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 00:00:22,960 --> 00:00:26,390 PROFESSOR: OK, let's start. 9 00:00:26,390 --> 00:00:30,600 So it's good to remind ourselves why 10 00:00:30,600 --> 00:00:34,390 we are doing what we are doing today. 11 00:00:34,390 --> 00:00:36,900 So we've seen that in a number of cases, 12 00:00:36,900 --> 00:00:41,750 we look at something like the coexistence line of gas 13 00:00:41,750 --> 00:00:47,070 and liquid that terminates at the critical point. 14 00:00:47,070 --> 00:00:50,620 And that in the vicinity of this critical point, 15 00:00:50,620 --> 00:00:54,730 we see various thermodynamic quantities and correlation 16 00:00:54,730 --> 00:00:57,570 functions that have properties that 17 00:00:57,570 --> 00:01:02,590 are independent of the materials that are considered. 18 00:01:02,590 --> 00:01:07,980 So this led to this concept of universality, 19 00:01:07,980 --> 00:01:14,720 and we were able to justify that by looking at properties 20 00:01:14,720 --> 00:01:17,230 of this statistical field. 21 00:01:23,650 --> 00:01:28,730 And we ended up with [INAUDIBLE] normalization 22 00:01:28,730 --> 00:01:32,320 group procedure, which classified 23 00:01:32,320 --> 00:01:35,780 the different universality classes according 24 00:01:35,780 --> 00:01:42,130 to the number of components of the order parameter, 25 00:01:42,130 --> 00:01:46,380 the thing that categorizes the coexisting phases, 26 00:01:46,380 --> 00:01:49,250 and the dimensionality of space. 27 00:01:49,250 --> 00:01:51,260 And that, in particular something 28 00:01:51,260 --> 00:01:56,950 like a liquid-gas system, would correspond to n equals to 1. 29 00:01:56,950 --> 00:01:59,400 Another example that would correspond to that 30 00:01:59,400 --> 00:02:04,950 would be, for example, a mixture of two metals 31 00:02:04,950 --> 00:02:06,320 in a binary alloy. 32 00:02:06,320 --> 00:02:09,729 You can have the different components mixed 33 00:02:09,729 --> 00:02:13,530 or phase separated from each other. 34 00:02:13,530 --> 00:02:21,220 So the normalization group method gave us the reason 35 00:02:21,220 --> 00:02:24,270 for there is this universality, but we 36 00:02:24,270 --> 00:02:26,940 found that calculating the exponent 37 00:02:26,940 --> 00:02:30,970 was a hard task coming from four dimensions. 38 00:02:30,970 --> 00:02:36,970 So the question is, given that these models or these numbers, 39 00:02:36,970 --> 00:02:40,390 the singularities here are universal, 40 00:02:40,390 --> 00:02:43,950 can we obtain them from a different perspective? 41 00:02:43,950 --> 00:02:48,210 And so let's say we are focused on this kind 42 00:02:48,210 --> 00:02:51,780 of liquid-gas system, which belong to this n 43 00:02:51,780 --> 00:02:54,860 equals to 1 universality class. 44 00:02:54,860 --> 00:02:57,440 So we can try to imagine the simplest model 45 00:02:57,440 --> 00:02:59,930 that we can try to solve that belongs 46 00:02:59,930 --> 00:03:02,220 to that universality class. 47 00:03:02,220 --> 00:03:06,570 And again, maybe thinking in terms of a binary alloy, 48 00:03:06,570 --> 00:03:09,520 something that has two possible values. 49 00:03:09,520 --> 00:03:11,550 In the liquid-gas, it could be cells 50 00:03:11,550 --> 00:03:15,090 that are either empty or filled with a particle. 51 00:03:15,090 --> 00:03:24,080 And so this binary model is this Ising model, 52 00:03:24,080 --> 00:03:26,860 where, at each side of a lattice, 53 00:03:26,860 --> 00:03:30,380 we put a variable that is minus plus 1. 54 00:03:30,380 --> 00:03:34,030 And so the idea is, again, if I take any one of these Ising 55 00:03:34,030 --> 00:03:36,790 models and I coarse-grain them, I 56 00:03:36,790 --> 00:03:39,190 will end up with the same statistical field, 57 00:03:39,190 --> 00:03:42,210 and it would have the same universality class. 58 00:03:42,210 --> 00:03:46,370 But if I make a sufficiently simple version of these models, 59 00:03:46,370 --> 00:03:50,840 maybe I can do something else and solve them in a manner 60 00:03:50,840 --> 00:03:54,450 that these critical behaviors can come up 61 00:03:54,450 --> 00:03:57,180 in an easier fashion. 62 00:03:57,180 --> 00:04:04,660 So let's say we are interested in two dimensions or three 63 00:04:04,660 --> 00:04:06,250 dimensions. 64 00:04:06,250 --> 00:04:09,080 I can draw two dimensions better. 65 00:04:09,080 --> 00:04:11,190 We draw a square lattice. 66 00:04:11,190 --> 00:04:15,580 On each side of it, we put one of these variables. 67 00:04:15,580 --> 00:04:19,690 And in order to capture this tendency 68 00:04:19,690 --> 00:04:24,150 that there is a possibility of coexistence where 69 00:04:24,150 --> 00:04:28,240 you have patches that are made of liquid or gas, 70 00:04:28,240 --> 00:04:31,740 or made of copper or zinc in our binary alloy, 71 00:04:31,740 --> 00:04:35,740 we need to have a tendency for things that are close to each 72 00:04:35,740 --> 00:04:39,700 other to be in the same state so that we can capture 73 00:04:39,700 --> 00:04:49,510 by a Hamiltonian, which is a sum over nearest neighbors, that 74 00:04:49,510 --> 00:04:54,250 gives an enhanced weight if they are parallel. 75 00:04:54,250 --> 00:04:58,210 And whatever that coupling is, once it 76 00:04:58,210 --> 00:05:05,660 is rescaled by kT, this combination, the energy divided 77 00:05:05,660 --> 00:05:12,830 by kT, we can parametrize by a dimensionless number k. 78 00:05:12,830 --> 00:05:15,550 And calculating the behavior of the system 79 00:05:15,550 --> 00:05:17,600 as a function of temperature, as the strength 80 00:05:17,600 --> 00:05:20,670 of the coupling in this simplified model, 81 00:05:20,670 --> 00:05:23,790 amounts to calculating the partition function 82 00:05:23,790 --> 00:05:28,740 as a function of a parameter k, which is a sum over, 83 00:05:28,740 --> 00:05:32,270 if I'm in a system that has n sites all to the n 84 00:05:32,270 --> 00:05:39,490 configurations, of a weight that tries to make variables that 85 00:05:39,490 --> 00:05:45,290 are next to each other to be in the same state. 86 00:05:45,290 --> 00:05:48,680 So clearly, what is captured here 87 00:05:48,680 --> 00:05:53,070 is a competition between energy-- energy 88 00:05:53,070 --> 00:05:57,190 would like everybody to be in the same state-- 89 00:05:57,190 --> 00:05:58,670 versus entropy. 90 00:05:58,670 --> 00:06:02,770 Entropy wants to have different states at each site. 91 00:06:02,770 --> 00:06:06,400 So you'll have a factor of 2 per site as opposed 92 00:06:06,400 --> 00:06:10,670 to everybody being aligned, which is essentially one state. 93 00:06:10,670 --> 00:06:14,330 And so that competition potentially 94 00:06:14,330 --> 00:06:17,520 could lead you to a phase transition between something 95 00:06:17,520 --> 00:06:21,080 that has coexistent at low temperature and something that 96 00:06:21,080 --> 00:06:24,650 is disordered at high temperatures. 97 00:06:24,650 --> 00:06:28,320 So now we have just recast the problem. 98 00:06:28,320 --> 00:06:32,725 Rather than having a partition function which was a functional 99 00:06:32,725 --> 00:06:36,680 integral over all configurations of the statistical field, 100 00:06:36,680 --> 00:06:38,960 I have to do this partition function, which 101 00:06:38,960 --> 00:06:41,790 is finite number of configurations, 102 00:06:41,790 --> 00:06:43,760 but it's still an interacting theory. 103 00:06:43,760 --> 00:06:48,990 I cannot independently move the variable at each site. 104 00:06:48,990 --> 00:06:52,300 So the question is, are there approaches 105 00:06:52,300 --> 00:06:55,470 by which I can calculate this? 106 00:06:55,470 --> 00:07:00,310 And one set of approaches is to start with a limit 107 00:07:00,310 --> 00:07:04,670 that I can solve and start expanding on that. 108 00:07:04,670 --> 00:07:08,660 And these expansions that are analogous to the perturbation 109 00:07:08,660 --> 00:07:13,240 expansions that we learned in 8.333 110 00:07:13,240 --> 00:07:16,320 about interacting systems, in this case 111 00:07:16,320 --> 00:07:20,240 are usually called series expansions. 112 00:07:20,240 --> 00:07:22,155 One would perform them on a lattice. 113 00:07:25,980 --> 00:07:31,740 Now, I kind of hinted at two limits of the problem 114 00:07:31,740 --> 00:07:35,220 that we know exactly what is happening, 115 00:07:35,220 --> 00:07:39,360 and those lead to two different series expansions. 116 00:07:39,360 --> 00:07:42,245 One of them is the low-temperature expansions. 117 00:07:49,980 --> 00:07:54,400 And here the idea is that I know what 118 00:07:54,400 --> 00:07:56,660 the system is doing at T equals to 0. 119 00:07:56,660 --> 00:08:01,210 At T equals to 0, I have to find the configuration that 120 00:08:01,210 --> 00:08:03,360 minimizes the energy. 121 00:08:03,360 --> 00:08:07,350 T equals to 0 is also equivalent to k going to infinity. 122 00:08:07,350 --> 00:08:11,570 I have to find a state that maximizes this weight, 123 00:08:11,570 --> 00:08:19,390 and that's obviously the case where all of the spins 124 00:08:19,390 --> 00:08:23,500 are either plus or minus. 125 00:08:23,500 --> 00:08:28,360 So all sigma i equals to plus 1 or all sigma i equals 126 00:08:28,360 --> 00:08:29,580 to minus 1. 127 00:08:32,970 --> 00:08:36,440 But for the sake of doing one or the other, 128 00:08:36,440 --> 00:08:44,900 let's imagine that they are all plus 129 00:08:44,900 --> 00:08:52,700 and that I am solving the problem for the generalization 130 00:08:52,700 --> 00:08:56,560 of square cube to d-dimensional lattice. 131 00:08:56,560 --> 00:08:59,790 After all, we were doing d dimensions in general. 132 00:08:59,790 --> 00:09:05,990 So in d dimensions, each spin will have d neighbors. 133 00:09:05,990 --> 00:09:10,330 And so if I ask, what is the weight that I 134 00:09:10,330 --> 00:09:15,100 will get at 0 temperature, essentially each spin 135 00:09:15,100 --> 00:09:20,360 would have d factors of k. 136 00:09:20,360 --> 00:09:26,590 So the weight that I would get at T equals to 0-- let's call 137 00:09:26,590 --> 00:09:34,210 that Z of T equals to 0-- is simply e to the dNk. 138 00:09:36,920 --> 00:09:39,170 There are N sites. 