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the existence of @xmath0-wave superconductivity in the one - band two - dimensional hubbard model has been established through a variety of theoretical methods at both weak  @xcite and strong coupling  @xcite , in other words for one - site interaction @xmath2 either much smaller or much larger than the bandwidth @xmath10 . generalizations of dynamical mean - field theory are particularly suited for the strong coupling limit , but they are also an excellent guide to the physics at weak to intermediate coupling . @xcite , these calculations suggest that pairing is maximized at intermediate coupling , where the on - site interaction @xmath2 is of order the bandwidth @xmath3 . some non - perturbative calculations based on weak coupling ideas even agree at intermediate coupling  @xcite with strong - coupling based approaches .    in all these approaches , spin fluctuations with either an antiferromagnetic or a singlet character  @xcite have been argued to drive the pairing . these spin fluctuations result from the presence of an on - site coulomb repulsion @xmath2 . at strong coupling , the characteristic energy scale of these fluctuations , the exchange interaction @xmath6 , is given by @xmath11 , and the @xmath0-wave gap symmetry adopted by the cooper pairs allows them to avoid the direct effect of the on - site repulsion @xmath2 . little attention has been paid so far to the effect of the nearest - neighbor coulomb repulsion ( or extended hubbard interaction ) @xmath5 that we expect to be detrimental to @xmath0-wave superconductivity . roughly speaking , in a simple bcs picture that does not take retardation into account , we expect the effective interaction to be the difference @xmath12 . in ordinary phonon - mediated superconductivity , the repulsion @xmath5 is replaced by a smaller pseudopotential @xmath13 to account for the fact that binding occurs at low frequencies through phonons while the coulomb interaction acts over a broad energy scale . this so - called anderson - morel mechanism  @xcite , leads to the following estimate for the coulomb pseudopotential @xmath14 where the debye frequency is @xmath15 and the fermi energy @xmath16 . one expects that in strongly - correlated superconductivity , the ratio @xmath17 must be replaced by a number closer to unity in which case this mechanism would no - longer be effective and superconductivity should disappear as soon as @xmath18 . this issue is crucial to understand high - temperature superconductors since that condition , or the weaker condition @xmath19 with @xmath20 the spin gap , is likely to be satisfied in these materials . from the value of the near - neighbor coulomb interaction with a relative dielectric constant of order 10 we estimate @xmath21 mev while @xmath22 mev  @xcite . so far , it has been shown using a variational wave - function approach for the @xmath23 model at strong coupling that superconductivity persists as long as  @xcite @xmath24 where @xmath25 is the doping . a large - n calculation gives superconductivity at least up to @xmath26 ,  @xcite while density matrix renormalization group ( dmrg ) calculations on hubbard or @xmath27 ladders  @xcite suggest that pairing can survive up to @xmath28 . @xcite at very small coupling ( @xmath29 ) , it has been argued  @xcite that pairing is destroyed as soon as @xmath30 . this weak coupling bound is close to the result of a flex calculation . @xcite    here we show that @xmath0-wave superconductivity in the one - band two - dimensional hubbard model at strong coupling is in fact more robust than expected . even for @xmath7 , as long as the inequality @xmath31 is satisfied , superconductivity persists this illustrates differences between pairing at weak and at strong coupling , especially in the pseudogap regime . the resilience of @xmath0-wave superconductivity to @xmath5 can be traced to the increase in the effective @xmath6 caused by @xmath5 at strong coupling , i.e. when at half - filling the system is a mott insulator . this increase in @xmath6 is visible in the dynamics of both the spin susceptibility and the cooper pair green s function which is enhanced by @xmath5 at low frequency . the pair - breaking effect of @xmath5 manifests itself at higher frequency . this leads overall to a sizeable range of values of @xmath5 where the order parameter is essentially independent of @xmath5 in the underdoped regime where a pseudogap appears . recall that at strong coupling , the pseudogap extends up to optimal doping , a sign that mott physics extends well beyond half - filling  @xcite . we use cellular - dynamical mean - field theory that allows us to study the strong - coupling limit and allows one to take into account both @xmath6 and @xmath5 as well as the effect of retardation . we first present the method and model and conclude with a discussion after the presentation of the results obtained from large scale numerical calculations . we start from the one - band extended hubbard model on a square lattice , @xmath32 where @xmath33 is the destruction ( creation ) operator for an electron of spin @xmath34 at site @xmath35 and @xmath36 is the corresponding number operator ( @xmath37 ) . we assume a band structure close to that of yba@xmath38cu@xmath39o@xmath40 , with nearest - neighbor hopping @xmath41 set to unity , diagonal hopping @xmath42 and third - neighbor hopping @xmath43 , unless otherwise indicated . this model is solved with a 4 site plaquette cellular dynamical mean field theory ( cdmft ) at @xmath44 using an exact diagonalization solver  @xcite that allows us to obtain real - time quantities without analytic continuation . this approach has been used to reveal the presence of @xmath0-wave superconductivity in the one - band hubbard model,@xcite and to study the pairing dynamics and retardation when @xmath45 . @xcite the results obtained for @xmath0-wave superconductivity with the plaquette are essentially identical to those obtained with larger clusters . @xcite    as in previous studies of the 2d hubbard model with this approach  @xcite , the plaquette is hybridized with a set of 8 bath orbitals , as illustrated in fig . [ fig : cluster ] , with 6 parameters to be determined through the cdmft self - consistency relation : the hybridizations @xmath46 between the bath and the cluster , the bath energies @xmath47 and the bath @xmath0-wave pairing parameters @xmath48 which , when nonzero , signal the presence of superconductivity and lead to a non - vanishing cluster @xmath0-wave order parameter @xmath49    the order parameter is extracted from the lattice green s function while the doping is measured on the cluster . the distance function is defined in ref .  . we use cutoff @xmath50 and fictitious inverse temperature @xmath51 .    while the extended interaction @xmath5 within the cluster can be treated exactly with an approach like cdmft , the coupling to neighboring clusters requires the hartree approximation . @xcite more specifically , the model hamiltonian eq . is modified by replacing the inter - cluster interaction by @xmath52 where @xmath53 denotes nearest - neighbor pairs within the plaquette and @xmath54 nearest - neighbor pairs across plaquettes . the mean field @xmath55 must be determined self - consistently ; in practice , it is treated like the six bath parameters wihtin the cdmft self - consistency loop and thus both the dynamical mean field ( represented by the bath parameters ) and the static hartree mean field are converged simultaneously .    to verify the accuracy of this approach , consider momentarily the half - filled , particle - hole symmetric extended hubbard model with @xmath56 . in that simple case , the hartree field @xmath55 is fixed by particle - hole symmetry . we find that the dynamical spin susceptibility at the antiferromagnetic wavevector @xmath57 has a dominant peak at a low frequency @xmath58 . the position of this peak is shown on fig . [ fig : chi_hf ] as a function of @xmath5 for two values of @xmath2 . in the large @xmath2 limit , the half - filled extended hubbard model should map to the heisenberg model with a super - exchange parameter @xmath9 . the energy denominator is easily understood by comparing the energy , at @xmath59 , between a configuration where all sites are exactly singly - occupied and another configuration in which one electron has vacated a site in order to doubly occupy a neighboring site . four @xmath5 bonds are lost , but three are gained . since we expect the frequency @xmath58 to scale like @xmath6 in the large-@xmath2 limit , fig . [ fig : chi_hf ] confirms that we obtain the correct scaling with @xmath5 . hence , our approach leads to the correct strong - coupling physics . to concentrate on the effect of short - range spin fluctuations on superconductivity , we ignore the possibility of long - range antiferromagnetic order predicted by cdmft at low doping ,  @xcite as well as the possibility of long - range charge density waves in the presence of @xmath5 . at weak coupling charge - density waves may occur for @xmath60 in two - dimensions . @xcite we found a similar result at half - filling for strong coupling . the @xmath0-wave order parameter for ybco hopping parameters is displayed on fig . [ fig : op ] for hole - doping at @xmath61 , 8 and 16 . in each panel , obtained for a given value of @xmath2 , we display the results for different values of @xmath5 . the mott transition at half - filling , that separates weak from strong coupling , occurs around @xmath62 . @xcite as observed before ,  @xcite in the weak coupling case ( @xmath61 ) superconductivity