 So I'd like to, let's see, this didn't quite work. Yeah, so thank you very much for the organizers to give me the opportunity to speak at this very nice conference. The title of my talk is Locality in Long Range Interacting AMO Systems. This is work that I did as a postdoc mostly when I was at NIST and the Joint Quantum Institute with some co-workers also at the Joint Center for Quantum Information at the University of Maryland. And my three collaborators on this project are all shown up here and I'll also tell you a little bit, even though I'm a theorist, about some experimental work that's been done at JQI along very similar lines, kind of studying the same ideas done by Chris Monroe's group down here. So a perfectly good alternative title to my talk that I didn't use because I was afraid nobody would come is mathematically rigorous bounds on the growth of entanglement in non-relativistic long range interacting quantum lattice systems or quantum spin systems. Now hopefully you'll see as I go along that the talk is not as technical as the subject matter seems and I hope you'll see that there's actually a lot of very simple physics kind of encoded in those words. What I'm going to do is to first give you a little bit of background and motivation. I'll tell you what I mean by locality in non-relativistic quantum systems and in particular I'll tell you what people already know in the case when interactions between particles are short range. So this is a relatively old story. I'll tell you then about the relevance of these kind of ideas to ongoing AMO experiments and then I'll move on to tell you part of the talk that's probably the most interesting for this conference, namely what happens to this story when you include long range interactions and how it gets modified. So I have to admit from the outset I'm not a mathematical physicist. So my introduction to mathematical physics is pretty basic and it's really just to give you some context for sort of what the corner of that field that's working on these kind of quantum problems is like. And I'll do that with just a simple example of the mathematical physics of Bose-Einstein condensation. So probably everybody here is familiar with at least sort of the physicist's perspective on Bose-Einstein condensation and I think that can be summed up something like this. So a long time ago, 100 years ago almost, Bose-Einstein proposed that for a non-interacting gas identical Bose-Einstein particles should condense at low temperature into the lowest energy state. Now it was thought that maybe this was just a pathology of non-interacting particles but it was shown with some amount of rigor by many people including the pioneering work of Boglubov just a couple of decades later that actually this phenomenon survived weak interactions so at least it was reasonably well understood for weakly interacting gases and by the end of the 20th century people had even created these kind of things in the laboratory. So I think the physicist's perspective on BEC is that it's more or less understood what BEC is, how it works and what its relation is to the physics of Bose fluids such as its role in superfluidity. Now I think this misses a very important perspective on the problem and that's the perspective of the mathematical physics community and a very important player and this is Elliot Leib and I hope there's at least a few big Lebowski fans in the audience who can recognize who this is but so he actually gave a talk in 1964 on BEC at the University of Colorado where BEC was made decades later and fortunately there's actually video of this talk it was put together into a movie by the Cohen brothers and I think on behalf of the mathematical physics community he kind of summed up the attitude of the physicist, you know, toward the physicist's perspective on BEC in the following way Yeah, well, you know, that's just like your opinion So I think, I mean, actually for a American English speaker who came to Italy I'm very kind of self-conscious because everybody thinks that Italian is one of the most beautiful languages and American English maybe not so much but I think this is kind of good evidence that actually it can be quite beautiful Now, okay, this is actually Elliot Leib he's maybe even slightly more eloquent than the dude so here's what he had to say about it he said, to be sure there have been attempts in the past to calculate the thermodynamic properties of liquid and solid helium but in view of the fact that we're still not overconfident about even low-lying excited states such calculations can hardly be the final word and here at the end, my favorite, humility therefore requires that we acknowledge that our mathematics is still some distance behind physical reality and hasty calculations and surmise cannot make it otherwise so if not something you'd probably hear from most physicists My only point here is that when it comes to the quantum mechanics of many body systems there's really not a whole lot that we know with the kind of mathematical rigor that this man is sort of famous for and other things we do know, he's actually responsible for relatively kind of a shockingly large percentage of them okay, so he's one of sort of the two heroes of this story along with somebody he worked with, Derek Robinson and so they did the pioneering work on, you probably can guess the name, Leib Robinson Bounce and basically what they showed is that there are speed limits to how fast entanglement can spread in quantum mechanical systems this was back in the 70s and before I go into any details, I just want to emphasize that really in