 No, no, no the speakers Okay, thanks. Okay, so our next speaker is Maya Verneuri From the Max Black Institute for Chemical Physics of Solids and the the North Sea International Physics Center and She will give us a Lecture and a tutorial on the role of crystalline symmetries in Ben theory hands-on Zidupak and IR E-wrap, sorry. Okay. Thank you Maya and the first yours Okay, thank you and Timo for the introduction and Being in charge of everything here. I can imagine it's not easy so yeah in in this talk I'm gonna talk about how to diagnose topology when we We will have crystalline symmetries basically when we have a crystal with symmetries and I'm gonna give an introduction to the Theoretical background and then we're gonna do a hands-on session on Zidupak and a new rep as and Timo said and I'm gonna do this together with Miquelira Ola and Inigo Robredo since these are the ones that have mainly developed and used these two codes and they have thought about Very interesting exercises Okay, so before The crystalline symmetries start playing an important important role in topological materials the periodic table of topology Can you hear me? the periodic table of topological materials was actually given by the tenfold way where they consider the most important anti-unitary symmetries time and reversal symmetry particle hole and chiral symmetry and here you have the different classes and The dimensions because topology is about symmetry and dimensionality the square of the of the main symmetries and the topological classification, so We can recognize some familiar topological phases here we have the churn insulator including the halte model and an integer quantum hole effect We don't need any symmetry and here below we have the 2d and 3d topological insulators and This image these topological phases were actually protected by one of these anti-unitary symmetries Um For before the way we could diagnose topology in a system was as was It has been introduced in the past day by calculating the Wilson loops of the non-Abelian very phase by using these days the displacement of the Vanier Char centers which is a way of calculating of course Whether a system is topological or not but with the inclusion of trial translation and Crystalline symmetries in particular inversion for centrosymmetric systems for one came propose the Parity criterion formula that says Okay, if we have a trivial insulator and a topological insulator If we go from one to the other one we face We see a phase transition and faces transition in van theory needs to happen by closing a gap so if we go from one Phase to another one we need to close a gap and reopen again and something need to happen in the In the in the bands to have these different phases. Well, what happens is that bands get inverted? so the parity criterion formula what claims is that if we consider the Again the the product of the eigenvalues at the high symmetry points the inversion eigenvalues and It's odd. We're gonna have a topological phase if it's even we're gonna have a trivial phase and this actually accelerates and Makes more efficient the diagnosis of topological materials Later on It was discovered in 2011 by Liam Fu that also point group symmetries can protect topological phases This was the case of the mirror turn insulator in Dindeluride So it will have mirrors in a system. You will have mirror symmetry in a system We're gonna have different mirror planes and in each mirror plane our wave function is gonna Have the I mean our bands are gonna have the eigenvalues of the mirror operation so if we define a surface space for each eigenvalue of the mirror symmetry for I or Minus I if we have a spin or recoupling or plus minus one if we don't We have a two-dimensional block that I have an alchemyltonian and is this two-dimensional block that I have an alchemyltonian We can define a turn number and this turn number protects at the political phase that actually displays Dear a cone in some particular surfaces is protected by mirror symmetry and one band Belongs to the eigenvalue plus I and the other one to the eigenvalue minus I What I want to say here is that the eigenvalue of the symmetry operators when we start dealing with crystalline symmetries I start playing an important role and calculated them becomes also very important so Keep Playing with crystal symmetries other type of topological phases were found like the hourglass fermion and the higher dirtiest And a new C2 index or a symmetry indicator was defined So the parity criterion of one cane is related to one band inversion But the C4 index is related with two band inversion because we can have two band inversion in different surface spaces And for example in the case of hourglass fermion It will have light symmetry what we get is like 2d rack cones in the surface And in the case of a higher 30 I when we have inversion what we get is a hinge states at the edges of Of our system in this in this case B smooth So crystalline symmetries add to the topological classification a lot of variety and it enrich the field but it was somehow a bit chaotic at the beginning because there was not like a clear classification of the crystal symmetries and This is what I'm gonna explain now and why the irrep code that Michael has developed becomes very important So if we consider the crystal structure of our 3d material We have a 230 space groups and the ingredients of the crystalline structure are you need? lattice translation point group operations non-symorphic symmetries and of course we have orbitals and atoms in some lattice positions so So we have our crystal and now I need to define a few concepts One is the concept of the atomic limit that was introduced by David on Tuesday But I want to bring it back again because it's very very important. So I David and Alex say Develop or describe in 2011 was the obstruction To an atomic limit when we are dealing with that topological phase and the atomic limit is defined or the trivial phase is defined by calculating the van yes states integrating over the block functions and It