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 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:22,210 --> 00:00:23,940 PROFESSOR: OK, let's start. 9 00:00:26,580 --> 00:00:29,970 So back to our Ising model. 10 00:00:35,870 --> 00:00:40,770 And for this lecture, focusing mostly on a square like this. 11 00:00:48,200 --> 00:00:50,765 At each side, we have a binary variable. 12 00:00:54,570 --> 00:01:00,110 And a weight that tries to keep neighboring 13 00:01:00,110 --> 00:01:05,790 the sides to be parallel. 14 00:01:05,790 --> 00:01:11,480 So every pair of neighboring sides 15 00:01:11,480 --> 00:01:14,980 is subject to a joined fate such as that. 16 00:01:14,980 --> 00:01:18,350 You're going to ignore the magnetic field. 17 00:01:18,350 --> 00:01:20,780 And to calculate a partition function, 18 00:01:20,780 --> 00:01:26,140 we have to sum over r 2 to the n configuration on both sides. 19 00:01:26,140 --> 00:01:30,900 And this will be a partition function that will [INAUDIBLE] 20 00:01:30,900 --> 00:01:37,070 on this one parameter k, which is some energy divided by kt. 21 00:01:37,070 --> 00:01:42,080 So then what we did was you we rewrote 22 00:01:42,080 --> 00:01:46,320 each one of these factors as a hyperbolic cosine 1 plus 23 00:01:46,320 --> 00:01:50,270 tangent and took out all of the factors 24 00:01:50,270 --> 00:01:54,260 of the hyperbolic cosine in the outside. 25 00:01:54,260 --> 00:01:56,670 And for the case of this square lattice, 26 00:01:56,670 --> 00:02:00,890 each side has two bonds going out of that. 27 00:02:00,890 --> 00:02:04,430 So there's 2 to the n, and I'll have to sum r 2 28 00:02:04,430 --> 00:02:10,030 to the n configurations product of what all bonds. 29 00:02:10,030 --> 00:02:15,870 These factors of 1 plus t sigma i sigma j, 30 00:02:15,870 --> 00:02:18,670 where this T is, of course, my shorthand 31 00:02:18,670 --> 00:02:20,100 for hyperbolic function. 32 00:02:25,680 --> 00:02:30,890 Now we saw that basically what we need to do 33 00:02:30,890 --> 00:02:35,620 is either we take 1-- that's the t right there-- 34 00:02:35,620 --> 00:02:40,080 or terms that have these factors of t sigma sigma. 35 00:02:40,080 --> 00:02:44,130 But in order to ensure that they will survive the summation 36 00:02:44,130 --> 00:02:48,780 over the two possibilities of sigma, each one of these terms 37 00:02:48,780 --> 00:02:51,454 has to be matched with another one. 38 00:02:51,454 --> 00:02:57,260 And so the most trivial diagram would be something like this. 39 00:02:57,260 --> 00:03:01,660 And then sum over the two possibilities at each side 40 00:03:01,660 --> 00:03:03,750 will give us a factor of n per side. 41 00:03:03,750 --> 00:03:09,660 So you have 2 to the n [INAUDIBLE] to the power of 2n. 42 00:03:09,660 --> 00:03:22,540 And then we have a sum over graphs which have zero, two, 43 00:03:22,540 --> 00:03:26,950 or four bonds emanating per site. 44 00:03:34,040 --> 00:03:36,850 So that this summation over the sigmas 45 00:03:36,850 --> 00:03:42,090 will give us a factor of 2 and another factor of 0. 46 00:03:42,090 --> 00:03:45,390 And then we just have to count t to the power 47 00:03:45,390 --> 00:03:47,944 of the number of 1's in the graph. 48 00:03:52,930 --> 00:03:56,400 Now we said that all of the exciting things 49 00:03:56,400 --> 00:04:03,520 have to do with this sum, which depends on this parameter t, 50 00:04:03,520 --> 00:04:04,040 of course. 51 00:04:07,390 --> 00:04:16,089 And this sum, as written here, has 52 00:04:16,089 --> 00:04:19,800 graphs such as the one that I indicated, but also 53 00:04:19,800 --> 00:04:23,460 potentially graphs that are more complicated which 54 00:04:23,460 --> 00:04:27,170 each have these joined pieces. 55 00:04:27,170 --> 00:04:33,360 And we've attempted to replace this sum s with another sum, s 56 00:04:33,360 --> 00:04:45,340 prime, which is the sum over gas of phantom loops. 57 00:04:45,340 --> 00:04:48,740 Let's call it multiple phantom loops. 58 00:04:55,330 --> 00:04:59,130 Essentially, if I allow these loops to basically go 59 00:04:59,130 --> 00:05:07,820 through each other, then I can exponentiate this and write it 60 00:05:07,820 --> 00:05:18,611 as a sum over all single phantom loops thereby removing, 61 00:05:18,611 --> 00:05:24,270 the factors of n squared, n cubed, et cetera, 62 00:05:24,270 --> 00:05:27,970 that will arise by moving the destroyed pieces 63 00:05:27,970 --> 00:05:30,670 all over the lattice. 64 00:05:30,670 --> 00:05:34,930 This sum, since I can essentially 65 00:05:34,930 --> 00:05:37,920 pick it apart particular loop and then slide it once 66 00:05:37,920 --> 00:05:41,122 over the lattice, is certainly extensive proportional to n. 67 00:05:44,290 --> 00:05:45,930 So this would be nice. 68 00:05:45,930 --> 00:05:50,030 We calculate this sum as prime last time, 69 00:05:50,030 --> 00:05:53,630 and we saw that it actually reproduced for us the Gaussian 70 00:05:53,630 --> 00:05:54,255 model. 71 00:05:54,255 --> 00:05:57,368 So this was equivalent to the Gaussian model. 72 00:06:01,120 --> 00:06:06,510 And in particular, because we were allowing 73 00:06:06,510 --> 00:06:09,880 this phantom condition, when we went to sufficiently 74 00:06:09,880 --> 00:06:13,700 low temperature or when t became sufficiently large, 75 00:06:13,700 --> 00:06:15,740 essentially the model was unstable 76 00:06:15,740 --> 00:06:17,550 and because I could just continue 77 00:06:17,550 --> 00:06:20,650 to put more and more of these loops. 78 00:06:20,650 --> 00:06:22,855 There was no condition that would 79 00:06:22,855 --> 00:06:24,490 say you should have a finite density. 80 00:06:24,490 --> 00:06:27,630 And then the density of the loops would go to infinity. 81 00:06:27,630 --> 00:06:29,885 And just like the Gaussian model, 82 00:06:29,885 --> 00:06:34,330 it doesn't make sense beyond some point. 83 00:06:34,330 --> 00:06:40,650 Now mathematically, it is clear to us 84 00:06:40,650 --> 00:06:46,600 that s is not equal to s prime because 85 00:06:46,600 --> 00:06:53,220 of two important reasons, one of which 86 00:06:53,220 --> 00:07:08,090 is obvious is multiple occupation of a bond that 87 00:07:08,090 --> 00:07:13,290 is in the original sum that I have up here. 88 00:07:13,290 --> 00:07:15,500 You can see that the contribution 89 00:07:15,500 --> 00:07:19,430 of each point connecting neighbors 90 00:07:19,430 --> 00:07:22,290 is either 1 or one factor of t. 91 00:07:22,290 --> 00:07:25,530 I cannot have more than that. 92 00:07:25,530 --> 00:07:30,310 But when I am calculating phantom loops 93 00:07:30,310 --> 00:07:32,130 and I have self-crossings, I have 94 00:07:32,130 --> 00:07:33,670 some things that are very trial. 95 00:07:33,670 --> 00:07:37,090 Like I can start and go back on myself. 96 00:07:37,090 --> 00:07:39,150 That completes a walk. 97 00:07:39,150 --> 00:07:42,140 I can have things that are more complicated. 98 00:07:42,140 --> 00:07:45,900 I could have a diagram such as this 99 00:07:45,900 --> 00:07:51,980 that still involves crossing something twice. 100 00:07:51,980 --> 00:07:58,540 Or something like this is another example. 101 00:07:58,540 --> 00:08:04,320 Now all of these are examples where I essentially 102 00:08:04,320 --> 00:08:09,780 continuously moved my chalk and drew a closed loop. 103 00:08:09,780 --> 00:08:12,250 But also from the exponentiation, 104 00:08:12,250 --> 00:08:14,430 I can also generate things that have 105 00:08:14,430 --> 00:08:18,070 multiple loops, such as this loop that does not intersect 106 00:08:18,070 --> 00:08:20,870 itself but may happen to intersect 107 00:08:20,870 --> 00:08:24,970 on a bond with another loop. 108 00:08:24,970 --> 00:08:30,400 So it is partly the presence of all of these things 109 00:08:30,400 --> 00:08:33,780 that allow multiple occupation that ultimately 110 00:08:33,780 --> 00:08:35,740 leads to the instability. 111 00:08:40,710 --> 00:08:45,206 But I also hinted last time, in response to a question, 112 00:08:45,206 --> 00:08:51,525 that there is another mistake that is involved not when there 113 00:08:51,525 --> 00:08:56,505 is intersection at the bond, but intersection at the site. 114 00:09:05,340 --> 00:09:07,470 This leads to over-counting. 115 00:09:16,000 --> 00:09:18,354 It's more subtle. 116 00:09:18,354 --> 00:09:21,722 Along the diagrams that I have in s, 117 00:09:21,722 --> 00:09:25,260 we certainly complete the good diagram 118 00:09:25,260 --> 00:09:33,350 such as this one where I have from this side 119 00:09:33,350 --> 00:09:37,180 four bonds going out farther than the usual bonds. 120 00:09:37,180 --> 00:09:40,030 So this is certainly OK. 121 00:09:40,030 --> 00:09:43,620 But when I want to represent that 122 00:09:43,620 --> 00:09:46,610 in terms of blocks on the lattice, 123 00:09:46,610 --> 00:09:48,460 I notice that I can do the following. 124 00:09:48,460 --> 00:09:51,750 Let's say in all cases, I start from here, 125 00:09:51,750 --> 00:09:59,240 I can start from there, I can go and do something like this, 126 00:09:59,240 --> 00:10:02,260 and come back to my starting point. 127 00:10:02,260 --> 00:10:13,070 Or I can do something like this, and back to the starting point. 