Xie Chen - Foliated Fracton and Beyond - IPAM at UCLA

we're starting the morning joe and told you about why um somehow we need to go to three spatial dimensions something more unusual like in one dimension there's no topological ordering two dimension there's topological order but probably not beyond and then starting in three dimension uh joanne showed you a model with all this fractal structure and and and all these weird properties of not having a string operator and all that so that really surprised a lot of people because these kind of phenomena was not expected in a physical system in physical systems we expect things like particle quasi-particle but the thing that joanne was talking about was like you create some particles but then individually they don't move because because you have to go through the the movie that joan was showing have to go through this fractal structure to separate four points away from each other and that's basically how how individual points can move but then if you try to just just take one of the points and try to move it it doesn't work okay so that's the a very very peculiar feature which has come to characterize um pretty much the whole set of what we call as frag tongue physics and um and and that model was discovered around 2010 2011 and it was so exotic i think jungkook mentioned that yesterday that um even until now i don't think physically we have a we have a good picture of uh of what's going on with all these fractal structures um but later on a couple years later when joe and went to mit they came up with some new models and one particular one is something called the x cubed model and this one turns out to be less exotic there's no fractal structure and they behave in a more controlled way as i will show you but still it has this weird property that there are point excitations which somehow don't move on their own or somehow they they don't move in the whole three-dimensional space okay so in that sense we also call this model a fractal model but this model looks much more regular than than the original house code there's no fractal structure a lot of properties look follow simpler rules and and this is the kind of model uh i want to focus on in this talk and and roughly uh within the fraction community we refer to these kind of model without a fractal structure we refer to them as the type one model and the one that joan was talking about in the morning we refer we refer to them as the type 2 model of course this is this is all phenomenological and if you try to get a rigorous definition i'll try not to rigorously define that but just that we observe that there are roughly two classes of models that we know um and so just to connect back to what nadi was talking about in the morning of course not he didn't get to the the fractal part yet he was talking a lot about the the xy model which is something non-frag tonic but my guess is that for the next part of his talk he's going to talk about something relate more related to the x-cube type of frag tongue where there's associated with subsystem symmetry which is plane-like or line like not fractal like and and and try to formulate um a continuum theory uh for those kind of models okay um so yeah so to get my talk started and to to motivate why we're even considering things like foliation let me first introduce to you and this famous model that that is now called the x-cube model it was first um proposed in one of the paper by saga vijay johan and don fu 2014 and i again i apologize um that i won't do a a good job with all the references i'll try to mention some of them but i'll probably miss most of them okay so the sql model is again a three-dimensional lattice model defined on cubic lags and the degrees of freedom will be put on the edges and the degrees of freedom are again cubist two-dimensional spins and the execute model is again one of the model that we call stabilizer code meaning that you can write it in the form that joan was introduced in the morning that the hamiltonian terms will be a tensor product of sigma x and sigma z and all the terms commute so so for xq model one of the terms will be 12 x around a single cube cancel product all together like this off of them but how much is even bigger bigger than what john was showing you in the morning that one was like an eight body so this one's top body but it looks nice right at least it's around the edges of cubes it looks a little bit more symmetric and uh and there are other terms at each vertex so at each vertex there are three more terms for disease and here about x and z i mean poly x and then each vertex there are three types of poly z operator each one involving four of the z's and they are oriented in three different phases okay in three different directions and you can check explicitly that all these terms commute so the model is exactly solvable uh very nice mathematical properties you can write it in terms of the polynomial formalism that joanne was introduced in the morning and tried to find out growth degeneracy excitations and things like that okay with that exercise you will find for example if you put the model on a three-dimensional taurus and ask what is the ground state degeneracy of this model turns out that the log of the ground safety generation follows a very simple formula so it's two of x plus two l y plus two and z this two l x plus two l y plus two l z minus three well where l xl yls and lc are the linear dimensions of the model in three directions so compared to house code mentioned in the morning this is a much more regular formula because it's a very simple linear function of the system size but on the other hand it's also very peculiar because for all the topological order that we know the ground state degeneracy is usually just a constant if you're given the model of three dimensional targets so that is saying that this model is definitely something unusual it's not one of the topological order that we know and of course now we know that this linear scaling of logarithmic determinancies indicating uh the fractal order of the confidence and that actually also saturate the bond that jorah was talking about that the log of the prostate degeneracy go okay so this is the ground state degeneracy and then we can go on to ask about what are the excitations of this model and it turns out that there are different ways we can try to make fractional excitations yes so for regular opportunities and constant correct expression loss of a function between conflict are there any other model at all um we don't know that function is quite complicated it's upper bounded by this linear um but we just don't know yeah yeah that's a great question um no but you can try to imagine that you might have something here anything in between oh yes if there's any other in-between constant attack spiritual function uh well so there is a rather bizarre by stitching things together at the intermediate scale but that's not physically you would want okay if you follow the polynomial formalism that i sketched you won't get anything between okay well that's not so your mathematic that's two that's phenomenal you know which is you know basically binary or you know continue so i just don't know whether this one total seem to be binary if they're constant or upper bound by expression okay that's it i'm just curious yeah yeah good question uh one one thing i want to mention just following up on that is that we well joanne already proved that but generically physically we don't expect ground state degeneracy to go beyond the scaling meaning exponentially linear system size if you have exponential in l squared or l cube that usually means you have local degrees of freedom that you didn't remove you can add some local term to remove it but intuition and joint has proven okay so let's now talk about fractional excitations so one way to make