 Okay, so we're moving to our second talk this afternoon. The speaker is Frank Paulman from Munich and he's going to tell us about quotient symmetry protected topological phenomena. Good. Thanks. And also thanks to the organizers for putting together this nice program and for giving me a hint to, for me to speak here. So, so I'm actually very happy about Nathan's talk because you already I'm showed that there are interesting phenomena that we find in symmetry protected topological phases. And what I want to show is that several of these phenomena can also survive even though the face itself is trivial. So so basically I want to show that certain topological phenomena, including edge modes and even topological phase transition can occur in trivial phases. So, so in the work that I'm presenting has been done in collaboration with Ruben Verizon, who was a student in my group in Munich, and Julian people who currently still is a student in my my group. So, so, so let me start slowly and I start basically reiterating a few things that Nathan already pointed out. So basically want to discuss or introduce the concept of different quantum phases and then show how we understand or distinguish different symmetry protected topological phases. So, so first of all, this is very general, but, but if we just talk about a gap quantum phase of matter. Well, we first of all talk about zero temperature. And, and we say that two Hamiltonians are in the same phase if there exists the path of gap Hamiltonians connecting them. In a, in a picture we would say that well we have now to Hamiltonian it's not, and age one, and we can now tune along some path given by by alpha and and the gap remains open so we can just do this without encountering any face transition, then we would say that these are in the same phase. And if there is an avoidable gap closing, then we would say that these are two different phases. So, of course, we might accidentally run across the face in a different position, but then we can find a different path, which, which is a gap then be on the same face but but if it's impossible to find any, any, any path without closing the gap or without encountering any kind of discontinuity, then, then we would say these are two different phases of matter. So and the kind of one of the prime examples for different topological phases is the the so called whole day in phase and while conventionally we find the whole day in phase in the spin one Heisenberg chain. And the, we can also think of a dimerized spin one half chain. And the spin one half chain or the dimerized been one half chain also has an SO three symmetry, where as the two cross the two is the subgroup of the symmetry time reversal symmetry translation symmetry so there's a bunch of symmetries in in in in this some showing here. And now, as we tune the parameter delta, in this case, we find that this system does undergo a phase transition at delta equals to zero. So, so really the translation variant Heisenberg chain is critical. So so so that's a critical point. And then we find that left and right of this critical point. There are two gap phases. So, and these gap phases cannot be distinguished by symmetry breaking, but instead they can be distinguished in terms of so called symmetry fractionization and this is something that Nathan already talked about, but let me just elaborate a little bit about how we can distinguish these two phases. And how, how, how, how basically, we can derive the topological invariant or label the two different phases. So, so, so we know that the Hamiltonian is gap, and we find now symmetric ground state. Which is now symmetric under certain symmetry operations. abstractly speaking this could be just G and age, but in practice this could be for example, global like spin rotations that are rotate all the spins by pi about the x axis or the axis. Right so so so have now a certain symmetry group under which the ground state and also the Hamiltonian is symmetric. And now what we are doing is we just now take a segment of our ground state and we apply the symmetry operation to it. Since it's a symmetry operation, it wouldn't change the the bulk, right because by definition if, if we just apply the symmetry operation to the state, it's invariant under the symmetry operation. However, it can act non trivially on on the edges. And what we can then do is we can then say that well, let us now represent the symmetry acting on the entire state in terms of operations acting only at the edges. And, and, and the assumption that we make now is that the symmetry acts, we were trivially on on the bike so we would say that inside like the the the local degrees of freedom have a have a linear representation of the of the symmetry. And if you stick to the example of spin rotations we would say that we can have like an integer spin rotation right so an integer spin rotation is a, and if you just take the, the, the algebra that's, that's a linear representation of the, of, for example, so the boundaries, in this case, however, might be a projective representation so productive representation is then a linear representation model or some face factor, and in particular sticking to the example of spin rotations, the, the boundary might transform in terms of half integer spin. Right so so and so in this gives us now a way how we can classify the different projective representation so we say that well, we have no certain symmetry group for symmetry group we can have certain different projective representations which are classified by sure multipliers or the second group homology, and then I can now classify can now just steal the labels that are used for classifying different projective representations and I can label different symmetry protected topological with it. So, so now we can apply exactly this to the aforementioned whole day model. So, so we have now here our dimerized Heisenberg chain, and we had this is the same face diagram that I showed before so we have the critical point where the model is a translation Heisenberg