1 00:00:00,060 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:21,650 --> 00:00:25,210 OK, let's start. 9 00:00:25,210 --> 00:00:28,600 So let's go back to our starting point 10 00:00:28,600 --> 00:00:30,720 for the past couple of weeks. 11 00:00:33,440 --> 00:00:39,130 The square lattice, let's say, where each side 12 00:00:39,130 --> 00:00:43,260 we assign a binary variable sigma, which is minus 1. 13 00:00:46,550 --> 00:00:52,420 And a weight that tends to make subsequent nearest 14 00:00:52,420 --> 00:00:55,750 neighbors [? things ?] to be parallel to each other. 15 00:00:55,750 --> 00:00:59,620 So into the plus k, they are parallel. 16 00:00:59,620 --> 00:01:03,400 Penalized into the minus k if they are anti-parallel. 17 00:01:03,400 --> 00:01:08,810 And of course, over all pairs of nearest neighbors. 18 00:01:08,810 --> 00:01:12,361 And the partition function that is obtained by summing over 2 19 00:01:12,361 --> 00:01:14,600 to the n configurations. 20 00:01:14,600 --> 00:01:17,560 Let's make this up to give us a function 21 00:01:17,560 --> 00:01:19,660 of the rate of this coupling, which 22 00:01:19,660 --> 00:01:23,850 is some energy provided by temperature. 23 00:01:23,850 --> 00:01:29,075 And we expect this-- at least, in two and higher dimensions-- 24 00:01:29,075 --> 00:01:32,600 to capture a phase transition. 25 00:01:32,600 --> 00:01:36,640 And the way that we have been proceeding with this 26 00:01:36,640 --> 00:01:41,840 to derive this factor as the hyperbolic cosine of k. 27 00:01:41,840 --> 00:01:49,440 1 plus r variable t, which is the hyperbolic [? sine ?] of k. 28 00:01:52,950 --> 00:01:55,710 Sigma i, sigma j. 29 00:01:55,710 --> 00:02:02,640 And then this becomes a cos k to the number 30 00:02:02,640 --> 00:02:07,760 of bonds, which is 2n on the square lattice. 31 00:02:07,760 --> 00:02:11,900 And then expanding these factors, 32 00:02:11,900 --> 00:02:15,270 we saw that we could get things that are either 33 00:02:15,270 --> 00:02:18,940 1 from each bond or a factor that 34 00:02:18,940 --> 00:02:22,990 was something like t sigma sigma. 35 00:02:22,990 --> 00:02:25,830 And then summing over the two values of sigma 36 00:02:25,830 --> 00:02:31,550 would give us 0 unless we added another factor of sigma 37 00:02:31,550 --> 00:02:33,520 through another bond. 38 00:02:33,520 --> 00:02:39,340 And going forth, we had to draw these kinds of diagrams 39 00:02:39,340 --> 00:02:44,060 where at each site, I have an even number of 1's selected. 40 00:02:44,060 --> 00:02:47,156 Then summing over the sigmas would give me a factor of 2 41 00:02:47,156 --> 00:02:49,200 to the n. 42 00:02:49,200 --> 00:02:55,881 And so then I have a sum over a whole bunch of configurations. 43 00:02:55,881 --> 00:02:57,380 [? There are ?] [? certainly ?] one. 44 00:02:57,380 --> 00:02:59,830 There are configurations that are composed 45 00:02:59,830 --> 00:03:04,860 of one way of drawing an object on the lattice 46 00:03:04,860 --> 00:03:08,060 such as this one, or objects that 47 00:03:08,060 --> 00:03:13,005 correspond to doing two of these loops, and so forth. 48 00:03:17,620 --> 00:03:23,150 So that's the expression for the partition function. 49 00:03:23,150 --> 00:03:27,140 And what we are really interested 50 00:03:27,140 --> 00:03:31,440 is the log of the partition function, which gives us 51 00:03:31,440 --> 00:03:33,960 the free energy in terms of thermodynamic quantities 52 00:03:33,960 --> 00:03:38,150 that potentially will tell us about phase transition. 53 00:03:38,150 --> 00:03:42,460 So here we will get a log 2 to the n. 54 00:03:42,460 --> 00:03:46,930 Well actually, you want to divide everything by n 55 00:03:46,930 --> 00:03:48,930 so that we get the intensive part. 56 00:03:48,930 --> 00:03:54,120 So here we get log 2 hyperbolic cosine squared of k. 57 00:03:56,680 --> 00:04:03,050 And then I have to take the log of this expression that 58 00:04:03,050 --> 00:04:05,540 includes things that are one loop, 59 00:04:05,540 --> 00:04:07,682 disjointed loop, et cetera. 60 00:04:07,682 --> 00:04:10,230 And we've seen that the particular loop 61 00:04:10,230 --> 00:04:14,640 I can slide all over the place-- so if you have a factor of n-- 62 00:04:14,640 --> 00:04:16,380 whereas things that are multiple loops 63 00:04:16,380 --> 00:04:19,420 have factors of n squared, et cetera, 64 00:04:19,420 --> 00:04:22,660 which are incompatible with this. 65 00:04:22,660 --> 00:04:29,970 So it was very tempting for us to do the usual thing 66 00:04:29,970 --> 00:04:32,980 and say that the log of a sum that 67 00:04:32,980 --> 00:04:37,540 includes these multiple occurrences of the loops 68 00:04:37,540 --> 00:04:43,620 is the same thing as sum over the configurations 69 00:04:43,620 --> 00:04:46,560 that involve a single loop. 70 00:04:46,560 --> 00:04:50,760 And then we have to sum over all shapes of these loops. 71 00:04:50,760 --> 00:04:54,530 And each loop will get a factor of t 72 00:04:54,530 --> 00:04:56,961 per the number of bonds that are occurring in that. 73 00:05:00,050 --> 00:05:04,160 Of course, what we said was that this equality does not 74 00:05:04,160 --> 00:05:11,110 hold because if I exponentiate this term, 75 00:05:11,110 --> 00:05:14,800 I will generate things where the different loops will 76 00:05:14,800 --> 00:05:17,540 coincide with each other, and therefore 77 00:05:17,540 --> 00:05:22,110 create types of terms that are not created in the original sum 78 00:05:22,110 --> 00:05:24,780 that we had over there. 79 00:05:24,780 --> 00:05:30,560 So this sum over phantom loops neglected the condition 80 00:05:30,560 --> 00:05:34,640 that these loops, in some sense, have some material to them 81 00:05:34,640 --> 00:05:39,330 and don't want to intersect with each other. 82 00:05:39,330 --> 00:05:43,050 Nonetheless, it was useful, and we followed up 83 00:05:43,050 --> 00:05:44,520 this calculation. 84 00:05:44,520 --> 00:05:48,075 So that's repeat what the result of this incorrect calculation 85 00:05:48,075 --> 00:05:49,260 is. 86 00:05:49,260 --> 00:05:54,997 So we have log of 2 hyperbolic cosine squared k, 1 over n. 87 00:05:57,710 --> 00:06:02,280 Then we said the particular way to organize 88 00:06:02,280 --> 00:06:06,430 the sum over the loops is to sum over 89 00:06:06,430 --> 00:06:09,190 there the length of the loop. 90 00:06:09,190 --> 00:06:12,400 So I sum over the length of the loop 91 00:06:12,400 --> 00:06:16,022 and count the number of loops that have length l. 92 00:06:16,022 --> 00:06:18,319 All of them will be giving me a contribution 93 00:06:18,319 --> 00:06:19,110 that is t to the l. 94 00:06:22,620 --> 00:06:26,890 So then I said, well, let's, for example, 95 00:06:26,890 --> 00:06:33,700 pick a particular point on the lattice. 96 00:06:33,700 --> 00:06:36,490 Let's call it r. 97 00:06:36,490 --> 00:06:44,380 And I count the number of ways that I can start at r, 98 00:06:44,380 --> 00:06:50,870 do a walk of l steps, and end at r again. 99 00:06:55,760 --> 00:06:59,780 We saw that for these phantom loops, 100 00:06:59,780 --> 00:07:03,110 this w had a very nice structure. 101 00:07:03,110 --> 00:07:05,680 It was simply what was telling me 102 00:07:05,680 --> 00:07:12,230 about one step raised to the l. 103 00:07:12,230 --> 00:07:16,450 This was the Markovian property. 104 00:07:16,450 --> 00:07:19,830 There was, of course, an important thing here 105 00:07:19,830 --> 00:07:24,510 which said that I could have set the origin of this loop 106 00:07:24,510 --> 00:07:28,870 at any point along the loop. 107 00:07:28,870 --> 00:07:31,747 So there is an over-counting by a factor of l 108 00:07:31,747 --> 00:07:35,730 because the same loop would have been constructed 109 00:07:35,730 --> 00:07:40,240 with different points indicated as the origin. 110 00:07:40,240 --> 00:07:44,640 And actually, I can go the loop clockwise or anti-clockwise, 111 00:07:44,640 --> 00:07:48,620 so there was a factor of 2 because of this degeneracy 112 00:07:48,620 --> 00:07:51,630 of going clockwise or anti-clockwise when 113 00:07:51,630 --> 00:07:52,460 I perform a walk. 114 00:07:55,040 --> 00:08:01,870 And then over here, there's also an implicit sum 115 00:08:01,870 --> 00:08:06,320 over this starting point and endpoint. 116 00:08:06,320 --> 00:08:09,250 If I always start and end at the origin, 117 00:08:09,250 --> 00:08:13,140 then I will get rid of the factor of n. 118 00:08:13,140 --> 00:08:17,780 But it is useful to explicitly include 119 00:08:17,780 --> 00:08:22,120 this sum over r because then you can explicitly 120 00:08:22,120 --> 00:08:26,010 see that sum over r of this object 121 00:08:26,010 --> 00:08:29,360 is the trace of that matrix. 122 00:08:29,360 --> 00:08:32,010 And I can actually [INAUDIBLE] the order, 123 00:08:32,010 --> 00:08:35,760 the trace, and the summation over l. 124 00:08:35,760 --> 00:08:40,909 And when that happens, I get log 2 cos squared k exactly 125 00:08:40,909 --> 00:08:43,080 as before. 126 00:08:43,080 --> 00:08:48,650 And then I have 1 over n. 127 00:08:48,650 --> 00:08:54,995 I have sum over r replaced by the trace operation. 128 00:08:57,510 --> 00:09:04,750 And then sum over l-- t, T raised to the l divided by 129 00:09:04,750 --> 00:09:17,470 l-- is the expansion for minus log of 1 minus t T. 130 00:09:17,470 --> 00:09:21,510 And there's the factor of 2 over there 131 00:09:21,510 --> 00:09:24,590 that I have to put over here. 132 00:09:24,590 --> 00:09:28,220 So note that this plus became minus 133 00:09:28,220 --> 00:09:32,290 because of the expansion for log of 1 minus 134 00:09:32,290 --> 00:09:34,640 x is minus x minus x squared over 2 135 00:09:34,640 --> 00:09:36,550 minus x cubed over 3, et cetera. 136 00:09:43,200 --> 00:09:52,690 And the final step that we did was 137 00:09:52,690 --> 00:10:00,380 to note that the trace I can calculate in any basis. 138 00:10:00,380 --> 00:10:04,500 And in particular, this matrix t is diagonalized 139 00:10:04,500 --> 00:10:07,920 by going to Fourier representation. 140 00:10:07,920 --> 00:10:13,580 In the Fourier representation, the trace operation 141 00:10:13,580 --> 00:10:16,750 becomes sum over all q values. 142 00:10:16,750 --> 00:10:19,800 Sum over all q values, I go to the continuum 143 00:10:19,800 --> 00:10:25,950 and write as n integral over q, so the n's cancel. 