document: the presence of impurities usually deeply modify the nature of the spectrum of a quantum system , and thus its coherence and transport properties . in the absence of interactions , if the impurity distribution is completely random , all states of the spectrum are exponentially localized in dimensions one ( 1d ) and two ( 2d ) , while a mobility edge exists in dimensions three ( 3d)@xcite . 
   if the impurity positions are correlated , as for instance if it exists a minimum distance between the impurities @xcite , some delocalized states can appear in the spectrum . 
   this was demonstrated in 1d in the context of the random dimer model ( rdm ) and of the dual random dimer model ( drdm ) @xcite . in 1d 
   , the effects of correlated disorder was studied in different physical contexts ( see for instance @xcite ) . in 2d , 
   the effect of correlations is almost unexplored , except for the case of a speckle potential @xcite , and for the case of pseudo-2d random dimer lattices with separable dimensions @xcite . 
   correlations in speckle potentials may mimic the presence of a mobility edge @xcite , but in the thermodynamic limit all states are localized @xcite . 
   random dimers introduce a set of delocalized states in pseudo-2d lattices @xcite as in 1d @xcite . from a statistical point of view 
   , the main difference between these two models is the decay of the correlation function that is algebraic for the first and exponential for the second . 
   this `` short - range '' feature of the random dimer model is at the basis of the delocalization mechanism .    in interacting systems , 
   the presence of disordered impurities gives rise to a remarkable richness of phenomena . 
   for instance , the condensate and the superfluid fraction are modified by the presence of the disorder @xcite , and this can shift the onset of superfluidity @xcite , and , on lattice systems , can induce exotic phases such as the bose glass @xcite .    in this work we study the effect of a short - range correlated disorder on a bose gas confined on a 2d square lattice . 
   first we introduce a 2d generalization of the drdm ( 2d - drdm ) . in such a model 
   , impurities can not be first neighbours and each impurity also modifies the hopping with its nearest neighbor sites . using a decimation and renormalization procedure @xcite , we show that , in the non - interacting regime , it exists a resonance energy at which the structured impurity is transparent and the states around this energy are delocalized . 
   it is remarkable that this resonance energy does not depend on the system dimensionality and it is the same as the drdm in 1d  @xcite . 
   then , we consider the case of a weakly interacting bose gas confined on such a potential . within a gutzwiller approach , 
   we show that the effect of the 2d - drdm is to drive the homogeneity of the ground state . 
   the disorder induces a non - monotonic behavior of the condensate spatial delocalization and of the condensate fraction as a function of the disorder strength , and enhances both in correspondence of the resonance energy of 2d - drdm single - particle hamiltonian . 
   we show that the dependence of such quantities on the interaction strength can be explained by including the effect of the healing length in the resonance condition discussion . 
   the manuscript is organized as follows . in sec . 
   [ sec : model ] , we introduce the 2d - drdm potential and we demonstrate its single - particle delocalization properties in the region of the spectrum around the resonance energy . 
   the effect of such a potential on a weakly - interacting bose gas is studied in sec . 
   [ sec : results ] , where we also introduce a suitable inverse participation ratio for our many - body system and study it for the case of the 2d - drdm potential and for an uncorrelated random disorder . 
   moreover , we compute the density distribution and the condensate fraction as functions of the disorder strength . our concluding remarks in sec . 
   [ sec : concl ] complete this work . 
   we consider the tight - binding single - particle hamiltonian @xmath0 where @xmath1 are the on - site energies , @xmath2 the first neighbor hopping terms , @xmath3 the number of sites and @xmath4 denotes the sum over first neighbor sites . 
   we focus on a 2d square lattice of linear dimension @xmath5 ( @xmath6 lattice sites ) , and compare the ordered lattice with @xmath7 and @xmath8 @xmath9 @xmath10 , as schematized in fig . 
   [ fig1](a ) with a lattice where we introduce an impurity at the site 0 , @xmath11 that modifies the hopping terms involving this site , @xmath12 [ fig . 
