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. 00:00:22.210 --> 00:00:23.940 PROFESSOR: OK, let's start. 00:00:26.580 --> 00:00:29.970 So back to our Ising model. 00:00:35.870 --> 00:00:40.770 And for this lecture, focusing mostly on a square like this. 00:00:48.200 --> 00:00:50.765 At each side, we have a binary variable. 00:00:54.570 --> 00:01:00.110 And a weight that tries to keep neighboring 00:01:00.110 --> 00:01:05.790 the sides to be parallel. 00:01:05.790 --> 00:01:11.480 So every pair of neighboring sides 00:01:11.480 --> 00:01:14.980 is subject to a joined fate such as that. 00:01:14.980 --> 00:01:18.350 You're going to ignore the magnetic field. 00:01:18.350 --> 00:01:20.780 And to calculate a partition function, 00:01:20.780 --> 00:01:26.140 we have to sum over r 2 to the n configuration on both sides. 00:01:26.140 --> 00:01:30.900 And this will be a partition function that will [INAUDIBLE] 00:01:30.900 --> 00:01:37.070 on this one parameter k, which is some energy divided by kt. 00:01:37.070 --> 00:01:42.080 So then what we did was you we rewrote 00:01:42.080 --> 00:01:46.320 each one of these factors as a hyperbolic cosine 1 plus 00:01:46.320 --> 00:01:50.270 tangent and took out all of the factors 00:01:50.270 --> 00:01:54.260 of the hyperbolic cosine in the outside. 00:01:54.260 --> 00:01:56.670 And for the case of this square lattice, 00:01:56.670 --> 00:02:00.890 each side has two bonds going out of that. 00:02:00.890 --> 00:02:04.430 So there's 2 to the n, and I'll have to sum r 2 00:02:04.430 --> 00:02:10.030 to the n configurations product of what all bonds. 00:02:10.030 --> 00:02:15.870 These factors of 1 plus t sigma i sigma j, 00:02:15.870 --> 00:02:18.670 where this T is, of course, my shorthand 00:02:18.670 --> 00:02:20.100 for hyperbolic function. 00:02:25.680 --> 00:02:30.890 Now we saw that basically what we need to do 00:02:30.890 --> 00:02:35.620 is either we take 1-- that's the t right there-- 00:02:35.620 --> 00:02:40.080 or terms that have these factors of t sigma sigma. 00:02:40.080 --> 00:02:44.130 But in order to ensure that they will survive the summation 00:02:44.130 --> 00:02:48.780 over the two possibilities of sigma, each one of these terms 00:02:48.780 --> 00:02:51.454 has to be matched with another one. 00:02:51.454 --> 00:02:57.260 And so the most trivial diagram would be something like this. 00:02:57.260 --> 00:03:01.660 And then sum over the two possibilities at each side 00:03:01.660 --> 00:03:03.750 will give us a factor of n per side. 00:03:03.750 --> 00:03:09.660 So you have 2 to the n [INAUDIBLE] to the power of 2n. 00:03:09.660 --> 00:03:22.540 And then we have a sum over graphs which have zero, two, 00:03:22.540 --> 00:03:26.950 or four bonds emanating per site. 00:03:34.040 --> 00:03:36.850 So that this summation over the sigmas 00:03:36.850 --> 00:03:42.090 will give us a factor of 2 and another factor of 0. 00:03:42.090 --> 00:03:45.390 And then we just have to count t to the power 00:03:45.390 --> 00:03:47.944 of the number of 1's in the graph. 00:03:52.930 --> 00:03:56.400 Now we said that all of the exciting things 00:03:56.400 --> 00:04:03.520 have to do with this sum, which depends on this parameter t, 00:04:03.520 --> 00:04:04.040 of course. 00:04:07.390 --> 00:04:16.089 And this sum, as written here, has 00:04:16.089 --> 00:04:19.800 graphs such as the one that I indicated, but also 00:04:19.800 --> 00:04:23.460 potentially graphs that are more complicated which 00:04:23.460 --> 00:04:27.170 each have these joined pieces. 00:04:27.170 --> 00:04:33.360 And we've attempted to replace this sum s with another sum, s 00:04:33.360 --> 00:04:45.340 prime, which is the sum over gas of phantom loops. 00:04:45.340 --> 00:04:48.740 Let's call it multiple phantom loops. 00:04:55.330 --> 00:04:59.130 Essentially, if I allow these loops to basically go 00:04:59.130 --> 00:05:07.820 through each other, then I can exponentiate this and write it 00:05:07.820 --> 00:05:18.611 as a sum over all single phantom loops thereby removing, 00:05:18.611 --> 00:05:24.270 the factors of n squared, n cubed, et cetera, 00:05:24.270 --> 00:05:27.970 that will arise by moving the destroyed pieces 00:05:27.970 --> 00:05:30.670 all over the lattice. 00:05:30.670 --> 00:05:34.930 This sum, since I can essentially 00:05:34.930 --> 00:05:37.920 pick it apart particular loop and then slide it once 00:05:37.920 --> 00:05:41.122 over the lattice, is certainly extensive proportional to n. 00:05:44.290 --> 00:05:45.930 So this would be nice. 00:05:45.930 --> 00:05:50.030 We calculate this sum as prime last time, 00:05:50.030 --> 00:05:53.630 and we saw that it actually reproduced for us the Gaussian 00:05:53.630 --> 00:05:54.255 model. 00:05:54.255 --> 00:05:57.368 So this was equivalent to the Gaussian model. 00:06:01.120 --> 00:06:06.510 And in particular, because we were allowing 00:06:06.510 --> 00:06:09.880 this phantom condition, when we went to sufficiently 00:06:09.880 --> 00:06:13.700 low temperature or when t became sufficiently large, 00:06:13.700 --> 00:06:15.740 essentially the model was unstable 00:06:15.740 --> 00:06:17.550 and because I could just continue 00:06:17.550 --> 00:06:20.650 to put more and more of these loops. 00:06:20.650 --> 00:06:22.855 There was no condition that would 00:06:22.855 --> 00:06:24.490 say you should have a finite density. 00:06:24.490 --> 00:06:27.630 And then the density of the loops would go to infinity. 00:06:27.630 --> 00:06:29.885 And just like the Gaussian model, 00:06:29.885 --> 00:06:34.330 it doesn't make sense beyond some point. 00:06:34.330 --> 00:06:40.650 Now mathematically, it is clear to us 00:06:40.650 --> 00:06:46.600 that s is not equal to s prime because 00:06:46.600 --> 00:06:53.220 of two important reasons, one of which 00:06:53.220 --> 00:07:08.090 is obvious is multiple occupation of a bond that 00:07:08.090 --> 00:07:13.290 is in the original sum that I have up here. 00:07:13.290 --> 00:07:15.500 You can see that the contribution 00:07:15.500 --> 00:07:19.430 of each point connecting neighbors 00:07:19.430 --> 00:07:22.290 is either 1 or one factor of t. 00:07:22.290 --> 00:07:25.530 I cannot have more than that. 00:07:25.530 --> 00:07:30.310 But when I am calculating phantom loops 00:07:30.310 --> 00:07:32.130 and I have self-crossings, I have 00:07:32.130 --> 00:07:33.670 some things that are very trial. 00:07:33.670 --> 00:07:37.090 Like I can start and go back on myself. 00:07:37.090 --> 00:07:39.150 That completes a walk. 00:07:39.150 --> 00:07:42.140 I can have things that are more complicated. 00:07:42.140 --> 00:07:45.900 I could have a diagram such as this 00:07:45.900 --> 00:07:51.980 that still involves crossing something twice. 00:07:51.980 --> 00:07:58.540 Or something like this is another example. 00:07:58.540 --> 00:08:04.320 Now all of these are examples where I essentially 00:08:04.320 --> 00:08:09.780 continuously moved my chalk and drew a closed loop. 00:08:09.780 --> 00:08:12.250 But also from the exponentiation, 00:08:12.250 --> 00:08:14.430 I can also generate things that have 00:08:14.430 --> 00:08:18.070 multiple loops, such as this loop that does not intersect 00:08:18.070 --> 00:08:20.870 itself but may happen to intersect 00:08:20.870 --> 00:08:24.970 on a bond with another loop. 00:08:24.970 --> 00:08:30.400 So it is partly the presence of all of these things 00:08:30.400 --> 00:08:33.780 that allow multiple occupation that ultimately 00:08:33.780 --> 00:08:35.740 leads to the instability. 00:08:40.710 --> 00:08:45.206 But I also hinted last time, in response to a question, 00:08:45.206 --> 00:08:51.525 that there is another mistake that is involved not when there 00:08:51.525 --> 00:08:56.505 is intersection at the bond, but intersection at the site. 00:09:05.340 --> 00:09:07.470 This leads to over-counting. 00:09:16.000 --> 00:09:18.354 It's more subtle. 00:09:18.354 --> 00:09:21.722 Along the diagrams that I have in s, 00:09:21.722 --> 00:09:25.260 we certainly complete the good diagram 00:09:25.260 --> 00:09:33.350 such as this one where I have from this side 00:09:33.350 --> 00:09:37.180 four bonds going out farther than the usual bonds. 00:09:37.180 --> 00:09:40.030 So this is certainly OK. 00:09:40.030 --> 00:09:43.620 But when I want to represent that 00:09:43.620 --> 00:09:46.610 in terms of blocks on the lattice, 00:09:46.610 --> 00:09:48.460 I notice that I can do the following. 00:09:48.460 --> 00:09:51.750 Let's say in all cases, I start from here, 00:09:51.750 --> 00:09:59.240 I can start from there, I can go and do something like this, 00:09:59.240 --> 00:10:02.260 and come back to my starting point. 00:10:02.260 --> 00:10:13.070 Or I can do something like this, and back to the starting point. 