Dominic Williamson: Anyons and matrix product operator algebras

well Dominica sitting up maybe a Justin owns the talk so session is Dominic William son and he'll talk about any ons and matrix product operator algebras okay great seems like it's on now yeah so I'm from the first Roddick group in Vienna and I'm it's a pleasure to be here thanks to the people at Microsoft for organizing this conference it's being great so far and I want to talk to you about some recent work we did connecting tensor networks basically to any ohms in a very kind of concrete way so the overview of the result is that we kind of start from some matrix product operators that are sort of inspired by symmetries that we've earlier earlier found in tens of networks that have topological order and then just sort of trying to use basic theory of tensor network States to get from this all the way to like the physical anions and actually constructing them very concretely in wave functions so the goal like the hope is then in future we can use this kind of representation to really like explore more quantum information any aspects of like an iam braided and young computing and other things like that great oh yeah and I'll give you a small overview of the talk so I'm gonna start by giving some background or there after the last talk maybe I can go quicker because we had a nice explanation of sort of a neon models and hopefully everyone would tune or that's nice lectures on the weekend I'll give a little brief overview of tenzin it works as well if you didn't but that should be good and then I'll get him to the results so cool let's start yeah okay so here's just a nice little mug shots of all my collaborators on this project the lion's share of the work I think was done by Nick Bolton can make he'll Marion who we're at Ghent also in Frank's group we also have the brac working on this and our supervisors used a hakama and Frank fist Radha so yeah all these guys need to optimize their SEO search because this is what you find if you google their names it's not great okay so I I think although 2016 was bad for a lot of reasons it was very good for topological physics there was recognition of some really great early work in the topological phases of meta by coastal it's Alice and Haldane and I think the really great thing about this for me is that you can really like trace the paths from their research to what's going on now like and it's sort of exploded so all the stuff we studied with topological order in 2d systems with a neon models and the sort of thing and like SPT and 1d you can kind of trace back to what these guys did yeah and perhaps one of the most interesting things that I'm sure they didn't they never expected when they were doing their work was the connection that was found between these kind of topologically ordered systems and so the quantum memories due to kitaev so I guess I'll just briefly say the idea kind of that because you have this topological degree of freedom that can't be accessed locally by any set of measurement then if you so just think about this you can rephrase this sort of condition of topological stability is just it being a code and I probably need more detail into that but yeah so I think Ellen the last thing that really was great in the pasta has been the experimental progress so the other group of Microsoft and Santa Barbara led by Friedman one of the other people who worked early on topological order for quantum computation has been sort of racing towards making these Meier on a qubits and other ideas sort of originally at least popularized by kitaev and I think I saw somewhere in an interview with Calvin joven one of the leading experimentalist that he's predicting 2017 to be the year of observing breeding which would be amazing I mean if not for topological quantum computation just for topological physics so I think it's a really exciting time to be working on this field cool so I guess the fundamental question of all this that is also just fundamentally it should be interesting to people in quantum information is just a classification of states problem so you can phrase it as a Hamiltonian problem or States but here I'll just stick to the state formulation so the question is really what states of matter can they possibly be I think this this line is ripped directly from the Nobel Prize announcement for like future work on this field and you can sort of phrase it in this way where you just think about some family of states so I've kind of been a bit sloppy on the board there but you have some family of states on though on lattices and you've got to pick the lettuce or sort of messy details but you want to sort of identify some Universal properties of these states and group them into equivalence classes what you don't care about sort of local details and the way we wash out the local details is by taking equivalence up to some constant depth local unitary so families of states equivalent up to constant depth local unitary and then one of the fundamental questions or probably the fundamental questions in topological phases of matter is what are the phases and how can we sort of come up with invariants such as if you give me some state or Hamiltonian I can sort of calculate something and tell you which phase it's in cool so the answer in 1d is kind of simple for bosonic systems at least there's sort of no topological order by this definition of course there's a lot of interesting things happening there in terms of spt but we won't talk more about that in 2d I think there isn't really a rigorous to the level of this community classification although the answer is known to condense matter theorists to their level of rigor and the answer is