## Field-Induced Switching of Ferro-Quadrupole Order Parameter in PrTi_{2}Al_{20}

#### Takigawa, Sakakibara, and Nakatsuji Groups

The series of Pr-based cubic compounds Pr*T*_{2}*X*_{20} (T = Ir, Rh, X = Zn; T = V, Ti, X = Al) has been actively studied recently to explore novel phenomena caused by multipoles of 4*f* electrons because the non-Kramers Γ_{3} doublet ground states in the crystalline electric field (CEF) has active quadrupole (*O*_{Z} = 3*z*^{2}-*r*^{2} and *O*_{X} = √3(*x*^{2}-*y*^{2})) and an octupole (*T*_{XYZ} = *xyz*) moments but no magnetic dipole moment [1]. Among this series, PrTi_{2}Al_{20} shows the highest transition temperature of 2 K into a simple ferro-ordered state of *O*_{Z} quadrupole moment.

We have performed ^{27}Ai-NMR and magnetization measurements on high quality single crystals of PrTi_{2}Al_{20} and unexpectedly observed discontinuous switching of quadrupole order parameter by applying small magnetic fields ** B** of a few tesla along [001] and [110] but not along [111] directions [2]. The symmetry of order parameters for different field directions were determined as shown in Fig. 1. For

**along [111], there is a single ordered phase of**

*B**O*

_{Z}type below

*T*

_{Q}= 2.2 K, which is independent of

**. Here we expect coexistence of three domains, where the symmetry axis**

*B***Z**of the quadrupole moment points along three equivalent <001> directions. The quadrupole order parameter can be conveniently represented as a two-dimensional vector with

*O*

_{Z}and

*O*

_{X}chosen to be the basis (rightmost panel of Fig. 1) [3]. For example, order parameters of the three domains for

**∥[111] is given by**

*B**O*

_{Z}= 3

*x*

^{2}–

*r*

^{2}= –

*O*

_{Z}/2 + √3

*O*

_{X}/2 and 3

*y*

^{2}–

*r*

^{2}= –

*O*

_{Z}/2 + √3

*O*

_{X}/2, corresponding to the polar angle of 0 and ±2π/√3.

For ** B** along [001], however, discontinuous jump of the NMR frequency occurs near 2 tesla with a finite range of field where two lines coexist, indicating a first order phase transition (Fig. 2a). From the analysis of NMR spectra and magnetic susceptibility, we conclude that while the high field phase has ferro-order of

*O*

_{Z}with

**Z**∥

**, the order parameter of the low field phase is closer to**

*B**O*

_{X}type (

*θ ~*±π/2). For

**along [110], the order parameter changes from**

*B**O*

_{Z}with

**Z**∥

**in low fields to**

*B*

*Z**~*–

**in high fields. The field-induced transition was also confirmed by measurements of heat capacity and magnetocaloric effect [4].**

*B*We have developed a Landau theory to identify symmetry allowed interactions between quadrupole and magnetic field, which can account for the experimentally observed switching of the order parameter [2]. Frist, the magnetic field induces mixing between Γ_{3} ground doublet and excited CEF levels, leading to the van-Vleck paramagnetism. This anisotropic Zeeman interaction, which is quadratic in ** B**, turns out to stabilize the order parameters observed in the high field phases both for

**∥[001] and**

*B***∥[110], leaving only a crossover as a function of temperature. On the other hand, we found that symmetry allows interaction between quadrupoles on neighboring Pr sites, which is responsible for the ferro-quadrupole order, to depend also quadratically on**

*B***for small**

*B***. This anisotropic interaction competes against the Zeeman interaction and can stabilize the order parameter of the low field phases. Therefore, the observed switching of the order parameter can be explained if the anisotropy of the quadrupole-quadrupole interaction wins the Zeeman interaction at low fields but becomes suppressed at high fields compared with the Zeeman interaction. We have indeed succeeded in reproducing the experimental phase diagram by assuming a non-monotonic field dependence of the anisotropic quadrupole interaction.**

*B*Although the mechanism for such a non-trivial field dependence of the quadrupole interaction is not understood yet, anomalous behavior of the NMR Knight shift in the quadrupole ordered states provides an important insight. As shown in Fig. 2(b), the Knight shift at Al sites, which probes the polarization of conduction electron spin, shows the identical temperature dependence to the magnetic susceptibility at a high field (6.6T). At a low field (1T), however, they show remarkable separation below the ordering temperature, indicating that the hybridization between 4*f* and conduction electrons (*c*-*f* hybridization) is strongly influenced by the quadrupole order. Since the quadrupole-quadrupole interaction is mediated by the *c*-*f* hybridization, this means that the quadrupole order modifies the quadrupole-quadrupole interaction via redistribution of 4*f* charge density, which in turn should affect the order itself. We suspect that such a feedback mechanism may be the key to understand the field-induced discontinuous transition.

##### References

- [1] T. Onimaru and H. Kusunose, J. Phys. Soc. Jpn.
**85**, 082002 (2016). - [2] T. Taniguchi
*et al*., J. Phys. Soc. Jpn.**88**, 084707 (2019). - [3] K. Hattori and H. Tsunetsugu, J. Phys. Soc. Jpn.
**83**, 034709 (2014). - [4] S. Kittaka
*et al*., J. Phys. Soc. Jpn.**89**, 043701 (2020).