David Aasen - Topological Defect Networks for Fracton Models - IPAM at UCLA

uh so the basic idea here is um to try and view fractal models as a network of defects sitting inside a three plus one dimensional tqft and hopefully this will make a lot more sense by the end of the talk but the basic idea is to sort of split up space in terms of this so-called stratification so we'll have volumes which are called three strata sort of planes that are called two strata and then lines which live at the intersections of these surfaces called one strata and then zero strata here and to each three strata we'll assign a three plus one d tqft to the two strata we'll assign a boundary condition over the adjacent three strata tqfts and so on so at the one strata will be a boundary condition over the two strata and three strata and similarly for the zero strata now a lot goes into this construction so i'll try to give the absolute minimum necessary background in order to understand these models and give a very explicit construction for the x-cube model from a defect network all right so before moving on i want to say thanks to my collaborators danny bulmash is in the audience and then it was also done in collaboration with abanov pram kevin slagle and dominic williamson and i should say i also have not properly referenced my slides i tried to get the important references in but i do apologize okay so first um i should talk about the motivation you know why are we interested in these fractal models well one reason to be interested in them is they realize exotic phases of matter so as jonghyun was talking about this morning and shea just previously they're saying we can have particle excitations which don't occur at the end of string operators which is a very exotic possibility for uh topological orders okay so they realize exotic phases of matter and they also test our conventional understanding of gapped phases so at least naively they don't appear like they can be realized as a three plus one dimensional tqft yet our naive intuition says that at zero temperature and zero energy all gap phases should behave like a topological quantum field theory so how do we reconcile that fact with uh fractal models um and then the other motivation is for potential applications in quantum computing so perhaps as a stable quantum memory i believe this was the original interest that motivated zhang wan to discover these models and feel free to ask questions at any time okay so before moving on i should review a little bit about topological order in two and three dimensions as these are the necessary ingredients in order to build our three plus one dimensional defect networks so when i say a two-dimensional topological order you should be thinking of some spin model in two dimensions or here's a more practical example which is the quantum hall effect consists of electrons confined to a plane in the presence of a strong magnetic field and a given filling fractions or electron densities sorry you'll realize a topological phase salient features of a topological order are fractalized excitations so these are excitations that cannot occur in our three plus one dimensional universe by themselves they're also called anions if we place the system on a manifold with a non-trivial topology it'll realize a robust ground state degeneracy and in particular there is no local order parameter that can actually detect which ground state you're in okay so that's sort of encoded in this equation here up to exponentially small terms all right and we know many theoretical examples of two plus one the topological orders and have a few uh practical or realistic experiments that actually uh realize these topological orders as well so one is the quantum hall effect which is presented here um a recently uh investigated phases of getav spin liquid which was theoretically proposed by lexicative over 20 years ago and has potentially non-abelian anions sitting inside of it and there are candidate materials which actually may realize it now more mathematical models are the toric code which shane mentioned in her talk and i'll actually explain in my talk now and then more general models are these quantum doubles or string net models which are exactly solvable community projector models that allow us to understand many universal properties of 2d topological order in three dimensions we have much less or much fewer experimental realizations of 3d topological orders but we do have many theoretical models so so-called digraph would engage theory and these walker wang models and so today i'll go over the torah code and z2 diagraph would engage theory in the context of these defect networks so in 3d generically what you expect are loop excitations that live at the boundary of membranes and particle excitations which can move freely through three-dimensional space and potentially have non-trivial statistics with the loop-like excitations okay so now i want to contrast uh three plus one e topological orders with fractal models or fraction phases um and uh in particular talk about the phenomenology of these models okay so one of the sort of key features of a fraction phase is that they realize mobility constraints on the particles so you could have particles which are restricted to planes or surfaces we call these plane ons and encountered them earlier today those surfaces don't necessarily need to be stacked straight in one direction through space but they could be curved and even self-intersect if you really want them to line-ons tend to well their mobility is only along paths or lines in three-dimensional space and a helpful way to think about them is living at the intersection of uh two surfaces okay so we call them linons or lineons and then the last type of particle you can have is a fractan which is an immobile particle and you can either think about