Dynamical exponent of a quantum critical itinerant ferromagnet: A Monte Carlo study

We consider the effect of the coupling between two-dimensional (2D) quantum rotors near an XY ferromagnetic quantum critical point and spins of itinerant fermions. We analyze how this coupling affects the dynamics of rotors and the self-energy of fermions. A common belief is that near a $q=0$ ferromagnetic transition, fermions induce an $\mathrm{\ensuremath{\Omega}}/q$ Landau damping of rotors (i.e., the dynamical critical exponent is $z=3$) and Landau overdamped rotors give rise to non-Fermi liquid fermionic self-energy $\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}{\ensuremath{\omega}}^{2/3}$. This behavior has been confirmed in previous quantum Monte Carlo (QMC) studies. Here we show that for the XY case the behavior is different. We report the results of large-scale quantum Monte Carlo simulations, which show that at small frequencies $z=2$ and $\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}{\ensuremath{\omega}}^{1/2}$. We argue that the new behavior is associated with the fact that a fermionic spin is by itself not a conserved quantity due to spin-spin coupling to rotors, and a combination of self-energy and vertex corrections replaces $1/q$ in the Landau damping by a constant. We discuss the implication of these results to experiments.