 part of the Bustin's lecture. Very good. So I first want to make a really compliment. It's really a pleasure how active you are. And I'm really surprised because you're getting these four lectures per day and still are really attentive. So it would be nice if you keep this pace for now. Right. So I want to just wrap up this section here. Now it's more the hopefully joyful part. It's not as technically demanding. We're just sort of getting the consequences of all our formal developments now. And just to remind you, the starting point is this situation. We just discussed so that equilibrium systems are protected by some specific discrete symmetry, which is absent or manifest explicitly broken in and out of equilibrium system. And the question that I want to follow now is let's assume we have such a situation. Let's assume we have this spread of couplings in the complex plane. So in the dissipative versus reversible dynamics, they are not tied to each other. But the system is confused and doesn't know what to thermalize to. Now what's the consequence of that in a specific model? And this model will be actually this exciton-polariton systems that I also introduced in the last lecture. So and to set the stage, I want to remind you of what is well known about phase transitions in equilibrium two-dimensional systems with a continuous symmetry. That's an important ingredient now. So we want a continuous symmetry. For example, simple most example is a U1 symmetry. And we discussed a lot about spontaneous symmetry breaking. Now I remind you, this is an effect that only is operative really sufficiently high spatial dimension whilst two dimensions is exactly the case where we cannot have spontaneous symmetry breaking into this Merman-Wagner theorem, presence of gapless modes for bits, stable condensation phenomenon and these gapless modes will actually lead to the destruction of such a condensate. Rather, what you see at low temperature, what replaces this really long-range ordered phase is what is known as an algebraically or quasi long-range ordered phase which is characterized by an algebraic decay of the correlation function, of the two-point correlation function. Think of it like that. So if we had really a condensate, so then if you have an expectation value at point x and one at point zero, so this correlation would just saturate independent of x if we have a condensate. Now the compromise that the two-dimensional systems do is they show still some strong correlation but it's not sort of a flat, it's not saturating as in the presence of a condensate. And then it is also well known so that in the two distinct phases, the low temperature phase at high temperature is replaced by a phase with a much more generic exponential decay of correlation function. That's a typical situation you usually have. So if you don't have much correlation in the system and high temperatures, the temperature leads to fluctuation that averages out the correlation that's typically an exponential behavior. Now also the responses are behaving qualitatively different in these two-dimensional systems whether you want symmetry, say Bosonic, Bose-Einstein scenario also. You have a superfluid response. So if you stir the system, you will see it with a response with a finite parameter here which is lost so the superfluid response is lost in the high temperature phase. And what interpolates between these two limiting cases, between these two qualitatively distinct phases of matter is the famous where the picture is that at low temperature the vortex degrees of freedom defects in the phase of this u1 symmetry here, defects in the phase are bound into pairs and do not show up in the long wavelength behavior and this algebraic decay is then just coming from smooth phase fluctuations in the problem while at high temperature these vortices they proliferate, they unbind and then this ordering or this quasi long range order is actually gone. And now we want to ask the question what the fate of such a scenario is in a driven open condensate, so a situation with the u1 symmetry but put place out of equilibrium and the compromise that I want to do here I focus mainly on the properties down here and the rest I won't get to it. So let's think about again so where do these long range correlations come from so just to give you a bit or maybe refresh your mind or to explain where this algebraic decay here comes from statement will be it roots really in the presence of gapless modes in this problem gapless spin wave fluctuations which do not cost energy to excite them and the action so for this spin wave is just described by the gradient of the phase variable squared so this is the kinetic energy that is contained in the phase variable and if I go to Fourier space well then you can see this q square here then to find the qualitative behavior of this function here we do what is known as a phase amplitude decomposition so we write the field in terms of a density like variable squared of that and the phase variable and the statement is so that the real the soft modes in this problem is the phase that can fluctuate a lot while the density only fluctuates mildly so we can approximate this density variable just by its mean by a mean density and then evaluating this correlation function here in this representation of the field comes down to just studying this correlation function of phases in the exponent here and if we make the assumption just to get an orientation that this is almost a Gaussian problem then you can use a few nice identities for evaluating such correlators here and you find that the expectation of the expectation of e to the i is the square of the argument here in the exponent that's a Gaussian integration identity and then you see we want to know this correlation function here here is how you calculate that Fourier transform this spin wave Green's function so which shows in particular this q square dependence down here and you see that this function here has this function as q goes to 0 there's an infrared divergence which shows up as a logarithmic dependence on the distance between the two points r is the absolute value of the distance and inserting now this logarithmic correlator here then you see e to the log of something just gives us this algebraic function so that's the reminder on equilibrium and now we want to understand how do the non-equilibrium conditions affect this low temperature where there is no temperature out of equilibrium not obviously how does the non-equilibriumness of this problem affect this result here and to this end in our way of thinking we do not need to start on the totally microscopic scale in the problem we can actually already move to this semi-classical limit for this problem because we are interested really in very very long wavelengths or very low frequency quantities which are by definition well below the noise level in our problem that's when we can take the semi-classical limit and then we are entitled to describe the whole problem this whole quantum problem in terms of the simplified Launce-Maur equation here so at this level we still want to do some simplification and because it's still a complicated equation with two fluctuating variables but we are guided now by the intuition that I was just telling you that this field can be well represented in terms of a density variable and a phase variable and if one does the calculation one sees actually that the time evolution for the density variable is very fast it damps out very quickly so this is a fast variable that we can integrate out adiabatically eliminate and what we end up with is an evolution equation just for the phase variable alone so this is a kind of an RG step if you like where we integrate out fast fluctuations in the problem so these are these density fluctuations but we keep everything that is still active on the largest distances in the problem and that's always the terms that come with a lot of gradients because as momentum goes to zero so this gradient, powers of gradient becomes power of momentum and that is still active in the long wavelength limit and it's