 I've been enjoying a lot so far and I think that most of the talks in the tutorial yesterday and also today maybe with the exception of Sean's were on using some new and interesting tools to solve some very old and difficult problems and today I'd like to try to talk about what I think is maybe a little bit of a new problem so I can use some old tools to try to understand something about it and this new problem is something which for lack of a better name I'm going to call a quantum ferroelectric metal and I'll explain in a moment what I mean by quantum ferroelectric metal it's something I've been thinking about with my collaborators here Rafa Fernandez, Vlad Kozi and Jonathan Rochman for the last maybe couple of years and if I have time I'm going to try to show some I think very interesting experiments in the group of Martin Grevin from the time when I was a postdoc at Minnesota so what do I mean when I say quantum ferroelectric metal so really what I mean is a quantum ferroelectric metal it should be a material which is metallic and it's also polar in the sense that it has broken inversion symmetry and should in some sense have dipole moments and at the same time it should have some kind of strong electronic correlations in some sense and this actually for many years was thought to be an impossibility to the point where if someone said ferroelectric metal in a seminar he got yelled at but in recent years people have discovered a real zoo of materials that have these properties they're typically doped semi-metals or semiconductors here are two kind of prototypical examples transium titanate it's a quantum power electric meaning it's a material that at low temperatures almost becomes ferroelectric when it's insulating and then you can play with it a bit and make it ferroelectric or molybdenum telleride which is a TMD which has also broken inversion symmetry and typically you can take these materials and dope them a little bit to make them a conducting and they're very easy to dope they're very easy to strain they're very easy to photo control which makes them a very useful application for applications in quantum materials and they have a huge number of fascinating properties both from basic science and applications point of view they typically host dipole moments have the strong spin orbit coupling almost I think maybe all of them have superconductivity at low temperatures and there are increasingly more and more signs of unconventional superconductivity in many of these materials turns of band structures you can find parabolic topological and non-topological dirac and vial band structures in various of these compounds and really all of this kind of goodness is a result of the fact that they have broken inversion symmetry and of course people have been studying metals with broken inversion symmetry already for quite some time in the context of you know non-centrosymmetric superconductors topological superconductivity and so forth but the problem that that we've been thinking about is the following one at the end of the day this process of inversion breaking is a dynamical process and what I mean is in many of these materials you can take them like these two and through some tuning parameter doping or pressure you can tune them in and out of the inversion broken state and that means that you go through a quantum critical point and so you're being driven somehow by dynamical fluctuations that break inversion symmetry that's something which usually comes with electric fields which doesn't make sense in the contents of a metal and so you really need to ask yourself what are these dynamical processes that break inversion in a metal and how do they affect the fermions in the system and can we use them for something useful so that is the the problem that I'm going to try to address in this talk and maybe convince you that it's an interesting problem to think about what some yes some no so the tip like I said there's typically typically semi-metals and semiconductors but of course typical isn't necessarily what's in the lab that's what's in bft so you'll often find that they are metallic and they're as grown states but this one's in titanate which is I think the only one of these materials that has been extensively studied there's really not enough experiment on these materials that one isn't insulating in its parent state and it can be doped to very very light doping we'll talk about it so really what we want to do is kind of construct a effective low energy theory for such a problem and the idea should be as follows take your your favorite version of an unconventional superconductor iron based cuprate uh uranium based and and we have this I think at least partial consensus in the community that at least qualitatively you can get an idea of what should go on in these materials by thinking of a quantum critical point between an ordered and disordered state then you have quantum critical fluctuations somehow dominate the phase diagram and hopefully they also give rise to the superconducting dome that appears at the bottom and I would like to try to construct a theory for the case of a material which breaks inversion symmetry and I do a similar kind of effective theory for these materials so typical questions I might ask is is this a Fermi liquid or a non-Fermi liquid what are the low temperature phases how does pairing occur in the system if at all and can I control it by external parameters this would be I think extremely important in the context of quantum materials it's also how we get paid because our grant is on quantum devices and materials okay so what I'm going to do now is I'll start with an introduction to introduce what exactly or what maybe our ferroelectrics and ferroelectric metals and how we should think of the problem then I'll go into a good deal of detail on what happens with a Fermi liquid that's a quantum ferroelectric metal this should it's not published here that should appear on the archive in the next day or two then I will try to show you this intriguing experiment to convince you that not everything I'm saying is theory la la land and then if I have time which I definitely want I'll try to talk a little bit about the case of the Dirac point I'm I really just put it here so I can tell you that it's interesting and there's superconductivity at a charge neutrality point but there's no way I'll get to it so let's go into the introduction instead and so in order to explain about a quantum ferroelectric metal I really need