:19,630 to effect of u of x and y. 986 01:17:19,630 --> 01:17:23,380 And then go to that appropriate continuum limit 987 01:17:23,380 --> 01:17:27,920 that describes the elasticity of these things. 988 01:17:27,920 --> 01:17:30,180 And it is going to look very much 989 01:17:30,180 --> 01:17:32,220 like that gradient of theta squared term 990 01:17:32,220 --> 01:17:35,970 that we wrote at the beginning, except that since this u is 991 01:17:35,970 --> 01:17:39,780 a vector, as we saw, even for an isotropic material, 992 01:17:39,780 --> 01:17:41,970 you will have the potential for having 993 01:17:41,970 --> 01:17:45,180 multiple elastic constants. 994 01:17:45,180 --> 01:17:49,110 But modeled on that, the conclusion that you would have 995 01:17:49,110 --> 01:17:53,430 is that as long as it is OK for me 996 01:17:53,430 --> 01:17:58,800 to make an expansion that is like the elastic theory-- 997 01:17:58,800 --> 01:18:03,170 some kind of a gradient of u expansion-- 998 01:18:03,170 --> 01:18:08,170 the conclusion would be that the correlations in u 999 01:18:08,170 --> 01:18:11,880 will grow logarithmically as a function of size. 1000 01:18:11,880 --> 01:18:16,670 And you will not have too long range of order, 1001 01:18:16,670 --> 01:18:19,640 but you will have some kind of a power of behavior, 1002 01:18:19,640 --> 01:18:23,730 such as the one that we have indicated over there. 1003 01:18:23,730 --> 01:18:26,290 On the other hand, when you go to very high temperature, 1004 01:18:26,290 --> 01:18:28,612 presumably this whole thing melts. 1005 01:18:28,612 --> 01:18:31,290 There is no reason to have correlations 1006 01:18:31,290 --> 01:18:35,520 beyond a few atoms that are close to you. 1007 01:18:35,520 --> 01:18:37,500 Add so typically at high temperature, 1008 01:18:37,500 --> 01:18:41,140 correlations will decay exponentially. 1009 01:18:41,140 --> 01:18:43,280 So this is low temperature expansion 1010 01:18:43,280 --> 01:18:47,200 is elastic theory expansion that we have written down 1011 01:18:47,200 --> 01:18:50,990 has to break down also in this case. 1012 01:18:50,990 --> 01:18:55,220 And a particular mechanism for its breakdown in two dimensions 1013 01:18:55,220 --> 01:18:58,590 is to create these topological defects, which 1014 01:18:58,590 --> 01:19:01,890 in the case of solid will correspond to these location 1015 01:19:01,890 --> 01:19:04,670 lines that, for example, correspond 1016 01:19:04,670 --> 01:19:10,040 to adding an addition row of particles here terminating 1017 01:19:10,040 --> 01:19:11,990 at some point. 1018 01:19:11,990 --> 01:19:15,620 And we can go through exactly the same kind of story 1019 01:19:15,620 --> 01:19:21,570 as we had before and conclude that these dislocations, 1020 01:19:21,570 --> 01:19:24,370 because of the competition between their energy 1021 01:19:24,370 --> 01:19:27,950 cost soaring logarithmically and their entropy gain growing 1022 01:19:27,950 --> 01:19:31,150 logarithmically, we need to unbind 1023 01:19:31,150 --> 01:19:33,740 at the critical temperature. 1024 01:19:33,740 --> 01:19:38,870 And so that provides a mechanism for describing 1025 01:19:38,870 --> 01:19:41,875 the melting of two-dimensional materials 1026 01:19:41,875 --> 01:19:45,690 in a language that is very similar to this, 1027 01:19:45,690 --> 01:19:51,380 except for the complications that have to do with this being 1028 01:19:51,380 --> 01:19:55,110 a vector rather than the scale of quantity. 1029 01:19:55,110 --> 01:19:59,650 And so what we find is that these topological charges 1030 01:19:59,650 --> 01:20:02,920 are different from minus 2 plus 2 pi. 1031 01:20:02,920 --> 01:20:06,930 The interactions between them is a particular version 1032 01:20:06,930 --> 01:20:12,900 of the Coulomb interaction, but that many of the other results 1033 01:20:12,900 --> 01:20:14,135 go through. 1034 01:20:14,135 --> 01:20:22,820 And we will get an idea of what happens when the solid melts 1035 01:20:22,820 --> 01:20:26,520 because of the unbinding of these locations. 1036 01:20:26,520 --> 01:20:28,150 But there is a puzzle that we will 1037 01:20:28,150 --> 01:20:31,530 find if we're not melting to a liquid, 1038 01:20:31,530 --> 01:20:34,090 but into something which is more like a liquid crystal. 1039 01:20:34,090 --> 01:20:37,670 So [INAUDIBLE] did they discover also something 1040 01:20:37,670 --> 01:20:39,675 about liquid crystals.