Unravelling the Edge Spectra of Non-Hermitian Chern Insulators

All right. Hello everybody, and welcome to today's webinar. My name is Ayla Strickland. I'm with the scientific group here at Wolfram Research, and I will just give you a quick introduction of today's presenter, James Bartlett. So James has just finished his Doctor of Philosophy in Physics at George Mason University, where he studied topological characterization of non hermitian block Hamiltonians. In his research, he investigated 1D and 2D non Hermitian lattice models to refine the understanding of the unique topological properties. Prior to his time at George Mason, he received his Bachelor of Science in Mechanical Engineering from Pennsylvania State University in 2012, immediately after he started working as an engineer at Naval Surface Warfare Center, Dahlgren Division, conducting computational multiphysics analysis using finite element analysis, computational fluid dynamics, and computational electromagnetics. Today, James will be discussing his research on 2D non Hermitian churn insulators, which is posted to ARXIV and currently in the review process for publication in Physical Review B. And with that, James, I'm handing over to you. Thank you so much.

All right, thank you. I'd like to thank you for the opportunity to speak on, on my research today and thank the participants. So, today I imagine there's a lot of various levels of background knowledge in this field, so I'll try to keep the technical details to a minimum and talk more big picture or sort of the big results that have come out of my research. So first I'll just start by briefly introducing what is a topological insulator switch. So topological insulators are these interesting materials where the interior of the material behaves as an insulator while the surface behaves as a conductor. And so these materials are important for several reasons. One reason is that the edge states experience edge or surface states experience dissipationless transport. So the spin up electrons are moving in one direction while the spin down electrons are moving in the opposite direction. This means there's no scattering or no loss due to heat. Another important property is that these these edger surface states also are robust against impurities. So these type of properties are why topological insulators have these promising potential applications and things like quantum computing or spintronics, where the electronic not only utilizes the charge of the electron, but but also the spin of an electron. So the reason these materials are called topological insulators are that they can be characterized by what's called a topological invariant. And so a topological invariant is a quantized value of an object that remains unchanged or remains constant under continuous deformations to the space. So one of the most well known examples of this is shown on the right, where a coffee mug is being continuously deformed into a doughnut. So throughout this deformation, the coffee mug to donut, there remains one hole. This hole can be thought of as a topological invariant. And so in the end we can say the coffee mug and the donut are topologically equivalent objects. Now, an important property of topological insulators is that we'll use when when characterizing these materials is what's called bulk boundary correspondence. And this is where we can look at bulk properties of the topological insulator and know what'll happen at the surface, or vice versa. So this is very important when we're defining topological invariance and characterizing these materials. So throughout this talk, I'm going to refer to topologically trivial phases or topologically non trivial phases. And simply trivial phases are where there are no edge states present. And then non trivial phases are where there will be edge states present. So there could be one pair of edge states, two pairs of edge states, and so on. They're all different non trivial topological phases. So before getting into the my research in non hermitian topological insulators, I want to briefly explain how traditionally topological insulators are characterized. So the first step is to write the real space Hamiltonian. We next Fourier transform this Hamiltonian into momentum space, or what we call in condensed matter K space. So this is using what's called Bloch's theorem. And that states that if you have a lattice with a piece periodic potential, or in other words, if you have a crystal, you can write the wave function as a plane wave E to the I k.r times a function U which has the same periodicity as the, as the lattice. So one of one of the important consequences of using Bloch's theorem and transforming into momentum space is you have to impose periodic boundary conditions. And so this means when you diagonalize to find the eigenvectors and eigenvalues of your momentum space Hamiltonian, you are no longer going to produce the edge states, you're only going to reproduce the bulk bands. And so this, this is okay because as I mentioned on the previous slide, topological insulators have this bulk boundary correspondence. So we're able to use the momentum space Hamiltonian in order to characterize the materials. One of the other powerful aspects of using Block's theorem and transforming into momentum space is we only need to look at a small subset of the lattice in order to get behavior of the wave functions in the whole lattice. And this small area is what's called the brilliant zone and typically it goes from 0 to 2 PI. So once we Fourier