happens when we have we can have a set of van yes states maximally localized on the atomic site whenever this happened if we have an insulator this insulator is gonna be trivial because the church is Absolutely localized in the atomic site the political phases is about what is happening in the church about what is happening in the gap? Okay, so the atomic limit is one comes out and now we're gonna Bridge it to another concept that comes from a group theory. So I want to define now. There is a question See for what is the symmetry indicator to detect a hot is C4 C4 equals to Okay, so now I'm gonna define another concept that is very important for this Construction of classification of topological crystalline symmetries, which is elementary van representation is EDR And it works in the following way So whenever we have a crystal structure within the same lattice Let's say the same space group. We can have different arrangements of the atoms We can have one atom per unit cell two atoms per unit cell in the honeycomb lattice Three atoms per unit cell in the Kagome and this multiplicity of the atoms in the unit cell is defined by the bike of position When a we have one atom a the most symmetric position in the cell to be we have two atoms be the second most symmetric Position in the cell and so on and so forth So the way function is not gonna transform in the same way where we have a triangular lattice Where we preserve C6 or we have a honeycomb lattice. Okay, so each arrangement determines different representations Now if we pick up for example the honeycomb lattice and we add orbitals We can add SS or PC orbitals or PX and PY So if we add S or PC orbitals the rotational symmetry is gonna be preserved in the XY plane But if we add PX and PY they transform into each other Which means that the S and PC Represent by themselves an irreducible representation, but the PX and PY they need to come together I mean they cannot be disentangled because they transform into each other all right, so If we consider The symmetries of the side that are called the side symmetry group Which are the symmetries that leave the point invariant and the orbitals that we add and the lattice with these three ingredients We can uniquely define the bands that we're gonna see in their reciprocal space and we can uniquely define them because the Way function is gonna transform in a particular way at the high symmetry in four points And this transformation is gonna be given by the little groups at the different high symmetry points. Okay, so it will have Honeycomb lattice S orbitals and the two wave I composition gamma one gamma four k3 and one and four are gonna defines the bands in reciprocal space and In the same way when we have PX and PY orbitals, we're gonna have also the corresponding Little groups at the high symmetry points the funding these bands Okay, so we have the atomic limit and we have the elementary one representation And what is the relation of these with topology the relation of these with topology comes from this paper from Michelle and Sack and other papers from Sack to where they claim that elementary band representations are actually connected in reciprocal space and They are connected in reciprocal space because actually what happens is an elementary band representation Defines a charge localizing the atomic side, which is actually the atomic limit So an elementary one representation described by these three ingredients orbital atomic site and a lattice What is actually describing is a set of maximally localized Bands that can be veneerizable respecting all the crystal symmetries, right? So Which means that? What we are doing by defining an elementary one representation. We are defining are actually an Atomic limit of a set of bands. Okay, so The way we are gonna classify the topological phases in Materials is by we are gonna classify all possible trivial phases And whenever our face that not belong to one of these trivial phases. It has to be topological So we have our lattice We see whether we can vaneerize it by Looking at whether we have an elementary band representation if the answer is yes We're not gonna have a political insulator and the charge is gonna be localized on the lattice on the contrary If we have our lattice and we cannot vaneerize it, which means we cannot define an elementary band representation These bands are gonna be disconnected in reciprocal space and the gap is gonna be topological Okay, and the charge is gonna be not localized at the atomic site Alright, so this is actually why We need to It's very important to actually Calculate the irreps. How do we calculate the irreps? So I think seeing these Ones in your life is not gonna harm at least and at least you know where things come from. Okay, so we have Honeycomb lattice with these generators To the bike of position and PC is painful orbitals in this site and We're gonna calculate the site symmetry group of G that leaves the Q invariant. Okay, so Let's say so the our generators are C2 C3 and M1 So if we apply C3 to our site this site We move it to this other one and by a lattice translation we can go to the original site So C3 is a symmetry operation of the site symmetry group If we apply mirror absolutely nothing happens so mirror is also a site of the site symmetry group and We apply C2 we move it here and there is no way we can go back to the original site by a combination of the lattice parameters So C2 is not gonna be part of the site symmetry group. Okay, so the site symmetry group is gonna be described by the point group C3v all right so What is the irrep that actually described these C3v of the lattice so what we need to do is calculate the matrix representation for each symmetry operation and then Sum up the traces when we sum up the traces we go to the crystallographic tables and we identify that actually is gamma 6 the irrep that actually is related to the traces of the of The symmetry operations of this point group. All right So now that we have the site symmetry group we need to consider the coset of Our system and the