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helical dirac fermions protected by topology and found at the surface of topological insulators ( ti)@xcite exhibit spin locked perpendicular to their momentum.@xcite doping with magnetic impurities can break time reversal symmetry and create massive dirac fermions as has been experimentally demonstrated@xcite in @xmath1 . a gap can also be introduced in thin films when the distance between top and bottom surfaces is of the order of the extend in space of the surface states . @xcite gapped dirac fermions arise in many other systems , for example in two dimensional membranes such as the dichalcoginide @xmath2 @xciteand silicene@xcite with buckled honeycomb lattice . in both these cases pseudospin plays the analogous role to the real spin of topological insulators . the electron - phonon interaction renormalizes quasiparticle dynamics and leads to important observable changes in electronic properties @xcite which illustrate the effects of many body renormalizations not captured in single particle theories . for the dirac electrons in graphene as an example , features observed in the density of states @xcite and in the dispersion curves measured in angular resolved photo emission spectroscopy @xcite have been interpreted as phonon structure . @xcitethe optical properties of graphene are also renormalized in a non trivial way.@xcite in a simple bare band picture there is no optical absorption in the photon energy region between the drude intraband contribution centered about @xmath3 and the interband onset at twice the value of the chemical potential @xmath4 . in reality the real part of the dynamic longitudinal conductivity is observed to be finite and almost one third its universal background@xcite value in this photon region.@xcite this absorption is due to many body renormalizations and is at least partially assigned @xcite to the electron - phonon interaction which provides boson assisted processes referred to as holstein processes . a phonon is created along with an electron - hole pair . understanding the transport properties of the dirac electrons on the surface of a topological insulator is important for possible device applications . at finite temperature the electron - phonon interaction is expected to be an important scattering chanel @xcite , x. zhu et al . @xcite have studied the surface phonons on the ( 001 ) surface of @xmath5 and in particular have found a giant kohn anomaly associated with a branch having a maximum of 18 thz . from measurements of the phonon self energy , the same group @xcite determined the size of the electron - phonon interaction report a coupling constant of 0.43 for a particular branch which is much larger than reported in angular resolved photo - emission spectroscopy . one such study by z. h. pan et al . @xcite gives a mass enhancement @xmath6 of 0.08 while another by r. l. hatch et al . @xcite found @xmath7 0.25 . a more recent report @xcite gives @xmath8 0.17 with characteristic phonon energy @xmath818 mev . the results of x. zhu et al . are further supported by an infrared study by a. d. laforge et al . @xcite which found a strong electron - phonon coupling to a 7.6 mev optical phonon while s. giraud et al . @xcite provide arguments for coupling to acoustic phonon with @xmath7 0.42 in their films and even larger in other geometries . very recently j. a. sobota et al . @xcite , using time - resolved photoemission spectroscopy , find evidence for coupling to a 2.05 thz surface phonon mode with the dirac electrons in @xmath5 . possible complications in the interpretation of optical pumping on time resolved data were discussed by s. ulstrup @xcite and need to be kept in mind . finally , the recent transport measurements of m.v . costache et al . @xcite were interpreted with strong coupling to a single optical phonon mode of energy @xmath8 8 mev .    