WEBVTT 00:00:00.500 --> 00:00:02.993 The following content is provided under a Creative 00:00:02.993 --> 00:00:03.719 Commons license. 00:00:03.719 --> 00:00:06.284 Your support will help MIT OpenCourseWare 00:00:06.284 --> 00:00:10.680 continue to offer high quality educational resources for free. 00:00:10.680 --> 00:00:13.632 To make a donation or view additional materials 00:00:13.632 --> 00:00:19.068 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:19.068 --> 00:00:21.460 at ocw.mit.edu. 00:00:21.460 --> 00:00:23.182 PROFESSOR: OK. 00:00:23.182 --> 00:00:24.904 Let's start. 00:00:24.904 --> 00:00:31.792 So we are back to calculating partition functions 00:00:31.792 --> 00:00:34.243 for Ising models. 00:00:34.243 --> 00:00:39.304 So again some in general hypercubic lattice 00:00:39.304 --> 00:00:44.370 where you have variables sigma i being 00:00:44.370 --> 00:00:49.818 minus plus 1 at each side with a tendency to be parallel. 00:00:49.818 --> 00:00:54.782 And our task is to calculate the partition function which 00:00:54.782 --> 00:00:59.246 for n sides amounts to summing over 2 00:00:59.246 --> 00:01:04.146 to the n binary configurations of a weight that 00:01:04.146 --> 00:01:10.434 favors near neighbors to be in the same state with a strength 00:01:10.434 --> 00:01:10.970 K. 00:01:10.970 --> 00:01:11.532 OK. 00:01:11.532 --> 00:01:16.590 So the procedure that we are going to follow 00:01:16.590 --> 00:01:19.408 is this high temperature expansion, 00:01:19.408 --> 00:01:27.078 that is writing this as hyperbolic cosine of k 1 00:01:27.078 --> 00:01:34.671 plus t standing for tan K sigma i, sigma j. 00:01:34.671 --> 00:01:40.479 And then as we discussed, actually this i 00:01:40.479 --> 00:01:47.819 would have to write as a product over all bonds. 00:01:47.819 --> 00:01:54.704 And then essentially to each bond on this lattice 00:01:54.704 --> 00:02:03.219 I would have to either assign 1 or t sigma i, sigma j. 00:02:03.219 --> 00:02:06.590 Binary choice now moves to bonds. 00:02:06.590 --> 00:02:12.000 And then we saw that in order to make sure 00:02:12.000 --> 00:02:15.246 that following the summation over sigma 00:02:15.246 --> 00:02:19.800 these factors survived we have to make sure that we construct 00:02:19.800 --> 00:02:24.757 graphs where out of each side go out an even number of bonds. 00:02:24.757 --> 00:02:27.669 And we rewrote the whole thing as 2 00:02:27.669 --> 00:02:32.402 to the n, cos K to the number of bonds, which 00:02:32.402 --> 00:02:35.660 for a hypercubic lattice is dn. 00:02:35.660 --> 00:02:40.004 And then we had a sum over graphs 00:02:40.004 --> 00:02:48.335 where we had an even number of bonds per side. 00:02:48.335 --> 00:02:53.525 And the contribution of the graph 00:02:53.525 --> 00:03:04.019 was basically t to the power of the number of bonds. 00:03:04.019 --> 00:03:10.649 So I'm going to call this sum over here s. 00:03:10.649 --> 00:03:15.959 After all, the interesting part of the problem 00:03:15.959 --> 00:03:19.755 is captured in this factor s, these 2 to the n cos K 00:03:19.755 --> 00:03:22.091 to the power of dn are perfectly well 00:03:22.091 --> 00:03:23.452 behaved analytical functions. 00:03:23.452 --> 00:03:27.700 We are looking for something that is singular. 00:03:27.700 --> 00:03:33.554 So this s I can either pick one from all of the factors-- 00:03:33.554 --> 00:03:38.510 so to the lowest order in t I would have to do to one. 00:03:38.510 --> 00:03:41.591 And then we saw that the next order correction 00:03:41.591 --> 00:03:43.625 would be something like a square, 00:03:43.625 --> 00:03:46.935 would be this t to the 4th. 00:03:46.935 --> 00:03:51.140 And then objects that are more complicated versions. 00:03:51.140 --> 00:03:58.462 And I can write all of those as a sum over kind of graphs 00:03:58.462 --> 00:04:04.424 that I can draw on the lattice by going around and making 00:04:04.424 --> 00:04:05.342 a loop. 00:04:05.342 --> 00:04:11.496 But then I will have graphs that will be composed of two loops, 00:04:11.496 --> 00:04:14.236 for example, that are disconnected. 00:04:14.236 --> 00:04:18.971 Since this can be translated all over the place 00:04:18.971 --> 00:04:22.832 this would have a factor of n for a lattice that 00:04:22.832 --> 00:04:25.951 is large for getting side effects and edge effects. 00:04:25.951 --> 00:04:29.321 This would have a contribution once I slide these two 00:04:29.321 --> 00:04:31.006 with respect to each other. 00:04:31.006 --> 00:04:33.614 That is, of the order of n squared. 00:04:33.614 --> 00:04:35.896 And then I would have things that 00:04:35.896 --> 00:04:41.449 would three loops and so forth. 00:04:41.449 --> 00:04:47.443 Now based on what we have seen before it 00:04:47.443 --> 00:04:52.879 is very tempting to somehow exponentiate this sum 00:04:52.879 --> 00:04:58.951 and write it as exponential of the sum 00:04:58.951 --> 00:05:04.264 over objects that have a single loop. 00:05:04.264 --> 00:05:12.488 And I will call this new sum actually s prime for reasons 00:05:12.488 --> 00:05:16.689 to become apparent because if I start 00:05:16.689 --> 00:05:19.930 to expand this exponential, what do I get? 00:05:19.930 --> 00:05:23.066 I will certainly start to get 1, then 00:05:23.066 --> 00:05:26.594 I will have the sum over single loop graphs 00:05:26.594 --> 00:05:33.560 plus one half whatever is in the exponent over here squared, 00:05:33.560 --> 00:05:40.728 1 over 3 factorial whatever is in the exponent, which is 00:05:40.728 --> 00:05:44.620 this sum cubed, and so forth. 00:05:44.620 --> 00:05:49.282 And if I were to expand this thing that 00:05:49.282 --> 00:05:52.390 is something squared, I will uncertainty 00:05:52.390 --> 00:05:55.033 get terms in it that would correspond 00:05:55.033 --> 00:05:57.865 to two of the terms in the sum. 00:05:57.865 --> 00:06:01.524 Here I would get things that would correspond to three 00:06:01.524 --> 00:06:03.786 of the terms in the sum. 00:06:03.786 --> 00:06:06.050 And the communitorial factors would certainly 00:06:06.050 --> 00:06:10.175 work out to get rid of the 2 and the 6, 00:06:10.175 --> 00:06:14.101 depending on-- let's say I have loop a, b, or c, 00:06:14.101 --> 00:06:16.757 I could pick a from the first sum 00:06:16.757 --> 00:06:19.759 here, b from the second, c from the third 00:06:19.759 --> 00:06:22.927 or any permutation thereafter that would amount to c. 00:06:22.927 --> 00:06:25.759 I would have a factor of c abc. 00:06:25.759 --> 00:06:32.791 But s definitely is not equal to s prime because immediately we 00:06:32.791 --> 00:06:39.315 see that once I square this term a plus b squared, in addition 00:06:39.315 --> 00:06:44.880 to a squared ab I will also get a squared and b squared, 00:06:44.880 --> 00:06:52.230 which corresponds to objects such as this. 00:06:52.230 --> 00:07:00.571 One half of the same graph repeated twice. 00:07:00.571 --> 00:07:01.071 Right? 00:07:01.071 --> 00:07:06.243 It is the a squared term that is appearing in this sum. 00:07:06.243 --> 00:07:11.490 And this clearly has no analog here in my sum s. 00:07:11.490 --> 00:07:16.790 I emphasize that in the sum s each bond can 00:07:16.790 --> 00:07:20.580 occur either zero or one time. 00:07:20.580 --> 00:07:26.564 Whereas if I exponentiate this you can see that s prime 00:07:26.564 --> 00:07:30.550 certainly contains things that potentially are repeated twice. 00:07:30.550 --> 00:07:34.600 As I go further I could have multiple times. 00:07:34.600 --> 00:07:39.820 Also when I square this term I could potentially 00:07:39.820 --> 00:07:46.660 get 1 factor from 1, another factor, which is also 00:07:46.660 --> 00:07:52.967 a loop from the second term, the second in this product of two 00:07:52.967 --> 00:07:56.771 brackets, which once I multiply them happen 00:07:56.771 --> 00:08:02.276 to overlap and share something. 00:08:02.276 --> 00:08:03.377 OK? 00:08:03.377 --> 00:08:05.580 All right. 00:08:05.580 --> 00:08:10.572 So very roughly and hand wavingly, 00:08:10.572 --> 00:08:19.740 the thing is that s has loops that avoid each other, 00:08:19.740 --> 00:08:21.030 don't intersect. 00:08:21.030 --> 00:08:25.545 Whereas s prime has loops that intersect. 00:08:25.545 --> 00:08:33.312 So very naively s is sum over, if you like, 00:08:33.312 --> 00:08:39.798 a gas of non intersecting loops. 00:08:39.798 --> 00:08:50.620 Whereas s prime is a sum over gas of what 00:08:50.620 --> 00:08:53.385 I could call phantom loops. 00:08:53.385 --> 00:08:56.150 They're kind of like ghosts. 00:08:56.150 --> 00:08:59.470 They can go through each other. 00:08:59.470 --> 00:09:00.973 OK? 00:09:00.973 --> 00:09:03.980 All right. 00:09:03.980 --> 00:09:08.572 So that's one of a number of problems. 00:09:08.572 --> 00:09:13.738 Now in this lecture we are going to ignore 00:09:13.738 --> 00:09:18.160 this difference between s and s prime and calculate s prime, 00:09:18.160 --> 00:09:21.798 which therefore we will not be doing the right job. 00:09:21.798 --> 00:09:25.906 And this is one example of where I'm not doing the right job. 00:09:25.906 --> 00:09:28.231 And as I go through the calculation 00:09:28.231 --> 00:09:31.611 you may want to figure out also other places where 00:09:31.611 --> 00:09:35.138 I don't know correctly reproduce the sum s in order 00:09:35.138 --> 00:09:38.000 to ultimately be able to make a calculation 00:09:38.000 --> 00:09:39.880 and see what it comes to. 00:09:39.880 --> 00:09:44.128 Now next lecture we will try to correct all of those errors, 00:09:44.128 --> 00:09:50.715 so you better keep track of all of those errors, 00:09:50.715 --> 00:09:58.403 make sure that I am doing everything right. 