 So what I would like to do in this talk is to describe quite funny physics that is possible in bilayer graphene at pretty low energy scale of several electron volts in terms of the electron excitation energies or Fermi level in bilayer graphene counted from the Dirac point. So what I'll do in this talk I'll start with an introduction and I'll have the road build from the high energy description of the electrons using the tie binding model down to two events which may happen or I think they do happen in bilayers. This is the elliptic transition in the electronic band structure which consists in the change of the topology of the of the electron Fermi surface which can be controlled and stimulated by strain in bilayers. And second I'll describe an alternative which results in exactly the same change in the band structure which consists in the phase transition in the electronic liquid itself. And the order in which I'll describe this two effects it's not in the order in which we actually published our work. We first understood the behavior of electrons in ideal pristine and perturbed bilayer and after we found that there is a plausible phase transition into the so-called nematic phase we looked at the effect of the strain on bilayers because to achieve such a regime where this phase transition would take place one has to work with very clean material with very high controllable carrier density to reach lower just scale for the Fermi level and therefore one has to work with suspended flakes. And when you suspend the flake you have to start worrying immediately about how much strain has been imposed on this few atoms in membrane when it is lying on some massive contacts and if contacts have moved then you should expect that there will be a little bit of strain acting in graphene. So I'll start from a big picture starting from a single layer and I need to do that to introduce a language and notations that will be used in the rest of the talk. So here we have the corner of the brilliant zone the part of the of the momentum space we're interested in the properties of the electrons. We have the honeycomb lattice and we have the AB hoping which is the main hoping that's determined the bent structure of electrons. When we look at the hoping from the subletters A into subletters B atoms for each atom A we have three B atoms surrounding it and the block states in the corner of the brilliant zone have such wave vectors that the phase of the block state on the three sides surrounding three B sites surrounding A site the sum of the three phases is equal to zero and as a result in the 2 by 2 Hamiltonian which describes the dispersion of the electrons in the vicinity of the brilliant zone corner the hoping element gamma node itself cancels out it only appears together with the momentum of the valley momentum counted from the brilliant zone corner which appears through the expansion of the phase factors in a small parameter the momentum multiplied by the lattice constant. So then you get the hey 2 by 2 Hamiltonian which has this form of the Dirac Hamiltonian and I'll use for a simplicity for compactness the notation this is not number pi it is an operator it is a sum of x and I y component of the electron momentum so now we have the Hamiltonian which describes how the plane waves how the block plane waves on the subletters A and subletters B of a single layer are coupled and this gives the linear dispersion of those Dirac cones close to the brilliant zone corner. Now we take two layers and according to what's known about graphite we have to arrange them into the Bernal stacking we have to consider the arrangements of atoms that appear of atoms in the two layers sit on the top of each other whereas for the other pair the middle of hexagon is above the atom and the other atom in the top layer is just above the middle of hexagon in the bottom layer and then we need to take into account the interlayer hoping which in the first instance we can introduce as the hoping between the two atoms which are on the top of each other this is the parameter gamma one according to Slanshevsky Weiss-McLeod parameterization and then we build up the 4 by 4 Hamiltonian now it is 4 by 4 because we have now not two but four atoms in the unit cell of the bar layer and we arrange the amplitudes of the electronic wave functions on the block functions on the subletters in such a way that the pair of subletters which appear on the top of each other appear here in one corner and the pair of subletters which appear not on the top of each other which appear on the top of or under the middle of the hexagon in the other layer they are collected in the top part of the four components spinner then after we do that and confine ourselves to the corner of the brilliant zone of graphene only then the intra-layer hoping of the electrons is now described by this pi and pi dagger operators which operate within each of the layers separately and the interlayer coupling is provided by this gamma one hoping which appears in its full glory not in the form of parameter multiplied by the small value of the moment. So when we diagonalize this Hamiltonian we get this bend structure with four bends in the brilliant zone corner where two bends are split apart so these are the bends based on the atoms which are sitting on the top of each other and there are two bends which touch each other and these are the bends also parabolic bends which correspond to the electronic wave functions which are mostly reside on the atoms which do not find the closest neighbors above or below in the other layer and then we can just concentrate on this low energy part of the electronic bend structure the energy scale less than 0.4 electron volts which is the value of gamma one and this energy scale will only have two bends and for the energy close to the neutrality condition we can simplify the description just looking on the two bends only and this would be described in terms of the 2 by 2 Hamiltonian which is now quadratic it's similar to the monolayer Hamiltonian in terms of its structure except that in the over diagonal parts describing the coupling of the subletuses now only two subletuses not all four and this happens after we eliminate the high energy part of the Hamiltonian by by three for wall transformation we get this quadratic pi dagger and square and pi square operators and now we have also the parabolic dispersion which is characterized by an effective mass which is about half of the effective mass of electrons in gallium arsenide this also can be transformed into information about london levels and I would like you to remember about the counting which is summarized in this transparency I like to remember it until the end of the talk because I'm going to use it if we have a monolayer where we have the Dirac spectrum then for the Dirac spectrum we have always one single london level at zero energy but when we count the and this is exactly a zero energy and then if we count the discrete degeneracies which appear in the system which has to take into account the electron spin and the existence of the second of two valleys then each of the zero energy london levels for the monolayer and I now say for the single Dirac cone in terms of the dispersion gives us the four full degenerate zero energy london level if we take the bilayer then at zero energy we'll have an additional degeneracy so that for the bilayer we'll have the eight full degenerate london level at zero energy so the rule is the following if we have the Hamiltonian where we have kind of corality in this