 urmins and in particular developed wind-wapping and gliders and a new powertrain and out of this came the right flyer. Okay and so this is kind of a metaphor for I think where we like today in Kilev's matter physics of quantum materials. At the beginning of this century quantum materials provide equally important insights into how we may understand and control the quantum physics of entanglement. And one of the key points here that is not understood very well today is a point that was made by Ralph Landauer who was an expert in mesoscopic IBM but also fascinated by the notion of quantum information and one of the things he claimed which I think is probably true is the information is physical. It's not a human thing it's actually something physical and if that is indeed true we don't really know what it is at the quantum level. Our colleagues up the hill use the word entanglement and quantum information in an interchangeable way but actually that's not a legal thing to do. So my main point is that when we come to quantum materials we have fully operational systems with entanglement and 10 to the 23 fully operational qubits. Okay this is a number that's comparable with a number of stars in the visible sky. Even including the James Webb telescope. And so as such we have an environment now where we have access to very new novel phases of matter. High temperature superconductors topological systems fractional quantum ball systems but we're also interested in their possible applications in the notion of quantum computers. Designer quantum materials such as twisted bilayer graphene okay and quantum phase transitions which are of course states which in some sense have infinite range entanglement. And so one of the questions that comes out of this is what new conceptual frameworks can unify quantum information and quantum field theory and many body physics. Okay the other important question to bear in mind even though we are 100 years into quantum mechanics is that like classical mechanics 100 years after Newton quantum mechanics is surely incomplete. Understanding information and entanglement is a key component of the next stage of discovery. And it's always a good point at this point for me to ask you what was not known what key concept of classical mechanics was unknown 100 years after Newton. Anyone know? You've all studied classical mechanics. I know well that's not fair. Anyone who I haven't told or who has studied the well I'll tell you it was the concept of energy okay. The concept of energy was completely unknown and was not established until the middle of the 19th century and it wasn't established by Hamilton or Lagrange but by the experimentalists very carefully checking the interchangeability of heat into motion or energy macroscopic motion energy. And the reason they were meeting this is they didn't understand the emergent nature of heat as a form of motion and so energy conservation appeared to break down in a practical sense until that was understood okay. And so we are no better than our classical forefathers and it's very likely that quantum mechanics today is incomplete. If you ask me what I think is the incomplete element of it well I can't tell you because I'm one of our time but a probable area to understand is the nature the physical nature of information okay. All right so just to rub this in I'm going to go on a little bit further and talk about harbingers. Harbingers of the quantum age okay. We all know this but it's good to think about it a little bit because there are other harbingers that we face today okay. So what were they well I've just taken a selective list of them why do hot things change colour, why does matter so hard, why does the sunshine okay. Why does matter so hard it can be rephrased as what is the origin of the reaction force that keeps you sitting in your chair okay. Newtonian physics gives a very nice description of the gravitational force pulling you down and it includes as a completely phenomenological object the reaction force because matter is hard but it doesn't know where that reaction force comes from and where your body made of bosons you would slip through the chair okay. So let me remind you about these things why do hot things change colour well this led to the concept of the photon and black body radiation and the hard work of this entire generation of experimentalists and physicists in the 19th going on into the 20th century and of course the peak in the black body radiation is literally the point where the frequency is of order the temperature in the appropriate units okay. Let's look at another example why is matter so hard the fact that a book Mutants Principia can sit on a table can't be understood in terms of classical mechanics okay and the understanding of this came from a very unlikely direction it came from the study of atomic emission lines okay. Who would have imagined that those two things would be connected and the understanding of matter's hardness grew out of atomic emission lines leading to the Pauli exclusion principle which is the origin of hardness and chemical diversity here's a good old Enrico Pauli writing to Aaron Schrodinger this is an English translation saying that with a heavy heart I've concluded being converted the idea that Fermi Dirac not Einstein Bose is the correct statistics he was disappointed by this but he worked on to write a note on Pauli paramagnetism and he spent the rest of his life dissing a condensed matter physics as squalid state physics or something to that effect dirt physics in fact the field prides itself in the interesting physics that comes in dirt physics and in fact it was probably perhaps Heitler and London who really understood this very carefully they combined Schrodinger's physics with the Pauli exclusion principle and they realized that they could then understand the hardcore repulsion between atoms in terms of the Pauli exclusion principle as you bring two helium atoms together the pushing the electrons up into the anti-symmetric level raises the energy some people say that it's got nothing to do with the Coulomb interaction actually that's not true the Pauli exclusion principle and the Coulomb interaction are intimately connected with one another because by forcing those electrons into the anti-symmetric state you reduce the electron charge between the two between the two ions and that causes increased repulsion between the protons okay so so but nevertheless it is a result of the exclusion principle and to think of my last example which is the question of and of course we can code that nowadays as a piece of algebra the anticommutation of the field operators okay and why does the sun sign so slowly this was a debate that which involved the greatest physics of their generation and here is a a copy of a publication in Macmillan's magazine from March 1862 what is Macmillan's magazine come on you all know what it is it's nature okay okay it was owned by Macmillan for more than a hundred years it's now owned by Springer okay and he made an estimate Lord Kelvin William Thompson and using the physics of time confidently estimated the age of the sun to be 20 million years he had a gravitational collapse model of the sun okay that's because he abandoned chemical energy because that gave it even shorter time period okay but this led to a great debate led to a call with geologists and with Charles Darwin who estimated a number closer to 500 billion years we now know that's a gross underestimate okay it's several billion years and of course later on it was discovered that the sun shines because of nuclear fusion but even with the understanding of nuclear fusion it wasn't understood why the proton proton collision rate was so incredibly slow okay accepting but that's the consequence of the weak interaction which is incredibly weak but why is the weak interaction so incredibly weak