 with normal transport in quantum many-body systems. Today's talk is about how one-dimensional quantum systems can concern these expectations. Now, it would be too simplistic to say that classically anomalous transport can't arise. In fact, there are several well-known mechanisms for how it does arise in classical systems. So, two classic ways to realize anomalous transporter through PDEs, deterministic PDEs. So examples of the fractional diffusion equation, which can be viewed as an appropriate scaling limit of non-Brownian walks. Another example is nonlinear diffusion of a kind that arises from averaging the Navier-Stokes equations. And actually both of these PDEs exhibit anomalous scaling that's neither diffusive nor ballistic. A different way to get anomalous transporters through fluctuation-denominator phenomena, and this is enhanced in low dimensions. So this, the recent understanding of this is kind of phenomenon goes by the name of nonlinear fluctuating hydrodynamics due to Van Buren and Schroen. And the basic idea is captured by a stochastic Burgess equation in one dimension, which exhibits anomalous spacetime scaling. So I'll be talking in more detail about two of these examples later on. Now, the last 10 years have seen an explosion of interest in anomalous transport in 1D quantum systems, motivated both by access to better 1D experiments as well as numerical advances, for example, the DNRG. So a well-known example of sub-diffusion is approaching the MBL transition from the ergodic side. I won't talk about that today, but you can read about it in a recent review article. The focus of today's talk will be superdiffusion. So some recent examples in one-dimensional metals at low temperature, you find a kind of anomalous, nonlinear diffusion of heat with a superdiffusive spreading exponent. In isotopic quantum magnets, you see integrability-protected superdiffusion in the KTZ class. This is characterized by a fixed, universal two-thirds exponent. And a very recent example is an exotic possibility in three-quarters scaling in the EZ-Phoenic 6V model that originates from having an infinite number of quasi-particle flavors. So these are some diverse mechanisms for anomalous transport in 1D quantum systems. In today's talk, I'm going to be focusing on two of the superdiffusive examples, namely nonlinear diffusion in metals and the KTZ physics and spin chain. So I'm going to argue that interacting one-dimensional metals that are thermalizing in all other respects, such as level statistics and charge transport, can exhibit superdiffusion of heat at low temperatures. And unlike some of the integrable examples, this type of superdiffusion actually persists at long times in nonintegral models. And should that will be observable in realistic experiments on quantum wires. And this is an unexpected violation of Fourier's law in a well-studied class of physical systems. So recall that one-dimensional metals are described by perturbed Luttinger liquids. So ideal 1D metals are described by the free Luttinger theory, which has divergent linear response transport coefficients because it's integrable and exhibits ballistic transport. The unperturbed Hamiltonian is just free bosons. By contrast, realistic 1D metals have finite transport coefficients because there are interactions or disorder that lead to scattering and to diffusion. And typical 1D interactions, the most interesting kind are the density wave instabilities, which show up as irrelevant vertex operators. Now a microscopic realization of this kind of physics that's easy to simulate is the XXZ model in a staggered field. So the staggered field breaks integrability and drives you to a Luttinger liquid phase, no matter how small the perturbation, and the low-energy field theory has exactly the form we just described as free bosons with an irrelevant perturbation. And previous work by Hoang Karash and Morne numerically verified that this model is non-integrable and moreover that the charge transport was diffusive with an exponent they could extract that was in agreement with the very old analytical results for this model. So we wanted to try to extend this analysis to the problem of heat transport in these systems. Now unfortunately, the thermal conductivity is not accessible analytically in these models. The exact arguments that give you the power law scaling for the charge conductivity simply don't apply and moreover it seems impossible to access the exponent numerically. Nevertheless, the minimal assumption is that it behaves like a power law. Very naively, you can justify this from Wiedem and Franz scaling. There's no reason why Wiedem and Franz scaling should hold in general, but this power law assumption still has predictive power and that's what we're going to test. And the predictive power arises because it implies that the energy density should satisfy a fast diffusion equation. So this is the type of nonlinear diffusion whose fundamental solutions exhibit super diffusive space time scaling and the fundamental solutions are so-called barren black paddle profiles. So to test this, we