document: zero - field hall effect in chiral @xmath0-wave superconductors ( scs ) has drawn much attention in literature recently . 
   @xcite because of the nature of broken time reversal ( @xmath5 ) symmetry , a nonzero hall conductivity can be possible in a chiral @xmath0-wave sc . 
   indeed , it has already been shown that spontaneous hall effect could arise from the intrinsic angular momentum of cooper pairs @xcite as well as from the spontaneous surface current . 
   @xcite more recently , hall conductivity due to impurity effect @xcite or to multiband sc structure @xcite was also studied , which could give possible explanation to the observed polar kerr effect in the superconducting state of sr@xmath6ruo@xmath7 . 
   @xcite    in this work , we address the zero - field hall effect in a chiral @xmath0-wave sc originating from another mechanism , namely the vortex dynamics near kosterlitz - thouless ( kt ) transition . in two - dimensional ( 2d ) superfluid ( sf ) or sc films , quantized vortices 
   are realized as topological defects in the condensates , whose dynamics has been one of the key ingredients in understanding 2d phase transition phenomena . 
   @xcite a few decades ago , kosterlitz and thouless @xcite suggested a static theory to relate a phase transition observed in superfluid @xmath8he film @xcite to vortex - antivortex pair unbinding process across a transition temperature @xmath9 . in this picture , the logarithmic vortex - antivortex interaction is screened by smaller pairs and is renormalized to @xmath10 of its bare value @xmath11 for temperature @xmath12 . 
   the length - dependent dielectric constant @xmath13 is used to describe the static screening of pair interaction . when @xmath14 , there exists a finite pair size @xmath15 such that the interaction becomes vanishingly small . 
   consequently , the pair unbinds and free vortices emerge ; superfluidity is then destroyed . 
   soon after that , ambegaokar , halperin , nelson , and siggia ( ahns ) @xcite combined this static theory with hall and vinen s dynamical description of vortex motion @xcite to give an analysis of the dynamical effect on the phase transition . 
   concisely speaking , the renormalization process in the static theory @xcite is truncated by vortex dynamics with a characteristic length @xmath16 instead of going to its completion . @xcite here @xmath17 is the diffusivity of vortex movement and @xmath18 is the driving frequency . 
   this results in broadening transition observed in @xmath8he sf films @xcite as well as in charged fermi systems such as high - temperature scs . @xcite 
   it is instructive to explore any physical consequence stemming from this kt transition in a broken @xmath5 symmetry state . 
   possible experimental candidates of broken @xmath5 symmetry state could be superconducting sr@xmath6ruo@xmath7 ( ref .  ) or @xmath19he - a phase thin film , @xcite in which pairing of chiral @xmath0-wave type is expected . 
   indeed , in literature some theoretical works have been done to investigate new features specific to scs with pairing of this type near kt transition . 
   @xcite    in this work , we consider a 2d @xmath0-wave pairing state with @xmath20-vector @xmath21 where @xmath22 is the unit vector normal to film surface , @xmath23 and @xmath24 denote the @xmath25 and @xmath26 component of the relative momentum @xmath27 of a cooper pair , and @xmath28 is the fermi momentum . 
   assuming isotropic fermi surface , two kinds of pairing fields can be obtained : @xmath29 and @xmath30 with asymptotic behavior at large @xmath31 being @xmath32 , @xmath33 , @xmath34 , and @xmath35 . 
   @xcite here @xmath36 is the spatial coordinate and @xmath37 is the modulus of the energy gap in the bulk . from these pairing fields , 
   we can identify two types of integer vortices called @xmath38vortex and @xmath39vortex respectively .    because of spontaneously broken @xmath5 symmetry , these two types of vortices are not equivalent . 
   @xcite in particular , their hall and vinen coefficients @xcite do not share the same value , i.e. , @xmath40 and @xmath41 ( see section ii a ) . 
   this results in a nonzero  convective " term in a vortex pair polarization fokker - planck equation in addition to the conventional diffusive terms , while in its @xmath42-wave counterpart such convective motion does not enter the dynamics . 
   @xcite the relative strength of convection is quantified by a convective ratio @xmath43 in this paper . 
   it is due to such distinct feature that pair polarization transverse to the driving force field becomes possible even without applied magnetic field . 
   a nonzero vortex - dynamics - induced hall conductivity @xmath44 then follows naturally . 
   the main result of this work is that in the bound pair dynamics description , we obtain nonvanishing hall conductivity @xmath44 and ac conductivity @xmath45 near the kt transition . 
   one of the interesting features in the hall conductivity is that strong positive peak and sign changes in @xmath46 are observed at suitable frequency region above the transition temperature , as well as above @xmath9 in temperature domain at fixed frequencies . on the other hand , @xmath47 is shown to have similar features as in ahns s results . 
   we note that the shapes of two length - dependent response functions @xmath48 and @xmath49 , which corresponds respectively to the longitudinal and transverse response of bound pairs with separation @xmath50 to external perturbation with frequency @xmath18 , play a determining role on the behavior of @xmath44 and @xmath45 . 
   we also discuss the contribution of free vortex motion and the resulting total conductivity tensor . 
   the paper is organized as follows : in section ii , we generalize ahns s vortex dynamics in the chiral @xmath0-wave context . to describe the vortex - antivortex bound pair dynamics , 
   the above - mentioned response functions @xmath48 and @xmath49 are derived from the fokker - planck equation governing the pair motion . 
   together with the free vortex contribution , we arrive at a matrix dielectric function @xmath51 which describes the total screening effect under time - dependent perturbation . in section iii 
   , we investigate the frequency and temperature dependence of the conductivity tensor @xmath52 constructed from @xmath51 , treating the bound pair and the free vortex contribution separately . 
   the behavior of total conductivity @xmath4 is also discussed . 
   a summary and remark are given in section iv . 
   finally , we discuss analytic expression of @xmath3 , @xmath53 , and @xmath54 in opposite limit of the convective ratio @xmath55 and @xmath56 in appendix a. 
   to construct a matrix dielectric function , we consider a neutral sf film system resembling that employed in ahns s dynamical theory , with film thickness of order the superconducting coherence length and linear dimension @xmath57 ( @xmath58 ) along @xmath25 ( @xmath26 ) direction . 