well that answer couldn't be provided by the particle physics of the 1960s or the early 1950s I should say and in fact by the end of the 1950s particle physics had largely given up on field theory and the clue to the next way forward came from a study of superconductors okay so and just as an aside a typical proton inside the sun waits nine billion years nine billion years before fusion which is good because it means that our sun is still very young okay right so it's only about five billion years yeah oh and so the short range nature of the electric weak force and the Meissner effect share a common mechanism and this is because this is because a gauge field acquires mass and is weakened okay and this weakening occurs via the Anderson Higgs mechanism in the case of superconductors is a charged pair condensate in the case of the electrical weak it's a hypercharged Higgs condensate which means almost nothing because we've no idea what the Higgs field actually is whether it's microscopic or whether it has an inner structure okay it's a pure phenomenology okay whereas the charge pair condensate is not a phenomenology it's related to what we understand about quantum mechanics at a microscopic level within condensed matter and of course these two things come with a lens scale there's a lens scale penetration depth of a superconductor there's also the electrical weak penetration depth of our universe which is separated by a good 11 orders of magnitude but really have the same common origin okay and Higgs based some of his early work on Anderson's ideas that you could extend the notion of of massive gauge fields from the case of a superconductor to the Yang-Mills theory and he built on that idea so these then are three examples of harbingers of the quantum age and in each case laboratory based studies were used to reveal deep concepts that apply equally to the laboratory and to the cosmos at large okay and so one of the things we should think about today is whether that still holds maybe it doesn't hold in which case you need to build bigger and bigger telescopes bigger and bigger accelerators or maybe there are things that can be discovered in the lab today that not only important for our understanding of quantum matter but perhaps important for revealing deep issues deep properties of quantum matter that are relevant perhaps to quantum information perhaps to our deeper understanding of the universe at large okay so that's the kind of thesis that I want to leave you with before going on to more detailed discussion okay there's a deep link between the laboratory and the cosmos and it applied in the in the 17th century it applied in the 20th century and it probably applies in the 21st century or did that deep link suddenly come to an end at the end of the 20th century very unlikely okay right so it's with that thesis that we should talk about talk about the situation today okay and so let's talk about some of the dark matter challenges of the solid state which perhaps are connected with principles of emergence that we need to understand and which perhaps will reveal emergent properties of the universe at large and of course dark matter is a coin of phrase that comes from astronomy and it's now very well established that there is some ephemeral dark matter that lives across our universe that is not lumpy but very very smooth and only interacts with our part of the universe via the energy momentum tense as far as we can tell okay that's what dark matter is but there's also dark energy and 94 percent of everything the background mass energy density of the vacuum is completely unaccounted for that is a truly mind boggling harbinger of our current age okay 24 percent is this dark matter that that forms this very smooth stuff that causes galaxies to have unusual rotation spectra and 70 percent is what's producing the acceleration of the expansion of the universe revealed in the high z the high z surveys of distant galaxies okay and it's completely it's not understood at all and a great deal of money is going into trying to detect the interaction of dark matter with our world and many of the theories that we put forward there such as the super fluid model of dark matter are inspired by condensed matter physics okay okay so this is a major challenge to 21st century physics but does that mean that there aren't equally important challenges in condensed matter physics so I would try to say that there are equally important challenges and one of those is just staring us in the face the fact that since 1987 we fail to understand this beautiful linear resistivity in the kubrigs by which I mean we have not developed a consensus theory that we can all use to create predictions and to move forward and understand in the context of the kubrigs the heavy fermions and in the context of the twisted bilayer graphene and other systems that display the same linear resistivity okay and this is the breakdown of the land our Fermi liquid in a fashion that we don't understand at this point okay and I'm going to tell you a bit today about my fascination with this strange insulator somerium hexaboride which I first got to know as an undergraduate as a graduate student rather and here you can see its resistivity as a function of temperature which shoots up at low temperatures and here you can see it as a function of one over temperature and it levels out at low temperatures this leveling out turns out to be quite fascinating but even more fascinating is the claim not entirely reproduced that this system exhibits bulk quantum oscillations which have been interpreted in terms of neutral fermi surfaces why then am I putting this up here well the reason I'm putting it up here is that some other systems some other insulators have also exhibited the same quantum oscillations and in parallel with it we have some peculiar solvable models of many body systems which seem to have certain features in common with these bulk fermi surfaces inside something resembling an insulator so these are two examples of dark matter challenges and these are only two selective ones these and many other examples suggest the potential for qualitatively new advances in our understanding of quantum matter let me take you through a few examples that are important and to remind you the fact that what we're really talking about here in many of these systems is the crossover between localized and delocalized behavior we heard this fascinating discussion this morning in on in under Chupacov's talk about when can you use the spin fluctuation itinerant picture and when can you use the localized moment picture and what do you do when you're halfway in between and what do you do when part of your system is localized and part of your system is delocalized this is a debate that began in the 1930s and is still going on today it was a debate started by Piles and Mott and then later Anderson and Slater and this is the battleground of our understanding at the moment this competition between localization and delocalization this here is a beautiful diagram of the periodic table reorganized by Comet Co and Smith in 1983 to show the central rows of the periodic table organized according to increasing localization vertically and of course due to the aptanide or lanthanide or the transition metal contraction localization they're running horizontally and the coloring here just shows that when you have what when you have elements the unfilled shells of the electron C form a nice metal down on the right hand bottom left hand side here and they form conventional superconductors but on the top right hand side if you go to materials such as dysprosium or galvanium their unfilled F shells form nice local moment magnets okay and of course when you take these elements and you make compounds out of them the crossover region is something of great interest and that's where all the