WEBVTT 00:00:00.060 --> 00:00:01.780 The following content is provided 00:00:01.780 --> 00:00:04.010 under a Creative Commons license. 00:00:04.010 --> 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. 00:00:22.120 --> 00:00:23.545 PROFESSOR: OK, let's start. 00:00:26.360 --> 00:00:32.360 So, we've moved onto the two-dimensional xy model. 00:00:36.910 --> 00:00:43.070 This is a system where, let's say 00:00:43.070 --> 00:00:49.410 on each side of a square lattice you put a unit vector that 00:00:49.410 --> 00:00:53.030 has two components and hence, can 00:00:53.030 --> 00:00:56.605 be described by an angle theta. 00:00:56.605 --> 00:01:01.130 So basically, at each side, you have an angle theta 00:01:01.130 --> 00:01:02.950 for that side. 00:01:02.950 --> 00:01:09.450 And there is a tendency for a neighboring spins to be aligned 00:01:09.450 --> 00:01:13.020 and the partition function can be 00:01:13.020 --> 00:01:18.670 written as a sum over all configurations, which 00:01:18.670 --> 00:01:22.780 is equivalent to integrating over all of these angles. 00:01:25.700 --> 00:01:32.490 And the weight that wants to make the two neighboring 00:01:32.490 --> 00:01:36.590 spins to be parallel to each other. 00:01:36.590 --> 00:01:39.900 So we have a sum over nearest neighbors. 00:01:39.900 --> 00:01:43.410 And the dot product of the two spins amounts 00:01:43.410 --> 00:01:47.890 to looking at the cosine of theta i minus theta j. 00:01:47.890 --> 00:01:51.390 And so, we have this factor over here. 00:01:55.460 --> 00:02:05.850 Now, if we go to the limit where k is large, 00:02:05.850 --> 00:02:10.210 then the cosine will tend to keep the angles close 00:02:10.210 --> 00:02:12.180 to each other. 00:02:12.180 --> 00:02:20.080 And we are tempted to expand this around the configurations 00:02:20.080 --> 00:02:22.680 where everybody's parallel. 00:02:22.680 --> 00:02:28.620 Let's call that nk by the factor of 2. 00:02:28.620 --> 00:02:33.530 And then, expanding with the cosine to the next order, 00:02:33.530 --> 00:02:40.550 you may want to replace this-- let's call this k0-- 00:02:40.550 --> 00:02:44.870 and have a factor of k, which is proportional 00:02:44.870 --> 00:02:48.890 to k0 after some lattice spacing. 00:02:48.890 --> 00:02:54.990 And integral of gradient of theta squared. 00:02:54.990 --> 00:03:01.150 So basically, the difference between the angles 00:03:01.150 --> 00:03:04.160 in the continuum version, I want to replace 00:03:04.160 --> 00:03:11.000 with the term that tries to make the gradient to be fixed. 00:03:11.000 --> 00:03:12.910 OK. 00:03:12.910 --> 00:03:18.940 Now the reason I put these quotes around the gradient 00:03:18.940 --> 00:03:22.440 is something that we noticed last time, which 00:03:22.440 --> 00:03:27.650 is that in principal, theta is defined up 00:03:27.650 --> 00:03:30.330 to a multiple of 2 pi. 00:03:30.330 --> 00:03:36.290 So that if I were to take a circuit along the lattice 00:03:36.290 --> 00:03:39.650 that comes back to itself. 00:03:39.650 --> 00:03:49.900 And all along, this circuit integrate this gradient 00:03:49.900 --> 00:03:55.260 of theta, so basically, gradient of theta would be a vector. 00:03:55.260 --> 00:03:59.330 I integrated along a circuit. 00:03:59.330 --> 00:04:03.370 And by the time I have come back and close the circuit to where 00:04:03.370 --> 00:04:08.450 I started, the answer may not come back to 0. 00:04:08.450 --> 00:04:16.985 It may be any integer multiple of 2 pi. 00:04:16.985 --> 00:04:17.485 All right. 00:04:22.480 --> 00:04:25.690 So how do we account for this? 00:04:25.690 --> 00:04:29.060 The way we account for this is that we 00:04:29.060 --> 00:04:38.280 note that this gradient of data I can decompose into two parts. 00:04:38.280 --> 00:04:41.240 One, where I just write it as a gradient 00:04:41.240 --> 00:04:44.440 of some regular function. 00:04:44.440 --> 00:04:46.950 And the characteristic of gradient 00:04:46.950 --> 00:04:51.340 is that once you go over a closed loop and you integrate, 00:04:51.340 --> 00:04:53.850 you essentially are evaluating this field 00:04:53.850 --> 00:04:57.420 phi at the beginning and the end. 00:04:57.420 --> 00:05:01.100 And for any regular single valued phi, 00:05:01.100 --> 00:05:04.800 this would come back to zero. 00:05:04.800 --> 00:05:09.360 And to take care of this fact the result 00:05:09.360 --> 00:05:11.770 does not have to come to zero if I integrate 00:05:11.770 --> 00:05:19.900 this gradient of theta, I introduce another field, u, 00:05:19.900 --> 00:05:26.190 that takes care of these topological defects. 00:05:34.390 --> 00:05:36.820 OK? 00:05:36.820 --> 00:05:43.370 So that, really, I have to include both configurations 00:05:43.370 --> 00:05:48.900 in order to correctly capture the original model that 00:05:48.900 --> 00:05:51.380 had these angles. 00:05:51.380 --> 00:05:54.890 OK, so what can this u be? 00:05:54.890 --> 00:06:02.100 We already looked at what u is for the case of one 00:06:02.100 --> 00:06:03.390 topological defect. 00:06:13.950 --> 00:06:17.560 And the idea here was that maybe I 00:06:17.560 --> 00:06:23.530 had a configuration where around a particular center, 00:06:23.530 --> 00:06:29.230 let's say all of the spins were flowing out, 00:06:29.230 --> 00:06:32.900 or some other such configuration such 00:06:32.900 --> 00:06:41.500 that when I go over a large distance r from this center, 00:06:41.500 --> 00:06:51.940 and integrate this field u, just like I did over there, 00:06:51.940 --> 00:06:55.620 the answer is going to be, let's say, 2 pi n. 00:06:55.620 --> 00:06:58.940 So there's this u. 00:06:58.940 --> 00:07:03.960 And I integrated along this circle. 00:07:03.960 --> 00:07:08.500 And the answer is going to be 2 pi n. 00:07:08.500 --> 00:07:15.460 Well, clearly, the magnitude of u times 00:07:15.460 --> 00:07:21.060 2 pi r, which is the radius of the circle 00:07:21.060 --> 00:07:25.960 is going to be 2 pi times some integer-- could 00:07:25.960 --> 00:07:29.890 be plus, minus 1, plus minus 2. 00:07:29.890 --> 00:07:34.010 And so, the magnitude of u is n over r. 00:07:43.390 --> 00:07:51.446 The direction of u is orthogonal to the direction of r. 00:07:51.446 --> 00:07:53.230 And how can I show that? 00:07:53.230 --> 00:07:55.850 Well, one way I can show that is I 00:07:55.850 --> 00:08:05.170 can say that it is z hat crossed with r hat, 00:08:05.170 --> 00:08:12.490 there z hat is the vector that comes out of the plane. 00:08:12.490 --> 00:08:16.260 And r hat is the unit vector in this direction. 00:08:16.260 --> 00:08:19.360 u is clearly orthogonal to r. 00:08:19.360 --> 00:08:24.530 The direction of the gradient of this angle is orthogonal to r. 00:08:24.530 --> 00:08:29.290 It is in the plane so it's orthogonal to this. 00:08:29.290 --> 00:08:36.730 And this I can also write as z hat 3 00:08:36.730 --> 00:08:43.830 crossed with the gradient of log of r, with some cut-off. 00:08:43.830 --> 00:08:47.800 Because the gradient of log of r will give me, 00:08:47.800 --> 00:08:51.410 essentially, 1 over R in the direction of r hat. 00:08:51.410 --> 00:08:53.540 And this is like the potential that I 00:08:53.540 --> 00:08:57.450 would have for a charge in two dimensions, 00:08:57.450 --> 00:09:01.640 except that I have rotated it by somewhat. 00:09:01.640 --> 00:09:12.070 And this I can also write as minus the curl of z hat 00:09:12.070 --> 00:09:16.655 log r over a with a factor of n. 00:09:20.120 --> 00:09:24.880 And essentially, what you can see 00:09:24.880 --> 00:09:30.610 is that the gradient of data for a field that 00:09:30.610 --> 00:09:33.860 has this topological defect has a part 00:09:33.860 --> 00:09:39.240 can be written as a potential gradient of some y, 00:09:39.240 --> 00:09:43.010 and a part that can be written as curl to keep track 00:09:43.010 --> 00:09:45.310 of these vortices, if you like. 00:09:45.310 --> 00:09:48.140 If you were to think of this gradient of data 00:09:48.140 --> 00:09:50.755 like the flow field that you would have in two dimensions. 00:09:50.755 --> 00:09:52.510 It has a potential part. 00:09:52.510 --> 00:09:58.480 And it has a part that is due to curvatures and vortices, which 00:09:58.480 --> 00:10:01.870 is what we have over here. 00:10:01.870 --> 00:10:03.950 OK. 00:10:03.950 --> 00:10:08.480 So this is, however, only for one topological defect. 00:10:08.480 --> 00:10:11.935 What happens if I have many such defects? 00:10:17.390 --> 00:10:22.940 What I can do is, rather than having just one of them, 00:10:22.940 --> 00:10:27.690 I could have another topological defect here, another one here, 00:10:27.690 --> 00:10:28.600 another one there. 00:10:28.600 --> 00:10:32.790 There should be a combination of these things. 00:10:32.790 --> 00:10:39.840 And what I can do in order to get the corresponding u 00:10:39.840 --> 00:10:45.080 is to superimpose solutions that correspond to single ones. 00:10:45.080 --> 00:10:48.180 As you can see that this is very much 00:10:48.180 --> 00:10:51.720 like the potential that I would have 00:10:51.720 --> 00:10:54.260 for a charge at the origin, and then 00:10:54.260 --> 00:10:57.220 taking the derivative to create the field. 00:10:57.220 --> 00:11:00.930 And you know that as long as things are linear, 00:11:00.930 --> 00:11:02.770 and there aren't too many of them, 00:11:02.770 --> 00:11:07.560 you can superimpose solutions for different charges. 00:11:07.560 --> 00:11:10.260 You could just add up the electric fields. 