144 00:10:25,950 --> 00:10:29,470 I will get 2 integration over q. 145 00:10:29,470 --> 00:10:33,900 These are essentially each one of them, qz and qz, 146 00:10:33,900 --> 00:10:34,780 in the interval. 147 00:10:34,780 --> 00:10:38,300 Let's say 0 to 2 pi or minus pi to 2 pi, doesn't matter. 148 00:10:38,300 --> 00:10:42,300 Interval of size of 2pi. 149 00:10:42,300 --> 00:10:44,120 So that's the trace operation. 150 00:10:44,120 --> 00:10:49,950 Log of 1 minus t. 151 00:10:49,950 --> 00:10:55,540 Then the matrix that represents walking along the lattice 152 00:10:55,540 --> 00:10:57,620 represented in Fourier. 153 00:10:57,620 --> 00:11:00,510 And so basically at the particular site, 154 00:11:00,510 --> 00:11:03,210 we can step to the right or to the left. 155 00:11:03,210 --> 00:11:07,695 So that's e to the i qx, e to the minus i qx, e to the i qy, 156 00:11:07,695 --> 00:11:09,770 e to the minus i qy. 157 00:11:09,770 --> 00:11:13,890 Adding all of those up, you get 2 cosine of qz 158 00:11:13,890 --> 00:11:15,814 plus cosine of qy. 159 00:11:21,590 --> 00:11:25,160 So that was our expression. 160 00:11:25,160 --> 00:11:29,300 And then we realized, interestingly, 161 00:11:29,300 --> 00:11:31,710 that whereas this final expression certainly 162 00:11:31,710 --> 00:11:37,090 was not the Ising partition function that we were after, 163 00:11:37,090 --> 00:11:39,980 that it was, in fact, the partition 164 00:11:39,980 --> 00:11:46,630 function of a Gaussian model where at each site 165 00:11:46,630 --> 00:11:50,060 I had a variable whose variance was 1, 166 00:11:50,060 --> 00:11:52,300 and then I had this kind of coupling 167 00:11:52,300 --> 00:11:54,900 rather than with the sigma variable, 168 00:11:54,900 --> 00:11:57,100 with these Gaussian variables that 169 00:11:57,100 --> 00:11:58,420 will go from minus to infinity. 170 00:12:05,190 --> 00:12:09,240 But then we said, OK, we can do better than that. 171 00:12:09,240 --> 00:12:15,950 And we said that log z over n actually 172 00:12:15,950 --> 00:12:19,880 does equal a very similar sum. 173 00:12:19,880 --> 00:12:24,310 It is log 2 hyperbolic cosine squared k, 174 00:12:24,310 --> 00:12:26,650 and then I have 1 over n. 175 00:12:31,740 --> 00:12:39,570 Sum over all kinds of loops where 176 00:12:39,570 --> 00:12:46,430 I have a similar diagram that I draw, but I put a star. 177 00:12:49,230 --> 00:12:54,080 And this star implied two things-- 178 00:12:54,080 --> 00:13:00,270 that just like before, I draw all kinds of individual loops, 179 00:13:00,270 --> 00:13:04,560 but I make sure that my loops never 180 00:13:04,560 --> 00:13:07,720 have a step that goes forward and backward. 181 00:13:07,720 --> 00:13:10,930 So there was no U-turn. 182 00:13:10,930 --> 00:13:15,030 And importantly, there was a factor of minus 1 183 00:13:15,030 --> 00:13:18,625 to the number of times that the walk crossed itself. 184 00:13:21,480 --> 00:13:25,490 And we showed that when we incorporate 185 00:13:25,490 --> 00:13:29,890 both of these conditions, the can indeed 186 00:13:29,890 --> 00:13:37,270 exponentiate this expression and get exactly the same diagrams 187 00:13:37,270 --> 00:13:40,100 as we had on the first line, all coming 188 00:13:40,100 --> 00:13:42,240 with the correct weights, and none 189 00:13:42,240 --> 00:13:45,660 of the diagrams that had multiple occurrences of a bond 190 00:13:45,660 --> 00:13:46,160 would occur. 191 00:13:49,410 --> 00:13:52,220 So then the question was, how do you 192 00:13:52,220 --> 00:13:55,590 calculate this given that we have 193 00:13:55,590 --> 00:14:01,860 this dependence on the number of crossings, which offhand may 194 00:14:01,860 --> 00:14:05,910 look as if it is something that requires memory? 195 00:14:05,910 --> 00:14:13,930 And then we saw that, indeed, just like the previous case, 196 00:14:13,930 --> 00:14:23,630 we could write the result as a sum over 197 00:14:23,630 --> 00:14:29,015 walks that have a particular length l. 198 00:14:29,015 --> 00:14:34,430 Right here, we have the factor of t to the l. 199 00:14:34,430 --> 00:14:42,520 Those walks could start and end at the particular point r. 200 00:14:42,520 --> 00:14:47,240 But we also specified the direction mu 201 00:14:47,240 --> 00:14:50,890 along which you started. 202 00:14:50,890 --> 00:14:56,200 So previously I only specified the origin. 203 00:14:56,200 --> 00:15:00,460 Now I have to specify the starting point as well 204 00:15:00,460 --> 00:15:02,670 as the direction. 205 00:15:02,670 --> 00:15:08,660 I have to end at the same point and along the same direction 206 00:15:08,660 --> 00:15:11,430 to complete the loop. 207 00:15:11,430 --> 00:15:18,490 And these were accomplished by having these factors of walks 208 00:15:18,490 --> 00:15:21,990 that are length l. 209 00:15:21,990 --> 00:15:27,635 So to do that, we can certainly incorporate this condition 210 00:15:27,635 --> 00:15:34,780 of no U-turn in the description of the steps that I take. 211 00:15:34,780 --> 00:15:38,020 So for each step, I know where I came from. 212 00:15:38,020 --> 00:15:42,100 I just make sure I don't step back, so that's easy. 213 00:15:42,100 --> 00:15:46,970 And we found that this minus 1 to the power of nc 214 00:15:46,970 --> 00:15:50,880 can be incorporated through a factor of e 215 00:15:50,880 --> 00:15:57,920 to the i sum over the changes of the orientation of the walker-- 216 00:15:57,920 --> 00:16:02,610 as I step through the lattice-- provided that I included also 217 00:16:02,610 --> 00:16:06,090 an additional factor of minus. 218 00:16:06,090 --> 00:16:09,315 So that factor of minus I could actually put out front. 219 00:16:09,315 --> 00:16:10,770 It's an important factor. 220 00:16:13,770 --> 00:16:16,990 And then there's the over-counting, 221 00:16:16,990 --> 00:16:21,950 but as before, the walk can go in either one of two directions 222 00:16:21,950 --> 00:16:25,930 and can have l starting points. 223 00:16:25,930 --> 00:16:31,500 OK, so now we can proceed just as before. 224 00:16:31,500 --> 00:16:35,680 log 2 hyperbolic cosine squared k, 225 00:16:35,680 --> 00:16:44,010 and then we have a sum over l here, 226 00:16:44,010 --> 00:16:49,380 which we can, again, represent as the log of 1 minus 227 00:16:49,380 --> 00:16:53,065 t-- this 4 by 4 matrix t star. 228 00:16:55,590 --> 00:16:59,130 Going through the log operation will change this sign 229 00:16:59,130 --> 00:17:01,670 from minus to plus. 230 00:17:01,670 --> 00:17:06,380 I have the 1 over 2n as before. 231 00:17:06,380 --> 00:17:16,290 And I have to sum over r and mu, which are the elements that 232 00:17:16,290 --> 00:17:19,829 characterize this 4n by 4n matrix. 233 00:17:19,829 --> 00:17:22,369 So this amounts to doing the trace log operation. 234 00:17:27,750 --> 00:17:37,350 And then taking advantage of the fact that, just as before, 235 00:17:37,350 --> 00:17:41,710 Fourier transforms can at least partially block 236 00:17:41,710 --> 00:17:46,030 diagonalize this 4n by 4n matrix. 237 00:17:46,030 --> 00:17:50,260 I go to that basis, and the trace 238 00:17:50,260 --> 00:17:52,200 becomes an integral over q. 239 00:17:55,890 --> 00:17:59,400 And then I would have to do the trace of a log of a 4 240 00:17:59,400 --> 00:18:01,580 by 4 matrix. 241 00:18:01,580 --> 00:18:05,240 And for that, I use the identity that a trace 242 00:18:05,240 --> 00:18:09,060 of log of any matrix is the log of the determinant 243 00:18:09,060 --> 00:18:12,060 of that same matrix. 244 00:18:12,060 --> 00:18:15,390 And so the thing that I have to integrate 245 00:18:15,390 --> 00:18:22,100 is the log of the determinant of a 4 246 00:18:22,100 --> 00:18:26,060 by 4 matrix that captures these steps 247 00:18:26,060 --> 00:18:28,680 that I take on the square lattice. 248 00:18:28,680 --> 00:18:32,800 And we saw that, for example, going 249 00:18:32,800 --> 00:18:37,305 in the horizontal direction would give me a factor of t 250 00:18:37,305 --> 00:18:39,040 e to the minus i qx. 251 00:18:41,670 --> 00:18:50,140 Going in the vertical direction-- up-- y qy. 252 00:18:50,140 --> 00:18:53,085 Going in the horizontal direction-- down. 253 00:19:00,880 --> 00:19:04,170 These are the diagonal elements. 254 00:19:04,170 --> 00:19:08,520 And then there were off-diagonal elements, 255 00:19:08,520 --> 00:19:14,240 so the next turn here was to go and then bend upward. 256 00:19:14,240 --> 00:19:19,600 So that gave me in addition to e to the minus i qx, 257 00:19:19,600 --> 00:19:22,520 which is the same forward step here. 258 00:19:22,520 --> 00:19:25,900 A factor of, let's call it omega so I 259 00:19:25,900 --> 00:19:28,335 don't have to write it all over the place. 260 00:19:28,335 --> 00:19:33,880 Omega is e to the pi pi over 4. 261 00:19:33,880 --> 00:19:38,080 And the next one was a U-turn. 262 00:19:38,080 --> 00:19:45,130 The next one was minus t e minus i qx omega star. 263 00:19:45,130 --> 00:19:50,840 And we could fill out similarly all of the other places 264 00:19:50,840 --> 00:19:53,560 in this 4 by 4 matrix. 265 00:19:53,560 --> 00:19:59,470 And then the whole problem comes down to having to evaluate a 4 266 00:19:59,470 --> 00:20:01,560 by 4 determinant, which you can do 267 00:20:01,560 --> 00:20:07,620 by hand with a couple of sheets of paper to do your algebra. 268 00:20:07,620 --> 00:20:11,520 And I wrote for you the final answer, 269 00:20:11,520 --> 00:20:16,660 is 1/2 2 integrals from minus pi 2 pi 270 00:20:16,660 --> 00:20:22,540 over qx and qy divided by 2 pi squared. 271 00:20:22,540 --> 00:20:29,030 And then the result of this, which is log of 1 272 00:20:29,030 --> 00:20:37,180 plus t squared squared minus 2t 1 minus t squared cosine of qx 273 00:20:37,180 --> 00:20:41,740 plus cosine of qy. 274 00:20:41,740 --> 00:20:47,410 This was the expression for the partition function. 275 00:20:57,460 --> 00:21:03,560 OK, so this is where we ended up last time. 276 00:21:03,560 --> 00:21:08,740 Now the question is we have here on the board 277 00:21:08,740 --> 00:21:19,300 two expressions, the project one and the incorrect one-- 278 00:21:19,300 --> 00:21:25,470 the Gaussian model and the 2-dimensionalizing one. 279 00:21:28,330 --> 00:21:33,780 They look surprisingly similar, and that 280 00:21:33,780 --> 00:21:39,500 should start to worry us potentially because we expect 281 00:21:39,500 --> 00:21:41,710 that many when we have some functional 282 00:21:41,710 --> 00:21:45,230 form, that functional form carries 283 00:21:45,230 --> 00:21:48,990 within it certain singularities. 