interesting stuff develops and so the cooperates what happened there okay yeah and here is the the contrast between waves and magnetic moments and of course we have systems such as the cerium heavy fermion systems the actinides iron based superconductors and the irradiates which are kind of at the crossover okay so these are electrons at the brink of localization and we know that new kinds of quantum matter develop at the brink of localization okay any questions yeah and here is another example of a strange metal which I love because cerium cobalt indium 5 is a miniature high tc system it if it were discussed in japan it would be called a bonsai high tc system it's a miniature version in japan they have these miniature trees which look just like full-sized trees but are actually just a hundred tons smaller and so cerium cobalt indium 5 is similar to that it has a tc of about two kelvin and instead of going up to a thousand kelvin up to 15 kelvin you see beautiful linear resistivity and there are some departures from linearity you can see at low temperatures in the in the ab plane okay and this is the thermal conductivity which also is linear okay and along the c-axis it's the most perfect linear resistivity we've ever seen the intercept down here is a hundredth of the maximum rise here which tells you that this linear resistivity has got nothing whatsoever to do with disorder or any such thing okay so this is an issue that you might want to consider if you're interested in syk models I would say but anyway okay and we know from applying in an illegal way the druida theory that this linear resistivity means we've got transport scattering time which is basically given by the inverse thermal energy the plankian scattering rate and but we don't really know where microscopically comes from and in the old days in fact I remember here being here when Henri Rückenstein introduced the marginal Fermi liquid theory roughly around 1990 here in Trieste it was a beautiful talk and it's frankly just as good a theory as the modern ADS CFT theories of of of plankia dissipation in the sense that neither fully gives us a microscopic theory at this point in my opinion and of course these things seem to be in many cases connected with quantum criticality certainly in the case of of tunable heavy electron systems you can tune them for a quantum critical point serum cobalt indium-5 aniterbium rhodium two silicon two are nice examples of this these are systems which in which the length scale and time scale of the quantum fluctuations diverge and we still don't understand how to describe this divergence of length and time scales in a metallic system we understand it pretty well I would say in the context of insulators but metals it's a big issue I would say okay and of course what we do see is a new kind of metal a strange metal with a linear resistivity seems to form in the quantum critical fan okay so so bringing that all together here's another example of such strange metal behavior in the ion-based superconductors it's not so perfect in terms of linearity but in these systems as in the case of the cooperates under some situations you can actually see an interchange between linear and t and linear in magnetic field and this is an ongoing developing mystery that is not fully understood at the moment okay and of course all of this it seems to be linked with high-temperature superconductivity in the cooperates the ion-base the 4f serums and nowadays you could probably also put into this the the flat band twisted Mare systems which also exhibit similar features so this then is a lineup of the dark matter challenges of the solid state it's the selective one but these are another example suggest the potential for qualitatively new advances in our understanding of quantum matter and this means we have to seek new conceptual ideas you can't keep going back to the old framework and start hanging everything on it you've got to spend part of your time thinking about new ideas that are risky but enable you to come up with new suggestions for experiment that might take you to a new world of understanding and we're interested in new forms of electron entanglement new kinds of quantum phase transition new kinds of broken symmetry the interplay with topology and the concept of fractalization it's on those last two topics that I want to spend the remainder of this talk okay questions anyone on the chat probably Andre there's five I can't I can't get to it I'm not going to try it so I'm going to talk about the curious state of Sumerian hexaboride okay and this will connect up a little bit with some of the things that that Alexei was talking about last week okay and this is the the material it's a cubic system and the little the blue spheres here are Sumerian Mayans and what's important about Sumerian Mayans is that they contain localize moments f electrons if you make the same material out of Lanthanum you have if you make the same thing out of Lanthanum you have a perfectly good metal a metal that used to be a great technological technological importance of the cathode in cathode ray tubes because it has a low work function but remarkably if you put the Sumerian in it it becomes an insulator at low temperatures whereas with Lanthanum it's a perfectly good it's a perfectly good metal at low temperatures and these are the local moments and it has this small insulating gap that develops at low temperatures and you can see the behavior better in the context of the inverse temperature plot here and you can estimate the size of the gap I forget where I've got it down here it's it's the gap is about uh it's not written down here the gap is about 20 20 getting wrong about two millivolts 20 Kelvin okay and so one very very naive interpretation of this system is as a condolatus um it's a gross oversimplification because there are valence fluctuations nevertheless it provides you with a cartoon understanding of the problem because if you have a condolatus then the interaction of the local moments with the electrons is a running coupling constant and it renormalizes so that it runs to strong coupling and if you're then so naive as to understand it that way you can then just solve the one impurity model with a very large j and that then gives you a singlet at each side okay if you then look at that model you can either pluck out an electron to create an unscreened local moment or you can add an electron to also create an unscreened local moment and when you do that those excitations cost an energy of order j and they produce a mobile mobile carriers and so this is in some sense like a regular band insulator but in which the spins behave as spin one half excitations converting the system to a filled band insulator they count a spin a half excitations and in some sense this is an early example of spin fractalization this is a model in which uh by conventional counting you would have a metal but if you count each spin one half as a particle as an electron you actually understand the system as a as an insulator and so it challenges us in two ways number one it looks as if we should think about this as a fractalized system but number two were then puzzled why neutral spins produce charged heavy fermions so two processes going on in this system which challenges one of which is the emergence of a gauge field and the other of which is is the locking of that gauge field to the external electromagnetic field via Higgs mechanism so it's quite a challenging system to understand okay anyway uh so uh the other interesting thing about this system is that it's got this plateau in the resistivity which never went away no matter how pure you made the samarium hexabroid this plateau never disappeared and uh no one understood but with the emergence of the idea of topological insulators it raised the interesting question whether whether this system