model, which is gapless and then we have these two gap phases. And if you now look at it, if we just look at the left hand side, we find that, well, at the, at the boundaries we have singlets and the singlets transform trivially under the symmetry and these are s equals to zero state at the boundary. So, so, so this would be the face factor is, is zero. So, so our x and our y commute. Right so so this is what we would say is now a trivial phase. So if we just take the other side of the face transition. Here we have the spin one half degrees of freedom localized at the edge right if you just take the extreme limit of dimerization. That's pretty clear that we have here just a single spin one half and the, the Z two cross Z two, or like the Z two rotations about the x or y axis, they're just represented in terms of the poly matrices the poly matrices anti commute. So we find that here, the face is, is pie. So now we have two different labels and these labels cannot change, unless we cross a face transition. So, and with these different labels or with the non trivial representation of the symmetry in terms of the edges, we get certain topological phenomena, in particular, based on these commutation relations we can directly deduce that we will have degenerate edge modes. And also, if we just look at the entanglement spectrum of the system, we find that they're necessarily degenerates. Good. So, so far, we have now, hopefully somewhat clean definition of what we mean by the, by an SPT phase. So now I want to just do the next step and I want to just destroy it. So, so let us now say that we have a Hamiltonian realizing a certain SPT phase. How can I trivialize it by trivializing I mean that I just can connect it to a trivial state without having to undergo a face transition. The way I mean one way on obvious way is that we just break the symmetry. So let's say we take the C2 C2 model, and we apply a staggered field to this, then the system becomes trivial, and all these nice features including the, the edge modes or degeneracies are immediately lifted right even if I apply a really tiny field. The edge modes are lifted and, and, and, and all these nice property go out of the window and if I look at these face diagram, and I just apply a staggered magnetic field. Then I would also point that there's no longer a phase transition. So, so the gap will be, we'll get small but we wouldn't hit an actual phase transition. So this is option number one, but there's another option which is maybe less obvious. So we can also extend the symmetry group. And if we extend the symmetry group, we can again find a path connecting the topological phase to the trivial phase. But the question is, if we do if we are doing this, to which extent to certain features that we learned from about from the topological phase, or certain how to which extent to certain topological phenomena actually survive. And this is what I want to discuss in in the following. I want to discuss it by by considering a very concrete model. And let me just walk you through the, the, the Hamiltonian. So, so I'm considering now a Hobart model. And so I have just a Hobart model, where we have now hopping between neighboring sites, but then on top of this, we have a staggered chemical potential. So the, so so the chemical potential is, is, is alternating so on on even sites say it's, it's it's lower than on on odd sites. And that's it's basically right so we have just a staggered chemical potential. And then we have a dimerized dimerized hopping to be allowed the hopping between neighboring sites to be the dimerized to be on say on on on even an odd bonds we have a different hopping strength. And at last we have the onsite Hobart interaction. So, so what I want to do now is I want to look at the phase diagram of this model, and I want to focus first on on a special limiting case. Let us now first consider the case where you, the Hobart interaction is is very large. The Hobart interaction is very large. We can apply perturbation theory, and we do get exactly the Hamiltonian that we had before, right so basically if you was infinitely large, the, the, the top of this phase diagram is exactly described by the Heisenberg Hamiltonian with a dimerized exchange interactions like so so so so for you infinity we exactly have the model that we had before. And so we know in the limit of you being infinity, that there is a phase transition between an SPT phase, and a trivial things. So but now if we just plot the entire phase diagram which we evaluated here using DMG, and it has also been evaluated by others before. What we actually find is that now, if we just lower you sufficiently that in that case, there is a path without actually crossing the phase transition. So, so there's a question in the chat, whether we can consider this extending of the symmetry as a central extension of the group, and maybe hold on for a bit I'm getting into the detail what we actually mean by extending the symmetry. So, so so now basically if we are considering now this, this, this full model model, we see that there is now a path that we can adiabatically connect them right so it's easiest to see even without doing numerics is we just start from the fully dimerized limits. And in that we can can see how we can reduce you to the non interacting limit. And once we are in the non interacting limit, this is basically just an SSH chain in the presence of a staggered chemical potential. And then we can actually directly show that there is a path where we never close the gap. So, so so now what we have shown that that there is actually a path between what used to be the SPT phase to what used to be the trivial phase so so it's, so so we have now actually trivialized to the, the SPT phase by extending the symmetry. And what basically happens in terms of the symmetry is that the, and how about model has an SU two symmetry, which in the limit of you goes to infinity becomes basically the quotient group of SU two and the fermion parity symmetry so so we have SU two mod the fermion