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 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,236 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,236 --> 00:00:17,861 at ocw.mit.edu. 8 00:00:21,500 --> 00:00:22,660 PROFESSOR: OK, let's start. 9 00:00:25,770 --> 00:00:31,060 So recapping what we have been doing, 10 00:00:31,060 --> 00:00:34,800 we said that many systems that undergo phase transition-- so 11 00:00:34,800 --> 00:00:38,830 there's some material that undergoes phase transition-- we 12 00:00:38,830 --> 00:00:41,130 could look at it and characterize it 13 00:00:41,130 --> 00:00:43,900 through a statistical field. 14 00:00:43,900 --> 00:00:49,290 But my analogy in the case of magnetization of magnet 15 00:00:49,290 --> 00:00:53,890 will be noted by m that varies as a function 16 00:00:53,890 --> 00:00:56,350 of the position on the sample. 17 00:00:56,350 --> 00:00:59,210 And it's a vector. 18 00:00:59,210 --> 00:01:01,950 And this vector has n components. 19 00:01:01,950 --> 00:01:04,879 And we said that basically we could distinguish 20 00:01:04,879 --> 00:01:09,470 different types of systems by the number of components of n. 21 00:01:09,470 --> 00:01:13,600 And for the case of things like liquid gas, 22 00:01:13,600 --> 00:01:17,060 we had a scale or density difference, 23 00:01:17,060 --> 00:01:18,980 which is one component. 24 00:01:18,980 --> 00:01:27,590 For the case of superfluid, we had the phase 25 00:01:27,590 --> 00:01:31,670 of a quantum mechanical wave function, which 26 00:01:31,670 --> 00:01:36,450 had, therefore, two components when we included the magnitude. 27 00:01:36,450 --> 00:01:45,860 And for the case of, say, [INAUDIBLE] ferromagnet, 28 00:01:45,860 --> 00:01:47,589 we had n equals 3. 29 00:01:52,080 --> 00:01:57,380 We said that basically all of these systems in the vicinity 30 00:01:57,380 --> 00:02:02,180 of the transition point where the field n of x 31 00:02:02,180 --> 00:02:05,970 is presumably fluctuating around a small quantity 32 00:02:05,970 --> 00:02:08,259 and the correlation lengths are large, 33 00:02:08,259 --> 00:02:12,640 we could describe in terms of weight that 34 00:02:12,640 --> 00:02:15,570 was constructed on the basis of symmetry 35 00:02:15,570 --> 00:02:20,240 and a form of locality which allowed 36 00:02:20,240 --> 00:02:24,970 us to express the weight in powers of m 37 00:02:24,970 --> 00:02:33,760 squared integrated in the vicinity of some point x. 38 00:02:33,760 --> 00:02:38,040 Then the connection between the different clients 39 00:02:38,040 --> 00:02:42,514 was captured through terms that involves gradients of m. 40 00:02:46,600 --> 00:02:51,850 And higher order derivatives are also possible. 41 00:02:51,850 --> 00:02:56,250 And that easy back to deviate from the symmetry axis, 42 00:02:56,250 --> 00:02:58,590 we could add a term that is h.m. 43 00:03:04,350 --> 00:03:07,860 So that was this statistical weight 44 00:03:07,860 --> 00:03:12,370 that we assigned to configurations of this field. 45 00:03:12,370 --> 00:03:15,810 Now we said that when you do measurements 46 00:03:15,810 --> 00:03:18,770 of these kinds of systems, for example, 47 00:03:18,770 --> 00:03:21,325 you will see singularities in heat capacity. 48 00:03:28,730 --> 00:03:33,580 And those in the vicinity of the phase transitions 49 00:03:33,580 --> 00:03:35,950 were characterized by an exponent alpha. 50 00:03:44,730 --> 00:03:50,250 Now the value of alpha, you can go and look at various system, 51 00:03:50,250 --> 00:03:54,000 you find for liquid gas systems many different versions 52 00:03:54,000 --> 00:03:57,820 of carbon dioxide, et cetera, and other systems 53 00:03:57,820 --> 00:04:00,600 that you would correspond to n equals 1, 54 00:04:00,600 --> 00:04:03,560 like binary mixtures such as the one that 55 00:04:03,560 --> 00:04:07,440 is in the first problem set, correspond 56 00:04:07,440 --> 00:04:17,430 to a value of alpha divergence that is roughly around 0.11. 57 00:04:17,430 --> 00:04:21,250 For the case of superfluid, we saw 58 00:04:21,250 --> 00:04:24,220 curves that described this lambda point. 59 00:04:24,220 --> 00:04:27,620 There is, again, a divergence, or the divergence 60 00:04:27,620 --> 00:04:29,010 is weaker than the [INAUDIBLE]. 61 00:04:29,010 --> 00:04:34,370 It is approximately a logarithmic divergence. 62 00:04:34,370 --> 00:04:37,520 Whereas for ferromagnets, there is a cost singularity. 63 00:04:37,520 --> 00:04:39,420 There's no divergence. 64 00:04:39,420 --> 00:04:42,400 And the singularity can be expressed 65 00:04:42,400 --> 00:04:45,610 in terms of a negative alpha. 66 00:04:45,610 --> 00:04:47,920 So there are these classes, depending 67 00:04:47,920 --> 00:04:53,860 on the value of this parameter n, which are all the same. 68 00:04:53,860 --> 00:04:58,300 And in our case, they are all described by this same field 69 00:04:58,300 --> 00:05:04,350 theory, with different number of components of this quantity n. 70 00:05:04,350 --> 00:05:09,140 So we asked whether or not we could get that result. 71 00:05:09,140 --> 00:05:15,600 So what we did was we said, OK, let's calculate the partition 72 00:05:15,600 --> 00:05:18,560 function that corresponds to this system 73 00:05:18,560 --> 00:05:22,260 by integrating over all configurations of this field. 74 00:05:25,340 --> 00:05:29,300 And this is actually just the singular part, 75 00:05:29,300 --> 00:05:32,400 because in the process of going from whatever 76 00:05:32,400 --> 00:05:36,850 microscopic variables we have to these variables that describe 77 00:05:36,850 --> 00:05:40,320 the statistical field, we have to integrate 78 00:05:40,320 --> 00:05:43,090 over many microscopic configurations. 79 00:05:43,090 --> 00:05:46,510 So there could be a non-singular part that emerges. 80 00:05:46,510 --> 00:05:50,960 But the singularities are due to the appearance of magnetization 81 00:05:50,960 --> 00:05:52,480 spontaneously. 82 00:05:52,480 --> 00:05:56,660 So they should be reflected in calculating the partition 83 00:05:56,660 --> 00:05:58,010 function of this component. 84 00:06:01,370 --> 00:06:05,830 Now what we did was, then, to say, 85 00:06:05,830 --> 00:06:08,490 OK, this is difficult thing. 86 00:06:08,490 --> 00:06:18,240 What I am going to do is do a subtle point approximation, 87 00:06:18,240 --> 00:06:22,415 which really amounted to finding the most probable state. 88 00:06:31,280 --> 00:06:39,280 And that most probable state corresponded to the m 89 00:06:39,280 --> 00:06:47,700 being uniform across the system, value m bar, that potentially 90 00:06:47,700 --> 00:06:55,640 would be directed along the magnetic field. 91 00:06:55,640 --> 00:06:58,650 But there's only the limit that magnetic field goes to 0, 92 00:06:58,650 --> 00:07:02,780 spontaneously select some kind of a direction. 93 00:07:02,780 --> 00:07:07,210 But of course, this m bar would be 0 94 00:07:07,210 --> 00:07:10,320 if you are above the transition, which 95 00:07:10,320 --> 00:07:13,180 in this most probable state occurs 96 00:07:13,180 --> 00:07:17,255 for t's that are positive at h equals 0. 97 00:07:20,340 --> 00:07:26,000 While for t negative, minimizing tm squared plus um 98 00:07:26,000 --> 00:07:30,920 to the fourth gave us a value of square root of minus t over 4. 99 00:07:39,370 --> 00:07:52,490 Then our z singular in the subtle point approximation 100 00:07:52,490 --> 00:08:00,240 evaluated as a function of t for h equals 0 101 00:08:00,240 --> 00:08:05,990 is simply related to the value of this most probable state 102 00:08:05,990 --> 00:08:08,700 at this particular point. 103 00:08:08,700 --> 00:08:14,480 And we found that the answer was exponential of minus because 104 00:08:14,480 --> 00:08:17,450 of the integration over space. 105 00:08:17,450 --> 00:08:18,695 But everything is uniform. 106 00:08:18,695 --> 00:08:21,740 It will be proportional to volume. 107 00:08:21,740 --> 00:08:24,870 And then multiplied by a function 108 00:08:24,870 --> 00:08:30,140 that was either 0, if you were looking at t positive. 109 00:08:30,140 --> 00:08:33,679 Whereas for t negative, substituting that value 110 00:08:33,679 --> 00:08:38,790 of h bar, gave us minus t squared over 16. 111 00:08:45,476 --> 00:08:45,975 Yeah. 112 00:08:55,640 --> 00:08:58,820 So essentially, it's a function. 113 00:08:58,820 --> 00:09:04,860 There is really no magnetization above the critical point. 114 00:09:04,860 --> 00:09:06,690 And you get 0. 115 00:09:06,690 --> 00:09:09,740 Below the critical point, what you 116 00:09:09,740 --> 00:09:13,550 have is this quadratic behavior in t. 117 00:09:13,550 --> 00:09:17,280 So if I were to take two derivatives of it, which 118 00:09:17,280 --> 00:09:19,950 would give me something that is proportional to the heat 119 00:09:19,950 --> 00:09:27,450 capacity-- so from here I would get a heat capacity evaluated 120 00:09:27,450 --> 00:09:33,570 in the subtle point method as a function of t for h 121 00:09:33,570 --> 00:09:39,545 equals 0, which would be either 0 or 1 over 8u, 122 00:09:39,545 --> 00:09:42,440 if I'm taking this derivative. 123 00:09:42,440 --> 00:09:48,440 So the prediction is that you have an alpha which is 0 124 00:09:48,440 --> 00:09:52,520 because there is no power law dependence. 125 00:09:52,520 --> 00:09:56,795 And what it really reflects is that there 126 00:09:56,795 --> 00:10:00,750 is a discontinuity in heat capacity. 127 00:10:00,750 --> 00:10:02,870 So none of the examples that I showed 128 00:10:02,870 --> 00:10:07,670 you aboved-- the liquid gas, the superfluid, ferromagnet-- 129 00:10:07,670 --> 00:10:09,820 have a discontinuous heat capacity. 130 00:10:09,820 --> 00:10:12,560 So this does not seem to work. 131 00:10:12,560 --> 00:10:15,320 On the other hand, this discontinuity 132 00:10:15,320 --> 00:10:21,323 is observed for superconductor transitions. 133 00:10:29,080 --> 00:10:32,070 So that's the state of the affairs. 134 00:10:32,070 --> 00:10:36,250 What we have to understand now is, first of all, 135 00:10:36,250 --> 00:10:39,060 why doesn't it work in general? 136 00:10:39,060 --> 00:10:43,620 Secondly, why does it work for superconductors? 137 00:10:43,620 --> 00:10:46,978 So that's the task for today. 138 00:10:46,978 --> 00:10:49,740 All right? 139 00:10:49,740 --> 00:10:53,780 So the one thing that is certainly 140 00:10:53,780 --> 00:10:57,220 a glaring approximation is to replace 141 00:10:57,220 --> 00:11:01,550 this integration over all configuration by just the one 142 00:11:01,550 --> 00:11:03,900 most probable state. 