might be an example of a strongly interacting topological insulator which we glibly called a condo in a topological condo insulator this is the work i did with max into zero kai sun victor glitzky and uh it turns out that uh to a good approximation this seems to explain a lot of the physics in this system let me show you the idea the idea is that there's a topological phase transition in which you take a d-band and cross it through a narrow band f electron system and this band crossing then changes the parity at the x points the three x points minus one cube gives you a topological insulator which then guarantees that you should get dirac cones on the surface in fact three of them if it all goes well and this seems to be born out by experiment uh uh for example uh you can see the the the dirac dispersion on the on the on the surface the hall constant derives just from the surface which you can tell from doing wedge measurements you get non-local the surface is conducting and you can even see the predicted bands on the one one one surface everything looks right for this to be a topological insulator and it has a remarkable bulk resistivity uh you can measure the bulk uh resistivity there's a problem here right because it's got conducting surfaces so how do you measure the bulk resistivity of a topological insulator uh if you do it conventionally your resistivity measurement is always shorted out by the surface and so the group at at University of Michigan came up with a non-local porbino measurement in which they injected the current and and measured the uh measured the voltage drop remotely and from this they could pull out the bulk resistivity and if you believe their results they say that the resistivity uh this is where it saturates uh in where the saturation occurs in the bulk resistivity in in the measured conventionally measured resistivity but if you extract out the bulk resistivity it goes on up through five more decades so this is a very good bulk insulator um but it turns out that that nice picture doesn't seem to explain all the properties and provides an an enduring mystery um because it seems to be an insulator with a firmy surface and this is where you have to try to make that Galilean extrapolation uh we're not quite sure how good the various experiments are we're not sure how all of them are reproducible so we've got to take uh our insights from the body of experimental results not just relying on one on their own okay so let's talk about that these are the two viewpoints about this system here is the resistivity here is the resistivity with this very nice plateau uh is that a topological condo insulator and is that the end of the story is a band topological insulator and uh and this is the uh body of results that supports that idea but let's but let's look at some of the weird properties here is the is the specific heat capacity of this system and uh you can see something fascinating here is lanthanum hexaboride the very same structure with lanthanum atoms and here is samarium hexaboride okay and you can see that there is an excess of specific heat even at the lowest temperatures this excess comes from the local moments okay uh is the entropy of the local moments but what's really fascinating is that the linear is a specific heat capacity this is C over T is linear at low temperatures it has a bulk linear specific heat there's about 10 times that of lanthanum hexaboride okay it's also an ac metal and a dc insulator here you see the optical convictivity measured optically and here you see the uh conductivity coming down at low temperatures but it never quite goes to zero that you don't see a nice gap here as you might have expected um and uh if you try to estimate the dielectric constant in this system it exceeds 600 here are the um here are the uh uh quantum oscillations in the magnetization uh uh um and uh you can see uh both low frequency oscillations and high frequency oscillations up here uh and uh the lichitz kasevich uh form is very nicely obeyed and one other thing you can see the torque in the bulk susceptibility so these are oscillations in the bulk susceptibility okay and this led to the Cambridge group uh claiming that there is a both small and large pockets uh in the Fermi surface pockets even though this thing is an insulator and these oscillations resemble lanthanum hexaboride there's a lot of controversy around this some groups have suggested that these low frequency oscillations here come from aluminium because one way of making smear and hexaboride is to grow it in a flux of aluminium however the measurements that were carried out here were made using uh smear and hexaboride that was in it going in an image furnace there's no aluminium there and so uh uh that explanation to be ruled out these low frequency oscillations have been reproduced by other groups but the high frequency ones corresponding to the large Fermi surface have seen by the Cambridge group have not been reproduced okay so these are the two viewpoints about this system one of which is it's a topological condor insulator a viewpoint that would have been regarded as radical some years ago but seems to be well established by many points here and then there are the bad actor aspects the linear specific heat the quantum oscillations the ac metal dc uh uh dc conductor that uh pose real problems okay so maybe only one of these pictures is right maybe there is a competing phase that exhibits in this system may be induced by the application of a field and it raises interesting question could there be two competing insulating phases with different kinds of spin fractalization so i'd like to end by talking about a model of condolatus with a neutral Fermi surface and uh i can do this in cartoon version but i can also do this with a little bit more uh detail perhaps i'll spend a bit more detail on this switched to a more another more detailed seminar talk that takes you into this but let me first begin by uh reminding you about the concept of fractalization um so fractalization is the emergence of excitations with fractal quantum numbers and the classic example of this are the domain walls the solitons of polyacetylene as established by sous free for an ego in 1980 when you have these domain walls you have charged zeros been one half excitations that are mobile very heavy the other classic example are anions in the fractal a quantum hall effect work of roba schrieffer and will check back in 1983 established the fact that these fractally charged excitations these zeros in the Loughlin wave function have fractal statistics okay but what we've heard about a lot in this in this meeting here is the spin version of fractalization and it's here where i think fractalization is is particularly well established the idea that magnons in a spin one half heisenberg chain fractalize into spinons and this is the the comparison of experiment with theory in the s of q and omega in a 1d spin one half heisenberg chain and so but another context in which this idea of fractalization is relevant is in the context of the condo lattice and this is a very unconventional view of the condo lattice that i'm now going to present the conventional view of the condo lattice is its firmly liquid behavior is related to the fact that on earth all condo lattices are derived from local moments that come from electrons but the condo lattice model as it stands is a mathematical model where the origin of the local moments is undefined it could be local moments as qubits it could be nuclear moments with a very strong hyperfine coupling or it could just be a a calculation you've given to your to your graduate student do a tensor network calculation on this and tell me what you find okay the point is the appearance of a large Fermi surface in the condo lattice is a signature of fractalization and it's something that we can understand