building this Hamiltonian which is one we have four full degenerate london level at zero energy if corality is doubled like we have in the bilayer we have the the eight full degenerate zero energy london level all other london levels in the spectrum they are four full degenerate both in the monolayer and in the bilayer and in the monolayer the spacing between levels is such that the largest separation is between the zeros and the next one and the rule of engagement is that the the magnetic field dependence is a square root dependence on the magnetic field and then the energy levels become denser and denser when you go up on the energy scale whereas in the bilayer the dispersion of london the fan plot looks very similar to what you have in usual semiconductors at high london level numbers it's almost linear in the number of the london level and it is also linear in the magnetic field now i'm going down to the low energy scales and at low energy scales which would correspond to that i would also be interested in london levels at relatively small magnetic fields i have to be more cautious and i have to take into account smaller effects which may appear due to couplings we have been ignored in the previous consideration so the the coupling that now has to be looked at in more details is the coupling between the atoms which are or subletters block states which do not appear on the top of each other this is described by the parameter gamma 3 in terms of logensky vice parameterization so what does this gamma 3 do if you look at the bilayer at the bilayer lettuce from the top and realize that this double site in the middle is now excluded from the consideration because it supports the split band states but it does not support the the low energy bands that we discussed on the previous few slides then the letters that you will see will be exactly like the honeycomb lettuce of the monolayer and the coupling gamma 3 on this lettuce is very similar to the coupling of gamma one gamma node on on a single layer so that its effect for the two subletters states that now are relevant for the low energy band structure will be exactly the same as the effect of gamma node coupling was in the monolayer the gamma 3 itself will disappear and it will only enter as the pre-factor in front of the momentum which appears after we take into account that the block states on the a subletters and the block states on b subletters have this rule of the phases that if we take any of this a atoms in one layer and look at three b tilde atoms in the other layer the amplitudes of the blocks functions on this b tilde sites will sum together to zero exactly in the brilliance on corner and only what's left will be proportional to the small value of the electron momentum counted from the brilliance on court that's why in this part we now get not gamma 3 itself but we get the much smaller contribution which is v3 pi v3 is determined is related to gamma 3 the same way as direct velocity is related to the intro layer hopping in a single layer and then what we need to do we just need to incorporate this into the two by two Hamiltonian describing the low energy electrons and then we can draw the sketches of the band structure which are shown here so the large energy scale the band structure has this two split bands and in two parabolic bands which touch each other in the brilliance on corner very close to this brilliance on corner the parabolic dispersion is already violated it's violated by the linear term which now appears in the Hamiltonian this term is permitted by the symmetry and therefore it is prescribed in principle to be included and then even more importantly at small momenta the linear term becomes as important as quadratic and this is reflected by the drastic change in the band structure which takes place in the small momentum interval around the corner of the brilliance on now what you see here is that if you start cutting the bands through at a fixed energy so we can say we vary the kera density in graphene and follow the change in the shape of the firm circle it becomes less and less similar to a circle it becomes triangulated so this is why the combination of these terms is called the trigonal warping causes trigonal warping of the band structure and even more after some critical energy scale the topology of the fermi line changes because now what you see is that at smaller energies the dispersion splits into separate diracons actually four of them one in the middle and three offside the three offside are asymmetric the one in the middle is symmetric and if you make energy cuts at different values of energies these are the constant energy lines this one the bigger one is at higher energy and the four pieces now separated correspond to a much smaller energy the change in the topology of the band structure we estimated happens at the energy of about one milli electron volt to do that we to make this estimate we took parameters which are known for bulk graphite there is no reason why they should be exactly the same in biolayer graphene because let us period the distance between layers is slightly different than it is in bulk graphite but nevertheless that's the best we can do to estimate and in terms of at what density such change in the band structure in the topology of the fermi line would happen we can convert this into the concentration of electrons of carriers in graphene electrons or holes and what you see here is relatively small 10 to the 10 inverse pure centimeter density where it happens so it means that if you want to see something of this sort the change in topology of the band structure you have to be able to control the carrier density in graphene with a very high precision and this is possible only if one works with suspended layers so this kind of change in the band structure is called Lyft's condition it was discussed in quite a lot of details in the physics of bulk metals and this material offers us a possibility to work with the system where many things can be changed in the bulk metal you would struggle to change carrier density to to a large amount and therefore changing the topology of the band structure by changing the density like we can do it in graphene would be quite difficult in graphene is due the last thing i would like to mention before i go into the physics of strains it is that now we can put on a magnetic field and we can look at the spectrum of lando levels which is shown here so what's shown in this plot is how the lando levels evolve when you go from the strong fields down to the weak fields so what do you see here at strong fields is that the fun plot with lando levels which depend linearly on the magnetic field at high fields there is the eightfold degenerate lando level at exactly zero energy and then at some relatively small range of magnetic fields the effect called magnetic breakdown takes place actually two lando levels come very close to each other to zero energy and then the lando level which was eightfold degenerate at high fields becomes lando level which is sixteenfold degenerate at the low fields why is that it is because now we have four diracons in the spectrum at low energies and if magnetic field is small then it means the inverse of magnetic lengths is also very small and therefore the formation of lando levels involve states in each pieces of the dispersions separately in each of these diracons separately and each of the diracons provides us four lando levels there are four diracons now at low energies in the spectrum therefore we'll have 16 lando levels at zero energy