   @xmath57 is very large and @xmath58 is large but finite . 
   vortex core motion relative to the local superfluid velocity leads to a magnus force @xmath59 @xcite @xmath60 in the above equation , @xmath61 and @xmath62 are the velocity of the @xmath63-th vortex core and the local superfluid flow excluding the diverging self - field of the @xmath63-th vortex respectively . 
   @xmath64 is the vorticity of the @xmath63-th vortex . 
   @xmath65 is the bare areal superfluid mass density , which is defined as the three - dimensional superfluid density integrated across the film thickness . 
   @xmath66 is the mass of the constituting particle , which is equal to the mass of a cooper pair . 
   the sf film is driven by a vibrating substrate . 
   a vortex core moving relatively to the substrate experiences a vorticity - dependent drag force @xmath67 @xcite @xmath68 where @xmath69 is the moving substrate velocity . @xmath70 and @xmath71 denote the vorticity - dependent drag coefficients originating from interactions with the substrate and with thermally excited quasiparticles and collective modes . here 
   @xmath72 ( @xmath73 ) @xmath74 ( @xmath75 ) for @xmath76 or @xmath77 ( @xmath78 ) for @xmath79 . 
   these quantities have been obtained for a three - dimensional clean @xmath42-wave sf / sc with isotropic fermi surface , @xcite and are related to the relaxation time @xmath80 for the caroli - degennes - matricon mode @xcite in the vortex core . due to 
   broken @xmath5 symmetry , the relaxation time @xmath81 for the mode in the @xmath39vortex and @xmath38vortex core , and thus the values of their drag coefficients , are different , resulting in the vorticity - dependent drag force @xmath67 on the vortex core . 
   these drag coefficients for 2d clean sf / scs with cylindrical fermi surface can be inferred from the three - dimensional result easily .    when two forces balance @xmath82 , the @xmath63-th vortex velocity is expressed by @xmath83 where @xmath84 here @xmath85 with @xmath86 being the boltzmann constant . 
   @xmath87 are fluctuating gaussian noise sources incorporated to bring the vortices to equilibrium . 
   their components satisfy @xmath88 . from ref . 
   , the local superfluid velocity @xmath89 is related to the spatial average superfluid velocity @xmath90 and the positions of individual vortices @xmath91 by @xmath92 . here 
   @xmath93 is the green function satisfying @xmath94 and boundary condition @xmath95 on the edges . 
   the function @xmath96 is localized in a region around @xmath97 with radius of the order of coherence length ( we could say that the function @xmath96 is a delta function in the coarse - grained scale ) . far away from the edges , @xmath98 . 
   the spatial average of @xmath99 turns out to be zero and thus @xmath100 represents the spatial average of @xmath89 . 
   finally , the time evolution of @xmath100 obeys @xmath101 reflecting the fact that the average superfluid velocity in the @xmath25 direction changes by @xmath102 when a vortex with @xmath103 moves across a strip with width @xmath104 .    in a charged system , 
   if we follow kopnin s description , @xcite the driving force on the vortex due to a transport current @xmath105 is a lorentz force @xmath106/c $ ] where @xmath107 and @xmath108 is the electron charge . 
   this force is balanced by the force from environment @xmath109 . if we set @xmath110 in eqs . 
   ( [ fd ] ) and ( [ one_vortex ] ) , the results derived in a neutral system can be carried over to a charged system by the translation @xmath111 and @xmath112 . 
   we consider the polarization of a test vortex - antivortex pair whose constituting vortices interact via a screened logarithmic interaction . 
   the pair is under the influence of an infinitesimal oscillating external field @xmath113 . 
   the langevin equation for their relative coordinate @xmath114 can be obtained by subtracting eq . 
   ( [ one_vortex ] ) from each other for opposite vorticity @xmath115 here @xmath116 and @xmath117 . 
   @xmath118 is one - half the dimensionless potential energy of the pair and the gaussian noise now satisfies @xmath119 . 
   the potential energy is given by @xmath120 in the above equation , the first term on the right hand side describes the logarithmic interaction screened by the kosterlitz dielectric constant @xmath121 . 
   @xmath122 in the second term is related to the energy required to create a pair with separation @xmath123 . in the last term , @xmath124 has dimension of velocity and acts as the perturbation . in the integration limit 
   @xmath123 is a length scale related to the size of a vortex core , and @xmath50 is the pair separation . 
   we can see in eq . 
   ( [ pair_eom ] ) that in addition to the conventional diffusive terms depending on @xmath125 , a convective term proportional to @xmath126 also enters the dynamical equation . while the strength of the former is proportional to the _ average _ of @xmath127 , that of the latter is related to the _ difference _ of @xmath128 between opposite vorticity . 
   we emphasize here that such a convective pair motion is one of the special features for a system with unequal opposite vortices , and is thus absent in an @xmath42-wave sf / sc since @xmath129 in that case . given this nonzero @xmath126 , 
   the pair polarization is tilted away from the direction of the force field @xmath130 , and has both longitudinal and transverse components even without applied magnetic field .    the fokker - planck equation corresponding to eq . 
