Graph Rewriting for Lattice Topological Invariants

Good morning everybody. My name is Silvio Pereira. I'm currently finishing my PhD at King's College London in photonics. And I will start my position as postdoc in the Max Planck Institute talking about chiral objects. And today I'm going to try to explain you a little bit more about fiber bundles, discrete rewriting rules and Skymius. And for that I'm going to use the Wolfram brick Gracias that was developed in the 2022 summer school in physics. So for that I will try to explain which are the main objectives that we wanted to obtain with this talk. Ideally, we live in a world in all our objects are continuous. We can go as close as possible between two states. Nevertheless, in many situations, doing some quantization, doing some discretization of our space has lead to many potential interesting applications. That's the case for example, of the Wolfram project. So in this case we are going to talk about geometry that normally is associated with continuous space, but we are going to try to cut into different pieces and try to see if we can discover some properties. In particular, we are interested in considering continuous function in this discrete space. And that's exactly the point of Gracias, which is an acronym of graphs, cycles and sky news and it means thank you in Spanish, that it's the country that I came from. So let's try to see and dip into which is the main topic for today. Our object will be fiber bundles that are a mathematical theory that it was developed in the 60s in parallel with a physics theory of gotcha calibration that again turn out to be the same thing. And the interesting thing of this connection is that objects that are developed in different fields at the end result to be the same idea, something similar in another scale. We're going to discover with this project that although we started in a pure mathematical scenario, suddenly it has really interesting connection to physics, in particular to the theory of Sky. So for that let's start to the consideration of our work. The idea is we are going to try to obtain properties from objects that are not appearing in the ingredients. That is called mesoscopic theory. We're going to have different parts and when we talk about all in junction, we're going to discover that we are still some new properties will arise and in particular we're going to try to discover that by using discrete lattice that we mentioned as fiber bundles. As a way of considering this idea, we are going to use Kyber geometry, a model that it was developed in the Wolfram language by Taliesin Baydon. That was my supervisor and advisor during the Wolfram project and it's available online on the following repository and completely compatible with the Golfar language. So let's start. Therefore, we need to do some configuration of all these ideas. And you have also this computational essay available in the handout, so you can also try by yourself later or during the presentation and some links to more interesting articles of it. So that as we mentioned before, we are going to talk about Kyber structures and for that we will recur fiber bundles that it's a topological mathematical or object that it could be understood as a hierecon. Let's have some basic structure that normally 2D, but it could be any dimension. When we imagine in a higher spaces. But let's imagine a 2D space or 1D space that we have that we can associate it as a line or as a plane. And in each of these planes we are going to have a fiber that it's the part of the fiber bundle that will correspond to to some quantity that we want to measure, some function. So idea, we have just in the two dimensional case, let me try to put the image over. Here we have some space that it's the bundle that corresponds to a fiber and a base. And then we obtain which different fibers that correspond to the bundle that is the cartesian product of this idea. This normally is exploited in the context of continuous function. But what happens if we consider instead of continuous function, discrete function? That's what we obtain that these quiver ideas. So, in one dimension we mentioned that it just aligned with several points. When we have that this line could be crossed between itself, we can obtain some cycle structures. We can have a triangle, a square or higher objects. Or we can start to obtain some two dimensional objects. In particular, we can have a triangular quiver in which we have two two preferential directions that they are related in a particular way. Or we can jump to higher directions and we can have a cubic quiver in which we can have instead of 2, 3 preferential axes. Moreover, we can have more exotic ideas in which the limits of these fiber bundles are not well defined. For example, in a Moebius strip we can have this entangle between the initial part and the final part. But. But in a twisted way we can have the projective pen, we can have the flame bottle or the torus extractor. All this will be seen in a later structure. But for us we need to remember that we are going to talk about fiber bundles that basically will be in our case a two dimensional place in which we can have some base that corresponding to the place where all the fibers will arise and fibers that correspond to some function that we evaluated in our case. Although this mathematically seems quite complicated, let's try to understand in a more simple way. So for that, let's try to consider which will be the basic idea that