parity. So so basically if you was infinitely large, then the, then the, the parity symmetry P becomes just a number which is is then same same minus one for for each side, and, and that's it. And if we have SU two for SU two, we know that there are no non trivial projective representation so the H two of SU two over you one is equals to zero, while H two of SO three is actually equal to Z two, so we get these two two phases. So, so, so what we basically have shown that is that the holding phase can be trivialized by adding charge fluctuations of the charge degrees of freedom. So, so, yes. So, so, so, so the way I mean I'm talking about how filling as you say, and so basically, if I just define my habit interaction display this ensures that the ground state is always at half filling. Yes. That's another question in the chat. Does color density show the parameter fire. No, no, no, thanks for this question. So, so, so the, the, the, the quantization of fire to having fire either. Zero or pie this only exists this distinction only the stick this distinction only exists for for you being infinitely large for any finite you this, this, this is no longer well defined as I'm showing in a, in a, in a moment, but let me just come to this so the, the, the, the color coding is really just really nice. So there's no no meaning to having these colors just to show that we can go without a discontinuity from one to the other phase. This is all all this means. Good. So, so, so, so but now we see that there is a path where we can go from one phase to the other without crossing a phase transition but we still we see that there is still a phase transition. In particular, we see that if we go down to a certain you, what what happened is we just have now the, the phase transition so this is a C equals to two to one critical point as we go here, down to some UC two. And if we just go below the C two and above UC one, then there is actually a first order transition between those two phases. So we're just commenting on what's going to happen or how we can explain this in a, in a moment. So, so let me let me know first discuss the stability of, of the edge modes. In particular, what we find is that the edge modes that we have for Delta, larger than than zero so so the these topologically protected edge modes, they are actually robust up to a certain certain point in this case diagram so basically if we look at the ground state degeneracy, we find that there is actually a four fold degenerate ground state degeneracy, all the way down to this dashed blue line. We have not a bike transition but a boundary transition to another state where where the, where the, where this feature, namely this 240 degeneracy disappears. And the way that we can understand it is that if we just look now how the symmetry acts on the on the boundary states that we find for the full Harvard model. The other relation is that our x times r y is equal to the fermion parity times our r y times times our x in the limit of you goes to infinity. The fermion parity is frozen to to minus one, and we get the r x times r y is equal to minus our y times r x, but, but the eigenvalue of PL cannot change immediately so we have a gap protecting. This this quantity and thus we find that the property of having the edge modes is actually so so so robust for a certain range of of you, a fairly easy way of seeing this is that if you just go to the completely limit if you go to the limit where delta is equals to one. So here we just have a completely like a completely decoupled spin on on on on the side, and then this you see that there is a level crossing exactly at you equals to to delta. Right so and, and if we now plot the low energy, the ground state and the low energy excitations of the full model for a finite delta so so now we are here at this, this dashed line. And what we see is that there is now this level crossing so for sufficiently large you, we find that there is a fourfold ground fourfold ground state degeneracy, and we can now identify the different ground states by having either kind of on the first and the last site and this is what this indicates that both up, both are down, one is up, one is down or the one is down or the other one is up. And if we, if we cross below this, then there's this kind of level crossing at the boundary states, and we see that the, the unique ground state is given where both, where both spins here are localized at the, at the left boundary So, so so so just to, to conclude so so so basically here, we have some, some cartoon state, which is a state that we would that this, this is a state that we would get for, for an infinite kind of a staggered field. Yes. Yes. No, no, no, no, if we have, and if you look at the Hamiltonian we have the staggered potential. This is very important because if we don't have this, if we don't have this, this delta. I mean, if we don't have this delta, then we basically have here some SSH chains and then we know that then then there are actually different topological phases and in that case, even if if if we, if we, if we wouldn't if we would set delta to zero. We have actually two distinct topological phases in terms of inversions like lettuce inversion symmetry. Yes. You say that it's clear that at some point there should be I think that you know, well, I think that I wanted to say that several years ago, I think that I think that's the point. Yes, exactly. So, so this you see to is an I think this is this has been looked at before. I mean, yes. I mean, what what what has been looked at before is this this line this line of delta equals to zero, this one has been looked at and it's known that there is this extended critical regime. Yes. But what we are just now basically arguing is that we are just transferring this to this exactly so so. We put these things together saying that well in the limit of you infinity that we do have SPT for you equals to zero we don't have SPT and now we are just putting this together and and one of the non trivial statements at least which wasn't obvious