what it would mean in terms of quantum Hall effect or should become the gas oscillations it would mean that field in factor eight would be the one which persists down to the lowest magnetic fields whereas all the other field in factors field in factor four three factor 12 and 16 and so on would very quickly disappear simply because they would have much smaller the much smaller activation energy so now about suspended graphene and about the regimes when one would expect delicious transition to take place you want to work with low densities therefore you want to have very clean structure very homogeneous carrier density and for that suspended graphene is the best that has been shown by a number of groups by now and as I said this is a situation when you have to start worrying about other past not the inhomogeneity but strain you put a flake onto a suspension and then if anything moves in your device you start deforming graphene a little bit and what we want to understand is what effect even the simplest variation of strain in the simplest version of strain homogeneous strain would have on the electron band structure at such small energy scale so how do we envisage strain so the flake has moved has deformed a little bit so that the lattice of graphene has reskilled in both layers in principle we can permit a little bit of shear between the two layers but the most interesting part is equally interesting part or I would say most interesting part is coming from the strain effect in each layer so that the strain affects the each of the layers separately or together sorry together so what is the effect of the strain if the effect of the strain consists in that now the parameters gamma node and gamma three are no more the same for three directions of the electron hole so that if we take gamma three for example and look at the hopping between the two layers in three directions from a atom into surrounding three b atoms then this hopping elements due to the due to the deformations of the lettuce are no more equal and because they're no more equal the consolation that we discussed earlier does not happen and as a result there will be contribution towards the electronic Hamiltonian which is proportional to the strain and it is proportional to the parameter gamma three itself this is the same thing as known in in the monolayer where the same effect is produced by the lack of the consolation between three terms which are proportional to gamma node this is not the only effect of the strain we have to worry about but there is one trivial which can be easily eliminated when we strain the structure the k points in the brilliant zone they also have to be moved so that's the first thing we do we first take into account the shape of the brilliant zone changes and then in the corner of the recalculated brilliant zone we do the expansion of the Hamiltonian into the subletters components so what do we have we have a four by four matrix this part the interlayer coupling of the two sides which are sitting on the top of each other we don't change it here but we can take into account the variation of this parameter upon strain it does not lead to any qualitative change in the Hamiltonian structure the off diagonal the anti-diagonal part on this Hamiltonian which describes the interlayer hopping is now affected by that the consolation between hops in three directions is no more happening so that we get a constant term which is proportional to the strain and it's also dependent on the direction of the strain and it's determined by how fast the parameter gamma node changes upon the variation of the distance between the carbon atoms and we also have the addition to the linear in momentum term in the block which resulted in this trigonal warping and here we have the addition which is proportional to the derivative of the parameter gamma 3 of the parameter characterizing the interlayer hopping and of course it also depends on the direction of the strain and this is not a very convenient form to work with and there is a simplification for this Hamiltonian which can be made by realizing that what stands here looks like a vector potential added to the electron momentum and if this vector potential is homogeneous it can be actually eliminated completely from this year if it's not homogeneous then one can generate magnetic fields the huge magnetic fields that Andrei mentioned which would be huge if gradients of such deformation is large so what we do here we split the this vector potential and by the way this vector potential would have the opposite sign in the opposite valleys so that it has nothing really to do with time inversion symmetry breakings purely let us deformation effect and then we can split this vector potential into two parts one would be kind of potential part and the other would have non-zero rotor and the one which can be separated into the gradient of some scalar field this part can be gauged out by this gauge transformation so that this would eliminate part of the vector potential part of this parameter a node if it depends on the coordinates and if it would be constant in space we would get rid of it completely but now we have to realize that we cannot get rid of a node and a3 simultaneously because they're different they're coming with different parameters and they're coming from different constants from different couplings interlayer coupling for a node and interlayer coupling for a3 therefore if we perform this gauge transformation we can eliminate in the case of homogeneous strain a node from the antidiagonal part but then what will happen that it will produce change but will not eliminate the the contribution from the strain due to the interlayer hopping and then what we're going to get we're going to get a non-zero addition to the linear term in the Hamiltonian in the part which previously was responsible for the trigonal warping so what it means it means that for homogeneous strain we simply move a little bit the dirac points the nominal dirac points zero for the momentum as we determine now for this pi for this new pi operator we move it a little bit away from the exactly brilliance on the corner and after that the Hamiltonian looks almost exactly the same as it was before we included the strain except that for in the block which characterizes the low-energy part the low-energy states we have a constant term which now has to be taken into account and if we project it on to the two-band model it means that in addition to the combination of quadratic in momentum and linear momentum terms that produce this band structure with the Liffchitz transition at very low energies we now need to modify it by taking into account some constant addition to the matrix where these two parameters w and w star arc is w is a complex number and real and imaginary parts of this complex number depend on the size and also on the direction of the axis along which we have stretched the structure so after we do that we now can draw many pictures and show how the band structure evolves if strain is applied but before it's nice to try to estimate the size of the effect and here is a little bit of trouble to estimate the size of this w the absolute value of w the absolute value of perturbation we can create in the Hamiltonian due to a strain of let's say one percent we need to know two parameters we need to know the logarithmic derivative of constant gamma node with respect to the change in the distance between carbons on the lettuce and also the same for gamma three the value of logarithmic derivative gamma node with