   [ fig1](b ) ] .     schematic representation of ( a ) the unperturbed hamiltonian ; ( b ) the hamiltonian in the presence of a single impurity ; ( c ) the effective hamiltonian after decimation of the site 0 in the hamiltonian ( a ) ; ( d ) the effective hamiltonian after decimation of the site 0 in the hamiltonian ( b ) . ]      with the aim of understanding the effect of the impurity , we consider the green s function @xmath13 projected on the subspace @xmath14 , including all sites except the site @xmath15 with coordinates @xmath16 . 
   using a decimation and renormalization technique @xcite , it can be shown that @xmath17 with @xmath18 & \phantom{bla}{\rm site\;of\;the\;site\;}0\\[1 mm ] h_{aa}&\phantom{bla}{\rm elsewhere } \end{array}\right.\ ] ] where @xmath19 . the effective hamiltonian for the unperturbed case in fig . [ fig1 ] ( a ) , 
   is schematically illustrated in fig . 
   [ fig1 ] ( c ) ; whereas the effective hamiltonian for the case with a single impurity in fig . 
   [ fig1 ] ( b ) , is illustrated in fig.[fig1 ] ( d ) . 
   the subspace @xmath14 does not `` feel '' the presence of the impurity if @xmath20 ( @xmath21 ) remains the same in the absence or in the presence of the impurity , namely if @xmath22 the condition ( [ condition ] ) is satisfied if @xmath23 . 
   if @xmath24 is an allowed energy of the system , namely if @xmath25 , at @xmath26 the impurity will not affect the eigenstate at this energy ( in the subspace @xmath14 ) .    if we add other impurities in the system , as the one in fig . 
   [ fig1 ] ( b ) , with the supplementary condition that on - site impurities can not occupy first neighbor sites ( fig . 
   [ fig2 ] ) , one can repeat the same argument as above , properly redefining the subspace @xmath14 , and one obtains exactly the same condition ( [ condition ] ) imposing that _ all _ the @xmath27 impurities do not perturb the system ( the subspace @xmath14 ) . 
   thus at @xmath26 , the impurities are transparent as in the 1d drdm  @xcite . indeed , with this procedure , we are defining a 2d - drdm , where at each `` isolated '' impurity correspond a structure of 4 hopping terms forming a cross , as shown in fig . 
   let us remark that this definition of the model provides the same condition ( [ condition ] ) independently from the dimensionality of the system @xcite . 
   however our model is fully 2d and the hamiltonian can not be mapped onto two 1d drdm as opposed to ref . 
   @xcite . 
   schematic representation of the 2d drdm . ]    with the aim of analyzing the localization properties of this model , we consider the inverse participation ratio ( ipr ) @xmath28 the symbol @xmath29 denotes the average over different disorder configurations , and @xmath30 the wavefunction on site @xmath31 and at energy @xmath32 . if @xmath33 is an eigenvalue of the system and @xmath34 is an extended state , then @xmath35 decreases as a function of @xmath5 . on the other side , 
   if @xmath36 is a localized state , then @xmath35 does not depend on @xmath5 ( if @xmath5 is larger than the localization length ) . in fig . 
   [ fig3 ] we show the behavior of @xmath37 and @xmath38 ( column left and right respectively ) , for the hamiltonian illustrated in fig . 
   [ fig2 ] .    [ cols="^,^ " , ]     we consider three set of parameters , ( i ) @xmath39 and @xmath40 , ( ii ) @xmath41 and @xmath42 , and ( iii ) @xmath43 and @xmath44 that give the same resonance energy , @xmath45 . in all the three cases , the curves @xmath46 collapse around @xmath26 meaning that the states are delocalized in this energy region . 
   moreover , due to the large strength of the disorder , the spectrum varies considerably for the cases ( ii ) and ( iii ) , and an energy gap appears in ( iii ) .    the inverse participation ratio , eq . 
