# Electric Quadrupolar Contributions in the Magnetic Phases of UNi4B

T. Yanagisawa  Department of Physics, Hokkaido University, Sapporo 060-0810,
Japan  H. Matsumori  Department of Physics, Hokkaido University, Sapporo
060-0810, Japan  H. Saito  Department of Physics, Hokkaido University, Sapporo
060-0810, Japan  H. Hidaka  Department of Physics, Hokkaido University,
Sapporo 060-0810, Japan  H. Amitsuka  Department of Physics, Hokkaido
University, Sapporo 060-0810, Japan  S. Nakamura  Institute for Materials
Research, Tohoku University, Sendai 980-8577, Japan  S. Awaji  Institute for
Materials Research, Tohoku University, Sendai 980-8577, Japan  D. I. Gorbunov
Hochfeld-Magnetlabor Dresden (HLD-EMFL) and Würzburg-Dresden Cluster of
Excellence ct.qmat, Helmholtz-Zentrum Dresden-Rossendorf (HZDR), 01328
Dresden, Germany  S. Zherlitsyn  Hochfeld-Magnetlabor Dresden (HLD-EMFL) and
Würzburg-Dresden Cluster of Excellence ct.qmat, Helmholtz-Zentrum Dresden-
Rossendorf (HZDR), 01328 Dresden, Germany  J. Wosnitza  Hochfeld-Magnetlabor
Dresden (HLD-EMFL) and Würzburg-Dresden Cluster of Excellence ct.qmat,
Helmholtz-Zentrum Dresden-Rossendorf (HZDR), 01328 Dresden, Germany  Institut
für Festkörper- und Materialphysik, TU Dresden, 01062 Dresden, Germany  K.
Uhlířová  Department of Condensed Matter Physics, Faculty of Mathematics and
Physics, Charles University, 121 16 Prague 2, Czech Republic  M. Vališka
Department of Condensed Matter Physics, Faculty of Mathematics and Physics,
Charles University, 121 16 Prague 2, Czech Republic  V. Sechovský  Department
of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles
University, 121 16 Prague 2, Czech Republic

###### Abstract

We present acoustic signatures of the electric quadrupolar degrees of freedom
in the honeycomb-layer compound UNi4B. The transverse ultrasonic mode $C_{66}$
shows softening below 30 K both in the paramagnetic phase and
antiferromagnetic phases down to $\sim 0.33$ K. Furthermore, we traced
magnetic field-temperature phase diagrams up to 30 T and observed a highly
anisotropic elastic response within the honeycomb layer. These observations
strongly suggest that $\Gamma_{6}$(E2g) electric quadrupolar degrees of
freedom in localized $5f^{2}$ ($J=4$) states are playing an important role in
the magnetic toroidal dipole order and magnetic-field-induced phases of UNi4B,
and evidence some of the U ions remain in the paramagnetic state even if the
system undergoes magnetic toroidal ordering.

The multipole formulation and its foundational concept in solid-state physics
have been developed by intensive research on $f$-electron systems [1, 2].
Recently, new theories based on the common language of ‘multipoles’ [3, 4] and
‘augmented multipoles’ [5, 6, 7, 8, 9], which are spatially extended
multipoles, have been evoked to construct a new framework for understanding
various physical phenomena that are related to spin-orbit interactions beyond
the differences in electron orbitals. In particular, the odd-parity augmented
multipoles, including magnetic/electric and toroidal ones [10, 11, 12], have
recently been extensively studied. Recent academic advances in understanding
augmented multipoles have been preceded by theory rather than experiment.
Therefore, it is necessary to demonstrate whether the new framework allows for
a unified understanding of spontaneous spatial inversion symmetry breaking in
metallic and insulating compounds. A major experimental challenge is to
demonstrate odd-parity multipole ordering by observing cross-correlation
phenomena and spontaneous spatial inversion-symmetry breaking in a suitable
compound [5, 8]. Among them we focus on the U-based honeycomb-layer compound
UNi4B [13, 14, 15, 16], which is considered to be a good candidate for
studying augmented odd-parity multipoles, magnetic toroidal dipoles (MTDs),
and the interplay with magnetoelectric phenomena [12, 17].

Figure 1:  Crystal and magnetic structure of UNi4B, reported for $B$ = 0 [18,
16]. (a) The pseudo-honeycomb network consists of 2/3 of U ions with $Cmcm$
lattice. Red, blue, and green circles indicate U and Ni or B on the layer at
$z$ = 0, 1/2, respectively. Dashed lines with open circles show the (pseudo)
Kagome layer of Ni atoms at $z\sim$ 1/4. (b) Colored backgrounds with pink-
rhomboidal and blue-rectanglar shapes denote antiferromagnetic unit cells in
hexagonal ($P6/mmm$) and orthorhombic ($Cmcm$) symmetry, respectively.