128 00:10:13,070 --> 00:10:18,300 Or I could have a diagram that would appear 129 00:10:18,300 --> 00:10:23,090 at the second order in the expansion of the exponential 130 00:10:23,090 --> 00:10:29,150 that I have that involves two loops that would correspond 131 00:10:29,150 --> 00:10:32,210 to the same geometry. 132 00:10:32,210 --> 00:10:37,190 So this thing that should have been counted once 133 00:10:37,190 --> 00:10:41,060 is really being counted three times. 134 00:10:41,060 --> 00:10:45,950 And so that's a mistake to be corrected. 135 00:10:45,950 --> 00:10:49,990 And actually, the reason for this factor of three 136 00:10:49,990 --> 00:10:52,675 goes back to this Gaussian model that I 137 00:10:52,675 --> 00:10:56,750 said that essentially what amounts to is 138 00:10:56,750 --> 00:11:04,160 that we are replacing Ising variables over here 139 00:11:04,160 --> 00:11:08,490 with these Gaussian variables, s. 140 00:11:08,490 --> 00:11:12,810 And so we arrange things such that the average of s 141 00:11:12,810 --> 00:11:17,440 squared was 1, so that the sides were reproducing 142 00:11:17,440 --> 00:11:19,850 the things that I had before. 143 00:11:19,850 --> 00:11:21,490 But then for something like this, 144 00:11:21,490 --> 00:11:25,630 I need to know the average s to the fourth. 145 00:11:25,630 --> 00:11:27,690 And if you use Wick's theorem, you 146 00:11:27,690 --> 00:11:30,205 can quickly see that the average of s to the fourth 147 00:11:30,205 --> 00:11:32,640 is three times average of s squared, 148 00:11:32,640 --> 00:11:34,700 so you get a factor of 3 here. 149 00:11:34,700 --> 00:11:37,740 That's the origin of the factor of 3. 150 00:11:37,740 --> 00:11:44,880 And indeed, if you had gone and done rather than one component, 151 00:11:44,880 --> 00:11:48,370 s, something that was n component, 152 00:11:48,370 --> 00:11:54,830 you could convince yourself that this becomes n plus 2. 153 00:11:54,830 --> 00:11:58,580 And that's consistent whenever you 154 00:11:58,580 --> 00:12:03,250 see something that has a loop having a factor of n, 155 00:12:03,250 --> 00:12:05,170 something that we had seen in we were 156 00:12:05,170 --> 00:12:08,760 doing this diagramatic expansion. 157 00:12:08,760 --> 00:12:12,030 OK, so that's a problem. 158 00:12:12,030 --> 00:12:15,020 So s is not equal to s prime. 159 00:12:17,830 --> 00:12:26,780 But I want to make the following very nice assertion 160 00:12:26,780 --> 00:12:47,420 that s is indeed the sum over multiple phantom loops, 161 00:12:47,420 --> 00:12:51,540 just like I had written for s prime with a couple 162 00:12:51,540 --> 00:12:54,375 of important constraints. 163 00:12:54,375 --> 00:12:56,900 The constraints are-- maybe I should write them 164 00:12:56,900 --> 00:12:59,850 in red-- with no U-turns. 165 00:13:04,970 --> 00:13:08,790 That is, you are not going to allow anything such as this 166 00:13:08,790 --> 00:13:14,200 or this when you would go forward 167 00:13:14,200 --> 00:13:16,545 and then immediately step backward. 168 00:13:16,545 --> 00:13:17,920 That's what I will call a U-turn. 169 00:13:17,920 --> 00:13:20,888 No U-turn is allowed. 170 00:13:20,888 --> 00:13:27,710 And more importantly, with a factor of minus 1 171 00:13:27,710 --> 00:13:36,000 to the number of crossings. 172 00:13:45,310 --> 00:13:48,630 So what do I mean by crossing? 173 00:13:48,630 --> 00:13:53,630 If you follow what I did in drawing this diagram, 174 00:13:53,630 --> 00:13:55,470 you can see that there was a path 175 00:13:55,470 --> 00:14:00,870 that I drew that never crossed itself, whereas here I 176 00:14:00,870 --> 00:14:03,700 kind of had to jump over where I was. 177 00:14:03,700 --> 00:14:06,280 I indicated that. 178 00:14:06,280 --> 00:14:10,550 So according to this rule, this diagram 179 00:14:10,550 --> 00:14:13,190 will get a factor of minus. 180 00:14:13,190 --> 00:14:16,640 These two diagrams don't have any self-crossings. 181 00:14:16,640 --> 00:14:18,970 They give factors of plus. 182 00:14:18,970 --> 00:14:22,800 And you can see that at least this particular diagram 183 00:14:22,800 --> 00:14:24,800 is resolved. 184 00:14:24,800 --> 00:14:29,130 And you can see that this continues 185 00:14:29,130 --> 00:14:31,725 to more complicated things. 186 00:14:31,725 --> 00:14:36,040 Let's say that I have a perfectly good diagram that 187 00:14:36,040 --> 00:14:39,975 is something like this that involves two crossings. 188 00:14:42,500 --> 00:14:48,390 And then I can break it roughly into two process, 189 00:14:48,390 --> 00:14:53,340 and each piece I can decompose as I had done before. 190 00:14:55,920 --> 00:15:09,460 Let's see, the left part I can either do this without crossing 191 00:15:09,460 --> 00:15:21,270 or I could cross or I could have this part as a separate loop 192 00:15:21,270 --> 00:15:22,160 from this half. 193 00:15:25,500 --> 00:15:29,655 And then I can join it on the other side 194 00:15:29,655 --> 00:15:34,380 with essentially any one of these things like this, which 195 00:15:34,380 --> 00:15:45,230 comes with a plus; this, which comes with a minus; 196 00:15:45,230 --> 00:15:53,580 or this one that comes with a plus. 197 00:15:53,580 --> 00:15:57,580 So you can see that this particular diagram 198 00:15:57,580 --> 00:16:05,490 of s in s prime would arise in nine possible ways. 199 00:16:05,490 --> 00:16:07,600 So I would have had an over-counting 200 00:16:07,600 --> 00:16:13,020 by a factor of a three-paired crossing or nine total, 201 00:16:13,020 --> 00:16:19,650 except that now when I assign these factors, 202 00:16:19,650 --> 00:16:23,115 some of these diagrams will come with minus, some of them 203 00:16:23,115 --> 00:16:24,450 will come with plus. 204 00:16:24,450 --> 00:16:28,210 But ultimately, only one will survive. 205 00:16:28,210 --> 00:16:29,240 This is it. 206 00:16:29,240 --> 00:16:32,690 All of them survive, then the net contribution is just 1, 207 00:16:32,690 --> 00:16:33,720 which is the correct on. 208 00:16:41,240 --> 00:16:41,840 Now let's see. 209 00:16:41,840 --> 00:16:55,270 So this removed problem B. so this was problem B resolved. 210 00:16:55,270 --> 00:17:01,420 Let's see about our problem A, which 211 00:17:01,420 --> 00:17:05,940 had to do with multiple occupation 212 00:17:05,940 --> 00:17:09,060 of the particular bond. 213 00:17:09,060 --> 00:17:13,670 So these are diagrams that will appear in s prime 214 00:17:13,670 --> 00:17:18,385 that have no counterpart in s. 215 00:17:18,385 --> 00:17:21,990 That is, let's say there is this bond that 216 00:17:21,990 --> 00:17:26,370 is occupied twice, so that would be 217 00:17:26,370 --> 00:17:29,890 a contribution by a factor of t squared. 218 00:17:29,890 --> 00:17:33,560 And then this part can go and join whatever it wants. 219 00:17:33,560 --> 00:17:36,390 This part can go and do whatever it wants. 220 00:17:39,370 --> 00:17:44,770 The point is that for every diagram such as this, 221 00:17:44,770 --> 00:17:49,670 I can construct a diagram where I leave everything out 222 00:17:49,670 --> 00:17:55,500 here exactly as it was, everything out 223 00:17:55,500 --> 00:18:00,310 here exactly as it was, except that the two terminals 224 00:18:00,310 --> 00:18:02,820 that I'll have to the left and the two terminals 225 00:18:02,820 --> 00:18:07,070 that I have to the right of the bond, rather than joining them 226 00:18:07,070 --> 00:18:10,850 this way, I'll join them like this. 227 00:18:14,580 --> 00:18:18,930 So this complicated diagram, as far as this bond is concerned, 228 00:18:18,930 --> 00:18:27,160 I can prescribe in two different ways using graphs of s prime. 229 00:18:27,160 --> 00:18:29,910 And one of them with respect to the other 230 00:18:29,910 --> 00:18:33,150 has an additional factor of minus 1, 231 00:18:33,150 --> 00:18:36,730 and so they give me this 0. 232 00:18:36,730 --> 00:18:41,070 And you can convince yourself that the same construction will 233 00:18:41,070 --> 00:18:45,430 hold if I have three terminals, four terminals-- it doesn't 234 00:18:45,430 --> 00:18:45,930 matter. 235 00:18:45,930 --> 00:18:50,470 I will do it for one pair and it will be OK. 236 00:18:50,470 --> 00:18:53,950 The only time that I wouldn't have been able to 237 00:18:53,950 --> 00:18:57,440 is if I have to terminals on one side and one terminal 238 00:18:57,440 --> 00:19:02,450 on the other side, which is these guys. 239 00:19:02,450 --> 00:19:05,850 And that's what I said no U-turns. 240 00:19:05,850 --> 00:19:08,820 So now that's taken care of first. 241 00:19:08,820 --> 00:19:11,620 So this problem A is now also resolved. 242 00:19:15,270 --> 00:19:18,580 So what do we have? 243 00:19:18,580 --> 00:19:25,900 We have now established that s-- let me just get a 0. 244 00:19:25,900 --> 00:19:45,550 Let's define this loop star to be the loops with no U-turns 245 00:19:45,550 --> 00:19:49,840 and minus 1 to the number of crossings. 246 00:19:49,840 --> 00:19:53,470 So this star is going to symbolize 247 00:19:53,470 --> 00:19:58,100 that these two constraints of no U-turns and minus 1 248 00:19:58,100 --> 00:20:00,630 to the power of number of crossings 249 00:20:00,630 --> 00:20:04,880 are imposed in construction and calculation of the contribution 250 00:20:04,880 --> 00:20:07,390 of these objects. 