Quantum order-by-disorder driven phase reconstruction in the vicinity of ferromagnetic quantum critical points

The formation of new phases close to itinerant electron quantum critical points has been observed experimentally in many compounds. We present a unified analytical framework to explain the emergence of new types of order around itinerant ferromagnetic quantum critical points. The central idea of our analysis is that certain Fermi-surface deformations associated with the onset of the competing order enhance the phase space available for low-energy quantum fluctuations and so, self-consistently lower the free energy. We demonstrate that this quantum order-by-disorder mechanism leads to instabilities toward the formation of spiral and $d$-wave spin-nematic phases close to itinerant ferromagnetic quantum critical points in three spatial dimensions.

Figure 1

Distortions of the Fermi surface (dashed lines) enhance the phase space for quantum fluctuations. (a) Uniform ferromagnet, (b) spiral, and (c) $d$-wave spin nematic. Quantum fluctuations correspond to excitations of pairs of particle-hole pairs of opposite spin and equal and opposite momenta.

Figure 2

Phase diagram of the uniform ferromagnet in mean-field theory. The transition between the uniform ferromagnet and the paramagnet is always second order.

Figure 3

Phase diagram of the uniform ferromagnet, including quantum fluctuations. Below the tricritical point (shown as a circle), quantum fluctuations drive the transition first order.

Figure 4

Phase diagram of the itinerant ferromagnet allowing for the possibility of spatially modulated ferromagnetism. Quantum fluctuations drive the formation of a spiral phase, which sets in below the tricritical point and preempts the first-order transition between the uniform ferromagnet and the paramagnet.

Figure 5

The evolution of the order parameters $M$ and $Q$ in the spiral phase. As we approach the tricritical point, the jumps in $M$ and $Q$ along the line of first-order transitions become smaller. At the tricritical point, $M=Q=0$. At the Lifshitz transition between the uniform ferromagnet and the spiral ferromagnet, $Q$ goes continuously to zero, while $M$ remains finite and behaves smoothly.

Figure 6

Phase diagram of the itinerant ferromagnet, allowing for the possibility for the formation of spiral and spin-nematic phases. At temperatures, which are about an order of magnitude smaller than the temperature of the tricritical point, a $d$-wave spin nematic forms between the spiral ferromagnet and the paramagnet.

Figure 7

Modification to the phase diagram of the spiral state with a weakly anisotropic dispersion. The onset of the spiral no longer coincides with the putative tricritical point of the fer-romagnet (circle). Instead, the spiral forms at a slightly higher temperature (square) and preempts a portion of the continuous transition between the uniform ferromagnet and the paramagnet (thin dashed line) as well as the first-order transition (thick dashed line). Note that, because of the anisotropy, the nature of the spiral-to-paramagnet transition changes. At higher temperatures, $M$ now behaves continuously, while at low temperatures, the transition is first order in $M$ as in the case of an isotropic dispersion.

Figure 8

Fluctuation contributions to $\alpha$: comparison of numerics (solid line) with leading low-temperature analytical dependence (dashed line).

Figure 9

Fluctuation contributions to $\beta$: comparison of numerics (solid line) with leading low-temperature analytical dependence (dashed line).