transform our Hamiltonian into momentum space, we then define a topological invariant on this Hamiltonian. And this invariant will tell us if we have a non trivial phase or a trivial phase. So an example of this, which I'll show on future slides, is what's called a churn number. And you can think of the churn number as the number of times the eigenvalue eigenvector wraps the two sphere in mathematics, or the block sphere as we call it in physics. So if it wraps once, we would say we have a churn number of 1. If it completely wraps it twice in one bruon zone, churn number of 2. And if it doesn't wrap it, it'd be a churn number of zero. So for those of you who are interested in sort of getting a good introduction on this topic, I listed a good review paper at the bottom, which should work walk step by step through how topological insulators are traditionally characterized. So now if anyone's picked up a quantum mechanics textbook, you've seen that it states the Hamiltonian of an isolated system must be Hermitian. So in other words, the Hamiltonian must be equal to its conjugate transpose. In physics terms, we write this as H equals H dagger. So there's several reasons why this constraint is imposed, but some of the main ones are if a Hamiltonian is Hermitian, it ensures that the energy spectra will be entirely real. It also a second reason is that if you look at the time evolution operator, it will be probability conservative. So these are two of the big reasons why this this Hermitian constraint is imposed. Now, if we want to look at a system that is now coupled to the environment rather than isolated, we're going to relax this constraint. And this is where we get now into what's called non hermitian physics. And so by relaxing this constraint, the Hamiltonian is no longer equal to its conjugate transpose. So H is no longer equal to H dagger. And again, this is if we want to study say for instance, a system coupled to the environment. Now it's no longer isolated. These effective non hermitian Hamiltonians are a way we can get information about phenomena such as decay, where now we only need to solve the time independent Schrodinger equation, rather than having to solve the more complicated time dependent Schrodinger equation. When using these effective non Hermitian Hamiltonians, we're going to come across a lot of new interesting phenomena. I've listed some of the ones we'll come across in this talk on the left. But I'm going to start with now introducing the complex energy spectrum. So as I mentioned on the previous slide, in Hermitian systems will always have an entirely real energy spectra. But generally in non Hermitian systems we'll have complex energy spectra. And so we can think about the real energy or the imaginary energy part as the gain or loss of energy of the system. So now that we have this complex energy spectra, the first thing we want to do is be able to define when we have a gapped system versus a gapless system. So I have some examples of different spectrums on the right. If we look at A, this is an example of a Hermitian system. So we have entirely real energies, we have two blue bands and then we say that this system is gapped if it does not cross a base point on the real axis. So if it does not cross this EF point, this is just your traditional way of defining an energy gap. Now if we go onto a non Hermitian system, we have two different types of gaps. So if we look at B, this is a non Hermitian system which with a point gap, so it is a, we say the system is point gapped if the energy does not cross a base point on the complex plane. So this point can be chosen anywhere in the complex plane. And as long as the energy does not cross that point, the system is point gapped. Now the second type of gap is what's called a line gap. And this you can see if you an example, this is shown in C. So we say that this system is line gaps because the energy is not crossing a line defined on the complex plane. So in summary, now that we have complex spectra, there's two different types of gaps. There's the point gap and the line gap. Whereas in the traditional Hermitian topological insulators you're only going to have one type of gap. And so again, for those who would like to do some more reading and get a good introductory review paper on this, I've listed a paper posted to ARXIV here on the bottom of the slide. So these non Hermitian systems have been experimentally realized a number of ways. So I've listed some of the main ways the these systems have been realized on the right and also some proposed potential applications for these non emission systems on the bottom right. So on the left I'm showing some images from these experimental realizations. So the top one, the top left is these coupled optical ring resonators. So you have these ring resonators connected by auxiliary rings. And these auxiliary rings will have different lengths in order for the light to acquire a different phase shift. And this will produce asymmetric hopping or in other words, for example, hopping to the right is decreased and hopping to the left is increased. On the bottom left I'm showing one of the other experimental realizations, what is what's called dilation method. And this is a really interesting way you can realize any non hermitian system. The idea is you dilate your non hermitian Hamiltonian into a larger Hermitian Hamiltonian by coupling it with an ancilla cubit. You can then evolve your larger Hermitian system with three time and then at the end you take a measurement and post select