in this paper we will emphasize the effect of the electron - phonon interaction on the spin texture of gapped dirac helical fermions . we will present for comparison , results for the familiar spectral density ( @xmath9 ) associated with quasiparticle renormalization which we compared with the much less familiar equivalent function which enters gap renormalizations ( @xmath10 ) . this is the essential quantity for the calculation of the z - axis ( perpendicular to the surface states ) properties . while only @xmath9 is needed in calculations of the density of states @xmath11 both @xmath12 and @xmath13 enter the dynamic conductivity @xmath14 . in section ii we present the necessary formalism for both the self energy associated with quasiparticle renormalizations @xmath15 and with the renormalization of the gap @xmath16 which determine the needed spectral functions @xmath9 and @xmath10 . numerical results are presented in section iii which deals with spectral densities , density of states and optical conductivity . in section iv we consider both in plane and out of plane spin texture . a summary and conclusions are given in section v. and @xmath17 in mev as a function of @xmath18 in mev . @xmath19 ( real part , solid black ) , and imaginary part ( dashed red ) gives the quasiparticle renormalization and @xmath17 ( real part , dotted blue ) , and imaginary part ( dash - dotted purple ) gives the gap renormalization . our choice of g=10 corresponds to a mass enhancement @xmath20.,width=307,height=307 ]    we begin with a model hamiltonian for the electronic states at the surface of a topological insulator which has the form    @xmath21+\delta\sigma_{z}+e_{0}(\mathbf{k})\label{ham}\ ] ]    where @xmath22 s are pauli spin matrices for real electron spin , @xmath23 is the fermi velocity of the dirac electrons , @xmath24 is momentum , @xmath25 is a gap which can be introduced for example by doping with magnetic impurities and @xmath26 is a quadratic term @xmath27 with @xmath28 the electron mass . the introduction of @xmath29 gives particle hole asymmetry . here we will assume that @xmath28 is very large , as a first approximation , and will ignore this term . the non interacting green s function takes on the form @xmath30 with @xmath31 the matsubara frequencies and @xmath32 is the non interacting chemical potential . the eigen energies are @xmath33 with @xmath34 for conduction and valence band respectively . these dispersion curves are shown schematically in fig . 1 with ( right frame ) and without ( left frame ) a gap .    the matrix ( @xmath35 ) green s function @xmath36 can be written in terms of the scalar @xmath37 as @xmath38 with the vector @xmath39 defined as @xmath40 we want to include an electron - phonon interaction . the simplest is the holstein model for coupling to a phonon mode of energy @xmath41 with matrix element between electronic and phonon assumed to be a constant ( @xmath42 ) . it is written as @xmath43 where @xmath44 creates an electron of momentum @xmath24 and spin @xmath45 and @xmath46 creates a phonon of momentum @xmath47 reduced to the first surface state brillouin zone and energy @xmath41 with @xmath42 coupling constant . the perturbing hamiltonian ( 5 ) provides a self energy to electron motion which has two components one proportional to the unit matrix @xmath48 and the other to @xmath49 . writing @xmath50 we get in lowest order perturbation theory @xmath51\label{sigmai}\end{aligned}\ ] ] and @xmath52\label{sigmaz}\end{aligned}\ ] ] where @xmath53 and @xmath54 are fermion and boson distribution functions @xmath55 $ ] respectively . here @xmath32 is the chemical potential which applies only to the electron distribution . note that @xmath56 in eq . ( 7 ) is directly proportional to the gap @xmath25 and will vanish for @xmath57 . in terms of the self energies @xmath58 and @xmath56 the interacting matrix green s function takes on the form @xmath59 with @xmath60 and @xmath61 in ref . ( 19 ) a factor @xmath62 was mistakenly introduced instead of @xmath56 in ( 9 ) and ( 10 ) . this leads to small numerical differences but has no qualitative significance . the spectral function associated with @xmath48 and @xmath49 matrix are @xmath63 which is the quantity measured in angular resolved photo emission spectroscopy arpes.@xcite further @xmath64\notag\nonumber \\   & = & -\frac{1}{\pi}im\{\frac{s[\delta+\sigma^{z}(\omega+i\delta)]g(k , s,\omega+i\delta)}{\sqrt{\hbar^{2}v_{f}^{2}k^{2}+(\delta+\sigma^{z}(\omega+i\delta))^{2}}}\}.\end{aligned}\ ] ] a considerable and instructive mathematical simplification of these complicated expressions for ( 11 ) and ( 12 ) result when the imaginary part of the z - component of the self energy @xmath65 is ignored . we get @xmath66^{2}+[im\sigma^{i}(\omega)]^{2}}\ ] ]    @xmath67a(k , s,\omega)}{\sqrt{\hbar^{2}v_{f}^{2}k^{2}+(\delta+re\sigma^{z}(\omega))^{2}}}\end{aligned}\ ] ]    with @xmath68 and @xmath69 where @xmath70 is a function of @xmath71 and @xmath18 . if we further take the imaginary part of @xmath19 to be zero ( 13 ) reduces to a dirac delta function but the real part of both @xmath72 and @xmath73 remain which renormalize the single particle energies and the gap . the probability of occupation of the state @xmath24 denoted by @xmath74 at temperature @xmath75 is given by@xcite @xmath76 and this is to be compared with the corresponding expression for the z - component of spin @xmath77 @xmath78 we can also calculate the average value of the square root of the sum of the squares of @xmath79 and @xmath80 component of spin which remains locked perpendicular to momentum but its magnitude is changed by the electron - phonon coupling @xmath81 with @xmath82g(k , s,\omega)}{\sqrt{\hbar^{2}v_{f}^{2}k^{2}+(\delta+\sigma^{z}(\omega))^{2}}}\}\ ] ] the density of electronic states @xmath11 follows from @xmath83 on integration over @xmath71 @xmath84 this quantity enters scanning tunneling microscopy experiments ( stm).@xcite the real part of the dynamic longitudinal conductivity which gives the absorption spectrum for light follows as @xcite @xmath85\notag\nonumber \\   &   & \times\lbrack a_{i}(\mathbf{k},\omega)a_{i}(\mathbf{k,}\omega+\omega)-a_{z}(\mathbf{k},\omega)a_{z}(\mathbf{k,}\omega+\omega)]\end{aligned}\ ] ] with @xmath86 and @xmath87 . note that both @xmath12 and @xmath13 enter this quantity even though it is an in - plane property . as a function of energy @xmath18 in mev for three values of momentum @xmath71 namely @xmath88 ( solid black ) , @xmath89 ( dotted blue ) and @xmath90 ( dashed red ) in units of the inverse lattice parameter @xmath91 . the top frame is for @xmath10 ( associated with the gap renormalization ) and the bottom frame is for @xmath9 associated with quasiparticle renormalization . vertical dotted black lines are at @xmath92 with phonon einstein energy set at 7.5 mev . the gap @xmath93 mev and the bare chemical potential is @xmath94 mev.,title="fig:",width=307,height=307 ]   as a function of energy @xmath18 in mev for three values of momentum @xmath71 namely @xmath88 ( solid black ) , @xmath89 ( dotted blue ) and @xmath90 ( dashed red ) in units of the inverse lattice parameter @xmath91 . the top frame is for @xmath10 ( associated with the gap renormalization ) and the bottom frame is for @xmath9 associated with quasiparticle renormalization . vertical dotted black lines are at @xmath92 with phonon einstein energy set at 7.5 mev . the gap @xmath93 mev and the bare chemical potential is @xmath94 mev.,title="fig:",width=307,height=307 ]     in inverse ev versus energy @xmath18 in ev . the dotted red line includes the electron - phonon interaction with @xmath95 in eq . ( 6)-(7 ) . the solid black line which is for comparison is for bare bands ( @xmath96 ) . the top frame has a gap @xmath97 mev while the bottom frame has a gap of 20 mev . , width=288,height=307 ]     in units of @xmath98 as a function of photon energy @xmath18 in mev . the dotted red line includes the electron - phonon interaction with @xmath95 in eq . ( 6)-(7 ) . the solid black line , which is for comparison , is for bare bands ( @xmath96 ) . the top frame has no gap @xmath57 and the bottom frame includes a gap @xmath93 mev . in both cases the bare chemical potential @xmath99 mev . the arrow identifies the phonon structure.,width=307,height=307 ]    in lowest order perturbation theory the chemical potential @xmath32 in eq . ( 6 ) and ( 7 ) is to be interpreted as having its non interacting , independent particle value and will be quoted in all the results to be presented in this paper . with electron - phonon coupling the chemical potential will in general be shifted from its free band value to @xmath100 . in fig . 2 we show our results for @xmath101 in mev as a function of energy @xmath18 in mev where we have analytically continued from matsubara to real frequencies . the solid black curve is @xmath102 with @xmath103 given by the dashed red curve while the dotted blue is for @xmath104 with its imaginary part @xmath105 given by the purple dash - dotted curve . first the real part of the quasiparticle self energy is not zero at @xmath106 but rather is of the order of @xmath107mev which reduces the chemical potential of the interacting picture by about 1/4 of its bare band value here taken to be 25 mev with the gap @xmath25 equal to 20 mev . with @xmath42 set to a value of 10 in eq . ( 6 ) and ( 7 ) , the mass enhancement parameter @xmath7 0.3 as can be seen in fig . the parameter @xmath6 is the slope of the real part of @xmath108 at @xmath3 , i.e. @xmath109 at @xmath3 . we have also taken the einstein phonon energy to be 7.5 mev which is representative of what is seen in many but not all experiments in @xmath5 and has been identified in the work of x. zhu et al . @xcite as a surface phonon . as discussed in the introduction