 First of all, I'd like to thank the organizer in particular, Professor Lauren Sieves, for giving me this opportunity to speak here. So this view graph shows the outline of my talk. First, I'd like to spend a few minutes in describing the future of electronic states in graphene, although you may already know. And then I'd like to talk about zero-mode anomalies appearing in various quantities, like diamagnetic susceptibilities, conductivity, and so on. Then I'd like to mention briefly about the special time-diverse symmetry and associated symmetry crossover. And then I'd like to talk about bioregraphene. Unfortunately, I don't have time to discuss phonons and electron-phono interactions in graphene, although very interesting. Now, before starting, let me mention the name of the collaborators. Mikito Koshino, who used to be in our group, but now moved to Tohoku universities. Yasunori Arimura and Masaki Noro, the former graduate students in our group, and Dr. Takeshi Nakanishi of AISD, AIST. Now, as you all know, in the graphene, the firm level lies in the so-called pi bands consisting of PZ orbital. And pi bands have a linear dispersion and cross at this k point, which is the corner point of such a hexagonal brian zone. Now, we have another corner point called k prime. The k and k prime are not equivalent, but they are ready to each other through the complex conjugate of the wave functions or a time-device operations. Now, the firm level lies here, so the electronic properties of the graphene and nanotubes, which is a road graphene sheet, are dominated by those states. Now, in this facility, the dispersion is like corns. As a result, electronic states are described by such a 2 by 2 matrix Schrodinger equation in the effective mass or k dot b scheme. Here, gamma is the steepness here or the velocity if you divide gamma by h bar. And kx and ky are the wave vector operators or derivatives. And fA and fB describe the amplitude at a side and b side in unit cell. Now, in terms of policy matrix, this is written like that, sigma dot k. And this is known as wireless equation for a neutrinos or a relativistic drug electron with vanishing rest mass. So you have a system in the relativistic limit. Now, this system also has some interesting singularities. Now, the equation motion looks like that, sigma dot k. And this means that spin sigma is quantized into the direction of the wave vector. It is known as a helicity of the neutrinos. And the wave function is written as a product of the spatial plane wave times the spin functions. And the spin, so it is given by spin rotation operators. Theta is the direction angle of the wave vectors. Now, the spin rotation operator changes sign, as you all know, if you rotate the spin once. And so all the state obtained by 180 degree rotation in one direction and the other direction have different signs. Of course, this spin is not a regular spin, but just a pseudo spin. And also the phase of the wave function is completely arbitrary. So you can choose the phase in such a way the wave function looks like that. This doesn't change the sign after 2 pi rotation, but then you have to take into account Barry's phase. Now, the Barry's phase is defined by this equation. You rotate the wave vector around the origin once, and you get extra phase. And this extra phase becomes non-trivial and changes the sign of the wave functions. And if this circle does not contain the origin, you immediately see that this phase disappears. So that means that you have a topological singularity at origin here. And this singularity leads to various interesting phenomena. One example is the presence of a Landau level at zero energy independent of the magnetic field. And this was noted by McLeer a very long time ago. And he was interested in the origin of very large diamagnetic susceptibility of a bulk graphite, a levitation, demonstrated by many people, including under Gaim. He calculated Landau levels and susceptibility in monographians. And he found out that the susceptibility has a delta function singularity. So it is infinite only at zero energy, and everywhere else vanishes. Now, this also leads to the so-called absence of bulk scattering. The bulk scattering corresponds to 180 degree rotation in the wave vector space. And usually, you have always timed reversal path, and you have to add these two contributions. But these two cancel each other because of this various phase, and you get a complete absence of bulk scattering. So on metallic nanotubes, it's a rolled graphene sheet. If one dimension systems, the metallic nanotube becomes a perfect conductor even in the presence of scatterers. Now, we have been working on these singularities, on the singularities in graphene for some time. And so I'd like to discuss these singularities on the diamagnetic susceptibility and transport quantities. Now, first, the diamagnetic susceptibilities. To understand the origin of very singular diamagnetic susceptibilities, we consider, for example, the response to spatially varying magnetic field, the magnetic field with wave vector Qs. Now, then, the susceptibility, of course, becomes dependent