to start by talking about what creates an insulating ferroelectric and typically the material is insular is a ferroelectric at low temperatures it works through the condensation typically not always of a soft phonon what's called the displacent transition and a toy a cartoon to think of it would be to think of an ionic lattice so you have ionic forces and then you have dipole forces between them and you have some soft polar mode which eventually condenses and deforms your lattice this creates a finite polarization in the system and a macroscopic electric field running through your system and a for many years it was believed and it's very often correct that this is a result of long-range coulomb forces essentially van der Waals forces between your lattice or inter interunit cell ionic forces and this is what causes the instability to formation of dipole moments and as a result it turns out that the optical mode that condenses must be transverse you can actually see that in this figure here this is showing the polar optical mode in strontium titanate which as you see has a longitudinal component that remains hard down to low temperatures this is in I believe an inverse centimeters and a transverse mode which dives down in frequency and becomes soft as you lower temperature actually in strontium titanate it never hits zero but you put a little bit of strain and it hits zero and the system becomes a ferroelectric this is well known under the name of the laden sax teller relation which tells you tells you that coulomb forces renormalize the longitudinal mode and force it to remain hard so only the transverse mode becomes soft and goes critical and this immediately causes trouble when you go to a metal because in the first place if you have a conducting system then you have a Fermi surface and so long-range coulomb is screened so you shouldn't even have a ferroelectric transition in such a system it should be screened out but even assuming that somehow I construct a material which does undergo such a transition at low temperature there's the question of how to couple to electrons the usual coupling to electrons is through electrostatic forces those go as the divergence of your phonon say you so they go as q dot the polarization but this is a transverse phonon so in the low temperature long wavelength limit q dot p just goes to zero so even if you did have such a material you would really have two independent subsystems an insulating subsystem that goes ferroelectric and a conducting system that knows nothing about the ferroelectric system but in practice it's clear that the systems are coupled because in all these materials you find the superconductivity it's very often somehow correlated with the quantum critical point of the polar phonon there's a good deal of indirect evidence and also there's a lot of other evidence so I'm not showing here of anomalous electronic properties some of those actually Mitya Maslow showed yesterday Stroncian titanate has extremely strange transport properties and so forth so the first step is actually can we even construct an interaction we need some way to couple the two systems together my electrons and my phonons and this is what you want to think of you have an insulator and you dope it with electrons and now you want to see how they speak to one another so what would be the properties of such a coupling so you're going to need a transverse optical phonon and you're going to need it inner and fermions and one thing to note is that you can have these you can have your system deform but you don't need to have an electric field you can just have some high energy conduction electrons screen out your polarization in the material but as an order parameter your phonon which breaks inversion symmetry is still a good object so you still have a phase transition in the second order lambda sense of the term you just don't have macroscopic fields in the material because something has screened them out so you can use your phonon transverse phonon which I will call eta as your order parameter and then you need to have a coupling which does break inversion for the fermions others not otherwise nothing interesting will happen but should not create macroscopic currents it cannot create an electric field or drive a current for your conduction electrons that would violate block theorem which says you can't have spontaneous currents and then you would kind of need to fine tune your material system to avoid block theorem which can be done but is a bad idea when you're working on a completely new system and actually several years ago people noticed that there is such a coupling and the idea is simple you just couple spin and charge you have to mix spin and charge together so that you don't create a charge current and the natural way to do it in this case is to take your phonon which is a vector and dot product it with k cross sigma to create a dynamical rush bar spin orbit coupling when your phonon condenses your material will become rush bar split if I think of a two-dimensional material and for most of my talk I will discuss only two-dimensional materials then basically one of two things can happen you can condense your phonon out of plane remember it's a 2d material but polarization and spin are always three-dimensional objects so I can condense out of plane that would create this sort of Zeeman splitting for my Fermi surfaces or I can condense in plane which would create this well-known rush bar splitting of my Fermi surfaces in plane so I have these two modes I'll actually call them the z-mode and the t-mode in a moment the reason why I'm talking about two dimension is just because the physics is much more transparent in two dimensions many of these materials are three-dimensional and maybe I'll briefly comment on 3d but it's just so much easier to do this in two dimensions so now before going into the details let me just try to give you the main message in pictures what are the main results of this model so here's the model in pictures I have an optical mode I call it eta it's transverse so it creates this sort of ionic deformation and it's always transverse so the direction of propagation is perpendicular to the polarization and it's coupled in 2d to some Fermi surface which I take to be circular and always keep in mind that what we know very well from strongly coupled boson fermion systems is that the the best coupling is when the boson scatter parallel to the Fermi surface so k Fermi will scatter when q is more or less