1 00:00:00,210 --> 00:00:05,040 Welcome back, everyone. This is the 21st lecture of the condensed matter course. 2 00:00:05,640 --> 00:00:10,320 When we left off last time, we were talking about magnetism and in particular, we were talking about feral magnetism. 3 00:00:11,280 --> 00:00:20,409 Feral. Magnetism and everything that we said about magnetism was really focussed on understanding what happens at zero temperature, 4 00:00:20,410 --> 00:00:27,040 all of the moments aligning with each other. But generally you can ask the question what happens to a power magnet as you raise the temperature? 5 00:00:27,040 --> 00:00:35,860 And we have a little bit of an intuition about what happens in power magnetism at Finite, which will be the subject of much of today's lecture. 6 00:00:37,840 --> 00:00:40,300 You know, at zero temperature, all the moments align. 7 00:00:40,570 --> 00:00:47,469 But then when you turn on the temperature, there's going to be some thermal fluctuations and the moments won't be completely aligned anymore. 8 00:00:47,470 --> 00:00:50,140 And at some point we might not have our magnetism at all. 9 00:00:50,380 --> 00:00:56,020 So in general, what we want to know is we would like to know as a function of temperature what is the magnetisation, 10 00:00:56,020 --> 00:01:01,540 some sort of equation of state, and maybe will also allow for an externally applied magnetic field. 11 00:01:01,810 --> 00:01:07,030 So this is what we're interested in finding out. Now, I should start with a bit of a caveat. 12 00:01:07,030 --> 00:01:15,909 In the last lecture, we spent a lot of time thinking about the idea of magnetic domains which are created due to long range dipolar interactions. 13 00:01:15,910 --> 00:01:20,920 And even though the local physics is to align all the moments in a ferromagnetic low temperature, 14 00:01:21,220 --> 00:01:27,190 there will be some global physics which will make some regions point up in some regions point down to minimise the global magnetic energy. 15 00:01:27,400 --> 00:01:31,150 So we're not going to think about that kind of physics today. We're only going to think sort of locally. 16 00:01:31,360 --> 00:01:38,139 So for us, a ferromagnetic at zero temperature would have all their moments aligned, at least for for the purpose of the day. 17 00:01:38,140 --> 00:01:42,190 And then we can put back into long range dipolar interactions later if we're interested. 18 00:01:42,700 --> 00:01:48,460 So we're going to use a very simple model and try to understand what happens to the magnetisation and find a temperature. 19 00:01:48,670 --> 00:01:54,639 We use the spin one half easing model, one half easing by spin, 20 00:01:54,640 --> 00:02:00,969 one half is what we mean is that the spin on each side will be some sigma i pointing the 21 00:02:00,970 --> 00:02:09,280 z direction where sigma I equals plus or minus one half and we'll write the Hamiltonian, 22 00:02:09,370 --> 00:02:14,560 which we've seen before. There will be a coupling to the external magnetic field. 23 00:02:15,370 --> 00:02:18,489 B Some overall sides sigma. 24 00:02:18,490 --> 00:02:25,840 I And here I've put the external magnetic field in the Z direction and then we have 25 00:02:25,840 --> 00:02:33,159 the coupling between neighbour spins minus one half J Some of our neighbouring spins. 26 00:02:33,160 --> 00:02:42,489 I am J Sigma I Sigma J So again the bracket notation means that I enjoy our neighbours and here we're concerned with our ferromagnetic. 27 00:02:42,490 --> 00:02:47,000 So we said J greater than zero. All right. 28 00:02:47,660 --> 00:02:51,770 So this is the the easing model, a fairly simple models write down. 29 00:02:51,770 --> 00:02:55,009 But as I mentioned before, it's actually quite difficult to solve. 30 00:02:55,010 --> 00:02:59,419 It is solvable in one dimension, in two dimensions. It's solvable without the magnetic field. 31 00:02:59,420 --> 00:03:06,049 And beyond that, it's not solvable at all. So that means we're going to have to resort to some sort of approximation method to get some 32 00:03:06,050 --> 00:03:10,220 sort of understanding of what happens to the magnetisation easing model of finite temperature. 33 00:03:12,100 --> 00:03:14,320 Now there's lots of approximation methods one can use, 34 00:03:14,590 --> 00:03:20,260 but the first resort of theories or first resort of scoundrels is what is known as mean field theory. 35 00:03:20,960 --> 00:03:31,090 I mean field theory, which we are going to use to try to estimate what happens to this Hamiltonian at finite temperature. 36 00:03:31,360 --> 00:03:44,470 So mean field theory can be approximately defined as whenever you approximate some operator or a quantity some operator. 37 00:03:47,660 --> 00:03:56,750 Or quantity with an expectation with an average or a mean. 38 00:04:00,050 --> 00:04:05,880 And this is used very, very commonly, not only in condensed matter physics, but also outside of condensed matter physics. 39 00:04:05,930 --> 00:04:12,020 This is sort of a very general statement. There are many things that you can start approximating as an expectation are an average. 40 00:04:12,350 --> 00:04:18,320 So we're going to use a particular variety of mean field theory known as molecular field theory. 41 00:04:24,590 --> 00:04:28,080 Very or also it's known as vice minefield theory. 42 00:04:28,680 --> 00:04:40,760 Vice mean field theory. This is the same guy, Vice, who was discovered the idea of domains in magnets as well and discussed them last time. 43 00:04:41,120 --> 00:04:44,479 And I have to admit, I don't know if it's I before or before either. 44 00:04:44,480 --> 00:04:47,240 So apologise about that. My spelling is not very good. 45 00:04:48,650 --> 00:04:56,240 So the idea of molecular field theory, it has basically two very simple steps that you can always follow. 46 00:04:56,540 --> 00:05:00,200 Step one is that you should treat one piece of the system. 47 00:05:00,470 --> 00:05:07,220 Exactly. You isolate one little piece of the system. It might be a single unit cell or it might be just a single site. 48 00:05:07,340 --> 00:05:10,909 And you to treat that. Exactly. So treat one piece of system. 49 00:05:10,910 --> 00:05:18,379 Exactly. Treat one piece of system. 50 00:05:18,380 --> 00:05:26,300 Exactly. Everything else should be averaged. 51 00:05:26,720 --> 00:05:30,620 All else is averaged. 52 00:05:35,030 --> 00:05:41,059 So we've actually seen something like this before. When we study the Einstein model of the solid, we treated one atom. 53 00:05:41,060 --> 00:05:46,490 Exactly. We treat it as quantum mechanics correctly, but we average the potential from all the other atoms. 54 00:05:46,610 --> 00:05:53,450 We made it into some rough average and it just became this approximate potential, well, that one atom is living in. 55 00:05:53,690 --> 00:05:58,849 So even though it is a very crude approximation in that one atom we treated very honestly, 56 00:05:58,850 --> 00:06:04,640 it was the effect of all the neighbours that we approximated. The second step of the field theory. 57 00:06:05,540 --> 00:06:10,010 Step two is self consistency. Self consistency. 58 00:06:12,950 --> 00:06:24,500 Which means all pieces. All pieces should be the same, should look the same in the end, should look the same at the end. 59 00:06:24,830 --> 00:06:30,260 And and so it shouldn't matter which part of the system we focus on to treat. 60 00:06:30,260 --> 00:06:34,220 Exactly. At the end of the day, the whole system should look homogeneous again. 61 00:06:34,430 --> 00:06:40,969 It shouldn't be anything special about the one piece we're going to look at. So for us, we will treat the easing from the magnet. 62 00:06:40,970 --> 00:06:47,000 And we're going to focus in on just one side, a one unit cell. In this case, these ferromagnetic unit cells, a single site. 63 00:06:48,080 --> 00:06:52,820 So the so we focus on, okay, so step one, 64 00:06:53,300 --> 00:07:02,780 look at one site look at one site site I for the easing ferromagnetic I'll write down the Hamiltonian for site I only 65 00:07:03,260 --> 00:07:12,470 so Hamiltonian for site I is going to be B Times Sigma I That's the coupling to the externally applied magnetic field. 