284 00:21:48,990 --> 00:21:52,210 And you say, well, these two functions, both of them 285 00:21:52,210 --> 00:21:56,220 are a double integral of log of something 286 00:21:56,220 --> 00:21:58,080 minus something cosine plus cosine. 287 00:22:00,690 --> 00:22:04,870 So after all of this work, did we 288 00:22:04,870 --> 00:22:08,450 end up with an expression that has the same singular 289 00:22:08,450 --> 00:22:12,500 behavior as the Gaussian model? 290 00:22:12,500 --> 00:22:18,160 OK, so let's go and look at things more carefully. 291 00:22:18,160 --> 00:22:25,120 So in both cases, what I need to do is to integrate A function 292 00:22:25,120 --> 00:22:30,600 a that appears inside the log. 293 00:22:30,600 --> 00:22:32,905 There is an A for the Gaussian, and then there 294 00:22:32,905 --> 00:22:38,000 is this object-- let's call it A star-- 295 00:22:38,000 --> 00:22:42,090 for the correct solution. 296 00:22:42,090 --> 00:22:47,710 So the thing that I have to integrate is, of course, q. 297 00:22:47,710 --> 00:22:51,160 So this is a function of the vector q, 298 00:22:51,160 --> 00:22:54,730 as well as the parameter that is a function of which I expect 299 00:22:54,730 --> 00:22:57,220 to have a phase transition, which is t. 300 00:23:01,930 --> 00:23:08,580 Now where could I potentially get some kind of a singularity? 301 00:23:08,580 --> 00:23:12,610 The only place that I can get a singularity 302 00:23:12,610 --> 00:23:19,400 is if the argument of the log goes to 0 because log of 0 303 00:23:19,400 --> 00:23:23,380 is something that's derivatives of singularities, et cetera. 304 00:23:23,380 --> 00:23:28,180 So you may say, OK, that's where I should be looking at. 305 00:23:28,180 --> 00:23:33,770 So where is it this most likely to happen 306 00:23:33,770 --> 00:23:36,080 when I'm integrating over q? 307 00:23:36,080 --> 00:23:40,450 So basically, I'm integrating over qx and qy 308 00:23:40,450 --> 00:23:41,980 over a [INAUDIBLE] [? zone ?] that 309 00:23:41,980 --> 00:23:48,110 goes from minus pi to pi in both directions. 310 00:23:48,110 --> 00:23:53,490 And potentially somewhere in this I encounter a singularity. 311 00:23:53,490 --> 00:23:57,030 Let's come from the site of high temperatures 312 00:23:57,030 --> 00:23:59,940 where t is close to 0. 313 00:23:59,940 --> 00:24:02,490 Then I have log of 1, no problem. 314 00:24:02,490 --> 00:24:06,410 As I go to lower and lower temperatures, 315 00:24:06,410 --> 00:24:08,320 the t becomes larger. 316 00:24:08,320 --> 00:24:12,100 Then from the 1, I start to subtract more and more 317 00:24:12,100 --> 00:24:14,050 with these cosines. 318 00:24:14,050 --> 00:24:16,300 And clearly the place that I'm subtracting 319 00:24:16,300 --> 00:24:20,110 most is right at the center of q equals to 0. 320 00:24:23,050 --> 00:24:27,120 So let's expand this in the vicinity of q 321 00:24:27,120 --> 00:24:32,220 goes to 0 in the vicinity of this place. 322 00:24:32,220 --> 00:24:37,200 And there what I see is it is 1 minus. 323 00:24:37,200 --> 00:24:41,470 Cosines are approximately 1 minus q squared over 2. 324 00:24:41,470 --> 00:24:45,910 So I have 1 minus 4t, and then I have 325 00:24:45,910 --> 00:24:51,070 plus t qx squared plus qy squared, which is essentially 326 00:24:51,070 --> 00:24:52,590 the net q squared. 327 00:24:52,590 --> 00:24:56,510 And then I have order of higher power z, qx and qy. 328 00:25:00,160 --> 00:25:01,700 Fine. 329 00:25:01,700 --> 00:25:06,360 So this part is positive, no problem. 330 00:25:06,360 --> 00:25:11,970 I see that this part goes through 0 331 00:25:11,970 --> 00:25:15,660 when I hit tc, which is 1/4. 332 00:25:15,660 --> 00:25:18,300 And this we had already seen, that basically this 333 00:25:18,300 --> 00:25:22,930 is the place where the exponentially increasing 334 00:25:22,930 --> 00:25:26,780 number of walks-- as 4 to the number of steps-- 335 00:25:26,780 --> 00:25:29,990 overcomes the exponentially decreasing 336 00:25:29,990 --> 00:25:33,510 fidelity of information carried through each walk, which 337 00:25:33,510 --> 00:25:35,460 was t to the l. 338 00:25:35,460 --> 00:25:40,470 So 4dt being 1, tc is of the order 1/4. 339 00:25:40,470 --> 00:25:42,600 We are interested in the singularities 340 00:25:42,600 --> 00:25:45,790 in the vicinity of this phase transition, 341 00:25:45,790 --> 00:25:50,780 so we additionally go and look at what 342 00:25:50,780 --> 00:25:57,160 happens when t approaches tc, but from this side above, 343 00:25:57,160 --> 00:26:01,440 because clearly if I go to t that is larger than 1/4, 344 00:26:01,440 --> 00:26:03,620 it doesn't make any sense. 345 00:26:03,620 --> 00:26:08,290 So t has to be less than 1/4. 346 00:26:08,290 --> 00:26:15,795 And so then what I have here is that I can write this as 4tc, 347 00:26:15,795 --> 00:26:21,490 so this is 4 times tc minus t. 348 00:26:21,490 --> 00:26:25,543 And this to the lowest order I can replace as tc-- q 349 00:26:25,543 --> 00:26:28,500 squared plus higher orders. 350 00:26:28,500 --> 00:26:32,650 This 4 I can write as 1 over tc. 351 00:26:32,650 --> 00:26:36,790 So the whole thing I can write as q 352 00:26:36,790 --> 00:26:48,852 squared plus delta t divided by tc, and there's a factor of 4. 353 00:26:53,190 --> 00:27:01,200 And delta t I have defined to be tc minus 10. 354 00:27:01,200 --> 00:27:05,650 How close I am to the location where this singularity takes 355 00:27:05,650 --> 00:27:08,030 places. 356 00:27:08,030 --> 00:27:12,830 So what I'm interested is not in the whole form 357 00:27:12,830 --> 00:27:15,400 of this function, but only the singularities 358 00:27:15,400 --> 00:27:17,150 that it expresses. 359 00:27:17,150 --> 00:27:27,230 So I focus on the singular part of this Gaussian expression. 360 00:27:27,230 --> 00:27:30,230 I don't have to worry about that term. 361 00:27:30,230 --> 00:27:31,670 So I have minus 1/2. 362 00:27:34,370 --> 00:27:37,180 I have double integral. 363 00:27:37,180 --> 00:27:41,760 The argument of the log, I expanded 364 00:27:41,760 --> 00:27:43,900 in the vicinity of the point that I 365 00:27:43,900 --> 00:27:47,320 see a singularity to take place. 366 00:27:47,320 --> 00:27:54,320 And I'll write the answer as q squared plus 4 delta t over tc. 367 00:27:54,320 --> 00:27:58,540 If I am sufficiently close in my integration to the origin 368 00:27:58,540 --> 00:28:04,430 so that the expansion in q is acceptable, 369 00:28:04,430 --> 00:28:08,140 there is an additional factor of tc. 370 00:28:08,140 --> 00:28:11,390 But if I take a log of tc, it's just a constant. 371 00:28:11,390 --> 00:28:13,660 I can integrate that out. 372 00:28:13,660 --> 00:28:17,376 It's going to not contribute to the singular part, just 373 00:28:17,376 --> 00:28:21,630 an additional regular component. 374 00:28:21,630 --> 00:28:25,060 Now if I am in the vicinity of q equals to 0 375 00:28:25,060 --> 00:28:30,440 where all of the action is, at this order, the thing 376 00:28:30,440 --> 00:28:34,050 that I'm integrating has circular symmetry so 377 00:28:34,050 --> 00:28:36,050 that 2-dimensional integration. 378 00:28:36,050 --> 00:28:39,250 I can write whether or not it's d qx, d qy 379 00:28:39,250 --> 00:28:46,325 as 2 pi q dq divided by 2 pi squared, 380 00:28:46,325 --> 00:28:47,740 which was the density of state. 381 00:28:50,960 --> 00:28:57,472 And this approximation of a thing being 382 00:28:57,472 --> 00:29:01,245 isotropic and circular only holds 383 00:29:01,245 --> 00:29:04,020 when I'm sufficiently close to the origin, 384 00:29:04,020 --> 00:29:08,730 let's say up to some value that I will call lambda. 385 00:29:08,730 --> 00:29:11,450 So I will impose some cut-off here, 386 00:29:11,450 --> 00:29:15,210 lambda, which is certainly less than one 387 00:29:15,210 --> 00:29:17,795 of the order of pi-- let's say pi over 10. 388 00:29:17,795 --> 00:29:19,550 It doesn't matter what it is. 389 00:29:19,550 --> 00:29:21,285 As we will see, the singular part, 390 00:29:21,285 --> 00:29:25,000 it ultimately does not matter what I put for lambda. 391 00:29:25,000 --> 00:29:28,820 But the rest of the integration that I haven't exclusively 392 00:29:28,820 --> 00:29:31,890 written down from all of this-- again, 393 00:29:31,890 --> 00:29:37,980 in analogy to what we had seen before for the Landau-Ginzburg 394 00:29:37,980 --> 00:29:39,880 calculation will give me something 395 00:29:39,880 --> 00:29:42,100 that is perfectly analytic. 396 00:29:42,100 --> 00:29:47,860 So I have extracted from this expression the singular part. 397 00:29:47,860 --> 00:29:52,590 OK, now let's do this integral carefully. 398 00:29:52,590 --> 00:29:54,914 So what do I have? 399 00:29:54,914 --> 00:29:59,440 I have 1 over 2 pi with minus 1/2, so I have minus 1 400 00:29:59,440 --> 00:30:02,690 over 4 pi. 401 00:30:02,690 --> 00:30:09,360 If I call this whole object here x-- so x is q 402 00:30:09,360 --> 00:30:11,760 squared plus this something-- then 403 00:30:11,760 --> 00:30:16,210 we can see that dx is 2 q dq. 404 00:30:19,280 --> 00:30:23,420 So what I have to do is the integral of dx log x, 405 00:30:23,420 --> 00:30:26,460 which is x log x minus x. 406 00:30:26,460 --> 00:30:35,870 So essentially, let's keep the 2 up there and make this 8 pi. 407 00:30:35,870 --> 00:30:50,570 And then what I have is x log x minus x itself, 408 00:30:50,570 --> 00:30:55,570 which I can write as log of e. 409 00:30:55,570 --> 00:30:59,416 And then this whole thing has to be evaluated between the two 410 00:30:59,416 --> 00:31:01,773 limits of integration, lambda and [INAUDIBLE]. 411 00:31:05,370 --> 00:31:09,795 Now you explicitly see that if I substitute 412 00:31:09,795 --> 00:31:14,900 in this expression for q the upper part of lambda, 413 00:31:14,900 --> 00:31:17,270 it will give me something like lambda 414 00:31:17,270 --> 00:31:20,480 squared plus delta t-- an expandable 415 00:31:20,480 --> 00:31:22,140 and analytical function. 416 00:31:22,140 --> 00:31:26,330 Log of a constant plus delta t that I can start analytically 417 00:31:26,330 --> 00:31:27,550 expanding. 418 00:31:27,550 --> 00:31:29,880 So anything that I get from the other cut-off 419 00:31:29,880 --> 00:31:33,660 of the integration is perfectly analytic. 