on the wave vector Q of the external magnetic field. Now, this shows an example of the chi of Q. At the Dirac point, chi of Q is just proportional to 1 over Q. So it becomes infinite in the long wavelength limit corresponding to the delta function singularities. Except the Dirac point, you have this curve. chi of Q as a function of Q of 2kf, Q divided by 2kf as universal functions. It furnaces up to 2kf and suddenly increases and then decays in proportion to 1 over Q. Now, this shows that the graphene does not have any response to slowly varying magnetic field when the wave vector is smaller than the typical larger than the typical wavelength is larger than the typical wavelength of the graphene, Fermi wavelength. Now, if you plot the chi of Q as a function of the energy, you get this curve here. No response here. And this area is independent of the wave vectors. So in the limit of a long wavelength limit, this becomes a delta function. Now, we can also consider effects of bandgap. If you place graphene on top of certain substrate, you can introduce the asymmetry, the A side and B side. Then you have effectively this kind of Hamiltonian, delta and minus delta. Then you have a gap. So the density of states vanishes within the gap like this. And the susceptibility, it becomes this step function. It is known zero, only inside the gap. Now, the susceptibility makes a discrete jump toward the paramagnetic direction when the Fermi level goes into the conduction band and the valence band. Now, this jump behavior, jump height can be understood by looking at the lambda level structure of these systems. This shows an example of such lambda levels. Now, the n equals 0 lambda level, which used to be at the direct point, now moves to the top of valence band for the k point and moves to the bottom of the conduction band for the k prime point. All other lambda levels are degenerate between k and k prime point. Now, if you look at this structure, you immediately notice that this is nothing but the lambda levels in vacuum. Now, in fact, if you calculate the effective Hamiltonian in the vicinity of the bottom of the conduction band, you have these Hamiltonians. The first term represent the kinetic energy determined by the effective mass proportional to the gap. Now, the second term represent the Schubert-Zermann energy and plus 4k point and minus 4k prime point. The presence of the Schubert-Zermann energy shows that we have a non-zero magnetic moment, even in the absence of magnetic field, but the direction perpendicular to the plane. But the direction is different between the k and k prime point. Now, in any case, if you have this Hamiltonian, you can easily calculate the susceptibilities. We have two terms, a polyparamagnetic susceptibility due to Schubert-Zermann energies and lambda diamagnetic susceptibility due to these first kinetic energies, lambda quantizations. Now, we know that this diamagnetic susceptibility is one-third of the polys susceptibility, so you have paramagnetic contributions. And in fact, this jump height toward this paramagnetic direction can be understood by the discrete jump in the density of states, the effective mass. Now, if you know that the susceptibility is independent of the energy within the band, and if you assume that the susceptibility should vanish when the system is empty or when the system is completely filled, then you can understand these kind of behaviors by these simple pictures. Now, this singularity also appears in the conductivities. The density of states is a linear function like this, so it vanishes at the zero energy. So the people call graphene a zero-gap semiconductor, but this is actually quite wrong. Now, this becomes clear if you calculate the conductivities. Now, you can use various formula, but here I'd like to use simple Einstein relations. The conductivity is a product of the diffusion constant times the density of states, and the diffusion constant is velocity squared times the relaxation time, and the velocity is constant, and the relaxation time is proportional to the inverse of the density of states through the final state in the scattering. As a result, the density of state disappears here, and you get a very simple expression for the connectivity. Now, here, doubly, the dimensionless parameter characterizing the strength of the scattering, u is the impurity strength, and n sub i the impurity densities. Now, this shows that the conductivity is independent of the electron concentration or the energy as long as you neglect possible dependence of this doubly on the electron concentration of the firm energy through the screening or the kinetic energy and so on. So you have a metal here. Now, at the direct point, of course, you cannot use this simple argument because the density of states vanishes, and you immediately see that the conductivity here is nearly universal, it takes universal values. So we can expect that the conductivity should exhibit such singular behaviors. Now, the similar singularity can appear in various quantities like transport coefficients in magnetic field. In this case, the field dependence is completely