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for the past two decades , extensive studies of high-@xmath0 cuprates have spotlighted many curious phenomena . it has been argued that most phenomena are attributable to the strong correlations among electrons , which play significant roles in these materials . however , since the discovery of sudden changes in the electron dispersion or `` kinks '' shown by the angle - resolved photoemission spectroscopy ( arpes ) ,  @xcite effects of electron - boson interactions on electronic self - energies have been recognized . while the kinks are now indisputable in cuprates ,  @xcite their origin as arising from electronic coupling to phonons  @xcite or magnetic excitations  @xcite remains unclear . recent scanning tunneling microscope study showed that the statistical distribution of energy of bosonic modes ( @xmath1 ) has meaningful difference between @xmath2 and @xmath3 materials . thus , it should be hard to exclude the possibility that electron - phonon interaction ( epi ) significantly affects the electronic states in cuprates .    in this study , we investigate the analysis upon the epi together with the electron - electron interaction ( eei ) on the basis of the two - dimensional ( 2d ) three - band hubbard holstein ( hh ) model . with the use of our three - band hh model , we can reproduce the situation of the real high-@xmath0 materials , in which epi mainly works on @xmath4 electrons at o sites . our model hamiltonian @xmath5 is composed of @xmath6 electrons at each cu site , @xmath4 electrons at o site , and lattice vibrations of o atoms . we consider only the on - site coulomb repulsion @xmath7 between @xmath6 electrons at each cu site as our eei . let us define that @xmath8 and @xmath9 represent the chemical potential and the number of all electrons , respectively . then , @xmath10 is divided into the non - interacting part , @xmath11 , the electron - electron interacting part @xmath12 , the phonon part @xmath13 , and the electron - phonon interacting part @xmath14 as @xmath15 here , @xmath16 and @xmath17 are the annihilation ( creation ) operator for @xmath6 and @xmath18 electrons of momentum @xmath19 and spin @xmath20 , respectively . the non - interacting part @xmath11 is represented by @xmath21 we take the lattice constant of the square lattice formed from cu sites as the unit of length . then , @xmath22 and @xmath23 , where @xmath24 is the transfer energy between a @xmath6 orbital and a neighboring @xmath18 orbital and @xmath25 is that between a @xmath26 orbital and a @xmath27 orbital . @xmath28 is the difference of energy levels of @xmath6 and @xmath4 orbitals . in this study , we take @xmath24 as the unit of energy . the residual parts are described as follows : @xmath29 @xmath30 and @xmath31 where @xmath7 is the on - site coulomb repulsion between @xmath6 orbitals , @xmath32 is the number of @xmath19-space lattice points in the first brillouin zone ( fbz ) , and @xmath33 is the electron - phonon matrix element , respectively . we consider that the half - breathing phonon mode ,  @xcite in which oxygen ions are vibrating along the @xmath34 or @xmath35 directions , is crucial for our problem . thus , ignoring the other phonon modes , we have the electron - phonon interacting part as eq . ( [ eq:8 ] ) . then , we introduced the unperturbed and perturbed green s functions , which are to be described in @xmath36 matrix form . the unperturbed green s function @xmath37 is derived from eq . ( [ eq:6 ] ) as @xmath38^{-1},\ ] ] where @xmath39 is a @xmath36 unit matrix . using the abbreviation of fermion matsubara frequencies , @xmath40 with integer @xmath41 and temperature @xmath42 , the perturbed green function @xmath43 is determined by the dyson equation : @xmath44 where @xmath45 is the self - energy expected to be a diagonal matrix . in order to estimate the @xmath6 electron self - energy @xmath46 in eq . ( [ eq:1 ] ) , we adopt the fluctuation exchange approximation  @xcite as follows : @xmath47 where @xmath48 , @xmath49 where @xmath50 with integer @xmath51 are boson matsubara frequencies , and @xmath52 in order to estimate the p@xmath53-electron self - energy @xmath54 in eq . ( [ eq:1 ] ) , we exploit the brillouin - wigner perturbation theory . we adopt the self - consistent one - loop approximation as follows : @xmath55 where @xmath56 and @xmath57 is the epi on p@xmath53 electron . our epi is determined as follows : @xmath58 , \label{eq:9}\ ] ] where @xmath59 and @xmath60 . @xmath61 is the specific phonon energy for the half - breathing