   ( [ pair_eom ] ) is given by @xmath131 where @xmath132 is the density of pairs per unit area of separation . 
   we take the time - independent state @xmath133 to be @xmath134 where @xmath135 . 
   now we follow the standard procedure , @xcite letting @xmath136 and keeping terms to first order in @xmath137 . in frequency space 
   , we obtain @xmath138 \nonumber\\ & & + 2 \overline c \nabla \cdot \left [ \left ( \frac{m^\ast}{\hbar } \hat z \times \delta \boldsymbol e \gamma_0 \right ) - \delta \gamma \hat z \times \nabla u_0 \right ] + 2 \overline d \nabla^2 \delta \gamma.\end{aligned}\ ] ] we employ the expansion @xmath139 , where @xmath140 is the angle measured from @xmath141 to @xmath142 in anti - clockwise sense . only @xmath143 with @xmath144 and @xmath145 
   are coupled to the external field . 
   we define a convective ratio @xmath146 , which measures the relative strength between convection and diffusion , and introduce an ansatz @xmath147 @xmath148 . 
   together with an approximation @xcite @xmath149 ( which is valid near the transition ) and a change of variable @xmath150 , the equations of @xmath151 in the ansatz corresponding to angular momentum @xmath152 are given by @xmath153 we are then able to write down the change of distribution function @xmath154 in terms of @xmath155 explicitly @xmath156 , \label{d_gamma}\end{aligned}\ ] ] where @xmath157 , \quad\quad \mathcal{g}_\perp(r ) = -\frac{i}{2}\left [ g_+ ( r,\omega ) - g_- ( r,\omega ) \right ] , \label{g_g}\end{aligned}\ ] ] and @xmath158 . in eq . 
   ( [ d_gamma ] ) , @xmath159 takes the role of pair - size - dependent response longitudinal to the driving field @xmath137 and @xmath160 is the transverse response function . in the limit of vanishing @xmath43 
   , @xmath159 reduces to the @xmath42-wave sf / sc result @xmath161 while @xmath160 becomes identically zero . in eq . 
   ( [ g_g ] ) , we find that @xmath162 depend on two quantities @xmath163 which describe pair motion with angular momentum @xmath164 respectively . 
   ( left column ) and @xmath160 ( right column ) are plotted as a function of @xmath165 for @xmath166 ( a ) @xmath167 , ( b ) @xmath168 , and ( c ) @xmath169 respectively . the real and imaginary part are indicated by blue dashed lines and red solid lines respectively . 
   @xmath159 is qualitatively similar to the @xmath42-wave sf / sc result . 
   @xmath160 is a new feature with a peak in the real part and a dip - recouping shape in the imaginary part around @xmath170.,scaledwidth=80.0% ]    we describe @xmath162 first . in fig . 
   [ mathcalg ] we plot @xmath159 and @xmath160 as a function of @xmath165 for fixed frequency @xmath171 for ( a ) @xmath172 , ( b ) @xmath173 , ( c ) @xmath174 . 
   blue dashed lines and red solid lines represent their real and imaginary parts respectively . in the left column of fig . 
   [ mathcalg ] , the longitudinal response function @xmath159 is qualitatively similar to the @xmath42-wave sf / sc result @xmath175 even for finite @xmath43 . 
   it has a step - function - like real part and a delta - function - like imaginary part concentrating near @xmath170 ( @xmath170 marks the position where @xmath176 changes sign ; see appendix a ) . 
   the response function @xmath177 is a new feature in this model . in the right column of fig . 
   [ mathcalg ] , @xmath178 $ ] has a peak structure . 
   this means that neither smaller nor larger pair gives response in transverse direction . 
   only pairs with characteristic pair size @xmath179 can give rise to the hall effect . 
   besides , @xmath180 $ ] shows negative dip shape for @xmath181 , and then recoups when @xmath182 . 
   it turns out in later section that such a dip - recouping antisymmetric shape in @xmath176 about @xmath170 plays a determining role in many features of the hall conductivity . 
   these features become more significant when @xmath43 increases . 
   ( left column ) and @xmath183 ( right column ) corresponding to angular momentum @xmath152 motion are plotted as a function of @xmath165 for @xmath166 ( a ) @xmath167 , ( b ) @xmath168 , and ( c ) @xmath169 respectively . 
   the real and imaginary part are indicated by blue dashed lines and red solid lines respectively . 
   asymmetry between the two columns becomes more prominent when @xmath43 increases.,scaledwidth=80.0% ]    in eq . 
   ( [ g_g ] ) , we can see that @xmath184 depends on the average of @xmath163 , i.e. , the average response with @xmath185 and @xmath186 angular momentum , and @xmath187 is related to the _ difference _ between them . 
   this means that any asymmetry between @xmath152 motion gives rise to nonzero transverse response . in fig . 
   [ capitalg ] we plot @xmath163 as a function of @xmath188 using the same set of @xmath18 and @xmath43 as in fig .  [ mathcalg ] . again , blue dashed lines and red solid lines represent their real and imaginary parts respectively . 
   we observe that asymmetry between @xmath152 motion grows with increasing @xmath43 . for small @xmath43 in fig . 
   [ capitalg](a ) , the motion described by @xmath189 and @xmath190 is only slightly asymmetric , and they look like the familiar curve @xmath191 . 
   when @xmath43 increases in fig . 
   [ capitalg](b ) and ( c ) , the asymmetry between the left and right column becomes more and more significant . 
   it is worth noting that , when we subtract from each other the real part of the functions for opposite @xmath192 , we can obtain the dip - recouping shape of @xmath193 . 
   equation ( [ eqt_for_g ] ) can be solved exactly to give the length - dependent response functions , but here we can employ approximate solution to @xmath194 in a manner similar to ref .   by neglecting @xmath195 and @xmath196 in eq . 
