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in low dimensional systems , the description of transport in terms of linear phenomenological laws , that works well in three dimensions , may not be possible . one example is the fourier law which relates the thermal current density @xmath8 to the thermal gradient @xmath9 by @xmath10 , where the @xmath11 is the thermal conductivity of the material . theoretical investigations of a number of translation invariant one - dimensional systems show that the fourier law is not followed in these systems @xcite . the ratio of the current density to the thermal gradient is no longer independent of the system size . nevertheless this ratio , again called ` thermal conductivity ' , is still useful in characterizing anomalous transport . the calculation of ` thermal conductivity ' in these systems can be problematic . ideally one would like to calculate the non - equilibrium stationary state ( nss ) measure of the system and calculate the expectation values of @xmath8 and @xmath9 using the nss measure . this requires the existence of local thermal equilibrium ( lte ) so that a local temperature @xmath12 is properly defined . a method generally used when considering a one dimensional system with its ends at two different temperatures @xmath13 and @xmath14 , is to calculate directly the current @xmath8 and the quantity @xmath15 , @xmath16 being the length of the system . another approach is to use the kubo - green formula which involves the time integral of the equilibrium current - current correlation function . the first formula does not require the existence of lte whereas the second does and it is not clear how the results from the two methods can be compared .    in 1-d systems with translation invariance , the ` conductivity ' , @xmath11 , shows a power - law dependence on the size of the system ; @xmath17  @xcite . a question that has been investigated extensively is whether @xmath18 is universal across a class of systems . the other related questions are about the number of such universality classes and the values of @xmath18 characterizing them . such an investigation has to take into account the difficulties posed by the considerations in the previous paragraph . hydrodynamic mode - coupling theory predicts two universality classes corresponding to @xmath19 and @xmath20 for one - dimensional momentum conserving systems  @xcite . specific models of one - dimensional lattice systems with pair - wise nonlinear interactions between neighboring particles have been studied extensively using numerical and analytical methods  @xcite . the analytical studies involve approximations at some stage , whereas numerical studies are handicapped by the system size and the time scales that can be achieved in computers which makes it difficult to give a definitive answer to the question of the universality of @xmath18 . results of these studies agree with the hydrodynamic results that the generic value of @xmath18 is @xmath21 ; when the leading order nonlinearity is cubic @xmath20  @xcite . however , for the so - called fermi - pasta - ulam chain , where the nonlinearity is quartic , there is still no consensus ; numerical simulations show different values of @xmath18  @xcite and analytic calculations , both classical and quantum , show @xmath22  @xcite . apart from the obvious theoretical interest in these problems there has also been a lot of experimental interest in studying heat transport at micro and nano scales @xcite . recent experimental studies on nanowires such as carbon nanotubes and boron - nitride nanotubes show a clear breakdown of the fourier law  @xcite . in order to attempt an understanding of realistic systems , it is essential to include all the modes of vibrations . theoretical results , both numerical and analytical , mentioned above have been obtained for only the longitudinal vibrations of the chain . it is natural to investigate how the exponent @xmath18 is affected by the transverse vibrations . chains with both longitudinal and transverse vibrations have only been studied in the work of wang and li  @xcite . the wang - li model contains two kinds of intractions ; a pairwise interaction between the neighboring particles and another interaction that depends on the bending of the chain . the harmonic approximation of the model yields modes whose frequencies @xmath23 depend on the wave - vector @xmath0 in the following manner . for small q , the longitudinal modes have @xmath24 whereas the transverse modes @xmath25 . wang and li used the mode - coupling analysis ( mca ) and molecular dynamics ( md ) simulations to study this system . their first important observation from md simulations is that the phonon dispersions are strongly renormalized from their bare values , particularly the transverse phonon dispersion gets renormalized to become linear in q. for this reason they use an effective hamiltonian method in which the form of the hamiltonian and its parameters are taken from the md simulations . for the conductivity exponent they obtain a generic value @xmath20 , but they also observe a value of @xmath22 in their md simulations which is attributed to crossover effects . when the parameter measuring the strength of transverse vibrations is sufficiently large , they also find that @xmath19 .    