UNi4B crystallizes in an orthorhombic structure (Space group; $Cmcm$,
$D_{2h}^{17}$, No. 63) as shown in Fig. 1(a) [18]. Below $T_{\rm N}$ = 20.4 K,
this compound orders antiferromagnetically (AFM) in a magnetic structure where
the magnetic moments are carried by two thirds of the U ions [UAFM sites in
Fig. 1(b) form vortices in each pseudo-honeycomb plane], and one third of the
U ions [UPM1 or UPM2 in Fig. 1(b)] remain in a paramagnetic (PM) state. [13,
15, 16] Assuming a hexagonal crystal structure ($P6/mmm$, $D_{6h}^{1}$, No.
191), an exotic magnetic structure was proposed from neutron scattering
experiments in earlier studies [16]. Since a slight deformation of the crystal
structure from hexagonal to orthorhombic symmetry and different site
occupations of Ni and B atoms have recently been reconfirmed by neutron and
resonant X-ray scattering studies as well as by 11B-NMR measurements [19, 20],
the previously proposed magnetic structure should be reconsidered based on the
orthorhombic space group.

On the other hand, Hayami et al. has pointed out that such vortex-type
magnetic structure in the (pseudo) honeycomb arrangement in UNi4B can be
understood in the framework of MTD order (see Fig. S1 in the supplemental
material (SM)) [12, 21]. Their theory has also predicted a new magnetoelectric
effect: current-induced magnetization, which can occur in ferro-toroidal
ordered metallic compounds, which has been experimentally confirmed in UNi4B
[17]. Recently, Yatsushiro and Hayami have reported on a theoretical
investigation of an atomic scale MTD by taking into account orbital degrees of
freedom with different parity [11]. This theory predicts that the orbital
degrees of freedom in an interorbital space play an important role in
stabilizing MTD order by odd-parity hybridization. However, the theory deals
with the tetragonal point group $C_{\rm 4v}$, and the contribution of the
orbital degrees of freedom such as even-parity electric multipolar moments to
the noncollinear magnetic order in UNi4B has not been investigated.

Another fascinating point and also an open question for this compound is a
specific-heat anomaly at $T^{*}$ $\sim 0.33$ K of unknown origin [22].
Previous studies have explained that the narrowness of the 0.33 K anomaly in
UNi4B may be an indication of glassy behavior caused by the geometrical
frustration of the paramagnetic U spins and their Kondo screening by the
conduction electrons, since AC magnetic susceptibility and $\mu$SR
measurements have shown no changes in the magnetic structure below $T^{*}$
[22]. Other possibilities, which have not been verified yet, are a nonmagnetic
multipolar order of the PM-1/3 U ions and/or Schottky peak due to level
splitting of the degenerate CEF ground state with low orthorhombicity. In
order to study the electrical multipole contribution, it is useful to measure
the elastic constants by ultrasound [23]. The elastic constants reflect the
coupling of the strain field caused by the ultrasonic wave to the electric
multipolar moments, which are described by orbital degrees of freedom of the
CEF state. In this study, we show evidence of an electric multipolar ground
state for UNi4B based on ultrasound results. We further analyzed the possible
contributions of electric quadrupoles to the noncollinear magnetic order and
to the low-temperature specific-heat anomaly (See Sect. C to G in the SM [21]
for experimental and analysis details).

Figure 2:  Elastic constants of UNi4B as a function of temperature. The insets
in each panel show enlarged views of the data below 50 K. Illustrations of the
distorted hexagon and/or rectangle indicate the lattice strain, which is
induced by the respective ultrasonic mode (See Table SI in the SM) [21, 24,
25] Figure 3:  (a) CEF level schemes used for the present analysis (see Table
SIII, SIV, and SV in the SM) [21]. Blue and red arrows indicate selection
rules where finite matrix elements exist for magnetic dipole $J_{x}$ and
electric quadrupole $O_{xy}$, respectively. (b) Temperature dependence of the
elastic constant $C_{66}$, compared with calculations based on the quadrupolar
susceptibility using the CEF level schemes shown in panel (a): Scheme 0
(green), Scheme 1 (orange), and Scheme 2 (blue) as shown by dashed and solid
curves. The dotted black and red curves show the background of the elastic
constant $C_{66}^{0}$ in the PM and AFM phases, respectively. The inset shows
an enlarged view of the low-temperature region on a log-$T$ scale.

Figure 2 shows the measured elastic constants of UNi4B as a function of
temperature. Here, the four ultrasonic modes are symmetrized using the
hexagonal point group $D_{6h}$. The ultrasound induces local strain and
rotation fields [21] in the solid sample as shown by the schematic
illustrations in each panel of Fig. 2. The local strain and rotation field
behave as conjugate fields for electric quadrupole or electric hexadecapole
moments [26]. Responses of the multipoles can be observed as sound-velocity
change and ultrasonic attenuation via electron-phonon interaction. By
comparing the temperature dependence of the four ultrasonic modes, it becomes
obvious that only the transverse ultrasonic mode $C_{66}$ exhibits a softening
below 30 K in the PM phase with a kink at $T_{\rm N}$ and keeps decreasing in
the antiferromagnetic (AFM) phase down to $\sim 0.33$ K [Fig. 3(b)].

