 So good afternoon everyone I would like to thank the organizer for giving me the opportunity to present our our work here. So the outline of my talk is the following. So I think I will talk a lot about the rather simple method exact organization. So the conceptual part will be very short but I will tell you a little bit about the possibilities like in terms of matrix dimensions and what kind of tricks one have to look at. So there will be some so that you have an idea if you're doing Hamiltonian truncation what are the limits of matrices you you in principle can achieve just to have an idea. And then I would like to to present you some results on our own study of Hamiltonian truncation of the five to the fourth year in one plus one dimension. So our history was that we we came across the paper by Lorenzo and Slava two bit more than two years ago and we realized that we had technology to to work on that as well so we tried to give our own own shot at it and so we will report about some of the results. And so the main point is basic that we're we're using somewhat larger Hilbert space sizes and we extrapolate and then you use some finite size scaling analysis to to come up with a with an independent estimate for the critical point I think which is in close agreement with the most recent ones by Slava and one with the previous speaker. And so and depending on how far I I get with this I might also talk a little bit about some torus energy spectra which you have obtained using lattice models so which are not true quantum field theories but which we tune to their to some believed conformal fixed points and we look at torus energy spectra and I might show you that that these torus energy spectra are actually useful as fingerprints in numerical simulations at least of lattice models we can use them as fingerprints to diagnose the CFT by just identifying what the spectrum is and matching it with the catalogue of established theories and in one of the applications it might also be that one can actually read off from the from the torus energy spectrum at least the the sign of of coupling constants for example of some perturbations which usually are not so easy to get so that that may one of the points there. So let me start by just giving a very short introduction or or a kind of a listing of possibilities for exact diagonalization so in exact diagonalization as we heard yesterday this is a conceptually simple method it's just matrix quantum mechanics as it has been called so we're trying to solve the time-dependent shooting equation H psi equals E psi and so we want to solve eigenvalues for this for these problems and we this problem lives in a Hilbert space of a certain dimension and the question is a bit under what circumstances can you read what type of matrices so if you're just working kind of with matrices which for some reason are completely full so you just have a matrix which say is her mission and there are a lot of entries everywhere so that's what we call a dense matrix then one would typically have to store the complete matrix obviously and that already put some constraints of what you can actually store simply as a numerical matrix and so if you have a matrix which is completely full and you want to solve it then there are ways using many of you know that lapak libraries or parallel ones is color park and then what you can do in terms of of linear dimensions of the matrix of the dimension is then for a is of the order of 10 to the 6 so roughly a million that's what you can do using supercomputers for complete diagonalization so so I think typically what one can do on a workstation is a few 10,000 so 10 to the 4 or a few of them but if there's really a super important problem you can you can put these matrices on a supercomputer and for example with libraries like scala pack on a distributed memory machine you you can diagonalize them up to roughly a million I mean we did something of the order of 800,000 and probably if one has a even larger machine or more computing time one can probably go to up a million roughly so that's just the the limit if you for some reason you have a completely full matrix that's about what you can do if and also if you want to have all all eigenvalues that that's about so the question is then is that all you can do so if you're doing for example Hamiltonian truncation can we treat larger problems if you're interested for example only in a in a part of the spectrum and so if you're only interested in low lying states a few of them then I guess also many of you know that there is a an algorithm which is very prominent which is called the Lanzos algorithm so that's an iterative method where you start with a random starting state in your Hilbert space and then you basically build a Krilov space which basically is a space of your starting state one power of the Hamiltonian applied to the starting state and then subsequent power and the Lanzos algorithm is a particular way to create to have a projection of the Hamiltonian into that Krilov subspace where the Hamiltonian has a three diagonal form so that by itself is not spectacular but it's interesting that if you look at at the spectrum of the of the projected Hamiltonian which just lives in a Hilbert in a Krilov space which is as large as a number of iterations you have done then the extremal eigenvalue so the bottom and the top one they converge exponentially quickly to the the true eigen state eigen energies of the of the complete problem so if we can use this Lanzos algorithm because we're only interested in states at the boundary of the spectrum that's already good because first of all it basically means that the number of matrix vector multiplications we have to do is not that large it does not it's not as large as the number of the dimension of your problem so that's that's already helpful yeah because the convergence is then quite large and then also if you since the only requirement which the algorithm asks from you from the physics kind of point of view is just to provide an application of the matrix of the Hamiltonian onto a vector it's not it's not requiring that you have your matrix lying in memory and it's going like like Gauss algorithms working on your matrix and update it in order to get the value out but it just requires you application of the Hamiltonian onto a vector and and so basically what the Lanzos algorithm really requires is basically something like two to three to four vectors in your Hilbert space I mean two to three to four that depends on what resize you want if you have a real symmetric matrix and you only want to know energies then two vectors are sufficient if you need an eigen vector in addition you need to have to allocate one vector more to get the result of the eigen vector and for stability reasons if you have a Hermitian problem and not a real symmetric one you might actually have three vectors to run energies and the fourth for the ground state so that's yeah and and so the question is what what have I done with the matrix now it now it depends on the on the physical problem what what can we do because in in principle I guess it's fastest if you can actually calculate the matrix and and store it in memory if that's still fits on your computer that's probably the still the fastest way to go but there might be applications where where but this is not feasible like for example yeah so we want to explore what what possibilities there are and so so there is this method which we call like on on the fly so if you calculate the matrix elements on the fly which means we have an efficient representation of our basis and we also know what different parts of the Hamiltonian how they act on our basis states and we do that in a very efficient way that's that's very helpful but of course that requires some investment that this is done very quickly because in each iteration you will do that job again because you somehow trade memory so that you're not using memory to save the matrix but you trade in for computing time but sometimes that's an that's a kind of a compromise you're willing to do yeah and and I mean for the problem I'm going to talk about later for the Hamiltonian truncation problem of of the five to the fourth theory we're actually I think as many others we're actually storing the matrix on a computer because at least right now it's still quite expensive to calculate the matrix elements also because there are a lot of them but we store them in a sparse matrix format so that's important I will tell you what the sparse matrix format I mean not the form itself but why do we have a sparse