perpendicular and now as I showed you before I have these two modes one I call the z-mode that's when eta the phonon crystallizes out of plane condenses out of plane and one which I call the t for transverse mode is when it condenses in the plane and therefore the spin must be out of plane and why must the spin be out of plane because it's k cross sigma so k and sigma are always perpendicular to one another and you see right now that there are an awful lot of vectors sitting on this slide and they all have to be perpendicular to one another and there's more than three of them and that means you can't do it so you're going to get frustration but this is a very interesting type of frustration it's only dynamical geometrical frustration and what I mean is it only occurs when you have dynamics when you have some momentum scattering with a finite q then you have four vectors that you need to put somehow perpendicular to one another and it doesn't work if you look at static properties you find the geometry is perfectly satisfied when you do dynamics you find this frustration and this as I will show you pretty interestingly and significantly impacts the critical properties of the system okay in terms of a phase diagram you know it it looks pretty similar to what we know and love in terms of unconventional superconductors you have some ordered state the ferroelectric state there's a second order transition and there's also a superconducting dome in a quantum critical region there is one additional detail it turns out that under certain circumstances the second order transition can become first order we'll go into that later but but the devil is in the details so it turns out that the different modes you can have either a non-fermil liquid or a marginal fermi liquid all the way down to the critical point you don't need to find you anything the system is just a marginal fermi liquid this first order transition is a result of something called quantum order by disorder it's basically because you have a vector and you have additional fluctuations in the system and you can tune it away using strain and when you look at superconductivity you find that basically s-wave and p-wave are essentially degenerate and all of this physics you can kind of pull out just from the coupling constant the reason why you have s-wave and p-wave is just because you have both charge and spin in your interaction the reason why you have quantum order by disorder or first order transitions is because you have a vector so you have something that looks like goldstone modes in the system and the reason why you can have both a non-fermil liquid and a marginal fermi liquid is because of this dot product because of trying to get too many objects to be perpendicular to one another so now let me try to go a little bit into you know the nitty-gritty details so here's the model it's kind of the simplest model you can write down you take electrons you take bosons and you put them together what you do is you write down effective action for your air time your boson it has some energy scale it's typically several MEVs say in strontium titanate and then it has a q squared plus omega squared i use q0 to denote a macho bar frequency so i've always worked with three or four vectors and q0 is just a macho bar frequency and you want to assume the system is more or less close to a critical point so r0 you know the inverse correlation length squared is some small number then i couple it using this k cross sigma to my fermions the fermions are just a fermi circle i didn't even write them down and i asked what i have to keep in mind is that this vector eta is a phonon projected onto the transverse sector so it's always transverse to the propagation direction i just have to keep track of this in my calculations you will not see any of that in the rest of the talk the the second thing to take away from this slide is that as usual in these models they're essentially cooked up to do so there's basically one effective coupling constant in the system it's the bare coupling square times the density of states divided by some typical energy so everything is going to scale with this one dimensionless coupling constant which i call g here and then what do we do yes no the u can be out of plane because polarization is 3d i'll just get q will be in plane okay so think of a lattice it can do this or it can do this this would be lamb waves right and acoustic these are optical so they're not exactly lamb waves but no problem more questions okay and so so now i have an action so i do what what you know what people do when they have an action they just start generating a responses let's keep to one loop and what do we expect normally in a you know in one of these quantum critical materials you're going to calculate the bosonic response and the fermionic response maybe with some vertex corrections which we typically drop away and what we expect to see the usual result is something like this i'm using the iron based on the math the critical point is an example first you find that the interactions can renormalize you all the way to the critical point so you can use the interaction strength to bring you to the qcp and then when you look at the boson dynamics you find the bosons are over damped this was discussed extensively yesterday in for example yorg's talk you get the lambda damping omega over q and therefore the system is over damped with z equals three and then you go back and you feed this into the fermions and you find that the fermions are also over damped and they create a non-fermin liquid at some typical energy scale omega g and in this case you have omega two third behavior this is just because it's a long wavelength problem this is also what shows up in many syks and then you also find in terms of pairing that pairing sets in always on the same energy scale that the non-fermin liquid behavior sets in so sigma is non-fermin liquid only on a scale of omega g or less and at the same scale pairing is going to set into your system and create superconductivity and it won't be bcs because it's a polynomial of some so what happens in our system well let's go back and look a little bit more careful at this question of geometric frustration i'll run through it quickly because i already showed the picture beforehand here's our coupling i've dropped a lot of indices to make things clearer so here's the model again and now if i ask myself about static order i want to let eta condense there are basically two configurations