66 00:07:12,860 --> 00:07:18,250 Then there's going to be the coupling to all of its neighbours, which I'll write this in this way. 67 00:07:18,260 --> 00:07:27,970 J is neighbour of I. I Sigma J time sigma i. 68 00:07:28,920 --> 00:07:34,440 So I've written out all the terms in the Hamiltonian that contain the spin sigma i. 69 00:07:35,350 --> 00:07:42,820 Right. Now, you may have thought here for a second that I may have dropped the factor of a half this written off top, but I haven't. 70 00:07:43,190 --> 00:07:48,669 Remember that the factor of a half of top is there, because in that sum, strictly speaking, 71 00:07:48,670 --> 00:07:56,950 there is signal one, sigma two, but there is also sigma two, sigma one. In this sum, if sigma I is sigma one, it does not occur the other way around. 72 00:07:56,950 --> 00:08:00,610 It only occurs once a sigma two single one would just be represented once in this Hamiltonian. 73 00:08:00,820 --> 00:08:08,580 So you don't need the factor of half when you write it this way. All right, so this looks a little complicated. 74 00:08:08,590 --> 00:08:12,190 Let's see if we can simplify it a little bit. Will right. 75 00:08:12,190 --> 00:08:16,780 Will define define an effective magnetic field. 76 00:08:17,080 --> 00:08:21,430 Be effective, be effective via the following equation. 77 00:08:21,940 --> 00:08:33,549 G b b effective equals g b the actual magnetic field, the external plus or minus the influence of the neighbours. 78 00:08:33,550 --> 00:08:47,260 J time sum over small j is a neighbour of ii of I sigma j So what we're doing is we're wrapping up 79 00:08:47,260 --> 00:08:53,620 the interaction with the neighbours and we're calling it part of the effective magnetic field. 80 00:08:54,280 --> 00:09:07,719 In terms of this quantity, we can rewrite the Hamiltonian for site I as just give you the be effective time Sigma II, which looks extremely simple. 81 00:09:07,720 --> 00:09:14,170 It just looks like a single spin sigma I coupled to an external effective magnetic field be effective 82 00:09:14,170 --> 00:09:18,370 and we know how to solve that we solve that before it is a single spin in a magnetic field. 83 00:09:18,400 --> 00:09:23,260 Should be easy, right? Well, okay, that's a little deceptive because we've sort of lied here. 84 00:09:23,470 --> 00:09:26,620 To be effective is not just a number here. It's not a constant. 85 00:09:26,800 --> 00:09:30,370 It actually has some degrees of freedom hidden in it. You should think of it as an operator. 86 00:09:30,610 --> 00:09:34,990 It has all these sigma JS in it, and that complicated life is going to make everything really complicated. 87 00:09:35,260 --> 00:09:41,410 So we can't really treat this simply as a spin coupled to a constant magnetic field. 88 00:09:42,790 --> 00:09:44,740 So that's where my field theory comes in handy. 89 00:09:45,010 --> 00:09:51,640 The rule of mean field theory is that you should approximate everything except this one spin in terms of an average. 90 00:09:51,940 --> 00:09:53,080 So step one, 91 00:09:53,560 --> 00:10:05,740 step one is approximate be effective by some average of the effective average would be effective by taking this average instead of being an operator, 92 00:10:05,740 --> 00:10:15,850 it's now just a number. So the Hamiltonian becomes one, the Hamiltonian becomes H, II is going to be expectation of B effective, 93 00:10:15,850 --> 00:10:23,890 which is now just a number couple of the Sigma I and that really is just a single spin coupled to a constant magnetic field. 94 00:10:24,520 --> 00:10:28,330 So we know how to solve this. We have done it before. We did it. We've done it last year. 95 00:10:28,630 --> 00:10:33,310 We did it in the last lecture. So I'll just write down the first step. 96 00:10:33,700 --> 00:10:44,170 You write down the partition function, it's either minus beta G, new B expectation B effective times plus one half plus. 97 00:10:44,320 --> 00:10:50,410 E to the plus give you the beta gbps expectation. 98 00:10:50,410 --> 00:10:53,950 B Effective times one half. 99 00:10:54,760 --> 00:11:04,270 How this looks familiar that we can differentiate this partition function in order to get expectation of Sigma I and the 100 00:11:04,270 --> 00:11:18,400 expectation sigma I then becomes minus one half hyperbolic tangent of beta g movie expectation and be effective times one half. 101 00:11:19,180 --> 00:11:21,850 Okay. So now we have an expression. 102 00:11:23,060 --> 00:11:31,710 For Sigma I for the expectation of Sigma I in terms of the external the effect of magnetic field the temperature and Gobi. 103 00:11:32,090 --> 00:11:34,640 So far, so good. Happy with this? All right, good. 104 00:11:34,970 --> 00:11:45,720 So now the last step is we have to figure out with this expectation and be effective is so expectation of right this way, be expectation. 105 00:11:45,740 --> 00:11:59,810 Be effective is just going to be, uh, be the external magnetic field minus j some over j is is neighbour of i. 106 00:12:02,920 --> 00:12:07,420 Of I. Times expectation of Sigma J. 107 00:12:08,600 --> 00:12:12,650 Okay so I just took the average of of be effective here. 108 00:12:13,750 --> 00:12:21,640 And the the final step of the minefield theory is that at the end of the day, all sites should look the same. 109 00:12:21,850 --> 00:12:26,690 So the average of sigma on site I should be the same as the average. 110 00:12:26,860 --> 00:12:31,419 So this is step to self consistency. Average of sigma on site. 111 00:12:31,420 --> 00:12:37,540 I should be the same as the average of sigma on site j. There is nothing special about site II versus idea, and we'll just call it Sigma. 112 00:12:38,240 --> 00:12:42,700 Okay, so we're just setting the average of Sigma the same on every every site. 113 00:12:44,080 --> 00:12:56,980 So if we do that, then our equation here with the tension, the hyperbolic tangent can be written as sigma equals minus one half hyperbolic tangent. 114 00:12:58,120 --> 00:13:01,900 And now I'll plug in our expression for be effective. 115 00:13:02,230 --> 00:13:05,970 Where? Well, actually. All right. 116 00:13:05,980 --> 00:13:10,990 Well, okay. Uh. Up here. 117 00:13:10,990 --> 00:13:16,149 This some here, I suppose. This some here. I can be the Times Sigma. 118 00:13:16,150 --> 00:13:19,780 Where Z is the number of neighbours. Sorry. Number of neighbours. 119 00:13:24,350 --> 00:13:33,649 Since we set Sigma expectations of Sigma J to be just this number Sigma and there's G terms in the sum Z terms in sum. 120 00:13:33,650 --> 00:13:36,980 Exactly what I just did. Sorry, was a little messy. Okay. 121 00:13:37,310 --> 00:13:51,180 All right. So this ends up coming out of minus beta jay z over two times sigma plus beta, JMU B B over two. 122 00:13:52,190 --> 00:13:56,930 So this is just a rewriting of this of the equation up there with the with the hyperbolic tangent. 123 00:13:57,170 --> 00:14:06,739 I plugged in the effective and I plugged in j time z four where I had j times expectation j times 124 00:14:06,740 --> 00:14:13,550 z times sigma is where I had j times sum over neighbours times expectation of Sigma J and yes. 125 00:14:14,090 --> 00:14:18,260 All right, all right. All right. So this is our our final result. 126 00:14:19,250 --> 00:14:25,970 And it is some sort of self-consistent equation for the magnitude of sigma, which is the expectation of the spin on site ie. 127 00:14:26,390 --> 00:14:33,500 And so in principle, we now have our solution for what sigma is as a function of the temperature, 128 00:14:33,800 --> 00:14:37,160 the coupling to the neighbours, the number of neighbours, external magnetic field. 129 00:14:37,760 --> 00:14:41,150 It is a complicated equation, but at least in principle we can solve it. 130 00:14:41,570 --> 00:14:46,340 The moment per side would then be minus g, b, times sigma. 