matrix so the reason is basically is that the Hilbert space as we have heard so the Hilbert space is typically typically exponential say in the volume or as we heard in the cutoff or something so the Hilbert space itself is is large is exponentially large but but on the other hand you can ask looking at the structure of having your Hamiltonian you can basically ask if you take any basis date you can ask how many terms are there in your Hamiltonian which can actually have a finite matrix element with this date another and then you're basically asking how many non-zero entries are there in a certain line row or column and this has to do if you write it down in in so second quantized form it basically amounts to the number of terms which which you have and there you can see that it starts to matter what your basis is and what the how the number of terms grow with your volume so and then there's a difference even though many problems which are more or less kind of local turn out to have matrices which are which are formally sparse which means so sparse basically means that the number of these entries here on the line they scale do not attain the finite fraction of your Hilbert space dimension if you scale up the problem so anything which is not not exponential qualifies as sparse but still there can be differences in the number of elements and so if you there are two cases perhaps one one can discuss so if you have a lattice problem we'd say spin spin operators so like a spin one half degree of freedom on each lattice site then basically and you look at some kind of a of an exchange term something like sigma plus sigma minus on ij on neighboring bonds where this is i and this is j so if you have a certain configuration there will be a spin flip between nearest neighbors but if your lattice is such that it has a finite number of bonds which typically a graph graph has then then the number of terms in your in your lattice like the number of nearest neighbor bonds will basically provide an upper bound to how many of diagonal matrix elements they can be because there are only as the number of bond possibilities to exchange two spins and even in some spin configurations there are not allowed to flip so the number of entries will be smaller but at least in in short range Hamiltonian system this number of bonds is proportional to the physical volume or in the lattice problem to the number of sites so that's quite mild then we really have a have basically situation where we have an exponentially large matrix but the number of often diagonal matrix elements is only proportion to the number of spins which which is a very modest a very small number however in the in the five to the fourth theory if you write down the Hamiltonian second quantization basically in five to the four you have like a sum over three independent momenta and then you will hit with with some bosonic creation or annihilation operators that does not matter but there is k1 k2 k3 and then here is the is the sum something that's such that I add up to zero so here you see you have three independent momenta so in principle if you act with this Hamiltonian if you count the number of terms this is basically case as the third power in the number of orbitals and and so that's still a sub exponential obviously but it still makes a difference when you have some of the order of 10 20 orbitals whether you have 20 or 20 to the cube that makes a difference obviously in the in the number of matrix elements you actually have to calculate so there are still formally sparse Hamiltonians but of course there are quite quite a lot of a lot of terms and so with with the with this I can perhaps show you a bit what are the number is a surprise to just give you an idea of what are the matrix matrix sizes one can achieve using current technology which also involves more or less serious parallelization efforts so if you have something like a spin problem so it's been one-half problem which has for example is total AC conservation which is like a U1 global U1 symmetry or something like that and you have lattice symmetries because for example you're working on a on the periodic regular square lattice yeah and say I said U1 symmetry so and then then like the the total Hilbert space for example of 50 spins is something which is which is intractable I would say but if you if you use this total U1 symmetry and you use the lattice symmetries which for example can you provide you up to like 200 discrete lattice symmetry operation something like 50 translation and also some point group operations then you actually can boil this down to a few hundred billion I think the simulations which which we done is something like three times ten to the eleven so these are now linear dimensions of the Hilbert space of problems which one can actually put on a supercomputer and calculate and so that's the that's a linear dimension so that's a linear dimension of the actual matrix problem which one one the non-trivial kind of connected part in which we believe the ground state is located so we want to calculate the lowest energy so that's like a matrix dimension which we can reach and I think yeah there might be other groups working on somewhat different problems where you can reach like 10 to the 12 but that's like the order of magnitude where I think just the computers become too small to go any further for the moment but that tells you like between the 10,000 you can do on a laptop and at least using complete diagonalization and what you can do for a kind of tailored code for specific problems that there are many orders of magnitude at least in Hilbert space size unfortunately for these exponentially scaling problems they usually do not translate in such an impressive growth in the number of spins or something but still I mean doing 50 spins is kind of a non-trivial result even though it's far away from the thermodynamic limit now if one goes to to Hamiltonians of this type here where one either stores the matrix or one recalculates them on the fly but then iteration takes a long time and one has to be a bit more modest and I think what we what we did is kind of to look where the limits are if you store the matrix for the five to the four theory in this one plus one-dimensional setting I think we can reach some or we did something like of the order of 10 to the 8 so 100 millions but so that's where we store the matrix and I think that's yeah that's what one can do using not completely crazy resources but it's still already somewhat demanding so that's for the for the five to the four two-dimensional theory if you store the matrix one can do of the order of 10 to the 8 that's the as the Hilbert space yeah depends how much resources your institute or department has I mean I mean this I think these machines that I was running on on say on the order of say 100 nodes or so with the very rich node has like 64 gigabytes or something which yeah it's not probably not the not the department cluster typically of that size so in my laptop I mean on your laptop I don't know how big your laptop is it depends on yeah yeah 10 to the 3 yeah no no not 10 to the 3 I mean if you store the matrix I think on the on the laptop you perhaps you can do a few a few hundred thousand or something I mean we in our code we also have a version which does not calculate the matrix elements or in principle we can we're not memory limited necessarily so we I think in a I mean actually this is problem of the five to the fourth theory at least on the formal level has some analogies to what people look at the when they start diffraction quantum Hall effects so if you go to lambda orbitals and so on you can write down Hamiltonian which look not exactly the same way but which also have like for both on or for fermion terms and so on but but which are written in these bases where you have to sum over many of the indices and then you you have in the very results like on in the few billion so 10 to the 9 or 10 to the 10 which are possible but but if you have a lot of these terms like having a number kind of number of orbital squared or the third power it becomes a limiting factor just the fact that that are a lot of matrix elements okay but but this gives you an idea like if you if you're trying to work work hard and you use some particular features of your problem I mean there is more than you than you what you can do just within say MATLAB or or Mathematica but these are like probably orders of magnitude which you can which you can this 10 to the 8 states are they a bit nice basis for the harmonic oscillators or what yeah yeah