coupling to specific optical phonons have been found in other experiments @xcite . of course there can also be coupling to acoustic phonons as studied by giraud et al .. @xcite these authors use a continuum model for the phonons and a deformation potential model to describe their coupling to the helical surface dirac fermions with due attention to the modifications brought about by helicity of the charge carriers . in our simplified model these complications enter only in determining the size of @xmath42 in eq . ( 6 ) and ( 7 ) which we set through consideration of experiment ( i.e. @xmath7 0.3 ) . in principle these could also be coupling to electronic modes but this is not considered in this work which deals with electron - phonon coupling alone . our choice of gap @xmath97 mev is representative of the work of chen et al . @xcite who found in arpes experiments a gap of @xmath110 mev in @xmath5 magnetically dopped with @xmath111 . we note prominent phonon structure at @xmath112 mev by choice , with additional structures at @xmath113 and at @xmath114 . for @xmath115 the imaginary part of the quasiparticle self energy is zero in the interval ( @xmath116 ) after which its absolute magnitude is finite and increases linearly with increasing value of @xmath18 . this dependence reflects the linear dependence on energy of the underlying bare density of states . also it needs to be negative because @xmath117 is a scattering rate , and hence positive . by contrast the imaginary part of the gap self energy is flat because the extra factor of @xmath118 appearing in ( 7 ) but not in ( 6 ) compensates for the density of states variarion which comes from the sum over @xmath119 . note also that both self energies have a finite imaginary part in the interval @xmath120 and @xmath121 . while the real part of the quasiparticle self energy can be both positive or negative , the real part of the gap self energy is everywhere negative and so decreases the bare gap @xmath25 at all @xmath18 considered here . its magnitude is everywhere of order 1/4 of the input value of @xmath25 with some @xmath18 dependence and is encoded with boson structure . these self energies have a profound effect on the corresponding spectral densities of eq . ( 11 ) and ( 12 ) as we see in fig . 3 where the top frame applies to @xmath122 ( conduction band ) and the bottom frame is for @xmath123 . for the range of values of @xmath71 shown namely , @xmath124 , @xmath125 and @xmath126 in units of the inverse of the lattice parameter ( @xmath91 ) , these two functions do not differ much from each other . this is traced to the fact that @xmath10 has the extra factor @xmath127 as compared with @xmath9 where we have , for simplicity , neglected renormalizations as these do not change the argument in an important way . as long as @xmath128 is small compared with @xmath25 the relevant additional factor is near one . for large @xmath71 however , larger than those shown in fig . 3 , @xmath10 will become much smaller in magnitude than @xmath9 by a factor of @xmath129 . returning to the top frame of fig . 3 we first note in the solid black curve for @xmath88 ( which is the momentum of the bottom of the bare conduction band ) a large , only slightly broaden and shifted in energy , quasiparticle peak centered approximately at @xmath130 mev below the fermi surface at @xmath106 . for the bare bands it would be at @xmath131 mev instead . but with correlations there are also additional features . there is a large increase in @xmath10 starting at @xmath132mev followed by a large boson structure extending down to @xmath133mev where it drops to zero . such a feature is also seen in the dashed red curve but it is much smaller . the lower cut off is identified to correspond to the energy @xmath134 . if we had shown the valence band contribution to the spectral density it would have a further boson structure at @xmath135 as in the self energy of fig . it is clear from this description that the effect of the electron - phonon interaction on the bare band is much more complicated than a simple constant shift in gap value and a slight shift in quasiparticle energies with small broadening . note also the clear phonon sidebands at energies above @xmath136 in all three curves . the black vertical dotted lines identify energies @xmath92 for easy reference . a different representation of these changes is given in fig . 