131 00:14:46,730 --> 00:14:51,440 Where are the sign? So this is the moment per per spin. 132 00:14:51,740 --> 00:14:59,270 This minus sign comes from the fact that electrons have negative charge, so the magnetic moment points in the opposite direction of the spin. 133 00:14:59,630 --> 00:15:02,690 Okay. So in principle, we've now solved our problem. 134 00:15:03,440 --> 00:15:13,850 However, this is sort of a complicated, transcendental equation, and it's not so obvious what its solution is with some equation four sigma. 135 00:15:14,420 --> 00:15:21,270 So we have to figure out how to solve it. And let's simplify to a an easy case first. 136 00:15:21,270 --> 00:15:25,370 So case. Case one hour. Just to set the case to be equals zero. 137 00:15:25,580 --> 00:15:29,750 External magnetic field is zero, in which case simplifying this equation. 138 00:15:29,750 --> 00:15:40,970 Moving a sign inside we get sigma is one half hyperbolic tangent of beta jay z sigma over two. 139 00:15:42,430 --> 00:15:49,720 Good. It's still a hard equation to solve because it's still a transcendental equation for Sigma. 140 00:15:50,080 --> 00:15:53,229 But there is a nice graphical way to solve this equation, 141 00:15:53,230 --> 00:16:04,570 which is to plot y equals sigma and y equals one half tense beta jay z sigma over two on the same plot. 142 00:16:05,700 --> 00:16:08,760 And where they intersect is the solution of this equation. 143 00:16:09,460 --> 00:16:13,180 That's what we're going to do. That's how we're going to solve this equation. They're also with me and we're happy. 144 00:16:13,590 --> 00:16:18,570 Yeah. Okay, good. So. So here we go. 145 00:16:20,290 --> 00:16:23,919 First we have to decide, you know, put it over here now. 146 00:16:23,920 --> 00:16:27,880 Okay. This right here. Why? Sigma over here. 147 00:16:28,210 --> 00:16:31,630 Temporary sigma. Sigma over here. 148 00:16:31,960 --> 00:16:35,160 Then we'll plot the first curve, which is y equals sigma like this. 149 00:16:36,700 --> 00:16:42,649 Y equals sigma. That is the first curve that we want to plot. 150 00:16:42,650 --> 00:16:49,220 And then the second curve is this hyperbolic tangent which asymptotes at one half and at minus one half. 151 00:16:49,940 --> 00:16:54,350 So we'll asymptote here and it will asymptote down here. 152 00:16:55,370 --> 00:17:02,840 But there's two possibilities as to what this plot looks like with the hyperbolic and hyperbolic. 153 00:17:04,190 --> 00:17:12,450 Hyperbolic tangent with this hyperbolic tangent looks like. One possibility is what occurs for a low temperature. 154 00:17:12,960 --> 00:17:17,580 Low temperature, what happens outside this new high temperature? First high temperature. 155 00:17:19,200 --> 00:17:27,180 High temperature is the first possibility. So in high temperature Beira is small and so the tense is a very gentle function. 156 00:17:27,240 --> 00:17:31,250 So it looks kind of like this. Okay. 157 00:17:31,510 --> 00:17:34,840 So this is a high temperature. High temperature. 158 00:17:38,790 --> 00:17:45,150 Can people happy with that? That's a hyperbolic tangent curve and it asymptotes to this dotted line. 159 00:17:46,420 --> 00:17:54,850 So what we see is these two curves, the hyperbolic tangent and the Y equal sigma curve intersect only at one point at sigma equals zero. 160 00:17:55,150 --> 00:18:00,850 So the only solution only solution is sigma equals zero. 161 00:18:01,660 --> 00:18:04,750 And that tells us that you do not have fair magnetism at high temperature. 162 00:18:04,960 --> 00:18:11,680 You in fact get R and equals magnetisation equals zero or aura magnetism rather than power. 163 00:18:11,680 --> 00:18:18,520 Magnetism. Power magnetism. Not very magnetism at high temperature. 164 00:18:22,150 --> 00:18:23,050 At high temperature. 165 00:18:25,530 --> 00:18:34,079 And this is actually quite typical that any firm magnet, if you raise it in temperature at some point, the pro magnetism will go away. 166 00:18:34,080 --> 00:18:39,209 And you have this sort of intuition that what that's coming from is that the moments are fluctuating 167 00:18:39,210 --> 00:18:43,890 thermally around so much that they eventually stop pointing in any particular direction, 168 00:18:43,890 --> 00:18:48,180 and they don't point in any if they're not pointing in any particular direction is not a fair magnet anymore. 169 00:18:48,300 --> 00:18:51,570 And in fact, it will be a power magnet. The show in a few moments. 170 00:18:52,860 --> 00:18:55,679 The other possibility is that we're at low temperature. 171 00:18:55,680 --> 00:19:06,060 So think about that possibility, low temperature, in which case the hyperbolic tangent is a much steeper curve like this. 172 00:19:07,950 --> 00:19:13,080 Because the argument of the hyperbolic tangent is larger because beta is larger at low temperature. 173 00:19:13,380 --> 00:19:19,200 And so in fact there's three solutions. So this is the low temperature of low temperature. 174 00:19:19,200 --> 00:19:24,299 So we have a solution here. We'll call that sigma, not a T with a solution at zero. 175 00:19:24,300 --> 00:19:27,810 And then there's a solution here which we'll call minus sigma, not of T. 176 00:19:28,710 --> 00:19:41,670 So we get 3 to 3 solutions, three solutions which are sigma equals zero, and then plus or minus sigma not of t which are non-zero. 177 00:19:42,060 --> 00:19:48,000 The fact that we have non-zero solutions for Sigma is telling us that we have a foul magnet and 178 00:19:48,000 --> 00:19:53,549 naturally the foul magnet can have it spins pointing up or it can have its pins pointing down. 179 00:19:53,550 --> 00:19:55,380 So these are the ferromagnetic solutions. 180 00:19:57,450 --> 00:20:05,700 And this is not surprising because we know that if you go down to zero temperature, this curve will be extremely steep and will intersect right here. 181 00:20:05,700 --> 00:20:08,660 As a matter of fact, when you get complete polarisation that sigma, 182 00:20:08,670 --> 00:20:13,020 the expectation of sigma is one half, meaning all the spins point in exactly the same direction. 183 00:20:13,410 --> 00:20:16,440 Okay, so that would be what you get at zero temperature. 184 00:20:16,620 --> 00:20:22,650 At higher temperature, the intersection moves down to a slightly lower point and you get slightly smaller 185 00:20:23,250 --> 00:20:27,360 sigma not of t so you get slightly smaller magnetisation but not zero magnetisation. 186 00:20:27,720 --> 00:20:35,640 Now we might wonder what about this guy? Here we have another solution which tells us we have an expectation of Sigma being zero. 187 00:20:36,000 --> 00:20:41,010 And this one actually is a little puzzling. It turns out it's a not a physical solution. 188 00:20:41,640 --> 00:20:46,830 Not physical. And you'll actually show that for one of your homework assignments. 189 00:20:47,280 --> 00:20:51,629 So it turns out what your show for your homework is that this self-consistent 190 00:20:51,630 --> 00:20:57,750 equation that we solve is actually the same thing as extra missing a free energy. 191 00:20:58,770 --> 00:21:02,730 And as you know, an extra mile is something you can either get a maximum or a minimum. 192 00:21:03,000 --> 00:21:06,480 So we're getting one maximum and two minimum of the free energies. 193 00:21:06,690 --> 00:21:12,870 The minimum are the free energies are the things that are realised physically and the maximum of the free energy is sort of mistake. 194 00:21:13,080 --> 00:21:18,719 It comes out of the equations, but it's not physical and it is much clearer when you actually write down the free energy 195 00:21:18,720 --> 00:21:22,230 which you'll do for homework and you'll see why it is that we don't keep that solution. 196 00:21:22,260 --> 00:21:27,540 So this one we sort of ignore and we only keep track of these ferromagnetic solutions. 197 00:21:29,160 --> 00:21:35,850 So we'd like to know at what temperature do we go from being a power magnet to being a feral magnet? 