not leading to a quick damp out of these fluctuations and then I mean so doing this step we arrive just by just doing it we arrive at this equation here and this is actually a pretty famous equation it is referred to as a Kardar or it goes back to Kardar, Parisi and Zang who studied a completely different problem which I will also tell you namely the roughening of surfaces so in our or of interfaces in our case this interphase is realized by the phase of this u1 symmetry that we were starting from in this exciton-polaritone condensate so and just to go through these terms so the time evolution of the phase is generated by phase diffusion so there's a z equals to dynamic exponent between time and space but then there's also this important non-linear term and there's of course noise that is inherited from the noise that we had in this problem originally okay and the main way how to read this equation now is the interesting insight so that this spin wave mode so this is called the spin wave so what usually behaves diffusively in such systems without a non-linear term now becomes non-linear so our Goldstone mode in the problem this u1 phase mode becomes non-linear by this Kpz term and what is more one can convince oneself that at least in dimensions d equals 2 and larger this non-linearity is strictly forbidden if the system is at equilibrium so going back to this picture here that we had if we are in such a situation at equilibrium then it is in dimension larger equal to ruled out that such a non-linear coupling occurs while I mean in an out of equilibrium problem there's absolutely nothing against it and you can think of Rg or coarse-graining as the statement that everything that can happen everything that is allowed by symmetry that will also happen that will be generated under Rg transformations under coarse-graining everything that's compatible with symmetry and this is a term that is compatible with u1 phase rotation symmetry but not compatible with equilibrium symmetry so out of equilibrium it will be present okay so maybe a bit of background for this Kpz equation where it really comes from it is actually a problem that is connected to the roughening of surfaces and the simple most physical situation that was also advocated in this paper here so in the original one is actually the growth of interfaces so here you have a situation where I have a plane, a tilted plane the tilt is crucial because when combined the tilt of the plane with a gravitational field you can see that there is a downhill acceleration so while these particles are falling from the top and such that the interface is growing there is clearly a downhill acceleration and that is something that breaks the conditions of equilibrium in a very intuitive way imagine I tilt this to angle zero then this interface just there is a balance of forces and the interface will just grow to the sky and in this case however when it's tilted there is a non-equilibrium drive in this problem in quite an intuitive way yeah ah very very very good um this is not the case I mean because the equation has too many theta is a real variable so we can't allow complex couplings because that will violate the reality of the phase right and it's so that's a high level argument this can't be the statement is just that I mean I basically decompose by doing this transformation I managed to decompose this complex value to the equation in the real and imaginary part and the imaginary part it just stays imaginary all the time it has to be like that yeah right so maybe another way of saying it yeah so it's a nice circumstance where this quite maybe more complicated picture with many many couplings that can be around the whole non-equilibrium character is now compressed into a single real parameter yeah a single parameter that measures whether the system is in or out of equilibrium yeah so it's it's just I mean this equation a real scalar field variable just doesn't allow for complex couplings if you write it more technically if you write it in the action you can symmetrize it in a way that will drop out so it just can't be yeah and but still I mean lambda equals zero means equilibrium lambda non-equal zero means non-equilibrium so it's a single parameter measure for the non-equilibriumness of this problem it's just very handy so we don't have to think about anymore about this spread we just have to see is this parameter zero or not and then we know equilibrium or not in the original yeah in this equation up here so right so they the G so the two body sector here yeah this has a real and imaginary part and the real part this is the elastic collisions imaginary two body loss as we said and the same now goes through for the single particle sector so there's a mu a rotating frame as we said yesterday and some loss rate for the principle I could add here also some here we only have coherent propagation but again as we said under RG everything that can happen happens so there is also a diffusion part so imaginary part coefficient of gradient square mm-hmm um um there is an intuitive argument in this um mm-hmm that's actually a good question so I always think about it can't be and mm I mean looking by the consequences yeah so it would so or maybe looking by this mm ah so but you also don't see it so nicely no I don't think I have now right away maybe I think in the background okay other questions yeah that's right I mean ah that yeah mm there is something going on like that so the real thing is lambda plus yeah so this is the standard growth term yeah but you can do a phase transformation to absorb this yeah so this is I mean this is saying it's like I mean this thing is a phase think of a phase and a phase you can always go into a co-moving frame so you move with the phase itself the interface picture you move with the interface you put yourself on the interface and you grow with it in this frame you don't see the linear growth the trivial one so that's why it's often left away because it's just a choice of rotating frame but what you cannot remove so they really they come together these guys yeah and this one you cannot remove by choosing a rotating frame more questions want to find out right so the the statement is we look at etn dt theta and then the first thing we see when we look at this equation so there is a term like a gamma loss minus gamma pump times n so this is really and plus then some higher order terms maybe n squared also gradients of theta the point is however so whenever this is finite here so the dominant effect from n is that it exponentially quickly finds its stationary state while all the terms here they always have to come with powers of gradient so that's these terms because the u1 phase rotation symmetry tells me that the equation must be invariant under the transformation of a constant phase shift right so and this can only this invariance must leave the equation of motion or the action invariant this can only be realized if the phase variable appears together with gradients so because then the gradient operation kills the constant alpha well I mean so when you know this so you say this adiabatic elimination so you say this variable finds its stationary state very quickly so let me just forget about this time dependence and then I can solve for n of function of theta and here is also some plus function of of n then you insert it here and that's how you eliminate it but this is the technical way to go it's really not complicated I mean maybe it sounds more complicated than this the conceptual thing that happens we have a fast variable this we can integrate it out before even thinking about the slow variable it's in the way that's fast you integrate it out and everything that's slow you want to keep it and then we have to study this in more detail we are effectively deriving a low energy or low frequency theory for this phase variable alone with a good reason so that the density variable is decaying very quickly more questions okay good right so this is this KBZ we indeed we remove we put ourselves into the frame of the growing interface itself and then the effect of the