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although a homogeneous neutron matter is one of the simplest nuclear systems , its importance can not be overemphasized since it constitutes the main component of neutron stars .
the density of the neutron matter forming neutron stars ranges from subnuclear densities ( in the inner crust ) up to extremely high values of @xmath4 ( expected in the center of the star ) , where @xmath5 is the saturation density@xcite .    in the regime of sufficiently low densities
one can perform very precise calculations since the neutron - neutron interaction is completely dominated by the scattering in @xmath6 channel , which is determined by two parameters only : the scattering length @xmath7 and the effective range @xmath8 .
indeed , at the densities @xmath9 ( @xmath10 ) the influence of other channels as well as of three - body forces is marginal and can be neglected@xcite .
the values of the scattering length and the effective range are well known from the low energy scattering experiments and read : @xmath11 , @xmath12@xcite . at the densities where @xmath13 the dilute neutron matter is an example of a strongly correlated system . moreover ,
the influence of the effective range can not be ignored except for very low densities where @xmath14@xcite .
it implies that only non - perturbative approaches are able to gain a reliable insight into physics of this system . the large class of such methods , which are known under the general name of quantum monte carlo ( qmc ) , has been applied for the dilute neutron matter@xcite . in spite of a considerable theoretical effort some of its static properties
are still vaguely known .
the open questions mostly concern the superfluid state , which is generated at sufficiently low temperatures .
for example the predicted value of the zero temperature energy gap differs significantly for various approaches , revealing a strong dependence on the details of the applied method ( see@xcite and references therein ) .
the same concerns the critical temperature of the superfluid - normal phase transition , where the well known bcs formula @xmath15 is believed to be valid .    here
we present the selected results concerning the superfluid state , obtained within the afqmc approach at non - zero temperatures@xcite for the dilute neutron matter of density @xmath16 ( @xmath17 ) .
this study can be regarded as a continuation of the investigations presented in the papers @xcite , where the details of the afqmc algorithm and the procedure of adjusting the interaction , have been discussed . in the present study we focused on two basic quantities which characterize a superfluid state : the critical temperature and the pairing gap .
in the case of superfluidity the convenient order parameter is specified by the long - distance behavior of the two - body density matrix @xmath18 where the field operators @xmath19 and @xmath20 obey the fermionic anticommutation relations .
such a long - distance behavior is directly related to the condensate fraction @xmath21 , which measure the fraction of cooper pairs forming a condensate@xcite .
this quantity can be efficiently computed within afqmc , provided the information concerning the one - body density matrix @xmath22 is also included .
the condensate fraction is given by:@xcite @xmath23 where @xmath24 stands for the number of particles and the correlation functions @xmath25 are defined as : @xmath26     as a function of temperature ( in units of fermi energy @xmath27 ) for three lattice sizes @xmath28 .
the inset presents the rescaled condensate fraction obtained for lattices @xmath29 ( squares ) and @xmath30 ( circles ) .
the intersection of the fitted lines determines the critical temperature of the superfluid - normal phase transition , @xmath31 . ]
the figure [ fig : nm_alpha ] presents the condensate fraction versus the temperature for three different lattice sizes @xmath28 .
it is clearly visible that the results for @xmath32 significantly deviate from those obtained for larger lattices .
it is due to the fact that for small lattices the limiting value of @xmath33 is rather poorly determined . indeed , 
the infinity  in this case is equal to the half of the box size and is only a few times larger than the interaction range .
consequently , it will generate a large systematic error when the lattice is too small .
the presented results indicate that this is exactly the case for @xmath32 and therefore these data were not included in the process of determination of the critical temperature .    in the thermodynamic limit
the condensate fraction has to vanish at the critical temperature @xmath2 .
however , in the case of calculations in the box the finite size effects come into play and smooth out all singularities typical for the phase transition .
still , one can estimate the critical temperature using the method based on the finite size scaling .
similar technique has been used to determine the critical temperature for cold atomic gases ( see@xcite for details ) .
the volume - dependent estimation of the critical temperature @xmath34 is obtained by finding the intersection of the rescaled condensate fraction @xmath35 for two different lattice sizes @xmath36 , where @xmath37 is the universal critical exponent and @xmath38 is the volume of the box . as @xmath39
the thermodynamic limit is recovered and the series @xmath34 converges to the critical temperature . to extract @xmath2
we have used two largest of the available lattices , namely @xmath40 .
