 very much. Can you see my screen? Yes. Thank you very much for the introduction and thank you to the organizer for organizing this conference and for allowing me to participate. So today I'm going to talk about Interaction-Resistant Metals in Multi-Component Fermi Systems, which is a work that I've done in collaboration with Matteo Fergaretto and Massimo Capone, which are from the condensed matter section of CISA. So at first I would like to give you a take-home message so in case you get bored doing the rest of my talk you already have something to take home. So if you have a model parameter lambda that you can tune and vary and in such a way that you jump from one insulator on the left, insulator A to another insulator, insulator B, then it's possible to have a metal in between and it's the quasi-degeneracy of two different insulators that can result into the formation of a metal. So this gives the outline of what I will talk about. In the first part of my talk the insulator A will be a multi-insulator and the insulator B will be an un-insulator and the model will be an Abarth-Kanamori Hamiltonian, while in the second part of my talk we will see the competition between a multi-insulator, a different kind of multi-insulator and a bend insulator. So this is the outline and let's begin with the first part. So I know that for sure you all know but just for the sake of clarity I would like to remind what is a single-band Abarth model. This is the Hamiltonian of a single-band Abarth model and you see that there is a kinetic term that favors hopping of particles between neighboring sites and an interaction terms which penalizes double or triple occupations. And the well-known phase transition that occurs in this model is the metal mode transition. That means that if you are at all-filling meaning that the number of fermions is exactly equal to the number of lattice sites then at low values of u over t the kinetic term prevails and elections are delocalized in the lattice while at high value of u over t elections get localized and the system is insulating in spite of the fact that traditional venturi would predict metallic behavior. But now we have a three-band Abarth model and you see that the Hamiltonian is considerably more complex and on top of the site index j and spin index sigma you have also an orbital index a which runs from one to three which are the bends in our system. And the interaction Hamiltonian is the one illustrated here and apart from the traditional Coulomb repulsion u there is also the Unz exchange coupling j which is responsible for the formation of Unz rules that I will show in a while. So this is the schematic picture of the three bands which can be imagined and has three parallel channels and now let's turn to the interaction part. The interaction part can be rewritten here as shown here on the right in terms of some quantum numbers nj which is the number of fermions at site j the local spin s and the local angular momentum l so s is proportional to is built in terms of Pauli matrices and lj is built in terms of the generators of the the symmetry group o3. So now we see the effect of this local interaction Hamiltonian on the arrangement of electrons in each side. So here we have three boxes which are the three orbitals of site j and we want to fill them with two electrons and there are a lot of ways to do it and for example I show two of them this is one way of doing it this is another way of doing it but if you do the math you will discover that there is an energy gap between them and the one on the bottom is favored and the energy gap is exactly proportional to to j which is the Unz and j exchange coupling. Another example in the case of three elections per site so we want to put three elections in these boxes is the one shown here so this is a possible configuration this is another possible configuration and we can see that there is an energy splitting between them so one is favored the other is dispervered so in summary we can say that the the und exchange coupling j favors local configuration having high spin and high orbital angular momentum and these are exactly the Unz rules that we studied in chemistry so when you fill the generate orbitals with some electrons they must be placed with with parallel spin and occupy the maximum possible number of orbitals. So what about the mode transition in this system? The mode transition is not as simple as before there is a whole let's say zoology of of situation that I try to summarize with these pictures so the critical value of u over t at which the system undergoes the meta interest insulator transition depends on the Unz coupling and depends also on the filling so the number of elections per site so these are three examples so a filling equal to one equal to two and equal to three and z is the quasi-particle spectral weight which is the degree of metallicity so z equal to one means perfect metal z equal to zero means insulator. So in the case of one electron per site j favors the metallic state you can see that increasing j the critical value of u at which you have a mode transition is pushed on the right so to very high values of u over t. Conversely if you have half filling so three elections per site then j favors an insulating state because the critical u is pushed to very low value of u over t so it's pushed to the left. In between so when the filling is two that will be our case the filling will have the j will play a Janus face role that means that on one hand j will push the critical value of u to the right so to high values on the other end j will spoil the metallicity at small values of u over t so this is called the Janus effect and in the in the next of my talk let's say in the in the following part I will want to clarify the mechanism behind the existence of metallic