we are going to study here. For that our fiber bundle will corresponding in a two dimensional space in which our realization will be just the corresponding values of a function evaluated on it. We are going to add an extra ingredient here. We are going to force that this function that we are evaluating as a fiber must be continuous, but here continuous in a discrete way. Because remember that this fiber bundle that is 4 by 3 discretization, we need to understand which is the concept of discrete here of continuous here. And for that the continuous. The continuous condition for continuous is that the jump that the function takes in each state that is going from left to right, it can be only plus, equal or minus one state. We can understand this as a. As a rubber band that it cannot be pushed too much far away. We cannot have a difference from here to here in just a single step. This idea of continuous it will be the one that we are going to consider in the full derivation of this, of this project. Okay? This is the most simple case in which we have what we call a linear case, in which we have a fiber and a bundle correspondent and we have the evaluation of this thing. We can have many different evaluation corresponding of different functions that we obtain in this fiber bundle. If you want to have a physical inside of this idea, it could be, for example, the potential of a parameter particular particle that as. As higher the position is, as much energy it has, we can jump up it. And the first idea that we are going to try is to classify all the structures that we can have between this object. And we are going to consider the transition between two states, if we still have some continuous deformation between both of them, we call it an omotomy. So, for example, in the previous case we can have, let's try to see here, all the corresponding transition in the given way. So we have now Instead of a 4x3, we have a 3x3. And basically in our consideration we are going to have the lower energy corresponding in the left hand side and then higher energy corresponding in the right hand side. And how can we transition from the left to the right is developed by this hypergraph that we develop over here each of the steps. It's analogous smooth transformation, anatomic transformation, in which what's one state, one part of our function it's perturbated. And then the resting object that arise from this idea is this graph that we are going to study in a further discussion. The idea about this thing is we are going to try to see which is the characterization of these objects and if they have some idea from physics world. So this was the case of a 3x3 object that we have from the minimum energy, the middle energy and the top energy that we can see that we can go from one place to another in different steps. So here the. Let's try to study objects that are in this way. The most simple case will be these linear cases in which we just have some dimensions, some number of points in the horizontal axis, some number of points in the vertical axis. And we are trying to see which are the graphs that arise from this idea. So, for example, when we have the corresponding case of a linear bundle, in which we have a general number of objects in the horizontal axis and only two steps in the vertical axis, we can represent by these graphs over here. Instead of using some level, we can use some colors. For example, in this case we just have a 3x2 element in which we have two different states and three different values for from each of these states. And here we can just see exactly the same case that we have a minimum state energy on the left middle energy here, and a higher energy. And we can see how we can transform from the lower to the high energy. We can have this for higher objects. And we can try to characterize the graph that appear in this case. So the good idea about this thing is that all this thing is just what we can consider a rubber band that there is. We can see that the object that it arises, we just need to rise each of the. Of the. Of the position. And the funny part is that we cannot rise twice in a row the same state, because therefore that will be not a continuous transformation. So that is, we just move one of each step the corresponding states over here. That corresponds to the idea of a rubber band. That was the number of axes and vertices of this object. It was already characterized in the OS surface. And we can see that nothing new was discovered here. Let's try to consider the opposite object. Now, we have a lot of different states, but only with on and off position. And we call that the serac machine, because indeed it was just a way of obtaining different hypercubes in different images. The idea is, when we only have on and off, when we have two dimensions, it will be just corresponding to a square. When we have three, three states, we just corresponding in a cube. And when we have higher dimension, we can obtain the different hypercubes. As you can imagine, again, the graphs that we are playing over here are already known in the literature and correspond basically to the number of edges and vertices of a cube of different dimensions. Okay, then for that we need to go for the next step, in which we have an unlimited number in the horizontal axis, and again an unlimited amount, arbitrary amount of position that these states can obtain. And that is, we call that a chess scheme, because indeed the graph that we obtain, or something similar, are