WEBVTT 00:00:00.070 --> 00:00:01.780 The following content is provided 00:00:01.780 --> 00:00:04.019 under a Creative Commons license. 00:00:04.019 --> 00:00:06.870 Your support will help MIT OpenCourseWare continue 00:00:06.870 --> 00:00:10.730 to offer high-quality educational resources for free. 00:00:10.730 --> 00:00:13.340 To make a donation, or view additional materials 00:00:13.340 --> 00:00:17.217 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.217 --> 00:00:17.842 at ocw.mit.edu. 00:00:22.790 --> 00:00:26.160 PROFESSOR: OK, let's start. 00:00:26.160 --> 00:00:30.410 So looking for a way to understand 00:00:30.410 --> 00:00:33.920 the universality of phase transitions, 00:00:33.920 --> 00:00:36.550 we arrived at the simplest model that 00:00:36.550 --> 00:00:37.935 should capture some of that. 00:00:37.935 --> 00:00:50.700 That was the Ising model, where at each site of a lattice-- 00:00:50.700 --> 00:00:52.820 and for a while I will be talking 00:00:52.820 --> 00:00:56.250 in this lecture about the square lattice-- 00:00:56.250 --> 00:00:58.920 you put a binary variable. 00:00:58.920 --> 00:01:06.840 Let's call it sigma i that takes two values, minus plus 1, 00:01:06.840 --> 00:01:08.540 on each of the N sites. 00:01:08.540 --> 00:01:11.480 So we have a total of 2 to the N possible configurations. 00:01:15.600 --> 00:01:19.820 And we subject that to an energy cost 00:01:19.820 --> 00:01:25.920 that tries to make nearest neighbors to be the same. 00:01:25.920 --> 00:01:29.830 So this symbol stands for sum over all pairs of nearest 00:01:29.830 --> 00:01:31.950 neighbors on whatever lattice you have. 00:01:31.950 --> 00:01:34.470 Here the square lattice. 00:01:34.470 --> 00:01:40.320 And the tendency for them to be parallel as opposed 00:01:40.320 --> 00:01:42.910 to anti-parallel is captured through 00:01:42.910 --> 00:01:49.050 this dimensionless energy divided by Kt parameter K. 00:01:49.050 --> 00:01:54.940 So then in principle, as we change K we could also 00:01:54.940 --> 00:01:57.420 potentially add a magnetic field. 00:01:57.420 --> 00:02:00.450 There could be a phase transition in the system. 00:02:00.450 --> 00:02:03.235 And that should be captured by looking at the partition 00:02:03.235 --> 00:02:07.130 function, which is obtained by summing over the 2 00:02:07.130 --> 00:02:11.760 to the N possibilities of this weight that 00:02:11.760 --> 00:02:16.690 is e to the sum over ij K sigma i sigma j. 00:02:20.300 --> 00:02:23.710 So that's easier said than done. 00:02:23.710 --> 00:02:28.130 And the question was, well, how can we proceed with this? 00:02:28.130 --> 00:02:32.720 And last lecture, we suggested two routes 00:02:32.720 --> 00:02:35.250 for looking at this system. 00:02:35.250 --> 00:02:38.780 One of them was to start by looking 00:02:38.780 --> 00:02:43.665 at a low-temperature expansion. 00:02:43.665 --> 00:02:51.820 And here, let's say, we would start 00:02:51.820 --> 00:02:54.860 with one of two possible ground states. 00:02:54.860 --> 00:02:58.100 Let' say all of the spins could, for example, 00:02:58.100 --> 00:02:59.565 be pointing in the plus direction. 00:03:06.510 --> 00:03:09.580 That is certainly the largest contribution 00:03:09.580 --> 00:03:11.240 that you would have to the partition 00:03:11.240 --> 00:03:13.480 function at 0 temperature. 00:03:13.480 --> 00:03:20.360 And that contribution is all of the bonds being satisfied. 00:03:20.360 --> 00:03:24.650 Each one of them giving a factor of e to the K. 00:03:24.650 --> 00:03:26.700 Let's say we are on a square lattice. 00:03:26.700 --> 00:03:29.510 On a square lattice, each site has 00:03:29.510 --> 00:03:32.070 two bonds associated with it. 00:03:32.070 --> 00:03:34.890 So this will go like 2N. 00:03:34.890 --> 00:03:39.240 Since we have N sites, we will 2N bonds. 00:03:39.240 --> 00:03:41.940 There is actually, of course, two possibilities. 00:03:41.940 --> 00:03:46.060 We can have either all of them plus or all of them minus. 00:03:46.060 --> 00:03:50.080 So there is a kind of trivial degeneracy of 2, 00:03:50.080 --> 00:03:53.470 which doesn't really make too much of a difference. 00:03:53.470 --> 00:03:58.110 And then we can start looking at excitations around this. 00:03:58.110 --> 00:04:02.060 And so we said that the first type of excitation 00:04:02.060 --> 00:04:05.480 is somewhere on the lattice we make one of these pluses 00:04:05.480 --> 00:04:06.650 into a minus. 00:04:06.650 --> 00:04:10.530 And once we do that, we have made 00:04:10.530 --> 00:04:15.140 4 bonds that go out of this minus site unhappy. 00:04:15.140 --> 00:04:20.560 So the cost of going from plus k to minus K, which 00:04:20.560 --> 00:04:27.700 is minus 2k, in this case is repeated four times. 00:04:27.700 --> 00:04:30.070 And this particular excitation can 00:04:30.070 --> 00:04:34.070 be placed in any one of N locations on the lattice. 00:04:34.070 --> 00:04:38.120 So this was this kind of excitation. 00:04:38.120 --> 00:04:42.400 And then we could go and have the next excitation 00:04:42.400 --> 00:04:44.666 where two of them are flipped. 00:04:44.666 --> 00:04:47.740 And we would have a situation such as this. 00:04:47.740 --> 00:04:53.530 And since this dimer can point in the x- or the y-direction 00:04:53.530 --> 00:04:58.300 on the square lattice, it has a degeneracy of 2. 00:04:58.300 --> 00:05:01.870 I have e to the minus 2K. 00:05:01.870 --> 00:05:04.850 And how many bonds have I broken? 00:05:04.850 --> 00:05:07.810 One, two, three, four, five, six. 00:05:07.810 --> 00:05:10.000 Times 6. 00:05:10.000 --> 00:05:12.270 And I can go on. 00:05:12.270 --> 00:05:17.800 So this was a situation such as this. 00:05:17.800 --> 00:05:21.520 A general term in the series would 00:05:21.520 --> 00:05:26.210 correspond to creating some kind of an island of minus 00:05:26.210 --> 00:05:27.455 in this sea of pluses. 00:05:30.600 --> 00:05:35.540 And the contribution would be e to the minus 2K 00:05:35.540 --> 00:05:39.550 times the perimeter of this island. 00:05:48.660 --> 00:05:51.685 So that would be a way to calculate 00:05:51.685 --> 00:05:58.550 the low-temperature expansion we discussed last time around. 00:05:58.550 --> 00:06:01.670 We also said that I could do a high temperature expansion. 00:06:09.610 --> 00:06:12.960 And for this, we use the trick. 00:06:12.960 --> 00:06:16.140 We said I can write the partition function 00:06:16.140 --> 00:06:20.370 as a sum over all these 2 to the N configuration. 00:06:20.370 --> 00:06:21.285 Yes. 00:06:21.285 --> 00:06:24.080 AUDIENCE: Is there a reason we don't have separate islands? 00:06:24.080 --> 00:06:25.080 PROFESSOR: Oh, we do. 00:06:25.080 --> 00:06:25.700 We do. 00:06:25.700 --> 00:06:30.990 So in general, in this picture I could have multiple islands. 00:06:30.990 --> 00:06:32.830 Yes. 00:06:32.830 --> 00:06:35.970 And what would be interesting is certainly 00:06:35.970 --> 00:06:40.490 when I take the log of Z, then the log of Z 00:06:40.490 --> 00:06:43.470 here I would have NK. 00:06:43.470 --> 00:06:45.440 I would have log 2. 00:06:45.440 --> 00:06:49.280 And then there would be a bunch of terms. 00:06:49.280 --> 00:06:53.300 And what we saw was those bunches of terms, 00:06:53.300 --> 00:06:57.500 starting with that one, can be captured 00:06:57.500 --> 00:07:00.970 into a series where the n comes out front 00:07:00.970 --> 00:07:05.380 and the terms in the series would be functions of e 00:07:05.380 --> 00:07:07.170 to the minus 2K. 00:07:07.170 --> 00:07:10.970 And indeed, when we exponentiate this, 00:07:10.970 --> 00:07:13.650 this would have single islands only. 00:07:13.650 --> 00:07:15.630 The exponential will have multiple islands 00:07:15.630 --> 00:07:17.482 that we have for the partition functions. 00:07:20.290 --> 00:07:23.800 So these terms, as we see, are higher powers of n 00:07:23.800 --> 00:07:25.452 because you would be able to move them 00:07:25.452 --> 00:07:26.960 in different directions. 00:07:26.960 --> 00:07:28.970 When you take the log, only terms that are 00:07:28.970 --> 00:07:30.300 linear and can survive. 00:07:35.310 --> 00:07:39.710 So this sum over bond configurations 00:07:39.710 --> 00:07:44.570 I can write as a product over bonds. 00:07:44.570 --> 00:07:48.330 And the factor of e to the K sigma i sigma 00:07:48.330 --> 00:07:53.260 j, we saw that we could capture slightly differently. 00:07:53.260 --> 00:07:55.830 So e to the K sigma i sigma j I can 00:07:55.830 --> 00:07:59.120 write as hyperbolic cosine of K 1 00:07:59.120 --> 00:08:05.660 plus hyperbolic tanh of K sigma i sigma j. 00:08:05.660 --> 00:08:11.670 And this t is going to be my symbol for hyperbolic tanh 00:08:11.670 --> 00:08:17.150 of K, so that I don't have to repeat it all over the place. 00:08:17.150 --> 00:08:20.220 So this is just rewriting of that exponential, 00:08:20.220 --> 00:08:23.260 recognizing that it has two possibilities. 00:08:23.260 --> 00:08:25.315 We took the cosh K to the outside. 00:08:28.150 --> 00:08:31.430 And if I'm, again, on the square lattice, 00:08:31.430 --> 00:08:35.909 I have 2N bonds so there will be 2N factors of cosh K 00:08:35.909 --> 00:08:38.470 that I will take into the outside. 00:08:38.470 --> 00:08:41.780 And then I would have a series, which 00:08:41.780 --> 00:08:47.140 would be these terms that I can start expanding in powers of t. 00:08:47.140 --> 00:08:50.580 And the lowest order I have 1. 00:08:50.580 --> 00:08:53.410 And then we discussed what kinds of terms are allowed. 00:09:01.510 --> 00:09:10.140 We saw that if I take just one factor of t sigma i sigma j, 00:09:10.140 --> 00:09:14.200 then I have a sigma i sitting here and sigma j sitting here. 00:09:14.200 --> 00:09:19.050 I sum over the two possibilities of sigma being plus or minus. 00:09:19.050 --> 00:09:21.180 It will give me 0. 00:09:21.180 --> 00:09:24.920 So there is a contribution order of t if I expand that, 00:09:24.920 --> 00:09:30.070 but summing over sigma i and sigma j would give me 0. 