a particular state and you're able to recover the time evolution of your your original non hermitian system. So this is an interesting way that non hermitian system has been realized. It's been done using qubits and it's also been done using nitrogen vacancy centers. But it shows an interesting connection between measurements or continued measurements and non hermitity. Okay, so now, now that I've sort of introduced some of the major topics, I'll get into what my research was concerning. And so what I was looking at was characterizing two dimensional non hermitian churn insulators in the slab geometry specifically. So the slab geometry is where you have a finite X direction and an infinite Y direction. So you have edges on the left and the right hand side of the lattice, and it's infinite in the Y direction. And so some of the motivations behind a why we did this research is first just to overall contribute to classifications of these non hermitian topological insulators. So we looked at three different ways of introducing non hermitian components to a Hermitian churn insulator. We also wanted to gain a better understanding of what's called the generalized Bruon zone in in 2D systems. So I'll talk about this in a few slides. But this is sort of the extension of the Bruan zone when you're dealing with non hermitian systems. And thirdly, we wanted to characterize the different edge spectrums in non Hermitian churn insulators. So in Hermitian or traditional churn insulators you'll get pairs of edge states where in the slab geometry, for instance, you would get one edge state on the left and one edge state on the right. In non Hermitian churn insulated you can get these scenarios where you have both edge states on one side of the lattice. So for instance, both edge states may be on the left hand side of the lattice, or both may be on the right hand side of the lattice. And so the model that we chose to use is the QWZ model. And we chose this model because it has a very straightforward clean cut phase diagram. So the Hermitian Hamiltonian is written here. So if I go back a slide to sort of talk about the lattice a bit, for the model we chose to study this QWZ model, for each of these dots there are two sites, A and B. So you'll have two sites per lattice location. And then you have hopping for instance in the X direction from A to A sites, B to B sites, A to B or B to A sites, and the same in the Y direction. So if we look at the H knot Hamiltonian, the sigmas are the Pauli matrices and then the sine and cosine terms are hopping terms. The M term is an on site term or the Dirac mass term. So for instance, if we look at this first term sine of kx sigma X, this is showing you the hopping from A to B in the X direction, A to B or B to A in the X direction. The sine of KY term likewise is A to B or B to A in the Y direction. And then these cosine terms, cosine kx and cosine ky are hopping from A to A or B to B in the X and Y directions. So to get this clean cut phase diagram, as I mentioned previously, we use what's called the Churn number. And I think the easiest way to visualize this, the mathematical way of writing it is written on the bottom right, is to just think about it as the number of times the eigenvectors wrap the Bloch sphere as they traverse one brilliant zone. So from 0 to 2 PI. So if we look at that, how many times the eigenvectors wrap this sphere, we see that if M is less than 2, they would wrap it one complete time. If M is greater than 2, the eigenvectors do not fully wrap the Bloch sphere. And so this, this gives a clear cut phase diagram where there's a transition at m equals 2. And so when the churn number is equal to 1. So when m is less than 2, this is the topologically non trivial region. And so we have a pair of edge states with one on the left hand side and one on the right hand side, when the churn number is equal to zero, that is the topologically trivial region. So there are no edge states present. So I mentioned that we looked at three different ways of introducing non hermitian terms into this QWZ model. So I listed the three different ways we did it in our paper. The first two are introducing imaginary magnetic fields. And then the last one, which is the one I'm going to focus on in these slides, is where we introduce a non reciprocal hopping term. So I T1 sine of KX Sigma Z. So what this is doing is this is introducing asymmetric hopping in the x direction. So you can think about it as hopping to the right is decreased and hopping to the left is increased. So first off, we want to compare the different phase diagrams for the periodic boundary conditions versus the slab geometry. So periodic boundary conditions is like an infinite lattice. And then the slab geometry is what I showed on on a previous slide. So just right off the bat we see that these two phase diagrams are drastically different. Now in the Hermitian case, we, with Hermitian topological insulators, these two phase diagrams would be identical due to the bulk boundary correspondence or bulk edge correspondence. So there, there is this breakdown of the bulk edge correspondence going on. So if you look on the left at the periodic boundary spectra phase diagram, we see that there's three different phases. There's the C1 phase in blue, which is this non trivial region, the trivial region with churn number equal to zero, which is shown in gray. And then a gapless region shown in red. And so this gapless region is where the energy bands are no longer separate. If we look at then on the slab