 Right, thanks so much. I'd like to thank the organizers for this opportunity to present some of our recent work. Since this is a conference partly to honor some of Pierce's contributions to Condense Matter, I kind of changed my title here to talk about some aspects of heavy fermion physics and frustrated magnetism. I guess Pierce's kind of has interests in both these and he's contributed a lot in collaboration with long-term collaborators like Premi who's here in the audience. And so what I wanted to, he's also had a lot of influence on the field through his pedagogical lectures and his textbook and I couldn't resist sharing a little Facebook post I had when he advertised his many body book a couple of years ago. And so it's been kind of learned a lot from his stuff. So what I'll talk about today is really thinking about what happens to magnetism when we get close to a MOT transition. And we'll see that this has kind of connections to two things that I'll discuss in the talk. But for motivation it's useful to think about the triangular lattice organic materials on which there have been a number of experiments where people can induce a MOT transition by using pressure to go from an incidating phase into a metallic phase. And depending on the kind of organic material one studies, one finds a variety of phases where you could either be inside a magnetically ordered phase at ambient pressure and then you pressurize it and it enters a conducting phase beyond some critical pressure or in some of these organics this MOT insulating phase which has no magnetic order, this paramagnetic phase gets pushed all the way down to zero temperature resulting in what seems to be a spin-liquid ground state which you can then again pressurize and drive through a MOT transition. So the question is, so there are various possible sequences for what happens to the spins as you do this and as you apply pressure, for instance, you could go from, as in one of these cases from an ordered phase into possibly through some kind of first order transition into a conducting phase or you could have novel magnetic orders or have some kind of intervening spin-liquid. So the spins can do various things and seems to depend to some extent on what the material properties are. And at least one of the driving forces for possible novel forms of magnetism or spin-liquid is the recognition that you not only have to deal with two spin interactions which are progressively probably getting longer ranged as you go towards the metallic phase but also that you're getting many body exchange interactions between spins as they are being driven into this phase where they're going to completely delocalize. So there have been kind of number of studies of this motivated partly by the organics but even motivated partly from experiments looking at Helium-3 on graphite going back to work from Claire Luria's group and Gregoire Miskvich where they had done some diagonalization studies but there's been work on effective spin Hamiltonians by Leszek Mortonich many years ago and a very recent pre-print from a Berkeley group where they've actually done DMRG on a Hubbard model where again as they go from take this for me on Hubbard model and do DMRG it looks like there is some kind of an intervening phase that they tentatively I think identify as a chiral spin-liquid phase but it's kind of at this point I think one needs to understand some of these results better. So the two questions that I would like to kind of ask in the stock which are kind of related are first could such kind of physics also play a role in heavy fermion systems and I'll kind of give you some motivation for why that might be interesting to think about and the second is going back to what I just mentioned of the organics but thinking about other types of spin liquids that might be driven by multiple spin interactions and in particular can one relate in some of these cases we find that we can sort of relate these spin liquids to some kind of interesting magnetically ordered phases and that will be kind of the second half of the talk. Okay, so let me begin with the first part and this is sort of an ongoing collaboration with my colleague Youngbeck at Toronto, Sangbin at KAIST and Youngbeck's student, Adarsh Patri and I've been collaborating quite extensively also with Simone Trebst and his two students, Frederick and Jan. So I'll present some numerical results which they've actually obtained during the course of this collaboration and we benefited a lot from discussions with Satharuna Katsuji and Kitasakai. So this is just some people, couple of them are out on the archive and a couple of them which are in preparation. So what is the conventional way to think about sort of the heavy fermion systems goes back to very early work by Doniak where the point is you want to think about some local moments coupled to conduction electrons and if this interaction between conduction electrons and the local moments is weak then you imagine integrating out the conduction electrons and asking what happens to the local moment degrees of freedom and typically you'll get some kind of two-body RKKV interaction driving some form of magnetic order and then as you keep increasing pressure eventually the spins completely hybridize with the conduction electron and at that point it's no longer legitimate to say I'm going