标题： Pr(TM)$_2$X$_{20}$ Kondo材料的多极序的朗道理论 显示英文标题

标题： Landau theory of multipolar orders in Pr(TM)$_2$X$_{20}$ Kondo materials

摘要： 一系列含有非克兰默Pr$^{3+}$ $4f^2$ 量子态的Pr(TM)$_2$X$_{20}$（TM=Ti,V,Rh,Ir 和 X=Al,Zn）Kondo材料，已被证明表现出如四极序和超导性等交织的序。 受这些实验的启发，我们提出并研究了一个多极序的朗道理论，以捕捉相图及其场依赖性。 在零磁场下，我们表明不同的四极态，或者四极和八极序的共存，可能导致具有多个对称性破缺的基态。 加热时，这些态可能会经历两步热转变进入对称的顺磁相，在中间相中部分恢复对称性。 对于非零磁场，我们展示了这些热相变的演变强烈依赖于磁场方向，这是由于自由能中的钟各向异性项。 我们的发现为Pr(TM)$_2$Al$_{20}$材料的实验结果提供了重要的见解。 我们提出了进一步的实验测试，以区分纯四极序与四极-八极序的共存。 显示英文摘要

摘要： A series of Pr(TM)$_2$X$_{20}$ (with TM=Ti,V,Rh,Ir and X=Al,Zn) Kondo materials, containing non-Kramers Pr$^{3+}$ $4f^2$ moments on a diamond lattice, have been shown to exhibit intertwined orders such as quadrupolar order and superconductivity. Motivated by these experiments, we propose and study a Landau theory of multipolar order to capture the phase diagram and its field dependence. In zero magnetic field, we show that different quadrupolar states, or the coexistence of quadrupolar and octupolar orderings, may lead to ground states with multiple broken symmetries. Upon heating, such states may undergo two-step thermal transitions into the symmetric paramagnetic phase, with partial restoration of broken symmetries in the intervening phase. For nonzero magnetic field, we show the evolution of these thermal phase transitions strongly depends on the field direction, due to clock anisotropy terms in the free energy. Our findings shed substantial light on experimental results in the Pr(TM)$_2$Al$_{20}$ materials. We propose further experimental tests to distinguish purely quadrupolar orders from coexisting quadrupolar-octupolar orders.