very clearly in the context of the large n limit of the condo lattice and the point is that or the strong coupling limit of the condo lattice condo singlets mean that we have an insulator and we can dope it and when we do that when we think about this system when it's undoped we have a Fermi surface of electrons and then we have the additional contribution to the Fermi surface coming from the local moments and the net combination of this leads to no Fermi surface we can connect this up with Oshikawa's theorem which basically shows in the condo lattice if you have a Fermi liquid then you will have a Fermi surface that counts the number of electrons and the number of spins and so this is the classic view of the condo lattice we think of the spins as fractalizing into an incompressible fluid of heavy fermions of f electrons and these things hybridized with the conduction electrons this little cartoon actually has a great deal of subtle subtlety beneath it this you can find this cartoon by the way in I think it's it's present in a in a paper by by Neville Mott from the late in the early 1970s but in the context of the condo lattice it's rather interesting because these are spins and to represent them as f electrons requires that you fractalize the spins okay such a system will have and in order to do that fractalization you're actually introducing an underlying gauge field that gauge field then interacts with the physical electrical magnetic gauge field and that subtle interaction involves a Higgs mechanism which drives the f electrons into becoming charged particles okay otherwise you wouldn't get an insulator okay so that's the condo insulator as viewed by Enrico Fermi okay but of course this decomposition of the spin operator is an assumption it's good in the larger limit but it's not guaranteed at spin one-half okay and so there was another way of doing this which is to think about the problem in the language of Majorana fermions and so the condo insulator if you think about the electrons the conduction electrons as Majorana fermions you have four Majorana fermions interacting with four Majoranas of the f electrons and so this is our picture of the condo lattice as viewed by Enrico Fermi let us now look at the condo insulator as viewed by Ettore Majorana so in that context we would be looking for a different description of the spins this is an equally good way of writing a spin operator in terms of a vector of Majorana fermions it's mathematically correct and in fact unlike the first way of doing it which involves a constraint which says that the number of f electrons is one this second way of doing it actually doesn't involve a constraint but it nevertheless involves an underlying gauge field because you can change the sign of these ateers here without changing the s this is underlying z2 gauge field behind this and so if this description becomes valid if you have a Majorana fractalization of the spins then what happens is that the spin is now represented by three Majoranas not four so let's look at what happens to this picture now come on yes so here you see i've got three f electrons or three Majorana fermions connecting up with three electrons right but the problem is that the electron fluid contains four Majorana fermions and so one gets left behind which means that you get a gapless Majorana band and this kind of situation was something that we've been playing with Alexis Fedeck myself and in the 1990s Eduardo Miranda for a long time we just plug in this perfectly good representation of the spins and no one behold we always got a gapless Majorana band out okay we even wrote a paper on this in 1993 or 1994 entitled why are our condensate is gapless and then we got confused by our mean field theory because it seemed to look like a superconductor and we rephrased the work and published it as a as an example of odd frequency triplet pairing okay so in the context of the smear mix boy this has led us to reconsider the situation okay and this is the model which Alexei told you about last week the Yao Li spin liquid in three dimensions with a condolatus and maybe i spend a little bit of time showing you a bit more details about that so um so let me show you this Yao Li model it was asked this morning in in the talk of Haiyong Qi are there higher examples of of spin liquids and Haiyong Qi told us yes but she said I don't regard them as very physical and I want to actually give an opposing viewpoint okay and first and foremost this is a way for us to look at a completely gapless spin liquid which is exactly solvable okay and we'll discuss later whether it might occur in reality okay so here's the model has this grotesque structure okay it's actually a cubic structure not at all obvious it's a it's actually a body centered cubic structure and at each site on the bcc lattice you have this little spiral of four atoms there's a chiral lattice okay and it has a uh this classic uh trivalent structure of a Kitayev model which makes it solvable and in this model the spins of the Kitayev model are now the orbitals of the Yao Li model they're labelled by the lambdas okay and the lambda lambda interacts just like it is in a conventional Kitayev model but now it is decorated by a spin interaction sigma dot sigma okay so there's the spins and there's the orbitals okay this model produces a spin liquid with gapless spin excitations with a Fermi surface okay and it's exactly solvable and why does that work out well the lambdas uh gap out and form bond variables just like in the Kitayev model the spin degrees of freedom actually fractalize as Majorana fermions and so uh you can actually rewrite this model as a gauge theory field and unlike the Kitayev model it's a z2 gauge field but now instead of having one Majorana fermion you have a triplet of them okay and these triplets define the uh spin degree of freedom when you flip a spin you don't create a vison in this system the visons uh this is the z2 gauge field and the visons live on these loops which in this interesting structure uh where is it oh it's come there's the vison okay it's this structure that it's a loop of of sites in this particular system and so lower low temperatures you can replace the uijs here by one a unit matrix in fact this system has a uh a z2 uh phase transition uh which runs at around point about uh one percent of k and below that temperature the visons have a mass and the fractalization is perfect that's the Yaoli model and it's interesting to us because we would like to and this is the Hamiltonian for it just to show it to you and it's interesting to us uh and here is the Fermi surface it's a Majorana Fermi surface um the Majorana unit cell is one half the electron unit cell in momentum space uh and it's a nice cube okay that's the Yaoli model and it's a very beautiful model and if we take it uh in three dimensions we have a beautiful example of a spin liquid a spin liquid made up of spin one fermions okay so now let's couple that up to electrons moving on exactly the same lattice and here they are and uh this is the conda coupling and uh and the conda coupling can be factorized in this interesting way where it becomes an attractive bilinear of chargey spinners and it's this chargey spinner that really fascinates us um if this thing condenses it means that you're binding a fractalized particle which of course you might think of as an imaginary object to an electron which you think of as a real object but in this system as I try to convince you the fractalized particles are real excitations and so if I they exist as real excitations we can now look at the possibility of binding pairing between electrons and fractalized excitations which allows us for the possibility of unusual quantum numbers a chargey spin one half condensate okay and so that's the model and once you've got this you can now proceed with conventional