geometry on the right, there are a lot more transition lines and more to topological phases. So we see that there's this blue non trivial region with churn number equal to one. And then we see the edge states for below one are localized one on the left and one on the right. And then above one you're getting both of the edge states localized. On the left hand side you get this trivial gray region where there are no edge states present. And then you get two different or three different gapless regions, two of them are in pink. And then the gold region is an interesting new gapless region. This is a region or a phase that is uniquely non hermitian. So this, this gold phase S is actually a phase where the energy is gapless, where the system is gapless. So the energy bands are not separate, but there is a hole in the spectra by which ed states traverse. So I'll show this on a few, in a few slides. But this is an interesting new uniquely non hermitian phase. So as I mentioned, these two phase diagrams are drastically different. And, and so since we no longer have this bulk edge correspondence, we're no longer going to be able to use the bulk spectra to determine what is happening at the edges. And so first we want to know what is causing this bulk edge correspondence breakdown. Well, first thing we do is look at these, the slab versus the bulk energy spectra. So that is what is shown on the left. In blue is the slab spectrum, and in red is the bulk spectrum. So the periodic lattice spectrum. We see that these two energies are, are very different. And then on the right is just the same blue slab energy plotted in three dimensions. So we see that there are two, what we call continuum bands and then two edge states that traverse this gap. And so this is what the slab geometry spectrum looks like. So now that we see that these two energy spectrums are completely different, what is causing this bulk edge correspondence breakdown and why are we getting this drastic difference in the two energy spectrums? Well, what's causing this drastic difference in the energy spectrums is what's called the non hermitian skin effect. And so this is where an extensive number of these bulk states are actually localizing to one hand side of the lattice. So if we look on the left, this is the traditional Hermitian topological insulator case. So we see that the, for the probability density, we see that all these bulk states are dispersed throughout the entire lattice. If we now introduce this non reciprocal, non hermitian term, we see on the right that these bulk eigenstates are now all localizing towards the one side of the lattice. And so this is what's causing our bulk boundary correspondence breakdown. And so in order to define this topological invariant or characterize these non hermitian topological insulators, we need to account for the fact that these bulk states are now all localizing to one side of the lattice. So we do that using what's called the generalized debris yuan zone concept. So this was laid out by this reference I have listed at the bottom. But the general idea is to take our bulk Hamiltonian H K, or in other words, our periodic boundary Hamiltonian H K and extend it so that our k values can now take on complex values. So it's going to be replaced by the variable P beta, which can now take on complex values. And so the idea is this imaginary component of beta is now accounting for the localization of these bulk eigenstates. So after, after we allow K to take on complex values, we now define a closed curve on the complex plane, which is what we call the generalized brilliant zone. This closed curve now when inputted back into H beta, will Reproduce the what we call the continuum bands and allow us to define a topological invariance and re establish bulk boundary correspondence. So if I go back slides, when we say continuum bands, what we mean are if you look at the slab geometry spectrum on the right, they are the two bands minus the edge states in the middle. So it's just the two structures on the left and the right minus not including the edge states traversing through the gap, that is what we refer to as the continuum band. So the idea to expand or continue H K to take on complex values and make H beta, establish a curve that this H beta will follow in order to reestablish bulk boundary correspondence. So how do we find this gbz? Not going to go into a lot of the technical details at this point. We go through it in our paper and as I mentioned, this reference at the bottom outlines this theory. But the main steps are you first solve the determinant of H beta minus E equals zero for beta solutions, these beta solutions will be complex. You then order the beta solutions by absolute value and you impose what is called the continuum condition. And so this continuum condition you choose the two middle solutions after ordering your beta solutions and you impose that the absolute values of these two solutions must be equal. And so this is a requirement for H beta to match that of the long chain in the limit L goes to infinity. Or in other words reproduce these continuum bands. And it's generally thought in you can think of it physically as a requirement necessary to form standing waves. So this H beta is in a true, isn't truly infinite. It's just a semi infinite system or semi finite system where it's a really large lattice, but it still has edges. So this continuum condition can be thought of as the condition necessary for forming these standing waves. So that that framework, I just sort