scaled by the firm energies. Here, h bar omega b is the effecting magnetic energy proportional to the lambda level separation. It's proportional to the scared b. And this shows that vanishing EF corresponds to infinite magnetic field or zero magnetic field depending on how you approach this zero energies. The similar singularity can appear also in the dynamical conductivities. Now, the dynamical conductivity sigma of omega consists of two terms, as has been discussed by Dr. Fedon Avoris. They drew the conductivity determined by the states in the vicinity of the firm energies and interband conductivity. Interband conductivity is universal, as was observed experimentally. But it turns out that the frequency dependence is completely scaled by the firm energy in this system. So again, zero firm energy means maybe infinite frequency conductivity or interband conductivity given by universal universal values. So you get these singular behaviors again. Now, this singularity should be removed by including level broadening effect. So we started working with these problems a long time ago, more than 10 years ago, 13 years ago. First, Guihon Shon calculated the density of states and conductivity using the so-called self-consistent bone approximation. Now, the density of states, this approximation is known to work quite well as long as the disorder is not so strong unless the localization effect is not important. It's important. Now, the linear density of states becomes larger with increasing disorder parameter w. And the density of states at the direct point becomes known zero, of course. Now, the Boltzmann conductivity is constant, independent of the energies. And if you take into account level broadening effect, the conductivity drops down to this universal value given by this expression, independent of the scattering strength. Now, you see very singular energy here. And this energy scale is shown here. It depends on this scattering disorder parameter in a singular manner. Now, the conductivity is, of course, measured in graphines. And this shows one example of the first experiment reported by Kostian of Osloff and Andre Geimes. And they observed this linear increase. And this linear increase means that the disorder parameter should decrease in proportion to the inverse of the electron concentrations. And you can think of various kind of scatters, a typical example that charges impurities with or without screening. And in fact, the simple calculations can show that you have a linear dependence here. The minimum conductivity, those are the correction of the minimum conductivity in these papers. Many of the data lie here, which is usually larger than the theoretical prediction. Maybe, in fact, the pi or something like that. So people called these missing pi problems, those are the experimental results of Philip Keeman co-workers, non-universal behaviors. There were lots of lots of theories on this subject. And I cannot follow all those theories. So recently, I extended my old calculation to the case of long-range scatters. First, I consider scatters with Gaussian potential with range t. Then you can study this dependence on d in a clear manner. And this shows an example of the calculated minimum conductivity as a function of the disorder parameter w. When the range is very small, here range is measured in units of lattice constant. And when the range is small, the minimum conductivity is essentially independent. But when the range is large, 20 or so, it's not so big, then the minimum conductivity increases with increasing a disorder parameter twice, three times, four times so you can get any value depending on your sample quality. Now, you can make calculations for charged impurities if you use Thomas Fermatab screening approximations. Now, then conductivity increases linearly as a function of electron concentration, of course. And of course, you have some kind of minimum conductivities. And minimum conductivity is larger than the minimum conductivity in the short-range case. So again, it is not universal. Now, one of the important questions is, of course, the dominant scattering mechanisms in graphene on certain substrates. Now, there have been lots of experiments. And here I show just a few examples of the recent experiments on the environmental screening effect. I think the first experiment was reported by the group of Maryland universities. They put very thin ice on top of the graphene and they observed slight increase of the conductivities. Now, these people from the Manchester University put liquid ethanol on top of graphene and directly constant varies between 25, 55, depending on these temperatures. So essentially, no change. A slight increase, maybe, like this. Those are the experimental data, unpublished data, of the Maryland University group. And those are the experimental results of other group, another group. This shows the experimental results reported by the French group. They measured the Shubnikov-Tohasa oscillation and analyzed their data, assuming certain relaxation times. And they compared this relaxation time with the transport relaxation time obtained by whole