 So welcome to this afternoon session on quantum criticality. So just a reminder that you are encouraged to ask questions even during the talk and to interrupt the speaker. So our first speakers for this session will be Premie Chandra and she will talk about quantum annealed criticality. Okay thank you very much. Can everyone hear me? Yes. Okay first of all thank you very much. It's wonderful to be back in Trieste. I have lots of good memories here and I haven't been here for a while so it's terrific to be back. So what is the interplay of quantum criticality with first order phase transitions? Today I'd like to tell you about a project that I've been working on with Gil Lonzoic who will be here later in the week, an experimentalist who has really motivated this to a large degree as you'll see with some recent experiments, Museo Contentino in Brazil and Paris Coleman who as you know will be here later in the week. So to make a long story short let me give you a nutshell of what we're going to talk about. There are a number of systems which have first order classical phase transitions and yet seem to indicate continuous quantum critical behavior. So the question that Gil came to us with is how is it, can we come up with a theoretical scenario where we have classically first order transitions but quantum mechanically a second order one. But before I begin, since this week is in many ways a celebration of a big birthday of peers who unfortunately can't be here, I probably have access to some of the earliest pictures of peers and as I was always told by my mentors to take advantage of my competitive edge, I might as well do that. So as you know, I think Andre read it this morning, he is extremely opinionated and wants you to ask lots of questions, unfortunately he can't be here right now. Now you may think that peers is only opinionated in the seminar room but one of the things we learned was that even in the sandbox, peers was very opinionated. He had a loud voice even then and he was telling people exactly what they needed to do with whatever they were playing with. All right, this is actually a picture more recently of peers with his dear mother Gloria and he sends his apologies for not being here right now. We were on a holiday with Gloria this past week and unfortunately she had a medical emergency. The good news is that she's fine, she needed to have some unexpected surgery. They're in Prague so being waylaid somewhere, Prague is not so bad. The doctors were wonderful, it all worked out with everything. I think she's due to be released today. She's eager to get going and peers will be here on Thursday. So he sends his regrets that he's not here right now and I thought that it might be fun given the occasion for me to talk about one of our most current projects. So that's what I'm going to do today. All right, and there'll be more pictures later on, but first some physics. All right, so once again the motivation for us is what is the interplay of quantum criticality with first-order transitions. As we'll see there are a number of experiments that indicate a line of first-order transitions that ends in a quantum critical point. Turns out that there's an old mechanism of coupling of the strain to the energy density where incompressible systems one always has a first-order transition. And what our merry gang set out to do is to ask whether what happens to the quantum generalization of this so-called Larkin-Pickin transition. As we'll see there are occasions where in fact you can have a quantum critical point even though it's always first-order classically. And in such a situation our proposed phase diagram temperature tuning parameter field we have two classical critical points. We have a quantum critical point and a sheet of first-order transitions that's bounded by those. And for those of you who are of the same as young as I am this may remind you of a rolling stone's cover. For those of you who don't know about that, whatever. But from last millennium there was a rolling stone's cover that looked a bit like this. Anyway, so let us continue. What's known since I was asked to give a little bit of a background in all this? Let's step back a minute. What's known about the interplay of quantum criticality with first-order transitions? Well, Gil with Andy Schofield who's here and two other Andes and more actually thought about this quite carefully in the context of strontium ruthonate. In that system they had a quantum first-order transition. And what they showed is as you increase the temperature, you get a trichritical point and then it turns into a continuous phase transition. So that's in some sense the flip of what we're talking about. And in that system, if you want to get a quantum critical point, you actually have to apply a field and there you go, have a quantum critical point. So this is work that's been around for a little while. In fact, there's a very nice review recently on metallic ferromagnets that discusses this. So we know that just because we have a classical second-order transition doesn't mean that quantum mechanically it has to be the same. And we're going to ask the other question. So marching right along, for those of you who are not so familiar, let's talk about the difference between classical and quantum phase transitions. And here I've taken a phase diagram from a very nice system that Gabriel Lepley, who's just come, has worked on. I think this is a nice way of summarizing it. For a classical phase transition, as you well know, thermal fluctuations are crucial, okay? The thermal fluctuations are crucial, and temperature is the tuning parameter, okay? However, in a quantum phase transition, we're at very low temperatures. And temperature isn't the important scale. Omega, there's a frequency, that's the important scale. And since there's a frequency involved, it also tells us that dynamics are involved. So we have another tuning parameter, not temperature. It can be pressure. It can be transverse field, as it is in this case. There's a number of different tuning parameters. And the point is that the effective dimensionality is not just the spatial dimension. It's D plus Z, where Z is the dynamical exponent due to the dispersion. So that tells us that we have to worry about dynamics as well. In a