198 00:21:36,180 --> 00:21:45,690 Just write that down. At what t at what t do we get? 199 00:21:49,040 --> 00:21:54,550 There are magnetism. Well, there's actually a name for that. 200 00:21:55,000 --> 00:22:01,810 D.C. is known as for less than D.C. You get feral magnetism from AG, 201 00:22:02,410 --> 00:22:09,490 and D.C. is known as the Curie Temperature Kerry Temp or the critical temperature critical temp. 202 00:22:13,130 --> 00:22:22,850 Although if you have feral magnetism. So we'd like to know when do we get at what is TSI or when do we get this magnetism at below what temperature? 203 00:22:23,640 --> 00:22:25,040 Well, the way to see that again, 204 00:22:25,040 --> 00:22:32,149 looking graphically is to realise that the transition between having one solution and having three solutions occurs when the 205 00:22:32,150 --> 00:22:41,030 hyperbolic tangent the function is exactly tangent to the y equals sigma line where it aligns up just exactly with the same slope. 206 00:22:41,450 --> 00:22:45,440 And the reason that that's the important point where, 207 00:22:45,530 --> 00:22:50,689 where the where you have the transition from power magnetism to magnetism is because of the 208 00:22:50,690 --> 00:22:55,820 hyperbolic tangent where any less steep you have only one solution and if any more steep, 209 00:22:56,030 --> 00:23:03,320 you have three solutions. So we have to look for the for the situation where the hyperbolic tangent has exactly the same slope as y equals sigma. 210 00:23:03,560 --> 00:23:16,010 Okay. So t from finding slope of tangent of tangent. 211 00:23:21,380 --> 00:23:31,790 Equals one. So if we just expand the Y equals equation here, the slope of which we're setting equal to one is better. 212 00:23:32,150 --> 00:23:37,100 Jay Z over four or just rearranging a little bit. 213 00:23:37,100 --> 00:23:45,380 We get KB T equals Jay Z over four, which is the critical value of the temperature. 214 00:23:46,030 --> 00:23:49,580 Is everyone still with me? Fairly. Fairly happy now? 215 00:23:50,180 --> 00:23:53,420 Yes, maybe. Okay. Maybe. All right. 216 00:23:53,600 --> 00:23:58,040 So let me just pause for a second to sort of. 217 00:23:59,610 --> 00:24:05,500 Okay. Well, it's time for chocolate, actually. So I suffer from one chocolate bar. 218 00:24:05,520 --> 00:24:14,819 What's the. What is the critical importance of. Think outside the box music is really good. 219 00:24:14,820 --> 00:24:17,970 He said that, as we say, he's married to Beyoncé, but it's close enough. 220 00:24:18,330 --> 00:24:22,500 Yeah, so. Oh, yeah, very good. 221 00:24:22,500 --> 00:24:27,130 Yeah. Okay, now Jay-Z is beside me managing Beyoncé. 222 00:24:27,330 --> 00:24:33,570 The importance of Jay-Z here is that it is. In fact, it is telling you that Beyonce's middle name. 223 00:24:36,030 --> 00:24:40,320 You. Not her last name. No, no, it's a sister's name. 224 00:24:41,040 --> 00:24:45,480 Yeah, right. No, no. Does anyone know Beyonce's baby's name? 225 00:24:47,530 --> 00:24:50,649 Yeah. Someone got it over there. Yes. Yeah. Okay. 226 00:24:50,650 --> 00:24:51,820 I have one for you also. All right. 227 00:24:52,660 --> 00:25:02,410 So those are the important of Jay-Z here is actually is the total amount of energy of coupling of the spin to its neighbours. 228 00:25:02,800 --> 00:25:10,750 And so we have the neighbours here and j is the the coupling strength from the spin to each of its neighbours. 229 00:25:11,050 --> 00:25:15,100 So when the temperature gets above the total amount of coupling strength to its neighbours, 230 00:25:15,370 --> 00:25:23,290 the fluctuation in the thermal fluctuations of that spin so overwhelmed the amount that it wants to stay in the same direction as its neighbours, 231 00:25:23,290 --> 00:25:29,040 which is what is keeping it a power magnet. It wants to have the same direction as its neighbours and when the energy of 232 00:25:29,350 --> 00:25:32,620 when the thermal energy is is greater than the energy holding it in place, 233 00:25:32,830 --> 00:25:37,030 you lose power, magnetism. Okay, that's the intuition behind this picture here. 234 00:25:37,450 --> 00:25:38,709 All right. All right. 235 00:25:38,710 --> 00:25:50,170 Anyway, so at least graphically, we can take this picture and we can solve for our sigma as a function of t just have to figure out where, 236 00:25:50,410 --> 00:25:55,899 where those two curves cross as we change t and we'll get a picture that looks kind of like this. 237 00:25:55,900 --> 00:25:59,500 Here is temperature. Here's sigma at zero. 238 00:25:59,500 --> 00:26:08,080 Temperature sigma is actually a half because of all the spins are aligned and then at TSI it goes to zero and above. 239 00:26:08,680 --> 00:26:16,870 Above T is exactly zero, but then it comes down kind of like this at a finite temperature as you changed. 240 00:26:17,080 --> 00:26:20,680 So this is it actually gives you the equation of state of how much magnetisation you 241 00:26:20,680 --> 00:26:26,140 get as a function of temperature and you can get it from this new field approximation. 242 00:26:26,320 --> 00:26:30,100 It is not exact, but it is fairly good. Okay. All right. 243 00:26:30,100 --> 00:26:36,040 So the last step is that I promised you that up in this region here, we have a power magnet, so we should. 244 00:26:37,990 --> 00:26:42,610 We actually convince ourselves that it really is a power magnet. And one way to do that is to calculate susceptibility. 245 00:26:43,090 --> 00:26:48,040 So things that we know is up in this region. If you do not apply a magnetic field sigma zero. 246 00:26:48,250 --> 00:26:51,790 If you apply a small magnetic field in Sigma is probably going be close to zero. 247 00:26:52,060 --> 00:26:59,469 So what we can do is we can take our original self-consistent equation and we can expand it for a small argument of the hyperbolic tangent, 248 00:26:59,470 --> 00:27:05,950 because everything inside there is by definition small C the sigma is smaller, b is small, everything is small. 249 00:27:06,250 --> 00:27:10,180 So we can write sigma equals. 250 00:27:10,210 --> 00:27:14,440 As you just writing this all out, I'm going to observe a couple of minus signs here. 251 00:27:15,100 --> 00:27:19,240 Beta C sigma over two minus beta. 252 00:27:19,900 --> 00:27:24,040 Jim, you b over to the magnetic field. 253 00:27:24,430 --> 00:27:29,260 Okay. And then we just rearrange this. 254 00:27:29,770 --> 00:27:35,530 So we solve for sigma as a functional magnetic field, we get sigma axial right this way. 255 00:27:35,530 --> 00:27:39,910 One minus G over for beta. 256 00:27:41,160 --> 00:27:46,920 Equals minus data. I give you B over four times a magnetic field. 257 00:27:47,310 --> 00:27:51,990 And this notice this quantity here is actually itsy bitsy. 258 00:27:52,650 --> 00:28:06,720 So with a little bit more rearranging, we get sigma equals minus one quarter grubby magnetic field B divided by k minus kb. 259 00:28:07,890 --> 00:28:14,320 T c. And then we want to actually convert this into some sort of susceptibility. 260 00:28:14,620 --> 00:28:21,610 So the magnetic moment is minus G, movie sigma and magnetisation. 261 00:28:22,210 --> 00:28:31,240 So this is the moment. Precisely. The Magnetisation is the moment perceived times rho the density of spin's density of spins, 262 00:28:33,400 --> 00:28:41,560 and then the susceptibility will be new, not by definition the m d b at zero. 263 00:28:44,150 --> 00:28:48,080 Which is just plugging in a couple lines up here. 264 00:28:48,350 --> 00:28:55,300 We then get one quarter row g b squared mu, 265 00:28:55,340 --> 00:29:12,169 not over KB times one over one minus t over t and this whole quantity here is actually the query susceptibility query for a spin one half. 266 00:29:12,170 --> 00:29:17,989 In other words, it is a susceptibility we would have calculated with just a free spin, one a half. 267 00:29:17,990 --> 00:29:24,320 In fact, it is exactly what we calculated when we calculated the magnetic susceptibility of a spin one half in the last lecture. 