 We talk about non-equilibrium dynamics in the 2D easing model using treatments on the drugs. Thanks a lot for the introduction and also thanks for giving me the opportunity to present my work in this conference. So by now we have a pretty clear understanding on how equilibration works in a classical setting. So if we take two liquids and mix them, then we know that after a certain time they will become this kind of homogeneous mass. And we also know or like have an intuition and also the laws to describe this type of dynamics. However, if we now go to the quantum case, as probably many of you know, there are still a lot of open questions in that regard. So for example, if we take such a two-dimensional spin lattice, we split it in two halves and we initialize one half with spin ups, the other half with spin downs, then actually it is not quite clear if the system would thermalize and if so, how it will happen. And so today I want to present my research on this topic and also answer a couple of other questions in such a two-dimensional spin lattice using a numerical technique called tensor networks. So we already got a great introduction to tensor networks last week by Miles, so I will keep it rather short. But then I will go a little bit more in depth on the specifics, namely the architecture that we use is called tree tensor network. Then I will present some benchmarks to hopefully convince you that our method can work in some cases to describe dynamics in two-dimensional systems. And then we will come to the fun part where we will actually look at two very interesting cases. Alright, so last week we already heard that for the sake of tensor networks we can think about tensors as just being very high-dimensional arrays of numbers. And the neat thing is that we can also use this graphical representation where we have the tensor as being a blob and the indices are basically just open legs. And now in this graphical notation tensor multiplication means we just connect two or more of these open legs. And this is a little bit reminiscent of the Einstein notation. And now if you think about it, the wave function for example of a spin one-half system is nothing else than just a collection of complex numbers. Because once we fix the basis we just need one complex number to describe basically the amplitude for each basis vector. And so in this sense we can arrange these complex numbers in this tensor form where we have one huge blob basically and now the number of physical sites we have are the number of open indices. And there we once again encounter the usual problem that the number of parameters grows exponentially with a system size. And a couple of smart people thought about can we do better? So instead of dealing with this one very high-dimensional tensor, we can actually split it up into a product of smaller ones. And it turns out that the number of parameters now goes from this exponential scaling to a polynomial one. Here you can see a graphical notation for a very prominent example which is called matrix product state where we split up this wave function tensor into n tensors and being the number of physical sites in our system. And these tensors are now connected with a bond that has a certain dimension chi. And this bond dimension chi is basically where this exponential complexity is hidden because in order to recover the full wave function we would need to scale the bond dimension once again exponentially with a system size. But the hope would be that by cutting off chi at some finite value we can actually reach a lot of physically interesting states and hopefully work with that. It turns out that chi is actually directly related to a physical quantity namely the entanglement entropy. And so from that one can derive that for representing ground states of one-dimensional local and gap-temal tonions matrix product states are extremely good. This is due to these states having aerial law of entanglement. However what we actually after is not finding ground states and also not one dimension but first of all we want to do time evolution to look at these thermalization processes and time evolution typically leads us towards high entanglement just because as time evolves correlations are being built up in the system. And furthermore in the two-dimensional lattice it is not quite clear what architecture to use because if we take this one-dimensional matrix product state there's no good way to map it onto this two-dimensional structure. And similarly for a different reason PAPS which are the generalization of matrix product states to two-dimensional lattices they are also not that great or there are some issues because of loops present in the architecture. All right so I want to stress that this is a very hard problem and I'm not saying that we have solved dynamics in two-dimensional quantum systems but at least we want to present an architecture that can work under certain circumstances and I will show that now. So the architecture that we use is called Triton's network and the name is I guess self-explanatory because you can see we have this tree structure and this can also be generalized to higher dimensions it can look like this in two dimensions for example and the hope would be that due to this hierarchical structure which actually respects the local nature of interactions in a typical Hamiltonian we can actually push a little bit further towards this higher entanglement regime when doing time evolution. To do time evolution we employ the so-called time-dependent variational principle where the idea is that our wave function lives on this tensor network manifold which is embedded in the full Hilbert space and now after doing one infinitesimal time step we solve the Schrodinger equation. This time step can potentially lead us out of our manifold so we always need to project back onto it using this projector P and this projector P can now be split up in different ways the math for example for Triton's network has been done in this paper here and one realizes that the usual concepts you may be familiar with from matrix product states like environments and orthogonality centers they also generalize to Triton's networks in this case we have bottom environments and top environments as well as orthogonality centers a further neat thing is that kind of the most expensive operation in this scheme are matrix-matrix multiplication as well as singular-validary compositions so we profit a lot from using a GPU over a CPU in this case you can see for one TDVP time step we're on the effect of 17 speed up okay are there any questions up to this point because this has been kind of the method support so having introduced the method we will go to the benchmarking part where I will hopefully convince you that our method can work and doing dynamics in two dimensions so we will look at the transfer field ising model you can see here and we will start with an initial state that is fully transversely polarized so we are deep in this power magnetic phase and now we will look at three different quenches our results are the blue lines so the Triton's network results for an 8x8 system and we have some benchmarking results available from your quantum state simulations for a 10x10 lettuce and now in the top row you can see the time evolution of the total magnetization of the