coset are actually the set of operation That's we need to add to the site symmetry group to reproduce the full group So the elements of the coset are gonna be C2 and the identity All right, so the multiplicity of the coset are gonna be It's gonna be 2 and this is why actually we have a Multiplicity 2 in the unit cell it's related to the fact that we have a 2v our atom place in the 2v by composition All right, so we have identified the the point group that actually leaves the The the Q invariant and we have identified the coset the composition and actually what we do is we want to induce the Reducible representation in the full space group. All right, and for that what we Do is define a matrix representation. So everything is about matrix that actually Under which our wave function transform in real space. So for each Let's say generator of the unit cell will have a different matrix. Well, I mean, I don't want to this is this is quite Let's say difficult to explain in such a short time But basically we have to define a matrix representation for its symmetry operator and we need to Fourier transfer to reciprocate space All right, and here we have the elements of the coset the composition and the translation and the generators There are books where you can see that So we go to reciprocate space We have our matrix representation now as a function of K and our matrix actually it has all the Operations, but it's gonna be debated in different block diagonals And what we do is again the same we calculate the traces of all the operators for our matrix representation for the different k points and we compare with the tables and what we see is that our At least Representation in reciprocal space at the gamma point is gonna be given by gamma 7 and gamma 8 because it's actually the ones that are produced By adding the traces the traces that we get in our matrix Okay, this is how we calculate the irreducible representation in and now we go back again to topology so We have our Real space three ingredients orbitals lattice Bike of position and with these we can define the Depends in reciprocal space by the irreducible representations at this high symmetry point okay, so All these has been tabulated in a bilbao crystallographic server in the section to political quantum chemistry so if you want to know what should be Let's say the the VR's the bands that are connected in reciprocal space That you should check for your system We go to this application band rep we add here the the space group and this is what we get So for every a space group in in reciprocal space We'll have this table. These are the bike of positions These are the irreducible representation in reciprocal space and these are the possible EBRs at the high symmetry points So these EBRs are gonna define a set of bands that are connected These EBRs are gonna define another set of bands that are connected for the different bike of positions So if you have your E reps at the high symmetry point You can tell whether they form an elementary band representation or not And also there is this function where we tell you whether your bands are decomposable or not because some of them They cannot be decomposed and they cannot actually display topological phases But also some of them can be decomposed. All right, so there is all this information here And if for example you want to Do it like in a faster way Mika will talk about how to generate the file that needs to be uploaded in the in the application Check topological map that can be calculated with your rep. It's called trace Text and you can get the results of all the symmetry indicators and the EREPs at the high symmetry point Okay, so basically what we want to know now in order to check whether our system is topological or not Is what I'm showing you in the in the screen now So we need to know at the high symmetry points What are the irreducible representation and with this information and And The bill bow crystallographic server we can tell whether our Our system is topological or not This I'm gonna skip so So basically The message that they want to send is that When we have crystalline symmetries We just need to calculate the wave function at the high symmetry points and be able to Identify the EREPs and with that information is enough at least to say whether your system is topological or not And this can be done with two codes C2 pack and EREP C2 pack Has the advantage that when we want to deal with Crystalline TI's not C2 but crystalline TI's as I told you before we need taken values of the symmetry operations Okay, so C2 pack what does is calculate taking values based Wilson loops And this is something that as far as I know vania tools so vania 90 cannot do Advantage it works for the FT codes vania 90 and tight binding models and calculates actually several invariants like the C2 mirror turn and the turn The turn topological invariant This is what C2 pack does and you will also use some exercises about how to calculate the Wilson loops based Again values based Wilson loop for 14th the right actually and then we have this other code that actually This is very convenient to do a proper symmetry analysis of the crystal It calculates the EREPs and the symmetry can value for every high symmetry points works for DFT and vania 90 and It also calculates some indices like like C2 and C4 and both of them have a the great advantage that There is no need to vania rise. It's just It's just the input of the ab initia calculation. So you can avoid all the projections and things like that Okay, so now I want to say a couple of things So a step and said well EREP has been made mainly developed by Mikkel and a step and it's Mikkel the one Who's gonna give you some some exercises so step and said that in order to get a position in the vast country you need to actually develop a code and Describe it with the vast fonts. These are the vast fonts and this is true But this is just a part of it. You also need to be Regional champion So it's the plan was the runner-up of the input quad championship and he can lift up to 134 