respect to the inter interatomic distance in a single layer has been post computed in dft and compared I would say determined from the electron-phonon coupling measured in Raman and this value is is a number we now can use but there is nothing known for sure about this parameter so what the best we can do is just to say that maybe this parameter is small or maybe it would not cancel here completely the effect of the strain coming from a single layer and we simply ignore it because we don't know it and if we just use the parameter we know we would estimate the size of this addition to the Hamiltonian the absolute value of w is about 60 electron volts for for strain of about one percent and now you see that this is comparable to the Liffritz transition energy we discussed before so that these two two effects have to be taken together on equal footing so that if you want to consider even small small strains we have to include them if you want to discuss seriously how the Liffritz transition looks like in in suspended play so I'm going now to show a sequence of figures demonstrating how switching on strain in some direction changes the electronic band structure and the way how I do it I take real and imaginary part of w so I don't really relate it directly to the direction of the strain which we know how to do instead I just follow what happens if you increase the real part of w or decrease the or increase the imaginary part of w so this is the same as increasing strain with excess which is somehow oriented in on the flake so this point is corresponds to zero strain and this is the case when the band structure undergoes Liffritz transition into four diracons three off side and one in the middle so now we start increasing strain in such a way that let's say the real part of w becomes positive and becomes larger and larger and the evolution in the band structure you see here is a following that the three diracons one in the middle and two off side would start moving toward each other merge together and then become a single diracons so you kind of get a two leg which is shown on the right hand side this is a dispersion which also has the Liffritz transition building it because at some energy when you go from high energies down to lower energies the single connected fermi line will split into two parts and you will end up with two diracons what if we change the strain in such a way that w will become bigger but negative so when we move along this axis of real part of w in the opposite direction what will happen is that this diracon in the middle and this off side will start moving toward each other at some critical value of strain they would merge and then they would generate a minimum in the dispersion local minimum the dispersion which will be lifted up higher and higher upon the increase of strain and then eventually vanish and will end up with a two leg dispersion two diracons left and the Liffritz transition which is now happening at the energy much higher than it was in unstrained graphene and with the value of Liffritz transition energy answer for critical density determined by the size of the strain what if we change the strain in such a way that a major part of w will increase with a zero real part then the other pair of diracons will start moving toward each other this one and that one they will eventually merge together at some critical value create a local minimum in the dispersion and then this local minimum the dispersion would again vanish upon increase of the strain leaving us with two legs of diracons one is hidden behind the one on the front and now we can the angle here tells in what direction the tail so if angle is zero angle is counted from the x axis w is real but to get purely imaginary well you have to figure out what angle it should be 90 degrees and what is shown here is the whole variety of things you can get you can get a region of strains small strains finite but small for which you would still have four diracons in the spectrum blue indicates the interval of strains the interval of failures w for which you would have two diracons left plus an additional minimum an additional minimum in the dispersion and red indicates the regime when you have only two diracons left and no local minimum in the dispersion and what's shown on the right hand side plot is how this left shift transition looks like in terms of the position of one whole singularity which is an indication that the electron Fermi energy has passed across the place where the in the in the band structure where the signal connected Fermi line splits into independent pieces each time you pass through the saddle point in the dispersion and this gives this divergence in the density of states in the system then what you can see that for high energies this density of states will saturate at a finite value but for the lower density of states for the lower energies below the below the Lyft condition you have density of states which rises linearly with the Fermi energy and well with the electron energy and this is the manifestation of the linear dispersion you have in those diracons now we know that there are diracons and we now know how to count London levels of zero energy in the spectrum so what's shown here in the middle of this slide is the sequence of london level spectra for three typical strain regimes the one in the middle one strain is too small to kill four dirac points the red one which is so large that only two dirac points are left in the spectrum and the blue intermediate one for which the spectrum has two dirac points two dirac two dirac cones and this is the example here two dirac cones and the and the additional dispersion minimum at some value of the moment and what you see in those panels for the london levels it is that if we go to the lowest magnetic field then when we have four diracons we have still this 16 degenerate zero energy london level when we have only two diracons left we have only eight degenerate london level at zero energy and this is because if of the each of these two diracons will give us four london levels due to the spin and valley degeneracy and now we can say what in the case of such a strong strain would be the what would what would be the longest living quantum whole effect state or shubnik of the gaza salation if we start going into lower density and structures where such feeling factor would be realized at at a lower magnetic field for the case when we have spectrum with a pair of diracons the the feeling factors which would be the strongest revivals would be feeling factor plus four and minus four instead of plus eight and minus eight as we discussed about the about the non-strained material and there is a very interesting feature that appears in intermediate strains which is that you see that there is a slender level that we are following from high fields down to the lowest fields and first this london level was evolving linearly with magnetic field and then it started to saturate saturate and then it got saturated and it has energy which is almost independent of the magnetic field and this is london level which got stuck in this dispersion minimum and therefore it cannot drop down when we start decreasing and decreasing magnetic field eventually london levels in this legs would drop down on the energy scale because there are dependence on the magnetic field the square root of magnetic field but there is a finite interval for which the excitation from the from the zero slender level to the next one would have the energy which is almost filled independent so you can say