mobilities. And they say that the ratio is about 2. Now, the calculations of this environmental screening effect is very simple. You just consider this kind of systems, graphene on top of silicon dioxide, some dielectric materials here. Then the effective dielectric constant becomes the average of these two dielectric constants. And so it looks like that. And calculation is very easy. You see it changed the dielectric material, constant of the top materials. And you see that the conductivity, of course, increases. But this amount of increase is not so big. This is mainly due to the reduction of the screening effect due to dielectric constant. So the increase of this dielectric constant is canceled by the decrease of the screening effect. Now, this shows the ratio of the two relaxation time, transport relaxation time, and use your lifetime relaxation time. This ratio lies between 2 and 3, even for the vacuum here. It's slightly larger than 2, but smaller than 3, like this. So this shows that the charging purity should be regarded as short-range characters in this system. This is quite in contrast to the situation in gallium arsenide, aluminum gallium arsenide heterostructures. There, this ratio is known to be very, very big. Now, with increasing dielectric constant, of course, because the screening becomes weaker, you increase effective range. In fact, this ratio becomes larger. And also, the minimum conductivity increases with increasing dielectric constant. Maybe it's due to the long range nature of the scatters. Now, there can be various other scattering mechanisms giving rise to the behaviors of the conductivity proportional to the electron concentration. And many people now are suggesting that you can have some resonance scatterings. And here I show one example of the results we made quite recently. Now, if you put certain strong and short-range scatterers, and if you have a gap, you always have some acceptor state above the valence band top. And if you increase the scattering strength, these acceptor states approach the center of the gap. And if you have attractive potential, then you have donors. And the situation is similar. But in graphene, you don't have a gap. So all those bound states become virtual bound states like this. So if you increase the potential strength, you have some resonances. But this resonance happens, you need very strong scatterers in this case. But the situation may change if you have two scatterers at A side and B side like that. In that case, you have very large splitting between bonding and anti-bonding states. And the anti-bonding states or bonding states can appear just at the direct point. And if you have such a resonance here, the effective scattering strength is proportional to the inverse of the energy, essentially. So you have the conductivity proportional to that electron concentration. Now, this shows that it's not so easy to get observed real zero-mode anomalies in molar graphene. But maybe the situation is different in different systems. And this shows the example of the interband-magnet absorption spectrum in epitaxial graphene measured by the group of Marek Potemski of Grenoble high-magnetic field lab. Now, they observe these interband absorptions as a function of the frequencies and as a function of the magnetic field. And according to their result, the critical lambda level index, where this lambda level structure disappears, it seems to be almost independent of the strength of the magnetic field. And this behavior is consistent with the scattering strength independent of the energy. In fact, the old calculation shows that the broadening of a lambda level is proportional to the effective magnetic energies and proportional to the scale of the disorder parameters. And the lambda level separation is, of course, proportional to the effective magnetic energy. So you have this critical lambda level index. N equal 10 gives W 1 over 80 or 1 over 100 or something like that. So maybe in this kind of systems, or you can really see the zero-mode anomalies at the Dirac point. Now, we have also suspended graphene. Graphene's embolonite tried substrate, CVD grown, and many different kinds of graphene. So of course, the situation can depend on those different kinds of systems. Now, the disorder, of course, affects this singularity in the diamagnetic susceptibilities. Now, if you assume broadening independent of the energy, you can expect that the delta function becomes a Lorentzian line shape like this. In fact, Hideo Fukuyama showed this result. But maybe the constant energy approximation, constant broadening approximation may not be valid because we have linear density of states and vanishing density of states at the Dirac point. So we did make similar calculation about the same time using the self-consistent bone approximation. The results are shown here. You have very singular energy dependence and decaying very slowly as a function of the firm energies. Now, quite recently, we extended this calculation also to the case of this long-range scatterers like scatterers with Gaussian potential with range t. Then you get this result. See