143 00:11:03,900 --> 00:11:07,190 But we did precisely that when we 144 00:11:07,190 --> 00:11:11,870 were talking about the subtle point method of integration 145 00:11:11,870 --> 00:11:17,110 in the previous class in 8 333. 146 00:11:17,110 --> 00:11:22,770 So let's examine why it was legitimate to do so 147 00:11:22,770 --> 00:11:24,690 at that point. 148 00:11:24,690 --> 00:11:29,020 So there we are evaluating essentially an integration 149 00:11:29,020 --> 00:11:31,360 that involved one variable. 150 00:11:31,360 --> 00:11:33,360 Let's call it m. 151 00:11:33,360 --> 00:11:42,410 And we had a large number that was appearing in the exponent. 152 00:11:42,410 --> 00:11:53,120 And we had some function that we were looking at, 153 00:11:53,120 --> 00:11:54,870 depending on the variable of integration. 154 00:11:57,640 --> 00:12:00,490 Now the most probable value of this 155 00:12:00,490 --> 00:12:04,500 occurs for some particular m bar. 156 00:12:04,500 --> 00:12:09,520 And what we can do, without essentially 157 00:12:09,520 --> 00:12:13,980 doing any approximation at this point, 158 00:12:13,980 --> 00:12:19,130 is to make a Taylor expansion of the function 159 00:12:19,130 --> 00:12:21,130 around its maximum. 160 00:12:21,130 --> 00:12:30,520 So the function I can write as psi evaluated at this extremum. 161 00:12:30,520 --> 00:12:33,180 But since I am looking at an extremum, 162 00:12:33,180 --> 00:12:35,630 if I make a Taylor expansion, the term 163 00:12:35,630 --> 00:12:38,650 that is proportional to the first derivative is absent. 164 00:12:38,650 --> 00:12:41,910 I'm expanding around an extremum. 165 00:12:41,910 --> 00:12:52,020 The term that is proportional to the second derivative evaluated 166 00:12:52,020 --> 00:12:57,600 at m bar will go with m minus m bar squared. 167 00:12:57,600 --> 00:13:00,370 And in principle, there are higher and higher order terms 168 00:13:00,370 --> 00:13:01,710 I can put in this expansion. 169 00:13:05,610 --> 00:13:10,410 Now the value at the most probable position, 170 00:13:10,410 --> 00:13:13,020 which is the subtle point value, is a constant. 171 00:13:13,020 --> 00:13:16,730 I can put it outside. 172 00:13:16,730 --> 00:13:20,410 So essentially, terminating here is exactly 173 00:13:20,410 --> 00:13:24,470 like what I was doing over there, more or less. 174 00:13:24,470 --> 00:13:28,140 But then I have fluctuations around 175 00:13:28,140 --> 00:13:30,540 this most probably state. 176 00:13:30,540 --> 00:13:33,840 So I can do the integration, let's say, 177 00:13:33,840 --> 00:13:38,100 in the variable delta n. 178 00:13:38,100 --> 00:13:44,176 I have the differential of delta m 179 00:13:44,176 --> 00:13:51,070 into the minus 1/2 psi double m bar 180 00:13:51,070 --> 00:13:57,060 m minus delta m bar delta m squared. 181 00:13:57,060 --> 00:13:59,680 And then I have higher order terms. 182 00:13:59,680 --> 00:14:02,790 And principle, those higher order terms I 183 00:14:02,790 --> 00:14:05,330 can start expanding over here. 184 00:14:14,600 --> 00:14:17,510 And I forgot the very important factor, 185 00:14:17,510 --> 00:14:21,530 which is that this whole thing is proportional to n. 186 00:14:21,530 --> 00:14:25,200 And indeed, all of these terms over here 187 00:14:25,200 --> 00:14:29,500 will also be proportional to n. 188 00:14:29,500 --> 00:14:30,870 OK? 189 00:14:30,870 --> 00:14:33,650 But the first term in the series is just 190 00:14:33,650 --> 00:14:36,080 the Gaussian integration. 191 00:14:36,080 --> 00:14:43,680 And so I know that the leading correction to the subtle point 192 00:14:43,680 --> 00:14:52,520 comes from this factor of root 2 pi n psi double prime of m bar. 193 00:14:52,520 --> 00:14:58,370 And then, in principle, there will be higher order terms. 194 00:14:58,370 --> 00:15:03,120 And if you keep track of how many factors of delta m 195 00:15:03,120 --> 00:15:05,160 allowed-- delta m cubed is certainly not 196 00:15:05,160 --> 00:15:07,890 allowed because of the evenness of what 197 00:15:07,890 --> 00:15:09,690 I'm integrating against. 198 00:15:09,690 --> 00:15:13,040 So the next order term will be delta m to the fourth. 199 00:15:13,040 --> 00:15:15,770 Evaluated against this Gaussian, it 200 00:15:15,770 --> 00:15:19,330 will give you something that is order of 1 over n squared. 201 00:15:19,330 --> 00:15:22,420 Multiplied by n, you will get corrections of the order of 1 202 00:15:22,420 --> 00:15:25,680 over n. 203 00:15:25,680 --> 00:15:29,770 So very systematically, we could see 204 00:15:29,770 --> 00:15:33,900 that if I called the result of this integration i, 205 00:15:33,900 --> 00:15:40,440 that log of i has a term that is dominated 206 00:15:40,440 --> 00:15:45,580 by the most probable value of the integrant. 207 00:15:45,580 --> 00:15:52,260 And then there are corrections, such as this factor of log n 208 00:15:52,260 --> 00:15:57,170 psi double prime m bar over 2 pi, 209 00:15:57,170 --> 00:16:02,190 and lower order corrections of order of 1 over n. 210 00:16:02,190 --> 00:16:09,400 Basically all of these terms in the limit of n 211 00:16:09,400 --> 00:16:13,300 being much larger than 1, you can ignore. 212 00:16:13,300 --> 00:16:16,340 And essentially, this term will dominate everything. 213 00:16:20,840 --> 00:16:24,660 So what we did over there kind of looks the same. 214 00:16:24,660 --> 00:16:29,700 So let's repeat that for our functional integral. 215 00:16:29,700 --> 00:16:31,970 So I have z. 216 00:16:31,970 --> 00:16:35,250 Actually it is the singular part of the partition 217 00:16:35,250 --> 00:16:40,790 function, which is obtained by integrating over 218 00:16:40,790 --> 00:16:44,560 all functions m of x. 219 00:16:44,560 --> 00:16:47,210 And for the time being, let's just focus on the h 220 00:16:47,210 --> 00:16:49,400 equals 0 part. 221 00:16:49,400 --> 00:16:53,450 So I have exponential of minus integral 222 00:16:53,450 --> 00:17:00,420 over x, t over 2m square, um to the fourth, 223 00:17:00,420 --> 00:17:05,595 k over 2 gradient of m squared, and so forth. 224 00:17:12,130 --> 00:17:18,700 And repeat what we did over there. 225 00:17:18,700 --> 00:17:22,530 So what we need over here was to basically pick 226 00:17:22,530 --> 00:17:27,819 the most probable state and then expand around 227 00:17:27,819 --> 00:17:30,280 the most probable state. 228 00:17:30,280 --> 00:17:37,300 So going beyond just picking the contribution of most probable 229 00:17:37,300 --> 00:17:41,670 state involves including these fluctuations. 230 00:17:41,670 --> 00:17:50,810 So let me write my m of x to be essentially m bar, 231 00:17:50,810 --> 00:17:53,130 but allowing a little bit of fluctuation. 232 00:17:53,130 --> 00:17:57,080 And we saw that we could divide the fluctuations 233 00:17:57,080 --> 00:17:59,100 into a longitudinal part. 234 00:17:59,100 --> 00:18:02,290 Let's call it e1 hat. 235 00:18:02,290 --> 00:18:09,610 And the transfers part, which is an n minus 1 component 236 00:18:09,610 --> 00:18:13,570 vector in the n minus 1 transfers directions. 237 00:18:17,280 --> 00:18:21,320 And then I substitute this over here. 238 00:18:21,320 --> 00:18:24,200 So what do I get? 239 00:18:24,200 --> 00:18:26,920 Just like here, I can pull out the term 240 00:18:26,920 --> 00:18:29,160 that corresponds to the subtle point. 241 00:18:29,160 --> 00:18:31,770 In fact, I had calculated it up there. 242 00:18:31,770 --> 00:18:37,646 So I have exponential of minus v, 243 00:18:37,646 --> 00:18:40,520 the value of this thing at the subtle point. 244 00:18:48,400 --> 00:18:52,960 And then I have essentially replaced the variable m 245 00:18:52,960 --> 00:18:56,040 with the integration over fluctuations. 246 00:18:56,040 --> 00:19:02,890 So now I have to integrate over the longitudinal fluctuations 247 00:19:02,890 --> 00:19:05,260 and the transfers fluctuations. 248 00:19:08,590 --> 00:19:15,720 And what I need to do is to expand this quantity up 249 00:19:15,720 --> 00:19:18,130 to second order. 250 00:19:18,130 --> 00:19:20,310 But that's exactly what we did last time, 251 00:19:20,310 --> 00:19:24,880 where you were looking at how the system was scattering. 252 00:19:24,880 --> 00:19:31,510 So we can rely on the result from last time 253 00:19:31,510 --> 00:19:34,290 for what the quadratic part is. 254 00:19:34,290 --> 00:19:39,130 So we saw that the answer could be written as minus k 255 00:19:39,130 --> 00:19:43,942 over 2, integral ddx. 256 00:19:43,942 --> 00:19:49,270 Well, actually, let's keep it this way. 257 00:19:49,270 --> 00:20:02,790 We have cl to the minus 2 plus phi l squared 258 00:20:02,790 --> 00:20:11,560 plus gradient of phi l squared. 259 00:20:11,560 --> 00:20:14,700 So this is what I did. 260 00:20:14,700 --> 00:20:21,300 What we had to do was to replace this function. 261 00:20:21,300 --> 00:20:24,035 The only part that has a contribution from variation 262 00:20:24,035 --> 00:20:29,260 in space, and hence contributes to gradient, comes from phi. 263 00:20:29,260 --> 00:20:32,830 So from here, we will get a k over 2 gradient 264 00:20:32,830 --> 00:20:33,790 of phi l squared. 265 00:20:37,090 --> 00:20:41,930 Then there is a contribution from t, 266 00:20:41,930 --> 00:20:44,350 and one that comes from expanding m 267 00:20:44,350 --> 00:20:47,720 to the fourth to quadratic folder, that 268 00:20:47,720 --> 00:20:51,190 are proportional to phi l squared. 269 00:20:51,190 --> 00:20:53,430 And the coefficient of both of them 270 00:20:53,430 --> 00:20:57,190 we combine to write as cl to the minus 2. 271 00:20:57,190 --> 00:21:01,070 And if I go back to what we had last time, 272 00:21:01,070 --> 00:21:09,510 our result was that k over cl squared 273 00:21:09,510 --> 00:21:17,120 was either t, if I was for t positive, 274 00:21:17,120 --> 00:21:20,055 or minus 2t if I was for t negative. 275 00:21:22,850 --> 00:21:30,050 Whereas, when I expanded the transfers component, 276 00:21:30,050 --> 00:21:35,750 what I got above tc, for t positive 277 00:21:35,750 --> 00:21:38,255 there is no difference between longitudinal transfers, 278 00:21:38,255 --> 00:21:40,380 so I had the same result. 279 00:21:40,380 --> 00:21:45,330 Below, there was no cost for these Goldstone modes, 280 00:21:45,330 --> 00:21:48,860 and the answer was 0. 281 00:21:48,860 --> 00:21:51,740 But essentially, I have a similar expression, then, 282 00:21:51,740 --> 00:22:05,050 to write for the transfers component. 