to me in the beginning was that that in fact the edge modes are robust right so we just find that this this property of these edge modes is in this is that these are protected for a finite range of parameters. But what do you mean. No, no, no, no. No, no, it's just the same so so so so in fact, this would maybe what you're referring to would be the case when delta is zero if you delta zero then you would have an SPT. And then you have a sushi for a model with fractional charges, but but but but we prevent this from happening by having this this ionic carbon model. Yes. And just because of the staggered chemical potential. So if you just, if you just were to make this potential very large then you find that the ground status is the one that the Dublin is on on the right on the left side. The, the, like, like you mean here on the right hand side. On the left side that it terminates with the at the icing transition. Probably I don't, I don't know any allergic I mean this is basically just reading off the results from from from from the numerics. And this argument, which which I just gave now for the degeneracy of the ground states we can directly apply also to the degeneracy of the entanglement spectrum in particular what we can do is we can now think of an infinite chain, which we cut at a given bond and put a partition into subsystem a everything left of this bond and subsystem be everything right of that bond, and we can then look at the spectrum of the reduced density matrix. And we find that that there is again a phase transition where are there basically the low energy states of the of the entanglement spectrum or equivalently the dominant eigenvectors of the reduced density matrix. Either have half integer or integer and spin. And then this is, I think also, at least intuitive, a nice understanding, because in the limit of large you there are no charge fluctuations, and the entanglement spectrum, either is only integer or half integer so we would find that either all energy states are degenerate, or all of them are non degeneracy. So, and, and if we if we allow charge fluctuations, then have like a half integers can can hop between the left and the right. And then at least the entanglement spectrum starts to mix and then there is again, just at a certain value of of of you, then there is a crossing of the integer and half integer levels. So as shown here. So, so so what I've shown so far is that the despite that those two phases can be now adiabatically connected that the edge modes remain robust over a certain range of parameters. Coming to the face transition to the face transition between the trivial and the SPT face then does not immediately get out after extending the symmetry group. And in particular, there are a few steps that I just want to mention here. So first of all, there is a duality symmetry between delta and minus delta. And so that if there is, if there is a direct transition between the two phases that always has to be at, at delta equals to zero. And we can see this duality symmetry from some modified translations where we just have now some symmetry D acting now on our line operators, which is in minus one to the power of n times CN right recall that the regular translations are no longer symmetry because of the staggered field, but these modified translations are then still a symmetry of the model. And based on this, we can also deduce that based on these modified translations that in the limit of you goes to infinity that we, we have a guaranteed lead Schultz-Martus theory, which guarantees a face transition at delta equals to zero. In particular, what we, what we use is that, and that in, in this case we have an onsite projective representation of the spin rotations right so so so for the translation invariant case, we have one spin per unit cell. And then we find that on each side, the rx and ry anti-commute and thus we can just immediately apply the deep Schultz-Martus theory. We can also extend this argument to, to finite you, and particularly if we have finite you, we actually find an emergent deep Schultz-Martus and theory, which, which, which enforces the stability of the, of this critical line further down of this face transition and let me just briefly tell you how we, how we can come to, to this conclusion. So, so, so, so I just pointed out, sorry, let me just come back to I pointed out, if you goes to infinity and if you don't have any charge fluctuations that then parity is a six to two minus one so we can just replace it by a number and that's why we find that they, that's onsite projective representation. So now, we argue, however, that the fermion parity string has a long range order as long as a fermionic parity operator is is kept in particular. We can now calculate a string order parameter where we just apply parity to a consecutive number of spins. We find that this scales as some constant smaller than one and larger than zero times e to the theta times the n minus m. So, so something which alternates and this theta can either be the zero or, or pi right so that we call in the limit of u infinity p is minus one and then we find that theta has to be pi because it's plus or minus one for even odd length of strings. So, so and if we have data equals to pi, this implies now some emergent anomaly that forbids unique and gap ground state so we can, for this emergent analogy we can show that by contradiction that if we assume a unique and gap ground state that we are running into a contradiction assuming that we have a pi here and thus, we can then prove that if we have this emergent anomaly that they are that that this yields a leap Schultz-Martis theory. And, and, and now there's basically two ways out of this. So there are two ways out of the system for kind of changing the fermion parity, it's either at the fermion parity gap closes, or it spontaneously breaks a symmetry, which destroys the these modified translation symmetry. And in our case, the system opts for the second option so it spontaneously breaks the symmetry. And this is