00:11:10.260 --> 00:11:14.600 So what I'm claiming is that I can write u 00:11:14.600 --> 00:11:25.280 as minus n curl of z hat, times some potential u of r, where 00:11:25.280 --> 00:11:32.920 psi of r is essentially the generalization of this log. 00:11:32.920 --> 00:11:39.000 I can write it as a sum over all topological defects. 00:11:39.000 --> 00:11:45.565 And I will have the n of that topological defect times log 00:11:45.565 --> 00:11:50.480 of r minus ri divided by a, where 00:11:50.480 --> 00:11:53.960 ri are the locations of these. 00:11:53.960 --> 00:11:59.350 So there could be a vortex here at r1 with charge n1, 00:11:59.350 --> 00:12:04.350 and other topological defect here at r2 with charge n2, 00:12:04.350 --> 00:12:06.960 and so forth. 00:12:06.960 --> 00:12:12.270 And I can construct a potential that basically looks 00:12:12.270 --> 00:12:17.460 at the log of r minus ri for each individual one, 00:12:17.460 --> 00:12:20.790 and then do this. 00:12:20.790 --> 00:12:21.650 OK? 00:12:21.650 --> 00:12:27.100 I will sometimes write this in a slightly different fashion. 00:12:27.100 --> 00:12:34.930 Recall that we had the Coulomb potential, which 00:12:34.930 --> 00:12:41.550 was related to log by just a factor of 1 over 2 pi. 00:12:41.550 --> 00:12:44.830 So the correct version of defining the Coulomb potential 00:12:44.830 --> 00:12:46.030 is this. 00:12:46.030 --> 00:12:51.330 So this I can write as the Coulomb potential, 00:12:51.330 --> 00:12:55.200 provided that I multiply the 2 pi ni. 00:12:55.200 --> 00:12:58.190 And I sometimes will call that qi. 00:12:58.190 --> 00:13:03.520 So essentially, qi is 2 pi and i is 00:13:03.520 --> 00:13:06.770 the charge of the topological defect. 00:13:06.770 --> 00:13:09.610 It can be plus or minus 2 pi. 00:13:09.610 --> 00:13:12.450 And then, the potential is constructed 00:13:12.450 --> 00:13:18.160 by constructing superposition of those charges divided 00:13:18.160 --> 00:13:21.611 or multiplied with appropriate Coulomb potential. 00:13:21.611 --> 00:13:22.110 OK? 00:13:26.520 --> 00:13:27.020 All right. 00:13:29.980 --> 00:13:39.190 So I can construct a cost for creating a configuration now. 00:13:39.190 --> 00:13:43.580 Previously, I had this integral gradient 00:13:43.580 --> 00:13:47.376 of theta squared in the continuum. 00:13:47.376 --> 00:13:54.400 And my gradient of theta squared has now 00:13:54.400 --> 00:13:59.920 a part that is the gradient of a regular, well-behaved 00:13:59.920 --> 00:14:04.890 potential, and a part that is this field 00:14:04.890 --> 00:14:13.200 u, which is minus-- oops. 00:14:13.200 --> 00:14:15.560 Then I don't need the ni's because I 00:14:15.560 --> 00:14:18.470 put the ni as part of the psi. 00:14:18.470 --> 00:14:25.760 Curl of z hat psi of r. 00:14:25.760 --> 00:14:28.070 So phi is a regular function. 00:14:28.070 --> 00:14:33.060 Psi with curl will give me the contribution 00:14:33.060 --> 00:14:36.430 of the topological defect involves 00:14:36.430 --> 00:14:39.500 both the charges and the positions 00:14:39.500 --> 00:14:42.022 of these topological defects. 00:14:42.022 --> 00:14:44.990 OK? 00:14:44.990 --> 00:14:48.090 And this whole thing has to be squared, of course. 00:14:48.090 --> 00:14:52.720 This is my gradient squared. 00:14:52.720 --> 00:14:55.810 And if I expand this, I will have three terms. 00:14:59.040 --> 00:15:01.540 I have a gradient of this 5 squared. 00:15:04.340 --> 00:15:09.980 I have a term, which is minus 2 gradient of phi dot producted 00:15:09.980 --> 00:15:14.540 with curl of z hat psi. 00:15:17.830 --> 00:15:25.800 And I have a term that is curl of z hat psi squared. 00:15:29.180 --> 00:15:29.680 OK? 00:15:36.000 --> 00:15:39.470 Again, if you think of this as vector, 00:15:39.470 --> 00:15:41.450 this is a vector whose components 00:15:41.450 --> 00:15:49.740 are the xy and the y phi, whereas this 00:15:49.740 --> 00:15:53.450 is a vector whose components are, let's say, 00:15:53.450 --> 00:15:57.380 dy psi minus dx psi. 00:15:57.380 --> 00:16:01.340 Because of the curl operation-- the x and y components-- one 00:16:01.340 --> 00:16:02.470 of them gets a minus sign. 00:16:02.470 --> 00:16:05.950 Maybe I got the minus wrong, but it's essentially 00:16:05.950 --> 00:16:06.650 that structure. 00:16:09.320 --> 00:16:14.880 Now you can see that if I were to do the integration here, 00:16:14.880 --> 00:16:18.220 there is a dx psi, dy psi. 00:16:18.220 --> 00:16:22.310 I can do that integration by parts and have, 00:16:22.310 --> 00:16:26.340 let's say phi dx dy psi. 00:16:26.340 --> 00:16:29.150 And then, I can do the same integration by parts here. 00:16:29.150 --> 00:16:33.270 And I will have phi minus dx dy psi. 00:16:33.270 --> 00:16:37.230 So if I do integration by part, this will disappear. 00:16:37.230 --> 00:16:41.260 Another way of seeing that is that the gradient 00:16:41.260 --> 00:16:42.990 will act on the curl. 00:16:42.990 --> 00:16:45.570 And the gradient of the curl of a vector is 0, 00:16:45.570 --> 00:16:48.830 or otherwise, the curl will act on the gradient 00:16:48.830 --> 00:16:50.150 with [INAUDIBLE]. 00:16:50.150 --> 00:16:56.890 So basically, this term does not contribute. 00:16:56.890 --> 00:17:03.360 And the contribution of this part and the part 00:17:03.360 --> 00:17:08.069 from topological defects are decoupled from each other. 00:17:08.069 --> 00:17:11.869 So there's essentially the Gaussian type of stuff 00:17:11.869 --> 00:17:14.579 that we calculated before is here. 00:17:14.579 --> 00:17:18.150 On top of that, there is this part 00:17:18.150 --> 00:17:20.969 that is due to these topological defects. 00:17:23.849 --> 00:17:29.170 Again, this vector is this squared. 00:17:29.170 --> 00:17:30.950 You can see that if I square it, I 00:17:30.950 --> 00:17:34.980 will get dy psi squared plus dx psi squared. 00:17:34.980 --> 00:17:37.580 So to all intents and purposes, this thing 00:17:37.580 --> 00:17:44.050 is the same thing as a gradient of psi squared. 00:17:44.050 --> 00:17:46.510 Essentially, gradient of psi and curl of psi 00:17:46.510 --> 00:17:49.980 are the same vector, just rotated by 90 degrees. 00:17:49.980 --> 00:17:52.890 Integrating the square of one over the whole space 00:17:52.890 --> 00:17:56.430 is the same as integrating the square of the other. 00:17:56.430 --> 00:17:58.430 OK? 00:17:58.430 --> 00:18:04.020 So now, let's calculate this contribution and what it is. 00:18:04.020 --> 00:18:11.250 Integral d2 x gradient of psi squared with K over 2 00:18:11.250 --> 00:18:16.280 out front-- actually you already know that. 00:18:16.280 --> 00:18:22.590 Because psi we see is the potential 00:18:22.590 --> 00:18:23.945 due to a bunch of charges. 00:18:26.690 --> 00:18:29.630 So this is essentially the electric field 00:18:29.630 --> 00:18:32.370 due to this combination of charges 00:18:32.370 --> 00:18:34.760 integrated over the entire space. 00:18:34.760 --> 00:18:37.530 It's the electrostatic image. 00:18:37.530 --> 00:18:41.090 But let's go through that step by step. 00:18:41.090 --> 00:18:45.320 Let's do the integration by parts. 00:18:45.320 --> 00:18:49.790 So this becomes minus k over 2. 00:18:49.790 --> 00:18:57.900 Integral d2 x psi the gradient acting on this 00:18:57.900 --> 00:19:01.500 will give me Laplacian of psi. 00:19:01.500 --> 00:19:06.250 Of course, whenever I do integration by parts, 00:19:06.250 --> 00:19:08.620 I have to worry about boundary terms. 00:19:15.360 --> 00:19:17.550 And essentially, if you think of what 00:19:17.550 --> 00:19:21.260 you will be seeing at the boundary, far, 00:19:21.260 --> 00:19:26.260 far away from there all of these charges are, let's say. 00:19:26.260 --> 00:19:30.120 Essentially, you will see the electric field 00:19:30.120 --> 00:19:34.910 due to the combination of all of those charges. 00:19:34.910 --> 00:19:42.170 So for a single one, I will have a large electric field 00:19:42.170 --> 00:19:44.750 that will go as 1 over r. 00:19:44.750 --> 00:19:48.480 And we saw that integrating that will give me the log. 00:19:48.480 --> 00:19:50.990 So that was not particularly nice. 00:19:50.990 --> 00:19:54.390 So similarly, what these boundary terms 00:19:54.390 --> 00:19:58.120 would amount to would give you some kind 00:19:58.120 --> 00:20:01.470 of a logarithmic energy that depends on the next charge 00:20:01.470 --> 00:20:03.470 that you have enclosed. 00:20:03.470 --> 00:20:17.070 And you can get rid of it by setting the next charge 00:20:17.070 --> 00:20:18.225 to be zero. 00:20:23.050 --> 00:20:28.390 So essentially, any configuration 00:20:28.390 --> 00:20:33.930 in which the sum total of our topological charges is non-zero 00:20:33.930 --> 00:20:38.270 will get a huge energy cost as we go to large distances 00:20:38.270 --> 00:20:40.140 from the self-energy, if you like, 00:20:40.140 --> 00:20:42.980 of creating this huge monopole. 00:20:42.980 --> 00:20:45.600 So we are going to use this condition 00:20:45.600 --> 00:20:48.140 and focus only on configurations that 00:20:48.140 --> 00:20:52.390 are charged topological charge neutral. 00:20:52.390 --> 00:20:53.530 OK. 00:20:53.530 --> 00:20:59.050 Now our psi is what I have over here. 00:20:59.050 --> 00:21:03.950 It is sum over i qi. 00:21:03.950 --> 00:21:06.930 This Coulomb interaction-- r minus ri. 00:21:10.740 --> 00:21:18.560 And therefore, Laplacian of psi is essentially 00:21:18.560 --> 00:21:21.300 taking the Laplacian of this expression 00:21:21.300 --> 00:21:25.590 is the sum over j qu. 00:21:25.590 --> 00:21:28.395 Laplacian of this is the delta function. 