00:09:58.403 --> 00:10:02.260 Seems good-- all right. 00:10:02.260 --> 00:10:10.340 So let's go and take a look at this s prime. 00:10:10.340 --> 00:10:22.183 So s prime, actually, log of s prime is a sum over graphs 00:10:22.183 --> 00:10:28.570 that I can draw on the lattice. 00:10:28.570 --> 00:10:31.474 And since I could exponentiate that, 00:10:31.474 --> 00:10:35.891 there was no problem in the exponentiation involved over 00:10:35.891 --> 00:10:36.432 here. 00:10:36.432 --> 00:10:40.760 Essentially I have to, for calculating the log, 00:10:40.760 --> 00:10:43.766 just calculate these singly connected loops. 00:10:43.766 --> 00:10:46.726 And you can see that that will work out fine 00:10:46.726 --> 00:10:48.510 as far as extensivity is concerned 00:10:48.510 --> 00:10:51.174 because I could translate each one of these figures 00:10:51.174 --> 00:10:55.900 of the loops over the lattice and get the factors of n. 00:10:55.900 --> 00:11:00.150 So this is clearly something that is OK. 00:11:00.150 --> 00:11:04.884 Now since I'm already making this mistake of forgetting 00:11:04.884 --> 00:11:07.802 about the intersections between loops 00:11:07.802 --> 00:11:13.486 I'm going to make another assumption that 00:11:13.486 --> 00:11:19.170 is in this sum of single loops. 00:11:19.170 --> 00:11:29.786 I already include things where the single loop 00:11:29.786 --> 00:11:37.266 is allowed to intersect itself. 00:11:37.266 --> 00:11:48.034 For example, I'm going to allow as a single loop entity 00:11:48.034 --> 00:11:54.681 something that is like this where this particular bond will 00:11:54.681 --> 00:11:57.687 give a factor of t squared. 00:11:57.687 --> 00:12:02.700 Clearly I should not include this in the correct sum. 00:12:02.700 --> 00:12:05.010 But since I'm ignoring intersections 00:12:05.010 --> 00:12:09.180 among the different groups and I'm making it phantom, 00:12:09.180 --> 00:12:12.672 let's make it also phantom with respect to itself, 00:12:12.672 --> 00:12:15.010 allow it to intersect itself [INAUDIBLE]. 00:12:15.010 --> 00:12:19.530 It's in the same spirit of the mistake 00:12:19.530 --> 00:12:22.930 that we are going to do. 00:12:22.930 --> 00:12:27.070 So now what do we have? 00:12:27.070 --> 00:12:33.280 We can write that log of s prime is 00:12:33.280 --> 00:12:40.170 this sum over loops of length l and then multiply it 00:12:40.170 --> 00:12:42.920 by t to the l. 00:12:42.920 --> 00:12:49.969 So basically I say that I can draw loops, let's say of size 00:12:49.969 --> 00:12:52.164 four, loops of size six. 00:12:52.164 --> 00:12:57.871 Each one of them I have to multiply by t to the l. 00:12:57.871 --> 00:13:09.500 Of course, I have to multiply with the number of loops 00:13:09.500 --> 00:13:12.857 of length l. 00:13:12.857 --> 00:13:13.980 OK? 00:13:13.980 --> 00:13:17.932 And this I'm going to write slightly differently. 00:13:17.932 --> 00:13:22.872 So I'm going to say that log of s prime 00:13:22.872 --> 00:13:26.628 is sum over this length of the loop. 00:13:26.628 --> 00:13:30.340 All the loops of length l are going 00:13:30.340 --> 00:13:35.477 to contribute a factor of t to the l. 00:13:35.477 --> 00:13:40.886 And I'm going to count the loops of length 00:13:40.886 --> 00:13:45.712 l that start and end at the origin. 00:13:45.712 --> 00:13:49.960 And I'll give that a symbol wl00. 00:13:49.960 --> 00:13:54.376 Actually, very soon I will introduce 00:13:54.376 --> 00:13:57.320 a generalization of this. 00:13:57.320 --> 00:14:01.736 So let me write the definition. 00:14:01.736 --> 00:14:06.128 I define a matrix that is indexed 00:14:06.128 --> 00:14:10.992 by two side of the lattice and counts 00:14:10.992 --> 00:14:18.569 the number of walks that I can have from one to the other, 00:14:18.569 --> 00:14:22.566 from i to j in l steps. 00:14:22.566 --> 00:14:36.727 So this is number of walks of length l from i to j. 00:14:36.727 --> 00:14:38.260 OK? 00:14:38.260 --> 00:14:43.408 So what am I doing here? 00:14:43.408 --> 00:14:50.272 Since I am looking at single loop objects, 00:14:50.272 --> 00:14:56.150 I want to sum over, let's say, all terms 00:14:56.150 --> 00:14:59.705 that contribute, in this case t to the 4. 00:14:59.705 --> 00:15:02.704 It's obvious because it's really just one shape. 00:15:02.704 --> 00:15:06.920 It's a square, but this square I could have started at anywhere 00:15:06.920 --> 00:15:09.131 on the lattice. 00:15:09.131 --> 00:15:15.027 And this slight factor, which captures the extensivity, 00:15:15.027 --> 00:15:19.592 I'll take outside because I expect this log of s 00:15:19.592 --> 00:15:20.600 to be extensive. 00:15:20.600 --> 00:15:22.616 It should be proportional to n. 00:15:22.616 --> 00:15:25.210 So one part of it is essentially where 00:15:25.210 --> 00:15:27.094 I start to draw this loop. 00:15:27.094 --> 00:15:31.224 So I say that I always start with the loops 00:15:31.224 --> 00:15:36.040 that I have at point zero. 00:15:36.040 --> 00:15:39.344 Then I want to come back to myself. 00:15:39.344 --> 00:15:43.887 So I indicate that the end point should also be zero. 00:15:43.887 --> 00:15:46.238 And if I want to get a term here, 00:15:46.238 --> 00:15:48.648 this is a term that is t to the fourth. 00:15:48.648 --> 00:15:52.220 I need to know how many of such blocks I have. 00:15:52.220 --> 00:15:52.720 Yes? 00:15:52.720 --> 00:15:55.960 AUDIENCE: Are you allowing the loop to intersect itself 00:15:55.960 --> 00:15:57.760 in this case or not? 00:15:57.760 --> 00:15:59.395 PROFESSOR: In this case, yes. 00:15:59.395 --> 00:16:01.684 I will also, when I'm calculating anything 00:16:01.684 --> 00:16:04.393 to do with s prime I allow intersection. 00:16:04.393 --> 00:16:08.144 So if you are asking whether I'm allowing something like this, 00:16:08.144 --> 00:16:09.200 the answer is yes. 00:16:09.200 --> 00:16:10.200 AUDIENCE: OK. 00:16:10.200 --> 00:16:11.200 PROFESSOR: Yeah. 00:16:11.200 --> 00:16:15.320 AUDIENCE: And are we assuming an infinitely large system so 00:16:15.320 --> 00:16:15.820 that-- 00:16:15.820 --> 00:16:16.392 PROFESSOR: Yes. 00:16:16.392 --> 00:16:16.964 That's right. 00:16:16.964 --> 00:16:20.110 So that the edge effects, you don't have to worry about. 00:16:20.110 --> 00:16:24.671 Or alternatively you can imagine that you have periodic boundary 00:16:24.671 --> 00:16:25.170 conditions. 00:16:25.170 --> 00:16:27.335 And with periodic boundary conditions 00:16:27.335 --> 00:16:30.985 we can still slide it all over the place. 00:16:30.985 --> 00:16:31.485 OK? 00:16:31.485 --> 00:16:34.588 But clearly then the maximal size of these loops, 00:16:34.588 --> 00:16:36.532 et cetera, will be determined potentially 00:16:36.532 --> 00:16:40.490 by the size of the lattice. 00:16:40.490 --> 00:16:44.994 Now this is not entirely correct because there 00:16:44.994 --> 00:16:47.246 is an over counting. 00:16:47.246 --> 00:16:51.715 This 1one square that I have drawn over here, 00:16:51.715 --> 00:16:54.802 I could have started from this point, 00:16:54.802 --> 00:16:57.010 to this point, this point. 00:16:57.010 --> 00:17:01.386 And essentially for something that has length l, 00:17:01.386 --> 00:17:05.769 I would have had l possible starting points. 00:17:05.769 --> 00:17:11.453 So in order to avoid the over counting I have to divide by l. 00:17:11.453 --> 00:17:16.595 And in fact I could have started walking along this direction, 00:17:16.595 --> 00:17:20.039 or alternatively I could have gone 00:17:20.039 --> 00:17:22.339 in the clockwise direction. 00:17:22.339 --> 00:17:24.796 So there's two orientations to the walk 00:17:24.796 --> 00:17:27.604 that will take me from the origin back 00:17:27.604 --> 00:17:30.080 to the origin in l steps. 00:17:30.080 --> 00:17:33.940 And not to do the over counting, I have to divide by 2l. 00:17:33.940 --> 00:17:34.440 Yes? 00:17:34.440 --> 00:17:38.528 AUDIENCE: If we allow walking over ourselves 00:17:38.528 --> 00:17:42.620 is it always a degeneracy of 2l? 00:17:42.620 --> 00:17:43.418 PROFESSOR: Yes. 00:17:43.418 --> 00:17:47.009 You can go and do the calculation to convince 00:17:47.009 --> 00:17:52.213 yourself that even for something as convoluted as that, 00:17:52.213 --> 00:17:55.420 you can... is too. 00:17:55.420 --> 00:17:56.628 PROFESSOR: OK. 00:17:56.628 --> 00:18:01.460 So this is what we want to calculate. 00:18:01.460 --> 00:18:06.046 Well, it turns out that this entity actually 00:18:06.046 --> 00:18:07.776 shows up somewhere else also. 00:18:07.776 --> 00:18:12.897 So let me tell you why I wanted to write a more general thing. 00:18:12.897 --> 00:18:16.769 Another quantity that I can try to calculate 00:18:16.769 --> 00:18:19.190 is the spin spin correlation. 00:18:19.190 --> 00:18:24.998 I can pick spin zero here and say spin r here, 00:18:24.998 --> 00:18:26.582 some other location. 00:18:26.582 --> 00:18:32.086 And I want to calculate what is the correlation 00:18:32.086 --> 00:18:34.942 between these two spins. 00:18:34.942 --> 00:18:35.660 OK? 00:18:35.660 --> 00:18:39.800 So how do I have to do that for the Ising model? 00:18:39.800 --> 00:18:42.775 I have to essentially sum over all configurations 00:18:42.775 --> 00:18:47.215 with an additional factor of sigma zero sigma 00:18:47.215 --> 00:18:58.508 r of this weight, e to the k, sum over ij, sigma i, sigma j, 00:18:58.508 --> 00:19:04.990 appropriately normalized, of course, by the partition 00:19:04.990 --> 00:19:05.920 function. 