 Thank you very much. Thank you also very much for giving me the opportunity to talk here. I have been asked to give a presentation of my recent work on quantum electrodynamics in 2 plus 1 space-time dimensions and in particular the possibility of spontaneous breaking of Lorentz symmetry in this theory. And this is basically a follow-up work on an old project done with Holger and Jens and Dietrich a couple of years ago. So why it all should one be interested in studying quantum electrodynamics in three space-time dimensions? Well initially this theory was just conceived as a toy model for in fact the theory of quarks quantum chromodynamics in four space-time dimensions. And this is because it's a much simpler theory than QCD on the technical level, but in terms of physics It's very similar in particular. It's it's also asymptotically free once we go to shorter distances to the end to the ultraviolet Then the gauge coupling the fact-defined structure constant goes to small values But in the infrared at larger distances then the gapless fermions interact strongly and in fact they might develop spontaneously Dynamically generated gap to to a chiral symmetry breaking in its compact version also QD3 might show a confinement transition However, it was then later realized that in fact QD3 is much more than just a playground for theories that more than just a toy model it describes real physics real materials and And in particular it has been derived as an effective low-energy theory for the super round taking states in the high-temperature Cooperates but also in very different other aspects of quantum condensed metaphysics this theory kind of pops up For example in in spin systems with just localized spins the elementary excitations being just Yes, I spin degrees bosonic spin degrees of freedom when there is strong frustration say on a triangular or a Kagami type of lettuce then this theory might Develop a spin liquid ground state with fractionalized excitations which turn out to be of fermionic nature and In particular in in certain cases then these fractionalized excitations can be Dirac fermions coupled to you one gauge fit QD3 very recently QD3 theory also popped up in In the fractional quantum Hall effect as in field theory for the Huffield Landau level state and a simultaneous development was to generalize the bosonic particle Bortex identity to fermionic systems with at its heart lying the conjecture that the dual description of Single Dirac fermion as it occurs for example on on the surface of a topological insulator is in fact again QD3 For all these applications the number of fermions that couples to the you engage field differs For example, it's n equals 2 in numbers of in terms of four component fermions in this high temperature corporate Application in order to make use of these of these effective theory with Therefore need to know the ground state as a function of and the number of fermions in fact this ground state t equals zero phase diagram was under debate the last couple of 30 years or so and The kind of common more or less scenario is is that we have two phases There is a lot at large n large number of fermions There is a conformal phase with a conformal fixed point interacting gapless fermions Which in condensed matter physics could be understood as a non-framing liquid however below a certain critical number of fermions the number of which is under debate Fermions are expected to develop a dynamical gap Carol symmetry breaking and in in condensed matter physics. This could be understood as a more insulator In this talk, I'm going to argue in a certain controlled limit of this theory that it's Impossible to have a direct transition from this conformal state to the current symmetry break and broken phase In particular, I would therefore distinguish between two different critical flavor numbers And I call them the conformal critical flavor number below which conformal symmetry the conformal state becomes unstable and the carl critical flavor number below which the carl symmetry breaking occurs Well in particular show that this carl critical flavor number is smaller than the conformal critical flavor number at least in a certain limit of this of this theory There's therefore an intermediate phase and I will argue that most most likely this intermediate phases characterized spontaneous Lorentz symmetry breaking Here's the outline of the talk I will start with epsilon expansion around the lower critical dimension of two space-time dimensions To show that there is a finite conformal critical flavor number for any finite epsilon Below which this conformal state becomes unstable But this conformal critical flavor number becomes arbitrarily large as long as we're going close to this lower critical dimension In the second step, I will then argue using some RG monogenicity arguments that We can find an upper bound for the possibility of carl symmetry breaking which in fact in the in this limit close to the low critical dimension Being just one in terms of four-component Dirac fermions There's therefore a large range of values of n below which Carl symmetry breaking is forbidden, but the conformal state