some sense what they showed it has a very simple intuitive picture, it's kind of natural to assume that it's true and it comes about already in classical systems so if you take just a material and you ask, you know, if I hit it at some point in space and time and I want to know how long does it take for a signal to propagate to some other point in space and time well, you know, of course there's the glib answer well it's not going to propagate fast from the speed of light, but this is not a very good answer because we know in materials there's a speed of sound and the key idea that gives rise to a speed of sound is basically that you have, when you look inside a material some kind of local interactions between the particles and those interactions occur on some kind of bounded natural energy scale so if you take the energy scale for two particles to talk to each other you take the length scale separating them you can combine this to form a velocity and this is, you know, up to sort of many details that are swept under the rug, the speed of sound and now you can think of this as sort of an effective light cone of course with a much reduced velocity and it's not an exact light cone in the sense that signals can propagate outside of it but for sort of, most purposes serves pretty well as a light cone now, in light of that picture what Lieben Robinson really did is they set out to answer the question of whether you can have a speed of sound or the equivalent of a speed of sound for entanglement and all throughout this talk I'm talking strictly about non-relativistic systems so where there isn't strictly speaking real locality or causality is however you want to call it okay, again before going into more details I want to give you a little bit of motivation that these kind of ideas are interesting that they're not purely interesting within the realm of, A, just kind of academic interest but also even within academic interest they have many applications outside of just this topic so one very kind of obvious application of such a bound is that it constrains your ability to send quantum information in a quantum computation kind of framework so if you have two points connected by some intermediate nodes that are all interacting locally these kind of bounds will set a limit on how fast you could do something like distribute entanglement between these two distant places somewhat more subtly but also a very beautiful connection is that these kind of bounds even control the equilibrium physics that's possible in sort of many different types of many body quantum systems and the argument goes something like this, I'll just give you one simple example so take the quantum ising model so I guess this is sort of like the quantum HMF model it's like one of the most canonical quantum models that you could have and the physics here is that of the competition between spins wanting to align with each other to form a ferromagnetic state and a transverse field that's trying to polarize them in the transverse direction and so if you plot phase diagram now as a function of this coupling so that's down here, what you find is that for small couplings the spins just align with this field we say this state is not magnetized because it doesn't spontaneously break any symmetry of the Hamiltonian and then as you turn up the field there is a second order phase transition at which the magnetization suddenly turns on and you get a state that spontaneously breaks the inversion symmetry on the z axis of this Hamiltonian either all pointing up or down and if you also look at the excitation gap between the ground state of this model and the first excited state that gap is generally finite throughout the phase diagram with the sole exception of the critical point right here so now imagine you want to create some kind of highly correlated ground state that's not sitting right here where this state is trivial or way out here where this state is trivial somewhere in between you could do that by preparing the system over here and now turning on your interactions and you would try to do it adiabatically and your ability to turn them on adiabatically this is guaranteed by the existence of a gap so if you now turn up the interactions you'll find that entanglement grows in the system but you know the time that you can do this in adiabatically by the adiabatic theorem scales like one over this gap and if you have a speed limit on how entanglement can spread that's going to tell you that the entanglement in your system is restricted to distance scales that are less than this quantity so this is a much more general feature of many body systems indeed if you take any kind of lattice system as long as you can connect adiabatically the ground state of the model to a trivial product state then by turning on interactions in a time t you'll find that the ground state develops correlations only over a length scale l that's just t times the lee robinson velocity the velocity that bounds entanglement growth now there are pretty profound consequences of this kind of thing so for one if you want to understand the classical complexity of simulating a many body spin system like now getting into a little bit dangerous grounds because unless I put lots of qualifiers in not everything I say will be strictly true but it's often the case that the computational complexity of simulating such a ground state of such a system classically scales exponentially with the entanglement entropy of a reduced density matrix of one of the subsystems so for example if I draw a line and cut off subsystem A from B then the entanglement