deposit is really just this non-linear term and right also from this geometric picture you can see I mean so you need to study a bit the geometry of this problem you will see that in the precisely in the case and there's an assumption to the tilted surface yeah if you do this then it's a geometric consideration that tells you if in the case that the interface is tilted there's a non-linear term maybe this is an intuition in the sense that I mean it goes away when you have no drive when you have really an interface that just moves up but no downhill acceleration that's maybe as intuitive as it gets from my view okay where this is for example has been roughly observed it's not perfect but in the defect growth in liquid crystals so you always have to think in such problems then in such surface growth problems where is the drive and in this case I mean one puts an electric field that pulls this interface to go larger and larger in a bacterial colony you have an interface here at the edge of it which is driven open systems they consume sugar for example or also the spreading of fire fronts is another example for example burning of paper which is driven by oxygen consumption chemical reactions also driven open systems so now let's connect the go back to our precise problem and the task that we now have we coarse grain the problem of the slowest variable in the problem the phase alone and then in the next step we now have to get really quantitative when we want to assess the question what is the impact of this non-equilibrium non-linearity when we really go to large asymptotically large macroscopic scales in the problem so until this point we got away with basically simple arguments and now we have to do a calculation to assess what is the impact of this non-linear term at long wavelengths in other words we have to do we have to track the flow the renormalization group flow of this parameter under renormalization group transformations by lowering and lowering the resolution and going to larger and larger distances in the problem so this is now the task of the lambda under renormalization group transformations and here is the result for that I mean this is analysis has been done ages ago in the context of KPC and this is the flow diagram where this x-axis here is actually the dimension of the system so one dimension two three and the idea is then so we we initialize the system here say slightly out of equilibrium with a small but finite value and then we ask the question so does this become larger under renormalization group transformations or does it become smaller if it becomes larger then the interpretation is that the importance of non-equilibrium effect becomes bigger and bigger and we expect quantitative modifications if it becomes smaller this tells us that the system in the long wavelengths limit effectively thermalizes so all the non-equilibrium nests of the problem then dies out so that's a neat feature that we package every non-equilibrium feature of this specific problem in a single parameter so we just need to follow this single parameter to say is this non-equilibrium or not at large distances and there is a very general trend for systems with gapless modes I claim that namely if you are in high spatial dimension and you initialize RG initialize on this mesoscopic scales here you initialize your problem with a weak perturbation out of equilibrium then actually the RG flow directs you down so we will asymptotically at very large distances the system does not remember although we broke equilibrium conditions on microscopic level it will not remember this at very large distances so this is one logical possibility that can happen and that is what happened in systems with gapless modes conversely if we go to low space dimensions like one or two you see really the opposite phenomenon so imagine I initialize on the mesoscopic scale my problem at very weak elongation from equilibrium then it will grow and grow under RG and find this what is known as strong coupling KBZ fixed point it's a fixed point that RG fixed point that's not very easy to characterize even nowadays it's still a challenge but it's widely accepted that there is northonomerically confirmed so that there is a fixed point in this problem which is very far from equilibrium fixed point where there's no detailed balance and something that you can notice is that under RG a length scale is actually generated where below which you would say the system is still pretty close to equilibrium and above this length scale I mean at least there is a crossover regime and very much above this length scale one would say okay we are very close to this strong non-equilibrium fixed point so at short distances so if we initialize it at weak a weak elongation from equilibrium we are very close to this equilibrium fixed point but at large distance then I mean beyond this crossover scale here length scale which is defined by this here going order one then we are close to this equilibrium fixed one and if you solve for this length scale from the RG flow of this problem here a weak initial non-equilibrium perturbation is actually an exponentially large length scale in these two dimensional systems in one dimension it's not the exponential okay and then we can with this understanding we can now come to the final conclusion so what is the fate of the long quasi long range or that low temperature or low noise level phase of this from our understanding of the RG flow at short distances at short enough distances at weak elongation from equilibrium we will still see the quasi long range order scaling that we are expecting for an equilibrium system but then beyond this length scale that is unavoidably generated in two dimensions we will sense the physics of the strong coupling fixed point and when we want to compute the scaling behavior of the correlation functions we have to insert actually the understanding that comes from what's the critical exponent say and what's the scaling behavior at this Kpz fixed point and if you do this then you find actually the correlation does no longer actually governed by an exponent so that is known from this strong coupling Kpz fixed point so it's a manifest non-equilibrium scaling behavior but in one short word it's interesting to notice that I mean it is impossible conceptually to put a system out of equilibrium and still expect quasi long range order so this quasi long range order is really strictly forbidden by this reasoning here in equilibrium nature of the problem so this is this summary here and there's also now beautiful experiments still in one dimension that really confirm that measured essentially the critical exponents of the Kpz universality class which are even exactly known in one dimension so they could confirm this mapping to Kpz from experiments so impressive I mean you still need huge systems to see the scaling behavior to resolve this in appropriate ways okay so I think with this I would leave it with a discussion of the low temperature phase of course there's interesting stories about what the fate of war disease in these two dimensional systems but I would now switch gears to the measurement part unless you right I mean so non Markovian I mean so my take on this is so that basically from a large enough distance everything looks Markovian so let's imagine so Markovian means that I mean your noise level is auto-correlated just auto-correlated now I take this function which is not a delta look at it from a very large distance it almost looks like a delta function and I mean mathematically so it is you can expand about this limit and then up to higher order gradient corrections you will not see much of this non Markovianity except for the case so when you do a calculation you typically don't find a delta function you find an exponential function so something like e to the minus t prime and some pre-factor that makes that you approach a delta function in the limit kappa to zero you have a highly non Markovian situation maybe if if you have you replace this exponential scaling here really by say a power law this is something that happens at