 Thank you Okay, so my my talk will be about Disordered systems of systems with quench disorder and so this is an ongoing work But this time we were interested so with Yvon Balog who is here and Mathieu Tissier in Paris we were interested by the dynamics in equilibrium meaning around equilibrium and the dynamics out of equilibrium of those systems Of course dynamics on large scales long times, so this is collective events and more specifically We want to focus on a class of systems Statistical physics condensed matter theory, so this is a broad spectrum of physical realizations that I will not detail For which the long-distance physics is dominated by Disorder disorder fluctuations rather than thermal fluctuations, and I will give some explanation for that so those systems are Models in a random field random field being random magnetic field For if it's a local magnetization, which is coupled to this field It could be a random chemical potential generally speaking It's a random source that is linearly coupled so to the local order parameter It could be also models O and models with a random anisotropy or it could be also a class of Interfaces so elastic manifolds In a random environment, so those are the systems we're interested in and The dominance of quench disorder means this that is when you consider equilibrium It means that the critical behavior is Controlled by a zero-temperature fixed point that is even if the critical behavior is at finite temperature. So this is for instance a Sketch of the phase diagram of a random field model above the The lower critical dimension, so we have temperature here You have disorder strength here, and this is the critical line separated paramagnetic from a ferromagnetic phase in magnetic language And then every point here except of course the pure is in case which is in here exactly zero disorder Every point here flows to zero temperature and finite disorder fixed point. So in that case Everything about criticality is terminated by the sample to sample the quench disorder fluctuations and In for such fixed points at zero temperature There's an additional exponent that controls how the temperature flows to zero So the renormalized temperature of course not the bare temperature But the renormalized temperature flows to zero to the fixed point and this gives an additional exponent that enters all sorts of Equalities between between critical exponents So this exponent theta is strictly positive, which means that indeed temperature is irrelevant now what is interesting for the dynamics is that The dynamics here I'm talking about equilibrium The dynamics the relaxation to equilibrium at finite temperature in in those cases That is around the critical point So the standard critical slowing down is in fact very different from usual So usually you have a power law relation between Time scale and length scale with the dynamical exponent z. Well here it turns out that just because The temperature is irrelevant. You have an exponential relation So I put the temperature here just for bookkeeping. It's not really Important an exponential relation between time the time for relaxing the system to equilibrium and the Length scale meaning that the relaxation to equilibrium is extremely slow so that's a special feature of that type of System which you know say sometimes it's sort of sort of glassy Dynamics now because it's so slow that means that there is also non-trivial physics at Very low temperature that is at very low temperature You can basically forget about relaxation to equilibrium and the only thing that you can still do Which is highly non-trivial in the presence of disorder is that you can drive your system, you know by applying An external field generally speaking an Force and then you drive your system out of equilibrium and this leads to in the case of Interfaces this leads to the physics of pinning of interfaces and depinning Transition and in the case more generally in the case of random fields random anisotropy You have a hysteresis phenomenon and out of equilibrium phase transitions. So this is general features of those models now Why do we need I will go pretty fast here because first I have only 20 minutes and I guess some of this we talked in in the past Why do we need? Non-perturbative functional renormalization group here So so the two terms in that case or I take them out as distinct Unless what we do generally in the field Well, the renormalization group in that case must be functional because if you think of the physics Which is related to the presence of all those impurities. Well, the physics is dominated by rare events rare Regents rare samples Which have the name at zero temperature of avalanches or shocks So these are collective events that take place at zero temperature and in real space at finite temperature you have low energy excitations that are Considered as or known as the droplets again That is something which is induced by the disorder and that you see in real space now. The problem is how to Keep the signature of those events in the renormalization group. You know, it's not easy To work directly with those objects because of course in the presence of random impurities quench disorder Well, you lose translational invariance. So that means that you know a generating functional here As a function of the source is a random object And of course, it's a whole mess if you try to renormalize a random object because you need to renormalize a Probability functional of the thing and that's so far basically impossible So what you do of course is that you can work with the cumulants of the random Functionals in general and those cumulants of course you can access In in a theory when you recover a translational invariance, so you're happy you're working with cumulants But the question is okay. I have cumulants, but then how it can I keep track of those events in real space that are pretty rare and The key point is that to describe this You need the dependence of the cumulants on their arguments because it's the singular Dependence of the cumulants on their arguments that capture the physics of those events So that means that to describe the physics you need to be functional Just to get that singular dependence and that signature of those rare events Now why do we need to be non-perturbative? Well first there is a general reason is that the standard perturbation theory just Breaks down completely and this is the problem that it will not address of the Dimensional reduction failure But more than that there is also this this point that for random fields and for random an isotropy models Well, you have a change of behavior So the physics drastically changes at a non-trivial Critical dimension and here for instance I plot a Phase diagram which would be for the random field on model So you have at the vertical axis is the number of components and from ising to large