00:19:05.920 --> 00:19:08.552 And I can make the same transformation 00:19:08.552 --> 00:19:12.901 that I have on the first line of these exponential factors 00:19:12.901 --> 00:19:18.130 to write this as a sum over sigma i, 00:19:18.130 --> 00:19:23.360 I have sigma zero, sigma r, and then product 00:19:23.360 --> 00:19:28.950 over all bonds of these factors of one plus t sigma i, sigma j. 00:19:28.950 --> 00:19:32.886 The factors of 2 to the n cos K will cancel out 00:19:32.886 --> 00:19:34.860 between the numerator and the denominator. 00:19:34.860 --> 00:19:41.324 And basically I will get the same thing. 00:19:41.324 --> 00:19:48.114 Now of course the denominator is the partition function. 00:19:48.114 --> 00:19:51.039 It is the sum s that we are after, 00:19:51.039 --> 00:19:55.580 but we can also, and we've seen this already how to do this, 00:19:55.580 --> 00:19:58.380 express the sum in the numerator graphically. 00:19:58.380 --> 00:20:02.176 And the difference between the numerator and the denominator 00:20:02.176 --> 00:20:06.217 is that I have an additional sigma sitting here, 00:20:06.217 --> 00:20:08.962 an additional sigma sitting there, 00:20:08.962 --> 00:20:15.110 that if left by themselves will average out to 0. 00:20:15.110 --> 00:20:18.894 So I need to connect them by paths 00:20:18.894 --> 00:20:23.151 that are composed of factors of t sigma sigma, 00:20:23.151 --> 00:20:28.816 originating on one and ending on the other. 00:20:28.816 --> 00:20:29.557 Right? 00:20:29.557 --> 00:20:38.202 So in the same sense that what is appearing in the denominator 00:20:38.202 --> 00:20:42.514 is a sum that involves these loops, 00:20:42.514 --> 00:20:47.418 the first term that appears in the numerator 00:20:47.418 --> 00:20:52.042 is a path that connects zero to r 00:20:52.042 --> 00:20:56.272 through some combination of these factors of t, 00:20:56.272 --> 00:21:01.230 and then I have to sum over all possible ways of doing that. 00:21:01.230 --> 00:21:04.898 But then I could certainly have graphs 00:21:04.898 --> 00:21:07.518 that involves the same thing. 00:21:07.518 --> 00:21:11.540 And the loop, there is nothing that 00:21:11.540 --> 00:21:17.910 is against that, or the same thing, and two loops, 00:21:17.910 --> 00:21:19.821 and so forth. 00:21:19.821 --> 00:21:25.280 And you can see that as long as, and only as long 00:21:25.280 --> 00:21:28.206 as I treat these as phantom objects 00:21:28.206 --> 00:21:33.149 that can pass through each other I can factor out this term 00:21:33.149 --> 00:21:37.692 and the rest of 1 plus 1, loop plus 2 loop, 00:21:37.692 --> 00:21:41.309 is exactly what I have in the denominator. 00:21:41.309 --> 00:21:49.265 So we see that under the assumption of phantomness, 00:21:49.265 --> 00:21:55.453 if phantom then this becomes really just 00:21:55.453 --> 00:22:02.810 the sum over all paths that go from 0 to r. 00:22:02.810 --> 00:22:07.978 And of course the contribution of each path 00:22:07.978 --> 00:22:12.874 is how many factors of t I have. 00:22:12.874 --> 00:22:13.479 Right? 00:22:13.479 --> 00:22:21.642 So I have to have a sum over the length of this path l. 00:22:21.642 --> 00:22:26.152 Paths of length l will contribute a factor of t 00:22:26.152 --> 00:22:27.505 to the l. 00:22:27.505 --> 00:22:30.346 But there are potentially multiple ways 00:22:30.346 --> 00:22:34.970 to go from i0 to r in l steps. 00:22:34.970 --> 00:22:37.709 How many ways? 00:22:37.709 --> 00:22:45.926 That's precisely what I call this 0w of lr. 00:22:45.926 --> 00:22:46.840 Yes? 00:22:46.840 --> 00:22:50.606 AUDIENCE: Why does a graph that goes 00:22:50.606 --> 00:22:56.000 from 0 to r in three different ways have [INAUDIBLE]? 00:22:56.000 --> 00:22:56.896 PROFESSOR: OK. 00:22:56.896 --> 00:23:01.824 So you want to have to go from 0 to r, 00:23:01.824 --> 00:23:05.852 you want to have a single path, and then 00:23:05.852 --> 00:23:09.740 you want that path to do something like this? 00:23:09.740 --> 00:23:10.880 AUDIENCE: Yeah. 00:23:10.880 --> 00:23:12.590 That doesn't [INAUDIBLE]. 00:23:12.590 --> 00:23:14.351 PROFESSOR: That's fine. 00:23:14.351 --> 00:23:17.873 If I ignore the phantomness condition 00:23:17.873 --> 00:23:21.990 this is the same as this multiplied 00:23:21.990 --> 00:23:26.687 by this, which is a term that appears in the denominator 00:23:26.687 --> 00:23:27.980 and cancels out. 00:23:27.980 --> 00:23:31.226 AUDIENCE: But you're assuming that you 00:23:31.226 --> 00:23:33.390 have the phantom condition. 00:23:33.390 --> 00:23:38.120 So this is completely normal. 00:23:38.120 --> 00:23:45.800 It doesn't matter. 00:23:45.800 --> 00:23:48.296 PROFESSOR: I'm not sure I understand your question. 00:23:48.296 --> 00:23:50.955 You say that even without the phantom condition 00:23:50.955 --> 00:23:52.380 this graph exists. 00:23:52.380 --> 00:23:54.320 AUDIENCE: With them phantom condition-- 00:23:54.320 --> 00:23:55.360 PROFESSOR: Yes. 00:23:55.360 --> 00:23:58.220 AUDIENCE: --this graph is perfectly normal. 00:23:58.220 --> 00:24:06.008 PROFESSOR: Even without the phantom condition 00:24:06.008 --> 00:24:12.498 this is an acceptable graph. 00:24:12.498 --> 00:24:13.796 Yeah. 00:24:13.796 --> 00:24:15.094 OK. 00:24:15.094 --> 00:24:16.400 Yeah? 00:24:16.400 --> 00:24:20.582 AUDIENCE: So what does phantomness mean? 00:24:20.582 --> 00:24:26.160 Why then we can simplify only a [INAUDIBLE]? 00:24:26.160 --> 00:24:27.682 PROFESSOR: OK. 00:24:27.682 --> 00:24:36.053 Because let's say I were to take this as a check 00:24:36.053 --> 00:24:40.435 and multiply it by the denominator. 00:24:40.435 --> 00:24:45.035 The question is, would I generate a series 00:24:45.035 --> 00:24:47.910 that is in the numerator? 00:24:47.910 --> 00:24:48.485 OK? 00:24:48.485 --> 00:24:53.678 So if I take this object that I have said is the answer, 00:24:53.678 --> 00:24:57.037 I have to multiply it by this object. 00:24:57.037 --> 00:25:03.049 And make sure that it introduces correctly the numerator. 00:25:03.049 --> 00:25:06.755 The question is, when does it? 00:25:06.755 --> 00:25:10.049 I mean, certainly when I multiply this by this, 00:25:10.049 --> 00:25:14.204 I will get the possibility of having a graph such as this. 00:25:14.204 --> 00:25:17.691 And from here, I can have a loop such as this. 00:25:17.691 --> 00:25:22.101 And the two of them would share a bond such as that. 00:25:22.101 --> 00:25:26.131 So in the real Ising model, that is not allowed. 00:25:26.131 --> 00:25:31.308 So that's the phantomness condition 00:25:31.308 --> 00:25:39.652 that allows me to factor these things. 00:25:39.652 --> 00:25:40.844 OK? 00:25:40.844 --> 00:25:43.240 All right. 00:25:43.240 --> 00:25:50.611 So we see that if I have this quantity that I have written 00:25:50.611 --> 00:25:55.660 in red, then I can calculate both correlation functions, 00:25:55.660 --> 00:26:01.130 as well as the free energy, log of the partition 00:26:01.130 --> 00:26:04.175 function within this phantomness assumption. 00:26:04.175 --> 00:26:09.167 So the question is, can I calculate that? 00:26:09.167 --> 00:26:12.309 And the answer is that calculating number 00:26:12.309 --> 00:26:16.089 of random walks is one of the most basic things 00:26:16.089 --> 00:26:20.430 that one does in statistical physics, 00:26:20.430 --> 00:26:24.380 and easily accomplished as follows. 00:26:24.380 --> 00:26:32.432 Basically, I say that, OK, let's say that I start from 0, 00:26:32.432 --> 00:26:38.665 actually let's do it 0 and r, and let's 00:26:38.665 --> 00:26:46.750 say that I have looked at all possible paths that have 00:26:46.750 --> 00:26:53.210 l steps and end up over here. 00:26:53.210 --> 00:26:57.190 So this is step one, step two, step number three. 00:26:57.190 --> 00:27:00.779 And the last one, step l, I have purposely 00:27:00.779 --> 00:27:03.049 drawn as a dotted line. 00:27:03.049 --> 00:27:06.681 Maybe I will pull this point further down 00:27:06.681 --> 00:27:10.394 to emphasize that this is the last one. 00:27:10.394 --> 00:27:14.615 This is l minus 1 is the previous one. 00:27:14.615 --> 00:27:21.124 So I can certainly state that the number of 00:27:21.124 --> 00:27:29.460 walks from 0 to r in l steps. 00:27:29.460 --> 00:27:34.320 Well, any walk that got to r in l steps, 00:27:34.320 --> 00:27:39.190 at the l minus one step had to be somewhere. 00:27:39.190 --> 00:27:39.811 OK? 00:27:39.811 --> 00:27:49.747 So what I do is I do a number of walks from 0 to r prime, 00:27:49.747 --> 00:27:58.150 I'll call this point r prime, in l minus one steps. 00:27:58.150 --> 00:28:07.830 And then times the number of ways, or number of walks 00:28:07.830 --> 00:28:12.833 from r prime to r in one step. 00:28:12.833 --> 00:28:18.134 So before I reach my destination the previous step 00:28:18.134 --> 00:28:22.189 I had to have been somewhere, I sum 00:28:22.189 --> 00:28:27.170 over all possible various places where that somewhere has to be, 00:28:27.170 --> 00:28:30.869 and then I have to make sure that I 00:28:30.869 --> 00:28:35.542 can reach from that somewhere in one step to my destination. 00:28:35.542 --> 00:28:38.919 That's all that sum is, OK? 00:28:38.919 --> 00:28:43.098 Now I can convert that to mathematics. 00:28:43.098 --> 00:28:50.098 This quantity is start from 0, take l steps, arrive at r. 00:28:50.098 --> 00:28:52.643 By definition that's the number. 00:28:52.643 --> 00:29:00.238 And what it says is that this should be a sum over r pi, 00:29:00.238 --> 00:29:08.194 start from 0, take l minus 1 steps, arrive at r prime. 00:29:08.194 --> 00:29:13.471 Start from r prime, take one step, 00:29:13.471 --> 00:29:22.300 arrive at your destination r, sum over r prime. 