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recent experimental work  @xcite in ultracold molecule physics has formed fermionic dipolar molecules of @xmath0k and @xmath1rb in their rovibrational and hyperfine ground state .
quantum degenerate dipolar systems have long been sought after because their long - range interactions makes them interesting for quantum computing applications and for understanding ordered phases that can arise from long range effects .
these systems are both complicated , but tunable , because the long range , anisotropic dipolar interaction can be manipulated with an external electric field  @xcite , and should give rise to rich physics with variety of different ordered phases of matter  @xcite .
the dipolar interaction is attractive if the dipole moments are aligned head to tail , and this can lead to undesirable recombination effects if the molecules have an exothermic reaction possible , as with @xmath2-@xmath3 molecules @xcite .
this can be mitigated by confining molecules to a narrow two - dimensional layer , and setting a strong electric field perpendicular to this layer , so that the dipole moments of molecules are aligned with the external field and always align head - to - head and tail - to - tail . in this case , the inter - molecule interactions are always repulsive .
note that we assume we can create a single isolated plane of dipoles , so there is no possibility to form superfluid pairing between planes , where the interactions are attractive again  @xcite .
we assume the dipolar molecules are loaded onto a single square optical lattice with the molecules fully polarized perpendicular to the plane .
since the @xmath2-@xmath3 molecules can be prepared in their lowest hyperfine nuclear spin state , the internal spin degree of freedom is effectively frozen out , and the molecules can be described as spinless fermions .
previous studies of spinless fermion models  @xcite mostly focused on short range interactions and used the simplification of going to the limit of large spatial dimensions .
these works show a rich phase diagram with charge - density - wave ordering and phase separation .
there also has been some work in two dimensions focused on stripe physics  @xcite .
more recent work , carried out in the context of ultracold dipolar molecules  @xcite , focused primarily on finding new exotic phases ( liquid crystal , smectic , or nematic ) , often achieved by tuning the direction of the external field with respect to the plane of molecules , but has not elucidated the details of the full phase diagram for the density - wave phases .
the paper is organized as follows : first , in sec .
[ sec : formalism ] we introduce the model and describe the possible density - wave orderings .
[ sec : method ] provides the details of the mean - field - theory formalism and the numerical calculations .
the results are presented in sec .
[ sec : results ] , and they are further discussed in sec .
[ sec : discuss ] , which are followed by the conclusions in sec .
[ sec : conclude ] .
we describe the rovibrational ground - state dipolar molecules in their lowest hyperfine state with the following hamiltonian : @xmath4 where @xmath5 is the fermion creation ( annihilation ) operator for a fermionic dipolar molecule at site @xmath6 , @xmath7 is the number operator , @xmath8 is the hopping amplitude between the adjacent sites , and @xmath9 is the long - range dipole interaction , with @xmath10 the position vector for the site @xmath11 . the lattice spacing is taken to be equal to one . with this model
, we have assumed that the hopping occurs only between nearest - neighbor sites , and that the molecules are always in the lowest level of the periodic optical lattice potential ( this second assumption is not always true when the interaction between the molecules is too large ) .
we have also assumed that the form for the dipole interaction is valid even when the distance between molecules is only one lattice spacing .    due to the bipartite nature of the square lattice ,
the hamiltonian can be rewritten in a particle - hole symmetric form and one can restrict to fillings that satisfy @xmath12 .
since there is no spin degree of freedom in this model , it can only support molecule density wave spatial order .
such order is described by its unit cell  a parallelogram spanned by the two fundamental translation vectors for a given density wave order .
the area of the unit cell ( @xmath13 ) gives the number of sites with independent average molecule density which is modulated periodically as the unit cell is tiled across the lattice .
the number of the independent order parameters for this density wave is equal to @xmath14 ( the average density , or filling , is not counted as an order parameter , and is fixed in the calculation ) .
since there is only a finite number of non - equivalent unit cells of a given area on a square lattice , there is only a finite number of possible density wave orders .
we have considered all such orders that have up to four different order parameters ( @xmath15 ) and hence this is a restricted phase diagram restricted to low - period phases only .
the corresponding fundamental translation vectors and the unit cells for the twelve candidate phases are shown in fig .  [
tab : clusters ] .
m1.6 in m1.6 in    [ cols="^,^,^,^",options="header " , ]     & & 0.1 +    another , less direct way to observe spatial order is time - of - flight measurements that use the expansion of the molecular cloud to infer the initial momentum distribution of the molecules .
