1 00:00:00,070 --> 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 --> 00:00:17,238 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,238 --> 00:00:17,863 at ocw.mit.edu. 8 00:00:22,030 --> 00:00:25,990 PROFESSOR: So we are going to switch directions. 9 00:00:25,990 --> 00:00:29,020 Rather than thinking about binary variables, 10 00:00:29,020 --> 00:00:35,400 these Ising variables, that were discrete, 11 00:00:35,400 --> 00:00:36,990 think again about a lattice. 12 00:00:40,240 --> 00:00:50,290 But now at each site we put a spin that has unit magnitude, 13 00:00:50,290 --> 00:00:52,520 but m component. 14 00:00:52,520 --> 00:01:03,440 That is, Si has components 1, 2, to n. 15 00:01:03,440 --> 00:01:12,150 And the constraint that this sum over alpha Si alpha squared 16 00:01:12,150 --> 00:01:13,690 is unit. 17 00:01:13,690 --> 00:01:19,285 So clearly, if I-- let's put it here explicitly. 18 00:01:19,285 --> 00:01:23,715 Alpha of 1 to n, so that's when I look at the case of n 19 00:01:23,715 --> 00:01:27,220 equals to 1, essentially I have one component. 20 00:01:27,220 --> 00:01:31,010 2 squared has to be 1, so it's either plus or minus. 21 00:01:31,010 --> 00:01:33,790 We recover the Ising variable. 22 00:01:33,790 --> 00:01:36,380 For n equals to 2, it's essentially 23 00:01:36,380 --> 00:01:40,760 a unit vector who's angle theta, for example, 24 00:01:40,760 --> 00:01:43,980 can change in three dimensions if we were exploring 25 00:01:43,980 --> 00:01:46,570 the surface of the cube. 26 00:01:46,570 --> 00:01:49,480 And we always assume that we have 27 00:01:49,480 --> 00:01:57,960 a weight that tends to make our spins to be parallel. 28 00:01:57,960 --> 00:02:03,310 So we use, essentially, the same form as the Ising model. 29 00:02:03,310 --> 00:02:06,680 We sum over near neighbors. 30 00:02:06,680 --> 00:02:10,325 And the interaction, rather than sigma I sigma 31 00:02:10,325 --> 00:02:13,690 j, we put it as this Si dot Sj, where 32 00:02:13,690 --> 00:02:19,030 these are the dot products of two vectors. 33 00:02:19,030 --> 00:02:23,860 Let's call the dimensionless interaction in front K0. 34 00:02:26,910 --> 00:02:35,200 So when we want to calculate the partition function, 35 00:02:35,200 --> 00:02:38,900 we need to integrate over all configurations 36 00:02:38,900 --> 00:02:46,906 of these spins of this weight. 37 00:02:52,370 --> 00:02:58,410 Now for each case, we have to do n components. 38 00:02:58,410 --> 00:03:01,510 But there is a constraint, which is this one. 39 00:03:04,880 --> 00:03:11,620 Now I'm going to be focused on the ground state. 40 00:03:11,620 --> 00:03:17,720 So when t equals to 0, we expect that spontaneously 41 00:03:17,720 --> 00:03:20,350 the particular configuration will be chosen. 42 00:03:20,350 --> 00:03:22,425 Everybody will be aligned to that configuration. 43 00:03:25,890 --> 00:03:32,840 Without loss of generality, let's choose aligned state 44 00:03:32,840 --> 00:03:36,840 to point along the last component. 45 00:03:36,840 --> 00:03:41,170 That is, all of the Si at t equal to 0 46 00:03:41,170 --> 00:03:47,100 will be of the form 0,0, except that the last component is 47 00:03:47,100 --> 00:03:51,810 pointing along some particular direction. 48 00:03:51,810 --> 00:03:58,260 So if it was two components, the y component would always be 1. 49 00:03:58,260 --> 00:04:02,470 It would be aligned along the y direction. 50 00:04:02,470 --> 00:04:05,790 Yes, question? 51 00:04:05,790 --> 00:04:08,700 AUDIENCE: What dimensionality is the lattice? 52 00:04:08,700 --> 00:04:09,950 PROFESSOR: It can be anything. 53 00:04:09,950 --> 00:04:13,910 So basically we have two parameters, as usual. 54 00:04:13,910 --> 00:04:16,880 n is the dimensionality of spin, and d 55 00:04:16,880 --> 00:04:21,339 would be the dimensionality of our lattice. 56 00:04:21,339 --> 00:04:24,830 In practice for what the calculations 57 00:04:24,830 --> 00:04:26,950 that we are going to be doing, we 58 00:04:26,950 --> 00:04:29,900 will be focusing in d that is close to 2. 59 00:04:36,880 --> 00:04:48,020 Now if the odd fluctuations at finite t, what happens 60 00:04:48,020 --> 00:04:51,900 is that the state of the vector is going to change. 61 00:04:51,900 --> 00:04:58,920 So this Si at finite temperature would no longer 62 00:04:58,920 --> 00:05:01,150 be pointing along the last component. 63 00:05:01,150 --> 00:05:04,270 It will start to have fluctuations. 64 00:05:04,270 --> 00:05:08,770 Those fluctuations will change the 0 from the ground state 65 00:05:08,770 --> 00:05:12,900 to some value I'll call pi 1, the next one pi 2, 66 00:05:12,900 --> 00:05:17,570 all of the big pi n minus 1. 67 00:05:17,570 --> 00:05:21,970 And since the whole entire thing is a unit vector, 68 00:05:21,970 --> 00:05:26,410 the last component has to shrink to adjust for that. 69 00:05:26,410 --> 00:05:31,020 So we would indicate the last component by sigma. 70 00:05:31,020 --> 00:05:37,050 So essentially this subspace of fluctuations around the ground 71 00:05:37,050 --> 00:05:40,795 state is captured through this vector 72 00:05:40,795 --> 00:05:45,790 pi that is n minus 1 dimensional. 73 00:05:45,790 --> 00:05:49,910 And this corresponds to the transverse modes 74 00:05:49,910 --> 00:05:52,500 that we're looking at when we were doing 75 00:05:52,500 --> 00:05:56,070 the expansion of the Landau-Ginzburg model 76 00:05:56,070 --> 00:06:00,120 around its symmetry broken state. 77 00:06:00,120 --> 00:06:02,920 In this case, the longitudinal mode, 78 00:06:02,920 --> 00:06:06,900 essentially, is infinitely stiff. 79 00:06:06,900 --> 00:06:09,870 You don't have the ability to stretch along 80 00:06:09,870 --> 00:06:12,500 the longitudinal mode because of the constraint 81 00:06:12,500 --> 00:06:14,530 that we have put over here. 82 00:06:14,530 --> 00:06:19,710 So if you think back, we had this wine bottle, 83 00:06:19,710 --> 00:06:22,450 or Mexican hat potential. 84 00:06:22,450 --> 00:06:25,380 And the Goldstone modes corresponded 85 00:06:25,380 --> 00:06:29,000 to going along the bottom, and how easy 86 00:06:29,000 --> 00:06:32,770 it was to climb this Mexican hat was 87 00:06:32,770 --> 00:06:36,010 determined by the longitudinal mode. 88 00:06:36,010 --> 00:06:40,170 In this case, the Mexican hat has become very, very stiff 89 00:06:40,170 --> 00:06:41,870 to climb on the sides. 90 00:06:41,870 --> 00:06:44,320 So you don't have the longitudinal mode. 91 00:06:44,320 --> 00:06:48,070 You just have these Goldstone modes. 92 00:06:48,070 --> 00:06:51,150 The cost to pay for that is that I 93 00:06:51,150 --> 00:06:54,100 have to be very careful in calculating the partition 94 00:06:54,100 --> 00:06:55,430 function. 95 00:06:55,430 --> 00:06:59,400 If I'm integrating over the n components 96 00:06:59,400 --> 00:07:06,865 of some particular spin, I have to make sure 97 00:07:06,865 --> 00:07:14,610 that I remember that this sum of all of these components is 1. 98 00:07:18,490 --> 00:07:22,720 So I have to integrate subject to that constraint. 99 00:07:22,720 --> 00:07:27,740 And the way that I have broken things down now, 100 00:07:27,740 --> 00:07:36,310 I'm integrating over the n minus 1 component of this vector pi. 101 00:07:36,310 --> 00:07:41,010 And this additional direction, d Sigma, 102 00:07:41,010 --> 00:07:44,310 but I can't do both of them independently. 103 00:07:44,310 --> 00:07:46,460 Because there's a delta function that 104 00:07:46,460 --> 00:07:51,720 enforces that sigma squared plus pi squared equals to 1. 105 00:07:51,720 --> 00:07:54,140 The pi squared corresponds to the magnitude 106 00:07:54,140 --> 00:07:58,490 of this n minus 1 component vector. 107 00:07:58,490 --> 00:08:05,610 And essentially, I can solve for this delta function, 108 00:08:05,610 --> 00:08:08,490 and really replace this sigma over here 109 00:08:08,490 --> 00:08:11,525 with square root of 1 minus pi squared. 110 00:08:14,740 --> 00:08:19,690 But I have to be a little bit careful in my integrations. 111 00:08:19,690 --> 00:08:21,800 Because this delta function I can 112 00:08:21,800 --> 00:08:24,800 write as a delta function of sigma 113 00:08:24,800 --> 00:08:28,433 plus or minus square root of 1 minus pi squared. 114 00:08:35,190 --> 00:08:41,059 And there is a rule that if I use this delta function 115 00:08:41,059 --> 00:08:45,410 to set sigma to be equal to square root of 1 minus pi 116 00:08:45,410 --> 00:08:48,210 squared, like I have done over here, 117 00:08:48,210 --> 00:08:53,600 I have to be careful that the delta function of a times x 118 00:08:53,600 --> 00:08:58,670 is actually a delta function of x divided by modulus of a. 119 00:08:58,670 --> 00:09:01,540 So essentially, I have to substitute something here. 120 00:09:01,540 --> 00:09:06,950 So this is, in fact, equal to the integration in the pi 121 00:09:06,950 --> 00:09:11,300 directions because of the use of this delta function 122 00:09:11,300 --> 00:09:14,520 to set the value of sigma, I have 123 00:09:14,520 --> 00:09:17,670 to divide by the square root of 1 minus pi squared. 124 00:09:21,970 --> 00:09:24,070 Which actually shortly we will write 125 00:09:24,070 --> 00:09:25,360 this in the following way. 126 00:09:35,020 --> 00:09:37,740 I guess there's an overall factor of 1/2 127 00:09:37,740 --> 00:09:40,670 but it doesn't really matter. 128 00:09:40,670 --> 00:09:42,410 So yes? 129 00:09:42,410 --> 00:09:44,290 AUDIENCE: So what do you do with the fact 130 00:09:44,290 --> 00:09:48,190 that there are two places where the delta function is done? 131 00:09:48,190 --> 00:09:51,020 PROFESSOR: I'm continuously connecting 132 00:09:51,020 --> 00:09:53,780 to the solution that starts at 0 temperature 133 00:09:53,780 --> 00:09:55,210 with a particular state. 134 00:09:55,210 --> 00:09:59,210 So I have removed that ambiguity by the starting 135 00:09:59,210 --> 00:10:02,740 points of my cube. 136 00:10:02,740 --> 00:10:08,070 But if I was integrating over all possibilities, 137 00:10:08,070 --> 00:10:10,720 then I should really add that on too. 138 00:10:10,720 --> 00:10:15,010 And really just make the partition function 139 00:10:15,010 --> 00:10:17,150 with the sum of two equivalent terms-- 140 00:10:17,150 --> 00:10:21,740 one around this ground state, one around another state. 