solution in the presence of large values of u and j so in spite of the fact that you and j tend to constrain electron mobility I will I will show you that instead you will have a nice metal and to do it I will consider our model Hamiltonian on the most on the simplest possible system non trivial circuit that you can build that means a three-site system with periodic boundary condition you can look at it as the simplest circuit that you can realize so let's focus on the atomic limit so t equal to zero for the moment and if over j if j over u is small then the system is in a multi-installating state that means exactly two electrons per site and that is the energy and even if you switch on the tunneling a little tunneling then open processes are suppressed not to pay the cost of triply occupied steps of triply occupied sites so you don't want to create triply occupations because u is very large so now we consider another limit j over u is very large in this case the system is in a uninsulating state and this is a disproportionate insulator that means that different sites post a different number of electrons this is the the energy of this configuration and when you switch on the tunneling open processes are suppressed not to pay the cost of locally violating own schools so not to lose the high spin and the high angular momentum of these configurations and if you plot the energy of these two atomic limits so the blue line and the red line and you compare it with the exact numerical with the energy of computed by means of the exact numerical diagonalization of the system then you see that you have a degeneracy at j over u equal to one-third so in at this value j over u equal to one-third the two insulators are degenerate and that's this is this is the point which which host interaction resilient metallicity so to provide you with another point of view here i show you the the local population the local population so you you can imagine to project your many body weight function say zero over the local configuration having n number of variance with the spin s and you measure the the amplitude the the probability of of finding such a configuration so you can see on the left that for sure you will find the two electrons per lattice size because we are in a multi-insulating state while on the on the right you have a one-third population in for n equal to zero and two-thirds of population for the purple line that means n equal to three and this is the hoontz insulator so the disproportionate insulator that i was mentioning before and in the what happens in in between so at j over u is equal to one-third is that various atomic multiplets are populated and this is another key to to observe the emergence of hoontz metallicity yes if you compute the if you if you look at this as a probability distribution because they are practically normalized to one you can you can see that the maximum of this the maximum entropy of this probability distribution is at j over u equal to one-third and this is exactly the point where hoontz metallicity arises so now i show you how to probe the conduction properties of our unknown system so classically if you want to to measure the resistance of something you apply a voltage and you measure the current so in my case i'm doing something very similar conceptually so i have my my small circuit the trimer with periodic boundary condition and i have a wave function the the blue download and i want to know if it is conductive or or insulating so what i do is simply to rotate the circuit around its axis and to measure the current which may flow or may not flow depending if it is metallic or insulating so to be more formal i switch on a power space proportional to e proportional to phi and i define the current the associated current operator which is simply the derivative of the Hamiltonian with respect to the applied flux and and i measure the expectation value of this operator in the in the ground state of the system so this is the result that that i find and on the horizontal axis there is the interaction u and on the vertical axis there is the hoontz and exchange coupling and this is the current you know this current is proportional to the to the degree of metallicity of the system that is also known as through the weight or the singular part of the dc electrical conductivity or to the quasi particle spectral weight z so different communities call these objects in different ways you can see that there are two big blue plateaus two big blue c's this one is the mod insulator high u small values of j and this other is the units insulator so high value of j but in between you see the current along this red line j over u is equal to one third is that is where you have persistent metallicity another way to look up this phenomena is through the the analysis of the many body spectrum of the system so i take the Hamiltonian and i find the first four thousand energy levels and you can see that you can recognize some bundles some branches of energy energy levels and now i show you in detail something so let's focus on the on the left part small values of j over u here it's the bundle of states corresponding to this configuration if you then the first excited band is the one where you start to violating hoontz rules so you no longer have a high spin and high orbital angular momentum but you start flipping locally the spins and then again the the following bundle is a is a bundle where you start flipping another local configuration and then only later only at more only at higher energies you find the the occurrence of charge excitation so here you are start breaking your insulator at high energies conversely on the on the other side for high values of j over u you have the first band is the one representing the hoontz insulator the first excited band is