simulated the staggered field xxc model with a localized Gaussian wave package initial condition and a handy diagnostic for this transport exponent is simply given by the logarithmic derivatives of the wave packet moment. And a non-trivial prediction of this anomalous diffusion model is power law collapse. In other words, these should converge to the same super diffusive exponent and the constraints on the conductivity exponent actually mean this exponent has to be greater than 2.3. Now, of course, this is a, if you start with a linear response equation, a nonlinear equation and look at its linear response for any nonzero bulk temperature, there will be a crossover time. So the limit of perfect super diffusion is when you have expansion into a ground state. That's just a consistency point. It wasn't really accessible in the numeric, this time scale, but in principle it's there. And when we simulated this for a low bulk temperature, we saw clear super diffusive rather than diffusive spreading of wave packets. So figure one shows the spreading of a Gaussian in time. One A is diffusive scaling, one B is super diffusive scaling and you see that the super diffusive collapse is much better. And moreover, looking at the moment in one C, it's clear that the actual scaling is not neither ballistic nor diffusive, it's something in between. And this is precisely the prediction of the simple nonlinear diffusion model. Furthermore, there seems to be some degree of universality in the long-term scaling of the wave packets. This is what's shown on the right-hand side. Interestingly, the scaling form is not consistent with the simple nonlinear PDE I wrote down earlier. The barren-black-patle profile is monotonic away from the center point, whereas you have this interesting doubly peak structure. And we believe this is because at short time, the wave packet spreads faster than it can thermalize. So it would be interesting to explore in future whether some modifications of GHD or some proximate integrability in the system could explain what we're seeing. So that concludes the first part of my talk and brings me to KPZ and isotropic magnets. Now, this is a phenomenon which is, various signatures of which have been observed for nearly 10 years. And what it amounts to is what you could call integrability-protected KPZ physics in 1D magnets with isotropic symmetry. And this is typically diagnosed through the long-time behavior of the spin-autocorrelation function at finite temperature and zero magnetization. What you see is a two-thirds exponent for integral models and what seemed to be a one-half exponent for nonintegral models, although Jacopo is going to explain what actually happens in non-integral models. And this basic picture has been verified in many numerical studies. I'm just citing a recent selection on this slide as many as would fit. But until recently, this interesting phenomenon was essentially unexplained. So the closest approach to a theoretical explanation of the two-thirds was the self-consistent derivation by Homan Serang, which has since been extended in important ways, that was based on generalized hydrodynamics. So previously, there were three fundamental questions about this phenomenon that hadn't been addressed. But the most glaring one was why the same phenomenon occurs for both quantum and classical systems. The second question hinting at some kind of universality was why are both integrability and isotropic symmetry necessary for this phenomenon to be stable at long times? And maybe worth noting, it's not just isotropic symmetry. It seems that any internal non-Abelian Lie group symmetry is actually sufficient. So there is a universal character to this phenomenon that needs explaining. And finally, why is the collapse to universal KPZ scaling functions observed numerically? Now, it's worth pointing out that evidence for this is much better in the classical case than the quantum case. This could be a matter of time scales or it could be pointing at something deeper. I think it's fair to say that's not clear yet. But the bottom line was that these questions reflected a basic lack of understanding of what was causing this phenomenon. So recently, we have proposed an explanation based on what you can think of as goldstone modes of GHD. They're soft modes of the magnetization that arise in certain local equilibrium states and are missed by standard hydrodynamic approaches. Now, I want to be careful when I say that. What I mean is that a naive application of non-linear fluctuating hydrodynamics as detailed in the review article by Schvon predicts simply diffusive spreading. Similarly, a naive application of GHD with a short wavelength cutoff in place also predicts purely diffusive spreading. So it seems that some ingredient missing from these standard theories and I'm going to argue that this ingredient is precisely these long wavelength coherent excitations of spin. In particular, these modes have a non-linear self-coupling and are separated in scale from these short wavelength hydrodynamic theories which allows for a channel for super diffusive