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 00:00:22,790 --> 00:00:26,160 PROFESSOR: OK, let's start. 9 00:00:26,160 --> 00:00:30,410 So looking for a way to understand 10 00:00:30,410 --> 00:00:33,920 the universality of phase transitions, 11 00:00:33,920 --> 00:00:36,550 we arrived at the simplest model that 12 00:00:36,550 --> 00:00:37,935 should capture some of that. 13 00:00:37,935 --> 00:00:50,700 That was the Ising model, where at each site of a lattice-- 14 00:00:50,700 --> 00:00:52,820 and for a while I will be talking 15 00:00:52,820 --> 00:00:56,250 in this lecture about the square lattice-- 16 00:00:56,250 --> 00:00:58,920 you put a binary variable. 17 00:00:58,920 --> 00:01:06,840 Let's call it sigma i that takes two values, minus plus 1, 18 00:01:06,840 --> 00:01:08,540 on each of the N sites. 19 00:01:08,540 --> 00:01:11,480 So we have a total of 2 to the N possible configurations. 20 00:01:15,600 --> 00:01:19,820 And we subject that to an energy cost 21 00:01:19,820 --> 00:01:25,920 that tries to make nearest neighbors to be the same. 22 00:01:25,920 --> 00:01:29,830 So this symbol stands for sum over all pairs of nearest 23 00:01:29,830 --> 00:01:31,950 neighbors on whatever lattice you have. 24 00:01:31,950 --> 00:01:34,470 Here the square lattice. 25 00:01:34,470 --> 00:01:40,320 And the tendency for them to be parallel as opposed 26 00:01:40,320 --> 00:01:42,910 to anti-parallel is captured through 27 00:01:42,910 --> 00:01:49,050 this dimensionless energy divided by Kt parameter K. 28 00:01:49,050 --> 00:01:54,940 So then in principle, as we change K we could also 29 00:01:54,940 --> 00:01:57,420 potentially add a magnetic field. 30 00:01:57,420 --> 00:02:00,450 There could be a phase transition in the system. 31 00:02:00,450 --> 00:02:03,235 And that should be captured by looking at the partition 32 00:02:03,235 --> 00:02:07,130 function, which is obtained by summing over the 2 33 00:02:07,130 --> 00:02:11,760 to the N possibilities of this weight that 34 00:02:11,760 --> 00:02:16,690 is e to the sum over ij K sigma i sigma j. 35 00:02:20,300 --> 00:02:23,710 So that's easier said than done. 36 00:02:23,710 --> 00:02:28,130 And the question was, well, how can we proceed with this? 37 00:02:28,130 --> 00:02:32,720 And last lecture, we suggested two routes 38 00:02:32,720 --> 00:02:35,250 for looking at this system. 39 00:02:35,250 --> 00:02:38,780 One of them was to start by looking 40 00:02:38,780 --> 00:02:43,665 at a low-temperature expansion. 41 00:02:43,665 --> 00:02:51,820 And here, let's say, we would start 42 00:02:51,820 --> 00:02:54,860 with one of two possible ground states. 43 00:02:54,860 --> 00:02:58,100 Let' say all of the spins could, for example, 44 00:02:58,100 --> 00:02:59,565 be pointing in the plus direction. 45 00:03:06,510 --> 00:03:09,580 That is certainly the largest contribution 46 00:03:09,580 --> 00:03:11,240 that you would have to the partition 47 00:03:11,240 --> 00:03:13,480 function at 0 temperature. 48 00:03:13,480 --> 00:03:20,360 And that contribution is all of the bonds being satisfied. 49 00:03:20,360 --> 00:03:24,650 Each one of them giving a factor of e to the K. 50 00:03:24,650 --> 00:03:26,700 Let's say we are on a square lattice. 51 00:03:26,700 --> 00:03:29,510 On a square lattice, each site has 52 00:03:29,510 --> 00:03:32,070 two bonds associated with it. 53 00:03:32,070 --> 00:03:34,890 So this will go like 2N. 54 00:03:34,890 --> 00:03:39,240 Since we have N sites, we will 2N bonds. 55 00:03:39,240 --> 00:03:41,940 There is actually, of course, two possibilities. 56 00:03:41,940 --> 00:03:46,060 We can have either all of them plus or all of them minus. 57 00:03:46,060 --> 00:03:50,080 So there is a kind of trivial degeneracy of 2, 58 00:03:50,080 --> 00:03:53,470 which doesn't really make too much of a difference. 59 00:03:53,470 --> 00:03:58,110 And then we can start looking at excitations around this. 60 00:03:58,110 --> 00:04:02,060 And so we said that the first type of excitation 61 00:04:02,060 --> 00:04:05,480 is somewhere on the lattice we make one of these pluses 62 00:04:05,480 --> 00:04:06,650 into a minus. 63 00:04:06,650 --> 00:04:10,530 And once we do that, we have made 64 00:04:10,530 --> 00:04:15,140 4 bonds that go out of this minus site unhappy. 65 00:04:15,140 --> 00:04:20,560 So the cost of going from plus k to minus K, which 66 00:04:20,560 --> 00:04:27,700 is minus 2k, in this case is repeated four times. 67 00:04:27,700 --> 00:04:30,070 And this particular excitation can 68 00:04:30,070 --> 00:04:34,070 be placed in any one of N locations on the lattice. 69 00:04:34,070 --> 00:04:38,120 So this was this kind of excitation. 70 00:04:38,120 --> 00:04:42,400 And then we could go and have the next excitation 71 00:04:42,400 --> 00:04:44,666 where two of them are flipped. 72 00:04:44,666 --> 00:04:47,740 And we would have a situation such as this. 73 00:04:47,740 --> 00:04:53,530 And since this dimer can point in the x- or the y-direction 74 00:04:53,530 --> 00:04:58,300 on the square lattice, it has a degeneracy of 2. 75 00:04:58,300 --> 00:05:01,870 I have e to the minus 2K. 76 00:05:01,870 --> 00:05:04,850 And how many bonds have I broken? 77 00:05:04,850 --> 00:05:07,810 One, two, three, four, five, six. 78 00:05:07,810 --> 00:05:10,000 Times 6. 79 00:05:10,000 --> 00:05:12,270 And I can go on. 80 00:05:12,270 --> 00:05:17,800 So this was a situation such as this. 81 00:05:17,800 --> 00:05:21,520 A general term in the series would 82 00:05:21,520 --> 00:05:26,210 correspond to creating some kind of an island of minus 83 00:05:26,210 --> 00:05:27,455 in this sea of pluses. 84 00:05:30,600 --> 00:05:35,540 And the contribution would be e to the minus 2K 85 00:05:35,540 --> 00:05:39,550 times the perimeter of this island. 86 00:05:48,660 --> 00:05:51,685 So that would be a way to calculate 87 00:05:51,685 --> 00:05:58,550 the low-temperature expansion we discussed last time around. 88 00:05:58,550 --> 00:06:01,670 We also said that I could do a high temperature expansion. 89 00:06:09,610 --> 00:06:12,960 And for this, we use the trick. 90 00:06:12,960 --> 00:06:16,140 We said I can write the partition function 91 00:06:16,140 --> 00:06:20,370 as a sum over all these 2 to the N configuration. 92 00:06:20,370 --> 00:06:21,285 Yes. 93 00:06:21,285 --> 00:06:24,080 AUDIENCE: Is there a reason we don't have separate islands? 94 00:06:24,080 --> 00:06:25,080 PROFESSOR: Oh, we do. 95 00:06:25,080 --> 00:06:25,700 We do. 96 00:06:25,700 --> 00:06:30,990 So in general, in this picture I could have multiple islands. 97 00:06:30,990 --> 00:06:32,830 Yes. 98 00:06:32,830 --> 00:06:35,970 And what would be interesting is certainly 99 00:06:35,970 --> 00:06:40,490 when I take the log of Z, then the log of Z 100 00:06:40,490 --> 00:06:43,470 here I would have NK. 101 00:06:43,470 --> 00:06:45,440 I would have log 2. 102 00:06:45,440 --> 00:06:49,280 And then there would be a bunch of terms. 103 00:06:49,280 --> 00:06:53,300 And what we saw was those bunches of terms, 104 00:06:53,300 --> 00:06:57,500 starting with that one, can be captured 105 00:06:57,500 --> 00:07:00,970 into a series where the n comes out front 106 00:07:00,970 --> 00:07:05,380 and the terms in the series would be functions of e 107 00:07:05,380 --> 00:07:07,170 to the minus 2K. 108 00:07:07,170 --> 00:07:10,970 And indeed, when we exponentiate this, 109 00:07:10,970 --> 00:07:13,650 this would have single islands only. 110 00:07:13,650 --> 00:07:15,630 The exponential will have multiple islands 111 00:07:15,630 --> 00:07:17,482 that we have for the partition functions. 112 00:07:20,290 --> 00:07:23,800 So these terms, as we see, are higher powers of n 113 00:07:23,800 --> 00:07:25,452 because you would be able to move them 114 00:07:25,452 --> 00:07:26,960 in different directions. 115 00:07:26,960 --> 00:07:28,970 When you take the log, only terms that are 116 00:07:28,970 --> 00:07:30,300 linear and can survive. 117 00:07:35,310 --> 00:07:39,710 So this sum over bond configurations 118 00:07:39,710 --> 00:07:44,570 I can write as a product over bonds. 119 00:07:44,570 --> 00:07:48,330 And the factor of e to the K sigma i sigma 120 00:07:48,330 --> 00:07:53,260 j, we saw that we could capture slightly differently. 121 00:07:53,260 --> 00:07:55,830 So e to the K sigma i sigma j I can 122 00:07:55,830 --> 00:07:59,120 write as hyperbolic cosine of K 1 123 00:07:59,120 --> 00:08:05,660 plus hyperbolic tanh of K sigma i sigma j. 124 00:08:05,660 --> 00:08:11,670 And this t is going to be my symbol for hyperbolic tanh 125 00:08:11,670 --> 00:08:17,150 of K, so that I don't have to repeat it all over the place. 126 00:08:17,150 --> 00:08:20,220 So this is just rewriting of that exponential, 127 00:08:20,220 --> 00:08:23,260 recognizing that it has two possibilities. 128 00:08:23,260 --> 00:08:25,315 We took the cosh K to the outside. 129 00:08:28,150 --> 00:08:31,430 And if I'm, again, on the square lattice, 130 00:08:31,430 --> 00:08:35,909 I have 2N bonds so there will be 2N factors of cosh K 131 00:08:35,909 --> 00:08:38,470 that I will take into the outside. 132 00:08:38,470 --> 00:08:41,780 And then I would have a series, which 133 00:08:41,780 --> 00:08:47,140 would be these terms that I can start expanding in powers of t. 134 00:08:47,140 --> 00:08:50,580 And the lowest order I have 1. 135 00:08:50,580 --> 00:08:53,410 And then we discussed what kinds of terms are allowed. 136 00:09:01,510 --> 00:09:10,140 We saw that if I take just one factor of t sigma i sigma j, 137 00:09:10,140 --> 00:09:14,200 then I have a sigma i sitting here and sigma j sitting here. 138 00:09:14,200 --> 00:09:19,050 I sum over the two possibilities of sigma being plus or minus. 139 00:09:19,050 --> 00:09:21,180 It will give me 0. 140 00:09:21,180 --> 00:09:24,920 So there is a contribution order of t if I expand that, 141 00:09:24,920 --> 00:09:30,070 but summing over sigma i and sigma j would give me 0. 142 00:09:30,070 --> 00:09:31,830 So the only choice that I have is 143 00:09:31,830 --> 00:09:36,810 that this sigma that is sitting out here I should square. 144 00:09:36,810 --> 00:09:39,220 And I can square it by putting a bond that 145 00:09:39,220 --> 00:09:42,380 corresponds to this sigma and that sigma. 146 00:09:42,380 --> 00:09:43,700 This became sigma squared. 