00:21:33.150 --> 00:21:35.700 So basically, that was the condition 00:21:35.700 --> 00:21:37.100 for the Coulomb potential. 00:21:37.100 --> 00:21:40.900 Or alternatively, you take 2 derivative of the log, 00:21:40.900 --> 00:21:45.390 and you will generate the delta function. 00:21:45.390 --> 00:21:47.430 OK? 00:21:47.430 --> 00:21:56.990 So, what you see is that you generate the following. 00:21:56.990 --> 00:22:12.210 You will get a minus k over 2 sum over pairs i and j, qi, qj. 00:22:12.210 --> 00:22:15.610 And then I have the integral over x or r-- 00:22:15.610 --> 00:22:17.240 they're basically the same thing. 00:22:17.240 --> 00:22:18.985 Maybe I should have written this as x. 00:22:23.860 --> 00:22:29.710 And the delta function insures that x 00:22:29.710 --> 00:22:33.462 is set to be the other i. 00:22:33.462 --> 00:22:35.890 So I will get the Coulomb interaction 00:22:35.890 --> 00:22:40.010 between ri minus rj. 00:22:40.010 --> 00:22:46.710 So basically, what you have is that these topological defects 00:22:46.710 --> 00:22:51.730 that are characterized by these integers n, or by the charges 00:22:51.730 --> 00:22:56.540 2 pi n, have exactly this logarithmic Coulomb 00:22:56.540 --> 00:22:57.500 interaction. 00:22:57.500 --> 00:22:58.770 in two dimensions. 00:22:58.770 --> 00:23:01.750 And as I said, this thing is none other 00:23:01.750 --> 00:23:03.990 than the electrostatic energy. 00:23:03.990 --> 00:23:06.180 The electrostatic energy you can write either 00:23:06.180 --> 00:23:08.930 as an integral of the electric field squared. 00:23:08.930 --> 00:23:11.640 Or you can write as the interaction 00:23:11.640 --> 00:23:16.900 among the charges that give rise to that electric field. 00:23:16.900 --> 00:23:19.620 OK? 00:23:19.620 --> 00:23:26.990 So, what I can do is I can write this as follows. 00:23:26.990 --> 00:23:32.160 First of all, I can maybe re-cast it in terms of the n's. 00:23:32.160 --> 00:23:36.820 So I will have 2 pi ni, 2 pi nj. 00:23:36.820 --> 00:23:43.330 So I will get minus 4pi squared k. 00:23:43.330 --> 00:23:46.010 There's a factor of one-half. 00:23:46.010 --> 00:23:49.870 But this is a sum over i and j-- so every pair is now 00:23:49.870 --> 00:23:51.700 counted twice. 00:23:51.700 --> 00:23:54.080 So I get rid of that factor of one-half 00:23:54.080 --> 00:24:00.180 by essentially counting each pair only once. 00:24:00.180 --> 00:24:04.450 So I have the Coulomb interaction 00:24:04.450 --> 00:24:08.610 between ri and rj, which is this 1 over 2 pi 00:24:08.610 --> 00:24:13.390 log of ri minus rj with some cut-off. 00:24:13.390 --> 00:24:17.320 And then, there's the term that corresponds to i equals 2j. 00:24:17.320 --> 00:24:30.340 So I will have a minus, let's say, 4pi squared k sum over i. 00:24:30.340 --> 00:24:33.040 And I forgot here to put n, i, and j. 00:24:36.470 --> 00:24:39.730 I will have ni squared. 00:24:39.730 --> 00:24:43.334 The Coulomb interaction at zero. 00:24:43.334 --> 00:24:44.250 ri equals [INAUDIBLE]. 00:24:46.970 --> 00:24:53.240 Now clearly, this expression does not make sense. 00:24:53.240 --> 00:24:58.640 What it is trying to tell me is that there 00:24:58.640 --> 00:25:06.340 is a cost to creating one of these topological charges. 00:25:06.340 --> 00:25:11.810 And all of this theory-- again, in order to make sense, 00:25:11.810 --> 00:25:15.220 we should remember to put some kind of a short distance 00:25:15.220 --> 00:25:16.670 cut-off a. 00:25:16.670 --> 00:25:18.550 All right? 00:25:18.550 --> 00:25:26.110 And basically, replacing this original discrete lattice 00:25:26.110 --> 00:25:31.040 with a continuum will only work as long as 00:25:31.040 --> 00:25:35.830 I keep in mind that I cannot regard things at the level 00:25:35.830 --> 00:25:40.640 of lattice spacing, and replace it by that formula, as we saw, 00:25:40.640 --> 00:25:41.800 for example, here. 00:25:41.800 --> 00:25:45.340 If I want to draw a topological defect, 00:25:45.340 --> 00:25:48.430 I would need right at the center to do something 00:25:48.430 --> 00:25:52.630 like this-- where replacing the cosines with the gradient 00:25:52.630 --> 00:25:55.120 squared kind of doesn't make sense. 00:25:55.120 --> 00:25:57.030 So basically, what this theory is 00:25:57.030 --> 00:26:03.440 telling me is that once you get to a very small distance, 00:26:03.440 --> 00:26:08.550 you have to keep track of the existence 00:26:08.550 --> 00:26:13.520 of some underlying lattice and the corresponding things. 00:26:13.520 --> 00:26:16.270 And what's really this is describing 00:26:16.270 --> 00:26:22.720 for you is the core energy of creating 00:26:22.720 --> 00:26:26.670 a defect that has object ni. 00:26:26.670 --> 00:26:27.470 OK. 00:26:27.470 --> 00:26:37.660 What do I mean by that is that over here, I 00:26:37.660 --> 00:26:43.485 can calculate what the partition function is for one defect. 00:26:43.485 --> 00:26:46.550 This we already did last time around. 00:26:46.550 --> 00:26:50.780 And for that, I can integrate out this energy 00:26:50.780 --> 00:26:56.650 that I have for the distortions. 00:26:56.650 --> 00:27:02.470 It's an integral of n over r squared. 00:27:02.470 --> 00:27:07.980 And this integration gave me this factor of e 00:27:07.980 --> 00:27:20.752 to the minus pi k log of r over a-- actually, that was 2 pi k. 00:27:20.752 --> 00:27:25.380 Actually, let's do this correctly once. 00:27:29.320 --> 00:27:32.100 I should have done it earlier, and I forgot. 00:27:32.100 --> 00:27:37.130 So you have one defect. 00:27:37.130 --> 00:27:45.460 And we saw that for one defect, the field at the distance r 00:27:45.460 --> 00:27:50.140 has n over r in magnitude. 00:27:53.830 --> 00:28:00.370 And then, the net energy cost for one of these defects-- 00:28:00.370 --> 00:28:05.120 if I say that I believe this formula starting 00:28:05.120 --> 00:28:14.380 from a distance a is the k over 2 integral from a. 00:28:14.380 --> 00:28:17.925 Let's say all the way up to the size of my system. 00:28:20.780 --> 00:28:29.630 I have 2 pi r dr from a shell at radius r 00:28:29.630 --> 00:28:32.880 magnitude of this u squared. 00:28:32.880 --> 00:28:37.830 So I have n squared over r squared. 00:28:37.830 --> 00:28:41.860 But then, I have to worry about all of the actual things 00:28:41.860 --> 00:28:45.320 that I have off the distance a. 00:28:45.320 --> 00:28:50.510 So on top of this, there is a core energy 00:28:50.510 --> 00:28:56.760 for creating this object that certainly explicitly depends on 00:28:56.760 --> 00:29:00.320 where I sit this parameter a. 00:29:00.320 --> 00:29:02.120 OK? 00:29:02.120 --> 00:29:06.740 This part is easy. 00:29:06.740 --> 00:29:13.510 It simply gives me pi km squared. 00:29:13.510 --> 00:29:16.200 And then I have the integral of 1 00:29:16.200 --> 00:29:21.340 over r, which gives me log of L over a. 00:29:26.290 --> 00:29:31.790 So if I want to imagine what the partition function of this 00:29:31.790 --> 00:29:38.560 is-- one defect in a system of size L-- 00:29:38.560 --> 00:29:46.160 I would say that z of one defect is Boltzmann weight responding 00:29:46.160 --> 00:29:47.660 to creating this entity. 00:29:47.660 --> 00:29:54.570 So I have e to the minus pi k m squared log of L over a. 00:29:54.570 --> 00:30:02.360 And then I have the core energy that corresponds to this. 00:30:02.360 --> 00:30:07.190 And then, as we discussed, I can place this anywhere 00:30:07.190 --> 00:30:10.750 in the system if I'm calculating the partition function. 00:30:10.750 --> 00:30:14.130 So there's an integration over the position 00:30:14.130 --> 00:30:16.570 of this that is implicit. 00:30:16.570 --> 00:30:19.480 And so, that's going to give me the square 00:30:19.480 --> 00:30:27.770 of the size of my system, except that I am unsure as 00:30:27.770 --> 00:30:33.330 to where I have placed things up to this cut-off a. 00:30:33.330 --> 00:30:37.310 So really, the number of distinct 00:30:37.310 --> 00:30:43.710 positions that I have scales like L over a squared. 00:30:43.710 --> 00:30:48.870 So the whole thing we can see scales like L 00:30:48.870 --> 00:30:52.790 over a 2 minus pi km squared. 00:30:55.470 --> 00:31:03.660 And then, there is this factor of e to the minus 00:31:03.660 --> 00:31:08.180 this core energy evaluated at the distance 00:31:08.180 --> 00:31:10.250 a that I will call y. 00:31:13.550 --> 00:31:15.480 Because again, in some sense, there's 00:31:15.480 --> 00:31:19.280 some arbitrariness in where I choose a. 00:31:19.280 --> 00:31:24.590 So this y would be a function of a, if we depend on that choice. 00:31:24.590 --> 00:31:28.190 But the most important thing is that if I 00:31:28.190 --> 00:31:34.830 have a huge system, whether or not this partition function, 00:31:34.830 --> 00:31:38.630 as a function of the size of the system, goes to infinity 00:31:38.630 --> 00:31:45.470 or goes to 0 is controlled by this exponent 2 minus pi k. 00:31:45.470 --> 00:31:49.040 Let's say we focus on the simplest of topological defects 00:31:49.040 --> 00:31:53.030 corresponding to n equal 2 minus plus 1. 00:31:53.030 --> 00:31:56.650 You expect that there is some potentially critical value 00:31:56.650 --> 00:32:02.940 of k, which is 2 over pi, that distinguishes the two 00:32:02.940 --> 00:32:06.830 types of behavior. 00:32:06.830 --> 00:32:09.460 OK? 00:32:09.460 --> 00:32:16.060 But this picture is nice, but certainly incomplete. 