00:09:30.070 --> 00:09:31.830 So the only choice that I have is 00:09:31.830 --> 00:09:36.810 that this sigma that is sitting out here I should square. 00:09:36.810 --> 00:09:39.220 And I can square it by putting a bond that 00:09:39.220 --> 00:09:42.380 corresponds to this sigma and that sigma. 00:09:42.380 --> 00:09:43.700 This became sigma squared. 00:09:43.700 --> 00:09:46.200 I don't have to worry about that. 00:09:46.200 --> 00:09:49.080 Then I can complete this so that that's 00:09:49.080 --> 00:09:52.600 squared and this so that that's squared. 00:09:52.600 --> 00:09:55.530 And so this was a diagram that contributed 00:09:55.530 --> 00:09:58.390 N this quantity t to the fourth. 00:10:01.090 --> 00:10:05.790 Well, what's the next type of graph that I could draw? 00:10:05.790 --> 00:10:07.510 I could do something like this. 00:10:11.730 --> 00:10:15.120 And that is, again, something that I 00:10:15.120 --> 00:10:18.783 can orient along the x-direction or along the y-direction. 00:10:18.783 --> 00:10:22.950 So there is a factor of 2 from the orientation. 00:10:22.950 --> 00:10:26.040 And that's how many factors of t. 00:10:26.040 --> 00:10:26.600 It is 6. 00:10:32.400 --> 00:10:35.890 And so in general, what do I have? 00:10:35.890 --> 00:10:39.130 I will have to draw some kind of a graph 00:10:39.130 --> 00:10:47.275 on the lattice where at each site, I either have 0, 00:10:47.275 --> 00:10:49.760 2, or potentially 4. 00:10:49.760 --> 00:10:54.350 There is no difficult with 4 bonds emanating from a site 00:10:54.350 --> 00:10:56.700 because that sigma to the fourth. 00:10:56.700 --> 00:11:00.800 Basically, those kinds of things actually 00:11:00.800 --> 00:11:04.850 will give me then the sum over-- possibilities of sigma 00:11:04.850 --> 00:11:09.040 will give me 2 to the number of sites. 00:11:09.040 --> 00:11:12.470 And then these graphs, basically will 00:11:12.470 --> 00:11:17.689 get a factor which is 2 to the number of bonds in graph. 00:11:24.100 --> 00:11:25.830 And then again, you can see that here I 00:11:25.830 --> 00:11:29.790 could have multiple loops, just like we discussed over there. 00:11:29.790 --> 00:11:34.210 Multiple loops will go with larger factors of N. 00:11:34.210 --> 00:11:38.310 The thing that we are interested is log Z, 00:11:38.310 --> 00:11:49.660 which is N log 2 cosine squared K. And then I 00:11:49.660 --> 00:11:52.643 have to take the log of this expression, 00:11:52.643 --> 00:11:58.170 and I'll call it g of t. 00:11:58.170 --> 00:11:59.974 Yes. 00:11:59.974 --> 00:12:05.710 AUDIENCE: How did we get rid of [INAUDIBLE] bonds? 00:12:05.710 --> 00:12:06.840 PROFESSOR: OK. 00:12:06.840 --> 00:12:16.790 So this e to the K sigma i sigma j has two possible values. 00:12:16.790 --> 00:12:23.410 It is either e to the K or e to the minus K. 00:12:23.410 --> 00:12:25.690 And I can write it, therefore, as e 00:12:25.690 --> 00:12:31.610 to the K plus e to the minus K over 2 plus e to the K minus 00:12:31.610 --> 00:12:37.240 e to the minus K over 2 sigma i sigma j, right? 00:12:37.240 --> 00:12:38.475 AUDIENCE: My question was-- 00:12:38.475 --> 00:12:40.710 PROFESSOR: Yeah, I know. 00:12:40.710 --> 00:12:45.390 And then, this became cosh K 1 plus what 00:12:45.390 --> 00:12:48.990 I call sigma i sigma j. 00:12:48.990 --> 00:12:54.760 So when I draw my lattice, for the bond that 00:12:54.760 --> 00:13:01.900 goes between sites i and j, in the partition function 00:13:01.900 --> 00:13:03.060 I have this contribution. 00:13:06.060 --> 00:13:10.850 This contribution is completely equivalent to this. 00:13:10.850 --> 00:13:12.700 And this is two terms. 00:13:12.700 --> 00:13:14.860 The cosh K we took outside. 00:13:14.860 --> 00:13:21.250 The two terms is either 1 or 1 line. 00:13:21.250 --> 00:13:24.220 There isn't anything that is multiple lines. 00:13:24.220 --> 00:13:27.110 And as I said, you can make separately 00:13:27.110 --> 00:13:33.770 an expansion in powers of K. This corresponds to nothing, 00:13:33.770 --> 00:13:37.990 or going forward and backward, or going forward, backward, 00:13:37.990 --> 00:13:41.959 et cetera, if you make an expansion in powers of K. This 00:13:41.959 --> 00:13:43.250 term corresponds to going once. 00:13:47.030 --> 00:13:52.140 Essentially, this captures all of the things that stay back 00:13:52.140 --> 00:13:56.045 to the same site and this is a re-summation of everything 00:13:56.045 --> 00:13:58.780 that steps you forward. 00:13:58.780 --> 00:14:03.920 So everything that steps you forward and carries information 00:14:03.920 --> 00:14:07.820 from this sites to that site has been appropriately 00:14:07.820 --> 00:14:09.060 taken care of. 00:14:09.060 --> 00:14:10.135 And it occurs once. 00:14:12.800 --> 00:14:15.835 And that's one reason. 00:14:15.835 --> 00:14:18.320 And again, if you have three things coming 00:14:18.320 --> 00:14:20.890 at a particular site, it's a sigma i q. 00:14:20.890 --> 00:14:23.710 So these are completely the only thing that happen. 00:14:26.930 --> 00:14:34.770 But in principle, this is one diagrammatic series. 00:14:34.770 --> 00:14:38.870 This is one diagrammatic series. 00:14:38.870 --> 00:14:41.640 But you stare at them a little bit 00:14:41.640 --> 00:14:46.890 and you'll see why I put the same g for both of them. 00:14:46.890 --> 00:14:50.600 On the square lattice, they're identify series. 00:14:50.600 --> 00:14:55.550 See that series had N something to the fourth power, 00:14:55.550 --> 00:14:58.500 2N something to the sixth power, N something 00:14:58.500 --> 00:15:00.980 to the fourth power, 2N something to the sixth power. 00:15:00.980 --> 00:15:04.150 The first two terms are identical. 00:15:04.150 --> 00:15:07.710 You can convince yourself that all the terms will be identical 00:15:07.710 --> 00:15:11.360 also, including something complicated, 00:15:11.360 --> 00:15:14.370 such as if I were to draw-- I don't know. 00:15:14.370 --> 00:15:20.750 A diagram such as this one, which has 2 or 4 per site. 00:15:20.750 --> 00:15:25.780 I can convert that to something that had spins plus out here, 00:15:25.780 --> 00:15:30.180 then it minus here, minus here, plus here. 00:15:30.180 --> 00:15:33.420 And it's a completely consistent diagram 00:15:33.420 --> 00:15:36.562 that I would have had in the low-temperature series. 00:15:36.562 --> 00:15:38.930 So you can convince yourself that there 00:15:38.930 --> 00:15:42.493 is a one-to-one correspondence between these two series. 00:15:42.493 --> 00:15:46.680 They are identical for the square lattice. 00:15:46.680 --> 00:15:50.770 As we will discuss, this is a property of the square lattice. 00:15:50.770 --> 00:15:53.360 So you have this very nice symmetry 00:15:53.360 --> 00:15:57.380 that you conclude that the partition function per site, 00:15:57.380 --> 00:16:00.820 the part that is interesting, either 00:16:00.820 --> 00:16:08.366 I can get it from here as K plus this function of e 00:16:08.366 --> 00:16:15.580 to the minus 2K or from the high temperatures series as log 2 00:16:15.580 --> 00:16:19.020 hyperbolic cosine squared K. Plus exactly 00:16:19.020 --> 00:16:25.630 the same function of tanh K. 00:16:25.630 --> 00:16:27.920 This part of it, actually, I don't really 00:16:27.920 --> 00:16:31.870 care because these are analytical functions. 00:16:31.870 --> 00:16:34.435 I expect this model to have a phase transition. 00:16:34.435 --> 00:16:37.197 The low-temperature and the high-temperature behavior 00:16:37.197 --> 00:16:38.030 should be different. 00:16:38.030 --> 00:16:41.550 It should be order at low temperature, [INAUDIBLE] 00:16:41.550 --> 00:16:42.860 at high temperature. 00:16:42.860 --> 00:16:45.570 There should be a phase transition and a singularity 00:16:45.570 --> 00:16:47.270 between those two cases. 00:16:47.270 --> 00:16:50.790 The singular part must be captured here. 00:16:50.790 --> 00:16:54.260 So these are the singular functions. 00:16:54.260 --> 00:16:57.490 So the singular part of the free energy 00:16:57.490 --> 00:17:01.640 has this interesting property that you can evaluate it 00:17:01.640 --> 00:17:06.819 for some parameter K, which is large in the low-temperature 00:17:06.819 --> 00:17:10.250 phase, or some parameter t, which 00:17:10.250 --> 00:17:13.589 is small in the high-temperature phase. 00:17:13.589 --> 00:17:20.069 And the two will be related completely to each other. 00:17:20.069 --> 00:17:22.530 They are essentially the same thing. 00:17:22.530 --> 00:17:25.970 So this property is called duality. 00:17:29.650 --> 00:17:33.400 And so what I have said, first of all, is 00:17:33.400 --> 00:17:39.460 that there is a relationship between, say, the coupling tanh 00:17:39.460 --> 00:17:47.910 K and a coupling that I can separately call K tilde, such 00:17:47.910 --> 00:17:51.900 that if I evaluate the high-temperature series at K, 00:17:51.900 --> 00:17:55.430 it is like evaluating the low-temperature series at K 00:17:55.430 --> 00:18:02.270 tilde, where K tilde of K is minus 1/2 log 00:18:02.270 --> 00:18:07.710 of the hyperbolic tanh at K. 00:18:07.710 --> 00:18:10.350 I can plot for you what this function looks like. 00:18:13.272 --> 00:18:18.740 So this is K. This is what K tilde of K looks like. 00:18:18.740 --> 00:18:22.700 And it is something like this. 00:18:22.700 --> 00:18:27.360 Basically, strong coupling, or low temperature, 00:18:27.360 --> 00:18:30.295 gets mapped to weak coupling, or high temperature, 00:18:30.295 --> 00:18:31.495 and vice versa. 