is strongest at half - filling when competition with antiferromagnetism is prohibited ( fig . [ fig : op]a ) . at larger coupling , the mott transition destroys superconductivity at half - filling ( figs . [ fig : op]b , c ) . the value of @xmath5 influences the order parameter in strikingly different ways at weak and at strong coupling . for @xmath61 , superconductivity has essentially disappeared at @xmath63 , in agreement with the upper limit @xmath64 found by weak coupling analysis . @xcite this is consistent with the fact that at weak coupling , @xmath5 always decreases the strength of spin fluctuations . @xcite by contrast , at strong coupling , @xmath5 can increase the strength of spin fluctuations through @xmath9 , and for the same ratio @xmath65 , the order parameter is still large for @xmath66 on fig . [ fig : op]b . in fact , it is barely influenced by @xmath5 in the pseudogap ( underdoped ) regime close to half - filling ( @xmath67 ) . on the lower panel , for @xmath68 , we have @xmath69 at @xmath70 and superconductivity still persists . the ratio @xmath71 is maximum at @xmath72 . we estimate that @xmath73 is the upper bound for the the appearance of @xmath0-wave superconductivity ( ignoring competing orders ) . the link between superconducting order parameter and spin fluctuations was demonstrated at @xmath45 by the correlation between the peaks in the imaginary part of the anomalous self - energy and the peaks in the spin fluctuation  @xcite . this correlation persists here ( not shown ) . to further investigate this link between spin fluctuations and superconductivity , let us thus first focus on the spin dynamics revealed by the the local spin spectral function @xmath74 illustrated on fig . [ fig : chi ] . panel ( a ) shows @xmath74 at @xmath66 , @xmath75 for the underdoped ( @xmath67 ) and overdoped ( @xmath76 ) regimes . the corresponding charge susceptibility is negligible on this scale . spin fluctuations are much smaller in the overdoped regime and their weight is spread in wavevector contrary to the underdoped case . panel ( b ) shows that in the underdoped regime ( @xmath67 ) where there is a pseudogap , a moderate extended interaction @xmath5 at strong coupling mainly shifts the spectrum of @xmath77 spin fluctuations to higher frequencies , as expected from the increase of @xmath9 caused by @xmath5 ; it decreases somewhat the overall spectral weight without affecting the spectral shape .    to show that low - frequency spin fluctuations are apparent in the pair dynamics , we compare the characteristic frequency @xmath58 of spin fluctuations , given by the position of the dominant peak of @xmath78 , with a characteristic frequency in the pair dynamics , as was done in ref .  . the pair dynamics can be studied through the integral @xmath79 where @xmath80 is the retarded gorkov function ( or anomalous green s function ) defined in imaginary time by @xmath81 with @xmath35 and @xmath82 nearest - neighbors . the infinite frequency limit of @xmath83 is equal to @xmath84 which in turn is proportional to the @xmath44 @xmath0-wave order parameter @xmath85 . @xmath86 is useful to estimate the frequencies relevant for binding . we call @xmath86 the cumulative order parameter .    in bcs theory , @xmath87 is a monotonically increasing function of @xmath88 that reaches its asymptotic value at the bcs cutoff frequency @xmath89 . @xcite in the eliashberg approach that includes retardation as well as the coulomb pseudopotential ,  @xcite the function overshoots its asymptotic value at frequencies near the main phonon frequencies before decaying to its final value because of pair breaking effects at higher frequencies . we define the characteristic frequency @xmath90 as the point where @xmath87 reaches half of its asymptotic value . if one imagines that @xmath91 is made of a single peak , then that peak would be located at @xmath90 . figure  [ fig : ig ] illustrates the cumulative order parameter @xmath87 for various values of @xmath5 . the top two panels are for two values of doping @xmath92 at @xmath66 , and the bottom one for @xmath68 in the underdoped regime . the asymptotic value is indicated by a horizontal line . following a sharp rise around @xmath93 , the function has a maximum and then decreases towards its asymptotic value , as in eliashberg theory . the link between the characteristic spin - fluctuation frequency @xmath58 and the characteristic frequency of the pair @xmath90 is summarized on fig . [ fig : u8 ] , which shows the evolution with doping of : ( a ) the position @xmath58 of the main af susceptibility peak , ( b ) its strength , and finally ( c ) the characteristic frequency @xmath90 , for four values of @xmath5 at @xmath66 . clearly , @xmath58 , @xmath94 and @xmath90 all decrease with doping . this shows that spin fluctuations and pair dynamics are strongly linked .    