to integrate out the conduction electrons that's no longer a meaningful thing to do when we are on this side of the phase diagram. And so the point is that there is this transition then between a magnetically ordered small Fermi surface state and this fully hybridized large Fermi surface state and it has been proposed by a number of people including Catherine who's here in the audience that one can think of this transition in terms of a more transition of the F electrons. You say that on this side it's as if the F electrons have become kind of metallic by hybridizing and here they have become localized and they've done something ordered or done something else. So here we kind of started to think about this motivated by experiments from Nakasuri's group on this class of heavy fermion systems. So these are systems where a prasadium has a 4F2 local moment degree of freedom and it's coupled with this complex set of conduction electrons coming from transition metals and aluminum that kind of form these cages around the local moment degrees of freedom and the condo coupling is of course doing various things in the problem it's going to scatter the electrons. In general if you go to lower temperature this might lead to various things such as possibly ordering of these local moments or this it might nucleate new types of metallic and or superconducting phases and one of the important things to understand is sort of what is happening at least in this class of compounds hopefully that will shed some light on the physics. So these systems one starts with the local moment degree of freedom is this 4F2 ion which has an orbital angular momentum phi and a spin angular momentum one which gets spin orbit coupled into a J equal to four spin degree of freedom and the crystal field splits this multiplet giving rise to a low energy doublet which is separated from all these excited multiplets by about 60 Kelvin energy gap and this low energy doublet is a non-cramers doublet because this is a 4F2 ion it's a non-cramers doublet which you can sort of write down explicitly in terms of these angular momentum components or one can write down effective pseudo spin half degrees of freedom which for reasons that will just notational convenience we've kind of chosen to work with spin up and spin down being these superpositions of this particular combination of doublets that I've written here, okay? So the interesting thing here is that unlike typical cramers doublet degrees of freedom this one has the X and Y, tau X and tau Y the pseudo spin components act as a quadrupolar degree of freedom so they even under time reversal and they have various lattice transformations that I'll touch upon later whereas the Z component of these pseudo spin acts is has a non-zero octopole moment so it's actually odd under time reversal, okay? So there have been a number of experiments on these systems in particular by changing the transition metal ions so for instance here is data on titanium whereas a function of temperature they find when they cool the system they find a single phase transition into a state which has been identified as having ferro quadrupolar order so this is the tau XY component of the pseudo spin ordering into a ferro state in this case when they replace titanium by vanadium they find that there are two fairly closely spaced transitions and it turns out that here they've identified that the quadrupolar order is not ferro order it's some kind of anti-ferro order the precise details of which are not yet known from the experiments at this point so that's what is known and one of the signatures that at least that the second compound the vanadium one is probably closer to this critical or to the space transition from and this kind of weak hybridization to the song hybridization that you're closer to that comes from some signatures which is still kind of I think a little bit tentative is they've identified signatures in the resistivity where there is something that looks like a square root T behavior which they kind of identify as possibly arising from two channel condo I think it's still a little bit unclear to me because it's over a limited range in temperature and the sign at least is not like this single ion condo sign of the square root it's a positive coefficient so it's not entirely clear how to interpret this data at this point but that's at least a tentative thing that they have kind of highlighted in their experimental work so the question that we would like to ask is the following so let's imagine that we are somehow on this side of the transition where it's legitimate to integrate out the conduction electrons you want to ask what happens if I start deep in the magnetically ordered side and kind of march towards the phase transition into the metallic large Fermi surface side you would expect exactly as in the organics you should expect that if these Fs are on the verge of delocalizing then these F moment interactions should also become progressively more complicated as you get closer and closer to this transition into delocalization and so what you would expect is that one must go beyond simple RKKY type interactions and look at sort of possibly more complex types of Hamiltonians and I'll turn to sort of reasons why this might be the case so for instance the simplest kind of terms you might write down in the hope of