entropy of the reduced density matrix of A this is really just sort of you know counting the bits of kind of classical uncertainty that you have about the state of A if you don't measure B and that all comes from these entangled bonds that cross this interface so it's proportional to the number of bonds that break this interface and that tells you right away that the entanglement entropy scales not with the volume of the system as you might expect for an entropy but actually just with the boundary area of subsystem A so this is what people call the area law and in 1D so people have actually used Matt Hastings in 2007 he used the Libra Robinson bound to prove that for gapped 1D systems for reasons related to this kind of adiabatic argument I gave but not exactly the same it turns out that the area law holds and the area law in 1D just says that the boundary of a subsystem is independent of the system size it's just a single point at the boundary of a 1D system and this has very deep connections to why we know in many cases how to efficiently simulate one-dimensional quantum systems classically okay so with all of that motivation let me now jump into describing in a little bit more detail what these Libra Robinson bounds actually are and what they say in the case of short range interactions so for simplicity they're quite a bit more general than this but let's just focus on lattice spin models so a bunch of spins lined up it could be in any dimension these could be spin one half or one any spin you want and I want you to keep in mind for what comes later this is really not just some abstraction in AMO we have many systems that to a good degree of approximation realize quantum Hamiltonians on such kind of spin models and you've already seen examples from yesterday and earlier in which you can do this with trapped ions or Rydberg atoms many different things so the kind of Hamiltonian you might consider is say something like two-body interactions so this is just the most generic Hamiltonian you could write with two-body spin-spin interactions and these coupling constants we're just going to constrain them to be you can take your pick either exponentially decaying or nearest neighbor or finite range we'll call all of those things short range they all count as short range for this purpose and that's the case that Libra Robinson treated so how do you understand locality in such a system well a very natural way to define it is to think about what would happen if you perturb the system locally so imagine you couple some operator A to a spin over here at time zero and position R equals zero and then you wait some time later and you want to measure the expectation value of an operator sitting out here so at a distance R away at a time T later so you know from very sort of standard many-body theory that this is defined by a causal response function and so basically this is just saying that the causal response is this function Q sub R so by first I take my state and I perturb it by evolving it under this operator A at time zero just a little bit and then I have this exacting on the bra and then at a time later B is evolving and the Heisenberg picture under this Hamiltonian I measure B and I subtract off what I would have measured if I hadn't perturbed the system so that's the causal response and you can see just after one line of algebra that this gives you a commutator between the operator A at time zero and the operator B at time T so it's really this commutator that captures the sort of causality or locality structure of the theory now it's not very convenient to deal with operators so it turns out that probably the most natural thing that you could try to bound is not the commutator itself but the operator norm of it so it doesn't matter if you don't know what this is it's basically just the largest eigenvalue of this operator but the important point is that because it's the largest eigenvalue it bounds from above all possible expectation values of this operator so it's a good measure of how big the operator is and in relativistic field theory you may be familiar with the idea that it's exactly this commutator that the theory is sort of constructed to ensure is equal to zero at space like separation so it's the identical vanishing of this outside of the light cone that you sort of get in a relativistic theory now in non relativistic theories it's in fact not going to vanish at any point in space in general after a finite time of evolution but you still expect given the qualitative arguments that gave you earlier that it could decay in space when you're sufficiently far away at a finite time and so what Lieben Robinson did is to bound this norm by some function that depends on distance and time and let me just give you well sorry so let me just say one final kind of technical piece is that from a function like this the way that this kind of defines the light cone is very simple you just say well let me insist that causality is almost enforced so this bound is less than some small number and this equation right here will define an inequality in the t and r plane for where you can have influences propagating so this defines a causal region so this is the analog of the light cone for a non relativistic system okay so the way Lieben and Robinson actually did this is as follows I'm just going to kind of sketch it because I don't want to go into all of the hairy details but they basically showed that you can write this kind of a quantity as a series that has a very natural physical interpretation so it's a power series in time and at every order in time you