00:32:16.060 --> 00:32:22.270 Because who said that there's any legitimacy in calculating 00:32:22.270 --> 00:32:24.390 the partition function that corresponds 00:32:24.390 --> 00:32:27.700 to just a single topological defect. 00:32:27.700 --> 00:32:31.680 If I integrate over all the configurations of my angle 00:32:31.680 --> 00:32:38.280 field, I should really be doing something 00:32:38.280 --> 00:32:43.700 that is analogous to this and calculating a partition 00:32:43.700 --> 00:32:47.270 function that corresponds to many defects. 00:32:47.270 --> 00:32:52.080 And actually what I calculated over here was in some sense 00:32:52.080 --> 00:32:54.960 the configuration of spins given that there 00:32:54.960 --> 00:32:59.170 is a topological defects that has the lowest energy. 00:32:59.170 --> 00:33:01.900 Once I start with this configuration-- let's 00:33:01.900 --> 00:33:04.270 say, with everybody radiating out-- 00:33:04.270 --> 00:33:07.620 I can start to distort them a little bit which 00:33:07.620 --> 00:33:13.060 amounts to adding this gradient of phi to that. 00:33:13.060 --> 00:33:15.940 So really, the partition function 00:33:15.940 --> 00:33:19.900 that I want to calculate and wrote down at the beginning-- 00:33:19.900 --> 00:33:23.680 if I want to calculate correctly, 00:33:23.680 --> 00:33:28.970 I have to include both these fluctuations 00:33:28.970 --> 00:33:31.900 and these fluctuations corresponding 00:33:31.900 --> 00:33:38.910 to an arbitrary set of these topological defects. 00:33:38.910 --> 00:33:44.820 And what we see is that actually, the partition 00:33:44.820 --> 00:33:49.460 functions and the energy costs of the two components 00:33:49.460 --> 00:33:51.200 really separate out. 00:33:51.200 --> 00:33:53.360 And what we are trying to calculate 00:33:53.360 --> 00:33:58.000 is the contribution that is due to the topological defects. 00:33:58.000 --> 00:34:02.400 And what we see is that once I tell you 00:34:02.400 --> 00:34:06.500 where the topological defects are located, 00:34:06.500 --> 00:34:12.790 the partition function for them has an energy component that 00:34:12.790 --> 00:34:16.560 is this Coulomb interaction among the defect. 00:34:16.560 --> 00:34:21.320 But there is a part that really is 00:34:21.320 --> 00:34:28.400 a remnant of this core energy that we 00:34:28.400 --> 00:34:31.610 were calculating before. 00:34:31.610 --> 00:34:35.969 So when I was sort of following my nose here, 00:34:35.969 --> 00:34:39.949 I had forgotten a little bit about the short distance 00:34:39.949 --> 00:34:40.940 cut-off. 00:34:40.940 --> 00:34:44.739 And then, when I encountered this C of zero, 00:34:44.739 --> 00:34:48.690 it told me that I have to think about the limit 00:34:48.690 --> 00:34:51.380 when two things come close to each other. 00:34:51.380 --> 00:34:56.360 And I know that that limit is constrained 00:34:56.360 --> 00:34:59.190 by my original lattice, and more importantly, 00:34:59.190 --> 00:35:02.560 by the place where I am willing to do 00:35:02.560 --> 00:35:10.140 self-averaging and replace this sum with a gradient. 00:35:10.140 --> 00:35:15.740 OK, so basically, this is the explanation of this term. 00:35:15.740 --> 00:35:18.280 So the only thing that we have established so far 00:35:18.280 --> 00:35:22.950 is that this partition function that I wrote down 00:35:22.950 --> 00:35:28.530 at the beginning gets decomposed into a part 00:35:28.530 --> 00:35:33.130 that we have calculated before, which was the Gaussian term, 00:35:33.130 --> 00:35:38.540 and is caused, really, the contribution due to spin waves. 00:35:38.540 --> 00:35:44.340 So this is when we just consider these [INAUDIBLE] modes, 00:35:44.340 --> 00:35:47.470 we said that essentially you can have an energy 00:35:47.470 --> 00:35:49.400 cost that is the gradient squared. 00:35:49.400 --> 00:35:54.900 So this is the part that corresponds to integral d phi 00:35:54.900 --> 00:36:00.150 into the minus k over 2 integral d2 x gradient of phi squared, 00:36:00.150 --> 00:36:03.305 where phi is a well-behaved, ordinary function. 00:36:11.300 --> 00:36:18.970 And what we find is that the actual partition function also 00:36:18.970 --> 00:36:25.100 has the contribution from the topological defects. 00:36:25.100 --> 00:36:31.670 And that I will indicate by ZQ. 00:36:31.670 --> 00:36:34.250 And Q stands for Coulomb gas. 00:36:39.260 --> 00:36:46.340 Because this partition function Z sub Q 00:36:46.340 --> 00:36:53.550 is like I'm trying to calculate this system of degrees 00:36:53.550 --> 00:37:00.080 of freedom that are characterized by charges n that 00:37:00.080 --> 00:37:04.270 can be anywhere in this two-dimensional space. 00:37:04.270 --> 00:37:08.300 And the interaction between them is governed by the Coulomb 00:37:08.300 --> 00:37:10.990 interaction in two dimensions. 00:37:10.990 --> 00:37:15.640 So to calculate this, I have to sum over 00:37:15.640 --> 00:37:19.040 all configuration of charges. 00:37:22.110 --> 00:37:26.520 The number of these charges could be zero, could be two, 00:37:26.520 --> 00:37:30.590 could be four, could be six, could be any number. 00:37:30.590 --> 00:37:33.540 But I say even numbers because I want 00:37:33.540 --> 00:37:37.040 to maintain the constraint of neutrality. 00:37:37.040 --> 00:37:39.450 Sum over ni should be zero. 00:37:39.450 --> 00:37:42.310 So I want to do that constrained sum. 00:37:42.310 --> 00:37:47.410 So I only want to look at neutral configurations. 00:37:47.410 --> 00:37:49.530 Once I have specified-- let's say 00:37:49.530 --> 00:37:57.310 that I have eight charges-- four plus and four minus-- well, 00:37:57.310 --> 00:38:01.540 there is a term that is going to come from here 00:38:01.540 --> 00:38:06.170 and I kind of said that the exponential of this term I'm 00:38:06.170 --> 00:38:08.700 going to call y. 00:38:08.700 --> 00:38:13.110 So I have essentially y raised to the power 00:38:13.110 --> 00:38:15.630 of the number of charges. 00:38:15.630 --> 00:38:19.190 Let's call this sum over i ni squared. 00:38:19.190 --> 00:38:21.730 And I'm actually just going to constrain ni 00:38:21.730 --> 00:38:23.480 to be minus plus 1. 00:38:23.480 --> 00:38:27.300 I'm going to only look at these primary charges. 00:38:27.300 --> 00:38:29.930 So the sum over i and i squared is just 00:38:29.930 --> 00:38:32.970 the total number of charges irrespective 00:38:32.970 --> 00:38:35.135 of whether they are plus or minus. 00:38:35.135 --> 00:38:39.390 It basically is replacing this. 00:38:39.390 --> 00:38:44.610 And then, I have to integrate over 00:38:44.610 --> 00:38:46.880 the positions of these charges. 00:38:46.880 --> 00:38:49.570 Let's call this total number n. 00:38:49.570 --> 00:38:56.400 So I have to integrate i1 2n d2 xi 00:38:56.400 --> 00:39:00.110 the position of where this charge is 00:39:00.110 --> 00:39:05.440 and then interaction which is exponential of minus 00:39:05.440 --> 00:39:11.610 4phi squared k sum over i less than j. 00:39:11.610 --> 00:39:17.130 And i and j the Coulomb interaction between location 00:39:17.130 --> 00:39:18.810 let's say xi and xj. 00:39:27.580 --> 00:39:32.240 Actually, I want to also emphasize that throughout, 00:39:32.240 --> 00:39:33.770 I have this cut-off. 00:39:33.770 --> 00:39:36.790 So when I was integrating over one, 00:39:36.790 --> 00:39:38.990 I said that the number of positions that I had 00:39:38.990 --> 00:39:42.760 was not L squared, but L over a squared to make it 00:39:42.760 --> 00:39:44.500 dimensionless. 00:39:44.500 --> 00:39:47.505 I will similarly make these interactions dimensioned 00:39:47.505 --> 00:39:49.180 as I divide by a squared. 00:39:53.310 --> 00:40:00.580 And so basically, this is the more interesting thing 00:40:00.580 --> 00:40:02.220 that we want to calculate. 00:40:08.172 --> 00:40:12.080 Also, again, remember I wrote this a squared down here, 00:40:12.080 --> 00:40:19.750 also to emphasize that within this expression, 00:40:19.750 --> 00:40:22.130 the minimal separation that I'm going 00:40:22.130 --> 00:40:26.720 to allow between any pair of charges is off the order of a. 00:40:26.720 --> 00:40:33.490 I have integrated out or moved into some continuum description 00:40:33.490 --> 00:40:38.055 any configuration in which the topological charges are 00:40:38.055 --> 00:40:39.908 less than distance a. 00:40:39.908 --> 00:40:40.408 OK? 00:40:43.800 --> 00:40:45.998 Yes? 00:40:45.998 --> 00:40:48.820 AUDIENCE: Essentially, when we were 00:40:48.820 --> 00:40:53.656 during [INAUDIBLE] it was canonical potential, 00:40:53.656 --> 00:40:57.002 [INAUDIBLE], canonical ensemble. 00:40:57.002 --> 00:41:00.030 And this is more like grand canonical ensemble? 00:41:00.030 --> 00:41:00.660 PROFESSOR: Yes. 00:41:00.660 --> 00:41:07.225 So, as far as the original two-dimensional xy model 00:41:07.225 --> 00:41:11.310 is concerned, I'm calculating a canonical partition function 00:41:11.310 --> 00:41:14.820 for this spin or angle degrees of freedom. 00:41:14.820 --> 00:41:20.590 And I find that that integration over spin angle 00:41:20.590 --> 00:41:23.830 degrees of freedom can be decomposed 00:41:23.830 --> 00:41:30.000 into a Gaussian part and a part that as you correctly point out 00:41:30.000 --> 00:41:36.670 corresponds to a grand canonical system of charges. 