the filling factor in both cases reads : @xmath41 . according to our experience ,
the above methodology is sufficient to estimate the critical temperature with an uncertainty smaller than @xmath42 ( in fact this procedure applied to the unitary gas provides the value of @xmath2 with the relative error smaller than @xmath43 ) .
finally , the estimated value of the critical temperature reads @xmath44 ( @xmath45 ) , where @xmath27 is the fermi energy ( see the inset of fig . [
fig : nm_alpha ] ) .
in order to determine the gap in the single particle spectrum , we have computed the spectral weight function @xmath46 . this quantity has been extracted from the imaginary time propagator @xmath47 through the analytic continuation@xcite .
the procedure is equivalent to solving the integral equation : @xmath48 where @xmath49 denotes the inverse of the temperature and @xmath50 is determined from the monte carlo calculations for the discrete set of values @xmath51 .
numerically , however , the above integral equation represents an ill - posed problem i.e. there is an infinite class of solutions for @xmath46 which satisfy eq .
( [ eqn : ap ] ) within uncertainties generated by monte carlo method .
therefore we have used two methods which were in particular designed to deal with such tasks : the truncated svd method and the maximum entropy method@xcite .
for the dilute neutron matter at selected temperatures : a )  @xmath52 , b )  @xmath53 , c )  @xmath54 , d )  @xmath55 .
the nonzero energy gap is present up to the critical temperature @xmath31 . ]
the spectral functions for selected temperatures ( below and above @xmath2 ) are presented in fig .
[ fig : nm_spectral ] .
our results admit the gapped solutions up to the critical temperature and above @xmath2 the gap vanishes .
one has to remember , however , that there is a finite resolution of both methods concerning the value of the gap .
namely , the gap can not be resolved if its value decreases below @xmath56 .
the value of the energy gap @xmath57 extracted from the spectral function at the temperature @xmath58 is expected to provide a good approximation of its value at zero temperature .
it reads : @xmath59 .    .
]    in fig .
[ fig : gap_comparison ] our results are compared to those obtained by other authors within _
ab  initio _
calculations for zero temperature .
we have found our results in agreement with the recent green function monte carlo calculations of gezerlis and carlson@xcite .
the calculations including other scattering channels together with the three body forces were performed by gandolfi _
et al.,_@xcite and predict larger paring gap . however , it is difficult to establish the source of this discrepancy since their method ( auxiliary field diffusion monte carlo ) has been constrained in order to avoid the fermionic sign problem and therefore can be regarded as a variational approach
.    an interesting result can be noticed by considering the ratio of the energy gap at zero temperature to the critical temperature .
namely , the ratio @xmath60 exceeds almost twice the well - known bcs value @xmath61 .
a similar situation has been encountered for the unitary gas , where the existence of the exotic `` pseudogap '' phase above @xmath2 was recently reported@xcite .
it indicates that the dilute neutron matter at this density is not a bcs - type superfluid .
we have performed the fully non - perturbative calculations for the dilute neutron matter of density @xmath62 at finite temperatures .
we focused on the basic quantities which characterize the superfluid state and extracted the value of the critical temperature and the pairing gap .
our results are free from uncontrolled approximations and are essentially exact with only uncertainties related to statistical errors and finite size effects .
the large value of @xmath60 suggests that the dilute neutron gas can not be described by the bcs theory and thus is not a bcs superfluid at low temperatures .
support from the polish ministry of science under contracts no .
n n202 328234 , n n202 128439 and by the unedf scidac collaboration under doe grant de - fc02 - 07er41457 is gratefully acknowledged .
calculations reported here have been performed at the interdisciplinary centre for mathematical and computational modelling ( icm ) at warsaw university .
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we report results of fully non - perturbative calculations , based on auxiliary field quantum monte carlo ( afqmc ) approach , for the dilute neutron matter at the density @xmath0 . 
 fundamental quantities which characterize the superfluid state : the single particle energy gap @xmath1 , and the critical temperature @xmath2 have been determined . 
 the large value of @xmath3 indicates that the system is not a bcs - type superfluid at low temperatures .