00:29:22.300 --> 00:29:23.289 OK? 00:29:23.289 --> 00:29:30.105 Now these are n by n matrices that are labeled by l. 00:29:30.105 --> 00:29:30.673 Right? 00:29:30.673 --> 00:29:37.549 So these, being n by n matrices, this summation over r prime 00:29:37.549 --> 00:29:40.439 is clearly a matrix multiplication. 00:29:40.439 --> 00:29:45.765 So what that says is that summing over r prime 00:29:45.765 --> 00:29:51.797 tells you that that is w 1, w of l minus 1, 0. 00:29:51.797 --> 00:29:56.359 And that is true for any pair of elements, starting 00:29:56.359 --> 00:29:57.703 and final points. 00:29:57.703 --> 00:29:59.943 So basically, quite generically, we 00:29:59.943 --> 00:30:03.800 see that w, the matrix that corresponds 00:30:03.800 --> 00:30:10.919 to the count for l steps, is obtained from the matrix that 00:30:10.919 --> 00:30:15.087 corresponds to the count for one step multiplying 00:30:15.087 --> 00:30:18.213 that of l minus 1 steps. 00:30:18.213 --> 00:30:23.064 And clearly I can keep going. wl minus 1, 00:30:23.064 --> 00:30:30.820 I can write as w 1, w of l minus 2, and so forth. 00:30:30.820 --> 00:30:33.760 And ultimately the answer is none other 00:30:33.760 --> 00:30:37.549 than the entity that corresponds to one step raised 00:30:37.549 --> 00:30:39.353 to the l power. 00:30:39.353 --> 00:30:43.412 And just to make things easier on my writing, 00:30:43.412 --> 00:30:52.560 I will indicate this as t to the l where t stands 00:30:52.560 --> 00:30:59.748 for this matrix for one set. 00:30:59.748 --> 00:31:00.950 OK? 00:31:00.950 --> 00:31:03.925 This condition over here that I said 00:31:03.925 --> 00:31:10.074 in words that allows me to write this in this nice matrix form 00:31:10.074 --> 00:31:13.734 is called the Markovian condition. 00:31:13.734 --> 00:31:20.330 The kind of walks that I have been telling 00:31:20.330 --> 00:31:24.320 you our Markovian in the sense that they only they 00:31:24.320 --> 00:31:28.598 depend on where you came from at the last step. 00:31:28.598 --> 00:31:33.989 They don't have memory of where you had been before. 00:31:33.989 --> 00:31:37.133 And that's what enables us to do this. 00:31:37.133 --> 00:31:42.399 And that's why I had to do the phantom condition because if I 00:31:42.399 --> 00:31:45.943 really wanted to say that something like this 00:31:45.943 --> 00:31:49.478 has to be excluded, then the walk must 00:31:49.478 --> 00:31:53.515 keep memory of every place that it had been before. 00:31:53.515 --> 00:31:54.015 Right? 00:31:54.015 --> 00:31:56.500 And then it would be non Markovian. 00:31:56.500 --> 00:32:01.230 Then I wouldn't have been able to do this nice calculation. 00:32:01.230 --> 00:32:06.559 That's why I had to do this phantomness assumption so 00:32:06.559 --> 00:32:15.062 that I forgot the memory of where my walk was previously. 00:32:15.062 --> 00:32:15.840 OK? 00:32:15.840 --> 00:32:17.260 Now... 00:32:17.260 --> 00:32:18.680 Yeah? 00:32:18.680 --> 00:32:20.100 Question? 00:32:20.100 --> 00:32:21.520 No? 00:32:21.520 --> 00:32:28.010 So this matrix, where you can go in one step, 00:32:28.010 --> 00:32:34.678 is really the matrix of who is connected to whom. 00:32:34.678 --> 00:32:35.496 Right? 00:32:35.496 --> 00:32:40.404 So this tells you the connectivity. 00:32:40.404 --> 00:32:47.766 So for example, if I'm dealing with a 2D 00:32:47.766 --> 00:32:57.146 square lattice the sides on my lattice are labeled by x and y. 00:32:57.146 --> 00:33:04.429 And I can ask, where can I go in one step 00:33:04.429 --> 00:33:08.069 if I start from x and y? 00:33:08.069 --> 00:33:15.461 And the answer is that either x stays the same and why y 00:33:15.461 --> 00:33:26.329 shifts by one, or y stays the same and x shifts by one. 00:33:26.329 --> 00:33:31.801 These are the four non zero elements 00:33:31.801 --> 00:33:37.547 of the matrix that allows you to go on the square lattice either 00:33:37.547 --> 00:33:39.385 up, down, right, or left. 00:33:39.385 --> 00:33:41.275 And the corresponding things that you 00:33:41.275 --> 00:33:45.430 would have for the cube or whatever lattice. 00:33:45.430 --> 00:33:47.380 OK? 00:33:47.380 --> 00:33:51.016 Now you look at this and you can see 00:33:51.016 --> 00:33:54.248 that what I have imposed clearly is such 00:33:54.248 --> 00:33:58.892 that for a lattice where every side looks 00:33:58.892 --> 00:34:03.351 like every equivalent side this is really 00:34:03.351 --> 00:34:08.734 a function only of the separation between the two 00:34:08.734 --> 00:34:09.319 points. 00:34:09.319 --> 00:34:12.829 It's an n by n matrix. 00:34:12.829 --> 00:34:17.208 It has n squared elements, but the elements really 00:34:17.208 --> 00:34:19.380 are essentially one column that gets 00:34:19.380 --> 00:34:23.284 shifted as you go further and further down 00:34:23.284 --> 00:34:26.760 in a very specific way. 00:34:26.760 --> 00:34:32.600 And whenever you have a matrix such as this translational 00:34:32.600 --> 00:34:37.408 symmetry implies that you can diagonalize it 00:34:37.408 --> 00:34:40.240 by Fourier transformation. 00:34:40.240 --> 00:34:46.848 And what do I mean by that? 00:34:46.848 --> 00:34:53.655 This is, I can define a vector q such 00:34:53.655 --> 00:35:00.310 that it's various components are things like e to the i 00:35:00.310 --> 00:35:03.740 q dot r in whatever dimension. 00:35:03.740 --> 00:35:06.860 Let's normalize it by square root of n. 00:35:06.860 --> 00:35:12.560 And I should be sure that that is an eigenvector. 00:35:12.560 --> 00:35:19.220 So basically my statement is that if I take the matrix t, 00:35:19.220 --> 00:35:27.050 act on q, then I should get some eigenvalue in the vector back. 00:35:27.050 --> 00:35:34.270 And let's check that for the case of the 2D system. 00:35:34.270 --> 00:35:47.530 So for the 2D system, if I say that x y t qx qy, what is it? 00:35:47.530 --> 00:35:54.778 Well, that is x y t x prime, y prime, 00:35:54.778 --> 00:35:58.271 the entity that I have calculated here, 00:35:58.271 --> 00:36:01.265 x prime, y prime qx, qy. 00:36:01.265 --> 00:36:06.410 And of course, I have a sum over x prime and y prime. 00:36:06.410 --> 00:36:07.870 That's the matrix product. 00:36:07.870 --> 00:36:13.382 And so again, remember this entity is simply 00:36:13.382 --> 00:36:21.650 e to the i qx x prime plus qy y prime, divided 00:36:21.650 --> 00:36:24.130 by square root of n. 00:36:24.130 --> 00:36:29.090 And because this is a set of delta functions, what 00:36:29.090 --> 00:36:30.578 does it do? 00:36:30.578 --> 00:36:34.746 It basically sets x prime either to x plus 1, or x minus 1, 00:36:34.746 --> 00:36:38.698 y prime either to y or y minus 1, y minus 1. 00:36:38.698 --> 00:36:44.606 You can see that you always get back your e to the i qx 00:36:44.606 --> 00:36:49.330 x plus qy y with a factor of root n. 00:36:49.330 --> 00:36:52.193 So essentially that by the delta functions 00:36:52.193 --> 00:36:56.290 just changes the x primes to x at the cost 00:36:56.290 --> 00:36:58.124 of the different shifts that you have 00:36:58.124 --> 00:37:02.094 to do over there, which means that you will get a factor of e 00:37:02.094 --> 00:37:06.462 to the i qx, e to the minus iqx, with qy not changing, 00:37:06.462 --> 00:37:11.052 or e to the i qy, and e to the minus i 00:37:11.052 --> 00:37:14.041 qy with the x component not changing. 00:37:14.041 --> 00:37:18.682 So this is the standard thing that you have seen, 00:37:18.682 --> 00:37:24.898 is none other than 2 cosine of qx, plus cosine of qy. 00:37:24.898 --> 00:37:33.310 And so we can see that quite generally in the d dimensional 00:37:33.310 --> 00:37:40.780 hypercube, my t of q is going to be 00:37:40.780 --> 00:37:52.403 the sum over all d components of these factors of cosine 00:37:52.403 --> 00:37:55.724 of q alpha. 00:37:55.724 --> 00:38:01.270 And that's about it, OK? 00:38:01.270 --> 00:38:05.630 So why did I bother to this diagonalization? 00:38:05.630 --> 00:38:11.090 The answer is that that now allows me to calculate 00:38:11.090 --> 00:38:13.754 everything that I want. 00:38:13.754 --> 00:38:20.414 So, for example, I know that this quantity that I'm 00:38:20.414 --> 00:38:25.440 interested in, sigma 0 sigma r, is 00:38:25.440 --> 00:38:34.865 going to be a sum over l, t to the l, then 0. 00:38:34.865 --> 00:38:44.146 wl is t to the l times r. 00:38:44.146 --> 00:38:46.030 Right? 00:38:46.030 --> 00:38:50.593 Now this small t, I can take inside here 00:38:50.593 --> 00:38:53.128 and do it like this. 00:38:53.128 --> 00:39:01.972 And if I want, I can write this as the 0 r component of a sum 00:39:01.972 --> 00:39:07.264 over l tt raised to the power of l. 00:39:07.264 --> 00:39:11.588 So it's a new matrix, which is essentially 00:39:11.588 --> 00:39:17.936 sum over l, small t times this connectivity matrix to the l th 00:39:17.936 --> 00:39:18.435 power. 00:39:18.435 --> 00:39:20.660 This is a geometric series. 00:39:20.660 --> 00:39:23.775 We can immediately do the geometric series. 00:39:23.775 --> 00:39:35.876 The Answer is 0, 1 over 1 minus tt going all the way to r. 00:39:35.876 --> 00:39:36.738 OK? 00:39:36.738 --> 00:39:43.270 And the reason I did this diagonalization is 00:39:43.270 --> 00:39:48.550 so that I can calculate this matrix element, because I don't 00:39:48.550 --> 00:39:51.832 really want to invert a whole matrix, 00:39:51.832 --> 00:39:54.968 but I can certainly invert the matrix when 00:39:54.968 --> 00:40:00.120 it is in the diagonal basis because all I have to do 00:40:00.120 --> 00:40:02.620 is to invert pure numbers. 