fraction is to take a string of edges and then add sigma x on it and then you can see that there are a bunch of vertex term that will be violated at the end of such terms for example if we add in in the direction i've got then i've color-coded the vertex term and then with this color coding the red term will be violated and similarly the blue term will be violated okay i will actually well in this direction is the blue and the green and in the third direction it will be the red and the green okay so this is very similar to what we see in topological phase that you apply a string operator create a pair of excitation and that's also said that this model that's type of string operation so didn't satisfy joann's rule of no string operator but this string operator is also very weird because the string operator implemented in different directions violate different terms meaning that they create actually different excitations so if you try to apply a string operator in this direction and then try to make a turn in a different direction then something actually has to be deposited at the corner because the excitation made in different directions they are different so at this end there's a blue and red but then at this end it's blue and the green so at the corner there's actually if we create a pair of anions then we can move them anywhere on the path but here if you try to drag it in one direction and change the direction something will be deposited at a corner meaning basically that the excitations don't want to change quote don't want to change directions and that's why we call these kind of excitations linears or linux and depending on how you want to pronounce it fractional excitations uh they only want to travel along a straight line so it's not in the most strict sense of fracture excitation which is the kind of hexadecimal at all but this this also is a kind of fractional excitation uh that does not move in the whole three-dimensional space it comes with some restricted mode okay so this is yes is the number of excitations the same as the ground state agent yeah good question um it depends on how you count so so for topological order uh there's a rule saying that uh the species of anion equals to the ground state degeneracy angular taurus right here it's more tricky than that uh i might come to talk about this later but but it's tricky let me just say that okay yeah okay so this is one type of excitation that you can make with x operation but then you can try to also do something with x with the z operator basically by operating z on a bunch of edges sticking out of a a rectangle shape and then the kind of cube terms that you violate would be at the corner okay so this is one of the cube this is the other and the third one and the fourth one so you see that every time you do this you make four cube excitations this is similar to joann's excitation in the morning you make four every time except that now it's not at the corner of a tetrahedron it's at a corner of more regular rectangle okay but because it's four at a time and there's no process which lets you do a two particle process so these kind of excitations cannot move by themselves either um they have to be sitting at a corner of a rectangle and the best you can do is to make the rectangle bigger and smaller and and move the four as a whole okay so this is what we call a fractal excitation meaning that individually they don't move okay okay uh great so this the the linear and the fractal the existence of such fractional excitations just indicate that there's something non-trivial going on with this model of course in new jersey is already indicating that but beyond that let me mention that there are also another type of excitation which we call plano which can be obtained by combining some of the linear excitation or fraction of citation for example if i take a double string for the line string operator like this if i take a double line and that double line will be able to turn corners without having anything deposited at the corner so if you take a double line meaning that if i take a what we call a linear dipole then the linear dipole actually can move in the plane that's perpendicular to the dipole direction so that becomes what we call a plane noun it's it moves in a two-dimensional plane so that's less restricted motion but it's still restricted right it's not like moving in the whole three-dimensional space um but it's closer to what we see in two-dimensional topological order something that moves into play and similarly if you just do the z string operator like this so you have two fractions stacked on top of each other which becomes a fractal dipole and that flat tongue dipole can also turn corners like this okay so the fractal dipole can also move in a two-dimensional plane making that a plane okay so so this model looks much better i don't know if it looks better to you but from a physicist's perspective from someone who has been playing with tariq code and other topological model previously this model looks more regular it looks more symmetric its property looks less exotic so the hope was that we can make connection between this model to some of the previous topological model that we know and have a better understanding of it okay and of course within the field of frag tongue people are trying to do everything with it and for us when i say i try to understand better about something one of the the key thing that we're trying to understand is in what sense this this particular model uh represent a kind of frag tongue order or represent a whole frag tongue face okay so spend some time explaining my motivation because this idea of order and face is arguably the most important idea in the whole of finance matter because these kind of models that we look at they're all toy models they're like 12-volt interaction everything commute but that never actually happened in a real material right or in uh in some even with with some more realistic model not to mention material not to mention experiment you won't really find models like this the the real thing will always be more messy and more ugly but we study these toy models anyway because the hope is that they represent the universal feature the universal property or order of a whole quantum antibody phase okay and that was the case for a topological order for topological order this is very well understood topological order for example we study code and we study the exactly solvable topological order and with the understanding that they're not going to show up exactly in a material but they represent the universal physics of a whole topological phase for example turtle represents the whole z2 topological phase which might show up in a material called harborsmith or some other spring liquid right so the hope with this kind of fractal model is similar that well eventually maybe we will be able to talk about realizing this kind of order in um in a in in a real material right um but unlike for topological order now we don't really know what we mean by fraction order because for topological order when we look at tariq code we know that there are certain key features that we need to focus on for example we know that the gramsci degeneracy on the taurus is equal to four so for example if we look at material or at least numerically if we look at a spring liquid model we can try to see if there's a full foreground state degeneracy on a taurus right and also we know that there are annual excitations and the anions becoming certain species and diffuse in a certain way and they have certain statistics among themselves and all these are universal for the whole face so if we have a let's say magnetic model we can also try to numerically or even experimentally if you're clever coming up with ways to do that and to see um the statistics or the existence of anions or the fusion of them