even for @xmath45 however , the increase in @xmath6 as measured by @xmath95 and @xmath90 in the underdoped regime , @xmath96 , does not correspond to an increase in the order parameter . as can be seen in fig . [ fig : op ] , for @xmath67 and @xmath76 the @xmath0-wave order parameter is essentially identical when @xmath45 and the increase in @xmath90 near half - filling manifests itself only in the maximum value of the cumulative order parameter @xmath87 in fig . [ fig : ig ] , not in the value of the order parameter . clearly the order parameter does not increase in the underdoped regime despite the increase in the characteristic spin fluctuation frequency @xmath95 because of another effect . that effect is the pseudogap . the pseudogap removes many of the states near the fermi energy that would otherwise be paired . the presence of the pseudogap can be seen by contrasting the single - particle local density of states at two dopings . in the overdoped regime ( @xmath76 ) the local density of states has no pseudogap in the normal state , as seen from the dashed red line on fig . [ fig : dos]b . only the superconducting gap is visible . by contrast , even at @xmath45 there is clearly a pseudogap in the underdoped regime , as can be seen from fig . [ fig : dos]a . the effect of superconductivity manifests itself only through the small more symmetric gap near @xmath97 . @xcite the pseudogap , as measured from the peak to peak distance , increases slightly with @xmath5 in the underdoped regime . the detrimental effect of the pseudogap on the superconducting order parameter was demonstrated in fig .  7 of ref .   where the bubble contribution to the pairing susceptibility decreases as we approach half - filling . this means that in the absence of interactions between the particles forming the pair ( represented by vertex corrections ) , self - energy effects disfavor superconductivity . this physics is also present at weak to intermediate coupling when the pseudogap is induced by long wavelength antiferromagnetic fluctuations  @xcite ( and not by the incipient short - range mott localisation that appears at strong coupling  @xcite ) . the effect of spin fluctuations and of near - neighbor repulsion @xmath5 on pairing is relatively straightforward in the overdoped regime ( @xmath98 ) . there , as seen on fig . [ fig : op]b and figs . [ fig : u8]a , c , the order parameter increases with the characteristic spin frequency @xmath58 and the characteristic pair frequency @xmath90 . like for the @xmath45 case , superconductivity disappears at finite @xmath5 when the amplitude @xmath94 of the lowest frequency peak in the spin fluctuations ( fig . [ fig : u8]b ) vanishes ,  @xcite as observed in experiment . @xcite the main effect of @xmath5 is to decrease the spin fluctuations , as happens systematically at weak coupling . @xcite    as we approach half - filling , the decrease of the order parameter for all values of @xmath5 ( figs . [ fig : op]b , c ) , despite the increase in the amplitude @xmath94 of the lowest peak in the spin fluctuations ( fig . [ fig : u8 ] ) , is mainly due to the increase of the pseudogap : it removes more and more states from the fermi energy as the doping @xmath92 decreases ( fig . [ fig : dos ] ) . note that @xmath5 causes opposite changes in the amplitude @xmath94 and in the characteristic frequencies ( figs  [ fig : u8 ] ) , which might explain the near independence of the order parameter with respect to @xmath5 in figs . [ fig : op]b , c . however , these changes are small on a relative scale and are probably not the main reason for the insensitivity of the order parameter to @xmath5 . the resilience of @xmath0-wave superconductivity to @xmath5 at strong coupling is best understood from the @xmath68 results for the cumulative order parameter @xmath86 shown in fig . [ fig : ig]c for @xmath67 . the largest value of @xmath86 scales roughly like @xmath9 , in other words @xmath5 increases the binding at low frequency , where the retardation is large . however , @xmath5 also has the expected pair - breaking effect : larger @xmath5 causes larger pair - breaking so that the low frequency increase in @xmath87 is essentially compensated by the time @xmath87 reaches its asymptotic value ( the order parameter ) .    to understand more deeply the dual role of @xmath5 , as both pair binding and pair breaking , it is helpful to return to the solution of the eliashberg equations in the electron - phonon case . figs .  7 , and 13 of ref . show that the imaginary part of the gap function remains positive for all frequencies when the coulomb pseudopotential vanishes . by contrast , one verifies from fig . 