141 00:10:21,740 --> 00:10:24,400 AUDIENCE: The product is supposed to be for a lattice 142 00:10:24,400 --> 00:10:28,390 site-- for the integration variable, not for the-- 143 00:10:28,390 --> 00:10:31,130 PROFESSOR: Right. 144 00:10:31,130 --> 00:10:34,080 So I did something bad here. 145 00:10:34,080 --> 00:10:39,602 So here I should have written-- so this is an n component 146 00:10:39,602 --> 00:10:43,762 integration that I have to do on each side. 147 00:10:43,762 --> 00:10:46,070 Now let's pick one of the sides. 148 00:10:46,070 --> 00:10:49,850 So let's say we pick the side pi. 149 00:10:49,850 --> 00:10:54,224 For that, I have a small n integration to do. 150 00:11:04,120 --> 00:11:05,090 What does it say? 151 00:11:05,090 --> 00:11:10,520 It basically says that if, for example, 152 00:11:10,520 --> 00:11:15,130 I am looking at the case of n equals to 2, 153 00:11:15,130 --> 00:11:20,570 then I have started with a state that 154 00:11:20,570 --> 00:11:23,600 points along this direction. 155 00:11:23,600 --> 00:11:26,410 But now I'm allowing fluctuations pi 156 00:11:26,410 --> 00:11:29,650 in this direction. 157 00:11:29,650 --> 00:11:34,560 And I can't simply say that the amount of these fluctuations, 158 00:11:34,560 --> 00:11:38,640 pi, is going let's say from minus infinity to infinity. 159 00:11:38,640 --> 00:11:42,820 Because how much pi I have changes 160 00:11:42,820 --> 00:11:45,730 actually whether it is small or whether it is large 161 00:11:45,730 --> 00:11:48,050 when I'm down here. 162 00:11:48,050 --> 00:11:53,170 And so there's constraints let's say on how big pi can be. 163 00:11:53,170 --> 00:11:55,530 Pi cannot be larger than 1. 164 00:11:55,530 --> 00:11:58,550 And essentially, a particular magnitude 165 00:11:58,550 --> 00:12:03,064 of pi-- how much weight does it has, it is captured by this. 166 00:12:08,790 --> 00:12:12,710 So that's one thing to remember when 167 00:12:12,710 --> 00:12:16,040 we are dealing with integration over unit spins, 168 00:12:16,040 --> 00:12:19,700 and we want to look at the fluctuations. 169 00:12:19,700 --> 00:12:24,610 The other choice of notation that I would like to do 170 00:12:24,610 --> 00:12:26,040 is the following. 171 00:12:26,040 --> 00:12:32,140 I said that my starting [INAUDIBLE] 172 00:12:32,140 --> 00:12:37,949 is a 0 sum over all nearest neighbors Si dot Sj. 173 00:12:41,653 --> 00:12:45,870 Now in the state where all of the spins 174 00:12:45,870 --> 00:12:50,070 are pointing in one direction, this factor is unity. 175 00:12:50,070 --> 00:12:56,110 So the 0 temperature state gets a factor of 1 here on each one. 176 00:12:56,110 --> 00:12:58,420 Let's say we are in a hyper cubic lattice. 177 00:12:58,420 --> 00:13:00,610 There are d bonds per site. 178 00:13:00,610 --> 00:13:04,690 So at 0 temperature, I would have NdK0, basically-- 179 00:13:04,690 --> 00:13:07,980 the value of this ground state. 180 00:13:07,980 --> 00:13:12,660 And then if I have fluctuations from that state, 181 00:13:12,660 --> 00:13:17,920 I can capture that as follows, as minus K0 over 2. 182 00:13:17,920 --> 00:13:27,596 It's a reduction in this energy, as sum over ij Si minus Sj 183 00:13:27,596 --> 00:13:28,096 squared. 184 00:13:33,530 --> 00:13:35,330 And you can check that. 185 00:13:35,330 --> 00:13:39,072 If I square these terms, I'm going 186 00:13:39,072 --> 00:13:43,060 to get 1, 1, 1/2, which basically reproduces 187 00:13:43,060 --> 00:13:45,900 this, which actually goes over her. 188 00:13:45,900 --> 00:13:54,410 And the dot product, minus 2 Si dot Sj, cancels this minus 1/2, 189 00:13:54,410 --> 00:13:56,628 basically gives you this exactly. 190 00:13:59,610 --> 00:14:02,700 So the reason I write it in this fashion 191 00:14:02,700 --> 00:14:08,700 is because very shortly, I want to switch from going and doing 192 00:14:08,700 --> 00:14:12,490 things on a lattice to going to a continuum. 193 00:14:12,490 --> 00:14:16,650 And you can see that this form, summing over 194 00:14:16,650 --> 00:14:19,340 the difference between near neighbors, 195 00:14:19,340 --> 00:14:24,500 I very nicely can go to a gradient squared. 196 00:14:24,500 --> 00:14:27,220 So essentially that's what I want to do. 197 00:14:27,220 --> 00:14:33,040 Whenever I have a sum over a site, 198 00:14:33,040 --> 00:14:39,260 I want to replace it with an integral over a space. 199 00:14:39,260 --> 00:14:42,350 And I guess to keep things dimensionless, 200 00:14:42,350 --> 00:14:45,450 I have to divide by a to get d. 201 00:14:45,450 --> 00:14:51,180 So I can call that the density that I have to include, 202 00:14:51,180 --> 00:14:54,040 which is also the same thing as the number of lattice 203 00:14:54,040 --> 00:14:56,342 points in the body of the box. 204 00:15:00,710 --> 00:15:06,560 So my minus beta H in the continuum 205 00:15:06,560 --> 00:15:10,350 goes over to whatever the contribution 206 00:15:10,350 --> 00:15:13,480 of the completely aligned state is. 207 00:15:13,480 --> 00:15:18,770 And then whatever the difference of the spins 208 00:15:18,770 --> 00:15:22,620 is, because of the small fluctuations, 209 00:15:22,620 --> 00:15:28,080 I will capture through an integration of gradient of S 210 00:15:28,080 --> 00:15:28,580 squared. 211 00:15:33,380 --> 00:15:39,550 And I call the original coupling that we're basically 212 00:15:39,550 --> 00:15:43,790 the strength of the interaction divide by KdK0. 213 00:15:43,790 --> 00:15:47,080 Clearly in order to get the coupling 214 00:15:47,080 --> 00:15:49,650 that I have in the continuum, I have 215 00:15:49,650 --> 00:15:52,175 to have this fact of a to the d. 216 00:15:52,175 --> 00:15:58,000 But then in the gradient, I also have to divide by distance. 217 00:15:58,000 --> 00:16:02,660 So there's something here, a factor of a to the 2 minus d 218 00:16:02,660 --> 00:16:06,860 that relates these two factors. 219 00:16:06,860 --> 00:16:09,310 It should be the other way around. 220 00:16:22,140 --> 00:16:24,359 It doesn't matter. 221 00:16:24,359 --> 00:16:25,150 AUDIENCE: Question. 222 00:16:25,150 --> 00:16:27,850 PROFESSOR: Yes. 223 00:16:27,850 --> 00:16:32,178 AUDIENCE: This step is only valid for a cubicle-- 224 00:16:32,178 --> 00:16:33,480 PROFESSOR: Yes. 225 00:16:33,480 --> 00:16:36,250 So if it was something like a triangular lattice, 226 00:16:36,250 --> 00:16:39,390 or something, there would be some miracle factors here. 227 00:16:39,390 --> 00:16:40,874 AUDIENCE: But I mean like writing 228 00:16:40,874 --> 00:16:42,290 the difference of the spin squared 229 00:16:42,290 --> 00:16:44,504 as the gradient squared. 230 00:16:44,504 --> 00:16:46,620 Like if it were a triangular lattice? 231 00:16:46,620 --> 00:16:48,100 PROFESSOR: Yeah, so the statement 232 00:16:48,100 --> 00:16:52,000 is that whatever lattice you have, 233 00:16:52,000 --> 00:16:54,960 what am I doing at the level of the lattice, 234 00:16:54,960 --> 00:16:56,970 I'm trying to keep things that are 235 00:16:56,970 --> 00:16:59,700 close to each other aligned. 236 00:16:59,700 --> 00:17:03,380 So when I go to the continuum, how is this captured, 237 00:17:03,380 --> 00:17:06,720 it's a term like a gradient squared. 238 00:17:06,720 --> 00:17:10,980 Now on the hyper cubic lattices, the relationship 239 00:17:10,980 --> 00:17:12,950 between what you put on the bonds 240 00:17:12,950 --> 00:17:16,270 of the hyper cubic lattice and what we've got in the continuum 241 00:17:16,270 --> 00:17:18,359 is immediately apparent. 242 00:17:18,359 --> 00:17:20,609 If you try to do it on the triangular lattice, 243 00:17:20,609 --> 00:17:21,630 you still can. 244 00:17:21,630 --> 00:17:23,876 And you'll find that at the end of the day, 245 00:17:23,876 --> 00:17:25,709 you will get the factor of square root of 3, 246 00:17:25,709 --> 00:17:26,890 or something like that. 247 00:17:26,890 --> 00:17:29,700 So there's some miracle factor that comes into play. 248 00:17:36,400 --> 00:17:39,640 And then at the end of the day, I also 249 00:17:39,640 --> 00:17:43,200 want to replace these gradient of a squared 250 00:17:43,200 --> 00:17:50,230 naturally in terms of essentially S has n components. 251 00:17:50,230 --> 00:17:53,680 N minus one of them are pi, and one of them is sigma. 252 00:17:53,680 --> 00:17:59,680 So this would be minus K/2 integral d dx. 253 00:17:59,680 --> 00:18:02,596 I have gradient of the pi component, 254 00:18:02,596 --> 00:18:05,998 and a gradient of the sigma component squared. 255 00:18:10,830 --> 00:18:31,080 So after integrating sigma using the delta functions, 256 00:18:31,080 --> 00:18:33,155 I'm going to the continuum limit. 257 00:18:38,540 --> 00:18:43,490 The partition function that we have to evaluate up 258 00:18:43,490 --> 00:18:50,430 to various non-singular factors, such as this constant 259 00:18:50,430 --> 00:18:56,680 over here, is obtained by integrating over 260 00:18:56,680 --> 00:19:01,030 all configurations of our pi field, 261 00:19:01,030 --> 00:19:05,636 now regarded as a continuously varying object 262 00:19:05,636 --> 00:19:07,310 under the dimensional lattice. 263 00:19:10,150 --> 00:19:22,060 And the weight, which is as follows, 264 00:19:22,060 --> 00:19:30,590 there is a gradient of pi squared, essentially 265 00:19:30,590 --> 00:19:33,490 this term over here. 266 00:19:33,490 --> 00:19:37,970 There is the gradient the other term. 267 00:19:37,970 --> 00:19:42,200 The other term, however, if I use the delta function 268 00:19:42,200 --> 00:19:50,780 in square root of 1 minus pi squared squared. 269 00:19:50,780 --> 00:19:54,940 And then there's this factor from the integration 270 00:19:54,940 --> 00:19:58,490 that I have to be careful of, which I can also 271 00:19:58,490 --> 00:20:04,580 take to the exponent, and write as, again, this density times 272 00:20:04,580 --> 00:20:07,310 log of 1 minus pi squared. 273 00:20:07,310 --> 00:20:09,632 There's, I think, a factor of [INAUDIBLE]. 274 00:20:23,820 --> 00:20:30,350 So the weight that I had started with, with S 275 00:20:30,350 --> 00:20:36,660 dot S was kind of very simple-looking. 276 00:20:36,660 --> 00:20:40,610 But because of the constraints, was 277 00:20:40,610 --> 00:20:48,010 hiding a number of conditions. 