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a quantum particle in a three - dimensional disordered lattice is either localized or diffusive ; this depends on the strength of the disorder @xcite . if the particles are localized within a finite localization length @xmath2 , one observes current over length scales @xmath3 , while no conductivity over length scales @xmath4 @xcite . if quantum transport is allowed over infinite length scales , particles are naturally assumed to diffuse over any arbitrary length scale . as was pointed out already in the original work of anderson @xcite , quantum particles with long - range hopping are diffusive in three - dimensional ( 3d ) lattices of infinite size , for any disorder strength . this absence of localization was later explained by levitov @xcite , who showed that diffusion occurs due to resonant transitions between distant lattice sites . the number of such resonances diverges with the lattice size , leading to particle transport over infinite length scales . however , if the lattice size is finite , the number of resonances is also finite . this raises the question : if the lattice size is reduced , can the system localize to length scales smaller than the lattice size or will the localization length always remain larger than the system size ?    to answer this question , we consider particles with dipolar hopping in 3d disordered lattices . dipolar hopping is long - range in 3d . for generality , we consider lattices with random on - site energies and random dilution . we show that as the lattice size is _ reduced _ , delocalized particles may , depending on the dilution and amount of disorder , become localized to length scales much smaller than the lattice size . this leads to a rather unusual phenomenon : particles can diffuse through a lattice of size @xmath0 , but not through a lattice of size @xmath1 . this means that quantum currents can be switched off by shrinking the size of the system .    for conventional disordered systems , the localization is typically characterized by computing certain properties as functions of the lattice size . for example , one can examine the distribution of nearest spacings of the hamiltonian eigenvalues , which must undergo a change from the wigner distribution to a poisson distribution near the diffusion  localization crossover . this change becomes sharper as the lattice size increases , corresponding to a phase transition in the infinite size limit . for particles with dipolar long - range hopping in 3d lattices , such conventional analyses are not possible as the parameters of interest are functions of the system size itself and the localization length diverges with the system size , in contrast to the case with nearest neighbour hopping . therefore , in order to examine the properties of particles with long - range hopping in finite - size lattices , we use the following approach :    * first , we compute the wavepacket expansion dynamics of particles placed in an individual site of the lattice to show that , for certain system parameters , a particle placed in the middle of the lattice does not reach the lattice edge over long times greatly exceeding the heisenberg time ; * second , we compute the inverse participation ratio for each eigenstate of the hamiltonian to illustrate that , for certain system parameters , all eigenstates are localized to parts of the lattice much smaller than the lattice size ; * third , we examine the fluctuations of the wavefunctions between instances of disorder to illustrate that these fluctuations become log - normal at strong disorder ; * fourth , we map out the localization - diffusion crossover region as a function of dilution and on - site disorder using the wavepacket dynamics described in ( i ) ; * finally , we combine our results with scaling theory from refs . @xcite in order to obtain the dependence of the localization - diffusion crossover region on the lattice size @xcite were originally introduced to explain delocalization of particles with long - range hopping . we consider particles with dipolar hopping in 3d disordered lattices , known to be diffusive over infinite length scales . we start by illustrating the absence of diffusion for such particles in finite - size lattices by computing the time evolution of wavepackets placed on an individual lattice site . for generality , we compute the wavepacket dynamics in a diluted , disordered lattice and characterize the localization - diffusion phase diagram as a function of both dilution and disorder strength , as was done for particles with short - range hopping in ref .  @xcite . the dynamics are governed by the following hamiltonian : @xmath5 where the operator @xmath6 removes the particle from site @xmath7 , @xmath8 is the energy of the particle in site @xmath7 , and @xmath9 is the amplitude for particle tunnelling from site @xmath10 to site @xmath7 . we introduce disorder by randomizing both the values of @xmath8 and @xmath9 , which makes eq . ( [ model ] ) relevant for both disordered lattices and amorphous systems . we consider a single particle in a 3d cubic lattice with @xmath11 sites per dimension . we randomize the values @xmath12 $ ] drawn from a uniform distribution and the values @xmath9 as in the site percolation model . for a given realization of disorder , we divide the lattice sites into two subsets @xmath13 and @xmath14 , with @xmath15 sites in the @xmath13 subset and @xmath16 in the @xmath14 subset . for a given value @xmath17 , the lattice sites are assigned to the subsets at random . the hopping amplitudes of the hamiltonian ( [ model ] ) are then defined as follows : @xmath18 where @xmath19 is the distance between sites @xmath7 and @xmath10 and @xmath20 determines the hopping range , with @xmath21 representing dipolar hopping . the value of @xmath22 is chosen such that @xmath23 for nearest neighbour ( nn ) sites . with @xmath9 thus defined , the hamiltonian ( [ model ] ) describes a particle with dipolar hopping in a disordered , diluted lattice with @xmath24 sites . the system parameters of interest are @xmath17 , @xmath25 , and @xmath11 .     