00:40:02.620 --> 00:40:11.013 So what is done is to go to the Fourier basis 00:40:11.013 --> 00:40:17.120 and rotate to the Fourier basis calculate this. 00:40:17.120 --> 00:40:20.630 It is diagonal in this basis. 00:40:20.630 --> 00:40:22.970 I have q r. 00:40:22.970 --> 00:40:25.895 And so what is that? 00:40:25.895 --> 00:40:30.348 These are these exponentials here evaluated at the origin, 00:40:30.348 --> 00:40:33.484 so it's just 1 over root n. 00:40:33.484 --> 00:40:36.295 This is 1 over root n. 00:40:36.295 --> 00:40:42.379 This is e to the i q dot r over root n. 00:40:42.379 --> 00:40:48.753 This is just the eigenvalue that I have calculated over here. 00:40:48.753 --> 00:40:58.005 So this entity is none other than a sum over q, e 00:40:58.005 --> 00:41:04.315 to the i q dot r divided by n, two 00:41:04.315 --> 00:41:07.261 factors of square root of n. 00:41:07.261 --> 00:41:10.698 1 over 1 minus t-- well, actually 00:41:10.698 --> 00:41:23.620 let's write it 2t-- sum over alpha cosine of q alpha. 00:41:23.620 --> 00:41:27.157 And then of course I'm interested in big systems. 00:41:27.157 --> 00:41:30.710 I replace the sum over q with an integral 00:41:30.710 --> 00:41:34.870 over q, 2 pi to the d density of states. 00:41:34.870 --> 00:41:39.256 In going from there to there, there's a factor of volume, 00:41:39.256 --> 00:41:43.325 the way that I have set the unit of length in my system. 00:41:43.325 --> 00:41:46.310 The volume is actually the number of particles 00:41:46.310 --> 00:41:47.600 that I have. 00:41:47.600 --> 00:41:50.180 So that factor of n disappears. 00:41:50.180 --> 00:41:54.655 And all I need to do is evaluate this factor 00:41:54.655 --> 00:42:00.211 of 1 minus 2t sum over alpha cosine of q alpha integrated 00:42:00.211 --> 00:42:05.700 over q Fourier transformed. 00:42:05.700 --> 00:42:07.370 OK? 00:42:07.370 --> 00:42:09.040 Yes? 00:42:09.040 --> 00:42:12.131 AUDIENCE: So I notice that you have a sum over q, 00:42:12.131 --> 00:42:15.692 but then you also have a sum alpha q alpha. 00:42:15.692 --> 00:42:16.400 PROFESSOR: Right. 00:42:16.400 --> 00:42:17.790 AUDIENCE: Is there a relationship 00:42:17.790 --> 00:42:20.300 between the q and the q alpha or not? 00:42:20.300 --> 00:42:20.990 PROFESSOR: OK. 00:42:20.990 --> 00:42:22.715 So that goes back here. 00:42:22.715 --> 00:42:26.456 So when I had two dimensions, I had qx and qy. 00:42:26.456 --> 00:42:26.956 Right? 00:42:26.956 --> 00:42:32.624 And so I labeled, rather than x and y, with q1 and q2. 00:42:32.624 --> 00:42:39.100 So the index alpha is just a number of spatial dimensions. 00:42:39.100 --> 00:42:46.040 If you like, this is also dq1, dq2, dqd. 00:42:46.040 --> 00:42:51.146 AUDIENCE: OK. 00:42:51.146 --> 00:42:53.700 [INAUDIBLE] 00:42:53.700 --> 00:42:55.824 PROFESSOR: OK. 00:42:55.824 --> 00:42:57.948 All right. 00:42:57.948 --> 00:43:03.258 So we are down here. 00:43:03.258 --> 00:43:05.382 Let's proceed. 00:43:05.382 --> 00:43:11.760 So what is going to happen? 00:43:11.760 --> 00:43:15.810 So suppose I'm picking two sides, 0 and r, 00:43:15.810 --> 00:43:18.960 let's say both along the x direction, 00:43:18.960 --> 00:43:21.804 some particular distance apart. 00:43:21.804 --> 00:43:25.794 Let's say seven, eight apart. 00:43:25.794 --> 00:43:32.879 So in order to evaluate this I would have an integral if this 00:43:32.879 --> 00:43:37.807 is the x direction of something like e to the iq x times r. 00:43:37.807 --> 00:43:44.709 Now when I integrate over qx the integral of e to the iqx times 00:43:44.709 --> 00:43:48.820 r would go to 0. 00:43:48.820 --> 00:43:51.682 The only way that it won't go to 0 00:43:51.682 --> 00:43:54.880 is from the expansion of what is in the denominator. 00:43:54.880 --> 00:43:57.624 I should bring on enough factors of e 00:43:57.624 --> 00:44:00.025 to the minus iqx, which certainly exist 00:44:00.025 --> 00:44:04.535 in these cosine factors, to get rid of that. 00:44:04.535 --> 00:44:09.440 So essentially, the mathematical procedure that is over here 00:44:09.440 --> 00:44:15.644 is to bring in sufficient factors of e to the minus i 00:44:15.644 --> 00:44:18.746 q dot r to eliminate that. 00:44:18.746 --> 00:44:22.878 And the number of ways that you are going to do that 00:44:22.878 --> 00:44:26.776 is precisely another way of capturing this entity, which 00:44:26.776 --> 00:44:33.196 means that clearly if I'm looking at something like this, 00:44:33.196 --> 00:44:39.178 and I mean the limit that t goes to be very, very small so 00:44:39.178 --> 00:44:41.760 that the lowest order in t contributes, 00:44:41.760 --> 00:44:46.390 the lowest order t would be the shortest path that 00:44:46.390 --> 00:44:48.250 joins these two points. 00:44:48.250 --> 00:44:53.578 So it is like connecting these two points with a string that 00:44:53.578 --> 00:44:54.910 is very tight. 00:44:54.910 --> 00:44:59.540 So that what I am saying is that the limit as t 00:44:59.540 --> 00:45:02.532 goes to 0 of something like sigma 0, 00:45:02.532 --> 00:45:12.110 sigma r is going to be identical to t to the minimum distance 00:45:12.110 --> 00:45:15.490 between 0 and r. 00:45:15.490 --> 00:45:19.146 Actually, I should say proportional, 00:45:19.146 --> 00:45:21.911 is in fact more correct. 00:45:21.911 --> 00:45:25.790 Because there could be multiple shortest paths 00:45:25.790 --> 00:45:30.580 that go between two points. 00:45:30.580 --> 00:45:31.540 OK? 00:45:31.540 --> 00:45:33.764 Now let's make sense. 00:45:33.764 --> 00:45:36.544 There's kind of an exponential. 00:45:36.544 --> 00:45:40.690 Essentially I start with the high temperature limit 00:45:40.690 --> 00:45:43.190 where the two things don't know anything about each other. 00:45:43.190 --> 00:45:46.201 So sigma 0, sigma r is going to be 0. 00:45:46.201 --> 00:45:50.788 So anything that is beyond 0 has to come from somewhere 00:45:50.788 --> 00:45:54.780 in which the information about the state of the site 00:45:54.780 --> 00:45:57.475 was conveyed all the way over here. 00:45:57.475 --> 00:46:01.600 And it is done through passing one bond at a time. 00:46:01.600 --> 00:46:04.885 And in some sense, the fidelity of each one 00:46:04.885 --> 00:46:09.940 of those transformations is proportional to t. 00:46:09.940 --> 00:46:13.988 Now as t becomes larger you are going 00:46:13.988 --> 00:46:18.542 to be willing to pay the costs of paths 00:46:18.542 --> 00:46:24.610 that go from 0 to r in a slightly more disordered way. 00:46:24.610 --> 00:46:28.150 So your string that was tight becomes 00:46:28.150 --> 00:46:30.600 kind of loose and floppy. 00:46:30.600 --> 00:46:34.030 And why does that become the case? 00:46:34.030 --> 00:46:37.845 Because now although these paths are longer and carry 00:46:37.845 --> 00:46:40.813 more factors of this t that is small, 00:46:40.813 --> 00:46:45.455 there are just so many of them that the entropy, 00:46:45.455 --> 00:46:49.911 the number of these paths starts to dominate. 00:46:49.911 --> 00:46:50.470 OK? 00:46:50.470 --> 00:46:56.640 So very roughly you can state the competition between them 00:46:56.640 --> 00:47:01.233 as follows, that the contribution of the path 00:47:01.233 --> 00:47:08.247 of length l, decays like t to the l, but the number of paths 00:47:08.247 --> 00:47:11.934 roughly grows like 2d to the power of l. 00:47:11.934 --> 00:47:14.608 And since we have this phantomness character, 00:47:14.608 --> 00:47:19.364 if I am sitting at a particular site, I can go up, 00:47:19.364 --> 00:47:20.573 down, right, left. 00:47:20.573 --> 00:47:24.210 So at each step I have in two dimensions 00:47:24.210 --> 00:47:29.127 a choice of four, in three dimensions a choice of six, 00:47:29.127 --> 00:47:32.270 in d dimensions a choice of 2d. 00:47:32.270 --> 00:47:35.366 So you can see that this is exponentially small, 00:47:35.366 --> 00:47:36.742 this is exponentially large. 00:47:36.742 --> 00:47:39.455 So they kind of balance each other. 00:47:39.455 --> 00:47:42.290 And the balance is something like e 00:47:42.290 --> 00:47:46.870 to the minus l over some typical l that we'll contribute. 00:47:46.870 --> 00:47:49.866 And clearly the typical l is going 00:47:49.866 --> 00:47:55.202 to be finite as long as 2dt is less than 1. 00:47:55.202 --> 00:47:59.458 So you can see that something strange has 00:47:59.458 --> 00:48:04.902 to happen at the value where tc is such 00:48:04.902 --> 00:48:09.060 that 2dtc is equal to 1. 00:48:09.060 --> 00:48:14.540 At that point the cost of making your paths longer 00:48:14.540 --> 00:48:19.868 is more than made up by increasing the number of paths 00:48:19.868 --> 00:48:23.216 that you can have, the entropy starts to be. 00:48:23.216 --> 00:48:28.860 And you can see that precisely that condition tells me 00:48:28.860 --> 00:48:34.152 whether or not this integral exists, right? 00:48:34.152 --> 00:48:38.556 Because one point of this integral where the integrand is 00:48:38.556 --> 00:48:41.384 largest is when q goes to 0. 00:48:41.384 --> 00:48:45.652 And you can see that as q goes to 0, 00:48:45.652 --> 00:48:49.576 the value in the denominator is 1 minus 2td. 00:48:49.576 --> 00:48:53.454 So there is precisely a pole when 00:48:53.454 --> 00:48:56.290 this condition takes place. 00:48:56.290 --> 00:48:59.839 And if I'm interested in seeing what 00:48:59.839 --> 00:49:04.910 is happening when I'm in the vicinity of that transition, 00:49:04.910 --> 00:49:09.838 right before these paths become very large, what I can do 00:49:09.838 --> 00:49:13.542 is I can start exploring what is happening 00:49:13.542 --> 00:49:16.548 in the vicinity of that pole. 00:49:16.548 --> 00:49:22.336 So 1 minus 2d, sum over alpha cosine of q alpha, 00:49:22.336 --> 00:49:25.400 how does it behave? 