in this paper , we study the quantum mechanical version of this model along the lines of  @xcite . in this procedure , one essentially calculates the second order self - energy of the phonons self - consistently , which enables us to obtain the wave - vector dependent relaxation rate @xmath26 . the thermal conductivity of the chain is then obtained through the use of kubo formula . the finite size of the chain is taken into account by putting on the time integral in the kubo formula an upper cutoff proportional to the size of the chain @xcite . this prescription yields @xmath27 , where @xmath28 governs the small - q behavior of the relaxation rate , @xmath29 . for the wang - li model , we find that in the second - order perturbation calculation , the relaxation rate @xmath30 for the transverse phonon diverges as @xmath31 , whereas the relaxation rate @xmath32 for the longitudinal phonons becomes constant . this makes the bare phonons invalid normal modes . these divergences basically arise due to the @xmath33-dispersion of the transverse modes . this necessitates a self - consistent study of the phonon dispersion for this system @xcite . such studies have been done on several systems , including one - dimensional systems as test cases for the procedure @xcite . here we follow the method of gtze and michel @xcite which provides a nice connection between the mori - zwanzig mode coupling technique and the perturbation theory . we show that this procedure indeed renormalizes self - consistently the transverse phonon dispersion to be linear , in qualitative agreement with the numerical results of md simulations @xcite . we further note that the result on the renormalized frequencies is independent of the bare frequencies , but does depend on the fact that the cubic interaction vanishes linearly with q. since the vanishing of the interaction with any of its wave - vector arguments is a consequence of the translation invariance , this result should be generic to translation invariant one - dimensional systems . we then perform a self - consistent analysis for the relaxation rates of the renormalized modes using the second order perturbation diagrams . this procedure leads to integral equations from which small - q dependence of the relaxation rates @xmath34 can be extracted . the main results of this analysis are : @xmath3 and @xmath4 . gratifyingly , this behavior matches the results derived by wang and li for the classical system using the mode - coupling analysis , which arrives at the results through a seemingly different mechanism . finally we use these results for understanding the behavior of conductivity with the system length @xmath16 . this paper is organized in the following manner . in sec . ii , we present the model and a calculation of the relaxation rates of the bare phonons . as mentioned above , this calculation shows that the modes are not well defined . accordingly in sec . iii , we present a self - consistent calculation for the frequencies of the modes , which shows that both the modes should disperse linearly at small wave - vectors . in sec . iv , we calculate the relaxation rates of the renormalized phonons using second order perturbation theory . in section v , we formulate equations for the relaxation rates using the self - consistent second order perturbation theory and obtain their behavior at small wave - vectors . in sec . vi , we turn to the calculation of thermal conductivity using the finite size procedure mentioned above . we close the paper with a summary and discussion of our results in sec . this model consists of a chain of equal mass point particles with two kinds of interactions . first is a pairwise interaction between nearest - neighbor particles which depends only on the absolute value of the distance between the particles and the second is a three particle interaction between neighbors that takes into account the bending of the chain . there are two transverse directions , but due to the axial symmetry of the chain the two directions are entirely equivalent . accordingly for understanding the qualitative role of the transverse vibrations on the relaxation of modes and transport , it suffices to consider just one transverse