251 00:20:07,390 --> 00:20:09,650 So then what do I have? 252 00:20:09,650 --> 00:20:16,760 I have that s is-- well, there's possibility of no loop. 253 00:20:16,760 --> 00:20:19,410 There is the one loop graphs. 254 00:20:19,410 --> 00:20:21,190 There's the two loops star graphs. 255 00:20:21,190 --> 00:20:24,770 My little loop star graphs. 256 00:20:24,770 --> 00:20:28,680 And just as in the case of s prime, 257 00:20:28,680 --> 00:20:37,095 I can exponentiate this as the sum of all one stars. 258 00:20:40,010 --> 00:20:42,960 And you may wonder what happens when 259 00:20:42,960 --> 00:20:45,350 I go to higher order terms. 260 00:20:45,350 --> 00:20:47,545 In higher order terms, potentially I 261 00:20:47,545 --> 00:20:50,870 will generate two of these things that 262 00:20:50,870 --> 00:20:55,270 cross when I go to a second order term. 263 00:20:55,270 --> 00:20:59,190 But then any intersection will involve two, four, 264 00:20:59,190 --> 00:21:00,840 and even number. 265 00:21:00,840 --> 00:21:02,950 Minus 1 to an even number is one, 266 00:21:02,950 --> 00:21:07,230 so there is really no additional interaction 267 00:21:07,230 --> 00:21:13,240 to worry about if multiple things are crossing each other. 268 00:21:19,750 --> 00:21:25,190 So we could exponentiate that s of t that we have over there. 269 00:21:25,190 --> 00:21:27,430 We are interested in taking the log 270 00:21:27,430 --> 00:21:30,670 of the expression for the partition function. 271 00:21:30,670 --> 00:21:39,270 I will do the factor of n log 2 hyperbolic cosine squared, 272 00:21:39,270 --> 00:21:43,490 because there are essentially two bonds per side. 273 00:21:43,490 --> 00:21:52,393 And then I have a sum over these loop stars. 274 00:22:01,110 --> 00:22:03,260 Essentially, I took the log of the expression 275 00:22:03,260 --> 00:22:04,160 that I had before. 276 00:22:07,370 --> 00:22:11,440 Now this sum-- well, let's sort of take 277 00:22:11,440 --> 00:22:14,422 care of this factor of n. 278 00:22:14,422 --> 00:22:17,640 So I divide by n. 279 00:22:17,640 --> 00:22:23,966 Log z divided by n is log of 2 hyperbolic cosines [INAUDIBLE] 280 00:22:23,966 --> 00:22:31,720 k plus-- well, the way to get rid of the factor of n 281 00:22:31,720 --> 00:22:36,500 in this sum is to fix one point of this loop 282 00:22:36,500 --> 00:22:40,710 so that it doesn't go all over the place. 283 00:22:40,710 --> 00:22:50,330 So let's say that I have the number of loops 284 00:22:50,330 --> 00:22:54,630 that start and end at the origin. 285 00:22:54,630 --> 00:22:59,740 I would have to sum over the length of those loops, 286 00:22:59,740 --> 00:23:04,630 and those things will give me a factor of t to the l. 287 00:23:04,630 --> 00:23:22,340 So I have defined this n star of l to be number of loop stars 288 00:23:22,340 --> 00:23:28,126 from 0 to 0 in l steps. 289 00:23:34,610 --> 00:23:37,070 So in place of this-- actually, maybe 290 00:23:37,070 --> 00:23:43,170 I will emphasize it's minus 1 to the number of crossings 291 00:23:43,170 --> 00:23:44,505 and no U-turns. 292 00:23:50,010 --> 00:23:55,860 Now gain, so what I have is my entire lattice. 293 00:23:58,490 --> 00:24:04,960 Let's say I have in loop that is of length 4. 294 00:24:04,960 --> 00:24:12,030 And now I have forced it to start and end at the origin 295 00:24:12,030 --> 00:24:17,590 so that I can put out this factor of n. 296 00:24:17,590 --> 00:24:21,810 But then I realized that I could have over-counted this 297 00:24:21,810 --> 00:24:28,670 because this loop could have been started from here, here, 298 00:24:28,670 --> 00:24:32,810 here, here and translated to the origin. 299 00:24:32,810 --> 00:24:36,280 So just as we saw for the case of the random walk loops 300 00:24:36,280 --> 00:24:43,960 before, there is this factor of l to correct. 301 00:24:43,960 --> 00:24:45,830 And then I'm talking about walks, 302 00:24:45,830 --> 00:24:50,070 I can either go clockwise or counterclockwise, 303 00:24:50,070 --> 00:24:56,110 so I have to divide by a factor of 2 to get rid of that. 304 00:25:00,759 --> 00:25:01,550 AUDIENCE: Question? 305 00:25:01,550 --> 00:25:03,200 PROFESSOR: Yes? 306 00:25:03,200 --> 00:25:05,632 AUDIENCE: So what happens if you have when 307 00:25:05,632 --> 00:25:07,090 you're doing your exponential loops 308 00:25:07,090 --> 00:25:09,465 and we have one loop nested inside another loop 309 00:25:09,465 --> 00:25:11,340 that you're multiplying together so that they 310 00:25:11,340 --> 00:25:12,630 share one of their edges. 311 00:25:12,630 --> 00:25:15,500 It seems like then they don't have to cross twice, 312 00:25:15,500 --> 00:25:19,770 and so you would still need something to cancel them out. 313 00:25:19,770 --> 00:25:23,930 PROFESSOR: OK, I should have maybe explained 314 00:25:23,930 --> 00:25:26,060 that graph a little bit more. 315 00:25:26,060 --> 00:25:30,730 But let's do it over here. 316 00:25:30,730 --> 00:25:40,660 So when I exponentiate s-- what am I calling it?-- 317 00:25:40,660 --> 00:25:47,630 s sum of the loops, among the terms that I will generate will 318 00:25:47,630 --> 00:25:50,940 certainly be something like this. 319 00:25:50,940 --> 00:25:55,800 And then you say this one is shared between the two of them. 320 00:25:55,800 --> 00:25:58,970 At the level of one loop graphs, there 321 00:25:58,970 --> 00:26:02,090 was a one-loop object that went like this. 322 00:26:09,040 --> 00:26:12,530 So the statement that I made here 323 00:26:12,530 --> 00:26:17,140 does not necessarily map the number of loops to each other, 324 00:26:17,140 --> 00:26:19,665 but it is correct, and the cancellation 325 00:26:19,665 --> 00:26:21,497 occurs at the level of one. 326 00:26:29,980 --> 00:26:33,320 And actually, I should have also indicated 327 00:26:33,320 --> 00:26:34,880 what's happening with the other graph 328 00:26:34,880 --> 00:26:40,320 that I have up there because I have this graph. 329 00:26:40,320 --> 00:26:44,250 So that's a one-loop graph, that cancels against this graph. 330 00:26:50,870 --> 00:26:56,030 So whatever you do, you can just follow the rule that I gave you 331 00:26:56,030 --> 00:26:58,556 and ensure that the cancellation works, of course. 332 00:26:58,556 --> 00:26:59,410 So Thank. 333 00:26:59,410 --> 00:27:00,320 You. 334 00:27:00,320 --> 00:27:02,360 I wanted to say this and I had forgot. 335 00:27:08,350 --> 00:27:09,860 OK, any other questions? 336 00:27:16,570 --> 00:27:24,700 OK, so essentially we are back to some extent of the formula 337 00:27:24,700 --> 00:27:28,620 that I had for ordinary random walks. 338 00:27:28,620 --> 00:27:35,810 And phantom loops these are partly phantom loops, 339 00:27:35,810 --> 00:27:42,080 but I have to take care of something like this. 340 00:27:42,080 --> 00:27:46,580 What did I go originally last lecture, rather than 341 00:27:46,580 --> 00:27:50,800 my ordinary graphs to these phantom loops? 342 00:27:50,800 --> 00:27:53,640 The reason was that for a phantom loops, 343 00:27:53,640 --> 00:27:57,060 I said that I had this Markovian condition. 344 00:27:57,060 --> 00:28:01,250 I could relate l step walks to l minus 1 step walks 345 00:28:01,250 --> 00:28:04,530 because there was no memory. 346 00:28:04,530 --> 00:28:08,330 I didn't have to know where I had crossed before. 347 00:28:08,330 --> 00:28:11,535 But it seems that in order to give a correct date 348 00:28:11,535 --> 00:28:14,940 to these new loops, I have to know 349 00:28:14,940 --> 00:28:16,800 how many times I cross myself. 350 00:28:20,880 --> 00:28:25,050 And c by itself is a non-Markovian thing. 351 00:28:25,050 --> 00:28:28,350 It requires memory. 352 00:28:28,350 --> 00:28:31,270 Expect that I don't need any c. 353 00:28:31,270 --> 00:28:36,080 I need only the parity of the number of crossings. 354 00:28:36,080 --> 00:28:40,410 And here is where there's a beautiful mathematical theorem 355 00:28:40,410 --> 00:28:46,305 that tells us here in this memory-like problem 356 00:28:46,305 --> 00:28:49,310 for something that is Markovian. 357 00:28:49,310 --> 00:29:01,700 And the statement is a theorem of Whitney's which 358 00:29:01,700 --> 00:29:05,330 states that the parity of a planar 359 00:29:05,330 --> 00:29:07,430 loop-- the thing that I've done here-- 360 00:29:07,430 --> 00:29:09,160 whether it's even or odd. 361 00:29:20,010 --> 00:29:23,550 Parity of the number of crossings 362 00:29:23,550 --> 00:29:40,990 is related to the total angle through which the tangent 363 00:29:40,990 --> 00:29:49,710 vector turns by the following. 364 00:29:52,920 --> 00:30:00,280 In C mod 2, which is the parity of the crossings, 365 00:30:00,280 --> 00:30:06,300 is 1 plus this total angle that I 366 00:30:06,300 --> 00:30:09,926 will call theta divided by 2 pi mod 2. 367 00:30:18,870 --> 00:30:23,700 So I'm not going to give a proof of this, 368 00:30:23,700 --> 00:30:27,710 but I will show you examples to make sure you understand what's 369 00:30:27,710 --> 00:30:35,000 happening by, let's say, comparing the following two 370 00:30:35,000 --> 00:30:47,020 graphs, one of which has no crossing and another one which 371 00:30:47,020 --> 00:30:52,720 is essentially the same thing but has a crossing. 