 Okay, welcome everyone. We are very happy to have Ryan Thorgan today in a similar series and so Ryan is a high-energy condensed matter physicist but also has a PhD in math so he's all over the place and today he's going to tell us about high berry phase and diabetical points. Please Ryan. Thank you. Yes, thanks for inviting me and it looks like this is a pretty intimate seminar. I mean I know we're on recording but there's not many of us actually here so please we can take this in any direction you'd like if you'd like to ask questions. Feel free to interrupt and yeah so I'm going to try to tell you about this topic of higher berry phase that is that has been dear to my heart for a while. I think my first encounter with the subject here's some bibliography of my papers. I think my first encounter with the subject was actually studying crystalline phases brought us into this realm so this paper with Dominic but mostly I'm going to tell you about about this work with with Potion and Anton about a higher berry phase from a effective field theory point of view and then maybe if I have time I'll talk about these recent works where we give some more lattice flavor construction of interesting families with a higher berry phase. So the slogan here is that we want to study systems and families and we want these families to be nice like for instance a family where every system has the same infrared fixed point or a family that is all of the nearby phases of a fixed point. We want to associate some topological invariance to these families using effective field theory and the goal is to try to understand what a generic phase diagram looks like and using these topological invariance to figure out global constraints on these shapes. So I'm going to start off by reviewing reviewing the quantum mechanical berry phase it's a very well known subject but hopefully a frame it in a way that makes it clear how we can generalize to higher berry phase which applies in many body systems and then a novelty of going beyond zero dimensions is now we can consider boundaries and they're interesting boundary phenomena associated to higher berry phase and there's even a bulk boundary correspondence which I'll explain and then I'll talk a little bit about how you include symmetries in the story the global symmetries and that connects us to subjects like the Dallas charge pump and there's a question already and then a little bit about a classification of the effective field theories that capture these things and then maybe if we have time I'll get into this construction of interesting families on the lattice okay so berry phase is it's a quantized adiabatic response of our system as a function of system parameters so there's also berry phase in band theory where you consider how the like free fermion states depend on momentum here I want to consider berry phase as a function of external parameters so the paradigmatic example of that is a spin one half in a magnetic field so this is a there's a two-state system so the Hilbert space is this two-dimensional space and there's we want to consider her mission matrices acting on this space and there's a three real-dimensional space of them which is spanned by these three-poly matrices sigma x sigma y sigma z it's not really important what these are it's just important that there are three of them and then we can consider um then we can consider this Hamiltonian where we choose some parameter h I don't think I said that h is so h like h is some element of r3 and to that element we define we define this Hamiltonian by just not product with this vector so this is a two by two matrix and for h non-zero any h non-zero there's going to be a unique smallest eigenvalue right so there's going to be unique ground state of this of this Hamiltonian and we can compute it and it only depends on if we if we look at spherical coordinates here for h where we have theta going from zero to two pi will be the latitude and then five will be the longitude from zero to two pi then we can then we can express this this ground state like so in the in the sigma z basis sometimes I'll just write x y z and you see this this expression is it's good everywhere except the south pole of this sphere when we have when we have theta equals pi then this term is going to be zero this term is going to be e to the minus i phi but phi is not defined at the poles right the longitude is not defined at the poles so this is this is not smooth there but it's smooth everywhere else well it's not really a problem if we want to express the ground state at the south pole we can we can exploit this fact that the we're only really interested in ground states up to phase factors so if we simply take our ground state psi and multiply by e to the i phi then we move this phase factor over and then this is going to be good at the south pole right because this term is just going to be zero but now it's not going to be good at the north pole for the same reason and it's a you can convince yourself it's actually impossible to find a ground a smooth ground state everywhere everywhere that there's a unique ground state so in other words everywhere except the origin and really what we want to say is that these two these two ground states which are really the same they form a section of a line bundle so there's there's a line bundle that has two patches on like everything but the south pole and everything but the but the north pole and this e to the i phi is the gluing function and you see e to the i phi has winding number one so this ground state is a section of a line bundle on the sphere with turn number one so the berry phase comes from studying a connection on this bundle and we can define it in terms of in terms of our local expressions for the ground state psi just by forming this forming this inner product with the derivative okay so here h is our parameter so you take you take derivatives in the parameter coordinate so this defines a this locally defines some one form on the parameter space and if we look at how how how it transforms when we when we exploit this phase ambiguity of the ground states if we rotate the phase by some parameter dependent function b shifts like a connection form so it defines a connection and for this example that I just showed you this has been one half it's actually the same as the vector potential of a magnetic monopole located at the origin so that's something you can show and it has this nice curvature in the in the radial direction and you see that the integral of the curvature is over any sphere around the origin is going to be two pi right so that's another way to see the turn number and we think about this point at the origin which is called a diabolical point because you have you have two states meeting but it's also sort of a devilish I suppose it's somehow it's like the monopole it's like the source of the berry curvature somehow and actually this this turn number shows us that we can't get rid of this point at the origin there's there's no way to get rid of it which is kind of funny right like if we if we draw the phase diagram of the system so we draw a picture in in h space which is r3 there's just going to be a single