kilos and 164 static something like that. So in case you need help with your baggages, you know who to call and Well This is cool is in memorial of our friend Alexey Soloyanoff. He was the developer Together with Dominic Ress of C2PAC when there were almost no codes available to calculate the topological invariance and even now It seems easy you vania rise Or a more easy to vania rise will have any tools back then around 2013 I think or 2014 there was almost no codes and it was really not Evident and we could not understand in the same way topology as we do now So this is part of the legacy of Alexey, but it's not all of it Alexey also together with David Um Formulate the obstruction to the C2 topological insulators that actually is the description of the atomic limit that this has been fundamental to develop of the crystalline symmetry space topological classification and No, not only that he also predicted the first another line in materials and the type to vile and Well, I mean he was an extraordinary and outstanding scientist, but he was also a Wonderful and caring friend, so I will join the organizers to say that we miss you Alexey. Thank you Now we're gonna start the real hands-on session with this theoretical background I hope I convince you that irreps and egg and values are very important. If not, I'm sure Mikkel or Inigo will do it Maybe while they set up if there are any questions from Participants in presence here raise your hand step and is there anyone having questions down at the Adriatico? Any questions here? No so far, okay So Okay, so Mikkel will conduct a session but in you and I can help with any problem that you might encounter okay, so So the first thing that we have to do is get the The script that is going to guide our session so we can find it in the github Opus of an entity here and in you code We go just to the to our sessions directory and here Okay, I'm going to start by by presenting here the it's a Python package that calculates irreversible representations of DFT bands So the question is why do we need Europe? So as Maya said we can determine the the topolo you can diagnose the topoly and classify topoly of our DFT materials our DFT calculations if we have the irreversible representations of balance bands, so Europe is a Code that calculates a Python code that calculates these irreversible representations of balance bands So Good, how does it work? How does it determine the irreversible representations? So it for that it uses traces, okay so When we calculate with DFT The wave function a wave function in our band. Okay, what we are What we mean is that we have the coefficients of the expansion of this block state In a particular basis. So for example in the case of bus quantum espresso these bases are plain waves, okay? so then when we want to To calculate how these state transforms we need to know how our bases transforms under symmetry operations of the space group Okay, and after that we can just take the The block here, so we calculate the the bracket and the traces So why do we need traces? every representation and in particular irreversible representations are Characterized for their traces. So remember that in our representation. We have a matrix for each Symmetry operation of our group. Okay, and then the set of traces of these matrices is Unique for each irreversible representation. So each irreversible representation has its own traces of symmetry operations, which is called character So what irreversible does is just read these The coefficients of this expansion of our block waves Calculate the traces and compare them with the traces of that are in the tables of irreversible representations Okay, so why is irreversible a good code? Why is it good to use irreversible? First of all it works for all 230 space groups with calculations that Include or don't include spin orbit coupling It can work with many DFT codes, BASPA, VNIT, quantum espresso and codes that have interfaces To one year 90 because it also reads the input files of one year 90 You can do your you can run your DFT calculation in any unit cell because then irreversible is going to Transform it to the unit cell used for the for the to write the tables of irrevers and It uses the same notation as with the Bilbao crystallographic server Which is a widely used server to to look the irrevers and it's open source, which means that it's free Okay, it's very easily installed We are not going to install it here because it's already installed for us, but it's very easy to install it You just have to write to type P-pistol irrevers in a command line Interface and it's done. Okay, you can also install it in a development mode if you want to develop the code or make some changes Okay, so we are going to start now with our first example. We are going to apply irrep So the first thing that we have to do is copy the We are in our home directory. Let me do this We have to copy the the material in our home directory and it's the second session Okay, so it's copying And we have it now. So what we are we are going to start with the IntelliWrite and we are going to calculate the reusable representations of balance bands and We are also going to check the calculation of the set 2 and set 4 numbers. So let's go to our Directory You see that there are many directors here. We are working with irrep IntelliWrite The part of more. Okay, so let me explain you that first When we want to run irrep, of course, we need the wave function So we we need to run a DFT calculations DFT calculation in order to to get these wave functions So the first step is always that to run a DFT calculation here. We have run a DFT Calculation with quantum espresso for you because it's it takes some time and we just want to focus on running irrep so we have Placed in the directory DFT all the The input files that you need to to run this calculation with quantum espresso and also the outputs in particular the out Directory here contains the wave functions so Let's start with irrep We can to its folder and to the first example, which is to calculate