 All right, so welcome back everyone to the after lunch session, which We're gonna since Pierce is now here We're gonna start with Alexei Tsevelek from Burkeven National Laboratory. I'm just gonna talk about miracles of Conor Heismarck's, please Well the the first miracle is this Probably the most important one and I'm So important that I almost forgot to thank the organizers for inviting me to this Excellent event and excellent place. I have not been here for four years, which is really almost a wasted time of my life So I will talk more about Pierce after five I will just Tell you that it's almost We have to Celebrate me and Pierce. We have to celebrate 30 years since we met We met in October 1998 Dbilisi 88 88 And spent a lot of good time here in Ateliesta so that You you you who can see here Antoine George, Andre Ruckenstein, Baskaran Even myself and We We have been meeting in the United States as well. This is Santa Barbara and I think this is an iconic Image of Pierce I I Immortalized this image putting it on the key short of Santa Barbara conference and For for me, it has all you'll always be this inquiring mind All right Now to business well, I will be Discussing One of the problems we started with spears long time ago and actually I think About 22 years ago. It's a problem of odd frequency Pairing and we he had this idea back then that Order order parameter, which is basically psi dot psi this kind of odd frequency Superconductor not a parameter can be formed in in in condolences where Localized spins core exists this Conduction electrons and we had a model but as we found out Very quickly odd frequency pairing was essentially a non-peturbative for phenomenon and I have never been so satisfied with How we treated this model and after many many years I found a version of the condor On the latest model where you can do everything rigorously They were I would claim that all steps Steps taken in my day derivation are under control And including non most importantly non-peturbative ones and Traditional view on the condor Lettices this is a picture taken from a review article by by I pierced and Andrew Nivedomsky that there is a competition be between condo screening and Heisenberg exchange interaction or induced archity why interaction and Depending on the degree of And what actually happens that if this interaction between the spins Increases it will this or that way the couple from the Conduction electrons so here or Condo screening takes over and and and and the Local moments are kind of sucked into a into the into the conduction band but on that side the Moments is essentially the the a couple from the electrons and leave their own life. They either order or It will be a spin liquid, but in any in any case. It's a competition Be be be it in the tool, but the scenario I am going to be be be discuss is about cooperation be be between between conduction electrons and local moments and this co operation will produce a special type of spin liquid which I Which is not possible to Produce without without conduction electrons and this state will The state which is formed supports odd frequency pairing and odd frequency charge density wave and there are many Remarkable features on top of this which I will hope I will have time to do discuss so the Core of this model the this model. I will be discussing is Is not one-dimensional, but it is it is made of Kind of one-dimensional from an array of coupled one-dimensional wires and this feature allows me to treat the most The the strongest interactions Non-peturbative so It is the same kind of construction people use now in Fractional quantum Hall effect the so-called wire construction so that since Now I will try to imitate pierce and play a Pedagogical role so so he he has always been insistent on being pedagogical and I will explain what is going on using the cartoons so the essential point how to How to create this unusual spin liquid we take Let's take One-dimensional electric gas Calculate to the spin on half Heisenberg chain. There is an exchange interaction between the be between the local moments and there isn't Condor anti-pharmagnetic contact interaction between the electrons and spins so Imagine now that we have many Such Any such letters, but they don't Interact with each other what will we have you have a two-dimensional brilliant zone and two Fermi surfaces there will be a fair me surface of Electrons and There will be a fair me surface of spin for 1d Spin one-half Heisenberg model this Fermi surface of spin is an absolutely rigorous thing the spin Excitations carry spin one-half and they are gapless So you may think about it as a Fermi surface so if the electronic Fermi surface is at different Position in the momentum space from the spin one There will be There may be only this kind of hybridization processes so This the the Spinons from here will strongly couple to Electronic degrees of freedom from here and vice versa. So this system will effectively be splitted into two independent parts so the Spectral gaps will open between this and that and this and that now What if you add Tunneling be be be be between the electronic chains. Well, the electron Fermi surface will be buckled and the nesting will not be perfect and We will get a Pockets as I particle pockets around this points only part of the Fermi surface will be Gap so this this kind of picture that's Essentially the basic basic mechanisms Which work here? But of course, there are many interesting d d details Which are related to the specifics of strong interactions in low in low the The dimensions first of all I will show you that This spin liquid pair states contains seats of all frequency pairing and As I as I told you about this Coa has a part cause a particle pockets If you have Sufficiently strong interchange tunneling the state will support a metallic state Where the Fermi surface volume is not related to the electron density? As a matter of fact in this particular model this Fermi surface volume will be zero They they they will be a complete complete compensation between Particle whole symmetry so and There is an interesting Correlation with the ideas about this small Fermi surface states with the ideas of seeing till such diff and water they They claim that if such states existing More than one dimension there must be fractionalized Excitations and yes that there are fractionalized excitations here now Let me go to the From cartoons to the technical D details so What do we do with this condom high condom high zebrick ladder? Well actually the What I have done was based on my Relatively old work with our own Zahar but then we missed some important D details in this walk so Eric as I said electron bandits in is in commensurate with the Latest and in this case What happens that they cannot be back scattering between spin excitations of the chain and Conduction electrons only forward scattering Is possible and after I solve this more model, I will go I will go further and look at the Array of the array of such letters so Essentially you you you have to imagine that there are One one dimensional every three wires What what on layer what the? as a layer is Coupled spin chains and they're coupled by so the one Dimension goes on Pindico to the to to the plane and they will be interchange tunneling and interchange exchange that's Will be the ultimate ultimate model. It's a little bit a little bit similar to striped Striped superconductor, but here Electron chains and spin chains are in the same plane, but here it will be two layers So let's go back to the core model. So what What we do we? Linearize the Spectrum near right