372 00:30:52,720 --> 00:31:06,540 So basically I put an arrow so that I 373 00:31:06,540 --> 00:31:11,750 can follow where the orientation of the bond, which 374 00:31:11,750 --> 00:31:16,180 is the location of this tangent s, as I step. 375 00:31:16,180 --> 00:31:20,950 Let's say from the origin, this is the first step, second step, 376 00:31:20,950 --> 00:31:25,450 third step, fourth step, so on and so forth. 377 00:31:25,450 --> 00:31:28,830 So let's do this for the upper graph. 378 00:31:28,830 --> 00:31:33,710 And what I will do is I will plot the angle. 379 00:31:36,290 --> 00:31:44,360 And the first step here, I start at 0 degrees pointing this way. 380 00:31:44,360 --> 00:31:47,960 So this is my step number one. 381 00:31:47,960 --> 00:31:51,200 At the next, I have gone to 90 degrees, 382 00:31:51,200 --> 00:31:53,340 so this is where I go to. 383 00:31:53,340 --> 00:31:57,490 At three, I'm back to pointing along 384 00:31:57,490 --> 00:31:59,770 the horizon in the direction. 385 00:31:59,770 --> 00:32:04,150 At four, I have gone back up here. 386 00:32:04,150 --> 00:32:10,462 At five, I go to 180 degrees. 387 00:32:10,462 --> 00:32:15,676 At six, I go all the way down. 388 00:32:15,676 --> 00:32:27,080 At seven, I go back to this horizontal. 389 00:32:27,080 --> 00:32:32,260 At eight, I go back to pointing down. 390 00:32:32,260 --> 00:32:33,720 And then I'm back to one. 391 00:32:37,150 --> 00:32:39,940 So if you follow what that tangent is doing, 392 00:32:39,940 --> 00:32:43,410 it's going woop, woop. 393 00:32:43,410 --> 00:32:47,260 At the end of the day, it has turned through 3 pi. 394 00:32:47,260 --> 00:32:51,395 So in this case, the total turn is 2 pi. 395 00:32:54,480 --> 00:32:59,060 1 plus the total turn divided by 2 pi in this case 396 00:32:59,060 --> 00:33:05,440 is 1 plus 2 pi over 2 pi, which is 2-- mod 397 00:33:05,440 --> 00:33:10,630 2 which is the same thing as 0. 398 00:33:10,630 --> 00:33:13,985 And of course, how many times has this thing crossed itself? 399 00:33:13,985 --> 00:33:16,316 Zero times. 400 00:33:16,316 --> 00:33:18,480 So let's see how it works if you go 401 00:33:18,480 --> 00:33:26,490 to apply the same set of rules to this other one. 402 00:33:26,490 --> 00:33:33,960 So again, one, two, three, four, five, six, seven, eight 403 00:33:33,960 --> 00:33:36,290 are my steps. 404 00:33:36,290 --> 00:33:42,610 One is pointing in the horizontal direction. 405 00:33:42,610 --> 00:33:46,250 Two goes vertical just as before. 406 00:33:46,250 --> 00:33:50,250 Three stays the same place, so two and three 407 00:33:50,250 --> 00:33:52,320 are at the same point. 408 00:33:52,320 --> 00:33:54,045 Four, I go back to horizontal. 409 00:33:56,700 --> 00:34:03,290 Five, I go down to minus 90 degrees. 410 00:34:03,290 --> 00:34:08,400 Six, I go all the way to 180 degrees 411 00:34:08,400 --> 00:34:10,920 and stay there at seven. 412 00:34:10,920 --> 00:34:14,719 Eight, I go back to minus 90 degrees, 413 00:34:14,719 --> 00:34:17,340 and then rejoin the origin. 414 00:34:17,340 --> 00:34:25,750 So this goes up, down, back, never completes a full turn. 415 00:34:25,750 --> 00:34:30,010 So in this case, theta is 0. 416 00:34:30,010 --> 00:34:36,989 1 plus theta over 2 pi is 1, and the number 417 00:34:36,989 --> 00:34:41,000 of crossings of his graph is 1. 418 00:34:41,000 --> 00:34:44,090 So you can go and repeat this for any graph 419 00:34:44,090 --> 00:34:48,380 that you like and convince yourself that this rule works. 420 00:34:54,100 --> 00:34:59,000 Well, how does that help us? 421 00:34:59,000 --> 00:35:01,280 Well, the diagrams that I have drawn here 422 00:35:01,280 --> 00:35:05,000 already tells you how it helps us 423 00:35:05,000 --> 00:35:10,800 because in order to find the total angle, 424 00:35:10,800 --> 00:35:16,280 all I need to do is to keep track of local changes. 425 00:35:16,280 --> 00:35:19,340 So essentially, as I go along, I carry a bag with me 426 00:35:19,340 --> 00:35:23,890 which adds the changing angle at every step. 427 00:35:23,890 --> 00:35:29,250 I don't need to know where I was 100 steps before. 428 00:35:29,250 --> 00:35:32,810 I just add another changing angle. 429 00:35:32,810 --> 00:35:36,430 By the time I get to the last stop, 430 00:35:36,430 --> 00:35:40,800 I figure out what my total angle is, and then I'm done. 431 00:35:44,870 --> 00:35:46,020 AUDIENCE: Yes? 432 00:35:46,020 --> 00:35:48,240 So is it really the total angle that 433 00:35:48,240 --> 00:35:50,520 matters, or is it more just the number of circles 434 00:35:50,520 --> 00:35:52,282 that you complete or not? 435 00:35:52,282 --> 00:35:53,740 PROFESSOR: They are the same thing. 436 00:35:53,740 --> 00:35:57,140 So if you prefer to say it in terms 437 00:35:57,140 --> 00:36:04,000 of the entanglement of your loop and the point at the origin, 438 00:36:04,000 --> 00:36:06,770 that's another way of saying it. 439 00:36:06,770 --> 00:36:12,050 This entity divided by 2 pi is a topological number, 440 00:36:12,050 --> 00:36:14,251 which counts essentially the number of times 441 00:36:14,251 --> 00:36:15,584 you have gone around the origin. 442 00:36:31,870 --> 00:36:38,530 So what I'm saying is that minus 1 to the power of the number 443 00:36:38,530 --> 00:36:44,090 of crossings-- this factor that I was after-- I can write 444 00:36:44,090 --> 00:36:50,030 as e to the i pi times the number of crossings. 445 00:36:50,030 --> 00:36:53,000 And it is evident that the only thing that is important 446 00:36:53,000 --> 00:36:58,300 here is the parity, so I can replace into the i pi 447 00:36:58,300 --> 00:37:02,830 the number of crossings with 1 plus this total angle divided 448 00:37:02,830 --> 00:37:03,590 by 2 pi. 449 00:37:06,190 --> 00:37:08,440 Which means I have e to the i pi, which 450 00:37:08,440 --> 00:37:11,110 is a factor of minus 1. 451 00:37:11,110 --> 00:37:17,850 And then I have e to the i theta over 2. 452 00:37:17,850 --> 00:37:23,020 And my statement is that this is the same thing as e 453 00:37:23,020 --> 00:37:30,530 to the i over 2, sum over the little bits of change of angle 454 00:37:30,530 --> 00:37:33,154 that people have as you go along there. 455 00:37:38,455 --> 00:37:47,510 So what I have to do is as I am walking 456 00:37:47,510 --> 00:37:51,240 around this square lattice, I better 457 00:37:51,240 --> 00:37:53,820 keep track of which direction I'm 458 00:37:53,820 --> 00:37:59,040 pointing so that I know from one step to the next step 459 00:37:59,040 --> 00:38:01,690 whether I change by 0 degrees, 90 degrees, 460 00:38:01,690 --> 00:38:04,260 minus 90 degrees, et cetera. 461 00:38:04,260 --> 00:38:05,940 So what do I do? 462 00:38:05,940 --> 00:38:07,325 I define a convention. 463 00:38:13,600 --> 00:38:24,125 So we are going to introduce orientation mu of step 464 00:38:24,125 --> 00:38:25,220 as follows. 465 00:38:28,050 --> 00:38:32,870 So let's say I'm at some point on the lattice i. 466 00:38:32,870 --> 00:38:36,605 Then I can increase the particle c along one of four directions. 467 00:38:39,150 --> 00:38:43,700 And I'm going to label them by mu 468 00:38:43,700 --> 00:38:47,936 being equal to 1, 2, 3, or 4. 469 00:38:47,936 --> 00:38:51,150 You can choose any notation you want. 470 00:38:51,150 --> 00:38:52,870 This will be the notation I will use. 471 00:39:00,490 --> 00:39:06,700 Secondly, I'm going to introduce the analog of the quantity 472 00:39:06,700 --> 00:39:11,140 that we had for the phantom random box, which was I 473 00:39:11,140 --> 00:39:15,550 introduced a set of matrices that were counting 474 00:39:15,550 --> 00:39:19,480 how many base I can go from one side 475 00:39:19,480 --> 00:39:23,550 to another side in l steps. 476 00:39:23,550 --> 00:39:30,320 So I will introduce the following notation. 477 00:39:30,320 --> 00:39:34,620 Something that involves the starred box that involves l 478 00:39:34,620 --> 00:39:36,720 steps. 479 00:39:36,720 --> 00:39:40,900 And I say that I start at some point xy and I 480 00:39:40,900 --> 00:39:44,085 end at some other point x prime y prime. 481 00:39:44,085 --> 00:39:46,380 So again, just counting how many ways 482 00:39:46,380 --> 00:39:49,550 I can go from one to another point. 483 00:39:49,550 --> 00:39:53,342 Except that I also want to keep track of these orientations. 484 00:39:57,680 --> 00:40:01,460 So this quantity is defined as follows. 485 00:40:01,460 --> 00:40:21,310 It is the sum over random walks that start at xy along m. 486 00:40:21,310 --> 00:40:24,710 That is, if this is my point, xy, 487 00:40:24,710 --> 00:40:27,490 and I'm looking at the second element of this, 488 00:40:27,490 --> 00:40:30,570 the next step I have to go up. 489 00:40:30,570 --> 00:40:34,180 If I have specified that mu equals to 1, 490 00:40:34,180 --> 00:40:38,340 it means that the first step I have to go to the right. 491 00:40:38,340 --> 00:40:41,410 OK, as I go further on the lattice, 492 00:40:41,410 --> 00:40:47,040 I ensure that I never have any U-turns, 493 00:40:47,040 --> 00:40:55,360 and I keep track of the factors of e 494 00:40:55,360 --> 00:41:02,910 to the i theta that changes as I take one step to the next step 495 00:41:02,910 --> 00:41:06,490 so that if I took my first step here and then next 496 00:41:06,490 --> 00:41:09,410 if I continued here, there would be no factor. 