in terms of these anions or super set selection sectors so this is kind of answering both questions at once you sort of look at consistent particle theories of topological excitations in these models and these provide you some kind of invariants of the phases and also tell you by looking at what possible of these models there can be it tells you what possible phases they can be yeah so I guess the core idea of this is looking at super selection sectors so here I'm denoting rows sort of the reduced density matrix kind of in this little circle region there and if we want to sort of look at the again we want to kind of wash out local details so we want to look at an equivalence class of kind of what reduced density matrices we can have here after we put in some excitations on top of the ground state upto local things we can do and here the the only restriction is that has to be unitary within that region so you can imagine the toy code you could have like 1a excitation and you kind of enclose this in the big region then this will be a distinct super selection sector from the vacuum because even though we can create other pairs of particles and particles or whatever you like there's always going to be an excess of one particle when we sort of do the overall counting and by taking this equivalence class yeah of these local reduced density matrices up to unit areas within a region we can kind of find that a discrete set of super selection sectors and this is so the starting point for building the ante on theory and just yeah one kind of key property of these things is that they really have some kind of they really change the correlation structure between the inside and the outside of this region so if I have a different sort of super selection sector within this region and then I kind of cut it out so everywhere around that the sort of state will still just be looking like the ground state locally but in fact these two states although they look the same are very different and you can kind of see this by analyzing entanglement properties okay so we abstract this kind of theory of super selection sectors into some diagrammatic calculus has also probably popularized by Kataya but goes back a bit further in terms of a more heavily categorical notions and the most basic thing you can do with these super selection sectors to just bring them together to form a bigger super selection sector which is described by this kind of fusion process and this will give you like some kind of algebra structure of the fusion and then I think the thing that actually makes it makes these super selection sectors so interesting for quantum computation or interesting at all is because the fusion the process is not actually associative it's only associative up to some kind of factor and you have this which way degree of freedom that sort of a topological encoded Hilbert space when you fuse three of these things and in inside this little piece of which way information you can encode if you can unicode some data and also when you look at the associativity you actually find this sub matrix acting on that degree of freedom the other thing you can do with these um super selection sectors in sort of space-time is start to braid them around each other and because like the path of one or on the other and 2d is sort of Homo topically non-trivial so you can't contract it you actually don't have to find just boson or fermions statistics you can find sort of more crazy statistics and even sort of unit areas that act within some larger Hilbert space and it's just sort of symbolized by this sort of crossing tenser which usually you write in some fusion basis so overall these things can't be anything they have to sort of fit together in a nice way and this sort of Pentagon equation is just the consistency for the F symbol and the key point when you get here is that the F symbol data and the fusion structure is kind of just it's sort of decoupled from this well it's sort of independent of this braiding structure so you can sort of solve this one first and then once you find these solutions you can look for different braided solutions on top of that and these are really this is sort of one level structure more than the Pentagon equation oh and just finally so when you solve these equations you rarely get a solution up to some gauge relations so one thing people like to do is look at gauge invariant put in quantities so the most popular ones of this s matrix and this T matrix so this is like the topological spin of the particle you can't give it a twist okay so now I'm too tense network background so yeah I guess the idea of tensor networks you can tell it's quite a good idea because it's come up quite a few times in different places before like exploding I guess in the work of the group of serac in the early 2000s but I guess earlier it came up in to me in this paper of Fanus snack to gala Verna as pure finitely correlated states where they were trying to take sort of continuum limits in a rigorous way and also it came up numerically in the work of a Steve white when he sort of created emoji although he didn't look at it that way it was later rephrase that's attention to work algorithm and I guess the big success of tensor networks is all in one dimension where like both theoretically and numerically it works really well so like numerically it works so well that it even works the systems that shouldn't work for like where the gap is closing but for gaps yeah spin chains you kind of it's just the go-to algorithm and in terms of theory there's been a lot of work by pres Garcia shook Estrada and serac as well as wolf I think sort of classic sort of characterizing beyond NSF de Gallivan of