283 00:22:11,020 --> 00:22:16,130 So this part amounts to essentially 284 00:22:16,130 --> 00:22:18,711 what I have over here. 285 00:22:18,711 --> 00:22:24,880 And in principle, I can put a whole bunch of other things 286 00:22:24,880 --> 00:22:30,470 that would correspond to higher order fluctuations, effects 287 00:22:30,470 --> 00:22:33,710 beyond the quadratic. 288 00:22:33,710 --> 00:22:37,420 But again, our anticipation is that, just 289 00:22:37,420 --> 00:22:41,200 like what is happening here, the leading correction 290 00:22:41,200 --> 00:22:47,340 to the subtle point will already come from the quadratic part. 291 00:22:47,340 --> 00:22:51,080 So let's evaluate that. 292 00:22:51,080 --> 00:22:53,460 So let's continue. 293 00:22:53,460 --> 00:22:59,155 So this is exponential of the subtle point phi energy. 294 00:23:05,460 --> 00:23:11,370 And then I have to do all of these integrations over phi 295 00:23:11,370 --> 00:23:12,560 l and phi q. 296 00:23:18,210 --> 00:23:23,350 Now what I can do, and I already did this also last time around, 297 00:23:23,350 --> 00:23:30,120 is we introduced an expansion of phi. 298 00:23:30,120 --> 00:23:34,120 We said each phi of x I can write 299 00:23:34,120 --> 00:23:37,510 as a sum over Fourier components-- 300 00:23:37,510 --> 00:23:44,790 e to the iq.x phi tilda of q, and with a root 301 00:23:44,790 --> 00:23:48,200 phi for normalization convenience. 302 00:23:48,200 --> 00:23:53,560 So I can certainly replace both phi l and phi t, 303 00:23:53,560 --> 00:23:58,610 just as I did last time, in terms of Fourier component. 304 00:23:58,610 --> 00:24:03,660 And then the integration over all configurations of phi 305 00:24:03,660 --> 00:24:05,720 is equivalent to integrating over 306 00:24:05,720 --> 00:24:10,760 all configurations of the phi tilda 307 00:24:10,760 --> 00:24:12,513 of q's, sll the Fourier amplitudes. 308 00:24:16,320 --> 00:24:19,280 But the advantage is that when we look at the Fourier 309 00:24:19,280 --> 00:24:23,680 amplitudes, the different q's are 310 00:24:23,680 --> 00:24:26,650 completely independent of each other. 311 00:24:26,650 --> 00:24:30,190 So this integration over here that 312 00:24:30,190 --> 00:24:33,430 was not the one-dimensional integration 313 00:24:33,430 --> 00:24:37,880 becomes a product of one-dimensional integrations 314 00:24:37,880 --> 00:24:42,410 when we go to the Fourier component representation. 315 00:24:42,410 --> 00:24:45,360 So now I have to integrate for each q. 316 00:24:45,360 --> 00:24:49,190 I have either phi l of q, or I have 317 00:24:49,190 --> 00:24:52,620 the n minus 1 component phi p of q. 318 00:24:52,620 --> 00:24:58,190 So these are whole bunch of one dimensional Gaussian 319 00:24:58,190 --> 00:24:59,710 integrations. 320 00:24:59,710 --> 00:25:03,720 Because when I look at what these rates are doing, 321 00:25:03,720 --> 00:25:11,800 I get e to the minus k over 2, q squared plus cl to the minus 2 322 00:25:11,800 --> 00:25:18,060 phi l of q squared for the longitudinal mode, 323 00:25:18,060 --> 00:25:22,620 and a very similar factor k over 2 q 324 00:25:22,620 --> 00:25:27,930 squared plus ct to the minus 2, phi t of q 325 00:25:27,930 --> 00:25:30,460 squared for the transfers vectors. 326 00:25:33,352 --> 00:25:35,830 I have a whole bunch of these different things. 327 00:25:39,080 --> 00:25:43,870 Now we can, again, follow like what we had before. 328 00:25:43,870 --> 00:25:51,200 The leading behavior is minus v t m bar squared over 2, u m bar 329 00:25:51,200 --> 00:25:53,510 to the fourth. 330 00:25:53,510 --> 00:26:00,150 And then I have a product of Gaussian integrations. 331 00:26:00,150 --> 00:26:04,860 For each one of these longitudinal modes, 332 00:26:04,860 --> 00:26:08,090 just like here, I will get a factor 333 00:26:08,090 --> 00:26:18,140 of 2 pi divided by k q squared plus cl 334 00:26:18,140 --> 00:26:23,040 to the minus 2 square root. 335 00:26:23,040 --> 00:26:26,950 And for each one of the transfers components, 336 00:26:26,950 --> 00:26:34,170 I will get 2 pi k q 2 plus ct to the minus 2. 337 00:26:34,170 --> 00:26:37,940 And there are n minus 1 of these. 338 00:26:37,940 --> 00:26:39,460 So I will get that factor. 339 00:26:42,250 --> 00:26:45,370 And then presume, again, there will 340 00:26:45,370 --> 00:26:50,110 be corrections due to higher orders that 341 00:26:50,110 --> 00:26:52,200 will be multiplying the whole thing. 342 00:27:00,220 --> 00:27:04,880 So the quantity that we are interested 343 00:27:04,880 --> 00:27:11,510 is, in fact, something like phi energy. 344 00:27:11,510 --> 00:27:13,710 So we take log of z. 345 00:27:13,710 --> 00:27:16,700 Let's look at the singular part. 346 00:27:16,700 --> 00:27:18,930 Let's divide it by volume, because we 347 00:27:18,930 --> 00:27:21,580 expect this to be an extensive quantity, 348 00:27:21,580 --> 00:27:25,156 just like this other result was proportional to n. 349 00:27:25,156 --> 00:27:28,010 And let's put a minus sign-- typically 350 00:27:28,010 --> 00:27:30,850 you have to change sign in any case-- 351 00:27:30,850 --> 00:27:36,890 so that the leading term then becomes this tm squared 352 00:27:36,890 --> 00:27:42,700 plus um to the fourth, which, let me remind you, 353 00:27:42,700 --> 00:27:46,250 is-- actually, let's just write it. 354 00:27:46,250 --> 00:27:50,140 tm bar squared over 2 plus u m bar to the fourth. 355 00:27:52,940 --> 00:28:00,122 And then when I take the log, this product over q 356 00:28:00,122 --> 00:28:04,162 will go to a sum over q. 357 00:28:07,858 --> 00:28:13,040 And the sum over q in the continuum limit, 358 00:28:13,040 --> 00:28:19,926 I will replace by v integral over q divided by 3 pi 359 00:28:19,926 --> 00:28:20,708 to the d. 360 00:28:23,640 --> 00:28:27,150 So then the next step of the process, 361 00:28:27,150 --> 00:28:29,540 I will have a sum over q which I replace 362 00:28:29,540 --> 00:28:31,830 with v times the integration. 363 00:28:31,830 --> 00:28:37,230 But the volume will go away, and what I'm left with 364 00:28:37,230 --> 00:28:39,230 is the integration. 365 00:28:39,230 --> 00:28:46,530 So I have the integral vdq 2 pi to the d. 366 00:28:46,530 --> 00:28:50,030 And I have the log of whatever is appearing over here. 367 00:28:53,810 --> 00:28:59,930 So what I have there is log of k q 368 00:28:59,930 --> 00:29:06,900 squared plus k cl to the minus 2 with 1/2. 369 00:29:06,900 --> 00:29:07,960 Why the 1/2? 370 00:29:07,960 --> 00:29:10,120 Because it's the square root. 371 00:29:10,120 --> 00:29:12,750 I take it to the exponential because it 372 00:29:12,750 --> 00:29:15,200 becomes one half of the log. 373 00:29:15,200 --> 00:29:17,160 In fact, it is in the denominator. 374 00:29:17,160 --> 00:29:18,440 So there's a minus sign. 375 00:29:18,440 --> 00:29:22,690 And the minus sign cancels the minus sign out here. 376 00:29:22,690 --> 00:29:25,580 And then the next term from the transfers component, 377 00:29:25,580 --> 00:29:31,310 I will get n minus 1 over 2, integral dbq 2 pi 378 00:29:31,310 --> 00:29:39,650 to the d log of kq squared plus kct to the minus 2. 379 00:29:39,650 --> 00:29:43,470 And presumably, if I go ahead with higher and higher order 380 00:29:43,470 --> 00:29:45,258 corrections, there will be other things. 381 00:29:45,258 --> 00:29:45,758 Yes, Carter. 382 00:29:45,758 --> 00:29:47,252 AUDIENCE: So [INAUDIBLE]. 383 00:29:52,740 --> 00:29:53,980 PROFESSOR: No. 384 00:29:53,980 --> 00:29:56,860 It's just, like the subtle point, 385 00:29:56,860 --> 00:30:00,790 I'm trying to calculate a systematic expansion 386 00:30:00,790 --> 00:30:02,420 around the subtle point. 387 00:30:02,420 --> 00:30:07,390 So I've calculated so far the lowest order term, 388 00:30:07,390 --> 00:30:11,120 although I haven't explicitly told you what its behavior is. 389 00:30:11,120 --> 00:30:14,790 Once I'm satisfied with what kind of connection that this, 390 00:30:14,790 --> 00:30:18,900 I need to go beyond and include higher and higher order terms, 391 00:30:18,900 --> 00:30:22,510 and maybe show you that they are explicitly unimportant, 392 00:30:22,510 --> 00:30:24,580 like they are in the ordinary subtle point, 393 00:30:24,580 --> 00:30:25,920 or that they are important. 394 00:30:25,920 --> 00:30:28,160 At this stage, we are agnostic. 395 00:30:28,160 --> 00:30:29,341 We don't say anything. 396 00:30:37,060 --> 00:30:39,330 One thing to note-- of course, there 397 00:30:39,330 --> 00:30:42,310 are all of these factors of 2 pi. 398 00:30:42,310 --> 00:30:46,900 Now if you go to a mathematician and show them 399 00:30:46,900 --> 00:30:51,570 a functional integral, they say it's an undefined quantity. 400 00:30:51,570 --> 00:30:54,190 And part of the reason for undefined quantity 401 00:30:54,190 --> 00:30:57,440 is, well, how many factors of 2 pi do you have? 402 00:30:57,440 --> 00:31:01,270 And what are the limits of this integration? 403 00:31:01,270 --> 00:31:03,970 So from the perspective of mathematics, 404 00:31:03,970 --> 00:31:05,680 a functional integral is something 405 00:31:05,680 --> 00:31:08,610 that is very sick and ill-behaved. 406 00:31:08,610 --> 00:31:11,190 In our case, there is no problem, 407 00:31:11,190 --> 00:31:14,110 because we know that our field, although I wrote it 408 00:31:14,110 --> 00:31:16,830 as a continuous function, it is really 409 00:31:16,830 --> 00:31:20,100 a continuous function that has a limited set of Fourier 410 00:31:20,100 --> 00:31:21,390 components. 411 00:31:21,390 --> 00:31:24,430 This product over q will not extend 412 00:31:24,430 --> 00:31:26,452 to arbitrary short wavelength. 413 00:31:26,452 --> 00:31:27,910 There's a characteristic wavelength 414 00:31:27,910 --> 00:31:31,370 which is the scale over which I did the coarse graining, 415 00:31:31,370 --> 00:31:33,710 and I don't have anything beyond that. 416 00:31:33,710 --> 00:31:36,110 So these are finite number of Fourier modes 417 00:31:36,110 --> 00:31:37,440 that I'm integrating here. 418 00:31:37,440 --> 00:31:41,080 There is a finite number of 2, 2 pi, et cetera, that one has. 419 00:31:45,780 --> 00:31:47,310 All right. 420 00:31:47,310 --> 00:31:50,320 So fine, so this is the behavior. 421 00:31:50,320 --> 00:31:55,640 Again, I have looked only as a function of t setting h 422 00:31:55,640 --> 00:31:58,230 equals to 0. 423 00:31:58,230 --> 00:32:01,300 I didn't include the effect of h. 424 00:32:01,300 --> 00:32:07,110 And let's explicitly look at what these things are for t 425 00:32:07,110 --> 00:32:12,850 positive that I will write above, and t negative 426 00:32:12,850 --> 00:32:14,402 that I will write below. 