for a particle initially at the centre site . @xmath26 is the diameter of a sphere containing no less than 90% of the particle density . the particle has @xmath27 and is in a disordered lattice with @xmath28 , @xmath29 and @xmath30 ( green , blue , and magenta , respectively ) . the horizontal dashed line shows the side - length of the cubic lattice ; the grey area highlights sizes smaller than this . the inset shows the same observables , but over shorter time scales . the values of @xmath26 are averaged over 1050 realizations of disorder . the heisenberg time for these systems is @xmath31 , as can also be seen from the inset . , title="fig : " ] +    we compute the wavepacket dynamics for a particle placed at the center of the cubic lattice . as described in ref @xcite , we use the eigenstates of the hamiltonian ( [ model ] ) to construct the time propagator and compute the wavefunction of the particle at extremely long times @xmath32 , far exceeding the heisenberg time ( typically @xmath31 for the systems considered here ) . to characterize the spread of the wavepacket in real space , we compute the diameter @xmath26 of the sphere enclosing at least 90% of the particle density as a function of time . the values of @xmath26 are averaged over 200 to 1050 realizations of disorder , as required for convergence and as specified in the figure captions . figure [ figurewavepackdyn ] shows the dependence of @xmath26 on time for @xmath33 and for different values of @xmath25 . as expected , for systems with low values of @xmath25 , the particle density expands to fill the entire lattice . however , for lattices with @xmath34 and @xmath35 , the particle density , on average , does not reach the lattice size and remains localized on a scale smaller than the lattice size .    ) computed for a point with strong disorder ( @xmath36 ; @xmath37 ) and a point with weak disorder ( @xmath38 ; @xmath39 ; @xmath40 ) . the ipr distributions for the point with strong disorder are computed for three lattice sizes with @xmath41 , @xmath42 and @xmath43 , illustrating a slow shift towards the diffusion regime recovered in the infinite size limit ( see figure [ figurescaling ] ) . the distributions are averaged over 1300 ( @xmath41 ) and 300 ( @xmath44 and @xmath45 ) instances of disorder . ]     for @xmath28 , @xmath29 , @xmath46 computed at time @xmath47 . the different colours refer to histograms at different values of @xmath48 . the values of @xmath2 and @xmath49 are obtained by scaling the computed histograms using the formula shown on the abscissa . @xmath2 corresponds to the localization length . the red curve shows @xmath50 , with @xmath51 the abscissa . the results are obtained using 3150 realizations of disorder . ]     for a particle with @xmath52 initially placed in the middle of a 3d lattice with @xmath53 after @xmath54 . the circles indicate the computed values , averaged over 200 to 600 realizations of disorder , as required for convergence . for points in the black area , @xmath55 , indicating the absence of transport . the white area corresponds to @xmath56 , indicating diffusion to the lattice edges in finite time . the grey area shows the crossover regime , where some values of @xmath57 allow diffusion to the lattice edges and some not . the dashed line in the lower panel is the @xmath53 line from figure [ figurescaling ] , determined as described in the text . ] the results of figure [ figurewavepackdyn ] illustrate that lattices with large @xmath25 do not permit diffusion of particles with dipolar hopping over long time scales . the slopes of the curves in figure [ figurewavepackdyn ] are zero ( within the numerical uncertainties ) , which indicates the absence of diffusion over infinite time . however , a numerical calculation of wavepacket dynamics can not strictly prove the absence of diffusion at infinite time . in order to confirm localization and gain a better understanding of the eigenstates of the system , we compute the inverse participation ratio ( ipr ) distribution of the entire set of the eigenstates of the hamiltonian ( [ model ] ) for two different dilutions and disorder strengths and at different system sizes .    figure [ figureipr ] shows these ipr distributions for a point with weak disorder ( @xmath38 ; @xmath39 ; @xmath45 ) and strong disorder ( @xmath36 ; @xmath37 ; @xmath58 ) . for states delocalized over the entire lattice , the ipr tends to @xmath59 . for states localized on a single lattice site , the ipr is one . figure [ figureipr ] shows that for weak disorder the ipr is narrowly distributed near zero , meaning that nearly all of the eigenstates are delocalized over most of the lattice . the ipr for strong disorder is markedly different , with all states having a much higher ipr .    