1 00:00:00,060 --> 00:00:02,500 The following content is provided under a Creative 2 00:00:02,500 --> 00:00:04,019 Commons license. 3 00:00:04,019 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,730 continue 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 00:00:21,291 --> 00:00:22,510 PROFESSOR: OK. 9 00:00:22,510 --> 00:00:23,245 Let's start. 10 00:00:26,730 --> 00:00:29,560 So for several lectures, we've been 11 00:00:29,560 --> 00:00:36,840 talking about the XY model, which 12 00:00:36,840 --> 00:00:44,340 is a collection of two component spins, 13 00:00:44,340 --> 00:00:46,120 for example defined on a square lattice, 14 00:00:46,120 --> 00:00:50,190 but it could be any type of lattice. 15 00:00:50,190 --> 00:00:53,680 And today we are going to compare and contrast 16 00:00:53,680 --> 00:00:56,530 that with the case of a solid. 17 00:00:59,440 --> 00:01:03,730 More specifically for simplicity, an isotropic solid. 18 00:01:09,180 --> 00:01:13,070 And a typical example of an isotropic solid in two 19 00:01:13,070 --> 00:01:16,970 dimension is the triangular lattice. 20 00:01:16,970 --> 00:01:18,210 So something like this. 21 00:01:29,090 --> 00:01:32,530 What both of these systems have in common 22 00:01:32,530 --> 00:01:37,350 is that there is an underlying continuous symmetry that 23 00:01:37,350 --> 00:01:39,020 is broken. 24 00:01:39,020 --> 00:01:45,290 In the case of the XY model, at low temperatures 25 00:01:45,290 --> 00:01:48,920 all of the spins would be pointing more or less 26 00:01:48,920 --> 00:01:49,865 in the same direction. 27 00:01:53,890 --> 00:01:57,900 Potentially with fluctuations, let's say, go 28 00:01:57,900 --> 00:02:00,090 around the direction where they're 29 00:02:00,090 --> 00:02:02,180 supposed to be pointing-- being, let's 30 00:02:02,180 --> 00:02:04,800 say, here in the vertical-- characterized 31 00:02:04,800 --> 00:02:06,550 by some kind of an angle. 32 00:02:06,550 --> 00:02:08,000 Let's call it theta. 33 00:02:11,500 --> 00:02:18,440 And essentially the uniform state at 0 temperature 34 00:02:18,440 --> 00:02:21,740 can be characterized by any theta. 35 00:02:21,740 --> 00:02:25,740 And because of the deformations, it's 36 00:02:25,740 --> 00:02:30,950 captured through these low energy Goldstone modes. 37 00:02:30,950 --> 00:02:35,100 And the probability of some particular deformation 38 00:02:35,100 --> 00:02:40,430 will be proportional at low temperature to the integral, 39 00:02:40,430 --> 00:02:43,920 let's say, in d dimensions of gradient, 40 00:02:43,920 --> 00:02:48,290 the change in theta within different spins squared. 41 00:02:48,290 --> 00:02:51,580 Of course, with some considerations as 42 00:02:51,580 --> 00:02:54,830 we discussed about topological defects, et cetera, 43 00:02:54,830 --> 00:02:58,450 being buried in this description. 44 00:02:58,450 --> 00:03:03,620 Now, similarly, away from 0 temperature 45 00:03:03,620 --> 00:03:08,910 the atoms of a solid will fluctuate around 46 00:03:08,910 --> 00:03:14,500 the position that corresponds to the minimum. 47 00:03:14,500 --> 00:03:17,050 And so the actual picture that you 48 00:03:17,050 --> 00:03:20,230 would see at any finite temperature 49 00:03:20,230 --> 00:03:24,860 would be distorted with respect to the perfect configuration 50 00:03:24,860 --> 00:03:29,190 and there would be some kind of a vector [INAUDIBLE] u 51 00:03:29,190 --> 00:03:36,160 at each location that would describe this fluctuation. 52 00:03:36,160 --> 00:03:43,020 Again, there is no cost in making a uniform distortion 53 00:03:43,020 --> 00:03:46,020 theta that is the same across space. 54 00:03:46,020 --> 00:03:48,750 And similarly here, there is no cost 55 00:03:48,750 --> 00:03:52,320 in uniformly translating things. 56 00:03:52,320 --> 00:03:56,300 So these would both imply that this row can [? continue ?] 57 00:03:56,300 --> 00:03:59,590 [INAUDIBLE] leads to Goldstone modes. 58 00:03:59,590 --> 00:04:02,920 And the corresponding energy cost 59 00:04:02,920 --> 00:04:09,050 in the case of the isotropically distorted system 60 00:04:09,050 --> 00:04:12,770 is typically returned as an integral involving 61 00:04:12,770 --> 00:04:15,460 elastic moduli, as we've seen. 62 00:04:15,460 --> 00:04:18,769 And the traditional way to write that 63 00:04:18,769 --> 00:04:32,920 is in this form where the same reason here 64 00:04:32,920 --> 00:04:35,210 that it wasn't theta that was appearing 65 00:04:35,210 --> 00:04:38,110 but gradient of theta, because a uniform theta 66 00:04:38,110 --> 00:04:40,140 does not make any cross. 67 00:04:40,140 --> 00:04:46,060 These strain fields are related derivatives of the distortion. 68 00:04:46,060 --> 00:04:48,280 But now we remember that this u is a vector 69 00:04:48,280 --> 00:04:56,890 and this strain field uij it was one half of the derivative 70 00:04:56,890 --> 00:05:00,110 of uj in the i direction derivative of ui 71 00:05:00,110 --> 00:05:01,630 in the j direction symmetrized. 72 00:05:05,770 --> 00:05:10,450 Actually, probably better to write this symmetrically 73 00:05:10,450 --> 00:05:12,590 as beta h equals to this quantity. 74 00:05:15,630 --> 00:05:21,270 And then the typical type of calculation 75 00:05:21,270 --> 00:05:25,500 that we did for the XY model was to go 76 00:05:25,500 --> 00:05:29,930 in terms of the independent spin wave modes. 77 00:05:29,930 --> 00:05:31,650 So we go to Fourier space. 78 00:05:31,650 --> 00:05:36,322 This becomes an integral ddq 2 pi to the d. 79 00:05:36,322 --> 00:05:39,900 The derivative here gave q squared. 80 00:05:39,900 --> 00:05:45,357 And then I had something like the Fourier transformed theta 81 00:05:45,357 --> 00:05:45,856 squared. 82 00:05:48,940 --> 00:05:53,880 Now I can also more clearly represent 83 00:05:53,880 --> 00:05:57,520 this in the Fourier description. 84 00:05:57,520 --> 00:06:04,930 I will get something like 1/2 integral vdq 2 pi to the d. 85 00:06:04,930 --> 00:06:11,510 And a little bit of work shows that this form Fourier 86 00:06:11,510 --> 00:06:16,190 transforms to something that is proportional to mu. 87 00:06:16,190 --> 00:06:18,940 Again, it has to be proportionality to q squared 88 00:06:18,940 --> 00:06:20,770 because the energy should go to 0 89 00:06:20,770 --> 00:06:24,310 as q goes to 0 for very long wavelength modes. 90 00:06:24,310 --> 00:06:30,140 And so we'll have something like q squared mu tilde of q 91 00:06:30,140 --> 00:06:30,880 squared. 92 00:06:30,880 --> 00:06:33,250 This is a vector squared. 93 00:06:33,250 --> 00:06:35,260 u is a vector. 94 00:06:35,260 --> 00:06:36,910 q is a vector. 95 00:06:36,910 --> 00:06:43,040 And there is another component that gives you mu plus lambda. 96 00:06:43,040 --> 00:06:47,300 And consistent with rotational symmetry, 97 00:06:47,300 --> 00:06:52,350 we can have a term that is q dotted with u tilde of q, 98 00:06:52,350 --> 00:06:53,540 the whole thing squared. 99 00:06:58,480 --> 00:07:01,630 And again, you can reason that you 100 00:07:01,630 --> 00:07:05,350 can describe the isotropic system in terms of just two 101 00:07:05,350 --> 00:07:09,310 elastic moduli is that the only rotational invariant 102 00:07:09,310 --> 00:07:12,710 quantities are q squared, u squared, and q.u, 103 00:07:12,710 --> 00:07:17,180 that those are the only terms that you can [? find. ?] 104 00:07:17,180 --> 00:07:19,210 Now the next thing. 105 00:07:19,210 --> 00:07:23,220 Once you know this form, you can immediately say something 106 00:07:23,220 --> 00:07:25,110 about the fluctuations. 107 00:07:25,110 --> 00:07:30,940 So you say that expectation value of theta tilde q theta 108 00:07:30,940 --> 00:07:38,180 tilde q prime, if I assume that this is a Gaussian theory 109 00:07:38,180 --> 00:07:40,310 and these are the only fluctuations 110 00:07:40,310 --> 00:07:42,620 that I'm considering, the answer is 111 00:07:42,620 --> 00:07:47,970 going to be 2 pi to the d delta function q plus q prime. 112 00:07:47,970 --> 00:07:53,410 And then I will get a factor of 1 over kq squared. 113 00:07:59,330 --> 00:08:04,900 The corresponding thing here is a bit more complicated. 114 00:08:04,900 --> 00:08:09,490 Because as we said, this quantity u tilde is a vector. 115 00:08:09,490 --> 00:08:12,990 And I can look at the correlation between, say, 116 00:08:12,990 --> 00:08:16,380 the i-th component of that vector along mode 117 00:08:16,380 --> 00:08:21,740 q, the j-th component of that vector along mu 2 prime 118 00:08:21,740 --> 00:08:24,750 and ask what this is. 119 00:08:24,750 --> 00:08:29,900 And again, only the same values of q 120 00:08:29,900 --> 00:08:33,350 are coupled together, so we get the usual formula over here. 121 00:08:37,620 --> 00:08:43,809 If this term was the only term in the story, that 122 00:08:43,809 --> 00:08:49,040 is if we were dealing with vectors q and u that 123 00:08:49,040 --> 00:08:50,850 are orthogonal to each other, then we 124 00:08:50,850 --> 00:08:55,680 would just get the 1 over q squared. 125 00:08:55,680 --> 00:09:02,030 And so there is actually a term that is like that. 126 00:09:02,030 --> 00:09:06,690 But because of this other term, there is the possibility 127 00:09:06,690 --> 00:09:11,060 that q and u are in the same direction, in which case, 128 00:09:11,060 --> 00:09:13,470 these two costs will add up and I 129 00:09:13,470 --> 00:09:15,960 will get something that is proportional to 2 130 00:09:15,960 --> 00:09:16,820 mu plus lambda. 131 00:09:16,820 --> 00:09:26,980 And so then I will get 2 mu plus lambda q squared. 132 00:09:26,980 --> 00:09:34,360 And here I will have qi qj. 133 00:09:34,360 --> 00:09:36,965 This becomes q to the 4th. 134 00:09:36,965 --> 00:09:41,835 Let me make sure that I did not write this incorrectly. 135 00:09:50,580 --> 00:09:53,100 Yes, I did write it incorrectly. 136 00:09:53,100 --> 00:09:57,025 It turns out that there is a mu plus lambda out here too. 137 00:10:03,330 --> 00:10:07,120 But essentially there's just a small complication 138 00:10:07,120 --> 00:10:13,710 that we get because of the fact that you have a vector u. 139 00:10:13,710 --> 00:10:17,200 But overall the scaling is something 140 00:10:17,200 --> 00:10:21,090 that goes like 1 over q squared, which is characteristics 141 00:10:21,090 --> 00:10:30,710 of Goldstone modes here with additional vectorial things 142 00:10:30,710 --> 00:10:32,990 to worry about. 