two double degeneracy at some point in this space and we might think this looks totally not generic we should be able to do some perturbation and get rid of this but actually you can't get rid of it and for for a very small perturbation it's easy to see that just because the Hamiltonian includes in it all of the two by two matrices so for a very small perturbation you can actually undo the perturbation by difumorphism it just moves this point around a little bit but more generally if you do a larger perturbation you might create more diabolical points but the rule is you'll only ever create them in pairs of opposite turn number so we might have a situation like this and the reason that is is because for a perturbation that is not blowing up at infinity or maybe that is decaying to zero at infinity on some large enough sphere this connection you know could be changed a little bit but the turn number can't jump and so the turn numbers inside have to add up because this big sphere is homologous to all of the little spheres around each of these diabolical points this is kind of a version of the intermediate value theorem if you if you like if you have a real function of a single variable which goes from minus infinity to infinity there's a single zero at the origin and small variations in the function move the zero around big variations can create zeros but sort of in opposite signed pairs where the where the index is is whether it's going down to up or up to down and this is kind of like the matrix version of this okay so this is what we want to do we want we want to now look for look for diabolical points in in the phase diagrams of many body systems and use things like very phase to give these kinds of topological constraints like saying that the points are stable or you can only create them in pairs statements like that are what we're interested in so physically you can you encounter the berry connection so I gave you a sort of abstract definition of it but you can you can physically encounter it in the adiabatic expansion so what we want to do is we want to let our parameters depend on time so I'm going to say I'm going to call sigma this thing that depends on time and say that the parameter h is going to be equal to this function of time and then if you have time varying parameters you have a time varying Hamiltonian you can you can compute transition amplitudes using this path ordered exponential of the Hamiltonian where we start with the ground state say it's a path we start with the ground state at the initial point of the path and then we we look at the the transition amplitude to the ground state at the end of the path and we can just say that the log of that defines something that will call the effective action of sigma and the idea is that if the energy gap is bounded from below so if we're not going arbitrarily close to this diabolical point if we're sort of maybe along some sphere of some fixed radius or something like that we expect that there's a good expansion of this effective action in terms of in terms of functionals of sigma integrals of functions of sigma and its derivatives which is sort of order by order in the derivatives so if we let sigma be very slowly varying with respect to the gap we expect that the most important contributions the largest contributions to this quantity are given by the ones with the smallest number of derivatives of sigma so if you just look at what sort of terms appear in the adiabatic expansion the first one depends on so this will be some b some function of sigma but not of its derivatives and the only place that derivative will appear is maybe a single derivative and a term like this is is is topological so here x is is like the whirlwind of the system if you like and so we can eliminate t and just express this as as the pullback of this form b and so we recognize this is exactly this is exactly the very connection and so the very the very connection here is the first term in the adiabatic expansion so you see it as as the variation of the phase of the ground state as you slowly vary it okay so this is a term this is a term in the effective action and you can sort of think of it as a what's the mean a written term so if if x if x is the boundary of some surface so x is it's a one manifold x is the boundary of some surface and sigma extends to this surface then because because b as turn number one db is the volume form on s2 and we can re-express this as an integral over the bounding surface now of the pullback of the volume form which is one of these west seminawit in terms that you're probably familiar with so we're going to we're going to use that to construct higher dimensional examples so here's here's the recipe so we're going to start with a system um can be any number of dimensions with the parameter space and we want to choose a subset of the parameter space for the system as well behaved so in the quantum mechanical example this was away from the diabolical locus maybe sort of away from a neighborhood of the diabolical points so that the gap is uniformly bounded from below and in this talk we're i'm only going to be talking about systems in the trivial phase so maybe it'll be like a subset of the parameter space where the system flows to a trivial phase although you could do this in general and then we're going to couple to background fields for the parameters and integrate out the original degrees of freedom form this effective action right all i mean is we define this effective action as some log of a partition function and then we're going to look for topological terms which could appear here topological terms such as this west seminawit in term and then wonder about what this says about our system so here is sort of a proof of concept where you kind of build if you start with a nonlinear sigma model with a wcw term we can certainly get a theory with a higher bary connection and that that works like this so so we're going to consider a sigma model where the target space is a d plus two sphere here little d is the space dimension so we're in d plus one dimensions x is our spacetime manifold and we're going to consider an action like this which has this spherical symmetry but it's not really important or we have some stiffness and we have we have maybe a polarizing field so this is going to play the role h here is some d plus three vector it's going to play the role of the magnetic field in the quantum mechanical bary phase so we're going to study this as a function of these two parameters but mostly of this parameter h and k is is an integer this is the level of this west seminawit in term so this term this term you need to again you need to you need to choose a d plus two manifold z whose boundary is spacetime and we also want you know we also we also want the map phi to extend to z and we might not be able to choose this data for general target spaces things like that the better the better way to talk about this term which will connect it to this bary connection is to think about it as the wholeonomy of of a connection on this d plus two sphere and well it won't be it won't be like the usual sort of connection that that we're used to engage theory it'll be