139 00:09:39,170 --> 00:09:43,155 Each one of them has d neighbors in d dimensions. 140 00:09:43,155 --> 00:09:46,550 Of course, each one of them has 2d neighbours, 141 00:09:46,550 --> 00:09:49,570 but then I have to count the number of neighbors per site. 142 00:09:49,570 --> 00:09:53,140 So basically, this bond is shared by two neighbors, 143 00:09:53,140 --> 00:09:56,610 so half of it contributes to this site. 144 00:09:56,610 --> 00:09:58,390 And there are two possibilities. 145 00:09:58,390 --> 00:10:03,700 So the partition function at T equals to 0 is simply this. 146 00:10:03,700 --> 00:10:08,000 It's just the contribution of the two ground states. 147 00:10:08,000 --> 00:10:14,060 Now, we are interested in the limit where T goes to 0. 148 00:10:14,060 --> 00:10:18,260 So at T equals to 0, I know what is happening. 149 00:10:18,260 --> 00:10:23,800 Now, what I will get as I allow temperature 150 00:10:23,800 --> 00:10:29,880 to be larger, at some cost I am able to flip 151 00:10:29,880 --> 00:10:34,270 some of these spins from, say, the plus to minus. 152 00:10:34,270 --> 00:10:46,280 And I will get, in this case, islands of negative spin 153 00:10:46,280 --> 00:10:47,345 in sea of plus. 154 00:10:51,500 --> 00:10:58,280 And these islands will give a contribution 155 00:10:58,280 --> 00:11:07,690 that is going to be exponentially small in k 156 00:11:07,690 --> 00:11:12,750 and something to do with the bonds that I have broken. 157 00:11:12,750 --> 00:11:18,930 And by broken, I mean gone from the high energy, well, 158 00:11:18,930 --> 00:11:23,630 highly satisfied plus-plus state to the unsatisfied plus-minus 159 00:11:23,630 --> 00:11:30,620 state, in fact, 2k times number of broken bonds. 160 00:11:40,130 --> 00:11:44,930 So we can very easily write the first few terms in this series. 161 00:11:44,930 --> 00:11:52,980 So let's make a list of the excitation, or island 162 00:11:52,980 --> 00:11:59,850 that I can make, how many ways I can 163 00:11:59,850 --> 00:12:07,300 make this, which I will call degeneracy, 164 00:12:07,300 --> 00:12:09,935 and the number of broken bonds. 165 00:12:20,030 --> 00:12:22,160 So clearly, the simplest thing that I 166 00:12:22,160 --> 00:12:29,790 can do in a sea of pluses is to make one island, which 167 00:12:29,790 --> 00:12:34,630 is simply a site that has been previously plus 168 00:12:34,630 --> 00:12:36,600 and now has gone to become a minus. 169 00:12:39,260 --> 00:12:48,790 And this particular excitation can occur any one of N places 170 00:12:48,790 --> 00:12:52,290 if I have a lattice of size N. And I'm 171 00:12:52,290 --> 00:12:56,470 going to ignore any corrections that I may have from the edges. 172 00:12:56,470 --> 00:13:00,350 If you want you can do that, and that'd be more precise. 173 00:13:00,350 --> 00:13:02,780 But let's focus, essentially, on things 174 00:13:02,780 --> 00:13:06,710 that are proportional to N. 175 00:13:06,710 --> 00:13:10,660 Then how many bonds have I broken? 176 00:13:10,660 --> 00:13:13,850 You can see that in two dimensions, 177 00:13:13,850 --> 00:13:16,160 I have broken four bonds. 178 00:13:16,160 --> 00:13:19,130 In three dimensions, it would have been six. 179 00:13:19,130 --> 00:13:23,530 So essentially, it is twice the number 180 00:13:23,530 --> 00:13:28,580 of dimensions that is taking place. 181 00:13:28,580 --> 00:13:30,760 And so the contribution to energy 182 00:13:30,760 --> 00:13:36,105 is going to be e to the minus 2d. 183 00:13:38,710 --> 00:13:41,950 And I went from plus k to minus k, 184 00:13:41,950 --> 00:13:44,940 so in fact, I would have to multiply by 2k. 185 00:13:49,030 --> 00:13:51,840 Now, the next thing that I can do, 186 00:13:51,840 --> 00:13:55,810 the lowest energy excitation is to put two minuses that 187 00:13:55,810 --> 00:13:58,655 are next to each other in this sea of pluses. 188 00:14:03,413 --> 00:14:05,900 OK? 189 00:14:05,900 --> 00:14:08,780 Now, you can see that in two dimension, 190 00:14:08,780 --> 00:14:14,290 I can orient this pair along the x-direction 191 00:14:14,290 --> 00:14:16,200 or along the y-direction. 192 00:14:16,200 --> 00:14:20,210 And in general, there would be d directions, so I would have dN. 193 00:14:23,140 --> 00:14:25,480 Roughly, you would say that the number of bonds 194 00:14:25,480 --> 00:14:33,590 that you have broken is twice what 195 00:14:33,590 --> 00:14:38,300 you had before if the two were separate. 196 00:14:38,300 --> 00:14:42,430 But there is this thing in between 197 00:14:42,430 --> 00:14:46,000 that is now actually a satisfied bond. 198 00:14:46,000 --> 00:14:50,400 So you can convince yourself that, actually, 199 00:14:50,400 --> 00:14:53,480 if the two of them were separate, 200 00:14:53,480 --> 00:14:58,530 these two minus excitations, I would have 4d. 201 00:14:58,530 --> 00:15:04,690 But because I joined them, essentially I have 2d minus 1 202 00:15:04,690 --> 00:15:08,020 from each one of them, and there's two of those. 203 00:15:08,020 --> 00:15:10,630 And of course, the next lowest excitation 204 00:15:10,630 --> 00:15:14,060 would indeed be to have two minuses that 205 00:15:14,060 --> 00:15:22,180 have no site, are totally separate from each other. 206 00:15:22,180 --> 00:15:25,790 And the contribution-- the number of these, 207 00:15:25,790 --> 00:15:28,220 well, this is something to count. 208 00:15:28,220 --> 00:15:32,070 The first one can any one of N places. 209 00:15:32,070 --> 00:15:37,830 The next one can be in any one of N minus 2d minus 1 places. 210 00:15:37,830 --> 00:15:41,710 It cannot be on the same one, and it cannot be in any 211 00:15:41,710 --> 00:15:43,370 of the 2d neighbors. 212 00:15:43,370 --> 00:15:45,810 And I should have double counting, 213 00:15:45,810 --> 00:15:49,590 so there is a factor of 2 here. 214 00:15:49,590 --> 00:15:54,230 And the cost of this is simply twice that, so this is 4d. 215 00:15:57,540 --> 00:16:05,630 So if I want to start writing a partition function expanded 216 00:16:05,630 --> 00:16:08,580 beyond what I have at 0 temperature, 217 00:16:08,580 --> 00:16:14,010 what I would have would be 2e to the dNk. 218 00:16:14,010 --> 00:16:16,500 There's zero temperature contribution. 219 00:16:16,500 --> 00:16:29,894 I would have 1 plus N e to the minus 4dk plus dN 220 00:16:29,894 --> 00:16:39,560 e to the minus 4 2d minus 1 k. 221 00:16:39,560 --> 00:16:43,650 And then from the other one that I have written, 222 00:16:43,650 --> 00:16:51,536 N N minus 2d minus 1 over 2e to the minus 8dk. 223 00:16:51,536 --> 00:16:55,180 And I can keep going and adding higher and higher order 224 00:16:55,180 --> 00:16:56,641 terms of the series. 225 00:16:56,641 --> 00:16:58,750 OK? 226 00:16:58,750 --> 00:16:59,380 OK. 227 00:16:59,380 --> 00:17:02,310 Once I have the partition function, 228 00:17:02,310 --> 00:17:05,339 I can start calculating the energy, which 229 00:17:05,339 --> 00:17:12,159 would be minus d log Z with respect to d beta. 230 00:17:14,859 --> 00:17:15,665 What is beta? 231 00:17:19,710 --> 00:17:26,430 Well, I said that this factor is something like 1 232 00:17:26,430 --> 00:17:31,130 over kT, which is beta and J. 233 00:17:31,130 --> 00:17:35,380 So assuming that I have a fixed energy 234 00:17:35,380 --> 00:17:39,600 and I'm changing temperature, and the variations of k 235 00:17:39,600 --> 00:17:43,370 are reflecting the inverse temperature beta, 236 00:17:43,370 --> 00:17:48,310 then I can certainly multiply here a J and a J, 237 00:17:48,310 --> 00:17:50,960 which is a constant. 238 00:17:50,960 --> 00:17:56,150 And all I need to do is to take a J d 239 00:17:56,150 --> 00:18:02,710 by dk of log of the expression that I have above there. 240 00:18:02,710 --> 00:18:03,210 OK? 241 00:18:03,210 --> 00:18:06,380 So let's take the log of that expression. 242 00:18:06,380 --> 00:18:16,530 I have log of 2 plus dNk from here. 243 00:18:16,530 --> 00:18:21,560 And then I have log of 1 plus terms in a series 244 00:18:21,560 --> 00:18:25,410 that I have calculated perturbity. 245 00:18:25,410 --> 00:18:29,500 Now, log of 1 plus a small quantity I 246 00:18:29,500 --> 00:18:32,220 can always expand as a small quantity 247 00:18:32,220 --> 00:18:34,550 minus x squared over 2. 248 00:18:34,550 --> 00:18:37,520 You may worry whether or not, with N ultimately 249 00:18:37,520 --> 00:18:41,910 going to infinity, this is a small quantity. 250 00:18:41,910 --> 00:18:43,970 Neglecting that for the time being, 251 00:18:43,970 --> 00:18:48,180 if I look at this as log of 1 plus a small quantity, 252 00:18:48,180 --> 00:18:55,190 from here I would get N e to the minus 4dk plus dN 253 00:18:55,190 --> 00:19:03,670 e to the minus 4 2d minus 1 k plus N N minus 1 254 00:19:03,670 --> 00:19:12,380 minus 2d over 2e to the minus 8dk, and so forth. 255 00:19:12,380 --> 00:19:18,970 But then remember that log of 1 plus x 256 00:19:18,970 --> 00:19:25,570 is x minus x squared over 2 plus x cubed over 3, and so forth. 257 00:19:25,570 --> 00:19:29,540 So if x is my small quantity, I will 258 00:19:29,540 --> 00:19:33,320 have a correction, which is minus x squared over 2. 259 00:19:33,320 --> 00:19:36,280 Let's just do it for this first term. 260 00:19:36,280 --> 00:19:43,250 I will get minus N squared over 2e to the minus 8dk, 261 00:19:43,250 --> 00:19:48,140 and there will be a whole bunch of higher-order terms. 262 00:19:48,140 --> 00:19:50,850 OK? 263 00:19:50,850 --> 00:19:53,300 Now, where am I going with this? 264 00:19:53,300 --> 00:19:59,380 Ultimately, I want to calculate various quantities that 265 00:19:59,380 --> 00:20:05,300 are extensive in the sense that they are proportional to N, 266 00:20:05,300 --> 00:20:07,840 and when I divide by N I will get something 267 00:20:07,840 --> 00:20:10,050 like energy per site. 268 00:20:10,050 --> 00:20:14,960 So if I do that, I have to divide this whole thing by N, 269 00:20:14,960 --> 00:20:17,710 I can see that here I have a term that 270 00:20:17,710 --> 00:20:22,980 is log T divided by N. In the N goes to infinity limit, 271 00:20:22,980 --> 00:20:26,960 it's a term that has order of 1 over N I can neglect. 