spin transport and the physical mechanism giving rise to this, of these long wavelength excitations coupled to a short wavelength quasi-particle bar is actually very similar to a mechanism which arose in a completely different context and namely 1D both gases. So the figure is a schematic illustration of these long wavelength coherent excitations on the left and the short wavelength fluctuating spins on the short length scale L. So how do these modes arise? Well, it's helpful to consider the simplest integral model with quantum integral model with F2 symmetry that's on a line namely the spin half-Heisenberg Hamiltonian. And as I'm sure everyone in this audience knows Data's Ansatz to solve this model will build eigenstates as a spin wave on a pseudo vacuum. So this is a reference state that's typically a product of highest weight states in your representation. And once you fix this, you forget about it, you build everything on top of it. Now something interesting happens when you have an internal linear symmetry. So say SU2 is in the case of Heisenberg, the direction of a pseudo vacuum inherits this symmetry. I can take an arbitrary direction on the sphere and build my excitations on top of that and I've solved the model. By contrast, when you have an applied field not all of these directions are allowed. Only pseudo vacuum directions parallel to the field yield a set of eigenstates. And this property of the beta ansatz is actually expressed in the TDA. So for example, the average of the magnetization has to lie along the pseudo vacuum. Now what this property means at the level of hydrodynamics is that there's a surprising inversion. So in the sense that if I coarse-grain my system into a bunch of fluid cells and one of them spontaneously forms a local magnetization over the course of some hydrodynamic flow then I've spontaneously broken the SU2 symmetry. In other words, I've picked a pseudo vacuum and I now ought to regard the pseudo vacuum as a dynamical degree of freedom. My personal view is that this vector degree of freedom is still missing from GHD but there ought to be some way, perhaps analogous to the Fermi liquid theory of spin half electron of keeping track of this vector degree of freedom. But failing that, one can write down an effective long wavelength dynamics for the pseudo vacuum. So one should consider states which are homogeneous on the short wavelength quasi-particle scale but exhibit long wavelength fluctuations of this vacuum and their effective Hamiltonian dynamics projected onto the vacuum sector is simply the Landau-Liffchitz equation. So then you can ask, okay, so there's this non-linear degree of freedom apparently missing from GHD, does it give rise to the observed K-P-Z scaling? And when you try to formulate the fluctuating hydrodynamics you find plausibly it does. So since I'm running low on time I'll just sketch the derivation. The key technical point is to eliminate this SU2 invariance, this arbitrariness in parametrizing the spin vector. So a nice trick to do that which I learned about from a paper from the 70s but is a century older, as Jacopo and collaborators pointed out, you can regard this spin as a tangent vector to a fictitious space curve. And the Freonet-Sarray equations for this curve essentially remove the gauge ambiguity in choosing this frame and they give you natural invariant coordinates which are known as the curvature in the torsion. Now there are various ways to argue that the curvature dynamics is relevant. I don't think, I think it's fair to say there's not consensus there but what you're left with is a single stochastic Burgers equation for the torsion. And from this it follows that K-P-Z scaling is natural to expect. So just to comment about why the curvature drops out, it's the, this is soft mode dynamics so the energy could be expected to be negligible. Finally, just to comment on why this should be integrability protected. Well, the key difference lies in the nature of the scalar bath. So the nature of the hydrodynamics at the short length scale. And in non-integral models there are only two scalar modes but through itself. So the variance of the scalar bath scale subsist extensively as I increase my fluid cells. And what this means is the distinction between fluctuating short wavelength modes and coherent long wavelength modes becomes disappeared at long wavelength. So this is a claim that non-linearity becomes irrelevant. As you'll hear in the next talk that the K actually renders the non-linearity marginally irrelevant. But by contrast in integral models this variance doesn't decay. One continues to have fluctuations of order one as the fluid cell is scaled up. And this is one way to see why why integrability protects KPZ physics in this case. So I'd just like to summarize some subsequent developments. This effective picture of a soft gauge mode being responsible for KPZ has been tested reasonably thoroughly at this point. Given a microscopic justification in the sense that solitons are the only thing it's been shown that Landau-Liffschitz solitons are the only thing that could