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during the last years important efforts have been devoted to study ultra - cold fermi gases , both experimentally and theoretically ( for a review see for example ref . @xcite ) . by means of feshbach resonance it is possible experimentally to adjust the interaction between atoms . the system can be tuned to a bcs state where fermions are weakly interacting , to the so called unitary regime where the two - body scattering length is infinite , or to form a bec condensate of bosons . theoretically , ultra - cold fermi gases have become an important testing ground for a host of many - body methods , as well as an avenue to studying new physics . the study of ultra - cold fermi gases is very intriguing , and the unitary limit became a very interesting many - body problem to solve for several reasons . these systems can be experimentally studied at very low temperatures of the order of 0.1 t@xmath1 , essentially zero temperature . they are very dilute , and their properties are independent of the form of the hamiltonian used to describe the system , the results given by any model satisfying the limits of large scattering length and small effective range must be identical . moreover these systems are very strongly interacting , methods based on perturbation theory , mean - field , or expansions of small parameters are not obviously convergent , and the calculation of the properties of the unitary fermi gas is very challenging . the properties scale with powers of the fermi momentum @xmath2 . the interparticle separation @xmath3 is approximately given by @xmath4 , and the dilute regime is guarantee by imposing @xmath5 , where @xmath6 is the effective range of the two - body interaction . the universal regime is given when @xmath7 , where @xmath8 is the scattering length , and the unitary limit correspond to the case of @xmath9 . in the limit of @xmath10 the only scale of the system is @xmath3 or more conveniently the fermi momentum @xmath11 . thus a natural unit for the energy if given by the fermi - gas energy @xmath12 , and there are no other free parameters . for this reason , these systems are called universal . ultra - cold fermi gases can also be realized with population imbalance , and the system can eventually exhibit different phases  @xcite where superfluid and normal phases may coexist or phase separate  @xcite , or other intriguing more exotic phases like loff or @xmath13-wave superfluidity could appear  @xcite . in the last few years a new direction of research has been undertaken : the realization of trapped two  component fermi gases with the mixture of fermions with different masses , in particular the @xmath14 mixture has been addressed  @xcite . the properties of systems with mass imbalance can be very different from the equal masses case ; for example , for majority light population the chandrasekhar clogston limit is very small and close to zero  @xcite . then , at unitarity , the two  components fermi gas with different masses may exhibit very different properties with respect to the equal mass case  @xcite , even in the bcs limit  @xcite . several predictions for equal mass fermi gases were made possible through quantum monte carlo techniques ( qmc ) , later confirmed by experiments  @xcite . the variational monte carlo ( vmc ) and the diffusion monte carlo ( dmc ) enable accurate calculations of the ground state of strongly interacting fermions at unitarity , and to study its properties  @xcite . in addition , the unbalanced two - component fermi gases with equal masses do not exhibit a sign problem using the auxiliary field monte carlo ( afmc )  @xcite , and this permits a very accurate calculation to benchmark other many - body calculations . experiments are possible at very low temperatures of the order of fractions of @xmath15 , and then the zero temperature properties can be directly obtained from experiments .    in this paper we will review several properties calculated using qmc methods , including the exact calculation of the energy , the role of the effective range , the bcs - bec crossover , and other properties including the contact parameter and the static structure factor . [ sec : ham ]    the afmc performed on a lattice through the use of auxiliary fields does not suffer of the fermion sign problem and permits to calculate exactly the energy of the unpolarized fermi gas  @xcite . the afmc method uses a bcs wave function as a trial wave function . in this section we describe the variational wave function and the dmc method in the continuum . in our study we model the ground state of the system in the continuum using the hamiltonian @xmath16 where @xmath17 and @xmath18 is the number of particles with mass @xmath19 and @xmath20 . since we want to model a system that is dilute , particles interact only in @xmath21-wave , and the potential @xmath22 is non - zero only between particles with opposite spin state or with different mass . several forms of the potential @xmath22 has been explored , and provided to be equivalent in the limit of small effective range  @xcite . in most of our studies we have used the pschl - teller already employed in several previous qmc calculations  @xcite : @xmath23 where @xmath24 is the reduced mass , and the parameters @xmath25 and @xmath26 tune respectively the effective range @xmath27 and the scattering length @xmath8 of the interaction . for this potential , the unitary limit corresponding to the zero - energy ground state between two particles is with @xmath28 and @xmath29 . most of the results presented in this work have been obtained by fixing @xmath30 but in several cases we considered different values to check effects due to the effective range of the potential . it is also possible to use other forms of the potential @xmath31 , and the results are universal in the limit of small effective range as we shall discuss in the next sections . the qmc calculations are performed using the jastrow - bcs form of the wave function of refs . @xcite : @xmath32    the antisymmetric part @xmath33 is a particle - projected bcs wave function including pairing correlations . it is given by @xmath34 \,.