1 00:00:00,060 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high-quality educational resources for free. 5 00:00:10,730 --> 00:00:13,340 To make a donation, or view additional materials 6 00:00:13,340 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:21,990 --> 00:00:25,410 PROFESSOR: OK, let's start. 9 00:00:25,410 --> 00:00:32,580 So last time we started thinking about phase transitions. 10 00:00:32,580 --> 00:00:35,310 We said that a simple example obtained 11 00:00:35,310 --> 00:00:42,260 by taking a piece of magnet such as iron, nickel 12 00:00:42,260 --> 00:00:44,605 and seeing what happens as a function of temperature. 13 00:00:47,800 --> 00:00:53,556 And there's a phase transition between a paramagnet 14 00:00:53,556 --> 00:00:57,240 at high temperature and a ferromagnet 15 00:00:57,240 --> 00:01:00,440 at low temperatures. 16 00:01:00,440 --> 00:01:05,880 This transition takes place at a characteristic Tc. 17 00:01:05,880 --> 00:01:12,490 And a nice way to describe what was happening thermodynamically 18 00:01:12,490 --> 00:01:17,490 was to also look at the space that included a magnetic field. 19 00:01:17,490 --> 00:01:20,370 And then it was clear that various thermodynamic 20 00:01:20,370 --> 00:01:27,650 properties had a singularity, had a discontinuity at the line 21 00:01:27,650 --> 00:01:32,330 h equals to 0 for all T less than or equal to Tc. 22 00:01:36,120 --> 00:01:42,440 Then we saw that if we looked at characteristic isotherms going 23 00:01:42,440 --> 00:01:47,450 from high temperatures going to low temperatures 24 00:01:47,450 --> 00:01:50,950 where there was a discontinuity, we more or less 25 00:01:50,950 --> 00:01:54,950 had to conclude by continuity that the one that 26 00:01:54,950 --> 00:01:59,750 goes along Tc, the magnetization as a function of field, 27 00:01:59,750 --> 00:02:06,830 has to come and hug the axis at 90 degree angle 28 00:02:06,830 --> 00:02:10,840 corresponding to having infinite susceptibility. 29 00:02:10,840 --> 00:02:14,700 We also said that once you have infinite susceptibility, 30 00:02:14,700 --> 00:02:16,560 you can pretty much conclude that you 31 00:02:16,560 --> 00:02:21,350 are going to have long-range correlations across the sample. 32 00:02:21,350 --> 00:02:23,400 So that if you make a fluctuation here, 33 00:02:23,400 --> 00:02:28,440 the influence is going to be felt at large distances away. 34 00:02:28,440 --> 00:02:29,590 OK? 35 00:02:29,590 --> 00:02:33,040 So given with that piece of knowledge, 36 00:02:33,040 --> 00:02:36,190 we said what we can do is to basically do 37 00:02:36,190 --> 00:02:37,490 some kind of averaging. 38 00:02:37,490 --> 00:02:40,290 I can take pieces of the sample. 39 00:02:40,290 --> 00:02:44,620 And at each piece, I can find locally 40 00:02:44,620 --> 00:02:49,930 what the magnetization is and have this field 41 00:02:49,930 --> 00:02:53,820 m as a function of x that varies from one point 42 00:02:53,820 --> 00:02:55,560 to another point. 43 00:02:55,560 --> 00:03:02,840 Presumably, close to this point either below or above, this m 44 00:03:02,840 --> 00:03:07,150 fluctuates across the sample over large distances 45 00:03:07,150 --> 00:03:10,380 and is typically small. 46 00:03:10,380 --> 00:03:13,230 So then we use those pieces of information 47 00:03:13,230 --> 00:03:15,210 to proceed as follows. 48 00:03:15,210 --> 00:03:18,360 We said all thermodynamic properties of the system 49 00:03:18,360 --> 00:03:22,870 can in principle be obtained by looking at a kind of partition 50 00:03:22,870 --> 00:03:25,750 function or Gibbs partition function that depends 51 00:03:25,750 --> 00:03:31,250 on temperature, let's say, which is obtained by tracing 52 00:03:31,250 --> 00:03:37,980 the Hamiltonian that governs all microscopic degrees of freedom 53 00:03:37,980 --> 00:03:39,300 that describe this system. 54 00:03:39,300 --> 00:03:45,650 The electrons, their spins, nuclei, all kinds of things. 55 00:03:45,650 --> 00:03:49,750 Now naturally, this I cannot do. 56 00:03:49,750 --> 00:03:56,980 But I can focus on this magnetization field close to Tc 57 00:03:56,980 --> 00:03:59,835 and say that each configuration of magnetization 58 00:03:59,835 --> 00:04:02,440 has some kind of a weight. 59 00:04:02,440 --> 00:04:04,960 And in principle, what I can do is 60 00:04:04,960 --> 00:04:08,890 I can subdivide all configurations 61 00:04:08,890 --> 00:04:11,680 of microscopic degrees of freedom 62 00:04:11,680 --> 00:04:17,680 that are consistent with a particular macroscopic weight. 63 00:04:17,680 --> 00:04:20,190 Macroscopic field m of x. 64 00:04:20,190 --> 00:04:26,230 And hence, in principle compute what that weight is. 65 00:04:30,500 --> 00:04:34,030 The analog of tracing over all degrees of freedom 66 00:04:34,030 --> 00:04:36,545 would now become integrating over 67 00:04:36,545 --> 00:04:40,140 all configurations of this magnetization 68 00:04:40,140 --> 00:04:43,700 that I indicate through this symbol 69 00:04:43,700 --> 00:04:46,250 of functional integration over all configurations. 70 00:04:49,520 --> 00:04:52,920 Now, clearly I can no more obtain this 71 00:04:52,920 --> 00:04:55,930 than I can do the original trace. 72 00:04:55,930 --> 00:04:57,650 So what did we do? 73 00:04:57,650 --> 00:05:02,260 We said I can guess what this is going to look like. 74 00:05:02,260 --> 00:05:09,670 Because in the absence of the field, 75 00:05:09,670 --> 00:05:12,280 it's a function that has rotational symmetry. 76 00:05:14,960 --> 00:05:21,680 So what I can do is I can write the log of that probability 77 00:05:21,680 --> 00:05:22,910 as something. 78 00:05:22,910 --> 00:05:25,100 So far I haven't done anything. 79 00:05:25,100 --> 00:05:28,130 I made the assumption that I can write it 80 00:05:28,130 --> 00:05:31,790 as an integral over space of some kind 81 00:05:31,790 --> 00:05:34,710 of a density at each point. 82 00:05:34,710 --> 00:05:39,460 So that was this kind of quasi-locality assumption. 83 00:05:39,460 --> 00:05:44,270 And then I would write anything that 84 00:05:44,270 --> 00:05:46,400 comes to my mind that is consistent 85 00:05:46,400 --> 00:05:49,080 with rotational symmetry. 86 00:05:49,080 --> 00:05:52,590 Now, since m is small in the vicinity of this point, 87 00:05:52,590 --> 00:05:55,550 it makes sense that I should make something 88 00:05:55,550 --> 00:05:58,010 like a Taylor expansion. 89 00:05:58,010 --> 00:06:01,770 So the Taylor expansion will start not with a linear term, 90 00:06:01,770 --> 00:06:05,260 which violates rotational symmetry, but something 91 00:06:05,260 --> 00:06:08,090 that is quadratic. 92 00:06:08,090 --> 00:06:11,330 And I can add any even power such as m 93 00:06:11,330 --> 00:06:17,440 to the fourth and higher order terms. 94 00:06:17,440 --> 00:06:22,680 And I can add all kinds of gradients that are consistent, 95 00:06:22,680 --> 00:06:25,680 again, with rotational symmetry. 96 00:06:25,680 --> 00:06:29,720 And the first of those terms is gradient of m squared. 97 00:06:29,720 --> 00:06:30,780 And there are many more. 98 00:06:39,210 --> 00:06:42,960 Of course, there can be an overall constant term. 99 00:06:42,960 --> 00:06:45,220 Maybe I will write it out front. 100 00:06:45,220 --> 00:06:52,030 Insert a Z regular, which means that you-- in the process, 101 00:06:52,030 --> 00:06:55,190 you have all kinds of other degrees of freedom 102 00:06:55,190 --> 00:06:58,340 that are not reflected in the magnetization. 103 00:06:58,340 --> 00:07:00,450 You're going to have phonon degrees of freedom. 104 00:07:00,450 --> 00:07:03,030 So there will be a contribution from the phonons 105 00:07:03,030 --> 00:07:05,480 to the partition function of the system. 106 00:07:05,480 --> 00:07:08,240 We are not interested in any of those things. 107 00:07:08,240 --> 00:07:12,930 We are interested in what becomes singular over here. 108 00:07:12,930 --> 00:07:15,850 And the reason I write that as Z regular 109 00:07:15,850 --> 00:07:19,590 is because it's presumably some benign function of temperature 110 00:07:19,590 --> 00:07:22,040 as I pass through this point. 111 00:07:22,040 --> 00:07:25,150 So it is, indeed, a function of temperature. 112 00:07:25,150 --> 00:07:28,840 It is also worth emphasizing that not only is 113 00:07:28,840 --> 00:07:33,990 this parameter that appears outside representing 114 00:07:33,990 --> 00:07:35,670 all kinds of other degrees of freedom 115 00:07:35,670 --> 00:07:39,100 a function of temperature, that these phenomenological 116 00:07:39,100 --> 00:07:41,540 parameters that I introduced here 117 00:07:41,540 --> 00:07:43,310 are also functions of temperature. 118 00:07:48,120 --> 00:07:53,840 Because the microscopic weight, the true Hamiltonian, 119 00:07:53,840 --> 00:07:58,280 is the one that is scaled by 1 over kt. 120 00:07:58,280 --> 00:08:00,980 Just because of analogy, I sometimes 121 00:08:00,980 --> 00:08:06,280 call this combination of what is happening here beta H or minus 122 00:08:06,280 --> 00:08:09,360 beta H as appearing in the exponent. 123 00:08:09,360 --> 00:08:15,130 But that, by no means, indicates that the coefficient here 124 00:08:15,130 --> 00:08:18,040 are scaling inversely with temperature 125 00:08:18,040 --> 00:08:21,070 like the true microscopic coordinates. 126 00:08:21,070 --> 00:08:26,250 Because in order to do this coarse graining, 127 00:08:26,250 --> 00:08:29,860 I have to integrate over a lot of different configurations. 128 00:08:29,860 --> 00:08:32,180 So there is energy associated with this. 129 00:08:32,180 --> 00:08:34,659 There is entropy associated with this. 130 00:08:34,659 --> 00:08:37,179 There is all kinds of complicated things 131 00:08:37,179 --> 00:08:40,340 that go into this-- these parameters. 132 00:08:40,340 --> 00:08:43,500 So that's important to remember. 133 00:08:43,500 --> 00:08:46,860 Finally, if I slightly go away from here just 134 00:08:46,860 --> 00:08:51,350 to explore the vicinity of having a finite magnetic field, 135 00:08:51,350 --> 00:08:53,930 then I can work in the ensemble where 136 00:08:53,930 --> 00:08:55,880 I have added the field here. 137 00:08:55,880 --> 00:08:58,060 And the weight here will be modified 138 00:08:58,060 --> 00:09:01,402 by an amount that is h dot m. 139 00:09:06,330 --> 00:09:17,650 So what is occurring here is this Landau-Ginzburg model 140 00:09:17,650 --> 00:09:22,400 that I right now introduced in the context 141 00:09:22,400 --> 00:09:25,760 of magnetic systems. 142 00:09:25,760 --> 00:09:28,020 But very shortly, I will introduce 143 00:09:28,020 --> 00:09:31,220 this in the context of super-fluidity. 144 00:09:31,220 --> 00:09:33,590 I can go back to the original example 145 00:09:33,590 --> 00:09:36,540 that we started with the liquid gas phenomena, 146 00:09:36,540 --> 00:09:39,780 replace m with some kind of a density difference, 147 00:09:39,780 --> 00:09:42,930 and then it would indicate a phase transition 148 00:09:42,930 --> 00:09:44,820 of the liquid gas system. 149 00:09:44,820 --> 00:09:47,390 So it is supposed to be very general, 150 00:09:47,390 --> 00:09:52,050 applicable to a lot of things because it is constructed 151 00:09:52,050 --> 00:09:57,890 on the basis of nothing other than symmetry principles. 152 00:09:57,890 --> 00:10:01,870 But at the cost of having no knowledge of how 153 00:10:01,870 --> 00:10:04,970 these phenomenological parameters depend 154 00:10:04,970 --> 00:10:08,210 on the microscopics of the system-- on temperature. 155 00:10:08,210 --> 00:10:11,710 Clearly, for example, even in the context of magnet, 156 00:10:11,710 --> 00:10:15,330 I would construct the same theory for iron, nickel, 157 00:10:15,330 --> 00:10:16,770 cobalt, et cetera. 