the irreps at maximal k points Okay, okay, so everyone is here, right? good so As I said, we have the the band structure and the wave functions calculated with quantum espresso and We are going to run it for for that first. We have to activate our Python environment We do it with this command Okay, you will see that it's activated now and we are going to run irrep So this is the command the way the one here that we are going to run Let me first comment on the parameters that we are setting in this command Okay, so you will see that there are some names or parameters that start by a dash like code here Okay, these are the actual names of the parameters that we can That we can tune when we run irrep So for example here, we are telling Europe that the DFT calculation was done with the with quantum espresso Okay, in prefix we are giving to it the path to the directory That contains as I said our model wave functions that we calculated and SMT is the The prefix argument that we said when running quantum espresso What's a cat a cat is the cutoff that irrep adopts For the way functions. Okay, so remember that our way functions are expanded in terms of plain waves So here we are getting rid of the coefficients that correspond to To plain waves of energy larger than 50 eb why because usually the information Most part of the information of our way function is is Is stored in the in the shortest way in the plane waves with shortest? Wave number Okay, so k points. This is a Usually a list of indices which indices so in our DFT calculation We we had many k points But we want the reserve representations of the first k point the 16th k point the k point number 31 and 46 Okay, we are just selecting the k points that we we are interested in and This kp names in similar it's telling Europe that the first k point is labeled as x the second k 16th k point is labelled as w and so on. How can we know the the levels that we have to set? So with Bilbao crystallographic serve There's a program here if you go to the first Toolbox space symmetry Kbeck and If you introduce here the number of your space group in this case is a face-enter cubic space group number 225 and hit enter You see here. There are the levels of the of the k points. Okay, so we are using just the same levels here That's how we can get the levels We get to know the levels with IV and we are Telling Europe that we just wanted reserve representations of the first 30 bands Which are the balance bands? We are we don't care about the conduction bands because they are not related to the topology of our material and We are telling Europe also to parse the the Fermi energy of more in the quantum espresso calculation And we are telling it also to To save the the output in our fight in this fight. So let's copy this this comment here We paste it in the The terminal and we run it take some seconds And now let's have a look at the at the output file. Okay So we don't want wrapping off of the output Okay, so first we have the it reprints the primitive vectors of our of the unit cell that we used in the DFT calculation Okay, in this case, it's a primitive unit cell then also information of the atoms that are in the in the Unit cell and the reciprocal lattice and after that you will find These two matrices the reviews in matrix and the vectors if you see what's what's this So as I say tell you I told you before We can run the DFT calculation in any unit cell as long as then we are able to transform this unit cell to the To the one that is used to write the irreversible representations in tables Okay, so irref automatically calculates the transformation to this To the unit cells so that you know, we don't we don't have to take care of it and This matrix is just express this transformation. Okay, so Yes, they express this the this equation here C1 C2 and C3 are there the matrices They are sorry the vectors that define the unit cell of the used in the tables a 1 a 2 and a 3 are the our DFT primitive vectors and M is the matrix that reviews it Shift you see is just expressing the shift of the of the origin of our cell with respect to the unit cell of the of tables Okay, after that you will find here a description of the space group. Okay, and each symmetry of variation of the space group and after This huge block of Information of the space group you will find you will find our main output Which are the traces of symmetry operations and the reservoir representations as you see so for example No, we have such a block for each of the k points. This is the k point zero point five zero zero point five And they receive all representations of of the bands. Okay, so for example, if we take the the band structure We can place the the irreversible representations on top of our bands So for example at the point X here, we see that the last balance band has irref X9 The the previous band has x8 the same happens for For can be done for w we have w7 w6 w7 the same order you see So this gives us knowledge about the reservoir representations of our balance bands and before closing this file Let me show you that there's a particular in symmetry operation, which is very interesting for us Which is this one the 25th operation. It's inversion. Okay, you see that the corresponding matrix is inversion This is going to be important later When we want to check the calculation of the set four and set two numbers, okay So, okay, we have the reservoir representations of our balance bands But we don't know yet whether they are topolical or not for that We have to take these irreversible representations and check if as Maya said if they can be written as as Combination linear combination of EBRs Okay, if they can be written with interior positive coefficients, then our bands are trivial And otherwise they are topolical so we have to do this work for that You will realize that in the directory where you run irrep. There's a file trace.txt, which was created by by irrep Okay, and it's going to It's going to be used now to to determine if our bands are topolical or not So we go to the below crystallographic