Figure 4:  Magnetic-field dependence of $C_{66}$ for (a) $H\parallel$
[$2\bar{1}\bar{1}0$] and (d) $H\parallel$ [$01\bar{1}0$] of UNi4B at selected
temperatures. $C_{66}$ vs. temperature under various magnetic fields for (b)
$H\parallel$ [$2\bar{1}\bar{1}0$] and (e) $H\parallel$ [$01\bar{1}0$]. In each
panel, vertical arrows indicate elastic anomalies, which correspond to phase
boundaries in the magnetic field-temperature phase diagram of UNi4B for (c)
$H\parallel$ [$2\bar{1}\bar{1}0$] and (f) $H\parallel$ [$01\bar{1}0$]. Dotted
and dashed lines are guides for the eyes. The relative change of the elastic
constant $C_{66}$ is highlighted by the color map from zero (red) to large
negative (blue) values.

The other ultrasonic modes, the longitudinal $C_{11}$, $C_{33}$, and
transverse $C_{44}$ modes, do not show such softening particularly below
$T_{\rm N}$. From the selection rules within the category of even-parity
multipoles, the results indicate that an electric quadrupole with
$\Gamma_{6}$(E2g) symmetry (in the hexagonal point group) is active in UNi4B.
In the first stage of the analysis, we consider the conventional localized
$5f$-electronic states with even-parity CEF levels and multipoles for the
analysis of the elastic responses. Here, we do not take into account the
contribution from odd-parity multipoles, because such effect only couples to
the elastic strain through cross-correlation with the application of
appropriate external fields with odd-parity space-inversion symmetry [6, 8].
As shown below, the current analysis mimics the experimental results well that
the odd-parity mixing inherent in the CEF is negligible. The softening in
$C_{66}$ in the paramagnetic phase could, however, not be reproduced by the
previously proposed CEF level scheme [Scheme 0 in Fig. 3(a)] [27] with the
localized $5f$-electronic state of U3+ having a hexagonal symmetry because the
$5f^{3}$ ($J=9/2$) state only shows $\Gamma_{6}$-quadrupolar excitation
($O_{xy}=\sqrt{3}(J_{x}J_{y}+J_{y}J_{x})/2$) in the off-diagonal elements
between the ground-state Kramers doublet and the excited levels that are
separated by an energy gap of over 600 K, as shown in Fig. 3(a) (for matrix
elements, see Sect. F in the SM [21]).

In order to reproduce the softening in the $C_{66}$ mode at higher temperature
than $T_{\rm N}$, we propose a different CEF model [Scheme 1 in Fig. 3(a)]
with a localized $5f^{2}$ ($J=4$) state of U4+, which has a pseudo-triplet
ground state. The CEF parameters of CEF Scheme 1 (Table SIV in the SM [21])
are set to reproduce simultaneously the elastic softening in $C_{66}$, no
elastic softening in $C_{44}$, and also the magnetic susceptibility below 50 K
at the same time (for details see Fig. S3 in the SM [21]). The coupling
constant $g^{\prime}_{\Gamma}$ of quadrupolar inter-site interactions is
described by the Hamiltonian $\mathcal{H}_{\rm
MM}=-\sum_{\alpha}g^{\prime}_{\Gamma_{6}}\left<O_{xy}\right>O_{xy}^{\alpha}$
for sublattices $\alpha$ and a $\Gamma_{6}$ symmetry quadrupolar moment
$O_{xy}$ (see Eqs. 17-21 in the SM) [28, 23, 21]. Our analysis reveals a
positive value $g^{\prime}_{\Gamma 6{\rm(PM)}}$ = +0.42 K, which strongly
suggests the presence of a weak but finite ferro-type quadrupolar interaction
in the PM phase of UNi4B. A possible relationship between the ferro-type MTD
order of this system and the ferro-quadrupolar interaction is nontrivial and
remains an open question. The softening of $C_{66}$ in the AFM phase can also
be analyzed using the same CEF parameters, since the PM-1/3 U ions (on the
UPM1 or UPM2 sites) located in the center of the pseudo-honeycomb plane are
not affected by the on-site electric/magnetic fields formed by the MTD moment.
Although the global inversion symmetry on the UPM1 and UPM2 sites is broken
due to the MTD order, the odd-parity CEF states and multipole will, however,
not be active on the PM-1/3 U sites when only considering the $J=4$ Hilbert
space and assuming weak orbital coupling between the U and Ni or B ions.

The inset in Fig. 3(b) shows the temperature dependence of $C_{66}$ on a
log-$T$ scale. Calculations of the quadrupolar susceptibility of the PM-1/3 U
ions using the CEF Scheme 1 (orange curve) [21] with hexagonal symmetry well
reproduces the softening in the AFM phase down to $\sim 0.33$ K. Here, the
contribution from the AFM ordered-2/3 U ions is assumed as constant background
and the CEF level scheme of the PM-1/3 U ions is not changed from PM phase.
Note that the intersite quadrupolar interaction for the AFM phase obtained by
our analysis is negative with $g^{\prime}_{\Gamma 6{\rm(AFM)}}=-0.045$ K,
which means the presence of antiferro-quadrupolar (AFQ) interaction in the AFM
phase. In order to reproduce the temperature dependence down to $T^{*}$ in the
present analysis, it is unlikely that $g^{\prime}_{\Gamma 6{\rm(AFM)}}$ takes
positive value. Therefore, we can conclude that in the ordered phase there is
an antiferro-quadrupolar interaction between the UPM’s [in Fig. 1(b)], which
is fundamentally the opposite to that for UAFM in the PM phase.