so so here in this 5 to the 4 this is so you let all them by K by the momentum yeah yeah I mean these are like really simple-looking Fox states I mean like that this like for example this representation I think you guys you use a similar I think there's like one part of their of the configuration stores the occupation of the the finite moment are going to the left there's a certain number of orbitals which which contain the occupations of the bosons in the in the branch going to the right and then there's a zero mode occupation so if you put numbers in here that specifies one one configuration in your Fox space and the way it is done is basically each of these configurations has a certain energy this age 0 which we heard in the previous talk and so you enumerate all of them which are below a certain and total energy cut off and this is the basis which you keep yeah okay so so I think that was the the blackboard part for the moment so let us now look at at some of the results which we have obtained so the problem I don't think I have to yeah so let us just start with a very simple plot and even though I'm going to show more than one figure I will take my time in in order to explain what's really on the figure and you you're encouraged to ask questions okay so so we start here now is so we're interested in the same problem as the the speaker before we're starting five to the fourth here in one plus one dimension so basically means putting the theory on a periodic ring in space and as I just explained there is like total momentum of linear momentum along the ring is conserved and also the parity of the number of bosons that's like even at the odd sector and so so here we're taking a very small system so the volume of one which is extremely small it's like ten times smaller than some of the plots we have seen in previous case and here we're using so the coupling constant here in this talk is called g4 it has the same role as I think as lambda before it's just a coupling constant in the problem and 0.1 so that's also very very small and so what is the purpose of this plot is to show the ground state energy so that's E0 the ground state energy and so this is just the offset with respect to some offset here and then these are the three different kind of cut-off treatments which we use and in terms of sophistication of the cut-off treatment we did not go beyond what of the first paper of Lorenzo and Slava so the raw basically just means that we're taking the the truncated problem and do not care about about any sophisticated treatment that's just kind of the diagonalizing of HLL I think it was called just the projection of the problem into the low energy subspace and then these these Ren and subleading are two kind of related versions to about this local renormalization and so here this is this is now the cut the ground state energy and here we plot it as a range over Emax so Emax is our cut-off in units of where the bear mass is one and so here we actually do a scan over quite an extensive range of Emax energies and so just to tell you here basically we go up to 120 which has to do that at some point the number of bosons which can which can be in the ground state can be 120 so that's like one of the basic states which we consider is the state where they can be up to 120 bosons in the in the in the zero mode I mean it's not zero but in the lowest in the zero momentum mode which is at energy one and so what we see here is basically that the ground state energy itself it does not have oscillations but but it actually stagnates in windows and then there is a somewhat abrupt change then it stagnates again and so on and if you use these treatments of the local renormalization and so on then you see that the ground state energy at least is missing these significant trend but however still feels these kind of abrupt changes here so there's some six or type of behavior which whose amplitude starts to damp out as you go to larger cut-offs but 120 is actually a pretty large cut-off I think for the volume and the coupling constant but but it's to show which is what was known to others but it's just to show that even if you use at least this level of cut-off treatment there are there are oscillations here remaining and one has to deal with them in some in some way and I think the the procedure outlined by the previous speaker has addressed a nice way how to get rid of them and even keeping small Hilbert spaces so the approach we're going to take is since we're not very sophisticated on that level we're just hitting hard in terms of of cut-off sizes and try to get meaningful results by extrapolating in the cut-off but using larger kind of cut-off for a given volume yeah yeah actually these blue lines that they are kind of the location if you look into the non interacting spectrum and you ask what kind of new matrix elements become active then there are new states I think where you can put two bosons in which are just hitting below the new cut-off so I think there are two I think are states where you put two bosons in I think at finite momentum compensating because within the zero momentum sector and they become available and apparently have a non-negligible effect on on it actually there's a second plot now where we look at this a bit more systematic the first line looks variational yes is there anything to be said about variational aspects of this method yeah I think that was discussed by the by the previous speaker sorry so I think it's variational because you you project the full problem into into a subspace and so that's what you get I mean this is not the mass but it's the ground state and or the vacuum energy this is supposed to be variational but the other treatments not as you as you can see no it was really not not fancy at all no the zero mode was was just treated like any other mode so here this is just kind of a pedagogic reason so what we do here is now we here in this plot we actually do now a Fourier transform so I mean that's like a simple reflex if you have signals like that where at least on this level where things are oscillating you start to to characterize these frequency and just analyze what they're doing so here we keep the coupling constant fixed but we changed the linear the volume of the of the problem and so we can see here there's like one type of processes which I have identified which apparently seem to scale so you see these red red plots so there there are some frequencies which obviously depend on volume and also this is like a fixed volume but here we change the G4 value so if there is no no G4 there are basically very small oscillations because they basically have zero kind of power spectrum but as you increase G4 you can see there are there is some interesting structure popping up I mean first of all there are some some frequencies like here but then also there are more important blobs developing and actually also that's something which we also saw is that here we have low frequency kind of oscillations and low frequency basically means these are like slow drifts and so they are kind of dangerous so it's not only it's not only important to get drift of the oscillations which we can clearly identify as having a period but there's also some long wavelength drift I mean long wavelengths actually means like energy drift in the in the cutoff so one has to be careful about about that and we did that actually in order to perhaps come up with a with a good guess how to get rid of these oscillations but but when we saw this kind of too complicated spectrum we said okay we don't care about it we just extrapolate and so that's what we what we do here but I mean it would be interesting to compare with the with your method whether if you do a similar analysis with a fine grid in whether it these structures are completely gone and you have really kind of taken care of of the effects which which which produce these oscillations or not okay so what we do now is okay it's a slightly different volume L equal 3 and the more substantial coupling G4 equals one and now what we do is that we you see we have still these three different type of result and now we're also plotting the mass gap which is the the energy difference the total energy difference between the vacuum energy in that volume minus the energy of the first odd state so the as I said there's a field parity in the problem so we can calculate the first kind of the the gap to the first excited state in the spectrum which which is at zero momentum but in the odd in the odd bosonic parity or particle sector okay so that's this quantity and what