278 00:20:48,010 --> 00:20:52,890 And if we explicitly look at those conditions 279 00:20:52,890 --> 00:20:59,056 and ask what is the weight of the fluctuations 280 00:20:59,056 --> 00:21:03,430 that I have to put around the ground 281 00:21:03,430 --> 00:21:05,550 state, these Goldstone modes, that 282 00:21:05,550 --> 00:21:08,780 is captured with this Hamiltonian. 283 00:21:08,780 --> 00:21:13,580 Part of it is this old contribution 284 00:21:13,580 --> 00:21:16,120 from Goldstone modes, the transfer 285 00:21:16,120 --> 00:21:18,290 modes that we had seen. 286 00:21:18,290 --> 00:21:23,720 But now being more careful, we see that these Goldstone modes, 287 00:21:23,720 --> 00:21:27,380 I have to be careful about integrating over them 288 00:21:27,380 --> 00:21:33,650 because of the additional terms that capture, essentially, 289 00:21:33,650 --> 00:21:38,520 the original full symmetry, full rotational symmetry, 290 00:21:38,520 --> 00:21:42,715 that was present in integration over S. Yes? 291 00:21:42,715 --> 00:21:45,355 AUDIENCE: The integration-- the functional integration, 292 00:21:45,355 --> 00:21:50,880 pi should be linked up to a sphere of 1. 293 00:21:50,880 --> 00:21:53,970 PROFESSOR: This will keep. 294 00:21:53,970 --> 00:21:59,300 So I put that constraint over here. 295 00:21:59,300 --> 00:22:03,110 And it's not just that it is limited to something, 296 00:22:03,110 --> 00:22:05,360 but for a particular value of pi, 297 00:22:05,360 --> 00:22:06,823 it gets this additional weight. 298 00:22:12,250 --> 00:22:15,470 So if you like, once I try to take my integrals 299 00:22:15,470 --> 00:22:19,808 outside that region, that factor says the weight as usual. 300 00:22:28,610 --> 00:22:32,205 So this entity is called the non-linear sigma model. 301 00:22:34,840 --> 00:22:40,065 And I never understood why they don't call it a non-linear pi 302 00:22:40,065 --> 00:22:40,908 model. 303 00:22:40,908 --> 00:22:45,090 Because we integrate immediately with sigma. 304 00:22:45,090 --> 00:22:46,363 That's how it is. 305 00:22:49,470 --> 00:22:56,730 So what you're going to do if we had, essentially, stuff 306 00:22:56,730 --> 00:22:59,870 that the first term not included any of the other things, 307 00:22:59,870 --> 00:23:02,730 we will have had the analysis of Goldstone modes 308 00:23:02,730 --> 00:23:06,370 that we had done previously. 309 00:23:06,370 --> 00:23:08,190 The effect of these things, you can 310 00:23:08,190 --> 00:23:12,360 see if I start making expansion in powers of pi, 311 00:23:12,360 --> 00:23:16,000 is to generate interactions that will 312 00:23:16,000 --> 00:23:18,650 be non-linear terms among the parts. 313 00:23:18,650 --> 00:23:21,440 So these Goldstone modes that we were previously 314 00:23:21,440 --> 00:23:26,480 dealing with as independent modes of the system, 315 00:23:26,480 --> 00:23:29,270 are actually non-linearly coupled. 316 00:23:29,270 --> 00:23:31,440 And we want to know what the effect of that 317 00:23:31,440 --> 00:23:33,990 is on the behavior of the entire system. 318 00:23:36,660 --> 00:23:41,240 So whenever we're faced with a non-linear theory, 319 00:23:41,240 --> 00:23:45,080 we have to do some kind of a preservative analysis. 320 00:23:45,080 --> 00:23:50,680 And the first thing that you may be tempted to do 321 00:23:50,680 --> 00:23:54,100 is to expand the powers of pi, and then 322 00:23:54,100 --> 00:23:57,070 look at the Gaussian part, and then the higher order 323 00:23:57,070 --> 00:23:58,900 parts, etc. 324 00:23:58,900 --> 00:24:00,370 That's a way of doing it. 325 00:24:00,370 --> 00:24:04,700 But there's actually another way that is more consistent, 326 00:24:04,700 --> 00:24:09,820 which is to organize the terms in this weight 327 00:24:09,820 --> 00:24:14,150 according to powers of temperature. 328 00:24:14,150 --> 00:24:17,710 Because after all, I started with a zero temperature 329 00:24:17,710 --> 00:24:19,000 configuration. 330 00:24:19,000 --> 00:24:21,610 And I'm hoping that I'm expanding 331 00:24:21,610 --> 00:24:23,540 for small fluctuation. 332 00:24:23,540 --> 00:24:26,710 So my idea is to-- I know the ground state. 333 00:24:26,710 --> 00:24:30,560 I want to see what happens if I go slightly beyond that. 334 00:24:30,560 --> 00:24:33,490 And the reason for fluctuations is temperature, 335 00:24:33,490 --> 00:24:45,410 so organize terms in this effective Hamiltonian 336 00:24:45,410 --> 00:24:50,540 for the pis in powers of temperature. 337 00:24:58,190 --> 00:25:02,980 And temperature, by this I mean the inverse of this coupling 338 00:25:02,980 --> 00:25:06,460 constant, K, because even, again, 339 00:25:06,460 --> 00:25:09,040 if I go through my old derivation, 340 00:25:09,040 --> 00:25:11,620 you can see that I go minus beta H, 341 00:25:11,620 --> 00:25:14,850 so K0 should be inversely proportional to temperature. 342 00:25:14,850 --> 00:25:16,470 K is proportion to K0. 343 00:25:16,470 --> 00:25:19,320 It should be inversely proportional to temperature. 344 00:25:19,320 --> 00:25:23,110 So to some overall coefficient, let's just 345 00:25:23,110 --> 00:25:26,760 define temperature [INAUDIBLE]. 346 00:25:26,760 --> 00:25:33,040 Now we see that at the level that we were looking 347 00:25:33,040 --> 00:25:37,220 at things before, from this term it's 348 00:25:37,220 --> 00:25:40,730 kind of like a Gaussian form, where 349 00:25:40,730 --> 00:25:44,470 I have something like K, which is the inverse temperature 350 00:25:44,470 --> 00:25:47,460 pi squared. 351 00:25:47,460 --> 00:25:54,730 So just on dimensional grounds, up to functional forms, etc. 352 00:25:54,730 --> 00:26:00,630 we expect pi squared to be proportional to temperature 353 00:26:00,630 --> 00:26:02,280 at the 0 order, if you like. 354 00:26:04,850 --> 00:26:07,260 Because, again, if temperature goes to 0, 355 00:26:07,260 --> 00:26:10,380 there's not going to no fluctuations. 356 00:26:10,380 --> 00:26:14,350 As I go away from 0 temperature, the average fluctuations 357 00:26:14,350 --> 00:26:14,960 will be 0. 358 00:26:14,960 --> 00:26:17,630 Average squared will be proportional to temperature. 359 00:26:17,630 --> 00:26:19,930 It all makes sense. 360 00:26:19,930 --> 00:26:25,230 So then if I look at this term, I 361 00:26:25,230 --> 00:26:28,420 see that dimensionally, it is inverse temperature 362 00:26:28,420 --> 00:26:30,670 pi squared is of the order of temperature. 363 00:26:30,670 --> 00:26:35,830 So this is dimensionally t to the 0. 364 00:26:35,830 --> 00:26:38,920 Whereas if I start to expand this, 365 00:26:38,920 --> 00:26:44,260 this log I can start to expand as minus pi squared 366 00:26:44,260 --> 00:26:52,210 plus pi to the 4th over 2 pi to the 6th over 3, and so forth. 367 00:26:52,210 --> 00:26:56,730 You can see that subsequent terms in this series 368 00:26:56,730 --> 00:27:01,070 are higher and higher order in this temperature. 369 00:27:01,070 --> 00:27:03,270 This will be the order of temperature-- temperature 370 00:27:03,270 --> 00:27:05,030 squared, temperature cubed. 371 00:27:05,030 --> 00:27:10,585 And already we can see that this term is small 372 00:27:10,585 --> 00:27:12,950 compared to this term. 373 00:27:12,950 --> 00:27:16,610 So although this is a Gaussian term, 374 00:27:16,610 --> 00:27:19,780 and I would've maybe been tempted 375 00:27:19,780 --> 00:27:22,550 to put it in the 0 order Hamiltonian, 376 00:27:22,550 --> 00:27:24,530 If I'm organizing things according 377 00:27:24,530 --> 00:27:29,420 to orders of temperature, my 0-th order will remain this. 378 00:27:29,420 --> 00:27:31,690 This will be the contribution to first order, 379 00:27:31,690 --> 00:27:33,780 2nd order, 3rd order. 380 00:27:33,780 --> 00:27:37,870 And similarly, I can start expanding this. 381 00:27:37,870 --> 00:27:43,610 Square root is 1 minus pi squared over 2. 382 00:27:43,610 --> 00:27:47,395 So then I take the gradient of minus pi squared over 2. 383 00:27:47,395 --> 00:27:50,290 I will get pi gradient of pi. 384 00:27:50,290 --> 00:27:54,500 You can see that the lowest order term in this expansion 385 00:27:54,500 --> 00:27:59,040 will be pi gradient of pi squared, and then 386 00:27:59,040 --> 00:28:02,350 higher order terms. 387 00:28:02,350 --> 00:28:07,020 And this is something that is order of pi to the 4th, 388 00:28:07,020 --> 00:28:12,170 so it gives the order of temperature squared multiplied 389 00:28:12,170 --> 00:28:13,710 by inverse temperatures. 390 00:28:13,710 --> 00:28:17,066 So this is a term that is contributing to order of T 391 00:28:17,066 --> 00:28:19,680 to 0, T to the 1. 392 00:28:19,680 --> 00:28:27,220 So basically, at order of T to 0, I have as my beta H0 393 00:28:27,220 --> 00:28:34,161 just the integral d dx K/2 gradient of pi squared. 394 00:28:38,000 --> 00:28:43,830 While at order of T the 1st power, 395 00:28:43,830 --> 00:28:50,170 I will have a correction which has two types of terms. 396 00:28:50,170 --> 00:28:59,350 One term is this K/2 integral d dx pi gradient of pi squared, 397 00:28:59,350 --> 00:29:04,190 coming from what was the gradient of sigma squared. 398 00:29:04,190 --> 00:29:08,220 And then from here, I will get a minus rho 399 00:29:08,220 --> 00:29:13,020 over 2 integral d dx pi squared. 400 00:29:13,020 --> 00:29:16,860 And then there will be other terms like order of T squared, 401 00:29:16,860 --> 00:29:22,140 U2, and so forth. 402 00:29:22,140 --> 00:29:28,430 So I just re-organized terms in this interacting Hamiltonian 403 00:29:28,430 --> 00:29:33,300 in what I expected to be powers of this temperature. 404 00:29:33,300 --> 00:29:39,290 Now here we-- one of the first things that we will do 405 00:29:39,290 --> 00:29:44,260 is to look at this and realize that we can decompose 406 00:29:44,260 --> 00:29:46,955 into modes by going to previous space, 407 00:29:46,955 --> 00:29:49,180 I do a Fourier transform. 408 00:29:49,180 --> 00:29:57,710 This thing becomes K/2 integral dd q divided by 2 pi to the d q 409 00:29:57,710 --> 00:30:03,230 squared pi theta of q squared. 410 00:30:03,230 --> 00:30:06,190 So let's write it as pi theta [INAUDIBLE]. 