268 00:29:24,590 --> 00:29:29,060 But then this whole thing gets multiplied by one over one minus t over t. 269 00:29:30,050 --> 00:29:37,370 So the susceptibility of our power magnet above the AC is a susceptibility of just a system of free spin, 270 00:29:37,370 --> 00:29:39,350 one halves which aren't coupled to each other. 271 00:29:39,530 --> 00:29:47,090 And the effect of the coupling in the easing model, at least with the mean field theory, is to multiply it by this factor of one minus t over t. 272 00:29:47,570 --> 00:29:57,530 Okay, now you may notice that at the critical temperature or the curie temperature, this factor diverges and that actually is physical, it is real. 273 00:29:57,830 --> 00:30:03,860 And the reason it's it's there is because we have to remember that that susceptibility tells you 274 00:30:04,100 --> 00:30:10,070 how much magnetisation you get when you turn on a little bit of magnetic field by definition. 275 00:30:10,220 --> 00:30:14,930 But when you go below the critical temperature, you get magnetisation even for zero magnetic fields, 276 00:30:15,090 --> 00:30:20,540 susceptibility must have gotten infinite somewhere along the line. So that is what this term is doing there. 277 00:30:22,310 --> 00:30:27,890 All right. So this is the general scheme of, ah, of minefield theory and how it works. 278 00:30:29,450 --> 00:30:37,550 Minefield theory, where was it? A minefield theory and how it works where we generally approximate some operator by an average. 279 00:30:38,780 --> 00:30:43,490 In this technique, a minefield can be used much, much more generally than what we did here. 280 00:30:43,730 --> 00:30:47,070 For homework, you'll do minefield theory calculation for an ANTIFERROMAGNETIC. 281 00:30:47,090 --> 00:30:52,219 Well, you could do it for a very magnet. You could do it for spins that aren't spin one half. 282 00:30:52,220 --> 00:30:59,750 The fact that we got a hyperbolic tangent in this in this whole story came from the fact that we were thinking about a spin one half. 283 00:30:59,750 --> 00:31:06,290 So it had only spin up and spin down. If we're thinking about spin one, we would have spin as the equals, one as equals, zero as equals minus one. 284 00:31:06,290 --> 00:31:10,700 We get something more complicated than a hyperbolic tangent. We get a so-called broken function. 285 00:31:10,940 --> 00:31:16,519 But the general themes of how you how you approach it treat one piece of the system exactly. 286 00:31:16,520 --> 00:31:20,239 And then insist on self consistency that remains the same. 287 00:31:20,240 --> 00:31:24,110 And the technique is used outside of magnetism in condensed matter. 288 00:31:24,350 --> 00:31:31,520 It is used for understanding liquid crystals, it is used for understanding fluid dynamics, is used in astrophysics, is used in atmospheric physics. 289 00:31:31,730 --> 00:31:38,450 Mean field theory is just a very general technique which is used for many, many things, and it is good to internalise the ideas of it. 290 00:31:39,470 --> 00:31:47,360 All right. We are going to use mine field theory for one more thing, which is to understand itinerant, feral magnetism. 291 00:31:47,690 --> 00:31:52,850 Itinerant, which is the next subject, an itinerant magnetism. 292 00:31:56,480 --> 00:32:02,540 And the word the word itinerant means wandering, basically, or sometimes it means lost. 293 00:32:03,380 --> 00:32:13,310 But what it really means is that the magnetic moment moments are carried are carried by mobile electrons. 294 00:32:14,240 --> 00:32:17,750 By mobile electrons. 295 00:32:20,140 --> 00:32:27,940 Now since we started talking about magnetism a couple lectures ago, we thought about putting magnetic moments on sides, 296 00:32:27,940 --> 00:32:31,240 and the magnetic moments can move and point in different directions. 297 00:32:31,390 --> 00:32:36,400 But we did not consider the possibility that the moments could actually hop from one side to another side. 298 00:32:36,700 --> 00:32:39,700 But in real magnets, like in iron, for example, 299 00:32:39,910 --> 00:32:47,230 is quite typical that the same things that are giving you magnetisation these electrons are actually moving around to in bands. 300 00:32:47,410 --> 00:32:50,410 And yet they managed to give us a magnetism. 301 00:32:51,380 --> 00:32:57,340 So as with other types of magnetism, itinerant magnetism is driven by interactions. 302 00:32:57,350 --> 00:33:02,600 It comes from the interaction between electrons from from interactions. 303 00:33:07,110 --> 00:33:15,270 So somehow we have to introduce interactions and treat them at least within some Enfield approximation in order to understand itinerant magnetism. 304 00:33:16,260 --> 00:33:22,650 Now of course, once you start introducing interactions between electrons, things get complicated very quickly. 305 00:33:22,860 --> 00:33:31,140 So we need a very simplified model which will represent the interaction between electrons and still give us the right, the right physics. 306 00:33:31,380 --> 00:33:34,560 So the model we're going to use is the so-called Hubbard model. 307 00:33:36,650 --> 00:33:43,030 Hubbard wrote this down in 1963 to try to understand itinerant magnetism. 308 00:33:43,360 --> 00:33:48,340 And since he wrote this down, the Hubbard model has taken on a huge life of its own. 309 00:33:49,180 --> 00:33:55,420 It is potentially it is probably the most important model of interacting electrons in all of physics. 310 00:33:55,810 --> 00:34:01,120 There have been, probably, without exaggeration, 100,000 papers written on the subject of the Hubbard model. 311 00:34:01,120 --> 00:34:07,029 It's that important to the field. And the reason it's so important is because it is exceptionally simple, and yet it is. 312 00:34:07,030 --> 00:34:14,260 Physics is quite complex. So by the way, Hubbard treated it in mean field theory is what we're going to do here. 313 00:34:14,440 --> 00:34:19,970 It is fairly simple. You get a good result, which helps you understand itinerant feral magnetism, which is what we're going to do in the model. 314 00:34:19,990 --> 00:34:33,880 It's pretty straightforward. What we do is we start by writing a Hamiltonian that is type binding, type binding type binding electrons with hopping. 315 00:34:37,630 --> 00:34:43,480 With some hopping t and this part of the Hamiltonian a tight binding electrons with some hopping. 316 00:34:43,660 --> 00:34:47,590 We have solved this before. We we looked at this earlier in the term. 317 00:34:47,590 --> 00:34:51,309 We got these cosine shaped bands. You can throw up the cosine shaped bands. 318 00:34:51,310 --> 00:34:53,850 If it's a Hatfield, then you get a metal if it's a full band again. 319 00:34:53,860 --> 00:35:00,219 INSULATOR But never in just this model here with just type binding electrons with hopping which you ever get from magnetism, 320 00:35:00,220 --> 00:35:05,410 you would never get a situation if you only have this type binding structure 321 00:35:06,070 --> 00:35:10,050 where you would ever have a different number of up moments from down moments, 322 00:35:10,060 --> 00:35:14,350 they would always come out. Even so, we have to add something to it, which is going to change the story. 323 00:35:14,590 --> 00:35:17,950 And what we're adding to it is the interaction term. 324 00:35:18,340 --> 00:35:26,680 Some overall size I parameter you use greater than zero times the number of electrons on site. 325 00:35:26,680 --> 00:35:30,850 I would spin up times the number of electrons on site I will spin down. 326 00:35:31,960 --> 00:35:43,060 Right. This is number of electrons on site I with spin up down to down. 327 00:35:43,810 --> 00:35:44,830 Oh, okay. 328 00:35:46,000 --> 00:35:54,850 So what this is doing is it is just telling you that you have an energy penalty of you whenever two electrons sit on the same atom, on the same side. 329 00:35:55,570 --> 00:35:59,110 And we've actually discussed this Hamiltonian before, we didn't use the word hybrid model, 330 00:35:59,410 --> 00:36:04,810 but I'll remind you, when we talked about Mott Insulators, Smart Insulators, 331 00:36:07,090 --> 00:36:13,330 we discussed a situation where the interaction between electrons was so strong that you could only put a single electron on a site, 332 00:36:13,780 --> 00:36:16,630 and then you can get a situation. If there is a single electron on a side, 333 00:36:16,900 --> 00:36:22,270 then everything gets frozen in a traffic jam because no one can hop to the neighbouring side because there's already someone sitting there. 