mode . as above , we ignore the effects in which eei and epi are coupled . thus , as shown by eqs . ( [ eq:2])-([eq:9 ] ) , in our formulation , eei and epi are completely decoupled . of course , this assumption is inadequate to analyze the case in which the characteristic energy due to eei is comparable to the phonon energy . however , as will be seen later , we actually treat the cases with rather high characteristic energy due to eei . hence , decoupling eei with epi should be justified in our analysis . the electron - phonon coupling constant @xmath62 does not depend on @xmath63 since @xmath64 and @xmath65 , where @xmath63 is the mass of an oxygen ion . in our model , thus , the isotope effect is reflected on the phonon energy in the electronic self - energies only , but not any changes in the strength . we need to solve eqs . ( [ eq:1])([eq:3 ] ) in a fully self - consistent manner . during numerical calculations , we divide the fbz into @xmath66 meshes . we prepare @xmath67 matsubara frequencies for temperature @xmath68 . as shown later , at this temperature , our calculation can reproduce the important behavior of electrons in normal state . moreover , to our knowledge , the situation will not be changed if we change @xmath42 to some extent . @xmath69 and @xmath70 , which are all common for our calculations . these values are chosen so that we can reproduce the typical fermi surface of bi@xmath71sr@xmath71cacu@xmath71o@xmath72 observed by arpes . @xcite @xmath73 , @xmath74 , and @xmath75 unless stated . the phonon energy is set as @xmath76 for @xmath77 material . .[table:1]number of doped holes . @xmath78 . [ cols="^,^,^,^,^",options="header " , ]     fermi surfaces for @xmath2 materials indicated in table  [ table:1].,width=196 ]    dispersion kinks along the nodal direction . momentum is measured from each fermi surface . arrows indicate the momenta at which kinks occur.,width=317 ]    doping dependence of kink energies . bars indicate the discretization error during analytic continuation.,width=317 ]    dispersion kinks along the nodal direction for @xmath79 . the number of doped holes for @xmath77 material is @xmath80 . the dashed lines show the ones for op in table  [ table:1].,width=317 ]    dispersion kinks along the nodal direction for @xmath81 . the number of doped holes for @xmath77 material is @xmath82 . the dashed lines show the ones for op in table  [ table:1].,width=317 ]    @xmath83 on @xmath84 , where @xmath85 for ( a ) , @xmath86 for ( b ) , @xmath87 for ( c ) , and @xmath88 for ( d ) , respectively.,width=317 ]    @xmath89 on the same @xmath90 points as in fig . [ figure:4].,width=317 ]    dispersion kinks along the nodal direction for @xmath91 and @xmath92 . they all correspond to op . momentum is measured from the fermi surface for @xmath93.,width=317 ]    we show the numbers of doped holes for our fully self - consistent solutions in table  [ table:1 ] . the numbers of doped holes both for @xmath2 and @xmath3 materials are exactly the same to three places of decimals and they correspond to four different hole - doped samples , lightly doped ( ld ) , underdoped ( ud ) , optimally doped ( op ) , and overdoped ( od ) , respectively . in fig . [ figure:1 ] , we show the color map of the one - particle spectrum at fermi level @xmath94 , where @xmath95 in order to indicate the fermi surfaces for @xmath2 materials . in eq . ( [ eq:4 ] ) , the pad@xmath96 approximation is exploited for analytic continuation . furthermore , we calculate the electronic dispersions of the antibonding band @xmath97 along the nodal direction indicated as the cut in fig . [ figure:1 ] for all our doping cases . @xmath97 is determined as the @xmath98 point on which @xmath99 has the maximum value at each energy level . due to this method , the curves of @xmath97 look like a series of line segments , as shown in fig .  [ figure:2 ] . we compare every @xmath97 of our solution with the one obtained by another fully self - consistent manner , in which the same calculation is performed , except for the epi . we show our results on these dispersions in fig . [ figure:2 ] , where we can easily recognize that the dispersion kinks along the nodal direction appear only when epi affects the @xmath18 electrons . the kink energies were slightly shifted by @xmath100 substitution . in fig . [ figure:3 ] , we detail how these kink energies shift depending on hole doping . these theoretically evaluated isotope shifts are at most @xmath101 , which are much smaller than the ones measured by another group s arpes experiment . @xcite furthermore , these isotope shifts are almost independent of hole doping while another group insists that they are critically affected . @xcite considering the energy and momentum resolutions in their experiment , it may be hard to detect the subtle isotope shifts and their dependence on hole doping shown in our model . let us now look at @xmath62 and @xmath28 dependences in the dispersion kinks along the nodal direction in detail . figure  [ figure:4 ] shows the energy dispersions for @xmath75 ( dotted lines ) and @xmath79 ( solid lines ) . there is no clear difference in the isotope effect between @xmath79 ( @xmath102 ) and @xmath75 ( @xmath103 ) , though the dispersion kinks for @xmath79 are distinctly shifted to the high binding energy compared with those for @xmath75 . on the other hand , the isotope shift for @xmath81 ( @xmath104 ) is slightly shrank compared to that for @xmath105 ( @xmath102 ) , though the dispersions depend on the @xmath28 considerably , as shown in fig . [ figure:5 ] . hence , our model calculation shows that the isotope shifts are not sensitive to @xmath62 and @xmath28 . considering that @xmath28 is closely related with eei in our three band hh model as discussed later , we can be fairly certain that the isotope shifts are determined by the relative strength between epi and eei . however , these changes of the isotope shifts are minute , thus , our discussions so far are valid regardless of @xmath62 and @xmath28 .    to clarify the epi effect on the @xmath18 electrons described above , we investigate the @xmath4 electron self - energy @xmath106 along the nodal direction . in fig .  [ figure:6 ] , we show @xmath83 on every six @xmath98 points along the nodal direction , located inside fermi surfaces . the energy where @xmath83 is maximal corresponds to the one of the dispersion kink and shifts upward by @xmath100 substitution , as shown in fig . [ figure:2 ] . the energy dependence of @xmath83 is definitely due to the epi introduced with the use of eqs . ( [ eq:1 ] ) , ( [ eq:3 ] ) , and ( [ eq:9 ] ) . thus , we can conclude that in our solutions for all doping levels from the ud to the od region , the dispersion kinks along the nodal direction are created _ only when the epi is included_.    hereafter , we will discuss why the magnetic ingredients hardly bring the dispersion kinks along the nodal direction . even when epi does not exist , the electrons in our results are exposed to the strong af fluctuation originating from the electronic correlation among @xmath6 electrons . in fig . [ figure:7 ] , we show @xmath89 on the same six @xmath98 points as in fig .  [ figure:6 ] . it is shown that there is no clear difference in @xmath89 between @xmath2 and @xmath3 materials . the energy dependence of @xmath89 is definitely due to the electronic correlation introduced with the use of eqs . ( [ eq:2])([eq:7 ] ) . we easily recognize that @xmath89 uniformly increases with the binding energy and has no maximal value up to @xmath107 even for ld case , in which the strong af fluctuation is expected to be grown . thus , along the nodal direction , the strong af fluctuation could cause the renormalization of the fermi velocity , however , it hardly promotes any anomalous behavior such as kink structure .    finally , we will discuss how our eei affects the dispersion for @xmath2 materials . when the on - site coulomb repulsion @xmath7 is changed , the fermi velocity is renormalized differently , but this would not change the kink energy so much since the kink energy is determined by the epi alone . [ figure:8 ] , we lay out our results for three different @xmath7s and they all correspond to op . their total doped holes @xmath108 are slightly different as : @xmath109 and @xmath110 for @xmath111 and @xmath92 , respectively . hence , the fermi momentum for @xmath112 moves inside ( or the fermi surface shrinks ) and the one for @xmath113 moves outside ( or the fermi surface enlarges ) , compared to the one for @xmath93 . these changes of the fermi momenta are small ; however , the dispersions at higher energy are quite affected , reflecting the binding energy dependence of @xmath89 , as shown in fig . [ figure:7 ] . therefore , the dispersion at higher energy could be changed a lot by the eei even if the fermi velocities are almost independent of them . by the analysis of our model , we can show that the dispersion kink along the nodal direction occurs due to epi . the isotope effect upon the electronic dispersion is shown near the kink of energy dispersion , not in the high binding energy portions . @xcite our evaluation of the subtle isotope shifts has been backed by the report of the recent arpes experiments  @xcite which show the lack of the unusual isotope effect in the high energy portion . @xcite fortunately for us , our scenario was possibly realized in further arpes experiment . @xcite in addition to that , we have investigated how eei effects on the nodal dispersion . it can hardly affect the kink and the nodal fermi velocity ; however , it can change the dispersion at higher energy . hence , epi and eei play different roles on the nodal energy dispersion , respectively .    