420 00:31:33,660 --> 00:31:35,410 I don't have to worry about it. 421 00:31:35,410 --> 00:31:39,980 If I'm interested in the singular part, 422 00:31:39,980 --> 00:31:44,760 I basically need to evaluate this as it's lower cut-off. 423 00:31:44,760 --> 00:31:47,040 So I evaluate it at q equals to 0. 424 00:31:47,040 --> 00:31:50,010 Well first off all, I will get a sign change 425 00:31:50,010 --> 00:31:53,560 because I'm at the lower cut-off. 426 00:31:53,560 --> 00:31:59,420 I will get from here 4 delta t over tc. 427 00:31:59,420 --> 00:32:04,155 And from here, I will get log of a bunch of constants. 428 00:32:04,155 --> 00:32:05,510 It doesn't really matter. 429 00:32:05,510 --> 00:32:08,438 4 over e delta t over tc. 430 00:32:14,800 --> 00:32:18,720 What is the leading singularity? 431 00:32:18,720 --> 00:32:24,010 Is delta t log t-- log delta t? 432 00:32:24,010 --> 00:32:27,146 And so again, the leading singularity 433 00:32:27,146 --> 00:32:30,830 is delta t log of delta t. 434 00:32:30,830 --> 00:32:35,685 There's an overall factor of 1 over pi. [? Doesn't match. ?] 435 00:32:35,685 --> 00:32:37,660 You take two derivatives. 436 00:32:37,660 --> 00:32:39,940 You find that the heat capacity, let's 437 00:32:39,940 --> 00:32:43,655 say that is proportion to two derivatives of log 438 00:32:43,655 --> 00:32:47,340 z by delta t squared. 439 00:32:47,340 --> 00:32:50,340 You take one derivative, it goes like the log. 440 00:32:50,340 --> 00:32:52,406 You take another derivative of the log, 441 00:32:52,406 --> 00:32:57,600 you find that the singularity is 1 over delta t. 442 00:32:57,600 --> 00:33:01,290 That corresponds to a heat capacity divergence 443 00:33:01,290 --> 00:33:04,460 with an exponent of unity, which is 444 00:33:04,460 --> 00:33:07,580 quite consistent with the generally Gaussian formula 445 00:33:07,580 --> 00:33:12,085 that we had in d dimensions, which was 2 minus t over 2. 446 00:33:17,690 --> 00:33:20,340 So that's the Gaussian. 447 00:33:20,340 --> 00:33:30,860 And of course, this whole theory breaks down 448 00:33:30,860 --> 00:33:35,430 for t that is greater than tc. 449 00:33:35,430 --> 00:33:38,000 Once I go beyond tc, my expressions 450 00:33:38,000 --> 00:33:39,310 just don't make sense. 451 00:33:39,310 --> 00:33:43,480 I can't integrate a lot of a negative number. 452 00:33:43,480 --> 00:33:46,300 And we understand why that is. 453 00:33:46,300 --> 00:33:49,690 That's because we are including all of these loops that 454 00:33:49,690 --> 00:33:52,480 can go over each other multiple times. 455 00:33:52,480 --> 00:33:55,640 The whole theory does not make sense. 456 00:33:55,640 --> 00:33:57,640 So we did this one to death. 457 00:34:00,830 --> 00:34:05,100 Will the exact result be any different? 458 00:34:05,100 --> 00:34:08,940 So let's carry out the corresponding procedure-- 459 00:34:08,940 --> 00:34:12,809 for A star that is a function of q and t. 460 00:34:15,460 --> 00:34:20,889 And again, singularities should come from the place 461 00:34:20,889 --> 00:34:23,960 where this is most likely to go to 0. 462 00:34:23,960 --> 00:34:27,650 You can see it's 1 something minus something. 463 00:34:27,650 --> 00:34:30,190 And clearly when the q's are close to 0 464 00:34:30,190 --> 00:34:32,719 is when you subtract most, and you're 465 00:34:32,719 --> 00:34:34,780 likely to become negative. 466 00:34:34,780 --> 00:34:38,000 So let's expand it around 0. 467 00:34:38,000 --> 00:34:44,960 This as q goes to 0 is 1 plus t squared squared. 468 00:34:44,960 --> 00:34:47,730 And then I have minus. 469 00:34:47,730 --> 00:34:51,260 Each one of the cosines starts at unity, 470 00:34:51,260 --> 00:34:57,330 so I will have 4t 1 minus t squared. 471 00:34:57,330 --> 00:35:02,090 And then from the qx squared over 2 qy squared over 2, 472 00:35:02,090 --> 00:35:06,330 I will get a plus t 1 minus t squared 473 00:35:06,330 --> 00:35:10,520 q squared-- qx squared plus qy squared-- plus order of q 474 00:35:10,520 --> 00:35:11,498 to the fourth. 475 00:35:20,310 --> 00:35:26,560 So the way that we identified the location of the singular 476 00:35:26,560 --> 00:35:32,125 part before was to focus on exactly q equals to 0. 477 00:35:32,125 --> 00:35:37,610 And what we find is that A star at q equals to 0 478 00:35:37,610 --> 00:35:41,660 is essentially this part. 479 00:35:41,660 --> 00:35:46,270 This part I'm going to rewrite the first end slightly. 480 00:35:46,270 --> 00:35:48,860 1 plus t squared squared is the same thing 481 00:35:48,860 --> 00:35:54,160 as 1 minus t squared squared plus 4t squared. 482 00:35:54,160 --> 00:35:56,020 The difference between the expansion 483 00:35:56,020 --> 00:35:57,970 of each one of these two terms is 484 00:35:57,970 --> 00:36:01,940 that this has plus 2t squared, this has minus 2t squared, 485 00:36:01,940 --> 00:36:04,200 which I have added here. 486 00:36:04,200 --> 00:36:09,600 And then minus 4t 1 minus t squared. 487 00:36:09,600 --> 00:36:12,240 And the reason that I did that is 488 00:36:12,240 --> 00:36:16,530 that you can now see that this term is twice 489 00:36:16,530 --> 00:36:20,230 this times this when I take the square. 490 00:36:20,230 --> 00:36:24,710 So the whole thing is the same thing as 1 minus t squared 491 00:36:24,710 --> 00:36:26,470 minus 2t squared. 492 00:36:31,270 --> 00:36:36,720 So the first thing that gives us reassurance happens now, 493 00:36:36,720 --> 00:36:40,600 whereas previously for the Gaussian model 1 minus 4t could 494 00:36:40,600 --> 00:36:44,470 be both positive and negative, this you can see 495 00:36:44,470 --> 00:36:45,525 is always positive. 496 00:36:48,150 --> 00:36:50,540 So there is no problem with me not 497 00:36:50,540 --> 00:36:53,170 being able to go from one side of the phase transition 498 00:36:53,170 --> 00:36:55,840 to another side of the phase transition. 499 00:36:55,840 --> 00:37:00,270 This expression will encounter no difficulties. 500 00:37:00,270 --> 00:37:05,670 But there is a special point when this thing is 0. 501 00:37:05,670 --> 00:37:13,200 So there is a point when 1 minus 2t c plus tc squared 502 00:37:13,200 --> 00:37:14,330 is equal to 0. 503 00:37:14,330 --> 00:37:17,210 The whole thing goes to 0. 504 00:37:17,210 --> 00:37:20,770 And you can figure out where that is. 505 00:37:20,770 --> 00:37:22,710 tc is 1. 506 00:37:22,710 --> 00:37:24,070 It's a quadratic form. 507 00:37:24,070 --> 00:37:30,560 It has two solutions-- 1 minus plus square root of 2. 508 00:37:30,560 --> 00:37:32,365 A negative solution is not acceptable. 509 00:37:41,131 --> 00:37:41,630 Minus. 510 00:37:45,244 --> 00:37:50,780 I have to recast this slightly. 511 00:37:50,780 --> 00:37:54,150 Knowing the answer sometimes makes you go too fast. 512 00:37:54,150 --> 00:38:00,210 tc squared plus 2tc minus 1 equals to 0. 513 00:38:00,210 --> 00:38:05,862 tc is minus 1 plus or minus square root of 2. 514 00:38:05,862 --> 00:38:09,110 The minus solution is not acceptable. 515 00:38:09,110 --> 00:38:12,630 The plus solution would give me root 2 minus 1. 516 00:38:15,860 --> 00:38:20,280 Just to remind you, we calculated a value 517 00:38:20,280 --> 00:38:23,900 for the critical point based on duality. 518 00:38:23,900 --> 00:38:28,520 So let's just recap that duality argument. 519 00:38:28,520 --> 00:38:32,250 We saw that the series that we had calculated, 520 00:38:32,250 --> 00:38:38,510 which was an expansion in high temperatures times k, 521 00:38:38,510 --> 00:38:42,980 reproduced the expansion that we had for 0 temperature, 522 00:38:42,980 --> 00:38:46,500 including islands of minus in a sea of plus, 523 00:38:46,500 --> 00:38:49,690 where the contribution of each bond was going from e to the k 524 00:38:49,690 --> 00:38:51,710 to e to the minus k. 525 00:38:51,710 --> 00:38:59,530 So there was a correspondence between a dual coupling. 526 00:38:59,530 --> 00:39:04,600 And the actual coupling, that was like this. 527 00:39:04,600 --> 00:39:08,750 At a critical point, we said the two of them 528 00:39:08,750 --> 00:39:11,470 have to be the same. 529 00:39:11,470 --> 00:39:14,900 And what we had calculated based on that, 530 00:39:14,900 --> 00:39:18,260 since the hyperbolic tang I can write in terms 531 00:39:18,260 --> 00:39:23,290 of the exponentials was that the value of e to the minus 532 00:39:23,290 --> 00:39:28,760 e to the plus 2kc was, in fact, square root of 2 plus 1. 533 00:39:31,630 --> 00:39:37,730 And the inverse of this will be square root of 2 minus 1, 534 00:39:37,730 --> 00:39:39,360 and that's the same as the tang. 535 00:39:39,360 --> 00:39:40,837 That's the same thing as the things 536 00:39:40,837 --> 00:39:43,380 that we have already written. 537 00:39:43,380 --> 00:39:47,450 So the calculation that we had done 538 00:39:47,450 --> 00:39:52,740 before-- obtain this critical temperature based 539 00:39:52,740 --> 00:39:56,390 on this Kramers-Wannier duality-- 540 00:39:56,390 --> 00:40:00,890 gave us a critical point here, which is precisely the place 541 00:40:00,890 --> 00:40:04,705 that we can identify as the origin of the singularity 542 00:40:04,705 --> 00:40:06,020 in this expression. 543 00:40:10,070 --> 00:40:11,475 So what did we do next? 544 00:40:11,475 --> 00:40:16,340 The next thing that we did was having identified up there 545 00:40:16,340 --> 00:40:19,530 where the critical point was-- which was at 14-- 546 00:40:19,530 --> 00:40:23,250 was to expand our A, the integrand, 547 00:40:23,250 --> 00:40:27,670 inside the log in the vicinity of that point. 548 00:40:27,670 --> 00:40:30,480 So what I want to do is similarly expand 549 00:40:30,480 --> 00:40:34,650 a star of q for t that goes into vicinity of tc. 550 00:40:38,410 --> 00:40:41,950 And what do I have to do? 551 00:40:41,950 --> 00:40:59,010 So what I can do, let's write it as t is tc plus delta t 552 00:40:59,010 --> 00:41:02,520 and make an expansion for delta t small. 553 00:41:07,760 --> 00:41:13,060 So I have to make an expansion or delta t small, first 554 00:41:13,060 --> 00:41:14,380 of all, of this quantity. 555 00:41:18,290 --> 00:41:23,916 So if I make a small change in t, 556 00:41:23,916 --> 00:41:26,650 I'll have to take a derivative inside here. 557 00:41:26,650 --> 00:41:31,780 So I have minus 2t minus 2, evaluate it at tc, 558 00:41:31,780 --> 00:41:33,810 times the change in delta t. 559 00:41:36,610 --> 00:41:42,260 So that's the change of this expression 560 00:41:42,260 --> 00:41:46,640 if I go slightly away from the point where it is 0. 