however , the momentum distribution data do not show prominent differences amongst the different ordered phases , so using this approach to detect the ordering is difficult .
the mft results show a diverse phase diagram with a number of distinct density - wave phases .
however , the mft is an approximate method , which neglects fluctuations and favors ordered states .
this brings to mind the question : _ how many of these density - wave phases are actually present in the exact solution _ ?    in the limit @xmath16 , the hamiltonian essentially describes a lattice gas model and can be solved numerically without approximation using classical monte carlo methods .
preliminary results support the existence of the checkerboard ( 2b ) and 4d phases , although strongly suppressed in temperature and filling , compared to the mft result .
since the mft neglects fluctuations , it is not surprising that the range of ordered phases is contracted in a more precise calculation .
however , some higher - period phases become more prominent in the mft solution for intermediate values of @xmath17 , so their presence can not be entirely ruled out by this @xmath16 study
. a more complete study will be the topic of further research .
also , the phase separation needs to be reviewed using better approximation techniques .
the presence of phase separation is tied to the existence of higher - period phases and the fact that the mft strongly stabilizes these phases at the corresponding commensurate fillings .
thus , phase separation is promoted by the mft just like the corresponding ordered phases are .
it is possible that including even higher order phases ( beyond 4 order parameters ) in a mft study would replace the phase - separated regions with higher - period ordered phases that are commensurate at the corresponding values of filling , yielding a devil s staircase - like structure .
another possible alternative to phase separation is the formation of a wigner crystal like state .
the reentrant behavior of the checkerboard phase at @xmath18 in the mft results is very likely an artifact of this method and should be reexamined using more precise techniques .
it is notable that the mft results do not favor stripes that are aligned with the lattice ( corresponding to phases 2a , 3a , 4a or 5a ) .
we found that these phases are either not stable or they are replaced by other phases with lower free energy . for small @xmath17 , good nesting of the fermi surface appears to be important to lower the energy of the system , while for large @xmath17 , strong repulsive interactions between adjacent sites prevents formation of such striped phases .
we also see no evidence for any nematic or smectic `` liquid - crystal '' phases .
another important question is , _ which of these density - wave phases can be reached and detected in experiment _ ?
the mft results show large entropy per particle when the checkerboard state develops at half filling for large interaction , but unfortunately the entropy is severely overestimated in this study , as shown by preliminary monte - carlo results .
the most easily observable features in experiment are the structure factors and the gap in the density of states at the commensurate values of filling .
these features are most pronounced for the checkerboard phase . in a trap environment
, a large gap at half filling should translate into large spatial regions of checkerboard order if the local density approximation is accurate .
of course , it is not clear if phases that are seen in a homogeneous system are stabilized or destabilized by the trap environment .
this is another question to be addressed in a future study .
we have studied the ordering of ultracold diatomic gases on a two - dimensional square optical lattice . we have formulated a model , based on spinless fermions with long range dipole interactions , and solved it using mean field theory .
we have found a number of possible density - wave orders that occur mostly around the corresponding commensurate values of filling and create gaps in the density of states at these values of the filling .
the phase separation is also often seen between different ordered phases in the mft results .
overall , the checkerboard phase is most stable near half filling and has the highest transition temperature . due to the lack of an energy scale associated with spin exchange , the transition temperatures increase with interaction strength .
different density wave phases show characteristic and distinct structure factors .
we have discussed the validity of these results and the limitations of the mft method used in this study .
this work was supported by a muri grant from air force office of scientific research numbered fa9559 - 09 - 1 - 0617 and from a grant of hpc resources from the arctic region supercomputing center at the university of alaska fairbanks as part of the department of defense high performance computing modernization program .
we also acknowedge support from the mcdevitt endowment fund .
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we model a system of ultracold fermionic dipolar molecules on a two - dimensional square lattice . assuming that the molecules are in their nondegenerate hyperfine ground state , and that the dipole moment is polarized perpendicular to the plane ( as in the recent experiments on @xmath0k-@xmath1rb molecules ) , we approximate these molecules as spinless fermions with long range repulsive dipolar interactions . 
 we use mean field theory to obtain the restricted phase diagram as a function of the filling , the strength of interaction and the temperature . 
 we find a number of ordered density wave phases in the system , as well as phase separation between these phases . 
 a monte carlo analysis shows that the higher - period phases are usually suppressed in the exact solution .