system for these three different quenches and as you can see they agree quite well and even for the most difficult case where we quench into the ferromagnetic phase we see okay the bond dimension 128 is not quite enough but by increasing the bond dimension we can improve on our approximation and then reach the benchmarking results we also have access to correlation functions so for example here you can see the light corresponding in the system and we also have access to entanglement properties I wanted to ask whether the bond dimension that you are indicating here is homogenous on the old tree or recent structure yeah basically due to this tree structure these are our physical indices they have dimension 2 so basically this bond has at most dimension 2 but then it basically is multiplicative so if you go to the next bond this would have dimension 4 because this is the whole information you can describe these two spins and so on and basically the place where we cut it off is the bond dimension here okay see thanks okay so this has been the benchmark and now we actually want to look at some interesting physics so we will stick to the transverse field ising model for the rest of the talk in this case I also added another part which is the longitudinal field and another way to think about the transverse field ising model is to write this interaction term by using this operator N which is the domain wall length operator and the idea here is basically if we have such a spin system 2D letters and we have clusters of spins then this operator would really measure the length of the domain wall separating these these spins and in a certain limit namely where the coupling is infinite the transverse field ising Hamiltonian can be reduced to the so-called PXP Hamiltonian and you may be familiar with it from for example quantum anybody's cars the idea here is that we only allow those transitions basically in the Hamiltonian which do not change the length of the domain wall so if you see here the longitudinal field term survives but for this transverse field term which actually flips spins we take only those that do not change the length of the domain wall maybe as an illustrative example if we look at this spin here it has two neighbors that are aligned in the same direction and two neighbors that are aligned in the opposite one so it has locally a domain wall length of 2 but if we flip it basically the orientation of the domain wall changes but the length does not so this is how we go from here to here basically alright and now I want to focus on a very specific initial condition namely we have a 16 by 16 spin lattice and we look at the corner of spins basically which is polarized in the same direction so for example if we take a corner like this we would initialize the spins inside the corner as looking in the positive in the action direction and the rest into the negative form here once again as a reminder the PXP Hamiltonian and it turns out that for these initial conditions the PXP Hamiltonian can be simplified even further namely it can be mapped to a one-dimensional non-interacting fermionic chain and the idea here is that if we have such a red corner configuration then each slope that goes upward can be mapped to a fermion and each slope that goes downward is a vacant site and basically the Hamiltonian that we get from this is given here the transverse field term so the spin flipping term becomes a hopping in the fermionic modes and now the interesting thing is that the longitudinal field now introduces a linear potential for our fermions and maybe you've heard that already if we had a linear potential for non-interacting fermions what we get is Vanja-Stark localization also known as like they're basically our block oscillations alright and now basically this whole thing one can solve it analytically this has been done in this paper here the exact form of this is not important right now but what is important is that we have some analytical predictions and so let's see how these Vanja-Stark localizations manifest themselves numerically so if we have no longitudinal fields basically we have no localization and what will happen is because on the corner will erode the blue line here is the analytical prediction I showed before however the interesting case is when we switch on the longitudinal field and as I said we have Vanja-Stark localization and what will happen is the corner will start to oscillate so we see around here so it starts to deteriorate this is where we reach half period and then from this point on basically the corner is rebuilding itself until we are at this point here so this has been the first example I want to show the second one being basically a different initial condition namely the following where we split up the the letters into two halves and initialize the spins on the left as spins up and on the right as spins down in this case we neglect the longitudinal field the Hamiltonian now I want to showcase two movies basically how the time evolution would look like for two extreme cases in the parameter regime on the left hand side you can see the case where the interaction strength is much much larger than the transverse field and actually because of the same analytical reasons I talked about before this domain wall is very stable it's actually stable in exponential times and this factor G over J on the other hand if we increase the transverse field we see that the letters pretty much almost thermalizes I also want to point out that the time scale here grows logarithmically so it kind of speeds up as time goes on and the question that we were asking ourselves is what happens in between so is there maybe some interesting physics going on there one quantity one can look at in that case is the so-called imbalance which is basically the difference in magnetizations of both halves so we start with a maximally imbalance state which means we have an imbalance of one and if the system thermalizes we will get an imbalance of zero and here you can see now our results for different transverse fields as well as different bond dimensions and one thing that kind of popped out to us is that for these intermediate values of G over J we have very long lift plateaus and a couple of questions that we were asking ourselves are can we describe the height and length of these plateaus for example in terms of this ratio of G over J what happens in the thermodynamic limits and is it maybe possible to derive some effective model to describe this type of dynamics so I won't be able to answer all of these questions today but at least I want to give you a taste on our progress on the last one so the idea is that to describe this type of dynamics we actually can separate it into two parts the first one being that we have this maximally imbalance state and if we now for a second neglect the interaction along the domain wall what will happen is that as time goes on the absolute value of the magnetization of both halves will kind of decrease and this will give us an effect which we call a kind of global reduction in contrast and the second effect is that spins along the domain wall will start to flip and so the domain wall will start to fluctuate and to describe this effect we came up with an effective model where the idea is that on each side in x direction we put a