334 00:36:22,270 --> 00:36:28,150 And even though you have something that looks like it should be a half filled band, you still have an insulator known as an insulator. 335 00:36:28,270 --> 00:36:32,229 So this is exactly the same physics. We're just going to allow ourselves to turn on and off that interaction. 336 00:36:32,230 --> 00:36:37,809 You make it as strong as we want or as weak as we want. Okay, so not too complicated. 337 00:36:37,810 --> 00:36:43,030 It looks like a very simple model, but again, it's one of these models. It looks simple, but isn't it solvable in one dimension? 338 00:36:43,030 --> 00:36:48,220 And absolutely nothing is known about it, despite 100,000 papers written on it in higher dimensions. 339 00:36:48,730 --> 00:36:54,610 Okay, maybe not absolutely nothing. But. But. But not as much as we would like, which is why people keep writing papers about it. 340 00:36:54,880 --> 00:37:04,930 Okay, so how do you treat this? Well, first thing we're going to do is we're going to write this interaction term as in the following form. 341 00:37:05,740 --> 00:37:18,820 We'll write it as you over four and up plus and down minus two over four and up of squared, minus and up, minus n down squared. 342 00:37:19,360 --> 00:37:23,950 So that is just an identity that is easy. We haven't done anything different here. 343 00:37:25,800 --> 00:37:29,190 What we're going to do now is we going to make the minefield, approximation minefield, 344 00:37:31,050 --> 00:37:35,580 which a minefield always somehow or other means take something that average 345 00:37:35,580 --> 00:37:38,940 where you're not allowed to do something that is incorrect by taking an average. 346 00:37:39,150 --> 00:37:43,550 And what we're going to do, which is not quite correct, is we're going to take an average of these terms. 347 00:37:43,560 --> 00:37:52,920 So we're going to take average of analysing down and square it, minus you over for average of and up, minus and down and square it. 348 00:37:53,480 --> 00:37:59,310 Okay. So that is the mean field approximation or it's one type of minefield approximation we're going to use and we see what it gives us. 349 00:37:59,670 --> 00:38:08,610 Well, first of all, this term here, average of up and down, that is just a total density, total number of electrons, total density. 350 00:38:09,510 --> 00:38:15,690 And that is fixed, probably fixed at the beginning of the day by the conditions that you have. 351 00:38:15,690 --> 00:38:18,060 So that is just going to give that first term is going to give us a constant. 352 00:38:18,390 --> 00:38:26,190 The interesting thing is this term here that end up minus and down is actually the moment for a site divided 353 00:38:26,190 --> 00:38:33,479 by the more magnetite and then it gets squared minus you over for a moment divided by the town square. 354 00:38:33,480 --> 00:38:37,620 Right, because the number of spin ups and downs is the magnetic moment on on the side. 355 00:38:38,190 --> 00:38:44,879 Okay. So we can rewrite this interaction term as some constant which we're not interested in, minus you over four. 356 00:38:44,880 --> 00:38:52,140 And then I'm going to rewrite this moment as magnetisation times the volume of the unit cell over new B squared. 357 00:38:52,380 --> 00:38:55,470 So Y equals volume of unit cell. 358 00:38:58,110 --> 00:39:01,319 And the reason I'm rewriting it that way is because I would like to have this thing 359 00:39:01,320 --> 00:39:06,000 is a magnetisation and magnetisation has dimensions of moment per unit volume. 360 00:39:06,000 --> 00:39:09,479 So I have to put in a the volume of unit cell to have the dimensions come out. 361 00:39:09,480 --> 00:39:19,440 Right. Okay. And the thing to notice about this is that the energy, the interaction energy gets smaller as the magnetisation gets larger. 362 00:39:20,280 --> 00:39:28,110 Why is that? Well, the intuition for that actually comes from an exclusion principle if all the spins were aligned. 363 00:39:28,500 --> 00:39:33,480 In other words, if the magnetisation were maximum, then there would be no interaction energy whatsoever. 364 00:39:33,660 --> 00:39:37,500 Because you can never have a situation where two electrons sat on the same side and you 365 00:39:37,500 --> 00:39:40,830 only have to pay this energy penalty when there are two electrons on the same side. 366 00:39:41,100 --> 00:39:46,770 So if you polarise all your spins in the same direction, you get the minimum possible interaction energy. 367 00:39:46,770 --> 00:39:51,509 And that is why it is that if you increase the magnetisation, you drop the interaction energy. 368 00:39:51,510 --> 00:40:00,870 Is that clear? Good. All right. So from this picture, it makes me think then, okay, once you turn on the interaction, all the spins want to align. 369 00:40:00,870 --> 00:40:04,499 But it is not so easy because aligning the spins cost you kinetic energy. 370 00:40:04,500 --> 00:40:09,209 And you have to you have to sort of let these two things fight it out with each other. 371 00:40:09,210 --> 00:40:15,030 So let's remind ourselves what happens to the kinetic energy as you change the magnetisation of a system. 372 00:40:15,390 --> 00:40:23,040 And now we studied this before. So if you recall poly power magnetism, which we did in the first week. 373 00:40:23,460 --> 00:40:32,360 Pauli Paramagnetic. We actually calculated, although maybe not in the same language. 374 00:40:32,660 --> 00:40:36,440 You'll recall we had the density of states of spin up versus energy kind of looks like this. 375 00:40:36,650 --> 00:40:43,070 And then we had the density of states for spin down versus energy and it looks exactly the same. 376 00:40:43,250 --> 00:40:47,540 And if there is no magnetisation, you fill them both up to the same level. 377 00:40:48,830 --> 00:40:57,590 Does it sound familiar from poly power magnetism? Now, if you want to make this person look identical, if you want to have a nonzero magnetisation, 378 00:40:57,800 --> 00:41:01,220 what you have to do is you have to take some of the spin downs and make them spin ups. 379 00:41:01,490 --> 00:41:07,040 So to do that, let's take a sliver of the spin downs and we have to move them up to higher energy up here. 380 00:41:08,000 --> 00:41:12,380 So it cost you energy. So these were supposed to be aligned with that. 381 00:41:12,650 --> 00:41:18,380 So it costs you energy to have a non-zero magnetisation kinetic energy to have a non-zero magnetisation. 382 00:41:18,890 --> 00:41:24,860 All right. So we save some interaction energy, but it costs kinetic energy, so we have to trade those off against each other. 383 00:41:25,220 --> 00:41:28,760 So in order to figure out how those are battle against each other, 384 00:41:28,910 --> 00:41:33,440 we have to figure out how much energy it actually costs you to have some non-zero magnetisation. 385 00:41:33,440 --> 00:41:39,319 So let's do that. So first we'll go back to the poly paramagnetic calculation. 386 00:41:39,320 --> 00:41:41,300 So we're going to set you equal to zero for a second. 387 00:41:41,310 --> 00:41:47,030 So now we're back to non interacting electrons and I'm going to write the free energy as a function of magnetism, 388 00:41:47,480 --> 00:41:53,240 as a function magnetisation per unit volume, and that will then be minus B. 389 00:41:53,900 --> 00:41:56,229 I think everyone is probably will accept that, 390 00:41:56,230 --> 00:42:04,160 that there should be a term like that in the free energy minus B that wants the MAGNETISATION to align with the with the external magnetic field. 