143 00:10:32,990 --> 00:10:36,141 Let me make sure I separate these two. 144 00:10:39,990 --> 00:10:44,410 Now one of the things that we have been concerned with 145 00:10:44,410 --> 00:10:51,170 is whether these fluctuations destroy long range order, so 146 00:10:51,170 --> 00:10:55,180 for long range order the information that we would like 147 00:10:55,180 --> 00:10:59,150 to have is that the entirety of the system 148 00:10:59,150 --> 00:11:01,980 is roughly pointing in the same direction, 149 00:11:01,980 --> 00:11:04,610 meaning that if I know that this beam here 150 00:11:04,610 --> 00:11:07,010 is pointed in the vertical direction, 151 00:11:07,010 --> 00:11:11,300 if I go very far away, it is still more or less 152 00:11:11,300 --> 00:11:15,030 pointed out-- pointing in the same direction. 153 00:11:15,030 --> 00:11:18,600 So for that, we could look, for example, 154 00:11:18,600 --> 00:11:22,350 as the degree which [? with ?] these two spins? 155 00:11:22,350 --> 00:11:23,175 Yes, question? 156 00:11:23,175 --> 00:11:25,496 AUDIENCE: [INAUDIBLE] q squared [INAUDIBLE]? 157 00:11:29,488 --> 00:11:30,720 PROFESSOR: This is 4. 158 00:11:30,720 --> 00:11:32,485 Is that what you're worried about? 159 00:11:32,485 --> 00:11:33,195 AUDIENCE: Yeah. 160 00:11:33,195 --> 00:11:34,195 It should be 2, I guess. 161 00:11:34,195 --> 00:11:37,244 PROFESSOR: No, because I have two powers of q out front. 162 00:11:37,244 --> 00:11:37,910 AUDIENCE: Right. 163 00:11:37,910 --> 00:11:39,326 PROFESSOR: So I think in the notes 164 00:11:39,326 --> 00:11:42,540 what I have is that I have actually 165 00:11:42,540 --> 00:11:44,110 written things this way. 166 00:11:44,110 --> 00:11:47,710 I put the factor of mu q squared out front. 167 00:11:47,710 --> 00:11:53,190 And then I had this as qi qj over q. 168 00:11:53,190 --> 00:11:54,290 And I think-- 169 00:11:54,290 --> 00:11:57,540 AUDIENCE: [INAUDIBLE] q squared [INAUDIBLE]? 170 00:11:57,540 --> 00:11:59,980 PROFESSOR: q squared. 171 00:11:59,980 --> 00:12:05,160 Let me one more time check to make sure that that's not-- 172 00:12:05,160 --> 00:12:06,610 actually there's a minus sign. 173 00:12:09,410 --> 00:12:11,790 All right. 174 00:12:11,790 --> 00:12:13,945 AUDIENCE: Wouldn't there be another factor of mu 175 00:12:13,945 --> 00:12:17,474 to compensate for the factoring out? 176 00:12:17,474 --> 00:12:18,330 PROFESSOR: Yeah. 177 00:12:18,330 --> 00:12:24,000 So let's check this in the following fashion. 178 00:12:24,000 --> 00:12:31,730 So I can look at modes where u and q are 179 00:12:31,730 --> 00:12:34,300 perpendicular to each other. 180 00:12:34,300 --> 00:12:36,980 When u and q are perpendicular to each other, 181 00:12:36,980 --> 00:12:40,025 I don't have this there. 182 00:12:40,025 --> 00:12:43,040 And so then what I should have is just a cost 183 00:12:43,040 --> 00:12:45,900 that is 1 over mu q squared. 184 00:12:45,900 --> 00:12:46,400 OK. 185 00:12:46,400 --> 00:12:48,710 So let's see if that comes about. 186 00:12:57,100 --> 00:13:01,000 What I can do is let's say imagine 187 00:13:01,000 --> 00:13:09,720 that I'm looking at a q that is oriented along the x direction. 188 00:13:09,720 --> 00:13:15,370 And I look at u's that are in the y direction. 189 00:13:15,370 --> 00:13:22,390 And I correlate two u's in the y direction. 190 00:13:22,390 --> 00:13:25,660 So I will definitely get a delta ij here. 191 00:13:25,660 --> 00:13:28,170 I would have a 1. 192 00:13:28,170 --> 00:13:34,720 Here I would get a factor of nothing, 193 00:13:34,720 --> 00:13:37,740 because this term is absent. 194 00:13:37,740 --> 00:13:40,250 The q's that I'm looking at don't 195 00:13:40,250 --> 00:13:44,700 have any component along the direction of u that I'm having. 196 00:13:44,700 --> 00:13:47,920 So this is absent and I will get a 1 197 00:13:47,920 --> 00:13:51,710 over mu q squared, which is consistent with that. 198 00:13:51,710 --> 00:13:59,250 Now let's look at the case where I look at u's that 199 00:13:59,250 --> 00:14:02,095 are aligned with the vector q. 200 00:14:02,095 --> 00:14:04,720 Let's say again in the x direction. 201 00:14:04,720 --> 00:14:09,180 Then, as far as the energy is concerned, 202 00:14:09,180 --> 00:14:12,620 both terms are present and the cost 203 00:14:12,620 --> 00:14:15,850 should be 2 mu plus lambda. 204 00:14:15,850 --> 00:14:18,160 So the answer that I should ultimately get 205 00:14:18,160 --> 00:14:20,560 should be 1 over 2 mu plus lambda. 206 00:14:20,560 --> 00:14:22,970 So let's check. 207 00:14:22,970 --> 00:14:28,130 So here q squared-- and cancels with this q squared. 208 00:14:28,130 --> 00:14:30,920 All of the q's are in the direction of the indices 209 00:14:30,920 --> 00:14:32,620 that I'm looking at. 210 00:14:32,620 --> 00:14:34,130 So that cancels. 211 00:14:34,130 --> 00:14:35,020 This is 1. 212 00:14:35,020 --> 00:14:39,685 So I will get 1 minus mu plus lambda over 2 mu plus lambda. 213 00:14:39,685 --> 00:14:47,110 So then I will get 2 mu plus lambda minus mu minus lambda. 214 00:14:47,110 --> 00:14:50,110 So I will get a mu in front that cancels this. 215 00:14:50,110 --> 00:14:53,040 And I will be left with just the 2 mu plus lambda. 216 00:14:53,040 --> 00:14:55,725 So we've checked that this is correct. 217 00:15:00,850 --> 00:15:01,540 All right. 218 00:15:01,540 --> 00:15:05,790 So this is a proof, if you like, by induction. 219 00:15:05,790 --> 00:15:09,830 You knew that the only forms that are consistent 220 00:15:09,830 --> 00:15:13,860 are delta ij and qi qj over q squared. 221 00:15:13,860 --> 00:15:17,110 So we can give them two coefficients, a and b, 222 00:15:17,110 --> 00:15:21,720 and then adjust a and b so that these two limits that I 223 00:15:21,720 --> 00:15:24,250 considered are reproduced. 224 00:15:24,250 --> 00:15:27,073 So-- OK. 225 00:15:27,073 --> 00:15:30,300 I didn't have to resort to my notes 226 00:15:30,300 --> 00:15:34,550 if I was willing to spend the corresponding three 227 00:15:34,550 --> 00:15:37,520 minutes here. 228 00:15:37,520 --> 00:15:42,400 So going back here, in order to see the degree of alignment 229 00:15:42,400 --> 00:15:46,370 of the two spins that we have over here, what we need to do 230 00:15:46,370 --> 00:15:49,260 is to do is to look something like cosine of, say, 231 00:15:49,260 --> 00:15:54,022 theta at location x minus theta at location x 232 00:15:54,022 --> 00:16:01,390 prime, which is non other than the real part 233 00:16:01,390 --> 00:16:06,220 of the expectation value of e to the theta at location 234 00:16:06,220 --> 00:16:08,810 x minus theta at location x prime. 235 00:16:12,680 --> 00:16:20,180 And assuming that everything is governed by this Gaussian rate, 236 00:16:20,180 --> 00:16:25,240 the answer is going to obtain by computing this Gaussian 237 00:16:25,240 --> 00:16:29,590 as exponential of minus 1/2 the average 238 00:16:29,590 --> 00:16:32,560 of theta x minus theta x prime squared. 239 00:16:35,680 --> 00:16:39,270 So then my task is to replace theta x 240 00:16:39,270 --> 00:16:44,870 and theta x prime in terms of these theta tilde in Fourier 241 00:16:44,870 --> 00:16:46,210 space. 242 00:16:46,210 --> 00:16:49,640 And then this will give me some expectation 243 00:16:49,640 --> 00:16:52,340 that involves this average. 244 00:16:52,340 --> 00:16:55,100 And when I complete that, eventually I 245 00:16:55,100 --> 00:16:58,563 need to Fourier transform 1 over q squared. 246 00:16:58,563 --> 00:17:01,970 And we've seen that the Fourier transform of the 1 over q 247 00:17:01,970 --> 00:17:05,060 squared is the Coulomb interaction. 248 00:17:05,060 --> 00:17:10,849 And so ultimately this becomes exponential of minus of 1 249 00:17:10,849 --> 00:17:11,920 over k. 250 00:17:11,920 --> 00:17:15,770 The Coulomb interaction as a function 251 00:17:15,770 --> 00:17:20,950 of x minus x prime with the appropriate cutoff 252 00:17:20,950 --> 00:17:23,040 included in this expression. 253 00:17:27,560 --> 00:17:31,740 And then the statement that we always make 254 00:17:31,740 --> 00:17:36,620 is that this entity at large distances, 255 00:17:36,620 --> 00:17:40,690 the Coulomb interaction saying three dimension, falls off as 1 256 00:17:40,690 --> 00:17:42,520 over separation. 257 00:17:42,520 --> 00:17:45,830 So in three dimensions, this at large distances 258 00:17:45,830 --> 00:17:49,210 goes to a constant 1 over k is something that 259 00:17:49,210 --> 00:17:50,930 is proportional to temperature. 260 00:17:50,930 --> 00:17:52,810 So after doing enough temperature, 261 00:17:52,810 --> 00:17:57,650 this entity at large distances goes to a constant. 262 00:17:57,650 --> 00:18:01,810 So there is some knowledge about the correlation 263 00:18:01,810 --> 00:18:04,440 between these things that is left. 264 00:18:04,440 --> 00:18:08,200 Whereas in two dimensions and below, 265 00:18:08,200 --> 00:18:11,440 when I go to sufficiently large distances 266 00:18:11,440 --> 00:18:14,720 the Coulomb interaction diverges. 267 00:18:14,720 --> 00:18:20,900 This whole thing goes to 0 and information is lost. 268 00:18:20,900 --> 00:18:25,740 So this is the in content that Goldstone modes 269 00:18:25,740 --> 00:18:30,860 destroy through long range order in two dimensions and below. 270 00:18:30,860 --> 00:18:34,560 Except that we saw that in two dimensions, 271 00:18:34,560 --> 00:18:38,430 because the Coulomb interaction grows logarithmically, 272 00:18:38,430 --> 00:18:41,340 this correlation fall off according 273 00:18:41,340 --> 00:18:49,440 to this description only as a power law with an exponent 274 00:18:49,440 --> 00:18:51,610 connected to k. 275 00:18:51,610 --> 00:18:55,710 And then the discussion that we had for the XY model 276 00:18:55,710 --> 00:18:57,850 was that at high temperatures you 277 00:18:57,850 --> 00:19:00,740 have exponential decay in two dimensions. 278 00:19:00,740 --> 00:19:04,240 At low temperature you have power law decay. 