147 00:09:43,700 --> 00:09:46,200 I don't have to worry about that. 148 00:09:46,200 --> 00:09:49,080 Then I can complete this so that that's 149 00:09:49,080 --> 00:09:52,600 squared and this so that that's squared. 150 00:09:52,600 --> 00:09:55,530 And so this was a diagram that contributed 151 00:09:55,530 --> 00:09:58,390 N this quantity t to the fourth. 152 00:10:01,090 --> 00:10:05,790 Well, what's the next type of graph that I could draw? 153 00:10:05,790 --> 00:10:07,510 I could do something like this. 154 00:10:11,730 --> 00:10:15,120 And that is, again, something that I 155 00:10:15,120 --> 00:10:18,783 can orient along the x-direction or along the y-direction. 156 00:10:18,783 --> 00:10:22,950 So there is a factor of 2 from the orientation. 157 00:10:22,950 --> 00:10:26,040 And that's how many factors of t. 158 00:10:26,040 --> 00:10:26,600 It is 6. 159 00:10:32,400 --> 00:10:35,890 And so in general, what do I have? 160 00:10:35,890 --> 00:10:39,130 I will have to draw some kind of a graph 161 00:10:39,130 --> 00:10:47,275 on the lattice where at each site, I either have 0, 162 00:10:47,275 --> 00:10:49,760 2, or potentially 4. 163 00:10:49,760 --> 00:10:54,350 There is no difficult with 4 bonds emanating from a site 164 00:10:54,350 --> 00:10:56,700 because that sigma to the fourth. 165 00:10:56,700 --> 00:11:00,800 Basically, those kinds of things actually 166 00:11:00,800 --> 00:11:04,850 will give me then the sum over-- possibilities of sigma 167 00:11:04,850 --> 00:11:09,040 will give me 2 to the number of sites. 168 00:11:09,040 --> 00:11:12,470 And then these graphs, basically will 169 00:11:12,470 --> 00:11:17,689 get a factor which is 2 to the number of bonds in graph. 170 00:11:24,100 --> 00:11:25,830 And then again, you can see that here I 171 00:11:25,830 --> 00:11:29,790 could have multiple loops, just like we discussed over there. 172 00:11:29,790 --> 00:11:34,210 Multiple loops will go with larger factors of N. 173 00:11:34,210 --> 00:11:38,310 The thing that we are interested is log Z, 174 00:11:38,310 --> 00:11:49,660 which is N log 2 cosine squared K. And then I 175 00:11:49,660 --> 00:11:52,643 have to take the log of this expression, 176 00:11:52,643 --> 00:11:58,170 and I'll call it g of t. 177 00:11:58,170 --> 00:11:59,974 Yes. 178 00:11:59,974 --> 00:12:05,710 AUDIENCE: How did we get rid of [INAUDIBLE] bonds? 179 00:12:05,710 --> 00:12:06,840 PROFESSOR: OK. 180 00:12:06,840 --> 00:12:16,790 So this e to the K sigma i sigma j has two possible values. 181 00:12:16,790 --> 00:12:23,410 It is either e to the K or e to the minus K. 182 00:12:23,410 --> 00:12:25,690 And I can write it, therefore, as e 183 00:12:25,690 --> 00:12:31,610 to the K plus e to the minus K over 2 plus e to the K minus 184 00:12:31,610 --> 00:12:37,240 e to the minus K over 2 sigma i sigma j, right? 185 00:12:37,240 --> 00:12:38,475 AUDIENCE: My question was-- 186 00:12:38,475 --> 00:12:40,710 PROFESSOR: Yeah, I know. 187 00:12:40,710 --> 00:12:45,390 And then, this became cosh K 1 plus what 188 00:12:45,390 --> 00:12:48,990 I call sigma i sigma j. 189 00:12:48,990 --> 00:12:54,760 So when I draw my lattice, for the bond that 190 00:12:54,760 --> 00:13:01,900 goes between sites i and j, in the partition function 191 00:13:01,900 --> 00:13:03,060 I have this contribution. 192 00:13:06,060 --> 00:13:10,850 This contribution is completely equivalent to this. 193 00:13:10,850 --> 00:13:12,700 And this is two terms. 194 00:13:12,700 --> 00:13:14,860 The cosh K we took outside. 195 00:13:14,860 --> 00:13:21,250 The two terms is either 1 or 1 line. 196 00:13:21,250 --> 00:13:24,220 There isn't anything that is multiple lines. 197 00:13:24,220 --> 00:13:27,110 And as I said, you can make separately 198 00:13:27,110 --> 00:13:33,770 an expansion in powers of K. This corresponds to nothing, 199 00:13:33,770 --> 00:13:37,990 or going forward and backward, or going forward, backward, 200 00:13:37,990 --> 00:13:41,959 et cetera, if you make an expansion in powers of K. This 201 00:13:41,959 --> 00:13:43,250 term corresponds to going once. 202 00:13:47,030 --> 00:13:52,140 Essentially, this captures all of the things that stay back 203 00:13:52,140 --> 00:13:56,045 to the same site and this is a re-summation of everything 204 00:13:56,045 --> 00:13:58,780 that steps you forward. 205 00:13:58,780 --> 00:14:03,920 So everything that steps you forward and carries information 206 00:14:03,920 --> 00:14:07,820 from this sites to that site has been appropriately 207 00:14:07,820 --> 00:14:09,060 taken care of. 208 00:14:09,060 --> 00:14:10,135 And it occurs once. 209 00:14:12,800 --> 00:14:15,835 And that's one reason. 210 00:14:15,835 --> 00:14:18,320 And again, if you have three things coming 211 00:14:18,320 --> 00:14:20,890 at a particular site, it's a sigma i q. 212 00:14:20,890 --> 00:14:23,710 So these are completely the only thing that happen. 213 00:14:26,930 --> 00:14:34,770 But in principle, this is one diagrammatic series. 214 00:14:34,770 --> 00:14:38,870 This is one diagrammatic series. 215 00:14:38,870 --> 00:14:41,640 But you stare at them a little bit 216 00:14:41,640 --> 00:14:46,890 and you'll see why I put the same g for both of them. 217 00:14:46,890 --> 00:14:50,600 On the square lattice, they're identify series. 218 00:14:50,600 --> 00:14:55,550 See that series had N something to the fourth power, 219 00:14:55,550 --> 00:14:58,500 2N something to the sixth power, N something 220 00:14:58,500 --> 00:15:00,980 to the fourth power, 2N something to the sixth power. 221 00:15:00,980 --> 00:15:04,150 The first two terms are identical. 222 00:15:04,150 --> 00:15:07,710 You can convince yourself that all the terms will be identical 223 00:15:07,710 --> 00:15:11,360 also, including something complicated, 224 00:15:11,360 --> 00:15:14,370 such as if I were to draw-- I don't know. 225 00:15:14,370 --> 00:15:20,750 A diagram such as this one, which has 2 or 4 per site. 226 00:15:20,750 --> 00:15:25,780 I can convert that to something that had spins plus out here, 227 00:15:25,780 --> 00:15:30,180 then it minus here, minus here, plus here. 228 00:15:30,180 --> 00:15:33,420 And it's a completely consistent diagram 229 00:15:33,420 --> 00:15:36,562 that I would have had in the low-temperature series. 230 00:15:36,562 --> 00:15:38,930 So you can convince yourself that there 231 00:15:38,930 --> 00:15:42,493 is a one-to-one correspondence between these two series. 232 00:15:42,493 --> 00:15:46,680 They are identical for the square lattice. 233 00:15:46,680 --> 00:15:50,770 As we will discuss, this is a property of the square lattice. 234 00:15:50,770 --> 00:15:53,360 So you have this very nice symmetry 235 00:15:53,360 --> 00:15:57,380 that you conclude that the partition function per site, 236 00:15:57,380 --> 00:16:00,820 the part that is interesting, either 237 00:16:00,820 --> 00:16:08,366 I can get it from here as K plus this function of e 238 00:16:08,366 --> 00:16:15,580 to the minus 2K or from the high temperatures series as log 2 239 00:16:15,580 --> 00:16:19,020 hyperbolic cosine squared K. Plus exactly 240 00:16:19,020 --> 00:16:25,630 the same function of tanh K. 241 00:16:25,630 --> 00:16:27,920 This part of it, actually, I don't really 242 00:16:27,920 --> 00:16:31,870 care because these are analytical functions. 243 00:16:31,870 --> 00:16:34,435 I expect this model to have a phase transition. 244 00:16:34,435 --> 00:16:37,197 The low-temperature and the high-temperature behavior 245 00:16:37,197 --> 00:16:38,030 should be different. 246 00:16:38,030 --> 00:16:41,550 It should be order at low temperature, [INAUDIBLE] 247 00:16:41,550 --> 00:16:42,860 at high temperature. 248 00:16:42,860 --> 00:16:45,570 There should be a phase transition and a singularity 249 00:16:45,570 --> 00:16:47,270 between those two cases. 250 00:16:47,270 --> 00:16:50,790 The singular part must be captured here. 251 00:16:50,790 --> 00:16:54,260 So these are the singular functions. 252 00:16:54,260 --> 00:16:57,490 So the singular part of the free energy 253 00:16:57,490 --> 00:17:01,640 has this interesting property that you can evaluate it 254 00:17:01,640 --> 00:17:06,819 for some parameter K, which is large in the low-temperature 255 00:17:06,819 --> 00:17:10,250 phase, or some parameter t, which 256 00:17:10,250 --> 00:17:13,589 is small in the high-temperature phase. 257 00:17:13,589 --> 00:17:20,069 And the two will be related completely to each other. 258 00:17:20,069 --> 00:17:22,530 They are essentially the same thing. 259 00:17:22,530 --> 00:17:25,970 So this property is called duality. 260 00:17:29,650 --> 00:17:33,400 And so what I have said, first of all, is 261 00:17:33,400 --> 00:17:39,460 that there is a relationship between, say, the coupling tanh 262 00:17:39,460 --> 00:17:47,910 K and a coupling that I can separately call K tilde, such 263 00:17:47,910 --> 00:17:51,900 that if I evaluate the high-temperature series at K, 264 00:17:51,900 --> 00:17:55,430 it is like evaluating the low-temperature series at K 265 00:17:55,430 --> 00:18:02,270 tilde, where K tilde of K is minus 1/2 log 266 00:18:02,270 --> 00:18:07,710 of the hyperbolic tanh at K. 267 00:18:07,710 --> 00:18:10,350 I can plot for you what this function looks like. 268 00:18:13,272 --> 00:18:18,740 So this is K. This is what K tilde of K looks like. 269 00:18:18,740 --> 00:18:22,700 And it is something like this. 270 00:18:22,700 --> 00:18:27,360 Basically, strong coupling, or low temperature, 271 00:18:27,360 --> 00:18:30,295 gets mapped to weak coupling, or high temperature, 272 00:18:30,295 --> 00:18:31,495 and vice versa. 273 00:18:35,120 --> 00:18:39,795 So it's kind of like this, that there 274 00:18:39,795 --> 00:18:46,490 is this axes of the strength of K going 275 00:18:46,490 --> 00:18:52,610 from low temperature, strong, to high temperature, weak. 276 00:18:52,610 --> 00:18:55,730 And what I have shown is that if I start with somewhere 277 00:18:55,730 --> 00:19:02,040 out here, it is mapped to somewhere down here. 278 00:19:02,040 --> 00:19:06,060 If I start from somewhere here, it 279 00:19:06,060 --> 00:19:09,140 would be mapping to somewhere here. 280 00:19:15,100 --> 00:19:22,440 So one question to ask is, well, OK, I start from here. 281 00:19:22,440 --> 00:19:24,820 I go here. 282 00:19:24,820 --> 00:19:30,890 If I put that value of k in here, do I go to a third point 283 00:19:30,890 --> 00:19:33,550 or do I come back to here? 284 00:19:33,550 --> 00:19:36,560 And I will show you that it is, in fact, 285 00:19:36,560 --> 00:19:41,620 a mapping that goes both ways. 286 00:19:41,620 --> 00:19:48,410 And a way to show that is like this. 287 00:19:48,410 --> 00:19:52,150 Let me look at the hyperbolic sine of 2K. 288 00:19:54,750 --> 00:19:58,950 Hyperbolic sines of twice the angles, or twice 289 00:19:58,950 --> 00:20:05,500 the hyperbolic sines of K hyperbolic cosine of K. 290 00:20:05,500 --> 00:20:07,915 All my answers are in terms of tanh K, 291 00:20:07,915 --> 00:20:12,650 so I can make this sine to become a tanh by dividing 292 00:20:12,650 --> 00:20:17,640 by cosh, and then making this cosh squared. 