00:18:35.120 --> 00:18:39.795 So it's kind of like this, that there 00:18:39.795 --> 00:18:46.490 is this axes of the strength of K going 00:18:46.490 --> 00:18:52.610 from low temperature, strong, to high temperature, weak. 00:18:52.610 --> 00:18:55.730 And what I have shown is that if I start with somewhere 00:18:55.730 --> 00:19:02.040 out here, it is mapped to somewhere down here. 00:19:02.040 --> 00:19:06.060 If I start from somewhere here, it 00:19:06.060 --> 00:19:09.140 would be mapping to somewhere here. 00:19:15.100 --> 00:19:22.440 So one question to ask is, well, OK, I start from here. 00:19:22.440 --> 00:19:24.820 I go here. 00:19:24.820 --> 00:19:30.890 If I put that value of k in here, do I go to a third point 00:19:30.890 --> 00:19:33.550 or do I come back to here? 00:19:33.550 --> 00:19:36.560 And I will show you that it is, in fact, 00:19:36.560 --> 00:19:41.620 a mapping that goes both ways. 00:19:41.620 --> 00:19:48.410 And a way to show that is like this. 00:19:48.410 --> 00:19:52.150 Let me look at the hyperbolic sine of 2K. 00:19:54.750 --> 00:19:58.950 Hyperbolic sines of twice the angles, or twice 00:19:58.950 --> 00:20:05.500 the hyperbolic sines of K hyperbolic cosine of K. 00:20:05.500 --> 00:20:07.915 All my answers are in terms of tanh K, 00:20:07.915 --> 00:20:12.650 so I can make this sine to become a tanh by dividing 00:20:12.650 --> 00:20:17.640 by cosh, and then making this cosh squared. 00:20:17.640 --> 00:20:22.400 So that is 2 hyperbolic tanh of K. 00:20:22.400 --> 00:20:27.730 And then, there's various ways to sort of remember 00:20:27.730 --> 00:20:31.350 the identity for hyperbolic cosine squared 00:20:31.350 --> 00:20:34.210 minus sine squared is 1. 00:20:34.210 --> 00:20:41.320 If I divide by c squared, it becomes 1 minus t squared 00:20:41.320 --> 00:20:43.680 is 1 over c squared. 00:20:43.680 --> 00:20:46.780 So the hyperbolic cosine squared here 00:20:46.780 --> 00:20:54.990 is the inverse of 1 minus hyperbolic tanh squared of K. 00:20:54.990 --> 00:21:00.710 Now, for tanh K, we have this identity, is 2 e to the minus 00:21:00.710 --> 00:21:02.640 2K tilde. 00:21:02.640 --> 00:21:04.700 1 minus the square of that. 00:21:08.010 --> 00:21:12.490 If I multiply both sides by-- e to the numerator 00:21:12.490 --> 00:21:20.398 and denominator by e to the plus 2K tilde, 00:21:20.398 --> 00:21:28.666 hopefully you recognize this as 1 over hyperbolic sine of K 00:21:28.666 --> 00:21:30.466 tilde 2K tilde. 00:21:33.800 --> 00:21:42.000 So the identity here that was kind of not very transparent, 00:21:42.000 --> 00:21:44.880 if I had made the change of variables 00:21:44.880 --> 00:21:48.950 to the hyperbolic sine of twice the angle, 00:21:48.950 --> 00:21:49.860 had this simple form. 00:21:55.830 --> 00:21:59.375 So then the symmetry between K and K tilde 00:21:59.375 --> 00:22:01.300 is immediately obvious. 00:22:01.300 --> 00:22:06.520 You pick one value of K, or sine K, and then the inverse. 00:22:06.520 --> 00:22:09.980 And the inverse of the inverse, you are back to where you are. 00:22:09.980 --> 00:22:14.640 So it is clear that this is kind of like an x to y0 1 over x 00:22:14.640 --> 00:22:15.670 mapping. 00:22:15.670 --> 00:22:20.050 x to 1 over x mapping would also kind of look exactly like this. 00:22:20.050 --> 00:22:22.070 In fact, if instead of K tilde and K 00:22:22.070 --> 00:22:26.190 I had plotted hyperbolic sine versus hyperbolic sine of twice 00:22:26.190 --> 00:22:29.056 the angle, it would have been just the 1 over x curve. 00:22:34.900 --> 00:22:44.600 Now, just to give you another example, if I had the function 00:22:44.600 --> 00:22:50.740 f of x, which is x 1 plus x squared. 00:22:50.740 --> 00:22:53.290 This function, if I divide by x squared, 00:22:53.290 --> 00:23:00.170 becomes x inverse 1 plus x inverse squared. 00:23:00.170 --> 00:23:03.560 So this is, again, f of x inverse. 00:23:03.560 --> 00:23:08.950 So if I evaluate this function for any value like 5, 00:23:08.950 --> 00:23:11.980 then I know the value exactly for 1/5. 00:23:11.980 --> 00:23:16.780 If I evaluate it for 200, I know it for 1/200, and vice versa. 00:23:16.780 --> 00:23:20.110 Our g function is kind of like that. 00:23:22.940 --> 00:23:26.070 Now, this function you can see that starts 00:23:26.070 --> 00:23:32.520 to go linearly increases with x, and then eventually it 00:23:32.520 --> 00:23:37.920 comes down like this must have one maximum. 00:23:37.920 --> 00:23:39.160 Where is the maximum? 00:23:42.360 --> 00:23:43.390 It has to be 1. 00:23:43.390 --> 00:23:46.950 I don't have to take derivatives of everything, et cetera. 00:23:46.950 --> 00:23:49.665 If there is one point which corresponds to the maximum, 00:23:49.665 --> 00:23:51.450 it's the point that maps to itself. 00:23:54.700 --> 00:23:58.250 Now, this function for the Ising model, 00:23:58.250 --> 00:23:59.730 I know it has a phase transition. 00:23:59.730 --> 00:24:02.190 Or, I guess it has a phase transition. 00:24:02.190 --> 00:24:04.920 So there is one point, hopefully one point, 00:24:04.920 --> 00:24:06.240 at which it becomes singular. 00:24:06.240 --> 00:24:08.080 I don't know, maybe it's three point. 00:24:08.080 --> 00:24:12.360 But let's say it's one point at which it becomes singular. 00:24:12.360 --> 00:24:16.590 Then I should be able to figure its singularity by precisely 00:24:16.590 --> 00:24:19.090 the same argument. 00:24:19.090 --> 00:24:25.400 So the function that corresponds to x going to 1 over x 00:24:25.400 --> 00:24:26.830 is this hyperbolic sine. 00:24:30.180 --> 00:24:35.840 So if there is a point which is the unique point that 00:24:35.840 --> 00:24:38.310 corresponds to the singularity, it 00:24:38.310 --> 00:24:42.860 has to be the point that is self-dual-- maps on to itself, 00:24:42.860 --> 00:24:44.790 just like 1. 00:24:44.790 --> 00:24:49.380 So sine of 2 Kc should be 1. 00:24:49.380 --> 00:24:50.520 And what is this? 00:24:50.520 --> 00:24:53.950 Hyperbolic sine we can write as e to the 2 Kc 00:24:53.950 --> 00:24:57.750 minus e to the minus 2 Kc over 2. 00:24:57.750 --> 00:25:03.610 We can manipulate this equation slightly 00:25:03.610 --> 00:25:12.940 to e to the 4 Kc minus 2 e to the 2 Kc minus 1 equals to 0, 00:25:12.940 --> 00:25:17.310 which is a quadratic equation for e to the 2 Kc. 00:25:17.310 --> 00:25:23.478 So I can immediately solve for e to the 2 Kc is-- 00:25:23.478 --> 00:25:29.305 This has a 2, so I can say 1 minus plus square root 00:25:29.305 --> 00:25:32.535 of square of that plus this. 00:25:32.535 --> 00:25:35.930 So that is a square root of 2. 00:25:35.930 --> 00:25:38.380 The exponential better be positive, 00:25:38.380 --> 00:25:40.180 so I can't pick the negative solution. 00:25:43.310 --> 00:25:44.660 And so we know-- 00:25:44.660 --> 00:25:46.070 AUDIENCE: Isn't it [INAUDIBLE]? 00:25:52.190 --> 00:25:54.020 PROFESSOR: Where did I-- OK. 00:25:54.020 --> 00:25:57.445 Multiply by e to the 2 Kc. 00:25:57.445 --> 00:26:00.300 AUDIENCE: Taking that as correct, I didn't check it-- 00:26:00.300 --> 00:26:02.140 PROFESSOR: Yes. 00:26:02.140 --> 00:26:03.595 OK. 00:26:03.595 --> 00:26:09.340 x squared minus 2b plus-- x plus c equals. 00:26:09.340 --> 00:26:12.220 Well, actually, this is-- so x is 00:26:12.220 --> 00:26:18.460 b minus plus square root of b squared minus ac. 00:26:18.460 --> 00:26:21.350 Our c is negative. 00:26:21.350 --> 00:26:22.380 So it's 1 plus 1. 00:26:26.310 --> 00:26:33.485 So the critical coupling that we have is 1/2 log of 1 00:26:33.485 --> 00:26:37.390 plus square root of 2. 00:26:37.390 --> 00:26:40.950 So we know that the critical point of the Ising model 00:26:40.950 --> 00:26:44.135 occurs at this value, which you can put on your calculator. 00:26:44.135 --> 00:26:46.775 And it is something like this. 00:26:50.060 --> 00:26:51.430 There is one assumption. 00:26:51.430 --> 00:26:54.790 Of course, there is essentially only one singularity. 00:26:54.790 --> 00:26:57.770 And there is one singularity in this free energy. 00:26:57.770 --> 00:27:01.661 But if it is, we have solved it for this case. 00:27:10.550 --> 00:27:20.390 So I think to emphasize this was discovered 00:27:20.390 --> 00:27:26.500 around '50s by Wannier, this idea of duality. 00:27:26.500 --> 00:27:30.530 And suddenly, you had exact solution for something 00:27:30.530 --> 00:27:33.830 like the square Ising model. 00:27:33.830 --> 00:27:38.350 Question is, how much information does it give you? 00:27:38.350 --> 00:27:42.880 First of all, the property of self-duality 00:27:42.880 --> 00:27:44.590 is that of the square lattice. 00:27:54.720 --> 00:27:59.050 So if I had done this on the triangular lattice, 00:27:59.050 --> 00:28:03.800 you would've seen that the low-temperature and 00:28:03.800 --> 00:28:06.900 high-temperature expansions don't match. 00:28:06.900 --> 00:28:11.010 It turns out that in order to construct 00:28:11.010 --> 00:28:14.420 the dual of any lattice, what you have to do 00:28:14.420 --> 00:28:20.620 is to put, let's say, points in the center of the units 00:28:20.620 --> 00:28:26.970 that you have and see what lattice these centers make. 00:28:26.970 --> 00:28:29.820 So when you try to do that for the triangular lattice, 00:28:29.820 --> 00:28:32.790 you will see that the centers form, actually, hexagonal 00:28:32.790 --> 00:28:35.550 lattice, and vice versa. 00:28:35.550 --> 00:28:40.060 However, there is a trick using duality 00:28:40.060 --> 00:28:43.380 that you can still calculate critical points of hexagonal 00:28:43.380 --> 00:28:45.770 and triangular lattice. 00:28:45.770 --> 00:28:48.130 And that you will do in one of the problem sets. 00:28:50.710 --> 00:28:52.960 OK, secondly. 00:28:52.960 --> 00:28:59.870 Again, that trick allows you to go beyond square lattice, 00:28:59.870 --> 00:29:02.630 but it turns out that for reasons 00:29:02.630 --> 00:29:05.950 that we will see shortly, it is limited. 00:29:05.950 --> 00:29:09.080 And you can only do these kinds of dualities 00:29:09.080 --> 00:29:12.150 to yourself for two-dimensional lattices. 00:29:17.880 --> 00:29:23.320 And what these kinds of mappings in general 00:29:23.320 --> 00:29:26.170 give for two-dimensional lattices 00:29:26.170 --> 00:29:30.990 is potentially, but not always, the critical value of Kc. 00:29:30.990 --> 00:29:33.520 And again, one of the things that you will see 00:29:33.520 --> 00:29:40.470 is that you can do this for other models in two dimension. 