susceptibilities as a function of energies. Now, the broadening is just determined by the range of scattering potential d. So it's a gamma over d. So this shows that the susceptibility becomes non-zero only for states which can be mixed with the Dirac point by disorders. Now, at low temperatures, the disorder becomes dominant in determining the transport. In the presence of disorder, all the symmetries are broken except that those related to time reversal. And if you make the time reversal operations, if you take the complex conjugate of the wave functions, e to the ikl becomes e to the minus ikl. So k becomes minus k, but minus k the same as k prime. So if you make the real time reversal operation, k fk becomes fk prime. fk prime becomes fk. And if you repeat this operation twice, you get plus signs. Now, within each k and k prime point, you have k dot p Hamiltonian, so in fact, shredding equations. So you can define a sort of time reversal operation, special time reversal operation, within each k and k prime point. Now, because of the presence of spin operators, you have to introduce this k matrix k, which is the square of k is equal to minus 1. As a result, if you repeat this operation twice, you get this minus sign. Now, depending on these signs, you have different symmetries. Now, if this sign is positive, you have the orthogonal symmetries. And the matrix element of each Hamiltonian can be chosen as real numbers. But if you have a symmetry, symplectic, then matrix element is given by quote on your real. And if you destroy the symmetries, the systems now have unitary symmetry. And the matrix element is always given by complex numbers. Now, this symmetry crossover plays important roles in reflection process. Consider such a reflection process, incident alpha and reflected beta bar. Bar means a time reversal. The time reversal process is incident beta and reflected alpha bar. So between these reflection coefficients, we have the symmetry relations. Actually, this sign, t squared or s squared appears at here. So between these coefficients, you have this sign difference. So in the case of the s matrix, this reflection coefficient matrix is completely anti-symmetric. So the diagonal element vanishes corresponding to the absence of backscatterings. Now, this s symmetry is not a real time reversal symmetry. So you can destroy the symmetry by various perturbations. First, the equi-energy line graphene is slightly deviated from the complete circle. This is known as the trigonal whooping. This effect can be incorporated by adding such higher order K dot p terms. Now, this h prime destroys these s symmetries. And if you have lattice distortions, you have some gauge field, effective vector potentials, then you destroy it again these s symmetries. Or you have local curvatures or optical phonon-like distortions or destroy these s symmetries. Now, if you have scatters, short-range scatters causing interval scattering between K and K prime point, the symmetry within each K and K prime point becomes irrelevant. So again, you have symmetry crossovers. Now, this symmetry plays very important rules in this transport at low temperatures. Now, like quantum corrections is a localization effect, or maybe in the conductance fluctuations. Now, if you have the usual orthogonal symmetries, you have negative quantum corrections. And if you apply a magnetic field, you destroy these quantum corrections so the connectivity increases. And or you have negative magneto-resistance. And if you have symplectic symmetry, you have positive quantum corrections called anti-localization corrections. And you destroy this term by magnetic field, so you have positive magneto-resistance. This crossover was discussed by Suzura nine years ago. And more complete theory was established by Ed Macian, the chairman here, and Professor Falcoz and co-workers. And there have been lots of experiments, I think. And in particular, this paper by T. Kornenko and co-workers in this paper, they demonstrated the real crossover depending on temperatures and the samples and so on. Now, bilayer graphene. Now, bilayer graphene was fabricated about the same time as the monolayer graphene and integer quantum whole effect was observed. Now, this shows the structure of the bilayer graphene. If you look at A site, nothing here. B site here, nothing here. And but here you have A1 and B1 and A2 here. Now, of course, you have interlayer interactions. The strongest is, of course, between B1 and A2 here. I cannot see because of this, but in any case, you can also have some next nearest interaction, next nearest neighbor interactions and so on. But if you stick to the, if you keep only the most important terms, the problem is simpler. So let's do that. Then you have four by four matrix Hamiltonians. This two by two represents the electron motion in the top layers. And this is a two by two. And the bottom layers, of course, this Hamiltonian was discussed first by Edmachian and Professor Falco. Now, the interlayer interaction between B1 and A2 appears here as delta, which is exactly equal to gamma 1, which is known to be about 0.4 electron volt in bulk