279 00:19:04,240 --> 00:19:06,310 Although there is no true long range order, 280 00:19:06,310 --> 00:19:09,160 there has to be a phase transition. 281 00:19:09,160 --> 00:19:11,040 And we saw that this transition was 282 00:19:11,040 --> 00:19:16,360 described by the unbinding of these topological defects. 283 00:19:16,360 --> 00:19:21,360 So we would like to state that something similar to that 284 00:19:21,360 --> 00:19:22,780 is going on over here. 285 00:19:25,580 --> 00:19:29,000 The only reason I went through this 286 00:19:29,000 --> 00:19:35,100 was to figure out what the analog of this entity 287 00:19:35,100 --> 00:19:39,180 should be in the case of the solid. 288 00:19:39,180 --> 00:19:44,765 So the analog of our theta is this vector u, the distortion. 289 00:19:47,790 --> 00:19:50,480 What should be the analog of e to the i theta? 290 00:19:54,150 --> 00:20:01,540 And the things that you should keep in mind is that theta, 291 00:20:01,540 --> 00:20:03,390 I can't really tell it's difference 292 00:20:03,390 --> 00:20:09,090 between theta and theta plus 2 pi, theta of 4 pi, et cetera. 293 00:20:09,090 --> 00:20:13,230 So this e to the i theta captures that. 294 00:20:13,230 --> 00:20:16,510 If theta goes to theta plus a multiple of 2 pi, e 295 00:20:16,510 --> 00:20:19,750 to the i theta remains invariant. 296 00:20:19,750 --> 00:20:24,470 So it's a good measure of ordering. 297 00:20:24,470 --> 00:20:27,680 It's better than theta itself. 298 00:20:27,680 --> 00:20:31,550 So similarly, there's a better measure 299 00:20:31,550 --> 00:20:34,910 of deviation from perfect order here 300 00:20:34,910 --> 00:20:38,510 for the case of the lattice. 301 00:20:38,510 --> 00:20:42,980 You say, well, why should I worry about 302 00:20:42,980 --> 00:20:48,836 that since I can measure what u is from the perfect position? 303 00:20:48,836 --> 00:20:52,880 Well, my answer is this point here. 304 00:20:52,880 --> 00:20:56,280 Did it come from here or from there? 305 00:20:56,280 --> 00:21:00,185 Should you use for u this or should you use that? 306 00:21:00,185 --> 00:21:02,910 You don't know. 307 00:21:02,910 --> 00:21:08,680 So the u that you have the to use similar to the theta 308 00:21:08,680 --> 00:21:12,920 that you're using is somewhat arbitrary. 309 00:21:12,920 --> 00:21:18,440 And the arbitrariness of it is given by unit vectors 310 00:21:18,440 --> 00:21:21,080 along the original lattice. 311 00:21:21,080 --> 00:21:26,570 That is, u and any u that is increased or decreased 312 00:21:26,570 --> 00:21:31,300 by some lattice vector of the original lattice 313 00:21:31,300 --> 00:21:32,360 is the same thing. 314 00:21:32,360 --> 00:21:35,230 You can't take them apart in the same sense that theta 315 00:21:35,230 --> 00:21:38,890 and multiple of 2 pi you cannot. 316 00:21:38,890 --> 00:21:43,980 So then that suggest that I can construct some kind of an order 317 00:21:43,980 --> 00:21:50,330 parameter, I call it row G, which is like this. 318 00:21:50,330 --> 00:21:52,790 This e to i something. 319 00:21:52,790 --> 00:21:57,200 It's e to the iu but u is a vector. 320 00:21:57,200 --> 00:22:04,490 And what I do I multiple by a G, which is an inverse lattice 321 00:22:04,490 --> 00:22:06,560 vector. 322 00:22:06,560 --> 00:22:10,830 So G is an inverse lattice vector. 323 00:22:14,950 --> 00:22:19,020 And essentially the inverse lattice vectors 324 00:22:19,020 --> 00:22:24,210 are defined to have the property that when you multiply them 325 00:22:24,210 --> 00:22:30,800 with any of the original lattice vectors-- which are, let's say, 326 00:22:30,800 --> 00:22:34,930 parametrized by two integers, m and n-- 327 00:22:34,930 --> 00:22:37,600 the answer is going to be some multiple of 2 pi. 328 00:22:44,300 --> 00:22:48,030 So you can see that irrespective of whether I choose 329 00:22:48,030 --> 00:22:53,120 u as distortion from one lattice position or any other lattice 330 00:22:53,120 --> 00:22:56,330 position that happens to be nearby, 331 00:22:56,330 --> 00:23:00,800 this measure is going to be giving you the same face. 332 00:23:03,940 --> 00:23:09,720 So in the same way as before for the spins where 333 00:23:09,720 --> 00:23:13,010 I asked whether spins at large distances 334 00:23:13,010 --> 00:23:18,270 are correlated long range, I can do the same thing here 335 00:23:18,270 --> 00:23:32,600 and define a long range correlations for a solid 336 00:23:32,600 --> 00:23:42,340 by looking at e to the iG dotted with, let's say, u at position 337 00:23:42,340 --> 00:23:45,190 x minus u at position x prime. 338 00:23:45,190 --> 00:23:49,340 And x and x prime are two points on the lattice. 339 00:23:49,340 --> 00:23:54,980 And again, the u's that I have are Gaussian distributed. 340 00:23:54,980 --> 00:23:56,960 So this I will complete the square. 341 00:23:56,960 --> 00:24:00,310 I will have exponential minus 1/2. 342 00:24:00,310 --> 00:24:04,020 Well, OK, let me be a little bit careful with this. 343 00:24:04,020 --> 00:24:08,920 I have G alpha G beta over 2 because I 344 00:24:08,920 --> 00:24:11,490 have two factors of G dot u. 345 00:24:11,490 --> 00:24:15,670 I will write them using Einstein's summation convention 346 00:24:15,670 --> 00:24:27,015 as G alpha times u alpha component times the u beta 347 00:24:27,015 --> 00:24:27,515 component. 348 00:24:36,160 --> 00:24:42,210 So once more what we can do is to then write 349 00:24:42,210 --> 00:24:47,170 these u's in terms of their Fourier vectors. 350 00:24:47,170 --> 00:24:51,180 So what I will have is exponential minus G alpha G 351 00:24:51,180 --> 00:24:52,480 beta over 2. 352 00:24:55,260 --> 00:25:02,675 I will need to integrate over q's. 353 00:25:06,830 --> 00:25:13,790 Because of these factors, I will get 354 00:25:13,790 --> 00:25:21,200 something like 2 minus 2 cosine of qx minus x prime. 355 00:25:29,590 --> 00:25:34,930 So whenever you go to Fourier space, you will get e to the iq 356 00:25:34,930 --> 00:25:38,100 x minus e to the iq x prime. 357 00:25:38,100 --> 00:25:40,080 Different one for here with different q 358 00:25:40,080 --> 00:25:42,350 primes, but when you take the average, 359 00:25:42,350 --> 00:25:45,360 q prime is set to minus q so then you 360 00:25:45,360 --> 00:25:47,180 get this form as usual. 361 00:25:47,180 --> 00:25:51,390 And then I will have the average Fourier transform use, which 362 00:25:51,390 --> 00:25:53,150 is what I wrote over there. 363 00:25:53,150 --> 00:26:02,890 I will get a 1 over mu q square delta i delta alpha beta minus 364 00:26:02,890 --> 00:26:12,938 q alpha q beta q squared mu plus lambda over mu plus lambda. 365 00:26:26,870 --> 00:26:32,090 There is a little bit of subtlety here. 366 00:26:32,090 --> 00:26:36,170 So I will do something this is not quite kosher 367 00:26:36,170 --> 00:26:38,740 but gives me the right answer. 368 00:26:38,740 --> 00:26:46,110 Certainly I can take this G alpha a beta inside here. 369 00:26:46,110 --> 00:26:49,200 When it multiplies delta alpha beta, 370 00:26:49,200 --> 00:26:50,870 I will simply get G squared. 371 00:26:53,600 --> 00:26:55,410 I really G alpha G alpha. 372 00:26:55,410 --> 00:26:56,210 Sum over alpha. 373 00:26:56,210 --> 00:26:58,990 It will give me G squared. 374 00:26:58,990 --> 00:27:05,360 Here what I will get is a G dot q 375 00:27:05,360 --> 00:27:11,110 squared because I will have two Gq's divided by q squared. 376 00:27:15,130 --> 00:27:20,270 Ultimately I have to integrate over all values of q, including 377 00:27:20,270 --> 00:27:21,170 all angle values. 378 00:27:23,990 --> 00:27:27,150 So there will be, as far as the angle is concerned, 379 00:27:27,150 --> 00:27:30,930 something like a cosine squared here. 380 00:27:30,930 --> 00:27:34,450 And the rather unconventional thing to do 381 00:27:34,450 --> 00:27:37,230 is to angular averages that first 382 00:27:37,230 --> 00:27:39,700 and write this as G squared over 2. 383 00:27:44,100 --> 00:27:46,520 Once you do that, then essentially 384 00:27:46,520 --> 00:27:51,730 you can see that this structure just gives you a constant. 385 00:27:51,730 --> 00:27:55,210 And what you will need to do is the usual Fourier 386 00:27:55,210 --> 00:27:58,990 transformation of 1 over q squared. 387 00:27:58,990 --> 00:28:03,300 So eventually you can see that the answer 388 00:28:03,300 --> 00:28:09,080 is going to be proportional to minus G squared. 389 00:28:09,080 --> 00:28:15,440 I will get a factor of 2 mu u from here. 390 00:28:15,440 --> 00:28:20,220 Once I have written this as G squared over 2, 391 00:28:20,220 --> 00:28:24,330 I will get 1 minus 1/2 of this quantity. 392 00:28:24,330 --> 00:28:28,230 So that becomes 2 mu plus lambda-- 393 00:28:28,230 --> 00:28:33,130 or 4 mu plus lambda minus mu minus lambda. 394 00:28:33,130 --> 00:28:41,880 So I will get a 3 mu plus lambda divided by 2 2 mu plus lambda. 395 00:28:41,880 --> 00:28:45,490 And so there's a factor of 4 here. 396 00:28:45,490 --> 00:28:48,830 So that's just the numerics that goes out front. 397 00:28:48,830 --> 00:28:52,750 The key to the whole thing will be the usual integration of 1 398 00:28:52,750 --> 00:28:54,160 over q squared. 399 00:28:54,160 --> 00:28:58,270 Just as we had before, I will get again the same Coulomb 400 00:28:58,270 --> 00:29:01,710 interaction as a function of x minus x 401 00:29:01,710 --> 00:29:04,920 prime with appropriate lattice cut-off 402 00:29:04,920 --> 00:29:06,920 introduced so that the short distance 403 00:29:06,920 --> 00:29:08,075 behavior is controlled. 404 00:29:14,350 --> 00:29:16,810 This whole thing has a name. 405 00:29:16,810 --> 00:29:18,595 It's called a Debye-Waller factor. 406 00:29:25,160 --> 00:29:25,710 Well, almost. 407 00:29:28,561 --> 00:29:29,060 In 3D. 408 00:29:34,030 --> 00:29:36,050 And the point is that if I'm looking 409 00:29:36,050 --> 00:29:39,056 at this whole thing in three dimensions, 410 00:29:39,056 --> 00:29:44,026 at large enough separation this will go to a constant, 411 00:29:44,026 --> 00:29:46,290 just as we had before. 