the one who already lost charge excitation so you are breaking your insulator and you are forming a metal and interestingly you can compute the energy gap of charge excitation on one side and on the other side and you can see that the charge gap collapses on both sides exactly at j over u equal to one third so this is a very nice and simple way to to look at the breaking down of the insulators and the emergence of metals and these properties of the many body spectrum is of course mirrored by the by the finite temperature properties so here we have the the specific heat of the system so first of all i compute the thermal expectation value of the energy and then i compute the specific heat and you know that the if you fix uh if you if you fix j over u you have a function so the specific heat is a function of temperature and this function as peaks exactly when a class of excitation unfreezes so it gets active so this is our thermometer let's say in logarithmic scale and we are working along this red line so at fixed j over u and we see that okay as very small the values of temperature then you have practically the ground state then you find the first peak in the specific heat and this corresponds to a loss of inter-site correlation then if you further increase temperature you start violating oon's rules so flipping of local spins and then only at high temperature you see the onset of charge excitations um so now let's comment something about charge correlators which is something that meanfield or meanfield methods techniques cannot capture so i want to to find the charge correlation between site e and site j and of course i can distinguish between the inter orbital and inter orbital contribution which are defined here in the bottom of this slide and what we observe from from the plot is that the total charge fluctuation are of course zero in the mod-insulating state but the total charge fluctuation increase with j over u but they are not maximal in the oon's metal region they are maximal in the oon's they are maximal in the oon's insulator region because of its disproportionate nature so we find a counter-intuitive phenomenon for which a charge excitation are stronger in an insulator than in a metal and this is a another another plot which is the plot about first neighbor charge correlations and you can see that again the total charge correlation are zero in the mod-insulating state they increase in absolute value increasing j over u and again they are stronger in in the oon's insulator phase rather than in the oon's metal which is something a little bit counter-intuitive okay now let's move to a second part which is more based on ultrafold atoms and and i will talk about the an su3 bar model with patterned potential so do you know what is this any idea about it yeah angelo adriano school barnacles a little hint this is the position on on the periodic table this is transium this is a chemical element which is very useful for what i will describe so it's an alkaline hurt metal and we like it because the electron in this in this element the electronic orbitals degrees of freedom are and the nuclear spin are decoupled so these these strontium atoms can be prepared in such a way to to to obey let's say the symmetry su n where n is 2i plus 1 and i is the nuclear spin of strontium which i'll show you in a while so in the in the case of strontium i is equal to nine nine albs and so this gives us with the 10 possible different flavors that you can see depicted here so different colors of fermions that can be obtained and of course you can work just with a subset of them but in principle you are offered with the 10 of them and so this is the amiltonian that is offered by this system and what what does it mean that the amiltonian is su n symmetric it means that all parameters in the amiltonian so the hopping t the upward repulsion u and the and the impossible presence of confining potential mu are independent on the flavor index so they just they do not depend on the color of this fermion so as i was saying the strontium offers with the 10 possible flavors but it's not the only element that can be used to to build su n symmetric systems so interbune offers six flavors and lithium offers three flavors and these are some references about experiments when where these things are done in practice so now i will focus on a on a system on an su3 system which has a super lattice so a patterned potential which makes some lattice deeper and some lattice shallower so in the case of mu j mu j equal to zero so not no super lattice i have just a regular lattice and in the limit of strong interaction so you u over t much much bigger than one and filling n equal to two two two and a multi-insulator is stabilized so you have a two fermions per lattice side very simple then we focus on another limit the limit in which you have a strong super lattice so there are a shallower shallower sides and deeper sides and the pattern is minus mu zero minus mu minus mu zero minus mu this is the building block of the of the super lattice and you can see that the fermions gather and and are packed in uh in the deeper sides so the the one having lowest energy and in this case you practically have a a bend insulator because you are you are you are filled whatever is possible and so this is a like let's say a trivial insulator so in this case in the first case you have no super lattice and the particle openings are suppressed due to due to you and so you have a multi-insulator in this other case you have a strong super lattice and particle openings are suppressed due to due to mu in the in between you have a an intermediate phase which is uh which uh emerges in the presence of a mild super lattice and particle openings are possible only between pairs of deep sides so it means that shallower sides always