graphite. Because of these interactions, two bands are pushed away. And you have effectively two by two matrix Hamiltonians, but power becomes two. But the Landau level structure. So you have two polybolic bands. But the Landau level structure is different. You have two degenerate Landau levels at zero energy, again, independent of the strength of the magnetic field. So again, you have singular susceptibilities, orbital susceptibilities, now logarithmic. Now, this shows the energy band. Two polybolic band touching together here's excited conduction band, excited valence band. Energy separation is 0.4 electron volt. It's parallel. Now, in bi-layer graphene, of course, there have been lots of works on the transport and various other quantities. And in particular, Mikuto Koshino applied the self-consistent bond approximation and showed that the minimum conductivity at the zero energies is, again, very close to universal. But actual value is twice as large as the minimum conductivity for monolegraphene. So quite recently, I extended this calculation to the case of long-range scatterers, again, such as Gaussian potentials. And here, I showed again the minimum conductivity as a function of the disorder parameters, W. Now, the range is measured in units of certain length scale corresponding to excited conduction band. Separation with delta. Now, if you have a short-range scatterers, again, the minimum conductivity is universal. But if you have a long-range scatterers, see the minimum conductivity becomes larger and larger with increasing disorder. So again, it depends on your sample qualities. Now, you can make calculations for the charged impurities also if you make Thomas Fermtab screening approximation. Then the conductivity increases when you go to the higher band because of the increase in the screening effect. And also, the minimum conductivity is increased slightly as a function of the impurity concentration. So again, it's not universal. Now, important feature of the biolographene is you can open up a gap, as you all know, I think. Now, if you have the potential difference between top layer and bottom layers, now you can open up a gap. Now, the gap is essentially given by this energy separation between top and bottom layers. Now, because you have such a Mexican heart-like dispersion, the effective gap is a little bit smaller. But essentially, the order of the gap is this potential difference between the top and bottom layers. Now, you have interesting behaviors in the density of states, looks like that. But if you calculate the conductivities, the conductivity doesn't exhibit such any features associated with this density of states. And it just drops, becomes very small inside the original gap. Now, you may also fabricate biolographene with A.A. stockings. Now, the biolographene with A.A. stocking looks like that. And in this case, the Hamiltonian now looks like that. You have interlayer interaction here, A1 and A2 and B1 and B2, so it's like this. This Hamiltonian is, however, not so interesting because it can be converted into this Hamiltonian by a simple unitary transformations. So you have two monolayer-like bands separated from each other by certain energies. And this energy is given by, is equal to the coupling. Or it can be modified by the applied electric field perpendicular to the layers. But in any case, it's just the combination of these two monolayer bands. So the density of states is very simple like that. Now, so you can make some calculations. So the density of states is not so interesting. If, of course, you have some accumulation of the density of states due to disorders. But if you look at the conductivities, now you have very interesting behaviors at the new Dirac point here, even for charged impurities. Now, in this case, you have a very large density of states. So you have large screenings. So effectively, the charged impurities become short-range here, again. So you have very sharp drop corresponding to zero-mode anomalies. So maybe in biographies with AS stacking, you can really see this zero-mode anomalies. Now, if you have the combination of monolayer and biographies, and you have an interface like a zigzag interface, or maybe armchair-like interface, you can enjoy another interesting physics. Now here, we just calculated the electron transmission through these boundaries interface. Now it turns out that this transmission strongly depends on the incident angle. And this shows an example here. So if the electron is in the k point, you have the maximum transmission in these directions. And the direction is opposite between k and k prime point. So if you make such a transmission, effectively, you can realize valepolarization, different population difference between k and k prime point. Then maybe you have some kind of magnetization or something like that if you have a band gap. Now, if you apply a magnetic field, you have the combination of the lambda levels in monolayers and biolayer graphene. They are completely different. So in between, you have some crossovers. Now, we found out that you have some flat regions here. And it should be regarded