it as living at intersections of three surfaces or the perspective we'll be taking later today is that they live in these uh volumes of our defect network all right so beyond the mobility constraints of the particles we also have the ground state degeneracy so the ground state the number of ground states in the system will grow exponentially in the system size which is not something that we would see in a three plus one e topological order and i will say there are many classes of exactly solvable models that do exist and one nice thing about this defect network construction is that it allows you to construct many more so really gives us a way to construct many of these fractal models very efficiently all right and typically these fractal phases come in two types which are distinguished by the mobility of the particles so your model will either have mobile particles or it will not if it has mobile particles such as linons or planons then we call it a type 1 fractal order and if it has no mobile particles then we'll call it a type 2 fractal order and typically those excitations will live at the boundary of fractal-shaped operators up to some gauge degrees of freedom are there any questions at this point okay so i did i said i did a poor job at referencing but i did want to mention a few of the important references so first um i think the first instance of these models being studied in the literature is by claudio shimon back in 2005 however it was not in the context of fractions it was in the context of studying quantum glass as shown here it wasn't until really jongwon introduced us to these models in his phd thesis when he was trying to develop stable quantum memories in three dimensions okay and then following uh jongwon's phd he went to mit and worked with liang fu and sagar vijay and gave us a nice understanding in term of these fractal phases in terms of gauging subsystem symmetries um okay and then there are many other references that i'm not referencing here there's a huge body of work that includes field theory constructions and gapless models in two dimensions and many many more okay this work is really really inspired by and based off of these three papers here so what these three papers showed well these were the first two the upper two and then it was reviewed very well in this casing that paper that's really where i learned it what they showed is that you could recover fraction models by taking layers of two plus 1d topological orders and adding an interaction along the intersection of those layers so that you drive the system through a phase transition do some perturbation theory and then find a fraction model at the end of the day okay and that's already very close to a defect picture because it's it's it's basically saying take layers of 2d topological orders and then make them interact uh with one another okay so this is i don't expect you to read this text but this is one paragraph from this cagnet paper which basically describes in words the defect picture so they're describing this figure 9 here which consists of 2d torque codes on the planes so i've only shown two foliation directions the xy plane and the yz plane but there's also xz planes that are not displayed and what they say is that they add an interaction to the hamiltonian which generates these four torque code anions and they view those as being attached by a loop sitting in three plus one d and then proliferate that over all three plus one d space or condense it as we would say in physics and if you look at this knowing about three plus one dtqfts such as the torah code model you can recognize these strings that are sitting inside these volumes as a 3d torcode degrees of freedom which are bound to 2d torque code degrees of freedom and so that's really where the inspiration of this defect network came from okay so before describing how we realize the x-cube model from a network of topological defects i want to review the execute model just discussed in chase talk so i'll try to be brief so this this model lives on the cubic lattice so these are the edges of a cubic lattice and on every edge we have one two-dimensional hilbert space as shown here and then we have four types of terms [Music] from from these four terms we construct this hamiltonian just the negative sum of those four terms the a term is a q so it acts on the boundaries of this cube here by poly x matrices and i've written poly x well poly x y and z matrices at the top of the slide okay so this is a 12 body term that acts on all the edges around a cube similarly there are three kinds of vertex terms here i've drawn the picture for one of them so this is a vertex term which lies in the xz-plane and we apply when we apply it on this vertex it hits these four terms with a z operator and then these two other terms are just uh symmetry rotated versions of that bxe operator okay and then you can analyze this model and you'll find this very exotic ground state degeneres c which is 2 to the lx plus 2 to the ly plus 2 to the lz minus 3. and this minus 3 is quite exotic so that's basically what distinguishes it from being decoupled to the torque codes we know that the 2d torque codes must talk to one another in some way in order to account for this minus three here all right let's discuss the fundamental excitations of this model so here's the four terms appearing in the hamiltonian the lattice on the upper right we should be able to understand the fundamental excitations by figuring out how these four terms are violated at finitely many points in our three-dimensional model all right so one way we can do that is by looking at a membrane which intersects a bunch of the edges of the cubic lattice and every time it intersects an edge in a plane we apply a sigma z operator z will commute with these sorry this laser pointer this sigma z will of course commute