158 00:10:16,770 --> 00:10:18,350 But presumably these coefficients 159 00:10:18,350 --> 00:10:22,770 will be very different from one system as opposed to the other. 160 00:10:25,400 --> 00:10:31,030 All right, so that's supposed to be, as far as we are concerned, 161 00:10:31,030 --> 00:10:33,380 sufficient to figure out what is going 162 00:10:33,380 --> 00:10:37,380 on in the vicinity of this transition. 163 00:10:37,380 --> 00:10:40,500 Maybe I should emphasize one more thing, 164 00:10:40,500 --> 00:10:44,730 which is that I said that after all what we are trying 165 00:10:44,730 --> 00:10:48,905 to figure out is the nature of singularities in free energy, 166 00:10:48,905 --> 00:10:52,110 in phase diagrams, et cetera. 167 00:10:52,110 --> 00:10:54,720 Yet, when I wrote this, I insisted 168 00:10:54,720 --> 00:10:58,420 on making an analytical expansion. 169 00:10:58,420 --> 00:11:01,740 And the reason making an analytical expansion here 170 00:11:01,740 --> 00:11:06,920 is justified is because to get this expansion, 171 00:11:06,920 --> 00:11:09,060 I summed over degrees of freedom-- 172 00:11:09,060 --> 00:11:12,760 I averaged degrees of freedom over some finite piece 173 00:11:12,760 --> 00:11:14,300 of my system. 174 00:11:14,300 --> 00:11:18,240 Maybe I took a 100 by 100 by 100 Angstrom cubed 175 00:11:18,240 --> 00:11:22,240 block of material and averaged the magnetization of spins 176 00:11:22,240 --> 00:11:25,010 in that area, et cetera. 177 00:11:25,010 --> 00:11:29,660 And the idea that we also encountered last semester 178 00:11:29,660 --> 00:11:31,680 is that as long as you are dealing 179 00:11:31,680 --> 00:11:35,610 with a finite system, all of the manipulations 180 00:11:35,610 --> 00:11:40,040 that you are doing involve analytic functions such as e 181 00:11:40,040 --> 00:11:45,390 to the minus beta H applied to a finite number of integrations. 182 00:11:45,390 --> 00:11:49,490 And you simply cannot get a singularity out of such 183 00:11:49,490 --> 00:11:50,770 a process. 184 00:11:50,770 --> 00:11:54,930 So the averaging process that goes in this coarse graining 185 00:11:54,930 --> 00:11:57,870 must give you analytical functions. 186 00:11:57,870 --> 00:12:00,560 That's not the origin of the singularity. 187 00:12:00,560 --> 00:12:02,750 The origin of the singularity must 188 00:12:02,750 --> 00:12:08,180 come from taking the size of the system, the volume 189 00:12:08,180 --> 00:12:12,230 of this piece of iron, essentially to infinity. 190 00:12:12,230 --> 00:12:16,060 That's what gives us the singularity. 191 00:12:16,060 --> 00:12:24,070 Also, because of that I expect that at the end of the day, 192 00:12:24,070 --> 00:12:28,650 this Z of T and h that I calculate 193 00:12:28,650 --> 00:12:31,400 will be something that is proportional 194 00:12:31,400 --> 00:12:35,000 to volume or something like that in the exponent. 195 00:12:35,000 --> 00:12:36,780 That is, I will have something like e 196 00:12:36,780 --> 00:12:43,640 to the-- from this component minus V some beta f regular. 197 00:12:47,380 --> 00:12:52,170 And actually, let me just forget about the beta 198 00:12:52,170 --> 00:12:54,910 and just write it as some other function regular. 199 00:12:54,910 --> 00:12:57,360 It's an extensive quantity. 200 00:12:57,360 --> 00:13:02,410 And the result of this integration 201 00:13:02,410 --> 00:13:05,020 of all of the field configurations 202 00:13:05,020 --> 00:13:08,890 should give me another contribution 203 00:13:08,890 --> 00:13:12,470 that is proportional to volume. 204 00:13:12,470 --> 00:13:18,060 And hopefully, all of the singularities that I expect 205 00:13:18,060 --> 00:13:19,950 will come from this piece. 206 00:13:23,800 --> 00:13:28,310 And I am going to expect those singularities to arise only 207 00:13:28,310 --> 00:13:30,340 in the limit where V becomes large. 208 00:13:33,390 --> 00:13:37,590 Now, last semester we saw a trick 209 00:13:37,590 --> 00:13:41,500 that allowed us to do integrations 210 00:13:41,500 --> 00:13:45,110 when the final answer was extensive. 211 00:13:45,110 --> 00:13:47,200 It was the saddle point result. 212 00:13:47,200 --> 00:13:52,140 Basically, we said that as we span the integration range, 213 00:13:52,140 --> 00:13:56,710 there is a part that corresponds to extremizing whatever you are 214 00:13:56,710 --> 00:14:01,010 integrating that gives you overwhelmingly 215 00:14:01,010 --> 00:14:05,450 larger weight than any other part of the integration. 216 00:14:05,450 --> 00:14:11,090 So let's try to, without justification-- 217 00:14:11,090 --> 00:14:14,150 and we'll correct this-- apply that same principle 218 00:14:14,150 --> 00:14:25,850 of a saddle point approximation to the functional integration 219 00:14:25,850 --> 00:14:28,260 that we are doing over here. 220 00:14:28,260 --> 00:14:34,910 That is, rather than integrating over all functions, 221 00:14:34,910 --> 00:14:38,100 let's find the extremum. 222 00:14:38,100 --> 00:14:41,270 Where is this function maximized, 223 00:14:41,270 --> 00:14:42,770 the integrand is maximized? 224 00:14:45,430 --> 00:14:50,660 So what I need to do is to find maximum of exponent. 225 00:14:54,580 --> 00:14:57,040 So what is happening over here. 226 00:15:00,700 --> 00:15:05,270 I am going to make one further statement here 227 00:15:05,270 --> 00:15:10,150 before I do that, which is that this term, the thing that 228 00:15:10,150 --> 00:15:13,370 is proportional to gradient of m, 229 00:15:13,370 --> 00:15:16,810 has something to do with the way spins 230 00:15:16,810 --> 00:15:19,360 know about their neighborhood. 231 00:15:19,360 --> 00:15:23,040 And if I am looking at something that is ferromagnet, 232 00:15:23,040 --> 00:15:27,160 the tendency is for neighbors to be in the same direction. 233 00:15:27,160 --> 00:15:29,940 That's how ferromagnetism emerges. 234 00:15:29,940 --> 00:15:36,640 And that corresponds to having a K that is positive. 235 00:15:39,710 --> 00:15:41,890 So that helps me a lot in finding 236 00:15:41,890 --> 00:15:44,860 the extremum of the function because it 237 00:15:44,860 --> 00:15:48,000 means that the extremum will occur when m of x 238 00:15:48,000 --> 00:15:50,530 is uniform across the system. 239 00:15:50,530 --> 00:15:55,260 Any variations in m of x will give you a cost from that 240 00:15:55,260 --> 00:15:58,410 and will reduce the probability. 241 00:15:58,410 --> 00:16:04,200 So with K positive, I know that the optimal solution 242 00:16:04,200 --> 00:16:08,020 is going to correspond to a uniform value 243 00:16:08,020 --> 00:16:10,200 across the system. 244 00:16:10,200 --> 00:16:11,610 What is this uniform value? 245 00:16:11,610 --> 00:16:15,120 If I put a uniform value, first of all, 246 00:16:15,120 --> 00:16:18,080 the integration will simply give me the volume. 247 00:16:18,080 --> 00:16:23,240 That's good because that's what I expect over here. 248 00:16:23,240 --> 00:16:26,180 So at the end of the day, what do I get? 249 00:16:26,180 --> 00:16:38,540 I will get that F singular is, in fact, the minimum of psi 250 00:16:38,540 --> 00:16:46,760 of m where psi of m is obtained by simply evaluating what 251 00:16:46,760 --> 00:16:51,290 is in the integrand at a uniform value of m. 252 00:16:51,290 --> 00:16:54,855 t over 2 m squared mu m to the fourth. 253 00:16:54,855 --> 00:16:59,510 Potentially higher orders, but no K. The K disappeared. 254 00:16:59,510 --> 00:17:00,968 Minus h m. 255 00:17:04,849 --> 00:17:14,329 Actually, I kind of expect also that the uniform solution, 256 00:17:14,329 --> 00:17:17,690 if I put a magnetic field, will point along the magnetic field. 257 00:17:17,690 --> 00:17:20,770 If there's an up field, the uniform solution 258 00:17:20,770 --> 00:17:21,980 will be along the field. 259 00:17:21,980 --> 00:17:23,650 If it's 0, then I don't know. 260 00:17:23,650 --> 00:17:26,250 If it is down field, it will be the opposite. 261 00:17:26,250 --> 00:17:30,620 So it makes sense that if I have a field, 262 00:17:30,620 --> 00:17:32,780 the uniform magnetization should point 263 00:17:32,780 --> 00:17:37,330 along it, which means that over here this dot product is really 264 00:17:37,330 --> 00:17:38,750 just h times m. 265 00:17:41,850 --> 00:17:47,280 So I reduced the complexity of the problem subject 266 00:17:47,280 --> 00:17:49,260 to this saddle point approximation 267 00:17:49,260 --> 00:17:54,170 to just minimizing some function. 268 00:17:54,170 --> 00:17:56,370 Now, it is important to note, since I 269 00:17:56,370 --> 00:18:00,520 was talking about analyticities, and singularities, and things 270 00:18:00,520 --> 00:18:05,450 like that, that whereas the function that is appearing here 271 00:18:05,450 --> 00:18:11,190 as psi of m as we discussed is completely analytical, 272 00:18:11,190 --> 00:18:15,000 the operation of finding the minimum of a function 273 00:18:15,000 --> 00:18:17,990 is something that introduces non-analyticity. 274 00:18:17,990 --> 00:18:21,920 So that's how we are going to get that structure that we 275 00:18:21,920 --> 00:18:24,240 have over there. 276 00:18:24,240 --> 00:18:28,240 Let's explicitly show how that occurs. 277 00:18:28,240 --> 00:18:33,110 So I'm going to show you the shape of this function 278 00:18:33,110 --> 00:18:37,260 in the space that is spanned by the parameters 279 00:18:37,260 --> 00:18:42,170 that I have here, which are t and h. 280 00:18:42,170 --> 00:18:43,910 We will come to u later on. 281 00:18:43,910 --> 00:18:48,540 But for the time being, since I want to show things in a plane, 282 00:18:48,540 --> 00:18:51,170 let's stick with t and h. 283 00:18:57,080 --> 00:19:03,830 So first of all, let's look at the regime where t is positive. 284 00:19:03,830 --> 00:19:04,880 So this is 0. 285 00:19:04,880 --> 00:19:08,610 To the right is t is positive. 286 00:19:08,610 --> 00:19:14,280 Right on the axis when h equals to 0, 287 00:19:14,280 --> 00:19:18,530 the function is m squared plus m to the fourth, et cetera. 288 00:19:18,530 --> 00:19:21,620 The coefficient of m squared is positive. 289 00:19:21,620 --> 00:19:24,260 So basically, right on this axis the function 290 00:19:24,260 --> 00:19:25,666 is like a parabola. 291 00:19:25,666 --> 00:19:32,280 So what I am plotting here are different forms of psi of m. 292 00:19:32,280 --> 00:19:35,120 So this is m squared is a parabola. 293 00:19:35,120 --> 00:19:37,387 If I go further out, m to the fourth 294 00:19:37,387 --> 00:19:39,880 is something to worry about. 295 00:19:39,880 --> 00:19:44,430 But if I'm really looking at case where the emerging 296 00:19:44,430 --> 00:19:47,915 magnetization is small, I can stick just 297 00:19:47,915 --> 00:19:49,891 with the lowest order term in the expansion. 298 00:19:49,891 --> 00:19:50,390 Yes? 299 00:19:50,390 --> 00:19:52,785 AUDIENCE: Are we plotting h or psi? 300 00:19:56,138 --> 00:19:59,970 The axis label was h. 301 00:19:59,970 --> 00:20:01,000 PROFESSOR: OK. 302 00:20:01,000 --> 00:20:05,651 So there is a two-parameter plane t h. 303 00:20:05,651 --> 00:20:10,600 At each plant in this parameter plane, I can plot what psi of m 304 00:20:10,600 --> 00:20:11,590 looks like. 305 00:20:11,590 --> 00:20:15,700 Unfortunately, I can't bring it out of the plane, 306 00:20:15,700 --> 00:20:18,580 so I am going to put it in the two dimensions. 307 00:20:18,580 --> 00:20:21,250 So this is why I wrote psi of m with green 308 00:20:21,250 --> 00:20:23,760 to distinguish it with the h that is in white. 