is unstable and finally I use mean field theory again This this this RG monogenicity arguments the f theorem as well as an independent susceptibility analysis to argue that this intermediate phase between Carl and KSB and and conf is governed by a vector order parameter V mu which gets a vacuum expectation value Here's the model Plane QED but now below four space-time dimensions Field strength tens of f coupled to Dirac fermions psi and psi bar via this covariant derivative capital D and this this Derivative involves the gauge charge if you like e and e has mass dimension e squared 4 minus d So that means below four space-time dimensions It's a relevant parameter towards the infrared and therefore goes to larger values of the coupling To the ultraviolet. However, it goes to smaller gauge coupling and therefore QED 3 is is is asymptotically free and super This theory enjoys the carl su2 and symmetry where Correality is in that respect in this three space-time dimensions just a consequent of the reducible Four-dimensional representation of the Clifford algebra. We use four component Dirac fermions Has explicitly the parody symmetry parody anomaly is absent and noture in Simon's term is generated Of course Lauren's symmetry and local you want gauge symmetry is also enjoyed by this theory Once we do our G However, new interactions which are not present in the initial action can be generated by the loop corrections Most of them at least close to the lower critical dimension will be irrelevant just by power counting however local for fermion terms are marginal in two space-time dimensions and Any interacting fixed points in in two plus epsilon dimensions Therefore, we have to take them into account because they can become irrelevant at such an interacting fixed point fortunately the number of of Such four fermion terms is strongly restricted by the large symmetry of our QD three action In fact what we can show is that any four fermion term can be written as a linear combination of these two old acquaintances This is G one which which which is the coupling in front of gross never type of interaction with With gamma three five being the product of gamma three and comma five the two left over gamma matrices which are not present in our kinetic term and G two With which is a tearing type of interaction known from the tearing model in two plus one dimensions the charge Has a particularly simple form for each flow and this is due to the word identity to the gauge gauge symmetry in fact to any loop order that the flow equation for the charge should always be Written in this form D minus four plus eta a times e squared e to a to arbitrary order At least is unknown, but we can compute it to leading order However independent of the order we calculate we therefore see that as long as we're in the perturbative regime It is small we always flow to larger values of e squared and finally the flow only stops if eta a is four minus D So that means that independent of the loop order Exactly eta a must be four minus D at any charged fixed point In fact, if if we now use this this information for the scaling of the photon propagator at such a charged fixed point Then the exponent in in Fourier space of the photon propagator must be two minus D And that means it becomes constant the photon propagator in two dimensions two space-time dimensions This is nothing but the well-known result of the Schringer model a massive photon in two dimensions In three dimensions, we recover also a known result that the photon propagator goes as one over absolute value of q And this is basically this is the fact exactly the result known from first and as well as a second order one over any expansion for this photo propagator and From this argument we now conjecture that's actually true to any order in the one over an expansion However once we now have such a strong charge e squared then new new couplings g1 and g2 will be generated by this box diagram So how does the flow look like once we have now this strong charge? Here is the flow for large in a large and limit infinite and limit Now projected onto the charge e squared and g2 tearing type of short-range coupling plane We have besides the Gaussian fixed point at the at zero coupling three interacting fixed points a Turing fixed point at at a non-vanishing short-range coupling which can be understood as ultraviolet completion of the Turing model and two charged fixed points if now the red line is the RG trajectory of pure q the pure QED action if we start the flow for say small gauge coupling and Initially vanishing short range interaction would end up at this fully attractive fixed point and since we're at large end That's precisely the conformal fixed point the fixed but which relates to the conformal state of the large and QED However, there's also a quantum critical point with one relevant direction and once we lower now this This fermion number n then these two fixed points approach each other in particular if we lower it towards Some conformal critical flavor number then they merge onto a single point and Below this conformal critical flavor number they disappear into the contract complex