497 00:41:09,410 --> 00:41:16,550 But if I went up, I would have a factor of e to the i pi 4. 498 00:41:16,550 --> 00:41:19,710 So I keep track of those factors. 499 00:41:19,710 --> 00:41:26,480 And then I want to end at x prime y prime 500 00:41:26,480 --> 00:41:30,370 and head along mu prime. 501 00:41:33,390 --> 00:41:40,020 Since I specified that my first step will then 502 00:41:40,020 --> 00:41:44,430 go along mu, when I reach the last step, 503 00:41:44,430 --> 00:41:47,060 I already know where I came from. 504 00:41:47,060 --> 00:41:49,470 But depending on which direction I 505 00:41:49,470 --> 00:41:54,340 specify I would turn to and head for my next step, 506 00:41:54,340 --> 00:41:56,790 I will get the changing angle. 507 00:41:56,790 --> 00:42:02,510 So I have to include the changing angle somewhere so 508 00:42:02,510 --> 00:42:05,170 that there are l changes in angle 509 00:42:05,170 --> 00:42:06,970 so the way that I have defined it, 510 00:42:06,970 --> 00:42:08,690 it will be essentially keeping track 511 00:42:08,690 --> 00:42:10,690 of where the next step is headed to. 512 00:42:13,420 --> 00:42:17,150 So again, if I were to draw a diagram, 513 00:42:17,150 --> 00:42:21,900 I'll start with this xy point, and I 514 00:42:21,900 --> 00:42:24,180 want to arrive at some point here-- 515 00:42:24,180 --> 00:42:25,860 let's say x prime y prime. 516 00:42:28,680 --> 00:42:32,030 And I want to do in l steps. 517 00:42:32,030 --> 00:42:35,180 And the first step, I go along the direction 518 00:42:35,180 --> 00:42:37,800 that is specified by mu. 519 00:42:37,800 --> 00:42:47,750 So this is step 1, and there's the 2, 3, 4, 5, 6, 7, 8, 9, 10. 520 00:42:47,750 --> 00:42:51,980 Let's see, the last one-- so this would step l minus 1. 521 00:42:51,980 --> 00:42:57,200 This is the last step, arrives me at point x prime y prime. 522 00:42:57,200 --> 00:43:02,320 But then I have to specify what is the direction of mu prime 523 00:43:02,320 --> 00:43:06,120 so that I keep track of the appropriate change of angle 524 00:43:06,120 --> 00:43:09,370 that I have to do over here as well. 525 00:43:09,370 --> 00:43:11,220 So this is the procedure. 526 00:43:20,690 --> 00:43:25,930 Now this walk, this quantity that I have defined for you 527 00:43:25,930 --> 00:43:31,070 here, has the Markovian property in that 528 00:43:31,070 --> 00:43:36,870 if I arrive at this point after l steps, 529 00:43:36,870 --> 00:43:42,620 then after l minus one steps, I was at one point-- 530 00:43:42,620 --> 00:43:46,630 x double prime, y double prime-- from which 531 00:43:46,630 --> 00:43:52,360 I took one step along some direction-- mu double prime-- 532 00:43:52,360 --> 00:43:54,670 and arrived at this. 533 00:43:54,670 --> 00:44:03,120 So I can write that I have to sum over all possible locations 534 00:44:03,120 --> 00:44:08,530 and orientations of the before [INAUDIBLE]. 535 00:44:08,530 --> 00:44:12,620 I start from the point xy, proceed along 536 00:44:12,620 --> 00:44:20,850 direction mu for a total of l minus 1 steps, 537 00:44:20,850 --> 00:44:26,740 landing on x double prime y double prime, 538 00:44:26,740 --> 00:44:29,760 and then going in direction to mu double prime. 539 00:44:32,340 --> 00:44:35,810 So then I have a walk that started at x double prime, 540 00:44:35,810 --> 00:44:38,520 y double prime along the direction in double 541 00:44:38,520 --> 00:44:46,260 prime, which is one step has to get me to my destination. 542 00:44:46,260 --> 00:44:48,640 And once I am at the destination, 543 00:44:48,640 --> 00:44:50,496 I head along the direction in prime. 544 00:44:56,730 --> 00:45:03,170 So this is clearly a matrix, a product. 545 00:45:03,170 --> 00:45:06,970 And what I have established is this matrix, w star 546 00:45:06,970 --> 00:45:12,110 of l-- which, by the way, is a 4n by 4n matrix, 547 00:45:12,110 --> 00:45:14,550 because our n points and 4 orientation, 548 00:45:14,550 --> 00:45:19,180 so it's 4n by 4n matrix-- and I have established 549 00:45:19,180 --> 00:45:23,010 that that is the product of the matrix that I have 550 00:45:23,010 --> 00:45:26,790 for one step, and the product of the matrix that I 551 00:45:26,790 --> 00:45:30,940 have for l minus 1 step. 552 00:45:30,940 --> 00:45:40,030 And this object that I will call t star essentially tells me 553 00:45:40,030 --> 00:45:44,520 something about the combined connectivity orientation 554 00:45:44,520 --> 00:45:47,770 that I have for square lattice. 555 00:45:47,770 --> 00:45:50,670 And since I can repeat this many times, 556 00:45:50,670 --> 00:45:55,745 I can see that I have the result that I want, that w star of l 557 00:45:55,745 --> 00:45:59,108 is simply t star raised to the power of l. 558 00:46:36,970 --> 00:46:39,672 Questions? 559 00:46:39,672 --> 00:46:40,658 Yes? 560 00:46:40,658 --> 00:46:43,100 AUDIENCE: Have we accounted for loops that [INAUDIBLE] 561 00:46:43,100 --> 00:46:45,016 we have the square loop of four steps and then 562 00:46:45,016 --> 00:46:47,992 the same square loop of eight steps that are exactly 563 00:46:47,992 --> 00:46:49,480 on top of the four-step one? 564 00:46:52,470 --> 00:46:58,334 PROFESSOR: OK, so you want me to take this and do it again? 565 00:46:58,334 --> 00:46:59,650 AUDIENCE: Yeah. 566 00:46:59,650 --> 00:47:01,710 PROFESSOR: OK, so I can certainly 567 00:47:01,710 --> 00:47:03,754 do something like this. 568 00:47:07,466 --> 00:47:08,860 AUDIENCE: I see. 569 00:47:08,860 --> 00:47:11,020 PROFESSOR: And of course, I can do the same thing 570 00:47:11,020 --> 00:47:15,505 over any of the bonds, but they are always [INAUDIBLE]. 571 00:47:20,550 --> 00:47:21,504 Yes? 572 00:47:21,504 --> 00:47:23,682 AUDIENCE: Doesn't [INAUDIBLE] become kind 573 00:47:23,682 --> 00:47:28,090 of like transfer matrix [INAUDIBLE]? 574 00:47:28,090 --> 00:47:30,030 PROFESSOR: No. 575 00:47:30,030 --> 00:47:32,990 It will reproduce the result that Onsager had, 576 00:47:32,990 --> 00:47:37,110 but the transfer matrix that Onsager had 577 00:47:37,110 --> 00:47:42,720 was it essentially going from column to column. 578 00:47:42,720 --> 00:47:46,580 Its size was 2 to the n times 2 to the n. 579 00:47:46,580 --> 00:47:48,585 This is 4n by 4n. 580 00:47:48,585 --> 00:47:52,396 It is vastly smaller matrix that I have to do. 581 00:48:08,710 --> 00:48:14,630 All right, so maybe we should just 582 00:48:14,630 --> 00:48:18,500 write down what this matrix t star is. 583 00:48:18,500 --> 00:48:24,810 So t star, I said, is this 4n by 4n matrix 584 00:48:24,810 --> 00:48:28,520 that tells me how going from some side 585 00:48:28,520 --> 00:48:35,270 xy I arrive at some other side x prime y prime. 586 00:48:35,270 --> 00:48:39,410 But it also has orientation information. 587 00:48:39,410 --> 00:48:45,340 So really, I should have four of these for this, 588 00:48:45,340 --> 00:48:46,840 and four of these for this. 589 00:48:46,840 --> 00:48:50,590 So it's actually a 4 by 4 matrix that I have 590 00:48:50,590 --> 00:48:54,010 once I keep track of orientations. 591 00:48:54,010 --> 00:48:57,620 So let me write down the 4 by 4 matrix explicitly. 592 00:49:02,290 --> 00:49:07,100 So here we have in mu, and this mu 593 00:49:07,100 --> 00:49:13,630 could be one, two, three, four-- which, again, specifically I 594 00:49:13,630 --> 00:49:18,450 have indicated as this, this, this, and this. 595 00:49:20,980 --> 00:49:25,570 And along the other direction, I can arrive at mu prime. 596 00:49:25,570 --> 00:49:27,820 Once I have arrived at the mu prime, 597 00:49:27,820 --> 00:49:33,620 I can either go forward, up, left, or down. 598 00:49:38,590 --> 00:49:45,140 So essentially what this says is start 599 00:49:45,140 --> 00:49:52,860 with a side xy, head in the horizontal direction. 600 00:49:52,860 --> 00:49:57,270 Since this is a one-step walk, after one step 601 00:49:57,270 --> 00:50:00,460 I will be arriving at some other point. 602 00:50:00,460 --> 00:50:05,320 Once you arrive at that next point, continue to [INAUDIBLE]. 603 00:50:08,550 --> 00:50:17,140 The next element says head to the right and then go up. 604 00:50:17,140 --> 00:50:21,060 The next element says go to the right 605 00:50:21,060 --> 00:50:24,750 and then start to the back. 606 00:50:24,750 --> 00:50:28,660 The next element says go to the right and then go down. 607 00:50:31,170 --> 00:50:34,420 Now we can construct the rest of them. 608 00:50:34,420 --> 00:50:55,160 Up, right, up, up, up, left, up, down, left, right, left, up, 609 00:50:55,160 --> 00:50:59,980 left, left, left, down. 610 00:50:59,980 --> 00:51:12,920 Lastly, down, right, down, up, down, left, down, down. 611 00:51:12,920 --> 00:51:16,930 So you do your aerobic exercises, 612 00:51:16,930 --> 00:51:24,880 and then in next stage is to actually write down the numbers 613 00:51:24,880 --> 00:51:27,550 that these correspond to. 614 00:51:27,550 --> 00:51:31,970 So first of all, you can see that-- I'm not sure 615 00:51:31,970 --> 00:51:33,700 whether I will have enough space, 616 00:51:33,700 --> 00:51:40,570 but let's hope that I do-- that in this first row 617 00:51:40,570 --> 00:51:46,900 of this matrix, your first step was always to the right. 