with these three b terms on the right which are all built out of poly z matrices um but it will anti-commute with some of the cubes that uh occur at the corners of this membrane operator at all places far away from the corners that will commute with the cube terms but the corners it only overlaps on one edge and so it violates that particular cube term and you find this particular membrane operator will create four excitations okay so those four excitations can be created and annihilated in quadrupoles and then you can ask what about their mobility so we could try to take this particle in the upper right corner of that square patch and try to move it and what you'll find is you won't be able to move it without creating additional excitations so that's why it has a mobility constraint every time you try to move it you'll create more and more excitations they're coming they're violations of these b operators okay so um so that's why isolated corners of this membrane operator these excitations created by this membrane operator are called fractions now pairs of fractans turn into plane ons in this model so one way to see that is we can just apply this membrane operator right next to it and we'll find that the two excitations that were previously on the right side of that membrane operator will move over in the system and similarly we can apply that membrane operator down into the powerpoint slide and we'll find that that pair of fractals can move back into the plane so fraction dipoles or pairs of fractans make up planons all right so what about the line odds uh that's the slide so we talked about how we can violate this cubed term that results in fractions and planons violating these vertex terms results in lines so here we apply a string operator which lives on the edges of a cubic lattice and we want it to commute with the cube term so we use the sigma x operators here and so if you apply them in a straight line you'll find that they locally commute with the hamiltonian however when we terminate this line it will violate one of these three terms and similarly if you try to turn a corner oh sorry the reference didn't show up this is from jongwon's paper and sagar and liang these pictures um so if you try to turn a corner you'll find that you actually create an excitation every time you try to turn a corner and that's why these are called line ons if we keep trying to turn corners it will generate more and more excitations we also notice that there are three types corresponding to the three different terms that we can violate and these live in the three coordinate directions of the cubic lattice okay moreover um one can check that those three line-ons that live in these three coordinate directions can all fuse to the vacuum so you can create triples of these line-ons out of the vacuum all right and because of this uh pairs of linons can be turned into plane ons so if you look at this picture when you try to turn that corner you'll exchange this mc line on and then violate no hamiltonian terms at this corner so line on dipoles also generate another flavor of planons okay and with that we get to the outline of my talk so we have this somewhat exotic lattice model and we want to understand from a tqft perspective how can we realize some of this phenomenology and these excitations okay so the outline will be review topological order in two plus one dimensions which i already did and then start talking about the torque code which is the z2 topological order and then defects and gap boundaries in two plus one dimensions as a warm-up example for defects in three plus one dimensions this will all be done with the torah code again and then we'll do torah code in three plus one dimensions and then get to the defect network construction for the x cube model as defects sitting inside a three plus one dimensional torque code all right any questions at this point all right so a little bit more about anions and topological order in two plus one dimensions so um the claim is that a topological the the universal data of a topological phase in two plus one dimensions is encoded in the exchange statistics and uh fusion rules of the excitations so what we're used to is having bosons and fermions we take a pair of bosons and exchange them pick up a plus sign if we take a pair of fermions and exchange them we'll pick up a minus sign anions are quite a bit different if you exchange a pair of anions you'll pick up a phase that's what we call abelian anions if we have non-abelian anions then you'll find that you pick up a unitary transformation on the degenerate ground states okay and then the braining statistics cannot change within a given phase which is why we expect this data to encode universal properties of a given gap phase of matter okay moreover um these systems are all locally look well for a given topological phase locally look identical the spectrum will look like this you'll have a ground state if there's no excitations if you have many excitations present you may have several ground states but it will always be finite and there'll be a finite gap to the lowest energy excited states [Music] yeah so the belief is that the uh universal properties of a topological order are described by a modular tensor category that's like the gauge invariant quantity that you expect [Music] any other questions yeah i'm not going to give a complete introduction topological order um okay so the ground states are locally indistinguishable which is you know why many people want to use them for building a quantum computer so how do we describe these systems mathematically well mathematically they're described by a unitary modular tensor category again i'm not going to give a definition of this i'm definitely not the right person but i will describe it in the way that we use it today modular tensor category will consist