411 00:30:10,970 --> 00:30:16,850 And again, as usual, we will end up 412 00:30:16,850 --> 00:30:22,987 needing to calculate averages with this Gaussian rate. 413 00:30:22,987 --> 00:30:29,655 And what we have here is that pi alpha of q1 pi beta of q2, 414 00:30:29,655 --> 00:30:33,150 we get this 0-th ordered rate. 415 00:30:33,150 --> 00:30:36,660 The components have to be the same. 416 00:30:36,660 --> 00:30:42,062 The sum of the two momenta has to be 0. 417 00:30:42,062 --> 00:30:46,972 And if so, I just get K q squared. 418 00:30:56,301 --> 00:31:01,040 Now I can similarly Fourier transform the terms 419 00:31:01,040 --> 00:31:04,310 that I have over here. 420 00:31:04,310 --> 00:31:08,790 So the interactions to one-- first one 421 00:31:08,790 --> 00:31:13,020 becomes rather complicated. 422 00:31:13,020 --> 00:31:16,630 We saw that when we have something that's is 423 00:31:16,630 --> 00:31:19,200 four powers of a field. 424 00:31:19,200 --> 00:31:21,990 And when we go to Fourier space, rather 425 00:31:21,990 --> 00:31:24,220 than having one integral over x, we 426 00:31:24,220 --> 00:31:26,750 ended up with multiple integrals. 427 00:31:26,750 --> 00:31:29,460 So I will have, essentially, Fourier transform 428 00:31:29,460 --> 00:31:32,640 of four factors of pi. 429 00:31:32,640 --> 00:31:35,580 For each one of them I will have an integration. 430 00:31:35,580 --> 00:31:42,160 So I will have dd q1, dd q2, dd q3. 431 00:31:42,160 --> 00:31:45,620 And the reason I don't have the 4th one 432 00:31:45,620 --> 00:31:51,990 is because of the integration over x, forcing the four q's 433 00:31:51,990 --> 00:31:55,350 to be added up to 0. 434 00:31:55,350 --> 00:32:07,885 So I will have pi alpha of q1, pi alpha of q2, 435 00:32:07,885 --> 00:32:14,130 Now note that this high gradient of pi 436 00:32:14,130 --> 00:32:16,670 came from a gradient of pi squared, 437 00:32:16,670 --> 00:32:20,490 which means that the two pis that go with this, 438 00:32:20,490 --> 00:32:23,290 carry the same index. 439 00:32:23,290 --> 00:32:27,620 Whereas for the next factor, pi gradient of pi, 440 00:32:27,620 --> 00:32:29,318 they came from different ones. 441 00:32:29,318 --> 00:32:35,662 So I have pi q3 I beta minus q1 minus q2 minus q3. 442 00:32:41,540 --> 00:32:44,490 Now if I just written this, this would've 443 00:32:44,490 --> 00:32:49,800 been the Fourier transform of my usual 4th order interaction. 444 00:32:49,800 --> 00:32:51,950 But that's not what I have because I 445 00:32:51,950 --> 00:32:55,510 have two additional gradients. 446 00:32:55,510 --> 00:33:02,060 And so for two of these factors actually 447 00:33:02,060 --> 00:33:04,620 I had to take the gradient first. 448 00:33:04,620 --> 00:33:07,410 And every time I take a gradient in Fourier space, 449 00:33:07,410 --> 00:33:10,476 I will bring a factor of I q. 450 00:33:10,476 --> 00:33:18,430 So I will have here I q1 dotted with Iq let's say 3. 451 00:33:24,820 --> 00:33:30,750 So the Fourier transform of the leading quartic interaction 452 00:33:30,750 --> 00:33:36,272 that I have, is actually the form that I have over here. 453 00:33:36,272 --> 00:33:38,090 There is a trivial term that comes 454 00:33:38,090 --> 00:33:40,070 from Fourier transforming. 455 00:33:40,070 --> 00:33:44,358 It's pi squared because then I Fourier transform 456 00:33:44,358 --> 00:33:52,976 that, I get simply pi alpha of q squared. 457 00:33:57,080 --> 00:33:59,304 Yes? 458 00:33:59,304 --> 00:34:00,720 AUDIENCE: Does it matter which q's 459 00:34:00,720 --> 00:34:04,120 you're pulling out as the gradient? 460 00:34:04,120 --> 00:34:07,170 PROFESSOR: You can see that these four pis over here 461 00:34:07,170 --> 00:34:10,449 in Fourier space appear completely interchangeably. 462 00:34:10,449 --> 00:34:12,280 So it really doesn't matter, no. 463 00:34:12,280 --> 00:34:15,865 Because by permutation and re-ordering these integration, 464 00:34:15,865 --> 00:34:18,290 you can move it into something else. 465 00:34:21,026 --> 00:34:26,270 No, there is-- I shouldn't-- I'll draw a diagram that 466 00:34:26,270 --> 00:34:30,634 corresponds to that that will make one constraint apparent. 467 00:34:30,634 --> 00:34:37,239 So when I was drawing interaction terms for m 468 00:34:37,239 --> 00:34:39,389 to the 4th tier for Landau-Ginzburg, 469 00:34:39,389 --> 00:34:44,070 and I have something that has 4 interactions, 470 00:34:44,070 --> 00:34:47,370 I would draw something that has 2 lines. 471 00:34:47,370 --> 00:34:50,670 But the 2 lines had 2 branches. 472 00:34:50,670 --> 00:34:54,704 And the branching was supposed to indicate that 2 of them 473 00:34:54,704 --> 00:34:57,420 were carrier 1 index, and 2 of them 474 00:34:57,420 --> 00:35:00,790 were carrying the same index. 475 00:35:00,790 --> 00:35:03,815 Now I have to make sure that I indicate 476 00:35:03,815 --> 00:35:05,540 that the branches of these things 477 00:35:05,540 --> 00:35:08,140 additionally have these gradients 478 00:35:08,140 --> 00:35:11,100 for the Iq's associated with them. 479 00:35:11,100 --> 00:35:14,750 And I make a convention the branch, 480 00:35:14,750 --> 00:35:20,600 or the q, that has the gradient on it, I will put a line. 481 00:35:20,600 --> 00:35:23,720 Now you can see that if I go back 482 00:35:23,720 --> 00:35:28,720 and look at the origin of this, that one of the gradients 483 00:35:28,720 --> 00:35:32,530 acts on one pair of pis, and the other acts 484 00:35:32,530 --> 00:35:35,000 of the other pairs of pis. 485 00:35:35,000 --> 00:35:38,902 So the other dashed line I cannot put on the same branch, 486 00:35:38,902 --> 00:35:41,620 but I have to put over here. 487 00:35:41,620 --> 00:35:45,110 So the one constraint that I have to be careful of 488 00:35:45,110 --> 00:35:48,900 is that these Iq' should pick one from alpha and one 489 00:35:48,900 --> 00:35:49,882 from beta. 490 00:35:58,007 --> 00:35:59,590 This is the diagrammatic presentation. 491 00:36:09,630 --> 00:36:16,000 So what I can do is to now start doing perturbation 492 00:36:16,000 --> 00:36:17,740 in these interaction. 493 00:36:17,740 --> 00:36:20,880 You want to do the lowest order to see 494 00:36:20,880 --> 00:36:25,630 what the first correction because of fluctuations 495 00:36:25,630 --> 00:36:30,590 and interaction of these Goldstone modes. 496 00:36:30,590 --> 00:36:34,210 But rather than do things in two steps, 497 00:36:34,210 --> 00:36:37,510 first doing perturbation, encountering difficulty, 498 00:36:37,510 --> 00:36:41,190 and then converting things to a normalization group, 499 00:36:41,190 --> 00:36:44,820 which we've already seen that happen, that story, in dealing 500 00:36:44,820 --> 00:36:49,380 with the Landau-Ginzburg model, Let's immediately 501 00:36:49,380 --> 00:36:51,670 do the perturbative renormalization group 502 00:36:51,670 --> 00:36:54,310 of this model. 503 00:36:54,310 --> 00:36:56,940 So what I'm supposed to do things 504 00:36:56,940 --> 00:37:01,600 is to note that all of these theories 505 00:37:01,600 --> 00:37:04,590 came from some underlying lattice model. 506 00:37:04,590 --> 00:37:07,640 I was carefully drawing for you the first lattice 507 00:37:07,640 --> 00:37:09,410 model originally. 508 00:37:09,410 --> 00:37:14,910 Which means that there is some cut off here, 509 00:37:14,910 --> 00:37:16,950 some lattice cut off. 510 00:37:16,950 --> 00:37:20,020 Which means that when I go to Fourier space, 511 00:37:20,020 --> 00:37:29,640 there is always some kind of a range of wave numbers or wave 512 00:37:29,640 --> 00:37:32,460 vectors that I have to integrate with. 513 00:37:32,460 --> 00:37:43,010 So essentially, my pi's are limited 514 00:37:43,010 --> 00:37:46,350 after I do a little bit of averaging, 515 00:37:46,350 --> 00:37:53,470 if you like that there is some shortest wavelength, 516 00:37:53,470 --> 00:37:58,200 and the corresponding largest wave number, lambda, 517 00:37:58,200 --> 00:38:01,310 in [INAUDIBLE]. 518 00:38:01,310 --> 00:38:04,820 And the procedure for RG, the first one, 519 00:38:04,820 --> 00:38:10,960 was to think about all of these pi modes, 520 00:38:10,960 --> 00:38:14,460 and brake them into two pieces. 521 00:38:14,460 --> 00:38:19,310 One's that we're responding to the short wavelength 522 00:38:19,310 --> 00:38:22,265 fluctuations that we want to get rid of, 523 00:38:22,265 --> 00:38:26,312 and the ones that correspond to long wavelength fluctuations 524 00:38:26,312 --> 00:38:30,800 that we would like to keep. 525 00:38:30,800 --> 00:38:36,895 So my task is as follows, that I have to really calculate 526 00:38:36,895 --> 00:38:41,010 the partition function over here, which in it's Fourier 527 00:38:41,010 --> 00:38:47,880 representation indicates averaging over all modes 528 00:38:47,880 --> 00:38:51,310 that's are in this orange. 529 00:38:51,310 --> 00:38:54,100 But those modes I'm going to represent 530 00:38:54,100 --> 00:38:58,570 as D pi lesser, as well as D pi greater. 531 00:38:58,570 --> 00:39:00,240 Each one of these pi's is, of course, 532 00:39:00,240 --> 00:39:03,550 an n minus on component vector. 533 00:39:03,550 --> 00:39:09,650 And I have a rate that i obtained 534 00:39:09,650 --> 00:39:14,792 by substituting pi lesser and pi greater in the expressions 535 00:39:14,792 --> 00:39:17,820 that I have up there. 536 00:39:17,820 --> 00:39:24,510 And we can see already that the 0-th order terms, as usual, 537 00:39:24,510 --> 00:39:28,124 nicely separates out into a contribution 538 00:39:28,124 --> 00:39:35,480 that we have for pi lesser, a contribution that we have 539 00:39:35,480 --> 00:39:43,080 for pi greater, and that the interaction terms will then 540 00:39:43,080 --> 00:39:49,520 involve both of these modes. 541 00:39:49,520 --> 00:39:52,540 And in principle, I could proceed and include 542 00:39:52,540 --> 00:39:53,920 higher and higher orders. 543 00:39:57,140 --> 00:40:03,697 Now I want to get rid of all of the modes that are here. 544 00:40:03,697 --> 00:40:08,170 So that I have an effective theory governing the modes 545 00:40:08,170 --> 00:40:11,150 that are the longer wavelengths, once I have gotten rid 546 00:40:11,150 --> 00:40:14,710 of the short wavelength fluctuations. 547 00:40:14,710 --> 00:40:19,400 So formally, once I have integrated over pi greater 548 00:40:19,400 --> 00:40:22,715 in this double integral, I will be 549 00:40:22,715 --> 00:40:27,080 left with the integration over the pi lesser field. 