to identify the point with strong disorder as in a localized regime , it is important to prove that the distributions have absolutely no states with the ipr close to @xmath60 . to do this , we analyzed the distributions in figure [ figureipr ] for each instance of disorder separately , to compute the minimum value of the ipr , @xmath61 , which represents the _ most extended _ eigenstate . we obtained the following results : of 1300 disorders for @xmath41 , the smallest ratio @xmath62 ; of 300 disorders for @xmath44 , @xmath63 ; of 300 disorders for @xmath45 , @xmath64 . this means that the _ most extended _ eigenstate of many disorder realizations is confined to less than one tenth of all lattice sites , proving that all eigenstates are confined to a volume much smaller than the lattice size . for comparison , the same ratio for the diffusive regime distribution in figure [ figureipr ] is @xmath65 . as the lattice size increases , the ipr distributions shift toward zero . these shifts should collapse the ipr distributions to a single line at zero in the limit of infinite size . however , figure [ figureipr ] shows that the approach of the distributions to that in the diffusive regime is very slow , suggesting that a localized regime could be expected to exist even for systems of large size . while the results of the previous section illustrate that all eigenstates of the strongly disordered hamiltonians are confined to a fraction of the lattice sites , the ipr values do not provide information on the shape of the wavefunctions . here , we demonstrate that the wavefunctions are exponentially localized by computing the fluctuations of the particle density between instances of disorder . we also determine the corresponding localization length . one of the characteristic features of anderson localization is the log - normal distribution of the particle density across disorder realizations @xcite . for example , as shown in section 2 of ref . @xcite , the transmission @xmath66 through a disordered region in the regime of anderson localization follows a log - normal distribution , in contrast to the diffusive case where @xmath66 follows a normal distribution . in the regime of anderson localization , @xmath67 thus follows a normal distribution . since @xmath67 is proportional to the extinction coefficient @xmath68 , the extinction coefficient also follows a normal distribution . the average of @xmath68 over instances of disorder is @xmath69 ( see eqn . 21 of ref . @xcite ) , where @xmath70 is the system length and @xmath2 is the localization length . the typical amplitude of an anderson - localized wavefunction @xmath71 , where @xmath51 is distance . for a specific instance of disorder , an anderson - localized wavefunction would then be @xmath72 . thus , at a specific value of @xmath51 , the magnitude of the wavefunction will vary log - normally between instances of disorder . this log - normal distribution is equivalent to the exponential decay of the wavefunction in real space . we plot in figure [ figurelognormalfluct ] the histograms of @xmath73 for the lattice with @xmath28 , @xmath29 , and @xmath46 . @xmath74 is obtained from wavepacket dynamics calculations at time @xmath47 . the histograms for different values of @xmath48 ( distance from the site of maximum density ) are scaled to lie on the same curve . the scaling is performed using the formula displayed on the abscissa of figure [ figurelognormalfluct ] . the parameter @xmath2 corresponds to the localization length . the values obtained are displayed in figure [ figurelognormalfluct ] . figure [ figurelognormalfluct ] shows that these scaled histograms all follow a gaussian distribution , illustrating the _ log - normal _ distribution of the particle density vs disorder realizations : @xmath75}.\end{aligned}\ ] ] this log - normal behaviour shows that particles with long - range hopping undergo exponential decay in finite - size systems . exponential decay is one key feature of anderson localization . moreover , the data in figure [ figurelognormalfluct ] show that the localization length ( see @xmath2 in figure [ figurelognormalfluct ] ) is much smaller than the side - length of the lattice ( @xmath76 ) . to understand the dependence of the diffusion / localization properties on the dilution and the on - site disorder , we repeat the wavepacket dynamics calculations of figure [ figurewavepackdyn ] for different values of @xmath17 and @xmath25 . the results are presented in figure [ figurephasediagram ] . figure [ figurephasediagram ] shows two regimes : the black area corresponds to what would be observed in any experiment as the localization regime ; the white area corresponds to the diffusive regime . to obtain these results , we first fix the lattice size and determine by a large number of computations whether a particle initially in the middle of the 3d lattice spreads to fill the entire lattice or forms a localized wavepacket that , after the long time @xmath77 , stays , on average , within a sphere of size @xmath26 smaller than the lattice size . we would like to emphasize that figure [ figurephasediagram ] is _ not _ a phase diagram as phase diagrams correspond to the thermodynamic limit and the present results are for a finite - size lattice . similarly , what we identify is a localization - diffusion _ crossover _ , not a localization - diffusion _ transition_.    