   ( [ eq : ipr ] ) , in two dimensions has the following asymptotic behavior  @xcite @xmath47 thus , the asymptotic behavior of the function @xmath48 is @xmath49 with @xmath50 for localized states , and @xmath51 for extended states . in fig . 
   [ fig : esponenti ] we have analyzed the exponent @xmath52 as a function of the energy for the set of parameters ( iii ) . 
   we observe a high - energy band of localized states that has been created by the disorder ; the original ( without noise ) band has been distorted , and the states at its boundaries are localized . 
   the center of the band , around @xmath24 , is mainly constituted of extended states . 
   the width of the feature around @xmath24 corresponds to the width of the resonance dip of the inverse participation ratio at this energy value ( fig . 
   [ fig3 ] ) .    ) as a function of the energy for @xmath43 and @xmath44 . 
   the exponent has been obtained using calculations for lattice - linear dimensions @xmath53 averaged over 20 realizations , the error bars correspond to the standard deviation of the fit of the data to eq . 
   ( [ eq : asympatty ] ) . 
   the vertical dashed line indicates @xmath24 . ] 
   these results confirm that our 2d extension of the drdm introduced by dunlap and collaborators in ref . 
   @xcite for 1d systems introduces a set of delocalized states even at higher dimensions . 
   we now consider the case of weakly interacting bosons in the presence of the potential defined in sec . 
   [ sec : model ] . 
   this system is described by the bose - hubbard hamiltonian in the grand canonical ensemble @xmath54 where @xmath55 is the creation operator defined at the lattice site @xmath31 , @xmath56 , @xmath57 the interparticle on - site interaction strength , and @xmath58 denotes the chemical potential fixing the average number of bosons . 
   we use a gutzwiller approach to find the ground state wavefunction for a given set of parameters and average number of particles . 
   the gutzwiller ansatz is given by the site product wavefunction in the occupation number representation @xmath59 where @xmath60 are the probability amplitudes of finding @xmath61 particles on site @xmath31 . 
   the ansatz provides an interpolating approximation correctly describing both the bose - condensed and mott - insulating phases for low and high @xmath57 , respectively , in dimensions larger than one . 
   in addition , the approximation becomes exact for all @xmath57 in the limit of infinite dimensions @xcite . 
   we minimize the average energy given by hamiltonian ( [ eq : bhgw ] ) as a function of the set of amplitudes @xmath60 with the normalization and average number of particle constraint for at least 30 disorder realizations for each set of parameters . 
   the minimization is done using standard conjugate - gradient and/or broyden - fisher techniques @xcite which provides reasonable performance for moderate lattice sizes .      to quantify the extent of delocalization of the ground state @xmath62 in the interacting regime 
   , we decompose it onto the localized basis @xmath63 , @xmath64 , representing the distribution of a homogeneous condensate with average density @xmath65 on the lattice @xcite . 
   we define the many - body ground - state ipr @xmath66 with respect to this basis as @xmath67 @xmath66 measures the homogeneity of the ground state in the condensation regime : the smaller @xmath66 the more spatially delocalized is the condensate .    in fig . 
   [ fig : igs_l20 ] we show the behavior of @xmath66 as a function of @xmath68 , by fixing @xmath69 , @xmath70 and @xmath71 , for several values of @xmath72 . 