309 00:20:27,960 --> 00:20:30,680 So this is supposed to be right on the axis. 310 00:20:30,680 --> 00:20:36,060 If I go up on the axis, what would psi of m look like? 311 00:20:36,060 --> 00:20:39,970 So now I have a term that is positive h m. 312 00:20:39,970 --> 00:20:45,510 So it starts linearly to go down, 313 00:20:45,510 --> 00:20:49,230 and then the m squared term takes over. 314 00:20:49,230 --> 00:20:51,985 So now the minimum is over here. 315 00:20:51,985 --> 00:20:54,280 Whereas, right on the axis-- again, 316 00:20:54,280 --> 00:20:57,750 by symmetry-- the minimum was at 0. 317 00:20:57,750 --> 00:21:01,820 If I go and look at what happens at the other side 318 00:21:01,820 --> 00:21:04,660 when h is negative, then the slope 319 00:21:04,660 --> 00:21:06,180 is in the opposite direction. 320 00:21:06,180 --> 00:21:08,980 So the function kind of looks like this. 321 00:21:08,980 --> 00:21:12,670 And the minimum has shifted over here. 322 00:21:12,670 --> 00:21:19,060 So if I were to plot now what the minimum m 323 00:21:19,060 --> 00:21:25,422 bar is as a function of the field h, 324 00:21:25,422 --> 00:21:32,830 as I scan from some positive t-- from h positive to h negative, 325 00:21:32,830 --> 00:21:37,020 what we have is that this sequence then 326 00:21:37,020 --> 00:21:41,250 corresponds to a curve in which, essentially, m 327 00:21:41,250 --> 00:21:43,583 is proportional to h. 328 00:21:43,583 --> 00:21:47,010 Something like this. 329 00:21:47,010 --> 00:21:49,236 So this is for t that is positive. 330 00:21:53,430 --> 00:21:55,910 Now, what happens if I try to draw 331 00:21:55,910 --> 00:22:00,320 these same curves for t that is negative? 332 00:22:00,320 --> 00:22:03,550 Let's stick first again to the analog of this curve, 333 00:22:03,550 --> 00:22:07,390 but now for h equal to 0. 334 00:22:07,390 --> 00:22:11,200 So for h equals to 0, in plotting psi 335 00:22:11,200 --> 00:22:15,320 if m, whereas previously the parabola had 336 00:22:15,320 --> 00:22:18,590 the positive coefficient, now the parabola 337 00:22:18,590 --> 00:22:20,070 has a negative coefficient. 338 00:22:20,070 --> 00:22:23,750 So it kind of looks like this. 339 00:22:23,750 --> 00:22:27,210 Now, one of the things that I know for sure in my system-- 340 00:22:27,210 --> 00:22:32,040 again, mathematics does not tell us that, physics tells us that. 341 00:22:32,040 --> 00:22:34,670 That I have a piece of iron and I 342 00:22:34,670 --> 00:22:37,730 know that the typical configurations of magnetization 343 00:22:37,730 --> 00:22:40,810 are some small number. 344 00:22:40,810 --> 00:22:45,130 So if the mathematics says that I have a function such as this, 345 00:22:45,130 --> 00:22:48,680 it is wrong because it would say that the extremum would 346 00:22:48,680 --> 00:22:51,920 go towards infinity. 347 00:22:51,920 --> 00:22:55,070 But that's where the u-term comes into play. 348 00:22:55,070 --> 00:22:59,250 So I need now to have a u-term with a positive sign 349 00:22:59,250 --> 00:23:04,520 to ensure that the function does not have an extremum that 350 00:23:04,520 --> 00:23:06,470 goes to minus plus infinity. 351 00:23:06,470 --> 00:23:08,920 Physics tells us that the function 352 00:23:08,920 --> 00:23:14,950 that I get if t is negative should have a positive u. 353 00:23:14,950 --> 00:23:16,875 Now, what happens if we take this curve 354 00:23:16,875 --> 00:23:19,750 and go to h positive? 355 00:23:19,750 --> 00:23:22,460 Well, now again, like we did over here, 356 00:23:22,460 --> 00:23:25,480 I start to shift it in one direction. 357 00:23:25,480 --> 00:23:31,510 In particular, what I find is that the minimum to the left 358 00:23:31,510 --> 00:23:38,860 goes up and the minimum to the right goes down. 359 00:23:38,860 --> 00:23:41,572 What did I do? 360 00:23:41,572 --> 00:23:42,991 Drew this poorly. 361 00:23:46,780 --> 00:23:48,285 I start to go like this. 362 00:23:57,678 --> 00:24:02,320 So now there is a well-defined minimum over here. 363 00:24:02,320 --> 00:24:15,750 If I go to negative fields, I will get the opposite 364 00:24:15,750 --> 00:24:16,800 Where I have this. 365 00:24:20,840 --> 00:24:25,560 So now if I follow a path that corresponds 366 00:24:25,560 --> 00:24:34,990 to these t negative structures, what do I get? 367 00:24:34,990 --> 00:24:41,010 What I get is that the extremum is at some positive value. 368 00:24:41,010 --> 00:24:44,680 So this is for t that is negative. 369 00:24:44,680 --> 00:24:47,440 As I scan h from positive values, 370 00:24:47,440 --> 00:24:48,850 I am tracking this minimum. 371 00:24:48,850 --> 00:24:52,550 But that minimum ends up over here at 0. 372 00:24:52,550 --> 00:24:56,380 So basically, I come to here. 373 00:24:56,380 --> 00:24:59,320 Whereas, if I come from the negative side, 374 00:24:59,320 --> 00:25:02,650 I am basically following the opposite curve. 375 00:25:02,650 --> 00:25:05,779 And then I have a discontinuity exactly at the h 376 00:25:05,779 --> 00:25:06,570 equals [INAUDIBLE]. 377 00:25:10,130 --> 00:25:16,490 Again, by continuity if I am exactly at t equals to 0, 378 00:25:16,490 --> 00:25:20,790 the function is kind of-- rather than the parabola 379 00:25:20,790 --> 00:25:24,450 is m to the fourth, and a little bit of thought 380 00:25:24,450 --> 00:25:28,510 convinces you that the shape of the curve kind of looks 381 00:25:28,510 --> 00:25:29,620 like this. 382 00:25:29,620 --> 00:25:34,492 We will quantify what that shape is shortly for t equals to 0. 383 00:25:37,170 --> 00:25:39,960 So what did we do? 384 00:25:39,960 --> 00:25:46,880 We were able to reproduce exactly the structure 385 00:25:46,880 --> 00:25:50,230 of the isotherms that we are getting 386 00:25:50,230 --> 00:25:55,600 for the case of the ferromagnet from this simple theory. 387 00:25:58,800 --> 00:26:03,710 In order to just match exactly this with what is going on 388 00:26:03,710 --> 00:26:07,810 with iron or nickel, what do I have to do? 389 00:26:07,810 --> 00:26:12,070 I just have to ensure that this parameter t goes 390 00:26:12,070 --> 00:26:16,340 to 0 at Tc of whatever the material is. 391 00:26:16,340 --> 00:26:20,310 Now, remember that I said that my t is really 392 00:26:20,310 --> 00:26:23,460 a function of temperature. 393 00:26:23,460 --> 00:26:26,360 So I can certainly make an expansion of it 394 00:26:26,360 --> 00:26:27,410 around any point. 395 00:26:27,410 --> 00:26:30,840 Let's see the Tc of the material. 396 00:26:30,840 --> 00:26:35,710 So what I require is that the first term in that expansion 397 00:26:35,710 --> 00:26:37,270 has to be 0. 398 00:26:37,270 --> 00:26:41,480 Then I will have something that is linear in T, and then 399 00:26:41,480 --> 00:26:48,630 quadratic in T minus Tc squared and so forth. 400 00:26:48,630 --> 00:26:51,840 So there is one condition that I have to impose, 401 00:26:51,840 --> 00:26:54,560 that that function of t-- which again, 402 00:26:54,560 --> 00:26:56,910 by all of the arguments that I mentioned, 403 00:26:56,910 --> 00:27:00,370 completely has to be an analytic function of temperature. 404 00:27:00,370 --> 00:27:03,580 Hence, expandable in a Taylor series. 405 00:27:03,580 --> 00:27:07,630 The 0 to order term in that Taylor series has to be 0. 406 00:27:07,630 --> 00:27:08,900 What else? 407 00:27:08,900 --> 00:27:11,260 That I can expand any other function. 408 00:27:11,260 --> 00:27:17,580 U of t is u0 plus u1 T minus Tc and so forth. 409 00:27:17,580 --> 00:27:26,980 K of t is k0 plus k1 T minus Tc and so forth. 410 00:27:26,980 --> 00:27:29,860 And I don't really care about any of these coefficients. 411 00:27:29,860 --> 00:27:33,670 The only things that I know are the signs. 412 00:27:33,670 --> 00:27:37,870 a has to be positive because the high temperature 413 00:27:37,870 --> 00:27:40,980 side corresponds to paramagnet. 414 00:27:40,980 --> 00:27:46,200 u0 has to be positive because I require the stability 415 00:27:46,200 --> 00:27:48,560 types of things that I mentioned. 416 00:27:48,560 --> 00:27:51,000 And k0 has to be positive because I 417 00:27:51,000 --> 00:27:55,780 want to have this kind of ferromagnetic behavior. 418 00:27:55,780 --> 00:28:00,650 Apart from that, I really don't know much. 419 00:28:00,650 --> 00:28:11,950 So the Landau-Ginzburg Hamiltonian with one condition 420 00:28:11,950 --> 00:28:16,370 reproduces the phenomenology of the magnet 421 00:28:16,370 --> 00:28:19,240 and all the other phase transitions, 422 00:28:19,240 --> 00:28:22,550 which as one of you-- I think it was David-- was pointing out 423 00:28:22,550 --> 00:28:31,930 to me, is very much like dealing with a branch cut singularity 424 00:28:31,930 --> 00:28:35,380 in the mathematical sense along this axis 425 00:28:35,380 --> 00:28:37,670 terminating at that point. 426 00:28:37,670 --> 00:28:41,300 So this branch cut singularity is a consequence 427 00:28:41,300 --> 00:28:45,310 of this minimization procedure of a purely 428 00:28:45,310 --> 00:28:47,630 analytical function. 429 00:28:47,630 --> 00:28:51,580 And the statement of universality at this level 430 00:28:51,580 --> 00:28:55,909 is that pick any analytical function 431 00:28:55,909 --> 00:28:56,950 and do this minimization. 432 00:28:56,950 --> 00:29:02,970 You will always get the same mathematical branch cut 433 00:29:02,970 --> 00:29:07,115 that we will now explain in terms of the exponents. 434 00:29:11,240 --> 00:29:17,030 So we said that experimentally these phase transitions, 435 00:29:17,030 --> 00:29:19,230 their universality was characterized 436 00:29:19,230 --> 00:29:22,580 by looking at singularities of various quantities 437 00:29:22,580 --> 00:29:29,140 and looking at the exponent, the functional forms. 438 00:29:29,140 --> 00:29:31,616 So let's first look at the magnetization. 439 00:29:38,460 --> 00:29:43,760 So if I do the extremization of this, what do I do? 440 00:29:43,760 --> 00:29:48,210 I have to have d psi by dm equals to 0. d psi by dm 441 00:29:48,210 --> 00:29:54,350 is tm bar plus 4u m bar cubed minus h. 442 00:29:59,360 --> 00:30:07,700 For h equals to 0 along the symmetry axis, 443 00:30:07,700 --> 00:30:09,270 then I have two solutions. 444 00:30:09,270 --> 00:30:14,350 Because that I can write as t plus 4u m bar squared 445 00:30:14,350 --> 00:30:17,160 times m bar equals to 0. 446 00:30:17,160 --> 00:30:20,260 If I state at that equation, I immediately 447 00:30:20,260 --> 00:30:26,240 see that the possible solutions are m bar is 0. 448 00:30:26,240 --> 00:30:28,800 And that's really the only solution 449 00:30:28,800 --> 00:30:32,420 that I have for any t that is positive. 450 00:30:35,500 --> 00:30:40,670 While for t negative, in addition to the solution at 0, 451 00:30:40,670 --> 00:30:43,400 which is clearly unphysical because it corresponds 452 00:30:43,400 --> 00:30:47,520 to a maximum and not a minimum, I have solutions at minus 453 00:30:47,520 --> 00:30:51,930 plus square root of minus t over 4u. 454 00:30:56,570 --> 00:31:01,180 And so if I were to plot this magnetization 455 00:31:01,180 --> 00:31:11,800 as a function of t, essentially I 456 00:31:11,800 --> 00:31:18,220 have a kind of coexistence curve because I have nothing above 457 00:31:18,220 --> 00:31:20,760 and below I have a square root singularity. 458 00:31:24,450 --> 00:31:32,250 So this corresponds to the exponent beta being 1/2. 459 00:31:32,250 --> 00:31:38,960 So that's the prediction [? from each. ?] 