  The Linear behavior of thermal transport has been widely explored, both
theoretically and ex?perimentally. On the other hand, the nonlinear thermal
response has not been fully discussed. In light of the thermal vector potential
theory [Phys. Rev. Lett. 114, 196601 (2015)], we develop a general formulation
to calculate the linear and nonlinear dynamic thermal responses. In the DC
limit, we recover the well-known Mott relation and the Wiedemann-Franz (WF) law
at the linear order response, which link the thermoelectric conductivity
{\eta}, thermal conductivity \k{appa} and electric conductivity {\sigma}
together. To be specific, the linear Mott relation describes the linear {\eta}
is proportional to the first derivative of {\sigma} with respect to Fermi
energy (for brevity we call the first derivative, the others are similar); and
the linear WF law shows the linear \k{appa} is proportional to the zero
derivative (i.e. the {\sigma} itself). We found there are higher-order Mott
relation and WF law which follow an order-dependent relation. At the second
order, the Mott relation indicates that the second order {\sigma} is
proportional to the zero derivative of the second order {\eta}; but the second
WF law shows that the second {\sigma} is proportional to the first derivative
of \k{appa}. At the third order, the derivative order in?creases once. Although
we only did explicit calculate up to the third order response, we can deduce
that the n-th order electric conductivity is proportional to the n-2-th
derivative of the n-th order thermoelectric conductivity for the nonlinear Mott
relation; and the n-th order electric conductivity is proportional to the
n-1-th derivative of the n-th order thermal conductivity for the nonlinear WF
law.