numbers And here it's the number of dimensions. So six is the upper critical dimensions Two is the lower critical dimensions for ising for the lower critical dimensions for for the Heisenberg and others And you see that there is a red line here, which goes down to about five point one for the ising case in terms of dimension That separates two different regions where the physics changes that is below here really you need the strong non-analyticity of the the Dependence the functional dependence of the cumulants you have a dominance of the avalanches You have dimensional reduction breakdown, and you have supersymmetry Breaking at the same time. So there is really a change of behavior and of course, it's a non-trivial dimension So there's no way even a very clever Protributive functional orgy will give you this directly Okay, so this is this is the motivation for using a non-pertributive functional orgy for those systems And from now on I will focus on the random field ising model. So the n n equals one version in a random field now The physics is that in and out of equilibrium. So consider here the system at zero temperature Okay, so this is the plane Fies the magnetization say and J is the the source the applied source and now you can consider The equilibrium problem So what you would do is just look at the ground state in it's at zero temperature and Look at you know What's the the behavior of the magnetization in the ground state and for that value of the disorder the magnetization? Which is the blue line has a discontinuous jump So there's a first-order transition We are then below the critical disorder and this is a first-order transition and the system has as the usual Z2 symmetry Now what you can also do? do is Consider the system at zero temperature, but now Drive it with the magnetic field So instead of waiting or trying to use tricks to equilibrate your system You say forget the equilibration. I drive the system by just ramping up for instance Here the magnetic field or if you wish now ramping down the magnetic field and following the magnetization Now, of course, I'm no longer in the ground state I will stay in metastable states some special metastable states of the system that are still locally stable But not the full ground state and I will have this hysteresis curve with an ascending branch When I ramp up the field and a descending branch when I ramp down the field And what happens is that if you if I do this very slowly so in a quasi static limit What happens is that there is also phase transition along the hysteresis branches So for instance at this disorder there is indeed a critical point here That will separate a region where the hysteresis is smooth with a region where it has also a jump a first order transition and as you see now the z2 symmetry is broken Locally because I'm in the field, but of course it is recovered On the other branch that is this branch has a symmetry compared to that branch So you have a critical point here, but you have also a critical point here But what we're interested in is say choose one of those branches and study the critical behavior of One of those branches, so this is something which is you know important for technology all those hysteresis and Business in disorder systems, but what is interesting here is that? it's a long there's been a long debate about what's the the nature of the critical point in and out of equilibrium and Simulations all sorts of clever simulations with pretty large system sizes Show that quite surprisingly When you look at the critical exponents and the scaling functions of the two systems the critical System out of equilibrium and a critical system in equilibrium You find you know very very close exponents and very very similar scaling functions and so the idea was that well maybe of course it's very different situations This one is out of equilibrium has no z2 symmetry Compared to the other one, but maybe they are controlled by the same fixed point And of course simulations will never answer this because you know, it's you would need a precision that is Enormous and even then we would not even possibly be convinced, but of course you can do an RG Calculation and see indeed if the two critical conditions flow toward the same fixed point Or if they flow to different fixed points and this is The main question that I will address today now There's also different effects of temperature in and out of equilibrium So as I said when you are around equilibrium you put a little bit of temperature and first the phase transition is not destroyed It's still there and you have this activated dynamic scaling with very slow relaxation When you put temperature Out of equilibrium on the hysteresis curve Well, of course in practice in an experiment you see nothing because as I said the time scale for relaxation is so large You see nothing But when you do statistical mechanics and you assume that you have an infinite time scale at your disposal to equilibrate or change Of course it changes everything. So if you want this Behavior out of equilibrium with phase transitions out of equilibrium It's the same for the pinning transition of interfaces is truly defined Rigorously only at zero temperature Okay, this is a problem of equilibrium versus metastability that is well known otherwise Now what's the model so we're looking at the dynamics of the random field ising model and Looking at you know long distance long time so we're using a field theoretical description And we're starting from a longevity equation where the field so the scalar field at Point x and time t obeyed the longevity equation Here where this is the force which is due to The standard for magnetic action for the standard 5-4 action for the pure system Then you have a term that does not depend on time, which is the random field and usually for simplicity take it as Gaussian With a variance here, which which gives the the bare strength of the disorder then you have an Applied source j of t and then you have thermal noise stochastic noise Which is also taken as usual as Gaussian with the variance which is given by the temperature Okay, so at zero temperature basically this term is not relevant Now j of t is is the applied source meaning that you have two settings You're looking at your system at equilibrium. Then of course the source is just constant It's what it is and actually to look at the phase transition you even Look at a source, which is equal to zero no source now for the quasi statically driven system at zero Temperature then your source actually is as I said slowly ramp up or down in time So the omega here the rate goes to zero, but it goes either and this is important It goes either to zero plus because you ramp up so even if your quest is static So it's almost a static calculation But you should keep track of the fact that you are going forward in a sense and You'll feel your