00:10:13.070 --> 00:10:18.300 Or I could have a diagram that would appear 00:10:18.300 --> 00:10:23.090 at the second order in the expansion of the exponential 00:10:23.090 --> 00:10:29.150 that I have that involves two loops that would correspond 00:10:29.150 --> 00:10:32.210 to the same geometry. 00:10:32.210 --> 00:10:37.190 So this thing that should have been counted once 00:10:37.190 --> 00:10:41.060 is really being counted three times. 00:10:41.060 --> 00:10:45.950 And so that's a mistake to be corrected. 00:10:45.950 --> 00:10:49.990 And actually, the reason for this factor of three 00:10:49.990 --> 00:10:52.675 goes back to this Gaussian model that I 00:10:52.675 --> 00:10:56.750 said that essentially what amounts to is 00:10:56.750 --> 00:11:04.160 that we are replacing Ising variables over here 00:11:04.160 --> 00:11:08.490 with these Gaussian variables, s. 00:11:08.490 --> 00:11:12.810 And so we arrange things such that the average of s 00:11:12.810 --> 00:11:17.440 squared was 1, so that the sides were reproducing 00:11:17.440 --> 00:11:19.850 the things that I had before. 00:11:19.850 --> 00:11:21.490 But then for something like this, 00:11:21.490 --> 00:11:25.630 I need to know the average s to the fourth. 00:11:25.630 --> 00:11:27.690 And if you use Wick's theorem, you 00:11:27.690 --> 00:11:30.205 can quickly see that the average of s to the fourth 00:11:30.205 --> 00:11:32.640 is three times average of s squared, 00:11:32.640 --> 00:11:34.700 so you get a factor of 3 here. 00:11:34.700 --> 00:11:37.740 That's the origin of the factor of 3. 00:11:37.740 --> 00:11:44.880 And indeed, if you had gone and done rather than one component, 00:11:44.880 --> 00:11:48.370 s, something that was n component, 00:11:48.370 --> 00:11:54.830 you could convince yourself that this becomes n plus 2. 00:11:54.830 --> 00:11:58.580 And that's consistent whenever you 00:11:58.580 --> 00:12:03.250 see something that has a loop having a factor of n, 00:12:03.250 --> 00:12:05.170 something that we had seen in we were 00:12:05.170 --> 00:12:08.760 doing this diagramatic expansion. 00:12:08.760 --> 00:12:12.030 OK, so that's a problem. 00:12:12.030 --> 00:12:15.020 So s is not equal to s prime. 00:12:17.830 --> 00:12:26.780 But I want to make the following very nice assertion 00:12:26.780 --> 00:12:47.420 that s is indeed the sum over multiple phantom loops, 00:12:47.420 --> 00:12:51.540 just like I had written for s prime with a couple 00:12:51.540 --> 00:12:54.375 of important constraints. 00:12:54.375 --> 00:12:56.900 The constraints are-- maybe I should write them 00:12:56.900 --> 00:12:59.850 in red-- with no U-turns. 00:13:04.970 --> 00:13:08.790 That is, you are not going to allow anything such as this 00:13:08.790 --> 00:13:14.200 or this when you would go forward 00:13:14.200 --> 00:13:16.545 and then immediately step backward. 00:13:16.545 --> 00:13:17.920 That's what I will call a U-turn. 00:13:17.920 --> 00:13:20.888 No U-turn is allowed. 00:13:20.888 --> 00:13:27.710 And more importantly, with a factor of minus 1 00:13:27.710 --> 00:13:36.000 to the number of crossings. 00:13:45.310 --> 00:13:48.630 So what do I mean by crossing? 00:13:48.630 --> 00:13:53.630 If you follow what I did in drawing this diagram, 00:13:53.630 --> 00:13:55.470 you can see that there was a path 00:13:55.470 --> 00:14:00.870 that I drew that never crossed itself, whereas here I 00:14:00.870 --> 00:14:03.700 kind of had to jump over where I was. 00:14:03.700 --> 00:14:06.280 I indicated that. 00:14:06.280 --> 00:14:10.550 So according to this rule, this diagram 00:14:10.550 --> 00:14:13.190 will get a factor of minus. 00:14:13.190 --> 00:14:16.640 These two diagrams don't have any self-crossings. 00:14:16.640 --> 00:14:18.970 They give factors of plus. 00:14:18.970 --> 00:14:22.800 And you can see that at least this particular diagram 00:14:22.800 --> 00:14:24.800 is resolved. 00:14:24.800 --> 00:14:29.130 And you can see that this continues 00:14:29.130 --> 00:14:31.725 to more complicated things. 00:14:31.725 --> 00:14:36.040 Let's say that I have a perfectly good diagram that 00:14:36.040 --> 00:14:39.975 is something like this that involves two crossings. 00:14:42.500 --> 00:14:48.390 And then I can break it roughly into two process, 00:14:48.390 --> 00:14:53.340 and each piece I can decompose as I had done before. 00:14:55.920 --> 00:15:09.460 Let's see, the left part I can either do this without crossing 00:15:09.460 --> 00:15:21.270 or I could cross or I could have this part as a separate loop 00:15:21.270 --> 00:15:22.160 from this half. 00:15:25.500 --> 00:15:29.655 And then I can join it on the other side 00:15:29.655 --> 00:15:34.380 with essentially any one of these things like this, which 00:15:34.380 --> 00:15:45.230 comes with a plus; this, which comes with a minus; 00:15:45.230 --> 00:15:53.580 or this one that comes with a plus. 00:15:53.580 --> 00:15:57.580 So you can see that this particular diagram 00:15:57.580 --> 00:16:05.490 of s in s prime would arise in nine possible ways. 00:16:05.490 --> 00:16:07.600 So I would have had an over-counting 00:16:07.600 --> 00:16:13.020 by a factor of a three-paired crossing or nine total, 00:16:13.020 --> 00:16:19.650 except that now when I assign these factors, 00:16:19.650 --> 00:16:23.115 some of these diagrams will come with minus, some of them 00:16:23.115 --> 00:16:24.450 will come with plus. 00:16:24.450 --> 00:16:28.210 But ultimately, only one will survive. 00:16:28.210 --> 00:16:29.240 This is it. 00:16:29.240 --> 00:16:32.690 All of them survive, then the net contribution is just 1, 00:16:32.690 --> 00:16:33.720 which is the correct on. 00:16:41.240 --> 00:16:41.840 Now let's see. 00:16:41.840 --> 00:16:55.270 So this removed problem B. so this was problem B resolved. 00:16:55.270 --> 00:17:01.420 Let's see about our problem A, which 00:17:01.420 --> 00:17:05.940 had to do with multiple occupation 00:17:05.940 --> 00:17:09.060 of the particular bond. 00:17:09.060 --> 00:17:13.670 So these are diagrams that will appear in s prime 00:17:13.670 --> 00:17:18.385 that have no counterpart in s. 00:17:18.385 --> 00:17:21.990 That is, let's say there is this bond that 00:17:21.990 --> 00:17:26.370 is occupied twice, so that would be 00:17:26.370 --> 00:17:29.890 a contribution by a factor of t squared. 00:17:29.890 --> 00:17:33.560 And then this part can go and join whatever it wants. 00:17:33.560 --> 00:17:36.390 This part can go and do whatever it wants. 00:17:39.370 --> 00:17:44.770 The point is that for every diagram such as this, 00:17:44.770 --> 00:17:49.670 I can construct a diagram where I leave everything out 00:17:49.670 --> 00:17:55.500 here exactly as it was, everything out 00:17:55.500 --> 00:18:00.310 here exactly as it was, except that the two terminals 00:18:00.310 --> 00:18:02.820 that I'll have to the left and the two terminals 00:18:02.820 --> 00:18:07.070 that I have to the right of the bond, rather than joining them 00:18:07.070 --> 00:18:10.850 this way, I'll join them like this. 00:18:14.580 --> 00:18:18.930 So this complicated diagram, as far as this bond is concerned, 00:18:18.930 --> 00:18:27.160 I can prescribe in two different ways using graphs of s prime. 00:18:27.160 --> 00:18:29.910 And one of them with respect to the other 00:18:29.910 --> 00:18:33.150 has an additional factor of minus 1, 00:18:33.150 --> 00:18:36.730 and so they give me this 0. 00:18:36.730 --> 00:18:41.070 And you can convince yourself that the same construction will 00:18:41.070 --> 00:18:45.430 hold if I have three terminals, four terminals-- it doesn't 00:18:45.430 --> 00:18:45.930 matter. 00:18:45.930 --> 00:18:50.470 I will do it for one pair and it will be OK. 00:18:50.470 --> 00:18:53.950 The only time that I wouldn't have been able to 00:18:53.950 --> 00:18:57.440 is if I have to terminals on one side and one terminal 00:18:57.440 --> 00:19:02.450 on the other side, which is these guys. 00:19:02.450 --> 00:19:05.850 And that's what I said no U-turns. 00:19:05.850 --> 00:19:08.820 So now that's taken care of first. 00:19:08.820 --> 00:19:11.620 So this problem A is now also resolved. 00:19:15.270 --> 00:19:18.580 So what do we have? 00:19:18.580 --> 00:19:25.900 We have now established that s-- let me just get a 0. 00:19:25.900 --> 00:19:45.550 Let's define this loop star to be the loops with no U-turns 00:19:45.550 --> 00:19:49.840 and minus 1 to the number of crossings. 00:19:49.840 --> 00:19:53.470 So this star is going to symbolize 00:19:53.470 --> 00:19:58.100 that these two constraints of no U-turns and minus 1 00:19:58.100 --> 00:20:00.630 to the power of number of crossings 00:20:00.630 --> 00:20:04.880 are imposed in construction and calculation of the contribution 00:20:04.880 --> 00:20:07.390 of these objects. 