460 00:31:38,960 --> 00:31:42,440 What about the shape of this green curve, the isotherm 461 00:31:42,440 --> 00:31:45,090 that you have at t equals to Tc? 462 00:31:45,090 --> 00:31:47,400 Well, t equals to Tc, in our language, 463 00:31:47,400 --> 00:31:51,110 corresponds to small t equals to 0, 464 00:31:51,110 --> 00:31:56,830 which means that I have to look at the equation 4u m bar cubed 465 00:31:56,830 --> 00:32:04,270 equal to h or m bar is proportional to-- 466 00:32:04,270 --> 00:32:06,060 well, let's write it. 467 00:32:06,060 --> 00:32:09,520 h over 4u to the power of 1/3. 468 00:32:12,210 --> 00:32:16,570 So this green curve that comes with infinite slope 469 00:32:16,570 --> 00:32:20,450 corresponds to a 1/3 singularity. 470 00:32:20,450 --> 00:32:25,150 The exponent delta was defined to be the inverse of this, 471 00:32:25,150 --> 00:32:29,160 so this corresponds to having an exponent delta that is 3. 472 00:32:38,800 --> 00:32:45,770 We had behavior of susceptibility characterized 473 00:32:45,770 --> 00:32:47,123 by another set of exponent. 474 00:32:54,990 --> 00:32:57,930 Now, the susceptibility, quite generally, 475 00:32:57,930 --> 00:33:01,730 is the response of the magnetization 476 00:33:01,730 --> 00:33:04,670 if you change the field. 477 00:33:04,670 --> 00:33:10,490 And typically, we were interested in the limit where 478 00:33:10,490 --> 00:33:14,460 we measured a response if you are just at h equals to 0 479 00:33:14,460 --> 00:33:17,070 and then you put a little bit more. 480 00:33:21,700 --> 00:33:24,830 The equation that we have that relates h and m bar 481 00:33:24,830 --> 00:33:29,995 is simply that h equals to tm bar plus 4u m bar cubed. 482 00:33:32,840 --> 00:33:37,060 Rather than taking dm by dh, let me 483 00:33:37,060 --> 00:33:42,020 evaluate its inverse, which is dh by dm. 484 00:33:42,020 --> 00:33:48,350 dh by dm is t plus 12u m bar squared. 485 00:33:51,460 --> 00:33:55,610 And so this inverse susceptibility, 486 00:33:55,610 --> 00:34:01,140 if I am for t positive, m bar is also 0. 487 00:34:01,140 --> 00:34:05,130 So the inverse susceptibility is t. 488 00:34:05,130 --> 00:34:14,219 If I am for t negative, m bar squared is minus t over 4u. 489 00:34:14,219 --> 00:34:17,530 I put a minus t over 4u here. 490 00:34:17,530 --> 00:34:19,820 And it becomes t minus 3t. 491 00:34:19,820 --> 00:34:23,120 So it becomes minus 2t. 492 00:34:23,120 --> 00:34:25,080 Again, nicely positive. 493 00:34:25,080 --> 00:34:27,889 Response functions have to be positive. 494 00:34:27,889 --> 00:34:32,179 If I were to plot this susceptibility, therefore 495 00:34:32,179 --> 00:34:35,400 as a function of temperature, or t 496 00:34:35,400 --> 00:34:40,320 that is related to T minus Tc, what do I get? 497 00:34:40,320 --> 00:34:48,520 I will get a divergence that is inverse of 1 over T minus Tc-- 498 00:34:48,520 --> 00:34:50,100 on both sides. 499 00:34:50,100 --> 00:34:55,260 So basically, I get something like this. 500 00:34:55,260 --> 00:34:59,390 And we said that the divergence of the susceptibility we 501 00:34:59,390 --> 00:35:02,870 characterize by exponent gamma. 502 00:35:02,870 --> 00:35:05,750 And as I had promised, we explicitly 503 00:35:05,750 --> 00:35:08,330 see that, in this case, the gammas 504 00:35:08,330 --> 00:35:12,300 on both sides of the transition are the same and equal 505 00:35:12,300 --> 00:35:13,060 to unity. 506 00:35:13,060 --> 00:35:17,050 The inverse vanishes linearly, so the susceptibility 507 00:35:17,050 --> 00:35:21,300 diverges with unit exponent. 508 00:35:21,300 --> 00:35:23,440 But actually, we are making-- we can 509 00:35:23,440 --> 00:35:27,070 make an additional statement here 510 00:35:27,070 --> 00:35:29,170 that experimentalists can go and check 511 00:35:29,170 --> 00:35:33,760 that I hadn't told you before. 512 00:35:33,760 --> 00:35:37,580 Now, you see in all of these other cases 513 00:35:37,580 --> 00:35:40,670 the only thing that I can say is universal side is 514 00:35:40,670 --> 00:35:41,990 the functional form. 515 00:35:41,990 --> 00:35:44,760 This is where the exponent beta came from. 516 00:35:44,760 --> 00:35:47,130 But the amplitude of what is happening there, 517 00:35:47,130 --> 00:35:50,270 or the amplitude of what is happening here, 518 00:35:50,270 --> 00:35:53,440 these are things that depend on you and all of these things 519 00:35:53,440 --> 00:35:56,250 that I have no idea about. 520 00:35:56,250 --> 00:35:59,780 Similarly here, because I don't know 521 00:35:59,780 --> 00:36:05,610 what the relationship between t minus Tc and the parameter t 522 00:36:05,610 --> 00:36:10,420 is, it involves this number a that I don't know of. 523 00:36:10,420 --> 00:36:13,630 But one thing that I notice is that the ratio of these two 524 00:36:13,630 --> 00:36:16,690 things is a pure number. 525 00:36:16,690 --> 00:36:19,730 So I say, OK, what you have is if you measure 526 00:36:19,730 --> 00:36:23,530 the susceptibility on the two sides of the transition, 527 00:36:23,530 --> 00:36:27,050 you will see amplitudes. 528 00:36:27,050 --> 00:36:31,560 And then T minus Tc absolute value 529 00:36:31,560 --> 00:36:34,540 to minus gamma plus or gamma minus. 530 00:36:34,540 --> 00:36:37,180 I have told you that the gammas are the same. 531 00:36:37,180 --> 00:36:39,730 I don't know what the amplitudes are, 532 00:36:39,730 --> 00:36:43,090 but I can tell for sure that the ratio of amplitudes-- 533 00:36:43,090 --> 00:36:46,230 if this is the theory that describes things-- 534 00:36:46,230 --> 00:36:49,190 is a pure number of 2. 535 00:36:49,190 --> 00:36:53,670 So that's another thing that you can go and say 536 00:36:53,670 --> 00:36:55,380 the experimentalists can check. 537 00:36:55,380 --> 00:36:57,560 They can check the divergence, and then 538 00:36:57,560 --> 00:37:01,100 see that the amplitude ratio is a universal object. 539 00:37:04,350 --> 00:37:08,640 OK, there's one other response function that I had mentioned. 540 00:37:08,640 --> 00:37:11,420 There was the exponent alpha that 541 00:37:11,420 --> 00:37:12,733 came from the heat capacity. 542 00:37:16,870 --> 00:37:19,316 So how do I calculate heat capacity? 543 00:37:28,260 --> 00:37:31,600 So the heat capacity, which is a function of temperature, 544 00:37:31,600 --> 00:37:37,130 let's focus only in the case where h equals to 0, 545 00:37:37,130 --> 00:37:41,155 is obtained by taking a temperature derivative 546 00:37:41,155 --> 00:37:43,725 of the internal energy of the system. 547 00:37:46,500 --> 00:37:48,820 Now, the energy, on the other hand, 548 00:37:48,820 --> 00:37:55,441 is obtained by taking a d by d beta of log of the partition 549 00:37:55,441 --> 00:37:55,940 function. 550 00:38:02,190 --> 00:38:09,580 Now, I have all of my answers in terms of these parameters t, u, 551 00:38:09,580 --> 00:38:11,300 et cetera. 552 00:38:11,300 --> 00:38:13,440 But I know that to lowest order, there 553 00:38:13,440 --> 00:38:17,730 is a linear relationship between small t 554 00:38:17,730 --> 00:38:20,440 and the real temperature that I have 555 00:38:20,440 --> 00:38:25,440 to put into these expressions. 556 00:38:25,440 --> 00:38:30,850 So in particular, I can do the following. 557 00:38:30,850 --> 00:38:34,615 I can say that something like d by dT d 558 00:38:34,615 --> 00:38:40,760 by d beta, where beta is 1 over kt, 559 00:38:40,760 --> 00:38:46,930 is something approximately when you 560 00:38:46,930 --> 00:38:53,090 look at the linear regime that is close to Tc of the order-- 561 00:38:53,090 --> 00:38:58,400 1 over beta is going to be kb t squared. 562 00:38:58,400 --> 00:39:03,220 And then I would have d by dT, and then another d by dT. 563 00:39:06,250 --> 00:39:08,840 I have to evaluate all of these things 564 00:39:08,840 --> 00:39:12,480 eventually in the vicinity of the critical point. 565 00:39:12,480 --> 00:39:17,620 Everything else is going to be a correction. 566 00:39:17,620 --> 00:39:20,740 So to lowest order, I will do that. 567 00:39:20,740 --> 00:39:28,420 And then I note that this is related-- derivatives 568 00:39:28,420 --> 00:39:32,280 between temperature and small t are 569 00:39:32,280 --> 00:39:35,570 related through a factor of a. 570 00:39:35,570 --> 00:39:38,637 So I do this and I put a squared up here. 571 00:39:41,910 --> 00:39:42,590 Doesn't matter. 572 00:39:42,590 --> 00:39:49,710 The only reason I do that is because through the process 573 00:39:49,710 --> 00:39:55,220 that we have described here, I get an idea for what log of Z 574 00:39:55,220 --> 00:39:55,925 is. 575 00:39:55,925 --> 00:40:02,780 So in particular, log of Z has a part 576 00:40:02,780 --> 00:40:09,080 that comes from all of the regular degrees of freedom 577 00:40:09,080 --> 00:40:15,066 and a part that comes from this additional minimization 578 00:40:15,066 --> 00:40:16,120 that we are doing. 579 00:40:16,120 --> 00:40:23,910 So we have a minus V times the minimum of the function, which 580 00:40:23,910 --> 00:40:30,370 is t over 2 m bar squared plus u m bar to the fourth 581 00:40:30,370 --> 00:40:31,790 when evaluated at h equals to 0. 582 00:40:35,690 --> 00:40:40,810 So this is log Z, which is some regular function 583 00:40:40,810 --> 00:40:44,130 of temperature, and hence t. 584 00:40:44,130 --> 00:40:48,560 Why is this part singular? 585 00:40:48,560 --> 00:40:53,720 Because for t that is positive, m is 0. 586 00:40:53,720 --> 00:40:56,865 So this is going to give me 0 contribution for t that 587 00:40:56,865 --> 00:40:59,010 is positive. 588 00:40:59,010 --> 00:41:02,840 Whereas, for t that is negative, I 589 00:41:02,840 --> 00:41:05,130 have to substitute the value of m bar 590 00:41:05,130 --> 00:41:10,770 squared as I found above, which is minus t over 4u. 591 00:41:10,770 --> 00:41:18,720 So I will get t over 2 times minus t over 4u. 592 00:41:18,720 --> 00:41:24,770 So that's going to give me minus t squared over 8u. 593 00:41:24,770 --> 00:41:28,290 Here I will have-- once I substitute that formula, 594 00:41:28,290 --> 00:41:30,630 plus t squared over 16u. 595 00:41:30,630 --> 00:41:33,975 The overall thing would be minus t squared over 16u. 596 00:41:37,910 --> 00:41:42,100 So I have to take two derivatives of this function. 597 00:41:42,100 --> 00:41:44,100 You can see that the function will give me 598 00:41:44,100 --> 00:41:50,876 0 above for t positive and it will give me a constant for t 599 00:41:50,876 --> 00:41:51,770 negative. 600 00:41:51,770 --> 00:42:00,180 So if I were to plot two derivatives of this function 601 00:42:00,180 --> 00:42:04,890 as T or T minus Tc is varied. 602 00:42:04,890 --> 00:42:06,900 Well, there is a background part that 603 00:42:06,900 --> 00:42:09,030 comes from all the other degrees of freedom. 604 00:42:09,030 --> 00:42:11,960 So there is basically some kind of behavior you would have had 605 00:42:11,960 --> 00:42:13,790 normally. 606 00:42:13,790 --> 00:42:18,100 What we find is that above t positive, 607 00:42:18,100 --> 00:42:21,820 that normal behavior is the only thing that you have. 608 00:42:21,820 --> 00:42:25,510 And when you are below, taking two derivatives of this, 609 00:42:25,510 --> 00:42:27,630 something is added to this. 610 00:42:27,630 --> 00:42:33,590 So basically, the prediction is that the heat capacity 611 00:42:33,590 --> 00:42:36,220 of the system as a function of temperature 612 00:42:36,220 --> 00:42:38,046 should have a jump discontinuity. 613 00:42:40,850 --> 00:42:43,910 Now, I said that in a number of cases 614 00:42:43,910 --> 00:42:48,040 we see that the heat capacity actually diverges 615 00:42:48,040 --> 00:42:51,370 and we introduce then exponent alpha 616 00:42:51,370 --> 00:42:53,750 to parametrize that divergence. 