412 00:29:46,290 --> 00:29:51,410 And I will conclude that this entity, as I 413 00:29:51,410 --> 00:29:55,000 look at two points that are very far apart, at least that's 414 00:29:55,000 --> 00:29:58,500 sufficiently low temperature and all of my mus and lambdas 415 00:29:58,500 --> 00:30:00,280 are scaled inversely with temperature, 416 00:30:00,280 --> 00:30:03,150 so this whole thing is proportional to temperature. 417 00:30:03,150 --> 00:30:07,450 Essentially, you can see that this information 418 00:30:07,450 --> 00:30:10,870 at large distance goes to a finite constant, 419 00:30:10,870 --> 00:30:14,250 telling me that there's long range orders preserved. 420 00:30:14,250 --> 00:30:16,880 Again, when I go to two dimensions, 421 00:30:16,880 --> 00:30:22,350 you would say that this, at large distances, 422 00:30:22,350 --> 00:30:24,510 grows logarithmically. 423 00:30:24,510 --> 00:30:27,780 And so the correlations will fall off, 424 00:30:27,780 --> 00:30:32,630 but fall of as a power law. 425 00:30:32,630 --> 00:30:38,780 Now for the case of a solid, usually 426 00:30:38,780 --> 00:30:47,960 what you do in order to probe for the order is 427 00:30:47,960 --> 00:30:56,400 you shine x-rays and you see what comes out. 428 00:30:56,400 --> 00:31:00,170 So essentially, you will see that there 429 00:31:00,170 --> 00:31:03,180 will be some scattering. 430 00:31:03,180 --> 00:31:08,740 And the way that the wave vector of the light has changed 431 00:31:08,740 --> 00:31:11,630 is characterized by a vector q. 432 00:31:11,630 --> 00:31:15,755 And we discussed this already, that the structure factor 433 00:31:15,755 --> 00:31:24,340 s of q is going to be related to expectation value of the phase 434 00:31:24,340 --> 00:31:26,280 that you get from the different points. 435 00:31:26,280 --> 00:31:31,070 So I will get a factor of e to the i q. 436 00:31:31,070 --> 00:31:37,900 the location of the different atoms, let's call them r, of x. 437 00:31:37,900 --> 00:31:40,140 I have to do a sum over x. 438 00:31:40,140 --> 00:31:43,010 I have to scatter from all of the atoms. 439 00:31:43,010 --> 00:31:45,740 There's some overall form factor that if I'm not 440 00:31:45,740 --> 00:31:48,070 really interested, but this whole thing 441 00:31:48,070 --> 00:31:49,245 has to be first squished. 442 00:31:53,850 --> 00:31:59,620 So once I square this, I will be getting averages 443 00:31:59,620 --> 00:32:02,060 between pairs of points. 444 00:32:02,060 --> 00:32:04,650 Because of translational symmetry 445 00:32:04,650 --> 00:32:08,770 I can relate to that sum over pairs of points 446 00:32:08,770 --> 00:32:11,370 to a center of mass that typically 447 00:32:11,370 --> 00:32:16,410 would pick up the number of points that I have. 448 00:32:16,410 --> 00:32:19,990 And then the expectation value of e 449 00:32:19,990 --> 00:32:23,960 to the something like sum over all points, e 450 00:32:23,960 --> 00:32:34,930 to the i q.r r of x minus r of 0 average. 451 00:32:34,930 --> 00:32:36,990 This average I can certainly take out here. 452 00:32:42,435 --> 00:32:43,409 Yes? 453 00:32:43,409 --> 00:32:44,170 STUDENT: Sorry. 454 00:32:44,170 --> 00:32:46,230 Is r what you're calling u before? 455 00:32:46,230 --> 00:32:47,600 PROFESSOR: No. 456 00:32:47,600 --> 00:32:52,910 So r is the actual location of where the particle is location. 457 00:32:52,910 --> 00:32:58,130 So r, let's say for a particle that at zero temperature 458 00:32:58,130 --> 00:33:00,500 was sitting at lattice point-- that is, 459 00:33:00,500 --> 00:33:05,520 let's say, in two dimensions labeled by m and n-- 460 00:33:05,520 --> 00:33:09,090 it is shifted by an amount that is u that I 461 00:33:09,090 --> 00:33:11,320 would assign to that point. 462 00:33:11,320 --> 00:33:16,900 So the r's that I put here are the actual physical position 463 00:33:16,900 --> 00:33:18,270 of the atom. 464 00:33:18,270 --> 00:33:20,210 Which is composed of where it was 465 00:33:20,210 --> 00:33:23,143 sitting at 0 temperature plus a little bit more. 466 00:33:30,880 --> 00:33:38,295 So when I do this, indeed I will get a sum over all positions. 467 00:33:41,850 --> 00:33:48,555 This could be a label mn, whatever, of e to the iq. 468 00:33:51,300 --> 00:33:53,890 The difference between the two r's 469 00:33:53,890 --> 00:33:57,930 would be, first of all, the difference between these two 470 00:33:57,930 --> 00:34:02,570 at 0 temperature, which would be some lattice 471 00:34:02,570 --> 00:34:05,060 vector of some variety. 472 00:34:05,060 --> 00:34:06,705 Let's call r 0 of x. 473 00:34:09,940 --> 00:34:15,230 And that quantity has [INAUDIBLE] fluctuations. 474 00:34:15,230 --> 00:34:23,080 The fluctuations go into e to the iq.u of x minus mu of 0. 475 00:34:32,170 --> 00:34:35,870 Now if I'm doing this at 0 temperature or very close to 0 476 00:34:35,870 --> 00:34:41,900 temperature where this is either 1 at 0 temperature or not too 477 00:34:41,900 --> 00:34:44,900 much fluctuating at high temperatures, 478 00:34:44,900 --> 00:34:47,929 I will essentially be adding a lot of e to iqr's. 479 00:34:51,520 --> 00:34:53,900 These are essentially phases. 480 00:34:53,900 --> 00:34:58,210 They can be positive, negative, all over the complex plane. 481 00:34:58,210 --> 00:35:02,400 And typically you would expect that for a randomly chosen q, 482 00:35:02,400 --> 00:35:08,410 the answer from adding all of these things will be 0. 483 00:35:08,410 --> 00:35:12,510 So the only time we're close to 0 temperature, 484 00:35:12,510 --> 00:35:18,450 you expect this to be something that is significant. 485 00:35:18,450 --> 00:35:36,470 Scattering is when q is one of these inverse lattice vectors. 486 00:35:36,470 --> 00:35:41,670 So if I was really doing this at 0 temperature, this would be 1. 487 00:35:41,670 --> 00:35:45,030 And the answer that I would get is essentially 488 00:35:45,030 --> 00:35:49,350 I would get these delta function back peaks 489 00:35:49,350 --> 00:35:53,665 at the locations that correspond to the inverse lattice spacing. 490 00:35:53,665 --> 00:35:56,215 And that's presuming something that you have seen 491 00:35:56,215 --> 00:35:58,470 in crystallography or whatever. 492 00:35:58,470 --> 00:36:02,120 You take a solid and you scatter light from it 493 00:36:02,120 --> 00:36:03,885 and you will back points. 494 00:36:13,120 --> 00:36:19,350 So what I expect is that when I have 495 00:36:19,350 --> 00:36:26,750 a q that is roughly close to G but not necessarily sitting 496 00:36:26,750 --> 00:36:30,870 exactly at that point, what I would get 497 00:36:30,870 --> 00:36:35,270 is a scattering that these proportional to an integral 498 00:36:35,270 --> 00:36:41,060 over the entire lattice of something 499 00:36:41,060 --> 00:36:50,590 like how far I am away from the right position. 500 00:36:50,590 --> 00:36:55,395 And I should really write this as a sum over lattice points, 501 00:36:55,395 --> 00:37:00,250 but I can approximate this by an integral. 502 00:37:00,250 --> 00:37:04,190 The integral again has the property that it will give me 0 503 00:37:04,190 --> 00:37:07,700 unless q is exactly sitting at G. 504 00:37:07,700 --> 00:37:11,160 And then for here, I evaluate this quantity 505 00:37:11,160 --> 00:37:14,240 when q is close to G according to what 506 00:37:14,240 --> 00:37:18,120 I have calculated up there. 507 00:37:18,120 --> 00:37:23,600 So I will get exponential of minus G 508 00:37:23,600 --> 00:37:29,160 squared-- What do I have? 509 00:37:29,160 --> 00:37:35,770 Over 4 mu, three mu plus lambda, 2 mu plus lambda. 510 00:37:35,770 --> 00:37:40,140 The Coulomb interaction as a function of x. 511 00:37:45,981 --> 00:37:46,480 Yes? 512 00:37:46,480 --> 00:37:49,012 STUDENT: Are you missing an overall factor of m? 513 00:37:49,012 --> 00:37:51,510 PROFESSOR: That's why I wrote the proportionality. 514 00:37:51,510 --> 00:37:56,340 Because what I'm really interested is what you see. 515 00:37:56,340 --> 00:38:02,160 And what you see is how things are varying as a function of q. 516 00:38:02,160 --> 00:38:04,230 So I'll tell you what the answer is. 517 00:38:04,230 --> 00:38:07,410 So you do the Fourier transform from this. 518 00:38:07,410 --> 00:38:14,640 And in d equals to 3, what you find at 0 temperature, 519 00:38:14,640 --> 00:38:18,700 as we said, you will get Bragg spots 520 00:38:18,700 --> 00:38:21,830 at the locations of G's that correspond 521 00:38:21,830 --> 00:38:29,990 to the inverse lattice vectors, which actually themselves 522 00:38:29,990 --> 00:38:31,030 form a lattice. 523 00:38:36,420 --> 00:38:40,040 So this is essentially the space of q. 524 00:38:40,040 --> 00:38:45,130 And these are the different values of G. 525 00:38:45,130 --> 00:38:49,290 Now what happens is that the strength of each one 526 00:38:49,290 --> 00:38:53,210 of these delta functions is then modulated 527 00:38:53,210 --> 00:38:57,620 by this factor evaluated at the corresponding G. 528 00:38:57,620 --> 00:39:00,890 And since it is proportional to G squared, 529 00:39:00,890 --> 00:39:03,750 you find that the spots that are close to the origin 530 00:39:03,750 --> 00:39:04,580 are well defined. 531 00:39:08,360 --> 00:39:15,210 As you go further and further away, the become very weaker. 532 00:39:15,210 --> 00:39:17,630 And at some point you cease to see them. 533 00:39:17,630 --> 00:39:21,290 So that's the-- the Debye-Waller factor 534 00:39:21,290 --> 00:39:25,910 describes how these things vanish as you 535 00:39:25,910 --> 00:39:27,380 go further and further away. 536 00:39:27,380 --> 00:39:30,420 And essentially, you find that they go and diminish 537 00:39:30,420 --> 00:39:33,400 as e to the minus G squared. 538 00:39:33,400 --> 00:39:39,960 But clearly, this kind of scattering from a crystal 539 00:39:39,960 --> 00:39:42,100 is very different from the scattering 540 00:39:42,100 --> 00:39:44,460 that you expect from an liquid, which 541 00:39:44,460 --> 00:39:51,860 is essentially you will have a bright spot at q close to 0 542 00:39:51,860 --> 00:39:54,320 and things that would symmetrically fall off, 543 00:39:54,320 --> 00:39:56,170 maybe with a couple of oscillations 544 00:39:56,170 --> 00:39:59,600 due to the size of the particles, at cetera. 545 00:40:04,120 --> 00:40:10,920 Now what happens in d equals to 2 is kind of interesting. 