293 00:20:17,640 --> 00:20:22,400 So that is 2 hyperbolic tanh of K. 294 00:20:22,400 --> 00:20:27,730 And then, there's various ways to sort of remember 295 00:20:27,730 --> 00:20:31,350 the identity for hyperbolic cosine squared 296 00:20:31,350 --> 00:20:34,210 minus sine squared is 1. 297 00:20:34,210 --> 00:20:41,320 If I divide by c squared, it becomes 1 minus t squared 298 00:20:41,320 --> 00:20:43,680 is 1 over c squared. 299 00:20:43,680 --> 00:20:46,780 So the hyperbolic cosine squared here 300 00:20:46,780 --> 00:20:54,990 is the inverse of 1 minus hyperbolic tanh squared of K. 301 00:20:54,990 --> 00:21:00,710 Now, for tanh K, we have this identity, is 2 e to the minus 302 00:21:00,710 --> 00:21:02,640 2K tilde. 303 00:21:02,640 --> 00:21:04,700 1 minus the square of that. 304 00:21:08,010 --> 00:21:12,490 If I multiply both sides by-- e to the numerator 305 00:21:12,490 --> 00:21:20,398 and denominator by e to the plus 2K tilde, 306 00:21:20,398 --> 00:21:28,666 hopefully you recognize this as 1 over hyperbolic sine of K 307 00:21:28,666 --> 00:21:30,466 tilde 2K tilde. 308 00:21:33,800 --> 00:21:42,000 So the identity here that was kind of not very transparent, 309 00:21:42,000 --> 00:21:44,880 if I had made the change of variables 310 00:21:44,880 --> 00:21:48,950 to the hyperbolic sine of twice the angle, 311 00:21:48,950 --> 00:21:49,860 had this simple form. 312 00:21:55,830 --> 00:21:59,375 So then the symmetry between K and K tilde 313 00:21:59,375 --> 00:22:01,300 is immediately obvious. 314 00:22:01,300 --> 00:22:06,520 You pick one value of K, or sine K, and then the inverse. 315 00:22:06,520 --> 00:22:09,980 And the inverse of the inverse, you are back to where you are. 316 00:22:09,980 --> 00:22:14,640 So it is clear that this is kind of like an x to y0 1 over x 317 00:22:14,640 --> 00:22:15,670 mapping. 318 00:22:15,670 --> 00:22:20,050 x to 1 over x mapping would also kind of look exactly like this. 319 00:22:20,050 --> 00:22:22,070 In fact, if instead of K tilde and K 320 00:22:22,070 --> 00:22:26,190 I had plotted hyperbolic sine versus hyperbolic sine of twice 321 00:22:26,190 --> 00:22:29,056 the angle, it would have been just the 1 over x curve. 322 00:22:34,900 --> 00:22:44,600 Now, just to give you another example, if I had the function 323 00:22:44,600 --> 00:22:50,740 f of x, which is x 1 plus x squared. 324 00:22:50,740 --> 00:22:53,290 This function, if I divide by x squared, 325 00:22:53,290 --> 00:23:00,170 becomes x inverse 1 plus x inverse squared. 326 00:23:00,170 --> 00:23:03,560 So this is, again, f of x inverse. 327 00:23:03,560 --> 00:23:08,950 So if I evaluate this function for any value like 5, 328 00:23:08,950 --> 00:23:11,980 then I know the value exactly for 1/5. 329 00:23:11,980 --> 00:23:16,780 If I evaluate it for 200, I know it for 1/200, and vice versa. 330 00:23:16,780 --> 00:23:20,110 Our g function is kind of like that. 331 00:23:22,940 --> 00:23:26,070 Now, this function you can see that starts 332 00:23:26,070 --> 00:23:32,520 to go linearly increases with x, and then eventually it 333 00:23:32,520 --> 00:23:37,920 comes down like this must have one maximum. 334 00:23:37,920 --> 00:23:39,160 Where is the maximum? 335 00:23:42,360 --> 00:23:43,390 It has to be 1. 336 00:23:43,390 --> 00:23:46,950 I don't have to take derivatives of everything, et cetera. 337 00:23:46,950 --> 00:23:49,665 If there is one point which corresponds to the maximum, 338 00:23:49,665 --> 00:23:51,450 it's the point that maps to itself. 339 00:23:54,700 --> 00:23:58,250 Now, this function for the Ising model, 340 00:23:58,250 --> 00:23:59,730 I know it has a phase transition. 341 00:23:59,730 --> 00:24:02,190 Or, I guess it has a phase transition. 342 00:24:02,190 --> 00:24:04,920 So there is one point, hopefully one point, 343 00:24:04,920 --> 00:24:06,240 at which it becomes singular. 344 00:24:06,240 --> 00:24:08,080 I don't know, maybe it's three point. 345 00:24:08,080 --> 00:24:12,360 But let's say it's one point at which it becomes singular. 346 00:24:12,360 --> 00:24:16,590 Then I should be able to figure its singularity by precisely 347 00:24:16,590 --> 00:24:19,090 the same argument. 348 00:24:19,090 --> 00:24:25,400 So the function that corresponds to x going to 1 over x 349 00:24:25,400 --> 00:24:26,830 is this hyperbolic sine. 350 00:24:30,180 --> 00:24:35,840 So if there is a point which is the unique point that 351 00:24:35,840 --> 00:24:38,310 corresponds to the singularity, it 352 00:24:38,310 --> 00:24:42,860 has to be the point that is self-dual-- maps on to itself, 353 00:24:42,860 --> 00:24:44,790 just like 1. 354 00:24:44,790 --> 00:24:49,380 So sine of 2 Kc should be 1. 355 00:24:49,380 --> 00:24:50,520 And what is this? 356 00:24:50,520 --> 00:24:53,950 Hyperbolic sine we can write as e to the 2 Kc 357 00:24:53,950 --> 00:24:57,750 minus e to the minus 2 Kc over 2. 358 00:24:57,750 --> 00:25:03,610 We can manipulate this equation slightly 359 00:25:03,610 --> 00:25:12,940 to e to the 4 Kc minus 2 e to the 2 Kc minus 1 equals to 0, 360 00:25:12,940 --> 00:25:17,310 which is a quadratic equation for e to the 2 Kc. 361 00:25:17,310 --> 00:25:23,478 So I can immediately solve for e to the 2 Kc is-- 362 00:25:23,478 --> 00:25:29,305 This has a 2, so I can say 1 minus plus square root 363 00:25:29,305 --> 00:25:32,535 of square of that plus this. 364 00:25:32,535 --> 00:25:35,930 So that is a square root of 2. 365 00:25:35,930 --> 00:25:38,380 The exponential better be positive, 366 00:25:38,380 --> 00:25:40,180 so I can't pick the negative solution. 367 00:25:43,310 --> 00:25:44,660 And so we know-- 368 00:25:44,660 --> 00:25:46,070 AUDIENCE: Isn't it [INAUDIBLE]? 369 00:25:52,190 --> 00:25:54,020 PROFESSOR: Where did I-- OK. 370 00:25:54,020 --> 00:25:57,445 Multiply by e to the 2 Kc. 371 00:25:57,445 --> 00:26:00,300 AUDIENCE: Taking that as correct, I didn't check it-- 372 00:26:00,300 --> 00:26:02,140 PROFESSOR: Yes. 373 00:26:02,140 --> 00:26:03,595 OK. 374 00:26:03,595 --> 00:26:09,340 x squared minus 2b plus-- x plus c equals. 375 00:26:09,340 --> 00:26:12,220 Well, actually, this is-- so x is 376 00:26:12,220 --> 00:26:18,460 b minus plus square root of b squared minus ac. 377 00:26:18,460 --> 00:26:21,350 Our c is negative. 378 00:26:21,350 --> 00:26:22,380 So it's 1 plus 1. 379 00:26:26,310 --> 00:26:33,485 So the critical coupling that we have is 1/2 log of 1 380 00:26:33,485 --> 00:26:37,390 plus square root of 2. 381 00:26:37,390 --> 00:26:40,950 So we know that the critical point of the Ising model 382 00:26:40,950 --> 00:26:44,135 occurs at this value, which you can put on your calculator. 383 00:26:44,135 --> 00:26:46,775 And it is something like this. 384 00:26:50,060 --> 00:26:51,430 There is one assumption. 385 00:26:51,430 --> 00:26:54,790 Of course, there is essentially only one singularity. 386 00:26:54,790 --> 00:26:57,770 And there is one singularity in this free energy. 387 00:26:57,770 --> 00:27:01,661 But if it is, we have solved it for this case. 388 00:27:10,550 --> 00:27:20,390 So I think to emphasize this was discovered 389 00:27:20,390 --> 00:27:26,500 around '50s by Wannier, this idea of duality. 390 00:27:26,500 --> 00:27:30,530 And suddenly, you had exact solution for something 391 00:27:30,530 --> 00:27:33,830 like the square Ising model. 392 00:27:33,830 --> 00:27:38,350 Question is, how much information does it give you? 393 00:27:38,350 --> 00:27:42,880 First of all, the property of self-duality 394 00:27:42,880 --> 00:27:44,590 is that of the square lattice. 395 00:27:54,720 --> 00:27:59,050 So if I had done this on the triangular lattice, 396 00:27:59,050 --> 00:28:03,800 you would've seen that the low-temperature and 397 00:28:03,800 --> 00:28:06,900 high-temperature expansions don't match. 398 00:28:06,900 --> 00:28:11,010 It turns out that in order to construct 399 00:28:11,010 --> 00:28:14,420 the dual of any lattice, what you have to do 400 00:28:14,420 --> 00:28:20,620 is to put, let's say, points in the center of the units 401 00:28:20,620 --> 00:28:26,970 that you have and see what lattice these centers make. 402 00:28:26,970 --> 00:28:29,820 So when you try to do that for the triangular lattice, 403 00:28:29,820 --> 00:28:32,790 you will see that the centers form, actually, hexagonal 404 00:28:32,790 --> 00:28:35,550 lattice, and vice versa. 405 00:28:35,550 --> 00:28:40,060 However, there is a trick using duality 406 00:28:40,060 --> 00:28:43,380 that you can still calculate critical points of hexagonal 407 00:28:43,380 --> 00:28:45,770 and triangular lattice. 408 00:28:45,770 --> 00:28:48,130 And that you will do in one of the problem sets. 409 00:28:50,710 --> 00:28:52,960 OK, secondly. 410 00:28:52,960 --> 00:28:59,870 Again, that trick allows you to go beyond square lattice, 411 00:28:59,870 --> 00:29:02,630 but it turns out that for reasons 412 00:29:02,630 --> 00:29:05,950 that we will see shortly, it is limited. 413 00:29:05,950 --> 00:29:09,080 And you can only do these kinds of dualities 414 00:29:09,080 --> 00:29:12,150 to yourself for two-dimensional lattices. 415 00:29:17,880 --> 00:29:23,320 And what these kinds of mappings in general 416 00:29:23,320 --> 00:29:26,170 give for two-dimensional lattices 417 00:29:26,170 --> 00:29:30,990 is potentially, but not always, the critical value of Kc. 418 00:29:30,990 --> 00:29:33,520 And again, one of the things that you will see 419 00:29:33,520 --> 00:29:40,470 is that you can do this for other models in two dimension. 420 00:29:40,470 --> 00:29:43,590 For example, the Potts model we can 421 00:29:43,590 --> 00:29:50,750 calculate the critical point through this kind of procedure. 422 00:29:50,750 --> 00:29:53,440 However, it doesn't tell you anything 423 00:29:53,440 --> 00:29:56,820 about the nature of the singularity. 