mode . in terms of the position @xmath35 and momentum @xmath36 vectors , the hamiltonian is written as @xmath37,\ ] ] where @xmath38 is the lattice constant , @xmath39 and @xmath40 is the unit vector along @xmath41 . the mass of the particles has been set to unity . this is a complicated many body problem . one assumes that a few lower - order terms in the taylor expansion of the nonlinear potential is enough to give the generic physical picture . also the low - temperature transport properties of the system are expected to be described by the low - energy modes of the system . so we set up the problem in terms of phonon modes . denoting the deviation from equilibrium position of the particles by @xmath42 we have @xmath43 and @xmath44 . fourier transform along the length of the chain is defined by @xmath45 where @xmath6 is the number of particles in the chain . the hamiltonian , considering nonlinearities up to cubic terms only , can be written as  @xcite @xmath46 where the summation indices in bold letters indicate that the summation is over all the polarization indices @xmath47 which takes values from @xmath48 and over all the wave - vectors @xmath49 , a notation we will follow hereafter . @xmath50 s are symmetric under exchange of pairs @xmath51 . translation invariance imposes two conditions on @xmath50 ; they vanish when any of the wave - vectors is set to zero and @xmath52 , where @xmath53 is unity when @xmath54 is a reciprocal lattice vector and zero otherwise . explicit forms for the @xmath50 s are obtained by expanding the potential term in eq . ( [ hamiltonianfull ] ) in taylor series and taking the fourier transforms . the harmonic terms are given by  @xcite @xmath55 note that the frequency of the longitudinal mode vanishes linearly with the wave - vector whereas for the transverse mode it vanishes quadratically . henceforth we choose units in which @xmath56 . the reflection symmetry in the transverse direction permits only two kinds of cubic terms to be nonzero , these being @xmath57 and @xmath58 . both these interactions are proportional to @xmath59 . these forms are generic when the potentials are taken to be functions of absolute values of relative displacements ( @xmath60)@xcite .    for the fermi - pasta - ulam chain  @xcite in which only the longitudinal vibrations are considered , the perturbation analysis is done around the bare phonon modes as they are well defined . but in this model we shall see that the bare phonon modes are not well defined , as the relaxation rates of the two kinds of phonons are not zero even for zero wave - vector mode . this requires that one works with the renormalized phonons as discussed below . we begin by calculating the relaxation rates of the bare phonons . for this purpose we first define the bare phonon operators @xmath61 in terms of which the quadratic part of the hamiltonian becomes diagonal . the hamiltonian can be written as @xmath62 where we have defined @xmath63 and @xmath64 .    . the bare phonon green s functions @xmath65 for the longitudinal phonons and @xmath66 for the transverse phonons are represented respectively by the solid lines and the dashed lines . the diagrams ( a ) and ( b ) are two contributions for the longitudinal phonons , while the diagram ( c ) gives the sole contribution for the transverse phonons.,width=566 ]    for a system with two phonon polarizations , in general one defines the single particle green s functions as the time - ordered expectation values @xmath67 \right\rangle$ ] , where the time dependence of operators is given by @xmath68 , @xmath69 being the imaginary time . the angular brackets correspond to thermal averages defined by @xmath70/tr[e^{-\beta h}]$ ] . the @xmath71 s are thus 2x2 matrices in polarization indices , but in the present case , the reflection symmetry in the transverse direction allows only the diagonal components to be nonzero . in the following we just write them as @xmath72 . the perturbation series for @xmath73 can be summed into the form @xmath74 , where @xmath75 s denote the usual matsubara frequencies and @xmath76 is the self - energy . the relaxation rate is defined as @xmath77 , and the on - shell relaxation rate is denoted by @xmath78 . we first evaluate the relaxation rate up to second - order in the perturbation series . the second - order diagrams for @xmath79 that contribute to the relaxation rate are given in figs . [ selfenergydiagrams ] ( a ) and ( b ) . their contribution is @xmath80 where @xmath81 is the following unperturbed green s function . @xmath82 where @xmath83 is the bose factor and @xmath84 . the relaxation rate is given by @xmath85 the sum is simplified by two conditions : the @xmath86-function contained in the potential @xmath87 and the @xmath28-function . therefore , we can write @xmath88 where @xmath89 is the solution of @xmath90 and the jacobian @xmath91 . if the sum on the rhs of eq . ( [ gamm2barephonon ] ) is finite we expect @xmath92 , as @xmath93 . we first note that the term corresponding to @xmath94 ( fig . 