546 00:40:10,920 --> 00:40:15,520 Because in d equals to 2, I'm doing a two 547 00:40:15,520 --> 00:40:19,510 dimensional integral, and this thing 548 00:40:19,510 --> 00:40:21,980 is growing as a power law. 549 00:40:21,980 --> 00:40:24,300 So basically what I end up having 550 00:40:24,300 --> 00:40:31,410 to deal with is something like e to the i q minus G x. 551 00:40:31,410 --> 00:40:37,270 And something that falls off as a over x to some power 552 00:40:37,270 --> 00:40:46,120 that I will simply cause eta G. So related to that combination. 553 00:40:46,120 --> 00:40:50,170 And when you integrate this, just 554 00:40:50,170 --> 00:40:52,450 dimensionally you can see that it 555 00:40:52,450 --> 00:40:58,140 is an integral that scales as x to the power of 2 minus eta. 556 00:40:58,140 --> 00:41:03,200 x is inverse to q minus G. So this is going to scale as 1 557 00:41:03,200 --> 00:41:07,893 over q minus g to the power of 2 minus eta. 558 00:41:11,280 --> 00:41:18,930 So as you go along some particular direction 559 00:41:18,930 --> 00:41:23,750 and ask what you see, you find that will 560 00:41:23,750 --> 00:41:30,860 see peaks close to the locations of the lattice vectors. 561 00:41:30,860 --> 00:41:36,410 So let's say they are equally spaced along the particular 562 00:41:36,410 --> 00:41:38,420 axis that I'm picking. 563 00:41:38,420 --> 00:41:41,750 In three dimension, you would have been seeing delta function 564 00:41:41,750 --> 00:41:45,980 at each one of these positions whose strength is governed 565 00:41:45,980 --> 00:41:48,010 by this Debye-Waller factor. 566 00:41:48,010 --> 00:41:50,040 In two dimensions, you will essentially 567 00:41:50,040 --> 00:41:54,752 see power laws close to each one of them. 568 00:41:54,752 --> 00:42:00,970 But as you go further, the exponent of the power law 569 00:42:00,970 --> 00:42:02,580 changes. 570 00:42:02,580 --> 00:42:06,030 So at some point you even cease to have a divergence. 571 00:42:06,030 --> 00:42:10,960 But you can essentially ignore anything 572 00:42:10,960 --> 00:42:12,360 like that [? happening. ?] 573 00:42:12,360 --> 00:42:16,400 So you can see that because we don't have true long range 574 00:42:16,400 --> 00:42:19,710 order into two dimensions because 575 00:42:19,710 --> 00:42:21,610 of these logarithmic divergences, 576 00:42:21,610 --> 00:42:24,440 et cetera, the traditional picture 577 00:42:24,440 --> 00:42:28,155 that we have of a solid that is characterized by Bragg 578 00:42:28,155 --> 00:42:32,950 scattering and delta function peaks is modified, 579 00:42:32,950 --> 00:42:34,790 but still it is something that looks 580 00:42:34,790 --> 00:42:37,850 very different from the liquid. 581 00:42:37,850 --> 00:42:42,680 So we expect that as we raise temperature 582 00:42:42,680 --> 00:42:46,240 there is a phase transition between scattering 583 00:42:46,240 --> 00:42:51,480 of this form and something that looks like liquid. 584 00:42:51,480 --> 00:42:55,120 And you can roughly guess that essentially it 585 00:42:55,120 --> 00:42:59,670 is going to be related with these etas becoming 586 00:42:59,670 --> 00:43:06,630 larger and larger so that these divergences disappear. 587 00:43:06,630 --> 00:43:08,130 But what is the mechanism? 588 00:43:10,860 --> 00:43:14,980 So let's go back and see what the mechanism was that we 589 00:43:14,980 --> 00:43:17,630 discovered for the XY model. 590 00:43:17,630 --> 00:43:23,050 So we are going to look at d equals to 2 from now on. 591 00:43:23,050 --> 00:43:26,780 That discussion so far was general. 592 00:43:26,780 --> 00:43:32,260 In d equals to 2, we said that we have topological defect. 593 00:43:38,980 --> 00:43:44,330 And the story here was that, as I have been discussing, 594 00:43:44,330 --> 00:43:47,870 the angle is undefined up to 2 pi. 595 00:43:47,870 --> 00:43:50,785 So it would configuration of angles that kind of radiate 596 00:43:50,785 --> 00:43:54,230 away from the origin. 597 00:43:57,640 --> 00:44:05,750 And the idea was that if I take a circuit 598 00:44:05,750 --> 00:44:09,720 and integrate around this circuit-- 599 00:44:09,720 --> 00:44:18,820 let's call this the s gradient of theta-- the answer does not 600 00:44:18,820 --> 00:44:21,560 need to come back to 0. 601 00:44:21,560 --> 00:44:25,194 It will be some multiple of 2 pi . 602 00:44:25,194 --> 00:44:26,110 Some integer multiple. 603 00:44:30,371 --> 00:44:35,100 And similarly here, we said that the value of u 604 00:44:35,100 --> 00:44:42,430 is undefined up to a lattice spacing. 605 00:44:42,430 --> 00:44:45,660 That leads to topological defects, which 606 00:44:45,660 --> 00:44:49,280 I find it very difficult to draw for the triangular lattice, 607 00:44:49,280 --> 00:44:51,745 so I'll draw it for the square lattice. 608 00:45:17,980 --> 00:45:22,500 So at each one of these positions 609 00:45:22,500 --> 00:45:23,980 there's a particle sitting. 610 00:45:26,710 --> 00:45:29,195 And you look out here. 611 00:45:29,195 --> 00:45:31,550 You have a perfect square lattice. 612 00:45:31,550 --> 00:45:33,080 You look out here. 613 00:45:33,080 --> 00:45:35,690 You have a perfect square lattice. 614 00:45:35,690 --> 00:45:38,770 But clearly it is not a perfect square lattice. 615 00:45:38,770 --> 00:45:42,670 And the analog of the circuit that I drew over here 616 00:45:42,670 --> 00:45:45,980 that encloses a singlularty-- there's clearly 617 00:45:45,980 --> 00:45:49,540 some kind of a defect sitting over here-- 618 00:45:49,540 --> 00:45:54,190 is that I can start with some point on the lattice 619 00:45:54,190 --> 00:45:57,310 and perform what is called a Burgers Circuit. 620 00:46:05,650 --> 00:46:09,640 What Burgers Circuit is is some walk 621 00:46:09,640 --> 00:46:12,150 that you would take on a lattice, which 622 00:46:12,150 --> 00:46:14,590 in a perfect lattice would bring you back 623 00:46:14,590 --> 00:46:16,690 to your starting point. 624 00:46:16,690 --> 00:46:20,460 So for example-- that was not a good starting point. 625 00:46:20,460 --> 00:46:24,520 Let me start from here. 626 00:46:24,520 --> 00:46:30,720 Let's say I take four steps up-- one, two, three, four. 627 00:46:30,720 --> 00:46:33,700 I take five steps to the right. 628 00:46:33,700 --> 00:46:36,910 One, two, three, four, five. 629 00:46:36,910 --> 00:46:38,920 I reverse my four steps. 630 00:46:38,920 --> 00:46:40,620 So I go four steps down. 631 00:46:40,620 --> 00:46:42,800 One, two, three, four. 632 00:46:42,800 --> 00:46:47,200 I reverse my five steps and I go five steps left. 633 00:46:47,200 --> 00:46:51,180 One, two, three, four, five. 634 00:46:51,180 --> 00:46:55,100 And you see that I did not end up where I was. 635 00:47:00,000 --> 00:47:03,570 I ended up here. 636 00:47:03,570 --> 00:47:07,580 And a failure to close my circuit 637 00:47:07,580 --> 00:47:09,370 has to be a lattice vector. 638 00:47:12,710 --> 00:47:26,310 And this is called d and it's called a Burgers Vector 639 00:47:26,310 --> 00:47:30,210 that characterize this defect, which is called a dislocation. 640 00:47:38,250 --> 00:47:45,070 So the uncertainty that I have in assigning a value for u 641 00:47:45,070 --> 00:47:50,800 because of being on a lattice or deforming a lattice 642 00:47:50,800 --> 00:47:55,430 gives rise to these kinds of ambiguity 643 00:47:55,430 --> 00:47:58,630 that are similar to the topological defects 644 00:47:58,630 --> 00:48:00,710 that you had over here. 645 00:48:00,710 --> 00:48:05,690 Whereas the defects here were characterized by an integer, 646 00:48:05,690 --> 00:48:07,760 the defects here are characterized 647 00:48:07,760 --> 00:48:10,410 by a lattice vector. 648 00:48:10,410 --> 00:48:15,310 And here the simplest things were, say, plus or minus. 649 00:48:15,310 --> 00:48:17,470 Clearly here on the square lattice 650 00:48:17,470 --> 00:48:21,530 the simplest things are plus b, minus b along the x direction 651 00:48:21,530 --> 00:48:24,280 and plus b minus b along the y direction. 652 00:48:24,280 --> 00:48:26,160 For example, on the triangular lattice 653 00:48:26,160 --> 00:48:30,560 it would be plus minus at any of the three directions 654 00:48:30,560 --> 00:48:32,010 that would define the lattice. 655 00:48:35,330 --> 00:48:41,470 So calculating the distortion field here was actually 656 00:48:41,470 --> 00:48:48,750 an easy matter because we said that the gradient of theta 657 00:48:48,750 --> 00:48:52,620 was something like 1 over r. 658 00:48:52,620 --> 00:49:04,110 And so the gradient of theta, which was 1 over r, 659 00:49:04,110 --> 00:49:09,370 I could in fact write-- we saw as vector 660 00:49:09,370 --> 00:49:14,640 that is orthogonal to the vector that is going out of the plane. 661 00:49:14,640 --> 00:49:17,420 So we call that a z hat. 662 00:49:17,420 --> 00:49:24,660 It was also orthogonal to the direction of motion 663 00:49:24,660 --> 00:49:28,760 away from the defect, which we could characterize 664 00:49:28,760 --> 00:49:32,410 by looking at the gradient. 665 00:49:32,410 --> 00:49:35,330 The gradient vector for this distortion 666 00:49:35,330 --> 00:49:39,040 is clearly in the radial direction. 667 00:49:39,040 --> 00:49:42,890 And the answer that we had here for one defect 668 00:49:42,890 --> 00:49:49,490 was ni log of r minus ri. 669 00:49:49,490 --> 00:49:51,870 And since I'm taking the gradient it doesn't matter. 670 00:49:51,870 --> 00:49:55,860 I can't put or not put the cut off. 671 00:49:55,860 --> 00:49:59,596 And if I had multiple ones, I would simply sum over it. 672 00:50:02,490 --> 00:50:08,820 So this was the contribution to the distortion 673 00:50:08,820 --> 00:50:15,950 that came from a collection of defects such as this. 674 00:50:15,950 --> 00:50:20,650 Now you can see that essentially in some continuum sense, 675 00:50:20,650 --> 00:50:25,440 all I'm doing here is similar. 676 00:50:25,440 --> 00:50:28,890 I can be taking a big circuit. 677 00:50:28,890 --> 00:50:33,530 And as I go along that big circuit, 678 00:50:33,530 --> 00:50:39,190 I can take a gradient-- So let's-- yeah. 679 00:50:39,190 --> 00:50:39,860 The s. 680 00:50:39,860 --> 00:50:44,500 gradient of the distortion field. 681 00:50:44,500 --> 00:50:46,300 Distortion field is, of course, a vector. 