i'm creating here two right angle triangles two right hand trios i can condense eta out of plane and put k and sigma like this or i can condense eta in plane and put sigma out of plane that would be sigma z if you want so there are basically three geometric constraints and i can obey them all but now when i go to dynamics i find five geometric constraints because i have to have q more or less perpendicular to k for me and also q more or less perpendicular to eta and then i see that for the z-mode i can do it because as long as the polarization is out of plane i can play with the rest of the vectors in plane but if i put my eta mode in plane and i must require q and eta to be perpendicular i don't have any way to make my Fermi vector uh transverse to to q and to eta and therefore i am going to get a suppression of the interaction so essentially in the t-mode the in-plane mode i have suppression of my effective coupling in terms of the results this is what happens if i look at the correlation length i find that they get renormalized differently so of course normally in a 2d system you would expect the two modes this t and z-modes to be split but even if we assume the lattice is somehow not important and they start together very close they will still be differently renormalized by the electron so only one of them will typically reach the critical point first and this is usually the z-mode and then you can use some tuning parameter to move away from the critical mode and go to the other one in terms of the boson dynamics you find that this z-mode it basically behaves like an eising ferromagnet so it has overdamped response pretty standard the t-mode because of the suppression of interaction is actually under damped all the way to the critical point it has omega over q squared landau damping this actually means that it's a ballistic mode and it remains coherent even at strong interactions so this i think is very different from what you normally expect and then when you go on your feed this back to the fermions you again find that the z-mode is essentially a conventional non-formy liquid with omega two-thirds behavior but the t-mode because the boson is under damped and because the coupling is suppressed essentially it remains a marginal form of liquid all the way down to the critical point this is true even if i tune my system so that rt is not g but zero the system will still be a marginal form of liquid as long as there's no superconductivity in the system so this is one very easy observable to look for you just have to do the very hard process of creating a two-dimensional peroelectric metal uh but people are working on this nowadays okay what about pairing so before telling you about pairings in this system let me give you yes because if you if you continue this to the real axis you'll get a real term no damping it will give you all it will you you have q squared minus omega squared minus omega squared over q squared so omega is a sharp mode okay so it's a propagating mode very good question and yeah it's kind of hard to see here because it's actually z equal two which is usually diffusive but it's not diffusive it's under damped and okay what about pairing so let me give you a two-minute crash course on unconventional pairing in a spin and charge coupled systems what do you do when you want to calculate pairing so you put some anomalous pairing function right on pairing vertex and then you calculate the correction this is basically summing up the ladder diagram by putting your favorite interaction and all the details are going to show up if you want in the form factor of the interaction now if i think excuse me of typical form factors this a phonon would be a density interaction so the form factor would be one this one's always attractive all types of superconductivity love phonons if i put a current or if you want a gauge field it's been known for a very long time that these are always repulsive to pairing and the reason is very simple it has to go from k to minus k and that adds a minus sign to the overall diagram so these are always repulsive when you put a spin term so say a ferromagnet then you find that it's always repulsive to singlet but it's attractive to some triplet modes you can go into the details okay it has to do with the fact that you take a spin term and you transpose it and you trace over it but the idea is pretty simple obviously ferromagnetism is bad for singlet coupling because it's even splits your modes it's also bad for some triplet coupling because it's even splits your modes but it enhances spin flip fluctuations in certain other modes and therefore it will enhance pairing in certain modes okay so now that's how you find out whether your system is attractive or not so now let's put k cross sigma and what you're going to see right away is first of all k is always repulsive to singlet and sigma is always repulsive to singlet so their product will be minus times minus so it's going to be attractive to singlet okay so the system is attractive to the singlet channel but it's also going to be attractive in the triplet channel it just is going to be attractive in different channels than you expected okay so here's a quick table if I look at this z-mode it's attractive to singlet and it's also attractive to kx sigma y minus ky sigma x so this is some nodeless you know phase changing mode and the t-mode is attractive to singlet and to to a doublet kx ky spin polarized triplet mode that means that in this mode you would see you could see chiral superconductivity nematic superconductivity and when you calculate to see you find that the leading order which means the splitting in between them is very small this is something that York also talked about briefly yesterday so leading order these two terms are degenerate okay so in these materials you really expect to see strong singlet and triplet pairing into dimensions in terms of the pairing temperature so for the z-mode because as I showed you it's a conventional non-fermy liquid you get the usual non-fermy liquid tc which is a polynomial this is the same g-squared that York showed yesterday and for the t-mode because of its weird frustration you find actually enhanced bcs it's one over the square root of the coupling constant and it's actually in a in a very narrow region around the critical point so it will be strong but it will be narrow and you have to probably tune your system pretty carefully to get to it and and then you could get chiral superconductivity yes no