host one fermion and you have a residual kinetic energy between pairs of neighboring deep sides but what is interesting is what is find what is found in between so along these red lines you have inter interaction resilient metals so when the first configuration is is the generate with the second or the second is the generate with the third then let's say everything is allowed and you have the presence of an interaction resilient metal um and you see that uh if you compute the energy of this configuration they turn the generate at mu over u equal to one and at mu over u equal to two so again to probe the conduction properties of this uh of this system let's say I use the same technique that I used before I rotate the the circuit around its axis and they measure the current and this is the result so along x you have along the horizontal axis you have u and along the vertical axis you have the depth of the of the super lattice and you you can recognize uh three plateaus here on the bottom right you have the motor insulator on the upper left corner you have a band insulator here in the center you have this strange intermediate phase where kinetic energy is demarised so it is localized between pairs of of deep sides but along the two red lines mu over u equal to one and mu over u equal to two then here you have interaction resilient metallici again this is this can be understood from looking at the evolution of local configuration with weight uh so computing the the weight the the probability of local configuration having nj number of fermions and what you find is that here at mu over u equal to one is two configuration of the generate and there are sites with one two and three fermions which are connected by open processes similarly on the other on the other transition lines so mu over two equal to two the generator and so you are offered with all possible configuration so both zero one two and three are uh all active at the same time and this is what offers the possibility of conduction so with this i come to to the conclusion let's say of my of my talk in the first part i we have so we have i try to explain that a metal arises from the competition between a moth and an insulator and this competition uh so these two terms sorry these two terms which both constrain electron mobility results in a metal in general of course if you want to probe the conduction property of the system you you can build a minimal circuit and rotate it and measure the current also known as through the weight while in the second part of my talk i tried to emulate this kind of physics by means of a platform of ultra cold atoms with a suit symmetry and the super lattice and by creating deeper sides and and uh shallower sides i've created the possibility to uh i i let these two let's say opposite tendency compete and the results is uh an interaction resilient metals as well so with this i i hope i finished and thank you for your attention thanks andria for this interesting talk i think that we have enough time for some questions comments i have a question i'm sure you explained it but in the very first slide you said it's uh there i mean in the take home message said that it's because of the quasi degeneracy between the two kinds of insulators and yes somehow i don't understand this quasi degeneracy like well in the in the if you switch on the tunneling uh you don't have a an atomic limit anymore so you cannot really talk about atomic insulators and you so i show you this slide that maybe helps just a second so here you see if you look at the blue line and the red line these are the energies of the atomic insulators and they are really degenerate at j over u equal to one third but if you consider the yellow line that is the exonomerical diagonalization of the system in the presence of hopping then of course you you you don't have a degeneracy you have a degeneracy of what was the original of you have a degeneracy of the corresponding atomic limit i hope this answers no no i i get it thanks i think there is another question from pierre yes so i suppose it's somehow related to this point so what bothers me a bit is that it feels like what you call um interaction resilient metal yes is the phase transition between the mod insulator and the other insulator or more generally two insulators exactly but but when when we call something a metal we we think of a phase and this is more fine-tuning than a phase yeah exactly it's it's the border between two phases but in the presence of a finite hopping in the presence of a finite hopping you don't have a line you have a stripe so because because of the presence of a of a tunneling here the the interaction resilient metal is not on a line on the red line is on a stripe centered uh about this line whose width is proportional to the tunneling but but so i don't understand like the the mod insulator and uh the other one was the hubbird insulator who insulator yes yeah the are there uh when including the the hopping are there the same phase or are they separated by a phase transition they are of course separated by a phase transition but so so it means that uh strictly speaking really your metal your interaction resilient metal is really on the on the phase transition it's just because you have a finite size or because you have no no it's not because you have a finite size this this metal is present also in the thermodynamic limit okay but i mean these are effects of the vicinity of the phase transition they are not yes they are not yes properties of the phase uh because i mean it's a it's a matter of terminology i guess sorry it i think it's a matter of terminology so i suppose it is but uh usually uh what i mean for for my understanding what it means for actual physical systems is that um it's especially a solid state system is difficult to have a