like a higher form connection so more more similar to a b field more similar to the b field in string theory or the c field you know there's there's a notion of uh u1 gauge field with arbitrary form degree and the definition is sort of inductive you can give an inductive definition very briefly the we'll say and this is this is a this is our definition we won't really use but a p form u1 connection it's a it's a collection of ordinary p forms on some fine enough open cover maybe an open cover by contractible opens so we'll call these b sub j such that on the overlaps so these uh on these pairwise intersections of of our opens that the b's differ by gauge transformation and a gauge transformation so that's this expression that the b's differ by gauge transformation the parameter of the gauge transformation c is itself a connection but of one less form degree so it's a p minus one form u1 connection so it's an inductive definition where you sort of work your way down now you want to express this p minus one form connection you'll have to do an open cover and blah blah and it bottoms out at one form connection which you already know what it is but you might even let it bottom out at a zero form connection and just say that a zero form u1 connection is just a map to u1 so when we go all the way to the end our constant you know our our zero form gauge transformations are just given by maps to u1 so there is such a connection so there's there i didn't write it here so there there exists some b some d plus one form connection on d plus on sd plus two such that db is the volume form and we're going to take the spherically symmetrical one and the spherically symmetrical one is going to define the very connection of this theory when we study it at large values of this parameter h okay so how do we see that it's pretty simple so if so if h is very very large then the field it wants to point in the direction of h or maybe there's a sign here maybe it wants to point whatever it wants to point in the direction of h so we can expand the field as as h plus a little bit and this little bit delta phi is going to be a gap fluctuation and the the mass squared of this is going to be of order the size of h so if h is very very large the corrections of these gap fluctuations is controlled and essentially we have that the effective action for sigma is is just some slightly renormalized original action that we are that we had where the stiffness now is going to be given by h but in particular it's going to have the same west amina witton term which now i've written as this uh halonomy of b and that's exactly the term we were looking for um in the effective action to say that there is a there's a higher berry connection i'm so close what is sigma here because there's no sigma it's effective oh sorry sigma here there's no sigma here so yeah um so we're taking h which is a parameter and expressing it as a function now which depends on both space and time so we can think about uh yeah sorry about that um i just want to distinguish between the the map to parameter space and then the coordinate parameter space itself but this time is is uh yeah so this time is just one of d plus one coordinates right it's not yeah yeah that's right for something extra at the end because in quantum mechanics you had one dimensional no i don't i don't add anything this is this is simply a map to um okay two x d plus two and this this is the one right you know sitting inside inside this h space where it's just a sphere of some very large radius cool so um yes we have this term in the in the effective action for sigma and we can encounter this term in the adiabatic expansion just like with uh just like with the the quantum mechanical berry phase but it's going to be ordered d plus one in derivatives it's not the leading term in the adiabatic expansion anymore we really have to see how in this case we're looking at the partition function rather than the phase of the wave function but we really have to see how the system responds to a parameter that is varying both in time and space and it has to be varying in all the coordinates so it's not leading in the adiabatic expansion but it is quantized there are there are churn numbers associated to this in this case the churn number is going to be the west eminuitan level k and these quantized terms are enough for us to protect diabolical points so they're enough for us to constrain these phase diagrams why is it not the leading one in the distance point well just because it's ordered d plus one in derivatives you could imagine something that is just order one in derivatives but depends on the metric right that uh that depends on the volume space for instance so for example if you have a if you have a ferromagnet right a lattice of spin one halves then you kind of have like you have a you have a berry connection that is like the volume times the the berry connection that we had previously so there are these like non-topological terms that are going to be in the adiabatic expansion and the topological ones you have to have as many derivatives as space-time coordinates just to write a form of the right degree so if I understand correctly you are focusing on on some particular term in in the expression for the partition function of this linear sigma model as yes as a function of this background field sigma exactly and you're saying there's going to be always this term okay this this this term that you wrote and okay and uh yeah now you yeah I don't yeah now I realize now I agree with you that you're saying that there are also other terms but now what's going to happen how you're going to focus on this term as opposed to the other terms something that I didn't don't clearly see yet yes well as long as this uh as long as this effective action is well defined so as so as long as we're in um a nice enough region say where we have a gap and the gap is uh maybe uniformly bounded from below then then all of the terms in as effective should depend kind of smoothly on other parameters so if we vary j for instance or we break the space if we break the spherical symmetry we we dimple our target space somewhat then as effective should be should be smooth and uh when it's not smooth is when we're going to have interesting things in the phase diagram like diabolical points so if I were to draw the phase diagram as a function of h at some fixed j and some non-zero k's and now this is a picture happening in in this h space so on some large sphere there's a trivial phase but with with a with a very connection and it can't just be trivial inside if it were trivial inside it would mean that this higher form connection which is defined on this sphere extends to the ball inside of it but because it has some non-zero churn number that can't happen so even though we might not understand the dynamics at small h maybe at h equals zero as a function of j we know that we know that there has to be there has to be something going on here it can't just be completely trivial phase inside so this is this is like a diabolical point but where you know there could be an island of a phase i'll show you some examples of diabolical points it's it's a little bit like and