server and we are going to upload this file where we hit on the black box topolical quantum chemistry check topological math Okay, so we are older, right and we have to To upload the trace.txt file We hit show and it's going to tell us whether our bands are topolical or not Okay, so let me anticipate the result. It's here. So it tells us We have a picture also there So It's going to tell us that our bands are topological. Okay, and they have Yeah, so it's the same result as here So it tells us that our 30 valence bands are topolical because they cannot be written as some of linear combination of EBRs and The topology is characterized by this is indicated by these symmetry based indicators Okay, so they set two and set four numbers are zero and they set eight numbers four which means that our material the interlure here can be a It's a mirror chair insulator. Okay, and it can be two different kinds of mirror chair insulators later on In your will use set to pack to to tell to determine which of the insulators Is the one of our face? Okay, so let me continue now by checking the the calculation of this set two and set four numbers So how can we do that? These numbers are related to the inversion eigenvalues Okay, they are obtained from inversion eigenvalues and irrep can calculate inversion eigenvalues That's why I Insisted before on the symmetry operation 25, which is inversion. So we are going to Do a calculation with Europe another calculation to check these numbers Okay, so we are here Okay in this path Europe inversion separate in this directory and here we are going to To run a command to to separate So we want at the end of day We want to cut to count the number of inversion odd parameters that we have okay in our balance funds So for that we are going to with Europe separate our bands into two subspaces one corresponding to odd behavior with respect to Inversion and the other one if the event subspace, okay So here we are telling Europe that we want to separate our balance bands based on eigenvalues of the 25th symmetry operation Which if you remember it's inversion, okay? The code is very there command is very similar to the previous one Just we are getting rid if you remember there was a k-point w now we are getting rid of it because It's it's a maximal k-point, but it's not left in body and under inversion Which means that if you apply inversion to it, it doesn't want it doesn't go to an equivalent K-point in every one so so Let's run the calculation. Okay, we copy this command here. We passed it in our terminal and we run the calculation Again, it takes some seconds Now we are going to open the the output file and we are going to check that the that our bands were separated by inversion Based on inversion so again if we escape the information, which is repeated. It's the same as in the previous calculation and we we reached the the blocks corresponding to our Ereps you see so it's telling us that we have separated our bands based on the 25th operation inversion and first it's listing the the bands corresponding to to even I am value of inversion and if you go down, okay, you will see that there are There's also a block corresponding to You see the inversion old I am values and a nice feature of here is that it automatically Counts the crammers the old crammers pairs for us So for example here at the point X we have three old crammers pairs Okay, with this information we are can apply the formulas to calculate the set to answer for numbers So the set two numbers calculated just with a full famous Fulcrum formulas which consists on counting the number of crammers Old all crammers pairs in all the time reversal invariant moment in our brilliance on okay so there are At gamma there are three old crammers pairs Yeah, here here at X. There are another three Old crammers pairs, but the X point appears three times in the in our brilliance also We have to multiply by three. There are seven old crammers per se at L But but there are four L points at the every answer. So We calculate this this multiplication. Okay, it's 40, but this set two numbers define modulo two So at the end of the day is equivalent to set two equal to zero as the blue or crystallographic server told us from irreps output We can apply also this formula to calculate the set for number. Okay. It's just okay, so Let me ask the question Okay, so Someone is asking what's the meaning of the set eight equal to four so It means that our material can be more it's a topolica insulator first and it can be It's a mirror chair in insulator. It can belong to two different phases of a mirror chain. So later, okay And a new was late. So at this point, we don't know which of the insulators is just by looking at the symmetry based indicators We will have to calculate the wisdom loops and in yours later going to use set to back to to tell us which of the insulator is Actually the one that we have Isim said is telling irrep that we want to to split the bands that we have in terms of The inversion eigenvalues. Okay, the 25th symmetry operation, which as I said, it's inversion So they are also asking what's If there is a fourfold the energy at a time in Maria moment that was three zero. So okay There can be symmetry operations which have three zero Even I mean if they are two fold if they correspond to two fold they generate Way functions or fourfold they generate with fancy. It doesn't matter So it's not important if there's a single symmetry operation with true zero What is important is the the set of traces corresponding to all symmetry operation. That's what characterizes the The irreversible representation. So in principle, there's not there's not any problem Okay, so if there was such a I mean nonsensical Result I would recommend to check the DFT calculation and improve its its convergence in principle and also check if we are Typing the command correctly But in principle, there's not any problem with having a just a trace of a symmetry operation equal to zero Good so As I said, we can calculate the set for number with this formula