happening along, but this is in this plot hard to see between UC one and UC two. And what we, what we actually do, we are now plotting here at along this, this line, we plot the, the, the string order, like we just plot what are going to design string order, which, which are differentiated between those two cases. So if the string order s plus is non zero, it tells us that a theta is zero. And if the s minus is non zero, it tells us that we are in this pi phase. And we see that indeed at this regime, where the break the symmetry like this coexistence region where this theta is no longer well defined we see that this is changing. So so we see that this is, this is the part where, where the, where the firm in parity is, is, is changing and this is where our deep shows matters theory and no longer holds and we can actually have. We can have a continuous path without crossing a phase transition. And this I, the, the, these ideas now I explained in detail for this model for this anionic and how about model, but exactly the same ideas generalize to, to general SPT model so the recipe is that we can say that well, let us now pick some SPT faith protected g, and then we can now extending the group g to a larger group g tilde, such that g is g mod h, and then we can find that the, that quantum numbers of this additional symmetry which in, in our case that we just looked in was was fermion parity. Those symmetry group and labels distinct representation and edge modes and remain reg modes and also transition remain robust as long as the excitations are charged that are charged under under age are gacked. So so it's a very general recipe that applies to that we can just use for for all SPT phases. So the same concept that we had here for for one dimensions where the SPT phase are classified by the second homology. We can also just do this to higher symmetries for example we can take a Z to two dimensional SPT and extended to Z for which is trivial, and then again trivialize an SPT phase for for two dimensions. So a couple of days ago or a week ago, there was actually a paper by Drew Potter and, and when, and it's not so long when I think, but, and, and, and they actually showed that the same idea also allow to to to find so called quotient protect quotient symmetry enriched phases right so so so the same ideas I extend to to to a number of different cases. So, so let me just very briefly now show a case where where we just have a realization of this SPT phase using exactly a Hubbard model of this type and this was done together with a group of a manual block where they amazing. And it builds to to actually design now for me how about letters so so so they have digital mirrors so that they can specifically design a letter says such such for example the this, this, this letter, and then depending on on how we basically visualize the model, they can now either tune into a into a SPT phase or they can tune into a trivial phase right so so so basically using the setup they can now exactly model this is face diagram that we are looking at at before. And then we limit ourselves to the limit of sufficiently large use so that we are deep in the in this regime of an SPT phase, and then they can using using quantum gas microscopy, they can actually directly measure, for example, non local order parameter they can measure string order parameters to distinguish different SPT phases so so the way that this can be done because the string order is in for the spin rotations and Z direction so they can just measure in the Z basis, and then by doing many snapshots, they can have directly measure the string order, and the string order then can distinguish the two different topological phases right so we can design a string order which should be non zero in the trivial phase, we can design a string order that should be a non zero in the other phase so we can just really tailor make the string orders to detect the different phases and then they can now be measured in in the setup, and also they can directly see the edge modes that are present in the in the non trivial in using these quantum gas mic microscopy. So, so this was just a short interview and now, before I'm concluding, I want to show one other way how we can like use these idea of of getting out out out symmetries, in particular, if we take now chains of that have a Z n cross the end symmetry, then the low energy symmetry, and so is the quotient group, which can stabilize actually direct trans continuous transitions, where we might not have expected them. And let me just briefly show the main idea and the main idea said, let us demonstrate this for the for a Z four cross the four model, where we can for the Z four cross the four we, we have four different topological phases, as opposed to the two that we had for the two cross the two. And, and, and we can now write down sort of toy Hamiltonians, which we, which was was done by the work by two and all. So, so you, these are generalizations of the, of the Z two cross the two cluster model to to hire groups of the n cross the end. And it is D is now labeling the face so so so for D equals to zero. This would be in the trivial faith but equals to one it's an SPT phase one and SPT phase two and three, depending on how we choose a D in this this Hamiltonian so this is nice to play around with and the symmetry generated are now the products of access. Where the product is over some subletters a and subletters B. And what was known before, and this is a work by two and others is that there are actually direct transitions between neighboring SPT phases so if you have a SPT phase D and D plus one, then there's in between a face transition with a continuous transition with C equals to two. And what wasn't really understood is what happens if you now have transitions between phases which are not neighboring right let's see that you have a tradition between HD and D plus one. And what you can actually show using the formalism that I just discussed that there is actually indeed one direct transition between those two with