fine tuning because you will have some disorder some imperfections and so what really matters is the property of a phase and not non-universal properties like what happens close to a fine tuning and so doesn't that mean that in terms of um let's say uh technological potency it is less uh it it will be less less used as a consequence i i don't think it's this comp i'm i'm let's say i'm not an expert of material science but i think that this concept of uns metal is rather rather rather robust in the in the in the community of solid state physics okay okay and as i said it's not specific of the line j over u equal to one third but it's enough to have a finite time a finite tunneling to allow you to have a persistent metal it's not something that disappears in the thermodynamic limit no no okay okay i sorry maybe i don't get fully your question no no but i think we can discuss later i think that uh answer the question in any case thank you very much okay there is another question from juliano please juliano thank you hi uh yeah so i wanted to ask um so if you look at the current along this uh j over u equal to one third line yes like how how does it decrease with the u over t because i guess like it decreases when you increase the interaction no so like what's the profile along the line well no no actually it is an an asymptote oh okay it's an asymptote so you in in in theory you should have you should have a metallicity even in the presence of infinite u and infinite mu and infinite j this is the the the thing which is amazing of uns metals okay that even if u and j are incredibly large you have a metal provided that you have provided that you have a finite tunneling uh i mean u over t going to infinity is equal to finite tunneling to zero tunneling right yes okay but in practice you never have uh u over t equal to infinity you have about one thousand but not infinite okay and just a little curious what's the this asymptotic value like uh what is it like compared to the maximum current sorry can can you repeat uh so like the asymptotic value to which the the current goes compared to the maximum current you have on uh on your bar it's a bit it's around 0.3 if i'm not wrong okay so one third of the of the conductivity that you had in the let's say non-interacting system which is not bad okay okay i think there is a question from the audience from vladan celebonovich uh did you apply your calculation to any real materials no unfortunately not uh but i can i can mention the a previous work from uh from my group that i was not part of it here on on on the right here on the right you see this is uh let's say a similar phase diagram uh with respect to the one that i plotted for a finite system so my colleagues obtained it with the dynamical mean field theory so in the limit of infinite dimension and better lattice so um you can see that uh i mean this is more quantitative this is more representative of real material with respect to to my simulation that was referred to to a simplest uh to the simplest circuit okay perfect so there is another question from arithra uh hi uh so i think it was already asked but uh like uh like this the the root rate that you get like is it a thermodynamic limit result or the finite size result can you speak sorry can you speak louder because i you are very very bad i don't know if the other people give you well for the for the drool bit uh is it the infinite size uh is the thermodynamic result or is it the finite size result that you get in the for the metallicity so you can can you repeat your question i i didn't hear you can i say that can you hear clearly yes thank you the root weight yes in the thermodynamic limit that you calculate or for the finite system for the finite system the drool the weight is done for a finite system so the minimal circuit that you can imagine and and does it depend on the on the system phase yes in principle it depends if you if you increase if you increase the the number of lattice sites so for example if you consider a ring with four five or six sites you will see that this drool the weight converts to the to an asymptotic value which is represented here in this slide on the red okay but i wanted to show you the minimal circuit i mean the the primer is the most essential circuit that allows you to understand the the the ingredients of the physics okay okay thank i think there is another question from pierre actually there was also a question from adriano but it's just that he raises his hand in real life and not virtually so okay so maybe let's do him first because it's twice that he gets his time and then then i'll be glad to ask him so i wanted to ask you this if i recall the phase diagram of the one d-hub and model just single band the easiest thing you can think about um tuning your filling the number of fermions that you have in the system you can actually have different kinds of insulators you have a multi insulator and our filling and then if you do the filling up some more you have a metallic phase where conduction is non-zero until you get to a band insulating phase where every site is doubly filled and then you don't conduct any more yes and in that case you actually have a conducting phase between insulators because it's simply the equivalent after you apply the vertical whole symmetry of the metallic phase that you would have between an empty lattice and a multi insulator sure definitely so then so can you interpret your results as in the one d case so it's the equivalent of what you will get in another let's say more simple and more basic to understand metallic phase just in terms maybe of a more complicated symmetry because there are three bands involved i would say no because in the example that you did which is perfectly correct in my opinion you are varying the filling so you are