550 00:40:31,517 --> 00:40:35,620 And the exponential gets modified as follows. 551 00:40:35,620 --> 00:40:41,230 First of all, if I were to ignore the interactions 552 00:40:41,230 --> 00:40:45,850 at the lowest order, the effect of doing 553 00:40:45,850 --> 00:40:49,663 the integration of the Gaussian modes that are out here, 554 00:40:49,663 --> 00:40:51,928 will, as usual, be a contribution 555 00:40:51,928 --> 00:40:58,230 to the free energy of the system coming from the modes 556 00:40:58,230 --> 00:41:00,250 that I integrated out. 557 00:41:00,250 --> 00:41:04,526 And clearly it also depends, I forgot to say, 558 00:41:04,526 --> 00:41:11,510 that the range of integration is now between lambda over b 559 00:41:11,510 --> 00:41:18,330 lambda, where b is my renormalization factor. 560 00:41:18,330 --> 00:41:20,520 Yes? 561 00:41:20,520 --> 00:41:23,355 AUDIENCE: Because you're coming from a lattice, does 562 00:41:23,355 --> 00:41:26,620 the particular shape of the Brillouin zone matter 563 00:41:26,620 --> 00:41:28,840 more now, or still not really? 564 00:41:28,840 --> 00:41:32,050 PROFESSOR: It is in no way different from what 565 00:41:32,050 --> 00:41:35,170 we were doing before in the Landau-Ginzburg model. 566 00:41:35,170 --> 00:41:38,230 In the Landau-Ginzburg model, I could have also started 567 00:41:38,230 --> 00:41:40,360 by putting spins, or whatever degrees 568 00:41:40,360 --> 00:41:41,900 of freedom on a lattice. 569 00:41:41,900 --> 00:41:45,204 And let's say if I was in hyper cubic lattice, 570 00:41:45,204 --> 00:41:48,340 I would've had Brillouin zones, such as this. 571 00:41:48,340 --> 00:41:50,730 And the first thing that we always said 572 00:41:50,730 --> 00:41:57,540 was that integrating all of these things 573 00:41:57,540 --> 00:42:01,775 gives you an additional totally harmless component 574 00:42:01,775 --> 00:42:05,890 to the energy that has no similar part in it. 575 00:42:05,890 --> 00:42:12,030 So we're always searching for the singularities that 576 00:42:12,030 --> 00:42:15,440 arise at the core of this integration. 577 00:42:15,440 --> 00:42:17,600 Whatever you do with the boundaries, 578 00:42:17,600 --> 00:42:24,037 no matter how complicated shapes they have, they don't matter. 579 00:42:31,290 --> 00:42:36,230 So going back to here. 580 00:42:36,230 --> 00:42:40,660 If we had ignored the interactions, 581 00:42:40,660 --> 00:42:43,620 integrating over pi greater would've 582 00:42:43,620 --> 00:42:46,620 giving me this contribution to the free energy. 583 00:42:46,620 --> 00:42:54,200 And, of course, beta H0 of pi lesser would've remained. 584 00:42:56,870 --> 00:43:02,500 But now the effect of having the interactions, as usual, 585 00:43:02,500 --> 00:43:05,450 it is like integrating into the minus 586 00:43:05,450 --> 00:43:10,160 u with the rate over here. 587 00:43:10,160 --> 00:43:15,120 So I would have an average such as this. 588 00:43:15,120 --> 00:43:19,030 And we do the cumulant expansion, as usual. 589 00:43:19,030 --> 00:43:23,588 And the first term I would get is the average 590 00:43:23,588 --> 00:43:32,550 of this quantity with respect to the Gaussian rate, 591 00:43:32,550 --> 00:43:38,770 integrating out the high component modes, high frequency 592 00:43:38,770 --> 00:43:40,702 modes, and high order corrections. 593 00:43:44,560 --> 00:43:45,170 Yes? 594 00:43:45,170 --> 00:43:48,130 AUDIENCE: So right here you're doing two expansions kind 595 00:43:48,130 --> 00:43:49,230 of simultaneously. 596 00:43:49,230 --> 00:43:52,147 One is you have non-linear model that you're 597 00:43:52,147 --> 00:43:54,630 expanding different powers and temperature. 598 00:43:54,630 --> 00:43:56,655 And then you further on expand it 599 00:43:56,655 --> 00:44:01,670 to cumulants to be able to account for that. 600 00:44:01,670 --> 00:44:08,293 PROFESSOR: No, because I can organize this expansion 601 00:44:08,293 --> 00:44:12,500 in cumulants in powers of temperature. 602 00:44:12,500 --> 00:44:18,420 So this u has an expansion that is u1, u2, etc. 603 00:44:18,420 --> 00:44:20,490 organized in powers of temperature. 604 00:44:20,490 --> 00:44:21,480 AUDIENCE: OK. 605 00:44:21,480 --> 00:44:24,780 PROFESSOR: And then when I take the first cumulant, 606 00:44:24,780 --> 00:44:27,627 you can see that the average, the lowest order term, 607 00:44:27,627 --> 00:44:28,127 will be-- 608 00:44:28,127 --> 00:44:31,606 AUDIENCE: The first cumulant is linear in temperature, 609 00:44:31,606 --> 00:44:32,886 and that's what you want? 610 00:44:32,886 --> 00:44:33,594 PROFESSOR: Right. 611 00:44:39,070 --> 00:44:44,040 So I'm being consistent also with the perturbation 612 00:44:44,040 --> 00:44:48,730 that I had originally stated. 613 00:44:48,730 --> 00:44:53,650 Actually, since I drew a diagram for the first term, 614 00:44:53,650 --> 00:44:57,920 I should state that this term, since we are now also thinking 615 00:44:57,920 --> 00:45:03,495 of it as a correction in u1, I have 616 00:45:03,495 --> 00:45:06,190 to regard it as 2 factors of pi. 617 00:45:06,190 --> 00:45:12,650 So I could potentially represent it by a diagram such as this. 618 00:45:12,650 --> 00:45:23,620 So diagrammatically, my u1 that I have to take the average 619 00:45:23,620 --> 00:45:27,130 is composed of these two entities. 620 00:45:38,940 --> 00:45:48,270 So what I need to do is to take the average of that expression. 621 00:45:48,270 --> 00:45:55,645 So I can either do that average over here. 622 00:45:55,645 --> 00:45:58,320 Take the average of this expression, 623 00:45:58,320 --> 00:46:00,760 or do it diagrammatically. 624 00:46:00,760 --> 00:46:03,348 Let us go by the diagrammatic route. 625 00:46:06,090 --> 00:46:09,400 So essentially, what I'm doing is 626 00:46:09,400 --> 00:46:18,370 that every line that I see over there that corresponds to pi, 627 00:46:18,370 --> 00:46:23,800 I am really decomposing into two parts. 628 00:46:23,800 --> 00:46:26,910 One of them I will draw as a straight line that corresponds 629 00:46:26,910 --> 00:46:31,070 to the pi lesser that I am keeping. 630 00:46:31,070 --> 00:46:33,370 Or I replace it with a wavy line, 631 00:46:33,370 --> 00:46:38,510 which is the pi greater that I would be averaging over. 632 00:46:41,500 --> 00:46:47,940 So the first diagram I had essentially something 633 00:46:47,940 --> 00:46:52,570 like this-- actually, the second diagram. 634 00:46:52,570 --> 00:46:56,070 The one that comes from rho pi squared. 635 00:46:56,070 --> 00:47:00,770 It's actually trivial, so let's go through the possibilities. 636 00:47:00,770 --> 00:47:08,600 I can either have both of these to be pi lessers-- sorry, 637 00:47:08,600 --> 00:47:09,570 pi greaters. 638 00:47:09,570 --> 00:47:11,870 So this is pi greater, pi greater. 639 00:47:11,870 --> 00:47:14,500 And when I have to do an average, 640 00:47:14,500 --> 00:47:16,275 then I can use the formula that I 641 00:47:16,275 --> 00:47:19,580 have in red about the average of 2 pi greaters. 642 00:47:22,790 --> 00:47:25,540 And that would essentially amount 643 00:47:25,540 --> 00:47:28,190 to closing this thing down. 644 00:47:28,190 --> 00:47:33,855 And numerically, it would gives me a factor of minus rho over 2 645 00:47:33,855 --> 00:47:40,680 integral d dK over 2 pi to the d in the interval between lambda 646 00:47:40,680 --> 00:47:42,990 over b, lambda . 647 00:47:42,990 --> 00:47:46,590 And I have the average of pi alpha pi 648 00:47:46,590 --> 00:47:51,060 alpha using a factor of delta alpha alpha. 649 00:47:51,060 --> 00:47:55,940 Summing over alpha will give me a factor of n minus 1. 650 00:47:55,940 --> 00:48:01,482 And the average would be something like K k squared. 651 00:48:01,482 --> 00:48:07,540 So I would have to evaluate something like this. 652 00:48:07,540 --> 00:48:09,910 But at the end of the day, I don't care about it. 653 00:48:09,910 --> 00:48:11,700 Why don't I care about it? 654 00:48:11,700 --> 00:48:16,210 Because clearly the result of doing this is another constant. 655 00:48:16,210 --> 00:48:19,430 It doesn't depend on pi lesser. 656 00:48:19,430 --> 00:48:29,260 So this is an addition to the free energy 657 00:48:29,260 --> 00:48:34,450 once I integrate modes between lambda over b to lambda, 658 00:48:34,450 --> 00:48:36,718 there is a contribution to the free energy that 659 00:48:36,718 --> 00:48:38,350 comes from this term. 660 00:48:38,350 --> 00:48:42,410 It doesn't change the rate that I 661 00:48:42,410 --> 00:48:47,650 have to assign to configurations of the pi lesser field. 662 00:48:47,650 --> 00:48:49,230 That's another possibility. 663 00:48:49,230 --> 00:48:53,910 Another possibility is I have one of them being a pi 664 00:48:53,910 --> 00:48:57,410 greater, one of them being a pi lesser. 665 00:48:57,410 --> 00:49:02,630 Clearly, when I try to get an average of this form, 666 00:49:02,630 --> 00:49:05,900 I have an average of one factor of pi 667 00:49:05,900 --> 00:49:08,943 with a Gaussian field that is even. 668 00:49:08,943 --> 00:49:10,655 So this is 0. 669 00:49:10,655 --> 00:49:13,190 We don't have to worry about it. 670 00:49:13,190 --> 00:49:19,640 And finally, I will get a term, which is like this. 671 00:49:19,640 --> 00:49:22,190 Which doesn't involve any integrations, 672 00:49:22,190 --> 00:49:24,920 and really amounts to taking that term that I 673 00:49:24,920 --> 00:49:28,190 have over there, and just making both of those pi 674 00:49:28,190 --> 00:49:30,010 to be pi lessers. 675 00:49:30,010 --> 00:49:33,610 So it's essentially the same form that will reappear, 676 00:49:33,610 --> 00:49:38,620 now the integration being from 0 to lambda over 2. 677 00:49:38,620 --> 00:49:45,180 So we know exactly what happens with the term on the right. 678 00:49:45,180 --> 00:49:51,270 Nothing useful, or important information emerges from it. 679 00:49:51,270 --> 00:49:56,510 If I go and look at this one however, 680 00:49:56,510 --> 00:50:02,250 depending on where I choose to put 681 00:50:02,250 --> 00:50:04,680 the solid lines or the wavy lines, 682 00:50:04,680 --> 00:50:07,840 I will have a number of possibilities. 683 00:50:07,840 --> 00:50:11,670 One thing that is clearly going to be there 684 00:50:11,670 --> 00:50:17,270 is essentially I put pi lesser for each one of the branches. 