this okay very good question this square root I saw that the damson was nodding and that's because in three-dimensional quantum criticality it's well known but this is what happens this is the same form as what happens in 3d quantum criticality the physical mechanism is completely different it has something to do with an enhanced because the boson is under damped its pairing fluctuations are strongly enhanced on the other hand the cooper logarithm is completely gone because of the suppression of coupling and what you get is essentially a log from the bosons this is you know something pretty strange okay it's very non bcs just has a form that looks a little bit like bcs okay so that's almost the entire picture but not quite and uh let me give you a couple of slides about quantum order by disorder this is a problem which was studied extensively in ferromagnets it was actually the original reason why we started looking at this problem because we were wondering whether ferroelectricity has quantum order by disorder the idea is as follows you can look at it here in this picture supposing I have a vector mode that condenses then I know that there's always a goldstone mode for example if it was a ferromagnet and I split my Fermi surfaces I would have soft phase modes that are between the two Fermi surfaces and a system will typically this costs energy fluctuation of energy and the system will like to remove the fluctuation of energy and it can do it either by preempting the second order transition via first order transition or by going into a spiral a finite density wave state okay with some finite momentum and basically what happens is that if you look here if I split my Fermi surfaces with some finite order which I'll call delta then there's a minimum momentum transfer for fluctuations I have to have some minimum momentum transfer to take my chiral Fermi on and move it to the other chirality in my Rashba split surfaces this gives an IR cutoff to fluctuations and basically what it does is it provides this cutoff which generates non-analytic terms in the free energy so you have to calculate all sorts of diagrams like this it's somewhat more involved than was done in ferromagnetism for technical reasons but you can do it and really what you find essentially is that it generates a negative non-analytic cubic term in the free energy which of course tells you the system will prefer to condense into a first order via a first order transition and it also turns out and this is probably a numerical art of you know just has to do with numbers but it's a big number that the z-mode is much much more unstable to the first order transition than than the t-mode so you know after cranking all of this out you can end up with kind of these sort of schematic phase diagrams in terms of the z-mode this is the picture I showed you beforehand you have a second order transition which eventually goes through a bi-critical point into a first order transition and when you go and just compare temperatures for superconductivity and this first order transition you find that that it probably probably rises above the first order transition but and this is a bit of a technical detail so I'll just say it as kind of an advertisement the mode that causes quantum order by disorder is actually the t-mode when you have z condensing it's the t-mode when you have t condensing it's the z-mode this again has to do with this weird transverse coupling and that means if you can control how split the two modes are you can control the extent of this for the first order transition so you can create a switch you can have it first order second order first order second order superconducting non-superconducting and so forth the t-mode looks more or less the same except probably the superconducting phase is buried under the first order transition but you can get rid of the first order transition anyway so it doesn't really matter okay so so far I've showed you some maybe interesting maybe strange properties and now let me tell you why I think this is also important and the reason is because we are doping a ferroelectric and ferroelectrics are very useful for applications point of view the reason is right here this is actually I think a calculation of a phase diagram for strontium titanate as a function of strain and what you see basically is that at zero strain it's essentially a para electric except for very small range which is experimentally incorrect experimentally it's a para electric down to zero temperature and then you play with the strain and it becomes ferroelectric and the reason is symmetry eta is an odd under inversion mode so it can only couple quadratically to strain of some form so it does so strain does not act as a field it acts as a mass enhancement or reduction and this means that you can play with the mass by playing with the strain just think of a simple problem allow for two types of strain volume preserving and probably volume non-preserving so if you want pressure and some d-wave and you'll get the phase diagram which will result from the fact that the z and the t-modes will just shift differently as you play with different types of strain so here's a schematic phase diagram it's quite busy but it basically shows you that through strain you can reach basically all of the phases that I talked about and to make things a little bit simpler I just plotted what uniaxial strain would look like you know it's just aligned through this phase diagram you see that it basically goes to the superconducting the first order and the second order phase transitions and this tells you that with appropriate engineering you can really create a quantum switch from these materials all right I'm going to skip most of 3d because you know I don't think I have much time I have a total of what 15 minutes 10 minutes don't remember when we started okay so let me then very very briefly tell you what happens in 3d the short answer is the same it's just more complicated because you have more dimensions to work with you know if you have some vector k then in transverse the k there are there's a whole plane so you have to you have to span the plane and play with all your vector products this is really technical the end result is more or less what you would expect with one surprise the system is a marginal Fermi liquid both without strain and with strain this is well known from three-dimensional systems three-dimensional systems are always marginal Fermi liquids even at the critical point because of