618 00:51:46,900 --> 00:51:52,120 So always, you will start from x and you end up at x plus 1 619 00:51:52,120 --> 00:51:54,140 while y does not change. 620 00:51:54,140 --> 00:51:59,350 So I will indicate that by x plus 1-- actually, 621 00:51:59,350 --> 00:52:00,380 what should I write it? 622 00:52:04,070 --> 00:52:06,450 Yeah, x prime, y prime. 623 00:52:06,450 --> 00:52:09,830 x prime has to be x plus 1. 624 00:52:09,830 --> 00:52:12,320 y prime has to be y. 625 00:52:12,320 --> 00:52:14,400 So I'll have to introduce the notation 626 00:52:14,400 --> 00:52:21,650 that x prime y prime xy means theta x x prime delta y 627 00:52:21,650 --> 00:52:23,116 y prime. 628 00:52:23,116 --> 00:52:29,630 So essentially, you just read off for x prime, y prime 629 00:52:29,630 --> 00:52:32,420 what the new points have to be. 630 00:52:32,420 --> 00:52:33,660 And I proceed forward. 631 00:52:33,660 --> 00:52:35,920 There's no change in face. 632 00:52:35,920 --> 00:52:38,440 The next one, I arrive at the same point, 633 00:52:38,440 --> 00:52:45,000 so x prime y prime is x plus 1 y. 634 00:52:45,000 --> 00:52:49,340 But now my tangent vector, my heading 635 00:52:49,340 --> 00:52:55,220 has shifted by 90 degrees, so I have to put a factor of e 636 00:52:55,220 --> 00:52:57,450 to the i pi over 4. 637 00:53:00,650 --> 00:53:04,550 The next one, I try to go back, but I've 638 00:53:04,550 --> 00:53:11,130 said U-turns are not allowed, so this matrix element is 0. 639 00:53:11,130 --> 00:53:18,900 The next matrix element, I have x prime y prime x plus 1 y. 640 00:53:18,900 --> 00:53:23,150 Now I have pinned it down, so minus 90 degrees-- 641 00:53:23,150 --> 00:53:26,154 the changing angle-- is minus i pi over 4. 642 00:53:30,450 --> 00:53:34,600 The second column, you can see that essentially 643 00:53:34,600 --> 00:53:39,450 the y-coordinate has to change by 1, increase by 1. 644 00:53:39,450 --> 00:53:45,810 So I have x prime, y prime being xy plus 1. 645 00:53:45,810 --> 00:53:48,350 And this element has a change of angle 646 00:53:48,350 --> 00:53:50,590 that corresponds to the minus 90 degrees, 647 00:53:50,590 --> 00:53:55,360 so this is e to the minus 5 pi over 4. 648 00:53:55,360 --> 00:53:59,330 The diagonal element continues to head straight, 649 00:53:59,330 --> 00:54:01,800 so there was no phase angle associated with that. 650 00:54:04,700 --> 00:54:08,680 The third element heads in the opposite direction, 651 00:54:08,680 --> 00:54:17,240 so the phase element for that is e to the minus i pi over 4. 652 00:54:17,240 --> 00:54:22,120 The fourth element is a U-turn, so it will be 0. 653 00:54:22,120 --> 00:54:26,330 The third column starts with a U-turn, which is a 0. 654 00:54:26,330 --> 00:54:29,930 The next one, you can see I have to step to the left, 655 00:54:29,930 --> 00:54:36,570 so x prime has to become x minus 1 while y does not change. 656 00:54:36,570 --> 00:54:40,660 The phase factor is e to the minus i pi over 4. 657 00:54:40,660 --> 00:54:43,215 Along with diagonal, there is no phase factor. 658 00:54:46,060 --> 00:54:52,290 And if I already had one e to the minus i over 4, 659 00:54:52,290 --> 00:54:57,130 I should have e to the i pi over 4 for the last one. 660 00:54:57,130 --> 00:55:02,440 And the last element corresponds to y decreasing by 1. 661 00:55:02,440 --> 00:55:07,930 So I start with x prime, y prime in xy minus 1. 662 00:55:07,930 --> 00:55:10,620 The first phase that I have to do, 663 00:55:10,620 --> 00:55:14,390 I can kind of read how things go diagonally. 664 00:55:14,390 --> 00:55:19,060 This is i pi over 4 to the i pi over 4. 665 00:55:19,060 --> 00:55:22,580 The next one has to be 0 because the 0's you can see, 666 00:55:22,580 --> 00:55:24,580 are proceeding diagonally. 667 00:55:24,580 --> 00:55:29,970 The next one would be x prime, y prime xy minus 1 668 00:55:29,970 --> 00:55:33,575 e to the i minus i pi over 4 and then 669 00:55:33,575 --> 00:55:38,340 x prime y prime xy minus 1 for the last time. 670 00:55:43,230 --> 00:55:46,890 So this keeps track of the changes in phase. 671 00:55:50,080 --> 00:55:56,750 So now the next thing that we did 672 00:55:56,750 --> 00:56:00,820 when we had the ordinary random walks was we 673 00:56:00,820 --> 00:56:06,870 took advantage of the translational invariance 674 00:56:06,870 --> 00:56:12,650 of the lattice to go to Fourier space and make diagonalization. 675 00:56:12,650 --> 00:56:18,222 And indeed, we can do that over here, too, but only partially 676 00:56:18,222 --> 00:56:22,920 in that this object has two sets of indices. 677 00:56:22,920 --> 00:56:27,680 There is the lattice coordinates and there's the orientation. 678 00:56:27,680 --> 00:56:34,300 But what I can certainly do is to diagonalize the the subspace 679 00:56:34,300 --> 00:56:37,260 that corresponds to positions. 680 00:56:37,260 --> 00:56:39,730 What do I mean by that? 681 00:56:39,730 --> 00:56:47,940 What I will do is I will introduce, let's say, qx, qy, 682 00:56:47,940 --> 00:56:54,682 xy-- these Fourier elements-- which are e to the i qx 683 00:56:54,682 --> 00:57:02,850 x plus qy y divided by square root of n just as before, 684 00:57:02,850 --> 00:57:06,590 without any orientation component. 685 00:57:06,590 --> 00:57:12,440 Then you can see that if I multiply this object to xq 686 00:57:12,440 --> 00:57:20,120 y xy, with this matrix that I have over here, 687 00:57:20,120 --> 00:57:30,480 xy t star x prime y prime, and sum over x and y, 688 00:57:30,480 --> 00:57:33,780 but leave the orientations unchanged. 689 00:57:33,780 --> 00:57:36,840 That is, basically I do this individually 690 00:57:36,840 --> 00:57:42,250 for each one of these 16 elements of this matrix 691 00:57:42,250 --> 00:57:45,237 that each one of them clearly depends on x, y, 692 00:57:45,237 --> 00:57:50,260 x prime, y prime, but also has some additional factors. 693 00:57:50,260 --> 00:57:52,930 In each case, what is going to happen 694 00:57:52,930 --> 00:57:59,660 is that because I'm shifting x or y by one step, 695 00:57:59,660 --> 00:58:03,725 I will get this factor back up to e to the i qx, e 696 00:58:03,725 --> 00:58:07,160 to the minus i qx, e to the i qy, e to the minus iqy 697 00:58:07,160 --> 00:58:10,800 exactly as I was doing before, except that I 698 00:58:10,800 --> 00:58:14,900 will have to do this for every single one of them. 699 00:58:14,900 --> 00:58:18,470 So you can see that essentially what this reproduces is 700 00:58:18,470 --> 00:58:23,020 a matrix that is four by four that depends on q, 701 00:58:23,020 --> 00:58:28,940 and then I will get xy qx of qy actually x prime y 702 00:58:28,940 --> 00:58:30,920 prime because I sum over x and y back. 703 00:58:34,610 --> 00:58:38,810 So what is this matrix, t star of q? 704 00:58:38,810 --> 00:58:44,190 It is very easily constructed from what I have over there. 705 00:58:44,190 --> 00:58:48,700 Because you can see that from the first one, what happens 706 00:58:48,700 --> 00:58:53,470 is that when I see x, I will change it 707 00:58:53,470 --> 00:58:58,350 to x prime, which is x minus 1. 708 00:58:58,350 --> 00:59:03,920 So from here, I will get a factor of e to the minus i qx. 709 00:59:03,920 --> 00:59:05,180 y and y prime are the same. 710 00:59:05,180 --> 00:59:07,830 I don't get anything from here. 711 00:59:07,830 --> 00:59:17,640 Next one, I will get e to the minus i qx plus i pi over 4 0 712 00:59:17,640 --> 00:59:23,030 e to the minus i qx minus i pi over 4. 713 00:59:23,030 --> 00:59:28,440 The next column, the y has been shifted by 1. 714 00:59:28,440 --> 00:59:34,630 So I will get e to the minus i cube y minus i 715 00:59:34,630 --> 00:59:43,840 pi over 4 e to the minus i qy e to the minus i qy plus i pi 716 00:59:43,840 --> 00:59:48,120 over 4 0. 717 00:59:48,120 --> 00:59:53,560 The next level, the third one, x prime is set to x minus 1, 718 00:59:53,560 --> 01:00:00,770 so I essentially get e to the i qx-- oops. 719 01:00:00,770 --> 01:00:03,850 Third element starts with 0, and then I'll 720 01:00:03,850 --> 01:00:12,820 have e to the i qx minus i pi over 4 e to the i qx, 721 01:00:12,820 --> 01:00:18,520 and then I'll have e to the i qx plus i pi over 4. 722 01:00:18,520 --> 01:00:29,110 The fourth element is e to the i cube y plus i pi over 4 0 723 01:00:29,110 --> 01:00:34,410 into the i qy minus i pi over 4 e to the i qy. 724 01:00:57,730 --> 01:01:06,590 So in the positions space, I have this 4n by 4n matrix 725 01:01:06,590 --> 01:01:09,600 where the different sides were connected to their neighbors 726 01:01:09,600 --> 01:01:12,020 with these phase factors. 727 01:01:12,020 --> 01:01:16,190 I have gone from coordinate to Fourier basis. 728 01:01:16,190 --> 01:01:18,700 I did this transformation. 729 01:01:18,700 --> 01:01:23,750 Now I have a matrix that is blocked diagonal. 730 01:01:23,750 --> 01:01:30,020 So for each value of the q, I have a four by four block. 731 01:01:30,020 --> 01:01:33,320 So in the q picture, imagine that you have this 4n 732 01:01:33,320 --> 01:01:35,670 by 4n matrix, and you have blocks of four 733 01:01:35,670 --> 01:01:37,540 for different q along the diagonal. 734 01:01:37,540 --> 01:01:38,650 Each one of them is this. 735 01:01:43,110 --> 01:01:49,296 So now let's go and calculate our partition function. 