1(a ) ) which involves only the longitudinal modes gives zero contribution as the condition @xmath95 can have only trivial solutions at which the self - energy itself becomes zero  @xcite .    the second term corresponding to @xmath96 ( fig . 1(b ) ) needs to be examined for singular behavior . the source for singularities are the jacobian @xmath97 and the number factors which for low frequencies have the form @xmath98 . for this term we have transverse mode frequencies which vanish quadratically with the wave - vector and hence are more singular . considering @xmath99 term , the small-@xmath0 solution is obtained to be @xmath100 and the jacobian @xmath101 . this yields a contribution to @xmath32 which diverges for small @xmath0 at finite temperatures and we have the leading behavior @xmath102 . the second - order diagram for @xmath103 is given in fig . the yields the relaxation rate as @xmath104 proceeding as in the analysis of eq . ( [ gamm2barephonon ] ) , we see that the term corresponding to @xmath105 is finite for @xmath106 . the above analysis shows that the bare phonon modes are not stable under scattering and thus are not good zeroth order approximations in the perturbation theory . therefore we follow the mode - coupling method as formulated by gtze and michel  @xcite to study how the bare phonon dispersions get affected by the nonlinearities . for completeness we first present a brief outline of the mode - coupling method  @xcite . the procedure requires evaluation of scalar products defined in the space of the dynamical operators ( denoted by @xmath107 ) as follows . @xmath108 \right \rangle , \quad\quad \im z\gtrless0 , \end{aligned}\ ] ] where @xmath109 . for us the key quantity is the static susceptibility @xmath110 . it is easily seen that in the harmonic approximation @xmath111 . the renormalized frequencies are the eigenvalues of the matrix @xmath112 for the nonlinear problem . it follows from the definition and the space reflection symmetry of the hamiltonian that @xmath112 is a real symmetric matrix which is an even function of @xmath0 . therefore , @xmath113 is diagonalizable with a complete and orthonormal set of eigenvectors @xmath114 obeying @xmath115 where the eigenvalues obey @xmath116 due to time - reversal symmetry .    in order to develop equations for these scalar products , we utilize the identity , @xmath117 \rangle$ ] which follows from eq . ( [ chidefinition ] ) . the lhs of this equation can be calculated using @xmath118 $ ] . for further calculations we define @xmath119 . taking ( i ) @xmath120 and ( ii ) @xmath121 respectively , we get the following equations which are exact . @xmath122 we write these equations in the matrix form , suppressing the @xmath0-dependence , as @xmath123^{-1 } \v{v},\ ] ] where @xmath124 and @xmath125 . to get a closure for this set of equations , we need to make some approximations . we do this in a self - consistent manner by expressing the higher order correlation functions @xmath126 in terms of @xmath113 matrices . the expression for @xmath127 is is given by @xmath128 the expectation contained in the above equation is evaluated using an effective quadratic hamiltonian given by @xmath129,\end{aligned}\ ] ] where the operators @xmath130 are the bosonic operators defined in the usual manner as @xmath131.\ ] ] here the frequencies @xmath132 and the polarization vector @xmath133 are defined in eq . ( [ renorm ] ) , and are to be determined self - consistently .    under this approximation using eq . (  [ chidefinition ] ) we get @xmath134 where we have used the compact notation @xmath135 . further @xmath136 and @xmath137 . ( [ dequation ] ) and ( [ aaeqexpanded ] ) together provide us the self - consistent equations for determining the renormalized frequencies @xmath138 . in order to obtain their solution , we first iterate them . starting with a diagonal @xmath139 in the rhs of eq . (  [ aaeqexpanded ] ) , we have @xmath140 and we see that only the diagonal terms survive . so this procedure gives a diagonal @xmath113 back if our starting @xmath113 is diagonal . putting this information in eq . ( [ dequation ] ) reduces it to @xmath141 in order to analyze the low wave - vector @xmath142 behavior of @xmath143 , we simplify the equations by noting that @xmath144 we first consider @xmath145 , whose expression contains two terms corresponding to scattering by longitudinal ( @xmath94 ) and transverse ( @xmath96 ) modes respectively . we may first suppose that the renormalized frequencies have