in ways you can okay um but for fracture um when we look at x cubed model it looks tricky right um for example if we try to use ground state degeneracy to define the kind of order we don't really know what we're talking about because the ground city degeneracy depends on system size so what exactly do we mean by the universe so grants the degeneracy are we looking at the linear coefficient in front of the the linear size or are we looking at this this um a constant of -3 and especially if we are dealing with a disorder system without translation in variance we can't even talk about l x l y and l z we don't even know what it means and in that case how do we even use that quantity to characterize the universal property of the face okay and and and it's actually a much bigger problem than that because for topological order usually we say that things in the same phase have the same ground city degeneracy because one of the definition of gapped phases that we we use a lot definition of gap phases is that you start from a gap model and if you can smoothly change your model smoothly evolve your model without closing gap and end up with a different model then these two models are in the same gap phase okay but that's guaranteed that in the process you won't close gap and you won't change the browser integers right but if that's the case then we look at xq model we would say that execute model with different system size would belong to different gap phase it's just according to definition right which is which is okay if you want to stick to the definition but which is also saying that maybe the definition is too strict because if i'm a microscopic creature living inside my execute model i have no idea how big the model is i see the the bulk physics will be exactly the same around me and uh and the in the sense that the the system total system size is just a boundary condition right if i'm ignorant of the boundary condition i would say that why do i need to care about that i see the same bulk physics i would like to call execute model with different system size to be in the same phase maybe it's a problem of my definition and not a problem of the model so what we thought that we we needed to do uh is that maybe we need to have a new definition of gap face that the the conventional definition just doesn't work as well it cares too about too much detail in this case that we might want to ignore some of the details and just look at things on a higher level so that we can talk about something that's more meaningful to uh to a microscopic creature that lives in the bulk of the system that captures the bulk physics instead of just boundary conditions okay so so this is what we set out to do this is what we hope to do yes um why do we have the assumption that the ground state degeneracy when you're translating the system from one phase to another you must stay the same oh sorry the statement is that if you stay within the same phase then the ground state degeneracy should be the same usually um before this this fractal work right so is that an assumption that's a it's a consequence of smooth evolving the hamiltonian without closing gap and if you require that then the ground state degeneracy cannot change and that's used roughly like a definition it's not completely rigorous it's a mathematical statement it's a continuous function it's a constant so you have to show that the degeneracy is a continuous flopping of the spectrum i see if that's true there's a cost yeah so consider everything i say rigorous with a quotation of a cap system i think this is okay um yeah thank you um right so this is the motivation and um and of course i had no idea how to do this until one day i run into um junghan and and who asked me about how to put the execute model on different manifold and starting from there we realized that this there's this this notion of foliation that actually plays a very important role uh in characterizing models like the execute model um so what i want to tell you about is how we were the idea of foliation the idea of foliation generalize the definition of gap phases so that it makes better sense for x cubed such that x cube of different system size will now belong which is something we would like to have and also allow us to to make connections between models of very different looking so models that originally look quite different now can be related under this new more general definition of fractal order so i'll try to show you examples of that and of course hopefully i will have time at the end of my talk to show you that even though we had a lot of fun with this foliation idea and we also know that other things beyond the fluid foliation framework of course the first thing that's beyond the foliation framework is house code house code has this fractal structure that doesn't belong anywhere i think the foliation framework but even for models with a more closer to layer um structure will be beyond foliation as well so hopefully at the end of the talk i'll be able to show you some examples like that just to show you now that we know how little we know now we only understand a little uh of some other okay so foliation so foliation of course is a term that comes i think it's used in geology geology and also topology which is referring to a layer structure in a bigger manifold for example i'm just drawing a cube here um within the cube might have layers so you have lower dimensional molecules cutting the higher dimensional manifold into slices and when people say foliation it means that these lower dimensional manifolds they won't intersect each other so they're parallel and um and that's pretty much it then you can imagine that you ask the question of give it a three-dimensional medical can you try to cut them up into parallel spices and what are the ways to do that okay so so of course this is a fascinating topic in topology and there are fascinating theorems about when you can or cannot do this um i will try to go into that direction first of all because i don't know that much exactly so we are very narrow-minded this is and and we care about a specific type of uh foliation uh that that has to do with what we want to do about xq models i'll just restrict my discussion to the kind of foliation structure that i i care about so why do we care about parallel layer structure in three-dimensional manifold and that's because what we realized is that in the xq model remember that we had this problem that the model on different system size will have if we look at two execute models one on a slightly smaller system size let's say this one is lx ly and lz compared to one on a slightly bigger system size which is lx ly lz plus one and the the the issue we had was that even though the model looked exactly the same and the bulk physics are all the same but because of the differencing system size the ground state degeneracy will be different and we will call them to be in different phases which is not what we want so we'll try to find ways to understand what's really different between these two models right and what we realize is that actually what's different is that we can just add a two-dimensional layer of thorough code so i didn't introduce two-dimensional total but i suppose maybe some of you have at least heard about it so just say two-dimensional torque code okay another exactly solved model a stabilizer model one of the textbook examples of two-dimensional topological order um and and and after inserting this two-dimensional target code into the left-hand side of the system and what we were able to show is that now we can smoothly connect these two sides without closing gap okay which is so so the arrow were originally allowed to do under the definition of gap phases but with the caveat that we add to one side of the equation something that's highly unusual something that's already highly