272 00:20:26,960 --> 00:20:30,940 But all of these other terms are proportional to N. 273 00:20:30,940 --> 00:20:37,630 And when I divide by N, I can drop these factors of N. 274 00:20:37,630 --> 00:20:40,470 Well, except that I have a couple of terms 275 00:20:40,470 --> 00:20:43,670 that, if I had left by themselves, 276 00:20:43,670 --> 00:20:46,330 potentially could have been order of N squared. 277 00:20:46,330 --> 00:20:49,460 I have N squared over 2, but fortunately, you 278 00:20:49,460 --> 00:20:52,870 can see that it cancels out over there. 279 00:20:52,870 --> 00:20:56,690 Now, the reason this happens, and also 280 00:20:56,690 --> 00:20:59,560 the reason this series is legitimate, 281 00:20:59,560 --> 00:21:03,280 is because we already did something very similar to that 282 00:21:03,280 --> 00:21:08,595 in 8.333 when we were doing these cumulant expansions. 283 00:21:08,595 --> 00:21:12,000 And when we were doing these cumulant expansions, 284 00:21:12,000 --> 00:21:16,660 we obtained the series for, then, the grand partition 285 00:21:16,660 --> 00:21:19,900 function, which was a whole bunch of terms. 286 00:21:19,900 --> 00:21:26,130 But when we took the log, only the connected terms survived. 287 00:21:26,130 --> 00:21:28,280 And the connected terms were the things 288 00:21:28,280 --> 00:21:30,190 that, because they had a center of mass, 289 00:21:30,190 --> 00:21:34,130 were giving you a factor that was proportional to volume. 290 00:21:34,130 --> 00:21:38,230 And here you expect that ultimately everything 291 00:21:38,230 --> 00:21:41,250 that I will get here, if I calculate, let's say, log Z 292 00:21:41,250 --> 00:21:44,650 properly and then divide by N, it 293 00:21:44,650 --> 00:21:48,100 should be something that is order of 1. 294 00:21:48,100 --> 00:21:52,240 It shouldn't be order of N, or N cubed, 295 00:21:52,240 --> 00:21:54,360 or any of these other terms. 296 00:21:54,360 --> 00:21:59,030 So essentially, the purpose of all of these higher-order terms 297 00:21:59,030 --> 00:22:04,550 is really to subtract off things such as this 298 00:22:04,550 --> 00:22:07,710 that would arise in the counting when 299 00:22:07,710 --> 00:22:11,080 we look at islands and excitations that 300 00:22:11,080 --> 00:22:12,250 are disconnected. 301 00:22:12,250 --> 00:22:15,310 So I could have something right here, something right here. 302 00:22:15,310 --> 00:22:18,930 So this would be, essentially, a product 303 00:22:18,930 --> 00:22:23,400 of the contributions of these different islands. 304 00:22:23,400 --> 00:22:29,630 As long as they are disconnected from some term in the series, 305 00:22:29,630 --> 00:22:33,300 there would be a subtraction that would get rid of that 306 00:22:33,300 --> 00:22:36,830 and would ensure that these additional factors of N, 307 00:22:36,830 --> 00:22:41,350 because I can move each island over the entire lattice, 308 00:22:41,350 --> 00:22:43,360 would disappear. 309 00:22:43,360 --> 00:22:51,960 So I have this series, and now I can basically 310 00:22:51,960 --> 00:22:54,080 take the derivatives. 311 00:22:54,080 --> 00:22:58,230 So I have minus J, and I take d by dk 312 00:22:58,230 --> 00:23:00,760 of the various terms that have survived. 313 00:23:00,760 --> 00:23:02,950 The first one is d. 314 00:23:02,950 --> 00:23:08,030 So dJ is essentially the energy pair site 315 00:23:08,030 --> 00:23:09,840 that I would have at 0 temperature. 316 00:23:09,840 --> 00:23:12,960 I have strength J, deeper site. 317 00:23:12,960 --> 00:23:14,980 And then the excitations will start 318 00:23:14,980 --> 00:23:17,310 to reduce that and correct that. 319 00:23:17,310 --> 00:23:25,690 And so from here, I would get minus 4d e to the minus 4dk. 320 00:23:25,690 --> 00:23:33,815 From here, I would get minus 4 2d minus 1 d 321 00:23:33,815 --> 00:23:39,380 e to the minus 4 2d minus 1 k. 322 00:23:39,380 --> 00:23:42,800 The N-squared terms disappeared. 323 00:23:42,800 --> 00:23:46,570 So I would have 2d plus 1 over 2. 324 00:23:46,570 --> 00:23:51,200 But then it gets multiplied by 8d when I take a derivative. 325 00:23:51,200 --> 00:24:02,830 So I will get plus 4d 2d plus 1 e to the minus 8dk, 326 00:24:02,830 --> 00:24:07,160 and so forth in the series. 327 00:24:07,160 --> 00:24:08,880 OK? 328 00:24:08,880 --> 00:24:14,360 So these terms that are subtraction, 329 00:24:14,360 --> 00:24:16,720 you can see that you can really easily 330 00:24:16,720 --> 00:24:22,200 connect to these primary excitations. 331 00:24:22,200 --> 00:24:25,720 If you like, this term corresponds 332 00:24:25,720 --> 00:24:30,790 to taking two of these and colliding them with each other. 333 00:24:30,790 --> 00:24:32,550 They cannot be on top of each other. 334 00:24:32,550 --> 00:24:34,750 They cannot be next to each other. 335 00:24:34,750 --> 00:24:37,250 And so there is a subtraction because a number 336 00:24:37,250 --> 00:24:39,570 of configurations are not allowed. 337 00:24:39,570 --> 00:24:42,710 So this is, in some sense, a kind 338 00:24:42,710 --> 00:24:47,710 of expansion in these excitations 339 00:24:47,710 --> 00:24:50,882 and the interactions among these excitations. 340 00:24:50,882 --> 00:24:52,210 OK? 341 00:24:52,210 --> 00:24:54,920 Now presumably, what is happening 342 00:24:54,920 --> 00:24:57,700 is that at very low temperature, you 343 00:24:57,700 --> 00:25:02,415 are going to get these individual simple excitations 344 00:25:02,415 --> 00:25:05,790 with a little bit of interaction between them. 345 00:25:05,790 --> 00:25:09,870 As you increase the temperature, the size of these islands 346 00:25:09,870 --> 00:25:11,280 will get bigger and bigger. 347 00:25:11,280 --> 00:25:13,850 They start to merge into each other. 348 00:25:13,850 --> 00:25:15,800 Configurations that you would see 349 00:25:15,800 --> 00:25:19,550 will be big islands in a sea. 350 00:25:19,550 --> 00:25:23,200 And presumably, the size of these islands 351 00:25:23,200 --> 00:25:26,330 is some measure of the correlation length 352 00:25:26,330 --> 00:25:29,980 that you have in this low temperature state. 353 00:25:29,980 --> 00:25:32,920 Eventually, this correlation length 354 00:25:32,920 --> 00:25:36,110 will hit the size of the system. 355 00:25:36,110 --> 00:25:40,160 And then the starting point, that you had a sea of pluses 356 00:25:40,160 --> 00:25:43,140 and you're exciting around it, it is no longer valid. 357 00:25:43,140 --> 00:25:48,170 If you like, that vacuum state has become unstable, 358 00:25:48,170 --> 00:25:51,830 and this series, the way that we are constructing it, 359 00:25:51,830 --> 00:25:55,970 ceases to go beyond that point. 360 00:25:55,970 --> 00:25:57,760 OK? 361 00:25:57,760 --> 00:26:01,670 So let's take another step. 362 00:26:01,670 --> 00:26:04,590 If I've calculated the energy, I could also 363 00:26:04,590 --> 00:26:12,310 calculate the heat capacity, which is d by dT. 364 00:26:12,310 --> 00:26:16,310 Actually, I expect the heat capacity to be extensive also, 365 00:26:16,310 --> 00:26:19,040 so I'll divide by N. So I will look 366 00:26:19,040 --> 00:26:23,210 at the heat capacity per site. 367 00:26:23,210 --> 00:26:26,170 I know that the natural units of heat capacity 368 00:26:26,170 --> 00:26:30,830 are kB, which has dimensions of energy divided by temperature. 369 00:26:30,830 --> 00:26:33,580 So I divide by kB. 370 00:26:33,580 --> 00:26:37,546 So here I will have kBT. 371 00:26:37,546 --> 00:26:47,010 But then I notice that kBT, these are related inversely 372 00:26:47,010 --> 00:26:55,000 to K, capital K. It is J over K. So I can write this as J 373 00:26:55,000 --> 00:27:06,340 over K, and d by d-- 1 over K will give me a minus K squared. 374 00:27:09,010 --> 00:27:14,530 I will have a factor of 1 over J, and the 1 over J 375 00:27:14,530 --> 00:27:18,330 actually cancels this factor of J here. 376 00:27:18,330 --> 00:27:20,670 So all I need to do-- well, actually, let me write 377 00:27:20,670 --> 00:27:27,520 it, J d by dK of this E over N. 378 00:27:27,520 --> 00:27:29,820 So the expression that I have above, I 379 00:27:29,820 --> 00:27:32,570 have to take another derivative with respect 380 00:27:32,570 --> 00:27:36,567 to K multiplied by minus K squared over J. 381 00:27:36,567 --> 00:27:38,990 The J's cancel out, and so I will 382 00:27:38,990 --> 00:27:42,630 have a series that will be proportional to K squared. 383 00:27:42,630 --> 00:27:45,800 Good, I made everything dimensionless. 384 00:27:45,800 --> 00:27:49,200 And then the first term that will contribute 385 00:27:49,200 --> 00:27:58,800 will be 16 d squared e to the minus 4dK, from here. 386 00:27:58,800 --> 00:28:04,640 And from here, I will get 16 2d minus 1 387 00:28:04,640 --> 00:28:11,680 squared d e to the minus 4 2d minus 1 K. 388 00:28:11,680 --> 00:28:19,416 And from here, I would get minus 32d 2d plus 1 e 389 00:28:19,416 --> 00:28:24,258 to the minus 8dK, and then so forth. 390 00:28:24,258 --> 00:28:26,520 OK? 391 00:28:26,520 --> 00:28:30,910 So you can see that this is something 392 00:28:30,910 --> 00:28:34,860 that is a kind of mechanical process, 393 00:28:34,860 --> 00:28:38,540 that in the '40s and '50s, without even 394 00:28:38,540 --> 00:28:43,420 the need for any computers, people could sit down and draw 395 00:28:43,420 --> 00:28:46,390 excitations, provide these terms, 396 00:28:46,390 --> 00:28:51,160 and go to higher and higher order terms in the series. 397 00:28:51,160 --> 00:28:54,190 Now, the reason that they were going to do this 398 00:28:54,190 --> 00:28:57,660 is that the expectation that if I 399 00:28:57,660 --> 00:29:01,810 look at this heat capacity as a function of something 400 00:29:01,810 --> 00:29:09,600 like temperature, which is e to the 1 over k, for example, 401 00:29:09,600 --> 00:29:12,780 then it starts at 0. 402 00:29:16,250 --> 00:29:21,920 And if we get corrections from these higher and higher order 403 00:29:21,920 --> 00:29:24,490 terms in the series-- I calculated 404 00:29:24,490 --> 00:29:27,770 the first few-- I don't know what 405 00:29:27,770 --> 00:29:31,750 will happen if I were to include higher and higher order terms. 