we do now is that we have all these three different approaches so like the raw which is still just the naive truncated version without any cut-off treatment and then we have Ren and subleading and now we are plotting this as a function as one over the cut-off squared because this is like this one behavior which you which you anticipate might also have logarithmic corrections but what we do here is scan up to quite large cut-off so here I think for this problem we went to a cut-off which was close to 100 for this for this volume which are already sizeable Hilbert space is probably for about 10 million or so not 100 but I think of the order of 10 millions or so and so what we do is actually since since we can see that as you go to larger and larger cut-offs the oscillation some are actually start to decrease so then we're actually fitting with I think with low order polynomials to these data in the window of of the largest cut-off values so I think it's starting from this red or green line here and then we're looking at what kind what is the value of the mass gap you get after extrapolation in the in the cut-off and we try to somehow use conservative error bars some actually we define the error bar as being as being the difference between the the raw extrapolated result and the other one from the two more sophisticated method and I mean if you use this more sophisticated method even beyond what we have done here the hope would be that this would be more precise but but our approach is actually to take rather small volumina and do extrapolation actually then the error is really quite small and then based on rather well converged results in finite volume we then try to do now the finite size analysis and and somehow determine the critical point that way yes I was going to ask how the renormalize things so you have a clear linear so one over epsilon one over e squared behavior for the raw result yes so I was wondering what the result of the emacs power is which the renormalized results are basically end up having very small slopes and I mean we're just fitting I think with linear or quadratic I think they're just linear slopes here in that in that window so it comes out that I mean I think it was this is the purpose of these random subleading method is at least they get rid of this large this large leading trend they obviously do that job but they do not get rid of the oscillations and and actually many cases at least if we really and at relatively small volume and that large cutoffs then there is also no drift remaining because what is a bit very some in our experience is that if these two methods here ren and subleading start to differ then and it's really I think we do would not trust extrapolations in a regime where at least already visually we see there's a difference between these methods and once the these two methods give the same result then we're more we believe more that these extrapolation starts to make sense but that's just empirical okay yeah and this is this is just an illustration of what we what we get now so now we keeping fixed g4 value which is of order which is one I think for this for this value and then we're calculating using this extrapolation procedure which I just shown for a l equal 3 for example we calculate the mass gap and so this this data point with its error bar is is the result which we get after this extrapolation in the in the cut sending the cutoff to infinity and then at this point I think it has also been recognized by by other authors it's actually important if you want to learn something about the thermodynamic limit it's actually important that that kind of the the simulations you do are taken take care of this apple planar corrections from the normal ordering whether your normal ordering in finite volume or in in infinite volume makes a difference and so we had to learn that first that this makes it I mean it's obvious that it makes a difference but but the finite size scaling behavior is actually rather irregular if you do not take that into account so even if you do very ridiculous volume of one for example I mean they start to make sense also with the rest of the else if you take the apple planar corrections into into account and then you get rather nice formulas which you then can also fit to luscious formulas for what is the finite volume behavior of the mass gap for massive theories and so here you see the fit over the range of l which we look at and typically we're going to see that later typically we're we can trust our results also closer to the critical point for for volume for volumes which are like four four up to five I think for four six and beyond it seems that we are not able to really get very careful very accurate results but but as we will show doing a very careful job at smaller volumes and using finite size analyzing the regular finite size behavior we're still able to actually get I think pretty accurate results so that's what we try to illustrate them here so this is now our attempt at the term the termination of the critical point so here I think what is a bit different to other studies at least in the Hamiltonian truncation business is that we're using several different proxies in order to to locate our critical point so the first one which we're using here that's that's basically based on CFT input so we expect that the critical point is an is an icing CFT so we know the spectrum of scaling dimensions we know that in this geometry in which we're simulating at the critical point we should find the spectrum of the icing CFT so so we're using the criteria to determine the the finite size critical point it's basically the coupling value where L divided by 2 pi times the mass at this system at this L attains crosses the value of 1 over 8 because if you look at the finite the the behavior of the mass gap on a very complicated G4 for for a kind of the mass gap for a fixed volume here then basically it goes down and has it has such a behavior and we're basically just saying okay if you multiply this with L times divided by 2 pi and then you check whether this when you when you reach this point here and that gives you the kind of for this L the critical point in G4 and so these are this collection of of red red points so you can see you can do that even for for volumes of one which I think one would probably believe this is nonsense to look at so small but actually they are part of a of a rather regular family of points and you can also see I mean at small volumes this crossing is determined quite accurately and as we go to large and larger volumina our extrapolation procedure becomes somewhat more fuzzy but there's still a window where this crossing is rather well defined and if the volumes get to like 5 or 6 our area becomes larger but but based on this crossing finite size critical point you're already quite in a quite a small window you see this is point 2.7 this is 2.8 and it gets a bit crowded but these are various different points with error bars from the literature and I think the point where we're converging here which is looks quite nice at least based on this proxy here is actually really within the most I think by now the most accurate method like the most kind of sophisticated Hamiltonian crankation study and also I think the MPS result is quite accurate and the and the most recent Monte Carlo results which you can see that these these results actually cluster around here but say at 2.76 or 77 I think that's the current I think our results here come pretty close to that and are confirming that that even from this approach but then we're also doing some some other things which are less accurate but which are in agreement so for example if you look at here at the larger window of G4 values this is also one over L squared we're looking for a different primary field of the CFD so we can ask where it's not the same mass gap it's a mass gap for the first kind of even even state at zero momentum so that one should have scaling dimension one so we're basically tracking this this finite size critical point and so this is way off on a given system size this is way off but but this has a large finite size effect but it's compatible to actually go and you can see it it's very steep but it's compatible with this point which we have here so we're not claiming that this is more accurate but at least it's not in it's not going to a wildly different in that sense at least it's supporting that is this estimate here is reasonable and it's not contradicted by the analysis of a diff of the crossing with a different primer kind of CFD field and another