 Thank you. Good afternoon ladies and gentlemen. Let me first take the opportunity to thank the organizers for putting together this very nice workshop and for having me here. So today I'm going to tell you about results we obtained for the two-dimensional Fermigas in a collaboration with Marianne Bauer and Mira Parish back then at Cambridge, as well as another collaboration with Igor Bötcher where we analyzed experimental cold atom data that were taken in the group of Selim Jochem in Heidelberg. So the motivation is as follows. Consider two types of fermions, spin up and spin down, non-relativistic, which are constrained to live in a two-dimensional plane. And there is an attractive interaction between unlike spins that tends to bind them. And what I will tell you about today is mostly concerned with the equation of state. That is the question, how does the density or the pressure depend on chemical potential, temperature and scattering length. And after that we are of course interested to go beyond and study transport and dynamical properties. So the Hamiltonian we are studying contains the kinetic term. And then we are considering the dilute limit where the spacing between particles is much, much bigger than the range of the interaction, say the size of the Lena Jones potential. So the interaction is well approximated by a contact term here, a purely local interaction. So this in the vacuum when you set the chemical potential to zero, the classical action is scale invariant with dynamical exponent 2 and the coupling is dimensionless. And still in the vacuum you can write down the exact beta function which is essentially minus the coupling squared. So the flow diagram is shown here on the right. The flow goes always to the left in the infrared to smaller coupling. So if you start with a repulsive coupling it will become small and the theory will be asymptotically free. While if you start on the attractive side it will flow to stronger attraction. So there is a logarithmically running coupling and it always an energy scale associated with a binding energy that breaks the scale invariance of the classical model. It's an example of a quantum anomaly. So this running coupling can be expressed as a scattering amplitude down here. So in three dimensions you would have a scattering amplitude that in the limit of low momentum or low energy would go towards a constant. In two dimensions that's different. So you have a logarithm of your energy with respect to the scattering length. So it keeps running as you go to lower energy and you see that this scattering amplitude has a bound state when the momentum is i over the scattering length. And this corresponds to the binding energy which goes like 1 over a squared. And if you now consider finite density the typical momentum scale will be set by the Fermi momentum. And then one divided by the logarithm of KFA will be the expansion parameter of perturbation theory. The phase diagram looks something like this. Let's start on the right side where you have a small binding energy and a weakly attractive Fermi gas. And in BCS theory you obtain a critical temperature towards super fluidity that goes like the exponential of the inverse coupling parameter. To the left you go to stronger binding and on the opposite limit you have a very large binding energy. That means the fermions are bound into tightly bound molecules and you have essentially a repulsive dilute gas of bosonic molecules. And Fischer and Thornbach studied the transition temperature to a super fluid phase in this dilute boson gas which goes like 1 over log log of density. And in between this is an artist rendering how these two regimes may be connected. So at high temperature you have a normal phase with exponentially decaying correlations. And at lower temperatures the standard picture is that you have a super fluid below a BKT transition with algebraic correlations. And this has recently been challenged by Jakob Zyk and Metzner who find that if you go to very long distances even the low energy phase of the boson gas will exhibit exponential correlations. But you can hear more about this later in the session by Pavel. So having a dilute gas means the range of the interaction is much smaller than any other length scale in the system. That means the properties of the system are universal and can be expressed in terms of the temperature with respect to Fermi temperature and the dimensionless combination KFA. Now a central quantity in the analysis of this system is the contact density which is essentially the probability to find up and down particles in the same place. And this was introduced by Sina Tan. Actually he put something on the archive in 2005 but people didn't really recognize it because it was also written in a quite unconventional way. And it took three years to publish these papers but then it made a splash and many people convinced themselves by their favorite field theoretic or OPE techniques that this is actually a good way of seeing things. So this contact density here also pops up when you take the derivative of your internal energy with respect to coupling then by the Feynman-Hellmann theorem you will get essentially this contact parameter. So here is a picture for the ground state energy the blue curve and it's normalized by the free Fermi gas. So here on the Fermi side it reaches one but then as you go to the left binding increases the energy becomes smaller and on the molecular side it approaches the two body binding energy which is displayed here is the orange curve. On the right hand side you see the derivative that is the contact which is small on the Fermi side and becomes exponentially large on the Bose side. This contact parameter determines not only the thermodynamics but because it's a short distance property of the gas it determines also the UV limit of correlation functions. And a famous example is given by the momentum distribution N of k which for large momenta goes like contact divided by k to the fourth all the way up to the cutoff set by the range of the interaction. So it extends over a wide window and it's of practical interest because this UV region starts already quite soon. So here is an experiment with cold atoms where they plot k to the fourth times Nk so if it goes to a constant that's the contact and you see it goes towards a constant already at twice the Fermi wave vector. So you reach the UV quite quickly and it's experimentally well accessible. And another famous relation is the energy formula given down here. It means the internal energy can be written as the sum of kinetic energy and the interaction energy which is now just a single number proportional to the constant and the kinetic energy is UV divergent as you see from this integration. It diverges logarithmically which is cancelled by the log lambda in here but the remaining part is just a regular piece for the interaction energy. And this formula down here has actually intrigued many people and they say well if you have a density functional you should be able if someone gives you the number c to just minimize it and find the right momentum distribution but it turns out that it's not so easy. And the value of this contact is I see it a bit like in the Landau Fermi liquid paradigm. If you make some measurement to determine the contact or some calculation you are able to make exact predictions about many other observables in one go. Let's turn to the pressure. If you have a scaling variant gas the pressure is just equal the internal energy however for the interacting gas there is a shift exactly by the contact again. So the contact quantifies the scaling variance breaking and if you now take the blue curve for the ground state energy and add back in the contact you get the pressure which interpolates between 