many body physics you can decouple this thing in terms of these two component spinners here they are and you can do a mean field theory on that or you can do just as we heard this morning you can do a cooper expansion in the diagrams and you can look at the pair susceptibility and here is the pair susceptibility so oney pair susceptibility it's logarithmically divergent which means that you have a cooper instability in this charge one e channel and so this is the phase diagram of this curious system the moment you turn on j at half filling you form this chargey condensate okay and let me just go one more step further here is the Fermi here is the Fermi surface of electrons which are split into Majorana fermions this then raises lots of interesting questions and I'll end with some of those questions here is here is a few more pictures describing what's going on here if you actually look at the pairing kernel of the electrons you have something very interesting because you can of course integrate out the Majorana fermions and look at the self-energy of the electrons when you do that it's it's a little bit unconventional but at first sight you might say well why are you making this big deal about these Majorana fermions it's a triplet superconductor accepting it has a number of curious features there's an onsite pairing kernel which is triplet which is impossible unless you have odd frequency pairing so onsite it's an odd frequency superconductor but it also has the property that you can set an electron moving along can plunge into the spin liquid and then come out an arbitrarily long distance away and so this self-energy functional here actually is kind of singular in momentum and frequency space it has zeros on the fermi surface of the Majorana fermions and this is because you can separate out the vx and the vy to arbitrarily long distances this self-energy factorizes into spinners okay and so I think and it also forms a pair density wave okay and returning to the Samaritan hexaboroid conundrum I'm sort of wrapping up now I wanted to talk about what it might mean and it's possible connections with Samaritan and other systems so if indeed you can have insulators that have quantum oscillations this is a paradox it really can't occur according to our standard understanding of metals and insulators in our standard model of metals and insulators the excitations are electrons okay and those electrons respond equally well to the electric and the magnetic field they can't undergo landau orbits without also conducting okay and so the possibility of an insulator with a fermi surface is an oxymoron it either means the experiments are nonsense or we have to come up with a completely new picture and even if you don't like what I'm talking about you can ask well what would we do from a phenomenological point of view one other way to try to get a the hasphenolaphan oscillation from a kind of spin liquid has been explored by chowdry et al he's not here this week but you can take a spin liquid and you can notice that the ring exchange is coupled to the external field and this will then give you quantum oscillations in my opinion they're not large enough to explain the experiments but it still provides me with a straw man to explain my thoughts and the point is that any such system that's an insulator has to have first of all the property that del dot j is zero it's an incompressible fluid okay but secondly it has to have the property that produces a magnetization and the curl of the magnetization is j so that's interesting how can you have a curl of the magnetization if del dot j is zero of course del dot j because zero doesn't necessarily mean that j is zero okay it just means that del dot j is zero it means the current is transverse okay so this then leads to the crazy idea that a phenomenological resolution of the smear and hexa boy conundrum which avoids having to worry that whether the experiments are valid or not and suggests that a novel fluid that allows divergence free circulating currents but with an insulating longitudinal conductivity okay to put that into better words let's look at the conductivity recall that in a metal or any bulk conductor you can divide the conductivity into two parts a longitudinal part and a transverse part in a cubic environment they're very simple to write down the longitudinal part first of all the transverse part has the property that if you take the dot product with q you get zero and therefore the divergence the currents are zero but the longitudinal part is the leftover bit so that is defined by qa q hat a times q hat b so this is a universal separation in a metal this separation is very valid and this is the term which connects with plasmas and this is the term that we measure when we do the optical conductivity and part of the urban legend of condensed matter theory is that as q goes to zero sigma longitudinal equals sigma transverse and so everyone is taught how to calculate sigma transverse and use it for sigma itself right so all the theories of localization are built around calculations of sigma transverse yet the thing you measure is sigma longitudinal you put leaves on and you look at the current going together they're probably the same but this is a system where you put where you put leaves on it you don't see any conductivity right it seems to be a very good insulator and if we accept tentatively that the experiments are correct then this suggests that sigma longitudinal has to be exponentially activated it's a good insulator but sigma transverse well maybe that's a druid of metal okay now anyone who looks at this will immediately tell me that this breaks all sorts of symmetries okay because in order to have sigma longitudinal sigma transverse be separate from one other you have to have a very long correlation level infinite if you take what I've written here too which would mean some kind of broken symmetry okay so it's a little bit of a wild thing but you could test it you see and so here is the optical conductivity delta gap in this system is 1.1 terahertz and we would normally expect the optical conductivity to have plummeted downwards here okay and yet it's not okay so could it be that perhaps you could go to lower frequencies and could you perhaps check sigma ac in the megahertz range somewhere down here to see how this thing evolves could this be could it be that this goes to a finite value even at zero okay I guess that's what I've done here I've drawn it in this is my imaginary notion of what might happen at low frequencies okay okay and such a fluid would decouple from phonons and will develop very long equilibration thermal equilibration times and one of the conundrums of this system is that no one's ever seen a thermal conductivity associated with its so-called neutral Fermi surface but on the other hand specific heat capacity has been measured those two things lie in contradiction to one another and it's perhaps because the thermal equilibration time for establishing a constant thermal gradient becomes very very long okay and now going back to my previous talk let me just end up here with where I was all right that one yep broader conclusions okay I think the broad conclusions which no one's gonna ever you may be skeptical of the things I told you in the last part of the talk and that's okay but history teaches us the physics in that lab and the cosmos are often intertwined and so this then provides a motivation for seeking answers into some of the things in the cosmos by looking at stuff in our everyday lab and no better place to look than in quantum matter and I think also there's a deep connection here with highly entangled systems and the possible connectivity with quantum information and I've told you about Samaritan Hexa Boyd an old mystery that brings together ideas of topology and fractalization in fact to understand it as a