one which is also I think popular in whatever condensed matter statistical mechanics is to actually take two different volumes so here volume is continuous I think in in lattice models one would typically use two different system sizes like a number of lattice sites but then you basically look for kind of for the gap to to go as one over L so you check whether this is this convert without imposing a value so these since we be typically take pairs of L's which are just by one length you need away from each other this is crossing together with the error bars which you have on the extrapolated mass gaps are much have a much larger so kind of error bar but but still if you look at the data points they actually seem to converge coming from the other side and also are approaching the same the same value so these are three different approaches as three different kind of finite size critical points but they all seem to go to the same critical point and which is I think which is among the most precise modern determinations of the critical point this five to the four theory then we also have have had a look as others at the critical energy spectrum so we also include some states at finite momenta and so the circles which are sometimes are hardly visible this is expected icing CFT results for a torus spectrum and the field data points with the error these are extrapolated results and so you can see that this the CFT tower of the of the torus is is rather well reproduced by the by this truncated Hamiltonian approach much better for other studies usually yeah that's true yeah okay and that's as I said before we also looked at a different observable which is the the central charge so the since we expect that the theory of the critical point is a CFT with central charge c equal one half so what we do is that that for the whole range of g4 values shown here we're actually looking at the energy density so the vacuum energy density divided by which means the total vacuum energy density divided by the by L the volume and this is expected to have quadratic in L corrections which are proportional to the central charge so what we basically do is that here for example like one color of data set for a data set includes a certain range of L values and then within these L values we calculate the ground state energy density and and a fit the linear slope to the data and the error bar is just some kind of of indication of the quality of the fit and then we're just plotting that so it's a result which depends on the series of system sizes you're looking at and and of course if you're in the in the kind of in the massive phase here there is no central charge so this approach the volume the vacuum energy density should converge exponentially so ultimately this quantity for large volumes should actually go to zero and indeed you see if you take two to six three to six four to six five to six you can see that somehow the trend is going down so this is not converging to a to a finite number and then one has to look at this reason at this real machine here and so I unfortunately I did not remove the last set because it really shows that some point results do not get precise but if you just focus on the one the systems which are smaller and where the cut-off extrapolation is more accurate you can really see there's a this effective C is increasing it has a maximum here and if you go to larger volume now here then you can actually see that it nicely converges to a maximum here which is which here this dotted line is point five so that's the expected central charge of that CFT so that actually looks quite nice if you disregard the red curve it's quite nice and so what you can do here is measure for example locate the maximum and so these are I think the pink one is the Monte Carlo results and I think these brown think is your most recent determination so we can see that the maximum also the crossing here of the derivative crosses zero is really well within the error bars of the other approaches and this is a confirmation that not which is not a surprise but we see the spectrum but also the central charge in these approaches is correctly reproduced okay yeah and then just to kind of show that technology is there we also dared to look into two plus one dimensions but this is very preliminary so this is not meant to be any any type of state-of-the-art calculation but it's just shown that that one can in principle do that from the kind of Hilbert space construction and the matrix element calculation point of view which is which is simple but still it's a little bit of work so what we're doing here just for registration purposes is that we're looking at so we're taking the same basis a free basis for mass one and we're actually quantizing the theory on a two-dimensional torus which basically means now and a torus of aspect ratio one which basically means that the grid of k points which we have now lie on a square lattice in momentum space and in principle is unbounded in all direction and that's like the at least the grid of k points for a two-dimensional square torus and then okay and then we start filling up states and then we have a small cut of e max of 10 or of 15 and the perturbation which we apply now is a simple mass perturbation so we just add a phi square term with a certain g2 value and that problem is obviously exactly solvable but within the approach it becomes slightly non-trivial and so here we're looking within a certain sector at the dependence of the the truncated results which are the black and the red symbols and the green one are the exact lines in that sector and so you can see for small perturbation there are reasonable results and at least here you can see that that if you increase the cut off at least there's a trend that going from black to red you converge towards the correct result and of course this is a trivial theory where there's nothing to be learned apart from the fact that there are truncation effects and they seem to go away as we increase the cut off but that as one explained there is a problem if you want to apply the phi to the fourth theory to this problem there are kind of problems with UV divergences in various sectors and we also did look at we also put the phi to the four perturbation in this but then weird things happen because then within a certain cut off if you start to increase a G4 there's suddenly like abrupt kind of changes in the ground state wave function and it seems to us that suddenly the ground state wave function lifts has a lot of population populates a lot of states which are at the at the cut off and so on so there are weird things happening but it seems that this might be a finite like a Hamiltonian truncation phenomenon which might be related to this kind of problems of UV divergences in the in the approach but it would clearly be or it's clearly required on one has that one has some feet filter ethical input in how to cure this problem and in order to make that the kind of a working method for two for two plus one dimensions yes I think it was really very small so this is just a demonstration calculation this was not an attempt at being accurate I would have to look it up but I don't know how large it is okay congratulations the first 3d from Italian truncation calculation yeah I think there are some stochastic results available but okay I would already be able to okay yeah so if there are no more questions on that part I just would like to briefly show you some results on the for lattice models on on torus energy spectrum because I mean it's kind of it's kind of related because if for example you choose I mean there are different ways to choose the k the momentum basis here now if you if you take a square torus that's what we did but of course you can also quantize them these field theories on a sphere I think that's what field theoretical or it minded people would would do more often we're more coming from lattice models or the torus is more familiar to us but say if you were able to put that program to work and go to the interacting fixed point of the 3d icing model in that formulation what is kind of the spectrum which we would expect on the on the torus so here it's it's it's a choice among many because you choose the torus but we could also have chosen the sphere but if you're simulating lattice models and we have heard there are very sophisticated numerical approaches at least to assess many observables in these in 2d we thought it would be interesting to explore how torus energy spectra actually look like so I think here I can skip this a bit so what is the what is