\ ] ] the operator @xmath35 is an anti - symmetrization operator , the unprimed coordinates are for spin - up or heavy particles and the primed are for spin - down or light particles , and @xmath36 for the unpolarized case . the pairing function is parametrized as @xmath37 + \beta(r ) \ , , \nonumber\\ \beta(r)&= & \tilde{\beta}(r)+\tilde{\beta}(l - r)-2 \tilde{\beta}(l/2)~ , \nonumber\\ \tilde{\beta } ( r ) & = & [ 1 + c b   r ] \ [ 1 - \exp ( -d b r ) ] \frac{\exp ( - b r ) } { dbr}~.\end{aligned}\ ] ] in the infinite volume limit @xmath38 is a function of one scalar variable , the product @xmath39 . the simulations use periodic boundary conditions in a finite volume . in this finite volume , the function @xmath40 has a range of @xmath41 , @xmath42 is the size of the simulation box , the value of @xmath43 is chosen such that @xmath40 has zero slope at the origin , and @xmath44 , @xmath45 and @xmath46 are variational parameters . note that if @xmath47 , and @xmath48 for @xmath49 , @xmath50 , where the latter is the wave function describing the non interacting fermi gas in the normal phase . as @xmath51 and @xmath33 are orthogonal , we can study the fermi gases both in the superfluid and in the normal phase  @xcite . the pairing wave function used here contains several free parameters that have been optimized by minimizing the energy of the system using vmc and following the strategy of ref . @xcite . in the bec side of the transition , when @xmath52 , we found that using the pairing function as described in ref . @xcite , i.e. the two  body wave function instead of the function of eq . [ eq : pairfn ] , gives lower energies . instead , in the bcs case for @xmath53 , the bcs wave function @xmath54 gives almost the same energy of @xmath51 as the pairing becomes less important to the equation of state . the polarized systems can be simulated by extending the wave function to include single - particle states for the unpaired particles  @xcite . note that the wave function previously described can not reproduce more exotic phases like loff or @xmath13-wave pairing . the jastrow @xmath55 includes a short- and a long - range part . the short - range term is given by @xmath56 where the function @xmath57 acts only between particles with different spin or mass , and it is obtained by solving the equation @xmath58 the parameter lambda is obtained by imposing the boundary condition @xmath59 , where the healing distance @xmath60 is a variational parameter , and the quenching @xmath61 is adjusted to be less than one only in the bec regime . the correct boundary conditions to the wave function are guaranteed by constraining @xmath62 where @xmath42 is the size of the simulation box , but we found that a typical good choice is given by taking @xmath63 . by solving eq . [ eq : jas ] we also assure the correct behavior of the wave function at small distances so that the jastrow function @xmath64 is defined to have @xmath65=0 at the origin . the calculation in the deep bec can be also improved by adding a short - range function @xmath57 between particles with same spin or mass , and solving the equation above with a repulsive interaction .    in order to precisely calculate the static structure function at low momenta , it is important to include the effect of phonons as long - distance correlations . in our calculation , we include a term @xmath66 \,,\ ] ] where @xmath67 and @xmath68 are new variational parameters  @xcite . the vmc algorithm is then used to optimize over wave functions of this form . diffusion monte carlo ( dmc ) is then used to project out the ground state from : @xmath69 where @xmath70 the factor @xmath71 is a constant used to control the normalization of the ground state . the propagation is achieved using a many - body propagator defined as @xmath72 and @xmath73    a good approximation of @xmath74 can be obtained using a trotter expansion , and iterating the above equation for small values of @xmath75 . note that for a zero - range interaction the exact two - body propagator could also be used to sample the many - body green s function  @xcite . in the case of fermions there is a sign problem that needs to be taken care . a common approach combined with the dmc method is the fixed - node approximation . the sampling of paths is restricted to regions where the trial wave function is positive , and the problem is then recasted into a bosonic problem ( without sign problem ) in a restricted subspace . this approximation results in providing an upper bound to the exact energy of the system , whose accuracy depends to the quality of the variational wave function . more details on the implementation of the vmc and dmc algorithms can be found in ref . the bcs - bec crossover has been calculated extensively over the past ten years . the unitary limit is very