here again IREP has counted the inversion odd and inversion even States more grammar space for us. We just apply the formula which keeps again a number that is zero because this set for number is defined Modulo 4 Okay, so remember that this is the result that the low crystallography Crystallographic server told us from the information of irreversible representations that we calculated with with Europe Okay, so are there any questions or any comments? maybe Yeah, maybe they're down In the other article is there any question or no so far Okay, so Okay, so let me explain the idea of this of this calculation now, okay It's another example which illustrates another application of Europe So as I I have shown now irreversible to to separate the bands based on inversion any values, but not on inversion also With respect to all the symmetry operations So we are going to illustrate these these Application in a different scenario here. Okay, so imagine that we have taken for example this material Okay, copper chubism of oxide and we have calculated its band structure along this path the gamma and x path Okay, and we say okay. There are there are crossings here So this band that is coming down is crossing the almost flat or horizontal bands But okay, are we sure that there are band crossings here? So we can zoom in of course, but I Can keep asking this question forever. Are you sure that there are crossings here? so a quantitative way of doing that is separating the bands in terms of Symmetry operation so because two bands that have different IM values of asymmetry operation They cannot have anti crosses between them if if they Yes, they are going to cross for sure. Okay, if they meet at an energy So if they had they belong to the same irreversible representation, then it would be very Much more difficult to have crossings between these two bands Okay, so in this material we are going to To check if these bands do really cross So we are going to use here for that the first thing that we have to do is okay We have our states Which can which could be the symmetry operation that is protecting our crossing So for that we have to to to determine that symmetry operation. Okay, so Let's go to To the other material that we have copper chubism of oxide There are some directors here again. We have done The DFT calculation for you with quantum espresso and the input files and output files are in the DFT directory We come to the irref directory And we are going to run irref here so The first thing is to get to know the the symmetry operation that may be protecting the the crossings So for that we are going to run irref, but with the only side only same level. This is telling irref to read the the Read information of the unit cell determine the space group and the symmetry operations and to stop after it does so So, okay, here are our symmetry operations like before and there's a particular Particularly interesting symmetry operation, which is this one 14. Why because we are interested in the bands that are connecting gamma and x. Okay and points at this line have Have the shape of zero as a component that can be different from zero between zero and zero point five and Again, the third component is zero. So such k vectors are left invariant by this operation. Okay Because this operation leaves invariant the the only component that these k points have So this operation is in is in the little group of these k points Good. So it's a uh candidate to be the symmetry that is protecting our crossings so Yeah So what if the crossing occurs somewhere not along the high symmetry line? Yeah, you mean, okay in a line with less symmetry So as long there are symmetries in your spatial crystal symmetries, I mean Apart from identity you can Still apply this procedure By using the symmetries that are left in general if you have a crossing at a General k point which only is left invariant by inversion. Sorry by identity Then the procedure is more subtle because there mean there are not crystal symmetries protecting our crossings Yeah, the crossings may exist, but They are not protected by a crystal symmetry Thank you. Yeah, thank you Good. So as I said the 14th symmetry operation It looks like a potential candidate to be to be the one that is protecting our crossing So let's check if that's the case. Okay So we are going to to copy the this command here And we are going to run it again We are telling you to separate our bands based on im values of the 14th symmetry operation Okay And we are telling it also to to generate files that may be used for band plotting Okay So let's copy this command We paste it in our command line And we run it Okay, so let me ask while it's running. Let me have a look at the zoom chat Yes, so someone is asking if we can apply irrep to calculate the residual representations of our conduction bands and the answer is yes There are two parameters, which for example appear in this In this command here that we are running just now. Okay, ib end and ib start start So these parameters can be used to to tell irrep Which are the bands we we are interested in so for which bands we want to calculate the residual representations In this case, we are telling irrep to calculate the the residual representations to the bands 129 130 31 and 32. Okay, so the bands that are in the range specified by this ib start and ib end Parameters so irrep is not valid just for the balance bands It can be used for any set of bands that that you have There's another question So there's another question Related to the above question of nico when i'm just interested in the symmetry levels at high symmetry k points Which symmetry level will the code write? This happens for example at x point of diamon for example, okay You mean which symmetry operation is going to Write so okay, that's that's a good question So in general irrep here for example, you see that it's writing Just a single form of asymmetry operation. Okay, but in the in our previous results. I'm going to show you now, okay so in our You just don't have to follow my steps now. Do you open