paper the structure of 1d matrix product states a 1d tense Network States which are called matrix product states okay so let's get to some more details usually we use this kind of notation which I'm going to use about the rest of the talk so hopefully you saw this in norbit's talk but we indicate these tensors which are just sort of arrays of complex numbers via these little boxes so if you stick in a label on each edge to fix it to fix it you'll find it just a complex number there we also use this kind of summation convention where if two lines are joined you sort of have a dummy index which is summed over in that tensor okay so now you can use this to create a state so in 1d you have these matrix product states which you just get by basically decomposing the coefficient in some basis where you've chosen the local basis on each side into a product of matrices so the reason this works really well is because computers are good at computer at computing products of matrices and this is what it looks like in the diagrammatic form ilysm okay and now onto a bit of structure of these things so if you think about what these matrices are doing let's maybe go back to this one as you hop along you're getting acted on by some guy from this set of matrices depending on the index down here the physical spin and as you move along the chain you can think of maybe I stay within some invariant subspace of this set of matrices so you can then just sort of rewrite just think of writing this mate these set of matrices in this basis of the invariant subspaces so if I have like some smallest invariant subspace I can kind of pull out that block and it'll be in irreducible algebra and then I might have another invariant subspace that sort of bigger than that one and that will be another block and then there might be some hopping between them and I can sort of continue this to get like an upper triangular block form for this set of matrices and then you realize because you're sort of tracing them at the end you have this trace here if you ever hopped when you were going along the chain that contribution will die so actually what you can do is just to get an equivalent matrix product state you just throw away all these off diagonal blocks sets that you just put zeros up here and you'll find the same state which is just given by this one the sum of these sort of irreducible States so there's a sort of the important building blocks for tensor Network States in 1d and now the second really important sort of result which I guess is now being called the fundamental theorem of matrix product States is that given a few you like nice conditions if you have these kind of to to such matrix product states that are equal for all lengths then there actually exists some local gage relation that relates the tenses which is I think still surprising to me but it's quite cool so you can see the converse that kind of obviously true I stick on this conjugated a tensor everywhere into a matrix product state all the XX inverses will just cancel and I'll be left with the same matrix products as if I just stuck in the a itself but this theorem is really proving the converse that if I have two seemingly unrelated tensors like just happen to give the same MPs for all sizes there's some local gauge relation so it's like a global to local thing that we will end up using later oh and the last thing I thought I should mention is just this sort of small generalization of matrix product states to matrix product operators so it's just the same idea except now you're just decomposing the coefficients of an operator into matrix product structure and of course if you kind of group the bra and ket labels back into a ket label go back to the theory of matrix product States so all the results that I just carry over directly and so going up a dimension things become more difficult both theoretically and numerically but there there is a natural generalization at least one of matrix product states the higher dimensions it's called projected entangled pair States and say on the square lattice in 2d these will be given by a fundamental building block that's like f5 index tensor which we then contract up on say a torus if this has periodic boundary conditions to find some state and the idea is sort of similar to matrix product states are they more difficult and these are the kind of states we're going to be looking at moving forward so the question you can ask is like what are the fundamental building blocks of these states another way of asking that is like is there a left inverse can I access the full virtual entanglement space by just acting on this one physical space so I should say the notation here I'm thinking of this a as a map from these virtual indices to this physical index and this pseudo inverse was being acting from this physical index to these virtual indices so if you kind of trace of this one through that's this guy here so you can ask this question can you access the full virtual space so this is a condition called injectivity and some work had previously been done sort of showing that injectivity excludes topological water basically so if we want to look at topological systems this is a really bad condition so yes kill that then in previous work of sir a cook and pres Garcia there was a generalization of this injectivity condition to some kind of group algebra and this allowed them to describe some topological phases it was then generalized by Bush shaper and later by Frank's group of the same collaborators as this paper - just a matrix product operator so we know this can't just be injective so it has to live in some subspace we want to characterize that subspace and we use the matrix product