00:49:25.400 --> 00:49:27.850 Each cosine I can start expanding around 00:49:27.850 --> 00:49:32.050 its q going to 0 as 1 minus q squared over 2, 00:49:32.050 --> 00:49:38.940 so you can see that this is going to be 1 minus 2td. 00:49:38.940 --> 00:49:43.466 And then I would have plus t q squared, 00:49:43.466 --> 00:49:46.612 because I will have q1 squared plus q2 squared plus qd. 00:49:46.612 --> 00:49:50.600 So this q squared is the sum of all the q's. 00:49:50.600 --> 00:49:55.048 And then I do have higher order terms. 00:49:55.048 --> 00:49:59.496 Order of q to fourth, and so forth. 00:49:59.496 --> 00:50:00.052 OK? 00:50:00.052 --> 00:50:07.031 So if I'm looking in the vicinity of t going to tc, 00:50:07.031 --> 00:50:10.565 this is roughly tc q squared. 00:50:10.565 --> 00:50:18.115 This is something that goes to 0 and once I factor out the tc, 00:50:18.115 --> 00:50:22.840 I can define a length squared in this fashion, 00:50:22.840 --> 00:50:23.881 inverse length squared. 00:50:23.881 --> 00:50:24.381 OK? 00:50:24.381 --> 00:50:31.070 PROFESSOR: You can see that this inverse length squared is 00:50:31.070 --> 00:50:42.133 going to be 1 over tc 1 minus 2d, which is 1 over tc times t. 00:50:42.133 --> 00:50:51.220 So you can see that this is none other than tc minus t. 00:50:51.220 --> 00:50:54.576 And if I'm looking at the vicinity of that point, 00:50:54.576 --> 00:50:58.491 I find that the correlation between sigma 0 sigma r 00:50:58.491 --> 00:51:01.576 is approximately the integral ddq 00:51:01.576 --> 00:51:09.680 2 pi to the d, Fourier transform of the denominator that I said 00:51:09.680 --> 00:51:16.080 is approximately 1 over tc times q squared plus z 00:51:16.080 --> 00:51:18.650 to the minus 2. 00:51:18.650 --> 00:51:20.938 We've seen this before. 00:51:20.938 --> 00:51:24.370 You evaluated this Fourier transform when 00:51:24.370 --> 00:51:27.240 you were doing Landau Ginzburg. 00:51:27.240 --> 00:51:32.140 So this is something that grows when 00:51:32.140 --> 00:51:36.340 you are looking at distances that 00:51:36.340 --> 00:51:39.924 are much less than this correlation 00:51:39.924 --> 00:51:43.200 length as the Coulomb power law. 00:51:43.200 --> 00:51:46.480 When you are looking at distances 00:51:46.480 --> 00:51:50.920 that are much larger than the correlation event, 00:51:50.920 --> 00:51:56.006 you get the exponential decay with this r 00:51:56.006 --> 00:52:02.990 to the d minus 1 over 2 factor. 00:52:02.990 --> 00:52:07.638 So what we find is that the correlation 00:52:07.638 --> 00:52:11.124 of these phantom loops is precisely 00:52:11.124 --> 00:52:16.676 the correlation that we had seen for the Gaussian model, 00:52:16.676 --> 00:52:17.770 in fact. 00:52:17.770 --> 00:52:21.610 It has a correlation length that diverges 00:52:21.610 --> 00:52:24.501 in precisely the same way that we 00:52:24.501 --> 00:52:28.640 had seen for the Gaussian model with the square root 00:52:28.640 --> 00:52:29.255 singularity. 00:52:29.255 --> 00:52:36.020 So this is our usual mu equals 1/2 type of behavior 00:52:36.020 --> 00:52:37.870 that we've seen. 00:52:37.870 --> 00:52:43.456 And somehow, by all of these routes, 00:52:43.456 --> 00:52:50.640 we have reproduced some property of the Gaussian model. 00:52:50.640 --> 00:52:54.600 In fact, it's a little bit more than that, 00:52:54.600 --> 00:52:59.000 because we can go back and look at what we 00:52:59.000 --> 00:53:01.946 had here for the free energy. 00:53:01.946 --> 00:53:06.860 So let's erase the things that pertain to the correlation 00:53:06.860 --> 00:53:15.188 length and correlations, and focus on the calculation 00:53:15.188 --> 00:53:26.040 that we kind of left in the middle over here. 00:53:26.040 --> 00:53:28.885 So what do we have? 00:53:28.885 --> 00:53:34.575 We have that log of S prime, the intensive part, 00:53:34.575 --> 00:53:39.662 is a sum over the lengths of these loops 00:53:39.662 --> 00:53:43.568 that start and end at the origin. 00:53:43.568 --> 00:53:48.225 And the contribution of a loop of length l 00:53:48.225 --> 00:53:50.840 is small t to the l. 00:53:50.840 --> 00:53:54.332 And since w of l is the connectivity matrix 00:53:54.332 --> 00:53:59.268 to the l power, it's really like looking at the matrix element 00:53:59.268 --> 00:54:00.696 of this entity. 00:54:00.696 --> 00:54:05.537 And of course, there is this degeneracy factor of 2l. 00:54:05.537 --> 00:54:11.010 And I can write this as 1/2 sum over l. 00:54:11.010 --> 00:54:18.552 Well, let's do it this way-- the 0 th, 00:54:18.552 --> 00:54:29.460 0 th element of sum over l dt to the l over l. 00:54:29.460 --> 00:54:33.600 And what is this? 00:54:33.600 --> 00:54:40.845 This is, in fact, the series expansion 00:54:40.845 --> 00:54:48.090 of minus log of 1 minus dt. 00:54:48.090 --> 00:54:56.298 So I can, again, go to the Fourier basis, 00:54:56.298 --> 00:55:07.370 write this as minus 1/2 sum over q0, 0 q log of 1 00:55:07.370 --> 00:55:12.520 minus t the eigenvalue t of q, and then q0. 00:55:12.520 --> 00:55:17.808 Each one of these is just the factor of 1 00:55:17.808 --> 00:55:21.048 over square root of n. 00:55:21.048 --> 00:55:28.176 The sum over q goes over to n integral over q. 00:55:28.176 --> 00:55:36.424 So this simply becomes minus 1/2 integral over q 2 pi 00:55:36.424 --> 00:55:48.580 to the d log of 1 minus t, this sum over alpha cosine of q 00:55:48.580 --> 00:55:55.790 alpha [INAUDIBLE] 2 that we had over here. 00:55:55.790 --> 00:56:00.020 And again, if I go to this limit where 00:56:00.020 --> 00:56:05.200 I am close to tc, the critical value of this t, 00:56:05.200 --> 00:56:08.200 and focus on the behavior as q goes to 0, 00:56:08.200 --> 00:56:10.300 this is going to be something that 00:56:10.300 --> 00:56:15.013 has this q squared plus c to the minus 2 type of singularity. 00:56:15.013 --> 00:56:17.997 And again, this is a kind of integral 00:56:17.997 --> 00:56:24.020 that we saw in connection with the Gaussian model. 00:56:24.020 --> 00:56:30.380 And we know the kind of singularities it gives. 00:56:30.380 --> 00:56:37.350 But why did we end up with the Gaussian model? 00:56:37.350 --> 00:56:39.441 Let's work backward. 00:56:39.441 --> 00:56:43.526 That is, typically, when we are doing 00:56:43.526 --> 00:56:47.846 some kind of a partition function of a Gaussian model-- 00:56:47.846 --> 00:56:54.610 let's say we have some integral over some variables phi i. 00:56:54.610 --> 00:56:58.086 Let's say we put them on the sides of a lattice. 00:56:58.086 --> 00:57:02.887 And we have e to the minus phi i some matrix m ij phi 00:57:02.887 --> 00:57:07.716 j over 2 sum over i and j implicit over there. 00:57:07.716 --> 00:57:09.472 What was the answer? 00:57:09.472 --> 00:57:13.606 Then the answer was typically proportional to 1 00:57:13.606 --> 00:57:18.480 over the determinant of this matrix to the 1/2, 00:57:18.480 --> 00:57:23.250 which, if I exponentiated, would be 00:57:23.250 --> 00:57:29.620 exponential of minus 1/2 logarithm of the determinant 00:57:29.620 --> 00:57:36.760 of this matrix. 00:57:36.760 --> 00:57:39.462 So that's the general result. And we 00:57:39.462 --> 00:57:42.936 see the result for our log of S prime 00:57:42.936 --> 00:57:47.656 is, indeed, the form of 1/2 minus 1/2 00:57:47.656 --> 00:57:50.936 of the log of something. 00:57:50.936 --> 00:57:55.528 And indeed, this sum over q corresponds 00:57:55.528 --> 00:57:58.175 to summing over the different eigenvalues. 00:57:58.175 --> 00:58:02.145 And if I were to express det m in terms 00:58:02.145 --> 00:58:05.268 of the product of its eigenvalues, 00:58:05.268 --> 00:58:08.180 it would be precisely that. 00:58:08.180 --> 00:58:12.348 So you can see that actually, what we 00:58:12.348 --> 00:58:15.995 have calculated by comparison of these two 00:58:15.995 --> 00:58:23.117 things corresponds to a matrix m ij, which 00:58:23.117 --> 00:58:33.159 is delta ij minus t times this single step connectivity matrix 00:58:33.159 --> 00:58:35.635 that I had before. 00:58:35.635 --> 00:58:38.730 So indeed, the partition function 00:58:38.730 --> 00:58:47.448 that I calculated, that I called Z prime or S prime, corresponds 00:58:47.448 --> 00:58:55.790 to doing the following-- doing an integral over phi i's 00:58:55.790 --> 00:58:57.330 from the delta ij. 00:58:57.330 --> 00:59:01.180 Essentially for each phi i, I would have a factor 00:59:01.180 --> 00:59:04.420 of minus phi i squared over 2. 00:59:04.420 --> 00:59:09.005 So essentially, I have to do this. 00:59:09.005 --> 00:59:13.600 And then from here, once it's exponentiated, 00:59:13.600 --> 00:59:30.980 I will get a factor of e to the sum over ij this t phi i phi j. 00:59:30.980 --> 00:59:39.320 So you can see that I started calculating Ising variables 00:59:39.320 --> 00:59:41.822 on this lattice. 00:59:41.822 --> 00:59:46.108 The result that I calculated for these phantom walks 00:59:46.108 --> 00:59:50.312 is actually identical if I had to replace the Ising variables 00:59:50.312 --> 00:59:52.736 with just quantities that I integrate 00:59:52.736 --> 00:59:56.648 all over the place, provided that I weigh them 00:59:56.648 --> 00:59:58.136 with this factor. 00:59:58.136 --> 01:00:01.608 So really, the difference between the Ising 01:00:01.608 --> 01:00:06.149 and what I have done here can be captured 01:00:06.149 --> 01:00:12.940 by putting a weight for the indirect integration per site. 01:00:12.940 --> 01:00:16.970 So if I really want to do Ising, the weight 01:00:16.970 --> 01:00:20.597 that I want to do for the Ising-- let's 01:00:20.597 --> 01:00:24.384 do it this way-- for phi has to have 01:00:24.384 --> 01:00:30.524 a delta function at minus 1 and a delta function at plus 1. 