427 00:32:14,402 --> 00:32:17,810 We saw that above, this is 0. 428 00:32:17,810 --> 00:32:20,510 Below, it is minus t squared over 16u. 429 00:32:24,020 --> 00:32:29,950 That this quantity kcl to the minus 2, it is t above 430 00:32:29,950 --> 00:32:33,660 and it is minus 2t below. 431 00:32:33,660 --> 00:32:39,820 This quantity kc to the minus t squared is t above and 0 below. 432 00:32:43,650 --> 00:32:46,710 Why do I bother to write that? 433 00:32:46,710 --> 00:32:52,250 Because I want to go and address this question of heat capacity. 434 00:32:52,250 --> 00:32:57,370 And we said that heat capacity is ultimately 435 00:32:57,370 --> 00:33:04,620 related to taking two derivatives of this log c 436 00:33:04,620 --> 00:33:08,850 singular with respect to temperature and beta 437 00:33:08,850 --> 00:33:09,740 and all of that. 438 00:33:09,740 --> 00:33:11,740 But let's write it as a proportionality. 439 00:33:11,740 --> 00:33:12,850 It goes like this. 440 00:33:15,716 --> 00:33:16,215 Yes? 441 00:33:19,470 --> 00:33:20,900 Yes? 442 00:33:20,900 --> 00:33:23,850 AUDIENCE: So the third line on the top board, 443 00:33:23,850 --> 00:33:30,020 you have under this continuous product over all elements of q. 444 00:33:30,020 --> 00:33:31,510 PROFESSOR: So this product over q 445 00:33:31,510 --> 00:33:33,818 goes all the way to the end of the line, yes. 446 00:33:33,818 --> 00:33:34,442 AUDIENCE: Yeah. 447 00:33:34,442 --> 00:33:38,387 So can [INAUDIBLE] be in the exponents, 448 00:33:38,387 --> 00:33:39,470 or are they still outside? 449 00:33:42,630 --> 00:33:44,593 PROFESSOR: What infinitesimals? 450 00:33:44,593 --> 00:33:46,485 AUDIENCE: d phi l and d phi t. 451 00:33:49,330 --> 00:33:51,140 PROFESSOR: OK, so what I have left out, 452 00:33:51,140 --> 00:33:53,285 and you're quite right, is the integral. 453 00:33:56,310 --> 00:34:02,730 So for each q, I have to do n integrations 454 00:34:02,730 --> 00:34:06,335 over this variable and this n minus 1 component. 455 00:34:06,335 --> 00:34:08,670 So I forgot the integral sign, so that's correct. 456 00:34:18,530 --> 00:34:20,100 All right. 457 00:34:20,100 --> 00:34:21,280 So what do we have? 458 00:34:24,040 --> 00:34:29,400 So for t positive, if I take two derivative of this with respect 459 00:34:29,400 --> 00:34:35,410 to t-- and actually there is a minus sign involved here, 460 00:34:35,410 --> 00:34:35,909 sorry. 461 00:34:38,719 --> 00:34:42,350 Above the transition, I will get 0. 462 00:34:42,350 --> 00:34:46,020 Below the transition, I will get this 1 over 8u. 463 00:34:46,020 --> 00:34:49,710 So this is the discontinuity that I had calculated before. 464 00:34:52,850 --> 00:34:56,400 Now above the transition, I have to take 465 00:34:56,400 --> 00:35:03,490 a derivative of log of tkq squared plus t with respect 466 00:35:03,490 --> 00:35:04,780 to t. 467 00:35:04,780 --> 00:35:07,960 Taking the derivative of log will give me 468 00:35:07,960 --> 00:35:11,010 1 over its argument. 469 00:35:11,010 --> 00:35:14,820 Taking the second derivative will give me 470 00:35:14,820 --> 00:35:16,740 the argument squared. 471 00:35:16,740 --> 00:35:20,310 Because of the minus sign, I forget about the minus sign. 472 00:35:20,310 --> 00:35:23,310 So two derivatives of this object with respect to t 473 00:35:23,310 --> 00:35:26,430 will bring down a factor of kq squared plus t squared. 474 00:35:29,030 --> 00:35:31,800 And I have to integrate that over q. 475 00:35:36,760 --> 00:35:43,540 And there is one from here, and there's n minus 1 from here. 476 00:35:43,540 --> 00:35:47,070 So there is a total of n over 2 of that. 477 00:35:54,070 --> 00:35:57,810 Below the transition, I have to take a derivative, 478 00:35:57,810 --> 00:36:02,090 except that plus thing is replaced with minus 2t. 479 00:36:02,090 --> 00:36:04,200 So every time I take a derivative, 480 00:36:04,200 --> 00:36:07,380 I will get an additional factor of 2. 481 00:36:07,380 --> 00:36:14,765 So rather than 1/2, I will end up with 2 integral over q 482 00:36:14,765 --> 00:36:21,950 2 pi to the d 1 over kq squared minus 2t squared 483 00:36:21,950 --> 00:36:24,360 from the longitudinal part. 484 00:36:24,360 --> 00:36:26,860 And the transfers part has no t dependence, 485 00:36:26,860 --> 00:36:28,137 so it doesn't contribute. 486 00:36:35,300 --> 00:36:41,090 So the entire thing, you can see, 487 00:36:41,090 --> 00:36:45,740 is what I had calculated at the subtle point. 488 00:36:45,740 --> 00:36:51,990 And to this order in expansions around the subtle point, which 489 00:36:51,990 --> 00:36:57,190 corresponds, essentially, only to the quadratic part, 490 00:36:57,190 --> 00:36:58,430 I have found a correction. 491 00:37:01,440 --> 00:37:06,070 And generically, we see that these corrections 492 00:37:06,070 --> 00:37:15,512 are proportional to an integral over q 2 pi to the d. 493 00:37:15,512 --> 00:37:17,690 I can actually pull out one factor 494 00:37:17,690 --> 00:37:24,220 of k squared outside so that the integral more looks more nice 495 00:37:24,220 --> 00:37:29,430 with some characteristic lengths scale, which is either 496 00:37:29,430 --> 00:37:32,000 coming from t or from minus 2t. 497 00:37:32,000 --> 00:37:34,160 So I can write it as cl squared. 498 00:37:39,640 --> 00:37:47,120 So in order to understand how important these corrections 499 00:37:47,120 --> 00:37:50,350 are-- and here the corrections were under control. 500 00:37:50,350 --> 00:37:54,920 So really, I'm asking question, are these corrections small 501 00:37:54,920 --> 00:37:57,270 compared to what I started in the same sense 502 00:37:57,270 --> 00:38:00,080 that log n is small compared to n? 503 00:38:00,080 --> 00:38:02,060 Well, what I need to do is to understand 504 00:38:02,060 --> 00:38:03,920 how this integral behaves. 505 00:38:03,920 --> 00:38:06,650 There is no factor of log n versus n, 506 00:38:06,650 --> 00:38:08,650 because you can see both of those terms 507 00:38:08,650 --> 00:38:10,921 have the factor of volume. 508 00:38:10,921 --> 00:38:14,770 So the issue is not that you have something like square, 509 00:38:14,770 --> 00:38:17,950 log of the volume that will give you small quantity. 510 00:38:17,950 --> 00:38:20,080 You have to hope that for some reason 511 00:38:20,080 --> 00:38:25,890 or other this whole integral here is not important. 512 00:38:25,890 --> 00:38:32,520 So if I look at the integrand-- well, I can do one more thing. 513 00:38:32,520 --> 00:38:37,260 I can note that it behaves as 1 over k squared. 514 00:38:37,260 --> 00:38:38,880 There is a combination that you will 515 00:38:38,880 --> 00:38:43,230 see appearing many, many times in this course. 516 00:38:43,230 --> 00:38:45,380 Because this is spherically symmetric, 517 00:38:45,380 --> 00:38:49,030 I can write it as some solid angle q 518 00:38:49,030 --> 00:38:51,340 to the d minus 1 with q. 519 00:38:51,340 --> 00:38:54,310 And so the whole thing is proportional to the ratio 520 00:38:54,310 --> 00:38:59,430 of solid angle divided by 2 pi to the d, which will occur so 521 00:38:59,430 --> 00:39:03,150 many times in this course that we will give it a name k sub d. 522 00:39:05,760 --> 00:39:09,230 And then the eventual integral is simply 523 00:39:09,230 --> 00:39:16,350 an integral over one variable q, q to the d minus 1. 524 00:39:16,350 --> 00:39:21,370 And then I have q squared plus c to the minus 2 squared. 525 00:39:26,910 --> 00:39:27,540 Yes? 526 00:39:27,540 --> 00:39:30,040 AUDIENCE: Should that 1 over kd be 1 over k squared? 527 00:39:34,840 --> 00:39:37,855 PROFESSOR: There is a q over k-- Yeah, that's right. 528 00:39:37,855 --> 00:39:39,090 I already had it. 529 00:39:39,090 --> 00:39:40,410 Yes, 1 over k squared. 530 00:39:40,410 --> 00:39:41,701 And then there's 1 over k. 531 00:39:41,701 --> 00:39:42,200 Thank you. 532 00:39:48,260 --> 00:39:52,770 So how does this integrand look like, 533 00:39:52,770 --> 00:39:56,120 the thing that I have to integrate? 534 00:39:56,120 --> 00:39:58,250 As a function of q, I have to integrate 535 00:39:58,250 --> 00:40:01,680 a function that at least that small q has 536 00:40:01,680 --> 00:40:06,320 no problem of singularity, divergence, et cetera. 537 00:40:06,320 --> 00:40:10,390 It is simply q to the minus something 538 00:40:10,390 --> 00:40:12,280 with the coefficient that is like c. 539 00:40:15,510 --> 00:40:21,131 At large distances, however, let's say three dimensions, 540 00:40:21,131 --> 00:40:27,906 it would fall off as q to the power of d minus 1 minus 4. 541 00:40:27,906 --> 00:40:30,960 At large q, I can ignore whatever is from here 542 00:40:30,960 --> 00:40:34,720 and just look at the powers of q. 543 00:40:34,720 --> 00:40:38,020 But then if I'm at sufficiently large dimension, 544 00:40:38,020 --> 00:40:42,030 the function will keep growing. 545 00:40:42,030 --> 00:40:45,100 So basically, depending on which dimensions 546 00:40:45,100 --> 00:40:49,040 you are, and the borderline dimension is clearly for, 547 00:40:49,040 --> 00:40:51,190 it's an integration that you can either 548 00:40:51,190 --> 00:40:55,130 perform without any difficulty going all the way to infinity 549 00:40:55,130 --> 00:41:00,010 in q, or you have to worry about the upper column. 550 00:41:00,010 --> 00:41:01,450 OK? 551 00:41:01,450 --> 00:41:07,600 So if you are in dimensions greater than 4, 552 00:41:07,600 --> 00:41:16,070 what you find is that this cf in dimensions that are larger 553 00:41:16,070 --> 00:41:22,640 than 4, as you go to larger and larger q, 554 00:41:22,640 --> 00:41:24,770 you are integrating something that 555 00:41:24,770 --> 00:41:27,550 is getting bigger and bigger. 556 00:41:27,550 --> 00:41:30,690 And you have to worry about that being infinity. 557 00:41:30,690 --> 00:41:33,110 Except, as I told you, we don't have any worries 558 00:41:33,110 --> 00:41:36,690 about infinity, because our q has 559 00:41:36,690 --> 00:41:41,160 to be cut off by the inverse of the character wavelength, which 560 00:41:41,160 --> 00:41:44,460 is the length scale over which I am doing the coarse grain. 561 00:41:44,460 --> 00:41:48,020 So let's call that cut off lambda, presumably 562 00:41:48,020 --> 00:41:53,510 this inverse of some kind of lattice-like spacing. 563 00:41:53,510 --> 00:41:55,060 It's not the lattice spacing. 564 00:41:55,060 --> 00:41:58,580 It's the coarse graining scale. 