of a list of objects which are our particles or excitations in the system military labels a b c and so on um in this case they live in two dimensions because we're doing two plus one dimensional topological order and so we have points that are labeled by the topological charge that they carry all right if we bring two of these topological charges together we can do a local measurement which will tell us what their local topological charge is and that's what we call a fusion rule okay and then there are several physical constraints such as there should be a trivial particle which is the collection of local excitations in the theory and generally we label it by the vacuum and all charges are invariant under fusion with the vacuum similarly there should always be an anti-particle for every particle right now usually we use a diagrammatic calculus to describe these systems and the diagrams that we build up are all built out of these trivalent vertices uh so the tensor product is just uh a way of encoding this fusion rule um so yeah some people just denoted with the times yes it's taking two two uh well separated anions close together and then performing a local measurement in order to determine their total topological charge so when they're far away there's no local measurement that can tell you what ground state they share any other questions okay and that that's basically this picture here so here you can think of this as a space-time picture where we have uh two particles a and b sitting in two-dimensional space and then as time goes on which you can think of as going up the slide those two anions come together and fuse into this anion c as determined by a measurement okay and these pictures we can think of as living inside a vector space which is labeled by the anions which sit at the line terminations okay so modular tensor category is a braided fusion category with additional properties the human category has this object called the f symbol which tells you how you can relate different ways of tensoring together these local hilbert spaces physically we can create these three anions in two different ways and we expect a unitary transformation to relate the associated hilbert spaces that unitary transformation we call the f symbol these f symbols are subject to some locality constraints by these diagrams which end up being encoded in the pentagon equation all right and yeah one disclaimer i have is that i will drop arrows on all these diagrams for the most part in my talk so last piece of data we need is the braiding which is just the process of taking a pair of these anions and pushing them around one another and we encode that data in another symbol which is called the r symbol okay so this um as far as we're concerned today this is just a linear relation on these diagrams a bunch of different particles given by like a cross b equals summation over matrices how do you know which particle they will fuse into uh you have to perform a local measurement in order to determine which particle it is um you can never say a priority that a and b are always going to fuse into a single particle like does it happen not not unless you know more information about the state so if you knew that a and a star were created from the vacuum then you for sure know that they'll fuse back to the vacuum but if you care create two particles far apart from one another and fuse the two that weren't pair created together you won't be able to tell what they fuse into until you've performed a measurement uh you need to know the super selection section that's correct any other questions okay so this is really where i want to get to there are some important gauge and variant quantities which we use to understand some of the excitations the one on the left is called the s matrix which encodes data about the exchange statistics of our anions you can tell there's a double grade there and the other one is a diagonal matrix that consists of these topological spin values which is morally telling you what happens when you rotate your system by two pi what phase your anion will pick up so we call theta the topological spin all right so now we can talk about well we can begin to talk about uh gap boundaries in two plus one dimensions and then we'll use what we learned here in order to build gap boundaries in three plus one dimensions all right so gap boundaries what are they useful for well they help us to describe boundaries from a topological order to another topological order um for example that topological order could be trivial or it could be non-trivial and you may have some exotic defect that lives between them okay and one general way to describe gap boundaries at least in the models that i'm discussing today is through uh anion condensation and so the idea is you have some topological order here um and you used to have this topological order on the right-hand side but then you added some local interactions which forced some of the particles to be isomorphic to one another so there are previously distinguishable particles but once we add those interactions they become isomorphic and in the mathematical language we condense this anion a okay and then between these two systems you'll have a gap boundary here whose data is also described by the anions that you're condensing okay and similarly a defect sitting inside a topological order can be thought of as a boundary for the folded system so if you take this two-dimensional topological order and fold it along that zigzag you'll find that it is some gapped boundary to the vacuum a gap boundary on top of a product of the system with itself and and so we should be able to describe that gap boundary as i just said by an algebra object or the anions that condense and so generically we should expect to be able to describe defects in 2 1d by some sort of condensation all right