spin n boson and now the y position of the domain wall is described by the z component of the boson and the corresponding Hamiltonian is given by this expression here it's now one dimensional and it also has this very weird looking absolute value interaction furthermore for the results to come we will focus only on this strip around the initial domain wall to get rid of boundary effects and so now this allows us to compare the results for our full two dimensional system with this one dimensional effective model so here you can see now this comparison for different values of the transverse fields as well as different models and as you can see they agree actually quite well so most of the features are there in both models but we can make it even better by now considering this effect I was talking about in the beginning with this global reduction in contrast which we can also simulate from the full model and so if we do that you see that the height and the length of the plateaus agrees now even well with this effective model even better alright so the last point which is yeah it's a fairly new result so what I still wanted to show it here is the question of how information actually spreads in this system so we have this imbalance in the middle this domain wall and in order for the system to thermalize this information that there's an imbalance in the middle it has to spread to the bulk basically of our system and to answer this question we looked at the so-called connected correlation function which is defined here and we basically take one spin along the domain wall this red one and we compute the connected correlation function of this spin with respect to all the other blue ones now we can make a plot like this where on the x-axis we have the sides which are basically these blue squares the red line represents our position of the spin with respect to which we calculate the correlation function and on the y-axis we have the time so this is the log of the connected correlation function and now we see that for two different values in the transverse field this spreading of the correlation actually behaves quite differently so for smaller g in this case minus 0.75 we see that so we go up to times of 100 and the correlation spread only very slowly on the other hand if we increase the transverse field we see that the correlation spreads much much faster and this only goes up to time 10 and if I would plot it up to time 100 you would basically just see yellow and so this kind of gives us a glimpse that something interesting is happening maybe at g-1 and this is something we want to investigate further alright so just to summarize we have this method of tritons and networks and tiny dependent variational principle which we can use to simulate dynamics in two-dimensional quantum systems and especially for these domain wall initial states we can actually reach quite long times which in turn allows us to observe these very long lift plateaus and then we came up with an effective model to describe the dynamics of these plateaus in terms of the global reduction in contrast and kind of fluctuating domain wall and furthermore we looked at the correlation spreading which gave us a hint that there's maybe some kind of transition happening at some intermediate value of in G over J thanks a lot for your attention thank you questions if you feel you don't need the microphone go ahead or why is that thank you very much can you go back to the slides where you were comparing the your results of imbalance from simulation with the one effective 1D model the ok here here in the second like picture it seems like with the comparison it seems like the comparison is actually good in the intermediate regime or at the late time and it's not very good in the initial time so normally I mean that we would normally what expect in tensor network based simulation is like the earlier times are actually better and I don't know like why is it that the comparison is better at late times and not very good at initial times so the way we do this plot is basically so we simulate the full two dimensional system as it's done on the left and then we simulate a second system where we have this initial state and we basically neglect all of the interaction in the middle and so this gives us basically a glimpse on how this global reduction contrast is happening but of course the exact dynamics at very early times may be a little bit different so it's kind of a little bit constructed so it's not supposed to be very accurate at early times because maybe the kind of the dynamics here is a little bit different but then the agreement it's important that it agrees at later times basically Mein you wanted to ask something can you go back to the final slide so no no no the summary one exactly so I'm looking at this long time plateaus when you plot the time axis on a log scale this reminds me a lot of phenomenon known as pre-thermalization is this lifetime then exponential in some quantity, some coupling quantity or is it just power law do you happen to know we've been investigating this question and we so this has been our kind of first assumption on why this is happening but it wasn't very fruitful because so there are some considerations how this pre-thermal plateau should look like and it should be described by a GGE so like a generalized Gibbs ensemble but the conserved quantities that define the GGE they should respect the Z2 symmetry but in this case we start with an initial state that is maximally so it breaks the symmetry and so we think that in this case this is actually not the reason why we observe these pre-thermal plateaus this is the 2D model like for this pre-thermal plateaus it should be described by a GGE remember correctly okay maybe you can chat about it question here so in the case of the coordinates when you look at the dynamics of the corner it seems to me that the next one when you compare the theoretical predictions with the numerical simulations it seems to me that there is a systematic discrepancy are you worried about that I'm worried because I was involved in the theoretical predictions so that's the point so I guess you are referring maybe to this point here yes so I think one of the reasons is that this whole consideration relies on this being the case but of course we are in a numerical setting so we have to set h to its finite value and also this this period here the smaller h the longer the period so we set h to a kind of intermediate value to be able to observe this whole period basically in a finite time and so I think this is where the discrepancies might come in we might look at that I think we haven't tried that yet one last question for me is about the expressivity of the treatments on networking in 2D I have not understood let's say I give you a state with an exact 2D area low for instance a 2D finite depth quantum circuit is there a way to represent it exactly with a finite bond dimension it's not a matter of scaling of area low it's a matter of absolute level my second question was can you extract efficiently with the treatments on network the entanglement entropy it is something which is complex numerically because of some orthogonality relations so also in 2D it should work like you no more questions so let's turn the speaker again and hold the speakers of this station