424 00:29:56,820 --> 00:30:00,560 So essentially, what we've shown is that on the K-axis, 425 00:30:00,560 --> 00:30:05,060 there is some point maybe that describes the singularity 426 00:30:05,060 --> 00:30:07,700 that you are going to have. 427 00:30:07,700 --> 00:30:11,390 But the shape of this singularity, the exponent 428 00:30:11,390 --> 00:30:12,890 can be anything. 429 00:30:12,890 --> 00:30:17,160 And this mapping does not tell you anything about that. 430 00:30:17,160 --> 00:30:19,130 It does tell you one thing. 431 00:30:19,130 --> 00:30:23,130 We also mentioned that the ratio of amplitudes 432 00:30:23,130 --> 00:30:27,780 above and below for various singular quantities 433 00:30:27,780 --> 00:30:30,740 is something that is universal because 434 00:30:30,740 --> 00:30:33,090 of these mappings from high temperatures 435 00:30:33,090 --> 00:30:34,260 to low temperatures. 436 00:30:34,260 --> 00:30:38,780 Although I don't know what the nature of the singularity is, 437 00:30:38,780 --> 00:30:42,860 I know that the amplitude ratio is [INAUDIBLE]. 438 00:30:42,860 --> 00:30:46,460 So there is some universal information 439 00:30:46,460 --> 00:30:50,210 that one gains beyond the non-universal location 440 00:30:50,210 --> 00:30:53,292 of the critical point, but not that much more. 441 00:30:57,861 --> 00:30:58,360 OK. 442 00:31:01,280 --> 00:31:03,498 Any questions? 443 00:31:03,498 --> 00:31:07,466 AUDIENCE: Is it possible to extract from this line 444 00:31:07,466 --> 00:31:08,954 a differential equation for g? 445 00:31:15,910 --> 00:31:17,110 PROFESSOR: Yes. 446 00:31:17,110 --> 00:31:22,710 And indeed, that differential relation 447 00:31:22,710 --> 00:31:24,280 you will use in one of the problem 448 00:31:24,280 --> 00:31:27,050 sets that I forgot to mention, and will 449 00:31:27,050 --> 00:31:30,900 be used to derive the value of the derivative, which 450 00:31:30,900 --> 00:31:34,196 is related to the energy of the system at the critical point. 451 00:31:34,196 --> 00:31:34,946 But you are right. 452 00:31:43,110 --> 00:31:45,710 This is such a beautiful thing that maybe we 453 00:31:45,710 --> 00:31:49,630 can try to force it to work in higher dimensions. 454 00:31:49,630 --> 00:31:52,775 So let's see if we were to try to go 455 00:31:52,775 --> 00:31:56,945 with this approach for the 3D Ising model what would happen. 456 00:32:02,885 --> 00:32:04,330 So what did we do? 457 00:32:04,330 --> 00:32:08,105 We wrote the low-temperature series, high-temperature series 458 00:32:08,105 --> 00:32:10,420 and compared them. 459 00:32:10,420 --> 00:32:14,980 Again, let's do the cubic lattice, 460 00:32:14,980 --> 00:32:17,690 which I will not really attempt to draw. 461 00:32:24,330 --> 00:32:29,090 That's the system that we want to calculate. 462 00:32:29,090 --> 00:32:30,990 So let's do the low T-series. 463 00:32:33,590 --> 00:32:37,830 Our partition function is going to start with the state 464 00:32:37,830 --> 00:32:41,010 where every spin is, let's say, up. 465 00:32:41,010 --> 00:32:43,720 Three bonds per site on the cubic lattice. 466 00:32:43,720 --> 00:32:46,520 So it's 3 NK. 467 00:32:46,520 --> 00:32:51,550 Again, the trivial degeneracy of 2 for the two possible all plus 468 00:32:51,550 --> 00:32:54,960 or all minus states. 469 00:32:54,960 --> 00:32:57,700 The first excitation is to flip a spin. 470 00:32:57,700 --> 00:33:02,526 So any one of N sites could have been flipped, creating a cube. 471 00:33:02,526 --> 00:33:08,595 A cube has 6 faces that go out, so there is essentially 472 00:33:08,595 --> 00:33:10,460 6 bonds that are broken. 473 00:33:10,460 --> 00:33:13,420 So basically, there is this minus 474 00:33:13,420 --> 00:33:17,750 that is in a box surrounded by plus. 475 00:33:17,750 --> 00:33:24,240 And as you can see, 6 plus minus 1 that go out of that. 476 00:33:24,240 --> 00:33:27,290 The next one would be when we have 2 minuses. 477 00:33:30,280 --> 00:33:36,001 And that can be oriented three ways in three dimensions. 478 00:33:36,001 --> 00:33:39,410 e to the minus 2K. 479 00:33:39,410 --> 00:33:41,220 1, 2 times 4. 480 00:33:41,220 --> 00:33:42,160 8 plus 2. 481 00:33:42,160 --> 00:33:42,900 Times 10. 482 00:33:45,630 --> 00:33:49,460 And so the general term in the series I 483 00:33:49,460 --> 00:33:59,100 have to draw some droplet of minuses in a sea of pluses. 484 00:33:59,100 --> 00:34:04,420 And then I would have e to the minus 2K times the boundary 485 00:34:04,420 --> 00:34:06,014 or the area of this droplet. 486 00:34:10,100 --> 00:34:12,380 Actually, droplets because there could 487 00:34:12,380 --> 00:34:15,179 be multiple droplets as we've seen. 488 00:34:15,179 --> 00:34:16,766 There's no problem with that. 489 00:34:20,933 --> 00:34:25,949 If I do the high T, follow exactly the procedure 490 00:34:25,949 --> 00:34:29,489 I had described before, partition function is going 491 00:34:29,489 --> 00:34:31,875 to be 2 to the number of sites. 492 00:34:31,875 --> 00:34:36,179 Cosh K to the power of the number of bonds and there 493 00:34:36,179 --> 00:34:38,540 are 3 bonds per site. 494 00:34:38,540 --> 00:34:41,250 So there's 3N there. 495 00:34:41,250 --> 00:34:45,739 And then we start to draw our diagrams. 496 00:34:45,739 --> 00:34:47,860 The first diagram is just exactly 497 00:34:47,860 --> 00:34:49,550 like what we had before. 498 00:34:49,550 --> 00:34:51,933 I have to make a square. 499 00:34:56,290 --> 00:35:04,760 And this square can be placed on any face of the cube. 500 00:35:04,760 --> 00:35:09,950 And there are 3 faces that are equivalent. 501 00:35:09,950 --> 00:35:14,810 The next type of diagram that I can draw has 6 bonds in it. 502 00:35:14,810 --> 00:35:18,940 So this could be an example of that. 503 00:35:18,940 --> 00:35:24,710 And if you do the counting, there are 18N of those. 504 00:35:24,710 --> 00:35:26,790 And so you go. 505 00:35:26,790 --> 00:35:28,760 And the genetic term in the series 506 00:35:28,760 --> 00:35:31,965 is going to be some find of a loop. 507 00:35:31,965 --> 00:35:37,860 Again, even number of bonds per site is the operative term. 508 00:35:37,860 --> 00:35:46,050 And then I have t to the power of the number of bonds 509 00:35:46,050 --> 00:35:48,478 making this closed loop. 510 00:35:48,478 --> 00:35:49,390 Or loops. 511 00:35:54,970 --> 00:35:56,840 So you stare at the series and you 512 00:35:56,840 --> 00:35:59,790 see immediately that there is no correspondence 513 00:35:59,790 --> 00:36:00,910 like we saw before. 514 00:36:00,910 --> 00:36:04,280 The coefficient here are 1N, 3N. 515 00:36:04,280 --> 00:36:06,480 Here, there are 1, 3, and 18N. 516 00:36:06,480 --> 00:36:08,580 Powers are 6, 10. 517 00:36:08,580 --> 00:36:10,520 Here are 4, 6. 518 00:36:10,520 --> 00:36:14,070 There's no correspondence between these two. 519 00:36:14,070 --> 00:36:18,770 So there is nothing that one could say. 520 00:36:18,770 --> 00:36:21,520 But you say, I really like this. 521 00:36:21,520 --> 00:36:26,590 So maybe I'll phrase the question differently. 522 00:36:26,590 --> 00:36:31,170 Can I consider some other model whose 523 00:36:31,170 --> 00:36:34,845 high-temperature expansion reproduces 524 00:36:34,845 --> 00:36:41,120 this low-temperature expansion of the Ising model? 525 00:36:41,120 --> 00:36:46,430 So this is the question, can we find 526 00:36:46,430 --> 00:36:59,640 a model whose high T expansion reproduces 527 00:36:59,640 --> 00:37:03,210 low T of 3D Ising model? 528 00:37:10,260 --> 00:37:15,140 So rather than knowing what the model is, 529 00:37:15,140 --> 00:37:18,040 now we are going to kind of work backward 530 00:37:18,040 --> 00:37:22,490 from this graphical picture that we have. 531 00:37:22,490 --> 00:37:28,440 So what would have been the analog thing over here, 532 00:37:28,440 --> 00:37:33,880 let's say that I had this picture of droplets 533 00:37:33,880 --> 00:37:36,050 in the 2D Ising model. 534 00:37:36,050 --> 00:37:41,000 I recognize that I need to make these perimeters out 535 00:37:41,000 --> 00:37:42,640 of something. 536 00:37:42,640 --> 00:37:49,170 And I know that I can make these things that are joined together 537 00:37:49,170 --> 00:37:55,820 to a procedure such as the one that we have over here. 538 00:37:55,820 --> 00:38:01,410 But the unit thing, there it was the elements 539 00:38:01,410 --> 00:38:04,280 that I had along the perimeter. 540 00:38:04,280 --> 00:38:06,580 What is the corresponding unit that I 541 00:38:06,580 --> 00:38:13,640 have for the low temperature series of the Ising model? 542 00:38:13,640 --> 00:38:21,350 I have e to the minus 2K to the power of the number of faces. 543 00:38:21,350 --> 00:38:25,140 So first thing is unit has to be a face. 544 00:38:31,970 --> 00:38:35,430 So basically, what I need to do is 545 00:38:35,430 --> 00:38:44,970 to have a series, which is an expansion in terms of faces, 546 00:38:44,970 --> 00:38:49,140 and then somehow I can glue these faces together, 547 00:38:49,140 --> 00:38:51,825 like I glued these bonds together. 548 00:38:54,820 --> 00:38:59,740 So we found our unit. 549 00:38:59,740 --> 00:39:02,910 The next thing that we need is some kind of a glue 550 00:39:02,910 --> 00:39:07,120 to put all of these LEGO faces together. 551 00:39:07,120 --> 00:39:10,400 So how did we join things together here? 552 00:39:10,400 --> 00:39:16,660 We had these sigmas that were sitting by themselves. 553 00:39:16,660 --> 00:39:18,970 And then putting two sigmas together, 554 00:39:18,970 --> 00:39:22,970 I ensured that when I summed over sigma, 555 00:39:22,970 --> 00:39:27,170 I had to glue two of the T's together. 556 00:39:27,170 --> 00:39:29,720 Can I do the same thing over here? 557 00:39:29,720 --> 00:39:34,100 If I put the sigmas on the corners of these faces, 558 00:39:34,100 --> 00:39:39,140 you can see it doesn't work because here I have three. 559 00:39:39,140 --> 00:39:49,010 So I'm forced to put the sigmas on the lines that 560 00:39:49,010 --> 00:39:49,982 join the faces. 561 00:39:52,640 --> 00:39:56,030 So what I need to do is, therefore, 562 00:39:56,030 --> 00:40:00,510 to have a variable such as this where 563 00:40:00,510 --> 00:40:07,490 I have these sigmas sitting on the-- let's call 564 00:40:07,490 --> 00:40:10,270 this a plaquette, p. 565 00:40:10,270 --> 00:40:12,680 And this plaquette will be having 566 00:40:12,680 --> 00:40:16,485 around it four different bonds. 