561 00:41:46,640 --> 00:41:47,314 Yes? 562 00:41:47,314 --> 00:41:48,480 AUDIENCE: I have a question. 563 00:41:48,480 --> 00:41:53,300 T here is [? tangent ?] of k where we calculated 564 00:41:53,300 --> 00:41:59,020 the [INAUDIBLE] to [? be as ?] temperature. 565 00:41:59,020 --> 00:42:01,850 PROFESSOR: Now all of these things 566 00:42:01,850 --> 00:42:05,000 are analytical functions of each other. 567 00:42:05,000 --> 00:42:12,790 So the k is, in fact, some unit of energy divided by kt. 568 00:42:12,790 --> 00:42:15,610 So Really, the temperature is here. 569 00:42:15,610 --> 00:42:19,800 And my t is tang of the above objects, 570 00:42:19,800 --> 00:42:24,260 so it's tang of j divided by kt. 571 00:42:24,260 --> 00:42:26,980 So the point is that whenever I look 572 00:42:26,980 --> 00:42:31,020 at the delta in temperature, I can translate 573 00:42:31,020 --> 00:42:35,440 that delta in temperature to a delta int times the value 574 00:42:35,440 --> 00:42:38,370 of the derivative at the location of this, 575 00:42:38,370 --> 00:42:40,330 which is some finite number. 576 00:42:40,330 --> 00:42:44,160 Of basically up to some constant, 577 00:42:44,160 --> 00:42:46,430 taking derivatives with respect to temperature, 578 00:42:46,430 --> 00:42:49,130 with respect to k, with respect to tang k, 579 00:42:49,130 --> 00:42:51,420 with respect to beta. 580 00:42:51,420 --> 00:42:55,190 Evaluate it at the finite temperature, 581 00:42:55,190 --> 00:42:57,500 which is the location of the critical point. 582 00:42:57,500 --> 00:43:00,980 They're all the same up to proportionality constants, 583 00:43:00,980 --> 00:43:04,300 and that's why I wrote "proportionality" there. 584 00:43:04,300 --> 00:43:07,000 One thing that you have to make sure-- 585 00:43:07,000 --> 00:43:10,930 and I spent actually half an hour this morning checking-- 586 00:43:10,930 --> 00:43:14,060 is that the signs work out fine. 587 00:43:14,060 --> 00:43:17,120 So I didn't want to write an expression for the heat 588 00:43:17,120 --> 00:43:19,350 capacity that was negative. 589 00:43:19,350 --> 00:43:21,710 So proportionalities aside, that's 590 00:43:21,710 --> 00:43:23,530 the one thing that I better have, 591 00:43:23,530 --> 00:43:26,174 is that the sign of the heat capacity, that is positive. 592 00:43:34,100 --> 00:43:39,750 So that's the expansion of the term that corresponds to q 593 00:43:39,750 --> 00:43:41,990 equals to 0. 594 00:43:41,990 --> 00:43:44,360 The term that was proportional to q squared, 595 00:43:44,360 --> 00:43:46,740 look at what we did in the above expression 596 00:43:46,740 --> 00:43:48,660 for the Gaussian model. 597 00:43:48,660 --> 00:43:52,980 Since it was lower order, it was already 598 00:43:52,980 --> 00:43:55,100 proportional to q squared. 599 00:43:55,100 --> 00:43:58,420 We evaluated it at exactly t equals to tc. 600 00:43:58,420 --> 00:44:04,750 So I will put here tc 1 minus tc squared q squared, 601 00:44:04,750 --> 00:44:06,250 and then I will have high orders. 602 00:44:10,920 --> 00:44:14,580 Now fortunately, we have a value for tc 603 00:44:14,580 --> 00:44:18,800 that we can substitute in a couple of places here. 604 00:44:18,800 --> 00:44:24,420 You can see that this is, in fact, twice tc plus 1, 605 00:44:24,420 --> 00:44:27,000 and tc plus 1 is root 2. 606 00:44:27,000 --> 00:44:29,650 So this is minus 2 root 2. 607 00:44:29,650 --> 00:44:32,320 I square that, and this whole thing 608 00:44:32,320 --> 00:44:35,425 becomes 8 delta t squared. 609 00:44:39,740 --> 00:44:49,790 This object here, 1 minus tc squared-- 610 00:44:49,790 --> 00:44:53,690 you can see if I put the 2 tc on the other side-- 1 minus tc 611 00:44:53,690 --> 00:44:55,620 squared' is the same thing as 2 tc. 612 00:45:00,460 --> 00:45:06,400 So I can write the whole thing here as 2 tc squared q squared. 613 00:45:09,530 --> 00:45:12,720 And the reason I do that is, like 614 00:45:12,720 --> 00:45:14,620 before, there's an overall factor 615 00:45:14,620 --> 00:45:18,690 that I can take out of this parentheses. 616 00:45:18,690 --> 00:45:22,760 And the answer will be q squared now 617 00:45:22,760 --> 00:45:31,210 plus 4 delta t over tc squared. 618 00:45:31,210 --> 00:45:34,130 It's very similar to what we had before, 619 00:45:34,130 --> 00:45:37,640 except that when we had q squared plus 4 delta t over tc, 620 00:45:37,640 --> 00:45:42,480 we have q squared plus 4 delta t over tc squared. 621 00:45:42,480 --> 00:45:45,700 Now this square was very important for allowing 622 00:45:45,700 --> 00:45:48,320 us to go for both positive and negative, 623 00:45:48,320 --> 00:45:52,720 but let's see its consequence on the singularity. 624 00:45:52,720 --> 00:45:59,190 So now log z of the correct form divided by n-- the singular 625 00:45:59,190 --> 00:46:04,520 part-- and we calculate it just as before. 626 00:46:04,520 --> 00:46:08,100 First of all, rather than minus 1/2 I have a plus 1/2. 627 00:46:11,020 --> 00:46:16,360 I have the same integral, which in the vicinity of the origin 628 00:46:16,360 --> 00:46:21,700 is symmetric, so I will write it as 2 pi qd 629 00:46:21,700 --> 00:46:27,110 q divided by 4 pi squared. 630 00:46:27,110 --> 00:46:29,300 And then I have log. 631 00:46:29,300 --> 00:46:33,160 I will forget about this factor for consideration 632 00:46:33,160 --> 00:46:38,245 of singularities 4 delta t over tc squared. 633 00:46:42,430 --> 00:46:50,010 And now again, it is exactly the same structure 634 00:46:50,010 --> 00:46:55,410 as x dx that I had before. 635 00:46:55,410 --> 00:46:57,500 So it's the same integral. 636 00:46:57,500 --> 00:47:11,470 And what you will find is that I evaluate it as 1/8 pi integral, 637 00:47:11,470 --> 00:47:19,820 essentially q squared plus 4 delta t over tc squared, 638 00:47:19,820 --> 00:47:26,700 log of q squared plus 4 delta t over tc squared 639 00:47:26,700 --> 00:47:32,140 over e evaluated between 0 and lambda. 640 00:47:32,140 --> 00:47:37,760 And the only singularity comes from the evaluation 641 00:47:37,760 --> 00:47:38,890 that we have at the origin. 642 00:47:43,330 --> 00:47:45,455 And so that I will get a factor of minus. 643 00:47:45,455 --> 00:47:48,980 So I will get 1/8 pi. 644 00:47:48,980 --> 00:47:52,620 Actually then, I substitute this factor. 645 00:47:52,620 --> 00:47:56,060 The 4 and the 8 will give me 2. 646 00:47:56,060 --> 00:47:59,510 And then I evaluate the log of delta t squared, 647 00:47:59,510 --> 00:48:01,820 so that's another factor of 2. 648 00:48:01,820 --> 00:48:05,760 So actually, only one factor of pi survives. 649 00:48:05,760 --> 00:48:12,710 I will have delta t over tc squared log of, let's say, 650 00:48:12,710 --> 00:48:15,033 absolute value of delta t over tc. 651 00:48:20,220 --> 00:48:24,420 So the only thing that changed was that whereas I 652 00:48:24,420 --> 00:48:29,160 had the linear term sitting in front of the log, now 653 00:48:29,160 --> 00:48:32,000 have a quadratic term. 654 00:48:32,000 --> 00:48:35,780 But now when I take two derivatives, and now 655 00:48:35,780 --> 00:48:39,900 we are sure that taking derivatives does not really 656 00:48:39,900 --> 00:48:44,230 matter whether I'm doing it with respect to temperature or delta 657 00:48:44,230 --> 00:48:45,301 t or any other variable. 658 00:48:48,250 --> 00:48:50,250 You can see that the leading behavior 659 00:48:50,250 --> 00:48:53,040 will come taking two derivatives out here 660 00:48:53,040 --> 00:48:56,170 and will be proportional to the log. 661 00:48:56,170 --> 00:49:03,410 So I will get minus 1 over pi log of delta t over tc. 662 00:49:07,640 --> 00:49:15,790 So that if I were to plug the heat capacity of the system, 663 00:49:15,790 --> 00:49:18,040 as a function of, let's say, this parameter t-- 664 00:49:18,040 --> 00:49:23,550 which is also something that stands for temperature-- but t 665 00:49:23,550 --> 00:49:26,890 goes between, say, 0 and 1. 666 00:49:26,890 --> 00:49:28,900 It's a hyperbolic tangent. 667 00:49:28,900 --> 00:49:33,830 There's a location which is this tc, which is root 2 minus 1. 668 00:49:36,590 --> 00:49:40,290 And the singular part of the heat capacity-- 669 00:49:40,290 --> 00:49:42,780 there will be some other part of the heat capacity 670 00:49:42,780 --> 00:49:46,160 that is regular-- but the singular part we see 671 00:49:46,160 --> 00:49:47,690 has a [INAUDIBLE] divergence. 672 00:49:55,800 --> 00:50:00,990 And furthermore, you can see that the amplitudes-- so 673 00:50:00,990 --> 00:50:03,460 essentially this goes approaching 674 00:50:03,460 --> 00:50:08,040 from different sides of the transition, A plus or A minus 675 00:50:08,040 --> 00:50:12,260 log of absolute value of delta a. 676 00:50:12,260 --> 00:50:15,280 And the ratio of the amplitudes, which we have also 677 00:50:15,280 --> 00:50:18,350 said is universal, is equal to 1. 678 00:50:18,350 --> 00:50:20,172 And you had anticipated that based 679 00:50:20,172 --> 00:50:21,380 on [? duality ?] [INAUDIBLE]. 680 00:50:37,600 --> 00:50:45,130 All right, so indeed, there is a different behavior 681 00:50:45,130 --> 00:50:46,785 between the two models. 682 00:50:46,785 --> 00:50:49,730 The exact solution allows us to go both above 683 00:50:49,730 --> 00:50:53,700 and below, has this logarithmic singularity. 684 00:50:53,700 --> 00:51:00,010 And this expression was first written down-- well, 685 00:51:00,010 --> 00:51:07,350 first published Onsager in 1944. 686 00:51:07,350 --> 00:51:10,870 Even couple of years before that, 687 00:51:10,870 --> 00:51:14,830 he had written the expression on boards of various conferences, 688 00:51:14,830 --> 00:51:17,270 saying that this is the answer, but he 689 00:51:17,270 --> 00:51:19,900 didn't publish the paper. 690 00:51:19,900 --> 00:51:24,960 The way that he did it is based on the transfer matrix network, 691 00:51:24,960 --> 00:51:26,680 as we said. 692 00:51:26,680 --> 00:51:32,590 Basically, we can imagine that we 693 00:51:32,590 --> 00:51:37,190 have a lattice that is, let's say, I parallel 694 00:51:37,190 --> 00:51:39,250 in one direction, l perpendicular 695 00:51:39,250 --> 00:51:41,390 in the other direction. 