have some coefficient that depends on your distance in space and what these coefficients are is you can think of it kind of like a path integral they're basically you know if this is the operator a and this is the operator b and they're sitting a distance r apart the nth coefficient is just obtained by summing all the different paths that you can take to get from a to b where this path is created by using links and these links here are just the matrix elements of the Hamiltonian so for a nearest neighbor Hamiltonian all of the paths would just go from one lattice site to the next so it turns out actually all of the hard work is in writing down this expression once you have it you can do this kind of a sum or at least you can bound this kind of a sum in just a few lines of algebra and what you find is Lieb and Robinson seminal result so this function for short range interactions is bounded by this functional form e to the vt minus r and v here is the Lieb Robinson velocity and generally this is just related to the energy scale associated with the short range spin-spin coupling in the model so if you take this function and do what I said you set it equal to a small number what you find is that this defines a linear light cone so very much like a relativistic system if you look at a fixed time at a cut through space what you find is that Lieb Robinson bound well it doesn't actually tell you very much useful inside the light cone so don't worry about what's going on in here but outside the light cone it tells you that you know for a fact that this quantity is going to decay as an exponential in space outside of the light cone okay so I want to move on now to describing sort of or giving a little bit of motivation for why these kinds of things are nice to study in AMO systems and then I'll talk about the generalization to long range interactions so I have a slide here that I feel a little bit guilty giving people have given many many beautiful you know talks about AMO physics but I think there's one thing that the AMO physics community often doesn't express to people that are outside the community and that's what we mean when we say that AMO systems are very cold and so if you compare for example an AMO system to a condensed matter system you can do a very simple calculation of the relevant energy scales by just thinking well let me think an AMO system is basically an atom moving around typically on length scales of a wavelength of light that defines a kinetic energy which is sorry actually I guess I did it in the other order but defines a kinetic energy that's like 10 to the minus 8 Kelvin if you put an electron in a lattice sort of a unicellular material you'd find a comparable energy the Fermi energy that's on the order of 10 to the 4 Kelvin so the relevant energy scales for cold atoms are like about 10 to the minus 12 colder than for real materials so you know people often say that ultra cold atoms can simulate the physics of something like a high temperature superconductor by putting atoms in a lattice and while that's true you should keep in mind if you make them 10 to the 12 times colder and if you look at what typical experimental temperatures are they're actually quite a bit hotter than what you need so the sort of state of the art in cooling say fermionic gases leaves us with metal with cold atom systems that are analogous to metals roughly in the sort of 1000 to 10,000 degree temperature range so they're really not that cold in this very important sense so what that means is that equilibrium is actually very hard to achieve at the kind of temperatures that people would like to see in cold atoms and the reason is simple it's just that to get to very low temperatures you have to wait very long times or you have to cool for very long times or find very powerful cooling methods and these things are always going to be susceptible to all of the dirt that people don't like to talk about that of course exists in AMO systems they're not perfectly clean and on some time scales that becomes relevant now that's not to say that things are hopeless I think there's a huge amount of beautiful work trying to push those temperatures down to get there and there's a lot of beautiful equilibrium physics to be seen I just want to emphasize for people that are outside the AMO community that it's really in many ways more natural to study non-equilibrium problems in cold atoms and I think that's why you've seen so many examples over the last week of AMO physicists talking about non-equilibrium physics using everything from Rydberg atoms to ions and the real benefit is that these low energy scales that make equilibrium physics hard they get traded in just by dimensional analysis those low time scales that make it very easy to observe on say millisecond or even longer time scales the dynamics that's going on in these systems and indeed I guess several years ago so there was a very beautiful experiment from Emmanuel Bloch's group in which he showed that you can take ultra-cold atoms, put them in an optical lattice so these atoms have short range interactions and you can kick the system and you can watch the growth of correlations or basically of entanglement in this system and you can see that it obeys a sort of Lee Robinson style light cone so very beautiful experiments what I'm really interested in in this talk is the additional knob that cold atoms give you they let you go beyond that picture and that is the ability to control the range of the interactions and that lets us kind of get out of the regime where the results of these guys already tell us what to expect