00:20:07.390 --> 00:20:09.650 So then what do I have? 00:20:09.650 --> 00:20:16.760 I have that s is-- well, there's possibility of no loop. 00:20:16.760 --> 00:20:19.410 There is the one loop graphs. 00:20:19.410 --> 00:20:21.190 There's the two loops star graphs. 00:20:21.190 --> 00:20:24.770 My little loop star graphs. 00:20:24.770 --> 00:20:28.680 And just as in the case of s prime, 00:20:28.680 --> 00:20:37.095 I can exponentiate this as the sum of all one stars. 00:20:40.010 --> 00:20:42.960 And you may wonder what happens when 00:20:42.960 --> 00:20:45.350 I go to higher order terms. 00:20:45.350 --> 00:20:47.545 In higher order terms, potentially I 00:20:47.545 --> 00:20:50.870 will generate two of these things that 00:20:50.870 --> 00:20:55.270 cross when I go to a second order term. 00:20:55.270 --> 00:20:59.190 But then any intersection will involve two, four, 00:20:59.190 --> 00:21:00.840 and even number. 00:21:00.840 --> 00:21:02.950 Minus 1 to an even number is one, 00:21:02.950 --> 00:21:07.230 so there is really no additional interaction 00:21:07.230 --> 00:21:13.240 to worry about if multiple things are crossing each other. 00:21:19.750 --> 00:21:25.190 So we could exponentiate that s of t that we have over there. 00:21:25.190 --> 00:21:27.430 We are interested in taking the log 00:21:27.430 --> 00:21:30.670 of the expression for the partition function. 00:21:30.670 --> 00:21:39.270 I will do the factor of n log 2 hyperbolic cosine squared, 00:21:39.270 --> 00:21:43.490 because there are essentially two bonds per side. 00:21:43.490 --> 00:21:52.393 And then I have a sum over these loop stars. 00:22:01.110 --> 00:22:03.260 Essentially, I took the log of the expression 00:22:03.260 --> 00:22:04.160 that I had before. 00:22:07.370 --> 00:22:11.440 Now this sum-- well, let's sort of take 00:22:11.440 --> 00:22:14.422 care of this factor of n. 00:22:14.422 --> 00:22:17.640 So I divide by n. 00:22:17.640 --> 00:22:23.966 Log z divided by n is log of 2 hyperbolic cosines [INAUDIBLE] 00:22:23.966 --> 00:22:31.720 k plus-- well, the way to get rid of the factor of n 00:22:31.720 --> 00:22:36.500 in this sum is to fix one point of this loop 00:22:36.500 --> 00:22:40.710 so that it doesn't go all over the place. 00:22:40.710 --> 00:22:50.330 So let's say that I have the number of loops 00:22:50.330 --> 00:22:54.630 that start and end at the origin. 00:22:54.630 --> 00:22:59.740 I would have to sum over the length of those loops, 00:22:59.740 --> 00:23:04.630 and those things will give me a factor of t to the l. 00:23:04.630 --> 00:23:22.340 So I have defined this n star of l to be number of loop stars 00:23:22.340 --> 00:23:28.126 from 0 to 0 in l steps. 00:23:34.610 --> 00:23:37.070 So in place of this-- actually, maybe 00:23:37.070 --> 00:23:43.170 I will emphasize it's minus 1 to the number of crossings 00:23:43.170 --> 00:23:44.505 and no U-turns. 00:23:50.010 --> 00:23:55.860 Now gain, so what I have is my entire lattice. 00:23:58.490 --> 00:24:04.960 Let's say I have in loop that is of length 4. 00:24:04.960 --> 00:24:12.030 And now I have forced it to start and end at the origin 00:24:12.030 --> 00:24:17.590 so that I can put out this factor of n. 00:24:17.590 --> 00:24:21.810 But then I realized that I could have over-counted this 00:24:21.810 --> 00:24:28.670 because this loop could have been started from here, here, 00:24:28.670 --> 00:24:32.810 here, here and translated to the origin. 00:24:32.810 --> 00:24:36.280 So just as we saw for the case of the random walk loops 00:24:36.280 --> 00:24:43.960 before, there is this factor of l to correct. 00:24:43.960 --> 00:24:45.830 And then I'm talking about walks, 00:24:45.830 --> 00:24:50.070 I can either go clockwise or counterclockwise, 00:24:50.070 --> 00:24:56.110 so I have to divide by a factor of 2 to get rid of that. 00:25:00.759 --> 00:25:01.550 AUDIENCE: Question? 00:25:01.550 --> 00:25:03.200 PROFESSOR: Yes? 00:25:03.200 --> 00:25:05.632 AUDIENCE: So what happens if you have when 00:25:05.632 --> 00:25:07.090 you're doing your exponential loops 00:25:07.090 --> 00:25:09.465 and we have one loop nested inside another loop 00:25:09.465 --> 00:25:11.340 that you're multiplying together so that they 00:25:11.340 --> 00:25:12.630 share one of their edges. 00:25:12.630 --> 00:25:15.500 It seems like then they don't have to cross twice, 00:25:15.500 --> 00:25:19.770 and so you would still need something to cancel them out. 00:25:19.770 --> 00:25:23.930 PROFESSOR: OK, I should have maybe explained 00:25:23.930 --> 00:25:26.060 that graph a little bit more. 00:25:26.060 --> 00:25:30.730 But let's do it over here. 00:25:30.730 --> 00:25:40.660 So when I exponentiate s-- what am I calling it?-- 00:25:40.660 --> 00:25:47.630 s sum of the loops, among the terms that I will generate will 00:25:47.630 --> 00:25:50.940 certainly be something like this. 00:25:50.940 --> 00:25:55.800 And then you say this one is shared between the two of them. 00:25:55.800 --> 00:25:58.970 At the level of one loop graphs, there 00:25:58.970 --> 00:26:02.090 was a one-loop object that went like this. 00:26:09.040 --> 00:26:12.530 So the statement that I made here 00:26:12.530 --> 00:26:17.140 does not necessarily map the number of loops to each other, 00:26:17.140 --> 00:26:19.665 but it is correct, and the cancellation 00:26:19.665 --> 00:26:21.497 occurs at the level of one. 00:26:29.980 --> 00:26:33.320 And actually, I should have also indicated 00:26:33.320 --> 00:26:34.880 what's happening with the other graph 00:26:34.880 --> 00:26:40.320 that I have up there because I have this graph. 00:26:40.320 --> 00:26:44.250 So that's a one-loop graph, that cancels against this graph. 00:26:50.870 --> 00:26:56.030 So whatever you do, you can just follow the rule that I gave you 00:26:56.030 --> 00:26:58.556 and ensure that the cancellation works, of course. 00:26:58.556 --> 00:26:59.410 So Thank. 00:26:59.410 --> 00:27:00.320 You. 00:27:00.320 --> 00:27:02.360 I wanted to say this and I had forgot. 00:27:08.350 --> 00:27:09.860 OK, any other questions? 00:27:16.570 --> 00:27:24.700 OK, so essentially we are back to some extent of the formula 00:27:24.700 --> 00:27:28.620 that I had for ordinary random walks. 00:27:28.620 --> 00:27:35.810 And phantom loops these are partly phantom loops, 00:27:35.810 --> 00:27:42.080 but I have to take care of something like this. 00:27:42.080 --> 00:27:46.580 What did I go originally last lecture, rather than 00:27:46.580 --> 00:27:50.800 my ordinary graphs to these phantom loops? 00:27:50.800 --> 00:27:53.640 The reason was that for a phantom loops, 00:27:53.640 --> 00:27:57.060 I said that I had this Markovian condition. 00:27:57.060 --> 00:28:01.250 I could relate l step walks to l minus 1 step walks 00:28:01.250 --> 00:28:04.530 because there was no memory. 00:28:04.530 --> 00:28:08.330 I didn't have to know where I had crossed before. 00:28:08.330 --> 00:28:11.535 But it seems that in order to give a correct date 00:28:11.535 --> 00:28:14.940 to these new loops, I have to know 00:28:14.940 --> 00:28:16.800 how many times I cross myself. 00:28:20.880 --> 00:28:25.050 And c by itself is a non-Markovian thing. 00:28:25.050 --> 00:28:28.350 It requires memory. 00:28:28.350 --> 00:28:31.270 Expect that I don't need any c. 00:28:31.270 --> 00:28:36.080 I need only the parity of the number of crossings. 00:28:36.080 --> 00:28:40.410 And here is where there's a beautiful mathematical theorem 00:28:40.410 --> 00:28:46.305 that tells us here in this memory-like problem 00:28:46.305 --> 00:28:49.310 for something that is Markovian. 00:28:49.310 --> 00:29:01.700 And the statement is a theorem of Whitney's which 00:29:01.700 --> 00:29:05.330 states that the parity of a planar 00:29:05.330 --> 00:29:07.430 loop-- the thing that I've done here-- 00:29:07.430 --> 00:29:09.160 whether it's even or odd. 00:29:20.010 --> 00:29:23.550 Parity of the number of crossings 00:29:23.550 --> 00:29:40.990 is related to the total angle through which the tangent 00:29:40.990 --> 00:29:49.710 vector turns by the following. 00:29:52.920 --> 00:30:00.280 In C mod 2, which is the parity of the crossings, 00:30:00.280 --> 00:30:06.300 is 1 plus this total angle that I 00:30:06.300 --> 00:30:09.926 will call theta divided by 2 pi mod 2. 00:30:18.870 --> 00:30:23.700 So I'm not going to give a proof of this, 00:30:23.700 --> 00:30:27.710 but I will show you examples to make sure you understand what's 00:30:27.710 --> 00:30:35.000 happening by, let's say, comparing the following two 00:30:35.000 --> 00:30:47.020 graphs, one of which has no crossing and another one which 00:30:47.020 --> 00:30:52.720 is essentially the same thing but has a crossing. 