   ( [ eqt_for_g ] ) . 
   then @xmath197 become @xmath198 where @xmath199 is a factor to fit the exact curve . 
   it is selected to be @xmath200 when @xmath201 and is of order @xmath202 for a wide range of @xmath43 . from eqs . 
   ( [ app_g1 ] ) and ( [ app_g2 ] ) we can see that for a small convective ratio @xmath203 , there are poles given by @xmath204 . 
   this means that the pair size with which a pair gives a strong response differs from the standard result @xmath205 by a length of order @xmath206 for motion with @xmath152 , creating the asymmetry demonstrated in fig .  [ capitalg ] . for a large convective ratio @xmath207 , such pole is removed for @xmath145 
   therefore , a broad and flat curve is expected [ fig . 
   [ capitalg](c ) right column ] .      in this subsection 
   , we introduce a susceptibility matrix @xmath208 for the bound pair polarization @xmath209 whose components are defined as @xmath210 they are so defined that the real parts of @xmath211 and @xmath212 are positive when @xmath213 . using eqs . 
   ( [ d_gamma ] ) and ( [ chib ] ) , we arrive at the expression @xmath214 the infinitesimal external field redistributes the pair polarization by an amount @xmath215 according to @xmath216 . the definition in eq . 
   ( [ chib ] ) means the external field tilts the polarization in clockwise direction when @xmath213 . 
   it should also be noted that the integration are performed from @xmath217 to @xmath15 where @xmath15 is a coherence length with behavior @xmath218 for @xmath12 and @xmath219 $ ] for @xmath14 . 
   @xmath220 is a non - universal positive constant of order unity . from eq . 
   ( [ chibb ] ) it is now clear that while the longitudinal polarization @xmath221 takes the role of the function @xmath222 in ahns s theory , the transverse polarization @xmath223 has no simple analogy in @xmath42-wave sf / scs and is specific to systems with finite convective ratios . integrating the two pair - size - dependent response functions with the weight function @xmath224 gives the susceptibility matrix elements from the bound pair contribution . 
   as for the free vortex contribution , from eq . 
   ( [ one_vortex ] ) we can obtain the equation for polarization @xmath225 for plasma of free vortices under the spatial averaged driving field @xmath226 . 
   the total free vortex density is given by @xmath227 with @xmath228 being the total number of free vortices and @xmath229 being the area of the film . in frequency space 
   , we have @xmath230.\end{aligned}\ ] ] if we define the susceptibility for free vortices as @xmath231 , we have @xmath232 where @xmath233 . 
   on the other hand , the free vortex density is related to the coherence length by @xmath234 where @xmath235 is a positive constant of order unity . 
   @xcite    together with the bound pair contribution , the total dielectric function can be obtained as @xmath236 and its inverse reads @xmath237 where @xmath238 $ ] and @xmath239 $ ] . 
   we are now in position to present our results on the hall conductivity and power dissipation due to this dielectric function . 
   it was discussed that the conductivity tensor @xmath4 in a charged system was related to the inverse dielectric function @xmath240 discussed above . 
   @xcite we can understand the relation by considering the total current under the influence of a driver coil electric field @xmath241 . 
   @xcite the vortex - modified total current @xmath242 is related to the field by @xmath243 where @xmath244 is the sheet kinetic inductance . 
   this relation is generalized to our model with a matrix dielectric function @xmath245 . 
   if we use the sign convention that the normal state electron hall conductivity with magnetic field pointing in @xmath22 direction is positive , @xmath52 and @xmath246 is related by @xmath247 in particular , we may write down @xmath248 , @xmath249 , and @xmath250 . among these expressions , the first two 
   are associated with the real and imaginary part of the hall conductivity , while the last one is related to the power dissipation @xmath251 because @xmath252 .    in the following subsections , 
   we investigate the frequency and temperature dependence of @xmath253 , @xmath254 , and @xmath255 respectively .       as a function of ( a ) @xmath256 and ( b ) @xmath257 . 
   the other parameters used are @xmath172 , @xmath258 and @xmath259 . in ( a ) , above the transition temperature , @xmath260 has peak structures and sign reversals . 
   @xmath261 has a sharp increase when frequency increases . in ( b ) , @xmath260 shows peak structure and sign reversal above @xmath262 and @xmath261 has a peak . 
   the features broaden and move to higher temperature when frequency increases.,scaledwidth=80.0% ]    we consider bound pair contribution to the hall conductivity first by ignoring the terms @xmath263 and @xmath264 in @xmath240 . 
   plots of the negative real part ( blue circles ) and imaginary part ( red squares ) of @xmath265 versus ( a ) @xmath256 and ( b ) @xmath266 are presented in fig . 
   [ hall ] . here , @xmath267 is the scaled frequency , and @xmath257 is the reduced temperature . in this and the following figures , we use the renormalization group flow equations @xmath268 and @xmath269,@xcite with initial conditions @xmath270/(1+t)$ ] and @xmath258 related to the bare interaction strength @xmath11 and chemical potential @xmath271 respectively . at transition temperature , 
   @xmath272 which is obtained numerically . here , it suffices to notice that @xmath273 is given by @xmath274 where @xmath275 is the renormalized interaction strength , and @xmath276 is related to the the number of bound pairs found with pair size @xmath277 . also , we use the convective ratio @xmath172 in the plots for illustration purpose . 
   we first focus on the discussion of the frequency dependence of @xmath265 at some fixed temperatures in fig . 