plane fixed point annihilation Now if we start the RG flow close to this Gaussian fixed point for the pure QED action the red line Then what happens is that we have a runaway flow always towards divergent G2 So the conformal state below this conformal critical flavor number is gone the questions of course What is this instability to? We can compute this conformal critical flavor number within this 2 plus epsilon expansion and what we find is that in fact It becomes arbitrarily large once epsilon is small enough Now we're now going to argue that this instability It's impossible to have chiral symmetry breaking Because the reason is that this instability happens at large and at such large value of and it should be impossible to have Well, as we'll see so many numbers of Goldstone volts Now we'll use RG monosinicity arguments to show that these these arguments are based on the simple observation that defective number of degrees of freedom that play a role in a certain system always decreases under the RG and And a simple example would be the Wilson Fisher o n fixed point where we have n critical modes But on either side of the transition we have less than n critical modes massless modes There are only n minus one massless Goldstone modes in the broken phase and there are even zero massless modes in the symmetric phase Of course to make use of any that that we need to somehow quantify this effective measure of Effective number of degrees of freedom But fortunately this has been done in one plus one dimension By some logic of showing that actually the central charge of the conformal address C Give such a measure of the effective number of degrees of freedom this C in fact He was able to show that C always decreases under the RG and becomes stable at any fixed point Particularly this means that at any ultraviolet fixed point this C must be larger than the C at the infrared fixed point and In a certain sense we can imagine the theory space as a landscape with with hills and and valleys and always Ultraviolet lives kind of for large valleys of C on top of the mountains and infrared fixed point live in the valleys And the flow always needs to be from from large valleys of C to small valleys of C Flow RG flow in a certain sense always goes downhill with respect to C central charge This theorem has recently been generalized to higher dimension Following a proposal by Cardi in three-plus one dimension the a normally coefficient so-called are Cardi's a theorem But only recently has been rigorously proven in four dimensions but then it Now is also known in three dimensions where a similar quantity which Yeah, gives this effective number of degrees of freedom is f So called sphere free energy which is obtained when the theory theory the conformal field theory is Conformally mapped to a three-sphere embedded in four four spacetime dimensions so very recently Between these three seemingly independent quantities Connection has has been possible to made by introducing this quantity f tilde the so-called generalized Sphere free energy f this generalized f f tilde in fact could we're able when we're able to show that this generalized f becomes Precisely a constant times the central charge in two dimensions It is exactly the sphere free energy f itself in three dimensions and is proportional again to the a normally coefficients in four dimensions We therefore have for any integer dimension that this f tilde fulfills RG monotonicity And it's that's a that's rigorous statement f tilde in any integer dimension at an ultra valid fixed point must be larger Then it's value at an infrared fixed point F tilde is is a quantity which is continuous in the spacetime dimension D therefore just from Analyticity we would expect that this holds this RG monotonicity also holds and continuous dimension not only an integer dimension There's no rigorous proof for this statement yet But there's various evidence in various different types of systems often to high-order napsal expansion that this f tilde fulfills RG monotonicity in any dimension also in fractional dimension In fact this f tilde has an plays an important role in quantum information theory There one wants to know the the so-called entanglement entropy which is a measure of how the states are entangled in a particular system this entanglement entropy is is defined by by dividing a system into a subsystem a and its complement a bar and computing the for Neumann entropy of the reduced so-called reduced density matrix which is obtained by tracing just over the states of the outer system a bar In fact one can compute this entanglement entropy for for different simple types of system for example This is a very simplest example where both a and a bar are just two level states Then if the whole system is in a product state say both spins pointing upwards That's a non entangled state then it turns out that this entanglement entropy is in fact is in exactly zero however, if if we're in a non product state in this EPR state for example Then entanglement entropy is non zero in fact one can show that Entanglement entropy scales linearly with the number of entangled states between a and a bar