the movements of a king in a chessboard. And the graph that we obtain is similar to the one that we saw at the beginning. And it is, we can see that for higher dimensions, we can see that it corresponds to the movement of a king in a chessboard of different dimensions. Funnily enough, you can play with a file by yourself that it's already in the handouts that the OS sequences associated with this distribution was already known. Nevertheless, the corresponding to the edges of this element wasn't included on the OS sequences, and we included recently part of the OS sequence. So, okay, we can move on and we can starting to discover more things. Because here we have talked about an initial state, our final state, and we didn't care about the things that happen between the final step and the initial step. We can be more naive and we can try to see what happened. Instead of a linear configuration, we can have some cyclic structure in which there is the initial case correspond to the final state, something like a Mobius tree. And that we can have this torus feature, this cycling feature in two elements, we can have in the horizontal axis or in the vertical axis. And that indeed will lead us to many different propositions corresponding to some intermediately cycle structures. We have some linear structure and a cycle structure corresponding to different considerations. And for that we just have two different possibilities. We can have that the cycle corresponding to the position that we have in the horizontal axis. And we are obtaining what we will call a cycle travel, because exactly in the same case that we have in the case when we have two states and we need to rise each of them individually. Now what we have is a cycle structure, a cylinder in which we have just placed a rubber band. And again, we can now not move in the same step two times the same the same point, and we just need to move step by step, the rubber band from going from the bottom to the top, and the inverse idea that indeed correspond when we have the top state corresponding to the lower state. It was what we Called and let me see what, let me see what we go here. Yes, sorry. So when we have the other possibility, we just obtain what we call the snake structure. That now what we have instead of a cycle in this horizontal axis, we have in the vertical axis. And we can see that it will correspond like the movements of a snake trying to move from a tree. So the idea here is that this example over here could start to have some reminders of some spin rotation. Because indeed we what we see is that the evolution of these states it's something that it's a cycle. We can start in some initial position, we starting to move out. And when we done a completely loop, we go to the initial state. This idea that in mathematics is quite trivial, it has very interesting implications in physics. Because whenever we have some phase that evolves from 0 to 2 PI, we just have some singularity. Because normally when we, when we have this thing, we have some very phase that it's obtaining in this trajectory. And therefore we're starting to carry some information in the literature. This is called a topological invariance. And it could have many implications in different fields. So the idea is with that is, we try to characterize furthermore how was the structure of these graphs. In general, we have mentioned that we can try to consider which is the structure in terms of vertices and edges on this structure. But we can also try to see which is the degree of correction. Because the ideas in all these scenarios we have seen that all the states are completely reachable one from each other. We can go from any state to other state in some finite amount of steps. So that means that the connected components of this structure they are always one. That's what we saw in this graphic. But that is we wanted to see how easy is from going from one state to. To the other. This physically could be associated of some idea of entropy of the system. Because if we can access to any of the system in a very fast way, this is a high entropic system and we need to do a lot of steps for going from one point to another. This is a highly ordered state system and therefore the entropy of the system will be lower. So later doing some of these ideas, trying to understand which was the different entropy of of all these graphs, we can move on and try to see what happens when we have this cycle feature not only in the horizontal axis, not only in the vertical axis, but at the same time in both dimensions. If you have played Spacing Butters sometime you will discover that this is basically what happened when you were playing that Left and right was exactly a cycle thing and also top and bottom was connected. So the idea is, let's try to see what happens when we have a fiber bundle in which the cycle is structured in the horizontal axis and in the vertical axis simultaneously. And it is we're starting to obtain really interesting things, because now there is some cases in which we are going to see that there are points that they are not reachable. And let me try to go here. Yes, in the lowest cases, all the connections are again completely reachable. But when we are starting to have higher number of points in the particular when we have 4 by 4 matrix, we obtain that although most of the states are connected, there are few points that they are not reachable that we cannot obtain from other things physically. That means that these