391 00:42:04,430 --> 00:42:06,440 But then the second term is less obvious, 392 00:42:06,740 --> 00:42:15,230 but it takes the form m squared mu not over two times the poly susceptibility, poly paramagnetic susceptibility. 393 00:42:16,100 --> 00:42:22,040 And I'll just remind you, we calculated this in the in the first week of the term with Chi Poly. 394 00:42:23,810 --> 00:42:30,590 Was high poly is new, not new B squared times the density of states at the Fermi Energy. 395 00:42:31,610 --> 00:42:38,840 Does that kind of sound familiar? Okay. Now, why is it that I wrote down this complicated term here, and why do I think this is correct? 396 00:42:39,110 --> 00:42:43,819 Well, first of all, this is sort of a tail or expansion in the magnetisation. 397 00:42:43,820 --> 00:42:46,490 So this is linear and magnetisation is quadratic in magnetisation. 398 00:42:46,730 --> 00:42:52,160 In principle, there would be higher times as well, which we're going to ignore because we're interested in small magnetisation in general. 399 00:42:53,030 --> 00:42:56,630 But why would I give it this term? Mu not over to Chi Pauli? 400 00:42:56,810 --> 00:43:01,790 Well, it turns out that it actually has to have this form and to see that it has to have this form. 401 00:43:02,030 --> 00:43:05,210 We know that the free energy should be should be minimised. 402 00:43:05,600 --> 00:43:16,930 So the steam has to equal zero. And that tells me that minus B plus m mu not over our chi poly has two equal zero. 403 00:43:17,300 --> 00:43:29,570 And if you just rearrange that, that tells me that the magnetisation is chi poly over mu not times the magnetic field, which is what defines Chi Poly. 404 00:43:29,570 --> 00:43:36,920 The definition of the poly susceptibility for our free electron gas is the magnetisation should be chi, poly times, the magnetic field. 405 00:43:36,920 --> 00:43:40,860 So that tells me the free energy has to have that form that I just wrote down. Okay. 406 00:43:41,330 --> 00:43:48,530 So a little bit of a trick you could go through the agony of of trying to actually calculate the magnetisation the 407 00:43:48,530 --> 00:43:53,030 energy is a function of magnetisation you get the same result you would discover it has to come out this way. 408 00:43:53,570 --> 00:44:06,020 Okay, so, so let's turn off setting, set the external field, set B equal to zero and turn, turn you on you not equal to zero. 409 00:44:07,010 --> 00:44:13,340 And then the free energy we get is going to be a combination of the two terms that we just calculated. 410 00:44:13,670 --> 00:44:18,890 So free energy volume is then, well, okay, we have this term here. 411 00:44:19,340 --> 00:44:30,530 I turned off B, so I just get m squared nu not over to chi poly but then also are just about to scroll off the top. 412 00:44:30,800 --> 00:44:40,370 There is minus u over four times while k v squared over new b squared. 413 00:44:41,330 --> 00:44:46,760 With this squared of times I'm going to get rid of a v magnetisation squared was v squared, 414 00:44:46,760 --> 00:44:51,770 but now it's going to be v because I'm writing the free energy per unit volume, not the free energy in this case. 415 00:44:51,770 --> 00:44:56,420 Okay. So I dropped a factor of a volume, so it's not like V squares is v. 416 00:44:57,230 --> 00:44:57,620 All right. 417 00:44:58,040 --> 00:45:12,770 So what this is, I guess we can factor out the M, so we have m squared and then new not over to chi poly minus u over for the over mu b squared. 418 00:45:15,080 --> 00:45:22,970 And this tells us there will be higher terms, two terms which are quadratic and magnetisation and so forth, which you're not interested in. 419 00:45:23,240 --> 00:45:30,380 But the key thing to realise here is if this thing in the bracket is less than zero, well magnetisation wants to be non-zero. 420 00:45:30,650 --> 00:45:34,760 Whereas if the thing in the brackets is greater than zero, then the magnetisation wants to be zero. 421 00:45:35,240 --> 00:45:41,480 Okay, so the criterion. So our criterion for magnetism. 422 00:45:43,370 --> 00:45:48,980 For Farrow, which is known as the stoner criterion. 423 00:45:50,420 --> 00:45:51,380 Stoner and Criterion. 424 00:45:54,900 --> 00:46:07,860 Is that this new not over too high poly minus u volume with you would sell over four more magnets on squared should be less than zero. 425 00:46:08,190 --> 00:46:15,990 So when you turn on the interaction strength strong enough, then the interaction strength wins and it wants to have this the spins polarise, 426 00:46:16,200 --> 00:46:18,780 whereas if the interaction is not strong enough, 427 00:46:19,110 --> 00:46:25,210 then in fact it is more beneficial to have an equal number of spin up since spin downs because you say kinetic energy that way. 428 00:46:25,470 --> 00:46:29,100 Actually, we can rewrite this in a just putting in the definition of chi poly. 429 00:46:29,100 --> 00:46:35,970 We can write this as our volume of the universal cell times. You times get f should be greater than two. 430 00:46:37,420 --> 00:46:46,870 And equivalent. Okay. So that is what we get out of this minefield approximation for figuring out when we have firm magnetism. 431 00:46:47,200 --> 00:46:51,310 But one of the things that's interesting about the Hubbard model is you can also get 432 00:46:51,850 --> 00:47:00,790 Hubbard ANTIFERROMAGNETIC Hubbard ANTIFERROMAGNETIC ISM and Thai Pharaoh magnetism. 433 00:47:03,820 --> 00:47:11,950 And this occurs when we have you as large, you large and one electron parasite. 434 00:47:11,980 --> 00:47:16,060 So this is in the Mott INSULATOR regime. Mott INSULATOR. 435 00:47:17,920 --> 00:47:23,410 So if you is large, it prevents you from having too electrons parasite or mostly prevents that. 436 00:47:23,770 --> 00:47:31,540 And then since it's one electron parasite, you have this traffic jam of electrons where you have exactly one electron on every side. 437 00:47:31,720 --> 00:47:37,180 And the way to convince yourself that this thing is going to be antiferromagnetic is to try the two possibilities. 438 00:47:37,190 --> 00:47:42,290 So case one is consider a ferromagnetic. Case, one is considered ferromagnetic. 439 00:47:44,380 --> 00:47:47,590 Ferromagnetic. So we have a bunch of spins. 440 00:47:48,040 --> 00:47:51,100 Electrons each with their spin sitting on each side. They're all aligned. 441 00:47:51,520 --> 00:47:55,090 And the energy of this situation is actually zero. 442 00:47:55,120 --> 00:47:58,840 The energy of the ground state is is exactly zero. 443 00:47:59,140 --> 00:48:07,270 And the reason zero is, well, you never pay any interaction energy, because due to the Pauli principle, no electrons sit on the same site ever. 444 00:48:07,450 --> 00:48:11,650 So you don't ever pay the price. You don't ever pay the new energy. 445 00:48:11,890 --> 00:48:19,060 And also, the electrons can't hop from one side to the other at all because they're in this traffic jam so they can't hop. 446 00:48:19,270 --> 00:48:27,100 So the total energy of hopping is also zero. Now the second possibility is that you have an antiferromagnetic case to antiferromagnetic. 447 00:48:28,540 --> 00:48:36,010 And this is a little more complicated because we have spin up, spin down, spin up, spin down and so forth. 448 00:48:36,460 --> 00:48:43,990 And here, well an electron can hop to the neighbouring side but it has to, it has to pay a price to do so. 449 00:48:44,230 --> 00:48:47,680 So you get spin up, spin down, you have zero on this side and then you have up, 450 00:48:47,680 --> 00:48:51,640 down on this site, which costs you energy you and then up on this side. 451 00:48:52,180 --> 00:48:56,650 So this is an excited state, which is a valid, excited state. It costs you some energy. 452 00:48:56,660 --> 00:49:00,880 You you might be large, but at least virtually you can make such a thing. 