conglolatus you have to fractalize the spins thank you yeah we have a lot of time left in fact that the fog is supposed to end at 340 so we have 24 minutes for the discussions so in the last part of the talk what breaks the symmetry like how do you choose the transverse direction known student direction yeah okay thank you so it is a broken symmetry stage no no because he did it with the microphone oh so with the microphone everyone can hear it online that's the great thing about the mic so but I will repeat it anyway the question is what chooses the direction and in fact the thing that chooses the direction is the spinner order parameter so the electrons components that are parallel to the spinner don't see it only the ones that the perpendicular spinner there are three components perpendicular to the spinner see it and they're the ones that gap out and so it is a broken symmetry state and so should be regarded from that point of view as a superconductor okay so how dare I even talk about that in the context in the context of the of samarik's boride and to be quite frank with you I don't quite know okay it there is one piece of insight however there that that we've been trying to hang on to which is the fact that a superconductor is a rather remarkable thing it has I used to think that a superconductor the most important part of it was the fact it has a superfluid stiffness and this point of view is emphasized very strongly for example in anderson's concepts and solids book that you need a stiffness set but it requires that one other component requires a topology topological resilience so for example an x y magnet is a superfluid you have superfluid flow of of spin and you can wind it up as much as you like but a heisenberg magnet ferromagnet or anti ferromagnet should not be regarded as a spin superfluid because when you wind up the spin it always unravels okay so if you had a superconductor with a spinner order parameter it would have unusual properties you couldn't wind up the spin to create a wind up the spinner to create a macroscopic super flow because every time you did it it would unwrap okay so this would be at the very best a superconductor that doesn't support a macroscopic current which of course is not a superconductor right okay right so what would it be so we thought and we proposed this experiment and it was done in at at at TFR in India we proposed that it would at least exhibit a Meissner phase at low at low pheoms and they did the experiment down to 10 to the minus five or six uh tesla now maybe lower 10 to minus nine tesla they saw nothing okay they were so embarrassed by seeing nothing that unlike vipersen and moorley they chose not to publish um uh and so our theory died at that moment as as a theory of samaranghex borai um but we are still worried that we maybe made a mistake somewhere along the way and that the Meissner phase doesn't exist due to aspects that we haven't fully understood at this point and so we're still optimistic that a revised version of our theory might yet come back to life yeah okay and we have we'll go one to three and there is an online question so yeah just a question you you briefly mentioned valence fluctuations so how do you think those might interfere with the picture that you've put forward yeah well i think it's just a question of coming up with the appropriate myirana uh slave boson description of uh hubbard operators um we've tried to invent such a thing using schringer bosons and myirana fumes we haven't quite got it to work properly um but i don't think it's ruled out um uh in a way in a way myirana fractalization spins ought to be better than than uh than uh conventional fractalization of spins in terms of fermions because the hillbett space of the myirana is is smaller than the hillbett space of of dirac electrons but it does seem that uh that's most the time the dirac fractalization is more stable in the case of this what we like about this yau lee spin liquid is it where it's guaranteed that this fractalization occurs so it raises a number of interesting questions and when i throw out there is would it be possible perhaps to find a version of the catechive model that has charge fluctuations a solvable model of the catechive model that has charge fluctuations i don't think it's ruled out such a beast may be out there to be found right just as the heisenberg model was solved in the 1930s and the hubbard model was then solved exactly in one dimension in the 1960s it might be that there there is a charge fluctuation analog of the catechive model that's out there okay if we had such a thing we could answer this question much more effectively so i have again a question about the transverse longitudinal um dichotomy that you mentioned again so when q equal to zero you cannot distinguish transverse longitudinal just because of momentum in zero but any experiment is done at the final momentum so what you are telling us that even if the momentum lies extremely small it still is larger than the scale where you see essentially the the distinction between longitudinal and transverse is that that would be the that would be the fallback the hardcore version is hard to go for because there's no phase transition in samarang hexaborate okay so uh but it might be that there's a long enough correlation then to sort of play these crazy games yeah here's a nice talk could could you just tell us what the status is on the hall effect in the samarang hexaborate thanks we've talked about you know sigma xx what about sigma xy um the hall effect that's been measured is is the surface hall effect so i don't know anything about the bulk hall effect of this system because it's short circuited by the surface um so it raises an interesting question whether the corbino geometry measurement could be used to measure the bulk or conductivity i don't know whether that's possible to do but it raises an interesting possibility but but oh that's a good question i don't know i don't have a prediction um but uh i suppose what you're saying is that i mean presumably the whole conductivity of the bulk is a transverse conductivity feature right so i think it would have a one over any oh i'm completely i'm i'm just looking at real time uh response if the transverse metal were a real concept then i suppose you would have a and it went all the way to zero frequency then you'd have then you'd have a sigma x y but i don't know whether you could measure that thing uh except optically uh you could measure optically of course gersh and i have discussed a little bit this possibility yeah okay so we have a one online question so let me just read it's hard to understand the question so let me just interpret it oh so it's um it's hard to see this well actually basically the year hemilltonian is spin spin and orbit orbit interaction uh probably weak compared with the skin orbit interaction so hemilltonian that you have written sigma dot the sigma lambda dot the lambda um i think they interpret that as a spin spin and orbit orbit interaction so is that weak compared to spin orbit interaction so maybe you can just elaborate a little bit about your hemilltonian yeah so so so i think they're talking about that so the the condo interaction i don't think i should do that should i no um the condo interaction is a completely normal kind of interaction it's a it's an interaction with the electron spins and the local moments but the yaw lee model has this uh interaction in which you have a heisenberg interaction of the spins multiplying a kiti of interaction of the orbital degrees of freedom okay and it is fair to say that no one has come up with a microscopic model that produces such an interaction okay um uh what i would like is for someone to say it's impossible because that would probably lead to it being discovered but at the moment we don't know either way okay so maybe it's uh