zero temperature for example and there's also very special circumstances in which there are conditions and it's not a power law of time but it's a power law in space and that comes maybe close to that so a scaling function for the noise that is something that could jeopardize this scenario but whenever it's in the class of the natural exponentially correlated system then it should collapse to this back to this in this Keldish language so this noise is the stuff that occurs sandwiched between these these fields so these two quantum fields what stands here is the noise we have here dT dx so and I mean so Markovian so just elaborating a bit more on this question is that when we have here just a constant decay level so in the field theory spirit again what can happen is allowed what is allowed will happen so maybe you have a gamma prime which comes with a dx squared or a gradient square and then a gamma prime prime with a gradient force so yes they exist these terms but they are of course these couplings here in the sense of the randomization group they are irrelevant of the noise in the sense it's colored in the sense of spatially slightly correlated but I mean it's irrelevant corrections in the RG sense so therefore the statement is I mean these details are shorter distance they don't matter for the long range wavelengths physics the only way out that I was pointing out is what happens if by chance this parameter is fine tuned to zero this is a different level and that's a more subtle problem but this would require fine tuning in a similar way that you have to fine tune temperature to zero to see quantum critical scaling there's actually really sharp analogy between scaling solutions of the action for a zero temperature quantum critical problem and a kind of scaling of the noise level in such a way then you can have analogs of quantum critical scaling also in this driven open context fine tuning for disordered electronic systems I would definitely recommend that because I mean with disordered systems you have another problem yeah so you I mean you can look at it like that I mean this order is like a bath with the difference from the bath as we are looking at here yes as you have in other words disordered problem is a kind of problem where you have random variables the flat correlations in time it's called quenched disorder yeah while the noise we are considering here is delta correlated in time so it's like this order is a kind of bath limit where there's which is quenched in time while we have the opposite limit of Markov level in time yeah what this order problem shares with the or the this order problem I mean this order is generated out and this can be easily done in the Keldish formalism and it's actually there could be some good reason to do this because I mean this is now a detail which I just tell you so that I mean when you want to average over disorder in an equilibrium partition function you run into problem of having to average a logarithm of the partition function and you cannot do this easily you have to sort to a replica you can average moments of logarithm of Z and then you run into the replica formalism in the Keldish formulation you can avoid that but I think that's really a bit goes in a different direction but the statement is yes you can describe disordered systems very efficiently with Keldish it's not this ballpark here there's a formal connection of you have some random variables but the overall formalism would be very convenient for that it was a discretion sorry see more some people call it like that this is really a smooth phenomenon so there's nothing sharp there's no I mean I take a sense of question but there's nothing I mean if I take a derivative along this I mean non-smooth that happening it's really a crossover phenomenon so this effect dominates at short distances because the RG flow has not grown enough and we are still close to equilibrium but at some scale the other effect starts to dominate because we're closer to the non-equilibrium fixed point good so then I switch gears 50 slides we start this lecture three right so it's really a change of gears also from my mind have to switch a bit now and this is the topic you get to know yesterday already and I want to look at it from a little bit a different angle also slightly different models but I think it's interesting to somehow look at the same at least ballpark of problems from different perspectives and here we will see how far we can move with this childish formalism and let's see how far we actually move in this lecture so okay so here is my mini introduction but I can I guess keep it brief my view on this measurement phase transition is this it's something measurement is a kind of physics question that usually comes from the small systems so we have this again this pattern now we want to go to large systems but so let's recall what measurements small systems where they declared are about you should remember from your quantum mechanics course so that almost on the axiomatic level one postulates two qualitatively distinct types of quantum evolution one is the deterministic evolution under Schrödinger equation so in it's integrated form it would be applying this unitary operation here as a function of time but then postulate of quantum mechanics is the measurement process and what then happens the wave function of the system behaves not deterministically but rather probabilistically in the sense that the wave function collapses gets projected into one of the Eigen states of the measurement observable so this P here is in the spectrum is a projector onto an Eigen state of the observable you're studying probabilistic element comes in here so this here is the born probability for this process to happen so for collapsing precisely into Eigen state number lambda and so yeah so this is kind of the probabilistic element of the measurement process and clearly this dynamics is quite interesting in the sense it will never end when the Eigen states when the Hamiltonian operator that generates a deterministic evolution and the measurement operator when they do not commute with one another so if they commute okay then the dynamics would be like this you collapse into an Eigen state of the measurement operator then because it commutes with the Hamiltonian this is also an Eigen state of the Hamiltonian and so the system will stay forever so this is what also called quantum non-demolition measurements I mean they don't commute so they don't share a set of the same set of Eigen states the evolution will always go on and this situation now we connect to many body physics is something that is very well known indeed in many body physics what is the physical consequence of two non-commuting operators in say the ground state of an equilibrium system this is exactly the situation that is discussed in quantum phase transitions so there just as a reminder you look at a Hamiltonian which consists of two terms which must not commute to have a quantum phase transition and you also require that H1 and H2 stabilize ground states with say a very different symmetry so or with very different qualitative properties at least think of an example this Bose-Harbert model here so this is I mean the H1 is realized by kinetic energy of bosons hopping on a lattice and H2 is the potential energy coming from on-site interactions and then there is a competition between these two guys the kinetic energy wants to delocalize the particles and that gives rise to a super fluid phase sorry that this regime here has F and conversely if the interaction dominates the particles they want to be localized have localized wave function no phase coherence in the system this is then this mod insulating states and the question of this measurement induced phase transitions it can also be formulated as this yeah let's put now two in a many body system two sets of operators into competition a Hamiltonian and measurement operators that both kind of want to push the system or want to have different structure in their eigenstates and then we ask so is there any phase transition and if so how do they look