685 00:50:17,270 --> 00:50:21,320 Essentially, when i write it here like this one, 686 00:50:21,320 --> 00:50:22,855 it is reproducing the integration 687 00:50:22,855 --> 00:50:25,960 that I have over there, except that, again, it 688 00:50:25,960 --> 00:50:30,990 only goes between 0 and lambda. 689 00:50:30,990 --> 00:50:34,080 And now I can start adding wavy lines. 690 00:50:34,080 --> 00:50:39,360 Any diagram that has one wavy, and I can put the wavy line 691 00:50:39,360 --> 00:50:43,560 either on that type of branch, or I 692 00:50:43,560 --> 00:50:46,110 can put it on this type of branch. 693 00:50:50,280 --> 00:50:55,930 It has only one factor of pi. 694 00:50:55,930 --> 00:50:59,340 By symmetry, it will go to 0, like this. 695 00:50:59,340 --> 00:51:00,990 There will be things that will have 696 00:51:00,990 --> 00:51:03,690 three factors of pi lesser. 697 00:51:20,650 --> 00:51:25,470 And all of these-- again because I'm 698 00:51:25,470 --> 00:51:29,440 dealing with an odd number of factors of pi 699 00:51:29,440 --> 00:51:34,290 greater that I'm averaging will give me 0. 700 00:51:34,290 --> 00:51:39,120 There's one other thing that is kind of interesting. 701 00:51:39,120 --> 00:51:46,600 I can have all four of these lines wavy. 702 00:51:46,600 --> 00:51:48,670 And if I calculate that average, there's 703 00:51:48,670 --> 00:51:51,650 a number of ways of contracting these four 704 00:51:51,650 --> 00:51:55,770 pi's that will give me nontrivial factors. 705 00:51:55,770 --> 00:52:02,460 But these are also contributions to the free energy. 706 00:52:02,460 --> 00:52:07,870 They don't depend on the pi's that I'm leaving out. 707 00:52:07,870 --> 00:52:11,630 So they don't have to worry about any of these diagrams 708 00:52:11,630 --> 00:52:12,150 so far. 709 00:52:14,720 --> 00:52:23,230 Now I dealt with the 0, 1, 3, and 4 wavy lines. 710 00:52:23,230 --> 00:52:26,870 So I'm left with 2 wavy lines and 2 straight lines. 711 00:52:26,870 --> 00:52:28,890 So let's go through those. 712 00:52:28,890 --> 00:52:33,085 I could have one branch be wavy lines 713 00:52:33,085 --> 00:52:35,595 and one branch be straight lines. 714 00:52:40,010 --> 00:52:44,770 And then I take the average of this object. 715 00:52:44,770 --> 00:52:48,210 I have a pi greater-- a pi greater here, 716 00:52:48,210 --> 00:52:51,230 and therefore I can do an average of two 717 00:52:51,230 --> 00:52:53,680 of those pi greaters. 718 00:52:53,680 --> 00:52:57,315 That average will give me a factor of 1 over K k squared. 719 00:53:00,690 --> 00:53:04,790 I have to integrate over that. 720 00:53:04,790 --> 00:53:08,470 But one of these branches had this additional dash thing 721 00:53:08,470 --> 00:53:12,290 that corresponds to having a factor of k. 722 00:53:12,290 --> 00:53:15,845 So the integral that I have to do 723 00:53:15,845 --> 00:53:20,780 involves something like this. 724 00:53:20,780 --> 00:53:26,010 And then I integrate over the entirety of the k integration. 725 00:53:26,010 --> 00:53:31,356 This is an odd power, and so that will give me a 0 also. 726 00:53:31,356 --> 00:53:33,070 So this is also 0. 727 00:53:37,460 --> 00:53:38,940 And there's another one that's is 728 00:53:38,940 --> 00:53:44,375 like this where I go like this. 729 00:53:52,740 --> 00:53:57,320 And although I do the same thing now with two 730 00:53:57,320 --> 00:54:00,820 different branches, the k integration is the same. 731 00:54:00,820 --> 00:54:02,225 And that vanishes too. 732 00:54:04,960 --> 00:54:08,350 So you say, is there anything that is left? 733 00:54:08,350 --> 00:54:09,340 The answer is yes. 734 00:54:09,340 --> 00:54:13,760 So the things that are left are the following. 735 00:54:13,760 --> 00:54:18,180 I can do something like this. 736 00:54:26,770 --> 00:54:29,331 Or I can do something like this. 737 00:54:37,440 --> 00:54:46,080 So these are the two things that survive and will be nontrivial. 738 00:54:46,080 --> 00:54:48,270 You can see that this one will be 739 00:54:48,270 --> 00:54:54,390 proportional to pi lesser squared, 740 00:54:54,390 --> 00:55:00,420 while this one is going to be proportional to gradient 741 00:55:00,420 --> 00:55:01,630 of pi lesser squared. 742 00:55:04,250 --> 00:55:10,825 So this one will renormalize, if you like, this coefficient. 743 00:55:13,660 --> 00:55:19,830 Whereas this one, we've modified and renormalize our coupling 744 00:55:19,830 --> 00:55:20,740 straight. 745 00:55:20,740 --> 00:55:23,310 So it turns out that that is really 746 00:55:23,310 --> 00:55:25,810 the more important point. 747 00:55:25,810 --> 00:55:29,100 But let's calculate the other one too. 748 00:55:29,100 --> 00:55:29,700 Yes? 749 00:55:29,700 --> 00:55:32,646 AUDIENCE: Why do we connect the ones down here with the loops, 750 00:55:32,646 --> 00:55:34,610 but left all the ends free in the ones. 751 00:55:34,610 --> 00:55:38,510 Was that just a matter of the case of how to write diagram, 752 00:55:38,510 --> 00:55:41,235 or does that signify something? 753 00:55:41,235 --> 00:55:42,610 PROFESSOR: Could you repeat that? 754 00:55:42,610 --> 00:55:43,780 I'm not sure I understand. 755 00:55:43,780 --> 00:55:47,906 AUDIENCE: So when we had two wavy line, both coming out 756 00:55:47,906 --> 00:55:50,930 one of the diagram, this line, they just stop. 757 00:55:50,930 --> 00:55:53,211 We connected them together when we 758 00:55:53,211 --> 00:55:55,640 were writing the ones on the bottom line. 759 00:55:55,640 --> 00:55:59,480 PROFESSOR: So basically, I start with an entity that 760 00:55:59,480 --> 00:56:07,180 has two solid lines and two wavy lines. 761 00:56:07,180 --> 00:56:11,640 And what I'm supposed to do is to do an integration-- 762 00:56:11,640 --> 00:56:19,070 an average of this over these pi greaters. 763 00:56:19,070 --> 00:56:26,920 Now the process of averaging essentially joins 764 00:56:26,920 --> 00:56:30,150 the two branches. 765 00:56:30,150 --> 00:56:33,616 If I had the momentum here, q1, and a momentum here, 766 00:56:33,616 --> 00:56:36,440 q2, if I had an index here, alpha, 767 00:56:36,440 --> 00:56:40,940 and an index here, beta, that process of averaging 768 00:56:40,940 --> 00:56:45,150 is equivalent to saying the same momentum has to go through, 769 00:56:45,150 --> 00:56:47,640 the same index has to go through. 770 00:56:47,640 --> 00:56:50,996 There is no averaging that is being done on the solid lines, 771 00:56:50,996 --> 00:56:53,365 so there is-- meaningless to do anything. 772 00:57:07,720 --> 00:57:18,410 So this entity means the following. 773 00:57:18,410 --> 00:57:29,980 I have K/ 2 let's call it legs 1, 2, 3, and 4. 774 00:57:29,980 --> 00:57:39,660 Integral q1 and q2, but q1 and q2 you can see explicitly 775 00:57:39,660 --> 00:57:40,936 are solid. 776 00:57:40,936 --> 00:57:46,648 So these are the integration from 0 to lambda over b. 777 00:57:46,648 --> 00:57:58,805 I have an integration over q3, which is over a wavy line. 778 00:57:58,805 --> 00:58:02,200 So it's between lambda over b and lambda. 779 00:58:06,080 --> 00:58:13,390 If I call this branch alpha and this branch beta, from here 780 00:58:13,390 --> 00:58:24,290 I have actually pi lesser alpha of q1 pi lesser beta of q2. 781 00:58:24,290 --> 00:58:26,300 I should have put them outside the integration, 782 00:58:26,300 --> 00:58:28,600 but it doesn't matter. 783 00:58:28,600 --> 00:58:36,970 And then here I had pi alpha of q3, pi beta of q4. 784 00:58:36,970 --> 00:58:41,575 But these also had these lines associated with them. 785 00:58:41,575 --> 00:58:49,840 So I have here actually an i q3, an i q4. 786 00:58:49,840 --> 00:58:59,210 Again, q4 has to stand for minus q1 minus q2 minus q3 from this. 787 00:58:59,210 --> 00:59:04,450 And then I had the pi pi here, which give me, 788 00:59:04,450 --> 00:59:09,150 because of the averaging, delta alpha beta. 789 00:59:09,150 --> 00:59:14,120 And then I will have an integration 790 00:59:14,120 --> 00:59:21,820 that forces q2 plus q4 to be 0. 791 00:59:21,820 --> 00:59:24,450 And then I have K q3 squared. 792 00:59:30,700 --> 00:59:36,382 Now q3 plus q4 is the same thing as minus q1 minus q3, if you 793 00:59:36,382 --> 00:59:39,530 like, because of that constraint. 794 00:59:39,530 --> 00:59:41,800 So I can take that outside the integration. 795 00:59:41,800 --> 00:59:44,840 There's no problem. 796 00:59:44,840 --> 00:59:51,520 I have one integration left, which is 1 over K q3 squared, 797 00:59:51,520 --> 00:59:54,530 but these two then become the same. 798 00:59:54,530 --> 00:59:56,155 These pi's I will take outside. 799 00:59:58,760 --> 01:00:02,940 I note that because of this constraint, q1 and q2 being 800 01:00:02,940 --> 01:00:10,004 the same, these two really become one integration that 801 01:00:10,004 --> 01:00:13,430 goes between 0 and lambda over b. 802 01:00:13,430 --> 01:00:17,010 And these indices have been made to be the same. 803 01:00:17,010 --> 01:00:21,995 So I have pi alpha of q, this q, squared. 804 01:00:28,530 --> 01:00:33,380 Then I have the integration from lambda over b to lambda. 805 01:00:33,380 --> 01:00:40,270 D d q3 2 pi to the d. 806 01:00:40,270 --> 01:00:43,270 Here I have i q3 i q4. 807 01:00:43,270 --> 01:00:47,530 But q4 was said to be minus q3. 808 01:00:47,530 --> 01:00:50,990 So the two i's and the minus cancel each other. 809 01:00:50,990 --> 01:00:54,910 And I will get a factor of q3 squared. 810 01:00:54,910 --> 01:00:58,425 And then here I have a factor of K q3 squared. 811 01:01:03,420 --> 01:01:08,680 So the overall thing is just that we 812 01:01:08,680 --> 01:01:13,370 can see that the K's cancel. 813 01:01:13,370 --> 01:01:21,840 I have one factor integral dd q 2 pi to the d. 814 01:01:21,840 --> 01:01:25,590 I have pi alpha of q squared. 815 01:01:25,590 --> 01:01:29,380 And these are q lessers. 816 01:01:29,380 --> 01:01:32,600 I integrated out this quantity. 817 01:01:32,600 --> 01:01:39,910 The q3's vanish, so I really have the integral of q3 818 01:01:39,910 --> 01:01:44,026 over 2 pi to the d. 819 01:01:44,026 --> 01:01:54,200 Now if I had done the integral of q3 2 pi 820 01:01:54,200 --> 01:02:03,370 to the d, all the way from 0 to lambda, what would I have done? 821 01:02:03,370 --> 01:02:05,970 If I multiply this volume here, that 822 01:02:05,970 --> 01:02:08,564 would be the number of modes. 