the same @xmath0-dependence as the bare frequencies , @xmath146 . then the use of eq . ( [ approxvn ] ) in eq . ( [ deqdiagonal ] ) shows that , as @xmath106 , the longitudinal contribution to @xmath145 behaves as @xmath33 whereas the transverse part gives rise to a divergence basically due to @xmath33-dependence of the transverse frequencies in the denominator . similarly the expression for @xmath147 also involves transverse mode frequencies in the denominator , which again give rise to a singular dependence on @xmath0 . if on the other hand we assume @xmath148 , all the integrals converge and we get @xmath149 . thus we have a self - consistent solution for @xmath150 in the @xmath106 limit , which is also diagonal . this is exactly what is seen in the md simulations of the classical system with the wang - li interaction  @xcite . from this analysis it is clear that the linear solution obtained here is independent of the form of bare frequencies . thus the linear @xmath0-dependence of the frequencies in one dimension seems to be a generic result . for further analysis we take a specific form for renormalized phonon frequencies , namely , @xmath151 which agrees with the self - consistent solution in the @xmath106 limit . we now study an effective hamiltonian given by eq . ( [ hcubicinnumberop ] ) and eq . ( [ adefinition ] ) in which @xmath152 is replaced by @xmath153 . the second - order relaxation rate for the renormalized modes are as before , @xmath154 first we estimate the low-@xmath0 form of @xmath155 . this proceeds exactly as the analysis of eq . ( [ gamm2barephonon ] ) . we have @xmath156 where @xmath157 are the solutions of @xmath158 and the jacobian is given by @xmath159 . since both @xmath160 the factors of @xmath0 in the denominator that arise due to number factors are canceled by the @xmath161 term . therefore if the jacobian @xmath162 does not go to zero at some @xmath157 , we get @xmath163 .    there are four cases for the energy - momentum conservation conditions due to four possible values of @xmath164 and @xmath165 . these have to be analyzed separately . we choose the brillouin zone @xmath166 $ ] . for @xmath99 , the energy conservation condition becomes @xmath167 . when @xmath168 there are two solutions @xmath169 and @xmath170 . but these correspond to no scattering and have to be excluded from the perturbation terms . thus as noted earlier the relaxation rate within this approximation is zero  @xcite . when @xmath171 nontrivial solutions are possible . [ figomegaequation](a ) shows plots of @xmath153 and @xmath172 with respect to @xmath0 for a representative set of values for @xmath54 and @xmath173 . this figure shows that , when @xmath174 , @xmath54 is a solution for the energy conservation condition @xmath175 when the value of @xmath0 corresponds to one of the intersections of the two graphs . for small-@xmath0 we get a solution @xmath176 with a finite value for the jacobian of the argument of the @xmath28-function . therefore we see that the contribution to @xmath177 from this case is proportional to @xmath33 if @xmath174 and zero otherwise .    ) can be satisfied . the figure on the left shows @xmath153 and @xmath172 in the same graph for @xmath178 and the figure on the right shows @xmath153 and @xmath179 for @xmath180 . dotted vertical lines mark the values of @xmath54 which is the solution for @xmath0 values corresponding to the intersections of the two graphs.,width=566 ]    for @xmath181 , the @xmath28-function condition is @xmath182 . [ figomegaequation](b ) shows lhs and rhs of this condition on the same graph . we see that for @xmath183 we have a nontrivial solution and in the small-@xmath0 limit we find a solution @xmath176 where the jacobian also is finite . therefore , for this case , the contribution to @xmath177 is proportional to @xmath33 for @xmath184 and zero otherwise . next we take @xmath185 . the energy conservation is given by @xmath186 . we have a nontrivial solution for @xmath171 which for @xmath187 is @xmath176 . again we find the contribution to @xmath177 is like @xmath33 for @xmath174 and zero otherwise . the fourth case @xmath188 has only trivial solutions . therefore , we have @xmath189    now we consider the relaxation rate @xmath190 . it has contributions coming from both transverse and longitudinal modes corresponding to @xmath96 and @xmath94 in eq . (  [ gammxrenormalized ] ) . the @xmath96 term again involves interaction between two @xmath191-phonons and one @xmath192-phonon as in the case for @xmath193 and its low wave - vetor behavior can be calculated in a similar way . as in the calculation of @xmath193 , for @xmath194 , there are nontrivial solutions satisfying the @xmath28-function except for @xmath195 . thus the corresponding contribution to @xmath196 is proportional to @xmath33 . the @xmath94 term involves three @xmath192-phonons and the contribution to the relaxation rate is zero because the @xmath28-function condition @xmath197 has no nontrivial solution . therefore @xmath198    thus , the renormalized phonon modes at low wave - vectors are slow decaying and well - defined to perform the perturbation analysis . the phonon relaxation rates of a chain with cubic nonlinearity but with only the longitudinal vibrations was studied earlier by us  @xcite . for this case , as mentioned in the above paragraph , the relaxation rate is zero in the second order as the two conservation conditions for energy and wave - vector can not be satisfied simultaneously . to overcome this difficulty we performed a self - consistent approximation which consists in replacing bare phonon green s functions in the second - order diagram ( fig.[selfenergydiagrams](a ) ) by full green s functions . this relaxes the strict energy conservation condition and allows for nonzero solutions for the relaxation rate through a solution of an integral equation . our aim in this section is to perform a similar analysis for the present case . for this purpose , one uses the spectral representation of the green s function , @xmath199 with the spectral function is given by @xmath200 where @xmath201 and @xmath202 . if we now replace @xmath203 by @xmath204 in eq . ( [ sigma ] ) , the corresponding equations for relaxation rate , eqs . ( [ gammxrenormalized ] ) and ( [ gammyrenormalized ] ) , are replaced by a set of self - consistent equations for @xmath205 and @xmath206 . we get @xmath207 we follow ref  @xcite in the analysis of these equations . we note that for small @xmath0 , the relaxation rate @xmath1 is very small and the spectral function @xmath208 can be approximated by a narrow peak at @xmath209 . also , @xmath210 . further we may replace @xmath211 where @xmath212 is a smooth function of @xmath213 . we use this approximation for the number factors @xmath214 in eq . ( [ selfconstfirst ] ) . we also replace @xmath215 and @xmath216 in the expression for @xmath208 to be used in eq . ( [ selfconstfirst ] ) . after these approximations , only the spectral functions depend on @xmath213 in eq . ( [ selfconstfirst ] ) and the integral can be performed using @xmath217}{((a - b)^2+\gamma_1 ^ 2+\gamma_2 ^ 2)^2 - 4\gamma_1 ^ 2\gamma_2 ^ 2}.\end{aligned}\ ] ] we note that @xmath218 because it depends on @xmath0 through @xmath153 which is even in @xmath0 and through @xmath87 factors which enters the perturbation terms as even function of @xmath0 only . then , @xmath219 and @xmath220 . we can write @xmath221 , with @xmath222 \nonumber \\ & & \times s_{+-}(qx , k i , q+k\,i),\end{aligned}\ ] ] where @xmath223}{[\chi^2_{s_1s_2}(k_1i_1,k_2i_2,k_3i_3 ) + \gamma^2_{i_2}(k_2)+\gamma^2_{i_3}(k_3)]^2 - 4\gamma^2_{i_2}(k_2)\gamma^2_{i_3}(k_3)}.\end{aligned}\ ] ] here we have introduced the definition : @xmath224 . similarly , for the transverse modes we get @xmath225|v_3(-qy ,- kx , q+k\,y)|^2 s_{+-}(qy , kx , q+k\,y ) \nonumber \\ & & + [ 1+n(\omr_x(k))]n(\omr_y(q - k ) ) |v_3(-qy , kx , q - k\,y)|^2 s_{-+}(qy , kx , q - k\,y ) \big\}.\end{aligned}\ ] ]    equations  ( [ selfconstgxa ] ) , ( [ selfconstgxb ] ) and ( [ selfconstgy ] ) have to solved self - consistently to obtain the small wave - vector behavior of the relaxation rates . we show that these equations have a solution @xmath226 the key feature that enables the solution of these equations is that for small @xmath0 , each of the equations can be written in the following schematic form @xmath227 here one factor of @xmath228 comes from the prefactor , which is basically a detailed balance factor and the other from the factor of @xmath229 . the integrand @xmath230 is not known as it as it depends on @xmath231 s . our strategy is to examine these integrands in eqs . ( [ selfconstgxa ] ) , ( [ selfconstgxb ] ) , ( [ selfconstgy ] ) at @xmath232 , by first assuming that @xmath233 . if the integral turns out to be finite , clearly @xmath234 is a self - consistent solution . if on the other hand the integral diverges , it presumably behaves for small @xmath0 as @xmath235 . so we assume that @xmath236 with @xmath237 , and examine the integral for small @xmath0 . a solution is achieved if a self - consistent value for @xmath28 can be obtained . in examining the singularities of the integrand in eq . ( [ fkq ] ) , a further simplification comes from noting that the bose factors @xmath238 give rise to factors @xmath239 for small frequencies , but these are canceled by the similar factors in interaction term @xmath240 . so all the singular behavior of the integrands come from the @xmath241-factors introduced in eq . ( [ sfactor ] ) . first , let us