like so and that's what you've seen yesterday so I go here very quickly so this has been pioneered in this random circuits so where I mean this really discussed it yesterday and where just as a reminder so the problem is very easily structured in the regimes where this competition this dimensionless parameter sends here in this problem it is the number of measurements per unit time versus the number of unit areas that you throw under the system per unit time it's clear that in the random circuits when G is 0 so no measurements then one has a non-indegrable chaotic evolution that leading to entanglement growth while conversely if you have no entangling dynamics the system the measured system collapses into a product state you see saturation and okay so let's just remind ourselves that I mean there is actually what these works you have really nicely shown is that at a critical coupling strength like in a quantum phase transition so there is a critical point at finite competition ratio which interpolates in this random circuits here between a volume that I mean it could also be logically it could just be that the critical point is at 0 so then a little bit of measurement and then everything collapses into product states but that's not the case so that's an interesting novel phase transition scenario that shows up in the entanglement entropy growth now to this lecture here I want to briefly introduce an alternative way of thinking about measurements instead of strong projective measurements weak continuous measurements that has the advantage that it lends itself to a path integral formulation of the problem with path integral as you've seen we always like an infinitesimal time step to be well declared and the projective measurement is really a hammer onto the system so we need to weaken this a bit then I mean I give you a little bit of phenomenology and then I hope I will still be able to at least give you a flavor of thinking for this problem okay so let's think first a bit about the strong projective versus weak continuous measurements so here is again what a projective measurement is about say now of a continuous variable we measure position position X and so this is the observable and this is the I mean a continuum formulation of the where this p is the projector onto position X so if I do an oops sorry if I do an actual measurement then I will strong projective measurement I will end up in a definite position X naught somewhere I will detect my particle and then let me just write the projector onto this state X naught that the post measurement position is X naught with certainty now we can realize so that nothing in nature really is perfectly sharp instead processes take time and there might be a bit smeared out so instead of this instantaneous collapse of the wave function we might think of this delta takes a finite amount of time here delta t or this is the time of observation and we replace a sharp delta function here by a sharp very sharp delta function by a Gaussian which still might be sharp but if we do so or it's just we can also represent it in this case and it's obtained for this time of observation basically compared and now you have to compare to something compared to other time scales in the problem is infinite so you give it infinite time to measure the system then it really project it approaches the limit of a projective measurement but we can also repeat this measurement many times in a row and then the dynamics that previously was looking like that so now resolving essentially each of these measurement processes in this limit looks rather like that so it really looks like a path integral already in the sense that there is a step of unitary evolution interpears by the amount of time but we repeat this very often so that we get eventually a continuum where in a continuum in time limit the Hamiltonian updates of the state and the measurement updates of the state they occur on equal footing okay so then discuss a bit the probabilistic character yeah you can I mean you can take other representations other approximations to the delta function it's a very good question which I also want to address I mean we do not expect so the question is what questions do you want to ask later on so if it's about does there exist a phase transition and what are its universal properties I would claim so that the details of the measurement process how you model this do not matter again by RG type arguments while I mean of course non-universal properties like where is the exact location of the critical point these will see the details of the measurement process it's really again it's this way of thinking this has nothing to do with equilibrium versus non-equilibrium it's just very general basic physics so details matter for non-universal properties but for universal one we don't expect it so there's a freedom in choosing the precise model this model is of course more easy to handle in tough integrals further good so let me connect this also a little bit to this long-term discussion that we had previously and the probabilistic character of these weak measurements so let's just declare so the probability to measure X on a state psi which I unravel here or which I write in a real-space representation and the limit delta t to zero this born probability is just apply this operator so there's a smooth and delta functions which I can also write as this trace here so this is just a rewriting where the A is again the kind of smear delta function that we're using here and now the measurement outcome is a Gaussian random variable why? because we can do the calculation the expectation value of this so first of all you see it already here it's a kind of Gaussian variable something quadratic in the exponent and more mathematically you can see you can look at what is the expectation value for measuring position X so what's the average position that you will measure then I just compute this X expectation with respect to this born probability distribution and doing the manipulations you see actually this is a very nice and interesting effect that the expectation value for the measurement of X naught coincides with the quantum mechanical expectation value of the operator X it's not totally trivial this statement so the expectation value under this probability distribution here of the measurement outcome X naught precisely is the expectation value of the quantum operator X furthermore so we have for a Gaussian variable we need and the variance I mean if you do a similar calculation here you find it to be determined by this and therefore with this in mind so this is a Gaussian distribution with characterised by expectation value and variance and this is the point where we can transit from this probabilistic formulation to a stochastic formulation of the problem as we did it at least in our minds when passing a deterministic formulation of the probabilistic problem into an explicitly stochastic equation which was this long-term equation same trick we can apply it here we just understand now that X naught is a random variable random Gaussian variable with mean quantum mechanical expectation value according to this calculation and variance which we can model and this variance the variance that scales with the square root of the time increment delta t so this is we pass from a probabilistic to a stochastic formulation of this problem you can also look at then the stochastic update of the way function and if we look at an infinitesimal time step update of the way function is given by this by this object here so here you see explicitly the noise increment so this is really like a fluctuation in a long-term equation stochastic force that acts on now a quantum way function you see here these expectation values around they come just from this normalization that stands downstairs so reducing the operators by the expectation values of this equation this is really nothing but taking into account the normalization of the way function and then you have just this stochastic update okay now let's do the same step as we did yesterday go from a single particle or