406 00:29:31,750 --> 00:29:37,210 But my expectation is that, say, at least at some point 407 00:29:37,210 --> 00:29:41,270 when this expansion from low temperature breaks down, 408 00:29:41,270 --> 00:29:45,460 I will have a divergence, let's say, of the heat capacity. 409 00:29:45,460 --> 00:29:47,710 Or maybe I calculated susceptibility 410 00:29:47,710 --> 00:29:50,160 or some other quantity, and I expect 411 00:29:50,160 --> 00:29:53,670 to have some singularity. 412 00:29:53,670 --> 00:30:00,940 And maybe by looking and fitting more terms in the series, 413 00:30:00,940 --> 00:30:04,990 one can guess what the exponent and the location 414 00:30:04,990 --> 00:30:07,890 of the singularity is. 415 00:30:07,890 --> 00:30:10,370 So you can see that, actually in this case, 416 00:30:10,370 --> 00:30:16,530 the natural variable that I am expanding is not K, 417 00:30:16,530 --> 00:30:22,050 but e to the minus 2dK-- sorry, e to the minus 2K 418 00:30:22,050 --> 00:30:26,850 because each excitation will have a number of broken bonds 419 00:30:26,850 --> 00:30:28,220 that I have to calculate. 420 00:30:28,220 --> 00:30:32,590 Each one of them makes a contribution like this. 421 00:30:32,590 --> 00:30:36,600 So maybe we can call this our new variable. 422 00:30:36,600 --> 00:30:40,830 And we have a series that has a function of this 423 00:30:40,830 --> 00:30:44,724 or some other variable, has a singularity. 424 00:30:44,724 --> 00:30:46,640 Actually, you should be able to, first of all, 425 00:30:46,640 --> 00:30:51,410 convince yourself that the nature of the singularity 426 00:30:51,410 --> 00:30:57,450 is not modified by any mapping that is analytical 427 00:30:57,450 --> 00:31:00,380 at the point of the singularity. 428 00:31:00,380 --> 00:31:03,820 So if the heat capacity as a function of k 429 00:31:03,820 --> 00:31:07,740 has a particular divergence, as a function of u 430 00:31:07,740 --> 00:31:10,435 it will have exactly the same divergence. 431 00:31:10,435 --> 00:31:20,140 In particular, we expect that as u approaches 432 00:31:20,140 --> 00:31:23,640 some critical value, the kinds of functions 433 00:31:23,640 --> 00:31:28,570 that we are interested have a behavior, a singular behavior, 434 00:31:28,570 --> 00:31:35,660 that is something like 1 minus u over uC. 435 00:31:35,660 --> 00:31:37,520 Let's say for the heat capacity, I 436 00:31:37,520 --> 00:31:42,820 would expect some kind of a singularity such as this. 437 00:31:42,820 --> 00:31:46,552 If I had a pure function such as this 438 00:31:46,552 --> 00:31:52,650 and I constructed an expansion in u, what do I get? 439 00:31:52,650 --> 00:31:58,240 I will get 1 plus alpha u over uC 440 00:31:58,240 --> 00:32:05,860 plus alpha alpha plus 1 over 2 uC squared u squared, 441 00:32:05,860 --> 00:32:08,110 and so forth. 442 00:32:08,110 --> 00:32:12,785 It's just a binary series expanded. 443 00:32:12,785 --> 00:32:18,970 The l term in the series would be alpha alpha plus 1 alpha 444 00:32:18,970 --> 00:32:26,700 plus l minus 1 divided by l factorial-- that's actually 445 00:32:26,700 --> 00:32:35,370 2 factorial-- uC to the power of l u to the l and so forth. 446 00:32:35,370 --> 00:32:35,870 OK? 447 00:32:38,730 --> 00:32:43,310 Now, typically, one of the ways that you look at series 448 00:32:43,310 --> 00:32:47,340 and decide whether it's a singular convergent series 449 00:32:47,340 --> 00:32:50,300 or what the behavior is is to look 450 00:32:50,300 --> 00:32:56,540 at the ratio of subsequent terms. 451 00:32:56,540 --> 00:33:02,010 So let's say that when I calculated my function 452 00:33:02,010 --> 00:33:07,240 C as a function of u, I constructed 453 00:33:07,240 --> 00:33:13,540 a series whose terms had coefficients 454 00:33:13,540 --> 00:33:15,862 that I will call al. 455 00:33:15,862 --> 00:33:17,640 OK? 456 00:33:17,640 --> 00:33:25,050 So here, if you had exactly this series, 457 00:33:25,050 --> 00:33:34,740 you would say that the ratio al divided by al minus 1 458 00:33:34,740 --> 00:33:39,170 is essentially the ratio of one of these factors compared 459 00:33:39,170 --> 00:33:41,190 to the previous one. 460 00:33:41,190 --> 00:33:43,650 And every time you add one of these factors, 461 00:33:43,650 --> 00:33:48,930 you add a term that is like this alpha plus l minus 1, 462 00:33:48,930 --> 00:33:53,970 l factorial compared to l minus 1 factorial has a factor of l, 463 00:33:53,970 --> 00:33:57,090 and then you have uC. 464 00:33:57,090 --> 00:34:05,820 And I can rewrite this as uC inverse, l divided by l is 1, 465 00:34:05,820 --> 00:34:12,559 and then I have minus 1 minus alpha divided by l. 466 00:34:12,559 --> 00:34:13,059 OK? 467 00:34:16,420 --> 00:34:22,310 So a pure divergence of the form that I have over here 468 00:34:22,310 --> 00:34:25,989 would predict that the ratio of subsequent terms 469 00:34:25,989 --> 00:34:28,630 would be something like this. 470 00:34:28,630 --> 00:34:31,699 And presumably, if you go sufficiently 471 00:34:31,699 --> 00:34:36,210 high in the series, in order to reproduce this divergence 472 00:34:36,210 --> 00:34:38,335 you must have that form. 473 00:34:38,335 --> 00:34:44,310 So what you could do as a test is to plot, 474 00:34:44,310 --> 00:34:47,550 for your actual series, what the ratio of these terms 475 00:34:47,550 --> 00:34:54,179 is as a function of 1 over l. 476 00:34:54,179 --> 00:34:59,560 So you can start with the ratio of the second to first term. 477 00:34:59,560 --> 00:35:02,640 You would be at 1/2. 478 00:35:02,640 --> 00:35:07,540 Then you would go 1/3, then you would go 1/4, 479 00:35:07,540 --> 00:35:10,590 you would have 1/5, and basically 480 00:35:10,590 --> 00:35:13,290 you would have a set of points. 481 00:35:13,290 --> 00:35:16,350 And you would plot what the location 482 00:35:16,350 --> 00:35:20,580 is for the first term in the series, the next term 483 00:35:20,580 --> 00:35:26,410 in the series, the next term in the series, and so forth. 484 00:35:26,410 --> 00:35:28,990 And if you are lucky, you would be 485 00:35:28,990 --> 00:35:32,980 able to then pass a straight line 486 00:35:32,980 --> 00:35:36,950 at large distances in the series. 487 00:35:36,950 --> 00:35:45,740 And the intercept of that extrapolated line 488 00:35:45,740 --> 00:35:50,660 would be your inverse of the singular point. 489 00:35:50,660 --> 00:35:55,000 And the slope of this line would give 490 00:35:55,000 --> 00:36:00,090 you 1 minus alpha or minus 1 minus alpha. 491 00:36:00,090 --> 00:36:02,220 OK? 492 00:36:02,220 --> 00:36:10,120 So there is really, a priori, not much reason 493 00:36:10,120 --> 00:36:13,280 to hope that that will happen because you 494 00:36:13,280 --> 00:36:18,560 can say that if I look at the series that is A 1 minus u 495 00:36:18,560 --> 00:36:26,320 over uC to the minus alpha, plus I add an analytic part, which 496 00:36:26,320 --> 00:36:36,120 is sum p equals 1 to, say, 52 of bl u to the l. 497 00:36:36,120 --> 00:36:43,990 For any bl in this function has exactly the same singularity 498 00:36:43,990 --> 00:36:46,070 as the original one. 499 00:36:46,070 --> 00:36:50,310 And yet the first 52 terms in the series, 500 00:36:50,310 --> 00:36:53,110 because of this additional analytical form, 501 00:36:53,110 --> 00:36:56,750 have nothing to do with the eventual singularity. 502 00:36:56,750 --> 00:36:59,340 They're going to be massing that. 503 00:36:59,340 --> 00:37:05,740 So there is no reason for you to expect that this should work. 504 00:37:05,740 --> 00:37:09,670 But when people do this, and they find that, let's say, 505 00:37:09,670 --> 00:37:18,270 for d equals to 2 up to some jumping up and down, 506 00:37:18,270 --> 00:37:22,010 they get a reasonable straight line. 507 00:37:22,010 --> 00:37:26,980 And the exponent that they get would correspond very closely 508 00:37:26,980 --> 00:37:31,060 to the alpha of 0, which is the logarithmic divergence that one 509 00:37:31,060 --> 00:37:31,970 gets. 510 00:37:31,970 --> 00:37:36,520 So this is, for d equals to 2, and then they repeat it, 511 00:37:36,520 --> 00:37:39,310 let's say, for d equals to 3, they 512 00:37:39,310 --> 00:37:44,520 get a different set of points. 513 00:37:44,520 --> 00:37:46,280 OK? 514 00:37:46,280 --> 00:37:49,050 Maybe not perfectly on a straight line, 515 00:37:49,050 --> 00:37:55,620 but you can still extrapolate and conclude from that 516 00:37:55,620 --> 00:38:00,770 that you'll have an alpha which is roughly 0.11 when 517 00:38:00,770 --> 00:38:05,160 d equals to 3, which is quite good. 518 00:38:05,160 --> 00:38:11,470 So for some reason or other, these lattice models 519 00:38:11,470 --> 00:38:16,990 are kind of sufficiently simple that, 520 00:38:16,990 --> 00:38:20,060 in an appropriate expansion, they 521 00:38:20,060 --> 00:38:24,800 don't seem to give you that much of a problem. 522 00:38:24,800 --> 00:38:28,680 And so people have gone and calculated series, 523 00:38:28,680 --> 00:38:31,500 let's say, this was in '50s and '60s, 524 00:38:31,500 --> 00:38:33,110 just by drawing things on hand. 525 00:38:33,110 --> 00:38:36,440 And maybe some primitive computers, 526 00:38:36,440 --> 00:38:40,930 you can go to order of 20 terms in this series, 527 00:38:40,930 --> 00:38:43,980 and then extrapolate exponents for various quantities. 528 00:38:46,702 --> 00:38:47,202 OK? 529 00:38:51,060 --> 00:38:54,340 But it's not as simple as that. 530 00:38:54,340 --> 00:39:00,640 And the reason I calculated the first three terms for you 531 00:39:00,640 --> 00:39:06,340 was to show you that what I told you here was clearly a lie. 532 00:39:06,340 --> 00:39:08,330 Why is that? 533 00:39:08,330 --> 00:39:12,530 Because of the three terms that I explicitly calculated for you 534 00:39:12,530 --> 00:39:17,204 in that series, the third one is negative. 535 00:39:17,204 --> 00:39:18,500 Right? 536 00:39:18,500 --> 00:39:22,535 So clearly, if I were to plot that, 537 00:39:22,535 --> 00:39:25,745 I will get something over here. 538 00:39:25,745 --> 00:39:27,490 Right? 