describing some of these experiments is to say well I'm going to integrate out the conduction electrons and write down the simplest symmetry allowed Hamiltonian on the diamond lattice which is where these local moments live and the simplest Hamiltonian then which is symmetry allowed at the nearest neighbor level takes the form where the quadruple order has some XY symmetric interaction but the octopolar can in general have a different coupling because it's not directly related in the same way this carries an octopole moment that forms a doublet under the so this carries a quadruple moment okay so the phases you might say well I can understand the titanium I just pick a negative sign for this it's ferromagnetic coupling with some lambda that's smaller than one then you can potentially use this to understand the ferro phase whereas the anti-ferro phase you would say well maybe the sign changes of this coupling and you can drive some kind of anti-ferro phase of these moments but it turns out that that's actually not adequate for reasons for various reasons and I kind of just sketch some of them here so for instance one of the important facts is in fact when you get this ferro orbital order in the titanium case it's known that it actually from NMR experiments it's known that it picks it doesn't have full XY symmetry actually you have to break the XY symmetry down and you have to favor only three of the possible angles in this XY plane and it turns out Hamiltonians which you know of this sort or even ones that will generalize these kinds of two-spin interactions are basically inadequate to describe the second thing which is also known from from at least some kind of land-out theory which I'll touch upon but for which there seems to be some evidence from ultrasound measurements with the elastic constant seems that any state which has anti-ferro quadrupolar order should have parasitic ferro quadrupolar order that accompanies it and again that's not something that would be contained in Hamiltonians of the sort and for instance the role of what these degrees of freedom do is not clear if you only focus on these quadrupolar degrees of freedom the question is can these octopolar degrees of freedom do something more interesting and what I'd like to point out is that multiple spin exchange interactions actually provides a potential route to kind of addressing all these issues. So the way one can we're going to write this down is to start by just analyzing sort of what are the symmetries of what how do these local moments transform under various lattice symmetries and for example under time reversal we know that this is odd the Z component and one can write down sort of various symmetries how these moments transform under a lattice symmetries and it turns out that there is actually an interesting and important third order three spin interaction that one can write down which involves taking three spins on the diamond lattice which are kind of neighbor neighbors to each other and have it such that you can have all three of them appearing with a tau plus so if I think about this in terms of a U1 symmetric some kind of Bosonic theory to begin with three particles appear and disappear and that's an allowed process in the problem. So it turns out that this is useful because this is precisely the kind of term which would lead to ordering so this just shows sort of Monte Carlo study of this Hamiltonian including the two spin interactions at the classical level and it has basically a thermal phase transition which is weakly first order as seen from this energy histogram and it's consistent with having this kind of a clock model rather than having the full xy symmetry as is expected based on this form of the Hamiltonian but and one can sort of explicitly see this by staring at the configurations and you see that at low temperature you pick out not the full manifold but you favor three of these states and these are precisely consistent with what NMR has observed. So you favor ordering into the tau y direction and it's 120 degree variance of that. So and this Hamiltonian if one can sort of then say well let me kind of do a long wavelength theory starting from here and it turns out that's precisely what we need that this three spin interaction will eventually lead to a cubic term in the Landau theory that pins the xy order into these three angles independent of the sign of this coefficient here you pick three of the possible directions on the circle. This problem also kind of helps with the antiferous side because if you have an antiferous quadrupole coupling and you add this cubic term it turns out it precisely produces this extra staggered, this canting so you start off with something that you think is perfectly staggered like in an antiferous magnet but in fact this cubic term produces a spontaneous canting and produces this weak parasitic ferro order so these two spins on the two sub lattices no longer point exactly opposite each other but they develop some canting leading to a ferro component which seems to have been observed indirectly from as I said from elastic constant measurements. And again one can sort of understand this from a Landau theory perspective where the square of the staggered moment can linearly couple to the ferrocodropolar order and so when you do this then any time you produce the staggered order