565 00:41:58,580 --> 00:42:02,010 So if I'm doing this, then this integral, 566 00:42:02,010 --> 00:42:05,500 I can really forget about what's happening here. 567 00:42:05,500 --> 00:42:06,930 Most of the integral contribution 568 00:42:06,930 --> 00:42:10,350 will come from the large lambda, and so the answer 569 00:42:10,350 --> 00:42:13,600 will be proportional to 1 over k squared 570 00:42:13,600 --> 00:42:16,093 and whatever this other cut off is 571 00:42:16,093 --> 00:42:20,336 raised to the power of t minus 4. 572 00:42:20,336 --> 00:42:21,920 It's proportion. 573 00:42:21,920 --> 00:42:24,670 I don't care about constants of proportionality, et cetera. 574 00:42:27,700 --> 00:42:32,430 However, if I am at dimensions that is 3 less than 4, 575 00:42:32,430 --> 00:42:36,670 any dimension less than 4, I can as well say the upper cut off 576 00:42:36,670 --> 00:42:40,440 go all the way to infinity, because the contribution that I 577 00:42:40,440 --> 00:42:44,180 get by replacing lambda to infinity 578 00:42:44,180 --> 00:42:47,150 is going to be very small. 579 00:42:47,150 --> 00:42:50,790 So then it becomes like a definite integral. 580 00:42:50,790 --> 00:42:53,700 And it becomes more like a definite integral 581 00:42:53,700 --> 00:42:57,370 if I scale q by c inverse. 582 00:42:57,370 --> 00:43:02,920 And then what you have to do is you have 1 over k squared. 583 00:43:02,920 --> 00:43:06,410 You have c inverse to the power of t minus 4 584 00:43:06,410 --> 00:43:09,980 or c to the power of 4 minus t, and then 585 00:43:09,980 --> 00:43:13,700 some definite integral, which is 0 to infinity dx, 586 00:43:13,700 --> 00:43:18,000 x to the d minus 1 divided by x squared plus 1 to the squared. 587 00:43:18,000 --> 00:43:20,340 I don't really care what the number is. 588 00:43:20,340 --> 00:43:23,606 It's just some number that goes in this proportionality. 589 00:43:23,606 --> 00:43:27,504 So this is what happens for d less than 4. 590 00:43:32,690 --> 00:43:38,397 So let's see what all of this means. 591 00:43:38,397 --> 00:43:41,180 So we are trying to understand the behavior of the heat 592 00:43:41,180 --> 00:43:49,270 capacity of the system as a function of this parameter t. 593 00:43:49,270 --> 00:43:55,820 And actually, only the part that corresponds 594 00:43:55,820 --> 00:43:57,466 to integrating the magnetization field. 595 00:43:57,466 --> 00:44:00,230 As I said, there's phonon contributions, 596 00:44:00,230 --> 00:44:02,700 all kinds of other phonon contributions 597 00:44:02,700 --> 00:44:05,020 that give you some kind of a background. 598 00:44:05,020 --> 00:44:08,870 Let's subtract that background and see what we have. 599 00:44:08,870 --> 00:44:14,410 So what we have is that from the subtle point part, 600 00:44:14,410 --> 00:44:17,130 we get this continuity. 601 00:44:17,130 --> 00:44:19,100 So let's draw the subtle point part. 602 00:44:19,100 --> 00:44:22,816 So the subtle point part is-- oops. 603 00:44:22,816 --> 00:44:25,140 Wrong direction. 604 00:44:25,140 --> 00:44:31,550 Above 0, it's 0. 605 00:44:31,550 --> 00:44:36,370 Below 0, it jumps to 1 over 8u. 606 00:44:36,370 --> 00:44:40,330 So it's a behavior such as this. 607 00:44:40,330 --> 00:44:44,120 So this part is the c of the subtle point. 608 00:44:50,770 --> 00:44:54,390 But to that, I have to add a correction. 609 00:44:54,390 --> 00:44:56,200 So let's look at the correction. 610 00:44:56,200 --> 00:44:58,390 First of all, if I'm looking at the correction 611 00:44:58,390 --> 00:45:03,790 above four dimensions, whether I'm above or below, 612 00:45:03,790 --> 00:45:08,100 I have to add one of these quantities. 613 00:45:08,100 --> 00:45:11,700 These quantities don't have any explicit dependence 614 00:45:11,700 --> 00:45:14,150 on t itself. 615 00:45:14,150 --> 00:45:17,510 So what happens is that if I add that, presumably there 616 00:45:17,510 --> 00:45:21,505 is a correction that I will get from below and a correction 617 00:45:21,505 --> 00:45:23,680 that I will get from above. 618 00:45:23,680 --> 00:45:30,760 So this is cf for d that is larger than 4. 619 00:45:30,760 --> 00:45:36,000 So what it certainly does is when I add this part to what 620 00:45:36,000 --> 00:45:42,450 I had before, I will change the magnitude of the discontinuity. 621 00:45:42,450 --> 00:45:43,430 But so what? 622 00:45:43,430 --> 00:45:47,220 The discontinuity itself was not something that was important, 623 00:45:47,220 --> 00:45:50,030 because u was not a universal number. 624 00:45:50,030 --> 00:45:55,260 So there was some singularity before, some singularity above. 625 00:45:55,260 --> 00:46:00,230 We see that the corrections for dimensions greater than 4 626 00:46:00,230 --> 00:46:03,620 do not change the qualitative statement that the heat 627 00:46:03,620 --> 00:46:07,800 capacity should have a discontinuity. 628 00:46:07,800 --> 00:46:12,580 But if I go to dimensions less than 4 629 00:46:12,580 --> 00:46:16,780 and I realize that my c goes like the square root of t-- 630 00:46:16,780 --> 00:46:22,060 there is the formulas for c over there, or t to the minus 1/2-- 631 00:46:22,060 --> 00:46:26,245 we find that this quantity is proportional to t 632 00:46:26,245 --> 00:46:29,715 to the minus 4 minus d over 2. 633 00:46:32,900 --> 00:46:38,850 So below four dimensions, what we get 634 00:46:38,850 --> 00:46:44,310 is that the correction that we calculated 635 00:46:44,310 --> 00:46:45,435 is actually divergent. 636 00:46:48,270 --> 00:46:53,570 So this is cf for d less than 4. 637 00:46:53,570 --> 00:46:57,360 There is a divergence as t goes to 0 that, let's say, 638 00:46:57,360 --> 00:47:00,030 if you're sitting three dimensions 639 00:47:00,030 --> 00:47:02,160 would be an exponent t to the minus 1/2. 640 00:47:07,140 --> 00:47:09,850 So you started with a subtle point prediction 641 00:47:09,850 --> 00:47:14,220 that the heat capacity should be discontinuous. 642 00:47:14,220 --> 00:47:16,640 You add the analog of these corrections 643 00:47:16,640 --> 00:47:19,340 to the subtle point calculation, and you 644 00:47:19,340 --> 00:47:24,620 find that the correction is much, much more important 645 00:47:24,620 --> 00:47:26,245 than the original discontinuity. 646 00:47:26,245 --> 00:47:30,530 It completely changes your conclusions. 647 00:47:30,530 --> 00:47:33,860 So once we go beyond this approximation 648 00:47:33,860 --> 00:47:36,840 that we did over here, the subtle point, 649 00:47:36,840 --> 00:47:40,240 and the difference between our problematic and the one 650 00:47:40,240 --> 00:47:44,705 that we did in 8 333 is that we don't have one variable 651 00:47:44,705 --> 00:47:46,510 that we are integrating. 652 00:47:46,510 --> 00:47:48,630 We are integrating over fluctuations 653 00:47:48,630 --> 00:47:51,170 over the entirety of the system. 654 00:47:51,170 --> 00:47:53,330 And we see that these fluctuations 655 00:47:53,330 --> 00:47:57,580 over the entirety of the system are so severe, at least close 656 00:47:57,580 --> 00:48:00,230 to the transition point, that they completely 657 00:48:00,230 --> 00:48:03,670 invalidate the results that you had from the subtle point. 658 00:48:03,670 --> 00:48:04,340 Yes? 659 00:48:04,340 --> 00:48:08,780 AUDIENCE: So obviously you have some high order [INAUDIBLE]. 660 00:48:08,780 --> 00:48:11,140 And here you're basically completing [INAUDIBLE]. 661 00:48:11,140 --> 00:48:12,540 PROFESSOR: Exactly. 662 00:48:12,540 --> 00:48:16,070 AUDIENCE: Is there an easy way to argue 663 00:48:16,070 --> 00:48:19,140 that for b greater than 4 there is no divergence 664 00:48:19,140 --> 00:48:22,850 lurking in the higher order terms? 665 00:48:22,850 --> 00:48:25,080 PROFESSOR: Actually, the answer is no. 666 00:48:25,080 --> 00:48:29,750 If I look at this integral that I have over here, 667 00:48:29,750 --> 00:48:31,610 it depends on t. 668 00:48:31,610 --> 00:48:34,760 If I take sufficiently high derivatives of it, 669 00:48:34,760 --> 00:48:37,520 I will encounter a singularity. 670 00:48:37,520 --> 00:48:41,810 So indeed, what I have focused here 671 00:48:41,810 --> 00:48:43,690 is at the level of the heat capacity. 672 00:48:43,690 --> 00:48:46,410 But if I were to look at the fifth derivative of the phi 673 00:48:46,410 --> 00:48:48,585 energy, I will see singularities. 674 00:48:48,585 --> 00:48:50,793 AUDIENCE: No, I'm talking about the second derivative 675 00:48:50,793 --> 00:48:52,342 for higher order terms. 676 00:48:56,590 --> 00:48:59,160 PROFESSOR: These higher order terms, the phis? 677 00:48:59,160 --> 00:49:00,090 OK, all right. 678 00:49:00,090 --> 00:49:02,100 So that was my next one. 679 00:49:02,100 --> 00:49:09,040 So you may be tempted to say, OK, I found the divergence. 680 00:49:09,040 --> 00:49:14,160 Let's say that the heat capacity diverges with exponent of 1/2. 681 00:49:14,160 --> 00:49:15,710 And no. 682 00:49:15,710 --> 00:49:19,470 The only thing that it says is that your starting point 683 00:49:19,470 --> 00:49:21,710 was wrong. 684 00:49:21,710 --> 00:49:23,930 Any conclusion that you want to make 685 00:49:23,930 --> 00:49:29,120 based on what we are doing here is wrong. 686 00:49:29,120 --> 00:49:33,050 There is no point in my going beyond and calculating 687 00:49:33,050 --> 00:49:35,040 the higher order term, because I already 688 00:49:35,040 --> 00:49:38,430 see that the lowest order correction is invalidating 689 00:49:38,430 --> 00:49:39,190 my result. 690 00:49:39,190 --> 00:49:42,200 AUDIENCE: So you [INAUDIBLE] conclude that mean field theory 691 00:49:42,200 --> 00:49:44,340 is good for bigger than 4. 692 00:49:53,870 --> 00:49:55,960 PROFESSOR: From what I have told you, 693 00:49:55,960 --> 00:50:00,000 I've shown you that the discontinuity in the heat 694 00:50:00,000 --> 00:50:04,030 capacity is maintained. 695 00:50:04,030 --> 00:50:07,870 It is true that if I look at sufficiently high derivatives, 696 00:50:07,870 --> 00:50:11,760 I may encounter some difficulty in justifying 697 00:50:11,760 --> 00:50:18,880 why d greater than 4 or less that 4 is making a difference. 698 00:50:18,880 --> 00:50:23,590 But certainly, as we will build on what we know later 699 00:50:23,590 --> 00:50:27,420 on in the course, I will be able to convince you 700 00:50:27,420 --> 00:50:29,220 that the mean field theory is certainly 701 00:50:29,220 --> 00:50:32,790 valid in dimensions greater than 4. 702 00:50:32,790 --> 00:50:39,270 But right now, I guess the only thing that we can say for sure 703 00:50:39,270 --> 00:50:43,880 is that the subtle point method cannot be applied when you are 704 00:50:43,880 --> 00:50:46,955 dealing with a field that is varying all over the space. 