be giving rise to this physics. And successfully applied to the Hubbard model in a paper which did a lot of other things. An exciting goal going forward would be to extend this idea to understand super-universality. Namely the observation of the same physics for higher internal symmetry groups. Very naively, they all contain an SU2 subgroup. So you could say, well, that's what's happening. But it will be good to clarify that further. So with that, thank you for listening. Thank you, Pierre. So we have some time for questions. If you have a question, just unmute yourself. Yeah, go ahead, is it? Wait, no. So recent. Okay. I have a question about this super-universality. I don't know these papers which were cited here, but so what is the claim that the symmetry group does not influence the scaling exponent or what? It appears that you get to two thirds. Yes. In brief, yes. I think it's been numerically checked for examples like SUN and SO5. But it seems that they're KTZ in all cases. So when you think about goldstone modes, then of course, if you have a bigger symmetry group, then you have more directions to make excitation. So I'm a bit surprised by this, that it does not enter at all. So I can speculate on why it may be that the... So, yeah, the Sangram and others gave arguments for why these higher order couplings can be irrelevant. I think something similar might be happening here, but there are these higher modes, but they may be irrelevant in a scaling sense. Okay, thanks. That's all I have to say. I have one question. So you want to mention that each of these higher groups contain an SC2 subgroup. So then essentially what if this picture, behold, you essentially need to somehow get an effective decoupling of the SC2 sectors on this high order? Quite, quite possibly. But it would be nice to make that rigorous and say that this cohomology class really leads to this nonlinear coupling. Can I ask a question? So here, but these exponents, these weird exponents you get, Lothi, could it be that, again, it's a long correction because there is some... No, because your perturbation is relevant, right? So it's not mildly relevant. Or at least that's... Oh, you mean it for the lottinger liquid? No. I think it is dangerously irrelevant. It's an irrelevant perturbation that's changing the class of the dynamics, but I think the mechanism is quite different from what's happening in your case, that the physical picture is just... Yeah, because it's nonlinear diffusion rather than a fluctuation effect that's changing. I still have no way to predict these exponents so far. So one could try... So the problem is there's no exact argument. With the charged conductivity, it's a very rigorous calculation. It's the self-energy. And if you want to look at a thermal conductivity, there's some stuff with the memory matrix, but as far as I know, this is not on the same rigorous footing as these Dyson series type arguments. So I don't know a good way to get that exponent. But maybe a show with a lower bound that is... Okay, it's a hard... Yeah, maybe worth adding that even if you don't know the exponent, it's the divergence that's important. Simply the fact that the diverges at low temperature will give you some kind of super diffusion. So I had a question about the first part of your talk. This is Robert Connick. Hi. So you mentioned the possibility of actually seeing the super diffusion in generic 1D wires. I was wondering, will phonons or how protected is this phenomena against phonons or disorder? Or how do you thought about it? So we think it's reasonably robust to... So for example, even in this simple case of X6E with a staggered field, as the Luttinger liquid, you have several kinds of irrelevant perturbation like fan curvature and even the umclap scattering. We haven't studied in detail what happens with more irrelevant terms that we would expect that as long as there is a dominant irrelevant term, there should be a clear super diffusion. Okay, any other question? Maybe I can ask one. So can you comment a bit on the numerical evidence for KpZ? I mean, they are like, especially in the quantum case, like not all profiles that you get from DMRGs match KpZ especially well in all cases and in particular with, you know, different symmetries. So do you have any thought on this? I mean, it could be very well done. Yeah, so, okay, so maybe... So the classical evidence seems to be reasonably convincing and all of the theory points to the fact that the underlying mechanism is classical. So it's the same between spin half X6Z and classical Landau liftships, namely these long wavelength coherent modes that as you mentioned that the numerical evidence on the quantum side really isn't very convincing for good bond dimensions and the tails don't collapse very well. And it's not yet clear, I think, whether this is a transient on the way to better agreement or whether there's genuinely a different scaling function that comes about from a more careful treatment of the fluctuating hydrodynamics. Yeah, thanks. All right, so if there's no more question, I suggest we thank here.