Discrete global symmetries and dynamics of emergent fermions

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.