and left Fermi points and use the but as a new station the but the new station procedure so I Found it more convenient to use non-abillion but as a new station the free the Hamiltonian of the free one-dimensional electron gas can be expressed in terms of this current Operators which are written here this this current operators Satisfy the cold so-called cuts moody algebra and it looks much more complicated than the Free fermions, but the advantage is that Increasing the complexity of on this initial state we win in win great see see Implicity when interactions are included. That's how this organization always works now it is Interesting thing that the spin one have Heisenberg model Can be D described in the same language Namely the Wow spin as you probably know spin one have Heisenberg model has this linear linear spectrum excitations right moving and late moving waves which are essentially and the main walls between be between the sea states with different orientation of the way of the nail order and This the dynamics of of this excitations can be Expressed in the same way as that the dynamics of the spin Subsector of a free electron gas. So essentially if you take Electron free electron gas Suppress the charge degrees of freedom. You will get essentially spin one half Heisenberg chain, but they this is very well So the but but as anization therefore the position for the for the Heisenberg model Is also in terms of the left and right currents and spins are expressed while in this way there is a smooth part of the spin operate and the staggered part and now for the Electrons there is also a smooth part of the modulation, but the Oscillating part oscillates with a different with a different wave factor and therefore when we Look at the conda exchange these terms drop out only the conda exchange in the lower energies becomes just a product of smooth parts of the modulation and that's exactly what what What corresponds to this? cartoon I Showed you B if all so The right moving guys Interact the right moving guys interact with the left moving guys and vice versa Only this interaction is marginally relevant and it is this so Essentially the spin sector splits in two parts was a Elect right moving electron spin interact with the left moving Heisenberg spin and vice versa and Then we Call Natan Andre to help us to solve this model we look at his paper circle 1980 and Here we are that's the Spectrum So there are two hybridized gapped Spins, so there are essentially two spin liquids fractionalize spin liquids both are made of electrons and spins and therefore in such Intuition if Every order per parameter field must include both electrons and spins And The odd the order parameter of the condor Heisenberg ladder looks like this and We see take Now one important point of course it's a So far in in wonder dimension. It's not a real order it is a Slowly slowly fluctuating field which which correlation functions Decay as power law in that sense. This is an this is an order parameter This guys Will order once I will bring this change to to to together and arrange the in Interaction, but the seeds of this order are Oh already in a already in a one-dimensional model now look at it This is a this is a superconducting pairing. It's a pair is Attached to this object made of Made of spins he essentially it's a bound state of the Electronic pair to to to the staggered magnetization of the hazard blockchain and staggered energy density what why it is What frequency pairing be be because you who can use the equation of motion for for the Electron of operator and psi psi dot will be exactly the product of spin to psi the times times the exchange exchange constant and the and the same with the odd frequency Now there is a very important aspect of this here that both this order to stagger it and the charge density wave includes oscillates with the wave vector which includes both conduction electrons and and and local ones. This is the so-called a shikawa theorem that in in in one dimension Fridl oscillations must include the wave vector of of of localized and delocalized electrons and and that's a Exactly what what happens here. So this two order parameters can be included into the single matrix form in this field is an S2 S2 S2 two matrix Essentially the S2 symmetry will be We will be broken when you include interactions between the electrons when if the interactions are small There is an approximate S2 symmetry. This slide just demonstrates how exactly you arrange this Amplitudes of the order parameter probably to be technical. Now there is a funny aspect of all this that since the order parameters include both spins and Electrons this order is robust against static disorder be because the static D D disorder present at Electron chains will not couple to the order parameter or or Likewise, if it is present only on the Heisenberg chain, it will not couple Again because these guys you see it they they include both electrons and spins so they Now Again, this is because chiral parts of spin operators pair these parts with opposite chirality from one dimension Therefore the perturbations cannot acquire a vacuum average and that's That's to leave the ground state So now Beyond 1d so well what you do now we Allow electrons to To tunnel between the chains and we also allow Pins to To have some kind of exchange interaction between the chain. So this this is the interchain tunneling and interchain exchange the most important part would be the coupling of the stagnant organizations and this kind of a Exchange will be always Generated as soon as the There is interchain tunneling in directive and Then There are two possibilities if this interchain coupling is Sufficiently weak smaller than the spin gap then if essentially There are no quagga they particles in the system. They're only The only fields which fluctuated low energies are Collective fields of of this of this level and then in this case we will Have some superconducting a charge density wave order at sufficiently small temperatures and above it will be a boss metal where the electron pairs or charge density wave Accentations will will tunnel between the chains and Nothing else this Okay Okay, so this book This is essentially the effective g g g his book Landau action for this object and In the in the Continuum limit the effective action is similar to helium 3 a There is superconducting phase there is charge density wave phase and there is something which Oscillates be be it wins the tool the Magnetic field where once it's a process One it exceeds hc1 will produce What is this and in their core there will be charge that density wave just one One thing about What happens if the interchain tunneling it exceeds the spin gap What happens is the following you you as as I said there will be quasi particle pockets and this can be Established very well if we look at the Non-petrobatics expression for the greens function of condor Heisenberg chain and this greens function Well, it's essentially Convoluted semen Which you you will we're talking About you see this square root so that and but once you Allow interchain tunneling the RPA greens function will produce bound states of spinons and holons which are quasi particles and when the Tulling matrix element exceeds This threshold you will get for me pockets Okay, and I think I Finish so Oh Conclusions and one may have a metallic state where for the surface volume is not related to the electron density that's essentially the state above the above the phase transition temperature and This I did not have time to Explain why why the state is not topologically topologically. No on three