4 where we show a false color plot of @xmath10 as a function for energy @xmath18 along the vertical axis and momentum @xmath71 along the horizontal as seen in eq . ( 12 ) and ( 14 ) , @xmath12 carries the sign of @xmath45 and is positive ( red ) in the renormalized conduction band and negative ( blue ) in the renormalized valence band . the phonon at @xmath92 is clearly identified as is the bottom of the renormalized conduction band at @xmath134 and the valence band phonon structure at @xmath135 . particularly striking is the large modification of the bare band dispersion curves in the region just above the renormalized conduction band minimum . the large renormalization effects seen in fig . 3 and fig . 4 can have a profound effect on certain quantities while at the same time have much more modest manifestations in others as we will now describe . in fig . 5 we show results for the density of states ( dos ) @xmath11 given in eq . the top frame is for a gap @xmath97 mev and the bottom for @xmath93 mev . the red dotted curve is the renormalized quasiparticle dos while the black solid curve is the bare band case shown for comparison . in both frames the bare chemical potential is set at @xmath137 mev , which is @xmath131 mev above the gap in the conduction band ( for the case @xmath93 mev ) . the energy variable @xmath18 is set so that @xmath3 corresponds to the fermi surface so that the bottom of the conduction band is at @xmath138 mev in bottom frame and @xmath139 mev in top frame with the top of the valence band at @xmath140 mev and @xmath141 mev respectively . in both cases the remaining gap between valence and conduction band has been very much reduced over its bare band value . the four phonon structures at @xmath142 and @xmath134 are clearly seen with the bottom of the conduction band given by the energy @xmath134 . by comparison , the corresponding boson structures in the real part of the dynamic longitudinal optical conductivity @xmath143 which is more closely related to a convolution of two dos factors and is given in eq . ( 22 ) , are much more modest as seen in fig . the top frame is for gapless dirac fermions ( @xmath57 ) and is included for comparison while the bottom frame is for the gapped case with @xmath93 mev . the solid black curves are for the bare band case with a small residual scattering rate included to broaden out the drude peak due to intraband transitions . this peak is large only at small @xmath18 and is centered at @xmath3 . the bare chemical potential is @xmath144 mev and we see the onset of a second absorption band coming from the interband transitions which start at @xmath4 . these transitions continue up to large energies and provide the so called universal background . in our units for @xmath143 which is @xmath98 , this background@xcite has a height of @xmath145 . as is clear in the top frame for @xmath57 this height is almost unaffected by the electron - phonon interaction ( red dotted curve ) . staying with the top frame we see however the appearance of the holstein processes above @xmath136 which provides significant phonon assisted absorption in the photon region above @xmath41 and below the main interband absorption edge at @xmath146 . the other feature to be noted is that , for the correlated case , the onset of the interband transitions has moved to lower energies and is now at twice the value of the interacting chemical potential . the lower frame which is for the gap fermion case has another important element . as is well known,@xcite when @xmath147 there is a peak in the interband transitions just above the threshold energy which persists up to a few @xmath25 above the threshold before the value of the background is reestablished at its universal value . in the clean limit we have the analytic result @xmath148\end{aligned}\ ] ] where @xmath149 is a heaviside function . we see that , just above the interband onset the conductivity is larger than its universal value of @xmath150 as is seen most clearly in the solid black curve of the lower frame of fig . 