that activation energy in this quantum whole effect state of feeling factor four would first go down then get saturated before it quickly drops down to zero whereas for all the other feeling factors the activation energy in the quantum whole effect would in this regime would be already so small that you would not expect the quantum whole effect to survive okay so i'm now going to the next item on my agenda and i would like to discuss what happens if the bilayer would be suspended cleaned very homogeneous but not strained and what i would like to describe is the pneumatic phase transition pneumatic phase of electrons which can form in such a nice nicely prepared bilayer graph this would be the effect which would develop intrinsically in the system and therefore it would not depend on the history of the device and it would be the same in a number of devices which can be studied either in one goal in a number of pieces of flake which can be studied in a single device or several flakes studied in different devices so first of all what is what is a pneumatic phase transition the pneumatic phase transition in the electronic liquid has been introduced by agonistian kielsen and fratkin and this is the transition in the electronic liquid which mimics the effect of unilateral deformation of the crystal it does not come from the deformation of the crystal it it is the transition which happens with the electrons it may then generate very small deformation of the crystal itself which would be difficult to detect using crystallography because that would depend on the strength of the coupling between electrons and the lattice deformations and if lattice is very rigid then for electrons it would be very difficult to change the lattice substantially so that this would lead to observable change in x-ray scattering and this pneumatic phase transition has been recently claimed in iron-based superconductors and it has not been seen as far as I know anywhere else so what we found by doing the renormalization group analysis of interactions of graphene it is that the transition mimicking the effect of lattice deformation is actually the most plausible phase transition in in bilayer graphene so how the story developed there were many suggestions of what kind of phase transition can happen in bilayers at zero magnetic field the case of a straw magnetic field was quite a clear cut because in the straw magnetic fields you have very high degeneracy of london levels if you know you have the eightfold degenerate london level at zero energy you know what's the degeneracy you know what are the quantum numbers you know electrons interact by Coulomb interaction and then there are very very natural suspects for the symmetry breaking to discuss the valley polarization and spin polarization of the electrons due to the electron exchange interaction for bilayers at zero magnetic field the story is not such a clear cut and it's much more difficult to do the analysis because of the the spectrum of electrons is continuous and there were several predictions which have been made by assuming or by liking this or that kind of the phase transition the there was a prediction of for electrically satonic insulator state by levittal and magnonald there was prediction of her magnetic state and there was discussion of some more complicated phases how would you find out theoretically which of the phase transition should happen not to suggest that it may happen if the interaction constant in the corresponding channel of fluctuations in the electronic system would be the strongest the way to answer this question is to analyze the interactions between electrons between fluctuations in the electronic system breaking all possible symmetries of the crystal analysis finding this interaction constants by doing the randomization group analysis finding them for electrons at low energies and then if we see that one of the constants grows the fastest when we go from high energies down to the low energies then this will tell us that this is the channel in which interaction would lead to a phase transition and what we found is that this channel of the interaction it actually is the one which leads to the pneumatic phase pneumatic phase transition mimicking the effect of unilateral strain so now i'm going to scare off all the experimentalist in zodias so i i ask for forgiveness but on the other hand i'll try to impress and use that we actually did some calculation so what what we did first of all we had to formalize the description the way to formalize the description starts with the group theory so we first of all need to understand what are all the irreducible representations of the symmetry group of the honeycomb lattice and for each of the irreducible representations we can attribute fluctuations in the electronic liquid which break the symmetry according to this irreducible representation and in interaction between fluctuations which have the same symmetry is characterized by the corresponding constants so we have one two three four five six seven eight of them and we don't know any of those constants before we do the calculation the only thing we know is that there is a strong Coulomb interaction between electron charges which is the most symmetric and does not break any symmetry at all so the way to do the calculation is the following starting from what we know and assuming that those symmetry breaking interactions were so small that we would not even possibly bother about their initial values to find out how the Coulomb interaction between electrons that we know how to describe at the short range scale would then sorry at the short range scale would then and at high energies across the entire band structure how this would trigger and produce interactions which start breaking the symmetry in the system so this is a weird way to write down the Hamiltonian using these matrices which act on the direct products of matrices which act on the subletters components of the electron states and also which act on the valley components and we need to do that because if we want to check whether or not there is some phase transition into a charge density state which would then oscillate in space with the vector of the brilliant zone corner then we need to take into account also everything about the valley structure of the electron wave functions then how the analysis is done well first of all we know the Coulomb interaction is there we know that it is strong and we know that when we have strong interaction strong Coulomb interaction in metals the first thing you should do is to screen it so that's what we do this wavy line the thick wavy line is the screen Coulomb interaction which we screen using the random phase approximation and then magic happens from a strong interaction we immediately get a weak interaction in which the weakness is controlled by a parameter 1 over n where n is a number of electron species in graphene n is equal to 4 because we have one quantum number for valley and one quantum number for spin so we have four different species of electrons that participate in screening and the more species of electrons you have the stronger screening is and the weaker the final interaction will be after it is screen so we take this screen interaction and then we use 1 over n as a small parameter and do 1 over n expansion and by now we can do it to higher orders and we did initially so that we know how to control the perturbation shear analysis of the renormalization of the interaction constants in all symmetry breaking channels so this process of renormalization