 Thank you. My name is Pavel Jakubtryk, and I would like to talk about the amplitude fluctuations in the Berezynsky-Kostalys-Taule's phase. I listed all the relevant collaborations Okay, how do I Out here So, okay different people were involved at different well kind of calculations, so I will try to make it clear who Who contributed to what during the talk? So as an introduction, let me recall the Merman-Wagner theorem, which tells us that No spontaneous breaking of continuous symmetries is possible in two dimensions and below at finer temperatures And I will be focusing on two dimensions So this theorem implies that in two dimensions, it's not possible to have true long range Order. There is a special situation where the symmetry of relevance as you want where Despite the absence of long-range order one may actually have a phase which is characterized by an algebraic decay of order parameter correlations and finite order parameter stiffness, and this is the so-called Berezynsky-Kostalys-Taule's phase. On the other hand at sufficiently large temperatures one expects the usual behavior where the correlations decay Exponentially and these two are separated by a very peculiar phase transition, which is the Berezynsky-Kostalys-Taule's transition and the special features are at least some are listed here so Okay, this is described as a transition driven by topological excitations of the vortex type The free energy is a smooth function of the parameters the stiffness Exhibits a universal jump and if you look at the singularity of the correlation length Approaching the transition from high temperatures you find that this is characterized by a singularity which is stronger than any power law So an essential singularity So the standard Okay, I should mention that I will for most of the talk I will not talk about the transition itself except for the last part. I will focus on the on the low-temperature phase and in particular the role of the amplitude fluctuation Or longitudinal fluctuations in this in this phase, which means I will focus on the temperature range, which is sufficiently low So in this regime it is commonly accepted that the Effective well the relevant excitations of the system are spin waves and that can be described in the terms of just well Effective Hamiltonian for phase fluctuations this kind of spin wave type and the vortices are become relevant only sufficiently high in temperature And the standard approach the kind of textbook approach tells us that the amplitude fluctuations are are not relevant And this is the point which I I want to well elaborate on within the context of functional Rg So it's not the first Work of course, so I listed all the relevant works Which I know which deal with the Bershinsky costalist Tau less systems from the functional Rg perspective and the setup which is provided by a By a five to the fourth type model, which is just all all two theory in two dimensions So the the most relevant paper for me at this point is this is this first one which is already 20 years old more than 20 years old and Well the last contribution which I listed here is a sort of refinement or a reveal kind of Rediscussion of this paper and I think we provided a little bit of Analytical understanding of this of this contact. So this these three papers deal with more computational approaches, which is the derivative expansion But what I want to emphasize is that these four contributions all have a common feature So the vortices are for well rather clear reasons not captured Which is well, which is a bit unlike the standard Treatment where they are believed to be actually the relevant Factor driving the transition on the other hand the amplitude fluctuations are kept while they are usually supposed to be like irrelevant in the and the Standard treatment. So the aim of this of this work done together with Walter Metzner is to carefully Examine the role of the amplitude mode There is a motivation behind this. So we are aware that in many systems for example in the context of Interacting both a gas but not only the due to the coupling between the the longitudinal and radio and transverse modes the Amplitude mass is actually renormalized at it in certain cases that may attain the zero value And this happens for example in the ground state superfluid In the interacting both a gas as is well established So we wanted to to carefully re-examined this kind of effects in the case of a of this particular system of D2 and N N2 So for the present purposes it is perfectly sufficient to to to focus on the five to the fourth theory Which has just a quarter coupling An order parameter expectation value and the Z factor and this is the initial action And we are always splitting the the order parameter field in this in this kind of linear basis So we have this sigma mode, which is the longitudinal and the transverse mode, which is denoted with pi so this radial or amplitude or Longitudinal mode is has this reputation of being rather Irrelevant or innocent and we are looking at the validity of this assumption within FRG So first we okay, I have the Framework here, but I will not talk much about it so the the the formalism we use uses based on veterinary equation and Well, we use two two truncations So for most of the talk I will talk about the most simple or almost most most simple Approximation, which would just builds a vertical expansion on top of the derivative expansion and parameterizes the flowing effective action using these four couplings This is very similar to what was done by Greta and Wetterich in 1995 On the other hand the subsequent the other calculation, which I will present in the last two slides uses the full Derivative expansion at second order, but this is a more like a computational thing. So I want to focus on this first of all So the flow equations Okay, so this kind of parameterization allows us to project the veterinary equation onto a set of either three coupled Partial differential equations, but in this case we obtain a set of for just flowing couplings Which can be studied to some extent also analytically well here one is one must Just resort to numerical Solutions So we start off by Reproducing the low temperature phase as I said we are now focusing on the low temperature phase the vortices are Irrelevant we take this for equations now. We follow the the standard Kind of approach we use the textbook and say okay the amplitude mode is not relevant So in the flow equations that we have here we just drop all the amplitude Fluctuation which means we drop all the turns which contain some massive Propagators and this Significantly simplifies the problem so that we can then study the these two equations which are coupled but still We can solve them for almost arbitrary cutoff and the infrared and the solution is here so an agreement with the Merlin Wagner CRM the order parameter vanishes and the vanishing is governed by the anomalous dimension on the other hand the Z Z factor diverges with the same exponent and it all happens in such a way that the stiffness which is just this product Attains the fixed point and we can Read off the value of the eta exponent which fully agrees With the standard costal is tau less formula. So this is the a way of Just reproducing the line of fixed points in the low temperature phase very simple Just take these equations and forget some terms on the right-hand side Okay, but we have dropped some terms this may namely those those amplitude or Longitudinal modes and then we want to ask about the consistency of this calculation. So we go back to the to the full set of equations Sorry, and we now look at the flow of the coupling Sorry of them of the