of course , our treatment of eei is just suited for _ weak coupling regime _ , and all of our parameter sets employed might be far from the ones for _ strong coupling regime_. if we investigate _ strong coupling regime _ with the use of another approach , the low energy structure corresponding to the superexchange @xmath114 could appear in the dispersion . however , our results presented here suggest that the structure should appear as a broad peak at higher energy due to the frequency dependence of the strong af fluctuation , which will grow into @xmath114 . as we all know , other works have already derive qualitatively similar conclusions on the basis of other models . some groups adopt @xmath115-@xmath114 models  @xcite and other groups do one - band hh models . @xcite furthermore , other groups have succeeded in explaining the arpes results . @xcite however , in our 2d three - band hh model , both the electron - electron interaction among the @xmath6 electrons and the epi on @xmath4 electrons are considered according to high-@xmath0 materials . we believe that it is important that quantitatively consistent results with the arpes experiments can be reproduced from such a model . the advantage will be when our analysis extends to the superconducting state , in which the @xmath4 electrons play important roles as well as @xmath6 electrons . the authors are grateful to h. iwasawa and t. yanagisawa for their stimulating discussions . the computation in this work was performed on intel xeon servers at neri in aist . a.  lanzara , p.  v. bogdanov , x.  j. zhou , s.  a. kellar , d.  l. feng , e.  d. lu , t.  yoshida , h.  eisaki , a.  fujimori , k.  kishio , j .- shimoyama , t.  noda , s.  uchida , z.  hussain , and z .- x . shen , nature ( london ) * 412 * , 510 ( 2001 ) . x.  j. zhou , t.  yoshida , a.  lanzara , p.  v. bogdanov , s.  a. kellar , k.  m. shen , w.  l. yang , e.  ronning , t.  sasagawa , t.  kakeshita , t.  noda , h.  eisaki , s.  uchida , c.  t. lin , f.  zhou , j.  w. xiong , w.  x. ti , z.  x. zhao , a.  fujimori , z.  hussain , and z .- x . shen , nature ( london ) * 423 * , 398 ( 2003 ) . d.  l. feng , n.  p. armitage , d.  h. lu , a.  damascelli , j.  p. hu , p.  bogdanov , a.  lanzara , f.  ronning , k.  m. shen , h.  eisaki , c.  kim , j .- shimoyama , k.  kishio , and z .- x . shen , phys . lett . * 86 * , 5550 ( 2001 ) . j.  f. douglas , h.  iwasawa , z.  sun , a.  v. fedorov , m.  ishikado , t.  saitoh , h.  eisaki , h.  bando , t.  iwase , a.  ino , m.  arita , k.  shimada , h.  namatame , m.  taniguchi , t.  masui , s.  tajima , k.  fujita , s.  uchida , y.  aiura , and d.  s. dessau , nature ( london ) * 446 * , e5 ( 2007 ) . h.  iwasawa , y.  aiura , t.  saitoh , h.  eisaki , h.  bando , a.  ino , m.  arita , k.  shimada , h.  namatame , m.  taniguchi , t.  masui , s.  tajima , m.  ishikado , k.  fujita , s.  uchida , j.  f. douglas , z.  sun , and d.  s. dessau , physica c * 463 - 465 * , 52 ( 2007 ) . h.  iwasawa , j.  f. douglas , k.  sato , t.  masui , y.  yoshida , z.  sun , h.  eisaki , h.  bando , a.  ino , m.  arita , k.  shimada , h.  namatame , m.  taniguchi , s.  tajima , s.  uchida , t.  saitoh , d.  s. dessau , and y.  aiura ( unpublished ) . w.  meevasana , n.  j. c. ingle , d.  h. lu , j.  r. shi , f.  baumberger , k.  m. shen , w.  s. lee , t.  cuk , h.  eisaki , t.  p. devereaux , n.  nagaosa , j.  zaanen , and z .- x . shen , phys . lett . * 96 * , 157003 ( 2006 ) .
we investigate the electronic dispersion of high-@xmath0 superconductor on the basis of the two - dimensional three - band hubbard model with the electron - phonon interaction together with the strong electron - electron interaction . in our model 
 , it is shown across the hole - doped region of high-@xmath0 superconductor that the electron - phonon interaction makes a dispersion kink , observed along the nodal direction , and that the small isotope effect appears on the electronic dispersion .