Fluctuation-induced pair density wave in itinerant ferromagnets

Magnetic fluctuations near to quantum criticality can have profound effects. They lead to characteristic scaling at high temperature which may ultimately give way to a reconstruction of the phase diagram and the formation of new phases at low temperatures. The ferromagnet UGe${}_2$ is unstable to $p$-wave superconducting order—an effect presaged by the superfluidity in ${}^3$He—whereas in CeFePO fluctuations drive the formation of spiral magnetic order. Here we develop a general quantum order-by-disorder description of these systems that encompasses both of these instabilities within a unified framework. This allows us to demonstrate that in fact these instabilities intertwine to form a pair density wave.

Figure 1

The phase diagram of Hamiltonian, Eq. (1), as a function of temperature and interaction strength. Our phase, the pair density wave, is highlighted by the red text. $g$ is the interaction strength, ${\nu }_{\text{F}}$ is the density of states at the Fermi surface, $T$ temperature, and $T_{\text{F}}$ the Fermi temperature. The magnetic background to this phase diagram is as determined in Ref. 35, allowing for a resummation of the leading divergent contributions to the free energy. At high temperatures, there is a continuous transition from ferromagnet to paramagnet. The temperature of this reduces with decreasing interaction strength until the red tricritical point. At this point, there is a Lifshitz transition from the ferromagnet into a magnetic spiral (green dotted line). Finally, the spiral magnetic order gives way to paramagnetism in a first-order transition. Superconductivity in the absence of magnetic order occurs below the blue dotted line. This transition temperature rises dramatically in the presence of magnetic order due to mode-mode coupling. Where superconductivity overlaps, the spiral magnetic order a Larkin-Ovchinnikov-Fulde-Ferrell-like pair density wave state is formed, highlighted by the red text.