539 00:39:27,490 --> 00:39:32,650 So what's going gone there is a different issue. 540 00:39:32,650 --> 00:39:37,510 And people have developed kind of methodologies and ways 541 00:39:37,510 --> 00:39:40,400 to look at series and guess what is going on 542 00:39:40,400 --> 00:39:45,560 and yet continue to extract exponents. 543 00:39:45,560 --> 00:39:58,460 So one potential origin for alternating signs-- 544 00:39:58,460 --> 00:40:03,050 and any series that has a divergence such as the one 545 00:40:03,050 --> 00:40:07,790 that I have indicated for you will have, eventually, signs 546 00:40:07,790 --> 00:40:13,640 that need to be positive-- has to do with the following. 547 00:40:13,640 --> 00:40:19,490 Let's say if I take a series, which is 1 over 1 minus z/2. 548 00:40:19,490 --> 00:40:20,610 OK? 549 00:40:20,610 --> 00:40:22,180 This is a very nice series. 550 00:40:22,180 --> 00:40:28,210 It's 1 plus 0/2 z squared/4, z cubed/8. 551 00:40:28,210 --> 00:40:32,520 You could apply this ratio test to this series 552 00:40:32,520 --> 00:40:36,815 and conclude that you have a linear divergence. 553 00:40:36,815 --> 00:40:41,200 Now, suppose I multiply that by 1 over 1 554 00:40:41,200 --> 00:40:45,630 plus z squared, which is a function that's 555 00:40:45,630 --> 00:40:49,360 perfectly well-behaved as a function of z. 556 00:40:49,360 --> 00:40:53,660 Yet if I multiply it here, I will get 1 minus z squared plus 557 00:40:53,660 --> 00:40:58,100 z to the fourth minus z to the sixth. 558 00:40:58,100 --> 00:41:05,020 And what it does is it kind of distorts 559 00:41:05,020 --> 00:41:07,520 what is happening over here. 560 00:41:07,520 --> 00:41:09,870 Actually, in this series you can see 561 00:41:09,870 --> 00:41:15,285 it kind of becomes ill-defined when z is of order of 1. 562 00:41:15,285 --> 00:41:19,260 It changes the signs, et cetera. 563 00:41:19,260 --> 00:41:25,580 But the function itself has a perfectly good singularity that 564 00:41:25,580 --> 00:41:27,910 appears at z equals to 2. 565 00:41:27,910 --> 00:41:31,050 And starting from an expansion from 0, 566 00:41:31,050 --> 00:41:35,110 there should be no problems along the line 567 00:41:35,110 --> 00:41:37,910 until you hit z of 2. 568 00:41:37,910 --> 00:41:41,530 What is the reason for these alternating signs? 569 00:41:41,530 --> 00:41:46,460 It is because you should be looking at the complex z plane. 570 00:41:46,460 --> 00:41:54,300 And in the complex z plane, you have poles at plus and minus i 571 00:41:54,300 --> 00:41:59,286 which are located closer to the origin than you have at 2. 572 00:41:59,286 --> 00:42:02,440 So basically, your series will start 573 00:42:02,440 --> 00:42:07,100 to have problems by the time you hit here, 574 00:42:07,100 --> 00:42:10,710 and that problem is reflected in the alternating behavior. 575 00:42:10,710 --> 00:42:13,116 It's also showing up over there. 576 00:42:13,116 --> 00:42:17,520 Yet it has nothing to do with going along the real axis 577 00:42:17,520 --> 00:42:20,986 and encountering the singularity that you are after. 578 00:42:20,986 --> 00:42:23,130 OK? 579 00:42:23,130 --> 00:42:27,330 So one thing that you can do is to say, well, 580 00:42:27,330 --> 00:42:31,740 who said I should use z as my variable? 581 00:42:31,740 --> 00:42:36,930 Maybe I can choose some other function v of z. 582 00:42:36,930 --> 00:42:38,080 OK? 583 00:42:38,080 --> 00:42:42,730 And then when I choose the appropriate thing, 584 00:42:42,730 --> 00:42:48,300 the singularity on the real axis will be pushed to v of 2. 585 00:42:48,300 --> 00:42:51,900 But maybe I chose appropriate function of v of z 586 00:42:51,900 --> 00:42:53,730 such that the other singularities 587 00:42:53,730 --> 00:42:59,030 are pushed very far away so that the first singularity that I 588 00:42:59,030 --> 00:43:01,076 encounter is over here. 589 00:43:01,076 --> 00:43:03,220 OK? 590 00:43:03,220 --> 00:43:06,010 And it turns out that if you take this series 591 00:43:06,010 --> 00:43:11,740 over here and rather than working with e to the minus k, 592 00:43:11,740 --> 00:43:16,630 we recast things in terms of tanh K-- let's call 593 00:43:16,630 --> 00:43:20,770 that v-- which is e to the K plus e to the minus 594 00:43:20,770 --> 00:43:24,120 K-- well actually, tanh K I can also write as e 595 00:43:24,120 --> 00:43:28,435 to the 2K minus 1 e to the 2K plus 1. 596 00:43:28,435 --> 00:43:30,830 I mean, it's just a transformation. 597 00:43:30,830 --> 00:43:35,220 So I can replace e to the minus 2K 598 00:43:35,220 --> 00:43:40,400 with some function v, substitute for u in that series, 599 00:43:40,400 --> 00:43:42,970 and I will have a different function 600 00:43:42,970 --> 00:43:46,250 as an expansion in powers of v. 601 00:43:46,250 --> 00:43:50,340 And once people do that, same thing happens as here. 602 00:43:50,340 --> 00:43:53,860 You'll find a function that all of its terms are, in fact, 603 00:43:53,860 --> 00:43:58,890 positive, and the things that I mentioned to you over here 604 00:43:58,890 --> 00:43:59,800 were applied. 605 00:43:59,800 --> 00:44:04,920 After such transformation, you get very nice behaviors. 606 00:44:04,920 --> 00:44:06,270 OK? 607 00:44:06,270 --> 00:44:08,590 So there seems to be some guesswork 608 00:44:08,590 --> 00:44:11,600 into finding the appropriate transformation. 609 00:44:11,600 --> 00:44:14,240 There are other methods for dealing with series 610 00:44:14,240 --> 00:44:17,800 and extracting singularities called 611 00:44:17,800 --> 00:44:23,680 Pade approximants, et cetera, which I won't go into. 612 00:44:23,680 --> 00:44:27,680 But there are kind of, again, clever mathematical tricks 613 00:44:27,680 --> 00:44:33,845 for extracting singularity out of series such as this. 614 00:44:33,845 --> 00:44:34,345 OK? 615 00:44:38,770 --> 00:44:45,060 So I'll tell you shortly why this tanh K is really 616 00:44:45,060 --> 00:44:48,240 a good expansion factor. 617 00:44:48,240 --> 00:44:51,590 It turns out that for Ising models, 618 00:44:51,590 --> 00:44:55,980 it's actually the right expansion factor 619 00:44:55,980 --> 00:44:58,670 if we go to the other limit of high temperatures. 620 00:45:03,576 --> 00:45:05,060 OK? 621 00:45:05,060 --> 00:45:12,220 So basically, now at T going to infinity, 622 00:45:12,220 --> 00:45:21,340 you would say that sigma i is minus or plus 1 623 00:45:21,340 --> 00:45:22,330 with equal probability. 624 00:45:27,830 --> 00:45:32,290 As T goes to infinity, this factor 625 00:45:32,290 --> 00:45:38,350 that encodes the tendency of spins to be next to each other 626 00:45:38,350 --> 00:45:42,280 has been scaled to 0, so I know exactly what 627 00:45:42,280 --> 00:45:43,940 is going on at infinite temperature. 628 00:45:43,940 --> 00:45:47,550 Basically, at each site, I have an independent variable 629 00:45:47,550 --> 00:45:50,440 that is decoupled from everything else. 630 00:45:50,440 --> 00:45:56,120 So I can start expanding around that for, say, the partition 631 00:45:56,120 --> 00:45:57,970 function. 632 00:45:57,970 --> 00:46:01,500 Let's think of it for a general spin system. 633 00:46:01,500 --> 00:46:04,210 So I will write it as a trace over, 634 00:46:04,210 --> 00:46:08,000 let's say, if I have Potts model rather than two values, 635 00:46:08,000 --> 00:46:12,220 I would have K values of something like e 636 00:46:12,220 --> 00:46:15,000 to the minus beta H, again, trying 637 00:46:15,000 --> 00:46:17,630 to be reasonably general. 638 00:46:17,630 --> 00:46:21,840 And the idea is that as you go to infinite temperature, 639 00:46:21,840 --> 00:46:25,260 beta goes to 0, and this function 640 00:46:25,260 --> 00:46:31,075 you can expand in a series 1 minus beta H plus beta squared 641 00:46:31,075 --> 00:46:33,897 H squared over 2, and so forth. 642 00:46:38,870 --> 00:46:41,820 Now, the trace of 1 is essentially 643 00:46:41,820 --> 00:46:45,900 summing over all possible states. 644 00:46:45,900 --> 00:46:47,440 Let's say the two states that you 645 00:46:47,440 --> 00:46:50,010 would have for the Ising model or however many 646 00:46:50,010 --> 00:46:55,160 that you have for Potts models at each site independently. 647 00:46:55,160 --> 00:46:58,940 So that can give me some partition function 648 00:46:58,940 --> 00:47:00,340 that I will call Z0. 649 00:47:00,340 --> 00:47:05,560 It is simply 2 to the n for the Ising model. 650 00:47:05,560 --> 00:47:09,130 But once I factor that, you can see 651 00:47:09,130 --> 00:47:12,430 that the rest of the terms in the series 652 00:47:12,430 --> 00:47:20,810 can be regarded as expectation values of this Hamiltonian 653 00:47:20,810 --> 00:47:23,270 with respect to this weight in which all 654 00:47:23,270 --> 00:47:25,920 of the degrees of freedom are treated 655 00:47:25,920 --> 00:47:28,530 as independent, unconstrained variables. 656 00:47:34,660 --> 00:47:37,090 And of course, the thing that I'm interested 657 00:47:37,090 --> 00:47:41,080 is log of the partition function. 658 00:47:41,080 --> 00:47:46,350 And so that will give me log of Z0, 659 00:47:46,350 --> 00:47:49,850 and then I have the log of this series. 660 00:47:49,850 --> 00:47:56,380 And then you can see that that series is a generating function 661 00:47:56,380 --> 00:47:59,590 for the moments of the Hamiltonian. 662 00:47:59,590 --> 00:48:03,350 So its log will be the generating function 663 00:48:03,350 --> 00:48:14,080 for the cumulant, so H to the l 0, the cumulant. 664 00:48:14,080 --> 00:48:16,480 So the variance at the second order 665 00:48:16,480 --> 00:48:18,334 and appropriate cumulant at higher orders. 666 00:48:44,220 --> 00:48:46,390 OK? 667 00:48:46,390 --> 00:48:51,500 So let's try to calculate this for the Ising model, where 668 00:48:51,500 --> 00:49:01,172 my minus beta H is K sum over i, j sigma i sigma j. 669 00:49:01,172 --> 00:49:03,420 OK? 670 00:49:03,420 --> 00:49:06,070 Then at the lowest order, what do I get? 671 00:49:06,070 --> 00:49:14,612 The average of beta H is K sum over i, 672 00:49:14,612 --> 00:49:22,110 j average of sigma i sigma j with this zeroed weight. 