705 00:50:50,490 --> 00:50:56,980 So we have this situation. 706 00:50:56,980 --> 00:51:00,730 On the other hand, you say, well, 707 00:51:00,730 --> 00:51:02,522 if it is so bad, why does it work 708 00:51:02,522 --> 00:51:03,980 for the case of the superconductor? 709 00:51:06,940 --> 00:51:11,290 So let's see if we can try to understand that. 710 00:51:11,290 --> 00:51:15,320 Again, sticking with the language of the heat capacity, 711 00:51:15,320 --> 00:51:21,140 we see that if I am, let's say, sitting in some dimensions 712 00:51:21,140 --> 00:51:26,760 below 4, to the lowest order I will 713 00:51:26,760 --> 00:51:31,390 predict that there is a discontinuity in the singular 714 00:51:31,390 --> 00:51:40,898 part and that the fluctuations lead to a correction 715 00:51:40,898 --> 00:51:42,235 where it should be divergent. 716 00:51:47,024 --> 00:51:49,130 Now it is mathematically correct. 717 00:51:49,130 --> 00:51:52,540 But let's see how you would go and see 718 00:51:52,540 --> 00:51:53,940 that in the experiments. 719 00:51:53,940 --> 00:51:57,890 So presumably in the experiment, in the analog of your t going 720 00:51:57,890 --> 00:52:03,560 to 0 is that you have a t that passes through tc. 721 00:52:03,560 --> 00:52:06,370 And what you are doing in the experiment 722 00:52:06,370 --> 00:52:09,140 is that you are making measurements, 723 00:52:09,140 --> 00:52:11,789 let's say, at this point, at this point, at this point, 724 00:52:11,789 --> 00:52:13,247 and then you are going all the way. 725 00:52:15,770 --> 00:52:18,380 Now we can see that there could potentially 726 00:52:18,380 --> 00:52:23,600 be a difference, depending on the amplitude of this term. 727 00:52:23,600 --> 00:52:27,490 If it is like that, and I can resolve things at this scale 728 00:52:27,490 --> 00:52:29,940 that I have indicated here, there's no problem. 729 00:52:29,940 --> 00:52:32,650 I should see the divergence. 730 00:52:32,650 --> 00:52:37,020 But suppose the amplitude is much, much smaller 731 00:52:37,020 --> 00:52:41,110 and it is something that is looking like this, 732 00:52:41,110 --> 00:52:43,320 and you are taking measurements that 733 00:52:43,320 --> 00:52:47,240 correspond to, essentially, intervals such as this, 734 00:52:47,240 --> 00:52:50,420 then you really integrate across this. 735 00:52:50,420 --> 00:52:53,740 You don't see the peak. 736 00:52:53,740 --> 00:52:55,820 You don't sufficient resolution. 737 00:52:55,820 --> 00:53:00,200 It's kind of searching for a delta function more or less. 738 00:53:00,200 --> 00:53:03,860 And so whether or not you are in one situation 739 00:53:03,860 --> 00:53:07,360 or another situation could tell you 740 00:53:07,360 --> 00:53:11,820 about the result of experimental observation. 741 00:53:11,820 --> 00:53:16,250 So how do I find out something about that? 742 00:53:16,250 --> 00:53:21,410 Well, I want the amplitude of this 743 00:53:21,410 --> 00:53:24,950 to be at least as large as the discontinuity 744 00:53:24,950 --> 00:53:27,330 for me to be able to state it. 745 00:53:27,330 --> 00:53:31,540 That is, I want to have a c that I 746 00:53:31,540 --> 00:53:35,910 have from the subtle point, which is a discontinuity that 747 00:53:35,910 --> 00:53:38,900 is of the order of one over 8u, so there's 748 00:53:38,900 --> 00:53:41,900 at a discontinuity heat capacity. 749 00:53:41,900 --> 00:53:48,015 This discontinuity should be of the order of this quantity 1 750 00:53:48,015 --> 00:53:54,830 over k squared c to the power of 4 minus t. 751 00:53:54,830 --> 00:54:00,240 But now it becomes kind of non-universal 752 00:54:00,240 --> 00:54:05,870 because I really want to compare things, compare amplitudes. 753 00:54:05,870 --> 00:54:12,330 I know that my c is predicted from the subtle point 754 00:54:12,330 --> 00:54:18,570 to go like t to the minus 1/2, where t is kind 755 00:54:18,570 --> 00:54:20,856 a rescaled version of temperature. 756 00:54:20,856 --> 00:54:25,370 So t is, let's say, tc minus t over tc. 757 00:54:25,370 --> 00:54:28,530 It is something that is dimensionless. 758 00:54:28,530 --> 00:54:30,600 And so all of the dimensions should 759 00:54:30,600 --> 00:54:33,940 be carried by some kind of a prefactor here, 760 00:54:33,940 --> 00:54:36,610 that is some kind of a landscape. 761 00:54:36,610 --> 00:54:40,260 So the correlation, then, is a length scale. 762 00:54:40,260 --> 00:54:43,060 There is some prefactor that is also a length scale, 763 00:54:43,060 --> 00:54:45,460 and then this reduced temperature 764 00:54:45,460 --> 00:54:49,960 that controls the functional divergence. 765 00:54:49,960 --> 00:54:55,730 Actually, I can read off what this c0 should depend on. 766 00:54:55,730 --> 00:55:03,240 You can see that c0 should scale like k square root of k. 767 00:55:09,050 --> 00:55:12,750 So then you can see that this object 768 00:55:12,750 --> 00:55:16,070 k scales like c0 squared. 769 00:55:16,070 --> 00:55:20,570 So this scales like 1 over c0 to fourth power. 770 00:55:20,570 --> 00:55:24,910 And this scales like c0 to the power of 4 minus t. 771 00:55:24,910 --> 00:55:28,090 And then I have this reduced temperature 772 00:55:28,090 --> 00:55:31,766 to the power of d minus 4 over 2. 773 00:55:37,200 --> 00:55:43,080 So you can see that for these things to be compatible, 774 00:55:43,080 --> 00:55:48,610 I should reduce my t to a value such 775 00:55:48,610 --> 00:55:53,400 that this divergence compensates for the combination 776 00:55:53,400 --> 00:56:01,220 c0 to the d delta csp, should be of the order 777 00:56:01,220 --> 00:56:05,210 of some minimal value of t. 778 00:56:05,210 --> 00:56:08,280 Let's call it tc. 779 00:56:08,280 --> 00:56:15,930 Actually, let's call it tg to the power of d minus 4 over 2. 780 00:56:20,440 --> 00:56:34,960 Or tg is of the order of delta csp c0 781 00:56:34,960 --> 00:56:38,250 to the d, the whole thing to the power of 2 782 00:56:38,250 --> 00:56:41,369 divided by d minus 4. 783 00:56:47,357 --> 00:56:50,040 Let me wrote that slightly better. 784 00:56:50,040 --> 00:56:57,660 So tg goes off the order of delta cp, delta csp 785 00:56:57,660 --> 00:57:03,510 to the power of minus 2 4 minus t, 786 00:57:03,510 --> 00:57:07,290 since we are going to be looking at dimensions such as 3, 787 00:57:07,290 --> 00:57:13,240 and then c0 to the power of minus 2 788 00:57:13,240 --> 00:57:17,752 divided by 2d divided by 4 minus d. 789 00:57:23,470 --> 00:57:28,700 So we can see that the resolution that you need, 790 00:57:28,700 --> 00:57:32,720 how close you have to go to the critical point, 791 00:57:32,720 --> 00:57:35,900 very much depends on this quantity c0. 792 00:57:35,900 --> 00:57:38,870 It does depend also on delta csp. 793 00:57:38,870 --> 00:57:43,230 But we can argue that that is a less important contribution. 794 00:57:43,230 --> 00:57:49,770 Let's focus, for the time being, on the dependence on this c0. 795 00:57:49,770 --> 00:57:54,060 So c0 presumably has something to do 796 00:57:54,060 --> 00:57:57,260 with the physics of the system that you are looking at. 797 00:57:57,260 --> 00:58:01,390 So then we are leaving the realm of things that were universal. 798 00:58:01,390 --> 00:58:05,900 And we have to think about the system under consideration. 799 00:58:05,900 --> 00:58:09,390 And we have to identify a length scale associated 800 00:58:09,390 --> 00:58:12,770 with the system that is under consideration. 801 00:58:12,770 --> 00:58:20,000 Now if I think about something like liquid gas, 802 00:58:20,000 --> 00:58:25,980 well, one kind of length scale that immediately comes to mind 803 00:58:25,980 --> 00:58:31,880 is the length scale over which the particles are interacting. 804 00:58:31,880 --> 00:58:35,290 Also I can look at the kind of phase diagrams 805 00:58:35,290 --> 00:58:40,990 that we were looking get, and there was some critical volume 806 00:58:40,990 --> 00:58:45,770 where this transition from one type of isotherm 807 00:58:45,770 --> 00:58:48,500 to another type of isotherm occurs, 808 00:58:48,500 --> 00:58:54,700 I can ask that critical volume how many angstroms it is. 809 00:58:54,700 --> 00:58:57,080 But again, everything here, we have 810 00:58:57,080 --> 00:59:00,410 to try to be as dimensionless as possible. 811 00:59:00,410 --> 00:59:04,350 So let's say this critical volume corresponds 812 00:59:04,350 --> 00:59:06,830 to how many particles. 813 00:59:06,830 --> 00:59:08,840 And let's take the cube root of that 814 00:59:08,840 --> 00:59:11,980 and convert it to a length scale over which 815 00:59:11,980 --> 00:59:15,820 these number of particles are confined in three dimensions. 816 00:59:15,820 --> 00:59:18,950 And what we find is, for liquid gas systems, 817 00:59:18,950 --> 00:59:24,980 that number c0 that you get in units of atomic spacing 818 00:59:24,980 --> 00:59:30,490 is of the order of 1 to 10 atomic spacings. 819 00:59:37,224 --> 00:59:38,186 Yes. 820 00:59:38,186 --> 00:59:43,010 AUDIENCE: Scale on which atoms interact with each other? 821 00:59:43,010 --> 00:59:44,800 PROFESSOR: Well, it could be. 822 00:59:44,800 --> 00:59:48,660 But for the case of, say, particles in this room, 823 00:59:48,660 --> 00:59:54,290 the range of interaction is not that different than the size 824 00:59:54,290 --> 00:59:56,180 of the particles coming together. 825 00:59:56,180 --> 00:59:58,510 It's maybe a few times that. 826 00:59:58,510 --> 01:00:02,060 So that's basically a few times of [INAUDIBLE] saying here. 827 01:00:02,060 --> 01:00:04,170 And I'm not going to argue whether it 828 01:00:04,170 --> 01:00:06,493 is twice that or 10 times that. 829 01:00:06,493 --> 01:00:07,742 It really makes no difference. 830 01:00:10,340 --> 01:00:11,870 The thing is that when I'm looking 831 01:00:11,870 --> 01:00:19,040 about the problem of superconductivity, 832 01:00:19,040 --> 01:00:23,130 this is the only place where we introduce 833 01:00:23,130 --> 01:00:25,050 a little bit of physics. 834 01:00:25,050 --> 01:00:28,410 When one is looking at something like aluminum 835 01:00:28,410 --> 01:00:32,400 that goes into being a superconductor, 836 01:00:32,400 --> 01:00:38,320 it is an ordering of bosons in the same sense 837 01:00:38,320 --> 01:00:41,050 that we have for liquid helium. 838 01:00:41,050 --> 01:00:43,640 But the difference is that what is ordering 839 01:00:43,640 --> 01:00:45,960 in superconductivity is not bosons, 840 01:00:45,960 --> 01:00:48,780 but it is fermions or electrons. 