11 of the same paper that the imaginary part of the gap function changes from positive to negative as frequency increases when the coulomb pseudopotential is finite . a sign change in the imaginary part of the gap function should lead to a similar behavior in the imaginary part of the gorkov function @xmath99 . in this case its integral @xmath87 , as defined in eq . ( [ f(w ) ] ) , should reach a maximum before decreasing to its asymptotic large frequency value . by contrast , when the imaginary part of the gap function remains positive at all frequencies , i.e. in the absence of coulomb pseudopotential , @xmath87 should increase monotonically to its asymptotic value , as in bcs theory . since we observe that @xmath87 has a maximum at finite frequency , we are in the case where the coulomb pseudopotential is important . comparison of figs .  11 and 13 of ref . also shows that the maximum value of the imaginary part of the gap function ( and that of its frequency integral , as one can check ) increases with the electron - phonon coupling constant . in our case , the increase in @xmath9 caused by @xmath5 similarly leads to an increase in the maximum value of the cumulative order parameter @xmath87 . in other words , we can surmise that the increase in @xmath5 leads to larger binding at low frequency , like an increase in the electron - phonon coupling constant , but it also leads to more pair - breaking at large frequency , coming from the coulomb pseudopotential - like effect of @xmath5 , and these two effects nearly cancel each other .    in the electron - phonon case , one also notes that the imaginary part of the gap function vanishes at about two to three times the maximum phonon frequency . this is where the cumulative order parameter @xmath87 would reach its asymptotic value . in the case of spin fluctuations , @xmath100 is a measure of the width of the spin - fluctuation spectrum . comparing fig . [ fig : ig]a for @xmath66 and fig . [ fig : ig]c for @xmath68 , one notices that the frequency for which @xmath87 reaches its asymptotic value is smaller by a factor of roughly two for @xmath68 where the @xmath45 value of @xmath6 is twice as small as for @xmath66 . the opposing effects discussed above conspire to leave the order parameter rather insensitive to @xmath5 in the presence of a pseudogap . the pseudogap induced by mott physics  @xcite is detrimental to superconductivity , but in its presence superconductivity is effectively protected from near - neighbour repulsion @xmath5 . indeed , in a doped mott insulator , short - range incipient localisation is strong enough to create a pseudogap , but while @xmath5 causes pair - breaking at high frequency , it also enhances spin fluctuations ( @xmath9 ) at low frequencies , thus compensating the pair - breaking effect . overall , retardation effects are crucial for the resilience of @xmath0-wave superconductivity to near - neighbor repulsion . we are indebted to d.j . scalapino , a. chubukov , j. carbotte and g. kotliar for useful discussions and to giovanni sordi for a careful reading of the manuscript . this work was supported by nserc ( canada ) , cfi ( canada ) , cifar , and the tier i canada research chair program ( a .- m.s.t . ) . computational facilities were provided by compute canada and calcul qubec . d.  snchal , cluster dynamical mean field theory , in _ theoretical methods for strongly correlated systems _ , edited by a.  avella and f.  mancini , volume 171 of _ springer series in solid - state sciences _ , chap . 341370 , springer , 2012 .
many theoretical approaches find @xmath0-wave superconductivity in the prototypical one - band hubbard model for high - temperature superconductors . at strong - coupling ( @xmath1 , where @xmath2 is the on - site repulsion and @xmath3 the bandwidth ) pairing 
 is controlled by the exchange energy @xmath4 . 
 one may then surmise , ignoring retardation effects , that near - neighbor coulomb repulsion @xmath5 will destroy superconductivity when it becomes larger than @xmath6 , a condition that is easily satisfied in cuprates for example . using cellular dynamical mean - field theory with an exact diagonalization solver for the extended hubbard model , 
 we show that pairing _ at strong coupling _ is preserved , even when @xmath7 , as long as @xmath8 . 
 while at weak coupling @xmath5 always reduces the spin fluctuations and hence @xmath0-wave pairing , at strong coupling , in the underdoped regime , the increase of @xmath9 caused by @xmath5 increases binding at low frequency while the pair - breaking effect of @xmath5 is pushed to high frequency . 
 these two effects compensate in the underdoped regime , in the presence of a pseudogap . while the pseudogap competes with superconductivity , the proximity to the mott transition that leads to the pseudogap , and retardation effects , protect @xmath0-wave superconductivity from @xmath5 .