696 00:51:41,390 --> 00:51:46,460 And then for the problems that you had to do, 697 00:51:46,460 --> 00:51:51,850 the transform matrix for one dimensional model, 698 00:51:51,850 --> 00:51:54,340 we can easy to do it for a [? ladder ?]. 699 00:51:54,340 --> 00:51:56,820 It's a 4 by 4. 700 00:51:56,820 --> 00:51:59,860 For this, it becomes a 2 to the l by 2 to the l matrix. 701 00:52:05,850 --> 00:52:08,140 And of course, you are interested in the limit where 702 00:52:08,140 --> 00:52:13,000 l goes to infinity so that you can come 2-dimensional. 703 00:52:13,000 --> 00:52:18,840 And so he was able to sort of look at this structure 704 00:52:18,840 --> 00:52:23,580 of this matrix, recognize that the elements of this matrix 705 00:52:23,580 --> 00:52:26,980 could be represented in terms of other matrices 706 00:52:26,980 --> 00:52:29,310 that had some interesting algebra. 707 00:52:29,310 --> 00:52:31,970 And then he arguably could figure out 708 00:52:31,970 --> 00:52:38,202 what the diagonalization looked like in general for arbitrary 709 00:52:38,202 --> 00:52:42,880 l, and then calculate log z in terms 710 00:52:42,880 --> 00:52:46,896 of the log of the largest eigenvalue. 711 00:52:46,896 --> 00:52:52,910 I guess we have to multiply by l parallel. 712 00:52:52,910 --> 00:52:58,210 And showed that, indeed, it corresponds to this 713 00:52:58,210 --> 00:53:01,190 and has this phase transition. 714 00:53:01,190 --> 00:53:05,310 And before this solution, people were not even sure 715 00:53:05,310 --> 00:53:08,440 that when you sum an expression such as that for partition 716 00:53:08,440 --> 00:53:12,770 function if you ever get a singularity because, 717 00:53:12,770 --> 00:53:14,950 again, on the face of it, it's basically 718 00:53:14,950 --> 00:53:16,780 a sum of exponential functions. 719 00:53:16,780 --> 00:53:19,020 Each one of them is perfectly analytic, 720 00:53:19,020 --> 00:53:20,670 sums of analytical functions. 721 00:53:20,670 --> 00:53:22,700 It's supposed to be analytical. 722 00:53:22,700 --> 00:53:29,160 The whole key lies in the limit of taking, say, l to infinity 723 00:53:29,160 --> 00:53:31,430 and n to infinity, and then you'll 724 00:53:31,430 --> 00:53:34,960 be able to see these kinds of singularities. 725 00:53:34,960 --> 00:53:37,520 And then again, some people thought 726 00:53:37,520 --> 00:53:39,320 that the only type of singularities 727 00:53:39,320 --> 00:53:42,890 that you will be able to get are the kinds of things 728 00:53:42,890 --> 00:53:45,200 that we saw [INAUDIBLE] point. 729 00:53:45,200 --> 00:53:48,210 So to see a different type of singularity 730 00:53:48,210 --> 00:53:53,000 with a heat capacity that was actually divergent and could 731 00:53:53,000 --> 00:53:55,240 explicitly be shown through mathematics 732 00:53:55,240 --> 00:53:59,650 was quite an interesting revelation. 733 00:53:59,650 --> 00:54:02,940 So the kind of relative importance 734 00:54:02,940 --> 00:54:07,170 of that is that after the war-- you 735 00:54:07,170 --> 00:54:12,430 can see this is all around the time of World War II-- Casimir 736 00:54:12,430 --> 00:54:18,950 wrote to Pauli saying that I have been away from thinking 737 00:54:18,950 --> 00:54:22,230 about physics the past few years with the war and all of that. 738 00:54:22,230 --> 00:54:24,890 Anything interesting happening in theoretical physics? 739 00:54:24,890 --> 00:54:28,290 And Pauli responded, well, not much except that Onsager 740 00:54:28,290 --> 00:54:32,040 solved the 2-dimensionalizing model. 741 00:54:32,040 --> 00:54:37,200 Of course, the solution that he has is quite obscure. 742 00:54:37,200 --> 00:54:43,140 And I don't think many people understand that. 743 00:54:43,140 --> 00:54:52,590 Then before, the form that people refer to was presented 744 00:54:52,590 --> 00:54:56,000 is actually kind of interesting, because this paper has 745 00:54:56,000 --> 00:54:58,950 something about crystal statistics, 746 00:54:58,950 --> 00:55:01,880 and then there's the following paper by a different author-- 747 00:55:01,880 --> 00:55:06,580 Bruria Kaufman, 1949-- has this same title 748 00:55:06,580 --> 00:55:10,130 except it goes from number two or something. 749 00:55:10,130 --> 00:55:13,500 So they were clearly talking to each other, 750 00:55:13,500 --> 00:55:19,950 but what she was able to show, Bruria Kaufman, 751 00:55:19,950 --> 00:55:22,650 was that the structure of these matrices 752 00:55:22,650 --> 00:55:25,490 can be simplified much further and can 753 00:55:25,490 --> 00:55:28,160 be made to look like spinners that 754 00:55:28,160 --> 00:55:30,930 are familiar from other branches of physics. 755 00:55:30,930 --> 00:55:33,550 And so this kind of 50-page paper 756 00:55:33,550 --> 00:55:37,510 was kind of reduced to something like a 20-page paper. 757 00:55:37,510 --> 00:55:45,240 And that's the solution that is reproduced in Wong's book. 758 00:55:45,240 --> 00:55:48,710 Chapter 15 of Wong has essentially a reproduction 759 00:55:48,710 --> 00:55:50,540 of this. 760 00:55:50,540 --> 00:55:53,430 I was looking at this because there aren't really 761 00:55:53,430 --> 00:55:58,100 that many women mathematical physicists, 762 00:55:58,100 --> 00:56:00,870 so I was kind of looking at her history, 763 00:56:00,870 --> 00:56:04,870 and she's quite an unusual person. 764 00:56:04,870 --> 00:56:07,200 So it turns out that for a while, 765 00:56:07,200 --> 00:56:11,100 she was mathematical assistant to Albert Einstein. 766 00:56:11,100 --> 00:56:17,200 She was first married to one of the most well known linguists 767 00:56:17,200 --> 00:56:18,970 of the 20th century. 768 00:56:18,970 --> 00:56:24,050 And for a while, they were both in Israel in a kibbutz 769 00:56:24,050 --> 00:56:27,890 where this important linguist was acting as a chauffeur 770 00:56:27,890 --> 00:56:29,970 and driving people around. 771 00:56:29,970 --> 00:56:35,400 And then later in life, she married briefly 772 00:56:35,400 --> 00:56:37,300 Willis Lamb, of Lamb shift. 773 00:56:37,300 --> 00:56:39,920 He turned out the term, 1949. 774 00:56:39,920 --> 00:56:44,680 She had done some calculation that if Lamb had paid attention 775 00:56:44,680 --> 00:56:47,500 to, he would have also potentially won 776 00:56:47,500 --> 00:56:50,080 a Nobel Prize for Mossbauer effect, 777 00:56:50,080 --> 00:56:52,210 but at that time didn't pay attention 778 00:56:52,210 --> 00:56:56,900 to it so somebody else got there first. 779 00:56:56,900 --> 00:56:58,850 So very interesting person. 780 00:56:58,850 --> 00:57:02,040 So to my mind, a good project for somebody 781 00:57:02,040 --> 00:57:04,650 is to write a biography for this person. 782 00:57:04,650 --> 00:57:08,010 It doesn't seem to exist. 783 00:57:08,010 --> 00:57:14,700 OK, so then both of these are based on this transfer matrix 784 00:57:14,700 --> 00:57:16,230 method. 785 00:57:16,230 --> 00:57:18,330 The method that I have given you, 786 00:57:18,330 --> 00:57:24,900 which is the graphical solution, was first 787 00:57:24,900 --> 00:57:31,320 presented by Kac and Ward in 1952. 788 00:57:31,320 --> 00:57:35,550 And it is reproduced in Feynman's book. 789 00:57:35,550 --> 00:57:42,260 So Feynman apparently also had one of these crucial steps 790 00:57:42,260 --> 00:57:44,830 of the conjecture with his factor of minus 1 791 00:57:44,830 --> 00:57:47,750 to the power of the number of crossings, 792 00:57:47,750 --> 00:57:52,860 giving you the correct factor to do the counting. 793 00:57:56,770 --> 00:58:04,360 Now it turns out that I also did not prove that statement, 794 00:58:04,360 --> 00:58:07,110 so there is a missing mathematical link 795 00:58:07,110 --> 00:58:10,450 to make my proof of this expression complete. 796 00:58:10,450 --> 00:58:14,050 And that was provided by a mathematician called Sherman, 797 00:58:14,050 --> 00:58:19,080 1960, that essentially shows very rigorously 798 00:58:19,080 --> 00:58:22,320 that these factors of minus 1 to the number of crossings 799 00:58:22,320 --> 00:58:28,680 will work out and magically make everything happen. 800 00:58:28,680 --> 00:58:30,840 Now the question to ask is the following. 801 00:58:34,090 --> 00:58:39,270 We expect things to be exactly solvable 802 00:58:39,270 --> 00:58:42,030 when they are trivial in some sense. 803 00:58:42,030 --> 00:58:44,310 Gaussian model is exactly solvable 804 00:58:44,310 --> 00:58:48,420 because there is no interaction among the modes. 805 00:58:48,420 --> 00:58:52,190 So why is it that the 2-dimensionalizing model 806 00:58:52,190 --> 00:58:54,311 is solvable? 807 00:58:54,311 --> 00:58:58,970 And one of the keys to that is a realization 808 00:58:58,970 --> 00:59:01,770 of another way of looking at the problem that 809 00:59:01,770 --> 00:59:14,870 appeared by Lieb, Mattis, and Schultz in 1964. 810 00:59:14,870 --> 00:59:20,310 And so basically what they said is let's 811 00:59:20,310 --> 00:59:23,090 take a look at these pictures that I 812 00:59:23,090 --> 00:59:26,880 have been drawing for the graphs. 813 00:59:26,880 --> 00:59:30,630 And so I have graphs that basically 814 00:59:30,630 --> 00:59:34,010 on a kind of coarse level, they look something like this maybe. 815 00:59:40,160 --> 00:59:46,950 And what they said was that if we look at the transfer 816 00:59:46,950 --> 00:59:50,630 matrix-- the one that Onsager and Bruria 817 00:59:50,630 --> 00:59:52,220 Kaufman were looking at. 818 00:59:52,220 --> 00:59:56,560 And the reason it was solvable, it looked very much 819 00:59:56,560 --> 01:00:00,060 like you had a system of fermions. 820 01:00:00,060 --> 01:00:05,420 And then the insight is that if we look at these pictures, 821 01:00:05,420 --> 01:00:09,870 you can regard this as a 1-dimensional system 822 01:00:09,870 --> 01:00:13,760 of fermions that is evolving in time. 823 01:00:13,760 --> 01:00:17,320 And what you are looking at these borderline histories 824 01:00:17,320 --> 01:00:20,740 of two particles that are propagating. 825 01:00:20,740 --> 01:00:23,160 Here they annihilate each other. 826 01:00:23,160 --> 01:00:25,140 Here they annihilate each other. 827 01:00:25,140 --> 01:00:26,861 Another pair gets created, et cetera. 828 01:00:30,090 --> 01:00:35,400 But in one dimensions, fermions you can regard two ways-- 829 01:00:35,400 --> 01:00:41,350 either they cannot occupy the same site or you can say, well, 830 01:00:41,350 --> 01:00:46,360 let them occupy in same site, but then I introduce these 831 01:00:46,360 --> 01:00:50,400 factors of minus 1 for the exchange of fermions. 