from a single degree of freedom that we observe now to many degrees of freedom and then it's just a good bookkeeping I will need this delta t later on but I don't see any so you can take this arbitrary small while a commutator relation is bounded by h-bar or something so what's that again spontaneous objective collapse okay let me maybe try to elaborate a bit so you don't get around this you can derive this I mean I now declared it as kind of a limit of of a softened delta function and this is a kind of heuristic way to go you could do this much more systematically but you don't get around the postulate of measurements in quantum mechanics as far as I understand it so and the point is this so you can derive this weak continuous measurements in the following way you take a system you want to observe you immerse this into an ancillar system that you allow to weakly entangle with the system you want to measure and then you projectively measure the ancillary system and then in this way you get only you get full information about the ancillar system but you only get little information about the the system you want to measure actually and that is described here so you don't get a full collapse of the wave function by just measuring the ancillar projectively you get a collapse of the ancillar but it's only weakly entangled with the system and so you get only little information about the system so this is a totally systematic physical way of deriving this equation from let's say at least from the first principle of the measurement postulate in quantum mechanics you can then have this more fundamentally if you but okay let's say this is in part a philosophical discussion in part maybe also some interesting example sorry sorry in which measurement the continuous projective versus projective you mean entanglement generation mm-hmm mm-hmm ah you want to know this connection will come will come will come it's essentially it's a limit the measurements so we average over all the measurement outcomes maybe with you can ask again when it comes and then you can ask if it's enough or you want to have more details okay good yeah because that's an important question I mean somehow if I take it technically what's the relation of this equation with the Lindblad equation so that's actually something we can figure out okay right now we're going from many so and that is really I mean after what we've said all the time it's not so difficult so here I restored actually a Hamiltonian so the infinitesimal generator of dynamics of Hamiltonian dynamics is the Hamiltonian operator himself herself and then we have these measurement contributions here where I put value onto this statement so that and then there's this noise update which really is like a Langevin equation now concept for a quantum okay and let's look at this and let's also make the connection to the collapse of the wave function which you have you hammer your projective measurement like a density and then you know after that hammer the system will be in Eigenstate of this those collapse of the wave function and to see this clearly let's switch off the Hamiltonian again for a second and let just the measurement dynamics govern the system and then you let's think about this noise term down here yeah so if this noise will always be a random force of the problem but there is one fixed where the wave function does not transform under time evolution anymore and that is precisely the case if the expect if Psi is indeed in an Eigenstate in this sense here of the local density operator yeah so if this condition is fulfilled so then although the noise is fluctuating this part will die away and then everything comes to rest so the wave will not evolve anymore under time and that means okay we have a dynamical fixed point or a dark state as it's also sometimes called the dark state of the measurement operators is precisely the Eigenstates of the measurement operators so indeed if the system is locally in an Eigenstate of particle number then this equation here will be obeyed and the dynamics comes to an end yeah and this is really a continuous collapse of the wave function so this is the probability as a for different times to measure x x naught some fixed x naught yeah and you can see here if you start with a short time with this black curve here and it's basically almost degenerate but if you wait for long enough time then you see how the system will find more and more the Eigenstates of these they become order one yeah and everything else dies away so this is a kind of in the upshot we have here really a continuous collapse of the wave function modeled in this week continuous measurement so that's maybe one part to this question one part for the answer we can say more questions for this okay and then what I already said so do we expect now that was the question up here a difference in the universal behavior for example or do we expect the phase transition that is present for a strong projective measurements is not present for week continuous measurement that was a discussion we already led so the details of the measurement process if you observe it very quickly or if you monitor it continuously that should not affect the universal properties of the phase transition in fact was also numerically verified very well so the next point so that I would also like to say a few words about is actually how we can extract information now in the perspective in this perspective on measurements and here we have this statement here so I mean I have a calculation but I think I won't show it in details I will put these little calculations and put them at the end of the file because we appreciate so much the fact that we have a smooth smoothness in the dynamics so then you can formulate easily a path integral because the measurement is something that is really a fundamentally different type of evolution which is not smooth so I mean the projective measurement it takes you from say some state in a path integral because there we want infinitesimal updates over time so we want to write it as e to the something small and this is not possible for projective measurements the theoretician would say this is very different on the microscopic level but I don't care about the on the macroscopic level and actually to be honest I'm maybe it's not totally true what I'm saying so whether there cannot be situations constructed where where the projective measurement gives also different universal physics from the week continuous I don't have a supers I mean except for this RG type argument one thing that speaks in favor of such an argument is the following so we can smoothly interpolate between these two limits weak strong projective and weak continuous limit so this is just taking this delta so these operators that I'm interpolating they are mutually always commuting so for sure we cannot expect the phase transition as the function of this interpolation whether the the universal properties of the phase transition might but I wouldn't know another example in equilibrium where such things matter but I mean the devil might be in the details but that is my best argument on this interpolation interpolating between different sets of commuting operators of no phase transition as a function of that present ah just relaxing okay that's also fair enough good okay so this little calculation is in some script here but let's just see what happens we can alternatively and I want to now make the connection is an update of density matrices so now we have to go from this vector evolution the stochastic vector evolution we have to construct the stochastic density matrix evolution so how do we do that well we just do the vector outer product here and we look at the state update of this product here and you maybe this I can easily give you so what you have to do is you now know d of psi psi and this will be certainly d psi times psi plus d psi and now are we done why not now I'm probably messing up all these vectors the point is so there is this d w thing and we said this scales square root so this was a variance this was not just a guess this was really