00:30:52.720 --> 00:31:06.540 So basically I put an arrow so that I 00:31:06.540 --> 00:31:11.750 can follow where the orientation of the bond, which 00:31:11.750 --> 00:31:16.180 is the location of this tangent s, as I step. 00:31:16.180 --> 00:31:20.950 Let's say from the origin, this is the first step, second step, 00:31:20.950 --> 00:31:25.450 third step, fourth step, so on and so forth. 00:31:25.450 --> 00:31:28.830 So let's do this for the upper graph. 00:31:28.830 --> 00:31:33.710 And what I will do is I will plot the angle. 00:31:36.290 --> 00:31:44.360 And the first step here, I start at 0 degrees pointing this way. 00:31:44.360 --> 00:31:47.960 So this is my step number one. 00:31:47.960 --> 00:31:51.200 At the next, I have gone to 90 degrees, 00:31:51.200 --> 00:31:53.340 so this is where I go to. 00:31:53.340 --> 00:31:57.490 At three, I'm back to pointing along 00:31:57.490 --> 00:31:59.770 the horizon in the direction. 00:31:59.770 --> 00:32:04.150 At four, I have gone back up here. 00:32:04.150 --> 00:32:10.462 At five, I go to 180 degrees. 00:32:10.462 --> 00:32:15.676 At six, I go all the way down. 00:32:15.676 --> 00:32:27.080 At seven, I go back to this horizontal. 00:32:27.080 --> 00:32:32.260 At eight, I go back to pointing down. 00:32:32.260 --> 00:32:33.720 And then I'm back to one. 00:32:37.150 --> 00:32:39.940 So if you follow what that tangent is doing, 00:32:39.940 --> 00:32:43.410 it's going woop, woop. 00:32:43.410 --> 00:32:47.260 At the end of the day, it has turned through 3 pi. 00:32:47.260 --> 00:32:51.395 So in this case, the total turn is 2 pi. 00:32:54.480 --> 00:32:59.060 1 plus the total turn divided by 2 pi in this case 00:32:59.060 --> 00:33:05.440 is 1 plus 2 pi over 2 pi, which is 2-- mod 00:33:05.440 --> 00:33:10.630 2 which is the same thing as 0. 00:33:10.630 --> 00:33:13.985 And of course, how many times has this thing crossed itself? 00:33:13.985 --> 00:33:16.316 Zero times. 00:33:16.316 --> 00:33:18.480 So let's see how it works if you go 00:33:18.480 --> 00:33:26.490 to apply the same set of rules to this other one. 00:33:26.490 --> 00:33:33.960 So again, one, two, three, four, five, six, seven, eight 00:33:33.960 --> 00:33:36.290 are my steps. 00:33:36.290 --> 00:33:42.610 One is pointing in the horizontal direction. 00:33:42.610 --> 00:33:46.250 Two goes vertical just as before. 00:33:46.250 --> 00:33:50.250 Three stays the same place, so two and three 00:33:50.250 --> 00:33:52.320 are at the same point. 00:33:52.320 --> 00:33:54.045 Four, I go back to horizontal. 00:33:56.700 --> 00:34:03.290 Five, I go down to minus 90 degrees. 00:34:03.290 --> 00:34:08.400 Six, I go all the way to 180 degrees 00:34:08.400 --> 00:34:10.920 and stay there at seven. 00:34:10.920 --> 00:34:14.719 Eight, I go back to minus 90 degrees, 00:34:14.719 --> 00:34:17.340 and then rejoin the origin. 00:34:17.340 --> 00:34:25.750 So this goes up, down, back, never completes a full turn. 00:34:25.750 --> 00:34:30.010 So in this case, theta is 0. 00:34:30.010 --> 00:34:36.989 1 plus theta over 2 pi is 1, and the number 00:34:36.989 --> 00:34:41.000 of crossings of his graph is 1. 00:34:41.000 --> 00:34:44.090 So you can go and repeat this for any graph 00:34:44.090 --> 00:34:48.380 that you like and convince yourself that this rule works. 00:34:54.100 --> 00:34:59.000 Well, how does that help us? 00:34:59.000 --> 00:35:01.280 Well, the diagrams that I have drawn here 00:35:01.280 --> 00:35:05.000 already tells you how it helps us 00:35:05.000 --> 00:35:10.800 because in order to find the total angle, 00:35:10.800 --> 00:35:16.280 all I need to do is to keep track of local changes. 00:35:16.280 --> 00:35:19.340 So essentially, as I go along, I carry a bag with me 00:35:19.340 --> 00:35:23.890 which adds the changing angle at every step. 00:35:23.890 --> 00:35:29.250 I don't need to know where I was 100 steps before. 00:35:29.250 --> 00:35:32.810 I just add another changing angle. 00:35:32.810 --> 00:35:36.430 By the time I get to the last stop, 00:35:36.430 --> 00:35:40.800 I figure out what my total angle is, and then I'm done. 00:35:44.870 --> 00:35:46.020 AUDIENCE: Yes? 00:35:46.020 --> 00:35:48.240 So is it really the total angle that 00:35:48.240 --> 00:35:50.520 matters, or is it more just the number of circles 00:35:50.520 --> 00:35:52.282 that you complete or not? 00:35:52.282 --> 00:35:53.740 PROFESSOR: They are the same thing. 00:35:53.740 --> 00:35:57.140 So if you prefer to say it in terms 00:35:57.140 --> 00:36:04.000 of the entanglement of your loop and the point at the origin, 00:36:04.000 --> 00:36:06.770 that's another way of saying it. 00:36:06.770 --> 00:36:12.050 This entity divided by 2 pi is a topological number, 00:36:12.050 --> 00:36:14.251 which counts essentially the number of times 00:36:14.251 --> 00:36:15.584 you have gone around the origin. 00:36:31.870 --> 00:36:38.530 So what I'm saying is that minus 1 to the power of the number 00:36:38.530 --> 00:36:44.090 of crossings-- this factor that I was after-- I can write 00:36:44.090 --> 00:36:50.030 as e to the i pi times the number of crossings. 00:36:50.030 --> 00:36:53.000 And it is evident that the only thing that is important 00:36:53.000 --> 00:36:58.300 here is the parity, so I can replace into the i pi 00:36:58.300 --> 00:37:02.830 the number of crossings with 1 plus this total angle divided 00:37:02.830 --> 00:37:03.590 by 2 pi. 00:37:06.190 --> 00:37:08.440 Which means I have e to the i pi, which 00:37:08.440 --> 00:37:11.110 is a factor of minus 1. 00:37:11.110 --> 00:37:17.850 And then I have e to the i theta over 2. 00:37:17.850 --> 00:37:23.020 And my statement is that this is the same thing as e 00:37:23.020 --> 00:37:30.530 to the i over 2, sum over the little bits of change of angle 00:37:30.530 --> 00:37:33.154 that people have as you go along there. 00:37:38.455 --> 00:37:47.510 So what I have to do is as I am walking 00:37:47.510 --> 00:37:51.240 around this square lattice, I better 00:37:51.240 --> 00:37:53.820 keep track of which direction I'm 00:37:53.820 --> 00:37:59.040 pointing so that I know from one step to the next step 00:37:59.040 --> 00:38:01.690 whether I change by 0 degrees, 90 degrees, 00:38:01.690 --> 00:38:04.260 minus 90 degrees, et cetera. 00:38:04.260 --> 00:38:05.940 So what do I do? 00:38:05.940 --> 00:38:07.325 I define a convention. 00:38:13.600 --> 00:38:24.125 So we are going to introduce orientation mu of step 00:38:24.125 --> 00:38:25.220 as follows. 00:38:28.050 --> 00:38:32.870 So let's say I'm at some point on the lattice i. 00:38:32.870 --> 00:38:36.605 Then I can increase the particle c along one of four directions. 00:38:39.150 --> 00:38:43.700 And I'm going to label them by mu 00:38:43.700 --> 00:38:47.936 being equal to 1, 2, 3, or 4. 00:38:47.936 --> 00:38:51.150 You can choose any notation you want. 00:38:51.150 --> 00:38:52.870 This will be the notation I will use. 00:39:00.490 --> 00:39:06.700 Secondly, I'm going to introduce the analog of the quantity 00:39:06.700 --> 00:39:11.140 that we had for the phantom random box, which was I 00:39:11.140 --> 00:39:15.550 introduced a set of matrices that were counting 00:39:15.550 --> 00:39:19.480 how many base I can go from one side 00:39:19.480 --> 00:39:23.550 to another side in l steps. 00:39:23.550 --> 00:39:30.320 So I will introduce the following notation. 00:39:30.320 --> 00:39:34.620 Something that involves the starred box that involves l 00:39:34.620 --> 00:39:36.720 steps. 00:39:36.720 --> 00:39:40.900 And I say that I start at some point xy and I 00:39:40.900 --> 00:39:44.085 end at some other point x prime y prime. 00:39:44.085 --> 00:39:46.380 So again, just counting how many ways 00:39:46.380 --> 00:39:49.550 I can go from one to another point. 00:39:49.550 --> 00:39:53.342 Except that I also want to keep track of these orientations. 00:39:57.680 --> 00:40:01.460 So this quantity is defined as follows. 00:40:01.460 --> 00:40:21.310 It is the sum over random walks that start at xy along m. 00:40:21.310 --> 00:40:24.710 That is, if this is my point, xy, 00:40:24.710 --> 00:40:27.490 and I'm looking at the second element of this, 00:40:27.490 --> 00:40:30.570 the next step I have to go up. 00:40:30.570 --> 00:40:34.180 If I have specified that mu equals to 1, 00:40:34.180 --> 00:40:38.340 it means that the first step I have to go to the right. 00:40:38.340 --> 00:40:41.410 OK, as I go further on the lattice, 00:40:41.410 --> 00:40:47.040 I ensure that I never have any U-turns, 00:40:47.040 --> 00:40:55.360 and I keep track of the factors of e 00:40:55.360 --> 00:41:02.910 to the i theta that changes as I take one step to the next step 00:41:02.910 --> 00:41:06.490 so that if I took my first step here and then next 00:41:06.490 --> 00:41:09.410 if I continued here, there would be no factor. 