requires the perturbation shear analysis of what happens with the interaction so this this dashed line characterizes interaction between electrons between fluctuations in the electron liquid which break symmetry in corresponding to one of the irreducible representations and this delta states that this is the change in the interaction constant if we eliminate from the consideration electrons in some interval of high energies reducing the band within which we describe the electrons down and down until we reach the low energy scale at which we expect some interesting physics related to electronic phase transition to take place and there are many diagrams which one has to take care of one has to renormalize properly the single particle weight of the electron states not to make mistakes and what else is included here the seek the the the full circles characterize the symmetry breaking interactions and the the the gray circle it characterizes the screened Coulomb interaction and what you see here and actually now I would like to turn to the next slide what you see here is that for the symmetry breaking interactions the purely Coulomb interaction is able to do something it's able to create a symmetry breaking interaction between two electrons in the second order in the screened Coulomb interaction and all this comes from the sub lattice structure of the electronic wave functions those those lines with arrows are green functions of the electrons which know everything about the sub lattice structure of the plane wave states of electrons in bilayer graphene and what you see here is that there is a pair of diagrams which in fact does not cancel and which gives a finite contribution towards one of the symmetry breaking interaction channels it doesn't give contribution to all of them it gives the contribution only to one and what it means that means that just the structure of the lattice the structure of the electronic states on the lattice plus Coulomb interaction is able to generate some symmetry breaking some interaction between symmetry breaking fluctuations in the electronic liquid but not between all of them only between one time so this has been noticed by Waffek and Young orders and we completed our work and the difference between Waffek and Young's paper and ours is that we took into account all possible interaction channels and as a result we had to work a little bit longer than Waffek who took only three interaction constants into account at the very beginning and before we were able to complete our work predictions of the other phase of the phase with the with the excitonic for electric excitonic insulator has appeared due to Levittor when he's student and here is an example of of that if one misses some diagrams we may end up with very strange predictions for example they initially calculated everything just using one diagram of this big zoo and they have discovered that there is some huge renormalization which is not even logarithmic which is log square which means that the phase transition they predicted would happen at much higher temperatures that actually anything can can happen in real life and then they found that if one includes two diagrams into consideration then this log square cancels out on the logarithmic correction appears and as you may see here this two diagrams they appear in the analysis only if there is already something in the already some finite interaction constant present in the system so that it can further increase but it cannot be generated if it's not there in contrast to this pair of diagrams which produce interaction is one of the symmetry breaking channels which is triggered only by more symmetric Coulomb interaction so the this slide is the last one which should scare experimentalists this horrendous line describes the renormalization group theory equations for all interaction constant I mentioned at the beginning and what you see here is that to renormalize one of the constants which breaks the symmetry which corresponds to the interaction between symmetry breaking fluctuations in the electronic liquid to get it renormalized to get it increased you need it to be already there and it is true for all the constants except for one and this is a constant which corresponds to the symmetry breaking from the irreducible representation E2 of the symmetry group of the crystal and this is a representation which describes the violation of symmetry when you displace one sub lattice with respect to the other or when you stretch the lettuce in one direction and then if we just throw away all the other constants which need to be finite to get renormalized and take only the one which is triggered by the Coulomb interaction then we solve the single rg equation for a single parameter and then we find at what energy scale the interaction constant becomes of the order of one and in this energy scale we should expect that the system is no more stable the symmetric electronic liquid is no more stable that it develops the phase which mimics the effect of the uniaxial strain so what is this phase this phase is parameterized by the order parameter which has two components it is similar to a director in pneumatic liquid so it looks like a vector but if you employ inversion this vector would not change and in terms of the way how this perturbation so how this order parameter appears in the electron Hamiltonian it is exactly the same way as the strain changes Hamiltonian and it can be parameterized using exactly the same parameter w as I described in the case of artificially strained bilayer so what this transition does it mimics the effect of unilateral strain with all the observable consequences I described before first of all you will have the density of states varied as a function f energy in such a way that there will be some Liffschitz transition and upon the phase transition you can get the state in which the dispersion will have an additional minimum in the dispersion and of course all the characteristics related to quantum Hall effect or should be of the gas oscillations for example that the field in factor four for the asymmetric state for the symmetric phase should be the most stable field in factor of low magnetic fields in contrast to fill in factor eight for the symmetric crystal it still stands so these are the predictions the question is what one can see in the experiment and here I'm showing the results that are not yet published which are due to the Manchester group so what's the top panel what's sorry what's the bottom panel in this slide shows is the effective density of thermally excited carriers electrons plus holes in monolayer graphene at exactly zero carrier density and you can understand that if we increase the temperature in the system we create electron-hole pairs just taking electrons from the valence band and putting them in the states in the conduction band and this quadratic dependence of the number of the electrons that are thermally electrons and holes which are thermally excited this quadratic dependence reflects the linear dependence of the electron density of states linear dependence give quadratic dependence in the number of effective carriers because you take the density of states integrated over energy up to the temperature and then from linear dependence you get quadratic in the bilayer if you look at the high energy range then you see linear dependence of the effective carrier density on the temperature which you should attribute to the constant density of states for a quadratic spectrum but then if you look in more details what