the area of figure [ figurephasediagram ] coloured in black corresponds to disorders where the average size @xmath26 of the wavepacket is smaller than the lattice size ; i.e. , where , on average , there is no diffusion to the lattice edges over the long time @xmath77 . we note that the absence of localization along the horizontal axis of figure [ figurephasediagram ] is consistent with the recent work of ref . @xcite that shows no localization for a 3d system with dipolar hopping on a diluted lattice and no on - site disorder ( @xmath78 ) . clearly , the off - diagonal disorder caused by the diluted lattice is insufficient to cause localization ; diagonal disorder is required . this also illustrates the absence of either a quantum or classical percolation threshold for a system of this size .    . the solid line corresponds to the dashed line in figure [ figurephasediagram ] , determined as discussed in the text . ] it is important to analyze how figure [ figurephasediagram ] changes with the size of the lattice . to do this , we combine the results of wavepacket calculations in figure [ figurephasediagram ] with the scaling analysis of levitov @xcite . from ref . @xcite , delocalization occurs through resonances defined by @xmath79 . the number of resonances diverges with the system size and delocalization must occur when @xmath80 where @xmath81 and @xmath82 is the index of a concentric sphere with radius @xmath83 . re - written as @xmath84 , where @xmath85 is some large constant , eq . ( [ condition ] ) produces a line in the @xmath86-phase diagram separating the diffusive and localized regimes : @xmath87 where we assumed that @xmath88 and @xmath1 and @xmath89 are unknown constants . to determine @xmath1 and @xmath89 , we first note that the probability distribution of the site energy differences is @xmath90,\ ] ] and calculate the probability of resonance between sites @xmath7 and @xmath10 as @xmath91 using eq . ( [ probability - resonance ] ) , we calculate the probability @xmath92 that the central site has _ no _ resonance in the entire system of a given size and draw the isoprobability lines for @xmath93 on the diagram from figure [ figurephasediagram ] . the isoprobability line with @xmath94 goes through the localization - diffusion crossover diagram inside the grey crossover regime , as shown in figure [ figurephasediagram ] . using this value of @xmath92 as characterizing the diffusion - localization crossover , we obtain the crossover lines for @xmath95 and fit eq . ( [ crossover - line ] ) to the resulting curves to obtain @xmath1 and @xmath89 . to verify the procedure , we compute the isoprobability lines for @xmath96 and @xmath97 , and obtain a perfect agreement with the values from eq . ( [ crossover - line ] ) . figure [ figurescaling ] shows the crossover lines for different lattice sizes . remarkably , figure [ figurescaling ] illustrates that the localization crossover may occur at finite values of @xmath25 for systems with a macroscopically large size . consider , for example , the case with @xmath98 , which corresponds to a 3d lattice with @xmath99 sites , which exceeds avogadro s number . in the infinite size limit , the crossover line tends to the @xmath100-axis , which corresponds to the absence of localization and is consistent with prior work @xcite . our results illustrate that the detection of the crossover for particles with dipolar hopping in 3d lattices is within reach of current experiments . this can be done with ions in rf - traps , ultracold magnetic atoms , ultracold rydberg atoms , or ultracold molecules trapped in optical lattices . an optical lattice typically contains @xmath101 sites per dimension . localization can be observed with internal excitations of the trapped atoms or molecules @xcite . if an optical lattice is partially populated with atoms or molecules , the empty lattice sites serve as impurities that scatter the excitations . the parameter @xmath17 in the diagram shown in figure [ figurephasediagram ] can thus be controlled by simply varying the number of atoms or molecules loaded in the optical lattice . the disorder in @xmath8 can be applied using an optical speckle potential , as in @xcite . for example , figure [ figurescaling ] shows that for molecules on an optical lattice with @xmath102 and a lattice population of 30 % @xcite , the crossover can be observed by varying the optical speckle potential from below to above @xmath103 . our results have , at least , three major applications . first , they can be used for the design of miniaturized quantum devices relying on the transport of small excitons that move via dipolar hopping . our results show how much disorder can be tolerated in such devices and what is the minimum length scale of such devices that allows high exciton mobility . second , our results show how one can engineer exciton transport on small length scales by doping molecular crystals with impurities . this is relevant for bulk heterojunction organic solar cells and similar light - harvesting devices that rely on the high mobility of excitons @xcite . third , our results are critical for the design of experiments aimed at detection of interaction - induced many - body localization ( mbl ) . mbl is predicted to hinder the return of a perturbed quantum system to equilibrium , thus precluding thermalization , conductivity , density equilibration or energy transport . it is the subject of numerous theoretical studies , many of which propose experiments for the observation of the localized phase . of particular interest is the localization phase transition in 3d dipolar systems , where excitations / spins hop between carrier particles due to dipolar interactions . a recent startling discovery @xcite shows that the localized phase can be induced by interactions between the particles . in order for such a phase transition to be observable , the system must be in the diffusive regime in the