00:41:36.670 --> 00:41:38.720 So the number of charges that are 00:41:38.720 --> 00:41:41.910 going to appear in the system I have not 00:41:41.910 --> 00:41:45.630 specified whether it is determined implicitly 00:41:45.630 --> 00:41:49.211 by how strong these parameter was. 00:41:49.211 --> 00:41:55.175 AUDIENCE: [INAUDIBLE] of canonical potential? 00:41:55.175 --> 00:41:58.080 PROFESSOR: y plays the role of E to the beta mu. 00:42:02.700 --> 00:42:05.650 The quantity that in 8333 we were 00:42:05.650 --> 00:42:09.750 writing as z-- E to the beta mu small z. 00:42:12.743 --> 00:42:13.243 OK? 00:42:19.240 --> 00:42:26.990 So, we thought we were solving the xy model. 00:42:26.990 --> 00:42:32.090 We ended up, indeed, with this grand canonical system, 00:42:32.090 --> 00:42:36.240 which is currently parametrized by two things. 00:42:36.240 --> 00:42:43.080 One is this k, which is this strength of the potential. 00:42:43.080 --> 00:42:46.260 The other is this y. 00:42:46.260 --> 00:42:49.590 Of course, since this system originally 00:42:49.590 --> 00:42:54.180 came from an xy model that went only one parameter, 00:42:54.180 --> 00:42:59.300 I expect this y to also be related to k. 00:42:59.300 --> 00:43:03.350 But just as an expression, we can certainly 00:43:03.350 --> 00:43:06.770 regard it as a system that is parametrized 00:43:06.770 --> 00:43:10.260 by two things-- the k and the y. 00:43:10.260 --> 00:43:12.120 For the case of the xy model, there 00:43:12.120 --> 00:43:15.530 will be some additional constraint between the two. 00:43:15.530 --> 00:43:19.900 But more generally, we can look at this system with its two 00:43:19.900 --> 00:43:21.460 parameters. 00:43:21.460 --> 00:43:29.110 And essentially, we will try to make an expansion in y. 00:43:29.110 --> 00:43:33.360 You'll say that, OK, presumable, I 00:43:33.360 --> 00:43:38.700 know what is going to happen when y is very, very small. 00:43:38.700 --> 00:43:47.210 Because then, in the system I will create only a few charges. 00:43:47.210 --> 00:43:50.110 If I create many charges, I'm going 00:43:50.110 --> 00:43:54.580 to penalize by more and more factors of y. 00:43:54.580 --> 00:43:57.420 So maybe through leading order, the system 00:43:57.420 --> 00:43:59.570 would be free of charge. 00:43:59.570 --> 00:44:01.410 And then, there would be a few pairs 00:44:01.410 --> 00:44:03.800 that would appear here and there. 00:44:03.800 --> 00:44:06.600 In fact, there should be a small density of them, 00:44:06.600 --> 00:44:09.870 even no matter how small I make y. 00:44:09.870 --> 00:44:12.170 There will be a very small density 00:44:12.170 --> 00:44:14.310 of these things that will appear. 00:44:14.310 --> 00:44:18.090 And presumably, these things will always 00:44:18.090 --> 00:44:20.490 appear close to each other. 00:44:20.490 --> 00:44:26.270 So I will have lots and lots of these pairs-- well, 00:44:26.270 --> 00:44:27.960 not lots and lots of these pairs-- 00:44:27.960 --> 00:44:33.270 a density of them that is controlled by how big y is. 00:44:33.270 --> 00:44:42.470 And as I make y larger-- so this is y becoming larger-- then 00:44:42.470 --> 00:44:46.930 presumably, I will generate more and more of these pairs. 00:44:46.930 --> 00:44:50.170 And once I have more and more of these pairs, 00:44:50.170 --> 00:44:54.480 they could, in principle, get into each other's way. 00:44:54.480 --> 00:44:56.530 And when they get into each other's way, 00:44:56.530 --> 00:44:59.890 then it's not clear who is paired with whom. 00:44:59.890 --> 00:45:06.450 And at some point, I should trade my picture 00:45:06.450 --> 00:45:11.240 of having a gas of pairs of these objects 00:45:11.240 --> 00:45:14.340 to a plasma of charges, plus and minus, 00:45:14.340 --> 00:45:18.380 that are moving all over the place. 00:45:18.380 --> 00:45:23.650 So as I tune this parameter y, I expect my system 00:45:23.650 --> 00:45:29.840 to go from a low density phase of atoms of plus-minus bound 00:45:29.840 --> 00:45:34.470 to each other to a high density phase where 00:45:34.470 --> 00:45:36.950 I have a plasma of plus and minuses 00:45:36.950 --> 00:45:38.230 moving all over the place. 00:45:38.230 --> 00:45:39.030 Yes? 00:45:39.030 --> 00:45:41.965 AUDIENCE: So, y is related to the core energy. 00:45:41.965 --> 00:45:42.800 PROFESSOR: Yes. 00:45:42.800 --> 00:45:46.370 AUDIENCE: And core energy is defined through [INAUDIBLE] 00:45:46.370 --> 00:45:49.960 direction at zero separation-- 00:45:49.960 --> 00:45:51.890 PROFESSOR: Well, no. 00:45:51.890 --> 00:45:54.590 Because the Coulomb description is only 00:45:54.590 --> 00:45:57.290 valid large separations. 00:45:57.290 --> 00:46:02.550 When I get to short distances, who knows what's going on? 00:46:02.550 --> 00:46:07.100 So there is some underlying microscopic picture 00:46:07.100 --> 00:46:10.320 that determines what the core energy is. 00:46:10.320 --> 00:46:12.800 Very roughly, yes, you would expect 00:46:12.800 --> 00:46:16.850 it to have a form that is of e to the minus k 00:46:16.850 --> 00:46:19.710 with some coefficient that comes from adding 00:46:19.710 --> 00:46:21.040 all of those interactions here. 00:46:23.610 --> 00:46:24.718 Yes? 00:46:24.718 --> 00:46:26.570 AUDIENCE: Just based on the sign convention, 00:46:26.570 --> 00:46:30.199 you're saying if you increase or decrease y, 00:46:30.199 --> 00:46:32.011 that it will go from a low density-- 00:46:32.011 --> 00:46:36.200 PROFESSOR: OK, so y is the exponential of something. 00:46:36.200 --> 00:46:42.010 y equals to zero means I will not create any of these things. 00:46:42.010 --> 00:46:45.800 y approaching 1-- I will create a lot of them. 00:46:45.800 --> 00:46:48.060 There's no cost at y equals to one. 00:46:48.060 --> 00:46:50.360 There's no core energy. 00:46:50.360 --> 00:46:52.730 I can create them as I want. 00:46:52.730 --> 00:46:56.814 AUDIENCE: So this would be like y equals minus epsilon c. 00:46:56.814 --> 00:46:57.730 Is that right? 00:46:57.730 --> 00:46:59.610 PROFESSOR: Yeah. 00:46:59.610 --> 00:47:00.510 Didn't I have that? 00:47:00.510 --> 00:47:03.650 You see in the exponential it is with the minus. 00:47:10.290 --> 00:47:10.790 OK. 00:47:10.790 --> 00:47:13.190 But in any case, that is the expectation. 00:47:13.190 --> 00:47:14.010 Right? 00:47:14.010 --> 00:47:17.330 So I expect that when I calculate, 00:47:17.330 --> 00:47:19.020 I create one of these defects. 00:47:19.020 --> 00:47:23.730 There is an energy cost which is mostly from outside. 00:47:23.730 --> 00:47:27.790 And then, there's an additional piece on the inside. 00:47:27.790 --> 00:47:30.247 So the exponential of that additional piece 00:47:30.247 --> 00:47:31.830 would be a number that is less than 1. 00:47:31.830 --> 00:47:32.734 AUDIENCE: [INAUDIBLE] 00:47:43.684 --> 00:47:44.350 PROFESSOR: Yeah. 00:47:44.350 --> 00:47:48.260 I mean, the original model has some particular form. 00:47:48.260 --> 00:47:51.240 And actually, the interactions of the original model, I 00:47:51.240 --> 00:47:52.730 can make more complicated. 00:47:52.730 --> 00:47:57.030 I can add the full spin interaction, for example. 00:47:57.030 --> 00:48:01.480 It doesn't affect the overall form much, 00:48:01.480 --> 00:48:05.120 just modifies what an effective k is, 00:48:05.120 --> 00:48:08.175 and what the core energy is independent. 00:48:11.420 --> 00:48:11.920 OK? 00:48:14.500 --> 00:48:15.380 All right. 00:48:15.380 --> 00:48:19.850 But the key point is that this system potentially 00:48:19.850 --> 00:48:25.520 has a phase transition as you change the parameter of y. 00:48:25.520 --> 00:48:29.270 And another way of looking at this transition 00:48:29.270 --> 00:48:35.460 is that what is happening here in different languages, 00:48:35.460 --> 00:48:38.680 you can either call it insulator or a dielectric. 00:48:42.840 --> 00:48:45.830 But what is happening here in different languages, 00:48:45.830 --> 00:48:50.780 you can either call, say, a metal or, as I said, 00:48:50.780 --> 00:48:51.550 maybe a plasma. 00:48:54.616 --> 00:48:59.960 The point is that here you have free charges. 00:48:59.960 --> 00:49:03.670 Here you have bound pairs of charges. 00:49:03.670 --> 00:49:08.090 And they respond differently to, let's say, 00:49:08.090 --> 00:49:10.150 an external electromagnetic field. 00:49:10.150 --> 00:49:15.850 So once we have this picture, let's kind of expand our view. 00:49:15.850 --> 00:49:18.120 Forget about the xy model. 00:49:18.120 --> 00:49:20.490 Think of a system of charges. 00:49:20.490 --> 00:49:25.536 And notice that in this low-density phase, 00:49:25.536 --> 00:49:29.220 it behaves like a dielectric in the sense 00:49:29.220 --> 00:49:32.730 that there are no free charges. 00:49:32.730 --> 00:49:37.900 And here, there will be lots of mobile charges. 00:49:37.900 --> 00:49:40.650 And it behaves like a metal. 00:49:40.650 --> 00:49:42.280 What do I mean by that? 00:49:42.280 --> 00:49:45.400 Well, here, if I, let's say, bring 00:49:45.400 --> 00:49:48.180 in an external electric field. 00:49:48.180 --> 00:49:54.360 Or maybe if I put a huge charge, what is going to happen 00:49:54.360 --> 00:50:00.290 is that opposite charges will accumulate. 00:50:00.290 --> 00:50:04.880 Or there will be, essentially, opposite charges for the field. 00:50:04.880 --> 00:50:08.590 So that once you go inside, the fact 00:50:08.590 --> 00:50:13.160 that you have an external electric field or a charge 00:50:13.160 --> 00:50:14.880 is completely screen. 