00:41:09.410 --> 00:41:16.550 But if I went up, I would have a factor of e to the i pi 4. 00:41:16.550 --> 00:41:19.710 So I keep track of those factors. 00:41:19.710 --> 00:41:26.480 And then I want to end at x prime y prime 00:41:26.480 --> 00:41:30.370 and head along mu prime. 00:41:33.390 --> 00:41:40.020 Since I specified that my first step will then 00:41:40.020 --> 00:41:44.430 go along mu, when I reach the last step, 00:41:44.430 --> 00:41:47.060 I already know where I came from. 00:41:47.060 --> 00:41:49.470 But depending on which direction I 00:41:49.470 --> 00:41:54.340 specify I would turn to and head for my next step, 00:41:54.340 --> 00:41:56.790 I will get the changing angle. 00:41:56.790 --> 00:42:02.510 So I have to include the changing angle somewhere so 00:42:02.510 --> 00:42:05.170 that there are l changes in angle 00:42:05.170 --> 00:42:06.970 so the way that I have defined it, 00:42:06.970 --> 00:42:08.690 it will be essentially keeping track 00:42:08.690 --> 00:42:10.690 of where the next step is headed to. 00:42:13.420 --> 00:42:17.150 So again, if I were to draw a diagram, 00:42:17.150 --> 00:42:21.900 I'll start with this xy point, and I 00:42:21.900 --> 00:42:24.180 want to arrive at some point here-- 00:42:24.180 --> 00:42:25.860 let's say x prime y prime. 00:42:28.680 --> 00:42:32.030 And I want to do in l steps. 00:42:32.030 --> 00:42:35.180 And the first step, I go along the direction 00:42:35.180 --> 00:42:37.800 that is specified by mu. 00:42:37.800 --> 00:42:47.750 So this is step 1, and there's the 2, 3, 4, 5, 6, 7, 8, 9, 10. 00:42:47.750 --> 00:42:51.980 Let's see, the last one-- so this would step l minus 1. 00:42:51.980 --> 00:42:57.200 This is the last step, arrives me at point x prime y prime. 00:42:57.200 --> 00:43:02.320 But then I have to specify what is the direction of mu prime 00:43:02.320 --> 00:43:06.120 so that I keep track of the appropriate change of angle 00:43:06.120 --> 00:43:09.370 that I have to do over here as well. 00:43:09.370 --> 00:43:11.220 So this is the procedure. 00:43:20.690 --> 00:43:25.930 Now this walk, this quantity that I have defined for you 00:43:25.930 --> 00:43:31.070 here, has the Markovian property in that 00:43:31.070 --> 00:43:36.870 if I arrive at this point after l steps, 00:43:36.870 --> 00:43:42.620 then after l minus one steps, I was at one point-- 00:43:42.620 --> 00:43:46.630 x double prime, y double prime-- from which 00:43:46.630 --> 00:43:52.360 I took one step along some direction-- mu double prime-- 00:43:52.360 --> 00:43:54.670 and arrived at this. 00:43:54.670 --> 00:44:03.120 So I can write that I have to sum over all possible locations 00:44:03.120 --> 00:44:08.530 and orientations of the before [INAUDIBLE]. 00:44:08.530 --> 00:44:12.620 I start from the point xy, proceed along 00:44:12.620 --> 00:44:20.850 direction mu for a total of l minus 1 steps, 00:44:20.850 --> 00:44:26.740 landing on x double prime y double prime, 00:44:26.740 --> 00:44:29.760 and then going in direction to mu double prime. 00:44:32.340 --> 00:44:35.810 So then I have a walk that started at x double prime, 00:44:35.810 --> 00:44:38.520 y double prime along the direction in double 00:44:38.520 --> 00:44:46.260 prime, which is one step has to get me to my destination. 00:44:46.260 --> 00:44:48.640 And once I am at the destination, 00:44:48.640 --> 00:44:50.496 I head along the direction in prime. 00:44:56.730 --> 00:45:03.170 So this is clearly a matrix, a product. 00:45:03.170 --> 00:45:06.970 And what I have established is this matrix, w star 00:45:06.970 --> 00:45:12.110 of l-- which, by the way, is a 4n by 4n matrix, 00:45:12.110 --> 00:45:14.550 because our n points and 4 orientation, 00:45:14.550 --> 00:45:19.180 so it's 4n by 4n matrix-- and I have established 00:45:19.180 --> 00:45:23.010 that that is the product of the matrix that I have 00:45:23.010 --> 00:45:26.790 for one step, and the product of the matrix that I 00:45:26.790 --> 00:45:30.940 have for l minus 1 step. 00:45:30.940 --> 00:45:40.030 And this object that I will call t star essentially tells me 00:45:40.030 --> 00:45:44.520 something about the combined connectivity orientation 00:45:44.520 --> 00:45:47.770 that I have for square lattice. 00:45:47.770 --> 00:45:50.670 And since I can repeat this many times, 00:45:50.670 --> 00:45:55.745 I can see that I have the result that I want, that w star of l 00:45:55.745 --> 00:45:59.108 is simply t star raised to the power of l. 00:46:36.970 --> 00:46:39.672 Questions? 00:46:39.672 --> 00:46:40.658 Yes? 00:46:40.658 --> 00:46:43.100 AUDIENCE: Have we accounted for loops that [INAUDIBLE] 00:46:43.100 --> 00:46:45.016 we have the square loop of four steps and then 00:46:45.016 --> 00:46:47.992 the same square loop of eight steps that are exactly 00:46:47.992 --> 00:46:49.480 on top of the four-step one? 00:46:52.470 --> 00:46:58.334 PROFESSOR: OK, so you want me to take this and do it again? 00:46:58.334 --> 00:46:59.650 AUDIENCE: Yeah. 00:46:59.650 --> 00:47:01.710 PROFESSOR: OK, so I can certainly 00:47:01.710 --> 00:47:03.754 do something like this. 00:47:07.466 --> 00:47:08.860 AUDIENCE: I see. 00:47:08.860 --> 00:47:11.020 PROFESSOR: And of course, I can do the same thing 00:47:11.020 --> 00:47:15.505 over any of the bonds, but they are always [INAUDIBLE]. 00:47:20.550 --> 00:47:21.504 Yes? 00:47:21.504 --> 00:47:23.682 AUDIENCE: Doesn't [INAUDIBLE] become kind 00:47:23.682 --> 00:47:28.090 of like transfer matrix [INAUDIBLE]? 00:47:28.090 --> 00:47:30.030 PROFESSOR: No. 00:47:30.030 --> 00:47:32.990 It will reproduce the result that Onsager had, 00:47:32.990 --> 00:47:37.110 but the transfer matrix that Onsager had 00:47:37.110 --> 00:47:42.720 was it essentially going from column to column. 00:47:42.720 --> 00:47:46.580 Its size was 2 to the n times 2 to the n. 00:47:46.580 --> 00:47:48.585 This is 4n by 4n. 00:47:48.585 --> 00:47:52.396 It is vastly smaller matrix that I have to do. 00:48:08.710 --> 00:48:14.630 All right, so maybe we should just 00:48:14.630 --> 00:48:18.500 write down what this matrix t star is. 00:48:18.500 --> 00:48:24.810 So t star, I said, is this 4n by 4n matrix 00:48:24.810 --> 00:48:28.520 that tells me how going from some side 00:48:28.520 --> 00:48:35.270 xy I arrive at some other side x prime y prime. 00:48:35.270 --> 00:48:39.410 But it also has orientation information. 00:48:39.410 --> 00:48:45.340 So really, I should have four of these for this, 00:48:45.340 --> 00:48:46.840 and four of these for this. 00:48:46.840 --> 00:48:50.590 So it's actually a 4 by 4 matrix that I have 00:48:50.590 --> 00:48:54.010 once I keep track of orientations. 00:48:54.010 --> 00:48:57.620 So let me write down the 4 by 4 matrix explicitly. 00:49:02.290 --> 00:49:07.100 So here we have in mu, and this mu 00:49:07.100 --> 00:49:13.630 could be one, two, three, four-- which, again, specifically I 00:49:13.630 --> 00:49:18.450 have indicated as this, this, this, and this. 00:49:20.980 --> 00:49:25.570 And along the other direction, I can arrive at mu prime. 00:49:25.570 --> 00:49:27.820 Once I have arrived at the mu prime, 00:49:27.820 --> 00:49:33.620 I can either go forward, up, left, or down. 00:49:38.590 --> 00:49:45.140 So essentially what this says is start 00:49:45.140 --> 00:49:52.860 with a side xy, head in the horizontal direction. 00:49:52.860 --> 00:49:57.270 Since this is a one-step walk, after one step 00:49:57.270 --> 00:50:00.460 I will be arriving at some other point. 00:50:00.460 --> 00:50:05.320 Once you arrive at that next point, continue to [INAUDIBLE]. 00:50:08.550 --> 00:50:17.140 The next element says head to the right and then go up. 00:50:17.140 --> 00:50:21.060 The next element says go to the right 00:50:21.060 --> 00:50:24.750 and then start to the back. 00:50:24.750 --> 00:50:28.660 The next element says go to the right and then go down. 00:50:31.170 --> 00:50:34.420 Now we can construct the rest of them. 00:50:34.420 --> 00:50:55.160 Up, right, up, up, up, left, up, down, left, right, left, up, 00:50:55.160 --> 00:50:59.980 left, left, left, down. 00:50:59.980 --> 00:51:12.920 Lastly, down, right, down, up, down, left, down, down. 00:51:12.920 --> 00:51:16.930 So you do your aerobic exercises, 00:51:16.930 --> 00:51:24.880 and then in next stage is to actually write down the numbers 00:51:24.880 --> 00:51:27.550 that these correspond to. 00:51:27.550 --> 00:51:31.970 So first of all, you can see that-- I'm not sure 00:51:31.970 --> 00:51:33.700 whether I will have enough space, 00:51:33.700 --> 00:51:40.570 but let's hope that I do-- that in this first row 00:51:40.570 --> 00:51:46.900 of this matrix, your first step was always to the right. 