C equals to two to one. So here is similar very similar to what we just discussed before. So H2 conserves now local Z two symmetry generated by X squared, and the symmetry flux has long range order so we have these string order parameters that we can design and the low energy sub space is actually Z two cross Z two. So we see that in the projected to the low energy subspace, the model reduces to a simple and cluster state model. And it's impossible to enter intermediate SPT phases, as long as we have this long range order which is protected by a gap. So we find it that there's now not a fine tuned critical point but again, a regime where we have the C equals to one critical point, which is an end which which which is an ending again in a spontaneous symmetry broken face so so the same idea that I discussed in some previous video for the eye on a cupboard model also applies to the CN cross CN. And in this case we can actually use it to argue that there should be direct continuous phase transitions between SPT phases which are not neighboring phases in this context. We can now come to the conclusion. So, so there are different ways of trivializing SPT phases, we can break the symmetry which immediately throws all the nice properties out of the window, or we can extend the symmetry. And if we just extending the symmetry group as shown for example for the eye on a cupboard model. We can actually find that certain features such as the edge modes, and also the critical point will survive for for for certain range of parameters. And this has, I think quite some experimental relevance because the, if even if you are thinking about a simple spin model in a real material it's probably realized in terms of some some effective low energy model for a Harvard model. Now, what it basically tells us that that certain of these features are relatively robust so which makes it more likely to actually observe it. And, and the second part is about the transitions and the work as I said I want to thank Ruben and Julian for this nice collaboration. The first part is already published and the second part about the CN cross CN is currently in preparation. So, so this I want to conclude and thank you for your attention. So, thank you, Frank for this nice talking for keeping time. We have time for questions. So, thank you. When you when you build the project symmetry for the for considering this edge states that you have. So you're large, the symmetry group of your chain. Yes, and you build this project symmetry group right. Well, no, first, I mean we have, we start off with a symmetry in this case, so three. Yeah, where we have projective representation so we have basically a projective representation with zero and pipe. So, and now, now we enlarge it so we just go from so free to as you to. And in this case, as you to has no non trivial projective representations. As you to have even number dimensional representations. So you are going. No, but we don't have projective representation so so my question is out of what you build the projective group. You have a original point group or some group right that's your system and then you out of the original group you build the projective symmetry group. Maybe this is the answer to your so so so let us say that you have an SPT phase protected by some group G. That could be the two cross the two. Okay. And now what you're doing is you just extending the group to a larger group, but you're extending in such a way that the original group G is your G tilde mod H. Yeah, I understand that. Okay, maybe the night. So my question is, so for instance, at the beginning you had this is all three group, right, then you enlarge it. Yes, and you get the SU two. Yes. Okay, so the SO three has, you know, all the number dimensional representations so they correspond to integer angular momentum if you want. And then the SU two is the symmetry group of the half integer angular momentum. And then it has even dimensional representation. This is what you get the genesis. Yes. Yeah, sure. So my question is, in that case, when you go from SO three to SU two that means that your system which originally has an angular momentum now has a half integer angular momentum. My question is, which is the source of that half angular momentum that you are adding to the original one. The source that we get this is for this how about more or less that we just add permanent parity into the game. But maybe we can talk in private with me I'm still not 100% sure I understand the question correctly. Frank, I think there's a question on the chat. Any other questions from the room. How can we deduce the Hubbard parameter value you which corresponds to the most stable phase. Well, I mean, the, the face is not stable. I mean, one of the points I want to make is that it's not a stable phase anymore like the face itself is, is trivial so we can without closing a gap, we can just go from the what used to be the SPT face to a trivial say so. It's not a phase anymore if you're asking about what, what is the you for which the topological phenomena are most robust that would be that you was infinity, because if you infinity we have a, we have an actual SPT face. So maybe this answers your question I don't know. The only question. Yeah, I mean the others I. Oh wait. Yes, right to the other thoroughly answered on the way. Yes, it's just a question about the dimensionality so here you're talking about one these systems. So do these ideas somehow generalized to say 2d I mean yes, so like we can the classification of 2d topological phases by enlarging local. It's quite slight right so so so the same concept you can generalize to to higher dimensions for example you can take a 2d SPT, which is classified by, by third comology, then you can just start from the two where you have a non trivial third you can extend it to the for and make it reveal. So so the same idea applies also to higher dimensions. And as shown by this paper by Drew Potter. Also, you can apply this to symmetry and rich topological phases. Thank you again.