   [ hall](a ) . in the low temperature phase @xmath278 , 
   both @xmath261 and @xmath260 are positive and increase steadily with frequency . 
   when temperature increases to the high temperature phase @xmath279 , @xmath260 shows a positive - valued peak followed by sign reversal at higher frequency indicated by the red horizontal double - headed bar . 
   meanwhile , @xmath261 increases sharply around the peak of @xmath260 . 
   these features move to higher frequency side when temperature increases further ( @xmath280 ) . for temperature dependence at fixed frequencies in fig . 
   [ hall](b ) , the sign reversal and peak structure in @xmath260 are also observed when temperature varies above @xmath9 . 
   meanwhile , @xmath261 shows a simple peak structure across @xmath9 . 
   when frequency increases , the structure broadens and moves to higher temperature region . 
   sign anomaly in hall conductivity has been observed in various superconducting systems such as high - temperature scs , @xcite and it is known that single vortex  upstream " motion or change of sign of charge carriers could give rise to such phenomenon.@xcite here in our model the sign change in @xmath260 stems from the vortex - antivortex pair unbinding process . for a vortex - antivortex pair moving  downstream " with same speed , total transverse electric field is canceled and no hall effect can be observed . 
   now that the constituting vortices are not equivalent to each other , they can respond differently under the influence of transport current , and a net transverse electric field follows .     under transport 
   current @xmath281 with convective ratio @xmath213 . 
   the black solid arrow is the transport current @xmath281 . 
   the blue and red solid arrow represent the pair polarization of pair with size @xmath181 and @xmath182 respectively . 
   the dashed arrows show the electric field generated by pair motion in the respective cases . 
   pairs with size @xmath181 contribute to positive @xmath260 and vice verse . 
   the net @xmath282 determines the overall sign of @xmath260 . 
   , scaledwidth=80.0% ]    from the dip - recouping shape of @xmath283 $ ] in fig . 
   [ mathcalg ] , the dip ( recouping ) region with pair size @xmath181 ( @xmath182 ) contributes to positive ( negative ) @xmath260 , since @xmath284/r$ ] . 
   the function @xmath285 is proportional to the number of pairs present in a ring with radius @xmath50 and width @xmath286 . 
   therefore , the sign of @xmath260 depends on whether there are more pairs with pair size smaller or larger than @xmath170 . 
   an interesting point to mention is that , upstream vortex motion is not required for negative @xmath260 ; instead , it is the asymmetric pair motion in opposite angular directions in fig . 
   [ capitalg ] that brings about the hall anomaly . 
   we interpret such result to be that the direction of electric field @xmath282 generated by a vortex pair is also pair - size - dependent . 
   this interpretation is illustrated in fig . 
   [ polarization ] . 
   the pair with @xmath181 polarizes in a such a way that @xmath282 points in the direction giving positive hall signal and vice verse . 
   the final sign of @xmath260 depends on the net @xmath282 caused by all the pairs . 
   versus @xmath188 for @xmath287 , @xmath288 , and @xmath289 . 
   the vertical arrow marks @xmath170 , the position where @xmath290 changes sign . @xmath291 . at @xmath292 , @xmath293 is indicated by the red double - headed arrow . 
   three representative situations concerning the sign issue of @xmath260 are @xmath294 increasing at @xmath170 , @xmath294 decreasing at @xmath170 , and @xmath295.,scaledwidth=80.0% ]    the weight function @xmath294 , which describes the probability of a pair having pair size @xmath277 , is plotted in fig .  [ y2 ] at different temperatures . 
   the other parameters used are @xmath296 , @xmath172 , and @xmath258 . for @xmath287 
   , @xmath294 is slightly decreasing at @xmath170 , so the dip - recouping shape in @xmath283 $ ] almost cancels each other in the integral in eq . 
   ( [ chibb ] ) , giving a weak positive @xmath260 . 
   if the temperature increases , for example @xmath297 , @xmath294 becomes increasing at @xmath170 . 
   more pairs are present in the recouping region and the sign change of @xmath260 thus occurs . 
   if the temperature increases further to @xmath298 , the coherence length @xmath15 characterizing the maximum pair size becomes comparable to @xmath170 . as a result , 
   only the dip part is integrated because the recouping part lies outside of the integration limit in eq . 
   ( [ chibb ] ) . 
   a strong positive - valued peak then replaces the sign reversal . in summary 
   , temperature controls the pair size distribution , which then determines the sign of @xmath260 . 
   since the positive slope of @xmath294 at @xmath179 only occurs at the high temperature phase , the claim that the sign reversal of @xmath260 is related to kt transition is justified .    to close this subsection , 
   we discuss the hall conductivity in the static limit . for low temperature phase @xmath278 , 
   hall conductivity diverges in zero - frequency limit . 
   log - log plots of @xmath260 and @xmath261 versus @xmath299 using linear fitting show that @xmath300 and @xmath301 with both @xmath302 and @xmath303 greater than zero but smaller than unity for the temperature range @xmath304 down to @xmath305 and frequency range @xmath306 to @xmath307 . 
   this mean that @xmath308 decreases slower than @xmath18 when frequency approaches zero , and as a result @xmath309 diverges at @xmath310 .       as a function of ( a ) @xmath256 and ( b ) @xmath257 . @xmath172 and @xmath258 . in ( a ) , @xmath311 increases with frequency when @xmath278 and is suppressed at small frequency when @xmath312 . in ( b ) , @xmath311 has peak structure around @xmath9 . the peak broadens and moves to higher temperature 
   when frequency increases.,scaledwidth=80.0% ]    in this subsection , we discuss the frequency and temperature dependence of @xmath3 . 
   again , we consider bound pair contribution here . in fig . 