that's how the variance looks like so we have to expand this equation to second order in del d v to produce terms that are really d t so by this argument so that is why we need here actually to really keep the second order differential change just to account for this scaling with with the time step and then what you produce is this equation here so here is a linear element in d w it comes with this funny anticommutator but if you look at this first term here you really directly recognize oh this is the structure of the Lindblad equation and so where this comes from well this d psi so this is plus and then this reduced measurement operator so some for density and with some strengths gamma halves or so so this is plus d w and so this thing and this all acting on psi so I insert this here this gives me a right action this will give me a left action yeah so this is the part in the Lindblad equation and this guy here this will produce the coupling of the two contours so the two this will give me a left and right action in this structure of this Lindblad almost Lindblad equation here so upshot what you can recognize here it's basically the Lindblad equation however with some additional stochastic fluctuations on top of it but what we just declared is that the expectation w is zero so therefore if you average this equation over realizations of the stochastic noise you will precisely recover the Lindblad equation and the physical interpretation of this is so this delta w is our vehicle to in a stochastic way describe the probabilistic nature of the measurement process so if you average over measurement outcomes of the measurement you can make this even more direct such a connection you can say or you can there's this unraveling of the master equation which so where you write a density matrix evolution of a stochastic as a stochastic evolution of pure state wave functions and these trajectories that come out they have more interpretation of strong measurements that's okay so that's good now we can understand a problem that surface is here in the following way now we have the evolution of this state projector here so this is a projector the row squares to row for this pure state evolution but it's pure state in the presence of noise of this stochastic element now let's look at usual observables I can I want to know the expectation of with this state row and then I have to do now I compute so to say conceptually I would like solve this equation for a given realization of the noise drawn from a Gaussian ensemble and then I would compute the expectation value of this operator by averaging over all the possible realizations of DW so this gives me then this tells me that I want to know actually the averaged row averaged over all noise realizations but there's now an immediate problem that you can see this average here will always look like a unit matrix this is clear why who thinks this is obvious I don't raise my hand so at least I mean but it's very easy to see so this equation here now there is a specific feature of the Linnblad operators being Hermitian themselves which is the case for measurements for measurements this is a special case of a pre Linnblad equation if you like so the Linnbladian contribution to it has it that the Linnblad operators are Hermitian that's not what we had yesterday or today even so we had always non-Hermitian operators for loss particle pumping here we have Hermitian operators that corresponds to measurement of Hermitian observables of Hermitian operators so therefore and now there is a special feature of Linnblad equations with Hermitian operators and that's really very easy to see so even some over L if I insert here something proportional to unit matrix minus observation squared you can see of course this just cancels out exactly and of course also the unit matrix commutes with any Hamilton operator so that is how you immediately see that at least the unit matrix here is a fixed point of the equation a dynamically fixed point and you can also I mean convince yourself I mean that or I think there is no totally precise statement but very generically this is also the only stable fixed point of the evolution when the Hamiltonian and the Linnbladian are sufficiently enough non-commuting with respect to each other so then I mean we take it now that there is a unit matrix solution and the unit matrix solution for the density matrix physically to the minus beta H where beta goes to 0 so this is an infinite temperature state infinite temperature unit matrix state okay very good so with this in mind here we immediately see that any of the standard observables do not work well or detecting anything in phase transition like and the solution is then that one can take non-linear in the state observables quote unquote we are leaving a little bit the standard quantum mechanics where observables are linear in the state expectation values linear in the state in this precise way here but we could allow ourselves to look at state dependent observables and they have this property that a function of the average is no longer the same the non-linear ones as the average of a function and examples of this phase transition it just is a highly non-linear function of the state and we just don't have at least this killer argument that there will be nothing observable in such a quantity but you can also be a little more modest and say okay I want something that is not arbitrary power in the state maybe it's enough to look at something that is quadratic in the state projector so this is average so this is non-linear in the state in that sense okay yeah sorry yeah we also do you can subtract something yeah but I mean the point is that this thing here has a chance of hosting still non-trivial information because it is not linear in the state and it's not efficiency in detecting something non-trivial is not immediately simple as that yeah so here I mean as an example let's just look at I mean just to also illustrate a bit more how this dynamics is running here so we can look here at our model of particles hopping and measured the density being monitored and the Hamiltonian being just hopping so let's operate this setup here on either of these sites so let's just look at how these statements that I made play out really in practice and what this can tell us about this measurement induced phase transition so let's look at the strong monitoring limit before so if you if you have no Hamiltonian at all there so then this dynamics will just be this continuous collapse right and then you see this thing here the particle is basically at all times pinned still pinned to the eigenstates of the measurement operators unless for very short excursion times where it transits so in a little bit like instantons and quantum mechanics so this is almost all the time in one of the eigenstates of the measurement operator but sometimes it's and then these plots here they have the interpretation basically of Rabi oscillations of a problem that you don't measure at all so then you would see it always going back and forth between these two different minima in energy here and this is the opposite limit in the sense that really here the system is always close to the eigenstates of the measurement operators and despite these massive visible differences that you can see easily on a computer but not in an experiment it is invisible in the linear averages these two situations here they will not show in the linear averages so in these guys but for example indeed if you look at such a connected we will see some signal some smooth signal and the idea is then if we go to a thermodynamic limit maybe this smooth signal becomes a sharp signal so it develops a non-analyticity as a function of system size that is how you see phase transitions also in equilibrium problems so that is the physical picture we are looking at such observables and we look whether such a signal here that sees something about the qualitative physics of the problem whether this can become sharp in the thermodynamic limit okay and this is now the model we studied I just leave it I think at this we