you immediately drive a small parasitic linear coupling to the uniform component to the ferro component. So most recently we've looked at what happens if you start generalizing this to the vanadium case where you might expect as you march towards this phase transition that more interactions might become important and in particular we think that it's possible that there might be some coupling between the X, Y quadrupolar degrees of freedom and the Ising octopolar degrees of freedom and the reason to at least kind of explore this is if we sketch the phase diagram as a function of let's say we sit here, you know let's say the second neighbor interaction we've explored a large window but let's say the second neighbor interaction is small the first neighbor interaction let's say is antiferromagnetic in the X, Y but let's turn on this force spin coupling that I just wrote down it turns out that once you produce some X, Y order then this force spin coupling if it becomes strong can actually drive Ising order of the moments to coexist. So what you can end up with are phases which we tentatively think might be relevant to the vanadium compound where as you cool the system you enter from a paramagnetic phase into a phase where the quadrupolar degrees of freedom order and when they order they can actually drive the secondary order parameter it's not slave to this it has to be an actual second transition where time reversal symmetry is broken and that can give rise to a second transition shown here in sort of specific heat numerics on this Hamiltonian where time reversal will be broken at the second transition and there is some evidence for this from recent magnetostriction measurements in these systems. So let me maybe spend a few minutes talking about a closely related problem as I mentioned which is you know chiral spin liquids which also arise from very similar physics and this is work done with my student Kiran Hikki who's did his PhD with me and then has now a postdoc in Simone Trepp's group and this was with Lucas Sanchio who was a postdoc at Perimeter and Zlatko who was also there at that time and the kind of question we were asking was how interaction effects in flat bands can drive interesting phases so we kind of started thinking about a very similar problem in some sense which was start with some problem where you know you want to think about this transition a more transition but in the presence of in band structures which might have some non-trivial band topology and so there has been a lot of work on thinking about this physics and flat bands where the physics might have some resemblance to fractional quantum Hall effect in landar levels but we were asking a question which is related to thinking about something like the Haldane model where you have churn number one and minus one so it's like a lattice version of the quantum Hall and you imagine filling up the lower band with spin up and spin down so that you're at a filling of one particle per site or one electron per site and ask what happens to the mott insulator what happens as you start cranking up interactions so you start with something that has a quantum Hall effect with 2e square over H and you want to drive up interactions to go into the mott phase just like so it's some kind of a metal insulator transition of some sort and again you want to ask what happens to the magnetism in the vicinity of this transition and there have been a lot of suggestions for what kind of phases would appear based on analogies with particles in landar levels and one expectation is that one should find some kind of a spin liquid phase, a chiral spin liquid which might be stabilized in the system in the mott regime so we started to look at this model and one way to kind of think about this is to again do what I mentioned earlier so of course if you're at very large U very deep in the mott phase two spin interactions dominate the physics and that will just give rise to some kind of effective J1, J2, Hamiltonian on the honeycomb lattice but once you becomes once you get smaller so that you're being driven into the conducting phase progressively higher spin interactions will become important and the next important leading term that plays a role here is a term that involves the scalar spin chirality so what you can do then is you can ask well I can start asking this question both from the strong coupling perspective and from just doing mean field theory of this fermion Hamiltonian you can ask well what are the phases you find in the system and in both cases we find that in contrast to a simple not simple but a chiral spin liquid phase what we find are magnetically ordered states from both these approaches and it's an interesting magnetic order so it's this what's called a tetrahedral state so it's a non-coplanar state of these spins which arranges in this manner so all the green points here would have spins pointing along this direction red points correspond to a different spin direction so overall the magnetic state you break spin rotational symmetry but you order into this non-coplanar state of these spins so in fact you don't find at least in the simple model we don't find a spin liquid phase but we find something that's very close in some sense because this has large spin chirality and in many ways this is almost liquid like in the sense if I look at any