interesting because perturbative methods can not be applied . in addition , there are no small parameters to expand , and thus using numerical calculations is the only way to accurately explore the properties of the system . the energy of these systems is given by @xmath76 where @xmath77 is called bertsch parameter . there have been several calculations of @xmath77 based on qmc techniques , and recently we used the afmc method to calculate exactly the energy of the system  @xcite , the results are summarized in fig . [ fig : lattice ] . since the lattice space provides a natural regularization of the hamiltonian , it is possible to test different models of kinetic energy or dispersion relations . in the figure , we show the results obtained using different regularization of the kinetic energy . all the results correspond the case of infinite scattering length , and in the case of @xmath78 and @xmath79 , the results are obtained at finite values of effective range and need to be extrapolated . results obtained using different number of particles are included in the figure . we note that finite size effects are basically absent , as using 38 , 48 or 66 fermions yields to the same energy . this fact is because the interaction is very short - range , and because pairing of the system is very strong . particles behave like bosons , and finite size effects associated to the kinetic energy vanish . the absence of sizeable finite size effects for more than @xmath8040 particles has been also shown with a dmc calculation  @xcite . by extrapolating the results to zero effective range gives the result of @xmath81  @xcite that is in excellent agreement with a subsequent experiment performed at mit  @xcite . the best upper bound estimate using the fixed - node dmc method is @xmath82  @xcite .     as a function of the lattice size for various particle numbers @xmath83 and hamiltonians , @xmath61 is the lattice spacing . see ref . @xcite for more details.,scaledwidth=70.0% ]    another quantity of interest is the dependence of the energy to the effective range : @xmath84 at unitarity the parameter @xmath21 has been calculated using dmc and afmc , and the different calculations yield @xmath85 and @xmath86  @xcite . in fig . [ fig : reff ] we show the results obtained using dmc with various potentials for the equal mass case , and for a mass ratio @xmath87 corresponding to the @xmath0 mixture . it is pretty interesting that the value of @xmath21 does not depend to the mass ratio of the two species .     as a function of @xmath88 calculated using dmc for the two - components with equal and unequal masses unitary fermi gas . in the case of equal masses , the results obtained using different two - body potentials are presented . see ref . @xcite for more details.,scaledwidth=70.0% ]    the knowledge of the energy dependence to the effective range is important to compare properties of neutron matter and cold atoms . at low - densities , neutron matter has similar properties to cold atoms : the interaction is mainly @xmath21-wave , the effective range is small , and the scattering length is large  @xcite . in order to make qualitative comparisons between cold atoms and neutron matter it is important to take into account the fact that the neutron - neutron interaction has a small but finite @xmath27 . using the estimate of @xmath21 as a function of @xmath89 opens the possibility to compare these two different systems . an example of this comparison is reported in fig . [ fig : nmat ] . in the figure we compare the cold atoms results corresponding to @xmath10 , the neutron matter equation of state calculated using the @xmath21-wave component of realistic nuclear forces  @xcite , and the results obtained by extrapolating the cold - atoms results to the effective range of neutron - neutron potential using eq . [ eq : reff ] . -wave part of a realistic neutron - neutron potential , and those obtained by extrapolating the cold - atoms results to large effective ranges . see the text for details . this figure is taken from ref . @xcite.,scaledwidth=80.0% ]    in the bcs - bec crossover , the two limits correspond to the bcs where fermions are weakly attractive and to the bec regime with very strong coupling . in the weak coupling limit , the energy of the interacting fermi gas is given by  @xcite @xmath90 where @xmath8 is the scattering length and k@xmath1 is the fermi momentum of the system . in the very strong coupling limit , when @xmath91 , the system becomes made of bosonic molecules , formed by one spin - up or heavy and one spin - down or light fermion , that are weakly repulsive . their energy is given by  @xcite @xmath92 \,,\ ] ] where @xmath93 is the boson - boson scattering length that can be obtained from a few - body calculation  @xcite , or even by fitting qmc results  @xcite . the bcs - bec crossover for a two - component @xmath0 fermi - fermi mixture is shown in fig . [ fig : crossover ] . in the figure , the blue and black points correspond to the qmc results obtained using two different values of @xmath27 in the two - body interaction . in the unitary limit , the energy can be expanded as @xmath94 the fit to qmc results , extrapolated to @xmath10 , gives @xmath95 , @xmath96 and @xmath97 . the values of @xmath98 and @xmath99 are very similar to the case of equal masses reported in ref . @xcite . in fig . [ fig : crossover ] we also show the two limits given by eq . [ eq : bcs ] and  [ eq : bec ] that agree with the qmc results . the crossover