00:50:14.880 --> 00:50:16.800 You won't see it. 00:50:16.800 --> 00:50:19.860 Whereas here, what is going to happen 00:50:19.860 --> 00:50:23.380 is that if you put in an electric field, 00:50:23.380 --> 00:50:26.170 it will penetrate into the system 00:50:26.170 --> 00:50:28.970 although it will be weakened a little bit 00:50:28.970 --> 00:50:32.220 by the re-orientation of these charges. 00:50:32.220 --> 00:50:38.650 Now, if you put a plus charge, the effect of that plus charge 00:50:38.650 --> 00:50:42.170 would be felt throughout, although weakened a little bit. 00:50:42.170 --> 00:50:47.790 Because again, some of these dipoles will re-orient in that. 00:50:47.790 --> 00:50:49.720 OK? 00:50:49.720 --> 00:50:56.380 So, this low-density phase we can actually 00:50:56.380 --> 00:51:06.050 try to parametrize in terms of a weakening of the interactions 00:51:06.050 --> 00:51:10.800 through a dielectric constant epsilon. 00:51:10.800 --> 00:51:13.140 And so, what I'm going to try to calculate 00:51:13.140 --> 00:51:18.580 for you is to imagine that I'm in the limit of low density 00:51:18.580 --> 00:51:25.640 or small y and calculate what the weakening is, what 00:51:25.640 --> 00:51:29.340 the dielectric function is, perturbatively in y. 00:51:29.340 --> 00:51:29.840 Yes? 00:51:29.840 --> 00:51:33.644 AUDIENCE: If you were talking about the real electric charges 00:51:33.644 --> 00:51:36.204 and the way to act on that [INAUDIBLE] 00:51:36.204 --> 00:51:38.060 real electric field or charge. 00:51:38.060 --> 00:51:42.214 But if we are talking about topological charges, what 00:51:42.214 --> 00:51:46.507 would be kind of conjugate force to that? 00:51:46.507 --> 00:51:47.480 PROFESSOR: OK. 00:51:50.640 --> 00:51:52.280 It's not going to be easy. 00:51:52.280 --> 00:51:54.710 I have to do something about say, 00:51:54.710 --> 00:51:58.530 re-orienting all of the spins on the boundaries, et cetera. 00:51:58.530 --> 00:52:00.550 So let's forget about that. 00:52:00.550 --> 00:52:03.780 The point is that mathematically, the problem 00:52:03.780 --> 00:52:06.620 is reduced to this system. 00:52:06.620 --> 00:52:11.040 And I can much more easily do the mathematics 00:52:11.040 --> 00:52:15.950 if I change my perspective and think about this picture. 00:52:15.950 --> 00:52:18.100 OK? 00:52:18.100 --> 00:52:22.890 And that's the thing you have to do in theoretical physics. 00:52:22.890 --> 00:52:25.650 You basically take advantage of mappings 00:52:25.650 --> 00:52:29.280 of one model to another model in order 00:52:29.280 --> 00:52:35.900 to refine your intuition using some other picture. 00:52:35.900 --> 00:52:38.370 So that's what we are going to do. 00:52:38.370 --> 00:52:44.820 So completely different picture from the original spin models-- 00:52:44.820 --> 00:52:52.330 imagine that you have indeed a box of this material. 00:52:52.330 --> 00:52:56.120 And this box of material has, because you're wise, 00:52:56.120 --> 00:53:00.130 more some combination of these plus and minus charges in it. 00:53:02.650 --> 00:53:07.320 And then, what I do is that I bring externally 00:53:07.320 --> 00:53:10.655 a uniform electric field in this direction. 00:53:13.870 --> 00:53:19.780 And I expect that once inside the material, 00:53:19.780 --> 00:53:24.220 the electric field will be reduced to a smaller value 00:53:24.220 --> 00:53:31.240 that I will call E prime because of the dielectric function. 00:53:31.240 --> 00:53:34.790 Now, if you ever calculated dielectric functions, 00:53:34.790 --> 00:53:37.250 that's exactly what I'm going to do now. 00:53:37.250 --> 00:53:38.970 It's a simple process. 00:53:38.970 --> 00:53:42.310 What you do, for example, is you do 00:53:42.310 --> 00:53:46.390 the analog of Gauss' theorem. 00:53:46.390 --> 00:53:53.150 Let's imagine that we draw a circuit such as this 00:53:53.150 --> 00:53:59.930 that is partly on the inside, and partly on the outside. 00:53:59.930 --> 00:54:06.070 So I can calculate what the flux of the electric field 00:54:06.070 --> 00:54:12.070 is through this circuit, the analog of the Gaussian pillbox. 00:54:12.070 --> 00:54:14.660 And so, what I have is that what is going on 00:54:14.660 --> 00:54:20.140 is E. If I call this distance to be L, 00:54:20.140 --> 00:54:26.710 the flux integrated through the entire thing 00:54:26.710 --> 00:54:30.872 is E minus E prime times f. 00:54:30.872 --> 00:54:35.040 So this is the integral of the divergence of the electric 00:54:35.040 --> 00:54:36.880 field . 00:54:36.880 --> 00:54:43.140 And by Gauss' theorem, this has to be charge enclosed inside. 00:54:49.588 --> 00:54:52.570 OK. 00:54:52.570 --> 00:54:55.810 Now, why should there be any charge enclosed 00:54:55.810 --> 00:55:01.045 inside when you have a bunch of plus and minuses. 00:55:01.045 --> 00:55:02.630 I mean, there will be some pluses 00:55:02.630 --> 00:55:05.190 and minuses out here as I have indicated. 00:55:05.190 --> 00:55:08.920 There will be some pluses and minuses that are inside. 00:55:08.920 --> 00:55:12.510 But the net of these would be zero. 00:55:12.510 --> 00:55:17.560 So the only place that you get a net charge 00:55:17.560 --> 00:55:20.934 is those dipoles that happen to be sitting right 00:55:20.934 --> 00:55:21.600 at the boundary. 00:55:26.230 --> 00:55:32.860 And then, I have to count how many of them are inside. 00:55:32.860 --> 00:55:35.270 And some of them will have the plus inside. 00:55:35.270 --> 00:55:38.550 And some of them will have the minus inside. 00:55:38.550 --> 00:55:41.640 And then, I have to calculate the net. 00:55:41.640 --> 00:55:50.490 The thing is that my dipoles do not have a fixed size. 00:55:50.490 --> 00:55:58.810 The size of these plus/minus molecules r 00:55:58.810 --> 00:56:00.890 can be variable itself. 00:56:00.890 --> 00:56:02.820 OK? 00:56:02.820 --> 00:56:05.670 So there will be some that are tightly bound to each other. 00:56:05.670 --> 00:56:10.160 There may be some that are further apart, et cetera. 00:56:10.160 --> 00:56:16.530 So let's look at pairs that are at the distance r 00:56:16.530 --> 00:56:20.350 and ask how many of them hit this boundary so that one 00:56:20.350 --> 00:56:23.990 of them would be inside, one of them will be outside. 00:56:23.990 --> 00:56:25.370 OK? 00:56:25.370 --> 00:56:30.270 So, that number has to be proportional 00:56:30.270 --> 00:56:36.960 to essentially this area. 00:56:36.960 --> 00:56:39.100 What is that area? 00:56:39.100 --> 00:56:44.730 On one side, it is L. On the other side, it is R. 00:56:44.730 --> 00:56:51.310 But if the dipole is oriented at an angle theta, 00:56:51.310 --> 00:56:56.250 it is, in fact r cosine theta. 00:56:56.250 --> 00:56:56.750 OK? 00:56:59.310 --> 00:57:05.320 So that's the number. 00:57:05.320 --> 00:57:12.360 Now, what I will have here would be the charge 2 pi. 00:57:12.360 --> 00:57:13.170 So this is qi. 00:57:19.880 --> 00:57:23.600 Actually, it could be plus or minus. 00:57:23.600 --> 00:57:25.750 The reason that there's going to be more 00:57:25.750 --> 00:57:31.500 plus as opposed to minus is because the dipole 00:57:31.500 --> 00:57:35.880 gets oriented by the electric field. 00:57:35.880 --> 00:57:45.090 So I will have a term here that is E to the E prime times q 00:57:45.090 --> 00:57:50.660 ir-- so that's 2 pi r. 00:57:50.660 --> 00:57:57.280 So this is qr again, times cosine of theta. 00:57:57.280 --> 00:58:00.150 So we can see that, depending on cosine of theta 00:58:00.150 --> 00:58:04.150 being larger than pi or less than pi, 00:58:04.150 --> 00:58:07.440 this number will be positive or negative. 00:58:07.440 --> 00:58:11.075 And that's going to be modified by this number also. 00:58:11.075 --> 00:58:14.300 And of course, the strength of this whole thing 00:58:14.300 --> 00:58:16.870 is set by this parameter k. 00:58:21.230 --> 00:58:28.480 And also how likely it is for me to have created 00:58:28.480 --> 00:58:36.840 a dipole of size r is controlled by precisely this factor. 00:58:36.840 --> 00:58:42.030 A dipole is something that has two cores. 00:58:42.030 --> 00:58:45.280 So it is something that will appear at order of y squared. 00:58:48.570 --> 00:58:51.050 And there is the energy, according 00:58:51.050 --> 00:58:54.810 to this formula, of separating two things. 00:58:54.810 --> 00:58:57.675 And so you can see that essentially, n of r-- 00:58:57.675 --> 00:59:01.320 maybe I will write it separately over here-- 00:59:01.320 --> 00:59:10.600 is y squared times E to the minus 4pi squared k. 00:59:10.600 --> 00:59:15.520 And from here, I have log of r divided by a. 00:59:15.520 --> 00:59:17.430 And then there's a factor of 2 pi 00:59:17.430 --> 00:59:19.610 because the Coulomb potential is this. 00:59:22.240 --> 00:59:27.960 So, this is going to be y squared 00:59:27.960 --> 00:59:33.430 a over r to the power of 2 pi k. 00:59:33.430 --> 00:59:38.720 The further you try to separate these things, the more 00:59:38.720 --> 00:59:40.300 cost you have to pay. 00:59:46.024 --> 00:59:48.900 OK. 00:59:48.900 --> 00:59:52.700 So if you were trying to calculate 00:59:52.700 --> 00:59:57.750 the contribution of, say, polarizable atoms 00:59:57.750 --> 01:00:02.150 or dipoles to the dielectric function of a solid, 01:00:02.150 --> 01:00:06.630 you would be doing exactly this same calculation. 01:00:06.630 --> 01:00:13.090 The only difference is that the size of your dipole 01:00:13.090 --> 01:00:16.580 would be set by the size of your molecule 01:00:16.580 --> 01:00:20.590 and ultimately, related to its polarizability. 