as we tune j the stiffness parameter the nature of this locus inside can change we can even break the spherical symmetry it won't it won't change the berry number out here although it might change the form of the connection you'll get you might get different diabolical loci inside but you always have to have something so it's a bit reminiscent of an anomaly if you like so an anomaly when you have a global symmetry implies that you can't have a trivial ground state but here we don't assume any kind of symmetry to get this conclusion so here's here's an example in a more detailed example of the of the same example which is that in in one plus one dimensions there's there's some critical parameter values where we get a cft and the cft we get is this se2 level one theory the central charge one cft and so the phase diagram if we like it has this it has this single point and so now d is one so now we're inside r4 the normal directions of this point correspond to four particular relevant operators of the cft and that turns out to be the so the cft has SO4 symmetry we can identify what h couples do as this SO4 vector of lightest vertex operators in the theory so i often use this compact boson notation with the two dual circle fields and they're these four vertex operators that transform as an SO4 vector and those give you the normal directions of this point but it's not a generic this is not a generic phase diagram because we can add if we if we deform this a little bit like if we if we break the SO4 symmetry then so the SO4 symmetry here is acting by rotations in this r4 and so it fixes the origin but if we break this symmetry say we add some other vertex operator so we can add this these charge two vertex operators and they're marginally relevant and if we do that this diabolical locus will will turn into something else and it turns out what we get is we get a somebody that's right we get a three-dimensional ball inside of this s4 where there's there's a twofold degenerate state on this three-dimensional ball so this is co-dimension one so it looks a bit like a first-order line although it's happening in four dimensions so it's a first-order three cycle it's not a cycle it has boundary the boundary is on some two-dimensional sphere where we have this sequence half ising CFT and this diabolical locus is generic so it's kind of funny like now we're moving in higher dimensions we're no longer going to get points uh generically speaking we're going to be getting these weird sort of blobs and other kinds of shapes that are nonetheless protected so you can you can tell this uh this doesn't have any particular symmetry none of these theories have anomalies the i6cf doesn't have any anomalies nonetheless it occurs inside this uh this big three-dimensional sphere surrounding it and there's no way to get rid of it you can change it but you can't get rid of it oh so here's a yeah this this more general situation on the right is really generic seems to be a very strong statement that's right that's right i'll i'll say a little bit more about uh what what generic should mean okay kind of like yeah actually i'm sorry but i'm still i'm still not fully yeah i'm still a bit confused about this very connection for field theory because you know quantum mechanics example was nice and easy and familiar but then um you reformulated in terms of the effective action and then you know the thing that i did understand is that you know for quantum mechanics we could think in terms of the effective action but we could also think in terms of this phase of the wave function and you know arguably this phase of the wave function is kind of walked down to earth and also easier to compute things concretely but for the field theory uh is this second interpretation of this um very connection is it available can we can we think of this some sort of phase of the ground state i don't know yeah it is available it is available although it restricts the sort of space times you can study so if we take if we take x if we take x to be you know some space y times maybe s one time then we can certainly consider consider maps to s as d plus two and uh what's going on here um right so so there will be some holonomy associated to this right so we can there will there will be some what if we uh you know b is a g plus one form right there's some holonomy associated to this map this is sigma i believe that that holonomy has the same interpretation as as the you have some ground state on y but now you have spatially varying parameters so you have to have the ground state as a function of the spatially varying parameters and then you also vary the parameter in time and you see how the phase of this way function transforms i believe that you can see that from the partition function point of view because the it's a gaps theory and so the you can see how the ground state it's going to be the most important um contribution to the partition function and then probably get the get the same conclusions studying that it's not a big limitation because as you said you know all your examples that you give them in on the next pages they refer to flat space so we don't actually care about some very complicated manifolds x do we that's right we don't we don't really care the the connection itself is kind of it's kind of um it should be locally defined that's the point of this effective action is that you get you get local terms local expressions um on x okay thanks yeah thanks for the question sorry maybe i can also ask yeah a simple question so what would happen if you seem to have a target space that is a sphere that is one dimension lower than your spacetime dimension right one higher one higher sorry yeah so what happens if you just consider an arbitrary sphere so you have an x d plus one two sp so uh you don't even have to consider spheres you can consider any space any space that uh has g plus one form connections on it um can um be the parameter space of some theory with the very phase or it can be the parameter space of a uh you know it like it can be the subset of parameter space where the system is say in a trivial phase right so that's fine but you need to have you need to have a d plus two cycles so so these d plus one form connections is very similar to line bundles it's classified by h2 these guys are classified by hd plus two so you really need to have uh if you want to have some some churn numbers some higher churn numbers um you need to have some cosmology and degree d plus two so most spheres are not going to actually yeah I see okay but other spaces can work yeah as long as your hd plus two's yeah tori are very useful for us right um your favorite target space I see okay thanks yeah thanks for the question I do spheres because it's kind of like easy to draw spheres and some things are sphere specific and I'll point that out okay so there there's also a free there's also uh free fermion versions of the same family that realize other sorts of diabolical points or other sorts of diabolical loci um let me not say too much about this example in the interest of time but if you have two complex fermions then there are many there are many different mass terms that you can write but