any questions about this slide yes so um if we agree that gapped boundaries of two-dimensional topological orders can be gap boundaries to vacuum can be described by condensing an algebra then we can take this system and like a book fold it on itself then we know that it's again a gap boundary to vacuum and so we expect that folded system to also be described by condensation okay or the defects to be described by condensation does that clear it okay uh any other questions i saw there is one yes you can diffuse with one another sorry the effects can be fused with one another they can be if you bring them close to one another but the so these defects won't necessarily be invertible okay um they cannot be braided as far as i know when they're not invertible does this correspond to what you said in the beginning that they somehow do not move but then you said in the beginning the defects are like other particles they do not move um yeah so in two plus one d all of the excitations that you could have on this defect can always be brought off the defect so that's special to two plus one d in three plus one dimensions that won't always be the case so you will end up with these mobility constraints so this mobility constraint has with the fact that they cannot degrade it or that's uh i don't know if it's related okay maybe we can discuss after any other questions all right so the mathematics of condensation is pretty complicated i just wanted at least complicated for me so here i just want to give you a flavor of it so the idea is we have some uh where is this thing there we go some braided fusion category c and then we find a new braided fusion category c mod a by condensing this object and by condensing i mean we add uh morphisms to our theory which make this object a isomorphic to the vacuum okay and physically this tells us that we are now treating this anion a as a particle which can be locally created from the vacuum and locally annihilated all right and i'll give an example of this in just a little bit okay and as i said condensation can be used to describe gap boundaries and defects and that will carry through to the three plus one dimensional system as well so one important thing to note is that we cannot condense an arbitrary anion so for example the vacuum always has to have trivial topological spin which is encoded in this diagram and it also has to have trivial exchange statistics so if we add an isomorphism between our particle a and the vacuum then we can use this diagram here to add this twist pull that dot around the twist and turn it into a twist of a and we'll you know again using this diagram or the fact that the left side is equal to the right side tells us that the a particle must have trivial topological spin there's a similar diagram you can draw for the exchange statistics of a it says it has to have trivial statistics as well okay so objects that braid um so also by studying diagrams like this you will find or discover that um objects which braid non-trivially with the particle you're trying to condense will become confined in the condensed theory and objects which braid trivially with it remain deconfined and still sit inside the condensed theory okay so that's what's described here so if we have any object b we can consider this process where we do a double braid of b with a this green line should be labeled a and if we find that the diagram is equal to the one on the right then b will remain deconfined in the condensed theory or remain an excitation of the condensed theory similarly if we find that the braid the double braid of a with b is not equal to the identity on a and b then b will become confined or it will turn into a defect of the condensed theory and i should say i'm assuming that a here is abelian which means i do not actually i'll never have lines that connect these two diagrams okay and then the last thing to note is that in the condensed theory the object b in the parent theory becomes isomorphic to be tensor a in the condensed theory any questions at this point all right so let's do an example and understand how some of this machinery works so here is the 2d tour code it lives on the square lattice in two spatial dimensions you can put on any lattice but square lattice is convenient so on each edge we have a qubit or a two dimensional hilbert space and then there are two types of terms that occur in the hamiltonian a plaquette term which consists of the product of sigma x's around a plucket and then a vertex term which consists of a product of sigma z's around a vertex and you can tell that whenever these uh terms overlap the um active terms in each plaquet or vertex term overlap on two edges at least which means they commute all right so then we can understand the excitations in this system it turns out there are two string operators which again correspond to ways of violating either the plucket term or the vertex term or both of them so if we have a string of sigma x operators that terminates at some point this will violate one of the vertex terms but no more similarly if we have a string of sigma z operators it will violate one of the plucket terms one of the pluck-out terms at each termination okay so those uh generate two of our three particles we call them e and m if we put these two particles e and m together that gives us a third kind of particle in the system which turns out to be a fermion okay and the e and m particles are bosons which means they are candidates for condensation okay and here i've just listed out the data of torque code so here are the fusion rules if you take e and m as the generators of the fusion rules you find a z2 times z2 fusion algebra the associators are all trivial and the r symbols are listed here the non-trivial r symbols okay so e and m are both condensable let's select one and condense it i'm going to take e so