Emergence of curved momentum-spacetime and its effect on cyclotron motion in the antiferromagnetic quantum critical metal

We show that anisotropic quantum corrections can dynamically give rise to curved momentum-spacetimes for quasiparticles in metals. In the (2 + 1)-dimensional antiferromagnetic quantum critical metal, a curved momentum-spacetime arises as the critical spin fluctuations generate red shift that dilates frequency of electron unevenly on the Fermi surface. As the disparity of the momentum-dependent red shift is controlled by the shape of the Fermi surface, the momentum-spacetime geometry that emerges at low energies depends on the bare nesting angle of the Fermi surface. With increasing nesting angle, the region in which electron motion is slowed down by critical spin fluctuations shrinks. On the other hand, the increasing nesting angle makes the red shift stronger near the hot spots due to the weakened screening of the interaction. These competing effects result in a nonmonotonic dependence of the cyclotron frequency of electron on the nesting angle of the Fermi surface. The red shift that becomes more singular at the hot spots with increasing nesting angle creates a possibility of realizing a momentum-space black hole horizon beyond a critical nesting angle: The electron motion becomes "perpetually" slowed down as it approaches a hot spot in the same way that the motion of a free falling object freezes near the event horizon of a black hole with respect to an asymptotic observer. However, the analogous horizon in momentum space does not lead to a vanishing cyclotron frequency because the metric singularity at the hot spots is cut off by thermal effects present above the nonzero superconducting transition temperature.