00:51:46.900 --> 00:51:52.120 So always, you will start from x and you end up at x plus 1 00:51:52.120 --> 00:51:54.140 while y does not change. 00:51:54.140 --> 00:51:59.350 So I will indicate that by x plus 1-- actually, 00:51:59.350 --> 00:52:00.380 what should I write it? 00:52:04.070 --> 00:52:06.450 Yeah, x prime, y prime. 00:52:06.450 --> 00:52:09.830 x prime has to be x plus 1. 00:52:09.830 --> 00:52:12.320 y prime has to be y. 00:52:12.320 --> 00:52:14.400 So I'll have to introduce the notation 00:52:14.400 --> 00:52:21.650 that x prime y prime xy means theta x x prime delta y 00:52:21.650 --> 00:52:23.116 y prime. 00:52:23.116 --> 00:52:29.630 So essentially, you just read off for x prime, y prime 00:52:29.630 --> 00:52:32.420 what the new points have to be. 00:52:32.420 --> 00:52:33.660 And I proceed forward. 00:52:33.660 --> 00:52:35.920 There's no change in face. 00:52:35.920 --> 00:52:38.440 The next one, I arrive at the same point, 00:52:38.440 --> 00:52:45.000 so x prime y prime is x plus 1 y. 00:52:45.000 --> 00:52:49.340 But now my tangent vector, my heading 00:52:49.340 --> 00:52:55.220 has shifted by 90 degrees, so I have to put a factor of e 00:52:55.220 --> 00:52:57.450 to the i pi over 4. 00:53:00.650 --> 00:53:04.550 The next one, I try to go back, but I've 00:53:04.550 --> 00:53:11.130 said U-turns are not allowed, so this matrix element is 0. 00:53:11.130 --> 00:53:18.900 The next matrix element, I have x prime y prime x plus 1 y. 00:53:18.900 --> 00:53:23.150 Now I have pinned it down, so minus 90 degrees-- 00:53:23.150 --> 00:53:26.154 the changing angle-- is minus i pi over 4. 00:53:30.450 --> 00:53:34.600 The second column, you can see that essentially 00:53:34.600 --> 00:53:39.450 the y-coordinate has to change by 1, increase by 1. 00:53:39.450 --> 00:53:45.810 So I have x prime, y prime being xy plus 1. 00:53:45.810 --> 00:53:48.350 And this element has a change of angle 00:53:48.350 --> 00:53:50.590 that corresponds to the minus 90 degrees, 00:53:50.590 --> 00:53:55.360 so this is e to the minus 5 pi over 4. 00:53:55.360 --> 00:53:59.330 The diagonal element continues to head straight, 00:53:59.330 --> 00:54:01.800 so there was no phase angle associated with that. 00:54:04.700 --> 00:54:08.680 The third element heads in the opposite direction, 00:54:08.680 --> 00:54:17.240 so the phase element for that is e to the minus i pi over 4. 00:54:17.240 --> 00:54:22.120 The fourth element is a U-turn, so it will be 0. 00:54:22.120 --> 00:54:26.330 The third column starts with a U-turn, which is a 0. 00:54:26.330 --> 00:54:29.930 The next one, you can see I have to step to the left, 00:54:29.930 --> 00:54:36.570 so x prime has to become x minus 1 while y does not change. 00:54:36.570 --> 00:54:40.660 The phase factor is e to the minus i pi over 4. 00:54:40.660 --> 00:54:43.215 Along with diagonal, there is no phase factor. 00:54:46.060 --> 00:54:52.290 And if I already had one e to the minus i over 4, 00:54:52.290 --> 00:54:57.130 I should have e to the i pi over 4 for the last one. 00:54:57.130 --> 00:55:02.440 And the last element corresponds to y decreasing by 1. 00:55:02.440 --> 00:55:07.930 So I start with x prime, y prime in xy minus 1. 00:55:07.930 --> 00:55:10.620 The first phase that I have to do, 00:55:10.620 --> 00:55:14.390 I can kind of read how things go diagonally. 00:55:14.390 --> 00:55:19.060 This is i pi over 4 to the i pi over 4. 00:55:19.060 --> 00:55:22.580 The next one has to be 0 because the 0's you can see, 00:55:22.580 --> 00:55:24.580 are proceeding diagonally. 00:55:24.580 --> 00:55:29.970 The next one would be x prime, y prime xy minus 1 00:55:29.970 --> 00:55:33.575 e to the i minus i pi over 4 and then 00:55:33.575 --> 00:55:38.340 x prime y prime xy minus 1 for the last time. 00:55:43.230 --> 00:55:46.890 So this keeps track of the changes in phase. 00:55:50.080 --> 00:55:56.750 So now the next thing that we did 00:55:56.750 --> 00:56:00.820 when we had the ordinary random walks was we 00:56:00.820 --> 00:56:06.870 took advantage of the translational invariance 00:56:06.870 --> 00:56:12.650 of the lattice to go to Fourier space and make diagonalization. 00:56:12.650 --> 00:56:18.222 And indeed, we can do that over here, too, but only partially 00:56:18.222 --> 00:56:22.920 in that this object has two sets of indices. 00:56:22.920 --> 00:56:27.680 There is the lattice coordinates and there's the orientation. 00:56:27.680 --> 00:56:34.300 But what I can certainly do is to diagonalize the the subspace 00:56:34.300 --> 00:56:37.260 that corresponds to positions. 00:56:37.260 --> 00:56:39.730 What do I mean by that? 00:56:39.730 --> 00:56:47.940 What I will do is I will introduce, let's say, qx, qy, 00:56:47.940 --> 00:56:54.682 xy-- these Fourier elements-- which are e to the i qx 00:56:54.682 --> 00:57:02.850 x plus qy y divided by square root of n just as before, 00:57:02.850 --> 00:57:06.590 without any orientation component. 00:57:06.590 --> 00:57:12.440 Then you can see that if I multiply this object to xq 00:57:12.440 --> 00:57:20.120 y xy, with this matrix that I have over here, 00:57:20.120 --> 00:57:30.480 xy t star x prime y prime, and sum over x and y, 00:57:30.480 --> 00:57:33.780 but leave the orientations unchanged. 00:57:33.780 --> 00:57:36.840 That is, basically I do this individually 00:57:36.840 --> 00:57:42.250 for each one of these 16 elements of this matrix 00:57:42.250 --> 00:57:45.237 that each one of them clearly depends on x, y, 00:57:45.237 --> 00:57:50.260 x prime, y prime, but also has some additional factors. 00:57:50.260 --> 00:57:52.930 In each case, what is going to happen 00:57:52.930 --> 00:57:59.660 is that because I'm shifting x or y by one step, 00:57:59.660 --> 00:58:03.725 I will get this factor back up to e to the i qx, e 00:58:03.725 --> 00:58:07.160 to the minus i qx, e to the i qy, e to the minus iqy 00:58:07.160 --> 00:58:10.800 exactly as I was doing before, except that I 00:58:10.800 --> 00:58:14.900 will have to do this for every single one of them. 00:58:14.900 --> 00:58:18.470 So you can see that essentially what this reproduces is 00:58:18.470 --> 00:58:23.020 a matrix that is four by four that depends on q, 00:58:23.020 --> 00:58:28.940 and then I will get xy qx of qy actually x prime y 00:58:28.940 --> 00:58:30.920 prime because I sum over x and y back. 00:58:34.610 --> 00:58:38.810 So what is this matrix, t star of q? 00:58:38.810 --> 00:58:44.190 It is very easily constructed from what I have over there. 00:58:44.190 --> 00:58:48.700 Because you can see that from the first one, what happens 00:58:48.700 --> 00:58:53.470 is that when I see x, I will change it 00:58:53.470 --> 00:58:58.350 to x prime, which is x minus 1. 00:58:58.350 --> 00:59:03.920 So from here, I will get a factor of e to the minus i qx. 00:59:03.920 --> 00:59:05.180 y and y prime are the same. 00:59:05.180 --> 00:59:07.830 I don't get anything from here. 00:59:07.830 --> 00:59:17.640 Next one, I will get e to the minus i qx plus i pi over 4 0 00:59:17.640 --> 00:59:23.030 e to the minus i qx minus i pi over 4. 00:59:23.030 --> 00:59:28.440 The next column, the y has been shifted by 1. 00:59:28.440 --> 00:59:34.630 So I will get e to the minus i cube y minus i 00:59:34.630 --> 00:59:43.840 pi over 4 e to the minus i qy e to the minus i qy plus i pi 00:59:43.840 --> 00:59:48.120 over 4 0. 00:59:48.120 --> 00:59:53.560 The next level, the third one, x prime is set to x minus 1, 00:59:53.560 --> 01:00:00.770 so I essentially get e to the i qx-- oops. 01:00:00.770 --> 01:00:03.850 Third element starts with 0, and then I'll 01:00:03.850 --> 01:00:12.820 have e to the i qx minus i pi over 4 e to the i qx, 01:00:12.820 --> 01:00:18.520 and then I'll have e to the i qx plus i pi over 4. 01:00:18.520 --> 01:00:29.110 The fourth element is e to the i cube y plus i pi over 4 0 01:00:29.110 --> 01:00:34.410 into the i qy minus i pi over 4 e to the i qy. 01:00:57.730 --> 01:01:06.590 So in the positions space, I have this 4n by 4n matrix 01:01:06.590 --> 01:01:09.600 where the different sides were connected to their neighbors 01:01:09.600 --> 01:01:12.020 with these phase factors. 01:01:12.020 --> 01:01:16.190 I have gone from coordinate to Fourier basis. 01:01:16.190 --> 01:01:18.700 I did this transformation. 01:01:18.700 --> 01:01:23.750 Now I have a matrix that is blocked diagonal. 01:01:23.750 --> 01:01:30.020 So for each value of the q, I have a four by four block. 01:01:30.020 --> 01:01:33.320 So in the q picture, imagine that you have this 4n 01:01:33.320 --> 01:01:35.670 by 4n matrix, and you have blocks of four 01:01:35.670 --> 01:01:37.540 for different q along the diagonal. 01:01:37.540 --> 01:01:38.650 Each one of them is this. 01:01:43.110 --> 01:01:49.296 So now let's go and calculate our partition function. 