the motivation since I mean since there are people from different topics I just would really would like to to mention that something which is well known to many but perhaps not to all of of the people in the audience is that if you're looking at the 1d problem kind of the space time you put it on the line so that's the space part the blue line and here you have time so it's like a cylinder and so there's a mapping between the kind of the R2 the complete plane to this and what is nice is that basically the scaling of the that the energies of your Hamiltonian problem which you're solving on the line is actually directly telling you this giving you the spectrum of scaling dimension of the CFT and I always found that very fascinating that by doing poor man ed on on lattice models I'm actually able to and not only me but others also to actually get kind of a look really into the we have a window into the CFT and see all these towers of thing that I find that very fascinating so the question was what happens in in higher dimensions in higher dimensions there is still a conformal mapping from R to the D to something but that something is actually the S the D minus one dimensional sphere times R so if you want to to have the property that the Hamiltonian energy spectrum gives you the energy spectrum kind of translates into the spectrum of scaling dimensions you have to quantize your your problem on the sphere but actually if you start writing down for example a transverse fieldizing model as a lattice model discretization on the sphere you have all kind of problems because the lattice I mean you cannot triangulate the sphere in a regular way there are defects they correspond to curvature so that you and it's difficult to get rid of them so it's not easy for kind of lattice model minded persons to actually do simulations on the sphere so it has some interest to actually understand what's going on on the torus but the problem is that the two torus so where you have two space dimensions that space dimensions wrapped up in a torus times R I mean it is not the same as the sphere and so the question I think is open or was open to what extent the energy spectrum of an interacting a CFT fixed point looks on a torus as compared to the sphere where due to the conformal bootstrap and so on we now have a very impressive knowledge about the spectrum of of low lying scaling dimensions and so what we were we're doing is to actually calculate numerically energy spectrum on tori so we were asking whether these there's a universe low energy spectrum and of course it is there is but also the question whether it's actually accessible numerically or whether it could also be that it's just too hard with the available methods to actually get get kind of results which are not finite size effects but really kind of feel too reticle results and then if it's accessible how it looks like and also at least from I find that curious to understand whether there's any kind of analogy or at least for some visible resemblance between the spectrum of scaling dimensions on the sphere and on the torus yeah so I think now I speed up a little bit so what we were doing in the first part is is taking an actual lattice model so the transverse field dicing model on different lattices and using exact organization and also complemented by quantum Monte Carlo results we are able to access some of the low lying gaps and so this is just to show you how an actual spectrum of such a lattice looks like so here is the coupling constant and so here you have different energy levels for different system sizes in some sectors so the question was a bit how to get something useful out of this whole mess and so this is now some finite size scaling results and so the situation is that we're looking at the square torus and this has been obtained by by taking a square lattice kind of a microscopic square lattice of up to 40 spins spin one half degrees of freedom and then so this one over n is one over the number of lattice sites and here what we're doing is we're focusing on certain symmetry sectors in our in our micro in our spectrum like the zero momentum sector and the full and the empty symbols they correspond to see to even and odd levels and the blue ones are at zero momentum here you see the corresponding case base grid so the the center is the blue part and the green one is the first momentum of so like 2 pi over L if you if you want and then we're doing so we calculate this so the the the non-red symbols are from exact diagonalization and the red symbols here they are is one set of gaps which is available from from quantum Monte Carlo and so you can see by using a phenomenological one over n scaling and and renormal setting the first non ground non vacuum level to one we actually see there is quite a regular finite size scaling behavior and then the extrapolated values here so these these blue ones then the the red ones we associate them to to be like the the CFT spectrum on that particular torus and then we actually did that game on the number of different lattices so there's another lattice which is not important what it is but it's different microscopic from the square lattice it's called the square octagon and if you do the same analysis on this and around a range of so this this model has a different critical points because some sites have a different coordination number so the microscopically it's a different model it has a different critical point but but the torus energy spectrum on the square torus for these simulations actually tend to give the same results within some numerical error and then if we actually put the theory on a different torus which is one which basically the unit cell is something like a hexagon which means it basically torus has a different modular parameter then we see there's some small difference in the sense that the the torus energy spectrum has some dependence on the the shape of the torus but but first of all from these numerics it seems that we're able to extract something which seems to be a stable spectrum which hopefully is something which corresponds to a spectrum of a of a particular CFT and then we previous but sigma t is 1.00 is that physical as a choice of normalization no that's a choice of normalization because since we're simulating lattice model our speed of light is not is not known beforehand and in some of the plots we actually made an attempt via different means to to determine the speed of light then kind of the and then we can renormalize it and get it out but otherwise we just here in these plots we just choose we set the first level to 1.00 and rescale the whole spectrum that way yes and natural guide to look at excited state would be the stress tensor the one corresponding to stress tensor with momentum to I guess or no I hear it's angle momentum to so we were looking for that but but it's actually not on that plot I mean we were looking for traces of the stress energy tensor but I think it's quite high up in the spectrum yeah and the also it splits up because on the square lattice like the L equals 2 of the sphere which has like 5 representation 5 it splits up into several representations of this of the symmetry group of the square Torah so it's a bit a delicate business what actually happens but we were looking for it and but it's not on this so would that have given you the speed of light in a natural way I think it's not it's not known at what location in energy it is not that simple no what you typically can do is basically something like that but you can just ask like where is the first state which at least quantum number wise could be a remnant of the stress energy tensor I mean yeah but I will show I will show why I dare to do that I mean no no I mean people insist that is there's no map and I know but but I still think I will show you in a second I still think at least at least for educational purposes it might still be good to actually remember that what what you should do what you expect on the sphere and what we get on the tourists it's not wildly different that is for the low-lying part one can then debate whether that's a useful kind of a comparison to make but at least I find that curious okay so here we then teamed up with with the set with said who's a student of super such death and they performed some at least in my opinion sophisticated epsilon expansion results so four minus epsilon so come coming from the three a fixed point to calculate the Torah spectrum in this epsilon expansion and so here here you see results for two different modular parameters so basically the situations