00:29:40.470 --> 00:29:43.590 For example, the Potts model we can 00:29:43.590 --> 00:29:50.750 calculate the critical point through this kind of procedure. 00:29:50.750 --> 00:29:53.440 However, it doesn't tell you anything 00:29:53.440 --> 00:29:56.820 about the nature of the singularity. 00:29:56.820 --> 00:30:00.560 So essentially, what we've shown is that on the K-axis, 00:30:00.560 --> 00:30:05.060 there is some point maybe that describes the singularity 00:30:05.060 --> 00:30:07.700 that you are going to have. 00:30:07.700 --> 00:30:11.390 But the shape of this singularity, the exponent 00:30:11.390 --> 00:30:12.890 can be anything. 00:30:12.890 --> 00:30:17.160 And this mapping does not tell you anything about that. 00:30:17.160 --> 00:30:19.130 It does tell you one thing. 00:30:19.130 --> 00:30:23.130 We also mentioned that the ratio of amplitudes 00:30:23.130 --> 00:30:27.780 above and below for various singular quantities 00:30:27.780 --> 00:30:30.740 is something that is universal because 00:30:30.740 --> 00:30:33.090 of these mappings from high temperatures 00:30:33.090 --> 00:30:34.260 to low temperatures. 00:30:34.260 --> 00:30:38.780 Although I don't know what the nature of the singularity is, 00:30:38.780 --> 00:30:42.860 I know that the amplitude ratio is [INAUDIBLE]. 00:30:42.860 --> 00:30:46.460 So there is some universal information 00:30:46.460 --> 00:30:50.210 that one gains beyond the non-universal location 00:30:50.210 --> 00:30:53.292 of the critical point, but not that much more. 00:30:57.861 --> 00:30:58.360 OK. 00:31:01.280 --> 00:31:03.498 Any questions? 00:31:03.498 --> 00:31:07.466 AUDIENCE: Is it possible to extract from this line 00:31:07.466 --> 00:31:08.954 a differential equation for g? 00:31:15.910 --> 00:31:17.110 PROFESSOR: Yes. 00:31:17.110 --> 00:31:22.710 And indeed, that differential relation 00:31:22.710 --> 00:31:24.280 you will use in one of the problem 00:31:24.280 --> 00:31:27.050 sets that I forgot to mention, and will 00:31:27.050 --> 00:31:30.900 be used to derive the value of the derivative, which 00:31:30.900 --> 00:31:34.196 is related to the energy of the system at the critical point. 00:31:34.196 --> 00:31:34.946 But you are right. 00:31:43.110 --> 00:31:45.710 This is such a beautiful thing that maybe we 00:31:45.710 --> 00:31:49.630 can try to force it to work in higher dimensions. 00:31:49.630 --> 00:31:52.775 So let's see if we were to try to go 00:31:52.775 --> 00:31:56.945 with this approach for the 3D Ising model what would happen. 00:32:02.885 --> 00:32:04.330 So what did we do? 00:32:04.330 --> 00:32:08.105 We wrote the low-temperature series, high-temperature series 00:32:08.105 --> 00:32:10.420 and compared them. 00:32:10.420 --> 00:32:14.980 Again, let's do the cubic lattice, 00:32:14.980 --> 00:32:17.690 which I will not really attempt to draw. 00:32:24.330 --> 00:32:29.090 That's the system that we want to calculate. 00:32:29.090 --> 00:32:30.990 So let's do the low T-series. 00:32:33.590 --> 00:32:37.830 Our partition function is going to start with the state 00:32:37.830 --> 00:32:41.010 where every spin is, let's say, up. 00:32:41.010 --> 00:32:43.720 Three bonds per site on the cubic lattice. 00:32:43.720 --> 00:32:46.520 So it's 3 NK. 00:32:46.520 --> 00:32:51.550 Again, the trivial degeneracy of 2 for the two possible all plus 00:32:51.550 --> 00:32:54.960 or all minus states. 00:32:54.960 --> 00:32:57.700 The first excitation is to flip a spin. 00:32:57.700 --> 00:33:02.526 So any one of N sites could have been flipped, creating a cube. 00:33:02.526 --> 00:33:08.595 A cube has 6 faces that go out, so there is essentially 00:33:08.595 --> 00:33:10.460 6 bonds that are broken. 00:33:10.460 --> 00:33:13.420 So basically, there is this minus 00:33:13.420 --> 00:33:17.750 that is in a box surrounded by plus. 00:33:17.750 --> 00:33:24.240 And as you can see, 6 plus minus 1 that go out of that. 00:33:24.240 --> 00:33:27.290 The next one would be when we have 2 minuses. 00:33:30.280 --> 00:33:36.001 And that can be oriented three ways in three dimensions. 00:33:36.001 --> 00:33:39.410 e to the minus 2K. 00:33:39.410 --> 00:33:41.220 1, 2 times 4. 00:33:41.220 --> 00:33:42.160 8 plus 2. 00:33:42.160 --> 00:33:42.900 Times 10. 00:33:45.630 --> 00:33:49.460 And so the general term in the series I 00:33:49.460 --> 00:33:59.100 have to draw some droplet of minuses in a sea of pluses. 00:33:59.100 --> 00:34:04.420 And then I would have e to the minus 2K times the boundary 00:34:04.420 --> 00:34:06.014 or the area of this droplet. 00:34:10.100 --> 00:34:12.380 Actually, droplets because there could 00:34:12.380 --> 00:34:15.179 be multiple droplets as we've seen. 00:34:15.179 --> 00:34:16.766 There's no problem with that. 00:34:20.933 --> 00:34:25.949 If I do the high T, follow exactly the procedure 00:34:25.949 --> 00:34:29.489 I had described before, partition function is going 00:34:29.489 --> 00:34:31.875 to be 2 to the number of sites. 00:34:31.875 --> 00:34:36.179 Cosh K to the power of the number of bonds and there 00:34:36.179 --> 00:34:38.540 are 3 bonds per site. 00:34:38.540 --> 00:34:41.250 So there's 3N there. 00:34:41.250 --> 00:34:45.739 And then we start to draw our diagrams. 00:34:45.739 --> 00:34:47.860 The first diagram is just exactly 00:34:47.860 --> 00:34:49.550 like what we had before. 00:34:49.550 --> 00:34:51.933 I have to make a square. 00:34:56.290 --> 00:35:04.760 And this square can be placed on any face of the cube. 00:35:04.760 --> 00:35:09.950 And there are 3 faces that are equivalent. 00:35:09.950 --> 00:35:14.810 The next type of diagram that I can draw has 6 bonds in it. 00:35:14.810 --> 00:35:18.940 So this could be an example of that. 00:35:18.940 --> 00:35:24.710 And if you do the counting, there are 18N of those. 00:35:24.710 --> 00:35:26.790 And so you go. 00:35:26.790 --> 00:35:28.760 And the genetic term in the series 00:35:28.760 --> 00:35:31.965 is going to be some find of a loop. 00:35:31.965 --> 00:35:37.860 Again, even number of bonds per site is the operative term. 00:35:37.860 --> 00:35:46.050 And then I have t to the power of the number of bonds 00:35:46.050 --> 00:35:48.478 making this closed loop. 00:35:48.478 --> 00:35:49.390 Or loops. 00:35:54.970 --> 00:35:56.840 So you stare at the series and you 00:35:56.840 --> 00:35:59.790 see immediately that there is no correspondence 00:35:59.790 --> 00:36:00.910 like we saw before. 00:36:00.910 --> 00:36:04.280 The coefficient here are 1N, 3N. 00:36:04.280 --> 00:36:06.480 Here, there are 1, 3, and 18N. 00:36:06.480 --> 00:36:08.580 Powers are 6, 10. 00:36:08.580 --> 00:36:10.520 Here are 4, 6. 00:36:10.520 --> 00:36:14.070 There's no correspondence between these two. 00:36:14.070 --> 00:36:18.770 So there is nothing that one could say. 00:36:18.770 --> 00:36:21.520 But you say, I really like this. 00:36:21.520 --> 00:36:26.590 So maybe I'll phrase the question differently. 00:36:26.590 --> 00:36:31.170 Can I consider some other model whose 00:36:31.170 --> 00:36:34.845 high-temperature expansion reproduces 00:36:34.845 --> 00:36:41.120 this low-temperature expansion of the Ising model? 00:36:41.120 --> 00:36:46.430 So this is the question, can we find 00:36:46.430 --> 00:36:59.640 a model whose high T expansion reproduces 00:36:59.640 --> 00:37:03.210 low T of 3D Ising model? 00:37:10.260 --> 00:37:15.140 So rather than knowing what the model is, 00:37:15.140 --> 00:37:18.040 now we are going to kind of work backward 00:37:18.040 --> 00:37:22.490 from this graphical picture that we have. 00:37:22.490 --> 00:37:28.440 So what would have been the analog thing over here, 00:37:28.440 --> 00:37:33.880 let's say that I had this picture of droplets 00:37:33.880 --> 00:37:36.050 in the 2D Ising model. 00:37:36.050 --> 00:37:41.000 I recognize that I need to make these perimeters out 00:37:41.000 --> 00:37:42.640 of something. 00:37:42.640 --> 00:37:49.170 And I know that I can make these things that are joined together 00:37:49.170 --> 00:37:55.820 to a procedure such as the one that we have over here. 00:37:55.820 --> 00:38:01.410 But the unit thing, there it was the elements 00:38:01.410 --> 00:38:04.280 that I had along the perimeter. 00:38:04.280 --> 00:38:06.580 What is the corresponding unit that I 00:38:06.580 --> 00:38:13.640 have for the low temperature series of the Ising model? 00:38:13.640 --> 00:38:21.350 I have e to the minus 2K to the power of the number of faces. 00:38:21.350 --> 00:38:25.140 So first thing is unit has to be a face. 00:38:31.970 --> 00:38:35.430 So basically, what I need to do is 00:38:35.430 --> 00:38:44.970 to have a series, which is an expansion in terms of faces, 00:38:44.970 --> 00:38:49.140 and then somehow I can glue these faces together, 00:38:49.140 --> 00:38:51.825 like I glued these bonds together. 00:38:54.820 --> 00:38:59.740 So we found our unit. 00:38:59.740 --> 00:39:02.910 The next thing that we need is some kind of a glue 00:39:02.910 --> 00:39:07.120 to put all of these LEGO faces together. 00:39:07.120 --> 00:39:10.400 So how did we join things together here? 00:39:10.400 --> 00:39:16.660 We had these sigmas that were sitting by themselves. 00:39:16.660 --> 00:39:18.970 And then putting two sigmas together, 00:39:18.970 --> 00:39:22.970 I ensured that when I summed over sigma, 00:39:22.970 --> 00:39:27.170 I had to glue two of the T's together. 00:39:27.170 --> 00:39:29.720 Can I do the same thing over here? 00:39:29.720 --> 00:39:34.100 If I put the sigmas on the corners of these faces, 00:39:34.100 --> 00:39:39.140 you can see it doesn't work because here I have three. 00:39:39.140 --> 00:39:49.010 So I'm forced to put the sigmas on the lines that 00:39:49.010 --> 00:39:49.982 join the faces. 