841 01:00:48,780 --> 01:00:51,480 And electrons have Coulomb repulsion. 842 01:00:51,480 --> 01:00:53,570 So what has to happen is that there 843 01:00:53,570 --> 01:00:56,850 is some mechanism, phonons or whatever, that 844 01:00:56,850 --> 01:01:01,960 gives an effective attraction between electrons and pairs 845 01:01:01,960 --> 01:01:04,960 them together into a Cooper pair. 846 01:01:04,960 --> 01:01:08,060 The characteristic size of a Cooper pair, 847 01:01:08,060 --> 01:01:12,820 because of the repulsion that you 848 01:01:12,820 --> 01:01:17,350 have between electrons, rather than being 1 to 10, say, 849 01:01:17,350 --> 01:01:28,350 angstroms, is c0 is suddenly of the order of 1,000 angstroms. 850 01:01:28,350 --> 01:01:32,520 Now note that if you are in three dimensions, 851 01:01:32,520 --> 01:01:35,110 this is something that is raised to the sixth power. 852 01:01:38,400 --> 01:01:44,140 So if I think of this after dividing by an atomic size 853 01:01:44,140 --> 01:01:46,860 or whatever, to a number that is of the order of, let's say, 854 01:01:46,860 --> 01:01:51,510 100 or even 1,000 and I raise it to the sixth power, 855 01:01:51,510 --> 01:01:53,670 you can see that the kind of resolution 856 01:01:53,670 --> 01:01:58,100 that you need when you raise something large to a huge power 857 01:01:58,100 --> 01:02:03,200 corresponds to t that is of the order of 10 to the minus 12, 858 01:02:03,200 --> 01:02:05,490 10 to the minus 15, et cetera. 859 01:02:05,490 --> 01:02:07,090 And that's just not the resolution 860 01:02:07,090 --> 01:02:08,640 that you have in experiment. 861 01:02:08,640 --> 01:02:12,880 So basically experiment will go over this 862 01:02:12,880 --> 01:02:14,760 without really seeing it. 863 01:02:14,760 --> 01:02:19,800 Essentially the units are so big that you 864 01:02:19,800 --> 01:02:25,210 don't have that many of them to fluctuate across the system. 865 01:02:25,210 --> 01:02:28,760 The effect of fluctuations is much diminished 866 01:02:28,760 --> 01:02:32,850 compared to superfluid helium or compared to liquid gas, 867 01:02:32,850 --> 01:02:35,110 where over the size of the system, 868 01:02:35,110 --> 01:02:39,860 you have many, many fluctuations that can take place. 869 01:02:39,860 --> 01:02:43,170 This condition, whether or not you're 870 01:02:43,170 --> 01:02:47,320 going to be able to see the effects of fluctuations 871 01:02:47,320 --> 01:02:49,910 and something that is [INAUDIBLE] field like, 872 01:02:49,910 --> 01:02:52,441 I'll call it t sub g, because it's called a Ginzburg 873 01:02:52,441 --> 01:02:52,940 criterion. 874 01:03:06,440 --> 01:03:12,150 So this basically answers the questions 875 01:03:12,150 --> 01:03:15,520 that we had over here. 876 01:03:15,520 --> 01:03:17,770 For all of our phase transitions, 877 01:03:17,770 --> 01:03:21,110 we constructed the Landau-Ginzburg theory, 878 01:03:21,110 --> 01:03:23,380 and we evaluated its consequences 879 01:03:23,380 --> 01:03:25,250 for phase transition, such as divergence 880 01:03:25,250 --> 01:03:29,120 of heat capacity using the subtle point method. 881 01:03:29,120 --> 01:03:31,360 We saw that the results worked extremely well 882 01:03:31,360 --> 01:03:36,000 for superconductors, but not for anything else. 883 01:03:36,000 --> 01:03:39,890 And the answer to that is that for superconductors, 884 01:03:39,890 --> 01:03:42,530 fluctuations are not so important. 885 01:03:42,530 --> 01:03:45,090 And the most probable state gives you 886 01:03:45,090 --> 01:03:47,410 a good idea of what is happening. 887 01:03:47,410 --> 01:03:49,990 Whereas for super helium, for liquid gas, 888 01:03:49,990 --> 01:03:53,320 et cetera, fluctuations are very important, 889 01:03:53,320 --> 01:03:56,550 and the starting point that is the subtle point, most probable 890 01:03:56,550 --> 01:04:00,695 state, is simply not good enough. 891 01:04:00,695 --> 01:04:01,665 Yes. 892 01:04:01,665 --> 01:04:05,545 AUDIENCE: So when you were giving us 893 01:04:05,545 --> 01:04:09,920 the system of different phase transitions [INAUDIBLE], 894 01:04:09,920 --> 01:04:12,096 you only talked about the critical exponents, 895 01:04:12,096 --> 01:04:18,370 because, for instance, there is a discontinuity of [INAUDIBLE] 896 01:04:18,370 --> 01:04:21,040 heat capacity for all phase transitions. 897 01:04:21,040 --> 01:04:24,372 But it's often masked with fixed singularity, right? 898 01:04:29,040 --> 01:04:31,340 PROFESSOR: Once you have a divergence, 899 01:04:31,340 --> 01:04:36,064 I don't know how you would be talking about a singularity. 900 01:04:36,064 --> 01:04:40,297 AUDIENCE: If you roughly measure the heat capacity further away 901 01:04:40,297 --> 01:04:42,920 from singularity, wouldn't it kind of 902 01:04:42,920 --> 01:04:47,300 converges left and right of two different values? 903 01:04:47,300 --> 01:04:48,010 PROFESSOR: OK. 904 01:04:48,010 --> 01:04:54,070 So if I draw a random function that has divergence, 905 01:04:54,070 --> 01:04:57,390 the chances are very, very good that, if I go a little bit 906 01:04:57,390 --> 01:04:59,200 further, the two of them will not 907 01:04:59,200 --> 01:05:00,990 be exactly at the same height. 908 01:05:00,990 --> 01:05:02,970 There will be an asymmetry. 909 01:05:02,970 --> 01:05:06,690 So are you talking about the asymmetry in amplitudes? 910 01:05:06,690 --> 01:05:10,230 Because I know the amplitudes are not symmetric. 911 01:05:10,230 --> 01:05:14,250 If I go very, very far away, then all kinds of other things 912 01:05:14,250 --> 01:05:15,110 come in to play. 913 01:05:15,110 --> 01:05:18,350 There's the phonon, heat capacity, et cetera. 914 01:05:18,350 --> 01:05:21,640 So the statement that you make, I 915 01:05:21,640 --> 01:05:24,390 have never heard before, in fact. 916 01:05:24,390 --> 01:05:26,290 But I'm trying to see whether or not 917 01:05:26,290 --> 01:05:28,150 it's even mathematically conceivable. 918 01:05:31,366 --> 01:05:36,390 AUDIENCE: Another question with this series 919 01:05:36,390 --> 01:05:38,580 you just wrote out with [INAUDIBLE] singularity, 920 01:05:38,580 --> 01:05:44,111 doesn't it give you that exponent for the singularity? 921 01:05:44,111 --> 01:05:44,694 PROFESSOR: No. 922 01:05:44,694 --> 01:05:47,630 AUDIENCE: It's a [INAUDIBLE] number. 923 01:05:47,630 --> 01:05:50,850 PROFESSOR: It is 1/2, yes. 924 01:05:50,850 --> 01:05:52,870 So there is a theory. 925 01:05:52,870 --> 01:05:54,920 There is a mathematical theory that 926 01:05:54,920 --> 01:05:58,270 has this 1/2 exponent divergence. 927 01:05:58,270 --> 01:05:59,880 What is that theory? 928 01:05:59,880 --> 01:06:06,550 It's a theory that is cut off at the Gaussian level. 929 01:06:06,550 --> 01:06:11,770 So if we had some system for which we were sure 930 01:06:11,770 --> 01:06:16,110 that when we write our statistical field theory, 931 01:06:16,110 --> 01:06:19,980 I can terminate at the level of Gaussian terms, 932 01:06:19,980 --> 01:06:22,170 m squared, gradient of m squared, et cetera. 933 01:06:22,170 --> 01:06:24,300 If such a theory existed, it would 934 01:06:24,300 --> 01:06:27,140 have exactly this divergence. 935 01:06:27,140 --> 01:06:31,305 But I don't see any reason for eliminating all those-- 936 01:06:31,305 --> 01:06:33,694 AUDIENCE: So we still have not found the reason 937 01:06:33,694 --> 01:06:36,210 why the actual experimental exponents are-- 938 01:06:36,210 --> 01:06:38,072 PROFESSOR: No, we have not found. 939 01:06:38,072 --> 01:06:39,940 Yes. 940 01:06:39,940 --> 01:06:43,265 AUDIENCE: So how do we interpret the larger 941 01:06:43,265 --> 01:06:46,800 signal of superconductity? 942 01:06:46,800 --> 01:06:50,260 Does that mean the correlation actually is longer? 943 01:06:50,260 --> 01:06:51,310 PROFESSOR: Yes, yes. 944 01:06:51,310 --> 01:06:54,340 AUDIENCE: But then why are we saying 945 01:06:54,340 --> 01:06:57,690 that the fluctuation there is not so important? 946 01:06:57,690 --> 01:07:00,566 We have longer correlation, then usually that 947 01:07:00,566 --> 01:07:02,440 means we have bigger fluctuation [INAUDIBLE]. 948 01:07:26,950 --> 01:07:30,710 PROFESSOR: OK, so let's see if we can unpack that. 949 01:07:30,710 --> 01:07:44,570 So our correlation length is some c0 t to the minus 1/2. 950 01:07:44,570 --> 01:07:49,520 And indeed, what that says is that 951 01:07:49,520 --> 01:07:54,130 at some particular same value of how far I am away 952 01:07:54,130 --> 01:08:00,860 from the critical point, the correlations are longer ranged. 953 01:08:00,860 --> 01:08:05,906 If I go and look at the amplitudes of the fluctuations 954 01:08:05,906 --> 01:08:21,649 that I have, then I am closer as a function of q 955 01:08:21,649 --> 01:08:25,460 to a situation such as this. 956 01:08:25,460 --> 01:08:33,330 So c0 is large c0. 957 01:08:33,330 --> 01:08:34,979 Inverse would be smaller. 958 01:08:34,979 --> 01:08:38,270 So that's correct. 959 01:08:38,270 --> 01:08:43,620 And then in real space, what it would mean 960 01:08:43,620 --> 01:08:49,439 is that if I look at my system, what I would have 961 01:08:49,439 --> 01:08:55,930 is that there are parts that are of the order of c0 t 962 01:08:55,930 --> 01:09:00,689 to the minus 1/2 that are doing the same thing. 963 01:09:14,496 --> 01:09:16,260 Let me understand your question. 964 01:09:16,260 --> 01:09:19,920 So it is true that the superconductor 965 01:09:19,920 --> 01:09:26,319 you have more correlations, and what that means is 966 01:09:26,319 --> 01:09:34,350 that the number of independent modes that you have that 967 01:09:34,350 --> 01:09:39,770 can contribute and fluctuate is less. 968 01:09:39,770 --> 01:09:46,290 And what we will see ultimately is 969 01:09:46,290 --> 01:09:52,470 that the reason for all of these exponents being different 970 01:09:52,470 --> 01:09:56,810 from what we have in superconductivity 971 01:09:56,810 --> 01:10:02,280 is that there is essentially a much more broader range 972 01:10:02,280 --> 01:10:04,700 of the influence that is contributing 973 01:10:04,700 --> 01:10:07,740 to the whole thing. 974 01:10:07,740 --> 01:10:12,670 So I'm not sure if I'm answering your question. 975 01:10:12,670 --> 01:10:15,790 Let's go back and think about your question. 976 01:10:15,790 --> 01:10:20,620 So basically for superconductor, certainly everything 977 01:10:20,620 --> 01:10:23,740 that we said, including being able to express it 978 01:10:23,740 --> 01:10:26,440 in terms of this statistical field theory, 979 01:10:26,440 --> 01:10:29,830 having large correlation lengths close to the critical point, 980 01:10:29,830 --> 01:10:32,480 all of that is correct. 