consider eq . ( [ selfconstgy ] ) , the equation for @xmath30 . at @xmath232 , the @xmath241-factors could give rise to singularities at @xmath169 if we set @xmath242 . for @xmath243 , in all the four terms in eq . [ selfconstgy ] , @xmath244 for small @xmath54 ; for example , in the third term , @xmath245 . using eq . ( [ gammaselfconst ] ) one finds that @xmath246 for small @xmath54 . since the integral is finite at @xmath232 , @xmath247 . we next consider @xmath32 which has been split into two terms @xmath248 and @xmath249 in eqs . ( [ selfconstgxa ] ) and ( [ selfconstgxb ] ) . the analysis of @xmath248 proceeds in a similar way as @xmath250 as @xmath251 in both the terms of eq . ( [ selfconstgxa ] ) and we get @xmath252 .    something different happens in eq . ( [ selfconstgxb ] ) for @xmath249 . it has again two terms for @xmath253 . the term @xmath94 involves only the longitudinal mode , and this situation is identical to the fpu-@xmath18 problem discussed in ref . we note that @xmath254\propto q k^2 $ ] as @xmath255 . using eq . ( [ gammaselfconst ] ) , for small @xmath0 and @xmath54 , we get @xmath256^{-1}$ ] . at @xmath232 , the corresponding integral is divergent . but the small @xmath0 divergence of the integral can be obtained by noting that the major contribution to the integral comes from small @xmath54 and small @xmath257 region . denoting the corresponding contribution to @xmath258 by @xmath259 , we get , following  @xcite , @xmath260 as is easily checked , eq . ( [ gamma1 ] ) is solved by taking @xmath261 . for the @xmath96 term @xmath262 $ ] for small @xmath0 and @xmath54 . the leading order behavior of @xmath241 in this limit is obtained using @xmath263 . we see that @xmath264 is not singular at @xmath169 for finite @xmath0 , @xmath265 . thus the limit @xmath266 of this sequence is finite and the corresponding contribution to @xmath267 is proportional to @xmath33 . we can now go back to the earlier equations and check if by setting @xmath268 . the earlier results for @xmath30 and @xmath248 are affected . the examination of the integrals shows that they remain finite . therefore to the leading order , @xmath269 and @xmath247 establishing the solution given by eq . ( [ gammaselfconst ] ) . in this section , we calculate the thermal conductivity of the system using the kubo - green formula . defining the thermal green s function for the total current operator @xmath97 as , @xmath270 \right \rangle,\ ] ] the kubo - green formula for thermal conductivity can be written as  @xcite @xmath271 where @xmath75 are the matsubara frequencies . the current operator , up to leading order , is given by @xmath272 where @xmath273 , @xmath274 . further , we neglect the cross - correlation between the x- and y- components of the current . under the approximation used in  @xcite that vertex corrections to the two particle green s function eq . ( [ defw ] ) has the same wave - vector dependence as @xmath275 for small @xmath54 , one arrives at the kinetic theory formula for conductivity , @xmath276 where @xmath277 is the relaxation time of single particle green s function and @xmath278 is proportional to the mode specific heat . we note that @xmath279 is well behaved for small @xmath0 and any possible divergence can come from @xmath280 only .    the formula given by eq . ( [ conductivitykinetictheoryformula ] ) is for an infinite chain and this quantity is divergent because @xmath281 has a non integrable singularity at @xmath232 . to obtain the conductivity @xmath5 for large but finite size chain of @xmath6 particles , we follow the prescription used by other workers in the area  @xcite . in this procedure one rewrites eq . ( [ conductivitykinetictheoryformula ] ) as a time integral in the following way . @xmath282 since @xmath283 , we get @xmath284 . one argues that when the boundaries of the system are connected to thermal baths , the correlation @xmath285 dies after a time scale of order @xmath6 due to interaction with the baths , i.e. the typical time it takes for the modes to travel across the chain . using in the time integral in eq . ( [ kuboformulanoneq ] ) such a cutoff yields @xmath286 it is to be noticed that the leading order divergence arise from the term corresponding to the @xmath191-modes . if the conduction was normal , conductivity would be finite and the contribution from these modes would be negligible compared to the contribution from the @xmath192-modes when the strength of the interaction giving rise to transverse oscillations , @xmath287 , is small . but here the conductivity itself is a divergent quantity and its divergence is determined by the slower relaxation of the transverse modes compared to the longitudinal modes . we have studied thermal transport in a chain of coupled particles