for me that i can is there any other question otherwise i can go with one clarification that i didn't say that it was unphysical i said precisely there's a recording so you can check i said it's a beautiful theory but it's uh artificial from a microscopic theorist oh that okay artificial very good and the reason for that is because the lambda is not a orbital angular momentum it is the uh two spins yes made out of so hematonian as you reply to the tomos question is hematonian is made of six spin interactions so it's not impossible but from a microscopic point of view let me rephrase that i shouldn't say artificial i should say it's quite unrealistic where it's hard to find it from a microscopic theory so so what i would comment on that is that samarium hex boride has a quartet of states okay four states just exactly the same number of states as the yaoli model okay and so uh most likely if you look at the interactions of those quartets between different sites it's not a yaoli model but i bet you that it contains a component that overlaps with the yaoli model in some sense but of course that would it's not a trivalent lattice and so you'd have to break some lattice symmetry to make it look that way nevertheless it has the right number of states right i mean the reason that i know a little bit about it because i was interested in realizing that's such a model so no one was interested in how to get that and one way to get it you might talk about the molecular spin orbit yes you have a three site which has been half then you can generate the right number of the state and that's how you can i should talk with you some more thank you so it's not like i'm you know i i think it's a beautiful theory and it'll be really nice to get some microscopics from out of this okay i think it's coffee time yeah i think it's uh anyone else oh yeah we have one because it's supposed and that we still have three minutes so i i have a quick question about this kondolaris model with a yaoli model yes you constructed so you worked with the marana bomeo representation but i presume uh you could rewrite the model using complex bomeo after you said uij equals to just one everywhere because the quadratic hammer in the end you have a quadratic Hamiltonian or marana so then then you could rewrite that quadratic Hamiltonian by transforming back to uh oh of course and i mean i so in fact um it went past very quickly there are various ways of going back to regular fermions that course the simplest one is just to is just to Fourier transform right and uh so yeah but the other way of doing it would be to i mean once you write marana fermions in momentum space oh dear once you write marana fermions in momentum space they of course become complex fermions yeah so my so so but on the other hand you could also rewrite uh you know marana fermions in terms of like in terms of uh four complex bomeo you could but it would be a projective Hamiltonian i know i understand but if you do so well then i think i i i think what happens is that you end up with the projected spin triplet bcs hammer time uh so you were certainly because but you'd have introduced that symmetry breaking by the direction in which you chose to extend your your free fermions in other words in other words you'd have to choose a direction which is akin to choosing a spinner in spin space uh and the spinner in spin space that you choose would be the direction which the marana fermions are projected out uh presumably but the point is that by the time you rewrite this way and uh to the standard hybridization say in field theory or something like that then basically by the time you couple these complex bomeons with the conduction electron yes that immediately uh induces the cooper pair channel in the triplet triplet uh there's precisely what you're discovering right there yeah it's it accepting the first way of doing it is totally artificial because there's no reason to expand the Hilbert space at this point because it's a perfect description of the physics okay because we actually saw uh we actually solved a similar model uh in two dimension uh when we couple guitar just uh you know ordinary guitar model for conduction electron honeycomb lattice and that's precisely what happens so it's not the same because in such a model a spin flip creates visons and so it interacts with the gauge fields whereas in the yawley model when you flip a spin you don't touch the visons you purely interact with the gapless uh marana fermions there is an orange orange in the body yes okay so i think i think that's here i'm going to you know just that i'll talk to you i think you can discuss more i'll probably have some debate over you as well uh so where we get the one sundra and then one online question then we'll end that okay good i wanted to draw a distinction between the condo model and the mixed valence model we all know that uh if there is particle hole asymmetry which is weak in the condo model it's irrelevant however in the mixed valence problem we not only have spin fluctuations with a given charge on the magnetic arm we also have charge fluctuation and uh it's an exactly solved model that if you go away from particle hole symmetry and stick to mixed valence the charge fluctuations are equally relevant and the the low energy excitation of the problem is not of the condo model it has a singularity at equal to zero this this is an exactly solved model both by wilson renormalization group with equivalent results obtained with theory jamachi and clemasir and me about 15 years ago it it's simply a fact and it is very interesting in relation to your talk that like most problems whose exact solution most impurity problems whose exact solution in the limit of low temperature meaning the ground state and the excitation spectra has both the ground state entropy and singularity in the spectrum log singularity in the fluctuation spectrum this model can be expressed equivalently in terms of a localized marijuana and a propagating marijuana so that is that that's simply an unarguable fact of the mixed valence problem uh what becomes hard is to maintain those symmetries when we go to the lattice so i'm sorry i won't say you can then take this mixed valence problem and try to make a lattice and then uh then the problem becomes uncertain and it's still however interesting to look at from the point of view of the experiments that you talked to thank you all right so i think that sounds more of comment uh so this is a last question from the chat um like to get us some ideas well have some ideas and help to give to cosmologists to solve their own problems uh huh whether the condensed metaphysics may have some ideas and help to give some you know help to cosmologists to solve their problems for example dark matter is an emergent gravitational many body path with a question mark okay now i got myself into trouble um i i'm not going to comment i i can't give uh advice to cosmologists but i think there is scope for inspiration in condensed matter physics from cosmology and we had a little fun recently in premichandral talk about that on friday i think friday yep uh using the idea of energy fluctuations as a pairing mechanism uh and uh one of the interesting things about strontium titanate is that the quantum critical transverse modes don't couple to the electrons unless you have a large amount of spin orbit coupling which is we believe absent in this system and this is a system where energy fluctuations of the background uh polarization fluctuations actually may produce a gravitational attraction between electrons that drives pairing um and so i'm not going to say that more about that if you're interested in this idea come back on friday okay so with that uh let's thank the spears again so we'll have a 30 minute uh actually it's about 34 minutes to break and back at 4 10 no again some some coffee