832 01:00:50,400 --> 01:00:54,060 So when two fermions cross each other in one dimension, 833 01:00:54,060 --> 01:00:56,610 their position has been exchanged 834 01:00:56,610 --> 01:00:59,830 so you have to put a minus 1 for crossing. 835 01:00:59,830 --> 01:01:03,180 And then when you sum over all histories for every crossing, 836 01:01:03,180 --> 01:01:05,680 there will be one that touches and goes away. 837 01:01:05,680 --> 01:01:09,860 And the sum total of the two of them is 0. 838 01:01:09,860 --> 01:01:14,310 So the point is that at the end of the day, 839 01:01:14,310 --> 01:01:18,450 this theory is a theory of three fermions. 840 01:01:23,050 --> 01:01:27,430 So we have not solved, in fact, an interacting 841 01:01:27,430 --> 01:01:28,700 complicated problem. 842 01:01:28,700 --> 01:01:30,100 It is. 843 01:01:30,100 --> 01:01:32,470 But in the right perspective, it looks 844 01:01:32,470 --> 01:01:37,240 like a bunch of fermions that completely non-interacting pass 845 01:01:37,240 --> 01:01:38,760 through each other as long as we're 846 01:01:38,760 --> 01:01:43,115 willing to put the minus 1 phase that you have for crossings. 847 01:01:49,090 --> 01:01:53,637 One last aspect to think about that you 848 01:01:53,637 --> 01:01:56,460 learn from this fermionic perspective. 849 01:02:03,900 --> 01:02:05,720 Look at this expression. 850 01:02:05,720 --> 01:02:09,960 Why did I say that this is a Gaussian model? 851 01:02:09,960 --> 01:02:15,140 We saw that we could get this, z Gaussian, 852 01:02:15,140 --> 01:02:24,020 by doing an integral essentially over weights 853 01:02:24,020 --> 01:02:27,470 that were continuous by i. 854 01:02:27,470 --> 01:02:36,300 And the weight I could click here as kij phi i phi j. 855 01:02:36,300 --> 01:02:41,490 Because in principle, if I only count an interaction once, 856 01:02:41,490 --> 01:02:43,260 let's say I count it twice. 857 01:02:43,260 --> 01:02:48,370 I could put a factor of 1/2 if I allow i and j to be some. 858 01:02:48,370 --> 01:02:51,280 Especially I said the weight that I have to put here 859 01:02:51,280 --> 01:02:53,955 is something like phi i squared over 2. 860 01:02:53,955 --> 01:03:00,610 So this essentially you can see z of the Gaussian 861 01:03:00,610 --> 01:03:07,470 ultimately always becomes something like z 862 01:03:07,470 --> 01:03:14,520 of the Gaussian is going to be 1 over the square root 863 01:03:14,520 --> 01:03:19,010 of a determinant of whatever the quadratic form is up there. 864 01:03:21,840 --> 01:03:25,870 And when you take the log of the partition function, 865 01:03:25,870 --> 01:03:31,010 the square root of the determinant becomes minus 1/2 866 01:03:31,010 --> 01:03:32,679 of the log of the determinant, which 867 01:03:32,679 --> 01:03:33,970 is what we've been calculating. 868 01:03:36,560 --> 01:03:39,700 So that's obvious. 869 01:03:39,700 --> 01:03:42,220 You say this object over here that you 870 01:03:42,220 --> 01:03:48,215 write as an answer is also log of some determinant. 871 01:03:51,150 --> 01:03:56,460 So can I think of these as some kind of a rock that 872 01:03:56,460 --> 01:04:01,170 is prescribed according to these rules on a lattice, 873 01:04:01,170 --> 01:04:04,980 give weights to jump according to what I have, 874 01:04:04,980 --> 01:04:07,081 then do a kind of Gaussian integration 875 01:04:07,081 --> 01:04:08,080 and get the same answer? 876 01:04:10,620 --> 01:04:14,420 Well, the difficulty is precisely 877 01:04:14,420 --> 01:04:19,310 this 1/2 versus minus 1/2, because when 878 01:04:19,310 --> 01:04:21,210 you do the Gaussian integration, you 879 01:04:21,210 --> 01:04:22,885 get the determinant in the denominator. 880 01:04:26,050 --> 01:04:37,230 So is there a trick to get the determinant in the numerator? 881 01:04:37,230 --> 01:04:40,390 And the answer is that people who 882 01:04:40,390 --> 01:04:49,450 do have integral formulations for fermionic systems 883 01:04:49,450 --> 01:04:52,530 rely on coherent states that involve 884 01:04:52,530 --> 01:04:57,560 anti-commuting variables, called Grassmann variables. 885 01:04:57,560 --> 01:05:00,970 And the very interesting thing about Grassmann variables 886 01:05:00,970 --> 01:05:06,810 is that if I do the analog of the Gaussian integration, 887 01:05:06,810 --> 01:05:08,910 the answer-- rather than being in the determinant, 888 01:05:08,910 --> 01:05:12,750 in the denominator-- goes into the numerator. 889 01:05:12,750 --> 01:05:17,930 And so one can, in fact, rewrite this partition function, 890 01:05:17,930 --> 01:05:22,290 sort of working backward, in terms 891 01:05:22,290 --> 01:05:28,230 of an integration over Gaussian distributed 892 01:05:28,230 --> 01:05:31,910 Grassmannian variables on the lattice, 893 01:05:31,910 --> 01:05:34,776 which is also equivalent to another way of thinking 894 01:05:34,776 --> 01:05:35,400 about fermions. 895 01:05:41,550 --> 01:05:43,680 Let's see. 896 01:05:43,680 --> 01:05:45,660 What else is known about this model? 897 01:05:48,270 --> 01:05:54,560 So I said that the specific heat singularity is known, 898 01:05:54,560 --> 01:05:57,992 so we have this alpha which is 0 log. 899 01:06:01,940 --> 01:06:05,280 Given that the structure that we have 900 01:06:05,280 --> 01:06:11,970 involves inside the log something that is like this, 901 01:06:11,970 --> 01:06:18,070 you won't be surprised that if I think in terms of a correlation 902 01:06:18,070 --> 01:06:21,445 length-- so typically q's would be an inverse correlation 903 01:06:21,445 --> 01:06:26,850 length, some kind of a q at which I will be reaching 904 01:06:26,850 --> 01:06:30,920 appropriate saturation for a delta t-- I will arrive 905 01:06:30,920 --> 01:06:35,710 at a correlation range that diverge as delta t 906 01:06:35,710 --> 01:06:38,130 to the minus 1. 907 01:06:38,130 --> 01:06:42,210 Again, I can write it more precisely as b plus b 908 01:06:42,210 --> 01:06:46,410 minus to the minus mu minus. 909 01:06:46,410 --> 01:06:52,840 The ratio of the b's this is 1, and the mus 910 01:06:52,840 --> 01:06:55,560 are the same and equal to 1. 911 01:06:55,560 --> 01:07:00,845 So the correlation length diverges with an exponent 1, 912 01:07:00,845 --> 01:07:04,590 and one can [INAUDIBLE] this exactly. 913 01:07:04,590 --> 01:07:10,820 One can then also calculate actual correlations 914 01:07:10,820 --> 01:07:18,450 at criticality and show that that criticality correlations 915 01:07:18,450 --> 01:07:21,760 decay with separation between the points 916 01:07:21,760 --> 01:07:27,270 that you are looking at as 1 over r to the 1/4. 917 01:07:27,270 --> 01:07:32,760 So the exponent that we call mu is 1/4 in 2 dimensions. 918 01:07:35,730 --> 01:07:38,240 Once you have correlations, you certainly 919 01:07:38,240 --> 01:07:41,145 know that you can calculate the susceptibility 920 01:07:41,145 --> 01:07:46,260 as an integral of the correlation function. 921 01:07:46,260 --> 01:07:52,390 And so it's going to be an integral b 2r over r to the 1/4 922 01:07:52,390 --> 01:07:54,710 that is cut off at the correlation length. 923 01:07:54,710 --> 01:07:59,370 So that's going to give me c to the power of 2 minus 1/4, 924 01:07:59,370 --> 01:08:01,370 which is 7/4. 925 01:08:01,370 --> 01:08:06,940 So that's going to diverge as delta t 2 to the minus gamma. 926 01:08:06,940 --> 01:08:08,140 Gamma is, again, 7/4. 927 01:08:13,940 --> 01:08:20,069 So these things-- correlations-- can 928 01:08:20,069 --> 01:08:23,720 be calculated like you saw already 929 01:08:23,720 --> 01:08:26,060 for the case of the Gaussian model 930 01:08:26,060 --> 01:08:27,720 with a appropriate modification. 931 01:08:27,720 --> 01:08:30,470 That is, I have to look at walks that 932 01:08:30,470 --> 01:08:33,470 don't come back and close on themselves, 933 01:08:33,470 --> 01:08:36,939 but walks that go from one point to another point. 934 01:08:36,939 --> 01:08:40,413 So the same types of techniques that is described here 935 01:08:40,413 --> 01:08:44,082 will allow you to get to all of these other results. 936 01:08:44,082 --> 01:08:44,910 AUDIENCE: Question? 937 01:08:44,910 --> 01:08:46,119 PROFESSOR: Yes? 938 01:08:46,119 --> 01:08:49,505 AUDIENCE: Do people try to experiment on the builds 939 01:08:49,505 --> 01:08:52,170 as things that should be [INAUDIBLE] to the Ising model? 940 01:08:52,170 --> 01:08:55,500 PROFESSOR: There are by now many experimental realizations 941 01:08:55,500 --> 01:08:59,220 of [INAUDIBLE] Ising model in which these exponents-- 942 01:08:59,220 --> 01:09:01,700 actually, the next one that I will 943 01:09:01,700 --> 01:09:05,060 tell you have been confirmed very nicely. 944 01:09:05,060 --> 01:09:08,710 So there are a number of 2-dimensional absorb, systems, 945 01:09:08,710 --> 01:09:12,910 a number of systems of mixtures in 2 dimensions 946 01:09:12,910 --> 01:09:14,090 that phase separate. 947 01:09:14,090 --> 01:09:17,662 So there's a huge number of experimental realizations. 948 01:09:22,090 --> 01:09:25,250 At that time, no, because we're talking about 70 years. 949 01:09:30,800 --> 01:09:32,740 So the last thing that I want to mention 950 01:09:32,740 --> 01:09:39,879 is, of course, when you go below tc, 951 01:09:39,879 --> 01:09:44,240 we expect that-- let's call it temperate-- 952 01:09:44,240 --> 01:09:48,540 when I go below temperature, there will be a magnetization. 953 01:09:48,540 --> 01:09:54,530 That has always been our signature of symmetry breaking. 954 01:09:54,530 --> 01:10:00,050 And so the question is, what is the magnetization? 955 01:10:00,050 --> 01:10:04,830 And then this is another interesting story, 956 01:10:04,830 --> 01:10:10,400 that around 1950s in a couple of conferences, 957 01:10:10,400 --> 01:10:14,310 at the end of somebody's talk, Onsager went to the board 958 01:10:14,310 --> 01:10:17,870 and said that he and Bruria Kaufman 959 01:10:17,870 --> 01:10:28,495 have found this expression for the magnetization of the system 960 01:10:28,495 --> 01:10:30,860 at low temperature as a function of temperature 961 01:10:30,860 --> 01:10:33,010 or the coupling constant. 