567 00:40:19,330 --> 00:40:24,071 And if I have the product of these four bonds-- 568 00:40:24,071 --> 00:40:29,311 again, these sigmas being minus plus 1-- 569 00:40:29,311 --> 00:40:34,380 I am forced to glue these sigmas in pairs. 570 00:40:34,380 --> 00:40:37,640 And I can join these things together, these squares, 571 00:40:37,640 --> 00:40:41,520 to make whatever shape that I like that would correspond 572 00:40:41,520 --> 00:40:45,060 to the shapes that I have over there. 573 00:40:45,060 --> 00:40:52,620 So what I need to do is to have for each face a factor of this. 574 00:41:01,137 --> 00:41:07,330 So this is the analog of this factor that I have over here. 575 00:41:07,330 --> 00:41:10,960 And then what I need to do is to do 576 00:41:10,960 --> 00:41:15,310 a product over all plaquettes. 577 00:41:15,310 --> 00:41:18,820 And I sum over all sigma tildes. 578 00:41:21,800 --> 00:41:27,112 And this would be the partition function of some other system. 579 00:41:27,112 --> 00:41:30,210 In this other system, you can see 580 00:41:30,210 --> 00:41:32,910 that if I make its expansion, there 581 00:41:32,910 --> 00:41:35,610 will be a one-to-one correspondence 582 00:41:35,610 --> 00:41:38,860 between the terms in the expansion of this partition 583 00:41:38,860 --> 00:41:42,857 function and the 3D Ising model partition function. 584 00:41:49,190 --> 00:41:51,960 Again, this kind of term we have seen. 585 00:41:51,960 --> 00:41:55,230 If I had put factors of cosh here, which don't really 586 00:41:55,230 --> 00:42:07,330 do much, I can re-express as e to the something-- k tilde 587 00:42:07,330 --> 00:42:13,140 sigma 1p sigma 2p sigma 3p sigma 4p. 588 00:42:13,140 --> 00:42:16,360 Essentially every time you see 1 plus t times 589 00:42:16,360 --> 00:42:19,814 some binary variable, you can rewrite it into this fashion. 590 00:42:23,190 --> 00:42:28,510 So what we have come up with is the following. 591 00:42:28,510 --> 00:42:32,390 That in order to construct the dual 592 00:42:32,390 --> 00:42:35,830 of the three-dimensional Ising model, what we do 593 00:42:35,830 --> 00:42:39,980 is you go all over your cube. 594 00:42:39,980 --> 00:42:46,630 On each bond of it, you put a variable that is minus plus 1. 595 00:42:46,630 --> 00:42:49,060 So previously for the Ising model, 596 00:42:49,060 --> 00:42:52,350 the variables were sitting on the sites. 597 00:42:52,350 --> 00:42:59,030 So Ising, these were site variables. 598 00:42:59,030 --> 00:43:05,200 Whereas, this dual Ising, these are the bond variables 599 00:43:05,200 --> 00:43:06,920 that are minus plus 1. 600 00:43:10,870 --> 00:43:14,990 In the case of the Ising, the interactions 601 00:43:14,990 --> 00:43:20,410 were the product of sites making on a bond. 602 00:43:20,410 --> 00:43:24,010 Whereas, for the dual Ising, the interactions 603 00:43:24,010 --> 00:43:25,530 are around the face. 604 00:43:25,530 --> 00:43:28,274 There's four of them that go around the face. 605 00:43:35,780 --> 00:43:42,720 But whatever this new theory is, we 606 00:43:42,720 --> 00:43:46,960 know that its free energy because of this relation 607 00:43:46,960 --> 00:43:50,375 is related to the free energy of the three-dimensional Ising 608 00:43:50,375 --> 00:43:50,874 model. 609 00:43:53,540 --> 00:43:56,030 Also, we know that the three-dimensional Ising model 610 00:43:56,030 --> 00:43:59,640 has a phase transition between a disordered phase 611 00:43:59,640 --> 00:44:03,290 and the magnetized phase at low temperature. 612 00:44:03,290 --> 00:44:05,110 There is a singularity. 613 00:44:05,110 --> 00:44:09,470 As I span the parameter K of the three-dimensional Ising model, 614 00:44:09,470 --> 00:44:11,206 there is a Kc. 615 00:44:11,206 --> 00:44:14,210 Now, I can find out what that Kc is 616 00:44:14,210 --> 00:44:16,380 because I don't have self-duality. 617 00:44:16,380 --> 00:44:20,830 But I know that as I span the parameter K of the Ising model, 618 00:44:20,830 --> 00:44:26,020 I'm also spanning the parameter K tilde of this new theory. 619 00:44:26,020 --> 00:44:29,230 And since the original model has a phase transition, 620 00:44:29,230 --> 00:44:33,030 this new model must also have a phase transition. 621 00:44:33,030 --> 00:44:35,840 So there exists a Kc for both models. 622 00:44:43,470 --> 00:44:44,210 You say, OK. 623 00:44:44,210 --> 00:44:46,185 Fine. 624 00:44:46,185 --> 00:44:49,150 But there is some complicated kind of Ising model 625 00:44:49,150 --> 00:44:53,830 that you have devised and it has a phase transition. 626 00:44:53,830 --> 00:44:55,810 What's the big deal? 627 00:44:55,810 --> 00:44:57,980 Well, the big deal is that this model is not 628 00:44:57,980 --> 00:45:03,690 supposed to have a phase transition because it 629 00:45:03,690 --> 00:45:07,060 has a different type of symmetry. 630 00:45:07,060 --> 00:45:09,740 The symmetry that we have for the Ising model 631 00:45:09,740 --> 00:45:13,490 is a global symmetry. 632 00:45:13,490 --> 00:45:23,660 That is, the energy of a particular state 633 00:45:23,660 --> 00:45:28,080 is the energy of the state in which all of the spins 634 00:45:28,080 --> 00:45:29,910 are reversed. 635 00:45:29,910 --> 00:45:33,240 Because the form of the energy is bilinear. 636 00:45:33,240 --> 00:45:36,700 If I take all of the sigmas from one configuration 637 00:45:36,700 --> 00:45:39,390 and make them minus in that configuration, 638 00:45:39,390 --> 00:45:41,680 the energy will not change. 639 00:45:41,680 --> 00:45:43,410 But I have to do that globally. 640 00:45:43,410 --> 00:45:46,380 It's a global symmetry. 641 00:45:46,380 --> 00:45:52,660 Now, this model has a local symmetry because what I can do 642 00:45:52,660 --> 00:45:56,310 is I can pick one of the sites. 643 00:45:56,310 --> 00:46:00,810 And out of this site, there are six off 644 00:46:00,810 --> 00:46:04,200 these bonds that are going out on which 645 00:46:04,200 --> 00:46:07,400 there is one of these sigma tilde. 646 00:46:07,400 --> 00:46:12,830 If I pick this site and I change the sign of all of these six 647 00:46:12,830 --> 00:46:19,470 that emanate from this site, the energy will not change. 648 00:46:19,470 --> 00:46:23,630 Because the energy gets contributions from faces. 649 00:46:23,630 --> 00:46:26,910 And you can see that for any one of the faces, 650 00:46:26,910 --> 00:46:30,220 there are two sigmas that have changed. 651 00:46:30,220 --> 00:46:33,110 So the energy, which is the product of all four of them, 652 00:46:33,110 --> 00:46:33,860 has not changed. 653 00:46:37,160 --> 00:46:41,920 So this model has a different form, 654 00:46:41,920 --> 00:46:44,730 which is a local symmetry. 655 00:46:44,730 --> 00:46:49,640 And in fact, it is very much related to gauge theories. 656 00:46:49,640 --> 00:46:54,300 It's a kind of discrete version of the gauge theories 657 00:46:54,300 --> 00:46:56,970 that you have seen in electromagnetism. 658 00:46:56,970 --> 00:46:58,550 Since there are two possibilities, 659 00:46:58,550 --> 00:47:00,240 it's sometimes called a Z2 gauge theory. 660 00:47:04,160 --> 00:47:09,760 Now, the thing about the gauge theories is that there is 661 00:47:09,760 --> 00:47:21,390 a theorem which states that local or these gauge theories, 662 00:47:21,390 --> 00:47:28,730 gauge symmetries, cannot be spontaneously broken. 663 00:47:39,570 --> 00:47:42,030 So for the case of the Ising model, 664 00:47:42,030 --> 00:47:46,130 we have this symmetry between sigma going to minus sigma. 665 00:47:46,130 --> 00:47:49,310 But yet, we know that if I go to low temperature, 666 00:47:49,310 --> 00:47:53,740 I will have a state in which globally all of the spins 667 00:47:53,740 --> 00:47:57,140 are either plus or minus. 668 00:47:57,140 --> 00:48:00,870 So there is a symmetry broken state 669 00:48:00,870 --> 00:48:04,340 which is what we have been discussing. 670 00:48:04,340 --> 00:48:08,210 Now, the reason that that cannot take place in these gauge 671 00:48:08,210 --> 00:48:11,490 theories, I will just sketch what is happening. 672 00:48:14,900 --> 00:48:16,570 Essentially, we have been thinking 673 00:48:16,570 --> 00:48:19,560 in terms of this broken symmetries 674 00:48:19,560 --> 00:48:24,590 by putting an infinitesimal magnetic field. 675 00:48:24,590 --> 00:48:29,930 And we saw that, basically, if I'm at temperatures of 1 676 00:48:29,930 --> 00:48:35,870 over K's that are below some critical value, 677 00:48:35,870 --> 00:48:40,090 then if h is plus, everybody would be plus. 678 00:48:40,090 --> 00:48:44,140 If h is minus, everybody would be minus. 679 00:48:44,140 --> 00:48:50,640 And the reason as you approach h goes to 0 from one site 680 00:48:50,640 --> 00:48:57,000 that you don't get average of 0 is because the difference 681 00:48:57,000 --> 00:48:59,610 between the energy of this state and that state 682 00:48:59,610 --> 00:49:05,640 as you go to 0 temperature is proportional to N times h. 683 00:49:05,640 --> 00:49:09,600 So although h is going to 0 with N being very large, 684 00:49:09,600 --> 00:49:15,800 the influence of infinitesimal h is magnified enormously. 685 00:49:15,800 --> 00:49:19,680 Now, for the case of these local gauge theories, 686 00:49:19,680 --> 00:49:22,150 you cannot have a similar argument. 687 00:49:22,150 --> 00:49:26,790 Because if I pick this spin, let's say one of these bond 688 00:49:26,790 --> 00:49:28,490 spins. 689 00:49:28,490 --> 00:49:35,440 And let's say is its average-- what is its average? 690 00:49:35,440 --> 00:49:38,382 As I said, the h going to 0. 691 00:49:38,382 --> 00:49:42,270 Well, the difference between a state 692 00:49:42,270 --> 00:49:46,010 in which it is plus or the state in which it is minus 693 00:49:46,010 --> 00:49:47,650 is, in fact, 6h. 694 00:49:47,650 --> 00:49:49,670 Because all I need to do is to pick 695 00:49:49,670 --> 00:49:52,180 a site that is close to that bond 696 00:49:52,180 --> 00:49:55,060 and flip all of the spins that are close to that. 697 00:49:55,060 --> 00:49:58,230 All of the K's are equally satisfied. 698 00:49:58,230 --> 00:49:59,890 The difference between that state 699 00:49:59,890 --> 00:50:04,030 and the one where there is a flip is just 6h. 700 00:50:04,030 --> 00:50:06,460 So that remains finite as h goes to 0. 701 00:50:06,460 --> 00:50:11,480 There is no barrier towards flipping those spins. 702 00:50:11,480 --> 00:50:16,400 So there is no broken symmetry in this system. 703 00:50:16,400 --> 00:50:21,810 So this can be proven very nicely and rigorously. 