823 01:02:08,564 --> 01:02:14,720 So this is, in fact, N/V, which is the quantity 824 01:02:14,720 --> 01:02:16,646 that I have called the density. 825 01:02:19,520 --> 01:02:21,590 But what I'm doing is, in fact, doing 826 01:02:21,590 --> 01:02:27,010 just a fraction of this integral from 0 to lambda over b. 827 01:02:27,010 --> 01:02:31,075 So if I do the fraction from 0 to lambda over b, 828 01:02:31,075 --> 01:02:36,100 then I will get 1 minus b to the minus d. 829 01:02:36,100 --> 01:02:42,320 Sorry, from lambda over b to lambda. 830 01:02:42,320 --> 01:02:46,480 Then if I had done all of the way from 0 to lambda, 831 01:02:46,480 --> 01:02:49,600 I will have had one, but I'm subtracting 832 01:02:49,600 --> 01:02:51,350 this fraction of it. 833 01:02:51,350 --> 01:02:57,650 So the answer is rho 1 minus b to the minus d. 834 01:02:57,650 --> 01:03:03,950 The overall thing here gets multiplied by rho 1 minus 835 01:03:03,950 --> 01:03:05,904 b to the minus d. 836 01:03:09,890 --> 01:03:14,720 It just would correct that factor of density that we have. 837 01:03:14,720 --> 01:03:17,810 We'll see shortly it's not something to worry about. 838 01:03:17,810 --> 01:03:21,934 The next one is really the more interesting thing. 839 01:03:21,934 --> 01:03:34,870 So here we have this diagram, which is K/2 integral from 0 840 01:03:34,870 --> 01:03:36,910 to lambda over b. 841 01:03:40,140 --> 01:03:43,400 Essentially, I will get the same structure. 842 01:03:48,730 --> 01:03:51,760 This time let me write the pi alpha lesser 843 01:03:51,760 --> 01:04:00,030 of q1 pi beta lesser of q2 outside the last configuration. 844 01:04:00,030 --> 01:04:07,070 I have the integral from lambda over b to lambda dd of q3 2 845 01:04:07,070 --> 01:04:09,860 pi to the 3d. 846 01:04:09,860 --> 01:04:13,460 And I will have the same delta function structure, 847 01:04:13,460 --> 01:04:21,600 except that now these factors of i q become i q1 times i q2. 848 01:04:21,600 --> 01:04:23,320 So I can put them outside already. 849 01:04:27,208 --> 01:04:30,210 And then I have here the delta function. 850 01:04:43,620 --> 01:04:47,520 So the only difference is that previously the q squared 851 01:04:47,520 --> 01:04:50,740 was inside the integration. 852 01:04:50,740 --> 01:04:54,140 Now the q squared is outside the integration. 853 01:04:54,140 --> 01:04:59,810 So the final answer will be K/2 integral 0 to lambda 854 01:04:59,810 --> 01:05:05,232 over d d d q 2 pi to the d. 855 01:05:05,232 --> 01:05:08,650 I have a q squared. 856 01:05:08,650 --> 01:05:15,724 pi of q lesser squared. 857 01:05:15,724 --> 01:05:18,560 And the coefficient of that would 858 01:05:18,560 --> 01:05:21,850 look like what I had before, except without this factor. 859 01:05:21,850 --> 01:05:24,830 So it's the integral from lambda over b 860 01:05:24,830 --> 01:05:31,873 to lambda dd of q3 divided by 2 pi to the d 1 over Kq3 squared. 861 01:05:47,920 --> 01:05:54,370 So once I do explicitly this calculation, 862 01:05:54,370 --> 01:06:00,400 the answer is going to be a weight that only depends 863 01:06:00,400 --> 01:06:04,360 on this pi lesser that I'm keeping 864 01:06:04,360 --> 01:06:09,450 and I would indicate that by beta H tilde, as usual. 865 01:06:09,450 --> 01:06:13,690 That depends of this pi lesser. 866 01:06:13,690 --> 01:06:18,410 And we now have all of the terms that 867 01:06:18,410 --> 01:06:20,840 contribute to this beta H tilde. 868 01:06:20,840 --> 01:06:23,740 So let's write them down. 869 01:06:23,740 --> 01:06:26,300 There's a number of terms that correspond 870 01:06:26,300 --> 01:06:30,110 to changes in the free energy. 871 01:06:30,110 --> 01:06:36,750 So we have a V delta f p at the 0-th order contribution 872 01:06:36,750 --> 01:06:40,960 of delta f p at the first order that are essentially 873 01:06:40,960 --> 01:06:46,410 a bunch of diagrams, both from here as well as from here. 874 01:06:46,410 --> 01:06:48,416 But we don't really care about them. 875 01:06:51,560 --> 01:07:03,720 Then we have types of terms that look like this. 876 01:07:03,720 --> 01:07:08,180 I can write them already after Fourier transformation 877 01:07:08,180 --> 01:07:09,090 in real space. 878 01:07:09,090 --> 01:07:12,080 So let me do that. 879 01:07:12,080 --> 01:07:17,990 I have integral dd x in real space, 880 01:07:17,990 --> 01:07:22,315 realizing that my cut off has been changed 881 01:07:22,315 --> 01:07:27,730 to ba-- things that are proportional to gradient 882 01:07:27,730 --> 01:07:31,722 of pi lesser squared. 883 01:07:31,722 --> 01:07:34,470 Now gradient of pi lesser squared 884 01:07:34,470 --> 01:07:36,820 is the things I in Fourier space becomes 885 01:07:36,820 --> 01:07:40,810 q squared pi lesser squared. 886 01:07:40,810 --> 01:07:42,470 And what is the coefficient? 887 01:07:45,050 --> 01:07:51,910 I had K/2, which comes from here. 888 01:07:51,910 --> 01:07:57,680 If I don't do anything, I just have the 0-th order modes 889 01:07:57,680 --> 01:08:00,530 acting on these. 890 01:08:00,530 --> 01:08:02,920 But I just calculated a correction 891 01:08:02,920 --> 01:08:09,440 to that that is something like this. 892 01:08:09,440 --> 01:08:14,060 So in addition to what I had before, 893 01:08:14,060 --> 01:08:21,747 I have this correction, K/2 times this integral. 894 01:08:21,747 --> 01:08:27,106 And I'm going to call the result of this integral 895 01:08:27,106 --> 01:08:39,439 to be I sub d-- see that this integral is 896 01:08:39,439 --> 01:08:43,580 proportional to 1/K. 897 01:08:43,580 --> 01:08:46,750 And when the integration-- let's call the result 898 01:08:46,750 --> 01:08:49,380 of the integral Id of b because it 899 01:08:49,380 --> 01:08:52,720 depends on both dimension of integration, 900 01:08:52,720 --> 01:08:55,870 as well as the factor V through here. 901 01:08:55,870 --> 01:08:59,455 So I have 1/K Id of b. 902 01:09:04,760 --> 01:09:10,050 So that's one type of that I have generated. 903 01:09:10,050 --> 01:09:14,950 I started from this 0-th order form. 904 01:09:14,950 --> 01:09:19,620 And I saw that once I make the expansion of this to the lowest 905 01:09:19,620 --> 01:09:22,640 order, I will get a correction to that. 906 01:09:22,640 --> 01:09:25,964 And actually, just to sort of think about it 907 01:09:25,964 --> 01:09:29,250 in terms of formulae, you see what 908 01:09:29,250 --> 01:09:34,140 happened was that the first term that I had over here 909 01:09:34,140 --> 01:09:39,490 was pi gradient of pi repeated twice. 910 01:09:42,200 --> 01:09:45,600 And what I did was I essentially did 911 01:09:45,600 --> 01:09:51,010 an average of these two pi's and got the correction to gradient 912 01:09:51,010 --> 01:09:57,785 of pi squared, which is what was computed here and [INAUDIBLE]. 913 01:09:57,785 --> 01:10:06,190 The next term that I have is this term by itself, 914 01:10:06,190 --> 01:10:12,860 not calculated with the pi lessers only-- so this object. 915 01:10:12,860 --> 01:10:24,458 So I will write it as K/2 I gradient of pi squared. 916 01:10:24,458 --> 01:10:26,266 There's no correction. 917 01:10:26,266 --> 01:10:32,040 The final term that I have is this term. 918 01:10:32,040 --> 01:10:41,600 So I have minus rho over 2 pi lesser squared. 919 01:10:41,600 --> 01:10:49,380 And then the correction that I have was exactly the same form. 920 01:10:49,380 --> 01:10:55,232 It was 1 minus 1 minus b to the minus b. 921 01:10:55,232 --> 01:10:59,130 The rho over 2 is going to be hung onto both of them. 922 01:10:59,130 --> 01:11:02,200 You can see that there's a rho and there's a 2. 923 01:11:02,200 --> 01:11:05,910 And so basically I have two add this to that. 924 01:11:05,910 --> 01:11:11,342 And to first order, this is the entire thing of the thing 925 01:11:11,342 --> 01:11:12,324 that I will get. 926 01:11:21,180 --> 01:11:26,520 So this however is just a course gradient. 927 01:11:26,520 --> 01:11:29,930 It's the first step of the RG. 928 01:11:29,930 --> 01:11:34,680 And it has to be followed by the next two steps of the RG. 929 01:11:34,680 --> 01:11:38,000 Where here, you look at you field 930 01:11:38,000 --> 01:11:41,340 and you can see that the field is much coarser 931 01:11:41,340 --> 01:11:45,310 because the short distance cut off rather than being a, 932 01:11:45,310 --> 01:11:47,541 has been switched to ba. 933 01:11:47,541 --> 01:11:53,710 So you define your x prime to x/b, so that you shrink. 934 01:11:53,710 --> 01:12:01,160 You get the same course pixel size that you had before. 935 01:12:01,160 --> 01:12:07,380 And you also have to do a change in pi. 936 01:12:07,380 --> 01:12:13,150 So you replace pi lesser with some factor zeta pi 937 01:12:13,150 --> 01:12:19,210 prime, so that the contrast will look fine. 938 01:12:19,210 --> 01:12:23,309 So once I do that, you can see that this effect 939 01:12:23,309 --> 01:12:27,240 of this transformation is that this coupling 940 01:12:27,240 --> 01:12:32,280 K will change to K prime. 941 01:12:32,280 --> 01:12:36,990 Because of the change of x with b x prime, 942 01:12:36,990 --> 01:12:41,330 from the integration, I will get b to the d. 943 01:12:41,330 --> 01:12:45,200 From the two derivatives, I will get V to the minus 2. 944 01:12:49,280 --> 01:12:53,130 From the fact that I have replace two pi lessers with pi 945 01:12:53,130 --> 01:12:58,420 primes, I will get a factor of zeta squared. 946 01:12:58,420 --> 01:13:07,410 And then I will have this factor K 1 plus 1/K Id of b. 947 01:13:14,370 --> 01:13:25,950 So this is the recursion formula that we will be dealing with. 948 01:13:25,950 --> 01:13:29,400 Now there is some subtleties that 949 01:13:29,400 --> 01:13:33,570 go with this formula that are worth thinking about. 950 01:13:33,570 --> 01:13:42,380 Our original system had really one coupling parameter K 951 01:13:42,380 --> 01:13:46,310 that because of the constraints of the full symmetry 952 01:13:46,310 --> 01:13:52,640 of this field, S, part of it became the quadratic part that 953 01:13:52,640 --> 01:13:54,340 was the free field theory. 954 01:13:54,340 --> 01:13:57,970 But part of it made the interactions. 