The calculated result using Scheme 1 deviates from the experimental data below
$T^{*}\sim 0.33$ K. This deviation indicates a small CEF splitting of the
ground-state doublet. Such splitting might occur for two reasons: 1) symmetry
lowering due to ordering of the remaining PM-1/3 U multipolar moments or 2)
the crystal structure essentially having lower symmetry. Since it has been
confirmed that the point group of the U ions in the present system has
orthorhombic symmetry, we have modified the CEF Scheme 1 by adding low
orthorhombicity (as CEF parameter $B_{2}^{2}$ and $B_{4}^{2}$ for Steven’s
operator $O_{2}^{2}$ and $O_{4}^{2}$ (see Eqs. 3-11 in the SM) [29, 30, 3,
21], respectively, have finite values) to split the ground-state non-Kramers
doublet with a gap of $\it{\Delta}\sim$ 0.79 K. This CEF Scheme 2 [Fig. 3(a),
Table SV in the SM] reproduces the leveling off of $C_{66}$ (blue curve) as
well as a Schottky-type specific-heat peak at $\it{\Delta}/$ 2.398 $\sim$ 0.33
K [22]. We, therefore, conclude that the mentioned $T^{*}$ transition, found
in specific heat, is a Schottky anomaly due to the orthorhombicity of the
crystal. It should be noted that we determined the magnetic-field dependence
of $T^{*}$ and its anisotropy for $H\parallel$ [$2\bar{1}\bar{1}0$] and
[$01\bar{1}0$] in the elastic constant $C_{66}$. Our results are roughly
consistent with those of earlier studies [16] (see Fig. S8 in the SM [21]).

The temperature and magnetic-field dependence of the elastic constant $C_{66}$
are shown in Figs. 4(a) and 4(b) for $H\parallel$ [$2\bar{1}\bar{1}0$] and in
Figs. 4(d) and 4(e) for $H\parallel$ [$01\bar{1}0$]. In Fig. 4(a), the data
with an asterisk are obtained in pulsed-magnetic-field measurements up to 60 T
(see Fig. S9 in the SM [21]). We obtained $C_{66}$ vs. $H\parallel$
[$01\bar{1}0$] up to 28 T, shown in Fig. 4(d), in static magnetic fields using
the cryogen-free hybrid magnet system equipped with a dilution refrigerator
[31]. Several elastic anomalies are observed, which are indicated by arrows;
$H^{*}$, $H_{\rm 1b}$, and $H_{\rm 3b}$ for $H\parallel$ [$2\bar{1}\bar{1}0$],
$H^{*}$, $H_{\rm 1a}$, $H_{\rm 2a}$, $H_{\rm 3a}$, and $H_{\rm 4a}$ for
$H\parallel$ [$01\bar{1}0$]. The data display both up and down sweeps of the
magnetic field. We observe hysteretic regions below $H_{\rm 3a}$. The elastic
responses in $C_{66}$ show a large in-plane (0001) anisotropy for $H\parallel$
[$2\bar{1}\bar{1}0$] and $H\parallel$ [$01\bar{1}0$], while the magnetization
does not show such strong anisotropies [16]. The positions of the elastic
anomalies are indicated as well in the magnetic field-temperature ($H$-$T$)
phase diagrams as shown in Figs. 4(c) and 4(f). Here, a number of phases are
distinguishable for $H\parallel$ [$01\bar{1}0$]; PM phase I, AFM phase II, the
spin-reoriented AFM phase III, the spin-flop phase IV with hysteresis, which
was previously evidenced by magnetization data [16], and a newly found unknown
phase V. The obtained phase boundaries are consistent with the previously
reported phase diagram [16] except for the high-magnetic-field region. On the
other hand, the $H$-$T$ phase diagram for $H\parallel$ [$2\bar{1}\bar{1}0$] is
quite different with less phases and completely different elastic responses.
The background red-white-blue color code in Figs. 4(c) and 4(f) represents the
relative changes in $C_{66}$ from larger to smaller stiffness. Remarkably, the
contour plot shows a significant difference in stiffness for $H\parallel$
[$01\bar{1}0$] and [$2\bar{1}\bar{1}0$], though no difference was detected in
magnetization. This new observation clearly indicates a possible contribution
of electric quadrupoles of the PM-1/3 U ions, which modifies the spin-
reorientation process as well.

In the low-magnetic-field and low-temperature regions for both $H\parallel$
[$01\bar{1}0$] and $H\parallel$ [$2\bar{1}\bar{1}0$], the blue color indicates
enhanced contributions from the $\Gamma_{6}$(E2g)-electric quadrupoles, i.e.,
this system has incoherent fluctuations of the electric quadrupoles due to the
CEF pseudo-doublet ground state. In general, the ground-state doublet
(including non-Kramers doublet) splits due to the Zeeman effect (with mixing
of excited-levels wave functions) in an external magnetic field or in the
internal fields produced by magnetic order, and the quadrupole degrees of
freedom become inactive, resulting in a hardening of $C_{66}$. Remarkably,
$C_{66}$ experiences a softening in phase IV for $H\parallel$ [$01\bar{1}0$]
and in the intermediate temperature range of phase III’ for $H\parallel$
[$2\bar{1}\bar{1}0$] compared with the changes in the other phases. This fact
suggests a reactivation of the quadrupole degrees of freedom with
$\Gamma_{6}$(E2g) symmetry above $\sim$12 T in the low-temperature region for
$H\parallel$ [$01\bar{1}0$].