solutions are local minima of our of our structure in the sense that we cannot reach it from a smooth transformation from the ground state. And basically that's what it's called as spin frozen spins, because here this position cannot be moved because they are frozen, as when we try to move one of them, we will lose the property of continuous that we mentioned before. So in the moment these states are freeze, which is quite interesting. But the idea is what happen when we have this thing in a higher mode, could these modes starting to be connected, as we see when we have a higher dimension, we can starting to see that this freezing states they starting to be connected in a very particular way, because we can associate them with having some topological feature. That is, normally our states are just randomly connected in a torus and they are just basically moving out up and down this position in our torus. But when we have a loop in the one of the in one of the circles in the other. Let me try to do some image over here. When we have a cycle or over here, or a cycle over here, we cannot untangle this structure to destroy it and obtain the trivial system. So when we starting to have bigger and bigger structures, for example, here we saw the 7 by 4 torus. We can start to see that several points are joined together, but they cannot be transformed in a smooth way to the ground state. This will correspond to null windy number at all. And this will correspond to two different rectangles in our in our torus corresponding to the positive and negative cells. If this is starting to get bigger and bigger, we can starting to see that we can obtain two loops, we can have three loops and so on so far. So from this very simple idea that we have mentioned about these fiber bundles, we can starting to obtain Some topology in the sense of we can have some objects that in the sense here perturbation means that we can jump from one state to the other in a smooth way. What it's called idea is there topologically protected in the sense of as far as we do some smooth perturbation to these objects, we cannot obtain to the ground state that correspond to no wind. So, okay, we have here talk about geometry, we have talked about topology, and now it's the turn to talk about a bit of physics. The idea is these objects in principle seem quite arbitrary because we cannot have any physical radiation of them. Nevertheless, we will discover that they can represent some very interesting idea in the context of magnetic fields. When we have some magnetic fields, that is the one that we have in the magnets, that we have some North Pole and South poles. There is some problem in physics that when we have something that goes from, from, from one position to other in a, in a discrete space, we have some singularity. Because all the things in physics try to be continuous as the one that we studied here. And we see that if here the things are pointing up and here the things are pointing down. When we starting to make a zoom and we starting to approach these two things close together, we cannot jump immediately from top to bottom. We need to have some evolution of the system that, that tries to entangle this discontinuity. This in magnetic context is what it's called domain wall. We have a domain wall with positive elements up top. We have a domain with lower elements of down spins. And we need to see how do the transition from one to the other. And in physics we can do it just in two different ways. We can starting to retain along this axis, what it's called a block domain wall. Or we can try to do the transformation in a nil way that it's in this other thing. And indeed this is something that we have already seen. When we take the projections of this thing in this plane or in this plane, we can obtain exactly these fiber boundaries that we have talked in the beginning, just corresponding to the nil and block example. For example, here it means that we have no evolution. For example, when we have this transition, there is no evolution in this plane. Or when we have this saw the transition, we can see that we have, we don't have any evolution in this plane. So we have done a interesting connection between these discrete fiber bundles and the domain walls. Okay, then that's what happened when we have some top structure and down structure. But when we have a singular point completely surrounded by structures that are down the evolution of the system. It takes the name of steria. And let me try to go over here. The idea is there is a very powerful object, a mathematical object, that it's the way that the system evolved. But for us, what we wanted to know is when we have a up point surrounded by down points, the way as the system tried to evolve from this point to this point, it's exterior. And we can do it again in two different ways, like doing like this or like that. And that give us the rise to two different types of ischemias that are an object that initially was developed in particle physics, then was adopted by condensed matter people trying to describe magnetic structure. And nowadays it's also understood in the context of optics, the ones that corresponding to my former studies, in my doctoral studies. So the idea is we have seen that when we have this discretization of our state, we can obtain in a very natural way the idea of scums. And now a very sensitive question arise why this is important. First of all, we can have a really important roadmap for trying to characterize