if you write to these four specific mass terms one is the usual su2 symmetric mass term and the other one are these chiral mass terms that that transform as a as su2 adjoint then we again get a four-dimensional space where you turn on any combination of these mass terms you get a trivial phase and there's going to be a higher very phase with turn number one on this on this boundary and this is an example where you can compute the partition function exactly and show that you get um this term the effective action but if you turn on other mass terms the diabolical point will change you want to just have the single point of the origin and you'll produce something like we had before where you'll have uh you'll have a two-dimensional sphere but it won't be the boundary of anything you just get a two-dimensional sphere you can perturb it further I think you can get rings in this example so what makes a phase diagram generic roughly speaking if we have our phase diagram uh what do we mean we have we have a system that depends on all these parameters and we get the phase diagram by labeling every parameter value by the infrared fixed point that it flows to so what the what the phase diagram is is a is a collection of of subspaces labeled by IR fixed points and if we turn on some generic perturbation you expect that it will be some at any point you know turn turn on some generic perturbation everywhere in the phase diagram at each point it'll look like a roughly generic perturbation and so you expect that if there are any relevant operators that are not already accounted for in the phase diagram that these things will just disappear so for instance if this loop if the if the boundary if the if the IR fixed point here say has two relevant operators only only one combination of them can be this normal direction and if we perturb by the other relevant operator we should be able to get rid of it so roughly speaking a generic phase diagram is one where the co-dimension of all of these strata is equal to the number of relevant operators there are some caveats about originally relevant operators and things like that but this is roughly what we're talking about so for instance sometimes sometimes you get totally generic looking things with with or no topological reason for instance if you have a two-parameter phase diagram you can certainly have a first order line you know where you have twofold degeneracy ending at some ending at some ising point so this is totally generic because this ising CFD has two relevant operators let's say d you know d equals one or two or something so in general what we're doing is you can think about there as being this abstract space of all the effective field theories that you could have or maybe of all of the systems that you're interested in and so this would be like the universal phase diagram you label all of those all of those objects by their IR fixed points if it's the space of effective field theories then you really have a flow you really have rg flow in this whole space and we're getting a stratification by the attractive basins in this flow and a generic phase diagram is a slice through this space which is transverse to all of the strata now it's a really wonky looking stratification it's very far from like any kind of nice mathematical stratification but nonetheless we can say that the generic phase diagram is a transversal slice so we're only going to hit things of finite co-dimension we're only going to hit things of co-dimension k if we have k coordinates you know k parameters to tune to get to them so what we're what we're doing when we find these higher bary phases is these are these are giving us non-contractable cycles in these strata so this is because this gives you an effective field theory on that part of the cycle which does not extend to the same theory on the inside like it does not smoothly extend to a target space that fills in that cycle so that cycle is actually non-contractable in this stratum and so far we've only discussed the stratum of the trivial phase which is an open stratum there's no relevant operators just co-dimension zero phases by the way are all open the open strata are the phases and these diabolical loci they're they're lower dimensional strata right higher co-dimension that punch holes they punch topological holes in in the bigger strata so that's what's going on in in these funny looking pictures where you have the diabolical point in the middle that's really like the stratum of this diabolical point it's like punching a hole through the other stratum so that's just sort of inspirational it's not really useful so I think I alluded to this before the to go back to this question of quantization of the bary number um to kind of motivate boundaries the the bary number over now I'm going to stick to a spherical cycle so this argument is really for spheres the bary number is going to be captured by the winding number of the partition function around a particular family of sphere partition functions so if we if we look at the partition function on uh spherical spacetime it doesn't really have to be a spherical spacetime we can do something with wave functions but let me do a spherical space spacetime and let me have this parameter field sigma depend on an s1 coordinate s so that this whole map has degree one and let me also do it so that so that sigma so that sigma zero equals sigma one are constant then what will happen is that you know for every s this will be say it's in a trivial phase it's some some non-zero number if s is slowly very slowly varying sorry if sigma is very slowly varying but the phase of this will wind and you can you can see that winding in the phase from this term in the effect of action and in this picture I've drawn an example of such a family where at s equals zero and s equals one your parameter value is just say here at the south pole of the sphere and then as you increase s the spacetime is traversing larger and larger spheres which end up wrapping all the way around and then becoming smaller and then going back to the constant at the end so this is a loop of partition functions so that z s equals zero equals z s equals one parameter fields themselves are just equal we lose together so we get a well-defined winding number and the winding number has to be quantized just because it's the number this kind of argument is a lot more delicate on non-spherical cycles I'll just mention that don't know how to do this kind of argument but what it shows is that if you have a boundary and what a boundary is going to be is a let's say that a boundary is the boundary condition should be defined for all the parameter values so let's say let's say we have such a boundary condition then it means that we can take our partition functions and now we can also define them on spacetimes with this with this boundary condition and they will depend on map sigma that are just maps from x into the parameter space and now we can unwind families so this so this spherical family this family of sphere partition functions if if instead it's a family of of ball partition functions the winding number is no longer well-defined and now I can create a two parameter family