682 00:50:46,300 --> 00:50:50,350 So it has components that I can write as the x component 683 00:50:50,350 --> 00:50:51,180 or the y component. 684 00:50:51,180 --> 00:50:55,560 So this alpha could be the x component of the distortion 685 00:50:55,560 --> 00:50:58,920 or the y component of the distortion. 686 00:50:58,920 --> 00:51:04,020 And once I complete the circuit, here the answer was 2 pi n. 687 00:51:04,020 --> 00:51:07,140 Here the answer is a lattice vector, 688 00:51:07,140 --> 00:51:11,665 which is this Burgers Vector that could be in any direction 689 00:51:11,665 --> 00:51:12,447 that you want. 690 00:51:15,370 --> 00:51:18,570 Now as far as mathematics is concerned, 691 00:51:18,570 --> 00:51:25,690 this line and this line for each component are exactly the same. 692 00:51:25,690 --> 00:51:29,960 So the solution that I can write for gradient of u 693 00:51:29,960 --> 00:51:33,870 I can just copy from here for each component. 694 00:51:33,870 --> 00:51:37,580 So I can write that gradient of the u that 695 00:51:37,580 --> 00:51:41,710 is due to a collection of dislocations 696 00:51:41,710 --> 00:51:46,360 is something like z hat crossed with curl 697 00:51:46,360 --> 00:51:54,480 of a sum over all of the potential locations 698 00:51:54,480 --> 00:51:56,640 off my defect. 699 00:51:56,640 --> 00:51:59,605 Rather than n, I have b over 2. 700 00:51:59,605 --> 00:52:07,611 And it's a vector log of minus ri over [INAUDIBLE]. 701 00:52:11,760 --> 00:52:17,533 So if I have a collection of these dislocations 702 00:52:17,533 --> 00:52:24,150 at different places on my lattice-- locations ri, 703 00:52:24,150 --> 00:52:28,010 strengths, b alpha which is a vector-- 704 00:52:28,010 --> 00:52:31,330 then the distortion field that they would generate 705 00:52:31,330 --> 00:52:33,830 is given by this. 706 00:52:33,830 --> 00:52:36,947 It's very just following the answers that the [INAUDIBLE]. 707 00:52:43,290 --> 00:52:51,780 So then what we did for the case of the overall system-- 708 00:52:51,780 --> 00:52:57,210 in order to find how the different topological defects 709 00:52:57,210 --> 00:53:02,070 interact with each was to calculate 710 00:53:02,070 --> 00:53:12,220 beta h, which was an integral of gradient of theta squared. 711 00:53:14,650 --> 00:53:15,150 Yes? 712 00:53:15,150 --> 00:53:20,131 STUDENT: So the b alpha, you have an index i [? also? ?] 713 00:53:20,131 --> 00:53:22,580 PROFESSOR: b alpha have index i also. 714 00:53:22,580 --> 00:53:24,068 Like like the ni have index i. 715 00:53:24,068 --> 00:53:25,930 So maybe I write it in this fashion. 716 00:53:25,930 --> 00:53:26,430 Yeah. 717 00:53:34,710 --> 00:53:38,960 So this is the cost that we have to evaluate, 718 00:53:38,960 --> 00:53:41,870 except that we have to be somewhat 719 00:53:41,870 --> 00:53:48,702 careful with the meaning of this gradient of theta 720 00:53:48,702 --> 00:53:52,160 in that the gradient of theta, as we said, 721 00:53:52,160 --> 00:53:58,370 has a contribution that is from regular spin 722 00:53:58,370 --> 00:54:03,200 waves, the Goldstone modes, that can be deformed back 723 00:54:03,200 --> 00:54:05,320 to everybody pointing in the same direction 724 00:54:05,320 --> 00:54:07,910 without topological defect. 725 00:54:07,910 --> 00:54:17,930 Plus a contribution due to this topological defect that are 726 00:54:17,930 --> 00:54:21,708 categorized in that case by ni. 727 00:54:21,708 --> 00:54:25,520 Now similarly here for the case of the solid, 728 00:54:25,520 --> 00:54:27,560 we have the beta h. 729 00:54:27,560 --> 00:54:31,330 Slightly more complicated integral d 2x. 730 00:54:31,330 --> 00:54:38,264 We have mu uij uij plus lambda over 2 uii ujj. 731 00:54:42,880 --> 00:54:46,970 And what we can do is to say that our strain field 732 00:54:46,970 --> 00:54:55,110 uij has a component that is like the Goldstone modes 733 00:54:55,110 --> 00:54:57,500 that we have been calculating so far, 734 00:54:57,500 --> 00:55:01,680 essentially treating everything as Gaussians. 735 00:55:01,680 --> 00:55:04,200 Except that I will write it as phi ij. 736 00:55:07,200 --> 00:55:14,430 And then a part that would come from the distortion field u 737 00:55:14,430 --> 00:55:16,480 bar that I calculated. 738 00:55:16,480 --> 00:55:20,660 So I take that distortion field and then take derivatives of it 739 00:55:20,660 --> 00:55:23,120 to symmetrize appropriately to construct 740 00:55:23,120 --> 00:55:24,790 the strain that sums from them. 741 00:55:28,490 --> 00:55:34,750 So we substituted this form over here. 742 00:55:34,750 --> 00:55:42,040 And we found that beta h that we got had a part that was simply 743 00:55:42,040 --> 00:55:45,080 relate to the Goldstone modes. 744 00:55:45,080 --> 00:55:49,220 This was the part that could be treating as a Gaussian. 745 00:55:49,220 --> 00:55:56,950 And then we had a part that corresponded 746 00:55:56,950 --> 00:56:05,620 to the interactions among these defects. 747 00:56:05,620 --> 00:56:11,782 And that part we saw had the character 748 00:56:11,782 --> 00:56:22,140 of charges ni and nj plus minus 1 749 00:56:22,140 --> 00:56:25,750 characterizing these topological defects 750 00:56:25,750 --> 00:56:28,350 having a Coulomb interaction [? between them ?]. 751 00:56:37,500 --> 00:56:42,810 And then, of course, this i less than j, 752 00:56:42,810 --> 00:56:46,610 we had to worry about what was happening when i was j. 753 00:56:46,610 --> 00:56:51,970 And that we considered to be the contributions 754 00:56:51,970 --> 00:56:57,390 of the core energy that, once exponentiated, 755 00:56:57,390 --> 00:57:01,350 we described as y. 756 00:57:01,350 --> 00:57:07,750 So it's beta h which is log of that would be log of y. 757 00:57:07,750 --> 00:57:14,400 And actually this k bar, by the way, was simply 2 pi k. 758 00:57:14,400 --> 00:57:20,680 And it was 2 pi k because each one of these charges is 2 pi n. 759 00:57:20,680 --> 00:57:22,400 So there's 2 pi 2 pi. 760 00:57:22,400 --> 00:57:24,050 But the Coulomb interaction really 761 00:57:24,050 --> 00:57:26,355 should be log divided by 2 pi. 762 00:57:26,355 --> 00:57:29,896 And so k bar becomes 2 pi k times this. 763 00:57:32,860 --> 00:57:40,422 So I can do the same thing over here. 764 00:57:40,422 --> 00:57:47,740 And what I find happens is that my beta h gets decomposed 765 00:57:47,740 --> 00:57:48,890 as follows. 766 00:57:48,890 --> 00:57:55,690 There will be a part that is simply the original expression 767 00:57:55,690 --> 00:58:01,680 now for the field that is well behaved 768 00:58:01,680 --> 00:58:03,470 and has no dislocation piece. 769 00:58:08,290 --> 00:58:10,920 And then you wouldn't be surprised 770 00:58:10,920 --> 00:58:13,520 because this structure is no different 771 00:58:13,520 --> 00:58:14,770 from the other structure. 772 00:58:14,770 --> 00:58:17,960 There's essentially a gradient of this distortion 773 00:58:17,960 --> 00:58:19,420 field squared. 774 00:58:19,420 --> 00:58:22,190 And you can see that if I take a second derivative 775 00:58:22,190 --> 00:58:27,290 of this distortion field or one derivative of this, effectively 776 00:58:27,290 --> 00:58:30,490 I will have two derivatives of a log, which 777 00:58:30,490 --> 00:58:32,170 will give me a delta function. 778 00:58:32,170 --> 00:58:35,730 So essentially these things do behave 779 00:58:35,730 --> 00:58:39,370 like what you would get from Coulomb type of potential. 780 00:58:39,370 --> 00:58:41,740 [INAUDIBLE] plus [INAUDIBLE] if the potential is 0. 781 00:58:41,740 --> 00:58:46,770 So not surprisingly, I we get a minus k bar sum 782 00:58:46,770 --> 00:58:52,600 over pairs of where these dislocations are located. 783 00:58:52,600 --> 00:59:00,440 And then I will have b at location i, b at location j. 784 00:59:00,440 --> 00:59:10,050 And then I would have a log of ri minus rj with some cut off. 785 00:59:10,050 --> 00:59:14,260 Except that these b's are actually vector. 786 00:59:14,260 --> 00:59:20,510 So this term is a dot product of the two b's. 787 00:59:20,510 --> 00:59:24,050 And it turns out that when you go through the algebra, 788 00:59:24,050 --> 00:59:28,170 there is another term, which is bi. 789 00:59:28,170 --> 00:59:41,820 ri minus rj, bj dotted with ri minus rj 790 00:59:41,820 --> 00:59:47,040 divided by ri minus rj squared. 791 00:59:51,190 --> 00:59:58,370 So the charges that we had in our original theory 792 00:59:58,370 --> 01:00:00,070 were scalar quantities. 793 01:00:00,070 --> 01:00:05,600 These defects were characterized by a scalar value of n. 794 01:00:05,600 --> 01:00:08,400 And they were interacting with something 795 01:00:08,400 --> 01:00:13,180 that was like the ordinary Coulomb potential. 796 01:00:13,180 --> 01:00:15,640 Whereas now we are looking at a system 797 01:00:15,640 --> 01:00:20,220 that is characterized by charges that are vectors. 798 01:00:20,220 --> 01:00:23,300 And it turns out that what we have here 799 01:00:23,300 --> 01:00:27,590 is the vectorial analog of the Coulomb potential. 800 01:00:27,590 --> 01:00:32,190 And again, if you think about an isotropic system and vectors, 801 01:00:32,190 --> 01:00:35,170 this is a vector that you can form, 802 01:00:35,170 --> 01:00:39,280 but b.r is another vector that you can form. 803 01:00:39,280 --> 01:00:41,440 And so both of them do appear. 804 01:00:41,440 --> 01:00:43,180 And once you go through the whole algebra 805 01:00:43,180 --> 01:00:46,680 and through the inverse Fourier transform, et cetera, 806 01:00:46,680 --> 01:00:49,370 you get this additional contribution 807 01:00:49,370 --> 01:00:51,050 to this vector Coulomb interaction. 808 01:00:57,470 --> 01:01:00,170 That's really is the only difference. 809 01:01:00,170 --> 01:01:01,830 And of course, you will again get 810 01:01:01,830 --> 01:01:09,220 a sum over all locations of the core energies 811 01:01:09,220 --> 01:01:11,497 for creating these defects. 812 01:01:17,840 --> 01:01:31,130 And the value of k bar here is related to these parameters 813 01:01:31,130 --> 01:01:39,536 by mu mu plus lambda divided by pi 2u plus lambda. 