00:39:52.640 --> 00:39:56.030 So what I need to do is, therefore, 00:39:56.030 --> 00:40:00.510 to have a variable such as this where 00:40:00.510 --> 00:40:07.490 I have these sigmas sitting on the-- let's call 00:40:07.490 --> 00:40:10.270 this a plaquette, p. 00:40:10.270 --> 00:40:12.680 And this plaquette will be having 00:40:12.680 --> 00:40:16.485 around it four different bonds. 00:40:19.330 --> 00:40:24.071 And if I have the product of these four bonds-- 00:40:24.071 --> 00:40:29.311 again, these sigmas being minus plus 1-- 00:40:29.311 --> 00:40:34.380 I am forced to glue these sigmas in pairs. 00:40:34.380 --> 00:40:37.640 And I can join these things together, these squares, 00:40:37.640 --> 00:40:41.520 to make whatever shape that I like that would correspond 00:40:41.520 --> 00:40:45.060 to the shapes that I have over there. 00:40:45.060 --> 00:40:52.620 So what I need to do is to have for each face a factor of this. 00:41:01.137 --> 00:41:07.330 So this is the analog of this factor that I have over here. 00:41:07.330 --> 00:41:10.960 And then what I need to do is to do 00:41:10.960 --> 00:41:15.310 a product over all plaquettes. 00:41:15.310 --> 00:41:18.820 And I sum over all sigma tildes. 00:41:21.800 --> 00:41:27.112 And this would be the partition function of some other system. 00:41:27.112 --> 00:41:30.210 In this other system, you can see 00:41:30.210 --> 00:41:32.910 that if I make its expansion, there 00:41:32.910 --> 00:41:35.610 will be a one-to-one correspondence 00:41:35.610 --> 00:41:38.860 between the terms in the expansion of this partition 00:41:38.860 --> 00:41:42.857 function and the 3D Ising model partition function. 00:41:49.190 --> 00:41:51.960 Again, this kind of term we have seen. 00:41:51.960 --> 00:41:55.230 If I had put factors of cosh here, which don't really 00:41:55.230 --> 00:42:07.330 do much, I can re-express as e to the something-- k tilde 00:42:07.330 --> 00:42:13.140 sigma 1p sigma 2p sigma 3p sigma 4p. 00:42:13.140 --> 00:42:16.360 Essentially every time you see 1 plus t times 00:42:16.360 --> 00:42:19.814 some binary variable, you can rewrite it into this fashion. 00:42:23.190 --> 00:42:28.510 So what we have come up with is the following. 00:42:28.510 --> 00:42:32.390 That in order to construct the dual 00:42:32.390 --> 00:42:35.830 of the three-dimensional Ising model, what we do 00:42:35.830 --> 00:42:39.980 is you go all over your cube. 00:42:39.980 --> 00:42:46.630 On each bond of it, you put a variable that is minus plus 1. 00:42:46.630 --> 00:42:49.060 So previously for the Ising model, 00:42:49.060 --> 00:42:52.350 the variables were sitting on the sites. 00:42:52.350 --> 00:42:59.030 So Ising, these were site variables. 00:42:59.030 --> 00:43:05.200 Whereas, this dual Ising, these are the bond variables 00:43:05.200 --> 00:43:06.920 that are minus plus 1. 00:43:10.870 --> 00:43:14.990 In the case of the Ising, the interactions 00:43:14.990 --> 00:43:20.410 were the product of sites making on a bond. 00:43:20.410 --> 00:43:24.010 Whereas, for the dual Ising, the interactions 00:43:24.010 --> 00:43:25.530 are around the face. 00:43:25.530 --> 00:43:28.274 There's four of them that go around the face. 00:43:35.780 --> 00:43:42.720 But whatever this new theory is, we 00:43:42.720 --> 00:43:46.960 know that its free energy because of this relation 00:43:46.960 --> 00:43:50.375 is related to the free energy of the three-dimensional Ising 00:43:50.375 --> 00:43:50.874 model. 00:43:53.540 --> 00:43:56.030 Also, we know that the three-dimensional Ising model 00:43:56.030 --> 00:43:59.640 has a phase transition between a disordered phase 00:43:59.640 --> 00:44:03.290 and the magnetized phase at low temperature. 00:44:03.290 --> 00:44:05.110 There is a singularity. 00:44:05.110 --> 00:44:09.470 As I span the parameter K of the three-dimensional Ising model, 00:44:09.470 --> 00:44:11.206 there is a Kc. 00:44:11.206 --> 00:44:14.210 Now, I can find out what that Kc is 00:44:14.210 --> 00:44:16.380 because I don't have self-duality. 00:44:16.380 --> 00:44:20.830 But I know that as I span the parameter K of the Ising model, 00:44:20.830 --> 00:44:26.020 I'm also spanning the parameter K tilde of this new theory. 00:44:26.020 --> 00:44:29.230 And since the original model has a phase transition, 00:44:29.230 --> 00:44:33.030 this new model must also have a phase transition. 00:44:33.030 --> 00:44:35.840 So there exists a Kc for both models. 00:44:43.470 --> 00:44:44.210 You say, OK. 00:44:44.210 --> 00:44:46.185 Fine. 00:44:46.185 --> 00:44:49.150 But there is some complicated kind of Ising model 00:44:49.150 --> 00:44:53.830 that you have devised and it has a phase transition. 00:44:53.830 --> 00:44:55.810 What's the big deal? 00:44:55.810 --> 00:44:57.980 Well, the big deal is that this model is not 00:44:57.980 --> 00:45:03.690 supposed to have a phase transition because it 00:45:03.690 --> 00:45:07.060 has a different type of symmetry. 00:45:07.060 --> 00:45:09.740 The symmetry that we have for the Ising model 00:45:09.740 --> 00:45:13.490 is a global symmetry. 00:45:13.490 --> 00:45:23.660 That is, the energy of a particular state 00:45:23.660 --> 00:45:28.080 is the energy of the state in which all of the spins 00:45:28.080 --> 00:45:29.910 are reversed. 00:45:29.910 --> 00:45:33.240 Because the form of the energy is bilinear. 00:45:33.240 --> 00:45:36.700 If I take all of the sigmas from one configuration 00:45:36.700 --> 00:45:39.390 and make them minus in that configuration, 00:45:39.390 --> 00:45:41.680 the energy will not change. 00:45:41.680 --> 00:45:43.410 But I have to do that globally. 00:45:43.410 --> 00:45:46.380 It's a global symmetry. 00:45:46.380 --> 00:45:52.660 Now, this model has a local symmetry because what I can do 00:45:52.660 --> 00:45:56.310 is I can pick one of the sites. 00:45:56.310 --> 00:46:00.810 And out of this site, there are six off 00:46:00.810 --> 00:46:04.200 these bonds that are going out on which 00:46:04.200 --> 00:46:07.400 there is one of these sigma tilde. 00:46:07.400 --> 00:46:12.830 If I pick this site and I change the sign of all of these six 00:46:12.830 --> 00:46:19.470 that emanate from this site, the energy will not change. 00:46:19.470 --> 00:46:23.630 Because the energy gets contributions from faces. 00:46:23.630 --> 00:46:26.910 And you can see that for any one of the faces, 00:46:26.910 --> 00:46:30.220 there are two sigmas that have changed. 00:46:30.220 --> 00:46:33.110 So the energy, which is the product of all four of them, 00:46:33.110 --> 00:46:33.860 has not changed. 00:46:37.160 --> 00:46:41.920 So this model has a different form, 00:46:41.920 --> 00:46:44.730 which is a local symmetry. 00:46:44.730 --> 00:46:49.640 And in fact, it is very much related to gauge theories. 00:46:49.640 --> 00:46:54.300 It's a kind of discrete version of the gauge theories 00:46:54.300 --> 00:46:56.970 that you have seen in electromagnetism. 00:46:56.970 --> 00:46:58.550 Since there are two possibilities, 00:46:58.550 --> 00:47:00.240 it's sometimes called a Z2 gauge theory. 00:47:04.160 --> 00:47:09.760 Now, the thing about the gauge theories is that there is 00:47:09.760 --> 00:47:21.390 a theorem which states that local or these gauge theories, 00:47:21.390 --> 00:47:28.730 gauge symmetries, cannot be spontaneously broken. 00:47:39.570 --> 00:47:42.030 So for the case of the Ising model, 00:47:42.030 --> 00:47:46.130 we have this symmetry between sigma going to minus sigma. 00:47:46.130 --> 00:47:49.310 But yet, we know that if I go to low temperature, 00:47:49.310 --> 00:47:53.740 I will have a state in which globally all of the spins 00:47:53.740 --> 00:47:57.140 are either plus or minus. 00:47:57.140 --> 00:48:00.870 So there is a symmetry broken state 00:48:00.870 --> 00:48:04.340 which is what we have been discussing. 00:48:04.340 --> 00:48:08.210 Now, the reason that that cannot take place in these gauge 00:48:08.210 --> 00:48:11.490 theories, I will just sketch what is happening. 00:48:14.900 --> 00:48:16.570 Essentially, we have been thinking 00:48:16.570 --> 00:48:19.560 in terms of this broken symmetries 00:48:19.560 --> 00:48:24.590 by putting an infinitesimal magnetic field. 00:48:24.590 --> 00:48:29.930 And we saw that, basically, if I'm at temperatures of 1 00:48:29.930 --> 00:48:35.870 over K's that are below some critical value, 00:48:35.870 --> 00:48:40.090 then if h is plus, everybody would be plus. 00:48:40.090 --> 00:48:44.140 If h is minus, everybody would be minus. 00:48:44.140 --> 00:48:50.640 And the reason as you approach h goes to 0 from one site 00:48:50.640 --> 00:48:57.000 that you don't get average of 0 is because the difference 00:48:57.000 --> 00:48:59.610 between the energy of this state and that state 00:48:59.610 --> 00:49:05.640 as you go to 0 temperature is proportional to N times h. 00:49:05.640 --> 00:49:09.600 So although h is going to 0 with N being very large, 00:49:09.600 --> 00:49:15.800 the influence of infinitesimal h is magnified enormously. 00:49:15.800 --> 00:49:19.680 Now, for the case of these local gauge theories, 00:49:19.680 --> 00:49:22.150 you cannot have a similar argument. 00:49:22.150 --> 00:49:26.790 Because if I pick this spin, let's say one of these bond 00:49:26.790 --> 00:49:28.490 spins. 00:49:28.490 --> 00:49:35.440 And let's say is its average-- what is its average? 00:49:35.440 --> 00:49:38.382 As I said, the h going to 0. 00:49:38.382 --> 00:49:42.270 Well, the difference between a state 00:49:42.270 --> 00:49:46.010 in which it is plus or the state in which it is minus 00:49:46.010 --> 00:49:47.650 is, in fact, 6h. 00:49:47.650 --> 00:49:49.670 Because all I need to do is to pick 00:49:49.670 --> 00:49:52.180 a site that is close to that bond 00:49:52.180 --> 00:49:55.060 and flip all of the spins that are close to that. 00:49:55.060 --> 00:49:58.230 All of the K's are equally satisfied. 00:49:58.230 --> 00:49:59.890 The difference between that state 00:49:59.890 --> 00:50:04.030 and the one where there is a flip is just 6h. 00:50:04.030 --> 00:50:06.460 So that remains finite as h goes to 0. 