we looked at the square Torah and the one with the hexagonal unit cell which we called triangular in these plots here and so here we actually calculated the speed of light so now it's taken into account and then you can see that the squares are epsilon expansion results and the the circles are exact organization QMC finite size extrapolated results and then we see that I mean there are not accurate to many digits but on the on the level of diversity of where the levels are the agreement on the respective locations between the extrapolated numerics and the epsilon expansion also has a has an error I guess but we don't know how large it is but at least this comparison between two methods which don't know nothing about each other seem to to give a reasonable account of what is the type of low energy spectrum which we expect for the icing CFT in two plus one dimension on a on a torus of two different modular parameters yeah and then and it's just just to give you an idea so if you that's the way we plot our tourist data and that's a synthetic plot which I made based on conformal bootstrap results and my understanding of how that would look like on the sphere so this is not an actual calculation this is like a model so I take the scaling dimensions which are known from conformal bootstrap and say what what are like the towers which would expect based on derivatives and so on so that's the spectrum which would expect on the on the on the sphere okay and then I mean here here that's the plot which perhaps at least show you why why I think it's instructive to make this comparison so here the O1 is actually the icing CFT which we just discussed and so here on the left to the right of this dotted line there is the the ED and the and the epsilon expansion results so that's on that side here and then we also explored the O2 critical point in some lattice model and O2 in the in the epsilon expansion and O3 as well so here we have a have like a list of the first fields the first fields popping up at zero momentum in those theories and and kind of labeled by kind of the charge they have under the global symmetry so here it's even and dot and for O2 you have like a linear angle momentum and here you have like the O3 representation so the different colors are different angle momentum channels of two and three and then what we what we took is the scaling dimensions from from the literature which basically means conformal bootstrap and here we have chosen to actually renormalize them that we put the epsilon field so the thermal field we set them to one in these units and then we actually see that that relative to each other they actually I mean it's not I'm not saying this is an exact statement but I find it more more than than just the coincidence that that these labels here so that's like the scaling dimension of the of the icing true scaling dimension of the icing CFT compares rather well at least within some limits and yeah and the same is true here for these other different fields which we have so there's there's often a good a good agreement at least on some on some level and and I say I mean as I know there is no exact state operator mapping so there's no it's not expected that these things coincide but but for us or at least for me if one is looking at the numerical problem say of a complicated CFT where the torus spectrum is not known and I would like to know what what should I look for if I have no other idea then I think it's fair to say I should look up what you guys or some other people did in terms of what is the operator content of a theory and then I would actually go forward and see whether in my torus spectrum I'm actually able to identify some at least some of the fields or in the which I would expect might show up in the torus even if I know don't know them beforehand so at least there is a there is a phenomenological or an empirical idea what I personally would look for but that's true in all the cases we don't know but I'd be that someone finds out at least how to what extent the scaling dimensions on the sphere get distorted how wildly how wild things can happen we don't know I mean there are things happening there are not the same but still it's it's remarkable that they at least match to within some some there's some loose correspondence it seems and just without going to the detail I'm just we're looking for some other motivation we were looking at this a slightly more complicated microscopic model and so this is now really a calculation where we had some motivation to study a microscopic model and we want to understand the critical behavior of that model so the particular model is a is a quantum Ashken Taylor model I mean I think I skipped the details if you want to know I can tell you but there are basically some in some coupled icing models with the transverse field and depending on the couplings you actually have really two microscopically coupled transfer silt icing models or you start coupling them and the question is what the phase diagram looks like and so this was a complementary approach on one hand we did a quantum Monte Carlo study on that particular quantum Ashken Taylor model and so if there's a certain microscopic coupling here if you tune that to zero it's as I said it's just two different two icing models and if you drive them with a transverse field there are just two copies of an icing model undergoing an icing transition but then if you start to couple the two icing models and it actually seems that there is a 3d x y transition here so so the number of phases is like this one large phase here which is paramagnetic there is one here where their symmetries are broken on both sub lattices and here there's one one situation where only one of the two sub lattices is ordered and the other one is disordered so this is known I think both from the structure of some two components five to the fourth theory with some cubic anisotropy so the phase diagram qualitatively is not a surprise as such I mean for quantitatively for that particular model it has had to be determined but one point which I would like to point out is that here along this line here you see this is an x y transition but if you look at the broken symmetries it's not that the problem has a continuous symmetry but there's some emergent x y a critical point even though the microscopic model actually has these cubic anisotropies which are non-zero but these are dangerously irrelevant perturbations and so for example if you go along these lines if you cross along this line in some cases you find a situation where where the broken symmetry states have peaks in these histograms in the corner of the square and here here there are on the on the main axis and so there's a there's something happening and you can associate that that's somehow the the sign of the cubic anisotropy changes and so there are different minima chosen and what I would like to point out is that there is some evidence that you can actually see that in the torus spectrum as you cross this line and the point is that if you look so what is shown here is that this is the the torus spectrum at the decoupled limit which you can understand from the torus spectrum of one copy and you just kind of take the product of the two theories and you you can understand that and so here this is now is a simple exact analyzation results on rather small lattices but what you what you can see here is that that the levels at the ising theory they start to to rearrange along that line so this is like a coupling value of 2.53 and 3.8 sorry but these are like different points along this line here and we're looking at the spectrum at the critical point and and the point I would like to make is that here for example these levels we set them to one and there are exactly degenerate by symmetry but here you can see that that the levels like the the triangle and the circle they are split in some way that triangle is on top and here they cross and here they order the other way around and I don't have an analytical argument but I I tend to believe that that this distortion of the spectrum is not just some random finite size effect but if one actually does something like like kind of something like conformal perturbation theory one would probably see that the finite size spectrum of such a theory if there's a dangerously irrelevant operator of the type like like this perturbation it would probably mean that they are split in one direction if the if the sign is one way and it splits the other way and so using ed I pretend to say that we can actually locate where the point is where this at