   [ diss ] , plots of @xmath311 versus ( a ) @xmath256 and ( b ) @xmath266 are presented . 
   the other parameters used are the same as those in fig . 
   [ hall ] . in fig . 
   [ diss](a ) , @xmath311 increases steadily with frequency at low temperature phase @xmath278 . 
   when @xmath312 , it is suppressed at small frequency . in fig . 
   [ diss](b ) , @xmath311 first increases with temperature , and is suppressed above @xmath9 , resulting in a peak structure . at higher frequency , the peak is broadened and shifts to higher temperature . 
   these features can be understood under the standard ahns theory : above @xmath9 , the coherence length @xmath15 becomes finite , and the integration limits can not cover the peak structure in @xmath313 at large @xmath188 . 
   thus , @xmath311 is suppressed at low frequency for @xmath312 in fig . 
   [ diss](a ) , and above some temperatures in fig . 
   [ diss](b ) . 
   indeed , it is not surprising that the result behaves similarly to its @xmath42-wave counterpart when we notice that the shape of @xmath314 is qualitatively similar to that of @xmath315 even for finite @xmath43 .    in zero - frequency limit in the low temperature phase , 
   @xmath316 decreases slower than @xmath18 when frequency approaches zero and thus the @xmath317 diverges . 
   this is found in log - log plots of @xmath311 versus @xmath18 that @xmath318 with @xmath319 but smaller than unity using linear fitting from @xmath304 to @xmath305 and @xmath320 to @xmath307 . 
   however , we can not interpret it to be the divergence of power dissipation because the dissipation also depends on the magnitude of the electric field in the superconducting bulk which is supposed to be vanishingly small in the static limit .       as a function of ( a ) @xmath256 and ( b ) @xmath266 . 
   the red squares and the blue circles represent the real part and imaginary part of @xmath321 respectively . @xmath322 and @xmath323 for illustration . 
   the other parameters used are the same as those in fig . 
   [ hall].,scaledwidth=80.0% ]     as a function of ( a ) @xmath256 and ( b ) @xmath266 . 
   the other parameters used are the same as those in fig . 
   [ totalhall].,scaledwidth=80.0% ]    having discussed the result due to bound pair dynamics , we study the total contribution from both bound pair and free vortex dynamics in this subsection . in order to discern the contribution of bound pair and free vortex dynamics to the total hall conductivity and power dissipation , we plot the total conductivity tensor @xmath321 and @xmath324 in figs .  [ totalhall ] and [ totaldiss ] as a function of ( a ) @xmath256 and ( b ) @xmath266 , as well as the @xmath325 , @xmath254 , and @xmath253 due purely to the free vortex motion in fig . 
   [ figfree ] . 
   we can compare these figures with those which take only the bound pair dynamics into account ( figs . 
   [ hall ] and [ diss ] ) . for the sake of illustration 
   , we use @xmath322 and @xmath323 for the weight of free vortex contribution .    for the frequency dependence in figs . 
   [ totalhall ] and [ totaldiss ] at temperature @xmath279 , we can see free vortex signal emerging at small frequency region . 
   if the temperature increases further at @xmath280 , the free vortex signal can merge with and outweigh the bound pair signal [ compared with figs . 
   [ hall ] and [ figfree](a ) @xmath280 ] . for @xmath278 ( 
   not shown ) , since there is no free vortex contribution , we expect the curves are the same as those in low temperature phase in figs .  [ hall ] and [ diss](a ) . as for the temperature dependence in figs . 
   [ totalhall ] and [ totaldiss](b ) , we can see for @xmath320 the free vortex signal appears at temperature very close to the bound pair signal and they almost merge into each other . when the frequency increases ( @xmath327 ) , this extra strong signal moves to higher temperature and becomes distinguishable from the bound pair signal . 
   the magnitude of the free vortex signal can be seen in fig . 
   [ figfree ] .    as discussed in the previous two subsections , 
   @xmath328 , @xmath329 , and @xmath330 diverge when @xmath310 . 
   however , it is only true in low temperature phase . in the case where @xmath312 , free vortex contribution dominates in the static limit @xmath331 . with such condition 
   , we have @xmath332 which is purely imaginary . from eq . 
   ( [ free ] ) , the diagonal part of @xmath333 and the off - diagonal part of @xmath334 are given by @xmath335 $ ] and @xmath336   $ ] respectively , and therefore both of them are proportional to @xmath337 . 
   this shows that @xmath338 is purely real , constant in frequency , and diverges when @xmath339 . as a result , 
   in the high temperature phase , both the hall conductivity and dissipation are protected from divergence in the zero - frequency limit but have strong signal when approaching @xmath9 from above .    , @xmath254 , and @xmath253 as a function of ( a ) @xmath256 and ( b ) @xmath266 . 
   the green triangles represent @xmath325 . the blue circles and 
   the red squares represent the negative real part and imaginary part of @xmath265 respectively . 
   the insets show the full plot range of @xmath325 . 
   the other parameters used are the same as those in fig . 
   [ totalhall].,scaledwidth=80.0% ]    finally , fig . 
   [ figfree ] shows @xmath325 , @xmath254 , and @xmath253 as a function of ( a ) @xmath256 and ( b ) @xmath266 including only the free vortex but not the bound pair contribution . 
   the frequency and temperature range are chosen to match those in figs . 
   [ totalhall ] and [ totaldiss ] . by eq . 
   ( [ free ] ) , we see that important features in the figure appear around the crossover from @xmath340 to the opposite limit @xmath341 . in fig . 
