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Quantum Lifshitz points in an altermagnetic metal

We predict the existence of two tri-critical quantum Lifshitz points in recently discovered $d$-wave altermagnetic metals subjected to an external magnetic field. These points connect a spatially modulated Fulde--Ferrell--Larkin--Ovchinnikov (FFLO) phase, a uniform polarized Bardeen--Cooper--Schrieffer (BCS) superconducting phase, and the normal metallic phase in a nontrivial manner. Depending on whether the FFLO state is primarily induced by the magnetic field or by $d$-wave altermagnetism, we classify the corresponding Lifshitz points as field-driven or altermagnetism-driven, respectively. Notably, the two types exhibit distinct behaviors: the transition from the FFLO phase to the polarized BCS phase is first-order near the field-driven Lifshitz point, as might be expected, whereas it becomes continuous near the altermagnetism-driven Lifshitz point. We further explore the effects of finite temperature and find that the altermagnetism-driven Lifshitz point is significantly more sensitive to thermal fluctuations.