as interface lambda levels. Now, this flat region interface lambda level should be can be proved by looking at the local density of states or maybe STS. Now, this shows the local density of states at A side and B side of the monolayer side, and A1, B1, and A2, and B2 in the biolayer side. So in fact, here, this structure is due to this flat band, flat lambda levels. And you can see many other features associated with this flat interface lambda levels. So let me summarize what I have talked about. In the effective mass approximation, the electron in graphene is a neutrinos or a relativistic Dirac electron with vanishing res mass. So this equation, the system has interest singularity, topological singularity associated various phase, causing various zero motor anomalies in the diamagnetic susceptibilities and various transport quantities. Now, we have interesting crossover due to time-diversal symmetries. And the biolayer under the biolayer graphene, of course, is very interesting. And you can have some interface. If you have an interface, again, you can have some interesting physics. So thank you very much for your attention. Question OK. Do you or do we understand the magnetism of graphite? The value, why it's so big or not? Do we need graphene for this or? I think the graphene is much better because you have a real monolayer graphite. Although I didn't talk about the, if you have a bilayers, you have logarithmic singularities, and if you have a trilayers, you have a monolayer plus just bilayers. So if you have many, many layers, essentially the susceptibility is dominated by the logarithmic singularities of bilayer graphene. So I think essentially the physics of very large diamagnetic susceptibility in bulk graphite should be the same as that in biolayer graphene. But you always have some stuck in fault or something like that, so we don't know. But this is a very strong statement because everyone thinks that graphite has susceptibility of the same origin as bismuth. And you are saying that it's not the case. Bismuth is also very diamagnetic, and it has very tiny pockets of electrons, and holes with small masses, and so on. So that's the question. Do we need graphene at all to understand graphite, or is there the physics is the same as in bismuth? No, the bismuth, we make some calculations for bismuth. The bismuth is nothing but the graphene with gap. In the bismuth, you have its 3D systems. So you have the wave vector in the direction of the magnetic field. So that the wave vector gives effective gap, and the gap increases with the wave vectors. So you have the combination of this with different gap. So you have this kind of a Mount Fuji-like dependence showing. And so we can explain the large dielectric constant in bismuth. Large diamagnetic susceptibility. President Scatteras can you comment on those? How would they see the picture that was one transparency? I believe you mentioned President Scatteras. What about, all right, its linear dependence on concentration, but what about minimum conductivity? Does it become universal, or non-universal, or whatever? I didn't get that one. The problem of this one, if you have a very large scattering, it's not so easy. I cannot use the self-consistent bone approximation. So I couldn't do any calculation on this. The self-consistent calculation is not so easy. Is this calculation basically? It's just one. You're taking up just a large value of W, you promised. No, no, no. Here, I just calculate the effective scattering potential. But how is the scattering difference from the earlier model when your earlier calculations, where you say you have a short range of potential? I mean, how does this differ? This is actually you. You prove it is strength. Uh-huh, uh-huh, OK. Did you also do these sort of calculations for the bilayer? No, no, no, I didn't do that. I didn't have time. So your calculations in the bilayer where you looked at the gap bilayer in the presence of disorder. Yes, that's a charged impurity. That was just for charged impurities. Or some gaussian, weak gaussians scattered. And would it be, you would not necessarily then be able to explain why it looks like it doesn't seem to be much of a transport gap in a lot of experiments? No, according to the calculations, the conductivity is very small within the original gap. So that calculation shows that you should see the real gap. But actually, the experiments, maybe not, do not show that. So we should look at a different type. Yes, maybe. There are different kind of scatterers playing dominant roles. I don't know. What should we look for in experiments if we want to study the monolayer bilayer edge states? Edge states? Is there a difference then between an edge state and your interface lambda level? But those are not the edge states, not edge states. It's a crossover between monolayer and bilayer. You still don't understand why you get these flat regions. Calculations, or if you look at these equations, you can really show that you should have some flat bands. But you still don't understand the real origin of that is flat lambda levels here. So let's thank Professor Ahn again.