dimensionality if you ask whether the system undergoes quantum order by disorder the answer is yes without strain but then you put strain and you can just get rid of it and make the system second order similar to the two-dimensional system in pairing it turns out in the absence of strain only singlet is attractive and and triplet is actually marginal neither attractive nor repulsive to leading order which means that you know non-universal sub-leading terms are what's going to determine whether it's attractive we expect that it's attractive but much more weakly than the singlet at the critical point but then you put strain and you basically make them almost degenerate again and this is quite surprising quite interesting because I think a lot of recent experiments showing signs of unconventional superconductivity in these materials occur in strain the in strain samples so we haven't done careful study of this but at least its suggestion all right so now I've I think I've spent a lot of time kind of showing some theoretical properties and if you fell asleep halfway through maybe now is a time to wake up because I'll try to show what I think is a beautiful experiment which I do not believe for a very long time but I sort of believe now and so I want to talk about strontium titanate strontium titanate is the poster child of a quantum ferroelectric metal it was realized that it was an unconventional superconductor back in the 60s until today people don't really know why it happens it has a very low density superconductivity you can basically tune the carrier density from 10 to the 17 to 10 to the 20 carriers per centimeter cubed and it was so superconductivity in almost all of these regions there is a good deal of indirect evidence for pairing as a result of the ferroelectric modes but if you go and do a calculation you find that you just do dft and you find out what the coupling of k cross sigma is you find that it's large but the Fermi surface is so small in these materials at low density that it's probably not enough to explain superconductivity but I'm showing you here is two kind of phase diagrams this is a phase diagram is a function of carrier density so in some insulating materials and some dope sorry materials you can have the the ferroelectric mode go down and then superconductivity rises out and there's possibly coexistence maybe more interesting is this three-dimensional phase diagram which shows that you can dope independently say with calcium to go to the critical point for ferroelectricity and then dope say with oxygen vacancies to change your carrier density and really control both the distance from the critical point and superconductivity in this system. What the group of Martin Graven did this was the her his grad student Sarjna Hamid and really pioneered by his postdoc Daniel Pech who has just opened his own group back in Croatia was to take the strontium titanate and squeeze it to death and I really mean to death they deformed it up to 10 the strontium titanate has beautiful elastic properties you can deform it up to 10 without it without it cracking and they took it and they squashed it between one and six percent they plastically deformed it something completely irreversible and of course this should totally kill superconductivity right huge disorder dislocations in homogeneities and then of course TC jumps by a factor of two so this is the pristine phase diagram dome superconducting dome of STO and here are the stars that they get by deforming various percentages so they manage to go up to a factor of two and the superconductivity is extremely inhomogeneous basically it really depends on whether you measure perpendicular or parallel to the direction of to the plane of the strain this is actually this this can be well understood classified it's not surprising at all and but something very strange is happening in this system and not only that but if you're going you measure resistivity you find the resistivity again is anisotropic and begins to drop quite strongly already at tens of Kelvin and it drops by orders of magnitude which is at least a hint that you might have superconducting fluctuations or small fluctuating small superconducting regions already at tens of Kelvin which is you know just insane TC here is a fraction of a Kelvin and when you do a whole measurements you find that the whole density is just completely flat so this is not a carrier density effect strangely enough the whole density just doesn't change what does happen if you look at neutrons you find this kind of very lovely figure where you see that all the Bragg peaks have kind of been smeared these are called asterisms and they are typically understood to be a sign of dislocations in the system what's happening is that your system has deformed into domains which are slightly tilted to one another and this kind of causes the Bragg piece to smear out into lines here is a Fourier transform of the Bragg diffraction pattern and here is just a numerical simulation of just by putting out dislocation wall that creates a series of these domains and you see that there's really a very good agreement in terms of the typical scales and the overall behavior so we believe that that's exactly what's happening in the system if you want a toy picture you have your system and it's sort of deforming into these series of tilted domains to one another which creates a series of dislocation walls running through the sim this the system these are burgers vectors for people who are familiar with elastic models but this is important because as I showed you before strain has a really strong impact on ferroelectric materials and basically what happens is that if you put a dislocation there's very strong strain near the dislocation and the case algebraically away and this tells you this is actually a calculation of the effect of a single dislocation on the ferroelectric mass as a function of distance and what you see here is that very very near the dislocation there's a huge change of the mass and the system must have ferroelectric order there and this will be very strong you will probably compete with superconductivity but the strain decays algebraically so very far away there's no strain and nothing interesting happens which tells you that there must be a region where the mass has to go the inverse mass must go to zero it's essentially a topological requirement so there's going to be a circle around the dislocation where fluctuations will be critically enhanced and where you expect