here's e um by looking at these braids what you'll find is that e braids non-trivially with m and psi so in the condensed theory these will become defects or they'll become confined excitations and we throw them away e braids trivially with itself and so that remains deconfined but it's really part of the condensate so it's proliferated over all of space so if we go to our picture here i have 2d tor code on the left which has the trivial particle psi e and m and then i have the condensed theory on the right hand side so in the condensed theory e is isomorphic to the trivial particle and psi and m are confined so they actually get removed said another way the e-excitation which is a finite energy excitation on the left side of the system becomes a no-cost excitation on the right-hand side because it's condensed the numbers of e's are not conserved on the right side of the system okay and if we try to take an m particle which is an excitation on the left side and move it to the right side we'll find that as we drag it over to the right side there'll be a linear cost in energy with distance from the boundary and so it becomes linearly confined and pushed back onto the boundary and because m fused with e is psi psi and m become isomorphic on this boundary okay any questions if not i will continue okay so now let's use condensation to understand defects in two plus one d topological order so here's the tor code on the left uh tor code on the right and then a gray region down the center which is a defect described by condensing these anions this is a diagonal condensation we're condensing the e from the left side with the e from the right side the m from the left side with the m from the right side and similarly for psi and so what this tells you is that if you were to bring an m particle close to this boundary you could pair create this pair of m's for free annihilate them on the left side and use that to shuttle m over to the right side so what you find is actually just the trivial invertible defect where these anions can freely pass through okay here's another example except now i'm condensing a slightly different set of bosons so here i'm condensing e with m and m with e and so if you do the same calculation what you'll find is that as you bring m across the bound across this defect it'll turn into e and as you bring e across this defect it'll turn into m but psi is left unchanged question and and really the right way to think about this is along that gray boundary i've folded the two systems up a little bit and i'm condensing across the one-dimensional boundary there yes it's uh not the same as condensing psi because it's an e uh from one torah code with an m from a different torah code good question though all right um so before building a interesting three-dimensional uh defect network i wanted to give a fairly trivial one which just consists of layers of 2d torque codes which was also mentioned in shadestalk um and so this is a trivial plain on model so each each layer consists of particles one e m and psi and we have one set of those per layer okay now in order to get a more interesting model what i'll do is fill in the bulk with a three-dimensional topological order and make that topological order interact with these uh two-dimensional layers via some condensation process so the three-dimensional topological order that i'm going to use for this today is just the 3d tor code model okay so the 3d tour code model lives on the cubic lattice and we have one qubit or a two dimensional hilbert space per site and then the hamiltonian looks very similar to the two detour code so there's a plaquette term for every plaquette which consists of sigma x matrices around the boundary of a pluck at and then on every vertex we have a product of sigma z matrices across on every edge that touches that vertex and one can again check that these two terms commute with one another all right what are relations of these terms well um if we want a term that violates the vertex term a natural choice is a string operator of sigma x's which is what i've shown here on the left so of course sigma x is commute with sigma x so you won't violate the plucket term with the string operator but whenever it terminates it will violate the vertex term that it's terminating adjacent to so here it's violating this vertex term similarly we can have violations of these plaquette terms which gen which are given by boundaries of membrane operators so here's a membrane operator which is intersecting the links of this cubic lattice and every time it intersects a link of that cubic lattice we apply a sigma z term sigma z will of course commute with this vertex term but it will anti-commute with this plucket term every time it crosses an odd number of these edges so here we violate a linear number of these plucket terms which is linear in the distance linear in the perimeter of the membrane operator all right in terms of pictures we can create these e particles in pairs out of the vacuum in order to distinguish it from the two plus one detour code i'm going to put a tilde on all the three-dimensional particles so here's e tilde being created in pairs similarly we have m tilde loop excitations which live at the boundaries of these membranes and i'll draw them like this and again those uh costs and energy which is proportional to the distance of the boundary of the membrane okay so let's put put some of these ideas together uh here we have a 2d torque code which is sitting inside two 3d torque codes which for now will be decoupled so we'll have a 3d torah code on the top and a 3d tor code on the bottom and i'll describe some additional local relations that i will add to the theory in order to view this 2d torah code as a defect sitting inside this three-dimensional torque code now because i'm fairly poor at drawing diagrams i will