981 01:10:32,480 --> 01:10:34,800 The only thing that is not correct 982 01:10:34,800 --> 01:10:43,060 is that the diversity of fluctuations here is less. 983 01:10:43,060 --> 01:10:47,130 And this lack of diversity of fluctuations compared 984 01:10:47,130 --> 01:10:51,620 to something like liquid gas gives you 985 01:10:51,620 --> 01:10:53,745 more subtle point, like exponents. 986 01:10:56,760 --> 01:11:05,720 AUDIENCE: So you mean the limit for my integrand with respect 987 01:11:05,720 --> 01:11:07,291 to q is smaller? 988 01:11:07,291 --> 01:11:07,790 [INAUDIBLE] 989 01:11:10,940 --> 01:11:14,500 So the q space I'm integrating is smaller. 990 01:11:14,500 --> 01:11:16,210 PROFESSOR: Yes. 991 01:11:16,210 --> 01:11:21,875 AUDIENCE: But if I calculate fluctuation function, 992 01:11:21,875 --> 01:11:22,716 something? 993 01:11:22,716 --> 01:11:23,340 PROFESSOR: Yes. 994 01:11:23,340 --> 01:11:28,060 So this is what I was trying to calculate here, yes. 995 01:11:28,060 --> 01:11:30,505 AUDIENCE: Then it should be larger than-- 996 01:11:30,505 --> 01:11:31,130 PROFESSOR: Yes. 997 01:11:31,130 --> 01:11:36,100 But it is larger for a smaller limit of q's. 998 01:11:36,100 --> 01:11:37,860 So I guess what you are saying is 999 01:11:37,860 --> 01:11:39,960 that if I look at the superconductor, 1000 01:11:39,960 --> 01:11:42,030 I will see something like this. 1001 01:11:42,030 --> 01:11:47,010 If I look at the liquid gas, I will see something like this. 1002 01:11:47,010 --> 01:11:50,560 AUDIENCE: And [INAUDIBLE] just intuitively interpret 1003 01:11:50,560 --> 01:11:53,280 what's the behavior of the heat capacity from this-- 1004 01:11:57,665 --> 01:11:59,560 PROFESSOR: [INAUDIBLE], because if you 1005 01:11:59,560 --> 01:12:03,160 look at something like this, and particular its t dependence-- 1006 01:12:03,160 --> 01:12:05,720 after all, everything that we are interested 1007 01:12:05,720 --> 01:12:10,880 is how things change as a function of t minus tc. 1008 01:12:10,880 --> 01:12:13,900 So presumably, when we do that for superconductor, 1009 01:12:13,900 --> 01:12:16,770 if you do some kind of a scattering experiment, 1010 01:12:16,770 --> 01:12:19,410 you will see some peak like this emerging, 1011 01:12:19,410 --> 01:12:21,890 but the peak never expanding as much 1012 01:12:21,890 --> 01:12:24,090 as it would do for these things. 1013 01:12:24,090 --> 01:12:27,130 You should be able, based on that, 1014 01:12:27,130 --> 01:12:30,350 to deduce that the range of wavelengths that 1015 01:12:30,350 --> 01:12:32,680 are fluctuating in the superconductor 1016 01:12:32,680 --> 01:12:35,860 is less compared to the liquid gas system. 1017 01:12:35,860 --> 01:12:42,730 And so there is not much range in the diversity of length 1018 01:12:42,730 --> 01:12:46,000 scales that are contributing to the fluctuations 1019 01:12:46,000 --> 01:12:46,945 in a superconductor. 1020 01:12:51,130 --> 01:12:54,460 AUDIENCE: So that explains why we have only very narrow peak 1021 01:12:54,460 --> 01:12:55,430 in a cp? 1022 01:12:59,410 --> 01:13:00,140 PROFESSOR: Yes. 1023 01:13:00,140 --> 01:13:02,950 You have to go very close in order 1024 01:13:02,950 --> 01:13:05,360 to expand the range of wavelengths. 1025 01:13:08,024 --> 01:13:11,170 But then you go a little bit one side or the other, 1026 01:13:11,170 --> 01:13:13,200 and then you are passed that range 1027 01:13:13,200 --> 01:13:14,920 that you can see very large wavelengths. 1028 01:13:17,500 --> 01:13:18,230 Yes. 1029 01:13:18,230 --> 01:13:20,420 AUDIENCE: So in our subtle point, the approximation 1030 01:13:20,420 --> 01:13:23,610 we found our maximum when we looked 1031 01:13:23,610 --> 01:13:25,855 at the second derivative, if we had considered more 1032 01:13:25,855 --> 01:13:28,640 derivatives, would we have captured those exponents? 1033 01:13:32,080 --> 01:13:34,970 PROFESSOR: So if we think about things, 1034 01:13:34,970 --> 01:13:40,230 and mathematical consistence, here we have a parameter. 1035 01:13:40,230 --> 01:13:45,360 And we can explicitly calculate higher and higher order terms 1036 01:13:45,360 --> 01:13:49,630 and how they are smaller and become more and more small 1037 01:13:49,630 --> 01:13:53,140 as the parameter becomes larger and larger. 1038 01:13:53,140 --> 01:13:58,140 Now what we have here is the following situation. 1039 01:13:58,140 --> 01:14:02,140 If I presumably stick at some value that is away 1040 01:14:02,140 --> 01:14:09,611 from the critical point, let's say t of 10 to the minus 1, 1041 01:14:09,611 --> 01:14:12,260 at that point I calculate subtle point. 1042 01:14:12,260 --> 01:14:13,940 And then I calculate fluctuations 1043 01:14:13,940 --> 01:14:17,140 around subtle point, and I add more and more term, 1044 01:14:17,140 --> 01:14:19,070 eventually, I think, I will converge 1045 01:14:19,070 --> 01:14:21,970 to some value for the heat capacity. 1046 01:14:21,970 --> 01:14:26,010 The problem is that I don't want to stick to one value of t. 1047 01:14:26,010 --> 01:14:29,990 I want to see what's the singularity as I approach 0. 1048 01:14:32,530 --> 01:14:37,150 Now we can see that the problem here 1049 01:14:37,150 --> 01:14:43,610 is that this correction gives a functional form that 1050 01:14:43,610 --> 01:14:45,750 is divergent. 1051 01:14:45,750 --> 01:14:54,130 And then I would say that if I go to from t of minus 1 to 10 1052 01:14:54,130 --> 01:15:00,870 to the minus 3, then I'm less sure about the first correction 1053 01:15:00,870 --> 01:15:03,680 and maybe I will do many, many more corrections, 1054 01:15:03,680 --> 01:15:05,880 and then I would get something else. 1055 01:15:05,880 --> 01:15:09,590 And presumably, the closer I get to t equals to 0, 1056 01:15:09,590 --> 01:15:13,130 I have to go further and further down in the series. 1057 01:15:13,130 --> 01:15:15,513 And so that becomes essentially useless. 1058 01:15:20,250 --> 01:15:23,140 Now we will actually do later on another version 1059 01:15:23,140 --> 01:15:27,020 of this problem, where we say the following. 1060 01:15:27,020 --> 01:15:31,030 What I did for you here was calculating essentially 1061 01:15:31,030 --> 01:15:33,220 Gaussian integrals. 1062 01:15:33,220 --> 01:15:36,390 And I know how to do Gaussian integrals. 1063 01:15:36,390 --> 01:15:40,165 And for Gaussian theory, this result is exact. 1064 01:15:40,165 --> 01:15:43,610 I will get alpha equals to 1/2. 1065 01:15:43,610 --> 01:15:47,550 Maybe what I can do, instead of doing subtle points 1066 01:15:47,550 --> 01:15:49,690 approximation, approach the problem 1067 01:15:49,690 --> 01:15:52,130 completely in a different fashion. 1068 01:15:52,130 --> 01:15:54,590 I will start with the Gaussian part, 1069 01:15:54,590 --> 01:15:59,800 and then I do a perturbation in all of these nonlinearities. 1070 01:15:59,800 --> 01:16:01,190 That's another approach. 1071 01:16:01,190 --> 01:16:06,190 You can say, OK, I know the problem for u equals 0, 1072 01:16:06,190 --> 01:16:10,570 and so let's say I got this result for u equals to 0. 1073 01:16:10,570 --> 01:16:13,170 And I want to calculate what the correction will 1074 01:16:13,170 --> 01:16:17,850 be in proportion to u, u squared, et cetera. 1075 01:16:17,850 --> 01:16:21,960 But what we find is that we start expanding in u 1076 01:16:21,960 --> 01:16:24,180 and calculate the first correction. 1077 01:16:24,180 --> 01:16:25,860 And the first correction, you'll find, 1078 01:16:25,860 --> 01:16:36,980 is proportional to uc to the power of t minus 4 over 2. 1079 01:16:36,980 --> 01:16:40,210 So exactly the same problem over here 1080 01:16:40,210 --> 01:16:43,490 reappears when we try to do preservation theory. 1081 01:16:43,490 --> 01:16:47,600 You think you are preserving around a small quantity, 1082 01:16:47,600 --> 01:16:51,140 but as you go to t equals to 0, you 1083 01:16:51,140 --> 01:16:55,810 find that the coefficient of the first term in the preservation 1084 01:16:55,810 --> 01:17:00,010 theory actually blows up. 1085 01:17:00,010 --> 01:17:03,920 So we will try a number of these methods 1086 01:17:03,920 --> 01:17:10,690 to try to extract the right answer out of this expression. 1087 01:17:10,690 --> 01:17:13,160 This expression is, in fact, correct. 1088 01:17:13,160 --> 01:17:15,385 The difficulty is mathematical. 1089 01:17:15,385 --> 01:17:19,830 We don't know how to deal with this kind of integration. 1090 01:17:19,830 --> 01:17:26,520 And I was just listening to the story of Oppenheimer and Pauli. 1091 01:17:26,520 --> 01:17:28,540 And Oppenheimer, when he was young, 1092 01:17:28,540 --> 01:17:33,530 goes to-- actually, not Pauli but [INAUDIBLE]. 1093 01:17:33,530 --> 01:17:36,270 And he says, I am working on some problem, 1094 01:17:36,270 --> 01:17:38,285 and I'm not having any progress. 1095 01:17:38,285 --> 01:17:41,120 He says is the problem, the difficulty, 1096 01:17:41,120 --> 01:17:43,560 mathematical or physical? 1097 01:17:43,560 --> 01:17:46,360 And Oppenheimer is flustered because he 1098 01:17:46,360 --> 01:17:48,590 didn't know the answer. 1099 01:17:48,590 --> 01:17:52,520 So here, we know the problem is mathematical, 1100 01:17:52,520 --> 01:17:56,570 because the physics is entirely captured here. 1101 01:17:56,570 --> 01:17:58,790 We haven't done anything. 1102 01:17:58,790 --> 01:18:01,790 Now the question, however, is whether 1103 01:18:01,790 --> 01:18:05,120 the mathematical problem will be resolved 1104 01:18:05,120 --> 01:18:08,820 by mathematical insights or physics insights. 1105 01:18:08,820 --> 01:18:11,500 And the interesting thing is that, in a number of cases 1106 01:18:11,500 --> 01:18:14,940 where the problem originates from physics, eventually 1107 01:18:14,940 --> 01:18:19,040 the mathematical solution is provided also by physics. 1108 01:18:19,040 --> 01:18:23,450 So ultimately, people develop this idea of a normalization 1109 01:18:23,450 --> 01:18:28,190 group that I will be developing for you in future lectures, 1110 01:18:28,190 --> 01:18:32,590 which is how to solve this mathematical problem, which we 1111 01:18:32,590 --> 01:18:34,820 have addressed from this perspective. 1112 01:18:34,820 --> 01:18:39,330 We will try to approach from the perturbative perspective. 1113 01:18:39,330 --> 01:18:43,045 And it just doesn't work until we introduce 1114 01:18:43,045 --> 01:18:46,470 a more physical way of looking at it.