least the leading dangerously irrelevant operator is actually a changing sign and so that might be one of the applications where by looking at Torah spectrum you can actually learn something about rather subtle questions which are not so easy to actually analyze in with with quantum Monte Carlo because sometimes you really large volumes but in spectroscopy it could be that some subtle questions could possibly be answered by looking at careful analysis but of course these needs to have a quantum field theoretical background otherwise it's just speculation but but it's at least in one dimension there are very prominent examples where you can see for example perturbations by marginal operators they lead to splitting of of CFT multiplets and if this marginal operator is vanishing you can see that the multiplet is restored so these things are well established in 1 plus 1d it would be interesting to understand whether similar results of course not in the same beauty but at least things like that could be could be have a theoretical foundation okay and with this I come to the end and I would like to acknowledge my collaborators so the first part on the five to the fourth theory was a master student project which I did together with Christoph Pernull so he did a lot of the work there on the numerical simulation parts and the part on the on the Torah spectrum which is published was a collaboration between Michael Schuler who's a PhD student in my group, Lule Polonri who was a postdoc in my group and and the epsilon expunge result as I mentioned were obtained by the Harvard by Subir and his his student and the last part on the quantum askin teller model that's an in-house project with michael and with Louis Paul and with this I would like to thank you all for your attention thanks and there's lots of questions yes please could you go back to the slide for the large the infinite volume limit for five four there was one result on there that you didn't comment on I just wanted to comment so which one the large volume that the large volume limit of five four you're doing the extrapolations for different methods are for the critical point yeah okay yeah there's an entry there for DLCQ that you didn't mark about and we'll be talking about that on Thursday DLCQ is way down below the 1.5 there I think this one here yeah that's a triangle sitting there that seems inconsistent with all the other yeah that's that's true yeah yeah we think we at least qualitatively understand the difference there I'm not sure if everyone's aware but we'll be talking about that on Thursday okay I also asked to go to this slide where you show the weight of each state in the function I think I didn't understand ah no that's just a Fourier transform of the of the dependence of the gap as a function of or here this one yes this is this is just the frequency so we consider this signal or I don't know whether it's the gap or the ground state energy dependence but but we're looking at data of that type and then we just want to understand what is the oscillation I mean here it's kind of clear we have already identified but this is something very basically only a single event is happening but if you go to larger interactions or larger volumes I mean these oscillations are much more messy they're not as nice stair steps as here so this was an attempt to take this data and just Fourier transform it and and let the Fourier transform tell us where the yeah in the energy so but do you have an idea which type of states yeah I think he or the ground state of the internet yeah here it's really I mean you can understand it by perturbation theory I guess no I mean here you at least in finite volume with the finite cutoff that there is like a regime at the perturbation I mean the interaction is so weak that that you can just do finite size perturbation theory so you understand like how is your wave function dressed and then it turns out that there are just some some matrix element of the complete second quantized Hamiltonian which are not allowed in in this setup with your truncation and if you shift if you shift one of them if you shift your cutoff then suddenly there are terms available or becoming possible and so here these blue these blue result is is one one these these lines here there are one caress one type of process which we have identified I'm not sure I think I said something I think could be that there's a new way to put in two more bosons or something like that while respecting the fact that it's an even sector and that you have you have to have zero momentum but but as you can see here there are more I mean this the situation is more complicated as you as you go to a larger interaction so it's not just one single process but that was what I'm hoping initially that if there were like a very clear oscillation structure we do because it could just fit the oscillations because we have understood what their origin was and then get rid of them by just eliminating them but it was not that if you could for example distinguish is it more important to go to high momentum but with low occupation numbers or with too high occupation numbers but still with low momentum with which type of state to represent is that an intuition for that no I don't have a simply enough intuition to like convey it to you but in our work we studied this question and definitely low occupation number kind of momentum is more important so you think these peaks is just like just two states what so this suggests that one should perhaps introduce different cut-offs in different equation number sectors but okay this has been around for a while but I don't think it has been implemented yet by anyone but it also depends a bit on the regime you want to go probably could be that at small interactions these effects are more important which you mentioned but on your hand if you want to go into the symmetry broken case at some point it also becomes clear that somehow the basis is really not appropriate to somehow describe a distorted harmonic oscillator I guess I mean at some point the zero mode also becomes important I guess and I mean I think the word papers looking at zero having this mini Hilbert space of the zero mode in addition that might also help in some cases but it's yeah we don't have an understanding where to say like what is the the simple thing you should improve there are a lot of things to improve but but I really see it as an analysis and I mean working very small volume right so even the zero momentum state has a large contribution yes bigger physical volume then the gaps are probably not as extreme yeah but I think what is one of the of the messages at least in this approach where we where we do kind of what was introduced like two three years ago in this cut-off treatment I think what the bottom line is we I mean of course we can set up the computer and run simulations for a equals 10 and we can also try to extrapolate but it somehow seems that these results are unreliable I mean we can do it they get energies but I don't think they are useful in like advancing our the precision of these results I think the approach we have taken after some exploration is to really stick with very small volumes but to do them very carefully and somehow try to do them a finite size results I think I think you showed results for a equals 10 and I think you get reasonable I think we cannot do that for a equals 10 okay we get also large Hilbert spaces but I think the quality of the results is really not good so I think for this approach here it's really important to be modest with volume but to do a careful job at small volume and if there's a substantial I mean a well-developed finite size behavior then we're actually able to track that but it's better to do a careful job at small volume and do a sloppy job at large volume and then it's not the volume which saves you I mean yeah I just said just to get a little bit of a sense and so how many states do you think you've converged the energies for I mean you've talked a lot about the ground state of the system or the first excited state of me how many states do you think are kind of within 10% error have converged I just want to have some sense of the reach of the method right I mean I think the CFT spectrum which we showed probably gives at least a lower limit to that I think yeah but okay that's right like a roughly like 10 states or so okay yeah okay well I think I personally have some more questions but I I restrict myself and I suggest others to restrict and so that we can proceed we can thank Andreas and we can proceed to the cocktail