happens at low temperatures and this is what's happening here then you see that this linear dependence get turned and it starts being more similar to what you see in the singular indicating that something happened to the density of states of electrons at low energies it got reduced as compared to what you would expect in the case of just a simple parabola even more one can look at the at the quantum hole effect the shubnik of the gas oscillations and there is already one published result by Jacobius group which has demonstrated that the filling factor four is the most persistent down to the lowest magnetic fields and also the same thing has been seen in the samples studied at Manchester except for one thing in Manchester they had different devices and in Manchester they measured the activation energies for this filling factor four and also for the other filling factors very careful so just for your to guide your eye I collected the predictions for the behavior of the activation energy and London levels and the dispersion for pneumatic phase and also for strain by layer graphene in which there is very particular regime I mentioned in the middle of the talk when the phase sorry when the band structure of by layer has two legs of dispersion cones plus a local minimum which was the result of that two dispersion cones two Dirac points came together kind of recombined split apart and produced this local minimum in the dispersion and for that case we identified that there is a London level at intermediate magnetic fields which gets stuck it's magnetic field dependence gets stuck it gets saturated and in terms of the excitation energy of the quantum hole effect this means that the activation energy of filling factor four which would be the most persistent filling factor in the quantum hole effect measurements would also get saturated before it eventually drops down and if you look at those diagrams which are the result of the experiment analysis of the experiment you see that this is exactly what is happening now the last question you may ask about this experiment is whether what has been observed was the result of a strain or it was a result of some intrinsic event like the pneumatic phase forming in the bilayer and there is a strong feeling on cost your side that it was the intrinsic effect in the bilayer because the same activation energy dependence on the magnetic field has been seen not in one sample but in many samples and it would be difficult to imagine that samples which have different history of preparation would have exactly the same value of the strain in post but if it would be the intrinsic event if it would be the phase transition what happens inside the high quality bilayer by itself then obviously it would be exactly the same exactly the same in all different structures and the last thing I would like to mention about this phase transition is that by the nature of the symmetry breaking by this order parameter by the director and by the nature of the honeycomb letters by the nature of the symmetry group of symmetry which is broken by this pneumatic order transition this phase transition has to be of the first order it is not a phase transition that continuously develops it happens and then you get a new system with a prescribed spectrum like in the for magnetic not like in for magnetic phase transition but like in in the phase transitions when you crystallize a liquid and then you get a crystal with all fixed parameters so that if this phase transition would take place then the band structure which electrons will have at low energy scale in the bilayer would be always the same independently of the details how the sample has been prepared just you need to make the sample clean enough and with the homogeneous carrier density so that there is a very strong case that the bilayers are the second system in which pneumatic phase of electronic liquid does exist in addition to to nictides and this is my last transparency and these are the conclusions all right questions please so the question was whether perpendicular electric field which we need to apply to to assess some finite concentration is strong enough to open the gap okay the answer is not it's not strong enough you make a simple estimate distance and what is in the land you go to micro volt regime opening the gap instead of four million electron volts or something what you see on this picture so you notice that the range of magnetic field is very small it's not like in in in your measurements with the same range of filling factors when you had tens of Tesla or 10 Tesla or 20 Tesla in those diagrams I have to say I switched off okay it's immediately reminded me yesterday's talk did not talk okay so I didn't notice the diagrams which would be responsible for renormalization of the Fermi velocity Fermi velocity you don't have Fermi velocity anymore it's parabolic dispersion at the dirac cones at very low concentration yeah so what we did we we we also studied renormalization of the effective mass and the renormalization of v3 as well so we renormalize them yeah we renormalize them but there were no bubble diagrams okay yeah so you think the phenomena of renormalization of v3 automatically contain the effects of renormalization we did we did we did include v3 and the mass in the renormalization group analysis so we know how they get renormalized and very little numerical factors are such that the mass and v3 renormalize very very little they just weird thing about uh prefactors so you can forget about their changes but that's what we do when we go from high energies down to the energies where the phase transition takes place after a phase transition takes place and you get this linear dispersions the analysis of what would have been happening at the lowest energies with the dispersion on those legs which actually is also an isotropic would require a separate step in the RG procedure which has nothing to do with whether or not phase transition takes place that would characterize details of the dispersion if one would be able to investigate them further shouldn't be done self consistently no no no there is no need for that would um temperature dependence help you distinguish between the just the strain effect and your pneumatic phase transition um it's a good question um I would say that uh the phase transition would uh would be killed by temperature so if you hit it up it it goes so that would really distinguish it from the strain effect yes except for one thing how would what what would you measure to distinguish because both measurements that I described relied on variation of the temperature so if you look at this temperature dependence then the change in the in the behavior may be also that at higher temperatures we we had this linear dependence just because parabolic spectrum and no phase transition at all yet lower down temperature phase transition happened and then we we got in the energy range where we have linear dispersion due to those two diracons for so that that's not easy because in all this measurements information is extracted from the temperature dependence the possibility may be in in doing the uh carrier density dependence but then for this kind of measurements it it's not easy because then the clear cut of what you call the effective density would be not so clear thank you at one point you showed density of states those little peaks in the density of states to do that I think the saddle points earlier just due to the strain effect is that right uh yeah yeah there what's the uh energy so this I remind you this is