704 00:50:21,810 --> 00:50:28,670 So we have now two statements about this Ising 705 00:50:28,670 --> 00:50:30,330 version of a gauge theory. 706 00:50:30,330 --> 00:50:34,480 First of all, we know that at low temperatures, 707 00:50:34,480 --> 00:50:38,150 still the average value of each bond 708 00:50:38,150 --> 00:50:41,730 is equally likely to be plus or minus. 709 00:50:41,730 --> 00:50:46,430 From that perspective of local values of these bond spins, 710 00:50:46,430 --> 00:50:50,710 it is as disordered as the highest-temperature phase. 711 00:50:50,710 --> 00:50:54,990 Yet, because of its duality to the three-dimensional Ising 712 00:50:54,990 --> 00:50:58,650 model, we know that it undergoes some kind 713 00:50:58,650 --> 00:51:01,785 of a singularity going from high temperatures 714 00:51:01,785 --> 00:51:03,450 to low temperatures. 715 00:51:03,450 --> 00:51:07,070 So there is probably some kind of a phase transition, 716 00:51:07,070 --> 00:51:09,930 but it has to be very different from any of the phase 717 00:51:09,930 --> 00:51:13,270 transitions that we have discussed so far because there 718 00:51:13,270 --> 00:51:18,440 is no spontaneous symmetry breaking. 719 00:51:18,440 --> 00:51:20,200 So what's going on? 720 00:52:00,750 --> 00:52:04,720 Now, later on in the course, we will 721 00:52:04,720 --> 00:52:09,520 see another example of this that is much less 722 00:52:09,520 --> 00:52:12,480 exotic than a gauge theory, but it 723 00:52:12,480 --> 00:52:16,010 has the same kind of principle applicable to it. 724 00:52:16,010 --> 00:52:20,690 There will be a phase transition without local symmetry breaking 725 00:52:20,690 --> 00:52:26,930 in something like a superfluid in two dimensions. 726 00:52:26,930 --> 00:52:30,820 So one thing that that phase transition and this one 727 00:52:30,820 --> 00:52:34,220 have in common is, again, the lack 728 00:52:34,220 --> 00:52:42,190 of this local-order parameter from symmetry breaking. 729 00:52:42,190 --> 00:52:46,560 And both of them share something that 730 00:52:46,560 --> 00:52:52,050 was pointed out by Wegner once this puzzle emerged, 731 00:52:52,050 --> 00:53:01,510 which is that one has to look at some kind of a global-- well, 732 00:53:01,510 --> 00:53:03,030 I shouldn't even call it global. 733 00:53:08,016 --> 00:53:09,890 It is something that is called a Wilson loop. 734 00:53:26,760 --> 00:53:29,440 So the idea is the following. 735 00:53:29,440 --> 00:53:32,580 That we have these variable sigma tilde 736 00:53:32,580 --> 00:53:36,280 that are sitting on the bonds of a lattice. 737 00:53:36,280 --> 00:53:42,810 Now, the problem is that with this local transformations, 738 00:53:42,810 --> 00:53:46,740 I can very easily make this sigma go to minus sigma. 739 00:53:46,740 --> 00:53:52,410 So that is not a good thing to consider. 740 00:53:52,410 --> 00:53:56,680 However, what was the problem? 741 00:53:56,680 --> 00:53:58,390 Let's say I pick this site. 742 00:53:58,390 --> 00:54:00,820 All of the sigmas that went out of that site 743 00:54:00,820 --> 00:54:03,030 I changed to minus themselves. 744 00:54:03,030 --> 00:54:04,700 That was the gauge transformation, 745 00:54:04,700 --> 00:54:07,280 but it became minus. 746 00:54:07,280 --> 00:54:12,950 But if I multiply the sigma with another sigma that goes out 747 00:54:12,950 --> 00:54:16,450 of that site, then I have cured that problem. 748 00:54:16,450 --> 00:54:19,750 If I changed this to minus itself, 749 00:54:19,750 --> 00:54:21,840 this changes, this changes. 750 00:54:21,840 --> 00:54:25,000 The product remains [INAUDIBLE]. 751 00:54:25,000 --> 00:54:27,100 But then I have the problem here. 752 00:54:27,100 --> 00:54:34,410 So what I do is I make a long loop. 753 00:54:34,410 --> 00:54:38,060 I look at the expectation value. 754 00:54:38,060 --> 00:54:41,820 So this Wilson loop is the product 755 00:54:41,820 --> 00:54:46,830 of sigma tilde around a loop. 756 00:54:51,720 --> 00:54:56,720 And what I can do is I can look at the average 757 00:54:56,720 --> 00:54:58,304 of that quantity. 758 00:54:58,304 --> 00:55:01,540 The average of that quantity is something 759 00:55:01,540 --> 00:55:05,260 that is clearly invariant to this kind of gauge 760 00:55:05,260 --> 00:55:06,770 transformation. 761 00:55:06,770 --> 00:55:10,610 So the signatures of a potential phase transition 762 00:55:10,610 --> 00:55:14,580 could potentially be revealed by looking at something like this. 763 00:55:17,680 --> 00:55:19,880 But clearly, that is a quantity that 764 00:55:19,880 --> 00:55:24,030 is always also going to be positive. 765 00:55:24,030 --> 00:55:25,990 So the thing that I am looking at 766 00:55:25,990 --> 00:55:29,627 is not that this quantity is, say, positive 767 00:55:29,627 --> 00:55:32,800 in one phase and 0 in the other phase. 768 00:55:32,800 --> 00:55:36,790 It is on how this quantity depends 769 00:55:36,790 --> 00:55:40,345 on the shape and characteristics of these loop. 770 00:55:42,960 --> 00:55:47,950 So what I can do is I can calculate this average, both 771 00:55:47,950 --> 00:55:53,410 in high temperatures and low temperatures, and compare them. 772 00:55:53,410 --> 00:56:05,193 So if I look at-- let's starts with, yeah, high temperatures. 773 00:56:05,193 --> 00:56:09,500 The high-temperature expansion. 774 00:56:09,500 --> 00:56:12,430 So I want to calculate the expectation 775 00:56:12,430 --> 00:56:18,550 value of the product of sigma tilde, let's call it i, 776 00:56:18,550 --> 00:56:22,580 where i belongs to some kind of a loop c. 777 00:56:22,580 --> 00:56:27,370 So c is all of these bonds. 778 00:56:27,370 --> 00:56:31,210 I want to calculate that expectation value. 779 00:56:31,210 --> 00:56:34,240 How do I calculate that expectation value? 780 00:56:34,240 --> 00:56:40,440 Well, I have to sum over all configurations 781 00:56:40,440 --> 00:56:48,900 of this product with a weight. 782 00:56:48,900 --> 00:56:50,760 What is my weight? 783 00:56:50,760 --> 00:56:55,800 My weight is this factor of product 784 00:56:55,800 --> 00:57:02,202 over all plaquettes of 1 plus t sigma tilde sigma tilde sigma 785 00:57:02,202 --> 00:57:05,191 tilde for the plaquette. 786 00:57:11,090 --> 00:57:13,400 So this is the weight that I have. 787 00:57:13,400 --> 00:57:16,560 I can put the hyperbolic cosine or not put it, 788 00:57:16,560 --> 00:57:18,970 it doesn't matter. 789 00:57:18,970 --> 00:57:21,715 But then this weight has to be properly normalized. 790 00:57:21,715 --> 00:57:25,510 It means that I have to divide by something 791 00:57:25,510 --> 00:57:28,830 in the denominator, which basically does not 792 00:57:28,830 --> 00:57:31,461 include the quantity that I am averaging. 793 00:57:46,400 --> 00:57:50,080 So the graphs that occur in the denominator 794 00:57:50,080 --> 00:57:52,960 are the things that we have been discussing. 795 00:57:52,960 --> 00:57:56,020 Essentially, I start with 1. 796 00:57:56,020 --> 00:57:59,520 The next term is to put these faces together 797 00:57:59,520 --> 00:58:02,860 to make a cube, and then more complicated shapes. 798 00:58:02,860 --> 00:58:05,520 That, essentially, every bond is going 799 00:58:05,520 --> 00:58:09,194 to be having some kind of a complement. 800 00:58:09,194 --> 00:58:12,280 Well, but what about the terms in the numerator? 801 00:58:12,280 --> 00:58:14,550 For the terms in the numerator, I 802 00:58:14,550 --> 00:58:17,970 have these factors of sigma tilde 803 00:58:17,970 --> 00:58:22,410 that are lying all around this loop. 804 00:58:22,410 --> 00:58:25,540 And I'm summing over the two possible values. 805 00:58:25,540 --> 00:58:30,610 So in order that summing over this does not give me 0, 806 00:58:30,610 --> 00:58:34,970 I better make sure that there is a complement to that. 807 00:58:34,970 --> 00:58:38,630 The complement to that can only come from here. 808 00:58:38,630 --> 00:58:42,610 So for example, I would put a face over here 809 00:58:42,610 --> 00:58:46,760 that ensures that that is squared and that is squared. 810 00:58:46,760 --> 00:58:48,305 But then I have this one. 811 00:58:48,305 --> 00:58:49,840 Then I will put another one here. 812 00:58:49,840 --> 00:58:52,155 I will put another one, et cetera. 813 00:58:55,400 --> 00:59:00,240 And you can see that the lowest-order term that I would 814 00:59:00,240 --> 00:59:08,110 get is this factor that characterize each one of them. 815 00:59:10,870 --> 00:59:15,688 Raised to the power of the area of this loop c. 816 00:59:19,150 --> 00:59:21,680 You can ask higher-order terms. 817 00:59:21,680 --> 00:59:25,550 You can kind of build a hat on top of this. 818 00:59:25,550 --> 00:59:27,790 This you can put anywhere, again, 819 00:59:27,790 --> 00:59:30,530 along this area of this thing. 820 00:59:30,530 --> 00:59:33,280 So the next correction in this series you can see 821 00:59:33,280 --> 00:59:37,450 is also going to be something that will be 1 plus t 822 00:59:37,450 --> 00:59:40,020 to the fourth times the area. 823 00:59:40,020 --> 00:59:43,880 The point is that as you add more and more terms, 824 00:59:43,880 --> 00:59:48,410 you preserve the structure that the whole thing is going 825 00:59:48,410 --> 00:59:53,180 to be proportional to the area of loop 826 00:59:53,180 --> 00:59:56,484 times some function of this parameter t. 827 01:00:01,180 --> 01:00:07,300 So what we know is that if I take the expectation 828 01:00:07,300 --> 01:00:11,340 value of this entity, then it's logarithm 829 01:00:11,340 --> 01:00:14,470 will be proportional in high temperatures 830 01:00:14,470 --> 01:00:16,144 to the area of this loop. 831 01:00:21,470 --> 01:00:25,680 Now, what happens if I try to do a low T-series 832 01:00:25,680 --> 01:00:26,700 for the same quantity? 833 01:00:34,090 --> 01:00:38,530 So I have to start with a configuration 834 01:00:38,530 --> 01:00:45,850 at low temperature that minimizes the energy. 835 01:00:50,930 --> 01:00:55,200 One configuration clearly is one where all of the sigmas 836 01:00:55,200 --> 01:00:55,970 are plus. 837 01:00:59,730 --> 01:01:03,570 And that will give me a term. 838 01:01:03,570 --> 01:01:07,180 If I am calculating the partition function 839 01:01:07,180 --> 01:01:09,630 in the denominator, there will be 840 01:01:09,630 --> 01:01:13,310 a term that will be proportional to e to whatever 841 01:01:13,310 --> 01:01:18,720 this K tilde is per face. 842 01:01:18,720 --> 01:01:22,112 And there are three faces of a cube. 843 01:01:22,112 --> 01:01:23,570 There are N cubed, so there will be 844 01:01:23,570 --> 01:01:30,210 3K tilde N for the configuration that is all plus. 845 01:01:30,210 --> 01:01:34,290 But this is not the only low-temperature configuration. 