limit of non - interacting particles . our work shows that this is not guaranteed and provides the conditions required to perform these experiments . particles with long - range hopping are known to be delocalized in lattices of infinite size . here , we have presented evidence that particles with dipolar long - range hopping may undergo exponential localization in 3d disordered lattices of finite size . our conclusions are based on three independent but mutually consistent results :    * first , we have demonstrated by rigorous quantum dynamics calculations that particles placed in an individual lattice site can not diffuse through lattices of finite size over long time scales . by itself , this is not sufficient proof of localization as this result does not rule out the possibility that particles can fill the entire lattice at infinite time . * to further confirm the presence of localization , we have calculated the inverse participation ratio for each eigenstate of the hamiltonian . our results show that no eigenstate occupies more than 10 % of the lattice for lattices of sufficiently small finite size . our results also show how the participation ratio changes with increasing system size towards a system with delocalized states . one may argue that this is not sufficient proof of localization as it does not rule out the possibility of states that occupy a small number of lattice sites that are spread throughout the lattice ( for example , every tenth site ) . however , the presence of such states would allow for significant transport through the lattice , which we do not observe in ( i ) .    the results in ( i ) and ( ii ) thus provide strong evidence for localization in finite - size lattices . however , the shape of the wavefunctions is still not clear . to better understand this ,    * we have computed the fluctuations of the wavefunctions between instances of disorder . our results illustrate that the wavefunction fluctuations follow a log - normal distribution , which corresponds to exponential localization of the wavefunctions . our results in figure [ figurephasediagram ] illustrate the crossover between what would be observed in any experiment as the localization regime and the diffusion regime . we have demonstrated that introducing diagonal disorder can bring the system to the localization regime , even if introducing dilution alone can not @xcite . furthermore , introducing diagonal disorder alone is sufficient to observe the localization regime . the crossover is shown to be within reach of current experiments . we would like to emphasize that the absence of localization with dilution alone ( x - axis of figure [ figurephasediagram ] ) shows that there is no classical or quantum percolation threshold for a system of this size . the lack of classical localization means that any localization induced by diagonal disorder must be due to quantum interference effects . we have combined our wavepacket calculations in figure [ figurephasediagram ] with the scaling theory from refs . @xcite to demonstrate how the crossover changes with the system size ( figure 5 ) . surprisingly , we find that one should expect to observe the absence of quantum transport even in systems with macroscopically finite size , providing the on - site disorder and the dilution are significant enough . these results are of practical interest as they can be used to inform the design of minituarized quantum devices , which always have finite size . we acknowledge useful discussions with andreas buchleitner . this work is supported by nserc of canada . c.  a. mller and d.  delande , _ disorder and interference : localization phenomena _ in _ les houches 2009 - session xci : ultracold gases and quantum information _ , edited by c. miniatura , l .- c . kwek , m. ducloy , b. gremaud , b .- englert , l. cugliandolo , a. ekert , k.  k. phua ( oxford university press , oxford 2011 ) .
non - interacting particles with long - range hopping are known to be delocalized in disordered systems of infinite size . 
 it is thus natural to assume that such particles can traverse any finite - size lattice . here 
 , we demonstrate that this is not generally true . 
 the delocalization mechanism is induced by resonances between distant lattice sites . 
 the number of such resonances diverges with the system size . for a finite - size lattice 
 the number of resonances is finite and may not be sufficient to result in delocalization . 
 we consider particles with long - range ( dipolar ) hopping in three - dimensional lattices with diagonal disorder and random dilution . 
 we compute the wavepacket dynamics of particles placed in an individual lattice site , the inverse participation ratios and the fluctuations of the wavefunctions with disorder . 
 we show that , for certain finite - size disordered lattices , particles remain localized within a finite volume much smaller than the lattice size , and that the wavefunctions exhibit the log - normal fluctuations characteristic of anderson localization . 
 we characterize the localization properties as functions of dilution and diagonal disorder . 
 we combine our results with scaling theory to obtain the size dependence of the localization  diffusion crossover . 
 our results indicate that particles with long - range hopping undergo exponential localization in lattices of finite size , even macroscopically finite . 
 our phase diagrams illustrate a rather unusual phenomenon : quantum particles can diffuse through a lattice of size @xmath0 , but not through a lattice of size @xmath1 .