homogeneous strain this is homogeneous strain homogenous like uh so you you you you you take you take a long uh flake or this shape you stretch it then forget about what's happening close to the end look in the middle okay or you take a very wide like like that and then stretch it look in the middle not near the top and bottom edge and and so these little peaks in the density of states where what are the energies like that they should appear what was the energy scale so the energy where peak appears depends on the strain so the higher the strain the further to higher energies you move the one posteriority so this give me see that's kind of give me numbers so in terms of energy uh e star is uh of the order of one mil electron volt yeah so the scale is here uh e star is about one millivolt we we we had to normalize it here to the uh to the unit of uh lift transition energy in the unperturbed bilayer so if you can create a homogeneous strain uh with few percent uh you would be able to see this at the scale of 10 mille electron volts that one percent is six um if you put on a strain um it's going to obviously affect the geometry around and therefore you're going to get a quite a big columbic repulsion because you're going to get as you strain this you're going to stretch this you're going to change the bond angles in your crystal so and that this this is the fact that we yeah taken to account the the hopping uh between the sides has changed now if if if if that's a case that's going to actually put a lot of energy into the system isn't it if you like when you when you strain when you're actually because you're basically you're pushing those electron if you like your electron um wave functions together you're going to get a repulsion well the the the first thing which you have to do is to overcome the directionality of bonds prescribed by sp2 hybridization which has nothing to do with the pi orbitals we discuss here but just just and to do that you need to put a lot of energy yeah that's the point I was making yes right you spent some time uh explaining uh how you got this quite elegant uh Hamiltonian including stray with gauge transformation and things can you do it uh from looking on your at your original Hamiltonian and using methods of invariance or yes yeah yeah that that's uh that's what you would do if you just yeah if you just look at this and if you if you go to the last slide yeah if you go to the last slide here and say that uh if we have broken the uh symmetry from c3v to c2v then it is permanent it will be characterized by a director and then you implement this director which belongs to e2 a irreducible representation of the symmetry group into Hamiltonian or just putting components of your strain in the places of some components of your momentum like latin latin jupy or picus putting strains in uh holes Hamiltonians yes but uh since you have this picture here you have this one whole singularity with a negative mass it's known that for other four-by-four Hamiltonians uh it is used for having um kind of emission of terahertz radiation because you have a cyclotron mass somewhere which is negative which gives you instead of cyclotron absorption thank you for this question yesterday night a worries almost convinced me not to give this talk today because he gave uh a fiery speech yesterday night you you're not there because that was only for uh invited speakers uh uh and he said we scientists we have to do only research that will improve competitiveness of american industry against chinese koreans and whoever else and he talked about it and convinced me that what i'm doing is absolutely useless and i shouldn't give the talk in the morning the only reason i came actually here i i felt responsible for obligations but uh now thank you very much so now you see uh if i don't i i do something useful so you can take bilayer you can strain it and whatever you need to do you will generate uh is it terahertz yeah it will be terahertz ten million volts thank you very much i'm going to go back to the maybe first slide that you had early on you don't have to bring it up but uh when you start with uh slanzis device you have certain overlap integrals and so on and we have values for those uh when you have a strain uh they're going to change and uh some of the symmetry that you that was originally put into slanzis device will be reduced so you'll have a representation that splits and you know what that is because you worked it out it's c2 it's not c6 uh so can you go back and now that you've done the calculation and you know which level because they don't all get so uh much perturbed just a couple that you said not all the um uh overlap integrals are strongly affected i that's what i got from your talk i'm not sure that that was correct so uh if you could uh tell the experimentalists a little bit in terms of the original hamiltonian which we already know and we have values for then some terms will not be changed much and maybe enough other terms will be but we can go and find out with a particular stretch uh those terms and how they've changed and give you the changed parameters under a certain amount of stress yeah so that that's the point i was trying to make when i uh oh i didn't get that clearly when i wanted to to say how big is the fact let's say of one percent of strength yeah you told us but yeah so what so what we know we know is absolute values of gamma node gamma 3 and gamma 1 very well now the change of gamma 1 will not produce any qualitative effect so i don't even buzzer about it uh the change of gamma node and gamma 3 related to the lower symmetry will result in in this additional in this additional perturbation in the hamiltonian which will change the bend structure so what do we know uh in in this expression we know pretty well from dft and from raman the value of the we have information about how a gamma node changes with the change of the distance between carbon atoms so that that that's what uh is very solid we don't know how gamma 3 changes there is no optical experiment which allowed to extract it uh at least i i don't know any of those uh for bilayers for biographed for graphite there is for biographed we try to extract this information and we struggled with two point the the interlayer separation is slightly different and this may actually change the value of this logarithmic derivative and second uh the the measurement itself i i i i wasn't sure i understand what's the uh what the process of extraction of the parameter was but so i i i i'm saying i don't know what this parameter is i don't want to claim it's very uh which that that ratio change of law gamma that that's your new parameter yeah that's parameter that we don't know that you don't know so you can turn it in the in the following way we know this so okay another a question related to that and then i'll shut up uh it uh in slanzuski vice there's a difference between the valence band and and conduction band a level land level separation and under strain that should change and maybe uh not symmetrically uh so my question is does it change symmetrically or not uh we're talking about very so let me go back that would be about the same size as changes in gamma three which is what you know what what i'm saying is that we're talking about very low energy scale so this uh electron holosymmetry is quadratic in the electron momentum so it it would be noticeable if we would look at small changes in the energies of london levels at high magnetic fields but in this part of the momentum space quantitatively this is very little