quantum phase transition, temperature is not the tuning parameter. It acts actually as a finite boundary effect. So let's continue on. Let's just do a quick review of the work that was done before. In that case, we had a T equals 0 first-order transition. And when do we get a first-order transition? We get a first-order transition when the quartic mode coupling in the effective action is negative, okay? So it's negative, it's originally negative. But what this group found is, as they increased the temperature, thermal fluctuations renormalized the quartic coupling, moving it from negative to zero at the tricritical point to positive. And so they had low temperatures, a first-order transition, then a tricritical point with wings, and then eventually a continuous phase transition. And this is from the review of Brando, Billets, Kirkpatrick, and Grosje. And there are many materials that show this type of behavior with this signature sort of winged phase diagram, okay? So that's where we are. Now, what is our experimental motivation? Our experimental motivation comes from ferroelectrics. I mentioned that the other phase diagram, the quantum critical endpoints, was motivated by metallic systems. We are going to be, for the moment, motivated by insulating systems. Now, ferroelectrics, like this poster, Child Ferroelectric Barium Titanate, are insulating materials where electromechanical coupling is crucial. And they're usually, at ambient pressure, they have first-order transitions, okay? So a first-order transition, here we have polarization as a function of temperature. This is just reminding us what a first-order transition is. It means that we have a metastable state, we have a very sharp specific heat, excuse me, polarization, like this. And in fact, what was shown in the, in barium titanate, with time-dependent x-ray diffraction, is that actually if you turn on a field, you eventually go out of this first-order transition and you have a classical critical point. So we have a field-induced classical critical point, but it's usually first-order, okay? All right, so what's the big deal? These are classically first-order materials. They're used for all kinds of interesting room temperature applications. So what's the big deal? Well, several people, including the Cambridge Group, also the Paris Group, decided to look at these materials at low temperatures. Now strontium titanate is a close cousin of barium titanate. But unlike barium titanate, it does not go ferroelectric. It has a dielectric susceptibility that gets big, big, big, but it never actually diverges, it saturates. However, if you take strontium titanate and replace, it's so close to a ferroelectric transition, that if you take strontium titanate and replace oxygen 16 by oxygen 18, you can drive it ferroelectric, okay? And so what's been done experimentally is one can actually drive it ferroelectric, and then apply pressure, either external or chemical pressure. And what's been found is that it actually has a quantum critical point here, okay? And for example, it has a diverging dielectric susceptibility. And this is a special mixture of 016 and 018. And it also has a diverging Grunheisen parameter to indicate that it really is a quantum critical point, okay? So what's this material doing, having quantum criticality when it's usually its first order? Turns out there are a number of other materials. There's even barium titanate. It turns out if you put barium titanate under pressure, then you actually, there's a lot of indication. This is still early days of the experiments, but there are a lot of indications that there's quantum critical behavior, and there are a number of other materials that are starting to be studied at low temperatures and high pressures, which seem to indicate first order transition. We know their first order classically, but when we apply pressure, and we of course as a collective apply pressure and go to low temperatures, they seem to have quantum critical points. So the question is what's going on? So this is our motivation. So we go back and ask ourselves about a known mechanism in system, compressible systems that gives us classical first order transitions. And the classical mechanism that we decided to look at is due to Larkin and Picken. And there we have the interaction of strain with a fluctuating order parameter. This was originally studied for magnetoelastic systems, but we're going to adapt it to handle compressible systems, both magnetic and ferroelectric. So let me just conceptually tell you what it's about before we jump into some more technical terms. So we have the interaction of the strain with the fluctuating critical order parameter. Let's take a clamped system first. We clamp the system so there's no strain. In that system we have a diverging specific heat, okay? Now we remove the clamping. And what they tell us under certain conditions which we're going to talk about is that we have a first order transition in the unclamped system, okay? And the Larkin Picken criteria for first order transition is that the effective modulus, and we'll get to what that is in a minute, has to be less than the jump in the specific heat in the clamp system over the transition temperature times the square of the strain derivative of Tc. All right, now what's this effective modulus? The effective modulus is related to the bare bulk and shear strain modulus. It's very important that this mu is finite. Shear strain is crucial in this picture. And physically this effective modulus is essentially the bare modulus times the ratio of the longitudinal of the square of the longitudinal sound over the shear sound of the transverse sound, okay? So that's the Larkin Picken criteria. So again, we have a coupling of the uniform strain to the energy density. That leads to a macroscopic instability of the critical point and a discontinuous phase transition. And so the question is, can we generalize this to the quantum case? Okay, now clearly we have to generalize the specific heat. That's not a very useful quantity at T equals zero. So what we set out to do was to re-derive Larkin Picken using correlation functions and response functions to the point that then we could look at the quantum version. So let's do that. So the simplest case of Larkin Picken is