and into much newer territory where it's less clear what should go on just to sort of give you a quick picture to jog your memory of what people have already talked about there's many AMO systems where you can have different interactions like 1 over R cubed or 1 over R to the 6 interactions of Rydberg atoms 1 over R cubed interactions of polar molecules you can even vary the interaction range in trapped ions and in cavity QED you can have all to all interactions very naturally so there have been in about a year ago there were two very nice papers one of which Christian Rus already talked about studying what happens to this kind of light cone physics when you take a system with long range interactions and disturb it I'm going to talk about the experiment that went on at the Joint Quantum Institute because I know about it much better and it hasn't already been talked about so this is an experiment from Chris Monroe's group and I'll give you the theorists kind of you know morally correct but wrong in lots of the details and the reminder of what's going on in these kind of experiments and how they replicate spin models so for Chris Monroe he uses as his spin degree of freedom he takes a bunch of uterbium ions they're trapped in a linear trap and they crystallize into this kind of pattern the spin degree of freedom for the spin model is just two different hyperfine states of this atom the way you get spin models qualitatively it's very simple if you shine a laser on an ion I imagine you fix all the other ones you can understand its response by thinking well it's just sort of in a harmonic potential formed by the springs around it and for technical reasons in their case they have to create this excitation with two lasers through an optically excited state but the important point is they can generate transitions that impart momentum so they kick this thing in the harmonic oscillator and at the same time they flip the spin so you go from down to up and you jump up with one quantum motion now if you shine a laser onto all the systems at the same time you get terms like this summed over the entire lattice and if you add now the propagation of the phonons due to the phonon-phonon coupling in the lattice you get a picture like this so it's very similar to QED where you get an interaction mediated by virtual particles so you can have spin flips that are mediated by virtual photons being sent back and forth now in exactly that way but with many more details to worry about in reality they can create Hamiltonians that look like this so they can create Ising models with or without a transverse field they can also create XY spin models so you can think of this as sort of like a flip-flop model where excitations are exchanged between different ions and for all of these models they can control the range of the interactions so they fall off like one over R to the alpha where alpha can go from 0 to 3 in principle so the experiment looks kind of like this so let's see they take their system at the initial time and they just optically pump all of the atoms into one spin state so they initialize it in some state that they know very well and then they apply an effective field in this spin space so they just rotate it into some non-equilibrium state after that rotation during this time they turn on their many-body Hamiltonian this say the XY model in the example that I'll show you that induces correlations to form between the spins so if some of the correlations have formed for some time T they then apply one final rotation and you can think of this rotation passively as just kind of rotating the axis in spin space along which they end up imaging so this final rotation just lets them read out along any direction in spin space that they want and in that way they can measure really basically any correlation function that you might write down it's numerically hard to measure or it's experimentally hard to measure very high-order correlation functions but this kind of connected correlation function is relatively straightforward and they see something like this when they do that so here's an example of their data where time is now plotted on this axis the correlation between this was an 11 ion chain so this is the ion on the left and the right side of the chain and these two different curves represent two different values of alpha that they chose and you see basically what you'd expect so for alpha relatively small this black curve the correlations turn on relatively quickly in time slightly bigger so for slightly shorter range of directions it takes some time for the correlations to actually develop across the chain and if you do exactly this experiment but now not just between the first and the last ion but between all different ions you can make a plot like this so now this whole plot is just a single cut in time and now we're doing this again at every position R and you can see how these correlations develop in space and time and you can see what the boundary is that controls where the correlations can spread to so clearly from these plots there does not appear to be any sort of good notion of a linear like home nor should there be, you shouldn't expect one to be because the Lieb-Robinson theorem only applies in its original form to systems with short range interactions and these systems by their definition of short range interactions are not short range for any power law so in the future I think while those experiments were relatively small scale and maybe don't tell you a lot that you can't learn by just doing simulations on a computer people will move eventually towards