magnetic field and your magnetizations are going up But you could also look at a different Case which is of course the descending branch and now omega would go to zero minus and in that case It would be the other way around the magnetic field Decreases albeit very slowly and the magnetizations also decreases So this is very important this sort of irreversible behavior that you have out of equilibrium, which is quite different from From equilibrium anyhow we induce we introduce copies replicas because This is something that we like to generate cumulants We use the Johnson the dominicis Martin CDRO's formalism to write this in terms of Generating functional with response field and what we get is a Dynamical action, okay, that has a term which is Independent of disorders the same that you would have for the dynamics of just the ising model or the 5-4 theory and a term Which is which is with two Carpese two cumulants a second cumulant which depends on the Bayer disorder you add an infrared regulator You do the usual trick you Fourier transform you get the scale dependent effective action You write down the exact equation that we all know here and and then we are in principle done except of course that we need an Approximation scheme and the approach approximation scheme here is derivative expansions so in gradients in time and in space, so this is again standard Has been done and we need also an expansion in cumulants as I said And so we need also to truncate in cumulants And so what we do is that at least for this model for other models who truncate at higher orders So you have a first cumulant with you see functions. So this this just local This is something that gives us the renomization of the wave function renomization a dynamical coefficient And what is important here is this second cumulant Renormalized cumulant of the random field that depends as you see on two arguments Again, you know we can use scaling dimensions and put all this in dimensionless forms and we get Then the flow equations that we have to look at and to solve The whole point which is which is central here in the dynamics is that Since we have at the fixed point at least A non-analytic a singular Action effective action. So remember the delta, which is the the second cumulant of the random field has a cusp When you go to the fixed point a cusp in the difference between the fields Okay, so when the two fields becomes equal there is a singular behavior So the function should be symmetric. It is symmetric, but it's now symmetric with a cusp Okay, and this is the singularity that keeps track of the avalanches in the problem When we are below that dimension five point one But the problem now is that the flow equations are ambiguous Because the flow equations need derivatives of this function Evaluated at the same for the same replicas and of course for the same replicas Well, it depends if you're on one side on the other So there is an ambiguity and that's a whole problem that is the same also in the case of interfaces in random environment for the Poturbative functional origin now The way to resolve this is that in equilibrium In fact as soon as you put a little bit of temperature in fact the cusp is rounded So the cusp here has a little rounding due to temperature of a width that goes to zero with temperature But it's rounded so basically it lifts the ambiguity So you can take derivatives and then you take the limit of temperature goes to zero It's not that easy. Okay, because it's it's a non-uniform convergence So you have to be careful But it lifts the ambiguity and indeed you go back to the zero temperature fixed point because again The temperature goes to zero and that's the way you get also activated dynamic scaling that I will not discuss now For hysteresis at t equals zero, so you cannot add temperature then there's no rounding So here you're in trouble But the way to solve this it's to remember that we had this infinitesimal rate of time for driving Which translates into an infinitesimal velocity for the change of the magnetization And again the fact that the system is irreversible That is when you drive up the magnetization goes up when you drive down the magnetization goes down Well, basically to make a very long story short I mean I must say that the first time we got we got it wrong, so it's not that easy But basically what what it does is that it selects one side or the other So even in the presence of a cusp the fact that you have this Irreversible behavior, you know with causality and everything makes that you can choose in Your flow equations Derivative on one side or the other and it's no longer ambiguous So you have flow equations now that you can solve non-ambiguous for equilibrium out of equilibrium and they are different and indeed what we find is That the fixed points are different In and out of equilibrium when you are below That special dimensions Where the cusp appears which is about 5.1 Now just to show you that they are different Let me just point this this function here that is the renormalization the wave function renormalization function as a function of the field So the red is in equilibrium So you see the nice Inversion symmetry the z2 symmetry as a function of the field for dimension 4.1 and the black one is the hysteresis Fixed point so you see they are clearly different the fixed points. It's no no way to to Confuse them on the other hand When you look here at the anomalous dimensions turns out there are two anomalous dimensions. Well the full lines are the our predictions for the equilibrium case and you see that the symbols here are our predictions So far we can't go much Down here. It's a kind of technical numerical bothering problem, but it's not nothing Fundamental you see that they are very very close. They are different though Okay, so there's a slight difference and when you see the exponent the other exponents. There's also a slight different those two so we have Basically, it's our answer We have different fixed points, but indeed for some reason The critical exponents look very similar Between equilibrium and out of equilibrium. So I guess this is this is the sort of answer that I guess you can reasonably have from that non-perturbative or G but that you couldn't have from any sort of Simulation because as you see the difference is so small and the simulation don't compute directly The the functions we're looking at anyhow. So those those are The conclusions let's say that to check this. So as you said the numbers are pretty good I didn't stress this for that model But we checked this on the other model of interfaces in the in a random environment where there are much more Many more data on that and again we get good results and good agreement with the existing data So we're pretty confident now on this result, but this you know There's still a way a long way to to go anyhow. So thank you for your