  Global symmetries that define the number of low energy degrees of freedom
have profound consequences on universal properties near topological quantum
critical points and in other gapless or nearly gapless states of emergent
fermions. We take a $Z_2$ global symmetry (such as time-reversal) as an example
to study its effect on thermodynamic and transport properties. Although the
thermal entropy density of $Z_2$ symmetric systems is simply twice of their
counterparts without any global symmetries or the $Z_1$ class, the temperature
dependence of thermal conductivity $\kappa$ is distinctly and drastically
different for different symmetries. For systems with dynamic exponent $z=1$, in
the $Z_2$ symmetric class, we have $\kappa\propto T^{-(d-1)}$ in the quantum
critical regime near weakly interacting fixed points, while for systems with no
global symmetries (i.e., the $Z_1$ class), we have $\kappa\propto T^{-(d+3)}$,
with $d$ being the spatial dimension. Only near strong coupling fixed points,
both cases with or without $Z_2$ global symmetries follow the same scaling
function, $\kappa \propto T^{d-1}$. These distinct scalings of thermal
conductivity can also appear in gapless surface Majorana states.

Strongly Correlated Electrons