   ( color online ) @xmath73 as a function of @xmath74 for @xmath69 , @xmath70 and @xmath71 particles per site . 
   the different curves correspond to different values of @xmath72 as indicated in the figure . 
   the filled symbols correspond to the 2d - drdm potential and the empty symbols correspond to the un - rand potential . ] 
   we compare the case of @xmath75 of correlated impurities @xmath27 with the one of the same percentage of uncorrelated impurities , where there is no restriction for the position distribution of the on - site impurities @xmath68 and no correlations between them and the additional hopping @xmath72 ( un - rand ) . 
   we note that due to the correlations present in the 2d - drdm the maximum percentage of allowed impurities is @xmath76 ( in this limit the system would be an ordered checkerboard ) . 
   we can observe that , in the case of the 2d - drdm potential , @xmath66 has a minimum as a function of @xmath68 , whose position depends on the value of @xmath72 . 
   this non - monotonic behavior is a signature of the resonance induced by the correlations of the disordered potential . 
   indeed , it disappears for the case of the un - rand potential and for large values of @xmath72 ( strong disorder ) . 
   the dip in the @xmath66 for the un - rand potential and weak disorder ( @xmath40 ) indicates that some drdm impurities may still statistically appear , in the absence of correlations . 
   the effect of such impurities is not fully destroyed by the other defects only if the strength of the disorder is weak .      in the perturbative regime for negligible interactions , one would expect that correlations modify the ground state if @xmath77 , @xmath78 being the ground - state energy per particle , which corresponds to @xmath79 in the weak disorder regime . 
   this condition , that can be written @xmath80 , \label{cond - pat}\ ] ] determines the location of the minimum of @xmath66 at @xmath81 for @xmath40 , @xmath82 for @xmath42 , and @xmath83 for @xmath44 . 
   however , in the limit of strong disorder , due to the interactions these values strongly differ from those shown in fig . 
   [ fig : igs_l20 ] . 
   in fact , we calculate @xmath66 for smaller values of @xmath65 and verify that the minimum location of @xmath66 depends on @xmath24 and that the shift observed is indeed an effect of the interactions . 
   the results are illustrated in fig . 
   [ fig : igs_l20_n ] , where we focus on the case @xmath42 . 
   ( color online ) @xmath73 as a function of @xmath74 for @xmath69 , @xmath84 and @xmath42 . 
   the different curves correspond to different values of the average density @xmath65 as indicated in the figure . 
   all the curves correspond to the 2d - drdm potential . 
   the vertical dashed line indicates the non - interacting resonance condition given in eq . 
   ( [ cond - pat ] ) . ]    by decreasing the value of @xmath65 , the minimum position @xmath85 of @xmath66 shifts from @xmath86 to about @xmath87 as expected by the perturbative argument . 
   this shift can be understood as follows . 
   the interactions introduce the so - called healing length @xmath88 @xcite that represents a coherence length over which the system feels the effect of an impurity , or in other words , the distance a site affects its neighborhood . for @xmath70 and @xmath65 from 20 to 5 , the value of @xmath89 ranges approximately from @xmath90 to 3 lattice spacing @xmath91 , which shows that , already for this @xmath57 value , the role of the interactions is important , effectively reducing the coherence length . to quantify this effect , we can partition the system into independent boxes of dimension @xmath92 , and use a mode - matching argument to determine their ground states : the condensate is more homogeneous if the lowest eigenvalue of each box is the same despite the presence of an impurity . 
   ( color online)boxes  of different sizes , in the presence and in the absence of an impurity . ] 
   therefore , this mode - matching argument fixes the value of @xmath68 . for the case @xmath70 and @xmath71 , @xmath93 , and this gives @xmath94 while for @xmath95 , @xmath96 , and we expect to find @xmath97 , in good agreement with the results showed in fig . 
   [ fig : igs_l20_n ] . 
   namely , the larger is @xmath89 , the better we recover the non - interacting condition eq . 
   ( [ cond - pat ] ) . 
   this effect is summarized in table [ tab : uloco ] . 
   ccc @xmath98 & fig . & @xmath68 +   + @xmath91 & [ figboxes](a)&@xmath99 $ ] + @xmath100 & [ figboxes](b ) & @xmath101 $ ] + @xmath102 & [ figboxes](c ) & @xmath103 $ ] +    we remark that this mode - matching condition is equivalent to match the resonance energy @xmath24 with the lowest eigenvalue of the unperturbed system of size @xmath92 . these simple arguments , 
   allow us to understand the shift of @xmath68 as a function of the interaction energy @xmath104 and the role of the structured impurities in the presence of the interactions . 
   we study the scaling behavior of @xmath73 with respect to @xmath5 . 
   analogously to the case of the single - particle ipr @xmath105 [ see eq . 