non-trivial but in a lower dimension code and then this equivalence relation can be established is the second trivial uh yes um so let me say that yeah let me clarify more on that so first of all my drawing is obviously condition it actually doesn't matter where i put this green layer uh it's anywhere in the middle of the system and when i inserted the layer it's totally decoupled just forcefully insert something without any coupling and then of course when i do the arrow i'll be changing the hamiltonian or into some operation between the decouple layer and the book and then basically there's some local degree of freedom being being mapped together and then will become the right-hand side but but the two sides will be unitarily equivalent to each other yes thanks and um and this makes a lot of sense right if you think about it because the two sides originally they have brought different ground state degeneracy but now we see that where the difference in ground state degeneracy comes from because this extra green layer um carries a ground city jersey on its own so if the system is on three dimensional taurus then the green layer is a two-dimensional taurus which has a bronze degeneracy of four and you take the log probably two so it exactly carries the extra bronchitis that you need and also and now um we can also trace back the origin of some of the fractional excitations that we saw before for example remember that we have these linear dipoles and fractal dipoles and these dipole excitation being pronounced meaning that they can move in two-dimensional planes and we can we can trace back to see that these kind of backstations uh they actually come from the two-dimensional torque code okay they were the excitation and excitation in the two-dimensional target code and because they come from these two-dimensional planes that's why they're moving a two-dimensional plane that's what they do of course this is this is um this this is just one layer and you can imagine that um instead of left side to the right hand side you can start from the right hand side starting from a pretty big execute model and go to the left hand side and keep doing that right and so you starting from the execute model and you can eventually decode a lot of layers of two-dimensional target code meaning that a lot of the the things inside the xu model can be accounted for by all the hidden two-dimensional layers for example the scaling of the ground state degeneracy having having this kind of form by the linear part they all come from the hidden two-dimensional layers because each layer will give you a a coefficient of four um and then you can do this in x y z direction so that's where the the linear scaling comes from and also the plain excitations and also i didn't mention there's some entanglement feature that also can come from the two-dimensional layers basically each two-dimensional layer can contribute something called topological entanglement entropy and if you calculate that for the three-dimensional book you will see a contribution coming from the sum of all the layers continuously by setting um yeah so so right so um so this is actually related to something i wanted so let me say that first and then i'll answer your question so so you can imagine that you start from a big chunk of x cube and decouple layers and layers and layers the question is eventually the reverse of your question is eventually do you get rid of everything or do you have something left over so so a rough way to say this is that you have something that's left over you cannot decouple everything and one indication is that for example you have this constant part of the ground degeneracy also that you have linear and fractional excitations you don't just have planar exhibitions meaning that the model is not just decoupled like uh sorry it's not just the couple target otherwise it will be too trivial it will be not interesting at all so the model even though it has a lot of decoupled layers hidden inside it's still after peeling off all these layers it still has a fractal or something like that so if you try to grow it from a core you have to supply the core and then add all the layers um maybe not [Music] explain that so my very high waving way of understanding why uh there's a minus three is that you can imagine that you keep peeling things off and if if you can peel things until the the last layer and everything is decoupled then you should just have two lx or 12 y12 but the reason that you have a minus three is because once you are left with one layer in each direction you can imagine and keep doing this in all three directions and until the end you are left with only one layer in three directions and then you cannot take them apart anymore it turns out that these three layers you can think of them as distillatory code but coupled along the intersection line and they're coupled [Music] and this is of course generalized in different ways and a couple layers or the the the string membrane but this is the very naive version of all that where you have torical in all three directions but then and they're coupled along the intersection line and one way to say the coupling is that the along the intersection line between the intersecting planes and they're combined or you can say that the the excitations the the planars coming from the two planes have to be created at the same time they have to travel along the intersection like at the same time so because of that the gets degeneracy so it goes from six to three yeah so this this picture actually works pretty well we we have played with xq model um or different manifolds but also later we have putting some effect in the in the valley smooth effect and we try to come across the degeneracy and and this picture helps us a lot to predict from safety [Music] yeah this two is the focal degeneracy that's carried by uh tutorial layers but i'll be careful to say whether i want to use them as universal quantities to characterize the face because this kind of depends on how you measure distance uh if you change your unit cell then the coefficient changes and also because i want the definition to be applicable to disorder system if you can't measure if you don't have translation invariants at all this will be hard to define so so in the end actually we don't try to use gross indeterminacy as universal problem that much for defining fractions unlike for top large color which is across right okay so so this is for uh xq model and and from this observation uh we made the definition of what we call foliated fractal order so remember that um so let me first say that um let me first give a go back a little bit say the definition of conventional gap faces conventional gap phases is like you have model a and you have model b and they're gapped and you want to ask if they're doing the same phase and then what you're usually allowed to do is to uh change the smoothly without closing gap i will write that but that's what the arrow indicates but actually it's a little bit more more general than that because what if the two systems are basically on different degrees of freedom what if when it's being modeled the other is electron model it looks totally different but we can still try to compare the phase and that can usually be done by adding some decoupled degrees of freedom but usually we require the couple degrees of freedom to be in the form of totally something totally trivial and usually it's a product state and just to make the the formulation more symmetric we can also do it on this side and we can just add product states on two sides and then after that we can match the degrees of freedom for example and then ask in this more general hilbert space can we smoothly deform the hamiltonian uh without closing gap and that's the conventional definition of gap faces okay so for foliated fracture order uh we need