larger systems so 1D systems can be made larger just by making cryogenic which I get to say just because I'm a theorist of course that's very challenging and also people study 2D systems of trapped ions that are already working with sort of hundreds of ions so these are relatively large systems for which classical computations are pretty intractable okay so let me now tell you what we can say about Lieb-Robinson bounds in the situation where interactions are long range so remember for short range we had this linear light cone bounding a causal region for long range interactions and long range here I'm going to define as a power law and it just has to be true that alpha is greater than D so any alpha greater than D I'm going to call all of that long range because it's not captured in here but it is captured by work in 2005 by Hastings and Coma and what they showed was that you can generalize the Lieb-Robinson bound and that's still a light cone by the definition I gave you but it has a very strange shape it doesn't look anything like the linear light cone it actually grows logarithmically in distance and if you think about it it's really kind of strange for two reasons one a logarithm means that the causal region is basically everywhere a log grows kind of as slow as anything can grow you can also think of this as implying that the maximum group velocity is kind of growing exponentially in time another strange feature is that this power lock exponent alpha sits out here you can take alpha to be as large as you want and it doesn't change the fact that this situation looks nothing like this situation it's still a log no matter how big alpha is now in the case when alpha is less than D it's also been shown by a nice paper in 2013 by people including one of our esteemed organizers here that the causal region just doesn't exist so you can construct models explicitly for alpha less than the dimension of space where there just is no well-defined causal region now even if we restrict our attention to alpha greater than D there would be some pretty strange consequences if the Hastings-Koma bound was well I don't want to say this right it's of course right, I mean they derived it and it's a mathematical result but if it's a good bound so if real physical systems could actually saturate it that would have very strange consequences but if you could distribute entanglement in a quantum system in a time that scales just with the logarithm of the size of the system it also kind of implies that entanglement in ground states of power law interacting systems even for very large power laws can have very long range entanglement and violate the area law which is not something that we generally see happen so the question that I want to ask in the remainder of the talk is can there actually be a logarithmic light cone and to understand the answer to that question let's look a little bit in detail at how this calculation goes and it's very similar to the original Lee Robinson calculation it again is related to this kind of a series the only difference really is that in these paths that you sum over these bonds no longer have to be nearest neighbor they can be any length you want and they should just be assigned a weight that's related to the size of the interaction matrix element at that distance so if you analyze this kind of a series carefully what you find is that because you pay so much for these long distance kind of very small bonds it's dominated by terms where you have one bond going almost the entire distance r which is about the size 1 over r to the alpha and all of the rest of the bonds kind of stay close to the starting point and because they stay close to the starting point you might see that you could understand the sum of all of the paths going from here to here just via a short range Lee Robinson bound because these are all comprised of kind of short-ranged interaction matrix elements and so they should give you a contribution that's like this e to the vt minus r only with r set equal to 0 because we're putting this point almost back near the origin so if you just multiply these two contributions out you get something that looks almost exactly like what Hastings and Coma derived a bound that goes like e to the vt over r to the alpha and it's setting this bound equal to a small constant to get the lycone that gives you this logarithmic lycone sorry there should be an alpha sitting out here so to understand what kind of goes what goes wrong in this derivation or what's missed by this derivation about the physics you should take a look at this kind of approximation in here or not approximation but this way of bounding the part of the path going from here to this point so we bounded this by a Lee Robinson bound a short-range Lee Robinson bound but if you notice the short-range Lee Robinson bound is e to the vt minus r it just doesn't make a lot of sense to apply it when your distance is within the lycone right it tells you very useful information out here that outside the lycone boundary this function to k is exponentially but inside the lycone we know that this bound was supposed to be a bound on the commutator of some spin operators right so they're just kind of operators with norm of order one so we can really impose a trivial bound that just says inside the lycone the commutator that we're after just can't be larger than something of order one so the problem is it's difficult technically to actually take that into account in reproducing these kind of bounds but the trick to doing it was sort of worked out in this paper from 2015 by me and some coworkers at