   [ figfree](a ) on the right panel , when observing from small frequency region @xmath340 , we see that all three quantities increase with frequency . 
   when frequency increases further , @xmath253 changes sign and then all the quantities decrease in magnitude with frequency , which means that it starts to be inefficient for the free vortex to respond to the perturbation when frequency is high . in fig . 
   [ figfree](b ) , similar features can be observed in the temperature domain . besides 
   , we can see that the main features in the curves emerge at a temperature closer to @xmath9 for small frequency ( left panel ) than for high frequency ( right panel ) . although the sign change in @xmath253 can also be observed in free vortex picture 
   , it does not arise from the response function @xmath342 mentioned before . 
   instead , it is attributed to the pole structure @xmath343 . 
   in this work , we have generalized ahns s vortex dynamics in the chiral @xmath0-wave superconducting state . 
   the behavior of the conductivity tensor near kt transition is investigated . 
   we show that the hall conductivity can be nonzero arising from the vortex - antivortex pair unbinding process or from free vortex motion even in the absence of magnetic field . 
   power dissipation is also predicted in this dynamical picture . in low temperature phase , 
   both the @xmath44 and @xmath344 diverge in the static limit . 
   but in high temperature phase , the contribution from free vortex motion gives finite results in the static limit . 
   we can distinguish the bound pair and free vortex contribution to the total conductivity tensor by comparing the figures from different contribution . 
   sign reversal and strong positive peak in @xmath46 are illustrated in some suitable frequency close to transition , as well as above the transition temperature at fixed frequencies . 
   on the other hand , the @xmath47 behaves in a fashion similar to that of the @xmath42-wave case . in bound pair description , 
   the induced hall conductivity is strongly influenced by the dip - recouping shape of transverse response function @xmath290 , which stems from the asymmetric angular response of pairs to the driving field . 
   the convective term in the fokker - planck equation , which originates from the broken @xmath5 symmetry nature of a chiral @xmath0-wave sc , is essential to this asymmetry . 
   all these results depend solely on the convective ratio @xmath43 , and are valid even without applied magnetic field .    in our work , temperature dependence of the drag coefficients 
   is omitted . 
   theoretical calculation of the drag coefficients has been performed microscopically for scs @xcite and for sf @xmath19he ( refs .  ) at temperatures considerably lower than the superconducting transition temperature by the use of quasiclassical theory . 
   generalization of such calculation in chiral @xmath0-wave pairing state around @xmath9 will be useful to estimate the strength of the convective ratio . 
   indeed , if we use the formulas of drag coefficients which only take into account the caroli - degennes - matricon mode at very low temperature @xmath345 $ ] and @xmath346 $ ] , @xcite we can arrive at the result @xmath347 and @xmath348 , which results in @xmath201 . however , we believe that the effect of temperature and delocalized excitations can contribute to the convective ratio . 
   in this appendix , we apply certain approximation to obtain analytic expression of @xmath311 , @xmath260 , and @xmath261 in the bound pair dynamics picture . 
   we first obtain analytic expression for susceptibility matrix elements @xmath211 and @xmath212 by employing an approximate solution @xmath349 and taking @xmath350 to be a constant function of @xmath50 . 
   we rewrite eq . 
   ( [ chibb ] ) in the form @xmath351 , \label{chi_para0 } \\ 
   \chi_\text{b}^{\perp } & = & -i\int_{a_0}^{\xi_+ } dr \frac{d \widetilde \epsilon}{dr } \left [ \left ( \frac { g_+ - g_-}{2 } \right ) + ix_0 \left ( \frac { g_+ + g_-}{2 } \right ) \right ] , \label{chi_perp0}\end{aligned}\ ] ] and further approximate the two functions appearing in the above integrands by @xmath352 \nonumber\\ & & + i \left[r i_2 \delta(r - r_2 ) - x_0 r i_1 \delta(r - r_1)\right ] , \\ 
   \frac { g_+ - g_-}{2 } & \approx & r i_1 \delta(r - r_1 ) + i r i_3 \delta(r - r_3 ) \nonumber\\ & & -ix_0 \left [ \left ( \frac {   1 } { 1+x_0 ^ 2}\right ) \theta ( r_4 - r ) + \text{sgn}(x_0 ) r i_1 \delta(r - r_1 ) \right],\end{aligned}\ ] ] where @xmath353 ^ 2 } - \frac{1 } { 1 + [ \omega r^2 / ( \lambda \overline d)+ x_0]^2 } \right ) \nonumber\\ & = &   \frac { x_0   \pi } { 4 + 4 x_0 ^ 2 } , \\ i_2 & \equiv & \frac{1}{2 } \int_0^\infty dr \frac{1}{r } \left ( \frac{\omega r^2 / ( \lambda \overline d ) } { 1 + [ \omega r^2 / ( \lambda \overline d)- x_0]^2 } + \frac{\omega r^2 / ( \lambda \overline d ) } { 1 + [ \omega r^2 / ( \lambda \overline d)+ x_0]^2 } \right ) \nonumber\\ & = & \frac { \pi } { 4 } , \\ 
   i_3 & \equiv & \frac{1}{2 } \int_0^\infty dr \frac{1}{r } \left ( \frac{\omega r^2 / ( \lambda \overline d ) } { 1 + [ \omega r^2 / ( \lambda \overline d)- x_0]^2 } - \frac{\omega r^2 / ( \lambda \overline d ) } { 1 + [ \omega r^2 / ( \lambda \overline d)+ x_0]^2 } \right ) \nonumber\\ & = & \frac{1}{4 } \left [ \cot^{-1 } \frac{2 x_0}{-1+x_0 ^ 2 } + \frac{\pi}{2 } \text{sgn}(x_0 ) \right].\end{aligned}\ ] ]    in the above expression , @xmath354 , @xmath355 , @xmath356 , and @xmath357 are some characteristic lengths defined as follows : @xmath354 , @xmath355 , and @xmath356 are the peak position of the integrand of @xmath358 , @xmath359 , and @xmath360 respectively . @xmath357 is chosen such that @xmath361 ^ 2 \ } = 1/ [ 2(1 + x_0 ^ 2 ) ] $ ] . 