of the mass of the longitudinal mass just assuming that this kind of scaling Hulls so what we find is that? That is That well this kind of power counting leads us to the conclusion that the mass will scale according to this power Which says that okay the actually the longitudinal mode is not really less relevant As compared to the goldstone mode, but it's equally relevant of course if provided we are deep in the infrared In the initial stages of the flow, this is all well Well, the mass is finite. So so the amplitude mode is really suppressed. So this leads us to a Kind of puzzling conclusion that the phase and Amplitude or pipes I should rather say longitudinal and transverse modes are equally relevant in the low energy well limit and then it may Suggest that the justification for dropping this amplitude mode Is not quite strong So we go back to the equations and now we solve the full flow. I mean including the the amplitude Mode so we look at the feedback of this, you know softened Longitudinal mode on the on the quasi long range order which and on the stiffness in particular We find and this was also observed in in the first paper many years ago that the fixed point is not really stable But instead acquires a very weak Well logarithmic flow So this and this leads to the ultimate collapse of the quasi long range order even if this occurs at very very low energy Scale so this plot here demonstrates the the collapse the sorry the the vanishing of the of the longitudinal mass and on the other hand I also plot here the stiffness and the anomalous exponent so you see if if I if I just keep the The goldstone fluctuations I obtain a kind of line of fixed points here But due to the presence of the amplitude mode Well this this whole thing ultimately well this fixed points here are not really fixed point but are shifted and this whole thing always collapses in the in the high-temperature phase and Okay, this is this is should be emphasized that this amplitude mode is not contained in the standard VKT theory, but Well, so to get the VKT theory you should just put vortices on top of the spin waves So instead of having the amplitude here you should add vortices and this would lead to the usual Costally stylus phase well here we have just something something different so this rises the natural question of the well stability of the Costally stylus phase and realistic systems with respect to this kind of amplitude fluctuations, so I I Shouldn't move on but not okay So this is a plot demonstrating the behavior of the correlation lens So the correlation lens since there's no transition must be fine it but in this calculation We can we can just find the its behavior in low temperatures. So you find that this is enormously large Well at at temperatures sufficiently low so for any practical purposes you can take you can just approximate but by by just infinity below some some temperature which should be somehow associated to what is described typically as a Costally stylus temperature So this is the summarizing picture of this calculation So as I said the well due to the interactions of the these two modes the the longitudinal mode becomes softened. So this its relevance is actually not any Lower than the relevance of the pine mode which fits back on which fits back on the On the stiffness and leads to the collapse of the quasi long range or that at any finer temperature even if at very very low scales so instead of the Instead of the true transition we we obtain a very sharp crossover So I'd like to emphasize that this all relies on the presence of the amplitude mode in the system and as far as we Understood and we looked at it with with a bit of care. So it does not contradict any result that would be like rigorously Established in more mathematical or mathematical physics Literature so now I move on to the other calculation, which is very natural to to do and also leads to Well, it's own picture of the transition. So now we just parametrize the The effective action with this most general form which has terms of Order at most q squared in the momentum representation Well similar Calculation was done also kind of long ago by Gustav and veteri and Yeah, so this I have to say that okay here, this is the numerical calculation. So the analytical understanding is as a bit limited And the pictures that we find and and also that well for technical reasons It's not possible to do this at very low temperatures. This is related to the singularity and countered in the flow So we are we are actually The picture that we are obtaining here is is a bit similar But also has differences as compared to the the one I presented below Before so the the presence of this quasi plateaus is just an omnipresent Features and we find that okay the kind of true BKT transit well behavior cannot be restored at any order in in fight but If we use the degree of freedom, which is provided by the leg regulator Then this is possible to to perform a procedure where we just fix temperature and then scan the regulator space in such a way that we obtain a true fixed point then we Shift the temperature and then we again look for such a regulator that the fixed point is The true really true fixed point is is attained at the end of the flow So this procedure leads us to the conclusion that this this is possible only for sufficiently low temperatures so if we are if we are above some Temperature which we identify here with the true costal stylist transition then we are not able to find any regulator Of course, we are searching in some restricted family that would lead us to the to the fixed point and quite strikingly to be the the the numerical values of the eta exponent also of the a stiffness face stiffness jump is very close to the to the standard costal stylist theory So for eta we obtained like zero point twenty four and the row as Jump was also very close to to the to the standard value which is two over pi So on the other hand if we if we insist on working with a fixed regulator it's never possible to to get a line of fixed point and Depending on the the choice we make we either end up with just one fixed point or or just a very similar picture to the one We had and in this low temperature like Calculation with we just get the quasi plateaus And that's it So that's that's the that's the summary So I want to somehow Emphasize for first of all that okay this this FRG Framework provides if you are just interested in recovering the costal stylist behavior So this is possible both in the low temperature phase, and this is very simple Just forget about close the eyes and forget about this amplitude mode Then this gives the perfect kind of picture on the other hand if you want to to get the behavior of the in the transition region Then you have to use this this full derivative expansion with a somewhat not a natural perhaps procedure which just well enforces the fixed point by by By tuning the regulator This gives a very accurate well agreement of the numbers Even though the mechanism that drives the transition is just completely different In this calculation and in the in the standard theory, so this this of course Suggests well kind of rejection of the costal stylist theory at all and somehow looking at the Well the possibility that okay this amplitude mode is Is the factor which which truly? destabilizes this this phase, but this this this can be Treated as as a well just hypothesis which really would require more more kind of careful Treatment so I think that's I still time We are still four minutes including the time for discussion. Ah four minutes including discussion So it's it's okay. So then I we can discuss. Yeah, so thank you very much