JQI and so what you do is you have to separate the long-distance physics from the short-range physics in a specific way so I'm just going to define now I feed out this cutoff in distance chi and this separates kind of the long-range tail of my Hamiltonian from some finite range so I'll call it a short-range part now the way that we take into account the fact that the contributions of the short-range Hamiltonian as they cause the system to evolve within its short-range lycone have this uniterity property where they just can't make the operators get any bigger right the operator this commutator that we're after bounding always stays the most of order one the way to build that structure into the theory from the outset is to immediately go into the interaction picture with the respect to this part of the Hamiltonian so we build the short-range Hamiltonian into the theory in a way that explicitly preserves its uniterity and so here formally if you have thought about this kind of stuff before you'll know what's being written here but it's just saying that the operators in the interaction picture they just evolve but the time evolution operator is now given by this long-range Hamiltonian piece that is also itself evolving under the short-range Hamiltonian and if you now plot kind of the kind of plot that I showed you before but now driving the bound in the interaction picture you're now expanding in this Leib Robinson series this time evolution operator and the terms that show up and it look a lot like they did for the case previously with one exception or two exceptions so one is that each of these bonds is basically one of these long-range Hamiltonians and the end points get kind of smeared out because they're being evolved under the short-range Hamiltonian so these operators leak out a little bit but they're constrained by a short-range Leib Robinson bound so they're very localized or they're exponentially localized you can think of them as almost being point-like and now the point is by making chi large in a suitable way we can keep this long-range part of the Hamiltonian relatively small we can just push this out until the integrated weight here is small at least if alpha is sufficiently large compared to D and so what we do is in this way we sort of operate a perturbative regime in this long-range Hamiltonian where we can't have very many bonds and we get away from this problem where lots of them pile up at short distance and cause this exponential time dependence that you saw in the Hastings and Coma bond so all of that said what you end up with at the end of the day if you do the calculation carefully and you also optimize over the location where you put this cut-off is you find a bound that looks like well a bound that gives rise to a causal region that looks like this so t is no longer a log of r but it can be shown to always go as a polynomial of r at least if alpha is larger than twice the dimensionality of space so a couple of power laws are shown here and the nice feature is that as you make alpha larger if you plot this power law exponent beta as a function of alpha it only makes sense for alpha greater than 2D as I said and it converges to 1 out for large alpha so as alpha becomes large you now recover this original Lieben-Robbins picture where you have a linear like going as you expect to so again this this rules out things like the ability to transfer information you know in just log r time in a quantum system it tells you it can only be a polynomial of r it also turns out that you can use this kind of improved bound to show that in power law interacting systems at least if they're gapped their correlations will decay as a power law which is something that people I think broadly expect and see in a lot of numerics but I think this is the only way I know of at least to get a rigorous proof that that should be the case for all quantum systems that satisfy these conditions the gap ground state so just as an outlook before I finish you can kind of divide now the state of what's known into a few different categories along this axis so for alpha less than D as I told you there are explicit examples where you don't have a causal region so we know that's true well for alpha between D and 2D we don't really know what's going on I apologize for this it should have been alpha equals 2D for alpha greater than 2D we know that there is a Lycone that's a polynomial so it's a polynomial in particular that becomes increasingly linear as you make the interactions shorter range and then you might expect although we don't know how to prove that for alpha sufficiently large beyond some kind of critical value you actually get back to truly short range character of the problem and you don't just pick up this linear Lycone asymptotically for large alpha but you get it exactly beyond some threshold I think this is sort of where some of our research is going but there's a lot left to be done to show this kind of a thing one of the nice results recently by one of my collaborators at JQI was just to kind of abandon this approach of driving rigorous bounds and show that at the very least this kind of picture is what you see if you look at the dynamics of non-relativistic quantum field theory so you do find that in general beyond some threshold you get a strictly linear Lycone for power law interactions and let me just wrap up by saying we're hiring postdocs so if you have any interest in living in Washington DC you should email me down here and I don't blame you if you want to wait until next year and see who your next door neighbors might be thanks for your time