814 01:01:44,970 --> 01:01:49,170 So this combination controls what you have. 815 01:01:57,790 --> 01:02:03,010 So the next thing that we did was 816 01:02:03,010 --> 01:02:08,640 to construct a perturbation. 817 01:02:08,640 --> 01:02:16,310 We essentially said that if I have these pairs of charges 818 01:02:16,310 --> 01:02:19,170 that can spontaneously appear at an energy 819 01:02:19,170 --> 01:02:26,790 cost but an entropy gain, they will effectively weaken 820 01:02:26,790 --> 01:02:28,850 the overall Coulomb potential. 821 01:02:28,850 --> 01:02:33,060 For example, if I have to test charges 822 01:02:33,060 --> 01:02:36,340 then there will be some polarization 823 01:02:36,340 --> 01:02:38,910 of the medium, some reorientation 824 01:02:38,910 --> 01:02:42,280 of these charges that appear, that weakens 825 01:02:42,280 --> 01:02:43,670 the effective charge here. 826 01:02:46,220 --> 01:02:48,840 And we could perturbatively calculate 827 01:02:48,840 --> 01:02:51,990 what the correction was, except that we 828 01:02:51,990 --> 01:02:54,480 found that that correction, which 829 01:02:54,480 --> 01:02:59,980 involves an integration over the separation of these things. 830 01:02:59,980 --> 01:03:02,460 There was an integration that was potentially 831 01:03:02,460 --> 01:03:07,610 divergent for us no matter how small we made the core 832 01:03:07,610 --> 01:03:09,910 energies here. 833 01:03:09,910 --> 01:03:13,770 And so what we did was eventually to construct an RG. 834 01:03:17,100 --> 01:03:24,600 And the RG was that the value of this parameter k 835 01:03:24,600 --> 01:03:30,010 bar-- actually more usefully its inverse that 836 01:03:30,010 --> 01:03:34,640 is related to temperature-- changed 837 01:03:34,640 --> 01:03:39,980 as a function of integrating out short distance 838 01:03:39,980 --> 01:03:42,580 degrees of freedom. 839 01:03:42,580 --> 01:03:46,641 It was something like 4 pi-- I think it was 4 pi cubed, 840 01:03:46,641 --> 01:03:47,890 it doesn't matter-- y squared. 841 01:03:52,005 --> 01:03:59,460 And that the scaling of to core energy 842 01:03:59,460 --> 01:04:05,820 itself was determined by 2 minus pi k, 843 01:04:05,820 --> 01:04:09,390 which in terms of this k bar becomes 2 minus k bar 844 01:04:09,390 --> 01:04:11,961 over 2 times y. 845 01:04:17,260 --> 01:04:20,770 And we can do exactly the same thing over here, 846 01:04:20,770 --> 01:04:23,670 except for the complication of having 847 01:04:23,670 --> 01:04:27,860 to deal with the charges that are vectorial. 848 01:04:27,860 --> 01:04:32,630 And you get that the strength of the Coulomb interaction, 849 01:04:32,630 --> 01:04:39,010 it's inverse that is related temperature 850 01:04:39,010 --> 01:04:41,670 gets re-normalized by exactly the same process. 851 01:04:41,670 --> 01:04:45,490 That is, there will be charges that will appear. 852 01:04:45,490 --> 01:04:49,120 All the mathematics you can see is clearly similar. 853 01:04:49,120 --> 01:04:52,190 And there will be a reduction, which then you 854 01:04:52,190 --> 01:04:56,340 can think of as an increase in temperature, which will 855 01:04:56,340 --> 01:04:58,840 be proportional to y squared. 856 01:04:58,840 --> 01:05:02,950 I don't know what the constant of proportionality is. 857 01:05:02,950 --> 01:05:07,360 And that dy by dl-- so essentially I 858 01:05:07,360 --> 01:05:10,230 focus on the simplest type of dislocations 859 01:05:10,230 --> 01:05:14,590 that I would have of unit spacing-- 860 01:05:14,590 --> 01:05:16,750 has exactly the same [INAUDIBLE], 861 01:05:16,750 --> 01:05:19,998 2 minus k bar over 2 y. 862 01:05:25,370 --> 01:05:28,926 It turns out that there is actually a difference. 863 01:05:28,926 --> 01:05:31,850 And the difference is as follows. 864 01:05:34,360 --> 01:05:39,020 That while we did not calculate the next correction 865 01:05:39,020 --> 01:05:44,430 in this series, I said that this correction came 866 01:05:44,430 --> 01:05:48,300 from essentially what would a single dipole do, 867 01:05:48,300 --> 01:05:51,880 which would be some contribution that is order of y squared. 868 01:05:51,880 --> 01:05:53,970 And I expect that there will be correction 869 01:05:53,970 --> 01:05:58,430 if I look at a configuration that has pairs of dipoles. 870 01:05:58,430 --> 01:06:01,270 And this will change this by order of y 871 01:06:01,270 --> 01:06:05,348 to the fourth and this by order of y cubed. 872 01:06:09,320 --> 01:06:14,050 Whereas if I look at something like a triangular lattice, 873 01:06:14,050 --> 01:06:20,110 you can see that if I have two dislocations out there 874 01:06:20,110 --> 01:06:22,450 as test dislocations pointing out 875 01:06:22,450 --> 01:06:25,430 in opposite directions, which would 876 01:06:25,430 --> 01:06:29,240 have some kind of a interaction such as this. 877 01:06:29,240 --> 01:06:31,430 And I ask how that interaction is 878 01:06:31,430 --> 01:06:36,696 modified by the presence of dipoles of dislocation 879 01:06:36,696 --> 01:06:39,240 in the medium. 880 01:06:39,240 --> 01:06:42,120 I could put a pair of dislocations 881 01:06:42,120 --> 01:06:44,660 integrate over the separation between them. 882 01:06:44,660 --> 01:06:47,480 I will get, essentially, the same type 883 01:06:47,480 --> 01:06:53,430 of mathematical structure that would ultimately give me this. 884 01:06:53,430 --> 01:06:57,010 But then the next order term in this series 885 01:06:57,010 --> 01:07:00,100 is not for dislocations because I 886 01:07:00,100 --> 01:07:03,500 can have a neutral configuration of three dislocations, 887 01:07:03,500 --> 01:07:06,820 such as this. 888 01:07:06,820 --> 01:07:09,310 And so once you do that, you will 889 01:07:09,310 --> 01:07:11,810 find that the next correction here 890 01:07:11,810 --> 01:07:16,196 is order of y cubed and order of y squared here. 891 01:07:23,340 --> 01:07:31,290 But generically, both of them have a phase diagram 892 01:07:31,290 --> 01:07:37,160 for RG flows that involves the inverse of the interaction 893 01:07:37,160 --> 01:07:38,210 of these charges. 894 01:07:38,210 --> 01:07:41,000 That is a temperature-like quantity. 895 01:07:41,000 --> 01:07:45,505 And there is a critical value of that inverse, which is 1/4. 896 01:07:49,950 --> 01:07:56,640 And what happens is that-- let's see, this other axis is y. 897 01:08:00,230 --> 01:08:04,260 Anything smaller than 1/4 y tends to go to 0. 898 01:08:04,260 --> 01:08:07,470 Anything larger, y tends to get larger. 899 01:08:07,470 --> 01:08:12,530 And there is going to be some kind of a separatrix 900 01:08:12,530 --> 01:08:20,170 that he describes the transition between flows that go down here 901 01:08:20,170 --> 01:08:23,729 and the flows that go away. 902 01:08:30,850 --> 01:08:40,590 And presumably, as I heat up my original system-- my lattice, 903 01:08:40,590 --> 01:08:44,180 different types of lattices-- at low temperature, 904 01:08:44,180 --> 01:08:47,870 I can figure out what this combination of mu an lambda is. 905 01:08:47,870 --> 01:08:51,740 That gives me some value of k. 906 01:08:51,740 --> 01:08:54,439 And there is some kind of the core energy for the vertices 907 01:08:54,439 --> 01:08:56,069 that I can calculate. 908 01:08:56,069 --> 01:08:58,600 So there will be some point in this phase diagram, which 909 01:08:58,600 --> 01:09:02,680 at low enough temperature I will go over here, which corresponds 910 01:09:02,680 --> 01:09:05,490 to a system that has an effective logarithmic 911 01:09:05,490 --> 01:09:08,069 interaction between dislocations. 912 01:09:08,069 --> 01:09:11,359 As I change the temperature, I will proceed around 913 01:09:11,359 --> 01:09:15,170 some trajectory such as this because everything will change. 914 01:09:15,170 --> 01:09:18,800 And presumably, at some point I will intersect this 915 01:09:18,800 --> 01:09:22,540 and I will go to the other phase. 916 01:09:22,540 --> 01:09:26,700 Now the characteristic of the other phase 917 01:09:26,700 --> 01:09:33,270 was that this parameter k was going to 0. 918 01:09:33,270 --> 01:09:35,590 It was essentially the logarithmic interaction 919 01:09:35,590 --> 01:09:36,820 was disappeared. 920 01:09:36,820 --> 01:09:44,500 And you can see that that parameter k going to 0 921 01:09:44,500 --> 01:09:49,370 means that mu, the shear modulus, has to go to 0. 922 01:09:49,370 --> 01:09:55,820 So basically this phase out here is definitely characterized 923 01:09:55,820 --> 01:09:57,520 by 0 shear modulus. 924 01:10:00,820 --> 01:10:04,960 So again, the two parameters mu and lambda, mu 925 01:10:04,960 --> 01:10:07,410 is the one that gives you the cost 926 01:10:07,410 --> 01:10:09,850 of trying to shear in a system. 927 01:10:09,850 --> 01:10:14,610 And essentially the presence of the resistance to shear 928 01:10:14,610 --> 01:10:17,440 is what defines a solid for you. 929 01:10:17,440 --> 01:10:22,450 So this is a solid to something that does not have rigidity 930 01:10:22,450 --> 01:10:22,950 transition. 931 01:10:26,390 --> 01:10:29,830 It turns out that there is one subtlety here. 932 01:10:29,830 --> 01:10:33,970 I don't know whether it's good to mention or not. 933 01:10:33,970 --> 01:10:48,660 But so in the case of alpha this superfluidity, for example, 934 01:10:48,660 --> 01:10:52,540 we said that this coupling strength as you 935 01:10:52,540 --> 01:10:58,130 approach the transition rose to a universal value. 936 01:10:58,130 --> 01:11:01,560 In these units it will be 4. 937 01:11:01,560 --> 01:11:04,669 But that it approaches that universal value 938 01:11:04,669 --> 01:11:05,960 with a logarithmic singularity. 939 01:11:10,390 --> 01:11:12,500 Sorry, with a square root singularly. 940 01:11:12,500 --> 01:11:13,270 Sorry. 941 01:11:13,270 --> 01:11:16,270 And we saw that this was experimentally observed 942 01:11:16,270 --> 01:11:19,210 in films of superfluid film. 943 01:11:19,210 --> 01:11:22,730 I also said that on the other side 944 01:11:22,730 --> 01:11:24,670 you have a finite correlations. 945 01:11:24,670 --> 01:11:27,740 The correlations will be decaying exponentially. 946 01:11:27,740 --> 01:11:33,900 And this correlation length has this very unusual signature, 947 01:11:33,900 --> 01:11:40,710 which is that it diverged as a square root of t minus tc 948 01:11:40,710 --> 01:11:42,660 in the exponent. 