where in in one of the parameters s it's doing like we had before it's it's trying to wrap around the sphere but in the other parameter r we're kind of pulling the we're using the boundary to pull it off of the sphere at the same time so the result is that at s equals what did I say here at s equals one this map has degree one this parameter map will factor through the map to the sphere and it'll have degree one but at at r equals zero it'll have degree zero and so the winding number changes so the only way for the winding number of the phase to change is for the phase to not be well-defined at some point in other words for the partition function to vanish and we ascribe this to a place where the boundary physics is not smooth where there's some some change in the boundary physics that that you have vanishing of the partition function and that's going to happen along some along some locus here on this sphere maybe it happens in some positive dimensional locus maybe it just happens at some points such that when you pull the boundary over this diabolical locus that's where the partition function is going to vanish and what we learn from generalizing this argument is that our phase diagram is going to look like this we're going to have some we're going to have some bulk diabolical points I don't know maybe there's three of them here um actually I already drew one here so this red point inside the sphere is supposed to be a bulk diabolical point and we're also going to for any choice of boundary condition there is going to be a boundary diabolical locus which ends on the bulk diabolical locus so we're going to get these sort of strings they might connect up pairs or they might run off to infinity or they can be more complicated shapes but they have to end on the they can only end on the bulk diabolical locus and the rule is that the complement of all of these guys so the complement of the of the bulk points and also these these boundary points that that that the berry connection has to be trivial there or it has to be it has to have zero berry number so this these boundary diabolical points they pop all of the non-contractable cycles in the phase diagram and it should remind you it should remind you of vile semimetals if you know about fermi arcs that's that's the statement that when you have quantum mechanical diabolical points in a band structure that when you study your material with boundary you get you get curves of boundary states you know metallic sort of fermi arcs of boundary states connecting the vile mints so this is like the field theory version of that good so here's an example back to the fermion so the so the fermion theory is nice because we can compute all sorts of things in particular there's a nice class of boundary conditions which are which are defined by interfaces so what we can do is we can say that we have to always kind of declare what which trivial phase we're going to consider a boundary to if you like or we can just say that we're going to create a boundary by having the system at some parameter value and then for all x less than zero we're going to we're going to tune the parameters to this some some fixed parameter value where the mass is is positive say and the and the ends these chiral masses are zero and then the actual boundary condition will be like the choices of these interfaces so if you like if we draw the we draw the phase diagram there's a diabolical point we're going to choose some point to be to be the trivial phase that's going to be this point and then our system we're going to have some bulk parameter value here and then along the boundary we're going to have some interface that connects them and you can already sort of see from this picture that as the as the bulk parameter the blue dot starts to wander around this loop is going to get it's going to get stuck on the diabolical point and in this system that exactly happens when we have a when we have a domain wall that goes from so here's some positive m to negative m and let's say n equals n equals zero everywhere then this is this is the famous problem that was studied by jackie even reby right so you forget about this term you just have a just have a massive domain wall then what you get in the Hilbert space of this theory is you get you get two localized modes on the wall let me call them psi up and psi down which are two complex zero modes so this this theory of this interface has a fourfold degeneracy it has it has four states of of the lowest energy depending on how you occupy these states and this fourfold degeneracy on the boundary we can we're going to consider this as a as this is our boundary diabolical locus so for for values of m that are less than zero and n which are zero that's like choosing a path that cuts right across going through this diabolical point and now we ask what happens when we turn n back on so we start to perturb this blue point a little bit well we can actually study that perturbatively in these zero modes so if i write the four states of the zero mode or if it's like chemistry notation so there's like there's there's an empty singlet and then there's then there's a then there's an there's a there's a se2 doublet here and then there's then there's a filled singlet and what you find when you turn back on n is that it acts as a magnetic field for this for this effective spin and what it's going to do is it's going to favor these states with non-zero spin and they're going to polarize in the direction of the field which is n so there are three n's it's just like a field and if we compute if we compute the berry phase of n sort of perturbatively for this interface system we find exactly the very phase example in quantum mechanics so we have this fun correspondence where here's our here's our bulk um diabolical point where on some sphere linking it we have we have a berry number you know berry phase with turn number whatever it is n then with boundary we're going to get some curve sort of near that curve if we look at spheres that are that are near the curve then we can kind of work perturbatively in just the boundary degrees of freedom we can forget about the bulk and what we'll find in that case is that there's that there's a boundary berry number and that is also equal to n so so this is the bulk of boundary correspondence now the reason why this is legal this cycle here is contractible right you can contract it you can pull it over this big sphere but you can't pull it over the big sphere without you know we forgot about the bulk you know we sort of integrated out the bulk in order to define this berry number so we did something somewhat illegal that once this sphere is large we it really comes back to bite us and the the winding numbers are actually all zero once you consider the bulk and the boundary together and that was what allows you to unlink this sphere cool so let me see um yeah in the in the remaining few minutes let me just quickly say what happens when you add symmetries so symmetries are pretty easy to add actually so if if we have a global symmetry that acts at every point in parameter space meaning that we're only going to study symmetric perturbations then we can couple to a background