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landau s fermi liquid theory @xcite can be considered as the `` standard theory '' of solid state physics . it predicts that the electronic properties of a ( normal ) metal follow , in the presence of coulomb correlations , a renormalized ( quasiparticle ) version of those for non - interacting electrons . among the most fundamental physical properties is the heat capacity which , at low temperatures , is predominated by the electronic degrees of freedom . for a normal metal , landau s fermi liquid theory @xcite predicts a linear increase of the specific heat capacity with temperature @xmath0 and a cubic term , more precisely a term @xmath1 , as the leading correction @xcite . for free electrons , the prefactor @xmath2 is directly proportional to the electronic density of states at the fermi level since , due to the pauli principle , only these electrons contribute . in the case of a fermi liquid , we merely need to introduce a quasiparticle renormalization factor @xmath3 to account for the coulomb interaction , which enhances the specific heat of a correlated metal by @xmath4 . however , this electronic contribution prevails only at temperatures much lower than the debye temperature . otherwise the cubic phononic term , which has a much higher prefactor because of its bosonic nature , dominates . as we have shown in a recent letter @xcite , this longstanding conception needs to be supplemented : for strongly correlated electrons , a clear kink in the temperature dependence of the specific heat appears , marking the abrupt change from one linear behavior to a second one with a reduced slope at higher temperatures . this can be shown numerically solving the hubbard model @xcite within dynamical mean field theory @xcite , using exact diagonalization as an impurity solver . quantitatively the same results can also been obtained analytically , following the derivation by abrikosov _ @xcite for the specific heat , taking the recently observed kinks in the self energy as a starting point @xcite . the experimental confirmation of these theoretical results is somewhat problematic since typically , at the kink temperature , the phononic contribution to the specific heat is already dominating . an exception in this respect is liv@xmath5o@xmath6 @xcite , the first heavy fermion system with @xmath7-electrons , since the kink temperature is here particularly low . and , indeed , the precise inspection of the experimental data shows a kink in the correct temperature range @xcite . in this proceeding , we will review the previous theoretical results and include additional data . in the presence of strong electronic correlations , the @xmath8-integrated spectral function @xmath9 shows a typical three peak structure with a lower and an upper hubbard band and a quasiparticle peak in between . using the dmft formula @xmath10 which relates green function and self energy at frequency @xmath11 , one can show @xcite that such a strongly correlated three peak structure necessarily entails a kink . note that the hybridization function @xmath12 in eq . ( [ eq : sigmacontributions ] ) is for a simple semi - circular density of states just @xmath13 ( @xmath14 : half the bandwidth ) . for other lattices , @xmath12 depends on the moments of the non - interacting density of states with a similar term proportional @xmath15 dominating . the kink argument is now as follows @xcite : the first three terms of the r.h.s of eq . ( [ eq : sigmacontributions ] ) yield an almost linear frequency dependence for the real part of the self energy over the entire region of the central quasiparticle peak . the derivative hence gives a quasiparticle renormalization factor @xmath16^{-1}\label{eq : z}\ ] ]    the last ( hybridization ) term however yields an additional contribution to @xmath17 which basically is proportional to @xmath18 . this real part can be directly obtained from the spectral function of the central quasiparticle peak @xmath19 via a kramers - kronig transformation . _ inside _ the overall width of the quasiparticle peak , there is a turning point in @xmath9 . hence , @xmath18 has a linear frequency dependence up to an maximum at the turning point @xmath9 . after this maximum changes in @xmath18 are minor . altogether this means that at low frequencies we have to add the slope obtained from the @xmath12 in eq . ( [ eq : z ] ) yielding an altogether smaller fermi liquid renormalization factor @xmath20 at low energies . in between the two regimes @xmath20 and @xmath21 , there is a kink at a frequency @xmath22 . in summary we hence have : @xmath23 with the constant @xmath24 providing for a continuous function . before we turn to the specific heat in the next section , let us note that the kink in the self energy directly leads to a kink in the energy - momentum dispersion relation of the correlated electrons . let us also remark here , that the result of generic kinks in the self energy of strongly correlated systems should be robust beyond dmft , as similar effects are to be expected in cluster @xcite and diagrammatic extensions @xcite of dmft . based on eq .  ( [ eq : sigma ] ) , we have developed an analytical theory for the specific heat on the basis of a formula by abrikosov , gorkov and dzyaloshinski ( agd ) for the entropy of a fermionic system at low temperatures@xcite : @xmath25 , \label{eq : abr2}\ ] ] this agd formula is ( through a low - temperature diagrammatic expansion ) based on the low frequency behavior of the self energy @xmath26 at zero temperature [ or the corresponding retarded and advanced green function in eq . ( [ eq : abr2 ] ) ] so that we can directly apply it with @xmath26 from eq . ( [ eq : sigma ] ) . let us note that @xmath27 is the density of states and the frequency integral has been rewritten through a dimensionless variable @xmath28 ( @xmath29 ) . the agd formula ( [ eq : abr2 ] ) can be easily differentiated w.r.t . @xmath30 , so that the specific heat @xmath31 can be computed directly . with some algebra ( see @xcite for more details ) and substituting substituting @xmath32 by @xmath33-functions , we obtain the final result @xmath34      d y \ ,   \frac{y^2   \mbox{e}^y } { ( \mbox{e}^y+1)^2}. \label{eq : final}\end{aligned}\ ] ]    knowing the two renormalization factors @xmath35 and @xmath36 and the kink frequency @xmath22 as well as the non - interacting density of states @xmath27 , eq . ( [ eq : final ] ) allows us to calculate the specific heat through a simple integral . if vice versa , the specific heat is known , e.g. , experimentally , we can employ eq . ( [ eq : final ] ) as a fit formula for obtaining @xmath35 , @xmath36 and @xmath22 . the latter ( fit formula ) approach had been taken for the experimental validation of the specific heat kinks in liv@xmath5o@xmath6 since lda+dmft calculations @xcite for liv@xmath5o@xmath6 @xcite could not provide for a fine enough resolution of the low frequency dependence of the self energy , even when using the projective quantum monte carlo method @xcite which is most appropriate for this purpose . as fig . 3 of @xcite shows the three fitting parameters provide for an excellent agreement of eq . ( [ eq : final ] ) and the specific heat of @xcite , including a clearly visible _ kink_. let us note that we did not subtract a phonon contribution for the specific heat since ( i ) it does not alter the existence or non - existence of a kink , ( ii ) it is still an order of magnitude smaller than the electronic contribution at the kink temperature and ( iii ) there is a considerable arbitrariness depending on the reference system chosen . [ fig:1 ] shows results of our final equation ( [ eq : final ] ) applied to the hubbard model with a semi - circular density of states with half bandwidth @xmath37 . as an input to eq . ( [ eq : final ] ) , @xmath35 , @xmath36 and @xmath22 have been fitted to the numerical renormalization data for the dmft self energy from ref . besides , we have also calculated the dmft specific heat directly , using exact diagonalization ( ed ) with 7 levels in the impurity bath . both results agree very well , and the minor deviations can be attributed to the numerical inaccuracy of the exact diagonalization study for which @xmath38 had to be obtained via a numerical differentiation of the total energy .    in the inset , we show the specific heat over a slightly extended temperature range . as the quasiparticle peak eventually ends , the specific heat has a maximum after the kink temperature . note that this maximum is not contained in our formula ( [ eq : final ] ) since it would require an extension of the self energy description eq . ( [ eq : sigma ] ) to higher energies where @xmath17 has a maximum at the frequencies between hubbard and quasiparticle bands and @xmath39 becomes important . obtained via the agd based equation ( [ eq : final ] ) [ solid lines ] and numerically by exact diagonalization of the dmft equations ( crosses ) . inset : behavior over an extended temperature range which shows a maximum after the kink temperature marking the end of the quasiparticle band . [ fig:1],width=340 ] landau s fermi liquid theory predicts a linear temperature dependence of the electronic specific heat capacity with temperature . we have shown this longstanding conception needs to be supplemented : for strongly correlated electrons , a clear kink in the temperature dependence appears , marking the abrupt change from one linear behavior to a second one with a reduced slope at higher temperatures . recent experiments on liv@xmath5o@xmath6 , an ideal material for studying the electronic specific heat , confirm our theory . a consequence of our findings is that materials with correlated electrons are more resistive to cooling at low temperatures ( where cooling is particularly difficult ) than expected from the behavior at intermediate temperatures . we thank r. bulla and m. nohara for making available their raw data . this work has been supported in part by the european - indian cooperative network monami .
the textbook knowledge of solid state physics is that the electronic specific heat shows a linear temperature dependence with the leading corrections being a cubic term due to phonons and a cubic - logarithmic term due to the interaction of electrons with bosons . 
 we have shown that this longstanding conception needs to be supplemented since the generic behavior of the low - temperature electronic specific heat includes a kink if the electrons are sufficiently strongly correlated .