846 01:01:34,290 --> 01:01:36,470 That is what we were discussing. 847 01:01:36,470 --> 01:01:40,780 Because I can pick a site out of N site 848 01:01:40,780 --> 01:01:42,630 and the 6 bonds that go out of it, 849 01:01:42,630 --> 01:01:45,110 I can make minus themselves. 850 01:01:45,110 --> 01:01:48,020 And the energy would be exactly the same. 851 01:01:48,020 --> 01:01:53,650 So whereas the Ising model, I had a multiplicity of 2. 852 01:01:53,650 --> 01:02:01,670 Here, there is a multiplicity of 2 to the N. 853 01:02:01,670 --> 01:02:06,950 So that's the lowest term in the low-temperature expansion 854 01:02:06,950 --> 01:02:08,500 of the partition function. 855 01:02:08,500 --> 01:02:10,090 I'm doing the partition function, 856 01:02:10,090 --> 01:02:11,381 which is the denominator first. 857 01:02:13,770 --> 01:02:16,090 Then, what can I do? 858 01:02:16,090 --> 01:02:19,520 Then, let's say I start with a configuration where all of them 859 01:02:19,520 --> 01:02:20,930 are pluses. 860 01:02:20,930 --> 01:02:23,450 There are, of course, 2 to the N gauge copies of that. 861 01:02:23,450 --> 01:02:25,780 So whatever I do to this configuration, 862 01:02:25,780 --> 01:02:28,610 I can do the analog in all of the others. 863 01:02:28,610 --> 01:02:32,740 But let's keep the copy where all of the sigmas are plus, 864 01:02:32,740 --> 01:02:37,270 and then I flip one of the sigmas to minus. 865 01:02:37,270 --> 01:02:40,190 Then, it's essentially-- think of a cube. 866 01:02:40,190 --> 01:02:43,850 There is a line that was plus and I made it minus. 867 01:02:43,850 --> 01:02:48,840 There are four faces going out of that that were previously 868 01:02:48,840 --> 01:02:52,920 plus K, now become minus K. So I will 869 01:02:52,920 --> 01:02:58,890 have 2K tilde times 4 because of these four things 870 01:02:58,890 --> 01:03:01,270 that are going out. 871 01:03:01,270 --> 01:03:04,966 And the bond I can orient in x-, y-, or z-direction. 872 01:03:04,966 --> 01:03:09,000 So there are 3N possibilities. 873 01:03:09,000 --> 01:03:13,750 And so I could have a series such as this in the denominator 874 01:03:13,750 --> 01:03:17,660 where subsequent terms would be to put more and more 875 01:03:17,660 --> 01:03:19,390 minus in this particular [INAUDIBLE]. 876 01:03:22,100 --> 01:03:26,670 Now, let's see how these series would affect the sum 877 01:03:26,670 --> 01:03:31,160 that I would have to do in order to calculate this expectation 878 01:03:31,160 --> 01:03:32,660 value. 879 01:03:32,660 --> 01:03:39,900 For any one of these configurations, 880 01:03:39,900 --> 01:03:42,330 since I am kind of looking at the ground state, 881 01:03:42,330 --> 01:03:45,120 let's say they're all pluses. 882 01:03:45,120 --> 01:03:51,270 Clearly, the contribution to this product will be unity. 883 01:03:51,270 --> 01:03:53,870 That does not change. 884 01:03:53,870 --> 01:03:56,540 But now, let's think about the configurations 885 01:03:56,540 --> 01:04:01,260 in which one of the bonds is made to flip. 886 01:04:01,260 --> 01:04:05,320 As long as that bond does not touch 887 01:04:05,320 --> 01:04:09,340 any of the bonds that are part of the loop, 888 01:04:09,340 --> 01:04:12,540 the value of the loop will remain the same. 889 01:04:12,540 --> 01:04:16,280 So let's say that the loop has P sites. 890 01:04:16,280 --> 01:04:22,590 So this is number of bonds in c. 891 01:04:22,590 --> 01:04:23,953 Let's call it Pc. 892 01:04:28,590 --> 01:04:38,047 For these, with this weight, the value of Wilson loop 893 01:04:38,047 --> 01:04:38,880 would still be plus. 894 01:04:41,520 --> 01:04:45,050 But for the times where I have picked one of these 895 01:04:45,050 --> 01:04:50,180 to become minus, then the product becomes minus. 896 01:04:50,180 --> 01:04:55,730 So for the remaining Pc times of this factor, 897 01:04:55,730 --> 01:04:59,390 e to the minus 2K tilde times 4, rather 898 01:04:59,390 --> 01:05:00,870 than having plus I will have minus. 899 01:05:06,410 --> 01:05:09,870 So you can see that-- what is this? 900 01:05:09,870 --> 01:05:11,190 N should be up here. 901 01:05:13,860 --> 01:05:20,340 That the difference between what is in the numerator 902 01:05:20,340 --> 01:05:25,480 and what is in the denominator of this low-temperature series 903 01:05:25,480 --> 01:05:29,200 has to do with the bonds that have 904 01:05:29,200 --> 01:05:34,070 been sitting as part of the Wilson loop. 905 01:05:34,070 --> 01:05:38,890 And if I imagine that this is a small quantity 906 01:05:38,890 --> 01:05:41,520 and write these as exponentials, you 907 01:05:41,520 --> 01:05:44,210 can see that this is going to start 908 01:05:44,210 --> 01:05:49,960 with e to the minus 2K tilde times 4 times 909 01:05:49,960 --> 01:05:53,605 the perimeter of this cluster. 910 01:05:53,605 --> 01:05:54,960 Of this loop. 911 01:05:57,680 --> 01:06:02,250 And you can go and look at higher and higher order terms. 912 01:06:02,250 --> 01:06:06,670 The point is that in high temperature, 913 01:06:06,670 --> 01:06:11,710 the property of the shape of the loop that determines 914 01:06:11,710 --> 01:06:15,806 this expectation value is its area. 915 01:06:15,806 --> 01:06:20,987 Whereas, in the low-temperature expansion, it is its perimeter. 916 01:06:25,090 --> 01:06:27,390 So you could, for example, calculate 917 01:06:27,390 --> 01:06:33,240 the-- for a large loop, the log of this quantity 918 01:06:33,240 --> 01:06:36,680 and divide it by the perimeter. 919 01:06:36,680 --> 01:06:39,910 And in the low-temperature phase, it would be finite. 920 01:06:39,910 --> 01:06:41,755 In the high-temperature phase, it 921 01:06:41,755 --> 01:06:44,055 would go to 0 because the area scale 922 01:06:44,055 --> 01:06:46,580 is bigger than the perimeter. 923 01:06:46,580 --> 01:06:49,900 So we have found something that is an analog of an older 924 01:06:49,900 --> 01:06:53,930 parameter and can distinguish the different phases, 925 01:06:53,930 --> 01:06:57,144 and it is reflected in the way that the correlations take 926 01:06:57,144 --> 01:06:57,644 place. 927 01:07:33,710 --> 01:07:38,730 Let's try to sort of think about some potential physics that 928 01:07:38,730 --> 01:07:41,860 could be related to this. 929 01:07:41,860 --> 01:07:47,230 Let's start with the gauge theory aspect of this. 930 01:07:47,230 --> 01:07:51,210 Well, the one gauge theory that you probably know 931 01:07:51,210 --> 01:07:56,010 is quantum electrodynamics, whose action 932 01:07:56,010 --> 01:07:59,540 you would write in the following way. 933 01:07:59,540 --> 01:08:02,845 The action would involve an integration over space 934 01:08:02,845 --> 01:08:05,760 as well as time. 935 01:08:05,760 --> 01:08:10,380 And by appropriate rescalings, you 936 01:08:10,380 --> 01:08:14,000 can write the energy that is in the electromagnetic field 937 01:08:14,000 --> 01:08:20,899 as d mu A mu minus d mu A mu squared, where 938 01:08:20,899 --> 01:08:33,450 A is the 4-vector potential out of which you can construct 939 01:08:33,450 --> 01:08:37,930 the electric field and magnetic field. 940 01:08:37,930 --> 01:08:41,880 And the reason this is a gauge theory 941 01:08:41,880 --> 01:08:55,750 is because if I take A and add to it some function of phi 942 01:08:55,750 --> 01:09:01,399 of x and t, as long as I take the derivative 943 01:09:01,399 --> 01:09:04,660 of this function, you can see that the change here 944 01:09:04,660 --> 01:09:08,540 would be d mu d mu minus d mu by d mu. 945 01:09:08,540 --> 01:09:10,160 There is really no change. 946 01:09:10,160 --> 01:09:14,500 And we know that basically you can 947 01:09:14,500 --> 01:09:20,010 choose whatever value of this phase-- 948 01:09:20,010 --> 01:09:24,073 this gauge fixing potential over here. 949 01:09:27,321 --> 01:09:30,880 Now, this is the electromagnetic field by itself. 950 01:09:30,880 --> 01:09:34,479 If you want to couple it to something like matter 951 01:09:34,479 --> 01:09:37,420 or electrons, you write something 952 01:09:37,420 --> 01:09:46,300 like i d bar, which is some derivative, 953 01:09:46,300 --> 01:09:50,864 and then you have e A bar. 954 01:09:50,864 --> 01:09:55,254 If there's a mass to this object, you would put it here. 955 01:09:55,254 --> 01:09:58,320 This would be something like psi bar psi. 956 01:09:58,320 --> 01:10:01,380 This would be something that would describe 957 01:10:01,380 --> 01:10:05,510 the coupling of this electromagnetic field 958 01:10:05,510 --> 01:10:08,801 to some charged particle, such as the electron. 959 01:10:14,720 --> 01:10:19,680 And this entire thing satisfies the gauge symmetry provided 960 01:10:19,680 --> 01:10:24,420 that once you do this, you also replace psi with e 961 01:10:24,420 --> 01:10:30,260 to the ie phi psi. 962 01:10:30,260 --> 01:10:37,140 So the same phi, if it appears in both, essentially the change 963 01:10:37,140 --> 01:10:40,190 in A that you would have from here 964 01:10:40,190 --> 01:10:42,390 will be compensated by the change 965 01:10:42,390 --> 01:10:45,590 that you would get from the derivative acting on this phase 966 01:10:45,590 --> 01:10:46,760 factor. 967 01:10:46,760 --> 01:10:51,030 And so the whole thing is not affected. 968 01:10:51,030 --> 01:10:55,470 What we have constructed in this model 969 01:10:55,470 --> 01:10:59,550 is kind of an Ising analog of this theory. 970 01:10:59,550 --> 01:11:03,350 Because the Hamiltonian that we have, 971 01:11:03,350 --> 01:11:07,730 which carries the weight after exponentiate 972 01:11:07,730 --> 01:11:11,740 of the different configurations, has a part 973 01:11:11,740 --> 01:11:19,540 which is the sum over all of the plaquettes of this sigma tilde 974 01:11:19,540 --> 01:11:25,020 sigma tilde sigma tilde sigma tilde, 975 01:11:25,020 --> 01:11:28,910 the four bonds around the plaquette. 976 01:11:28,910 --> 01:11:32,870 We could put some kind of a coupling here if we want to. 977 01:11:32,870 --> 01:11:36,980 And the analog of this transformation 978 01:11:36,980 --> 01:11:41,650 that we have-- well, maybe it will become more apparent if I 979 01:11:41,650 --> 01:11:47,500 add the next term, which is the analog of the coupling 980 01:11:47,500 --> 01:11:48,290 to matter. 