project this into the two-dimensional plane so in this picture this thick black line here represents the two-dimensional torque code which is coming in and out of the powerpoint slide and similarly for the 3d torque code on top and bottom okay so we want to understand the defect that i'm building in terms of what can be locally created so of course in the bottom of this picture i compare create e tilde anions for free or topological excitations for free similarly for the top of the picture and we can also build these loop excitations that are at the boundary of membrane operators similarly in this picture we can pair create m particles of the 2d to our code and pair create e particles of the 2d torah code now in order to get a non-trivial fractal on physics we have to couple these systems together in some non-trivial way all right so on the left here um i have shown a 2d tour code with a certain type of boundary condition over the adjacent 3d torque codes which is just the blank space here the boundary condition is such that the m-tilde loop excitations can only terminate on the m excitations of the 2d tor code okay so this is kind of like a flex condensing boundary glued in with a 2d torque code if you prefer that language the other kind of condensation we will require is that pairs of e tilde particles from the adjacent 3d torque codes become isomorphic to the e tilt the excitation of the 2d torah code okay are there any questions about that boundary condition okay so let's analyze what that boundary condition does for this defect inside the 3d tor code so here we have an isolated e-tilde particle and we want to move this e-tilde particle from the lower 3d torah code to the upper 3d tor code applying this relation here that's what's being condensed on this boundary so we can create this triple of excitations out of the vacuum here for free and then annihilate the bottom two etilda excitations and find an e tilde in the upper 3d torah code with an e station left over okay so we can shift uh an etild excitation from the lower 3d torque code to the upper 3d tour code at the expense of leaving behind a 2d excitation all right any questions about that okay so now we want to uh really build our non-trivial defect and we're going to foliate space with 2d torque codes so here we have a layer of 2d torque codes sitting in the z direction and then here a layer of 2d torque codes sitting in the x direction and then again a layer of two detour codes sitting in the y direction and then behind them on all of the volumes that these cut out we have a 3d torque code sitting there at each interface we introduce the boundary condition that i just discussed on the previous slide okay so again because i can't draw these 3d diagrams very well i'm going to project them into the plane and analyze the physics there so here you can imagine that i've looked just off the xy plane and i have intersected a layer of 3d tour code cubes which are described by these plaquettes here and then each edge is describing a 2d torah code that's coming out of the slide towards you okay and then again i place this boundary condition on each 2d torque code okay is that picture clear great okay so let's analyze the mobility of our isolated excitations so here i have an e tilde excitation which is isolated to this volume of 3d tor code and it's very important that i have this boundary condition on every edge and let's ask what happens if we try to move it say in this direction well we saw that we can move this excitation over as long as we find an excitation using this rule here and similarly we can keep moving e tilde however at each point when we move it we'll find we leave behind an ex citation so what that tells us is that in isolation these e tilde particles are confined to their respective 3d torah codes they cannot move freely in the system they are fractions all right now let's consider a pair of e tilde excitations that's what we have here and move that pair of util to excitations over by one unit cell okay so we move it over by one unit cell and we can only do that as long as we leave these e excitations behind just shown here uh but those ex citations they actually live on the same 2d tour code so those excitations can just pair annihilate by moving them together inside that 2d tor code so that tells us that these e tilde excitations pairs of them are actually planons by symmetry they can also move in the perpendicular direction or directions that that is transverse to their dipole moment and similarly they can be pulled apart in the vertical direction um using a similar process sure um [Music] so in this picture one line represents the same 2d tour code [Music] uh in this picture no they only interact through the three-dimensional torah code so if you allowed interaction would um potentially there's probably there's probably there's many choices that you can place for the one-dimensional boundary condition and it it may be possible to give it 3d mobility but i'm not sure off the top of my head yeah there's something like uh if you look at the this defect network where you just have three detour codes on every three cell and then choose generic boundary conditions generic but you know standard digraph winning boundary conditions on the two and one cells you find that there's something like three thousand models that you can write down and i've only analyzed a handful of them okay any other questions all right so what about the m particles well we can create a loop of m m tilde i should say inside one of these 3d torah codes and then push it into the boundary and we have a pair of m particles sitting apart from one another and we may think that this forms a plane on so let's try moving them apart what you find is if you start moving them apart you'll violate a linear number of plecat terms as you start pushing them apart