these physical objects in a mathematical way, because we can talk about some energy corresponding to each of the states that has been described here. And we can try to discover why exactly these SK are a physical solution corresponding to a minimum energy system that appears in nature. This is proof, because the transition that we obtained for creating this scheme in a system is continuous. It's smooth. And normally smooth things are physical. And therefore they are the more cheap in energetical terms. So we have proven in a very naive way mathematically why skirmiums are a good approximation to these discontinuities that we have and how the system naturally evolves to isolate singularity. Okay then, so we have this characterization, but now we are going to go up a step further. Have talked about spin up and spin down components, but we can also talk about light. When we have a dipole that is oscillating, it could be represented in a very powerful way by an arrow pointing up, which is the transition the normal plane corresponding to the different charges that are displayed over there. Normally for emitting light, we have a dipole that is the most simple way to emitting light. That is basically just a plus charge and a minute charge that they are starting to oscillate that changing similarly the charge. And it has a very particular profile pattern. And it can allow us to do many interesting things. In particular, with dipoles, we can try to obtain some control of the light in the sense of we can have a Dipole that is starting to rotate like that. And we have. Let me try to go here only emission in one direction. And it started to rotate in the other direction. We have emission in the other direction. That was a really famous article that tried to obtain some nano routing of the light. And indeed it has lot of different components. To understand that we are not interested here. But the idea is when we have a dipole, we can start to control how it rotates. And we have emission in one and other directions. So the idea that we have now it's okay, we took that dipole and we try to instead of having just a single point that corresponding to a single case, we try to obtain an array of different dipoles and try to see what happened. And we create some new types of light emission that it was with dipoles in a structured way along a ring. That tried to mimic the idea of a nil of a block, a nil domain wall. But. But trying to join the initial and final part over here. And indeed, although this again tries to be a quite mathematical idea, in the simple case that it was when we have all the dipoles oriented in a ring in a different way, we discovered that we can describe a object that is the toroidal dipole. That it was a third family, despite the electric and magnetic dipole or only described by these formats. And we come and we prove that the toroidal dipole equals nothing else. That a ring of magnetic dipoles displayed in a circle. And all of this using this idea of discrete fiber bundles. And again, what is the point of doing these things? Well, indeed we can do really interesting things. Because when we have a source that can allow us to move and control the light emission in one another direction, we can try to put two of them. And we can start to create logic gates with these electric dipoles that works only with light with no electricity. So we can have something that is called optical logic gates. What is the improvement of that? When we have an optical object, we need just one single photon to obtain these ideas. Electric things by the other side corresponding to conducting electrons, that corresponding to to millions of electrons going one place to another. The effectiveness and the power consumption of these objects are completely in different orders of magnitude and can allow us to construct computers that works with light. Furthermore, further ideas could be explored in the context of not exotic pins. But I'm afraid that will be for another time. Now that I have two minutes left, I will try to go through all the questions that we have in the chat for that let me try to go over here. And let me try to go to some of the Q and A questions and yes, can you from Rohit Yossi can you please share your presentation notebook? I hope it's in the handouts and if not it's available on the link that I have provided in the handout that is linktrip Cinua Let me try to share this thing over here and in principle here you can have all the information over here and from here that's all from my side. Thank you so much for your kind attention and let me try to change this here and well I have a lot of acknowledge for this project but I encourage you to visit the Wolfrank community post in which you can play around for yourself for this idea and in the linktree access that I have provided you can test these things in a more scientific way with all the papers that were written related to this project. Finally, I wanted to thank Stephen Wolfram for the opportunity to participate in the 2022 edition of the Summer School and it was a three weeks very intense project that allowed me to create this project that now it has been replicated and expanded in many other different disciplines from physics because that was in the context of physics project about the hypergraphs consideration but now we have moved from magnetisms in the context of magnetic sk and now it has been moved from the context of optics from to thank you so much for your kind attention.