gauge field and terms for the background gauge field will also appear in the effective action and there will be terms that couple the parameter fields to the background gauge fields and so these are sort of invariance of now geosymmetric families and the example of an example of such term if we're in one plus one dimensions where our parameter space is a circle we can have a term that and maybe maybe maybe gz1 we can we can have a term that couples the gauge field a to the winding density d sigma and what this tells us is that if we if we say wind around the target space once in the time direction so let's let's consider let's consider a torus if we wind wind around s1 in the time direction then and say we don't vary the parameters spatially then this integral is going to split it's going to be a time integral sorry space integral of a and then the time integral d sigma so this part is just going to be one and we're going to be left with this term which is a which is a charge q wilson line which stretches across the system and the interpretation of this is that varying the parameter in the circle pumps q units of charge of u1 charge across the system so this family is is a quantized charge pump this is this known as the thallus charge pump and this kind this gives us another simple picture of why we have boundary diabolical loci because if this s1 family is occurring outside of some diabolical point then if we have a boundary and we go around the cycle we're going to pump charge and charge is going to accumulate at the boundary and so you can't have a smooth boundary because the because the charge at the ground the charge of the at the boundary is going to change so you have to have some level crossing somewhere and that level crossing is going to be where this boundary diabolical locus is so it's going to be it's going to look like two states of opposite charge crossing in energy okay and you can do that with the fermion and you can you can classify these things so let me I think let me let me end there and I'll end on this on this sort of question slide I think so I didn't tell you about the classification but you can sort of guess from like these terms that we were writing that it's easy to it's easy to classify these terms using using comology can we construct can we construct all of the families in the classification that's one question I think the answer to that is yes although we don't know how to do it in general then a question is for those families what do some of the diabolical loci look like we wonder also if there are topologically nontrivial families which are not captured by the effective action so there's some indications from lattice constructions that there might be more general kinds of thallus pumps that I don't know how to express as an effective action that would be interesting to understand and also in this talk I only talked about the trivial phase so cycles in the in that component but for topologically ordered phases you can do things involving anions if you have ground state degeneracy there are these vacuum crossing phenomena where you go around a cycle and the ground states get permuted or if you have a gap gapless phase then I don't even know what the possibilities are so that's all very interesting and yeah let me let me end there thank you thanks Ryan are there any questions can I ask one sure use this so go ahead so the parameter space that you consider in principles say if you have a field theory you can couple a parameter to any operator right that's right so you have an infinite dimensional space and yes but but most of them are going to be irrelevant right irrelevant in the RG sense or in the RG sense and you only need to look at the relevant ones for this well the idea is that if you if you have n parameters then theories with k relevant operators are going to appear along n minus k dimensional subspaces of your of your phase diagram for these n parameters so when you're studying when you're studying generic perturbations then you can't control which operator you're coupling to if you have symmetries then you can I mean this is often an issue for studying systems on the lattice that you know we want we flow to some CFT we want to study some specific CFT operator it can be hard to isolate it if you just write down some perturbation that breaks all of the symmetries then usually it's just some generic combination of all of the operators I don't know if that does that answer your question yeah yeah I see okay thank you yeah the kind of space that I tried to talk about with with this stratification this universal space is like it's maybe not well defined maybe only slices through it are well defined it might only be like a mathematical abstraction but it can be I don't know we can sort of study it co-dimension by co-dimension so I think this is a nice program maybe for I don't know for CFT people to study yeah CFTs with a small number of relevant operators I think are especially interesting thank you the questions maybe I'll ask what happens if you impose any kind of supersymmetry constraint on the theory does it simplify or I know or makes more complex analysis what would happen uh yeah so I don't think I'm qualified to answer that question but I think it's an interesting question there are some there are some papers I think David Tong has a paper about where you get some kind of supersymmetric berry connection but it's the quantum mechanical berry connection and it's related to like these TT star equations and you get like a supersymmetric model you get some kind of super connection I'm not exactly sure it's been a long time since I tried to look at that what I don't know is whether supersymmetry you know can it can it enter into the story the way that other global symmetries do and I'm not sure about that be really interesting be really interesting to know okay like you can't couple to gauge fields right it's all right so that's my usual trick a couple to gauge fields study the effective gauge theory but it still restricts the perturbations which are available to you so probably it gives you some special some special families which are not generic without supersymmetry sorry why can't you couple to gauge fields uh for the supersymmetry generator well because it has a spin index but we do that in super gravity you couple to the gravity now which is like a a spin tree half gauge field yeah that's right but I don't I mean maybe this is just my ignorance I don't know how to think about that it's like a connection on some bundle and like in the sense this one has to deal with a super gravity and one generally gravity right it's not a kft anymore something different yeah but is it okay if it's just background super gravity because I don't I don't ever want to make this gauge field dynamical then like all kinds of uncontrolled things will happen I don't know maybe maybe there's a way to do that I'm sure people have studied topological terms that you can write in super gravity and then you could ask if these are associated to like supersymmetry protected topological phases or something you know and then you would add parameters so I know in supersymmetries you have sorry go ahead there