superconductivity to be very strong if you add a wall this is london lift sheets problem you just get screening so instead of long-range strain you get exponentially decaying strain with two kinds of sheets running parallel to the dislocation wall where superconductivity should be strongly enhanced so we did this and I didn't believe it even though I wrote the theory and we sent it into nature materials and surprise surprise they didn't believe us either and I was ready to give up but the experimenter said no we are going to prove that this is what's happening in the system and they sent it for Raman to do Raman scattering why Raman scattering because Raman scattering is only coupling to polar modes in the presence of broken inverse and symmetry they wanted to prove that the dislocations were in fact inducing local ferroelectricity which is so obvious that you don't need to prove it but the referees didn't believe and so they did it and this is what you get this is an insulating squashed sample and what you see here is an optical to mode which only exists in the deformed sample so yay definitely broken inversion symmetry in the system but when I saw this I got super excited because what you see here is a much bigger feature some very low energy huge spectral weight and for anybody who works in unconventional superconductivity this is a real trigger because this was shown in an iron based superconductors to be exactly what appears in Raman when you approach a critical point when you approach an ematic critical point you get this huge enhancement of spectral fluctuations which becomes stronger and softer and this is also true in the dope systems you see the same huge enhancement of spectral weight it looks a little bit different because there's some subtraction of background and the mode becomes soft it perfectly obeys QA wise behavior or very well obeys QA wise behavior so actually what happened is that kind of by chance I think for the first time these guys image directly the inelastic quantum critical fluctuations of the polar mode if you have order and you have quantum critical fluctuations then by the topological argument that you find you must have regions of enhanced superconductivity I am basically out of time right or five minutes I speak too fast okay and so so when I saw this let me say when I saw this I finally was going to believe that maybe maybe something is happening it's important in terms of what I was saying beforehand because it tells you that you can take a material you can dope it into a metallic state and it does not kill ferroelectricity and it also does not kill the ferroelectric fluctuations and that means that just find any coupling I don't care which take my k cross sigma take some other coupling I don't care what you are going to get strong electron phonon coupling in the system strongly enhanced by critical fluctuations and so everything I said will at least qualitatively be correct all right in one minute let me tell you something about the dirac material so you can do the same game for the dirac material and take a dirac mode cone and couple it to basically this k cross sigma now you have to use gamma matrices but what happens is that upon creating order your cone splits into two vile vile cones and then the question is you know what happens at this kind of transition this can occur for example in let tell a ride possibly in molydyte tell a ride which is a choose me a type 2 vile material and so again you try to calculate various things and when you calculate the bosonic response you find something which is quite surprising already at one loop order this is definitely something that I do not know happening in typical Fermi liquid system usually does not happen already at one loop order you find a strong negative logarithmic correction which means the system is always unstable to creating a finite momentum a ferroelectric density wave and then this already happens like I said at one loop it only happens for ferroelectric coupling and it only happens in three dimensions in two dimensions the sign is wrong if I remember correct and what this means is that you have some you know some renormalized critical point to a finite ferroelectric density wave order and then you go and you calculate pairing and you find something rather strange normally when you have a electron hole order say like ferroelectricity and superconductivity they fight they're always competing because they're kind of fighting for phase space on the Fermi surface but it turns out that when you calculate for this specific case what happens to the bosonic fdw propagator as you increase delta it becomes softer the system wants delta superconductivity helps the system become more and more ferroelectric and the reason basically is because it's at the charge of trality point by creating ferroelectric order you're kind of creating a larger phase space for pairing fluctuations and the converse is also true the fdw instability drives pairing so they like one another and they drive one another together and when you go and you calculate for example the free energy of the system you find that the free energy is basically unstable to both ferroelectricity and superconductivity which tells you that the system should somehow undergo first order phase transition into a coexisting ferroelectric and superconducting state this is all occurring precisely at the charge and trality point okay this is kind of a phase diagram showing a hysteresis loop that you would get you would start say at low temperature at some point you would hit superconductivity and and ferroelectricity then you could go out and you wouldn't lose ferroelectricity and superconductivity until you reach some other point in the system and here are some open questions that I think are quite interesting for example what happens to this instability upon doping like I told you this one loop thing is very strange okay and it definitely goes away upon dumping and what about electric screw me on here I have some spin charge material clearly I should be able to create electric vortices this is the problem we're working on the question of transporting these materials is a huge open question that Mitya was talking about a little bit yesterday and finally I think another important question is what exactly is the structure near dislocations but there'll be magnetism there you know what is the spin texture and so on and so forth so let me slash my summary slide and I think that will end here all right so there