corresponding to the @xmath0 mixture is qualitatively similar to the equal masses case  @xcite . the two different set of points correspond to the results obtained using different values of @xmath27 . the dashed lines correspond to the perturbative calculation of weakly attractive fermi gas in the bcs , and of the weakly repulsive bose gas in the bec.,scaledwidth=70.0% ] one of the most intriguing properties of strongly interacting fermi gases is the contact parameter . shina tan showed that several quantities , including the pair distribution function , momentum distribution , and the static structure factor in the limit of small distances or high momenta are fully described by the contact parameter  @xcite . also the equation of state is related to the contact parameter . this particular property is very interesting because the contact can be calculated in different ways , and , more important , can be measured in different experiments ( see for example refs . @xcite ) .    in ref . @xcite several observables related to the contact have been calculated using qmc methods . we note that , apart the fixed - node approximation , any other observable depends to the choice of @xmath100 . within dmc , we have computed observables with @xmath101 where @xmath102 and @xmath103 note that the energy is a special case , because the operator @xmath104 commutes with the propagator @xmath74 , and in this case it s easy to see that @xmath105 . the fact that several operators basically describe the same contact parameter is a proof of tan s relations but also of the very good quality of the trial wave function employed in our calculations . we have calculated the contact parameter in the bcs - bec crossover using the adiabatic relation that relates the contact to the equation of state : @xmath106 the contact parameter is shown in fig . [ fig : contact ] , and compared with recent measurements and with the result obtained by enss _ et al . _ @xcite . in the inset we show the equation of state . the experimental measurements based on the bragg spectroscopy are in a very good agreement with our calculations , both at unitarity and in the bec limit  @xcite .     obtained from the equation of state calculated using dmc ( shown in the inset ) . the experimental points of refs . @xcite and the result of ref . @xcite are also shown.,scaledwidth=70.0% ]    one of the quantities related to the contact is the static structure factor @xmath107 , that corresponds to the sum - rule of the density - response function : @xmath108 the density - response function @xmath109 has been measured by means of the bragg spectroscopy  @xcite . at high momenta , this quantity is related to the contact through the relation @xmath110.\ ] ] at low momenta @xmath107 is related to the long - wavelength phonons in the system . the quantity @xmath107 has been calculated using qmc for different values of @xmath89 in ref . @xcite , and the agreement with experimental data is within 1% . the calculation of @xmath107 at small @xmath111 requires extra care . due to the phonon dispersion , we found it very important to include long - range correlations in the variational wave function as discussed in sec . [ sec : ham ] . in fig [ fig : sq ] we show the vmc results for @xmath107 obtained by including ( or not ) the long range correlations in the jastrow . we note that when the dmc algorithm is also used , the expectation value of @xmath107 is less dependent to the presence of long - range correlations though even in this case the accuracy of the calculated s(q ) is much greater with the improved trial function . it would be very interesting to measure the behavior of @xmath107 at small momenta , where collective modes are expected to be important .    . the results have been obtained with the vmc , with and without long - range correlations in the jastrow . , scaledwidth=70.0% ] in this paper we present recent results obtained using qmc methods for two - component strongly interacting fermi gases . the energy and its dependence on the product of the effective range and fermi momentum have been precisely determined theoretically and compared to experiment across the bcs - bec crossover . other properties related to the contact have also been computed , in particular the static structure factor , and the results are in very good agreement with experimental data . future directions include dynamic response of the unitary fermi gas , inhomogeneous systems and the transitions from three- to two - dimensions , and multi - component fermi gases . 10 url # 1#1urlprefix[2][]#2 giorgini s , pitaevskii l  p and stringari s 2008 _ rev . phys . _ * 80 * 12151274                wille e , spiegelhalder f  m , kerner g , naik d , trenkwalder a , hendl g , schreck f , grimm r , tiecke t  g , walraven j  t  m , kokkelmans s  j  j  m  f , tiesinga e and julienne p  s 2008 _ phys . lett . _ * 100 * 053201
in recent years quantum monte carlo techniques provided to be a valuable tool to study strongly interacting fermi gases at zero temperature . 
 we have used qmc methods to investigate several properties of the two - components fermi gas at unitarity and in the bcs - bec crossover , both with equal and unequal masses corresponding to the @xmath0 fermi mixture . in this paper we present several recent qmc results , including the energy at zero and finite effective range , the contact parameter and the static structure factor , which , at low momentum , depends strongly on the phonons in the unitary fermi gas .