operator to do that which was general enough to sort of describe all known chiral topological order in 2 plus 1 D so we'd a few more conditions than just this matrix product operator being the local sort of restriction on the entanglement structure we also if we wanted to describe some kind of topological property we wanted this pulling through condition such that this thing behaves a bit like a Wilson loop if you put it into your state it can kind of freely move so it's not locally detectable and with a few more technical conditions this was a set of axioms for this kind of class of matrix product states so this is where the idea to study these matrix product operators came from they were kind of a relevant symmetry in these topological models that are known as string net models ok so now on to the results so the idea in this section is that we just want to start we've got to kind of forget about the 2d stuff a bit fern for now and just think about this matrix product operator we're going to look at its structure and then try to see what arises from that and identify some sort of structure we can see as an ante on theory by just using sort of ideas for matrix from matrix product states so the first thing to note is because we're always considering these sort of projectors to be a set of periodic boundary conditions we can use this normal form this breaks up this projector into some sum of blocks and these like blocks are kind of the building the building blocks of this big thing so if you now think about what the fact that this is a projector means it means for these individual blocks that they form in algebra so because you have this sort of global to local reduction thing you know that the blocks on the left and the right side of this have to match and so if you do a bit of reshuffling you see that the product of any two blocks on this side can be identified with some linear combination of blocks on the right hand side and since this is true for all sizes there's this local reduction tensor which will become important later that actually reduces this product to the single guy in it's possible outcomes now I want to yeah so this is all just from the theory of MPs but to actually get anywhere we needed to make one more sort of assumption that was true for all the models we knew and still not sure how restrictive this is but we basically need that there exists a form of this MPs or a matrix product operator such that this condition holds which is just a stronger version of the last condition and it's kind of saying that we throw away some subspace here that sort of doesn't matter once we reduce down to here and it just was never there in the first place so there there are ways this could fail but it's true for all the interesting matrix product operators we know and one way to see why this is important is by thinking about this sort of fixed point peps so if you have this peps with this entanglement structure locally you can make a fixed point version by just taking the local tensa to be this little loop of NPR and then your fixed point state is just a whole bunch of these little pieces of matrix product operator tied together and if we want this state itself to have this sort of topological symmetry we need some condition like this and in fact the zipa condition I showed you in the previous slide ensures this is true so it tells us that the state we build out of this thing which is meant to describe topological order is indeed topological itself it's kind of good and once we have this condition we can we actually find that we recover an F symbol so this X guy behaves like a fusion vertex that I showed you in the sort of algebraic description of anions and now if you look at fusing a product of three MP O's in two different ways so first you can fuse a and B and then the C or first B and C and then with a you actually find that there's some vector space there because this X there actually may be multiple X's doing the same fusion this result in the middle if you kind of do the reduction all at once it's a degree of freedom and so as well and so this matrix can actually act on that degree of freedom as well as these two degeneracy indices to see how you actually derive this I have a little messy slide but hopefully you can follow the general idea you look at this reduction of three tenses to 1 and the other reduction of the same product of three tenses to one you then bring in the inverse to this right hand side which gets rid of it there but gives you this kind of commutator on this on the right hand side of the equation you then use this sort of injectivity property of these irreducible matrix product operators to get rid of the MPO itself and then upon taking the trace of this little line here and this bit here you'll find this actually gives you the F symbol so it kind of comes out directly you can then look at products of four of these guys to get back the Pentagon equation I didn't show it again because conceptually it's basically the same as in the fusion category language and so the result is really here just starting from these matrix product operators that build up the full projector these are like the objects of some tensor category you have these morphisms which are just basically matrices acting on the set of virtual indices of these things and we have a notion of tensor products in the fusion category sense which is not like just a sort of normal Hilbert space tensor product but a more complicated tensor product that's given by just taking local products of these matrix product operators with open boundaries so of course if you kind of close this boundary you're back to actually just having an algebra so of course it has