to generalize this definition and from from from this observation here we know what we need how we need to generalize it basically we want to replace these extra product states by two dimensional topological layers so so if we have a system a and we want to compare it to system b and we know they're gapped we just want to know if they have the same gap fraction order then what we allow ourselves to do is to add two dimensional layers decouple two-dimensional layers decoupled between the layers and also from a and then on this side we also add layers but we can add different number of layers and we can add different species of layers so let me just color it a little bit just to indicate that this might be different topological order but after doing that if we can smoothly connect the two sides by changing the hamiltonian without closing gap then we say that a and b are in the same place and of course the first motivation that that led us to to define this is because we want execute model with different system size to be in the same phase and that is well captured by this definition right because i know that for one xq model you add a layer okay and you're allowing the two dimensional layers to be not necessarily just codes and allow the two-dimensional layers they don't have to be tortured they can be anything oh yeah so we allow the two-dimensional layers to be any gap or two-dimensional stuff and it got two-dimensional topological if you don't love symmetry so in other words view as part of a 3d system these two systems grow like non-trivially yeah great point that if we just have decoupled two-dimensional layers of topological order we would call that a trivial foliated fractal order and if you think about it if you just stack two dimensional layers you already have a lot of non-trivial feature if you consider it as a three-dimensional system because the ground state degeneracy will will increase with linear system size and there will be particle moving a two-dimensional plane with restricted motion if you consider it as a three-dimensional model but people of course knew this but they never took it out to be something unusual because it's just a stacking of two-dimensional things that's what it how it should be right but of course now we understand that we can think of this as something trivial within a more non-trivial class so if you just have a stack and that's a um that's a trivial foliated fracture order and also another common i want to make is that now we see why we need to go to three spatial dimension um because as joan mentioned there's no one-dimensional topological order so if you try to start one dimensional chains and not much happens so in two dimensional gap phases we don't see a fractal order like this at least not gap rectangle like this can add some product products like on a bench after we add some trivial bundle and we can just use the case theory to classify the computer but in this for a screen is the [Music] no short answer is no i don't know i don't know but but hopefully something like this yeah yeah thank you oh [Music] me be more yes about the arrows so these arrows one one is smoothly changing the hamiltonian without closing gap without closing gap or oh we can say that this is a finite depth local unitary transformation this is something related to the the qca that people will talk about so so i won't try to go into detail so me saying one and two are equivalent is kind of tricky but i'll just say that for now i'll just use them as interchangeable definitions but once we go to qca we'll we can talk about whether they're exactly equivalent or not finite apps local unitary circuit yes uh yeah generically we do yes but for circuit it's usually hard to consider exponentially decaying tail i'm again i'm not being rigorous to the sense of mathematical immigrants okay good great so um yes so so this this definition you see that at least it solves the first problem of that we want to have xq model of different sizes being the same place but actually brings more benefits being that we now can see that some of the originally very different looking models um are actually related uh using using this relation okay and just let me let me uh introduce to you another model which is in another paper by johan and saga vijay and lengthu which they call the maruana checkerboard model checkerboard okay so as the name suggests this is a fermionic model it's not a spin model at all it's uh degrees of freedom i'm around the modes so of course if you just look at it looks very different it is on cubic lattice with one varana mode per lattice side okay so degrees of freedom look totally different they're not on the edges and and and the maruana operator let me denote by gamma and the hamiltonian term is just eight gammas around the cube okay you multiply all eight gammas around the cube that's an even fermionically even operator then that's your hamiltonian term it's an interacting term but that's okay okay and and the the and of course um one thing that i need to mention is that the hamiltonian term is not on every cube because if it's on every cube then not all the terms commute but to have terms commute with each other we can only have the terms on a checkerboard pattern meaning every other cube and then you can check that all the hamiltonian terms commute very nicely again it's kind of like a stabilizer but this is of course the marauder stabilizer model with fermioni degrees of freedom so all the hamiltonian terms commute again you can calculate ground state degeneracy you can talk about fraction excitation you can compute entanglement you can do everything you want and and and and and we can do that okay so so first of all the ground state degeneracy um well the ground state degeneracy actually looks exactly the same as in the xq model which is 2lx plus 2ly plus 2lz minus 3. and i want to mention that since this model is on a checkerboard pattern so the translation is by two cubes in one step okay so the single unit cell contains eight cubes because the because of the the hamiltonian pattern but if you take that unit cell then the ground state degeneracy looks exactly the same as x cube model and then you can ask how or how do the excitations look like well one of the ways to make excitations is to apply a string operator like this you apply gamma gamma gamma gamma gamma and if you do this i'll try to draw it you actually violate a pair of cubed terms at the end at each end okay so at this end you violate two terms and at this end you also violate two terms okay and so this is this is like the lineal excitation actually it is indeed a linear excitation because if you try to drag this and make it turn i'm sorry my drawing is wrong sorry it's here because it's the checkerboard pattern for the hamiltonian term so at each end you only violate two of the terms and if you actually try to drag this and make a turn something will be left at the corner and then you will make some more excitations uh here okay so this is actually a lean young excitation um on the other hand we also have fractal excitation and the fractal excitation are just created by doing this by doing gammas um on a rectangle at each of the points and then you would have one of the terms being violated i will just try to draw one but then you actually have four here here here you will have one at each corner and so individually these cube terms are like fractal excitations okay so individual cubes were like were like fractional excitations that they cannot move by themselves but then you see that um if you put two of the cubes together they can move as a linear and if you put the cubes in different ways for example if you put them with a little bit of separation um see if i can draw them nicely with a bit of separation and this pair actually gives you a play now this one is something that can move in