6 which is the bare band result . the horizontal straight line segment indicate @xmath145 . we note that , with electron - phonon ( dotted red curve ) , the magnitude of the absorption in the region of the edge still remains above the universal background value but now there is also a small phonon structure highlighted by the vertical arrow . no such structure is seen in the top frame for @xmath57 . it is the variation with energy of the background ( in the presence of a finite gap ) which allows for the phonon structure to be revealed . as a function of momentum @xmath71 normalized to @xmath151 . the solid black line is for @xmath96 ( no electron - phonon interaction ) and is for comparison with the dotted blue curve which includes the self energy with coupling @xmath95 ( eq . ( 6)-(7 ) ) . , width=307,height=307 ]     as a function of momentum @xmath71 normalized to its value at the fermi surface @xmath151 . the solid black curve is for comparison and involves the bare bands ( @xmath96 ) with gap @xmath93 mev . the dotted blue has the same gap but includes the electron - phonon renormalizations with coupling @xmath95 in eq . ( 6)-(7 ) . the dashed red is for a smaller value of gap @xmath97 mev . , width=307,height=307 ]     of eq . ( 18 ) as a function of momentum @xmath71 normalized to its value at the fermi surface @xmath151 ( short dashed red curve ) with the probability of occupation of state @xmath74 of eq . ( 17 ) ( long dashed black curve ) . the heavy solid black line gives the case of bare bands for comparison . , width=307,height=307 ]    the z - axis spin texture is determined from the spectral density @xmath152 of eq . results are presented in fig . 7 for @xmath77 as a function of momentum @xmath71 normalized to its value at the fermi surface ( @xmath151 ) . @xmath77 involves an integral over frequency @xmath18 of the overlap of @xmath152 and the fermi function @xmath153 as given in eq . ( we only show results for the conduction band @xmath154 . for the valence band @xmath155 and @xmath77 will change sign . the gap @xmath93 mev and the chemical potential @xmath94 mev as before in section iii . the solid black curve is the bare band result for temperature @xmath156 . we see an abrupt drop to zero at @xmath157 because of the thermal factor which here is basically a heaviside function with cut off at the fermi surface . by contrast the dotted blue curve includes the electron - phonon renormalization discussed at length in the previous section . now the magnitude of @xmath77 at @xmath88 is reduced below its bare band value . more importantly the jump at @xmath158 is smaller than it is in the solid curve and further , there are finite tails beyond this momentum which are entirely due to many body correlation effects that go beyond a bare band description . they can be taken as representative of other correlation effect such as those due to electron - electron interactions rather than electron - phonon . compared to the modifications found in the spectral densities , the density of states and the conductivity , these effects are certainly not as spectacular . mathematically this results from the fact that @xmath77 involves an integration over energy @xmath18 while it is this energy dependence which reflects most directly the details of the correlation effects . the in - plane spin texture is also changed , although the perpendicular spin momentum locking remains . in fig . 8 we show results for the momentum dependence ( in units of @xmath158 ) of @xmath159 defined in eq . ( 19 ) . it gives the magnitude of the in - plane spin at momentum @xmath71 . without interactions it would be equal to @xmath160 and is shown as the solid black curve for a gap of @xmath161 mev and temperature @xmath162k . it starts at zero for @xmath88 and rises monotonically with a sharp cut off at @xmath157 at zero temperature . when the electron - phonon interaction is included through eq . ( 20 ) we get the dotted blue curve which follows closely the solid black curve with the important difference that it has tails beyond @xmath163 characteristic of correlation effects . the dashed red curve is for the same value of @xmath42 but now the gap has been reduced to @xmath164 mev . in this case the curve rises more sharply out of @xmath88 , flattens before showing a sharp drop at @xmath163 and the correlation tails beyond this drop off are now considerably larger . the correlation tails seen in @xmath77 as well as in @xmath159 are very similar to the much more familiar case@xcite of the momentum distribution in a fermion system with electron - phonon interaction . the probability of occupation of the state @xmath71 at zero temperature denoted by @xmath74 is given in eq . ( 17 ) and in the bare band picture is a step function @xmath165 . as shown in fig . 