   ( [ eq : asympatty ] ) ] , we expect that @xmath106 with @xmath50 for a condensate localized on few sites , and @xmath51 for a homogeneous extended condensate . the behavior of @xmath73 for different values of @xmath5 is shown in fig . 
   [ fig : igs_l ] . 
   ( color online ) @xmath73 as a function of @xmath74 for @xmath42 , @xmath70 and @xmath71 particles per site . 
   the different curves correspond to different values of @xmath5 as indicated in the figure . 
   the filled symbols correspond to the 2d - drdm potential and the empty symbols correspond to the un - rand potential . ] 
   we observe that the minima , corresponding to different system sizes , collapse all together , meaning that the ground state corresponds to a spatial homogeneous condensate in the parameter regime where the correlations are dominant . at lower values of @xmath68 , @xmath73 scales as @xmath107 , and larger values of @xmath68 , @xmath73 scales as @xmath108 , with @xmath109 and @xmath110 @xmath111 . 
   this sort of `` super - delocalization '' , in the low @xmath68 region , is determined by the large value of @xmath72 that compensates , in the structured impurities , the effect of the site defect . 
   indeed , we observe an analogous behavior for the un - rand potential . 
   for such a potential , where the effect of @xmath72 is no more dominant , all the curves collapse together . 
   thus we expect that in this region the effect of the uncorrelated impurities on the ground state density distribution does not depend on the system size .      with the aim of characterizing the ground state configurations in the different regions , we show in figs . 
   [ fig : denlatt2.1][fig : denlatt15.1 ] the spatial density distribution @xmath61 for @xmath69 , @xmath71 at @xmath112 ( fig . 
   [ fig : denlatt2.1 ] ) , @xmath113 ( fig .  [ fig : denlatt6.6 ] ) , and @xmath114 ( fig .  [ fig : denlatt15.1 ] ) together with a pattern showing the locations of impurities . 
   ( color online ) lattice density plots together with site and bond impurities locations for @xmath42 , @xmath115 and drdm disorder ( top ) and un - rand ( bottom).,title="fig : " ] + ( color online ) lattice density plots together with site and bond impurities locations for @xmath42 , @xmath115 and drdm disorder ( top ) and un - rand ( bottom).,title="fig : " ]    ( color online ) same as fig . 
   [ fig : denlatt6.6 ] for @xmath116.,title="fig : " ] + ( color online ) same as fig . 
   [ fig : denlatt6.6 ] for @xmath116.,title="fig : " ]    ( color online ) same as fig . 
   [ fig : denlatt6.6 ] for @xmath117.,title="fig : " ] + ( color online ) same as fig . 
   [ fig : denlatt6.6 ] for @xmath117.,title="fig : " ]    the addition of a hopping term @xmath72 favors the delocalization of the density both for the 2d - drdm and un - rand disorders . 
   however , in the case of the 2d - drdm , it is more beneficial as it tends to partially compensate the decrease in the density caused by the site impurity , reducing the decrease by means of the structured disorder . 
   for small values of @xmath68 ( see fig .  [ 
   fig : denlatt2.1 ] ) , in the region where the effect of @xmath72 is dominant , the density in the impurity regions is even larger with respect to the density elsewhere . for large values of @xmath68 ( see fig .  [ 
   fig : denlatt15.1 ] ) , the effect of both types of disorder is similar as the change in the on - site energies dominates . 
   this limit gives rise to a strongly depleted density at the impurity location plus a rather uniform background . 
   the largest differences among the 2d - drdm and un - rand results are seen at the minimum of @xmath66 ( see fig.[fig : denlatt6.6 ] ) , where we can clearly observe a more homogeneous density spread over the lattice ( lower @xmath66 ) , and a consequently larger delocalization for the 2d - drdm than for the un - rand potential . 
   the density behavior determines the condensate fraction @xmath118 , as shown in fig . 