a plane sorry my drawing is messy but this this pair of um frag tongue sometimes it's a linen sometimes it's a plain one depending on how you combine them and this is what uh in jawan's paper they call it a sorry what did you call the hierarchy i forgot dimensional hierarchy that you take um frag tongs and put them together become linens and then you put them in a different way pump becomes uh plain nouns and so on and so forth okay right so this model it shares some similarity with xq model right but it also looks different because first of all it's for meonic there's some premium degrees of freedom and then um and then there's this dimensional dimensional hierarchy thing that you take a fraction you put fraction fraction together it becomes a lineal and then you take fraction fraction together it becomes a noun while in the in the xq model remember that the plain and fractant looks kind of separated from each other because sorry the linear and fractant look separated because linens are in the x sector and the fraction is in the z sector they kind of don't talk to each other while in the marina model they're all marunas there's no difference between x and z it's all myron operator so it looks like in the maruna case the the the fractal and linear are more intimately connected and another way to see the difference is that if you think about the statistics between the linear strings of the miranda model it looks different from the case in the um in the sql model where the string operator is a bunch of x right in the execute model the linear string operator are just a bunch of x so even if they might cross each other and they touch each other they will always commute with each other but in the marina model since these string operators are made of marunas and when they touch each other they might cross at a marana operator so if you choose the the length of the string well then they might anti-commute with each other if you have an even length string in in two directions and they they cross each other at one point uh and then these two strings will anti-commute with each other it's seeming to seemingly to indicate that the linear excitations will have different statistics in these two cases usually when we see that we will say oh they have different order right if we see that in the top large quarter we will definitely say that that indicates different order yeah but here we're more relaxed in our definition because we allow the addition of extra two-dimensional place and what happens is that this kind of this kind of commutation or anti-commutation is exactly what happens in a two-dimension so in two-dimensional hormonal you can have something uh that like a semi-acceleration that by semionic excitation and then you bring it in one direction or in the other direction uh then those two will anti-commute with each other so this is indicating that the difference between these two models probably can be captured by adding something two-dimensional so this all these observations motivated us to actually set out to find the equivalence between these models and we spent quite some time on it and eventually we were able to show that indeed if you start from the xq model and on this side you start from the miranda checkerboard model and then there's indeed a way to relate them in the way of finite apps local unitary circuit um if we supply both sides with some extra two dimensional topological order and actually i think we need to add some fermion moles because one is the fermionic model and the other is a spin model so we might also need to add some decouple from your modes um but that's implicit already um so by doing this we can establish their equivalence and the key of course is what are these layers and it turns out these layers are the two-dimensional double-semi model double signal and the reason we choose two-dimensional double semions is exactly what i was talking about here that because if you supplement this kind of braiding process by adding underneath it a two-dimensional double samyang then you can bind the double semion string into the linear string operator of the original model and then change the change the commutation relation from plus one to minus one so if we uh add a two-dimensional double semion a bunch of layers into the system then there's no difference between these two sides whether the linear operator commute or anti-community with each other it's just a matter of definition which one you call your elementary linear string operator does that make sense what's the other side oh it's both doubles i'm sorry both of them [Music] oh yeah i think we can actually yes oh yeah yeah yeah yeah maybe that's what we did i forgot so i think one of the hamiltonian terms you can treat it as a feminine party and just project on onto that then it becomes a bosonic model yeah this already so we actually we did this for uh the spin checker board first and for the spin checkerboard the degrees of freedom are already on the vertices so we so what happens if i remember correctly is that if you start from the checkerboard and you because the unit cell has twice the the size the twice the linear size it's actually the vertex is like center of edge but eventually we were able to match the degrees of freedoms yeah but this is of course only required because we were trying to derive the local unitary transformation if you more generally there's no reason we want to match degree of freedom exactly like that yeah so i won't try to show you um the details of the the mapping i'll just ask you to trust me that it did it does exist that this definition allows us to connect seemingly different looking models and say oh yeah and we did the exercise for a bunch of models and we found that oh they're pretty much all x-cube which is fun but also in a way uh kind of disappointing because we started from a bunch of models and then they all look the same but of course now this has been couple years old and and by now of course we already understand that um now these models being equivalent to x cube doesn't mean that things are not exciting and that's only because we were only looking at a very small subset of models and if we allow ourselves to to have a a more general mindset and explore more models then we can have things definitely uh beyond uh foliation okay so that's uh the third part of my my talk so this is a perfect question which is are you classified fully [Music] um so oh so we have a definition and then um the the first few models we explored um seem to be equivalent to x cube but of course you can ask do you have anything beyond that and the answer is yes so within the defoliated framework there are called the twisted foliated fracture order just like double samuel's twisted version of x so we have been able to show these are models actually proposing some other paper but and then we look at it from a foliated perspective um and then we were able to show that they do have the foliation structure first of all so they're not beyond foliation but secondly the foliated order is different from x cubed meaning that you cannot relate x cubed and that model by just adding two dimensional layers and we we're putting some effort to to showing that but that is not a stabilizer that is some more twisted model with facebook i know you are not an investment so if it's just the best theorem or you have some you know words of quantitative food what i mean is you have an excuse you have another job and you suspect young people and you try your best to prove here yes that doesn't mean they are different so so let me tell you more about what we proved looking for a universe yes uh of course let me say that i'm not a mathematician so everything i prove needs to be scrutinized but what we did let me say what we did so we we took um a model which we know is foliated and what we look at is actually just the