Quantum criticality in a two-component two-dimensional nematic metal

Nematic fluctuations alter the low-energy electronic behavior and reshape the mean-field phase diagram. We study a two-dimensional nematic metal with a two-component internal degree of freedom. Near the quantum critical point (QCP), longitudinal nematic fluctuations generate the non-Fermi-liquid self-energy $\propto \mathrm{\mid }\omega {\mathrm{\mid }}^{2\mathrm{/}3}$, whereas transverse fluctuations give a subleading logarithmic, marginal-Fermi-liquid correction. Nevertheless, the transverse sector dominates the correction to the phase diagram through a quantum order-by-disorder mechanism: the emergent nematic background reshapes the electronic spectrum, alters the decay of nematic bosons, and produces a negative singular free-energy correction. This correction reconstructs the phase diagram, drives the transition first order, and is described by a scaling function whose two asymptotic limits define two critical regimes with free-energy corrections scaling as $R^{8\mathrm{/}3}$ and $R^3$, respectively, where $R$ is the order-parameter magnitude; their powers follow from the competition between the characteristic momentum of normal-state nematic fluctuations and that of the ordered-state background. These two regimes are experimentally distinguishable through nematic-susceptibility hysteresis measurements. The corrected phase diagram shows that the enlarged nodal $\alpha$ phase masks the otherwise bare QCP, making it inaccessible; consequently, the logarithmic self-energy correction is by no means negligible at low energies.