   they are given respectively by @xmath362 , @xmath363 , and @xmath364 . 
   one more characteristic length worth mentioning is the pair size @xmath365 at which @xmath290 changes its sign . within the approximation described above , we find that @xmath366 
   . then we can obtain the quantity @xmath367 mentioned in the main text .    finally , 
   using @xmath368 , we obtain analytic expression for the components of the susceptibility matrix for @xmath369 @xmath370 where some shorthand notations are introduced : @xmath371 and @xmath372 . from the expression above , we notice that @xmath373 ( @xmath374 ) is ( anti- ) symmetric in @xmath43 as it should be . 
   we can also identify the contribution from the dip and recouping part of @xmath290 to be @xmath375 and @xmath376 in the imaginary part of eq . 
   ( [ chi_perp ] ) .    [ [ analytic - expression - of - im - epsilon_parallel-1-imepsilon_perp-1-and - re - epsilon_perp-1 ] ] analytic expression of @xmath311 , @xmath260 , and @xmath261 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~    in this subsection , @xmath311 , @xmath260 , and @xmath261 for nonzero @xmath43 are calculated in two limiting cases @xmath377 and @xmath378 . for @xmath377 , 
   we take @xmath379 , @xmath380 , and @xmath381 at the opposite limit @xmath382 , we take @xmath383 , @xmath384 , and @xmath385 besides , we neglect the @xmath43 dependence of @xmath386 and @xmath387 in the expansion . 
   such approximation is valid for @xmath388 and all temperature near @xmath9 because @xmath389 is almost independent of @xmath43 . 
   it is also valid for @xmath382 in low temperature phase because the renormalization almost go to completion at large @xmath43 and @xmath387 can be treated as independent of @xmath390 . 
   but it is not valid for @xmath382 in high temperature phase for two reasons : first , @xmath387 is increasing at large @xmath43 and second , @xmath389 are very likely to be greater than @xmath15 so that the integrations in eqs . 
   ( [ chi_para0 ] ) and ( [ chi_perp0 ] ) give the trivial results @xmath391 and @xmath392 . 
   expanding using small @xmath43 , the imaginary part of @xmath393 is approximately given by @xmath394 where we have used @xmath395 in the last step . 
   the presence of the second term acts as a suppression of dissipation when @xmath43 is switched on . 
   if we neglect the correction term proportional to @xmath396 , the expression reduces to the @xmath42-wave sf / scs result . in the opposite limit @xmath378 , the imaginary part of @xmath393 to the leading order in the smallness of @xmath397 
   is given by @xmath398 assuming @xmath399 . 
   surprisingly , the leading order of @xmath311 at large @xmath43 has the same form as its small @xmath43 counterpart . 
   the only difference is that the characteristic length is changed from @xmath400 to @xmath401 . 
   this can be understood if we recall that the shape of @xmath402 has no qualitative change for finite @xmath43 . 
   following similar procedure we can obtain analytic expression for @xmath403 and @xmath404 . for small @xmath43 , 
   we obtain @xmath403 to the first order in @xmath43 @xmath405     } { ( \widetilde \epsilon_4 ^ 2 +    k_0 ^ 2 \pi^8 y_4 ^ 4)^2 } \nonumber\\ & \approx & x_0 \widetilde \epsilon_4^{-2 } k_0 \pi^4 \left [ \left(y_1 ^ 2 - y_4 ^ 2\right ) + 4 \widetilde \epsilon_4^{-1 } k_0 \pi^3 y_4 ^ 4 \right].\end{aligned}\ ] ] we have used the condition @xmath406 in the last line . from the expression , the behavior depends on whether @xmath407 or @xmath408 dominates . from numerical results , @xmath407 dominates at a wide range of temperature in the low temperature phase . only when approaching the transition temperature from below and away from the vicinity @xmath409 , the @xmath410 becomes more significant . on the other hand , @xmath411 we have used @xmath412 in the second line . for large @xmath43 , we assume @xmath413 and obtain to the leading order @xmath414 and @xmath415 it has to be mentioned that , although we have used the condition @xmath406 extensively , it is only valid in the temperature range @xmath416 . 
   above such temperature , numerical results show that @xmath417 can be greater than @xmath386 at small frequency region .
summary: we discuss hall effect and power dissipation in chiral @xmath0-wave superconductors near kosterlitz - thouless transition in the absence of applied magnetic field . in bound pair dynamics 
   picture , nonzero hall conductivity emerges when vortex - antivortex bound pair polarization has a component transverse to the direction of external perturbation . 
   such effect arises from the broken time reversal symmetry nature of a chiral @xmath0-wave superconducting state and does not require an applied magnetic field . 
   a frequency - dependent matrix dielectric function @xmath1 is derived to describe the screening effect due to the pair polarization . 
   quantities related to the hall conductivity and power dissipation , denoted as @xmath2 and @xmath3 , are investigated in frequency and temperature domain . 
   the imaginary part of the former can show peak structure and sign reversal as a function of frequency close to transition temperature , as well as in the temperature domain at various fixed frequencies . 
   the latter shows peak structure near transition temperature . 
   these features are attributed to pair - size - dependent longitudinal and transverse response function of bound pairs . 
   consequences due to free vortex dynamics and the resulting total conductivity tensor @xmath4 are also discussed .