617 00:42:53,750 --> 00:42:56,740 Since we don't have this divergence, 618 00:42:56,740 --> 00:43:00,350 people have resorted to indicating this 619 00:43:00,350 --> 00:43:02,590 with alpha equals to 0. 620 00:43:02,590 --> 00:43:06,390 But since alpha equals to 0 is ambiguous, 621 00:43:06,390 --> 00:43:11,630 putting a discontinuity in addition 622 00:43:11,630 --> 00:43:14,084 to be precise about what is happening. 623 00:43:18,130 --> 00:43:24,140 So the predictions of the saddle point method 624 00:43:24,140 --> 00:43:26,970 applied to this field theory are the set 625 00:43:26,970 --> 00:43:29,550 of exponents and functional forms 626 00:43:29,550 --> 00:43:33,612 beta equals to 1/2, gamma as being 1, et cetera. 627 00:43:33,612 --> 00:43:36,940 A nice set of predictions. 628 00:43:36,940 --> 00:43:42,200 And of course, the test is, do they agree? 629 00:43:42,200 --> 00:43:46,360 It turns out that there is one and only one case where 630 00:43:46,360 --> 00:43:49,310 you do the experiments and you get these precise exponents. 631 00:43:49,310 --> 00:43:52,320 And that's something like a superconductor. 632 00:43:52,320 --> 00:43:55,230 And the picture that I showed you last time 633 00:43:55,230 --> 00:43:58,300 for the gas, et cetera, corresponds 634 00:43:58,300 --> 00:44:02,630 to totally different set of exponents. 635 00:44:02,630 --> 00:44:07,680 So at this point, we have to face one of two alternatives. 636 00:44:07,680 --> 00:44:11,480 One, the starting point is wrong. 637 00:44:11,480 --> 00:44:14,640 We put everything we could think of in the starting point. 638 00:44:14,640 --> 00:44:17,930 Maybe we forget something, but it seems OK. 639 00:44:17,930 --> 00:44:21,270 The other is, maybe we didn't do the analysis right when 640 00:44:21,270 --> 00:44:24,120 we did the saddle point approximation. 641 00:44:24,120 --> 00:44:26,540 And we'll gradually build the case 642 00:44:26,540 --> 00:44:31,010 that that is, indeed, the case and that we should treat 643 00:44:31,010 --> 00:44:35,270 the problem in a slightly better fashion. 644 00:44:35,270 --> 00:44:37,861 Any questions? 645 00:44:37,861 --> 00:44:38,360 OK. 646 00:44:38,360 --> 00:44:39,932 Yes? 647 00:44:39,932 --> 00:44:42,840 AUDIENCE: Just to remind me, the saddle point approximation 648 00:44:42,840 --> 00:44:47,466 was saying m was continuous, a continuous number 649 00:44:47,466 --> 00:44:49,610 across the substrate? 650 00:44:49,610 --> 00:44:51,400 PROFESSOR: The saddle point approximation 651 00:44:51,400 --> 00:44:55,890 is to evaluate the functional integral, which corresponds 652 00:44:55,890 --> 00:44:58,980 to looking at all configurations, 653 00:44:58,980 --> 00:45:03,460 replacing that integration with the value of the integrand 654 00:45:03,460 --> 00:45:06,560 at the point that is most probable. 655 00:45:06,560 --> 00:45:11,200 In this case, the most probable point was the uniform case. 656 00:45:11,200 --> 00:45:14,490 But maybe in some other case, the most probable configuration 657 00:45:14,490 --> 00:45:15,640 would be something else. 658 00:45:15,640 --> 00:45:20,640 The saddle point is to replace the entire functional integral 659 00:45:20,640 --> 00:45:24,730 with just one value of the integral. 660 00:45:24,730 --> 00:45:25,883 Yes? 661 00:45:25,883 --> 00:45:30,240 AUDIENCE: So your second claims that analysis is probably wrong 662 00:45:30,240 --> 00:45:30,740 somewhere. 663 00:45:30,740 --> 00:45:33,650 It is most likely when we are trying 664 00:45:33,650 --> 00:45:38,320 to compute the energy of the system from all of field m 665 00:45:38,320 --> 00:45:41,296 and we just assumed something incorrectly, 666 00:45:41,296 --> 00:45:45,380 and that's why we get incorrect exponents? 667 00:45:45,380 --> 00:45:46,110 PROFESSOR: No. 668 00:45:46,110 --> 00:45:51,500 My claim is that up to the place that I say Landau-Ginzburg, 669 00:45:51,500 --> 00:45:54,920 I have been extremely general. 670 00:45:54,920 --> 00:45:57,370 It may be that I missed something, 671 00:45:57,370 --> 00:46:00,310 but I will convince you that that's not the case. 672 00:46:00,310 --> 00:46:04,080 Then the line below that says saddle point approximation. 673 00:46:04,080 --> 00:46:07,738 My claim is that that's where the error came. 674 00:46:07,738 --> 00:46:12,112 AUDIENCE: Also, we can do a similar kind of analysis 675 00:46:12,112 --> 00:46:15,060 for liquid gas transition in critical [INAUDIBLE]? 676 00:46:15,060 --> 00:46:15,945 PROFESSOR: Yes. 677 00:46:15,945 --> 00:46:18,020 AUDIENCE: And then, how would it be 678 00:46:18,020 --> 00:46:21,436 reasonable to assume uniform density? 679 00:46:21,436 --> 00:46:25,680 Because I guess the whole point of behavior-- 680 00:46:25,680 --> 00:46:30,010 PROFESSOR: We did exactly that approximation in 8.3.3.3. 681 00:46:30,010 --> 00:46:34,990 I wrote down some theory for the liquid gas transition 682 00:46:34,990 --> 00:46:37,776 out of which came the van der Waals equation. 683 00:46:37,776 --> 00:46:38,490 AUDIENCE: Yes. 684 00:46:38,490 --> 00:46:40,110 PROFESSOR: And the assumption for that 685 00:46:40,110 --> 00:46:43,790 was that the density in the grand canonical ensemble, 686 00:46:43,790 --> 00:46:47,040 in the grand canonical ensemble was uniform. 687 00:46:47,040 --> 00:46:48,970 So that you either got the density 688 00:46:48,970 --> 00:46:52,180 for the liquid or the density for the gas. 689 00:46:52,180 --> 00:46:56,040 But I made there the saddle point approximation also. 690 00:46:56,040 --> 00:46:59,409 I assumed that there was a uniform density that was-- 691 00:46:59,409 --> 00:46:59,950 AUDIENCE: OK. 692 00:46:59,950 --> 00:47:02,225 So was just likely to be the point where-- 693 00:47:02,225 --> 00:47:02,850 PROFESSOR: Yes. 694 00:47:02,850 --> 00:47:04,070 AUDIENCE: --something breaks up. 695 00:47:04,070 --> 00:47:04,695 PROFESSOR: Yes. 696 00:47:09,990 --> 00:47:16,250 So before doing that, let me point out 697 00:47:16,250 --> 00:47:20,480 to something interesting that happened. 698 00:47:20,480 --> 00:47:35,040 And it's just a matter of terminology. 699 00:47:35,040 --> 00:47:41,890 Note that we constructed the Landau-Ginzburg Hamiltonian 700 00:47:41,890 --> 00:47:46,400 for h equals to 0 on the basis that we 701 00:47:46,400 --> 00:47:49,200 should have rotational symmetry. 702 00:47:49,200 --> 00:47:53,870 Nonetheless, even for h equals to 0, what we find 703 00:47:53,870 --> 00:47:57,670 are solutions where the magnetization is pointing 704 00:47:57,670 --> 00:48:00,090 in one direction or the other. 705 00:48:00,090 --> 00:48:04,400 So it is possible to have the state that 706 00:48:04,400 --> 00:48:08,030 emerges as a result of a weight that has some symmetry 707 00:48:08,030 --> 00:48:09,990 to not have that symmetry. 708 00:48:09,990 --> 00:48:13,880 So the symmetry is spontaneously broken 709 00:48:13,880 --> 00:48:19,400 and the direction in space is selected. 710 00:48:19,400 --> 00:48:22,480 Now, of course, what that means is 711 00:48:22,480 --> 00:48:25,990 that if you apply the rotation operation to one 712 00:48:25,990 --> 00:48:29,000 of these ground states, then you will 713 00:48:29,000 --> 00:48:31,650 generate another equally good ground state. 714 00:48:31,650 --> 00:48:35,090 You can take everything that is pointed along the z-axis 715 00:48:35,090 --> 00:48:37,460 and make them all point along the x-axis. 716 00:48:37,460 --> 00:48:40,410 That's an equally good ground state. 717 00:48:40,410 --> 00:48:44,890 So essentially, you have a manifold of possible states. 718 00:48:44,890 --> 00:48:48,330 And making a change from one state deforming 719 00:48:48,330 --> 00:48:53,670 to another ground state does not cost you any energy. 720 00:48:53,670 --> 00:49:04,980 So one consequence of that is that slow deformations 721 00:49:04,980 --> 00:49:07,965 should cost little energy. 722 00:49:12,040 --> 00:49:14,180 What do I mean by that? 723 00:49:14,180 --> 00:49:17,250 So let's imagine that I start with a state 724 00:49:17,250 --> 00:49:24,360 where after I minimize, I find that all of my magnetizations 725 00:49:24,360 --> 00:49:27,940 are pointing up. 726 00:49:27,940 --> 00:49:35,860 Now, as I said, I could rotate everybody into this direction 727 00:49:35,860 --> 00:49:40,360 and the formation of my state would cost no energy. 728 00:49:43,070 --> 00:49:45,940 That's a uniform deformation. 729 00:49:45,940 --> 00:49:50,180 What if I took a deformation that is very slow? 730 00:49:50,180 --> 00:49:59,320 So I gradually rotate from one to the other state. 731 00:49:59,320 --> 00:50:04,050 Then in the limit where the wavelength of this deformation 732 00:50:04,050 --> 00:50:07,680 becomes of the order of the size of your system, 733 00:50:07,680 --> 00:50:10,470 you should have no energy cost. 734 00:50:10,470 --> 00:50:13,740 And it kind of makes sense that in the limit where 735 00:50:13,740 --> 00:50:20,090 you have long wavelengths, you should have little energy cost. 736 00:50:20,090 --> 00:50:31,400 So you should have slow wavelength, no energy 737 00:50:31,400 --> 00:50:42,120 distortions or modes, called Goldstone modes. 738 00:50:50,260 --> 00:51:06,780 But you can only have this for a broken continuous symmetry 739 00:51:06,780 --> 00:51:09,370 such as what I have depicted there, 740 00:51:09,370 --> 00:51:12,670 where all orientations are equally likely. 741 00:51:12,670 --> 00:51:15,010 But if I had the liquid gas system, 742 00:51:15,010 --> 00:51:19,330 the density was either above average or below average. 743 00:51:19,330 --> 00:51:22,320 If I had uniaxial magnet, the spin 744 00:51:22,320 --> 00:51:25,240 would be either pointing up or down. 745 00:51:25,240 --> 00:51:29,830 Then I can't deform slowly from one to the other. 746 00:51:29,830 --> 00:51:33,330 So for discrete symmetries, you don't have these modes. 747 00:51:33,330 --> 00:51:36,481 For continuous symmetries, you have these modes. 748 00:51:39,430 --> 00:51:44,560 And actually, we've already seen one set of those modes. 749 00:51:44,560 --> 00:51:46,910 These were the phonons. 750 00:51:46,910 --> 00:51:49,490 When in the first lecture I was constructing 751 00:51:49,490 --> 00:51:53,390 this theory of elasticity, I said 752 00:51:53,390 --> 00:51:58,910 if we take the whole deformation and move it uniformity, 753 00:51:58,910 --> 00:52:00,750 there is no cost. 754 00:52:00,750 --> 00:52:03,750 And then we were able, based on that, 755 00:52:03,750 --> 00:52:07,700 to conclude that long wavelength phonons have little cost. 756 00:52:07,700 --> 00:52:10,760 And we wrote their dispersion, relation, et cetera. 757 00:52:10,760 --> 00:52:14,480 So phonons are an example of Goldstone modes. 758 00:52:14,480 --> 00:52:19,730 These kinds of rotations of spins in a magnet-- magnons 759 00:52:19,730 --> 00:52:22,930 are another example of these modes. 760 00:52:22,930 --> 00:52:25,730 But something else that we said, therefore in the first lecture, 761 00:52:25,730 --> 00:52:29,100 is something that we should start to think about. 762 00:52:29,100 --> 00:52:34,610 Which is that we said that because these modes exist 763 00:52:34,610 --> 00:52:39,345 and they have so little energy cost, if I 764 00:52:39,345 --> 00:52:44,930 am at some finite temperature, I will be able to excite them. 765 00:52:44,930 --> 00:52:50,010 So I know for sure that if I'm at finite temperature, 766 00:52:50,010 --> 00:52:52,950 there are at least these fluctuations that 767 00:52:52,950 --> 00:52:54,960 are going on in my system. 768 00:52:54,960 --> 00:52:57,000 And maybe in lieu of that, I should 769 00:52:57,000 --> 00:53:02,280 be wary of assuming that only the state where everything 770 00:53:02,280 --> 00:53:04,474 is uniform is the thing that is contributing. 