and Output file you will see that here for example it's writing the Two matrices for the rotational part and also for the spinor rotation and so on that's because The unit cell that we use for the DFT calculation is not identical to the unit cell that we used for that Is used for to write the irrepressible representations in the tables So irrep has calculated the transformation matrix to this Conventional unit cell, okay Then it's going to it has printed the the symmetry operations represent no the symmetry operations expressions for both in the DFT unit cell and in the In the unit cell of the conventional of the alpha hour tables So it's going to give the description of both expressions of the symmetry operation So let's have a look now at the output file that was generated by our Last run of irrep. Okay, remember that we wanted to calculate to separate our Our bands based on the fourth in symmetry operation And you see that it it has been done. So if we skip the the symmetry operations description You see it's the eigenvalue model bands have been separated according to plus one and minus one eigenvalues of the fourth in symmetry operation and more interestingly irrep has generated some Files that are used or that can be used as we specified with the with the argument plot bands here It has also created or generated the files that we can pass to new plot in order to to plot our band Our bands, okay separated by this inversion eigenvalue. So there's a file here prepared for you Run new plot, which only contains it just contains a A new plot command you can run it Okay, and it's going to generate the It's going to generate the the files more a file with the A file with the With the plot of the bands. Okay, so It's just this file that I showed you before this one in the right. So here the The red band the red bands correspond I can make this bigger So the red bands correspond to to eigenvalue minus one of of the 14 symmetry operation, which is 14 symmetry operation is a glide Reflection, okay with respect to the x-plane and also in blue the the bands with plus one eigenvalue So it's clear that these bands are crossing they cannot be any discontinuity between these bands here So that was another the illustration of another applicability of irrep And Okay, so that's from my side. That's The presentation of irrep. I encourage you to To to use irrep of course and also to collaborate in it if you want By reporting us bugs Giving us your feedback and if you want to contribute it has its own it have repository Uh, so I encourage you to to do so and I want to finish this by thinking To step on for his patience and for let me let me work his scheme, which is a pleasure And now I'm free. I'm Ready to take questions if there are any That's a question in the zoom chat Can you please tell again how the labels in flure 2 are written? Okay, yes, so I guess that okay these labels. Yes, so these are their real representations of our balance bands so how can we Identify them so Okay, so let me show you the the output file from which we got them Okay, so they are here in our output file So for example, let's let's do it Let's do it for the gamma point So for example, we see that the The last valence band, which is the one at the bottom. Okay, I'll tell you more about this minus 1.7 So this one has irreplaceable representation gamma bar 11 Okay, the next valence band has A super representation gamma 8 and so forth and this is the way in which it's done for example for l Okay So for example for l We can see that the last valence band has irreplaceable representation l 9 Okay, like it's indicated here and l 4 l 5 for the previous one So you can do this for all the k points. Okay And that's the way in which you can Uh, you can place your irreplaceable representations on top of a band structure Okay, there's another question by Raphael. Is it possible to calculate the irreplaceable representation for magnetic symmetry operations? So that's a very interesting question. Also, uh At this point we we we haven't yet Implemented the magnetic just the magnetic groups in europe So it's something that We hope to do at some point, but we still haven't done so so at this point It just works for the non magnetic spacers 230 spacers But in a future we hope to do to do so. Okay, any other question? Yeah, maybe the last question How can we see the band inversion for this example of typotopological material? I think was uh, bismuth salinate so That's okay, that's our our questions. I would question. So we are sure that this uh A kind of band inversion which makes our topological bands our valence bands topological So probably some uh, there will be an irrep at some point and maybe at l because the gap is quite small here, but I cannot say tell it for sure There will be a K point at which the our bands have been inverted and there's going to be content There's going to be a A new rep at a conduction band, which if we exchange it with certain irreps of the valence band is going to give us valence bands that are trivial so the way in which you could do that is just Calculate the irrepressible representations also for the at least first conduction bands Okay, then let's say smaller energy of conduction bands and check which are the exchanges that Could give you trivial bands Switch over so that will be the topological material, right? Yeah, exactly. So you have to uh, check the the exchanges that give you trivial or topological bands You can use for that also the tables of the topology of the below crystallographic server in which you have the elementary band representation so that you can see Which exchanges give you bands that can be written as a combination of elementary band representations Yes, I think it is nicer than you can see which bands invert other than just You know uploading a file to to the server and then it tells oh you have something interesting going on Yeah, that's fine. That's a that would be a cool future. Yeah Like yeah, thank you Hey, so I think we should stop here For the lunch break. Let's thank again our speakers