01:01:49.296 --> 01:01:51.790 So what do we have? 01:01:51.790 --> 01:02:00.030 We have that log z over n is log 2 hyperbolic cosine 01:02:00.030 --> 01:02:11.480 squared of k plus 1/2 sum over l. 01:02:14.700 --> 01:02:20.837 Sum over l of these loops that go back 01:02:20.837 --> 01:02:21.920 all the way to themselves. 01:02:24.800 --> 01:02:30.140 So this is t to the l over l. 01:02:30.140 --> 01:02:36.125 This is loop star of length l. 01:02:39.140 --> 01:02:42.855 I want to relate loop star of length 01:02:42.855 --> 01:02:50.330 l to this w star of length l, but I 01:02:50.330 --> 01:02:56.530 want to start and end at the origin. 01:02:56.530 --> 01:03:00.530 So I start from the origin and end at the origin. 01:03:04.300 --> 01:03:06.900 But I have to be careful, because let's say 01:03:06.900 --> 01:03:11.510 I make a loop such as this and I end up at the origin. 01:03:11.510 --> 01:03:14.450 I have to get the right phase factor. 01:03:14.450 --> 01:03:18.480 So if I started along direction mu, 01:03:18.480 --> 01:03:22.940 when I get back to the starting point, I cannot turn this way, 01:03:22.940 --> 01:03:25.480 this way and get the right phase factor. 01:03:25.480 --> 01:03:27.860 I have to go and head in the same direction as mu. 01:03:32.100 --> 01:03:35.800 So I have to head to the same direction. 01:03:42.180 --> 01:03:47.010 Now I can certainly do this as a summation 01:03:47.010 --> 01:03:50.540 also over the starting point. 01:03:50.540 --> 01:03:52.970 Instead, I have to start and end at 0. 01:03:52.970 --> 01:03:55.820 I could have started and ended at any point, 01:03:55.820 --> 01:03:59.250 and then I do a sum of xy and mu, 01:03:59.250 --> 01:04:08.560 but then I better divide by the n, right? 01:04:08.560 --> 01:04:12.120 And the reason I do that is because now you 01:04:12.120 --> 01:04:19.920 can see that the structure of this is like a trace, 01:04:19.920 --> 01:04:20.760 and I like that. 01:04:23.380 --> 01:04:31.970 Actually, I made a mistake when I did this because, you see, 01:04:31.970 --> 01:04:36.040 the factor that I had to really include is minus 1 01:04:36.040 --> 01:04:37.165 to the number of crossings. 01:04:39.970 --> 01:04:44.670 But my w l star is just keep track of e 01:04:44.670 --> 01:04:47.880 to the i delta thetas. 01:04:47.880 --> 01:04:51.435 Actually, I should have put in here a delta theta over 2. 01:04:51.435 --> 01:04:54.100 So I had forgotten that. 01:04:54.100 --> 01:04:58.830 But we can see that that factor is different from minus 1 01:04:58.830 --> 01:05:02.451 to the nc to the minus sign. 01:05:02.451 --> 01:05:05.720 So there was that minus sign that I had forgotten. 01:05:05.720 --> 01:05:09.770 And actually, I better make this a minus sign. 01:05:09.770 --> 01:05:15.330 So what was plus before becomes a minus because of this factor 01:05:15.330 --> 01:05:16.470 that I have over here. 01:05:19.390 --> 01:05:22.270 So let's write this again. 01:05:22.270 --> 01:05:34.280 This is log 2 hyperbolic cosine squared of k minus 1 over 2n. 01:05:34.280 --> 01:05:38.915 And then this sum over xy mu is like a trace. 01:05:41.670 --> 01:05:44.110 And what is it that I'm tracing? 01:05:44.110 --> 01:05:52.051 I'm tracing a sum over l, t t star to the l divided by l. 01:05:56.001 --> 01:05:56.500 Yes? 01:05:56.500 --> 01:06:01.660 AUDIENCE: So we'll also sum it over mu [INAUDIBLE] 4? 01:06:01.660 --> 01:06:07.270 PROFESSOR: No, because I don't know 01:06:07.270 --> 01:06:10.200 which direction my first step is. 01:06:10.200 --> 01:06:13.230 So what I'm doing is I'm summing over all always 01:06:13.230 --> 01:06:17.780 of starting step from the origin, head in direction mu. 01:06:17.780 --> 01:06:21.360 Then I have to make sure that I come back to mu. 01:06:21.360 --> 01:06:24.690 Now it is true that I sum over mu, 01:06:24.690 --> 01:06:26.940 but I already took care of that when 01:06:26.940 --> 01:06:31.770 I divided by 2l, because let's say 01:06:31.770 --> 01:06:34.910 you look at the diagrams of length 4 01:06:34.910 --> 01:06:37.390 that I generate through this procedure. 01:06:37.390 --> 01:06:40.960 Starting from here, depending on which direction I go, 01:06:40.960 --> 01:06:44.825 I will generate this or this or this, or this. 01:06:44.825 --> 01:06:47.545 These are precisely the four diagrams 01:06:47.545 --> 01:06:51.940 that I generate depending on which starting point I pick. 01:06:51.940 --> 01:06:56.170 So it is truly there is an over-counting, 01:06:56.170 --> 01:06:58.934 but that's an over-counting that you've already taken care of. 01:07:08.000 --> 01:07:10.770 Now again, we did this last time. 01:07:10.770 --> 01:07:15.450 This is the series for minus log of 1 01:07:15.450 --> 01:07:19.060 minus 2 t star-- matrix t star. 01:07:21.960 --> 01:07:26.240 So this is the same thing as log of 2 hyperbolic cosine 01:07:26.240 --> 01:07:37.510 squared of k plus now 1 over 2n trace of log of 1 01:07:37.510 --> 01:07:39.430 minus t t star. 01:07:51.200 --> 01:07:57.150 Now I said that my matrix was blocked diagonal 01:07:57.150 --> 01:08:00.530 when I went to look at the q basis. 01:08:00.530 --> 01:08:03.240 And I can take the trace in any basis, 01:08:03.240 --> 01:08:05.705 whether it's in coordinate basis, in momentum basis. 01:08:05.705 --> 01:08:08.120 Trace is trace. 01:08:08.120 --> 01:08:11.170 So now focus on what the trace will look 01:08:11.170 --> 01:08:13.960 like if I go to the Fourier basis. 01:08:13.960 --> 01:08:17.149 I have these four by four blocks, 01:08:17.149 --> 01:08:19.649 and then I calculate the trace, I 01:08:19.649 --> 01:08:22.470 will calculate the trace of one four by four. 01:08:22.470 --> 01:08:26.270 And the other four by four, I go over all q's. 01:08:26.270 --> 01:08:32.420 So basically this can be written as a sum over q's. 01:08:32.420 --> 01:08:36.500 Log of 1 minus t. 01:08:36.500 --> 01:08:41.520 This four by four matrix t star of q, and the trace of that. 01:08:47.430 --> 01:08:53.880 And finally-- oops, I forgot a factor of 1 over 2n here. 01:08:53.880 --> 01:09:00.670 The sum over q I'm going to replace with an integral over q 01:09:00.670 --> 01:09:03.660 times n over 2 pi squared. 01:09:03.660 --> 01:09:05.910 So the final answer here is going 01:09:05.910 --> 01:09:09.630 to look like log 2 hyperbolic cosine squared 01:09:09.630 --> 01:09:14.609 root of k plus 1/2. 01:09:14.609 --> 01:09:18.819 The 1 over n I will get rid of when I write the sum over q 01:09:18.819 --> 01:09:21.124 as n integral d2 of q. 01:09:21.124 --> 01:09:26.102 So I have integral d2 of 1 divided by 2 pi squared. 01:09:29.370 --> 01:09:30.240 One more step. 01:09:35.450 --> 01:09:45.000 Trace of a log of any matrix I can write-- 01:09:45.000 --> 01:09:49.700 let's say we find a basis in which the m is diagonal. 01:09:49.700 --> 01:09:54.900 Then it becomes a sum over alpha log of lambda 01:09:54.900 --> 01:10:01.470 alpha, where lambda alphas are diagonal values of this. 01:10:01.470 --> 01:10:03.870 But sum over logs is the same thing 01:10:03.870 --> 01:10:12.021 as log of the product over alpha of lambda alpha. 01:10:12.021 --> 01:10:15.300 The product of eigenvalues of the matrix you 01:10:15.300 --> 01:10:18.670 also recognize to be the determinant. 01:10:18.670 --> 01:10:22.840 So this is the log of the determinant of the matrix. 01:10:22.840 --> 01:10:26.230 So this is a very useful, famous identity 01:10:26.230 --> 01:10:29.640 that trace log is the same thing as log determinant 01:10:29.640 --> 01:10:31.020 that I will use. 01:10:31.020 --> 01:10:34.230 And rather than calculate the trace log, 01:10:34.230 --> 01:10:39.090 I will write it as the log of the determinant, 01:10:39.090 --> 01:10:42.200 and I will explicitly write down for you 01:10:42.200 --> 01:10:45.390 the determinant of which four by four matrix. 01:10:45.390 --> 01:10:50.310 It is simply 1 minus p times the elements of that matrix. 01:10:50.310 --> 01:10:55.980 So it's 1 minus t e to the minus i qx minus t 01:10:55.980 --> 01:11:05.540 e to the minus i qx plus i pi over 4 0 minus t 01:11:05.540 --> 01:11:11.880 e to the minus i qx minus i pi over 4. 01:11:11.880 --> 01:11:15.350 Second element, second row-- minus t 01:11:15.350 --> 01:11:23.250 e to the minus i qy minus i pi over 4 1 minus t 01:11:23.250 --> 01:11:30.240 e to the minus i qy minus t e to the minus i qy plus i pi 01:11:30.240 --> 01:11:36.020 over 4 minus-- oops, 0 for the last element here. 01:11:36.020 --> 01:11:37.900 It's a U-turn. 01:11:37.900 --> 01:11:47.770 The third thing is 0 minus t e to the i qx minus i 01:11:47.770 --> 01:11:55.530 pi over 4 1 minus t e to the i qx diagonal element minus t e 01:11:55.530 --> 01:11:59.590 to the i qx plus i pi over 4. 