981 01:11:48,290 --> 01:11:55,110 If I put a spin and then sigma tilde ij sj. 982 01:11:55,110 --> 01:12:01,940 So again, imagine that we have our cubic lattice, 983 01:12:01,940 --> 01:12:07,100 or some other lattice, in which we have these variables sigma 984 01:12:07,100 --> 01:12:10,680 tilde that are sitting on the bonds. 985 01:12:10,680 --> 01:12:14,730 And the first term is the product around the face. 986 01:12:14,730 --> 01:12:17,840 And the second term I put these variables 987 01:12:17,840 --> 01:12:19,620 s that are sitting here. 988 01:12:24,770 --> 01:12:30,350 And I have made a coupling between these two s's. 989 01:12:30,350 --> 01:12:32,830 So if the sigma tildes were not there, 990 01:12:32,830 --> 01:12:36,730 I could make an Ising model with s's being plus or minus, 991 01:12:36,730 --> 01:12:40,080 which are coupled across nearest neighbors. 992 01:12:40,080 --> 01:12:42,830 What I do is that the strength of that coupling I 993 01:12:42,830 --> 01:12:47,010 make to be plus or minus, depending 994 01:12:47,010 --> 01:12:51,770 on the value of the gauge field, if you like. 995 01:12:51,770 --> 01:12:55,310 This Ising gauge field that is sitting over there. 996 01:12:55,310 --> 01:13:00,030 Now, the analog of these symmetries that we have for QED 997 01:13:00,030 --> 01:13:01,700 is as follows. 998 01:13:01,700 --> 01:13:08,860 I can pick a particular s i and change its value to minus. 999 01:13:08,860 --> 01:13:15,725 And I can pick all of the sigma tildes that go out of that i 1000 01:13:15,725 --> 01:13:22,390 to the neighbours and simultaneously make them minus. 1001 01:13:22,390 --> 01:13:25,580 And then this energy would not change. 1002 01:13:25,580 --> 01:13:30,240 First of all, let's say if I pick this site 1003 01:13:30,240 --> 01:13:33,970 and change its face from being plus or minus, 1004 01:13:33,970 --> 01:13:38,795 then the bonds that-- the sigma tildes that go out of it 1005 01:13:38,795 --> 01:13:42,460 will change their values to minus themselves. 1006 01:13:42,460 --> 01:13:46,820 Since each one of the faces contains two of them, 1007 01:13:46,820 --> 01:13:51,880 the value of the energy from here is not changed. 1008 01:13:51,880 --> 01:13:57,400 Since the couplings to the neighboring s's involve 1009 01:13:57,400 --> 01:13:59,800 the sigma tildes that sit between them, 1010 01:13:59,800 --> 01:14:02,170 and I have flipped both s and sigma tilde, 1011 01:14:02,170 --> 01:14:05,070 those do not change irrespective of what 1012 01:14:05,070 --> 01:14:08,240 I do with the face of all the other ones. 1013 01:14:08,240 --> 01:14:13,970 So I have made an Ising, or binary version, 1014 01:14:13,970 --> 01:14:16,260 of this transformation, constructed 1015 01:14:16,260 --> 01:14:21,280 a model that has, except being Ising symmetry, 1016 01:14:21,280 --> 01:14:23,735 a lot of the properties that you would 1017 01:14:23,735 --> 01:14:25,230 have for this kind of action. 1018 01:14:29,930 --> 01:14:37,530 Now again, continuing with that, this difference between why 1019 01:14:37,530 --> 01:14:42,080 we see an area rule or a perimeter rule 1020 01:14:42,080 --> 01:14:47,600 has some physical consequence that is worth mentioning. 1021 01:14:47,600 --> 01:14:51,430 And this, again, has not much to do 1022 01:14:51,430 --> 01:14:53,700 with the main thrust of this course, 1023 01:14:53,700 --> 01:14:57,690 but just a matter of overall education. 1024 01:14:57,690 --> 01:14:59,290 It is useful to know. 1025 01:15:02,500 --> 01:15:07,160 So in this picture, where one of the dimensions 1026 01:15:07,160 --> 01:15:14,990 corresponds to time, imagine that you 1027 01:15:14,990 --> 01:15:21,320 create a kind of a Wilson loop which 1028 01:15:21,320 --> 01:15:25,640 is very long in one direction that I 1029 01:15:25,640 --> 01:15:29,670 want to think of as being time. 1030 01:15:29,670 --> 01:15:36,680 And the analog of this action that we have discussed 1031 01:15:36,680 --> 01:15:43,425 is to create a pair of charges, separate them by a distance x, 1032 01:15:43,425 --> 01:15:50,350 propagate them for a long time t, and then [INAUDIBLE]. 1033 01:15:50,350 --> 01:15:53,430 And ask, what is the contribution of a configuration 1034 01:15:53,430 --> 01:15:58,940 such as this to their action on average? 1035 01:15:58,940 --> 01:16:06,420 And so you would say that if particles are at a distance x, 1036 01:16:06,420 --> 01:16:11,130 they are subject to some kind of a potential v of x. 1037 01:16:11,130 --> 01:16:13,720 And if this potential has been propagated 1038 01:16:13,720 --> 01:16:17,090 in time for a length or a duration 1039 01:16:17,090 --> 01:16:24,710 that I will call T, that the effect that it has 1040 01:16:24,710 --> 01:16:33,910 on the system is to have an interaction such as this 1041 01:16:33,910 --> 01:16:39,310 in the action propagated over a time T. 1042 01:16:39,310 --> 01:16:42,274 So I should have something like this. 1043 01:16:45,230 --> 01:16:48,730 So this should somehow be related 1044 01:16:48,730 --> 01:16:54,540 to this average of the Wilson loop in manner 1045 01:16:54,540 --> 01:16:58,530 to not be made precise, but very rough just 1046 01:16:58,530 --> 01:17:01,190 to get the general idea. 1047 01:17:01,190 --> 01:17:07,970 And so what we have said is that the value of this Wilson loop 1048 01:17:07,970 --> 01:17:10,313 has different behaviors at high T 1049 01:17:10,313 --> 01:17:15,780 and low T in its dependence on shape. 1050 01:17:15,780 --> 01:17:20,480 And that high temperatures, it is proportional to the area. 1051 01:17:20,480 --> 01:17:24,790 So it should be proportional to xT. 1052 01:17:24,790 --> 01:17:28,820 Whereas, in low temperature, it is proportional to perimeter. 1053 01:17:28,820 --> 01:17:36,600 So it should be proportional to x plus T. 1054 01:17:36,600 --> 01:17:43,285 So if I read off the form of V of x from these two dependents, 1055 01:17:43,285 --> 01:17:48,360 you will see that V of x goes proportionately 1056 01:17:48,360 --> 01:17:53,660 to x in one regime. 1057 01:17:53,660 --> 01:17:58,190 And once I divide by T, the leading 1058 01:17:58,190 --> 01:18:01,130 coefficients-- so this is essentially 1059 01:18:01,130 --> 01:18:07,460 I want to look at the limit where T is becoming very large. 1060 01:18:07,460 --> 01:18:11,460 So this thing when I divide by T goes to a constant. 1061 01:18:15,900 --> 01:18:19,720 And I can very roughly interpret this 1062 01:18:19,720 --> 01:18:22,120 as the interaction between particles 1063 01:18:22,120 --> 01:18:29,660 that are separated by x via this kind of theory. 1064 01:18:29,660 --> 01:18:31,890 And I see that for particles that 1065 01:18:31,890 --> 01:18:36,900 are separated by x in that kind of theory, 1066 01:18:36,900 --> 01:18:42,070 there is a weak coupling-- high temperature 1067 01:18:42,070 --> 01:18:49,430 corresponds to weak coupling-- where the further apart I 1068 01:18:49,430 --> 01:18:55,310 go, the potential that is bringing them together 1069 01:18:55,310 --> 01:18:59,510 becomes linearly stronger. 1070 01:18:59,510 --> 01:19:01,490 So this is what is called confinement. 1071 01:19:07,000 --> 01:19:13,970 Whereas, in the other limit of low temperatures, 1072 01:19:13,970 --> 01:19:18,950 or strong coupling, what you find 1073 01:19:18,950 --> 01:19:22,990 is that the interaction between them 1074 01:19:22,990 --> 01:19:25,175 essentially goes to a constant. 1075 01:19:25,175 --> 01:19:27,050 The potential goes to a constant, 1076 01:19:27,050 --> 01:19:29,210 so the force would go to 0. 1077 01:19:29,210 --> 01:19:31,720 So they are asymptotically free. 1078 01:19:38,500 --> 01:19:43,800 So if I start with this kind of theory 1079 01:19:43,800 --> 01:19:47,770 and try to interpret it in the language of quantum field 1080 01:19:47,770 --> 01:19:52,050 theory as something that describes interaction 1081 01:19:52,050 --> 01:19:57,690 between particles, I find that it has potentially two phases. 1082 01:19:57,690 --> 01:20:01,110 One phase in which the particles of the theory 1083 01:20:01,110 --> 01:20:04,950 are strongly bind together, like quarks 1084 01:20:04,950 --> 01:20:07,830 that are inside the nucleus. 1085 01:20:07,830 --> 01:20:09,940 And you can try to separate the quarks, 1086 01:20:09,940 --> 01:20:11,390 but they would snap back. 1087 01:20:11,390 --> 01:20:13,680 You can't have free quarks. 1088 01:20:13,680 --> 01:20:19,120 And then there is another phase where essentially the particles 1089 01:20:19,120 --> 01:20:20,930 don't see each other. 1090 01:20:20,930 --> 01:20:24,810 And indeed, quarks right inside the nucleus 1091 01:20:24,810 --> 01:20:26,260 are essentially free. 1092 01:20:26,260 --> 01:20:29,430 We can sort of regard them as free particles. 1093 01:20:29,430 --> 01:20:33,780 So this theory actually has aspects 1094 01:20:33,780 --> 01:20:38,730 of what is known as confinement and asymptotic freedom 1095 01:20:38,730 --> 01:20:41,550 within quantum chromodynamics. 1096 01:20:41,550 --> 01:20:43,590 The difference is that in this theory, 1097 01:20:43,590 --> 01:20:47,720 there is a phase transition and the two behaviors 1098 01:20:47,720 --> 01:20:49,625 are separated from each other. 1099 01:20:49,625 --> 01:20:53,590 Whereas in QCD, it's essentially a crossover 1100 01:20:53,590 --> 01:20:55,980 from one behavior to another behavior 1101 01:20:55,980 --> 01:20:58,790 without the phase transition. 1102 01:20:58,790 --> 01:21:04,140 So we started by thinking about these Ising models. 1103 01:21:04,140 --> 01:21:06,620 And we kind of branched into theories 1104 01:21:06,620 --> 01:21:10,170 that describe loops, theories that describe droplets, 1105 01:21:10,170 --> 01:21:13,930 theories that describe gauge couplings, et cetera. 1106 01:21:13,930 --> 01:21:17,850 So you can see that that nice, simple line of the partition 1107 01:21:17,850 --> 01:21:22,180 function that I have written for you has within it 1108 01:21:22,180 --> 01:21:25,010 a lot of interesting complexity. 1109 01:21:25,010 --> 01:21:27,870 We kind of went off the direction 1110 01:21:27,870 --> 01:21:31,260 that we wanted to go with phase transitions, 1111 01:21:31,260 --> 01:21:33,910 so we will remedy that next time, coming 1112 01:21:33,910 --> 01:21:39,350 back to thinking about how to think in terms of the Ising 1113 01:21:39,350 --> 01:21:43,975 model, and try to do more with understanding the behavior 1114 01:21:43,975 --> 01:21:47,550 and singularities of this partition function.