to be associative and the algebra you get by taking these periodic boundary conditions is just this sort of thing called the fusion ring or the fusion algebra of this fusion category sort of a common concept yeah so this is like the first step there's still no braiding here though so this is not actually identifying the onions of the theory yet so how do we actually construct a neon re-identified a neon in the Pips that's sort of the next step and you could make a guess that maybe this a neon is just sort of but you get this by putting in some linear combination of the blocks of this algebra into your kept state this turns out to have totally the wrong properties to describe super selection sectors though for one thing these things are not sort of locally orthogonal it also doesn't give you the right number of anions in sort of known models so one way to sort of think about the resolution to this which was initially just I think a guess of Nick but it turned out to be connected to a lot of stuff people have done previously in the theory of sort of 2d topological order and category theory and sort of planar algebras is to put in this extra loop and you can kind of think of what this extra loop is doing is although this anyone theory is not greater just sort of imagine it was braided and we have some kind of an eon sitting there we want to project this into a definite charge sector there might be some superposition there but by braiding around we can measure which exact anyone is there and by putting in the correct sort of superposition of encircling onions we can really project onto a chaud sector so that's kind of the inspiration for this guess so here this is all just the same matrix product operator but here we have this tensor and this really contains all the topological information so it's just a for index tensor living in the virtual spaces of this MPO and once you define this you now have a secondary algebra so you can the first thing you can do is you can expand this tensor out in some basis which identify these blue guys because it's kind of too hot to draw this every time but this blue your eyes are just given by some six index object where two of them are degeneracy indices the other three indices are those of the matrix product operator and virtual indices and it's you can just decompose it into two of these fusion tenses so yeah maybe just remember that the first three letters in this description of it the a B and C indices that kind of go through the tensor and this last index D is the one that goes around and then the degeneracy indices are not too important for understanding the general structure so when you start to like look at the properties of this these NPOs first of all they form an algebra which is not immediately obvious but because this sort of MPO is itself a projector you can kind of reduce everything to just some contraction of these two tenses and then they actually go back to some expansion in the original basis so I'm just denoting different objects in the algebra about different colors and we take a product of two you end up back in the algebra so it's closed there's good you can take linear combinations and also it turns out to be a star algebra so if you take some complex conjugation of this thing acting the outside to the inside and then there are some more tricky properties I didn't mention which have to do with orientations which are actually end up being important but I don't have time to cover them and you use these kind of properties of bending lines from one orientation to the other you find that it's actually closed under this emission conjugation operation too so it's a sea star algebra and now the thing to do with si star algebra is to see what it's block decomposition is so it has some kind of block decomposition and we want to start looking for the projectors on two different blocks these projectors will give you like a good set of quantum numbers for this for kind of measurements on two discreet and definite charged sectors and so that this was actually mostly due to Mikkel and he used a constructive version which he programmed up of the art and wedding Byrne theorem to basically just find these central idempotence yes so the idea is here's the conditions just written in terms of the matrix product operators and I've denoted these solutions as having a white dot in them so the little blobs of the white dot are these special elements of the algebra that are actually central idempotence you also need to make sure that irreducible otherwise we'll have a superposition of any ones and this is I think where the hard work is but once you do this you have this complete set of idempotence that sort of decompose the identity and your algebra and these give you your set of quantum numbers for the charged States okay and once you have that so this is just this block decomposition in terms of these central idempotence tells you the anions but now we wanted to extract some topological information from them the first thing that's easiest to extract is the topological spin so if you look at what doing a 2 pi rotation actually means for this little idempotent you can reduce it to just some little loop around this tensor and if you actually just you can so do this by hand just compute what happens here it ends up actually giving you the topological spin times the original guy you can also do fusion with these things and this is somewhat reminiscent of this pair of pants decomposition if you kind of think of each of these as being some boundary hole and this as between them being some surface this is like some morphism space from this outside ring to these two inside rings so we've constructed a second algebra