01:00:30.524 --> 01:00:35.718 Rather than doing that, I have calculated 01:00:35.718 --> 01:00:40.912 a w that corresponds to the Gaussian 01:00:40.912 --> 01:00:50.420 where the weight for each phi is basically a Gaussian weight. 01:00:50.420 --> 01:00:55.160 And if I really wanted to do the Landau Ginzburg, 01:00:55.160 --> 01:01:01.720 all I would need to do is to add here a phi to the 4th. 01:01:01.720 --> 01:01:05.640 The problem with this Gaussian-- the phantom system 01:01:05.640 --> 01:01:09.070 that I have-- is the same problem 01:01:09.070 --> 01:01:12.104 that we had with the Gaussian model. 01:01:12.104 --> 01:01:16.344 It only gives me one side of the phase transition. 01:01:16.344 --> 01:01:20.632 Because you see that I did all of these calculations. 01:01:20.632 --> 01:01:24.412 All of these calculations were consistent, as long 01:01:24.412 --> 01:01:30.980 as I was dealing with t that was less than tc. 01:01:30.980 --> 01:01:34.610 Once I go to t that is greater than tc, 01:01:34.610 --> 01:01:37.520 then this denominator that I had became negative. 01:01:37.520 --> 01:01:39.370 It just doesn't make sense. 01:01:39.370 --> 01:01:41.220 Correlations negative don't make sense. 01:01:41.220 --> 01:01:45.105 The log, the argument that I have to calculate here, 01:01:45.105 --> 01:01:49.155 if t is below-- is larger than 1 over 2d, 01:01:49.155 --> 01:01:53.440 then it doesn't make sense. 01:01:53.440 --> 01:01:56.531 And of course, the reason the whole theory doesn't make sense 01:01:56.531 --> 01:01:58.628 is kind of related to the instability 01:01:58.628 --> 01:02:01.414 that we have in the Gaussian model. 01:02:01.414 --> 01:02:03.404 Essentially, in the Gaussian model 01:02:03.404 --> 01:02:06.088 also, when t becomes large enough, 01:02:06.088 --> 01:02:09.864 this phi squared is not enough to remove 01:02:09.864 --> 01:02:14.158 the instability that you have for the largest eigenvalue. 01:02:14.158 --> 01:02:17.504 Physically, what that means is that we 01:02:17.504 --> 01:02:20.074 started with this taut string. 01:02:20.074 --> 01:02:23.476 And as we approached the transition, 01:02:23.476 --> 01:02:26.320 the string became more flexible. 01:02:26.320 --> 01:02:29.944 And in principle, what this instability is telling 01:02:29.944 --> 01:02:35.302 me is that you go below the transition of t greater 01:02:35.302 --> 01:02:39.320 than tc, and the string becomes something 01:02:39.320 --> 01:02:44.492 that can go over and over itself as many times, 01:02:44.492 --> 01:02:46.796 and gain entropy further and further. 01:02:46.796 --> 01:02:49.100 So it will keep going forever. 01:02:49.100 --> 01:02:51.950 There is nothing to stop it. 01:02:51.950 --> 01:02:55.833 So the phantomness condition, the cost that you pay for it, 01:02:55.833 --> 01:02:58.852 is that once you go beyond the transition, 01:02:58.852 --> 01:03:00.636 you essentially overwhelm yourself. 01:03:00.636 --> 01:03:04.870 There's just so much that is going on. 01:03:04.870 --> 01:03:12.640 There is nothing that you can do. 01:03:12.640 --> 01:03:17.080 So that's the story. 01:03:17.080 --> 01:03:23.550 Now, let's try to finally understand some of the things 01:03:23.550 --> 01:03:29.040 that we had before, like this other critical dimension of 4. 01:03:29.040 --> 01:03:31.280 Where did it come from, et cetera? 01:03:31.280 --> 01:03:36.340 You are now in the position to do things and understand 01:03:36.340 --> 01:03:36.970 things. 01:03:36.970 --> 01:03:43.466 First thing to note is, let's try 01:03:43.466 --> 01:03:52.760 to understand what this exponent mu equal to 1/2 means. 01:03:52.760 --> 01:03:59.900 So we said that if I think about having information 01:03:59.900 --> 01:04:04.898 about my site at the origin, that 01:04:04.898 --> 01:04:10.571 has to propagate so that further and further neighbors start 01:04:10.571 --> 01:04:15.880 to know what the information was at site sigma 0-- 01:04:15.880 --> 01:04:22.900 that that information can come through these paths that 01:04:22.900 --> 01:04:26.029 fluctuate, go different distances, 01:04:26.029 --> 01:04:34.741 and eventually, let's say, reach a boundary that is at size r. 01:04:34.741 --> 01:04:43.045 As we said, the contribution of each path decays exponentially, 01:04:43.045 --> 01:04:49.580 but the number of paths grows exponentially. 01:04:49.580 --> 01:04:53.792 And so for a particular t that is smaller 01:04:53.792 --> 01:04:57.068 than the critical value, I can roughly 01:04:57.068 --> 01:05:00.804 say that this falls off like this, 01:05:00.804 --> 01:05:05.700 so that there is a characteristic length, l bar. 01:05:05.700 --> 01:05:12.368 This characteristic l bar is going to be minus 1 01:05:12.368 --> 01:05:15.152 over log of 2dt. 01:05:15.152 --> 01:05:23.592 And 2dt I can write as 2d tc plus t minus tc. 01:05:23.592 --> 01:05:28.926 2 d tc is, by construction, 1. 01:05:28.926 --> 01:05:37.308 So this is minus 1 over log of 1 plus something 01:05:37.308 --> 01:05:47.320 like 2d, which is 1 over t minus tc, t minus tc over tc. 01:05:47.320 --> 01:05:53.365 Now, log of 1 plus a small number-- so if my t goes 01:05:53.365 --> 01:05:56.628 and approaches tc-- this log will behave 01:05:56.628 --> 01:05:59.436 like what I have over here. 01:05:59.436 --> 01:06:04.584 So you can see that this diverges as t minus tc 01:06:04.584 --> 01:06:07.786 to the minus 1 power. 01:06:07.786 --> 01:06:15.255 I want it, I guess, to be correct-- tc minus t, 01:06:15.255 --> 01:06:19.078 because t is less than tc. 01:06:19.078 --> 01:06:24.019 But the point is that the divergence is linear. 01:06:24.019 --> 01:06:29.662 As I approach tc, the length of these paths 01:06:29.662 --> 01:06:36.830 will grow inversely to how close I am. 01:06:36.830 --> 01:06:39.485 Now what are these paths? 01:06:39.485 --> 01:06:44.795 I start from the origin, and I randomly take steps. 01:06:44.795 --> 01:06:51.035 And I've said that the typical steps that I will get 01:06:51.035 --> 01:06:54.490 will roughly have length l bar. 01:06:54.490 --> 01:06:58.338 How far have these paths carried the information? 01:06:58.338 --> 01:07:03.226 These are random walks, so the distance over which they 01:07:03.226 --> 01:07:06.304 have managed to carry the information, 01:07:06.304 --> 01:07:10.921 c, is going to be like the square root 01:07:10.921 --> 01:07:13.570 of the length of these walks. 01:07:13.570 --> 01:07:18.658 And since the length of the walks grows like t minus tc, 01:07:18.658 --> 01:07:24.478 this goes like tc minus t to the minus 1/2 power. 01:07:24.478 --> 01:07:31.022 So the exponent mu of 1/2 that we have been thinking about 01:07:31.022 --> 01:07:36.566 is none other than the 1/2 that you have for random walks, 01:07:36.566 --> 01:07:40.256 once you realize that what is going on 01:07:40.256 --> 01:07:44.846 is that the length of the paths that carry information 01:07:44.846 --> 01:07:51.167 essentially diverges linearly on approaching this. 01:07:51.167 --> 01:07:56.560 So that's one understanding. 01:07:56.560 --> 01:08:01.860 Now, you would say that this is the Gaussian picture. 01:08:01.860 --> 01:08:06.274 Now I know that when we calculated things 01:08:06.274 --> 01:08:14.722 to order of epsilon, we found that mu was 1/2 plus something. 01:08:14.722 --> 01:08:16.839 It became larger. 01:08:16.839 --> 01:08:19.679 So what does that mean? 01:08:19.679 --> 01:08:26.509 Well, if you have these paths, and the paths cannot cross each 01:08:26.509 --> 01:08:31.489 other-- it comes here, it has to go further away, 01:08:31.489 --> 01:08:36.341 because they are really non phantom-- then they will swell. 01:08:36.341 --> 01:08:40.400 So the exponent mu that you expect to get 01:08:40.400 --> 01:08:42.655 will be larger than 1/2. 01:08:42.655 --> 01:08:48.520 So that's what's captured in here. 01:08:48.520 --> 01:08:53.054 Well, how can I really try to capture that more 01:08:53.054 --> 01:08:53.679 mathematically? 01:08:53.679 --> 01:08:57.928 Well, I say that in the calculations that I did-- 01:08:57.928 --> 01:09:01.675 let's say when I was calculating the correlation functions sigma 01:09:01.675 --> 01:09:05.883 0 sigma r, in the approximation of phantomness, 01:09:05.883 --> 01:09:11.160 I included all paths that went from 0 to r. 01:09:11.160 --> 01:09:15.174 Among those there were paths that were crossing themselves. 01:09:15.174 --> 01:09:21.013 So I really have to subtract from that a path that 01:09:21.013 --> 01:09:23.719 comes and crosses itself. 01:09:23.719 --> 01:09:26.683 So I have to subtract that. 01:09:26.683 --> 01:09:31.623 I also had this condition that I had the numerator 01:09:31.623 --> 01:09:35.236 and denominator that cancel each other, which really means 01:09:35.236 --> 01:09:39.617 that I have to subtract the possibility of my path 01:09:39.617 --> 01:09:45.169 intersecting with another loop that is over here. 01:09:45.169 --> 01:09:53.399 And we can try to incorporate these as corrections. 01:09:53.399 --> 01:09:57.239 But we've already done that, because if I Fourier transform 01:09:57.239 --> 01:10:02.365 this object, I saw that it is this 1 over q squared 01:10:02.365 --> 01:10:03.739 plus k squared. 01:10:03.739 --> 01:10:07.410 And then we were calculating these u perturbative 01:10:07.410 --> 01:10:12.536 corrections, and we had diagrams that kind of looked like this. 01:10:12.536 --> 01:10:22.480 Oops, I guess I want to first draw the other diagram. 