planar so we kind of only care about the playing down in a certain direction and we compare the plane now in that model with the plane down in execute model and the nice thing about plane is that well that's a billion model so they form certain group and xq model is similar it's a billion so it forms a certain group but there's also a locality structure to the plane nouns because this is a three-dimensional model you can imagine a plane now be localized in different heights of the model so by taking this group of planar and also considering the locality structure we can prove that i might remember it wrong but but i've let me double check on that but what i remember we proved is that for this twisted model there's certain global constraints among the plain noun which you cannot generate by doing local things on the x-cube side well they actually also have some global constraint but a global constraint of different global constraints and because we are only allowed to do things locally we cannot match the two with each other okay okay what i'm looking for is that you take your different citations you throw away the dimensions or dimensions like that which is a highlighted effect how do you realize those two what do you see that is like a part of the kind of sorry i didn't get a question well what i'm saying is that i'm looking for proof mathematically which is you can define your fleet and then we know that degeneracy but i assume that the approaching other two dimensions will become finally mentioned oh yeah i think it has a mathematical structure um and then you show that those two models have a potion which are not always working yeah this is this is a great question because the way i talk about the order is by portioning out plotting out the plain nouns but then when i try to prove that the twisted one is twisted i were actually looking at the plain ones because because we actually don't have an example of twisted foliated order which looked different than x x cubed that when you mod out the plane nodes if you look at the fractal it looks pretty much the same at least for the ones that we can prove that has a foliation structure so we somehow have to go back to planos uh we're not throwing the way we go back to planos and by looking at planos we see they're different but but there might be foliate fracking order which are already different at the fractal level meaning that you can throw away the plane now uh we just didn't um the example we found was not of that kind um but that would be actually very interesting yeah [Music] that would be a very nice way i just don't know how to use that i will know what what i should say it's the constant which may be tricky whether it that the coefficient if you take the minimal integer uh it's tricky yeah i would love to have something like that yeah thank you that's great questions okay uh right so so so without going to i actually don't want to spend too much time on this twisted uh foliated order if you're interested i can definitely tell you more about it after but i do want to talk more about beyond foliation just because i think this is showing how little we know and if we try to establish a framework it's very important to know that there are things beyond and what it might look like okay um so for the the beyond foliation examples um we have to go go beyond stabilizer models because for the auto stabilizer models we look at they're pretty pretty much x-cube um i don't know if uh joanne has approved for anything but well definitely not the type 2 one not house code or anything with a fractal structure those are definitely beyond foliation but they're way beyond they're they're beyond my current capability of saying anything about them but the the ones i consider beyond foliation still are pretty regular they have kind of a layer structure and they have linens and planos so things that moving a line or moving a plane and then fractions showing up at the corner of of things so it's not that wild and they they look closer in form and to x cube but then we also know that they're not they're they don't have a foliation structure okay and um and so if we actually go beyond stabilizer model we have to find a different framework and the the formulation we choose to play with is this chainsawman theory and i don't have time to give you a full introduction on transcendent theory um but i'll try to give you the minimum information for you to to to know what i'm talking about and to to trust that i'm saying something meaningful um so this this transcendent theory of course is uh where we're restricted to a billion u1 transcendent theory um and usually it's uh defining two plus one d and they have been very popular wildly widely used uh to describe two-dimensional obedient topological order um and the lagrangian is written as as this so this is of course a shorthand notation and and there are a lot of possibilities a lot of variations that you can play with this kind of transcendent term in particular you can have so these these a's are what we call u1 gauge field this is why this is called a u1 transamine gauge theory and why it's called a billion because u1 is a billion group u1 gauge field and one variation we can play on this term is that we can have not just one type of u1 gauge field we can have multiple of them and we can have a superscript to this kind of uh u1 gauge field this i will take value in like one two three and then we can have coupling between a different species of these kind of u1 gauge fields and then the coefficient excuse me we'll have will be like excuse me hey i j and then we sum over ing okay of course the the first case where we just have one species of the u1 gauge field uh correspond to the case where this this coefficient k is just a number being one and what we know about this two-dimensional transamine gauge theory is not this k if you think of it as a matrix with index i and j this has to be a symmetric integer matrix meaning that all the entries have to be integer and then kij has to be equal to kji but that's pretty much all the requirement we have for the transsalmon theory and with a different k you can always write down with a different symmetric integer matrix or you call it symmetric by linear form maybe people know a more accurate expression for it um we always have a legitimate uh u1 transamin gauge theory and and that will describe um some kind of topological order or if if k is non-singular non-singular okay post like k is equal to one this is the integer called [Music] k is equal to two by two matrix zero two two zero and then this is the rhetorical so effective field theory way of describing two dimensionals okay so what we did is to take this formulation and try to say okay let's use it to describe three plus one d fraction physics and of course if we try to use it for three plus one d we need to [Music] generalize in some way one of the ways we can try to generalize it is to take not just one or two or three species of e1 gauge fields we can try to take infinite number of gauge fields and imagine that somehow they have a structure of being in different layers and then we can try to write down a similar lagrangian taking the same form except that now i and j takes value in from minus infinity to plus infinity and then with the same column so now this k matrix actually become an infinite dimensional matrix infinite dimensional bi-linear right so this is continuous in the thing in two plus one in the third direction it's actually a discrete label this is like a mystery okay so in some way this looks more three plus one even though it's kind of anaesotropic um so let's say we take the simplest k let's just take a to b diagonal diagonal like this it's infinite dimensional because it goes indefinitely and so b if we're not on the diagonal everything's zero and this has a very simple physical