In summary, we conclude that the electric-quadrupole degrees of freedom play a
crucial role in the low-temperature properties of UNi4B, leading to
anisotropic $H$-$T$ phase diagrams and a newly revealed field-induced phase V.
The observed softening of the $C_{66}$ elastic constant can be well explained
by quadrupolar-strain interactions. The corresponding CEF analysis results in
a new level scheme (Scheme 2) taking into account the established orthorhombic
symmetry and the $5f^{2}$ ($J=4$) state of uranium ions. Furthermore, in this
level scheme the puzzling specific-heat anomaly at $\sim 0.33$ K can be
understood as a Schottky anomaly due to a small level splitting of the non-
Kramers ground-state doublet. Moreover, our results confirm that some of the U
ions stay disordered in the MTD ordered phase. Further theoretical
considerations would be of interest to clarify the quadrupolar contributions
in the toroidal order and anisotropic elastic response in the newly
established magnetic-field-temperature phases.

###### Acknowledgements.

We thank C. Tabata, S. Hayami and H. Kusunose for fruitful discussions. The
present research was supported by JSPS KAKENHI Grant Nos. JP17K05525,
JP15KK0146, JP15K13509, JP15H05885, JP15K21732, and the Strategic Young
Researcher Overseas Visits Program for Accelerating Brain Circulation from
JSPS. This study was partly supported by Hokkaido University, Global Facility
Center (GFC), Advanced Physical Property Open Unit (APPOU), funded by MEXT
under “Support Program for Implementation of New Equipment Sharing System”
Grant No. JPMXS0420100318. A part of this work was performed at the High Field
Laboratory for Superconducting Materials, Institute for Materials Research,
Tohoku University (Project No. 17H0063). We also acknowledge the support of
the Hochfeld-Magnetlabor Dresden at HZDR, a member of the European Magnetic
Field Laboratory (EMFL), and the Deutsche Forschungsgemeinschaft (DFG) through
SFB 1143 (Project No. 247310070) and the Würzburg-Dresden Cluster of
Excellence on Complexity and Topology in Quantum Matter–$ct.qmat$ (EXC 2147,
Project No. 390858490). The sample used in HZDR has been grown and
characterized in the Materials Growth and Measurement Laboratory MGML which is
supported within the program of Czech Research Infrastructures (Project No.
LM2018096).