   [ fig : n0_l ] . 
   ( color online ) condensate fraction @xmath119 as a function of @xmath74 for @xmath42 , @xmath70 and @xmath71 particles per site . 
   the different curves correspond to different values of @xmath5 as indicated in the figure . 
   the filled symbols correspond to the 2d - drdm potential and the empty symbols correspond to the un - rand potential . ]    in correspondence of the minimum of the function @xmath66 , we observe that the condensate fraction @xmath119 does not depends of the system size , in the presence of the 2d - drdm potential . 
   the resonance condition minimizes the fluctuations with respect the chosen homogeneous basis @xmath63 and fixes @xmath119 . at lower value of @xmath68 
   , we observe a `` super - delocalization '' ( @xmath73 scales as @xmath107 ) , and for both the 2d - drdm and the un - rand potentials , the large value of @xmath72 enhances the coherence and @xmath119 increases with system size .    at larger values of @xmath68 , where @xmath73 scales as @xmath108 , the 2d - drdm impurities create holes in the system , and @xmath119 decreases with system size . for the case of the un - rand potential 
   , one can observe a monotonic behavior of @xmath119 as a function of @xmath68 . 
   as for the case of the 2d - drdm , the region where all the curves @xmath73 collapse together corresponds to a region where @xmath119 does not depend on the system size . 
   the difference with the 2d - drdm is a larger decrease of @xmath119 in this region . for 2d - drdm , only one value of @xmath68 has this peculiarity , and the maximum position of the condensate fraction foregoes this point . 
   let us remark that the minimum of @xmath120 corresponds to the minimum deviation with respect a homogeneous condensate , and , because of border effects , this target state is not necessarily the one that ensures a maximum value of @xmath119 in finite systems . 
   the predicted condensate fraction enhancement for the drdm at low @xmath68 , being it very small , could be very difficult to be measured . 
   however the non - diminishing of the coherence in a range of about @xmath121 should be observable , and could be directly compared with the result for the un - rand where the decrease of the coherence should be sizable . 
   in summary , we introduce a correlated disorder model that is the natural extension of the drdm in 2d . 
   we show that , in the non - interacting regime , such a disorder introduces some delocalized states if the resonance energy characterizing these structures belongs to the spectrum of the unperturbed system . in the presence of weak interactions 
   , the 2d - drdm drives the density spatial fluctuations . by means of a mode - matching argument that includes the effect of the interactions , 
   we show that the resonance energy is at the origin of these phenomena . 
   a direct consequence is a non - monotonic behavior of the condensate fraction as a function of the disorder strength , and its enhancement for values close to the resonance condition . 
   this work shows that short - range correlations in a disordered potential can modify and enhance the coherence of a many - body system in the weak interacting regime . 
   such effects could be measured in the context of ultracold atoms with an accurate measurement of the density and coherence , via , for instance , a fringes contrast interference experiment . 
   our results could also be extended to homogeneous systems provided one is able to engineer suitable impurities that are transparent for a given energy .
summary: we introduce a two - dimensional short - range correlated disorder that is the natural generalization of the well - known one - dimensional dual random dimer model [ phys . 
   rev . 
   lett * 65 * , 88 ( 1990 ) ] . 
   we demonstrate that , as in one dimension , this model induces a localization - delocalization transition in the single - particle spectrum . 
   moreover we show that the effect of such a disorder on a weakly - interacting boson gas is to enhance the condensate spatial homogeneity and delocalisation , and to increase the condensate fraction around an effective resonance of the two - dimensional dual dimers . 
   this study proves that short - range correlations of a disordered potential can enhance the quantum coherence of a weakly - interacting many - body system .