varying the number of carriers in in the in the lattice and on top of that you are violating the commensurability condition so if you basically if you want to make a multi insulate a multi insulator you need to have a commensurate number of particles of course if you have a non commensurate number of particles you are somehow cheating i mean you can have a huge interaction you but you will never have a multi insulator there is nothing which stabilizes it in my case the situation is different because i'm not varying the number of particles and i am that's why i'm always allowing the system to form a multi insulator so the u is very large the number of particles is always commensurate the system is expected to undergo a multi localization but it doesn't because of the other terms that determine the competition between different tendencies so you're not making holes by taking away particles from a band insulator you're making holes by making say the band insulator less convenient so particles want to do another kind of insulator and there are some that are left around exactly exactly okay thank you please pierce go ahead with your question so i suppose it goes also in the same direction my question is to what extent would you say that this existence of an interaction resilient metal regime exist next to any mod insulator phase transition well if you have a regular mod insulator if you have a regular metal mod insulator phase transition you have on one side a metal on the other side you have an insulator so there is no concept of this interaction resilient metal if i understood your question so between of course two isolating phase one being a mod insulator well can you make an example of another transition between i mean a lot of insulating topological phases next to mod insulating phases this i don't know i i mean i cannot answer your question i don't know if it is a general okay so the question the answer is i don't know if it is a general concept but it is provided by the simultaneous present presence of different number of fermions at different sides this is the basic condition which allows you to have to have a persistent metallicity so it's not enough for two insulators to be degenerate but the degenerate the generacy must be in such a way that different sides host different number of fermions that are connected by opening and that's why you have a matter but in general if on one end you have one kind of insulator on the other end you have another kind of insulator but the connection which the the state which links them is not connected by by hoping the configuration the local configurations are not connected by hoping then you cannot expect an interaction resilient metal okay to my knowledge i say this is my answer so for example between a charged density wave and a mod insulator you would expect this yes but not between i don't know charged density wave phase and spin density wave phase something like this exactly exactly or for example if you yes yes yes i cannot make a better example yes okay thanks other questions have a very trivial and stupid question in this figure that you show what is this white or gray region i mean i'm just trying to understand why and this in this gray region there is another another effect which which i'm not an expert of so there is the so this is actually can is an answer to pierre question this is another insulator this is another kind of insulator and you can see pierre but there is no so in the gray in the gray area is it we have another insulator that is neighboring to the blue area which is the hoon's insulator so here in between there is no interaction resilient metal because the two configurations the gray one and the blue one are not connected by hoping so the so to to answer to answer shatka shatka question here we have another kind of insulator which is due to the fact that you want to maximize the the local spin and the local and the local angular momentum so the local configuration change arrangement and the fermions get packed in a in a different way okay thanks so just to like again conclude it's not really true that any kind of two insulators will have one metallic transition in between for example between this gray and blue region no no no it's not a general statement at all here in the take-home message i wrote i wrote can result into a metal it's not for sure okay thank you there are specific conditions to let's say to to meet if you want this to be true thanks other questions not let me thank again all the speakers of this morning session shata manuele and andrea for the very interesting talks so i think that in 15 minutes we have the the photo session so we can recombine the new zoom link i think yeah everybody should have received an email with a separate zoom link it's not the same as this one the rules are a bit different i think everybody will will be visible so that we can actually take the picture so we can meet there at 12 the machinery should be in place uh so we can actually take the photo and then we can go for lunch in the meantime maybe we can have a snack before i'm sure many of you are starving as i am yes but don't hesitate to try to connect a bit earlier even if we only take the photo after 12 just because this way we we know when everybody is here and and available so that it will get quicker this way perfect see you later yes see you see you very soon so amongst the spectators that are still there do you all have the have the link for the for the photo if uh if not send me a message and i give you the link so i'm receiving no message so i deduce that everybody has the the link for the photo so i see a thanks but i don't know if it's a thanks please send me the link or a thanks uh i have the link all right i guess everybody has a link so see you in the in the photo session right now