really look at this hopping here and the measurement as a repetition so this is the Hamiltonian if you leave this alone govern the dynamics it will lead to a volume law growth of entanglement as you've heard yesterday if we just have the measurement operators on no Hamiltonian we get of course a product state so area law entanglement but we have to keep now in mind these nonlinear and state correlators to extract information now how does this look for in this problem here is the the phase diagram for this problem of fermions hopping on a lattice so here we put this parameter is the measurement strength versus the hopping strength and the most important axis is the one in the thermodynamic limit so 1 over L so here this axis here is thermodynamic limit so this is a variable that happens at no measurements at all is actually an unstable point and it's immediately replaced by a logarithmic growth of the entanglement entropy and this is an extended phase for this problem with a logarithmic grow of the entanglement and then at some point we get a sharp cutoff and we transit into an area law so it is also a measurement log law to area law transition okay so here are some details and I think this is an important or interesting figure so that gives a hint what kind of phase transition this might be namely we look here at the coefficient of the logarithmic growth of the entanglement entropy in this phase here and what you see this coefficient here at some point drops really and this is something that might remind some people of costalus phase transition where instead of the coefficient of such a logarithmic term in the entanglement entropy such a behavior is observed for the superfluid stiffness so there is a what is called universal jump of the superfluid stiffness which is exactly happening at the costalus phase transition and that is BKT scaling to observables in this problem one sees oh this is indeed looks like a BKT scaling and maybe in the last five minutes I will give you a flavor on how to how to see this coming out why this should be coming out of this problem and in which in particular also in which degrees of freedom for this problem so to approach this problem we want to work starting from the week we measure at side so around here very weak weak in the sense of gamma over J is a very small parameter in this problem now then you can look at how does the dynamics of say this particle number expectation value look when you look at it on a computer and then you see here actually there is a structure really of left and right moving ballistically propagating fermions in the game and to model this we can go and patch this preon zone for the we have a dominant Hamiltonian now which is just hopping and then we have occasionally these measurements or weak weak amount of measurements in the problem so we can go and patch this preon zone here into many little patches each of which has a linear dispersion and you can see from this picture here that there is actually a dominant velocity and this will be our dominant patch and essentially in the spirit of a low energy theory there would be more words to spend one can reduce this problem now to looking at these patches here one by one and the most dominant one is the red one in the middle of the center of the preon zone but this situation is something that you might know from your model of fermions goes into a description of a continuum Dirac model description so the Dirac model is something pretty simple it is you have two flavors of fermions which in this case are realized by left and right moving particles and there is a matrix sigma z sandwich between these spinners which where the minus sign of motion of these particles left and right and there is just a linear dispersion which basically you read off from these patches here so we linearize around this for every of these patches okay so this is the kind of how one can take a continuum limit for this problem and we end up with this Dirac fermions and then from the fermionic representation of the problem into a bosonic representation by the bosonization dictionary where this free Dirac problem here transforms into a free Lachinger liquid description and these Lachinger variables that occur they are the phase fluctuations and the density fluctuations so this is the bosonized version of this Dirac and of course also our local measurement operators through this bosonization dictionary and then you see actually two terms a local density of fermions every fermion is comes with a left and right mover so there's actually two combinations that show up one is called a current operator in this Dirac theory if you bosonize this by using the bosonization table you get linear so this is density fluctuations and then there is also this contraction here which gives us a nonlinear term nonlinear in this density fluctuations and then one has to think about what's actually the importance of this nonlinear term you can clearly see that there is a competition in this bosonic description these operators of variable phi they don't commute with this actually give rise to a phase transition and the intuition is really so that this bosonic theory describes the long wave lengths hydrodynamic fluctuations of the conserved fermions that are propagating in this problem and what you can also learn from that the problem setting is now a nonlinear cosine nonlinear term okay right and so here is the way so now comes this replica construction so we have to do that because we want to represent correlators which are nonlinear in the state so we can rewrite this correlation function here in particular as a product of traces of row and then average over the noise so we can describe these correlators by introducing replicas for the density matrices like this it's completely analogous to infox space or in introducing operators infox space via tensor products of say single side Hilbert spaces so it's very same idea and also the operators that act on these replicas they get and then you can rewrite nicely this correlator here in this replicated formulation and the beautiful feature about this is that this correlator here that is nonlinear in the original density matrix becomes now linear in the replicated density matrix so what we eventually want to have is an analysis can be done and then you can end up with the evolution equation for this replicated density matrix in the very same spirit as we had it for the single measured wave function and the dynamics looks like that so it is two decoupled this piece here is two decoupled lindblad operators that are not going to have the problem to infinity but then there is also a coupling that comes between these different replicas and that is a very important term that will determine the physics of this problem and of course there is still also the stochastic forcing term but if we take the average of this equation here then this term will go away and it has no counterpart in single replica evolution this replica coupling and for Gaussian theories an interesting insight is that these that we can like the Celtic coordinates we can introduce new coordinates for this problem new fields new operators which are the average coordinate the center of Mars and the relative coordinate and when the equation of motion for for the replicated system into a single equation of motion decoupled a single equation of motion for the average coordinate and another independent equation for the relative coordinate and the upshot of this is that the absolute coordinate only describes heating and the relative was exactly possible due to this coupling of the different replicas to each other and the physics of this phase transition and I think I'm coming to an end so the physics of this phase transition is really encoded in this relative replica fluctuations they look like if you include the non-linearities they look like replicas fluctuations that undergoes a BKT phase transition and that makes peace with the numerical observations that we had previously sorry that was in the end very fast anyways thank you very much for your attention and I happy if you want