of introduced this gbz, it was proposed in one dimension. And so there is this question of can this be expanded into two dimensions or three dimensions and so on. There's that's still an open question at this point. So for our system, since it's a 2D system, we need to sort of adapt this 1D framework for our system, since we're looking at the slab geometry, we have an infinite y direction and finite in the x direction. So we only actually need to replace kx with beta and we can keep KY as Ky. We then treat KY as a good quantum number. And so we can basically input a value for Ky, solve our H beta and then input a new value for Ky and so on. So we'll do the same method where we impose this continuum condition and then as we vary ky, we'll get a series of curves. If you have enough fidelity, you can produce what we call a GBZ surface. So an example of this is shown on the right. And so this was calculating for each KY value, the GBZ curve. And as you can see, this GBZ is not a smooth surface. So you see there's these sharp cusps and you can actually encounter singularities as well. You can get infinities or the GBZ can go to zero. And we sort of talk about that in our paper, but I'm not going to get into those different scenarios for this talk. But now that we have the gbz, we can now use this surface, input it back into H beta and completely characterize the topological phase diagram, find all the different topological phases and produce that diagram for the, the slab geometry. So we first do this for the the gapped regions. So this blue region here and the light gray region. So in order to find when we go from the non trivial blue region or trivial gray region to the gapped, which is in dark gray, we do this by finding when the continuum bands close. So in other words, now that we have our GBZ and our H beta, we can diagonalize that to get the energy spectra and just numerically find when this gap closes. This in certain systems could be done analytically. But for instance, in our model we had to numerically find it. If we then want to find when we get the non trivial versus trivial gapped regions. So this blue region versus this light gray region, we again use the churn number. So this is the wrapping of the eigenvectors around the block sphere. But now that we're, we're dealing with non hermitian systems, we need to use bi orthogonal normalization. So this is sort of established that we need to account for both the right and the left eigenvectors in our normalization. So after normalizing using this biarthogonal normalization, we solve for the eigen. We can then look at the wrapping and we see for instance this left sphere is this C1LR region. And we see that it wraps once. And this sphere on the right is the C1LL region. And we see that that also wraps once. So these are what's giving each number of one in the trivial region, this great light gray t region on the right, I'm not showing it, but that would show the eigenvectors do not fully wrap the block sphere. So there's, there's one More thing we want to do to fully characterize these different gapped regions and it is determining when the edges edge states are localized on the left and right hand side of the lattice, and when the edge states are both localized on the left hand side of the lattice. So we show analytically how to do this in our paper. So for those of interested, those interested, you can go and see how, how to analytically solve for this. But numerically we can see what's happening. And I show that on the left. So we're looking at the probability densities of the edge states. So below the transition line for T1 equal to 0.9, we just see that we have one edge state in red on the left hand side of the lattice and one edge state in blue on the right hand side of the lattice at the transition line. So when T1 equals to 1, we see that this edge state on the right in blue is now dispersed throughout the entire lattice. And then after the transition or above the transition line, when T1 equals to 1.1, we now see that this blue edge has now shifted to the left hand side of the lattice. And so in our paper we show how you can find this analytically. And then you can produce these localization diagrams on the right. And so once you find these, these transition lines, you can then determine the different regions where the edge states will be in the left or right, or both on the left or both on the right, and so on and so forth. And so this is interesting in that this only happens in non Hermitian topological insulators. So in Hermitian topological insulators you will get edge states on in pairs with one on the left and one on the right. But in our non Hermitian system, we're now getting these different scenarios where the edge states within the same topological phase are shifting sides of the lattice. Okay, and then the last step is then to characterize these different gapless phases. So in order to find when the different gapless transitions occur. So from the pink to the gold regions, you find this by again using the GBZ and plugging that back into H beta and then looking for when the imaginary energy gap closes. So if you looked at the imaginary energy Ky plane, if you just numerically find when that the two bands have a gap closing, that is where those transition lines occur. So as I mentioned before, this gold region, this phase S prime is this uniquely non Hermitian topological or non trivial phase. And so it is where we no longer have separation of the energy spectrum or the two bands. So we have a gapless