771 00:53:04,474 --> 00:53:05,640 What about the fluctuations? 772 00:53:09,410 --> 00:53:13,760 So let's think about these fluctuations that 773 00:53:13,760 --> 00:53:18,590 are easiest and most easily generated 774 00:53:18,590 --> 00:53:22,770 and look at their thermal excitations and consequences 775 00:53:22,770 --> 00:53:25,210 for the phase in phase transition. 776 00:53:25,210 --> 00:53:28,920 And let's do that in the context of superfluid. 777 00:53:35,130 --> 00:53:39,590 So we saw the problem of super fluidity 778 00:53:39,590 --> 00:53:42,840 towards the end of 8.3.3.3. 779 00:53:42,840 --> 00:53:47,540 You had helium that was an ordinary liquid. 780 00:53:47,540 --> 00:53:51,480 We cooled it below 2.8 degrees and suddenly it 781 00:53:51,480 --> 00:53:56,420 became a new form of matter that has this ability 782 00:53:56,420 --> 00:53:59,380 to flow through capillaries, et cetera. 783 00:53:59,380 --> 00:54:01,390 And we pointed out that there was 784 00:54:01,390 --> 00:54:04,720 some kind of quite likely quantum origin 785 00:54:04,720 --> 00:54:06,900 to that because of the similarities 786 00:54:06,900 --> 00:54:10,930 that it showed to Bose-Einstein condensation. 787 00:54:10,930 --> 00:54:13,620 And basically, it was in this context 788 00:54:13,620 --> 00:54:20,970 that Landau introduced something like this The theory 789 00:54:20,970 --> 00:54:25,720 that we write down where he chose as order 790 00:54:25,720 --> 00:54:33,810 parameter as the analog of the m of x that we have over there, 791 00:54:33,810 --> 00:54:38,060 a complex function psi of x. 792 00:54:38,060 --> 00:54:42,050 And very roughly, you can regard this-- 793 00:54:42,050 --> 00:54:48,650 and again, this is very rough-- as overlap of wave 794 00:54:48,650 --> 00:54:58,030 function with the ground state at position 795 00:54:58,030 --> 00:55:00,790 x in some coarse-grained sense. 796 00:55:03,490 --> 00:55:06,670 Now, anything quantum mechanical we 797 00:55:06,670 --> 00:55:09,460 saw has an amplitude and a phase. 798 00:55:09,460 --> 00:55:14,030 So this is actually a number plus a phase. 799 00:55:14,030 --> 00:55:18,050 Or if you like, it has a real part and an imaginary part. 800 00:55:18,050 --> 00:55:23,780 And there is no way that we know anything about the phase. 801 00:55:23,780 --> 00:55:26,480 The phase is not an unobservable. 802 00:55:26,480 --> 00:55:30,620 So the probability that when we scan the system 803 00:55:30,620 --> 00:55:35,210 we have identified some psi of x that the probability should 804 00:55:35,210 --> 00:55:37,700 depend on the phase is meaningless. 805 00:55:37,700 --> 00:55:39,500 It's not an observable. 806 00:55:39,500 --> 00:55:42,310 So this functional should only depend 807 00:55:42,310 --> 00:55:45,920 on things like absolute value of psi. 808 00:55:45,920 --> 00:55:51,080 If, like Landau, we assume that it is a local form, 809 00:55:51,080 --> 00:55:54,100 then the kinds of terms that we can write 810 00:55:54,100 --> 00:55:58,750 are absolute value of psi squared, absolute value of psi 811 00:55:58,750 --> 00:56:01,140 to the fourth power. 812 00:56:01,140 --> 00:56:07,000 And the tendency for the order to expand across the system you 813 00:56:07,000 --> 00:56:11,640 would put through a term such as gradient of psi squared. 814 00:56:15,320 --> 00:56:18,340 Now, for the case of the superfluid, 815 00:56:18,340 --> 00:56:21,750 there is no physical field that corresponds to the h. 816 00:56:21,750 --> 00:56:25,300 That just you don't have that field. 817 00:56:25,300 --> 00:56:29,386 You can convince yourself that if you write psi 818 00:56:29,386 --> 00:56:36,700 to be psi 1 plus psi 2, i psi 2-- real and imaginary part. 819 00:56:36,700 --> 00:56:41,060 And put that in this formula, that corresponds exactly 820 00:56:41,060 --> 00:56:44,270 to the theory that we wrote over there as long 821 00:56:44,270 --> 00:56:47,530 as we choose a two-component magnetization. 822 00:56:47,530 --> 00:56:49,031 So these two theories are identical. 823 00:56:56,110 --> 00:57:05,690 Now, if I look at this system for t negative 824 00:57:05,690 --> 00:57:12,430 and try to find a minimum of the functional 825 00:57:12,430 --> 00:57:17,540 that I have over there, then the shape 826 00:57:17,540 --> 00:57:23,970 of the function functional psi-- poor choice of notation. 827 00:57:23,970 --> 00:57:27,650 Psi both being the wave function as well as the function 828 00:57:27,650 --> 00:57:28,950 that I have to extremize. 829 00:57:28,950 --> 00:57:30,920 But let's stick with it. 830 00:57:30,920 --> 00:57:36,220 It has a minimum that goes along a circuit. 831 00:57:36,220 --> 00:57:40,110 So basically, take this picture that we 832 00:57:40,110 --> 00:57:43,940 have over here that corresponds to essentially one direction 833 00:57:43,940 --> 00:57:46,230 and rotate it. 834 00:57:46,230 --> 00:57:49,400 And what you will get is what is sometimes 835 00:57:49,400 --> 00:57:54,340 called the wine bottle type of shape, or the Mexican hat 836 00:57:54,340 --> 00:57:56,910 potential, or whatever. 837 00:57:56,910 --> 00:57:59,100 But essentially, it means that there 838 00:57:59,100 --> 00:58:03,770 is a ring of possible ground states. 839 00:58:03,770 --> 00:58:10,170 So the minimum occurs for psi of x 840 00:58:10,170 --> 00:58:14,220 having some particular magnitude which corresponds 841 00:58:14,220 --> 00:58:17,610 to the location of this ring-- how far 842 00:58:17,610 --> 00:58:19,400 away it is from the center. 843 00:58:19,400 --> 00:58:22,610 And that will be given by the square root of t formula 844 00:58:22,610 --> 00:58:24,020 that we have up there. 845 00:58:24,020 --> 00:58:32,900 But then there is a phase that is something 846 00:58:32,900 --> 00:58:33,750 that you don't know. 847 00:58:39,190 --> 00:58:42,040 Now, let's ask the question. 848 00:58:42,040 --> 00:58:45,880 Suppose I allow this phase to vary 849 00:58:45,880 --> 00:58:50,420 from one part of the sample to another part of the sample. 850 00:58:50,420 --> 00:58:55,180 So that's the analog of this slow distortion 851 00:58:55,180 --> 00:58:57,750 that I was making up there. 852 00:58:57,750 --> 00:59:01,160 So essentially, as I go from one part of the sample 853 00:59:01,160 --> 00:59:04,140 to another part of the sample, I slowly 854 00:59:04,140 --> 00:59:07,570 move around this bottom of this with potential. 855 00:59:10,320 --> 00:59:15,780 And I ask, what is the cost of this distortion that I 856 00:59:15,780 --> 00:59:18,060 impose on the system? 857 00:59:18,060 --> 00:59:28,700 If I calculate beta H for psi bar e the i theta x, what I get 858 00:59:28,700 --> 00:59:33,940 is whatever I have put over here, 859 00:59:33,940 --> 00:59:40,700 such as this function which minimizes the-- which 860 00:59:40,700 --> 00:59:43,810 is the location of the minimum. 861 00:59:43,810 --> 00:59:49,950 But because of the variation in theta that I have allowed, 862 00:59:49,950 --> 00:59:51,850 there is a cost. 863 00:59:51,850 --> 00:59:57,370 So let's write that as beta H0 plus this additional cost. 864 00:59:57,370 --> 01:00:00,390 The additional cost comes from this term. 865 01:00:00,390 --> 01:00:04,540 If I simply put psi bar in to the i theta over there, 866 01:00:04,540 --> 01:00:09,780 I will get an integral d dx k psi 867 01:00:09,780 --> 01:00:14,420 bar squared over 2 gradient of theta squared. 868 01:00:21,450 --> 01:00:27,000 So there is an additional energy cost. 869 01:00:27,000 --> 01:00:29,700 This is very similar to the energy cost 870 01:00:29,700 --> 01:00:31,170 that we had for phonons. 871 01:00:31,170 --> 01:00:32,870 Because if I Fourier transform, you 872 01:00:32,870 --> 01:00:35,640 can see that I get a k squared just 873 01:00:35,640 --> 01:00:38,070 like we got for the case of phonons. 874 01:00:38,070 --> 01:00:40,770 And just like for the case of phonons, 875 01:00:40,770 --> 01:00:43,470 I expect that at some finite temperature, 876 01:00:43,470 --> 01:00:46,850 these kinds of modes are thermally excited. 877 01:00:46,850 --> 01:00:51,220 So in reality, I expect that if I'm at some finite temperature, 878 01:00:51,220 --> 01:00:54,695 this phase will fluctuate across my system. 879 01:00:57,820 --> 01:01:04,290 And maybe I should note that whereas I'm here 880 01:01:04,290 --> 01:01:08,040 thinking in terms of thermal fluctuations, 881 01:01:08,040 --> 01:01:11,330 by appropriate boundary conditions one can establish 882 01:01:11,330 --> 01:01:15,200 a gradient of theta that is uniform across the system. 883 01:01:15,200 --> 01:01:18,770 And that actually corresponds to a superfluid flow. 884 01:01:18,770 --> 01:01:22,700 So the case of a superfluid flow can 885 01:01:22,700 --> 01:01:25,920 be regarded by having a gradient of theta 886 01:01:25,920 --> 01:01:28,700 being proportional to velocity, and then this 887 01:01:28,700 --> 01:01:31,070 is something like the kinetic energy. 888 01:01:31,070 --> 01:01:33,460 But that's a different story. 889 01:01:33,460 --> 01:01:35,370 Let's just stick with the fact that this 890 01:01:35,370 --> 01:01:39,180 is the cost of making these distortions. 891 01:01:39,180 --> 01:01:43,460 And I want to know, what's the probability of having 892 01:01:43,460 --> 01:01:45,410 one of these distorted shapes? 893 01:01:45,410 --> 01:01:46,232 Yes. 894 01:01:46,232 --> 01:01:47,216 AUDIENCE: Question. 895 01:01:47,216 --> 01:01:49,600 When you're introducing the psi as another parameter 896 01:01:49,600 --> 01:01:54,590 and you call it overlap of [INAUDIBLE] what is boundaries 897 01:01:54,590 --> 01:01:58,070 and what values this sort of parameter can take? 898 01:01:58,070 --> 01:02:02,090 So I basically wonder if we have this Mexican hat-shaped 899 01:02:02,090 --> 01:02:08,140 potential with minima on the ring of radius at 1, 900 01:02:08,140 --> 01:02:12,380 can the value of potential-- of the other parameter principally 901 01:02:12,380 --> 01:02:13,540 be further than that? 902 01:02:16,417 --> 01:02:17,000 PROFESSOR: OK. 903 01:02:17,000 --> 01:02:21,940 So again, here we are trying to phenomenologically explain 904 01:02:21,940 --> 01:02:27,220 an observation that there is a transition between a case where 905 01:02:27,220 --> 01:02:30,960 there is no super fluidity and right when 906 01:02:30,960 --> 01:02:33,830 a certain small amount of super fluidity 907 01:02:33,830 --> 01:02:36,130 has been established in the system. 908 01:02:36,130 --> 01:02:38,720 The question that you asked over here is legitimate, 909 01:02:38,720 --> 01:02:40,750 but you could have asked it also for the case 910 01:02:40,750 --> 01:02:42,870 of the magnetization. 911 01:02:42,870 --> 01:02:46,450 So you could have said, why not to have 912 01:02:46,450 --> 01:02:49,870 that potential with the minimum somewhere else? 913 01:02:49,870 --> 01:02:52,630 But that does not explain the phenomena 914 01:02:52,630 --> 01:02:54,320 that we are trying to explain. 915 01:02:54,320 --> 01:02:56,880 The phenomena that we are trying to explain 916 01:02:56,880 --> 01:03:01,280 is the observation that I go from having nothing 917 01:03:01,280 --> 01:03:03,840 to having a little bit of something. 918 01:03:03,840 --> 01:03:06,070 And I choose the mathematical form 919 01:03:06,070 --> 01:03:09,961 that is capable of describing that. 920 01:03:09,961 --> 01:03:11,460 AUDIENCE: My question basically was, 921 01:03:11,460 --> 01:03:13,668 when we were talking about magnetization, if you take 922 01:03:13,668 --> 01:03:16,030 a piece of metal, you can magnetize it 923 01:03:16,030 --> 01:03:20,057 from 0 to a pretty large value. 