1 and 0. So it's always positive and becomes 0 for the weakly repulsive Bose gas. Now this big binding energy is in a sense a trivial contribution so one might just subtract this two-body binding energy to obtain many body observables. So here this blue curve is now the many body part of the ground state energy which interpolates between 1 and 0 and if you take the derivative of it you get the many body contact which quantifies the local correlations on top of the trivial two-body part. And you see that this is peaked or is maximum in this crossover region between the Fermi and the Bose sides. And these Monte Carlo results then in the asymptotic limits on the Fermi side and the Bose side agree for example with Fermi liquid theory which gives you a certain prediction around the right hand side limit. So now we want to go one step beyond and compute the equation of stated finite temperature and in vacuum the gas is completely described by this repeated particle-particle scattering the T matrix which can be viewed as the propagator for a pair of fermions and then the second step is to compute a fermion ex-health energy which physically corresponds to the process that a fermion scatters off a virtual molecule. So the T matrix can be computed we start with the bare scattering amplitude that you have seen before and add in medium effects basically that these internal lines are fermions that can only excite outside the Fermi C and you see a bound state of the molecule and then a continuum of higher energy states with various branch cuts here and because of this somewhat complicated frequency dependence we choose to keep the whole frequency dependence and not just use a derivative expansion that would focus on the bound state here only. Then we do the second step plug this into compute the fermion self energy and at lower temperatures and stronger binding it is actually advantageous to iterate this procedure and obtain a self consistent solution where we take the self energy and plug it into the internal lines of the self energy of the T matrix again and we do this on a grid of 400 momentum and 400 Matsubara frequency values and then iterate to find a numerical solution and which is then analytically continued to real frequency and you get for instance the spectral function here in the crossover region just above Tc where you see that the spectral weight is reduced near the chemical potential and you see these Bugolubov shadow bands appearing. Then we can integrate over horizontal cuts to get the density of states and you see that there is a substantial suppression of spectral weight already at three times the critical temperature so which we call pseudo gap and we can integrate this further to finally get the number density the thermodynamic quantities and here is the number density normalized by the density of the non interacting Fermi gas as a function of chemical potential for different fixed values of the binding energy. So you see when you come from the high temperature limit the density is renormalized substantially by a big factor and then as you go to even higher densities or even lower temperatures it becomes smaller again and this was first surprising as a bit because if you look at the corresponding case in three dimensions for the unitary Fermi gas you see that the same quantity the density is monotonically going up but it can be understood here because you always have this additional scale of binding energy in the problem. So this is a grand canonical is using grand canonical variables if you go to canonical variables and take the same trace here you start in the high temperature strongly interacting region go to low temperature still a strong interaction but then if you go further you go to lower temperature but weak interaction and this is why the density renormalization is then going down again. And this has been there have been recent experiments with cold atoms and the Zilin Jochen group in Heidelberg and you go by myself analyze these data which took us quite some effort and these are the colorful points here. So this is the same quantity that is plotted here for weak intermediate and strong binding and notice the difference in the vertical scale. So here it's single digit double and here you have renormalization in the hundreds and the solid lines are these Latin Jawad calculations of the equation of state and they fit reasonably well and what you also notice here is that as you go to stronger binding this peak in the interaction is actually a scaling peak. It shifts to the left for larger binding energy and it becomes much higher and it shifts to the left rather by approximately half the binding energy but it grows the height of the peak grows exponentially with the binding energy and this can actually be understood rather easily by looking at the high temperature virial expansion. So if you start here with the virial expansion you have the free Fermigas and then higher virial coefficients that capture the effects of binding and if you're on the Fermi side then all these corrections are small and you just have the usual virial expansion of the free Fermigas so beta mu is the relevant variable in the fugacity. If instead you go to the Bose side then the second virial coefficient is goes like exponential of the binding energy. If you plug this in the first term doesn't count but the second term incorporates the chemical potential of two fermions shifted by the binding energy which is exactly the chemical potential of one boson. So you make you smoothly go into the virial expansion for the Bose gas limit so the chemical potential of the bosons is a good variable that interpolates over the whole crossover from the Fermi to the Bose. And now you it is plausible that if the bosonic chemical potential becomes larger than zero you start to occupy states. So this means for the fermion chemical potential you reach minus the binding energy half and this is actually what is found both in experiment and Latentra Ward and Monte Carlo calculations. There's a small offset here which we observe of roughly 0.7 and the peak height or the density is by the same argument given by the exponential of half the binding energy which again agrees by all different methods that we have available. So let me summarize in the two dimensional attractive Fermi gas the scale invariance is always broken by this binding energy and the scale invariance breaking is quantified by this contact parameter which is at the center of a number of exact universal relations for the unitary Fermi gas. And I would like to stress that this Latentra Ward formalism or the self consistent calculation actually satisfies these exact relations also exactly even if the left hand side of the exact relation is approximate and the right hand side is approximate but their relation is exact. And we find a large renormalization of the density and this density driven crossover. So if you would now translate it back to say an atomic gas in a trapping potential then the outer regions where the density is low behave like a classical gas as you go inside you pass a strongly interacting region and then in the middle where the density is highest it becomes a more weekly interacting quantum degenerate gas. As an outlook we are presently working on quantifying the pseudo gap effect in this crossover and separately we are looking at transport calculations in particular spin diffusion and we're interested in going also into the low temperature phase. And for this it certainly makes a lot of sense to use an RG and we are presently looking into how to parametrize the vertices and in particular the frequency dependence to capture effects like the pseudo gap but still be able to do the analytical continuation and with this I'm looking forward to your questions.