01:11:59.590 --> 01:12:06.725 Final row-- minus t e to the i cube y plus i 01:12:06.725 --> 01:12:13.050 pi over 4, 0 for the U-turn, minus t e 01:12:13.050 --> 01:12:19.770 to the i qy minus i pi over 4 diagonal turn 1 minus t e 01:12:19.770 --> 01:12:23.620 to the i cube y, and that's it. 01:12:26.180 --> 01:12:28.730 And that's the answer. 01:12:28.730 --> 01:12:34.330 So calculating the partition function 01:12:34.330 --> 01:12:38.180 of the 2-dimensionalizing model is 01:12:38.180 --> 01:12:43.780 reduced to calculating this four by four determinant, which 01:12:43.780 --> 01:12:46.930 we can do by hand. 01:12:46.930 --> 01:12:48.000 I won't do it here. 01:12:48.000 --> 01:12:50.560 I will write the answer. 01:12:50.560 --> 01:13:00.120 So the log of z over n is log 3 hyperbolic cosine squared 01:13:00.120 --> 01:13:10.660 of k plus 1/2 integral d2 q 2 pi squared. 01:13:10.660 --> 01:13:13.860 Log, you take the determinant. 01:13:13.860 --> 01:13:19.680 What you find is 1 plus t squared squared minus 2 t 1 01:13:19.680 --> 01:13:24.330 minus t squared cosine of qx plus cosine of qy. 01:13:29.870 --> 01:13:33.940 If you want, you can write it in a slightly different way 01:13:33.940 --> 01:13:39.250 by taking the cosine squared inside this logarithm 01:13:39.250 --> 01:13:43.360 and doing a little bit of algebra. 01:13:43.360 --> 01:13:46.870 We get log 2 plus 1/2. 01:13:46.870 --> 01:13:53.510 Explicitly, these are integrals that go from 0 to 2 pi, 01:13:53.510 --> 01:13:56.270 because that's the range of q vectors 01:13:56.270 --> 01:13:59.110 that are allowed by this transformation. 01:13:59.110 --> 01:14:00.970 I'll have a q in the x direction. 01:14:00.970 --> 01:14:03.350 I'll have a q in the y direction. 01:14:03.350 --> 01:14:08.230 2 pi squared, each one of them goes in the range 0 to 2 pi. 01:14:08.230 --> 01:14:10.376 I have a log. 01:14:10.376 --> 01:14:16.090 And once I take this cosine squared inside, 01:14:16.090 --> 01:14:18.720 it becomes cos to the fourth. 01:14:18.720 --> 01:14:22.780 You write this t sine squared divided by cos squared. 01:14:22.780 --> 01:14:25.860 You can see that the cos to the fourth will cancel this. 01:14:25.860 --> 01:14:28.670 You will get cos squared plus sine squared, 01:14:28.670 --> 01:14:31.840 which is the same thing as hyperbolic cosine of twice 01:14:31.840 --> 01:14:34.160 the angle squared. 01:14:34.160 --> 01:14:40.200 And the other terms conspires to give you the sine of 2kx 01:14:40.200 --> 01:14:45.340 and then cosine of qx plus cosine of qy. 01:14:45.340 --> 01:14:55.196 And so this is the partition function 01:14:55.196 --> 01:14:56.550 of the 2-dimensionalizing model. 01:14:59.250 --> 01:15:01.850 You can do a little bit more manipulations, 01:15:01.850 --> 01:15:05.350 write this integral in terms of special functions, 01:15:05.350 --> 01:15:09.740 but you won't gain much. 01:15:09.740 --> 01:15:12.970 So this is the answer. 01:15:12.970 --> 01:15:18.000 I want you to absorb and appreciate this derivation. 01:15:18.000 --> 01:15:21.020 And next time we look at this and see 01:15:21.020 --> 01:15:24.030 what it means for the similarities in the phase 01:15:24.030 --> 01:15:26.050 behavior of the 2-dimensionalizing. 01:15:26.050 --> 01:15:27.954 Yes? 01:15:27.954 --> 01:15:30.120 AUDIENCE: [INAUDIBLE] in the log, it's cos 01:15:30.120 --> 01:15:32.340 squared minus [INAUDIBLE]? 01:15:32.340 --> 01:15:34.480 PROFESSOR: Yes. 01:15:34.480 --> 01:15:36.940 But both of them went to twice diagonal. 01:15:36.940 --> 01:15:39.710 Here everything is in terms of k. 01:15:39.710 --> 01:15:43.150 And once I took the hyperbolic cosine squared inside then 01:15:43.150 --> 01:15:48.720 did the manipulations, they became twice the length. 01:15:48.720 --> 01:15:52.340 AUDIENCE: What's the subscript up there [INAUDIBLE]? 01:15:52.340 --> 01:15:54.956 PROFESSOR: 2. 01:15:54.956 --> 01:15:56.330 There should be a subscript, yes. 01:16:04.834 --> 01:16:07.000 AUDIENCE: Is there an extension to higher dimensions 01:16:07.000 --> 01:16:08.765 where you just do a summation over cosine? 01:16:08.765 --> 01:16:10.140 PROFESSOR: You would wish, right? 01:16:10.140 --> 01:16:11.181 I mean, that's actually-- 01:16:11.181 --> 01:16:12.680 [LAUGHTER] 01:16:12.680 --> 01:16:15.240 PROFESSOR: And quite a number of people 01:16:15.240 --> 01:16:17.780 have come with that conjecture, including 01:16:17.780 --> 01:16:19.640 myself when i was a graduate student 01:16:19.640 --> 01:16:22.140 and I didn't know better. 01:16:22.140 --> 01:16:28.960 If you sort of write things in terms of not only 01:16:28.960 --> 01:16:33.560 if you make kx and ky to be different, 01:16:33.560 --> 01:16:38.060 then this takes the form of cos 2kx cos 2ky. 01:16:38.060 --> 01:16:40.950 This becomes, in some sense, a very nice version 01:16:40.950 --> 01:16:42.820 of 2kx and 2ky. 01:16:42.820 --> 01:16:46.370 There is a way natural-- and the thing that is nice 01:16:46.370 --> 01:16:50.000 is that if you put any one of the 2kx's to 0, then 01:16:50.000 --> 01:16:53.300 you have reduced the formula for the 1-dimensionalizing model, 01:16:53.300 --> 01:16:54.930 as you should. 01:16:54.930 --> 01:16:56.700 And then the natural thing would be 01:16:56.700 --> 01:16:59.970 to write a similar product in three dimensions, 01:16:59.970 --> 01:17:02.460 and you did when said one of k's equals to 0. 01:17:02.460 --> 01:17:05.030 You get the corrected dimensionalizing model. 01:17:05.030 --> 01:17:08.804 So it passes a number of test, yet it's unfortunately not 01:17:08.804 --> 01:17:09.304 correct. 01:17:14.244 --> 01:17:16.220 AUDIENCE: So the problem is conditional 01:17:16.220 --> 01:17:21.320 in this three dimensions is counting the loops? 01:17:21.320 --> 01:17:24.530 PROFESSOR: What we relied on heavily 01:17:24.530 --> 01:17:27.980 was this factor of minus 1 to the product 01:17:27.980 --> 01:17:30.260 of the number of crossings. 01:17:30.260 --> 01:17:34.050 And if you think about it as a topological entity, 01:17:34.050 --> 01:17:38.300 these crossings only make sense in two dimensions. 01:17:38.300 --> 01:17:43.350 So you don't have the basic tool to go in three dimensions. 01:17:43.350 --> 01:17:45.750 And actually, what those minus signs 01:17:45.750 --> 01:17:48.820 mean I will explain next time. 01:17:48.820 --> 01:17:51.817 Has something to do with fermionic character of this 01:17:51.817 --> 01:17:52.317 [INAUDIBLE]. 01:17:59.705 --> 01:18:00.830 So it's a beautiful result. 01:18:00.830 --> 01:18:04.100 You should appreciate it. 01:18:04.100 --> 01:18:05.020 Yes? 01:18:05.020 --> 01:18:07.780 AUDIENCE: [INAUDIBLE] this Thursday? 01:18:07.780 --> 01:18:09.940 PROFESSOR: I'll go through the history, too. 01:18:09.940 --> 01:18:13.920 The person who first derived this field energy 01:18:13.920 --> 01:18:19.260 was Onsager with this transfer matrix method that I described. 01:18:19.260 --> 01:18:24.500 This way of doing it in terms of graphs came much later. 01:18:24.500 --> 01:18:27.860 And a number of people that were involved 01:18:27.860 --> 01:18:29.290 that, including Feynman. 01:18:29.290 --> 01:18:32.520 In fact, in Feynman's book, there's 01:18:32.520 --> 01:18:35.740 a very nice derivation along these lines that you can see. 01:18:38.430 --> 01:18:41.600 The connections to fermions and the number of people Mattis, 01:18:41.600 --> 01:18:46.410 Schultz, Lieb, et cetera, came up with. 01:18:46.410 --> 01:18:50.140 One thing that-- well, OK, guess I have a few minutes. 01:18:50.140 --> 01:18:52.650 I can say a few things. 01:18:52.650 --> 01:18:59.510 So this result dates back to around 1950s. 01:18:59.510 --> 01:19:04.580 And I know a generation of physicists 01:19:04.580 --> 01:19:08.530 who are now about to retire or have retired in the past 10 01:19:08.530 --> 01:19:12.490 years or so who were very young when these things were 01:19:12.490 --> 01:19:13.940 introduced. 01:19:13.940 --> 01:19:16.600 And as far as I can see, all through their life, 01:19:16.600 --> 01:19:18.990 they did versions of this. 01:19:18.990 --> 01:19:23.400 So you can sort of do versions of the 2-dimensionalizing model 01:19:23.400 --> 01:19:26.610 and you can make the interactions to be different. 01:19:26.610 --> 01:19:28.980 You can play the different types of interactions. 01:19:28.980 --> 01:19:31.500 You can make the boundaries to be 01:19:31.500 --> 01:19:33.630 different, periodic, et cetera. 01:19:33.630 --> 01:19:36.240 There's been variants that you can find, 01:19:36.240 --> 01:19:38.640 and there are some people who seem 01:19:38.640 --> 01:19:40.540 to have done that throughout their career. 01:19:43.290 --> 01:19:47.900 But then after that, we had the renormalization group, 01:19:47.900 --> 01:19:51.180 and a totally different perspective.