736 01:01:49,296 --> 01:01:51,790 So what do we have? 737 01:01:51,790 --> 01:02:00,030 We have that log z over n is log 2 hyperbolic cosine 738 01:02:00,030 --> 01:02:11,480 squared of k plus 1/2 sum over l. 739 01:02:14,700 --> 01:02:20,837 Sum over l of these loops that go back 740 01:02:20,837 --> 01:02:21,920 all the way to themselves. 741 01:02:24,800 --> 01:02:30,140 So this is t to the l over l. 742 01:02:30,140 --> 01:02:36,125 This is loop star of length l. 743 01:02:39,140 --> 01:02:42,855 I want to relate loop star of length 744 01:02:42,855 --> 01:02:50,330 l to this w star of length l, but I 745 01:02:50,330 --> 01:02:56,530 want to start and end at the origin. 746 01:02:56,530 --> 01:03:00,530 So I start from the origin and end at the origin. 747 01:03:04,300 --> 01:03:06,900 But I have to be careful, because let's say 748 01:03:06,900 --> 01:03:11,510 I make a loop such as this and I end up at the origin. 749 01:03:11,510 --> 01:03:14,450 I have to get the right phase factor. 750 01:03:14,450 --> 01:03:18,480 So if I started along direction mu, 751 01:03:18,480 --> 01:03:22,940 when I get back to the starting point, I cannot turn this way, 752 01:03:22,940 --> 01:03:25,480 this way and get the right phase factor. 753 01:03:25,480 --> 01:03:27,860 I have to go and head in the same direction as mu. 754 01:03:32,100 --> 01:03:35,800 So I have to head to the same direction. 755 01:03:42,180 --> 01:03:47,010 Now I can certainly do this as a summation 756 01:03:47,010 --> 01:03:50,540 also over the starting point. 757 01:03:50,540 --> 01:03:52,970 Instead, I have to start and end at 0. 758 01:03:52,970 --> 01:03:55,820 I could have started and ended at any point, 759 01:03:55,820 --> 01:03:59,250 and then I do a sum of xy and mu, 760 01:03:59,250 --> 01:04:08,560 but then I better divide by the n, right? 761 01:04:08,560 --> 01:04:12,120 And the reason I do that is because now you 762 01:04:12,120 --> 01:04:19,920 can see that the structure of this is like a trace, 763 01:04:19,920 --> 01:04:20,760 and I like that. 764 01:04:23,380 --> 01:04:31,970 Actually, I made a mistake when I did this because, you see, 765 01:04:31,970 --> 01:04:36,040 the factor that I had to really include is minus 1 766 01:04:36,040 --> 01:04:37,165 to the number of crossings. 767 01:04:39,970 --> 01:04:44,670 But my w l star is just keep track of e 768 01:04:44,670 --> 01:04:47,880 to the i delta thetas. 769 01:04:47,880 --> 01:04:51,435 Actually, I should have put in here a delta theta over 2. 770 01:04:51,435 --> 01:04:54,100 So I had forgotten that. 771 01:04:54,100 --> 01:04:58,830 But we can see that that factor is different from minus 1 772 01:04:58,830 --> 01:05:02,451 to the nc to the minus sign. 773 01:05:02,451 --> 01:05:05,720 So there was that minus sign that I had forgotten. 774 01:05:05,720 --> 01:05:09,770 And actually, I better make this a minus sign. 775 01:05:09,770 --> 01:05:15,330 So what was plus before becomes a minus because of this factor 776 01:05:15,330 --> 01:05:16,470 that I have over here. 777 01:05:19,390 --> 01:05:22,270 So let's write this again. 778 01:05:22,270 --> 01:05:34,280 This is log 2 hyperbolic cosine squared of k minus 1 over 2n. 779 01:05:34,280 --> 01:05:38,915 And then this sum over xy mu is like a trace. 780 01:05:41,670 --> 01:05:44,110 And what is it that I'm tracing? 781 01:05:44,110 --> 01:05:52,051 I'm tracing a sum over l, t t star to the l divided by l. 782 01:05:56,001 --> 01:05:56,500 Yes? 783 01:05:56,500 --> 01:06:01,660 AUDIENCE: So we'll also sum it over mu [INAUDIBLE] 4? 784 01:06:01,660 --> 01:06:07,270 PROFESSOR: No, because I don't know 785 01:06:07,270 --> 01:06:10,200 which direction my first step is. 786 01:06:10,200 --> 01:06:13,230 So what I'm doing is I'm summing over all always 787 01:06:13,230 --> 01:06:17,780 of starting step from the origin, head in direction mu. 788 01:06:17,780 --> 01:06:21,360 Then I have to make sure that I come back to mu. 789 01:06:21,360 --> 01:06:24,690 Now it is true that I sum over mu, 790 01:06:24,690 --> 01:06:26,940 but I already took care of that when 791 01:06:26,940 --> 01:06:31,770 I divided by 2l, because let's say 792 01:06:31,770 --> 01:06:34,910 you look at the diagrams of length 4 793 01:06:34,910 --> 01:06:37,390 that I generate through this procedure. 794 01:06:37,390 --> 01:06:40,960 Starting from here, depending on which direction I go, 795 01:06:40,960 --> 01:06:44,825 I will generate this or this or this, or this. 796 01:06:44,825 --> 01:06:47,545 These are precisely the four diagrams 797 01:06:47,545 --> 01:06:51,940 that I generate depending on which starting point I pick. 798 01:06:51,940 --> 01:06:56,170 So it is truly there is an over-counting, 799 01:06:56,170 --> 01:06:58,934 but that's an over-counting that you've already taken care of. 800 01:07:08,000 --> 01:07:10,770 Now again, we did this last time. 801 01:07:10,770 --> 01:07:15,450 This is the series for minus log of 1 802 01:07:15,450 --> 01:07:19,060 minus 2 t star-- matrix t star. 803 01:07:21,960 --> 01:07:26,240 So this is the same thing as log of 2 hyperbolic cosine 804 01:07:26,240 --> 01:07:37,510 squared of k plus now 1 over 2n trace of log of 1 805 01:07:37,510 --> 01:07:39,430 minus t t star. 806 01:07:51,200 --> 01:07:57,150 Now I said that my matrix was blocked diagonal 807 01:07:57,150 --> 01:08:00,530 when I went to look at the q basis. 808 01:08:00,530 --> 01:08:03,240 And I can take the trace in any basis, 809 01:08:03,240 --> 01:08:05,705 whether it's in coordinate basis, in momentum basis. 810 01:08:05,705 --> 01:08:08,120 Trace is trace. 811 01:08:08,120 --> 01:08:11,170 So now focus on what the trace will look 812 01:08:11,170 --> 01:08:13,960 like if I go to the Fourier basis. 813 01:08:13,960 --> 01:08:17,149 I have these four by four blocks, 814 01:08:17,149 --> 01:08:19,649 and then I calculate the trace, I 815 01:08:19,649 --> 01:08:22,470 will calculate the trace of one four by four. 816 01:08:22,470 --> 01:08:26,270 And the other four by four, I go over all q's. 817 01:08:26,270 --> 01:08:32,420 So basically this can be written as a sum over q's. 818 01:08:32,420 --> 01:08:36,500 Log of 1 minus t. 819 01:08:36,500 --> 01:08:41,520 This four by four matrix t star of q, and the trace of that. 820 01:08:47,430 --> 01:08:53,880 And finally-- oops, I forgot a factor of 1 over 2n here. 821 01:08:53,880 --> 01:09:00,670 The sum over q I'm going to replace with an integral over q 822 01:09:00,670 --> 01:09:03,660 times n over 2 pi squared. 823 01:09:03,660 --> 01:09:05,910 So the final answer here is going 824 01:09:05,910 --> 01:09:09,630 to look like log 2 hyperbolic cosine squared 825 01:09:09,630 --> 01:09:14,609 root of k plus 1/2. 826 01:09:14,609 --> 01:09:18,819 The 1 over n I will get rid of when I write the sum over q 827 01:09:18,819 --> 01:09:21,124 as n integral d2 of q. 828 01:09:21,124 --> 01:09:26,102 So I have integral d2 of 1 divided by 2 pi squared. 829 01:09:29,370 --> 01:09:30,240 One more step. 830 01:09:35,450 --> 01:09:45,000 Trace of a log of any matrix I can write-- 831 01:09:45,000 --> 01:09:49,700 let's say we find a basis in which the m is diagonal. 832 01:09:49,700 --> 01:09:54,900 Then it becomes a sum over alpha log of lambda 833 01:09:54,900 --> 01:10:01,470 alpha, where lambda alphas are diagonal values of this. 834 01:10:01,470 --> 01:10:03,870 But sum over logs is the same thing 835 01:10:03,870 --> 01:10:12,021 as log of the product over alpha of lambda alpha. 836 01:10:12,021 --> 01:10:15,300 The product of eigenvalues of the matrix you 837 01:10:15,300 --> 01:10:18,670 also recognize to be the determinant. 838 01:10:18,670 --> 01:10:22,840 So this is the log of the determinant of the matrix. 839 01:10:22,840 --> 01:10:26,230 So this is a very useful, famous identity 840 01:10:26,230 --> 01:10:29,640 that trace log is the same thing as log determinant 841 01:10:29,640 --> 01:10:31,020 that I will use. 842 01:10:31,020 --> 01:10:34,230 And rather than calculate the trace log, 843 01:10:34,230 --> 01:10:39,090 I will write it as the log of the determinant, 844 01:10:39,090 --> 01:10:42,200 and I will explicitly write down for you 845 01:10:42,200 --> 01:10:45,390 the determinant of which four by four matrix. 846 01:10:45,390 --> 01:10:50,310 It is simply 1 minus p times the elements of that matrix. 847 01:10:50,310 --> 01:10:55,980 So it's 1 minus t e to the minus i qx minus t 848 01:10:55,980 --> 01:11:05,540 e to the minus i qx plus i pi over 4 0 minus t 849 01:11:05,540 --> 01:11:11,880 e to the minus i qx minus i pi over 4. 850 01:11:11,880 --> 01:11:15,350 Second element, second row-- minus t 851 01:11:15,350 --> 01:11:23,250 e to the minus i qy minus i pi over 4 1 minus t 852 01:11:23,250 --> 01:11:30,240 e to the minus i qy minus t e to the minus i qy plus i pi 853 01:11:30,240 --> 01:11:36,020 over 4 minus-- oops, 0 for the last element here. 854 01:11:36,020 --> 01:11:37,900 It's a U-turn. 855 01:11:37,900 --> 01:11:47,770 The third thing is 0 minus t e to the i qx minus i 856 01:11:47,770 --> 01:11:55,530 pi over 4 1 minus t e to the i qx diagonal element minus t e 857 01:11:55,530 --> 01:11:59,590 to the i qx plus i pi over 4. 