phase. But as you can see, there's a hole in the imaginary energy plane and we have these two edge straights traversing this hole. So this is a new type of phase, uniquely non hermitian and is something that would be an interesting thing to study further in future papers and better understand these new gapless phases with edge states. Also we didn't get to defining any topological invariants for these different gapless regions. So that's another area that requires some, some future studies. So in summary, what our research was was we were providing an algebraic procedure where we can rely reliably characterize the edge state behavior of these two dimensional non hermitian churn insulators. So we looked at three different ways of introducing non hermicity. We showed how you can calculate the GBZ surface for these slab geometries, showed how you can find the different edge mode locations, how you can calculate these churn numbers, then also show this uniquely non hermitian topological phase where it is gapless with edge states. As I mentioned, we only looked at the slab geometry. And so another future step would then be moving on to the square geometry. This becomes much more complicated as now you have open edges in the X and the Y. And so now you need H beta X and H beta Y rather than just one beta. And so this makes finding the GBZ much more complicated. So that, that's all the slides I have, but I'll quickly go through some example calculations in the Mathematica notebook provided showing some of the ways we did our calculations. Okay, so this is just some sample calculations of some of the things we did in our research. So first I'm going to show how you can find the energy spectra of one of our models, actually the H2 model when introducing random perturbations. So this again is our H2 model where we're introducing random perturbations. And the idea behind introducing these random perturbations is to test the robustness of these edge states. And so H2 is again I didn't show this model explicitly, but it's just instead of the non reciprocal term, it it's an imaginary energy term in the imaginary magnetic field in the x direction. And so we're looking at a lattice size of 50 unit cells in the X direction and infinite in the Y. Again and we're setting random perturbations to go from negative 0.2 to 0.2. So we can create our random perturbation matrix and then perform a loop for the different kys, creating a Hamiltonian, our non hermitian Hamiltonian for our systems, adding the random perturbation and then diagonalizing this, this matrix in order to get our energy spectra and look to see if the edge states are robust. So we can plot that in 3D. And so you see that we get this energy spectrum here, which is gapless. And we see that these edge states are surviving this perturbation. And so we can conclude that these are, these are robust edge states. We can also look at it for just the KY versus real energy plane. And it is easy to see that these edge states still exist despite the random on site perturbation. This is an example of how to visualize the churn number, as I mentioned before, of how many times it wraps the block sphere. Again, this is H2 model. Some of these are just simplifications which we, we walk through in the paper. So I'm not going to go through in detail these simplifications, but the basic idea is you're solving for the eigenvectors of your H beta and then you're parameterizing it to, to be shown on the block sphere, so getting your thetas and phis to show it on the block sphere. And so we can basically do a list line plot in order to plot this, these eigenvectors trajectories over 1GBZ on the block sphere. And as you can see in this case this would be a non trivial region because the eigenvectors are wrapping the block sphere one time over one gbz. Lastly, I just want to, I just included a, a rough computation of the GBZ for the H3 model. This is using one of the algorithms we talk about in our paper, which is the self interaction algorithm. So this has a lot of different sort of analytical steps which, which we go through step by step in the paper. So I'm not going to go through this right now, but I'll be happy to answer questions if anyone wants to ask about that. The idea is you input a radius for your complex number, find a guess radius, you find intersection points, and then if those intersection points meet certain conditions, they would be part of the gbz. That's sort of the general idea behind this particular algorithm. And so you can, you can do that, you can find the intersection points either analytically or you could, you could code it in to numerically find these points, and then if they meet certain conditions, you would add them to your GBZ curve. And in the end you'll get something along these lines depending upon your precision and tolerance levels. So these different algorithms all have different strengths and Work weaknesses for the gbz. For instance, this, this algorithm is good at finding cusps or areas where the curve comes to a point, whereas there's another algorithm we talk about that is good at resolving elliptical shapes, elliptical GBZs. And so for your research, if you were to use this gbz, you may need to combine the different algorithms or, or mess around with your precision in order to get the amount of points you want to have a high resolution GBZ curve. And so that is all I have for today. So thank you again for allowing me the opportunity to talk. And at this point I'll ask or I'll answer any questions anyone may have.