01:00:20.590 --> 01:00:23.720 And rather than having this Coulomb interaction, 01:00:23.720 --> 01:00:26.755 you would have some dissociation energy or something 01:00:26.755 --> 01:00:31.800 else, or the density itself would come over here. 01:00:31.800 --> 01:00:36.630 So, the only final step is that I 01:00:36.630 --> 01:00:43.470 have to regard my system having a composition 01:00:43.470 --> 01:00:46.110 of these things of different sizes. 01:00:46.110 --> 01:00:52.000 So I have to do an integral over r, as well as orientation. 01:00:52.000 --> 01:00:57.060 So I have to do an integral over E theta. 01:00:57.060 --> 01:00:58.910 Of course, the integration will go 01:00:58.910 --> 01:01:03.678 from a through essentially, the size of the system or infinity. 01:01:07.670 --> 01:01:11.760 I forgot one other thing, which is 01:01:11.760 --> 01:01:18.240 that when I'm calculating how many places I can put this, 01:01:18.240 --> 01:01:21.570 again, I have been calculating things 01:01:21.570 --> 01:01:26.020 per unit area of a squared. 01:01:26.020 --> 01:01:29.800 So I would have to divide all of these places 01:01:29.800 --> 01:01:34.629 where r and L appear by corresponding factors of a. 01:01:37.552 --> 01:01:38.052 OK? 01:01:42.950 --> 01:01:45.650 So, the last step of the calculation 01:01:45.650 --> 01:01:48.145 is you expand this quantity. 01:01:48.145 --> 01:01:50.120 It is 1. 01:01:50.120 --> 01:01:52.670 For small values of the electric field, 01:01:52.670 --> 01:02:01.000 it is 2 pi r E prime cosine of theta k plus higher order 01:02:01.000 --> 01:02:03.240 terms. 01:02:03.240 --> 01:02:07.160 And then, you can do the various integrations. 01:02:07.160 --> 01:02:12.570 First of all, 1 the integration against 1 01:02:12.570 --> 01:02:14.960 will disappear because you are integrating 01:02:14.960 --> 01:02:18.740 over all values of cosine of theta. 01:02:18.740 --> 01:02:21.870 Integral of cosine of theta gives you zero. 01:02:21.870 --> 01:02:25.220 Essentially, it says that if there was no electric field, 01:02:25.220 --> 01:02:30.400 there was no reason for there to be an additional net charge 01:02:30.400 --> 01:02:32.920 on one side or the other. 01:02:32.920 --> 01:02:35.780 So the first term that will be non-zero 01:02:35.780 --> 01:02:38.930 is the average of cosine theta squared, which 01:02:38.930 --> 01:02:41.160 will give you a factor of one-half. 01:02:41.160 --> 01:02:50.970 And so, what you will get is that E minus E prime times 01:02:50.970 --> 01:03:04.400 L is-- well, there's going to be a factor of L. 01:03:04.400 --> 01:03:07.420 The integral of the theta cosine of theta 01:03:07.420 --> 01:03:16.340 squared-- the integral of cosine of theta squared 01:03:16.340 --> 01:03:20.780 is going to give you 2 pi, which is the integration times 01:03:20.780 --> 01:03:21.900 one-half. 01:03:21.900 --> 01:03:24.790 So this is the integral the theta cosine 01:03:24.790 --> 01:03:28.350 square theta will give you this. 01:03:28.350 --> 01:03:31.000 So we did this. 01:03:31.000 --> 01:03:33.700 We have two factors of y. 01:03:33.700 --> 01:03:36.330 y is our expansion parameter. 01:03:36.330 --> 01:03:38.510 We are at a low density limit. 01:03:38.510 --> 01:03:42.310 We've calculated things assuming that essentially, I 01:03:42.310 --> 01:03:47.050 have to look at one value of these diplodes. 01:03:47.050 --> 01:03:50.860 In principal, I can imagine that there will be multiple dipoles. 01:03:50.860 --> 01:03:54.560 And you can see that ultimately, therefore, potentially, I 01:03:54.560 --> 01:03:59.830 have order of y to the fourth that I haven't calculated. 01:03:59.830 --> 01:04:02.325 OK, so we got rid of the y squared. 01:04:05.340 --> 01:04:11.230 We have a factor of E prime on the expansion here. 01:04:23.550 --> 01:04:27.595 This factor is bothering me a little bit-- let me check. 01:04:27.595 --> 01:04:29.796 No, that's correct. 01:04:29.796 --> 01:04:31.940 OK, so I have the factor of k. 01:04:37.300 --> 01:04:41.010 I have a factor of 2 pi here that came from the charge. 01:04:41.010 --> 01:04:44.435 I have another 2 pi here-- so I have 4pi squared. 01:04:51.470 --> 01:04:54.900 I think I got everything except the integration 01:04:54.900 --> 01:05:04.110 over a to infinity dr. There is this r dr, which 01:05:04.110 --> 01:05:07.650 is from the two dimensional integration. 01:05:07.650 --> 01:05:11.495 There was another r here, and another r here. 01:05:11.495 --> 01:05:15.360 So this becomes r to the three. 01:05:15.360 --> 01:05:18.400 From here, I have minus 2 pi k. 01:05:20.930 --> 01:05:23.650 And then, I have the corresponding factors 01:05:23.650 --> 01:05:27.230 of a to the power of 2 pi k minus 4. 01:05:34.200 --> 01:05:34.700 OK. 01:05:37.690 --> 01:05:44.690 So you can see that the L's cancel. 01:05:44.690 --> 01:05:52.750 And what I get is that E-- once I 01:05:52.750 --> 01:05:56.210 take the E prime to the other side-- 01:05:56.210 --> 01:06:10.010 becomes E prime 1 plus I have 4pi cubed k y squared-- again, 01:06:10.010 --> 01:06:16.290 y squared is my small expansion parameter. 01:06:16.290 --> 01:06:22.460 And then, I have the integral from a to infinity, the r, r 01:06:22.460 --> 01:06:26.140 to the power of 3 minus 2 pi k, a 01:06:26.140 --> 01:06:33.050 to the power of 2 pi k minus 4, and then order of y squared, 01:06:33.050 --> 01:06:34.366 y to the fourth. 01:06:37.852 --> 01:06:40.840 OK. 01:06:40.840 --> 01:06:51.450 So basically, you see that the internal electric field 01:06:51.450 --> 01:06:57.700 is smaller than the external electric field 01:06:57.700 --> 01:07:03.610 by this factor, which takes into account the re-orientation 01:07:03.610 --> 01:07:08.080 of the dipoles in order to screen the electric field. 01:07:08.080 --> 01:07:10.210 And it is proportional in some sense 01:07:10.210 --> 01:07:15.000 to the density of these dipoles. 01:07:15.000 --> 01:07:18.610 And the twist is that the dipoles that we have 01:07:18.610 --> 01:07:21.850 can have a range of sizes that we 01:07:21.850 --> 01:07:24.940 have to integrate [INAUDIBLE]. 01:07:24.940 --> 01:07:31.170 So typically, you would write E prime to the E over epsilon. 01:07:31.170 --> 01:07:36.580 And so this is the inverse of your epsilon. 01:07:36.580 --> 01:07:45.500 And essentially, this is a reduction in everything 01:07:45.500 --> 01:07:49.110 that has to do with electric interactions because 01:07:49.110 --> 01:07:53.280 of the screening of other things. 01:07:53.280 --> 01:07:56.120 I can write it in the following fashion. 01:07:56.120 --> 01:08:02.150 I can say that there is an effective k-- that's 01:08:02.150 --> 01:08:08.870 called k effective-- which is different from the original k 01:08:08.870 --> 01:08:11.950 that I have. 01:08:11.950 --> 01:08:14.250 It is reduced by a factor of epsilon. 01:08:16.760 --> 01:08:23.960 So we were worried when we were doing the nonlinear sigma 01:08:23.960 --> 01:08:26.090 model that for any [INAUDIBLE], we 01:08:26.090 --> 01:08:31.689 saw that the parameter k was not getting modified because 01:08:31.689 --> 01:08:37.090 of the interactions among the spin modes, and that's correct. 01:08:37.090 --> 01:08:39.080 But really, at high temperatures, 01:08:39.080 --> 01:08:40.460 it should disappear. 01:08:40.460 --> 01:08:42.870 We saw that the correlations had to go away 01:08:42.870 --> 01:08:47.029 from power law form to exponential form. 01:08:47.029 --> 01:08:50.220 And so, we needed some mechanism for reducing 01:08:50.220 --> 01:08:52.399 the coupling constant. 01:08:52.399 --> 01:08:57.140 And what we find here is that this topological defects 01:08:57.140 --> 01:09:00.910 and their screening provide the right mechanism. 01:09:00.910 --> 01:09:03.540 So the effective k that I have is 01:09:03.540 --> 01:09:06.660 going to be reduced from the original k 01:09:06.660 --> 01:09:08.080 by the inverse of this. 01:09:08.080 --> 01:09:10.240 Since I'm doing an expansion in y, 01:09:10.240 --> 01:09:14.704 it is simply minus 4pi cubed ky squared, 01:09:14.704 --> 01:09:20.870 integral a to infinity dr, r to the 3 minus 2 pi k, 01:09:20.870 --> 01:09:28.210 a to the 2 pi k minus 4, plus order or y to the fourteenth. 01:09:28.210 --> 01:09:28.710 OK. 01:09:31.920 --> 01:09:36.550 Now actually, in the lecture notes that I have given you, 01:09:36.550 --> 01:09:42.210 I calculate this formula in an entirely different way. 01:09:42.210 --> 01:09:49.080 What I do is I assume that I have two topological defects-- 01:09:49.080 --> 01:09:53.689 so there I sort of maintain the picture of topological defect. 01:09:53.689 --> 01:09:58.290 And their interaction between them 01:09:58.290 --> 01:10:03.440 is this logarithmic interaction that has coefficient k. 01:10:03.440 --> 01:10:06.050 But then, we say that this [INAUDIBLE] interaction 01:10:06.050 --> 01:10:13.640 is modified because I can create pairs of topological defect, 01:10:13.640 --> 01:10:15.860 such as this, that will partially 01:10:15.860 --> 01:10:19.060 screen the interaction. 01:10:19.060 --> 01:10:22.120 And in the notes, we calculate what 01:10:22.120 --> 01:10:25.610 the effect of those pairs at lowest order 01:10:25.610 --> 01:10:29.970 is on their interaction that you have between them. 01:10:29.970 --> 01:10:32.860 And you find that the effect is to modify 01:10:32.860 --> 01:10:35.370 the coefficient of the logarithm, which 01:10:35.370 --> 01:10:38.670 is k, to a reduced k. 01:10:38.670 --> 01:10:42.230 And that reduced k is given exactly by this form. 01:10:42.230 --> 01:10:44.771 So the same thing you can get different ways. 01:10:44.771 --> 01:10:45.270 Yes? 01:10:45.270 --> 01:10:47.635 AUDIENCE: What if k is too small-- 01:10:47.635 --> 01:10:48.910 PROFESSOR: A-ha. 01:10:48.910 --> 01:10:50.280 Good. 01:10:50.280 --> 01:10:55.410 Because I framed the entire thing as if I'm 01:10:55.410 --> 01:10:58.364 doing a preservation theory for you 01:10:58.364 --> 01:11:02.140 in y being a small parameter. 