important question is how does Entanglement entropy scale with the size of this subsystem with a Say with a radius r for trivially gap state. There's there's a Yeah, well-known law in this field the so-called area law that the entanglement entropy always should scale in three dimensions and three spatial dimensions as r squared and in two plus one dimensions in our case It should scale linearly. It should scale proportionally to r however for for non-trivial conformal for example phases This area law is violated in particular in two plus one dimensions There one can show that there's a constant shift with with this constant usually called gamma It was recently shown that this gamma is in fact nothing but the sphere free energy and similar relations also hold in one plus one and three plus one dimensions we therefore have that these Universal coefficients and the scaling of this entanglement entropy are again monotonous quantities under renormalization group a full fulfill RG monotonicity arguments and these quantities Should be larger at any ultraviolet fixed point then their values at the infrared fixed point I'm now going to assume that this theorem not only holds in integer dimensions But also into in two plus that's on dimensions non-integer dimensions if we assume so then we we can make Strong statements about possible RG phases. Sorry infrared phases We can compute for example the f tilde at the ultraviolet fixed point and not too surprisingly It's proportional to n the number of gapless fermionic states and in this theorem theory For the other possible candidate infrared phases for example the conformal state we in fact find Find that it's also goes linearly within the number of Gapless fermions, but there's a constant shift and this constant shift is negative such that in fact F uv is always larger than f con and this is reassuring because then our proposed flow from this ultraviolet Gaussian fixed point to the conformal QED fee a QD 3 fixed point is in fact Perfectly consistent with this generalized theorem However, we already know that this this Conformal fixed point annihilates and the conformal state becomes unstable and the question is is it possible then to have chiral symmetry breaking? chiral symmetry breaking with the breaking pattern u2n going to u n times u n would have two n squared massless goldstone nodes and therefore f tilde also goes with n squared the number of goldstone nodes and Therefore for large and it's impossible that f uv is larger than f chi sb and according to this f Theorem it's impossible to have a flow from the ultraviolet fixed point to such a chiral symmetry breaking infrared phase if n is large In fact, we find that it's always larger f chi sb is always larger than f uv if n is larger than one in terms of four component iraq fermions one four component iraq fermions therefore Assuming this generalized f theorem is strict upper bound for the possibility of chiral symmetry breaking close to the lower critical dimension However, we knew Know already that the conformal critical flavor number is very large close to the lower critical dimension and therefore it's impossible Near and below this lower this conformal critical flavor number that that this this phase This instability is towards chiral symmetry breaking The phase below this conformal critical fiber number cannot exhibit chiral symmetry breaking that must be a novel intermediate phase At least it's not yeah, it's not chirally breaking So what can it be? There's a beautiful theorem by wafa and witten showing that q d3 should have first an unbroken u n times u n symmetry at least And therefore u to n going to u n times u n is the only possible chiral breaking pattern No other breaking patterns with say less than than n 2 n squared massless goldstone modes is forbidden by this wafa-witten theorem Secondly the wafa-witten theorem also tells us that the spectrum in their fact of q d3 should have should be gapless Should be have a gapless spectrum And this rules out plain parity symmetry breaking by which the the fermions would acquire A mass a gap, but there wouldn't be any goldstone modes and the spectrum would not no longer be gapless So what else can it be Well insight into the problem can be gained by again looking at the flow When we look at the flow we see within this epsilon expansion. It is always towards diverging g2 But both e squared the charge at g1 this crossover type of interaction remain finite So effectively on a mean field level this this theory in the infrared looks as if there were gapless fermions Dirac fermions coupled via g2 type of tearing type of interaction On a mean field level this theory however it can be solved What we find is it's a free energy for a vector order parameter v mu Which relates to the to the u1 current So this what is this is going? Trying to tell us is that for large g2 This vac vector order parameter composite vector order parameter acquires a vacuum expectation value However, if a vector order parameter acquires vacuum expectation value This means it selects a direction in two plus one dimensional space time and therefore it spontaneously breaks the Lorentz invariance