955 01:13:57,970 --> 01:14:00,020 But because of this vertical symmetry, 956 01:14:00,020 --> 01:14:02,740 to form of that interaction was fixed 957 01:14:02,740 --> 01:14:08,030 and had to be proportional to K. Now 958 01:14:08,030 --> 01:14:13,950 if we do our normalization group correctly, to full symmetry 959 01:14:13,950 --> 01:14:19,720 that we had has to be maintained at all levels. 960 01:14:19,720 --> 01:14:24,260 Which means that the functional form that I should end up with 961 01:14:24,260 --> 01:14:28,630 should have the same property in that the higher order 962 01:14:28,630 --> 01:14:32,640 coefficient should be related to the lower order 963 01:14:32,640 --> 01:14:39,150 coefficient, exactly the same way as we had over there. 964 01:14:39,150 --> 01:14:45,380 And at least at this stage, it looks like that did not happen. 965 01:14:45,380 --> 01:14:48,730 That is, we got the correction to this term, 966 01:14:48,730 --> 01:14:53,110 but we didn't get the correction to this term. 967 01:14:53,110 --> 01:14:54,970 We shouldn't be worried about that right 968 01:14:54,970 --> 01:15:01,950 here because we calculated things consistently corrections 969 01:15:01,950 --> 01:15:06,040 to order of T. And this was already 970 01:15:06,040 --> 01:15:12,400 a term that was order of T. So the real check 971 01:15:12,400 --> 01:15:17,100 is if you go and calculate the next order correction, 972 01:15:17,100 --> 01:15:21,070 you better get a correction to this term at next order 973 01:15:21,070 --> 01:15:23,820 that matches exactly this. 974 01:15:23,820 --> 01:15:25,770 People have done that, and have checked that. 975 01:15:25,770 --> 01:15:27,690 And that indeed is the case. 976 01:15:27,690 --> 01:15:31,316 So one is consistent with this. 977 01:15:31,316 --> 01:15:34,240 There are other kinds of consistency checks 978 01:15:34,240 --> 01:15:36,700 that have happened all over the place, 979 01:15:36,700 --> 01:15:40,756 like the fact that this came back 1 minus 1 980 01:15:40,756 --> 01:15:45,940 came out to be b to the minus d, so that the density is 981 01:15:45,940 --> 01:15:49,030 the same as before, consistent with the fact 982 01:15:49,030 --> 01:15:52,490 that you shrunk the lattice after RG so 983 01:15:52,490 --> 01:15:56,110 that the pixel size was the same as before is 984 01:15:56,110 --> 01:15:58,450 a consequence of that. 985 01:15:58,450 --> 01:16:01,780 You may worry that that's not entirely the case 986 01:16:01,780 --> 01:16:04,610 because when I do this, I will have also 987 01:16:04,610 --> 01:16:07,370 a factor of zeta squared. 988 01:16:07,370 --> 01:16:12,000 But it turns out that zeta is 1 plus order of temperature, 989 01:16:12,000 --> 01:16:13,640 as we will shortly see. 990 01:16:13,640 --> 01:16:16,360 So I gain consistent-- everything's 991 01:16:16,360 --> 01:16:20,750 consistent at this level that we've calculated things. 992 01:16:20,750 --> 01:16:25,555 And the only change is this factor. 993 01:16:25,555 --> 01:16:28,376 Now the one thing that we haven't calculated 994 01:16:28,376 --> 01:16:32,310 is what this zeta is. 995 01:16:32,310 --> 01:16:36,878 So to calculate zeta, I note the following 996 01:16:36,878 --> 01:16:42,465 that I start with a unit vector that 997 01:16:42,465 --> 01:16:44,770 is pointing at 0 temperature along this direction. 998 01:16:47,530 --> 01:16:50,660 Now because of fluctuations, this 999 01:16:50,660 --> 01:16:55,230 is going to be kind of rotating around this. 1000 01:16:55,230 --> 01:16:59,750 So there is this vector that is rotating. 1001 01:16:59,750 --> 01:17:04,160 If you average it over some time, what you will see 1002 01:17:04,160 --> 01:17:08,630 is that the average in all of these direction is 0. 1003 01:17:08,630 --> 01:17:12,020 The variance is not 0, but the average is 0. 1004 01:17:12,020 --> 01:17:13,750 But because of those fluctuations, 1005 01:17:13,750 --> 01:17:17,730 the effective length that you see in this direction 1006 01:17:17,730 --> 01:17:18,370 has shrunk. 1007 01:17:21,100 --> 01:17:26,940 How much has it shrunk by is related to this rescaling 1008 01:17:26,940 --> 01:17:29,520 factor that I should chose. 1009 01:17:29,520 --> 01:17:32,250 And so it's essentially average of something 1010 01:17:32,250 --> 01:17:34,820 like 1 minus pi squared. 1011 01:17:38,590 --> 01:17:41,730 But really it is the pi lesser squares 1012 01:17:41,730 --> 01:17:44,740 that I'm averaging over. 1013 01:17:44,740 --> 01:17:49,820 Which the lowest order is 1 minus 1/2 the average of pi 1014 01:17:49,820 --> 01:17:58,660 lesser squared, which is 1 minus 1/2 . 1015 01:17:58,660 --> 01:18:02,980 Now this is an N minus 1 component vector. 1016 01:18:02,980 --> 01:18:08,210 So each one of the components will give you one contribution. 1017 01:18:08,210 --> 01:18:13,426 The contribution that you get for one of these 1018 01:18:13,426 --> 01:18:17,275 is simply the average of pi squared, which 1019 01:18:17,275 --> 01:18:26,330 is 1 over K k squared, which I have to integrate over K's 1020 01:18:26,330 --> 01:18:29,138 that lie between lambda over b and lambda. 1021 01:18:32,810 --> 01:18:39,570 And you can see that this is 1 minus 1/2 N and minus 1. 1022 01:18:39,570 --> 01:18:43,080 It is inversely proportional to K. 1023 01:18:43,080 --> 01:18:45,990 And then the integration that I have to do 1024 01:18:45,990 --> 01:18:49,330 is precisely the same integration as here. 1025 01:18:49,330 --> 01:18:52,725 So it is, again, this Id of b. 1026 01:19:01,460 --> 01:19:06,750 So let me write the answer, say a couple of words about it, 1027 01:19:06,750 --> 01:19:09,720 and then we will deal with it next time. 1028 01:19:09,720 --> 01:19:13,875 So K prime is going to be b to the d 1029 01:19:13,875 --> 01:19:19,002 minus 2-- a new interaction parameters. 1030 01:19:19,002 --> 01:19:26,486 It is one factor of 1 plus 1/K Id of b. 1031 01:19:29,320 --> 01:19:33,600 And then there's one factor of this square of zeta. 1032 01:19:33,600 --> 01:19:44,110 So that gives you N minus 1 over K Id of b times K. 1033 01:19:44,110 --> 01:19:49,320 So we'll analyze this more next time around. 1034 01:19:49,320 --> 01:19:55,830 But I thought I would give you the physical reason for how 1035 01:19:55,830 --> 01:19:59,110 this interaction parameter changes. 1036 01:19:59,110 --> 01:20:00,870 Let's say we are in two dimensions. 1037 01:20:00,870 --> 01:20:04,454 So let's forget about this factor. 1038 01:20:04,454 --> 01:20:08,410 In two dimensions, we can see that there is one factor that 1039 01:20:08,410 --> 01:20:13,390 says at finite temperature, we are going to get weaker. 1040 01:20:13,390 --> 01:20:16,400 The interaction is going to get weaker. 1041 01:20:16,400 --> 01:20:19,100 And the reason for that is precisely what 1042 01:20:19,100 --> 01:20:21,340 I was explaining over here. 1043 01:20:21,340 --> 01:20:25,110 That is, you have some kind of a unit vector, 1044 01:20:25,110 --> 01:20:28,050 but because of its fluctuations, you 1045 01:20:28,050 --> 01:20:30,410 will see that it will loop shorter. 1046 01:20:30,410 --> 01:20:33,650 And it is less likely to be ordered. 1047 01:20:33,650 --> 01:20:36,720 The more components it has to fluctuate, 1048 01:20:36,720 --> 01:20:39,185 the shorter it will look like. 1049 01:20:39,185 --> 01:20:41,260 So there is that term. 1050 01:20:41,260 --> 01:20:44,950 So if this was the only effect, then K 1051 01:20:44,950 --> 01:20:46,950 will become weaker and weaker. 1052 01:20:46,950 --> 01:20:49,950 And it will have disorder. 1053 01:20:49,950 --> 01:20:52,720 But this effect says that it actually 1054 01:20:52,720 --> 01:20:56,470 gets stronger because of the interactions 1055 01:20:56,470 --> 01:20:58,750 that you have among the modes. 1056 01:20:58,750 --> 01:21:02,830 And to show you this to you, we can experiment with it 1057 01:21:02,830 --> 01:21:04,190 yourself. 1058 01:21:04,190 --> 01:21:06,780 So this is a sheet of paper. 1059 01:21:06,780 --> 01:21:09,560 This bend is an example of a Goldstone mode 1060 01:21:09,560 --> 01:21:11,990 because I could have rotated this sheet 1061 01:21:11,990 --> 01:21:14,200 without any cost of energy. 1062 01:21:14,200 --> 01:21:19,520 So this bend is a Goldstone mode that costs very little energy. 1063 01:21:19,520 --> 01:21:22,840 Now this paper has the kinds of constraints 1064 01:21:22,840 --> 01:21:24,660 that we have over here. 1065 01:21:24,660 --> 01:21:27,460 And because of those constraints, 1066 01:21:27,460 --> 01:21:30,530 if I make a mode in this direction, 1067 01:21:30,530 --> 01:21:33,970 I'm not going to be able to bend it in the other directions. 1068 01:21:33,970 --> 01:21:37,630 So clearly the mode that you have this direction, 1069 01:21:37,630 --> 01:21:39,840 and this direction, are coupled. 1070 01:21:39,840 --> 01:21:43,960 That's kind of an example of something like this. 1071 01:21:43,960 --> 01:21:50,530 Now while it is easy to do this bend because of this coupling, 1072 01:21:50,530 --> 01:21:54,250 if thermal fluctuations have created modes 1073 01:21:54,250 --> 01:21:57,330 that are shorter wavelength, and I have already 1074 01:21:57,330 --> 01:22:00,310 created those modes over here, then you 1075 01:22:00,310 --> 01:22:01,720 can experiment yourself. 1076 01:22:01,720 --> 01:22:06,700 You'll see that this is harder to bend compared to this. 1077 01:22:06,700 --> 01:22:10,480 You can see this already. 1078 01:22:10,480 --> 01:22:14,520 So that's the effect that you have over here. 1079 01:22:14,520 --> 01:22:17,510 So it's the competition between these two. 1080 01:22:17,510 --> 01:22:19,660 And depending on which one wins, and you 1081 01:22:19,660 --> 01:22:22,810 can see that that depends on whether n 1082 01:22:22,810 --> 01:22:26,274 is larger than 2 or less than 2. 1083 01:22:26,274 --> 01:22:29,340 So essentially for n that is larger than 2, 1084 01:22:29,340 --> 01:22:31,090 you'll find that this term wins. 1085 01:22:31,090 --> 01:22:33,740 And you get disorder in two dimension. 1086 01:22:33,740 --> 01:22:36,130 And is less than 2, you will get order 1087 01:22:36,130 --> 01:22:40,490 like we know the Ising model can be order. 1088 01:22:40,490 --> 01:22:41,950 But there's other things that can 1089 01:22:41,950 --> 01:22:45,736 be captured by this expression, that we will look at next time.