in which particles can have longitudinal as well as transverse vibrations . particles interact via anharmonic potentials proposed by wang and li  @xcite . for this model , the mode frequencies in the harmonic approximation have the following dispersions at small wave - vectors : for the longitudinal mode @xmath288 ; for the transverse mode @xmath25 , where @xmath0 denotes the wave - vector . we first show that due to quadratic dispersion of the transverse mode , the relaxation rates for both the modes become too large to leave them as meaningful quasi - particles . so we calculate their renormalized frequencies taking into account the anharmonic interactions in a self - consistent approximation . we find that this procedure yields the linear dispersion for both kinds of modes irrespective of their bare forms . we then perform a calculation of the relaxation rates for the renormalized phonon in a self - consistent approximation based on the second - order perturbation diagrams . this calculation yields that for the longitudinal phonons , the relaxation rate @xmath3 , while for the transverse mode @xmath289 . we then use these results to calculate the thermal conductivity @xmath5 of a chain of @xmath6 particles . the finite size is incorporated by using a cutoff of the order of @xmath6 in the time integral over current - current correlation function that occurs in the kubo formula for the conductivity . this yields a contribution to @xmath5 from longitudinal phonons to be @xmath290 , that from transverse phonons to be @xmath291 . the latter contribution dominates giving @xmath7 for large n.    99 f. bonetto , j. l. lebowitz and l. rey - ballet in _ mathematical physics _ ( world scientific , london 2000 ) , p. 128 . s. lepri , r. livi and a. politi , phys . * 377 * , 1 ( 2003 ) . g. r. lee - dadswell , b. g. nickel and c. g. gray , phys . e * 72 * , 031202 ( 2005 ) . o. narayan and s. ramaswamy , phys . lett . * 69 * , 200601 ( 2002 ) . santhosh . g and deepak kumar , phys . e * 77 * , 011113 ( 2008 ) . s. lepri , r. livi and a. politi , europhys . lett . * 43 * , 271 ( 1998 ) . t. mai , a. dhar and o. narayan , phys . . lett . * 98 * , 184301 ( 2007 ) . a. perverzev , phys . e * 68 * , 056124 ( 2003 ) . s. lepri , phys . e * 58 * , 7165 ( 1998 ) . g and deepak kumar , phys . e * 76 * , 021105 ( 2007 ) . s. volz , _ microscale and nanoscale heat transfer _ , topics in applied physics ( springer , new york , 2007 ) , vol . 107 . c. w. chang , d. okawa , h. garcia , a. majumdar and a. zettle , phys . * 101 * , 075903 ( 2008 ) . p. kim , l. shi , a. majumdar and p. l. mceuen , phys . 87 * , 215502 ( 2001 ) . j. hone , m. whitney , c. piskoti , and a. zettle , phys . b * 59 * , r2514 ( 1999 ) . c. w. chang , a. m. fennimore , a. afanasiev , d. okawa , t.ikuno , h. garcia , a. majumdar and a. zettle , phys . rev . lett . * 97 * , 085901 ( 2006 ) . j. s. wang and b. li , phys . e * 70 * , 021204 ( 2004 ) . a. a. maradudin and a. e. fein , phys . rev . * 128 * , 2589 ( 1962 ) . w. gtze and k. h. michel , _ dynamical properties of solids _ edited by g. k. horton and a. a. maradudin , north - holland publishing company , 1974 , vol . 1 , chapter 9 . h. horner , _ dynamical properties of solids _ edited by g. k. horton and a. a. maradudin , north - holland publishing company , 1974 , vol . 1 , chapter 8 . g. k. horton and e. r. cowley , _ physics of phonons _ , ( springer , karpacz 1987 ) , p50 . a. cuccoli , v. tongetti , a. a. maradudin , a. r. mcgurn and r. vaia , phys . b * 46 * , 8839 ( 1992 ) . e. s. freidkin , g. k. horton and e. r. cowley , phys . b * 52 * , 3322 ( 1995 ) . j. scheipers and w. schirmacher , z. phys . b * 103 * , 547 ( 1997 ) . g. d. mahan , _ many - particle physics , third edition _ , ( kluwer academic / plenum publishers , new york ) p
we study thermal transport in a chain of coupled atoms , which can vibrate in longitudinal as well as transverse directions . 
 the particles interact through anharmonic potentials upto cubic order . 
 the problem is treated quantum mechanically . 
 we first calculate the phonon frequencies self - consistently taking into account the anharmonic interactions . 
 we show that for all the modes , frequencies must have linear dispersion with wave - vector @xmath0 for small @xmath0 irrespective of their bare dispersions . 
 we then calculate the phonon relaxation rates @xmath1 , where @xmath2 is the polarization index of the mode , in a self - consistent approximation based on second order perturbation diagrams . 
 we find that the relaxation rate for the longitudinal phonon , @xmath3 , while that for the transverse phonon @xmath4 . 
 the consequence of these results on the thermal conductivity @xmath5 of a chain of @xmath6 particles is that @xmath7 . 
 v#1 # 1#1^