924 01:03:20,057 --> 01:03:20,640 PROFESSOR: No. 925 01:03:20,640 --> 01:03:23,577 AUDIENCE: If we are interested in something not too large-- 926 01:03:23,577 --> 01:03:24,160 PROFESSOR: No. 927 01:03:24,160 --> 01:03:25,660 AUDIENCE: And as I say, [INAUDIBLE]. 928 01:03:25,660 --> 01:03:27,750 PROFESSOR: What is your scale? 929 01:03:27,750 --> 01:03:28,805 What is very large? 930 01:03:31,440 --> 01:03:34,662 For a magnet, there is a maximum magnetization-- 931 01:03:34,662 --> 01:03:38,415 AUDIENCE: In this case, I mean that spontaneous magnetization 932 01:03:38,415 --> 01:03:43,930 for a magnet would be lower than saturation, no? 933 01:03:43,930 --> 01:03:45,180 PROFESSOR: What is saturation? 934 01:03:45,180 --> 01:03:47,487 AUDIENCE: When all spins are same direction. 935 01:03:47,487 --> 01:03:48,070 PROFESSOR: OK. 936 01:03:48,070 --> 01:03:52,200 So you have a microscopic picture in mind. 937 01:03:52,200 --> 01:03:54,350 Now, the place that we are is far 938 01:03:54,350 --> 01:03:56,110 from that saturation magnetization. 939 01:03:56,110 --> 01:03:59,320 Similarly, in this case, presumably 940 01:03:59,320 --> 01:04:04,380 if I go to 0 temperature, there is some uniformity. 941 01:04:04,380 --> 01:04:08,642 And if I call this an overlap, the maximum of it will be 1. 942 01:04:08,642 --> 01:04:09,266 AUDIENCE: Yeah. 943 01:04:09,266 --> 01:04:11,120 But basically, I just don't understand-- 944 01:04:11,120 --> 01:04:14,150 what is overlap of wave function? 945 01:04:14,150 --> 01:04:15,650 PROFESSOR: Well, that's why I didn't 946 01:04:15,650 --> 01:04:17,190 want to go into that detail. 947 01:04:17,190 --> 01:04:19,830 But basically, the overlap of two wave functions 948 01:04:19,830 --> 01:04:24,040 would be the psi 1 star psi 2 of x. 949 01:04:24,040 --> 01:04:26,650 And if you are thinking about the ground state, 950 01:04:26,650 --> 01:04:29,460 let's say that I have normalized this function 951 01:04:29,460 --> 01:04:32,940 to have a maximum of 1. 952 01:04:32,940 --> 01:04:35,776 The point is that what that maximum is, 953 01:04:35,776 --> 01:04:41,680 is folded into all of these parameters-- a, u, et cetera-- 954 01:04:41,680 --> 01:04:45,912 and is pretty irrelevant to the nature of the transition. 955 01:04:45,912 --> 01:04:46,453 AUDIENCE: OK. 956 01:04:50,150 --> 01:04:52,000 PROFESSOR: OK? 957 01:04:52,000 --> 01:04:54,960 So the probability of a particular configuration 958 01:04:54,960 --> 01:05:00,030 of theta across the system is given by this formula. 959 01:05:00,030 --> 01:05:03,440 I can unpack that a little bit better just 960 01:05:03,440 --> 01:05:08,040 like we did for the case of phonons 961 01:05:08,040 --> 01:05:12,000 by writing theta in terms of Fourier modes. 962 01:05:12,000 --> 01:05:18,600 e to the i q dot x theta of q, which for the time being, 963 01:05:18,600 --> 01:05:22,650 I assume I have discretized appropriate values of q. 964 01:05:22,650 --> 01:05:28,460 I choose this normalization root V in this context. 965 01:05:28,460 --> 01:05:32,070 If I substitute that over here, what 966 01:05:32,070 --> 01:05:36,920 I find is that beta H as a function of the collection 967 01:05:36,920 --> 01:05:41,700 of theta q's is, again, some beta H0, 968 01:05:41,700 --> 01:05:44,230 which is not important. 969 01:05:44,230 --> 01:05:46,440 I put gradient of theta. 970 01:05:46,440 --> 01:05:48,810 Once I do gradient of theta, you can 971 01:05:48,810 --> 01:05:51,200 see that I get a factor of iq. 972 01:05:51,200 --> 01:05:53,640 I will have two of them, so the answer 973 01:05:53,640 --> 01:05:58,510 is going to be proportional to q squared. 974 01:05:58,510 --> 01:06:02,260 Let's call this combination k psi bar squared k bar 975 01:06:02,260 --> 01:06:05,350 so that I don't have to write it again and again. 976 01:06:05,350 --> 01:06:07,690 k bar over 2. 977 01:06:07,690 --> 01:06:12,760 And then theta of q squared. 978 01:06:12,760 --> 01:06:14,890 OK? 979 01:06:14,890 --> 01:06:20,800 So the probability of some particular combination 980 01:06:20,800 --> 01:06:28,050 of these Fourier amplitudes is proportional to exponential 981 01:06:28,050 --> 01:06:29,480 of this. 982 01:06:29,480 --> 01:06:34,820 And therefore, a product of independent 983 01:06:34,820 --> 01:06:36,670 Gaussian-distributed quantities. 984 01:06:46,037 --> 01:06:47,023 q squared. 985 01:06:51,480 --> 01:06:53,349 Yes. 986 01:06:53,349 --> 01:06:57,870 AUDIENCE: When you plug in your summation of your-- well, 987 01:06:57,870 --> 01:07:00,785 your Fourier series into the gradient 988 01:07:00,785 --> 01:07:03,740 and then you square that, why don't you 989 01:07:03,740 --> 01:07:08,390 get interactions between the different Fourier amplitudes? 990 01:07:08,390 --> 01:07:10,180 PROFESSOR: OK, let's do it explicitly. 991 01:07:10,180 --> 01:07:21,150 I have integral d dx gradient of theta gradient of theta. 992 01:07:21,150 --> 01:07:28,220 Gradient of theta is iq sum-- OK. 993 01:07:28,220 --> 01:07:39,110 Is i sum over q q e to the i q dot x theta tilde of q. 994 01:07:39,110 --> 01:07:40,900 And I have to repeat that twice. 995 01:07:40,900 --> 01:07:47,160 So I have i sum over q prime q prime into the i q prime dot 996 01:07:47,160 --> 01:07:51,330 x theta tilde of q prime. 997 01:07:51,330 --> 01:07:55,150 So basically, this went into that. 998 01:07:55,150 --> 01:07:58,330 This went into that. 999 01:07:58,330 --> 01:08:00,720 OK? 1000 01:08:00,720 --> 01:08:03,360 So I have a sum over q, sum over q prime, 1001 01:08:03,360 --> 01:08:05,330 and an integral over x. 1002 01:08:05,330 --> 01:08:13,590 What is the integral over x of e to the i q plus q prime dot x? 1003 01:08:13,590 --> 01:08:20,450 It is 0, unless q and q prime add up to 0. 1004 01:08:20,450 --> 01:08:22,240 This is delta function. 1005 01:08:22,240 --> 01:08:23,930 So you put it there. 1006 01:08:23,930 --> 01:08:25,729 Only one sum survives. 1007 01:08:25,729 --> 01:08:28,960 Actually, I had introduced the normalizations that 1008 01:08:28,960 --> 01:08:33,450 were root V. The normalizations get rid of this factor of V. 1009 01:08:33,450 --> 01:08:36,180 That's why I had normalized it with the root V. 1010 01:08:36,180 --> 01:08:39,620 And I will get one factor of q remaining. 1011 01:08:39,620 --> 01:08:45,500 Since q prime is minus q iq iq prime becomes q squared. 1012 01:08:45,500 --> 01:08:49,020 And then I have theta tilde of q theta tilde 1013 01:08:49,020 --> 01:08:51,410 of minus q, which gives me this. 1014 01:09:00,720 --> 01:09:04,140 So each mode is independently distributed according 1015 01:09:04,140 --> 01:09:07,085 to a Gaussian, which immediately tells me 1016 01:09:07,085 --> 01:09:11,696 the average of theta of q tilde is, of course, 0. 1017 01:09:11,696 --> 01:09:16,300 Let's be careful and put the tildes all over the place. 1018 01:09:16,300 --> 01:09:22,250 While the average of theta tilde of q squared 1019 01:09:22,250 --> 01:09:26,240 is 1 over k bar q squared. 1020 01:09:26,240 --> 01:09:32,870 Again, all that says is that as you go to long wavelength modes 1021 01:09:32,870 --> 01:09:36,240 and q goes to 0, the fluctuations 1022 01:09:36,240 --> 01:09:41,340 become larger because the energy cost is smaller. 1023 01:09:41,340 --> 01:09:44,490 OK, so that's understandable. 1024 01:09:44,490 --> 01:09:47,209 But now, let's look at what is happening in real space. 1025 01:09:57,360 --> 01:10:02,990 I pick two points, x and x prime. 1026 01:10:02,990 --> 01:10:06,340 And I ask, how do the fluctuations 1027 01:10:06,340 --> 01:10:09,750 vary from one point to another point? 1028 01:10:09,750 --> 01:10:12,605 So I'm interested in theta x. 1029 01:10:15,540 --> 01:10:29,150 Let's say minus theta. 1030 01:10:29,150 --> 01:10:32,400 Let's calculate the following first, 1031 01:10:32,400 --> 01:10:35,740 theta of x, theta of x prime just because the algebra 1032 01:10:35,740 --> 01:10:36,710 is slightly easier. 1033 01:10:39,500 --> 01:10:44,380 Now, theta of x I can write in terms of theta tilde of q. 1034 01:10:44,380 --> 01:10:53,236 So this becomes sum over q q prime e to the iq dot x 1035 01:10:53,236 --> 01:10:56,765 e to the iq prime dot x prime. 1036 01:10:56,765 --> 01:10:58,706 There is a factor of 1 over V that 1037 01:10:58,706 --> 01:11:02,240 comes from the normalization I chose. 1038 01:11:02,240 --> 01:11:05,350 And then the average of theta tilde 1039 01:11:05,350 --> 01:11:10,970 of q theta tilde of q prime. 1040 01:11:10,970 --> 01:11:15,180 But we just established that the different modes 1041 01:11:15,180 --> 01:11:18,100 are independent of each other. 1042 01:11:18,100 --> 01:11:21,490 So basically, this gives me a delta function 1043 01:11:21,490 --> 01:11:26,060 that forces q and q prime to add up to 0. 1044 01:11:26,060 --> 01:11:30,450 If they do add up to 0, the expectation that I get 1045 01:11:30,450 --> 01:11:32,690 is 1 over k bar q squared. 1046 01:11:35,200 --> 01:11:42,470 And so what I find is that this becomes related 1047 01:11:42,470 --> 01:11:49,420 to a sum over q 1 over V e to the iq dot x 1048 01:11:49,420 --> 01:11:54,480 minus x prime divided by k bar q squared. 1049 01:11:57,360 --> 01:12:00,960 If I go to the continuum limit where the sum over 1050 01:12:00,960 --> 01:12:05,290 q I replace with an integral over q, 1051 01:12:05,290 --> 01:12:10,570 then I have to introduce the density of states. 1052 01:12:10,570 --> 01:12:15,080 And so then I find that theta of x theta of x 1053 01:12:15,080 --> 01:12:25,265 prime is 1 over k bar integral d dq 1054 01:12:25,265 --> 01:12:32,145 2 pi to the d the Fourier transform of 1 over q squared. 1055 01:12:37,340 --> 01:12:40,895 Now, the Fourier transform of 1 over q squared 1056 01:12:40,895 --> 01:12:45,310 is something that appears all over physics. 1057 01:12:45,310 --> 01:12:48,180 So let's give it a name. 1058 01:12:48,180 --> 01:12:54,730 So we're going to call the integral d dq 2 pi to the d 1059 01:12:54,730 --> 01:13:01,730 e to the iq dot x divided by q squared. 1060 01:13:01,730 --> 01:13:04,900 And let's put a minus sign in front of that. 1061 01:13:04,900 --> 01:13:08,370 And I'll give it the name the Coulomb potential 1062 01:13:08,370 --> 01:13:09,720 in d dimensional space. 1063 01:13:14,810 --> 01:13:19,770 And for those of you who haven't seen this, 1064 01:13:19,770 --> 01:13:22,520 the reason this is the Coulomb potential 1065 01:13:22,520 --> 01:13:24,880 is because if I take two derivatives 1066 01:13:24,880 --> 01:13:31,470 and construct the Laplacian of that function-- 1067 01:13:31,470 --> 01:13:34,140 so I take two derivative with respect to x here, 1068 01:13:34,140 --> 01:13:38,650 the two derivatives will go inside the integral. 1069 01:13:38,650 --> 01:13:43,810 And what they do is they bring down 1070 01:13:43,810 --> 01:13:48,680 two factors of iq divided by q squared. 1071 01:13:51,420 --> 01:13:54,330 The minus sign disappears, q squared over q squared 1072 01:13:54,330 --> 01:13:55,470 goes to 1. 1073 01:13:55,470 --> 01:14:00,290 Fourier transform of e to the iq x is simply the delta function. 1074 01:14:04,990 --> 01:14:12,070 So this Cd of x is the potential that would emerge from a unit 1075 01:14:12,070 --> 01:14:16,100 charge at the origin at a distance x. 1076 01:14:21,070 --> 01:14:25,890 So again, for those who have forgotten this or not seen it, 1077 01:14:25,890 --> 01:14:31,210 let's calculate it explicitly using Gauss' theorem. 