should be a supersymmetric version of the gravitational change I want yes yeah for example and then you should have something like okay so there are families that are so the theta angle is an interesting s1 family where as you as you if you have a 4d theory with a theta angle and you go around you go around the the theta it's like you pump a trance simons term to the boundary right because theta equals 2 pi can be written as d of trance simons term so that's also kind of like a thallus pump and yeah so you could have a supersymmetric version of that but I also wonder like you know in the susie like theories you often have interesting modular spaces of cfts so modular spaces are sort of rare in cfts generally speaking so my understanding but yeah so maybe you want to study like the connections naturally to find connections on this modular space in this kind of like very framework and normally you need like adiabaticity so you need some you need some gap in order to have the adiabatic expansion make sense but maybe you don't need that in a supersymmetric theory maybe like you can make sense of it okay thank you yeah thanks for the question maybe I would like to ask you a question about something quite old one of the first occurrences of topological terms in condense matter physics was this beautiful halden conjecture which was eventually proved and relating the possibility to to have a alternance between half and half and integer spin chains the half integers chains being critical and integer chains being massive would you have a new perspective on this all problems of spin chains with this approach in terms of the particular point in this parameter spaces yeah so thanks that's that's a that's a really good question I believe that what you what you are getting in in that case is for any spin you'll get you'll get some bary number at some at some large value of the polarizing field so basically you'll still have this large three-dimensional sphere where you polarize the system and I can't tell you what happens at the origin which is where the halden conjecture happens right okay but I can tell you that there's got to be something going on inside the sphere and I believe what what will happen if we if we preserve if we preserve the let me get a little drawing thing here yeah so we have this large sphere and then say we're in the integer spin case maybe spin one then with this if you preserve spin symmetry for instance then then you'll get some some topological phase in the inside which we know is a different phase from far away and so the diabolical locus here will be some some gapless points along us along a two-dimensional sphere separating this sort of island of the phase inside from outside okay but I can't tell you that this happens I can just tell you that if it's gapped then there must be some gapless points sort of nearby I see okay okay yeah it's a good question thank you I also have a related question so okay so one can summarize all this story from some practical perspective is that you found this very nice way to show that something non-trivial happens inside from doing some relatively simple calculations at least for you okay I'm not sure I understand to the point that I could repeat myself but at large fields where things are under control so and so for example you know we know that some theories have massless have CFT have CFTs other theories are massive so some of these cases you can explain using this new way of understanding things right but but but but but you know what I didn't quite understand is that you know presumably not every CFT not the existence of every CFT can be explained using this method so presumably there are some other massless fixed points some other non-trivial fixed points that have some dynamical origin but cannot be explained in terms of finding some deformation with non-trivial very very face at infinity at some large deforming field so could you like could you say like which fixed points one could hope to explain using this mechanism of which not like do we know a priori yeah what is your point to do yeah for example easing easing the fact that the easing CFT the fact that the the phi to the fourth theory in 3d has a second or first transition and not first or first transition it's a non-trivial fact could you ever hope to explain this fact using this sort of tricks or is it like not topological and so hence cannot be explained this in this way right so I think there's a number of things to say about this so so there's there's the nearby like any CFT we can define the nearby phase diagram which is which is this we just consider it with as many parameters as we have relevant operators and we just draw everything that happens near this point and in each of those nearby phases and maybe phase boundaries you can you can sort of study the topology of of if you like you draw a sphere around the point and you look at the link of the phase diagram and if you have topology there then you can you can you can hope to have some some very phase something like the easing CFT does not have that yeah the nearby phase diagram of the easing CFT looks like this where you have the CFT at this point in this first order line there's no topology going on out here if you look at this link it's just an interval and a point and at least the methods that I told you there's no sort of smooth component with any topology maybe you might like to say that the jump here is topologically protected well we could try to end complex magnetic field maybe that's true we still still still doesn't help we can't close the circle right but I think what can happen in these situations is that you might not even be guaranteed this gapless point inside like you can always have you can always have what what you're studying be kind of occluded by another phase which is just separated by a first order transition right but you might have some really interesting looking phase diagram you know with with some nearby phases right but all of it you might not see any of it because there's just this island of something else of lower energy within first order boundary so it could be everything here is is like gapped and I don't know how to rule those things out I think that in the space of diabolical low low side like there will be gapless points always like I do think that if you have topology out here in this nearby phase diagram that there's some way you can get rid of this island to reveal the critical point but that's like that's like a belief that comes from kind of like an aesthetic perspective I don't know how you would argue it but this really gets into the details of like hard problems like you said usually fine determining whether transitions continuous or not is usually very hard problem yeah okay well still seems to be very powerful I think it's like a nice way of thinking like to think about to think about the whole phase diagram and sort of try to think about the constraints of genericness yeah I would I would advocate the way of thinking above the techniques so anyone else questions last round the bar's closing well if not let's thank Ryan again for a beautiful talk thanks thanks so much thank you yeah thanks for having me