## References

  * [1] P. Santini, S. Carretta, G. Amoretti, R. Caciuffo, N. Magnani, and G. H. Lander, Multipolar interactions in $f$-electron systems: The paradigm of actinide dioxides, Rev. Mod. Phys. 81, 807 (2009).
  * [2] Y. Kuramoto, H. Kusunose, and A. Kiss, Multipole Orders and Fluctuations in Strongly Correlated Electron Systems, J. Phys. Soc. Jpn. 78, 072001 (2009).
  * [3] H. Kusunose, Description of Multipole in f-Electron Systems, J. Phys. Soc. Jpn. 77, 064710 (2008).
  * [4] Y. Kuramoto, Electronic Higher Multipoles in Solids, Prog. Theor. Phys., Suppl. 176, 77 (2008).
  * [5] S. Hayami, Y. Yanagi, and H. Kusunose, Bottom-up design of spin-split and reshaped electronic band structures in antiferromagnets without spin-orbit coupling: Procedure on the basis of augmented multipoles, Phys. Rev. B 102, 144441 (2020).
  * [6] S. Hayami, M. Yatsushiro, Y. Yanagi, and H. Kusunose, Classification of atomic-scale multipoles under crystallographic point groups and application to linear response tensors, Phys. Rev. B 98, 165110 (2018).
  * [7] S. Hayami and H. Kusunose, Microscopic Description of Electric and Magnetic Toroidal Multipoles in Hybrid Orbitals, J. Phys. Soc. Jpn. 87, 033709 (2018).
  * [8] H. Watanabe and Y. Yanase, Group-theoretical classification of multipole order: Emergent responses and candidate materials, Phys. Rev. B 98, 245129 (2018).
  * [9] M.-T. Suzuki, H. Ikeda, and P. M. Oppeneer, First-principles Theory of Magnetic Multipoles in Condensed Matter Systems, J. Phys. Soc. Jpn. 87, 041008 (2018).
  * [10] M. Yatsushiro and S. Hayami, Odd-Parity Multipoles by Staggered Magnetic Dipole and Electric Quadrupole Orderings in CeCoSi, J. Phys. Soc. Jpn. 89, 013703 (2020).
  * [11] M. Yatsushiro and S. Hayami, Atomic-Scale Magnetic Toroidal Dipole under Odd-Parity Hybridization, J. Phys. Soc. Jpn. 88, 054708 (2019).
  * [12] S. Hayami, H. Kusunose, and Y. Motome, Toroidal order in metals without local inversion symmetry, Phys. Rev. B 90, 024432 (2014).
  * [13] J. A. Mydosh, G. J. Nieuwenhuys, S. A. M. Mentink, and A. A. Menovsky, Unusual magnetic ordering in the intermetallic compound UNi4B, Phil. Mag. B 65, 1343 (1992).
  * [14] S. A. M. Mentink, H. Nakotte, A. de Visser, A. A. Menovsky, G. J. Nieuwenhuys, and J. A. Mydosh, Reduced-moment antiferromagnetism in single-crystal UNi4B, Physica B 186-188, 270 (1993).
  * [15] S. A. M. Mentink, A. Drost, G. J. Nieuwenhuys, E. Frikkee, A. A. Menovsky, and J. A. Mydosh, Magnetic Ordering and Frustration in Hexagonal UNi4B, Phys. Rev. Lett. 73, 1031 (1994).
  * [16] S. A. M. Mentink, G. J. Nieuwenhuys, H. Nakotte, A. A. Menovsky, A. Drost, E. Frikkee, and J. A. Mydosh, Magnetization and resistivity of UNi4B in high magnetic fields, Phys. Rev. B 51, 11567 (1995).
  * [17] H. Saito, K. Uenishi, N. Miura, C. Tabata, H. Hidaka, T. Yanagisawa, and H. Amitsuka, Evidence of a New Current-Induced Magnetoelectric Effect in a Toroidal Magnetic Ordered State of UNi4B, J. Phys. Soc. Jpn. 87, 033702 (2018).
  * [18] Y. Haga, A. Oyamada, T. D. Matsuda, S. Ikeda, and Y. Onuki, Crystal structure of frustrated antiferromagnet UNi4B, Physica B 403, 900 (2008).
  * [19] C. Tabata, unpublished.
  * [20] R. Takeuchi, Y. Kishimoto, H. Kotegawa, H. Harima, Y. Homma, F. Honda, A. Nakamura, Y. Shimizu, D. X. Li, D. Aoki, and H. Tou, 11B-NMR Investigation for Crystal Structure in Antiferromagnet UNi4B, JPS Conf. Proc. 29, 013001 (2020).
  * [21] See the Supplemental Material (attached) for (A) Schematic illustration of the magnetic toroidal order, (B) Indexing a hexagonal system with four axes, (C) Experimental details, (D) Formulation of multipolar susceptibility, (E) Definition of multipolar moments and equivalent operator expressions, (F) Matrix elements of magnetic dipole and electric quadrupoles, (G) Consistency test for CEF calculations of the magnetic susceptibility, (H) Elastic constant $C_{66}$ and calculated quadrupolar susceptibility at low temperatures ($<$ 1 K), (I) Anisotropy of the $H-T$ phase diagram and the 330 mK anomaly, (J) Elastic response of the $C_{66}$ mode under pulsed magnetic fields above 30 T.
  * [22] R. Movshovich, M. Jaime, S. Mentink, A. A. Menovsky, and J. Mydosh, Second Low-Temperature Phase Transition in Frustrated UNi4B, Phys. Rev. Lett. 83, 2065 (1999).
  * [23] B. Lüthi, Physical Acoustics in the Solid State (Springer, Berlin, 2006).
  * [24] V. Dohm and P. Fulde, Effective spin models for spin-phonon chains by flow equations, Z. Physik B 21, 369 (1975).
  * [25] T. Goto, A. Tamaki, T. Fujimura, and H. Unoki, Quadrupolar Response and Rotational Invariance of Singlet Ground State System; HoVO4, J. Phys. Soc. Jpn. 55, 1613 (1986).
  * [26] R. Kurihara, K. Mitsumoto, M. Akatsu, Y. Nemoto, T. Goto, Y. Kobayashi, and M. Sato, Critical Slowing Down of Quadrupole and Hexadecapole Orderings in Iron Pnictide Superconductor, J. Phys. Soc. Jpn. 86, 064706 (2017).
  * [27] A. Oyamada, M. Kondo, T. Itou, S. Maegawa, D. X. Li, and Y. Haga, Spin dynamics in a triangular antiferromagnet UNi4B, J. Phys.: Conf. Ser. 145, 012044 (2009).
  * [28] P. Thalmeier and B. Lüthi, Handbook on the Physics and Chemistry of Rare Earths, Vol. 14, edited by J. K. A. Gschneider and L. Eyring (North-Holland, Amsterdam, 1991) p. 311.
  * [29] K. W. H. Stevens, Matrix elements and operator equivalents connected with the magnetic properties of rare earth ions, Proc. Phys. Soc. London Sect. A 65, 209 (1952).
  * [30] M. Hutchings, Point-charge calculations of energy levels of magnetic ions in crystalline electric fields (Academic Press, New York, 1964) pp. 227 - 273.
  * [31] K. Watanabe, G. Nishijima, S. Awaji, K. Takahashi, K. Koyama, N. Kobayashi, M. Ishizuka, T. Itou, T. Tsurudome, and J. Sakuraba, Performance of a Cryogen-Free 30 T-Class Hybrid Magnet, IEEE Trans. Appl. Supercond. 16, 934 (2006).