from one another and so what this tells you is that the m particles are actually confined in this model okay but there are still non-trivial excitations associated with these m strings so in particular if you have an m string which connects um these 2d torque codes here and here then you'll find that this op this excitation which uh costs less and less energy as you push it into the corner but never zero cannot be removed by any of these uh local excitations here and so therefore it's a non-trivial topological excitation and it has a mobility along the intersection of the two planes that it's touching okay so short m strings are line-ons in this model and for the same reason as before line on dipoles as shown here so short strings of line-ons straddling these two 2d torque codes create plane ons in the transverse direction okay what about the fusion rules of these line-ons well here's a 3d picture of this 3d torque code volume and inside that 3d picture we create a loop of m tilde which is a local excitation we can create it for free out of the vacuum and then push that m tilde into the torah codes that are living at the boundaries of this cube here and there we find three line-on types that live on the three coordinate axes just like we had in the torah code sorry x-cube model okay yeah and this picture down here is just showing how one line-on can be through a similar process transformed into the two line-ons on the other coordinate directions okay so this shows the phenomenology of these of the x-cube model is uh identical to the phenomenology of this defect network i just described um it turns out you can very explicitly write down a communicator model for this hamiltonian in fact down many of them it's not unique they all give the same physics and um if you make a judicious choice then you can show that it's actually unitarily equivalent to this execute model we do that explicitly in our paper if you want to find more details okay so that's the end of that example are there any questions about that yeah you could well i'm not sure that's out of my pay grade so um yeah i guess the yeah i i shouldn't comment pardon hold these games yeah i agree so there it's more like a a three category for the three strata and a module three a bi-module category for the three categories on the two strata and then a higher module category on the one strata roughly so i'm waiting for mathematicians to define that for me and then i'll apply it later but you know it it if you go to the literature it's hard to pin down definitions that are practical and you know useful for doing computations yeah that's the right what we idea reduce one dimension but the key boundary condition all the horizontal and conditions on this sure harmony to choose that that's right so yeah so you can ask what yeah what did he get so yeah what kind of yeah so so it's pretty subtle actually um there are natural choices you can make for the two strata boundary conditions and the one strata boundary conditions for the zero strategic boundary conditions you choose um an item potent in this endomorphism algebra and that's basically amounts to projecting onto a certain topological charge coming from the uh adjacent three two and one strata so there's not many choices for what you can place at the zero strata yeah in two in two dimensions it's the same so at the zero strata you can project onto the vacuum e m or psi well i'm thinking [Music] you can do it in a way so that there is no degeneracy i think you you can definitely do it in a way so there is degeneracy as well but then it you know it won't be a gap phase but it won't be good yeah it will be extensive degeneracy okay uh i try to decide if one dimension of that right next to us do you think you have excuse me uh i would have thought is four because you'll have like four majoranas yeah colleen [Music] [Music] yeah i feel like it uh maybe i haven't called it that but it could be that uh you know mathematicians would want to call it a morita context the defects have to be invertible uh and when you define the tqft from a merida context it's about two categories [Music] [Music] is the same as modular so it's kind of next stage okay yeah i think and it was proposed by johnson freya to describe this topology called uh two-dimensional topological orders maybe this is the definition you are looking for maybe it makes good sense to me it would be great yeah okay i guess one subtle thing is that in the tqft we would often fluctuate over all the defects in this case i want them to be frozen okay so what are the benefits and drawbacks of the defect network well one benefit is that it devised this complicated fractal physics into more tractable tqft pieces that we can analyze by hand or on a computer it separates the microscopic details you know when i was describing the defect network i didn't actually have to refer to a lattice model in order to define it it could be defined purely diagrammatically and it may be able to help us construct new topological orders and also help characterize them okay and then i think i'll skip this slide but it's just reiterating the same pictures that i've been talking about so on the three strata we're assigning three plus one e tqft the two strata we're assigning uh bi modules over those three plus one dtqfts and then on the one strata sort of a higher order module and then the zero strata are essentially determined by pinning a topological charge okay and with that i'd like to say thanks and if you want more details on this specific instruction you can check out these papers [Applause] you

Channel: Institute for Pure & Applied Mathematics (IPAM)

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Keywords: ipam, math, mathematics, ucla

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Length: 61min 47sec (3707 seconds)

Published: Fri Sep 10 2021