962 01:10:33,010 --> 01:10:39,440 But they never wrote the solution down until in 1952, 963 01:10:39,440 --> 01:10:43,560 actually CN Yang published a paper 964 01:10:43,560 --> 01:10:45,730 that has derived this result. 965 01:10:45,730 --> 01:10:50,330 And since this goes to 1 at the critical point-- why 966 01:10:50,330 --> 01:10:53,760 duality [INAUDIBLE] 2k [INAUDIBLE] 2k-- 967 01:10:53,760 --> 01:10:56,000 [? dual ?] plus 1. 968 01:10:56,000 --> 01:10:58,430 This vanishes with an exponent beta, 969 01:10:58,430 --> 01:11:02,112 which is 1/8, which is the other exponent in this series. 970 01:11:10,890 --> 01:11:14,206 So as I said, there are many people 971 01:11:14,206 --> 01:11:18,640 who since the '50s and '60s devoted their life 972 01:11:18,640 --> 01:11:22,790 to looking at various generalizations, extensions 973 01:11:22,790 --> 01:11:24,880 of the Ising model. 974 01:11:24,880 --> 01:11:26,860 There are many people who try to solve it 975 01:11:26,860 --> 01:11:30,300 with a finite in 2 dimensions, a finite magnetic field. 976 01:11:30,300 --> 01:11:31,690 You can't do that. 977 01:11:31,690 --> 01:11:33,610 This magnetization is obtained only 978 01:11:33,610 --> 01:11:36,970 at the limit of [INAUDIBLE] going to 0. 979 01:11:36,970 --> 01:11:40,150 And clearly, people have thought a lot 980 01:11:40,150 --> 01:11:44,780 about doing things in 3 dimensions or higher dimensions 981 01:11:44,780 --> 01:11:48,000 without much success. 982 01:11:48,000 --> 01:11:51,680 So this is basically the end of the portion 983 01:11:51,680 --> 01:11:56,690 that I had to give with discrete models and lattices. 984 01:11:56,690 --> 01:12:00,400 And as of next lecture, we will change our perspective 985 01:12:00,400 --> 01:12:02,160 one more time. 986 01:12:02,160 --> 01:12:04,720 We'll go back to the continuum and we 987 01:12:04,720 --> 01:12:10,110 will look at n component models in the low temperature 988 01:12:10,110 --> 01:12:14,370 expansion and see what happens over there. 989 01:12:16,984 --> 01:12:17,983 Are there any questions? 990 01:12:20,970 --> 01:12:23,860 OK, I will give you a preview of what 991 01:12:23,860 --> 01:12:28,880 I will be doing in the next few minutes. 992 01:12:28,880 --> 01:12:36,100 So we have been looking at these lattice models in the high t 993 01:12:36,100 --> 01:12:39,770 limit where expansion was graphical, 994 01:12:39,770 --> 01:12:42,015 such as the one that I have over there. 995 01:12:42,015 --> 01:12:44,430 But the partition function turned out 996 01:12:44,430 --> 01:12:47,960 to be something of two constants, as 1 997 01:12:47,960 --> 01:12:51,720 plus something that involves loops and things that 998 01:12:51,720 --> 01:12:55,210 involve multiple loops, et cetera. 999 01:12:55,210 --> 01:12:58,445 This was for the Ising model. 1000 01:12:58,445 --> 01:13:02,165 It turns out that if I go from the Ising model 1001 01:13:02,165 --> 01:13:04,190 to n component space. 1002 01:13:04,190 --> 01:13:07,280 So at each side of the lattice, I 1003 01:13:07,280 --> 01:13:11,430 put something that has n components 1004 01:13:11,430 --> 01:13:20,998 subject to the condition that it's a unit vector. 1005 01:13:20,998 --> 01:13:25,280 And I try to calculate the partition function 1006 01:13:25,280 --> 01:13:30,350 by integrating over all of these unit vectors 1007 01:13:30,350 --> 01:13:34,643 in n dimension of a weight that is just 1008 01:13:34,643 --> 01:13:40,330 a generalization of the Ising, except that I would have 1009 01:13:40,330 --> 01:13:44,090 the dot product of these things. 1010 01:13:44,090 --> 01:13:49,000 And I start making appropriate high temperature expansions 1011 01:13:49,000 --> 01:13:50,200 for these models. 1012 01:13:50,200 --> 01:13:53,590 I will generate a very similar series, 1013 01:13:53,590 --> 01:13:56,615 except that whenever I see a loop, 1014 01:13:56,615 --> 01:13:59,280 I have to put a factor of n where 1015 01:13:59,280 --> 01:14:02,240 n is the number of components. 1016 01:14:02,240 --> 01:14:04,250 And this we already saw when we were 1017 01:14:04,250 --> 01:14:06,320 doing Landau-Ginzburg expansions. 1018 01:14:06,320 --> 01:14:10,650 We saw that the expansions that we had over here 1019 01:14:10,650 --> 01:14:14,770 could be graphically interpreted as representations 1020 01:14:14,770 --> 01:14:17,550 of the various terms that we had in the Landau-Ginzburg 1021 01:14:17,550 --> 01:14:18,840 expansion. 1022 01:14:18,840 --> 01:14:25,240 And essentially, this factor of n is the one difficulty. 1023 01:14:25,240 --> 01:14:30,010 You can use the same methods for numerical expansions 1024 01:14:30,010 --> 01:14:34,080 for all n models that are these n component models. 1025 01:14:34,080 --> 01:14:38,170 You can't do anything exactly with them. 1026 01:14:38,170 --> 01:14:42,640 Now the low temperature expansion for Ising 1027 01:14:42,640 --> 01:14:49,650 like models we saw involved starting with some ground 1028 01:14:49,650 --> 01:14:57,270 state-- for example, all up-- and then including excitations 1029 01:14:57,270 --> 01:15:04,400 that were islands of minus in a sea of plus. 1030 01:15:04,400 --> 01:15:09,570 And so then there higher order in this series. 1031 01:15:09,570 --> 01:15:13,980 Now for other discrete models, such as the Potts model, 1032 01:15:13,980 --> 01:15:15,680 you can use the same procedure. 1033 01:15:15,680 --> 01:15:18,600 And again, you see that in some of the problems 1034 01:15:18,600 --> 01:15:21,250 that you've had to solve. 1035 01:15:21,250 --> 01:15:26,190 But this will not work when we come to these continuous spin 1036 01:15:26,190 --> 01:15:33,590 markets, because for continuous spin models, 1037 01:15:33,590 --> 01:15:38,515 the ground state would be when everybody is pointing in one 1038 01:15:38,515 --> 01:15:45,280 direction, but the excitations on this ground state 1039 01:15:45,280 --> 01:15:49,070 are not islands that are flipped over. 1040 01:15:49,070 --> 01:15:58,770 They are these long wavelength Goldstone modes, 1041 01:15:58,770 --> 01:16:02,580 which we described earlier in class. 1042 01:16:02,580 --> 01:16:07,226 So if we want to make an expansion 1043 01:16:07,226 --> 01:16:12,840 to look at the low temperature [INAUDIBLE] for systems with n 1044 01:16:12,840 --> 01:16:16,670 greater than 1, we have to make an expansion involving 1045 01:16:16,670 --> 01:16:20,010 Goldstone modes and, as we will see, 1046 01:16:20,010 --> 01:16:22,110 interactions among Goldstone modes. 1047 01:16:22,110 --> 01:16:25,730 So you can no longer regard at that appropriate level 1048 01:16:25,730 --> 01:16:29,590 of sophistication as the Goldstone modes maintain 1049 01:16:29,590 --> 01:16:31,770 independence from each other. 1050 01:16:31,770 --> 01:16:36,160 And very roughly, the main difference between discrete 1051 01:16:36,160 --> 01:16:40,500 and these continuous symmetry models is captured as follows. 1052 01:16:40,500 --> 01:16:46,510 Suppose I have a huge system that is L by L 1053 01:16:46,510 --> 01:16:51,654 and I impose boundary conditions on one side where all of this 1054 01:16:51,654 --> 01:16:56,850 spins in one direction, and ask what 1055 01:16:56,850 --> 01:17:02,960 happens if I impose a different condition on the other side. 1056 01:17:02,960 --> 01:17:08,010 Well, what would happen is you would have a domain boundary. 1057 01:17:08,010 --> 01:17:10,370 And the cost of that domain boundary 1058 01:17:10,370 --> 01:17:13,950 will be proportional to the area of the boundary 1059 01:17:13,950 --> 01:17:16,000 in d dimensions. 1060 01:17:16,000 --> 01:17:23,420 So the cost is proportional to some energy 1061 01:17:23,420 --> 01:17:26,580 per bond times L to the d minus 1. 1062 01:17:30,710 --> 01:17:34,930 Whereas if I try to do the same thing for a continuous spin 1063 01:17:34,930 --> 01:17:43,360 system, if I align one side like this, the other side like this, 1064 01:17:43,360 --> 01:17:48,940 but in-between I can gradually change from one direction 1065 01:17:48,940 --> 01:17:51,740 to another direction. 1066 01:17:51,740 --> 01:17:54,050 And the cost would be the gradient, 1067 01:17:54,050 --> 01:17:59,830 which is 1 over L squared times integrated 1068 01:17:59,830 --> 01:18:01,900 over the entire system. 1069 01:18:01,900 --> 01:18:05,570 So the energy cost of this excitation 1070 01:18:05,570 --> 01:18:13,390 will be some parameter j, 1 over L-- which 1071 01:18:13,390 --> 01:18:18,120 is the shift-- squared-- which is the [? strain ?] squared-- 1072 01:18:18,120 --> 01:18:19,845 integrated over the entire volume, 1073 01:18:19,845 --> 01:18:24,760 so it goes L to the d minus 2. 1074 01:18:24,760 --> 01:18:31,000 So we can see that these systems are much softer 1075 01:18:31,000 --> 01:18:33,990 than discrete systems. 1076 01:18:33,990 --> 01:18:36,930 For discrete systems, thermal fluctuations 1077 01:18:36,930 --> 01:18:41,862 are sufficient to destroy order at low temperature 1078 01:18:41,862 --> 01:18:47,660 as soon as this cost is of the order of kt, which for large L 1079 01:18:47,660 --> 01:18:51,850 can only happen in d [INAUDIBLE] 1 and lower. 1080 01:18:51,850 --> 01:18:56,930 Whereas for these systems, it happens in d of 2 and lower. 1081 01:18:56,930 --> 01:18:59,940 So the lower critical dimension for these models. 1082 01:18:59,940 --> 01:19:03,370 With continuous symmetry, we already saw it's 2. 1083 01:19:03,370 --> 01:19:06,715 For these models, it is 1. 1084 01:19:06,715 --> 01:19:09,520 Now we are going to be interested in this class 1085 01:19:09,520 --> 01:19:11,070 of models. 1086 01:19:11,070 --> 01:19:14,380 I have told you that in 2 dimensions, 1087 01:19:14,380 --> 01:19:17,400 they should not order. 1088 01:19:17,400 --> 01:19:22,210 So presumably, the critical temperature-- if I continuously 1089 01:19:22,210 --> 01:19:24,700 regard it as a function of dimension, 1090 01:19:24,700 --> 01:19:30,600 will go to 0 as a function of dimension as d minus 2. 1091 01:19:30,600 --> 01:19:34,610 So the insight of Polyakov was that maybe we 1092 01:19:34,610 --> 01:19:38,570 can look at the interactions of these old Goldstone modes 1093 01:19:38,570 --> 01:19:41,830 and do a systematic low temperature expansion that 1094 01:19:41,830 --> 01:19:47,056 reaches the phase transition of a critical point systematically 1095 01:19:47,056 --> 01:19:48,830 ind minus 2. 1096 01:19:48,830 --> 01:19:51,970 And that's what we will attempt in the future.