.

Supplemental Material for
Electric Quadrupolar Contributions
in the Magnetic Phases of UNi4B

T. Yanagisawa, H. Matsumori, H. Saito, H. Hidaka, H. Amitsuka, S. Nakamura, S.
Awaji, D. I. Gorbunov, S. Zherlitsyn, J. Wosnitza, K. Uhlířová, M. Vališka, V.
Sechovský

## A. Schematic Illustration of the Magnetic Toroidal Order

Fig. S1 shows a schematic illustration of the magnetic toroidal order in UNi4B
with hexagonal structure. Green and red arrows indicate the magnetic dipolar
moment and the magnetic toroidal moment, respectively. The spheres indicate
the U ions, with the blue-colored ions representing the one-third of the U
ions that maintain a paramagnetic state in the AFM phase. This vortex-like
magnetic structure can be reproduced theoretically by assuming Heisenberg-
exchange interactions between spins on the honeycomb arrangement (similar to
the 120 degree three-sublattice Néel order in a triangular lattice) using an
appropriate parameter space. Thus, the possible contribution of orbital
degrees of freedom on this one-third of U ions is not required in this
framework.

Figure S1:  Scheme of the ferro-magnetic-toroidal order of UNi4B.

## B. Indexing A Hexagonal System with Four Axes

The four axes, ${\bf a_{1}}$, ${\bf a_{2}}$, ${\bf a_{3}}$, and ${\bf c}$, of
the hexagonal unit cell are indicated in Fig. S2. When the trace operator
${\bf T}$ is written as ${\bf T}=h{\bf a_{1}}+k{\bf a_{2}}+l{\bf a_{3}}+m{\bf
c}$, the high-symmetry axes are described by the four-digit Weber indices as
[$hklm$]. Since the symmetry of the axes ${\bf a_{1}}$, ${\bf a_{2}}$, and
${\bf a_{3}}$ is equivalent and follows the relationship ${\bf a_{1}}+{\bf
a_{2}}+{\bf a_{3}}={\bf 0}$, the combination of the indices with relative
prime integral numbers is also limited as $h+k+l=0$.

Figure S2:  Direction indices in the hexagonal system with four-digit indices.

## C. Experimental Details

Two single crystals of UNi4B with parallel [$2\bar{1}\bar{1}0$],
[$01\bar{1}0$], and [0001] facets were prepared. Here, we used the hexagonal
point-group symmetry with a four-axis index for describing lattice directions,
since the orthorhombicity of the present crystal is negligibly low ($\sim
0.02^{\circ}$ deviation from $60^{\circ}$ of the hexagonal lattice) and cannot
be distinguished using conventional X-ray diffractometry. Sample 1, which was
grown by Czochralski technique, with dimensions of $3.94\times 1.96\times
1.33$ mm3, was used for measurements under static magnetic fields up to 17 T
at Hokkaido University and up to 28 T using the cryogen-free hybrid magnet
system at High Field Laboratory for Superconducting Materials, IMR Tohoku
University. Sample 2, which was grown by the floating-zone technique at
Charles University, with dimensions of $2.23\times 1.67\times 1.80$ mm3, was
used for pulsed-magnetic-field studies up to 60 T with pulse durations of 150
ms at the Dresden High Magnetic Field Laboratory (HLD) at HZDR. No obvious
sample dependence was observed at least with X-ray diffraction, magnetization,
and ultrasonic measurements. Ultrasound was generated and detected using
LiNbO3 transducers with a thickness of 100 $\mu$m, which were bonded on well-
polished sample surfaces with RTV silicone or superglue. Sound-velocity
measurements were performed by applying a conventional phase-comparative
method [23] using a digital storage oscilloscope. The sound velocities
$v_{ij}$(m s-1) have been converted to elastic constants $C_{ij}$ (J m-3)
using the formula $C_{ij}=\rho v_{ij}^{2}$. Here, $\rho$ = 10.88 (g cm-3) is
the density of the sample calculated from the lattice constants $a=4.952$Å and
$c=6.954$Å.

## D. Formulation of Multipolar Susceptibility

We start from the Hamiltonian for a hexagonal CEF [30] and an elastic-strain-
mediated perturbed state [28],

$\mathcal{H}=\mathcal{H}_{\rm CEF}+\sum_{\Gamma}\frac{\partial\mathcal{H}_{\rm
CEF}}{\partial\epsilon_{\Gamma}}\epsilon_{\Gamma}.$ (1)

Here, $\epsilon_{\Gamma}$ is the symmetrized strain with point-group symmetry
$\Gamma$, which is induced by ultrasound. The hexagonal CEF Hamiltonian taking
Zeeman effect and tiny orthorhombic distortions into consideration is written
as