453 00:49:00,890 --> 00:49:06,459 And so in second order perturbation theory, during this sort of hopping process, you get a shift in the energy, 454 00:49:06,460 --> 00:49:12,520 which is some over all possible excitations excitation hopping Hamiltonian ground 455 00:49:12,520 --> 00:49:19,990 state squared over energy zero minus energy excited state and this is less than zero. 456 00:49:20,350 --> 00:49:23,740 So in fact, the change in energy is less than zero. 457 00:49:24,070 --> 00:49:32,080 So Antiferromagnetic energy for Antiferromagnetic is less than energy referring magnet. 458 00:49:35,030 --> 00:49:39,800 When you have one electron parasite in this in this might insulator limit. 459 00:49:40,010 --> 00:49:43,520 Now, this might seem a little complicated, but let me rephrase what we just did. 460 00:49:44,300 --> 00:49:51,470 If you have a ferromagnetic so in the case of a ferromagnetic, each electron is basically sitting in an infinitely tall well. 461 00:49:52,070 --> 00:49:56,809 You can't hop to the left, you can have to the right. That is absolutely forbidden by the exclusion principle. 462 00:49:56,810 --> 00:49:59,840 So you should think of it as an electron sitting in an infinitely tall well. 463 00:50:00,170 --> 00:50:07,190 Whereas for an antiferromagnetic you have an electron in a finite well. 464 00:50:08,360 --> 00:50:11,930 So the height of the well is only you. It can hop to the neighbouring side. 465 00:50:12,140 --> 00:50:18,140 It costs an energy you, but it can do it. It is not forbidden by Pauli and the energy of the anti. 466 00:50:18,140 --> 00:50:22,970 For a magnet with a finite barrier is lower than the energy of the Fuhrer magnet with the infinite barrier. 467 00:50:23,210 --> 00:50:30,020 So therefore Antiferromagnetic is favoured. So this is sort of one of the things that makes the anti for our magnetism a will. 468 00:50:30,320 --> 00:50:34,280 So it makes the hybrid model quite interesting that it can give you anti-foreign magnetism. 469 00:50:34,280 --> 00:50:35,540 It can give you foreign magnetism. 470 00:50:35,690 --> 00:50:40,999 It can give you lots of other physics as well, including superconductivity, charge, density waves and all sorts of other things, 471 00:50:41,000 --> 00:50:44,960 which is why there have been a hundred thousand papers written on this hybrid model. 472 00:50:45,110 --> 00:50:49,610 You're not responsible for all of that, I promise. You're supposed to know something about itinerant power magnetism. 473 00:50:49,610 --> 00:50:53,230 But that's all. Okay. So sit still. 474 00:50:53,660 --> 00:50:57,380 We're going to go over on this lecture because we actually have come to the end of the course. 475 00:50:57,590 --> 00:51:00,350 So I just want to wrap everything up. We're going to cancel tomorrow's lecture. 476 00:51:00,680 --> 00:51:05,299 This is this is actually on schedule, which is it's actually planned to be 21 lectures. 477 00:51:05,300 --> 00:51:08,810 But I leave an extra lecture in case of disasters or running behind. 478 00:51:08,810 --> 00:51:14,840 And we didn't run behind. We did everything exactly as planned. So we are actually finished the course more or less. 479 00:51:15,620 --> 00:51:20,120 A couple of organisational comments before before the closing remarks. 480 00:51:20,840 --> 00:51:24,020 First of all, obviously no lecture tomorrow. That is the first comment. 481 00:51:24,260 --> 00:51:32,210 The second comment is in no one or not, no one, but very few people actually use the message board heavily this this term that that's okay. 482 00:51:32,420 --> 00:51:35,629 I don't know why it is people didn't like it that that's fine. 483 00:51:35,630 --> 00:51:38,060 But you know, this message board is still up and running. 484 00:51:38,300 --> 00:51:43,400 And I think as it gets closer to exams, people will get more enthusiastic about using it and asking questions. 485 00:51:43,610 --> 00:51:47,060 That is fine. That is what it's meant to be, therefore. So don't hesitate to use it. 486 00:51:47,270 --> 00:51:52,700 Ask questions. Your heart's content. I will try to answer as many of them as I can as you revise for your exams. 487 00:51:52,970 --> 00:52:00,080 The only thing that I want to ask you to do is let me know what you want to have for the revision lectures. 488 00:52:00,080 --> 00:52:07,730 We have two revision lectures in the fourth week of 20 term and I am always completely at a loss as to what the most useful thing to talk about is. 489 00:52:08,450 --> 00:52:10,819 If you think I should try to just give an overview to do that, 490 00:52:10,820 --> 00:52:15,890 if there are particular questions from that from previous year's exams that you want to go over, let me know. 491 00:52:16,100 --> 00:52:19,460 I'll try to assemble something that people will find useful. 492 00:52:19,910 --> 00:52:22,910 All right. So just a couple of closing ideas for the course. 493 00:52:23,690 --> 00:52:28,999 When we started seven and a half weeks ago, I promised that that by the end of the course, 494 00:52:29,000 --> 00:52:33,160 we'd actually know something more about the world around us. And I think we genuinely do. 495 00:52:33,170 --> 00:52:41,540 We know things like why diamonds transparent and and why ions a magnet and why solids expand at higher temperature and many, many more things. 496 00:52:41,540 --> 00:52:47,449 And some of them very esoteric and some of them less esoteric things like quantum why quantum effects are important 497 00:52:47,450 --> 00:52:53,330 for heat capacity or how a transistor works and and how you determine the crystal structure of a material in X. 498 00:52:53,930 --> 00:52:55,549 For those of you who go on in science, 499 00:52:55,550 --> 00:53:02,750 I hope this forms a good foundation for for future learning and discovering more things along the same lines or even in other fields. 500 00:53:03,170 --> 00:53:06,499 For those of you do, don't go on to science. That is fine too. 501 00:53:06,500 --> 00:53:12,079 I hope that at least in a good and an entertaining exploration of the natural world, 502 00:53:12,080 --> 00:53:16,280 and I hope it leaves you with a better appreciation for the subject at the very least. 503 00:53:16,910 --> 00:53:21,050 I do realise that there is an awful lot of material in this course and I apologise about that. 504 00:53:21,320 --> 00:53:27,620 It is more or less a fixed amount of material fixed by the exams, the exam committee and the syllabus. 505 00:53:27,620 --> 00:53:30,740 So we don't really have that much freedom to remove material. 506 00:53:31,400 --> 00:53:34,790 So I realise that a lot of people view this as drinking from a firehose. 507 00:53:35,210 --> 00:53:39,590 It's just so much information coming, coming at you. At the same time it's hard to absorb. 508 00:53:39,920 --> 00:53:45,590 But ironically, the the biggest problem with the course is that there is actually not enough in it. 509 00:53:45,590 --> 00:53:48,979 And I don't say that because I feel like you need to learn more for your exams. 510 00:53:48,980 --> 00:53:54,860 I say that because condensed matter is such an enormous subject that we have hardly scratched the surface. 511 00:53:55,070 --> 00:53:57,950 It seems such a shame to leave it here, but this is all we have time for. 512 00:53:58,580 --> 00:54:06,559 We could easily do a year of lectures or more on material physics or quantum effects or statistical effects or device physics, 513 00:54:06,560 --> 00:54:11,450 or any one of a dozen types of things that we just barely mentioned at all. 514 00:54:11,750 --> 00:54:17,450 And as well as many, many things that we didn't even get to superconductors, superfluids, fractionalised, particles, so forth and so on. 515 00:54:17,730 --> 00:54:22,550 I hope I really hope many of you have the opportunity to go on and learn about those things in the future, 516 00:54:23,960 --> 00:54:27,890 and hopefully it will be as entertaining or even more entertaining as what you've learned so far. 517 00:54:28,460 --> 00:54:33,320 Finally, I just want to thank you all for having come to lectures and having looked enthusiastic throughout the term. 518 00:54:33,980 --> 00:54:34,580 Thank you very much.