interpretation because if you look at the the lagrangian each a field only couples to itself meaning that the layers are actually decoupled right so physically this just means that we have a pure stack of 2d since each of them is mu equal 1 3 fractional from the top not talking to each other and i mentioned before this is already a fractal in a sense but it's kind of trivial kind of foliated factor it's just a stack of two-dimensional things and then we say okay we can try to make this slightly more non-trivial by by not taking k to the diagonal or even block diagram in case block diagonal is fun as blocks it's still kind of trivial because you can take each of the block think of it as a two-dimensional system so that's still kind of trivial but we want k to be somehow more intrinsically coupled between the layers so so you can just make it just add the next diagonal and what we can do for example is that we can add 1 to the next diagonal and because this is symmetric so we have to add like this and everywhere that i'm not writing is 0. okay so this is still legitimate and it can the size can it goes to infinity the two wall um as the size goes to infinity it really goes off oh yeah this is uh me trying to say there's period of boundary condition but if you think of infinite size it [Music] doesn't thank you um so this is the k matrix and um and the question is what are we looking at right what is the real physics that we can try to read and of course i won't try to prove anything but just quote some of the results no about the how of the k matrix gives you the ground state degeneracy of the topological order of the two dimensional terms but of course now we have three dimensional taurus so we actually have a periodic k matrix and then we have close the direction in the other two direction as well but we can similarly calculate graphs integers in that way degeneracy and then we can also calculate statistics from the k matrix basically here anions would correspond to column vectors let's say anyone unit vectors there's a single one at a certain location this is the eyes location the different anions will be labeled by different vectors again my mathematical expression is not the most not the best way but for simplicity let me just say that these are integer vectors and then the statistics the braiding statistics between the anion label by v and then sorry the label the annual label by i and the any label by j is given by 2 pi times v i transpose k inverse v j and then the topological spin of the individual anion is pi inverse vi okay this is of course something well understood but usually it's applied to finite dimensional k matrix and now we're trying to apply it to infinite dimensional k matrix of course the way we apply it to infinite dimensional k matrix is by taking a finite dimensional version of it and then scale it up okay and here's the result five more minutes and i can show you why this is beyond foliation and the result is here of course this very nice calculation uh carried out by the details you can shoes him so for this um so forth is what we call the one three one matrix the determinant goes like this one three one okay so where n is the number of layers uh or the size of the matrix is uh n by n and determinant follows a a pretty crazy formula okay and if you think about it what we call the thermodynamic limit if you take the limit of n goes to infinity this roughly skills as this that's the first indication why this is not foliated because the ground city degeneracy goes roughly as an exponential but the base of the exponential is an irrational number and if you have a system that's foliated that grows by adding two-dimensional logical order into it and every time you add a two-dimensional order the ground state degeneracy will multiply by integer factor so if if a system is foliated the ground state degeneracy will multiply by any so we'll go like exponential but with the base of the exponential being an integer or root of integer right the system size not well but here because it's an irrational number meaning that there's no integer multiple we can take up this number so that it turns into an integer this just means that it's not possible to grow the system increase n by adding decoupled two-dimensional topological order into the system okay and then we can ask about statistics by looking at of course the entries in um the k inverse matrix and this goes like i think this only this is in the limit of large large season size okay so the entries in the k matrix the entries in the k matrix decays exponentially with the separation between i and j meaning that if you go off diagonal in a k matrix the entries decay but they never decay to exactly zero just exponentially decays um and then it also decays right so this cannot happen in a foliated system either because remembering 40th is they come from the individual layers so so the statistics would have at least finite range the statistic wouldn't go too far once you have somehow sold the layer into the box it won't have a statistic that decay exponentially that's something that's pretty crazy i should say that this is not the exponential decay of correlation length in a generic um not exactly solved model we can have a finite correlation length and the exponential um the correlation can decay exponentially but here this i'm not talking about the chaos correlation i'm talking about decay of statistics um which is which is a different thing and it's a pretty weird thing and uh and usually i've never seen this kind of thing happen analyzing this model okay and finally i have oh i'm already at time so let me just wrap it up by saying that the one thing that the fusion group also looks very weird the fusion group is a product between two uh between two cyclic groups so this is a billion of course the diffusion group will be a a product of a billion groups of uh of cyclic groups um but there are only two generators and each of the generator will have exponential order and here fn means fibonacci number people at the ends people energy number so the fibonacci series of course is a growing exponentially so the order of the generator grows exponentially with n so one of them is of order fn the other is of order 5 fn and that generates a group and that can't happen with a foliated order as well because in foliated order all the pronouns come from two-dimensional planes and two-dimensional topological order we know that all the anions have finite order they can't have infinite order like this okay so this is a very simple well in terms of transcendent formulation it looks like a very simple model yet it's a it's very weird if you think about it from uh from a fractal perspective and especially from rg perspective because the the thing we're trying to do uh with uh xq model is basically asking how to do renormalization group transformation uh on the model and joven had a at a similar construction for the house code uh that you can ask what happens when i increase the system size by a factor of two or actually by a factor of eight and and in this case we know that you need to add two-dimensional layers and in joann's case he needs to add some three-dimensional more even more exotic fractal models but it's understood what's needed what extra is needed if you want to increase the system size um but for this one well i'll just say that it's beyond foliation so even if it has a layer structure you can't increase the system size by adding some two-dimensional layers and uh and just it just again just shows how how little we understand about the whole fractal thing okay and i'll just stop there [Applause] you

Channel: Institute for Pure & Applied Mathematics (IPAM)

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Keywords: ipam, math, mathematics, ucla

Id: M6E8dK6RYH8

Length: 90min 55sec (5455 seconds)

Published: Fri Sep 10 2021