949 01:11:42,660 --> 01:11:45,190 It was giving rise to these essentially singularities. 950 01:11:49,480 --> 01:11:54,425 While everything kind of looks identical between this RG 951 01:11:54,425 --> 01:11:58,930 and the other RG, it turns out that once you 952 01:11:58,930 --> 01:12:03,780 keep track of these corrections, these corrections actually 953 01:12:03,780 --> 01:12:05,790 are important. 954 01:12:05,790 --> 01:12:09,750 And for the vectorial version of the Coulomb gas, 955 01:12:09,750 --> 01:12:17,660 you find that they shear modulus that is finite in this phase 956 01:12:17,660 --> 01:12:20,160 reaches its final value, which is not 957 01:12:20,160 --> 01:12:22,590 universal because the thing that the universal is 958 01:12:22,590 --> 01:12:26,380 that combination that involves both mu and lambda. 959 01:12:26,380 --> 01:12:32,410 But rather than stc minus t to the 1/2, 960 01:12:32,410 --> 01:12:35,260 stc minus t to some exponent that 961 01:12:35,260 --> 01:12:42,566 is called mu bar, which is 0.36963 [INAUDIBLE]. 962 01:12:42,566 --> 01:12:45,440 So this is a kind of subtle thing 963 01:12:45,440 --> 01:12:48,500 that is buried in these recursion relationships 964 01:12:48,500 --> 01:12:50,480 once you go to a higher order. 965 01:12:50,480 --> 01:12:53,410 And accompanying that is a correlation length that 966 01:12:53,410 --> 01:13:03,480 also diverges with some behavior dominated by the same exponent. 967 01:13:10,780 --> 01:13:16,960 So once people understood the original [? Koster ?] 968 01:13:16,960 --> 01:13:20,040 [INAUDIBLE] transition, it was kind 969 01:13:20,040 --> 01:13:23,600 of a very natural next step for them 970 01:13:23,600 --> 01:13:26,460 to think about these locations in a two dimensional 971 01:13:26,460 --> 01:13:30,920 solid and say that in the same manner 972 01:13:30,920 --> 01:13:35,750 that the unbinding of these defects 973 01:13:35,750 --> 01:13:39,680 got rid of the residual long range order that 974 01:13:39,680 --> 01:13:42,900 was left in the XY model. 975 01:13:42,900 --> 01:13:46,560 That this kind of ordering that we said 976 01:13:46,560 --> 01:13:50,030 exists for the two dimensional solid, 977 01:13:50,030 --> 01:13:56,240 this appears as a result of unbinding of dislocations 978 01:13:56,240 --> 01:13:59,500 and you get a liquid. 979 01:13:59,500 --> 01:14:02,820 Turns out that that's not correct. 980 01:14:02,820 --> 01:14:06,490 And it was pointed out by Bert Halperin 981 01:14:06,490 --> 01:14:12,020 I think that if you look at a solid, 982 01:14:12,020 --> 01:14:13,465 there's two things that you. 983 01:14:13,465 --> 01:14:16,890 You have the translational order, 984 01:14:16,890 --> 01:14:19,510 but there's also orientational order. 985 01:14:19,510 --> 01:14:23,110 That is, you can look, for example, 986 01:14:23,110 --> 01:14:27,370 at a collection of spins here and the bonds 987 01:14:27,370 --> 01:14:30,800 are pointing along the x and y directions 988 01:14:30,800 --> 01:14:33,662 and you can look down here and the bonds are pointing 989 01:14:33,662 --> 01:14:34,745 in the x and y directions. 990 01:14:38,000 --> 01:14:43,410 And from the picture that I drew over here, 991 01:14:43,410 --> 01:14:47,420 you can kind of see that once I insert this dislocation which 992 01:14:47,420 --> 01:14:50,190 corresponds to an additional line, 993 01:14:50,190 --> 01:14:54,110 the positions here are distorted. 994 01:14:54,110 --> 01:14:57,660 Once I have many of these inserted, 995 01:14:57,660 --> 01:14:59,725 it is kind of obvious that I will 996 01:14:59,725 --> 01:15:05,045 lose any idea of how the position of this lattice point 997 01:15:05,045 --> 01:15:08,210 and something out there is related. 998 01:15:08,210 --> 01:15:10,310 It's not so obvious that inserting 999 01:15:10,310 --> 01:15:13,510 lots and lots of these lines will 1000 01:15:13,510 --> 01:15:19,530 remove the knowledge that I have about 1001 01:15:19,530 --> 01:15:23,970 the orientation of the bonds. 1002 01:15:23,970 --> 01:15:29,040 And the insight that we will follow up-- 1003 01:15:29,040 --> 01:15:32,790 and maybe it's probably a good idea to do it next time-- 1004 01:15:32,790 --> 01:15:40,700 is that actually once the dislocations unbind, 1005 01:15:40,700 --> 01:15:45,730 we will lose this kind of ordering. 1006 01:15:45,730 --> 01:15:47,960 If I do this in two dimensions, these 1007 01:15:47,960 --> 01:15:51,730 are power law decays at low temperatures. 1008 01:15:51,730 --> 01:15:54,080 Once the dislocations unbind, this 1009 01:15:54,080 --> 01:15:58,890 becomes an exponential decay, as I would expect. 1010 01:15:58,890 --> 01:16:05,500 But I can also define locally some kind of an orientation. 1011 01:16:05,500 --> 01:16:07,940 Let's again call it theta. 1012 01:16:07,940 --> 01:16:14,510 But theta is, let's say, the orientation of the bond that 1013 01:16:14,510 --> 01:16:18,150 connects two neighboring spins. 1014 01:16:18,150 --> 01:16:23,520 Now what I can do is to look at the correlation of orientations 1015 01:16:23,520 --> 01:16:25,700 at different positions. 1016 01:16:25,700 --> 01:16:33,380 And it turns out that if I'm doing something 1017 01:16:33,380 --> 01:16:44,810 like a triangular lattice, I can't just pick the angle 1018 01:16:44,810 --> 01:16:49,170 and correlate the angle from one point to another point 1019 01:16:49,170 --> 01:16:54,980 because let's say on the square lattice the angle itself, 1020 01:16:54,980 --> 01:16:57,210 I don't know whether it was coming 1021 01:16:57,210 --> 01:16:59,790 from a bond that was originally in the x direction 1022 01:16:59,790 --> 01:17:01,490 or in the y direction. 1023 01:17:01,490 --> 01:17:05,310 So this is unknown up to a factor of 90 degrees. 1024 01:17:05,310 --> 01:17:08,930 In the triangular lattice it is an unknown up 1025 01:17:08,930 --> 01:17:13,540 to a factor of 60 degrees. 1026 01:17:13,540 --> 01:17:19,670 So what I should do is I should define e to the 6 i theta 1027 01:17:19,670 --> 01:17:29,090 at each location as a measure of the orientational order. 1028 01:17:29,090 --> 01:17:33,150 You can see that, again, if I'm at 0 temperature, 1029 01:17:33,150 --> 01:17:37,470 I can reorient the lattice any way that I like. 1030 01:17:37,470 --> 01:17:42,070 This phase would be the same across the entire system. 1031 01:17:42,070 --> 01:17:44,930 So what I'm interested is to find out 1032 01:17:44,930 --> 01:17:53,460 what is the expectation value of this object when 1033 01:17:53,460 --> 01:17:57,115 I look at points that are further apart. 1034 01:17:57,115 --> 01:17:59,990 And again, clearly if I'm at 0 temperature 1035 01:17:59,990 --> 01:18:02,300 these thetas are the same. 1036 01:18:02,300 --> 01:18:05,960 This will go to 1 at finite temperature. 1037 01:18:05,960 --> 01:18:12,140 Because of the fluctuations it will start to move a bit. 1038 01:18:12,140 --> 01:18:16,600 And what we can show-- and you will do it next time-- 1039 01:18:16,600 --> 01:18:19,985 is that as long as you are in the 0 temperature phase 1040 01:18:19,985 --> 01:18:25,545 you will find that this goes to a constant, 1041 01:18:25,545 --> 01:18:29,090 even in two dimensions. 1042 01:18:29,090 --> 01:18:34,620 But once the dislocations unbind, what we will show 1043 01:18:34,620 --> 01:18:37,770 is that it doesn't decay exponentially, 1044 01:18:37,770 --> 01:18:43,710 but then it starts to decay as 1 over x to some power, 1045 01:18:43,710 --> 01:18:45,142 let's call it eta theta. 1046 01:18:48,790 --> 01:18:51,480 And then at some higher temperatures, 1047 01:18:51,480 --> 01:18:53,670 it eventually starts to decay exponentially. 1048 01:18:58,410 --> 01:19:02,420 So there is the original solid in two 1049 01:19:02,420 --> 01:19:07,210 dimension that we started that has true long range 1050 01:19:07,210 --> 01:19:11,560 order in the orientations. 1051 01:19:11,560 --> 01:19:15,810 And that is reflected in these Fourier 1052 01:19:15,810 --> 01:19:17,570 transforms that we make. 1053 01:19:17,570 --> 01:19:21,900 So we said that when we do the Fourier transform and look 1054 01:19:21,900 --> 01:19:24,550 at the x-ray scattering, let's say from a two 1055 01:19:24,550 --> 01:19:30,820 dimensional crystal, you were seeing these kinds of pictures. 1056 01:19:30,820 --> 01:19:32,900 And then these were becoming weaker 1057 01:19:32,900 --> 01:19:36,680 as we went further and further out. 1058 01:19:36,680 --> 01:19:39,700 But now you can see that this picture clearly 1059 01:19:39,700 --> 01:19:44,720 has a very, very defined orientational aspect to it. 1060 01:19:44,720 --> 01:19:49,000 Now once we go to this intermediate phase, 1061 01:19:49,000 --> 01:19:53,490 all of these peaks disappear. 1062 01:19:53,490 --> 01:19:59,960 But what you find is that when you look at this, rather 1063 01:19:59,960 --> 01:20:02,570 than getting a ring that is uniform, 1064 01:20:02,570 --> 01:20:09,395 you will a see a ring that has very well- defined variation. 1065 01:20:09,395 --> 01:20:16,945 One, two, three, four, five, six. 1066 01:20:16,945 --> 01:20:21,040 Six-fold symmetry to it. 1067 01:20:21,040 --> 01:20:25,490 So this phase has no knowledge of where 1068 01:20:25,490 --> 01:20:30,490 the particles are located, but knows the orientations. 1069 01:20:30,490 --> 01:20:33,060 It's a kind of a liquid crystal. 1070 01:20:33,060 --> 01:20:34,500 It actually has a name. 1071 01:20:34,500 --> 01:20:36,290 It's called hexatic. 1072 01:20:36,290 --> 01:20:44,370 But it's one of the family of different types of materials 1073 01:20:44,370 --> 01:20:47,370 that have no translational order but some kind 1074 01:20:47,370 --> 01:20:49,390 of orientational order that are hexatics. 1075 01:20:53,882 --> 01:21:00,000 So the transition between this hexatic phase 1076 01:21:00,000 --> 01:21:04,580 to this fully disordered phase, it turns out in two dimension 1077 01:21:04,580 --> 01:21:08,500 to be another one of these dislocation-- 1078 01:21:08,500 --> 01:21:12,120 well, topology defect unbinding transition. 1079 01:21:12,120 --> 01:21:16,340 So maybe we will finish with that next time around.