673 00:49:22,110 --> 00:49:24,340 But as I emphasized, at zeroed weight, 674 00:49:24,340 --> 00:49:27,960 every site independently can be plus or minus. 675 00:49:27,960 --> 00:49:33,430 Because of the independence, I can do this. 676 00:49:33,430 --> 00:49:37,390 And then since each site has equal probability to be 677 00:49:37,390 --> 00:49:40,170 plus or minus, its average is 0. 678 00:49:40,170 --> 00:49:41,970 So basically, this will be 0. 679 00:49:46,263 --> 00:49:48,180 OK? 680 00:49:48,180 --> 00:49:51,670 So the first thing that can happen in that series-- 681 00:49:51,670 --> 00:49:54,560 if I go to the next order. 682 00:49:54,560 --> 00:50:00,000 So at next order, beta H squared would involve K 683 00:50:00,000 --> 00:50:10,130 squared sum over i, j K, l sigma i sigma j sigma K sigma l. 684 00:50:10,130 --> 00:50:13,850 And I have to take an average of this, 685 00:50:13,850 --> 00:50:16,690 which means that I have to take an average of something 686 00:50:16,690 --> 00:50:19,080 like this. 687 00:50:19,080 --> 00:50:19,580 OK. 688 00:50:19,580 --> 00:50:23,630 And you would say, well, again, everything is 0. 689 00:50:23,630 --> 00:50:26,630 Well, there is one case where it won't be 0-- 690 00:50:26,630 --> 00:50:29,340 if these two pairs are identical. 691 00:50:29,340 --> 00:50:30,730 Right? 692 00:50:30,730 --> 00:50:41,540 So this is going to give me K squared sum over pair i, 693 00:50:41,540 --> 00:50:44,960 j being the same as K, l. 694 00:50:44,960 --> 00:50:46,830 Then I will get, essentially, sigma i 695 00:50:46,830 --> 00:50:48,780 squared sigma j squared. 696 00:50:48,780 --> 00:50:50,690 Sigma i squared is 1. 697 00:50:50,690 --> 00:50:52,420 Sigma j squared is 1. 698 00:50:52,420 --> 00:50:54,510 So basically, I will get 1. 699 00:50:54,510 --> 00:50:56,600 And this is going to give me K squared 700 00:50:56,600 --> 00:50:59,233 times the number of bonds. 701 00:50:59,233 --> 00:51:01,000 OK? 702 00:51:01,000 --> 00:51:02,890 So you can see that I can start thinking 703 00:51:02,890 --> 00:51:05,350 of this already graphically. 704 00:51:05,350 --> 00:51:12,610 Because what I did over here, I said that on my lattice 705 00:51:12,610 --> 00:51:19,690 this sum says you pick one sigma i sigma j. 706 00:51:19,690 --> 00:51:23,910 If I were to pick the other sigma i sigma j over here, 707 00:51:23,910 --> 00:51:25,680 the average would be 0. 708 00:51:25,680 --> 00:51:30,780 I am forced to put two of them on top of each other. 709 00:51:30,780 --> 00:51:34,350 If I go to three, there is no way 710 00:51:34,350 --> 00:51:38,580 that I can draw a diagram that involves 711 00:51:38,580 --> 00:51:46,730 three pairs in which every single site occurs twice, 712 00:51:46,730 --> 00:51:48,120 which is what I need. 713 00:51:48,120 --> 00:51:53,120 Because a single site appearing by itself or three times 714 00:51:53,120 --> 00:51:56,430 will give me sigma i cubed is the same as sigma i. 715 00:51:56,430 --> 00:52:00,080 It will average to 0. 716 00:52:00,080 --> 00:52:05,380 So the next thing that I can do is to go to level four. 717 00:52:05,380 --> 00:52:08,260 At the level of four, I can certainly 718 00:52:08,260 --> 00:52:09,870 do something like this. 719 00:52:09,870 --> 00:52:12,830 I can put all four of them on top of each other, 720 00:52:12,830 --> 00:52:16,740 and then I get a K to the fourth contribution. 721 00:52:16,740 --> 00:52:23,060 Or I could put a pair here, and if they're here, 722 00:52:23,060 --> 00:52:27,410 for log Z that would be unacceptable because that will 723 00:52:27,410 --> 00:52:30,840 get subtracted out when I calculate the variance. 724 00:52:33,400 --> 00:52:34,720 It's not a connected piece. 725 00:52:34,720 --> 00:52:36,490 It's a disconnected piece. 726 00:52:36,490 --> 00:52:38,660 But I could have something like this, 727 00:52:38,660 --> 00:52:40,610 two of them turned like this. 728 00:52:40,610 --> 00:52:42,390 So that's four. 729 00:52:42,390 --> 00:52:45,960 But really, the one that is nontrivial and interesting 730 00:52:45,960 --> 00:52:50,050 is when I do something like this, like a square. 731 00:52:50,050 --> 00:52:54,270 So I go here sigma 1 sigma 2, sigma 2 sigma 3. 732 00:52:54,270 --> 00:52:57,070 That sigma 2 has been repeated twice and becomes 733 00:52:57,070 --> 00:52:59,560 sigma 2 squared and goes away. 734 00:52:59,560 --> 00:53:02,770 Sigma 3 sigma 4, sigma 3 repeated twice, 735 00:53:02,770 --> 00:53:07,246 sigma 4 repeated twice, sigma 1 repeated twice, [INAUDIBLE]. 736 00:53:07,246 --> 00:53:08,070 OK? 737 00:53:08,070 --> 00:53:11,622 So you can see that this kind of expansion 738 00:53:11,622 --> 00:53:16,270 will naturally lead you into an expansion in terms 739 00:53:16,270 --> 00:53:19,220 of loops on a lattice. 740 00:53:19,220 --> 00:53:23,370 So the natural form of high temperature expansions 741 00:53:23,370 --> 00:53:26,257 are these closed strings or loops, 742 00:53:26,257 --> 00:53:28,340 if you like, that you have to draw on the lattice. 743 00:53:31,480 --> 00:53:37,360 Now, it's also clear that the thing that 744 00:53:37,360 --> 00:53:45,720 goes between two sites, that I'm indicating by K, in all cases 745 00:53:45,720 --> 00:53:48,140 is likely to be repeated by putting 746 00:53:48,140 --> 00:53:51,030 more and more things on top of each other 747 00:53:51,030 --> 00:53:52,645 without modifying the effect. 748 00:53:52,645 --> 00:53:57,750 So I can go here to 4 and things like actually 3 and things 749 00:53:57,750 --> 00:53:58,760 like that. 750 00:53:58,760 --> 00:54:01,540 So basically, you can see that I should really 751 00:54:01,540 --> 00:54:08,040 do a summation over the contribution of 2, 4, et 752 00:54:08,040 --> 00:54:11,620 cetera all on top of each other, or 1, 3, 753 00:54:11,620 --> 00:54:16,440 5 on top of each other, and call them new variables. 754 00:54:16,440 --> 00:54:23,280 So when we were doing the cluster expansion for particles 755 00:54:23,280 --> 00:54:25,870 interacting, we encountered this thing 756 00:54:25,870 --> 00:54:29,870 that we thought v was a good variable to expand it. 757 00:54:29,870 --> 00:54:32,010 But then because of these repeats, 758 00:54:32,010 --> 00:54:35,860 we decided that e to the minus beta v minus 1 759 00:54:35,860 --> 00:54:39,190 was a good variable to expand it. 760 00:54:39,190 --> 00:54:41,280 So a similar thing happens here. 761 00:54:41,280 --> 00:54:45,740 And for the Ising model, it is a very natural thing 762 00:54:45,740 --> 00:54:51,120 to recast this series in a slightly different way. 763 00:54:51,120 --> 00:54:58,120 You see that the contribution of each bond to the partition 764 00:54:58,120 --> 00:55:02,350 function, and by a bond I mean a pair of neighboring sites, 765 00:55:02,350 --> 00:55:07,030 is a factor e to the K sigma i sigma j. 766 00:55:07,030 --> 00:55:07,530 OK? 767 00:55:10,130 --> 00:55:13,560 Now, since we are dealing with binary variables, 768 00:55:13,560 --> 00:55:17,110 this product, sigma i sigma j, can only take two values. 769 00:55:17,110 --> 00:55:20,930 It's either plus K or minus K depending 770 00:55:20,930 --> 00:55:24,160 on where things are aligned or misaligned. 771 00:55:24,160 --> 00:55:26,700 So I can indicate the binary nature 772 00:55:26,700 --> 00:55:28,140 of this in the following fashion. 773 00:55:28,140 --> 00:55:33,690 I can write this as e to the K plus e to the minus K over 2 774 00:55:33,690 --> 00:55:41,760 plus sigma i sigma j e to the K minus e to the minus K over 2. 775 00:55:41,760 --> 00:55:47,460 So that when I'm dealing with sigma sigma being plus, 776 00:55:47,460 --> 00:55:49,100 I add those two factors. 777 00:55:49,100 --> 00:55:50,730 e to the minus K's disappear. 778 00:55:50,730 --> 00:55:54,000 I will get e to the K. When I'm dealing 779 00:55:54,000 --> 00:55:57,190 with this thing to the minus, the e to the K's disappear, 780 00:55:57,190 --> 00:56:00,470 and I will get e to the minus K. So it's 781 00:56:00,470 --> 00:56:03,000 correct rewriting of that factor. 782 00:56:03,000 --> 00:56:04,885 The first term you, of course, recognize 783 00:56:04,885 --> 00:56:08,640 as the hyperbolic cosine of K, the second one 784 00:56:08,640 --> 00:56:12,740 as the hyperbolic sine of K. And so I 785 00:56:12,740 --> 00:56:16,930 can write the whole thing as hyperbolic cosine 786 00:56:16,930 --> 00:56:24,345 1 plus hyperbolic tanh of K sigma i sigma j. 787 00:56:24,345 --> 00:56:24,845 OK? 788 00:56:29,740 --> 00:56:37,270 So this tanh is really same thing as here. 789 00:56:37,270 --> 00:56:40,390 It's the high-temperature expansion variable. 790 00:56:40,390 --> 00:56:43,620 As K goes to 0 at high temperature, 791 00:56:43,620 --> 00:56:46,150 tanh K also goes to 0. 792 00:56:46,150 --> 00:56:50,620 And it turns out that a much nicer variable to expand 793 00:56:50,620 --> 00:56:55,830 is this quantity tanh K. And so that I don't have to repeat it 794 00:56:55,830 --> 00:57:00,140 throughout, I will give it the symbol t. 795 00:57:00,140 --> 00:57:04,570 So small t stands not for reduced temperature anymore, 796 00:57:04,570 --> 00:57:07,980 but for hyperbolic tanh of K. 797 00:57:07,980 --> 00:57:11,115 So my partition function now, Z-- maybe I'll 798 00:57:11,115 --> 00:57:13,760 go to another page. 799 00:57:49,920 --> 00:58:00,580 So my partition function is a sum over the 2 to the N binary 800 00:58:00,580 --> 00:58:10,630 variables e to the K sigma i sigma j sum over all bonds. 801 00:58:10,630 --> 00:58:16,150 I can write that as a product of these exponential factors over 802 00:58:16,150 --> 00:58:17,910 [INAUDIBLE]. 803 00:58:17,910 --> 00:58:28,910 Each of these exponential factors I can write as cosh K 1 804 00:58:28,910 --> 00:58:31,150 plus t sigma i sigma j. 805 00:58:35,270 --> 00:58:39,570 All the factors of cosh K I will take to the outside. 806 00:58:39,570 --> 00:58:43,610 So I will get cosh K raised to the power 807 00:58:43,610 --> 00:58:47,030 of the number of bonds that I have in my lattice 808 00:58:47,030 --> 00:58:49,330 because each bond will contribute 809 00:58:49,330 --> 00:58:52,440 one of these factors. 