00:50:06.460 --> 00:50:11.480 There is no barrier towards flipping those spins. 00:50:11.480 --> 00:50:16.400 So there is no broken symmetry in this system. 00:50:16.400 --> 00:50:21.810 So this can be proven very nicely and rigorously. 00:50:21.810 --> 00:50:28.670 So we have now two statements about this Ising 00:50:28.670 --> 00:50:30.330 version of a gauge theory. 00:50:30.330 --> 00:50:34.480 First of all, we know that at low temperatures, 00:50:34.480 --> 00:50:38.150 still the average value of each bond 00:50:38.150 --> 00:50:41.730 is equally likely to be plus or minus. 00:50:41.730 --> 00:50:46.430 From that perspective of local values of these bond spins, 00:50:46.430 --> 00:50:50.710 it is as disordered as the highest-temperature phase. 00:50:50.710 --> 00:50:54.990 Yet, because of its duality to the three-dimensional Ising 00:50:54.990 --> 00:50:58.650 model, we know that it undergoes some kind 00:50:58.650 --> 00:51:01.785 of a singularity going from high temperatures 00:51:01.785 --> 00:51:03.450 to low temperatures. 00:51:03.450 --> 00:51:07.070 So there is probably some kind of a phase transition, 00:51:07.070 --> 00:51:09.930 but it has to be very different from any of the phase 00:51:09.930 --> 00:51:13.270 transitions that we have discussed so far because there 00:51:13.270 --> 00:51:18.440 is no spontaneous symmetry breaking. 00:51:18.440 --> 00:51:20.200 So what's going on? 00:52:00.750 --> 00:52:04.720 Now, later on in the course, we will 00:52:04.720 --> 00:52:09.520 see another example of this that is much less 00:52:09.520 --> 00:52:12.480 exotic than a gauge theory, but it 00:52:12.480 --> 00:52:16.010 has the same kind of principle applicable to it. 00:52:16.010 --> 00:52:20.690 There will be a phase transition without local symmetry breaking 00:52:20.690 --> 00:52:26.930 in something like a superfluid in two dimensions. 00:52:26.930 --> 00:52:30.820 So one thing that that phase transition and this one 00:52:30.820 --> 00:52:34.220 have in common is, again, the lack 00:52:34.220 --> 00:52:42.190 of this local-order parameter from symmetry breaking. 00:52:42.190 --> 00:52:46.560 And both of them share something that 00:52:46.560 --> 00:52:52.050 was pointed out by Wegner once this puzzle emerged, 00:52:52.050 --> 00:53:01.510 which is that one has to look at some kind of a global-- well, 00:53:01.510 --> 00:53:03.030 I shouldn't even call it global. 00:53:08.016 --> 00:53:09.890 It is something that is called a Wilson loop. 00:53:26.760 --> 00:53:29.440 So the idea is the following. 00:53:29.440 --> 00:53:32.580 That we have these variable sigma tilde 00:53:32.580 --> 00:53:36.280 that are sitting on the bonds of a lattice. 00:53:36.280 --> 00:53:42.810 Now, the problem is that with this local transformations, 00:53:42.810 --> 00:53:46.740 I can very easily make this sigma go to minus sigma. 00:53:46.740 --> 00:53:52.410 So that is not a good thing to consider. 00:53:52.410 --> 00:53:56.680 However, what was the problem? 00:53:56.680 --> 00:53:58.390 Let's say I pick this site. 00:53:58.390 --> 00:54:00.820 All of the sigmas that went out of that site 00:54:00.820 --> 00:54:03.030 I changed to minus themselves. 00:54:03.030 --> 00:54:04.700 That was the gauge transformation, 00:54:04.700 --> 00:54:07.280 but it became minus. 00:54:07.280 --> 00:54:12.950 But if I multiply the sigma with another sigma that goes out 00:54:12.950 --> 00:54:16.450 of that site, then I have cured that problem. 00:54:16.450 --> 00:54:19.750 If I changed this to minus itself, 00:54:19.750 --> 00:54:21.840 this changes, this changes. 00:54:21.840 --> 00:54:25.000 The product remains [INAUDIBLE]. 00:54:25.000 --> 00:54:27.100 But then I have the problem here. 00:54:27.100 --> 00:54:34.410 So what I do is I make a long loop. 00:54:34.410 --> 00:54:38.060 I look at the expectation value. 00:54:38.060 --> 00:54:41.820 So this Wilson loop is the product 00:54:41.820 --> 00:54:46.830 of sigma tilde around a loop. 00:54:51.720 --> 00:54:56.720 And what I can do is I can look at the average 00:54:56.720 --> 00:54:58.304 of that quantity. 00:54:58.304 --> 00:55:01.540 The average of that quantity is something 00:55:01.540 --> 00:55:05.260 that is clearly invariant to this kind of gauge 00:55:05.260 --> 00:55:06.770 transformation. 00:55:06.770 --> 00:55:10.610 So the signatures of a potential phase transition 00:55:10.610 --> 00:55:14.580 could potentially be revealed by looking at something like this. 00:55:17.680 --> 00:55:19.880 But clearly, that is a quantity that 00:55:19.880 --> 00:55:24.030 is always also going to be positive. 00:55:24.030 --> 00:55:25.990 So the thing that I am looking at 00:55:25.990 --> 00:55:29.627 is not that this quantity is, say, positive 00:55:29.627 --> 00:55:32.800 in one phase and 0 in the other phase. 00:55:32.800 --> 00:55:36.790 It is on how this quantity depends 00:55:36.790 --> 00:55:40.345 on the shape and characteristics of these loop. 00:55:42.960 --> 00:55:47.950 So what I can do is I can calculate this average, both 00:55:47.950 --> 00:55:53.410 in high temperatures and low temperatures, and compare them. 00:55:53.410 --> 00:56:05.193 So if I look at-- let's starts with, yeah, high temperatures. 00:56:05.193 --> 00:56:09.500 The high-temperature expansion. 00:56:09.500 --> 00:56:12.430 So I want to calculate the expectation 00:56:12.430 --> 00:56:18.550 value of the product of sigma tilde, let's call it i, 00:56:18.550 --> 00:56:22.580 where i belongs to some kind of a loop c. 00:56:22.580 --> 00:56:27.370 So c is all of these bonds. 00:56:27.370 --> 00:56:31.210 I want to calculate that expectation value. 00:56:31.210 --> 00:56:34.240 How do I calculate that expectation value? 00:56:34.240 --> 00:56:40.440 Well, I have to sum over all configurations 00:56:40.440 --> 00:56:48.900 of this product with a weight. 00:56:48.900 --> 00:56:50.760 What is my weight? 00:56:50.760 --> 00:56:55.800 My weight is this factor of product 00:56:55.800 --> 00:57:02.202 over all plaquettes of 1 plus t sigma tilde sigma tilde sigma 00:57:02.202 --> 00:57:05.191 tilde for the plaquette. 00:57:11.090 --> 00:57:13.400 So this is the weight that I have. 00:57:13.400 --> 00:57:16.560 I can put the hyperbolic cosine or not put it, 00:57:16.560 --> 00:57:18.970 it doesn't matter. 00:57:18.970 --> 00:57:21.715 But then this weight has to be properly normalized. 00:57:21.715 --> 00:57:25.510 It means that I have to divide by something 00:57:25.510 --> 00:57:28.830 in the denominator, which basically does not 00:57:28.830 --> 00:57:31.461 include the quantity that I am averaging. 00:57:46.400 --> 00:57:50.080 So the graphs that occur in the denominator 00:57:50.080 --> 00:57:52.960 are the things that we have been discussing. 00:57:52.960 --> 00:57:56.020 Essentially, I start with 1. 00:57:56.020 --> 00:57:59.520 The next term is to put these faces together 00:57:59.520 --> 00:58:02.860 to make a cube, and then more complicated shapes. 00:58:02.860 --> 00:58:05.520 That, essentially, every bond is going 00:58:05.520 --> 00:58:09.194 to be having some kind of a complement. 00:58:09.194 --> 00:58:12.280 Well, but what about the terms in the numerator? 00:58:12.280 --> 00:58:14.550 For the terms in the numerator, I 00:58:14.550 --> 00:58:17.970 have these factors of sigma tilde 00:58:17.970 --> 00:58:22.410 that are lying all around this loop. 00:58:22.410 --> 00:58:25.540 And I'm summing over the two possible values. 00:58:25.540 --> 00:58:30.610 So in order that summing over this does not give me 0, 00:58:30.610 --> 00:58:34.970 I better make sure that there is a complement to that. 00:58:34.970 --> 00:58:38.630 The complement to that can only come from here. 00:58:38.630 --> 00:58:42.610 So for example, I would put a face over here 00:58:42.610 --> 00:58:46.760 that ensures that that is squared and that is squared. 00:58:46.760 --> 00:58:48.305 But then I have this one. 00:58:48.305 --> 00:58:49.840 Then I will put another one here. 00:58:49.840 --> 00:58:52.155 I will put another one, et cetera. 00:58:55.400 --> 00:59:00.240 And you can see that the lowest-order term that I would 00:59:00.240 --> 00:59:08.110 get is this factor that characterize each one of them. 00:59:10.870 --> 00:59:15.688 Raised to the power of the area of this loop c. 00:59:19.150 --> 00:59:21.680 You can ask higher-order terms. 00:59:21.680 --> 00:59:25.550 You can kind of build a hat on top of this. 00:59:25.550 --> 00:59:27.790 This you can put anywhere, again, 00:59:27.790 --> 00:59:30.530 along this area of this thing. 00:59:30.530 --> 00:59:33.280 So the next correction in this series you can see 00:59:33.280 --> 00:59:37.450 is also going to be something that will be 1 plus t 00:59:37.450 --> 00:59:40.020 to the fourth times the area. 00:59:40.020 --> 00:59:43.880 The point is that as you add more and more terms, 00:59:43.880 --> 00:59:48.410 you preserve the structure that the whole thing is going 00:59:48.410 --> 00:59:53.180 to be proportional to the area of loop 00:59:53.180 --> 00:59:56.484 times some function of this parameter t. 01:00:01.180 --> 01:00:07.300 So what we know is that if I take the expectation 01:00:07.300 --> 01:00:11.340 value of this entity, then it's logarithm 01:00:11.340 --> 01:00:14.470 will be proportional in high temperatures 01:00:14.470 --> 01:00:16.144 to the area of this loop. 01:00:21.470 --> 01:00:25.680 Now, what happens if I try to do a low T-series 01:00:25.680 --> 01:00:26.700 for the same quantity? 01:00:34.090 --> 01:00:38.530 So I have to start with a configuration 01:00:38.530 --> 01:00:45.850 at low temperature that minimizes the energy. 