01:10:22.480 --> 01:10:26.920 And then we had a diagram that was like this. 01:10:26.920 --> 01:10:30.489 You remember when we were doing these phi 01:10:30.489 --> 01:10:33.081 to the 4th calculations, the corrections 01:10:33.081 --> 01:10:38.484 that we had for the propagator, which was related to the two 01:10:38.484 --> 01:10:42.524 point correlation function, were precisely these diagrams, where 01:10:42.524 --> 01:10:47.634 we were essentially subtracting factors that were set by u. 01:10:47.634 --> 01:10:52.250 Of course, the value of u could be anything, 01:10:52.250 --> 01:10:56.275 and you can see that there is really a one to one 01:10:56.275 --> 01:10:56.900 correspondence. 01:10:56.900 --> 01:11:00.752 Any of the diagrams that you had before really 01:11:00.752 --> 01:11:04.179 captures the picture of one of these paths 01:11:04.179 --> 01:11:08.337 trying to cross itself that you have to subtract. 01:11:08.337 --> 01:11:13.732 And you can sort of put a one to one mathematical correspondence 01:11:13.732 --> 01:11:15.420 between what is going on here. 01:11:15.420 --> 01:11:15.920 Yeah. 01:11:15.920 --> 01:11:18.104 AUDIENCE: So why can't we have the path 01:11:18.104 --> 01:11:19.742 in the first correction you drew? 01:11:19.742 --> 01:11:21.668 Because aren't we allowed to have 01:11:21.668 --> 01:11:24.062 four bonds that attach to one site 01:11:24.062 --> 01:11:26.960 when we're doing the original expansion? 01:11:26.960 --> 01:11:29.687 PROFESSOR: OK, so I told you at the beginning 01:11:29.687 --> 01:11:32.719 that you should keep track of all of my mistakes. 01:11:32.719 --> 01:11:35.005 And that's a very subtle thing. 01:11:35.005 --> 01:11:39.199 So what you are asking is, in the original Ising model, 01:11:39.199 --> 01:11:45.019 I can draw perfectly OK a graph such as this that has 01:11:45.019 --> 01:11:47.444 an intersection such as this. 01:11:47.444 --> 01:11:51.684 As we will show next time-- so bear with me-- 01:11:51.684 --> 01:11:55.019 in calculating things while in the phantom condition, 01:11:55.019 --> 01:11:59.149 this is counted three times as much as it should. 01:11:59.149 --> 01:12:03.815 So I have to subtract that, because a walk that comes here 01:12:03.815 --> 01:12:06.300 can either go forward, up, or down. 01:12:06.300 --> 01:12:09.090 There is some degeneracy there that essentially, this 01:12:09.090 --> 01:12:11.786 has done an over counting that is important, 01:12:11.786 --> 01:12:15.980 and I have to correct for when I do things more 01:12:15.980 --> 01:12:17.640 carefully next time around. 01:12:17.640 --> 01:12:18.140 Yes. 01:12:18.140 --> 01:12:20.072 AUDIENCE: When you did the Gaussian model, 01:12:20.072 --> 01:12:22.559 we never had to put any sort of requirement 01:12:22.559 --> 01:12:24.360 on the lattice being a square lattice. 01:12:24.360 --> 01:12:25.340 PROFESSOR: No. 01:12:25.340 --> 01:12:27.491 AUDIENCE: Didn't we have to do that here when 01:12:27.491 --> 01:12:28.690 you did those random walks? 01:12:28.690 --> 01:12:33.140 PROFESSOR: No, I only use the square condition or hypercube 01:12:33.140 --> 01:12:36.522 condition in order to be able to write 01:12:36.522 --> 01:12:37.926 this in general dimension. 01:12:37.926 --> 01:12:40.734 I could very well have done triangular, FCC, 01:12:40.734 --> 01:12:44.648 or any other lattice. 01:12:44.648 --> 01:12:57.480 The expression here would have been more complicated. 01:12:57.480 --> 01:13:06.560 So finally, we can also ask, we have a feel 01:13:06.560 --> 01:13:11.100 from renormalization group, et cetera, 01:13:11.100 --> 01:13:15.707 that the Gaussian exponents, like mu equals to 1/2, 01:13:15.707 --> 01:13:18.940 are, in fact, good provided that you 01:13:18.940 --> 01:13:20.875 are in sufficiently high dimension-- 01:13:20.875 --> 01:13:23.197 if you are above four dimensions. 01:13:23.197 --> 01:13:33.384 Where did you see that occurring in this picture? 01:13:33.384 --> 01:13:42.019 The answer is as follows. 01:13:42.019 --> 01:13:51.109 So basically, I have ignored the possibility of intersections. 01:13:51.109 --> 01:13:58.652 So let's see when that condition is roughly good. 01:13:58.652 --> 01:14:03.098 The kind of entities that I have as I 01:14:03.098 --> 01:14:08.683 get closer and closer to tc in the phantom case 01:14:08.683 --> 01:14:11.519 are these random walks. 01:14:11.519 --> 01:14:15.139 And we said that the characteristic of a random walk 01:14:15.139 --> 01:14:19.934 is that if I have something that carries l steps, 01:14:19.934 --> 01:14:30.296 that the typical size in space that it will grow scales 01:14:30.296 --> 01:14:35.006 like l to the 1/2. 01:14:35.006 --> 01:14:41.770 So we can recast this as a dimension. 01:14:41.770 --> 01:14:49.240 Basically, we are used to say linear objects having 01:14:49.240 --> 01:15:00.059 a mass that grows-- what do I want to do? 01:15:00.059 --> 01:15:07.717 Let's say that I have a hypercube 01:15:07.717 --> 01:15:18.657 of size L. Let's actually call it size R. Then 01:15:18.657 --> 01:15:22.816 the mass of this, or the number of elements 01:15:22.816 --> 01:15:31.650 that constitute this object, grow like R to the d. 01:15:31.650 --> 01:15:35.731 So if I take my random walk, and think of it 01:15:35.731 --> 01:15:39.194 as something in which every step has unit mass, 01:15:39.194 --> 01:15:44.034 you would say that the l is proportional to mass 01:15:44.034 --> 01:15:48.404 so that the radius grows like the number of elements 01:15:48.404 --> 01:15:51.979 to the 1/2 power or the mass of the 1/2 power. 01:15:51.979 --> 01:15:55.081 So you would say that the random walk, 01:15:55.081 --> 01:16:01.412 if I want to force it in terms of a relationship between mass 01:16:01.412 --> 01:16:06.691 and radius, that mass goes like radius squared. 01:16:06.691 --> 01:16:14.579 So in that sense, you can say that the random walk 01:16:14.579 --> 01:16:28.170 has a fractal or Hausdorff dimension of 2. 01:16:28.170 --> 01:16:32.967 So if you kind of are very, very blind, 01:16:32.967 --> 01:16:38.300 you would say that this is like a random walk. 01:16:38.300 --> 01:16:41.065 It's a two dimensional thing. 01:16:41.065 --> 01:16:42.730 It's a page. 01:16:42.730 --> 01:16:49.770 So now the question is, if I have two geometrical entities, 01:16:49.770 --> 01:16:51.690 will they intersect? 01:16:51.690 --> 01:16:58.676 So if I have a plane and a line in three dimensions, 01:16:58.676 --> 01:17:00.884 they will barely intersect. 01:17:00.884 --> 01:17:05.409 In four dimensions, they won't intersect. 01:17:05.409 --> 01:17:10.557 If I have two surfaces that are two dimensional, in three 01:17:10.557 --> 01:17:13.380 dimensions, they intersect in a line. 01:17:13.380 --> 01:17:16.521 In four dimensions, they would intersect in a point. 01:17:16.521 --> 01:17:19.701 And in five dimensions, they won't generically intersect, 01:17:19.701 --> 01:17:24.579 like two lines generically don't intersect in three dimensions. 01:17:24.579 --> 01:17:31.146 So if you ask, how bad is it that I ignored 01:17:31.146 --> 01:17:36.699 the intersection of objects that are inherently random walking 01:17:36.699 --> 01:17:40.124 in sufficiently high dimensions, I 01:17:40.124 --> 01:17:43.549 would say the answer geometrically 01:17:43.549 --> 01:17:50.189 would be in intersection is generic if d 01:17:50.189 --> 01:17:56.989 is less than 2 df, which is 4. 01:17:56.989 --> 01:18:02.587 So we made a very drastic assumption. 01:18:02.587 --> 01:18:08.054 But as long as we are above four dimensions, it's OK. 01:18:08.054 --> 01:18:12.088 There's so much space around that 01:18:12.088 --> 01:18:17.183 statistically, these intersections, this non 01:18:17.183 --> 01:18:23.601 phantomness, is so entropically constraining that it never 01:18:23.601 --> 01:18:24.100 happens. 01:18:24.100 --> 01:18:28.033 You can ignore it, and the results are OK. 01:18:28.033 --> 01:18:31.969 But you go to four dimensions and below, you 01:18:31.969 --> 01:18:34.503 can't ignore it, because generically, these things 01:18:34.503 --> 01:18:36.313 will intersect with each other. 01:18:36.313 --> 01:18:40.441 That's why these diagrams are going to blow up on you, 01:18:40.441 --> 01:18:43.509 and give you some important contribution that would swell, 01:18:43.509 --> 01:18:47.019 and give you a value of mu that is larger than the 1/2 01:18:47.019 --> 01:18:52.739 that we have for random walks. 01:18:52.739 --> 01:18:57.119 So that's the essence of where the Gaussian model was, 01:18:57.119 --> 01:19:00.955 why we get mu equals to 1/2, why we 01:19:00.955 --> 01:19:03.555 get mu's that are larger than 1/2, what 01:19:03.555 --> 01:19:06.570 the meaning of these diagrams is, what four dimensions 01:19:06.570 --> 01:19:07.392 is special. 01:19:07.392 --> 01:19:11.913 All of it really just comes down to central limit theorem 01:19:11.913 --> 01:19:15.493 and knowing that the sum of a large number of variables 01:19:15.493 --> 01:19:19.822 that has a square root of n type of variance and fluctuations. 01:19:19.822 --> 01:19:23.579 And it's all captured by that. 01:19:23.579 --> 01:19:29.177 But we wanted to really solve the model exactly. 01:19:29.177 --> 01:19:35.322 It turns out that we can make the conditions that 01:19:35.322 --> 01:19:39.586 were very hard to implement in general dimensions 01:19:39.586 --> 01:19:43.329 to work out correctly in two dimensions. 01:19:43.329 --> 01:19:46.489 And so the next lecture will show you 01:19:46.489 --> 01:19:49.649 what these mistakes are, how to avoid them, 01:19:49.649 --> 01:19:54.600 and how to get the exact value of this sum in two dimensions.