810 00:58:52,440 --> 00:59:02,045 And then I have this sum over sigma i product over bonds. 811 00:59:07,980 --> 00:59:12,980 So this is the product of 1 plus t factors. 812 00:59:12,980 --> 00:59:18,485 So for each-- maybe I'll do it over here. 813 00:59:28,170 --> 00:59:33,810 So for each i, j, I have to pick one of these factors. 814 00:59:33,810 --> 00:59:37,780 I can either pick 1, nothing, or I 815 00:59:37,780 --> 00:59:43,715 can pick a factor of t sigma i sigma j. 816 00:59:43,715 --> 00:59:44,215 OK? 817 00:59:47,430 --> 00:59:49,390 So the first term in this series-- 818 00:59:49,390 --> 00:59:54,160 since it's a series in powers of t, the first term in the series 819 00:59:54,160 --> 00:59:56,800 is to pick 1 everywhere. 820 00:59:56,800 --> 01:00:01,230 The next term is to pick one factor at some point. 821 01:00:01,230 --> 01:00:04,320 But then when I pick that factor, 822 01:00:04,320 --> 01:00:08,460 that term in the series, I have to sum over sigma i. 823 01:00:08,460 --> 01:00:11,870 And when I sum over sigma i, since this sigma i can 824 01:00:11,870 --> 01:00:16,414 be plus or minus with equal probability, it will give me 0. 825 01:00:16,414 --> 01:00:17,670 OK? 826 01:00:17,670 --> 01:00:22,590 So I cannot leave this sigma i by itself. 827 01:00:22,590 --> 01:00:26,545 So maybe I will pick another higher-order term 828 01:00:26,545 --> 01:00:31,490 in the series that has a t, a sigma i that would make this 829 01:00:31,490 --> 01:00:36,734 into a sigma i squared, and then I will have a sigma K here. 830 01:00:36,734 --> 01:00:38,090 OK? 831 01:00:38,090 --> 01:00:42,540 Now, note it was kind of similar to what I was doing here. 832 01:00:42,540 --> 01:00:45,660 But here I could pick as many bonds 833 01:00:45,660 --> 01:00:52,860 as I like on as many factors of K. Now what has happened here 834 01:00:52,860 --> 01:00:57,310 is, effectively, I have only two choices. 835 01:00:57,310 --> 01:01:00,330 One choice is having gone many, many times, 836 01:01:00,330 --> 01:01:04,830 so summing all of the terms that had 2, 4, et cetera. 837 01:01:04,830 --> 01:01:08,420 That's what gives you the cosh K. Or including something 838 01:01:08,420 --> 01:01:12,150 like this, sum of 1, 3, 5, et cetera. 839 01:01:12,150 --> 01:01:15,410 That's what gives you tanh K. But the good thing 840 01:01:15,410 --> 01:01:17,590 is that it's really now a binary choice. 841 01:01:17,590 --> 01:01:21,550 You either draw one line, or you don't draw anything. 842 01:01:21,550 --> 01:01:23,030 OK? 843 01:01:23,030 --> 01:01:30,530 So again, your first choice is to somehow complete the series 844 01:01:30,530 --> 01:01:33,970 by drawing something like this. 845 01:01:33,970 --> 01:01:40,430 And quite generically-- OK, so after that has happened, 846 01:01:40,430 --> 01:01:42,280 then this is sigma i squared. 847 01:01:42,280 --> 01:01:43,300 This is sigma j squared. 848 01:01:43,300 --> 01:01:47,320 These are all-- they have gone to 1. 849 01:01:47,320 --> 01:01:49,420 And then you do the sum over sigma i, 850 01:01:49,420 --> 01:01:52,240 you will get a factor of 2. 851 01:01:52,240 --> 01:01:57,800 So the answer is going to be 2 to the number of sites, 852 01:01:57,800 --> 01:02:03,005 N, cosh K to the power of the number of bonds. 853 01:02:07,610 --> 01:02:10,500 And then I would have a series, which 854 01:02:10,500 --> 01:02:22,810 is the sum over all graphs with even number of bonds 855 01:02:22,810 --> 01:02:26,320 per site like here. 856 01:02:26,320 --> 01:02:31,710 So I either have 0 bond going here, or I can have two bonds. 857 01:02:31,710 --> 01:02:34,520 I could very well have something like this, four bonds. 858 01:02:34,520 --> 01:02:36,790 That doesn't violate anything. 859 01:02:36,790 --> 01:02:40,740 So all I need to ensure in order that sigma i does 860 01:02:40,740 --> 01:02:45,430 not average to 0 is that I have an even number per site. 861 01:02:45,430 --> 01:02:47,800 And then the contribution of the graph 862 01:02:47,800 --> 01:02:51,550 is t to the number of bonds in the graph. 863 01:02:55,600 --> 01:02:58,100 And at this stage when I'm calculating a partition 864 01:02:58,100 --> 01:03:02,000 function, there is no reason why I could not 865 01:03:02,000 --> 01:03:05,060 have disconnected graphs. 866 01:03:05,060 --> 01:03:07,190 For the partition function, there is no problem. 867 01:03:07,190 --> 01:03:10,305 Presumably, when I take the log, the disconnected pieces 868 01:03:10,305 --> 01:03:12,560 will go away. 869 01:03:12,560 --> 01:03:14,010 OK? 870 01:03:14,010 --> 01:03:14,579 Yes? 871 01:03:14,579 --> 01:03:16,745 AUDIENCE: Where does the 2 to the N come from again? 872 01:03:16,745 --> 01:03:17,680 PROFESSOR: OK. 873 01:03:17,680 --> 01:03:25,170 So at each site, I have to sum over sigma i. 874 01:03:25,170 --> 01:03:28,130 So sigma i is either minus 1 or plus 1. 875 01:03:28,130 --> 01:03:33,400 What I'm doing is sum over sigma i sigma i to some power. 876 01:03:33,400 --> 01:03:36,530 And this is either going to give me 2 or 0 877 01:03:36,530 --> 01:03:42,520 depending on whether P is even or P is odd. 878 01:03:42,520 --> 01:03:45,415 All right? 879 01:03:45,415 --> 01:03:45,915 OK? 880 01:03:49,810 --> 01:03:58,640 So you can try to calculate general terms for this series. 881 01:03:58,640 --> 01:04:04,540 Let's say we go to hypercubic lattice, which 882 01:04:04,540 --> 01:04:07,250 is what we were doing before. 883 01:04:07,250 --> 01:04:12,940 The number of bonds per site is d. 884 01:04:12,940 --> 01:04:16,060 So this, for the hypercubic lattice, the number of bonds 885 01:04:16,060 --> 01:04:17,930 will be dN. 886 01:04:17,930 --> 01:04:20,620 You could do this calculation for a triangular lattice. 887 01:04:20,620 --> 01:04:23,740 You don't have to stick with FCC lattice. 888 01:04:23,740 --> 01:04:27,780 You don't have to stick with these hypercubic lattices. 889 01:04:27,780 --> 01:04:35,770 The first diagram that you can create is always the square. 890 01:04:35,770 --> 01:04:37,130 OK? 891 01:04:37,130 --> 01:04:42,370 And in d dimensions, one leg has a choice of d direction. 892 01:04:42,370 --> 01:04:44,740 The next one would be d minus 1. 893 01:04:44,740 --> 01:04:51,320 So this would be d d minus 1 over 2 t to the fourth. 894 01:04:51,320 --> 01:04:54,190 But you could start it from any site on the lattice 895 01:04:54,190 --> 01:04:57,030 so you would have something like this. 896 01:04:57,030 --> 01:05:00,110 The next term that you would have in the series 897 01:05:00,110 --> 01:05:06,800 is something that involves, let's say, six bonds. 898 01:05:06,800 --> 01:05:11,420 So the next term will be N t to the 6. 899 01:05:11,420 --> 01:05:15,470 And I think I sometimes convince myself 900 01:05:15,470 --> 01:05:18,285 that the numerical factor was something like this, 901 01:05:18,285 --> 01:05:19,347 but doesn't matter. 902 01:05:19,347 --> 01:05:20,680 You could calculate out of this. 903 01:05:20,680 --> 01:05:20,935 Yes? 904 01:05:20,935 --> 01:05:22,351 AUDIENCE: What if we have diagrams 905 01:05:22,351 --> 01:05:25,402 of order of t squared, just [INAUDIBLE] there and back? 906 01:05:25,402 --> 01:05:26,810 PROFESSOR: OK. 907 01:05:26,810 --> 01:05:29,502 Where would I get the t squared from here? 908 01:05:33,310 --> 01:05:34,740 OK? 909 01:05:34,740 --> 01:05:37,900 So from this bond, I have this factor, 910 01:05:37,900 --> 01:05:41,570 1 plus t sigma i sigma j. 911 01:05:41,570 --> 01:05:44,090 There is no t squared. 912 01:05:44,090 --> 01:05:48,620 I would have had K squared, K to the fourth, et cetera. 913 01:05:48,620 --> 01:05:52,280 But I re-summed all of them into hyperbolic cosine 914 01:05:52,280 --> 01:05:54,450 and the hyperbolic sine. 915 01:05:54,450 --> 01:05:55,050 So this-- 916 01:05:55,050 --> 01:05:57,740 AUDIENCE: So [INAUDIBLE] taking this product 917 01:05:57,740 --> 01:06:01,730 along all the bonds, you can kind of go along the same bond. 918 01:06:01,730 --> 01:06:04,580 PROFESSOR: We already summed all of those things 919 01:06:04,580 --> 01:06:07,809 together into this factor t. 920 01:06:07,809 --> 01:06:08,350 AUDIENCE: OK. 921 01:06:08,350 --> 01:06:09,320 PROFESSOR: Yeah? 922 01:06:09,320 --> 01:06:10,290 OK? 923 01:06:10,290 --> 01:06:11,760 Yeah, it's good. 924 01:06:11,760 --> 01:06:18,360 And that's why this tanh is such a nice variable. 925 01:06:18,360 --> 01:06:20,480 OK? 926 01:06:20,480 --> 01:06:24,130 So there is actually the nicer series 927 01:06:24,130 --> 01:06:27,810 to work with in terms of trying to extract exponent 928 01:06:27,810 --> 01:06:30,120 is this high-temperature series in terms 929 01:06:30,120 --> 01:06:32,810 of these new diagrams, et cetera. 930 01:06:32,810 --> 01:06:37,680 But I'm not going to be doing diagrammatics. 931 01:06:37,680 --> 01:06:44,770 What I will be using this high-temperature series 932 01:06:44,770 --> 01:06:47,510 is the following. 933 01:06:47,510 --> 01:06:52,270 One, to show that in a few minutes 934 01:06:52,270 --> 01:07:02,910 we can use it to exactly solve the one-dimensional Ising model 935 01:07:02,910 --> 01:07:05,870 and gain a physical understanding of what's 936 01:07:05,870 --> 01:07:15,482 happening, and 2, to re-derive Gaussian model. 937 01:07:19,730 --> 01:07:23,470 Turns out that there is a close connection between all 938 01:07:23,470 --> 01:07:26,500 of these loops that you can draw on a lattice 939 01:07:26,500 --> 01:07:29,220 through some kind of a path integral way of thinking 940 01:07:29,220 --> 01:07:31,660 about it with the Gaussian model. 941 01:07:31,660 --> 01:07:35,650 And that we actually will use as a stepping stone towards where 942 01:07:35,650 --> 01:07:42,724 we are really headed, which is the exact solution 943 01:07:42,724 --> 01:07:44,544 of the 2D Ising model. 944 01:08:11,930 --> 01:08:13,960 OK? 945 01:08:13,960 --> 01:08:20,069 So the 1D Ising model. 946 01:08:23,779 --> 01:08:28,870 And actually, the method is sufficiently powerful 947 01:08:28,870 --> 01:08:34,380 that we can compare and contrast two cases, one 948 01:08:34,380 --> 01:08:40,100 when you have open chain. 