01:11:02.140 --> 01:11:03.850 OK? 01:11:03.850 --> 01:11:07.720 But now, we see that no matter how 01:11:07.720 --> 01:11:20.510 small y is, if k is in fact less than 2 over pi-- 01:11:20.510 --> 01:11:24.890 so this has dimensions of r to the 4 minus 2 pi k. 01:11:24.890 --> 01:11:32.240 So if k is less than 2 over pi-- which incidentally is something 01:11:32.240 --> 01:11:41.920 that we saw earlier-- if k is less than that, 01:11:41.920 --> 01:11:43.940 this integral diverges. 01:11:46.750 --> 01:11:53.310 So I thought I was controlling my expansion 01:11:53.310 --> 01:11:58.950 by making y arbitrarily small, but what 01:11:58.950 --> 01:12:04.990 we see is that no matter how small I make y, 01:12:04.990 --> 01:12:09.440 if k becomes too small, the perturbation acuity 01:12:09.440 --> 01:12:12.600 blows up on me. 01:12:12.600 --> 01:12:16.990 So this is yet another example of a singular perturbation 01:12:16.990 --> 01:12:20.810 theory, which is what we had encountered when we were 01:12:20.810 --> 01:12:23.310 doing the Landau-Ginzburg model. 01:12:23.310 --> 01:12:27.070 We thought that our co-efficient of phi to the fourth u 01:12:27.070 --> 01:12:29.020 was a small parameter. 01:12:29.020 --> 01:12:32.810 You are making an expansion naively in powers of u. 01:12:32.810 --> 01:12:35.380 And then we found an expression in which 01:12:35.380 --> 01:12:38.230 the coeffecient-- the thing that was multiplying u 01:12:38.230 --> 01:12:41.240 at the critical point was blowing up on us. 01:12:41.240 --> 01:12:44.490 And so the perturbation theory inherently 01:12:44.490 --> 01:12:47.210 became singular, despite your thinking 01:12:47.210 --> 01:12:50.840 that you had a small parameter. 01:12:50.840 --> 01:12:55.200 So we are going to use the same trick 01:12:55.200 --> 01:13:04.300 that we used for the case of the Landau-Ginzburg model-- 01:13:04.300 --> 01:13:15.350 this is deal with singular perturbations 01:13:15.350 --> 01:13:18.080 by renormalization group. 01:13:22.020 --> 01:13:28.110 So what we see is that the origin of the problem 01:13:28.110 --> 01:13:31.520 is the divergence that we get over here when 01:13:31.520 --> 01:13:34.670 we try to integrate all the way to infinity 01:13:34.670 --> 01:13:37.880 or the size of the system. 01:13:37.880 --> 01:13:43.460 So what we do instead is we said, OK, let's not 01:13:43.460 --> 01:13:45.890 integrate all the way. 01:13:45.890 --> 01:13:51.400 Let's replace the short distance cut-off 01:13:51.400 --> 01:13:58.270 that we had with something that is larger-- ba-- 01:13:58.270 --> 01:14:03.470 and rather than integrating all of a to infinity, we integrate 01:14:03.470 --> 01:14:10.450 only over short distance fluctuations between a and ba. 01:14:10.450 --> 01:14:14.450 This is our usual [INAUDIBLE]. 01:14:14.450 --> 01:14:19.790 So what we therefore get is that the k effective is 01:14:19.790 --> 01:14:25.580 k 1 minus 4pi q ky squared integral 01:14:25.580 --> 01:14:37.530 from a to ba br r to the 3 minus 2 pi k a to the 2 pi k minus 4. 01:14:37.530 --> 01:14:42.820 And then, I have to still deal with 4pi cubed ky 01:14:42.820 --> 01:14:46.660 squared, integral from ba infinity dr 01:14:46.660 --> 01:14:53.840 r to the 3 minus 2 pi k, a to the 2 pi k minus 4, 01:14:53.840 --> 01:14:57.770 plus order of y to the fourth. 01:14:57.770 --> 01:15:00.728 OK? 01:15:00.728 --> 01:15:01.228 All right. 01:15:04.690 --> 01:15:14.990 So, you can see that the effect of integrating this much 01:15:14.990 --> 01:15:22.500 is to modify the decoupling to a new value which 01:15:22.500 --> 01:15:29.450 depends on b, which is just k minus 4pi cubed k squared 01:15:29.450 --> 01:15:37.730 y squared, a to ba br r to the 3 minus 2 pi k. 01:15:37.730 --> 01:15:42.471 a to the 2 pi k minus 4. 01:15:42.471 --> 01:15:42.971 OK? 01:15:46.430 --> 01:15:52.060 And then, I can rewrite the expression for k 01:15:52.060 --> 01:16:00.130 effective to be this k tilde. 01:16:00.130 --> 01:16:07.530 And then, whatever is left, which is 4pi cubed k 01:16:07.530 --> 01:16:14.220 squared y squared integral ba to infinity dr 01:16:14.220 --> 01:16:24.647 r to the 3 minus 2 pi k, a to the 2 pi k minus 4, order of y 01:16:24.647 --> 01:16:25.230 to the fourth. 01:16:29.530 --> 01:16:35.480 You see that k has been shifted through this transformation 01:16:35.480 --> 01:16:39.700 by an amount that is order of y squared. 01:16:39.700 --> 01:16:45.320 So at order of y squared in this new expression, 01:16:45.320 --> 01:16:53.470 I can replace all the k's that are carrying with k tilde 01:16:53.470 --> 01:16:56.625 and it would still be correct for this order. 01:16:59.950 --> 01:17:05.320 Now I compare this expression and the original expression 01:17:05.320 --> 01:17:07.750 that I had. 01:17:07.750 --> 01:17:12.510 And I see that they are pretty much the same expression, 01:17:12.510 --> 01:17:19.120 except that in this one, the cut-off is ba. 01:17:19.120 --> 01:17:21.290 So I do step two of origin. 01:17:21.290 --> 01:17:32.480 I define my R prime to be dr so that my new cut-off will 01:17:32.480 --> 01:17:35.300 be back to a. 01:17:35.300 --> 01:17:38.270 So then, this whole thing becomes 01:17:38.270 --> 01:17:45.850 k effective is k tilde minus 4pi cubed k 01:17:45.850 --> 01:17:49.740 tilde squared y squared. 01:17:49.740 --> 01:17:53.430 Because of the transformation that I did over here, 01:17:53.430 --> 01:17:59.760 I will get a factor of b to the 4 minus 2 pi k tilde, 01:17:59.760 --> 01:18:05.040 integral from a' to infinity-- the r prime-- r prime to the 3 01:18:05.040 --> 01:18:11.950 minus 2 pi k tilde, a to the 2 pi k tilde, minus 4 01:18:11.950 --> 01:18:13.629 plus order of y to the fourteenth. 01:18:20.120 --> 01:18:27.820 So we see that the same effective interaction 01:18:27.820 --> 01:18:32.810 can be obtained from two theories that 01:18:32.810 --> 01:18:39.780 have exactly the same cut-off, a, except that in one case, 01:18:39.780 --> 01:18:42.100 I had k and y. 01:18:42.100 --> 01:18:47.082 In the new case, I have this tilde or k prime at scale b. 01:18:47.082 --> 01:18:49.390 And I have to replace where I had 01:18:49.390 --> 01:18:53.920 y with y with this additional factor. 01:18:53.920 --> 01:18:58.370 So the two theories are equivalent provided 01:18:58.370 --> 01:19:03.430 that I say that the new interaction at scale b 01:19:03.430 --> 01:19:11.440 is the old interaction minus 4pi cubed k squared y squared. 01:19:11.440 --> 01:19:14.570 This integral is easy to perform. 01:19:14.570 --> 01:19:16.000 It is just the power law. 01:19:16.000 --> 01:19:21.190 It is b to the fourth minus 2 pi k minus 1. 01:19:21.190 --> 01:19:28.820 And then, I have 4 minus 2 pi k order of y to the fourth. 01:19:28.820 --> 01:19:33.910 And my y prime is y-- from here I 01:19:33.910 --> 01:19:38.116 see b to the power of 2 minus pi k. 01:19:43.462 --> 01:19:44.434 AUDIENCE: [INAUDIBLE] 01:19:48.322 --> 01:19:50.090 PROFESSOR: This k squared? 01:19:50.090 --> 01:19:51.550 AUDIENCE: Oh, I see. 01:19:51.550 --> 01:19:52.050 Sorry. 01:19:58.525 --> 01:19:59.400 PROFESSOR: All right. 01:20:03.820 --> 01:20:13.200 So, our theory is described in terms of two parameters-- 01:20:13.200 --> 01:20:17.780 this y and this k. 01:20:17.780 --> 01:20:22.970 or let's say, it's inverse-- k inverse, 01:20:22.970 --> 01:20:25.910 which is more like temperature. 01:20:25.910 --> 01:20:29.100 And what we will show next time is 01:20:29.100 --> 01:20:34.950 that these recursion relations, when I draw it here, 01:20:34.950 --> 01:20:38.170 will give me two types of behavior. 01:20:38.170 --> 01:20:41.470 One set of behavior that parameterizes 01:20:41.470 --> 01:20:47.700 the low temperature dilute limit that corresponds 01:20:47.700 --> 01:20:51.890 to flows in which y goes through zero. 01:20:51.890 --> 01:20:55.400 So that when you look at the system at larger and larger 01:20:55.400 --> 01:20:59.830 lens scales, essentially it becomes less and less depleted 01:20:59.830 --> 01:21:04.030 of these excitations. 01:21:04.030 --> 01:21:07.030 So, once you have integrated the very essentially, 01:21:07.030 --> 01:21:09.960 you don't see any excitations. 01:21:09.960 --> 01:21:12.210 And then there's another phase, which 01:21:12.210 --> 01:21:18.810 as you do this removal of short distance fluctuations. 01:21:18.810 --> 01:21:22.550 You tend to flow to high temperatures 01:21:22.550 --> 01:21:24.690 and large densities. 01:21:24.690 --> 01:21:29.451 And so, that corresponds to this kind of face. 01:21:29.451 --> 01:21:33.150 Now the beauty of this whole thing 01:21:33.150 --> 01:21:38.600 is that these recursion relations are exact and allow 01:21:38.600 --> 01:21:43.650 us to exactly determine the behavior of these space 01:21:43.650 --> 01:21:46.140 transition in two dimensions. 01:21:46.140 --> 01:21:49.830 And that's actually one of the other triumphs 01:21:49.830 --> 01:21:54.040 of renormalization group is to elucidate 01:21:54.040 --> 01:21:56.950 exactly the critical behavior of this transition, 01:21:56.950 --> 01:22:00.000 as we will discuss next time.