If for example v mu has a temporal component from mu equals zero then in a condensed matter system This this uh such a state could experiment and reveal it safe by a spontaneous Formation of some type of charge order and it's therefore perfectly consistent both with Waffa Witten theorem as well as with the f theorem In fact, this this conclusion can independently on also obtained by by an Susceptibility analysis now again controlled in two plus epsilon dimensions We add to the action small infinitesimally small symmetry breaking terms seeds magnetic fields if you like of chiral Yeah, we break chiral symmetry parody symmetry some type of cacally order or Lorentz symmetry And then we compute the flow by these diagrams to leading order in in in these small Masses and get this exponents x which are related by the usual scaling form of free energy to the susceptibility exponents gamma for these different types of orders In fact, what we find at this quantum critical point that there is a unique order which Which turns out to have a positive susceptibility exponent gamma and this gamma is is is exactly one It's the gamma which is related to the Lorentz symmetry breaking delta mu mass or order parametration This again corroborates our conclusion In two plus epsilon dimensions now that this instability is towards the spontaneous breaking of the Lorentz symmetry Now all that I've said so far was controlled only in the limit of small epsilon Questions, of course, what happens in the physical case of d equals three space-time dimensions That's a difficult problem because we're at finite and now and and and d equals three Beyond above the lower critical dimension Strong coupling and it's difficult to answer However, by the f theorem we can at least find this More or less strict upper bound for the possibility of carol symmetry breaking again And what by comparing these f quantities and what you may now find is that in d equals three A rigorous upper bound would be that carol symmetry breaking is not allowed to occur above 4.4 in terms of four component ferments To make a prediction for the conformal critical flavor number Is a little bit harder, but using different methods ranging from our old work Using a functional rg as well as just naïve extrapolating these two plus epsilon results And and and and as well as a one-loop Expansion a fixed dimension we get values between four and ten so as long as this carol Critical carol critical flavor number does not exactly saturate the this upper bound and this conformal critical flavor number Does not saturate these say lower bound of four Then there may be an even in d in d equals c in the physical dimension an intermediate range of values Where carol symmetry breaking is not yet allowed, but the conformal state is unstable But of course to nail on in particular this number higher order calculations are necessary So here's the phase diagram in in d spacetime dimension d and fermion number n plane Most of what i've said is Is related to this upper left corner and there we can find a fully controlled approximation An expansion in in two plus epsilon dimensions And there we can compute this conformal critical flavor number and can show explicitly via the susceptibility analysis That the the instability is towards lower symmetry breaking in fact This is the only instability which is consistent with both wafa witten and the f theorem In d equals three this conformal critical flavor number becomes smaller and the possibility Of the upper bound for the carol critical flavor number becomes larger And so we don't do not yet know what quite happens in d equals three, but at least from these from these approaches We suggest that Maybe also this intermediate phase also the physical dimension So let me conclude Three-dimensional qed has a conformal ground state if the number of fermions n is larger than a certain conformal critical flavor number And in two plus epsilon dimension this conformal critical flavor number has a particular value But it becomes very large once we're going to the lower critical dimension Because this conformal critical flavor number is so large the common scenario that this instability is below this This Conformal state is towards carol symmetry breaking is inconsistent with the generalized f theorem And this is definitely to exist as long as we're in within the ups and expansion The reason is that where there are just too many goldstone modes for this carol symmetry breaking to occur The only possibility that is consistent Both with wafa witten as well as with the f theorem is a conformal critical Below this conformal critical flavor number is that spontaneous lower symmetry breaking occurs And in fact it's precisely this conclusion that one's led to by an independent mean field as well as a susceptibility analysis Which confirmed this existence of the law and symmetry breaking phase In d equals three We we do not yet know much, but at least the extrapolation as well as a Proturbative expansion in in fixed d equals three suggests that there is a finite window of law and symmetry breaking phase also in the physical dimension Thank you very much for your attention