1078 01:14:31,210 --> 01:14:34,310 The potential due to a unit charge 1079 01:14:34,310 --> 01:14:37,820 is going to be spherically symmetric, so it's only 1080 01:14:37,820 --> 01:14:40,180 a function of the magnitude of x. 1081 01:14:40,180 --> 01:14:44,750 It doesn't depend on the orientation. 1082 01:14:44,750 --> 01:14:51,033 And Gauss' law states that the integral of Laplacian 1083 01:14:51,033 --> 01:14:59,230 over volume is the same as the integral of the analog 1084 01:14:59,230 --> 01:15:03,590 of the electric field, which is the gradient over the surface. 1085 01:15:03,590 --> 01:15:07,930 So I have the surface integral of gradient. 1086 01:15:14,540 --> 01:15:17,440 Now, for the case that we are dealing with, 1087 01:15:17,440 --> 01:15:21,930 the left-hand side Laplacian is a delta function. 1088 01:15:21,930 --> 01:15:26,650 So when we integrate that over the sphere, I simply get 1. 1089 01:15:26,650 --> 01:15:28,640 So this gives me 1. 1090 01:15:28,640 --> 01:15:32,865 What do I get on the right-hand side? 1091 01:15:32,865 --> 01:15:34,854 It's just like the flux of the electric field 1092 01:15:34,854 --> 01:15:35,895 that you have calculated. 1093 01:15:35,895 --> 01:15:41,910 It is the magnitude of the electric field 1094 01:15:41,910 --> 01:15:45,930 times the surface area. 1095 01:15:45,930 --> 01:15:49,350 And I am doing this generally in d dimensions. 1096 01:15:49,350 --> 01:15:51,810 So the surface area in d dimension 1097 01:15:51,810 --> 01:15:55,280 grows like x to the d minus 1. 1098 01:15:55,280 --> 01:16:00,300 And then there is a factor such as 2 pi, 4 pi, et cetera, 1099 01:16:00,300 --> 01:16:02,220 which is the solid angle that you 1100 01:16:02,220 --> 01:16:05,160 would have to put in d dimensions. 1101 01:16:05,160 --> 01:16:10,790 And to remind you, the solid angle in d dimensions 1102 01:16:10,790 --> 01:16:17,029 is 2 pi to the d over 2 d over 2 minus 1 factorial. 1103 01:16:45,170 --> 01:16:48,143 So the magnitude of this derivative dC 1104 01:16:48,143 --> 01:16:54,900 by dx following from that is simply 1105 01:16:54,900 --> 01:17:00,000 1 divided by x to the d minus 1, or x to the 1 1106 01:17:00,000 --> 01:17:03,100 minus d divided by Sd. 1107 01:17:08,990 --> 01:17:12,310 So this is generalizing how you would calculate 1108 01:17:12,310 --> 01:17:14,830 Gauss' law in three dimensions. 1109 01:17:14,830 --> 01:17:19,090 So now I just integrate that and I find that the d dimensional 1110 01:17:19,090 --> 01:17:24,450 Coulomb potential is x to the 2 minus d 1111 01:17:24,450 --> 01:17:29,720 divided by 2 minus d Sd. 1112 01:17:29,720 --> 01:17:32,592 And of course, there could be some constant of integration. 1113 01:17:37,100 --> 01:17:42,670 So it reproduces the familiar 1 over x law in three dimensions. 1114 01:17:42,670 --> 01:17:45,680 But the thing that is important to note 1115 01:17:45,680 --> 01:17:49,590 is how much this Coulomb potential 1116 01:17:49,590 --> 01:17:52,980 depends on dimensions. 1117 01:17:52,980 --> 01:17:58,090 It determines these angle-- angle fluctuation correlations. 1118 01:17:58,090 --> 01:18:00,300 And typically, here in this context 1119 01:18:00,300 --> 01:18:03,580 we want to know something about large distances. 1120 01:18:03,580 --> 01:18:06,793 If I make a fluctuation here, how far away 1121 01:18:06,793 --> 01:18:10,110 is the influence of that fluctuation felt? 1122 01:18:10,110 --> 01:18:14,460 So I would be interested in the limit of this 1123 01:18:14,460 --> 01:18:19,723 when the separation is large-- goes to infinity. 1124 01:18:19,723 --> 01:18:22,370 And we can see that the answer very much 1125 01:18:22,370 --> 01:18:24,630 depends on the dimensions. 1126 01:18:24,630 --> 01:18:30,140 So we find that for d that is greater than 2, 1127 01:18:30,140 --> 01:18:33,170 like the Coulomb potential in three dimensions, 1128 01:18:33,170 --> 01:18:36,150 you basically go to a constant. 1129 01:18:36,150 --> 01:18:40,920 While in d less than 2, it is something 1130 01:18:40,920 --> 01:18:43,325 that grows as a function of distance 1131 01:18:43,325 --> 01:18:49,550 as x to the 2 minus d 2 minus d Sd. 1132 01:18:49,550 --> 01:18:53,450 And actually, write at the borderline dimension 1133 01:18:53,450 --> 01:18:58,150 of d equals to 2, it also grows at large distances. 1134 01:18:58,150 --> 01:19:00,370 If you do the integration correctly, 1135 01:19:00,370 --> 01:19:05,853 you will find that it is 1 over 2 pi log x over some distance 1136 01:19:05,853 --> 01:19:06,839 or [INAUDIBLE]. 1137 01:19:14,250 --> 01:19:17,240 Now, what you are really interested-- actually, 1138 01:19:17,240 --> 01:19:21,790 this thing that I wrote down is not particularly meaningful. 1139 01:19:21,790 --> 01:19:23,410 The thing that you are interested in 1140 01:19:23,410 --> 01:19:26,040 is what I had originally written, 1141 01:19:26,040 --> 01:19:30,890 which is that if I look at the angle that I have at x and then 1142 01:19:30,890 --> 01:19:33,440 I go far away-- because the angel itself is not 1143 01:19:33,440 --> 01:19:38,030 an observable, but angle differences are. 1144 01:19:38,030 --> 01:19:41,150 So the average of this quantity will be 0. 1145 01:19:41,150 --> 01:19:43,130 But there will be some variance to it, 1146 01:19:43,130 --> 01:19:44,530 so I can look at this quantity. 1147 01:19:47,810 --> 01:19:51,420 And that quantity-- I can expand this-- 1148 01:19:51,420 --> 01:19:56,820 is twice the average of theta at some particular location. 1149 01:19:56,820 --> 01:19:59,740 Presumably, it doesn't matter which location I look at. 1150 01:19:59,740 --> 01:20:03,990 So it's the variance locally that you have in the angles. 1151 01:20:03,990 --> 01:20:08,940 And then minus twice theta of x theta of x prime, 1152 01:20:08,940 --> 01:20:12,535 which is the quantity that I calculated for you above. 1153 01:20:15,150 --> 01:20:20,290 So all I need to do is to take the Coulomb potential that I 1154 01:20:20,290 --> 01:20:24,140 calculated, multiply it by a factor of 2, 1155 01:20:24,140 --> 01:20:28,030 divide by a factor of k bar that I basically 1156 01:20:28,030 --> 01:20:30,750 indicated as part of the definition. 1157 01:20:30,750 --> 01:20:36,870 So this object is going to be 2 x to the 2 1158 01:20:36,870 --> 01:20:43,620 minus d divided by k bar 2 minus d Sd. 1159 01:20:43,620 --> 01:20:47,070 And actually, the reason I do this is because now I 1160 01:20:47,070 --> 01:20:51,310 can indicate the overall constant as follows. 1161 01:20:54,170 --> 01:20:59,490 Remember that all of our statistical field theories 1162 01:20:59,490 --> 01:21:02,250 are obtained by averaging. 1163 01:21:02,250 --> 01:21:06,240 And I shouldn't believe any of these formulas 1164 01:21:06,240 --> 01:21:09,260 when I look at very short wavelengths. 1165 01:21:09,260 --> 01:21:12,770 So I shouldn't really believe any answer 1166 01:21:12,770 --> 01:21:16,910 that I got from those formulas when the points x prime and x 1167 01:21:16,910 --> 01:21:19,680 come too close to each other. 1168 01:21:19,680 --> 01:21:24,330 So there is something of the order of a lattice spacing, 1169 01:21:24,330 --> 01:21:28,340 averaging distance, et cetera, that I'll call a. 1170 01:21:28,340 --> 01:21:33,140 And by the time I get to a, I expect that my fluctuations 1171 01:21:33,140 --> 01:21:35,340 vanish because that's the distance over which 1172 01:21:35,340 --> 01:21:37,320 I'm doing the average. 1173 01:21:37,320 --> 01:21:40,870 So I manage to get rid of whatever this constant is 1174 01:21:40,870 --> 01:21:46,140 by substituting [INAUDIBLE] with the scale over which I expect 1175 01:21:46,140 --> 01:21:49,370 my theory to cease to be valid. 1176 01:21:49,370 --> 01:21:52,490 But again, what I find is that if I 1177 01:21:52,490 --> 01:21:59,130 look at the fluctuation between two points at large distances, 1178 01:21:59,130 --> 01:22:02,540 if I am in dimensions greater than 2, 1179 01:22:02,540 --> 01:22:04,680 this thing eventually goes to a constant. 1180 01:22:08,460 --> 01:22:11,330 Which means that if I'm in three dimensions, 1181 01:22:11,330 --> 01:22:16,080 the fluctuation in phase between one place and another place 1182 01:22:16,080 --> 01:22:20,520 are not necessarily small or large because I don't know what 1183 01:22:20,520 --> 01:22:22,180 the magnitude of this constant is, 1184 01:22:22,180 --> 01:22:23,800 but they are not getting bigger as I 1185 01:22:23,800 --> 01:22:26,440 go further and further along. 1186 01:22:26,440 --> 01:22:30,690 Whereas, no matter what I do in d that is less than 1187 01:22:30,690 --> 01:22:38,600 or equal to 2, this thing at large distances blows up. 1188 01:22:38,600 --> 01:22:41,340 So I thought that I had a system where 1189 01:22:41,340 --> 01:22:47,030 I had broken spontaneous symmetry and all of my spins, 1190 01:22:47,030 --> 01:22:51,470 all of my phases were pointing in one direction. 1191 01:22:51,470 --> 01:22:55,910 But I see that when I put these fluctuations, no matter 1192 01:22:55,910 --> 01:23:00,500 how small I make the amplitude, the amplitude doesn't matter. 1193 01:23:00,500 --> 01:23:04,820 If I go to far enough distances, fluctuations 1194 01:23:04,820 --> 01:23:08,020 will tell me that I don't know what the phase is from here 1195 01:23:08,020 --> 01:23:12,090 to here because it has gone over many multiples of 2 pi, 1196 01:23:12,090 --> 01:23:15,740 so that it has become divergent. 1197 01:23:15,740 --> 01:23:23,350 So what that really means is that because of fluctuations, 1198 01:23:23,350 --> 01:23:26,740 you cannot have long-range order. 1199 01:23:26,740 --> 01:23:38,370 So destroy continuous long-range order 1200 01:23:38,370 --> 01:23:43,390 in dimensions that are less than or equal to 2. 1201 01:23:43,390 --> 01:23:45,230 This is called the Mermin-Wagner theorem. 1202 01:23:51,390 --> 01:23:55,490 So you shouldn't have any, for example, super fluidity, 1203 01:23:55,490 --> 01:23:59,840 magnetization, anything in two dimensions of one dimension. 1204 01:23:59,840 --> 01:24:04,110 If you go to long enough, you will 1205 01:24:04,110 --> 01:24:07,430 see that fluctuations have destroyed your order. 1206 01:24:07,430 --> 01:24:12,590 So we can see already how important fluctuations are. 1207 01:24:12,590 --> 01:24:15,700 This d equal to 2 is called the lower critical dimension. 1208 01:24:21,210 --> 01:24:27,760 It is this phenomena of symmetry breaking, ordering, 1209 01:24:27,760 --> 01:24:29,750 phase transition, et cetera, that we 1210 01:24:29,750 --> 01:24:34,010 are discussing for continuous systems-- 1211 01:24:34,010 --> 01:24:36,580 for continuous symmetry breaking can only 1212 01:24:36,580 --> 01:24:40,810 exist in three dimensions but not in two dimensions. 1213 01:24:40,810 --> 01:24:43,780 We'll see that for discrete symmetries, 1214 01:24:43,780 --> 01:24:47,090 you can have ordering in two dimensions, but not one 1215 01:24:47,090 --> 01:24:48,240 dimensions. 1216 01:24:48,240 --> 01:24:51,990 So there, the lower critical dimension is 1. 1217 01:24:51,990 --> 01:24:52,720 Yes. 1218 01:24:52,720 --> 01:24:54,382 AUDIENCE: And does this hold for any n? 1219 01:24:54,382 --> 01:24:55,590 This example you were doing-- 1220 01:24:55,590 --> 01:24:55,950 PROFESSOR: Yes. 1221 01:24:55,950 --> 01:24:57,575 AUDIENCE: --it just has two components. 1222 01:24:57,575 --> 01:25:00,960 PROFESSOR: Any n. n equals to 2, 3, 4, anything. 1223 01:25:00,960 --> 01:25:03,800 We'll see later on, towards the end of the course, 1224 01:25:03,800 --> 01:25:07,210 that there is a slight proviso for the case of n equals to 2, 1225 01:25:07,210 --> 01:25:09,746 but that we'll leave for later.