858 01:11:59,590 --> 01:12:06,725 Final row-- minus t e to the i cube y plus i 859 01:12:06,725 --> 01:12:13,050 pi over 4, 0 for the U-turn, minus t e 860 01:12:13,050 --> 01:12:19,770 to the i qy minus i pi over 4 diagonal turn 1 minus t e 861 01:12:19,770 --> 01:12:23,620 to the i cube y, and that's it. 862 01:12:26,180 --> 01:12:28,730 And that's the answer. 863 01:12:28,730 --> 01:12:34,330 So calculating the partition function 864 01:12:34,330 --> 01:12:38,180 of the 2-dimensionalizing model is 865 01:12:38,180 --> 01:12:43,780 reduced to calculating this four by four determinant, which 866 01:12:43,780 --> 01:12:46,930 we can do by hand. 867 01:12:46,930 --> 01:12:48,000 I won't do it here. 868 01:12:48,000 --> 01:12:50,560 I will write the answer. 869 01:12:50,560 --> 01:13:00,120 So the log of z over n is log 3 hyperbolic cosine squared 870 01:13:00,120 --> 01:13:10,660 of k plus 1/2 integral d2 q 2 pi squared. 871 01:13:10,660 --> 01:13:13,860 Log, you take the determinant. 872 01:13:13,860 --> 01:13:19,680 What you find is 1 plus t squared squared minus 2 t 1 873 01:13:19,680 --> 01:13:24,330 minus t squared cosine of qx plus cosine of qy. 874 01:13:29,870 --> 01:13:33,940 If you want, you can write it in a slightly different way 875 01:13:33,940 --> 01:13:39,250 by taking the cosine squared inside this logarithm 876 01:13:39,250 --> 01:13:43,360 and doing a little bit of algebra. 877 01:13:43,360 --> 01:13:46,870 We get log 2 plus 1/2. 878 01:13:46,870 --> 01:13:53,510 Explicitly, these are integrals that go from 0 to 2 pi, 879 01:13:53,510 --> 01:13:56,270 because that's the range of q vectors 880 01:13:56,270 --> 01:13:59,110 that are allowed by this transformation. 881 01:13:59,110 --> 01:14:00,970 I'll have a q in the x direction. 882 01:14:00,970 --> 01:14:03,350 I'll have a q in the y direction. 883 01:14:03,350 --> 01:14:08,230 2 pi squared, each one of them goes in the range 0 to 2 pi. 884 01:14:08,230 --> 01:14:10,376 I have a log. 885 01:14:10,376 --> 01:14:16,090 And once I take this cosine squared inside, 886 01:14:16,090 --> 01:14:18,720 it becomes cos to the fourth. 887 01:14:18,720 --> 01:14:22,780 You write this t sine squared divided by cos squared. 888 01:14:22,780 --> 01:14:25,860 You can see that the cos to the fourth will cancel this. 889 01:14:25,860 --> 01:14:28,670 You will get cos squared plus sine squared, 890 01:14:28,670 --> 01:14:31,840 which is the same thing as hyperbolic cosine of twice 891 01:14:31,840 --> 01:14:34,160 the angle squared. 892 01:14:34,160 --> 01:14:40,200 And the other terms conspires to give you the sine of 2kx 893 01:14:40,200 --> 01:14:45,340 and then cosine of qx plus cosine of qy. 894 01:14:45,340 --> 01:14:55,196 And so this is the partition function 895 01:14:55,196 --> 01:14:56,550 of the 2-dimensionalizing model. 896 01:14:59,250 --> 01:15:01,850 You can do a little bit more manipulations, 897 01:15:01,850 --> 01:15:05,350 write this integral in terms of special functions, 898 01:15:05,350 --> 01:15:09,740 but you won't gain much. 899 01:15:09,740 --> 01:15:12,970 So this is the answer. 900 01:15:12,970 --> 01:15:18,000 I want you to absorb and appreciate this derivation. 901 01:15:18,000 --> 01:15:21,020 And next time we look at this and see 902 01:15:21,020 --> 01:15:24,030 what it means for the similarities in the phase 903 01:15:24,030 --> 01:15:26,050 behavior of the 2-dimensionalizing. 904 01:15:26,050 --> 01:15:27,954 Yes? 905 01:15:27,954 --> 01:15:30,120 AUDIENCE: [INAUDIBLE] in the log, it's cos 906 01:15:30,120 --> 01:15:32,340 squared minus [INAUDIBLE]? 907 01:15:32,340 --> 01:15:34,480 PROFESSOR: Yes. 908 01:15:34,480 --> 01:15:36,940 But both of them went to twice diagonal. 909 01:15:36,940 --> 01:15:39,710 Here everything is in terms of k. 910 01:15:39,710 --> 01:15:43,150 And once I took the hyperbolic cosine squared inside then 911 01:15:43,150 --> 01:15:48,720 did the manipulations, they became twice the length. 912 01:15:48,720 --> 01:15:52,340 AUDIENCE: What's the subscript up there [INAUDIBLE]? 913 01:15:52,340 --> 01:15:54,956 PROFESSOR: 2. 914 01:15:54,956 --> 01:15:56,330 There should be a subscript, yes. 915 01:16:04,834 --> 01:16:07,000 AUDIENCE: Is there an extension to higher dimensions 916 01:16:07,000 --> 01:16:08,765 where you just do a summation over cosine? 917 01:16:08,765 --> 01:16:10,140 PROFESSOR: You would wish, right? 918 01:16:10,140 --> 01:16:11,181 I mean, that's actually-- 919 01:16:11,181 --> 01:16:12,680 [LAUGHTER] 920 01:16:12,680 --> 01:16:15,240 PROFESSOR: And quite a number of people 921 01:16:15,240 --> 01:16:17,780 have come with that conjecture, including 922 01:16:17,780 --> 01:16:19,640 myself when i was a graduate student 923 01:16:19,640 --> 01:16:22,140 and I didn't know better. 924 01:16:22,140 --> 01:16:28,960 If you sort of write things in terms of not only 925 01:16:28,960 --> 01:16:33,560 if you make kx and ky to be different, 926 01:16:33,560 --> 01:16:38,060 then this takes the form of cos 2kx cos 2ky. 927 01:16:38,060 --> 01:16:40,950 This becomes, in some sense, a very nice version 928 01:16:40,950 --> 01:16:42,820 of 2kx and 2ky. 929 01:16:42,820 --> 01:16:46,370 There is a way natural-- and the thing that is nice 930 01:16:46,370 --> 01:16:50,000 is that if you put any one of the 2kx's to 0, then 931 01:16:50,000 --> 01:16:53,300 you have reduced the formula for the 1-dimensionalizing model, 932 01:16:53,300 --> 01:16:54,930 as you should. 933 01:16:54,930 --> 01:16:56,700 And then the natural thing would be 934 01:16:56,700 --> 01:16:59,970 to write a similar product in three dimensions, 935 01:16:59,970 --> 01:17:02,460 and you did when said one of k's equals to 0. 936 01:17:02,460 --> 01:17:05,030 You get the corrected dimensionalizing model. 937 01:17:05,030 --> 01:17:08,804 So it passes a number of test, yet it's unfortunately not 938 01:17:08,804 --> 01:17:09,304 correct. 939 01:17:14,244 --> 01:17:16,220 AUDIENCE: So the problem is conditional 940 01:17:16,220 --> 01:17:21,320 in this three dimensions is counting the loops? 941 01:17:21,320 --> 01:17:24,530 PROFESSOR: What we relied on heavily 942 01:17:24,530 --> 01:17:27,980 was this factor of minus 1 to the product 943 01:17:27,980 --> 01:17:30,260 of the number of crossings. 944 01:17:30,260 --> 01:17:34,050 And if you think about it as a topological entity, 945 01:17:34,050 --> 01:17:38,300 these crossings only make sense in two dimensions. 946 01:17:38,300 --> 01:17:43,350 So you don't have the basic tool to go in three dimensions. 947 01:17:43,350 --> 01:17:45,750 And actually, what those minus signs 948 01:17:45,750 --> 01:17:48,820 mean I will explain next time. 949 01:17:48,820 --> 01:17:51,817 Has something to do with fermionic character of this 950 01:17:51,817 --> 01:17:52,317 [INAUDIBLE]. 951 01:17:59,705 --> 01:18:00,830 So it's a beautiful result. 952 01:18:00,830 --> 01:18:04,100 You should appreciate it. 953 01:18:04,100 --> 01:18:05,020 Yes? 954 01:18:05,020 --> 01:18:07,780 AUDIENCE: [INAUDIBLE] this Thursday? 955 01:18:07,780 --> 01:18:09,940 PROFESSOR: I'll go through the history, too. 956 01:18:09,940 --> 01:18:13,920 The person who first derived this field energy 957 01:18:13,920 --> 01:18:19,260 was Onsager with this transfer matrix method that I described. 958 01:18:19,260 --> 01:18:24,500 This way of doing it in terms of graphs came much later. 959 01:18:24,500 --> 01:18:27,860 And a number of people that were involved 960 01:18:27,860 --> 01:18:29,290 that, including Feynman. 961 01:18:29,290 --> 01:18:32,520 In fact, in Feynman's book, there's 962 01:18:32,520 --> 01:18:35,740 a very nice derivation along these lines that you can see. 963 01:18:38,430 --> 01:18:41,600 The connections to fermions and the number of people Mattis, 964 01:18:41,600 --> 01:18:46,410 Schultz, Lieb, et cetera, came up with. 965 01:18:46,410 --> 01:18:50,140 One thing that-- well, OK, guess I have a few minutes. 966 01:18:50,140 --> 01:18:52,650 I can say a few things. 967 01:18:52,650 --> 01:18:59,510 So this result dates back to around 1950s. 968 01:18:59,510 --> 01:19:04,580 And I know a generation of physicists 969 01:19:04,580 --> 01:19:08,530 who are now about to retire or have retired in the past 10 970 01:19:08,530 --> 01:19:12,490 years or so who were very young when these things were 971 01:19:12,490 --> 01:19:13,940 introduced. 972 01:19:13,940 --> 01:19:16,600 And as far as I can see, all through their life, 973 01:19:16,600 --> 01:19:18,990 they did versions of this. 974 01:19:18,990 --> 01:19:23,400 So you can sort of do versions of the 2-dimensionalizing model 975 01:19:23,400 --> 01:19:26,610 and you can make the interactions to be different. 976 01:19:26,610 --> 01:19:28,980 You can play the different types of interactions. 977 01:19:28,980 --> 01:19:31,500 You can make the boundaries to be 978 01:19:31,500 --> 01:19:33,630 different, periodic, et cetera. 979 01:19:33,630 --> 01:19:36,240 There's been variants that you can find, 980 01:19:36,240 --> 01:19:38,640 and there are some people who seem 981 01:19:38,640 --> 01:19:40,540 to have done that throughout their career. 982 01:19:43,290 --> 01:19:47,900 But then after that, we had the renormalization group, 983 01:19:47,900 --> 01:19:51,180 and a totally different perspective.