9 long dashed black curve for @xmath162k , @xmath95 and a gap @xmath93 mev , @xmath74 is considerably reduced from value one throughout the occupied states . it still has a finite discontinuous jump at @xmath163 with correlation tail for @xmath166 . the short red dashed curve is @xmath77 previously calculated and presented here for comparison with @xmath74 . this shows the qualitatively similar effect of the electron - phonon interaction on these two quantities . we calculated the effect of electron - phonon coupling on a system of helical gapped dirac fermions . a simple holstein model with coupling to a single einstein phonon of energy @xmath41 was used . for massive dirac fermions two self energies need to be introduced . there is the familiar quasiparticle renormalizarion which changes the bare band energies to dressed dispersion curves and provides damping . in addition there is a second self energy directly associated with modifications of the gap . it is energy ( @xmath18 ) dependent and complex with the real part directly modulating the magnitude of the bare band gap . both these self energies enter the quasipaticle spectral density @xmath83 with @xmath71 momentum @xmath18 energy and @xmath34 giving conduction and valence band respectively . in direct analogy , a second spectral density @xmath152 associated with the gap channel is also introduced . the first function @xmath83 can be measured directly in angular resolved photo - emission spectroscopy ( arpes ) . its average over momentum @xmath71 determines the electronic density of states @xmath11 measured in scanning tunneling spectroscopy ( stm ) . the second @xmath152 enters along with @xmath83 the expressions for the dynamic longitudinal optical conductivity . it also determines the out of plane spin texture . both @xmath167 and @xmath12 are functions of @xmath18 and are encoded with sharp phonon structures at @xmath142 and @xmath134 . these phonon structures manifest directly in the density of states and in a somewhat different way in the optical conductivity . the out of plane and in plane spin textures are modified by the electron - phonon interaction . but these quantities involve an integration over energy of an overlap of spectral density and thermal factor . consequently the manifestation of phonon - electron coupling in these quantities is more subtle and not as direct . nevertheless important corrections to a bare band picture arise . just as the probability of occupation of a state of momentum @xmath71 is reduced from one for @xmath71 less than @xmath151 , and extended ( in momentum ) tails appear for @xmath166 , the magnitude of @xmath77 follows the same trend . a similar picture applies for the magnitude of the in plane spin component . while the spin remains locked in the direction perpendicular to momentum , correlation tails appear beyond @xmath163 . while the calculations are for the explicit case of the electron - phonon interaction , they serve to illustrate how many body interactions in general modify the bare band picture . this work was supported by the natural sciences and engineering research council of canada ( nserc ) and the canadian institute for advanced research ( cifar ) . this material is also based upon work supported by the national science foundation under the nsf epscor cooperative agreement no . eps-1003897 with additional support from the louisiana board of regents . jpc initiated the work and wrote much of the text , zl did all the calculations . 10 m. z. hasan and c. l. kane , rev . mod . phys . * 82 * , 3045 ( 2010 ) . xuetao zhu , l. santos , r. sankar , s. chikara , c. . howard , f. c. chou , c. chamon , and m. el - batanouny , phys . lett . * 107 * , 186102 ( 2011 ) ; xuetao zhu , l. santos , c. howard , r. sankar , f. c. chou , c. chamon , and m. el - batanouny , phys . rev . lett . * 108 * , 185501 ( 2012 ) .
the metallic surface states of a topological insulator support helical dirac fermions protected by topology with their spin locked perpendicular to their momentum . they can acquire mass through magnetic doping or through hybridization of states on opposite faces of a thin sample . in this case 
 there can be a component of electron spin oriented perpendicular to the surface plane . 
 the electron - phonon interaction renormalizes the dynamics of the charge carriers through their spectral density . 
 it also modifies the gap channel and a second spectral function enters which , not only determines the out of plane spin component , but also comes into in - plane properties . while the out of plane spin component is decreased below the fermi momentum ( @xmath0 ) , the in plane component increases . 
 there are also correlation tails extending well beyond @xmath0 . 
 the angular resolved photo - emission line shapes aquire holstein side bands . 
 the effective gap in the density of states is reduced and the optical conductivity aquires distinct measurable phonon structure even for modest value of the electron - phonon coupling .