01:00:50.930 --> 01:00:55.200 One configuration clearly is one where all of the sigmas 01:00:55.200 --> 01:00:55.970 are plus. 01:00:59.730 --> 01:01:03.570 And that will give me a term. 01:01:03.570 --> 01:01:07.180 If I am calculating the partition function 01:01:07.180 --> 01:01:09.630 in the denominator, there will be 01:01:09.630 --> 01:01:13.310 a term that will be proportional to e to whatever 01:01:13.310 --> 01:01:18.720 this K tilde is per face. 01:01:18.720 --> 01:01:22.112 And there are three faces of a cube. 01:01:22.112 --> 01:01:23.570 There are N cubed, so there will be 01:01:23.570 --> 01:01:30.210 3K tilde N for the configuration that is all plus. 01:01:30.210 --> 01:01:34.290 But this is not the only low-temperature configuration. 01:01:34.290 --> 01:01:36.470 That is what we were discussing. 01:01:36.470 --> 01:01:40.780 Because I can pick a site out of N site 01:01:40.780 --> 01:01:42.630 and the 6 bonds that go out of it, 01:01:42.630 --> 01:01:45.110 I can make minus themselves. 01:01:45.110 --> 01:01:48.020 And the energy would be exactly the same. 01:01:48.020 --> 01:01:53.650 So whereas the Ising model, I had a multiplicity of 2. 01:01:53.650 --> 01:02:01.670 Here, there is a multiplicity of 2 to the N. 01:02:01.670 --> 01:02:06.950 So that's the lowest term in the low-temperature expansion 01:02:06.950 --> 01:02:08.500 of the partition function. 01:02:08.500 --> 01:02:10.090 I'm doing the partition function, 01:02:10.090 --> 01:02:11.381 which is the denominator first. 01:02:13.770 --> 01:02:16.090 Then, what can I do? 01:02:16.090 --> 01:02:19.520 Then, let's say I start with a configuration where all of them 01:02:19.520 --> 01:02:20.930 are pluses. 01:02:20.930 --> 01:02:23.450 There are, of course, 2 to the N gauge copies of that. 01:02:23.450 --> 01:02:25.780 So whatever I do to this configuration, 01:02:25.780 --> 01:02:28.610 I can do the analog in all of the others. 01:02:28.610 --> 01:02:32.740 But let's keep the copy where all of the sigmas are plus, 01:02:32.740 --> 01:02:37.270 and then I flip one of the sigmas to minus. 01:02:37.270 --> 01:02:40.190 Then, it's essentially-- think of a cube. 01:02:40.190 --> 01:02:43.850 There is a line that was plus and I made it minus. 01:02:43.850 --> 01:02:48.840 There are four faces going out of that that were previously 01:02:48.840 --> 01:02:52.920 plus K, now become minus K. So I will 01:02:52.920 --> 01:02:58.890 have 2K tilde times 4 because of these four things 01:02:58.890 --> 01:03:01.270 that are going out. 01:03:01.270 --> 01:03:04.966 And the bond I can orient in x-, y-, or z-direction. 01:03:04.966 --> 01:03:09.000 So there are 3N possibilities. 01:03:09.000 --> 01:03:13.750 And so I could have a series such as this in the denominator 01:03:13.750 --> 01:03:17.660 where subsequent terms would be to put more and more 01:03:17.660 --> 01:03:19.390 minus in this particular [INAUDIBLE]. 01:03:22.100 --> 01:03:26.670 Now, let's see how these series would affect the sum 01:03:26.670 --> 01:03:31.160 that I would have to do in order to calculate this expectation 01:03:31.160 --> 01:03:32.660 value. 01:03:32.660 --> 01:03:39.900 For any one of these configurations, 01:03:39.900 --> 01:03:42.330 since I am kind of looking at the ground state, 01:03:42.330 --> 01:03:45.120 let's say they're all pluses. 01:03:45.120 --> 01:03:51.270 Clearly, the contribution to this product will be unity. 01:03:51.270 --> 01:03:53.870 That does not change. 01:03:53.870 --> 01:03:56.540 But now, let's think about the configurations 01:03:56.540 --> 01:04:01.260 in which one of the bonds is made to flip. 01:04:01.260 --> 01:04:05.320 As long as that bond does not touch 01:04:05.320 --> 01:04:09.340 any of the bonds that are part of the loop, 01:04:09.340 --> 01:04:12.540 the value of the loop will remain the same. 01:04:12.540 --> 01:04:16.280 So let's say that the loop has P sites. 01:04:16.280 --> 01:04:22.590 So this is number of bonds in c. 01:04:22.590 --> 01:04:23.953 Let's call it Pc. 01:04:28.590 --> 01:04:38.047 For these, with this weight, the value of Wilson loop 01:04:38.047 --> 01:04:38.880 would still be plus. 01:04:41.520 --> 01:04:45.050 But for the times where I have picked one of these 01:04:45.050 --> 01:04:50.180 to become minus, then the product becomes minus. 01:04:50.180 --> 01:04:55.730 So for the remaining Pc times of this factor, 01:04:55.730 --> 01:04:59.390 e to the minus 2K tilde times 4, rather 01:04:59.390 --> 01:05:00.870 than having plus I will have minus. 01:05:06.410 --> 01:05:09.870 So you can see that-- what is this? 01:05:09.870 --> 01:05:11.190 N should be up here. 01:05:13.860 --> 01:05:20.340 That the difference between what is in the numerator 01:05:20.340 --> 01:05:25.480 and what is in the denominator of this low-temperature series 01:05:25.480 --> 01:05:29.200 has to do with the bonds that have 01:05:29.200 --> 01:05:34.070 been sitting as part of the Wilson loop. 01:05:34.070 --> 01:05:38.890 And if I imagine that this is a small quantity 01:05:38.890 --> 01:05:41.520 and write these as exponentials, you 01:05:41.520 --> 01:05:44.210 can see that this is going to start 01:05:44.210 --> 01:05:49.960 with e to the minus 2K tilde times 4 times 01:05:49.960 --> 01:05:53.605 the perimeter of this cluster. 01:05:53.605 --> 01:05:54.960 Of this loop. 01:05:57.680 --> 01:06:02.250 And you can go and look at higher and higher order terms. 01:06:02.250 --> 01:06:06.670 The point is that in high temperature, 01:06:06.670 --> 01:06:11.710 the property of the shape of the loop that determines 01:06:11.710 --> 01:06:15.806 this expectation value is its area. 01:06:15.806 --> 01:06:20.987 Whereas, in the low-temperature expansion, it is its perimeter. 01:06:25.090 --> 01:06:27.390 So you could, for example, calculate 01:06:27.390 --> 01:06:33.240 the-- for a large loop, the log of this quantity 01:06:33.240 --> 01:06:36.680 and divide it by the perimeter. 01:06:36.680 --> 01:06:39.910 And in the low-temperature phase, it would be finite. 01:06:39.910 --> 01:06:41.755 In the high-temperature phase, it 01:06:41.755 --> 01:06:44.055 would go to 0 because the area scale 01:06:44.055 --> 01:06:46.580 is bigger than the perimeter. 01:06:46.580 --> 01:06:49.900 So we have found something that is an analog of an older 01:06:49.900 --> 01:06:53.930 parameter and can distinguish the different phases, 01:06:53.930 --> 01:06:57.144 and it is reflected in the way that the correlations take 01:06:57.144 --> 01:06:57.644 place. 01:07:33.710 --> 01:07:38.730 Let's try to sort of think about some potential physics that 01:07:38.730 --> 01:07:41.860 could be related to this. 01:07:41.860 --> 01:07:47.230 Let's start with the gauge theory aspect of this. 01:07:47.230 --> 01:07:51.210 Well, the one gauge theory that you probably know 01:07:51.210 --> 01:07:56.010 is quantum electrodynamics, whose action 01:07:56.010 --> 01:07:59.540 you would write in the following way. 01:07:59.540 --> 01:08:02.845 The action would involve an integration over space 01:08:02.845 --> 01:08:05.760 as well as time. 01:08:05.760 --> 01:08:10.380 And by appropriate rescalings, you 01:08:10.380 --> 01:08:14.000 can write the energy that is in the electromagnetic field 01:08:14.000 --> 01:08:20.899 as d mu A mu minus d mu A mu squared, where 01:08:20.899 --> 01:08:33.450 A is the 4-vector potential out of which you can construct 01:08:33.450 --> 01:08:37.930 the electric field and magnetic field. 01:08:37.930 --> 01:08:41.880 And the reason this is a gauge theory 01:08:41.880 --> 01:08:55.750 is because if I take A and add to it some function of phi 01:08:55.750 --> 01:09:01.399 of x and t, as long as I take the derivative 01:09:01.399 --> 01:09:04.660 of this function, you can see that the change here 01:09:04.660 --> 01:09:08.540 would be d mu d mu minus d mu by d mu. 01:09:08.540 --> 01:09:10.160 There is really no change. 01:09:10.160 --> 01:09:14.500 And we know that basically you can 01:09:14.500 --> 01:09:20.010 choose whatever value of this phase-- 01:09:20.010 --> 01:09:24.073 this gauge fixing potential over here. 01:09:27.321 --> 01:09:30.880 Now, this is the electromagnetic field by itself. 01:09:30.880 --> 01:09:34.479 If you want to couple it to something like matter 01:09:34.479 --> 01:09:37.420 or electrons, you write something 01:09:37.420 --> 01:09:46.300 like i d bar, which is some derivative, 01:09:46.300 --> 01:09:50.864 and then you have e A bar. 01:09:50.864 --> 01:09:55.254 If there's a mass to this object, you would put it here. 01:09:55.254 --> 01:09:58.320 This would be something like psi bar psi. 01:09:58.320 --> 01:10:01.380 This would be something that would describe 01:10:01.380 --> 01:10:05.510 the coupling of this electromagnetic field 01:10:05.510 --> 01:10:08.801 to some charged particle, such as the electron. 01:10:14.720 --> 01:10:19.680 And this entire thing satisfies the gauge symmetry provided 01:10:19.680 --> 01:10:24.420 that once you do this, you also replace psi with e 01:10:24.420 --> 01:10:30.260 to the ie phi psi. 01:10:30.260 --> 01:10:37.140 So the same phi, if it appears in both, essentially the change 01:10:37.140 --> 01:10:40.190 in A that you would have from here 01:10:40.190 --> 01:10:42.390 will be compensated by the change 01:10:42.390 --> 01:10:45.590 that you would get from the derivative acting on this phase 01:10:45.590 --> 01:10:46.760 factor. 01:10:46.760 --> 01:10:51.030 And so the whole thing is not affected. 01:10:51.030 --> 01:10:55.470 What we have constructed in this model 01:10:55.470 --> 01:10:59.550 is kind of an Ising analog of this theory. 01:10:59.550 --> 01:11:03.350 Because the Hamiltonian that we have, 01:11:03.350 --> 01:11:07.730 which carries the weight after exponentiate 01:11:07.730 --> 01:11:11.740 of the different configurations, has a part 01:11:11.740 --> 01:11:19.540 which is the sum over all of the plaquettes of this sigma tilde 01:11:19.540 --> 01:11:25.020 sigma tilde sigma tilde sigma tilde, 01:11:25.020 --> 01:11:28.910 the four bonds around the plaquette. 01:11:28.910 --> 01:11:32.870 We could put some kind of a coupling here if we want to. 01:11:32.870 --> 01:11:36.980 And the analog of this transformation 01:11:36.980 --> 01:11:41.650 that we have-- well, maybe it will become more apparent if I 01:11:41.650 --> 01:11:47.500 add the next term, which is the analog of the coupling 01:11:47.500 --> 01:11:48.290 to matter. 