is one question in the chat from Andrei Chubukov does this permanent self-energy I guess this relates to the part about the two-dimensional criticality does this further self-energy going as omega log omega appear only when the mass of T boson is tuned to zero so this is the first question no no it appears it appears ah so there's a crossover it's purely omega log omega the critical point and then if you put some finite distance then there'll be a crossover scale at the lowest frequencies it's good for me like whether the higher frequencies omega log omega no it's if it has spin orbit then it's it maintains inversion symmetry so actually to get k cross sigma you need to have microscopically probably some spin orbit coupling but you always have two degenerate states so called sigma just two orbitals and you get the same physics was that clear so you need to have two orbitals you can call them spin you can call them something else for example in strontium titanate there is spin orbit coupling and the lowest two bands are j three-half and they happen to be mz minus three-half and plus three-half so it's basically a spin degree of freedom time reversal is maintained you hear explicitly maintained that's right that's right so you that's why the spin should be thought of as some effective spin but that's correct it just so happens that in the most interesting material it really is and then there is another question from andrei which is on superconductivity you said that cooper log is suppressed how do you get this exponential of minus one to the square root g normally this holds when you get both a cooper log and an extra log from a boson which results in a g log squared in the cooper channel yeah so what you get is something completely different you get that the boson goes as one over square root of something that gives the log so it's one over square root of g that's from lambda damping and then it's one over omega which gives you a log so it's nothing to do with the log squared in 3d it's really just from the boson yeah it's it's really non-trivial and it took me a while to believe it maybe i don't believe it now but at least i'm claiming i do about the electric skirmills what is the order parameter again again about the skirmills could you explain about the skirmills what yeah so we haven't done what is the order parameter yeah this eta so this eta would go around in circles the k cross sigma the rush bus spin orbit would kind of do something like this so something which would be electric there is a rotational invariance for eta right yes yes so you break it and then you play and what is the aminianality of your oh well i didn't do it so i can't say i have i have the critical theory for both two and three systems i in dirac yes it's 3d in iraq it's 3d and 2d it's not all this does more or less goes away so these are real skirmills not not not not baby skirmills right i guess so i can tell you what we have people looking at you just take that we were looking at hexagonal lattices and the hope that in the presence of a lattice symmetry you break down the rotational symmetry just to a discrete symmetry and then you can look for triple q states and then then you have a skirmian without going into all this the other stuff in the case of the pneumatic system this is said truth that some of the beautiful critical physics that we love is being suppressed when you include acoustic phonons correct what's the situation for your case because you also have a structural degree of freedom if you were to include acoustic phonons would it give you a non-analytic long-range interaction of your balls onto which then would spoil all the beauty so i don't think it would spoil the beauty in two in two ways first of all to some extent i did include the truth stick phonons i just didn't include the fluctuations because i put a strength tensor explicitly in my system i could add q dependence to it i said i could now add q and omega dependence and i know it would happen i would get larkin picking instabilities but these are typically very low temperature so if my electronic scale is high enough i can hope that they're not there i don't know of any measurement of larkin picking instabilities in these materials i definitely agree that if your system has a larkin picking instability then you know i underwent the first order purely bosonic transition so what are you even talking about sir when there is a impurity in the system like when in the polar only case um the impurity in the system creates an electric field inside in case of puller and so what will happen in this case any comment on that yeah so the short answer is i don't know um but uh the the point is you have conduction electrons so presumably on a Thomas Fermi screening scale that should be screened out um beyond that we really don't know the answer it's a definitely a good question including the questions i think even more interesting is what happens when you have two of them do they create some kind of FFLO like interaction we don't know that yeah the first question probably you've answered but how do you experimentally distinguish between your z-modes and the t-modes and the second one i did not say it's very hard the real problem in this system is that measuring anything is very hard because it's a ferroelectric that's being screened okay and actually one of the motivations here is that i would say that one of the only ways that you can really accurately measure things is by looking at collective modes because those will just show up in spectral space it turns out that because one is overdamped the one is under damped they have very different collective modes i didn't go into the details um and andre and i have another paper on a different system which basically kind of shows what are these possible modes and and these are probably the best ways the another way is to look at optical conductivity that's been calculated by a premi chandra's group by abhishek kumar and others and finally one fantastic way to do it by mistake is to squeeze the material very hard because then you locally break polar symmetry while leaving the bulk inversion symmetric and that's how you can measure things directly as i showed you second question is that in the 3d case does the dynamical prosthesis still exist no okay less so this very strong dynamical prosthesis frustration is a 2d thing yeah it's there in 3d but it's a lot less clear to see where it comes up and this is why 90 percent of the talk wasn't wasn't 2d thank you