it's turned out that this also has a fusion structure that's not just the same as the multiplication structure of the of the operators and here that this tensor is just the X tensor it also has a braiding structure so if you look at you just want to sort of solve this equation for an unknown here and it turns out that the solution is actually the same as just this item pertinent which is actually a well-known thing in the condensed matter literature so this knowing this equation tells you how to do braiding as well so now we've recovered this full structure of the onion theory we have fusion and the braiding and in principle we can actually look at complicated processes of raiding the and fusing these anions within a tensor network so here I've just I'm trying to draw like a cool circuit going on in terms of a topological quantum computation but I got a bit lazy because it would took a long time to draw us so I've just drawn like one braid but in principle what we want to start doing is looking at putting these excitations into a 2d tenser Network State for the ground state of the model that hosts these excitations and then starting to braid them so here I just brought these around clockwise um one full braid and we'll find we'll start to find these matrices on the virtual level and sorry I didn't actually draw in the tenses explicitly but each crossing of the red line with the black line should be an MPO tensor and so if we start to do more complicated things what we'll find is we'll have this big network that's just sort of like a tensor network on the virtual level that tells you what's going on with this topological quantum computation in terms of the correlation degrees of freedom of the state this is very reminiscent of helmet measurement basic quantum computational works and peps in measurement based quantum computation and peps after doing the measurements you can kind of see exactly the circuit arising on the virtual level and here after doing a bunch of braiding we don't have to pick a definite fusion basis we can just immediately see some kind of almost quantum circuit which you can lift to a quantum circuit it's sort of a unitary on a subspace and hopefully we can understand some more structure or some kind of improvements on topological quantum computation using this approach from the correlation degrees of freedom okay so to summarize this results of this matrix product operate approach we first looked at these individual blocks coming from the symmetry of a tensor Network state that has topological order and we found that they actually form a fusion category which I called C we then found this auxilary operator that you need to decompose sort of excitations into definite charge sectors and we found that these were the physical anions of the theory and actually mathematically this thing is called the double of this category so there's some process in category theory that is called taking the double of the drinfeld Center and it turns out that doing this whole calculation in terms of tensor networks was just really doing this on the category Theory side and that of course agrees with the known answer of what the excitation should be in a model that has this kind of symmetry so if you take a string net model with this as the sort of input category like these are string labels then it's known that the out the output anions are this double and if you take our matrix product operator for the symmetry of that state then the output algebra is this double of the input ok so here's some examples first the toric code which everyone knows and loves so it's I have to admit this is kind of overkill to do the toric code this was previously contained in the formalism of G injectivity but it's a nice example to get a feel for what's happening so here the matrix product operators are actually just tensor products so if I have this matrix product operator projector there's just an index running around the virtual level that's coordinating whether to be in the identity or the X locally so I'll have like a sum of the identity with the X so I have the full projector and now these values of the crossing tensor give you the different particles and maybe we can go through them so like just to get a feel for it so if I have 0 going in both directions that means that I have nothing coming out of this particle no line it's just a line of identities and around here it's just identity if I have a 1 going this way it means it's a ring of X's around there so this is just projecting onto the support of this matrix product operator which we knew was our vacuum state because that's the sort of that's a symmetry of the the peps locally without any excitations this excitation is now projecting orthogonal to that the M excitation has a line of exes coming out but is projecting onto the identity plus X in this sector so that you think that is a flux because it's kind of like a line of exes coming out and then the e/m particle has this projector onto the 1 minus X state while also having a line of flux and if you calculate the topological spins you see what you expect these are all bosons and this one's a fermions so this is in half okay now a more involved example so you can do this for all the string nets and for the Fibonacci Theory the MPO kind of looks like this so the basics of this theory is that it has fusion rule single non-trivial particle and the fusion rule is that it can go back to its self older vacuum it has a set of estimates which I didn't bother to write down because I don't think you'll get anything out of that but the way that the MPO works is it's basically just given by an F symbol so this is from our previous work and it's kind of a bit tricky