So you had mentioned in the beginning that one possible realization for a non hermitian system is using a bigger Hermitian system. Yes, that's the same in quote trick that we use to go from a density matrix description of a system to a pure state description where we first add a couple qubits, make it a pure state and then trace out are the number of needed ancilla qubits a different number than we need to make a pure state.

The number of ancilla qubits needed is one for no matter the size of your non hermitian system.

Oh, okay. So that's actually much smaller than for generic going from density matrix to a.

Yes. Yeah. So using sort of this dilation method. Yeah. However big your non hermitian matrix is, you would only need one insulin qubit. Yeah. To make it hermitian and evolve it.

And from a purely numerical point of view, I assume this is clearly preferable.

Yes. Yeah, yeah.

And the 1 qubit is also enough if the, if the non emission system would have continuous degrees of freedom and not be a letter system.

I, I believe so. I, I believe the. I believe generally you can always use just one qubit to dilate a, any Hermitian Hamiltonian into any non Hermitian Hamiltonian into her mission Hamiltonian. I see the one. The one thing about this dilation method though is it does require post selection in order to recover your non hermitian system. So that you can't necessarily set up your system in any way experimentally. You have to sort of have this post selection where you only select a particular state of the ancilla to recover your evolution. Yeah.

I'm not sure how other questions from participants come. I don't want to take any of them away. Okay, then I have another one. So for classical brilliant zones there is the notion of higher order brilliant zones. For UMKLA processes and related things, is that equivalent for the non harmonician case.

I'm not sure. I'm not sure if that's known yet. For the non hermitian case. Yeah, right. Yeah. So the, the generalized Briuan zone at this point is well understood. For these 1D chains, there, there is some general algorithms, which I, I meant, I mentioned in the paper, that you can always find for 1D cases, this, this GBZ. Now, going into two dimensions or higher dimensions, this becomes much more complicated to find this gbz, and it is still a question of is it even generalizable to two dimensions or higher dimensions. I believe it is, but there's still that question. And there's no general algorithm to find higher dimensional GBZs at this point. Thank you.

Let me think. That was a very nice talk.

Thank you.

Quite a good overview over the determination channel.

Thank you. Yes.