949 01:08:40,100 --> 01:08:50,240 So this is a system that is composed of sites 1, 2, 3, 4, 950 01:08:50,240 --> 01:08:55,710 N minus 1, N. On each one of them I have an Ising variable. 951 01:08:58,689 --> 01:09:11,270 And if I follow my nose, it's a Z is 2 to the number of sites 952 01:09:11,270 --> 01:09:17,010 cosh K to the power of the number of bonds. 953 01:09:17,010 --> 01:09:21,420 Actually, clearly with open systems, the number of bonds 954 01:09:21,420 --> 01:09:24,380 is 1 less than the number of sites. 955 01:09:24,380 --> 01:09:26,890 So I can be extremely precise. 956 01:09:26,890 --> 01:09:30,240 It is N minus 1. 957 01:09:30,240 --> 01:09:34,560 And then I have to draw all graphs 958 01:09:34,560 --> 01:09:39,710 that I can on this lattice that have an even number of bonds 959 01:09:39,710 --> 01:09:43,540 emanating from each site. 960 01:09:43,540 --> 01:09:44,547 Find one. 961 01:09:44,547 --> 01:09:46,937 [LAUGHTER] 962 01:09:46,937 --> 01:09:47,899 OK. 963 01:09:47,899 --> 01:09:51,560 Since you won't have one, that stands. 964 01:09:51,560 --> 01:09:53,630 So you can take the log of that. 965 01:09:53,630 --> 01:09:57,455 You have this free energy, whatever you like. 966 01:09:57,455 --> 01:09:59,650 We can't. 967 01:09:59,650 --> 01:10:06,650 1 is essentially not the zeroth order term in this series. 968 01:10:06,650 --> 01:10:07,792 Yes? 969 01:10:07,792 --> 01:10:08,960 That was the question. 970 01:10:08,960 --> 01:10:10,881 OK. 971 01:10:10,881 --> 01:10:11,380 All right? 972 01:10:14,080 --> 01:10:17,310 You can use the same thing, same technology, 973 01:10:17,310 --> 01:10:19,640 to calculate spin-spin correlation. 974 01:10:19,640 --> 01:10:23,055 So I pick spins m and n on this chain. 975 01:10:23,055 --> 01:10:28,000 Let's say this is spin m here, and somewhere here I 976 01:10:28,000 --> 01:10:29,180 put spin n. 977 01:10:29,180 --> 01:10:33,090 And I want to know the average of that quantity. 978 01:10:33,090 --> 01:10:35,240 What am I supposed to do? 979 01:10:35,240 --> 01:10:41,730 I'm supposed to sum over all configurations with the weight 980 01:10:41,730 --> 01:10:48,216 sigma i sigma i plus 1 product over all-- well, actually, 981 01:10:48,216 --> 01:10:49,720 we can be general with this. 982 01:10:49,720 --> 01:10:54,980 Let's call it product over all bonds, which, in this case, 983 01:10:54,980 --> 01:10:57,930 are near neighbors, sigma i sigma j. 984 01:10:57,930 --> 01:11:03,580 That weight I have to multiply by sigma m sigma n. 985 01:11:03,580 --> 01:11:09,450 And then I have to divide by the partition function 986 01:11:09,450 --> 01:11:11,892 so that this is appropriately weighted. 987 01:11:11,892 --> 01:11:12,391 OK? 988 01:11:15,160 --> 01:11:20,960 So I can do precisely the same decomposition over here. 989 01:11:20,960 --> 01:11:27,730 So I will have 2 to the N cosh K to the number of bonds. 990 01:11:27,730 --> 01:11:33,380 In fact, this I can do in any dimensions. 991 01:11:33,380 --> 01:11:44,280 It's not really what I would have only in one dimension. 992 01:11:44,280 --> 01:11:47,010 And the partition function, you have seen, 993 01:11:47,010 --> 01:11:51,190 is the sum over all graphs, where 994 01:11:51,190 --> 01:11:58,590 t to the number of bonds in graph is called g. 995 01:12:01,270 --> 01:12:06,700 Now I can do the same kind of expansion that I did over here. 996 01:12:06,700 --> 01:12:13,050 If I multiply with an additional sigma m sigma n, 997 01:12:13,050 --> 01:12:15,980 it is just like I have already a sigma m 998 01:12:15,980 --> 01:12:17,300 and a sigma n somewhere. 999 01:12:20,470 --> 01:12:22,360 And when I sum over sigmas, I have 1000 01:12:22,360 --> 01:12:29,260 to make sure that these things don't average to 0. 1001 01:12:29,260 --> 01:12:33,050 So what I need to do is to draw graphs 1002 01:12:33,050 --> 01:12:35,540 that have an even number at all sites 1003 01:12:35,540 --> 01:12:39,220 and an odd number at these two sites. 1004 01:12:39,220 --> 01:12:39,720 All right? 1005 01:12:39,720 --> 01:12:52,230 So this is sum over g with even number except on m and n, 1006 01:12:52,230 --> 01:12:54,880 where you have to have an odd number, 1007 01:12:54,880 --> 01:12:57,821 and t is subset of graphs. 1008 01:13:01,120 --> 01:13:01,830 OK? 1009 01:13:01,830 --> 01:13:08,110 So if I do this for the 1D model, 1010 01:13:08,110 --> 01:13:20,480 sigma m sigma n, I have to draw graphs that have, essentially, 1011 01:13:20,480 --> 01:13:24,830 an odd number. 1012 01:13:24,830 --> 01:13:27,210 Essentially, sigma m and sigma n should 1013 01:13:27,210 --> 01:13:32,330 be the origins or ends of lines. 1014 01:13:32,330 --> 01:13:38,350 And clearly, I can draw a graph that connects these two. 1015 01:13:42,520 --> 01:13:47,210 And so what I will get is t to the number of steps 1016 01:13:47,210 --> 01:13:50,250 that I have to make between the two of them. 1017 01:13:50,250 --> 01:13:53,720 The rest of the it is going to be the same, 2 to the N cosh K 1018 01:13:53,720 --> 01:13:56,700 to the N minus 1 in the numerator and denominator, 1019 01:13:56,700 --> 01:13:58,633 they cancel each other. 1020 01:13:58,633 --> 01:14:00,480 OK? 1021 01:14:00,480 --> 01:14:04,740 So you can see explicitly that this 1022 01:14:04,740 --> 01:14:11,980 is a function that decays since t is less than 1 1023 01:14:11,980 --> 01:14:15,440 as I go further and further out. 1024 01:14:15,440 --> 01:14:17,620 And that it is a pure exponential. 1025 01:14:17,620 --> 01:14:20,630 So you remember that we said in general you would have a power 1026 01:14:20,630 --> 01:14:23,730 law line in front that would have an exponent [? theta. ?] 1027 01:14:23,730 --> 01:14:25,950 And when we did r of g, I told you, well, 1028 01:14:25,950 --> 01:14:29,070 [? theta ?] came out to be 1 such that you have pure 1029 01:14:29,070 --> 01:14:29,900 exponential. 1030 01:14:29,900 --> 01:14:32,280 Well, here is the proof. 1031 01:14:32,280 --> 01:14:37,910 And furthermore, from this we see that c is minus 1 1032 01:14:37,910 --> 01:14:41,470 over log of the hyperbolic tanh of K. 1033 01:14:41,470 --> 01:14:43,390 And if you expand that, you will find 1034 01:14:43,390 --> 01:14:46,420 that as K goes to infinity, it has 1035 01:14:46,420 --> 01:14:50,250 precisely that e to the 2K divergence 1036 01:14:50,250 --> 01:14:52,710 that we had calculated. 1037 01:14:52,710 --> 01:14:56,570 So you can see that calculating things 1038 01:14:56,570 --> 01:15:01,080 using this graphical method is very simple. 1039 01:15:01,080 --> 01:15:03,830 And essentially, the interpretation of t 1040 01:15:03,830 --> 01:15:07,590 is that it is the fidelity with which information 1041 01:15:07,590 --> 01:15:10,300 goes from one site to the next site. 1042 01:15:10,300 --> 01:15:13,790 And so the further away you go every time, 1043 01:15:13,790 --> 01:15:16,440 you lose a factor of t in how sure 1044 01:15:16,440 --> 01:15:20,310 you are about the nature of where you started with. 1045 01:15:20,310 --> 01:15:22,900 And so as you go further, you have this exponential decay. 1046 01:15:26,360 --> 01:15:28,370 OK? 1047 01:15:28,370 --> 01:15:32,816 And the other thing that we can do at no cost 1048 01:15:32,816 --> 01:15:34,585 is periodic boundary conditions. 1049 01:15:39,680 --> 01:15:45,680 So we take, again, our spins 1, 2, 3, 1050 01:15:45,680 --> 01:15:53,450 except that we then bend it such that the last one comes 1051 01:15:53,450 --> 01:15:56,755 and gets connected to the first one. 1052 01:15:56,755 --> 01:15:59,030 OK? 1053 01:15:59,030 --> 01:16:05,590 So what's the partition function in this case? 1054 01:16:05,590 --> 01:16:09,910 It is 2 to the N. 1055 01:16:09,910 --> 01:16:11,810 The number of bonds, in this case, 1056 01:16:11,810 --> 01:16:13,960 is exactly the same as the number of sites. 1057 01:16:13,960 --> 01:16:16,040 It's one more than before, so I get 1058 01:16:16,040 --> 01:16:26,870 to cosh K raised to the power of N. And then is it just one? 1059 01:16:26,870 --> 01:16:30,190 There is one thing that goes all the way around, 1060 01:16:30,190 --> 01:16:36,220 so I have 1 plus t to the N. So this is an exponentially small 1061 01:16:36,220 --> 01:16:40,740 correction as we go further and further out. 1062 01:16:40,740 --> 01:16:45,700 You can kind of regard that as some finite-size interaction. 1063 01:16:45,700 --> 01:16:53,841 I can similarly calculate sigma m sigma n, the expectation 1064 01:16:53,841 --> 01:16:54,340 value. 1065 01:16:59,576 --> 01:17:01,970 OK? 1066 01:17:01,970 --> 01:17:05,400 And in the denominator from the partition 1067 01:17:05,400 --> 01:17:12,700 function, I have this factor of 1 to the t to the N. 1068 01:17:12,700 --> 01:17:14,820 In the numerator, again, you should 1069 01:17:14,820 --> 01:17:16,615 be able to see two graphs. 1070 01:17:16,615 --> 01:17:20,760 We can either connect this way or we can connect that way. 1071 01:17:20,760 --> 01:17:25,410 So you'll have t to the power of n minus m, 1072 01:17:25,410 --> 01:17:28,710 but you don't know which angle is the smaller one, so you'll 1073 01:17:28,710 --> 01:17:31,867 have to also include the other one. 1074 01:17:35,220 --> 01:17:37,620 OK? 1075 01:17:37,620 --> 01:17:40,920 So again, if we take N to infinity and these two 1076 01:17:40,920 --> 01:17:43,080 sufficiently close, you can see that all 1077 01:17:43,080 --> 01:17:47,120 of these finite-size effects, boundary effects, 1078 01:17:47,120 --> 01:17:49,090 et cetera disappear. 1079 01:17:49,090 --> 01:17:51,520 But this is, again, a toy model in which 1080 01:17:51,520 --> 01:17:55,330 to think about what the effects of boundaries is, et cetera. 1081 01:17:55,330 --> 01:17:59,090 You can see how nicely this graphical method 1082 01:17:59,090 --> 01:18:03,790 can enable you to calculate things very rapidly. 1083 01:18:03,790 --> 01:18:08,600 We'll see that, again, it provides the right tools 1084 01:18:08,600 --> 01:18:14,010 conceptually to think about what happens in higher dimensions.