correlation function that doesn't directly involve this broken spin rotational symmetry it's perfectly uniform across the lattice so if you look at nearest neighbor spin correlations it would be equal on all three neighboring bonds or if you look at any spin multi-spin correlations for example they all respect all the lattice symmetries except time reversal and so now what we asked ourselves can one sort of then just imagine melting this order so it's a non-coplanar state it has large chirality the only problem is that it's actually magnetically ordered but it turns out you can sort of perturb the system very gently so imagine we have these two spins that are parallel we turn on a very small anti-ferromagnetic coupling that would tend to kind of take these spins which are lined up and try to disallow that and it turns out that as you keep cranking up this third neighbor coupling you can actually drive this tetrahedral order into this tidal spin liquid which I was mentioning and evidence for this comes partly from exact diagonalization studies where you see that there is on this torus geometry there's an isolated doublet which is what you expect for the nu equal to half Laughlin liquid of bosons which is what this state is related to and there is also additional evidence from DMRG looking at the edge entanglement spectrum showing that this is an SU2 level one edge theory that comes out from looking at the entanglement spectrum if within DMRG so this idea then we thought how general is this idea so can one take these non-coplanar states and imagine disordering them by frustration and quantum fluctuations can one disorder them and is that a way to actually access spin liquids that people have found in other lattice geometries and we find that in fact it seems to be very generic so in particular we are aided by this very nice work by Greg Barmisch which Laura, Missio and Claire where they classified what they called regular magnetic orders so you look at all possible classical magnetic orders on different lattices and you ask that this magnetic order should be such that if you do any lattice operation followed by spin rotations and they took spin inversion but that's not a real symmetry in the quantum case but so let's if we restrict ourselves to asking which magnetic orders are such that if you do a lattice transformation followed by some global spin rotation then the state is left invariant so this is in some sense a classical analog of PSG type constructions where you want to say well you want to do some operations on your spartans and then follow it up at some gauge transformation this is kind of an analog of that in the classical problem in some sense and so they have classified all these regular states and the simplest one for example might be a ferromagnet or a state like this a collinear nail antiferromagnet these would be examples of regular orders but if you look at the non-coplanar orders in fact it turns out that there are very few so in particular there is a single one on each of these lattices so if you look at the honeycomb lattice the Kagome and the triangular it turns out that there's a single non-coplanar order that is a regular magnetic order and we now have numerical evidence so we had numerical evidence on the honeycomb which I showed you earlier there is work from Andreas Lauchli looking at a very closely related problem on the triangular lattice more recently and they also find that they can sort of drive this transition here using second neighbor coupling between a very similar tetrahedral phase and spin liquid on the triangular lattice which is this like this Kalmeyer Lauffelin type liquid but with SU2 symmetry and we also have some diagonalization and DMRG on this again pointing to this connection between this ordered state and the spin liquid and more recently with Simon I've been doing some variational Monte Carlo calculations showing that something very similar happens on the Kagome where you start with an octahedral state where spins point towards the various spin axes plus minus x, plus minus y, plus minus z that's again a regular non-coplanar order and you can drive it into the chiral spin liquid which was initially one of the first chiral spin liquids that people had found in DMRG studies so it looks like that it's probably like a very generic mechanism that you can start with these non-coplanar magnetic orders and either by frustration, materials you might imagine if you have a non-coplanar order maybe apply pressure or tune something to actually drive the system into this chiral spin liquid phase and we are still thinking about some aspects of this phase transition between these ordered phases and the spin liquid phases. So with that I'll conclude and put up my summary here so one general message to take away from the first part might be that these multi-spin interactions which I mentioned the context of heavy fermions might actually be generically important to think about especially if you're driving your heavy fermion system close to a transition into the large Fermi surface state then such multi-spin couplings might give rise to interesting ordered phases or maybe even spin liquid phases and then in these chiral spin liquids it turns out that one can view them fruitfully as having descended from various types of non-coplanar phases. So thank you for your attention.