Research Group for Materials Physics for Heavy Element systems
Group Leader: Shinsaku KAMBE
Members: Yoshinori Haga, Naoyuki Tateiwa, Yo Tokunaga, Hironori Sakai, Etsuji Yamamoto,Katsunori Kubo, Taisuke Hattori and Nonoka Higa

In heavy element systems, valence fluctuations, the Kondo
effect, and the RKKY interaction compete with one another.
Because of this, exotic behaviors such as quantum critical points,
heavyfermions,
non-Fermi
liquids,
anisotropic
superconductivity and multipolar ordering appear when suchcompetition is strong. Recently, it has become clear that theseexotic behaviors for 5f-electron systems are different from thosefor 4f-electrons. This is because electrons with different spin andorbital character can coexist in 5f actinide systems, in contrast to
the case of 4f electrons. By means of advanced experimental andtheoretical approaches, our research group tries to clarify these
exotic behaviors due to the “many-fold” character of both 4 f and
5f compounds, including transuranium.
In this mid-term project, we also try to break new ground suchas topological and spintronic aspects in these compounds.
Odd Parity rank-5 ordering in URu2Si2 [1]Fig. 1 Schematic electron distribution at the U site for P4/nnc state,
which has 422 (D4) local symmetry. This type of ordering corresponds toelectric dotriacontapolar order in the present case, which is a strong
candidate for the hidden ordered state of URu2Si2.

The heavy fermion superconductor URu2Si2 undergoes a second
order phase transition at T0 ∼ 17.5 K. Since the order parameterof the transition has not ever been clearly identified, such orderhas been termed “hidden order”. Identifying the hidden orderparameter is one of the challenging topics in condensed matterphysics at present. In this study, 29Si and 101Ru NMR
measurements were performed on high–quality, single–crystal
URu2Si2 samples with a residual resistivity ratio RRR∼70. Our
results showed that the Si and Ru sites exhibited 4-fold electronic
symmetry around the c-axis in the hidden-order state. Since theU and Si sites were aligned along the c-axis, we concluded further
that the electronic state shows 4-fold symmetry around the U site
below the hidden–order transition. From this observed local
symmetry, possible space groups for the hidden–order state arethought to be P4/nnc or I4/m, based on group theoreticalconsiderations. Since the order vector is considered to be Q =(001), the hidden–order state is then found to be P4/nnc with rank
5 odd-parity, i.e. electric dotriacontapolar order. In the P4/nnc

case, the point group at the U site is 422 (D4). Figure 1 shows anelectronic state for the totally symmetric representation of the 422
point group A1u:xyz(x2-y2). Now confirmation of this orderedstate is in progress.
Superconductivity with quadrupole degrees of freedom [2, 3]
A cubic system with the Γ3 crystalline electric field (CEF) stateis an ideal system to investigate multipole physics, since it does
not have dipole but has higher-order multipoles such as
quadrupole and octupole. In addition, superconductivity has been
observed in PrT2X20 (T = Ir, Rh, X = Zn; T = Ti, V, X = Al)systems with the Γ3 CEF ground state. To approach these
phenomena theoretically, we considered a model composed of felectrons with the total angular momentum j = 5/2. The j = 5/2
states split into Γ7 and Γ8 levels under a cubic CEF. When we
assumed an antiferromagnetic interaction between the Γ7 and Γ8orbitals, the f 2 ground state was the Γ3 state composed of singlets
between these orbitals. Thus, we employed this model as one of
the simplest models to describe the Γ3 state. From this model, we
have succeeded in deriving multipole interactions for the Γ3systems [4]. In this study, we investigated superconductivity ofthe same model. In multiorbital systems, we can expect
anisotropic superconductivity originating from the orbital
anisotropy. For example, d-wave spin-triplet state in a model foreg orbitals on a square lattice [5]. We applied random phaseapproximation for superconductivity in a multiorbital system tothe present model and obtained a d-wave spin-singlet state. Thispairing state is naturally expected from the f 2-Γ3 CEF groundstate. Such anisotropic superconductivity in a cubic system canlower the symmetry of the system and coexist with quadrupoleorder as shown in Fig. 2. We will further explore the coexisting
phase theoretically to understand experimental phase diagrams ofthe Γ3 systems.

Fig. 2 Phase diagram expected for a superconductor with quadrupole
degrees of freedom. x denotes a parameter such as external pressure andT denotes a temperature. Superconductivity (SC) occurs inside thequadrupole ordered phase.

References[1] S. Kambe et al., Phys. Rev. B 97, 235142 (2018).
[2] K. Kubo, J. Phys. Soc. Jpn. 87, 073701 (2018).
[3] K. Kubo, AIP Adv. 8, 101313 (2018).
[4] K. Kubo and T. Hotta, Phys. Rev. B 95, 054425 (2017).
[5] K. Kubo, Phys. Rev. B 75, 224509 (2007).