01:11:48.290 --> 01:11:55.110 If I put a spin and then sigma tilde ij sj. 01:11:55.110 --> 01:12:01.940 So again, imagine that we have our cubic lattice, 01:12:01.940 --> 01:12:07.100 or some other lattice, in which we have these variables sigma 01:12:07.100 --> 01:12:10.680 tilde that are sitting on the bonds. 01:12:10.680 --> 01:12:14.730 And the first term is the product around the face. 01:12:14.730 --> 01:12:17.840 And the second term I put these variables 01:12:17.840 --> 01:12:19.620 s that are sitting here. 01:12:24.770 --> 01:12:30.350 And I have made a coupling between these two s's. 01:12:30.350 --> 01:12:32.830 So if the sigma tildes were not there, 01:12:32.830 --> 01:12:36.730 I could make an Ising model with s's being plus or minus, 01:12:36.730 --> 01:12:40.080 which are coupled across nearest neighbors. 01:12:40.080 --> 01:12:42.830 What I do is that the strength of that coupling I 01:12:42.830 --> 01:12:47.010 make to be plus or minus, depending 01:12:47.010 --> 01:12:51.770 on the value of the gauge field, if you like. 01:12:51.770 --> 01:12:55.310 This Ising gauge field that is sitting over there. 01:12:55.310 --> 01:13:00.030 Now, the analog of these symmetries that we have for QED 01:13:00.030 --> 01:13:01.700 is as follows. 01:13:01.700 --> 01:13:08.860 I can pick a particular s i and change its value to minus. 01:13:08.860 --> 01:13:15.725 And I can pick all of the sigma tildes that go out of that i 01:13:15.725 --> 01:13:22.390 to the neighbours and simultaneously make them minus. 01:13:22.390 --> 01:13:25.580 And then this energy would not change. 01:13:25.580 --> 01:13:30.240 First of all, let's say if I pick this site 01:13:30.240 --> 01:13:33.970 and change its face from being plus or minus, 01:13:33.970 --> 01:13:38.795 then the bonds that-- the sigma tildes that go out of it 01:13:38.795 --> 01:13:42.460 will change their values to minus themselves. 01:13:42.460 --> 01:13:46.820 Since each one of the faces contains two of them, 01:13:46.820 --> 01:13:51.880 the value of the energy from here is not changed. 01:13:51.880 --> 01:13:57.400 Since the couplings to the neighboring s's involve 01:13:57.400 --> 01:13:59.800 the sigma tildes that sit between them, 01:13:59.800 --> 01:14:02.170 and I have flipped both s and sigma tilde, 01:14:02.170 --> 01:14:05.070 those do not change irrespective of what 01:14:05.070 --> 01:14:08.240 I do with the face of all the other ones. 01:14:08.240 --> 01:14:13.970 So I have made an Ising, or binary version, 01:14:13.970 --> 01:14:16.260 of this transformation, constructed 01:14:16.260 --> 01:14:21.280 a model that has, except being Ising symmetry, 01:14:21.280 --> 01:14:23.735 a lot of the properties that you would 01:14:23.735 --> 01:14:25.230 have for this kind of action. 01:14:29.930 --> 01:14:37.530 Now again, continuing with that, this difference between why 01:14:37.530 --> 01:14:42.080 we see an area rule or a perimeter rule 01:14:42.080 --> 01:14:47.600 has some physical consequence that is worth mentioning. 01:14:47.600 --> 01:14:51.430 And this, again, has not much to do 01:14:51.430 --> 01:14:53.700 with the main thrust of this course, 01:14:53.700 --> 01:14:57.690 but just a matter of overall education. 01:14:57.690 --> 01:14:59.290 It is useful to know. 01:15:02.500 --> 01:15:07.160 So in this picture, where one of the dimensions 01:15:07.160 --> 01:15:14.990 corresponds to time, imagine that you 01:15:14.990 --> 01:15:21.320 create a kind of a Wilson loop which 01:15:21.320 --> 01:15:25.640 is very long in one direction that I 01:15:25.640 --> 01:15:29.670 want to think of as being time. 01:15:29.670 --> 01:15:36.680 And the analog of this action that we have discussed 01:15:36.680 --> 01:15:43.425 is to create a pair of charges, separate them by a distance x, 01:15:43.425 --> 01:15:50.350 propagate them for a long time t, and then [INAUDIBLE]. 01:15:50.350 --> 01:15:53.430 And ask, what is the contribution of a configuration 01:15:53.430 --> 01:15:58.940 such as this to their action on average? 01:15:58.940 --> 01:16:06.420 And so you would say that if particles are at a distance x, 01:16:06.420 --> 01:16:11.130 they are subject to some kind of a potential v of x. 01:16:11.130 --> 01:16:13.720 And if this potential has been propagated 01:16:13.720 --> 01:16:17.090 in time for a length or a duration 01:16:17.090 --> 01:16:24.710 that I will call T, that the effect that it has 01:16:24.710 --> 01:16:33.910 on the system is to have an interaction such as this 01:16:33.910 --> 01:16:39.310 in the action propagated over a time T. 01:16:39.310 --> 01:16:42.274 So I should have something like this. 01:16:45.230 --> 01:16:48.730 So this should somehow be related 01:16:48.730 --> 01:16:54.540 to this average of the Wilson loop in manner 01:16:54.540 --> 01:16:58.530 to not be made precise, but very rough just 01:16:58.530 --> 01:17:01.190 to get the general idea. 01:17:01.190 --> 01:17:07.970 And so what we have said is that the value of this Wilson loop 01:17:07.970 --> 01:17:10.313 has different behaviors at high T 01:17:10.313 --> 01:17:15.780 and low T in its dependence on shape. 01:17:15.780 --> 01:17:20.480 And that high temperatures, it is proportional to the area. 01:17:20.480 --> 01:17:24.790 So it should be proportional to xT. 01:17:24.790 --> 01:17:28.820 Whereas, in low temperature, it is proportional to perimeter. 01:17:28.820 --> 01:17:36.600 So it should be proportional to x plus T. 01:17:36.600 --> 01:17:43.285 So if I read off the form of V of x from these two dependents, 01:17:43.285 --> 01:17:48.360 you will see that V of x goes proportionately 01:17:48.360 --> 01:17:53.660 to x in one regime. 01:17:53.660 --> 01:17:58.190 And once I divide by T, the leading 01:17:58.190 --> 01:18:01.130 coefficients-- so this is essentially 01:18:01.130 --> 01:18:07.460 I want to look at the limit where T is becoming very large. 01:18:07.460 --> 01:18:11.460 So this thing when I divide by T goes to a constant. 01:18:15.900 --> 01:18:19.720 And I can very roughly interpret this 01:18:19.720 --> 01:18:22.120 as the interaction between particles 01:18:22.120 --> 01:18:29.660 that are separated by x via this kind of theory. 01:18:29.660 --> 01:18:31.890 And I see that for particles that 01:18:31.890 --> 01:18:36.900 are separated by x in that kind of theory, 01:18:36.900 --> 01:18:42.070 there is a weak coupling-- high temperature 01:18:42.070 --> 01:18:49.430 corresponds to weak coupling-- where the further apart I 01:18:49.430 --> 01:18:55.310 go, the potential that is bringing them together 01:18:55.310 --> 01:18:59.510 becomes linearly stronger. 01:18:59.510 --> 01:19:01.490 So this is what is called confinement. 01:19:07.000 --> 01:19:13.970 Whereas, in the other limit of low temperatures, 01:19:13.970 --> 01:19:18.950 or strong coupling, what you find 01:19:18.950 --> 01:19:22.990 is that the interaction between them 01:19:22.990 --> 01:19:25.175 essentially goes to a constant. 01:19:25.175 --> 01:19:27.050 The potential goes to a constant, 01:19:27.050 --> 01:19:29.210 so the force would go to 0. 01:19:29.210 --> 01:19:31.720 So they are asymptotically free. 01:19:38.500 --> 01:19:43.800 So if I start with this kind of theory 01:19:43.800 --> 01:19:47.770 and try to interpret it in the language of quantum field 01:19:47.770 --> 01:19:52.050 theory as something that describes interaction 01:19:52.050 --> 01:19:57.690 between particles, I find that it has potentially two phases. 01:19:57.690 --> 01:20:01.110 One phase in which the particles of the theory 01:20:01.110 --> 01:20:04.950 are strongly bind together, like quarks 01:20:04.950 --> 01:20:07.830 that are inside the nucleus. 01:20:07.830 --> 01:20:09.940 And you can try to separate the quarks, 01:20:09.940 --> 01:20:11.390 but they would snap back. 01:20:11.390 --> 01:20:13.680 You can't have free quarks. 01:20:13.680 --> 01:20:19.120 And then there is another phase where essentially the particles 01:20:19.120 --> 01:20:20.930 don't see each other. 01:20:20.930 --> 01:20:24.810 And indeed, quarks right inside the nucleus 01:20:24.810 --> 01:20:26.260 are essentially free. 01:20:26.260 --> 01:20:29.430 We can sort of regard them as free particles. 01:20:29.430 --> 01:20:33.780 So this theory actually has aspects 01:20:33.780 --> 01:20:38.730 of what is known as confinement and asymptotic freedom 01:20:38.730 --> 01:20:41.550 within quantum chromodynamics. 01:20:41.550 --> 01:20:43.590 The difference is that in this theory, 01:20:43.590 --> 01:20:47.720 there is a phase transition and the two behaviors 01:20:47.720 --> 01:20:49.625 are separated from each other. 01:20:49.625 --> 01:20:53.590 Whereas in QCD, it's essentially a crossover 01:20:53.590 --> 01:20:55.980 from one behavior to another behavior 01:20:55.980 --> 01:20:58.790 without the phase transition. 01:20:58.790 --> 01:21:04.140 So we started by thinking about these Ising models. 01:21:04.140 --> 01:21:06.620 And we kind of branched into theories 01:21:06.620 --> 01:21:10.170 that describe loops, theories that describe droplets, 01:21:10.170 --> 01:21:13.930 theories that describe gauge couplings, et cetera. 01:21:13.930 --> 01:21:17.850 So you can see that that nice, simple line of the partition 01:21:17.850 --> 01:21:22.180 function that I have written for you has within it 01:21:22.180 --> 01:21:25.010 a lot of interesting complexity. 01:21:25.010 --> 01:21:27.870 We kind of went off the direction 01:21:27.870 --> 01:21:31.260 that we wanted to go with phase transitions, 01:21:31.260 --> 01:21:33.910 so we will remedy that next time, coming 01:21:33.910 --> 01:21:39.350 back to thinking about how to think in terms of the Ising 01:21:39.350 --> 01:21:43.975 model, and try to do more with understanding the behavior 01:21:43.975 --> 01:21:47.550 and singularities of this partition function.