but you basically get that each of these indices going through the tensor I just copied twice so you get two copies of each degree of freedom two different legs and then the overall weight of this MPO is given by the F symbol the fusion tenses are also also given by F symbols in a similar way the algebra is spanned by these elements so I think it's seven dimensional and yeah just remember the first three entries of this other lines going through into the the a neon and then the last entry is the loop going around the onion so here we can see the vacuum particle is again just a projector on to one plus tau going around the onion which is the vacuum sector all the MPO of the vacuum and then these other particles we get a complex conjugate pair with um conjugate spins that we identify as tau and tau bar and then we get their product telltale bar which actually has is a two dimensional block in the algebra this this sort of approach also works for non modular theories so there's an example with s3 and the paper which I didn't put on a slide because it just has too much information so we have like the double of s3 and in principle it can work given any sort of input category there's just a process you have to run and you'll find these outputs which are the anions although identifying the can become a bit trickier when there's a lot of them okay so the things we want to do in future with this some of which we've already started is including fermions which you might think is a small tribute like sort of it won't change the general structure that much but it turns out to actually make things a lot more interesting we also want to look at symmetry in enriched topological order which is connected to how transversal gates act on topological codes more on actually how this virtual level representation might help us think about topological quantum computation and looking at things like domain walls between different topological orders and then of course going to high dimensions which may be too tricky okay so thanks yeah the overall picture is that we have some symmetry of a tensor Network state and from this we can extract the annuum theory which tells us everything about the topological order in that state and thanks your attention questions how do pattern continuing already small new looks like how the Hamiltonian so given a tensor network state you automatically have a Hamiltonian and a lot of the time I was just talking about the matrix product operator but there's this kind of chain of inclusions I guess from this matrix product operator if we could construct a pets which was this tensor I showed where you have the pests and so just being a small loop of the matrix product operator that peps will have a parent Hamiltonian so if you like you can think of that parent Hamiltonian and how that works is it just looks at a small patch of the peps so like for sites in of the pepsin square lattice and it just projects under the support subspace of the temp of the map from the virtual degrees of freedom to the physical degrees of freedom so that that's kind of the parent Hamiltonian you associate to an MPO yep thank you I guess so what are these excitation a cannula they're exact eigenstates of the pallet Hamiltonian so it's a bit tricky because we actually if you notice we change the Hilbert space slightly when we think about this and this counter doesn't matter when you're thinking up through the selection sectors because you can do I Samba trees but if you really want to make if use those an N sets what you want to do is put in I kind of skipped this detail but maybe I should show just explicitly you want to put in some different tensor here that is joined to this index so this this kind of a MPR here just identifies the subspace on which the excitation lives there can also be some local degrees of freedom involved with the a neon and other other things that are just not important topologically but if you wanted to use it in practice you would want to introduce a six index tensor here that's different to the other tenses that reflects the properties of the onion and as long as it's on the support subspace of this guy it will be some description of that anyone I don't know if that answers the question and and that thing should be it should be possible for that to be an exact ik eigenstate yeah there can you get any modular tensor category in that way so say if I give you fusion routes can you find tensors that make that work yes so the way this works is that only the output engines are only non Carolinians because they always go through this double construction the only results we find always back through this double construction and this if the input theory was modular is just a copy of the state and it's time reverse which tells you that's definitely non chiral but even in this case where it's not modular and this double is a bit more complicated it's still not modular so the only things we can find it doubles and that's kind of a I think that's more fundamental than the approach we took if we generalize their approach slightly it would still be doubles but of course looking for chiral things there's another ongoing project I know like Norbert shook and Ignacio so active worked a lot on this and I'm making progress just very interesting but it seems to be that you have to start from a slightly different place to get their other MPO symmetries are important so if there's no further questions at this point then let's conclude the session and that's like Dominic and both speakers of this session [Applause]

Channel: Microsoft Research

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Length: 37min 44sec (2264 seconds)

Published: Tue Jan 31 2017