isotropic elasticity and a scalar order parameter, okay? Then we write down our interaction Hamiltonian and we're going to couple the order parameter square, the energy density to a volumetric strain. And when we do that, you might say, well, what's the big deal? This is isotropic elasticity. We have our elastic degrees of freedom, our Gaussian. Even I remember how to do a Gaussian integral. What's the big problem? Turns out, and this is emphasized in the Larkin Picken paper and then in great detail in the renormalization group paper by Bergman and Halpern, we have to be very, very careful. Why do we have to be careful? You might say, I thought I knew Gaussian integrals was the only one I could do. The reason we have to be careful is the following. When we write down the strain, and this is like one or two lines in the original Larkin paper, which Bergman and Halpern expanded, thank goodness, for the rest of us to understand, we have to separate out the strain, the q equals zero boundary condition part from the part that is due to fluctuating atomic displacements. We have to separate it out. Now, that's another way of saying that is you can't think of all of the strain as a gradient. The boundary conditions don't come across as a gradient. So you have to separate out the boundary conditions and the part that's due the finite q, which is fluctuating atomic displacements. Okay, so when we integrate out the elastic degrees of freedom, this part, for the most part, gives us short range interactions between the critical modes. This part, which is q equals zero, gives us an infinite range interaction between critical modes. And you might say, well, it's an infinite range attractive interaction. So what's the big deal? Well, remember, q equals zero isn't included in this. So in order to perform our Gaussian integral, we have to actually, we have to complete, include a q equals zero term, adding and subtracting it to be able to do our Gaussian integral. That part gives us an infinite, yes. The voice is not loud enough, so. Oh well. When q is zero, this is zero. No, this, no, no, no. So this is q equals, this is the q equals zero term. Now I have q finite. But when I do, so this is all finite. And now when I do my Gaussian integral, I have to include another term to fulfill the full Gaussian integral. When I do that, it gives me an infinite range repulsive term. So in order to do this properly, I have to make sure that overall I have an infinite range attractive term. Okay. And it's that that's responsible for driving the first order transition. Yes. Is that if you added a q equal to zero term, since it goes to like q, this will be zero. Oh, no, all I'm saying. No, I'm not to say, sorry. I just wanted to say that when we actually go through the Gaussian integral, we'll have to add this term. This term, when I do the Gaussian integral, will give me an infinite range. You have to go through it a little bit. It will give me an infinite range repulsive interaction. And so just doing this is not enough. I have to make sure that the two terms, yes, that's all. Sorry, I didn't say it very well. Apologies. I can show you more in detail. I understand what you're saying. You're saying that there's a q term. Yeah, sorry. Okay, so what you have to do is separate the two terms. You do the Gaussian integral and what you find is that there's an infinite range of interaction between critical modes that you can describe as due to a classical strain field. Okay? And that's what is responsible for driving it first order. So what happens when we do the quantum generalization of Larkin-Picken? When we do the quantum generalization, we sum over all possible space-time configurations. So once again, we have boundaries we're only interested in static boundaries. So we have q equals zero and no time. But then for the other term, now we have finite q and finite frequency, okay? And so now this is our summation. And when we do our integration, we have to be very careful again about the infinite range repulsive and attractive interaction, but it's overall attractive. So that's very good. And so we end up with an effective action where we're integrating over space and time. We're in three dimensions here, where we have a Lagrangian and we have this term here where phi is the classical strain field that's uniform in both space and time. Okay, now this is pretty straightforward. We can write down a very simple, in the simplest case, a very simple phi four Lagrangian. And now what happens? Well, we have a partition function of the clamp system and we ask, what's the renormalized modulus? Well, the renormalized modulus will just be one over v times the second derivative of f and we'll see that it'll be the effective modulus minus a correction term. And you notice that this correction term is a correlation function. Now, when we look at the classical limit, which means that we don't have time in this, we have temperature. In the classical limit, we get this lambda squared, lambda is the coupling constant that's the change in TC with strain times this. This is going to be kappa minus delta kappa and you might recognize this as energy fluctuations. It's basically specific heat. So what we've done is we've been able to recover in the classical case the Larkin-Pickin criteria but in terms of correlation functions. Good. And you notice that this term actually diverges, okay? And so as a result, this kappa tilde, which is kappa minus delta kappa is driven negative, which means the system is unstable. So that's why. And so the reason is a little bit counterintuitive. We have a divergence of these energy fluctuations and so we get a first order transition because the system has a macroscopic instability. So that's a reformulation of the Larkin-Pickin criteria. And again, we have kappa tilde going to zero, less than zero, which means we have this instability. We're able to recover the regular Larkin-Pickin criteria and as the specific heat diverges, we get a first order transition. This was of course first studied by Larkin and Pickin but the approach that we're taking is very much following what was done from a renormalization group approach later by Bergman and Halpern. The idea is to identify a critical point and then to check its stability, okay? It turns out to be technically much simpler than following all the renormalization group flows. All right, onward and upward. We can also get all this diagrammatically. So we can get the quartic term, we can write as a bare term plus the interaction do this classical strain field. We can write a Dyson equation for the strain propagator. So we'll have the dressed version bare plus this Dyson equation. We can write this then as equation for the renormalized bulk modulus. This gives us the same as what we've had before. Here we have the susceptibility is basically the Fourier transform of these correlations. And again it's the infrared behavior here that's going to be most important, okay? We want to check the infrared behavior if it's diverging then we have a first order transition. All right, so in order to check what's going on from a quantum mechanical standpoint we can look at the infrared behavior by doing a Gaussian approximation. So we can do a Gaussian approximation to the correlation functions. And when we do that, what we're interested in is looking at the corrections to the renormalized bulk modulus, okay? And in this case we have a Q to the D minus one, DQ, D new the frequency. We know that the susceptibility scales one over Q squared, the frequency scales as Q to the Z. So when we put all that together we find that limit T going to zero of the corrections goes as Q to the D plus Z over Q four. And so we see right away that it will not be singular for D plus Z is greater than four. So in other words, we will not have the corrections to the renormalized bulk modulus or not universal if we are above the upper critical dimension. So in other words the zero point fluctuations, sorry, it's two fluctuations, toughen the system against the macroscopic instability present classically and we can get quantum criticality. So essentially what's happening is that as we increase the dimensionality at T equals zero our system is liberated for if you will the infrared slavery of the strain. Okay. So generalized Larkin-Pickin results this is for T not equal to zero. We always have the renormalized kappa less than zero. So we always have a classical first order transition when T goes to zero. It's non-universal for D plus Z is greater than four. We have possible quantum criticality which has to be checked for particular settings. And so this is the type of phase diagram that we have. We have then a quantum critical point. We have two classical critical points and we have this first order sheet of transitions bounded by these lines. So how do we link to experiment? Of course we can't say anything overriding because this is not universal but we started off by talking about fur electrics so let's go back to fur electrics. Fur electrics have as equals one so they're in the marginal dimension. Okay. But to date experimentally logarithmic corrections have not yet been observed. So they're pretty, so experimental probes for example like the Grünheisen ratio loss hysteresis all of that would be wonderful. We'd expect to get a jump. We won't have in our case we will do not expect in this pure case of being isotropic elasticity and scale order parameter we don't expect a tricritical point. We expect a jump in the bulk modulus, a jump in the longitudinal sound. Okay. So future work, this is just the beginning. Future work, multi-component order parameters. Okay. As you might guess for this Larkin pick and the specific heat exponent is crucial. For multi-component order parameters the specific heat exponent alpha is negative so we expect then a tricritical point of some sort. We did not have that. We had a positive alpha for the scalar order parameter. Elastic anisotropy, that's ever present. That's something that we have to think about. Domain dynamics, that's something that for example Peter Littlewood has emphasized a lot recently theoretically in fur electrics. We've always in our treatment we just generalized explicitly what Larkin pick and have done. We could put on different kinds of boundary conditions. That's something to do. Disorder, specific physical settings. Another possibilities order order transitions. We know of cases where one has a first order phase transition that seems to end in a quantum critical point that's not between order and disorder it's between two ordered states. The other is of course metallic systems. After Larkin and pick and wrote their paper together pick and actually wrote a subsequent paper trying to understand to generalize these ideas to metallic systems. And so that's something to do next to see whether we can understand that and then generalize it for the quantum case. So that's where we are. The idea is that hopefully with this type of approach we can open up new lattice sensitive settings for the expiration of new exotic quantum phases. Gil is very excited about these systems because he tells me that because they're so lattice sensitive you can get a broad temperature range for much lower pressures than in magnetic systems. And so he's excited about this. Then of course these are insulating but we know that you can add spin, add charge, you can get all kinds of, you can use them as building blocks. So that's what we've been up to. I'm not quite done because I have my pictures. Okay, all right. Well I know I have two, two, three. Okay, so that's the end of this. But now we get to, so as you know, this is the reason I thought I would talk about this is because this is my most recent project with peers. And since part of this week is the celebration of peers, I thought I'd bring it up. So anyway, as you know, peers as many of you know, peers and I have had fun and adventures for many years since last millennium. So I tried my best to look at my laptop and see some pictures. Not all pictures are digitized. I know some of you don't realize that. But anyway, I found some old pictures. This is a picture from last millennium when peers were shocked that I grew up in the US and had never been to Disneyland. So actually, he and his father took me to Disneyland. So that's last millennium when we were even younger than we are now. Then of course there's been lots of frustration along the way, other projects. And more recently, and hopefully more adventures, this is recently on a hike in Aspen. So thank you very much. I'm within time.