$\displaystyle\mathcal{H}_{\rm CEF}$ $\displaystyle=$ $\displaystyle
B_{2}^{0}O_{2}^{0}+B_{2}^{2}O_{2}^{2}+B_{4}^{0}O_{4}^{0}+B_{4}^{2}O_{4}^{2}+B_{4}^{4}O_{4}^{4}$
(2)
$\displaystyle+B_{6}^{0}O_{6}^{0}+B_{6}^{2}O_{6}^{2}+B_{6}^{4}O_{6}^{4}+B_{6}^{6}O_{6}^{6}$
$\displaystyle+g_{J}\mu_{\rm B}\sum_{i=x,y,z}J_{i}H_{i}.$

Here, $B_{m}^{n}$ are the CEF parameters and $O_{m}^{n}$ are the Stevens
operators. [29, 30]

$\displaystyle O_{2}^{0}$ $\displaystyle=$ $\displaystyle 3J_{z}^{2}-J(J+1)$
(3) $\displaystyle O_{2}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(J_{+}^{2}+J_{-}^{2})$ (4) $\displaystyle O_{4}^{0}$
$\displaystyle=$ $\displaystyle
35J_{z}^{4}-30J(J+1)J_{z}^{2}+25J_{z}^{2}-6J(J+1)$ (5)
$\displaystyle+3J^{2}(J+1)^{2}$ $\displaystyle O_{4}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\\{[7J_{z}^{2}-J(J+1)-5](J_{+}^{2}+J_{-}^{2})$ (6)
$\displaystyle+(J_{+}^{2}+J_{-}^{2})[7J_{z}^{2}-J(J+1)-5](J_{+}^{2}+J_{-}^{2})]\\}$
$\displaystyle O_{4}^{4}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(J_{+}^{4}+J_{-}^{4})$ (7) $\displaystyle O_{6}^{0}$
$\displaystyle=$ $\displaystyle
231J_{z}^{6}-315J(J+1)J_{z}^{4}+735J_{z}^{4}+105J^{2}(J+1)^{2}J_{z}^{2}$ (8)
$\displaystyle-525J(J+1)J_{z}^{2}+294J_{z}^{2}-5J^{3}(J+1)^{3}$
$\displaystyle+40J^{2}(J+1)^{2}-60J(J+1)$ $\displaystyle O_{6}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\\{[33J_{z}^{4}-18J_{z}^{2}J(J+1)-123J_{z}^{2}+J^{2}(J+1)^{2}$
(9) $\displaystyle+10J(J+1)+102](J_{+}^{4}+J_{-}^{4})$
$\displaystyle+(J_{+}^{4}+J_{-}^{4})[33J_{z}^{4}-18J_{z}^{2}J(J+1)$
$\displaystyle-123J_{z}^{2}+J^{2}(J+1)^{2}+10J(J+1)+102]\\}$ $\displaystyle
O_{6}^{4}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\\{[11J_{z}^{2}-J(J+1)-38](J_{+}^{4}+J_{-}^{4})$ (10)
$\displaystyle+(J_{+}^{4}+J_{-}^{4})[11J_{z}^{2}-J(J+1)-38]\\}$ $\displaystyle
O_{6}^{6}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[(J_{x}+iJ_{y})^{6}+(J_{x}-iJ_{y})^{6}]$ (11)

The numerical values of $B_{m}^{n}$, which were used in the present analysis,
are listed in TABLE SIII, SIV, and SV. The second term of Eq. (1) is explained
in terms of electric multipole–strain interaction. Particularly for a rank-2
multipole (quadrupole), this term is written as

$\displaystyle\mathcal{H}_{\rm MS}$ $\displaystyle=$ $\displaystyle-g_{\Gamma
1}\bigl{[}O_{B}\epsilon_{B}+O_{2}^{0}\epsilon_{u}\bigr{]}-g_{\Gamma
5}\bigl{[}O_{yz}\epsilon_{yz}+O_{zx}\epsilon_{zx}\bigr{]}$ (12)
$\displaystyle-g_{\Gamma
6}\bigl{[}O_{xy}\epsilon_{xy}+O_{2}^{2}\epsilon_{v}\bigr{]},$

where $g_{\Gamma}$ are the coupling constants for the rank-2 multipoles.
$O_{\Gamma}$ and $\epsilon_{\Gamma}$ are quadrupole operators and symmetrized
strains, respectively. They are listed in TABLE SII, and the quadrupole
operators are defined in section E.

Table SI:  Symmetry, symmetrized strains and rotation, even parity multipoles,
and elastic constants in hexagonal notation.

Symmetry (D6h) | Strain and Rotation | Multipole | Elastic Constant
---|---|---|---
$\Gamma_{1}$$\oplus$$\Gamma_{6}$(A1g$\oplus$E2g) | $\epsilon_{xx},\epsilon_{yy}$ | - | $C_{11}$
$\Gamma_{1}$(A1g) | $\epsilon_{zz}=\epsilon_{B}/3-\epsilon_{u}/\sqrt{3}$ | - | $C_{33}$
$\Gamma_{5}$(E1g) | $\epsilon_{yz}$ | $O_{yz}=\sqrt{3}(J_{y}J_{z}+J_{z}J_{y})/2$ | $C_{44}$
| $\epsilon_{zx}$ | $O_{zx}=\sqrt{3}(J_{z}J_{x}+J_{x}J_{z})/2$ | $C_{44}$
$\Gamma_{6}$(E2g) | $\epsilon_{xy}$ | $O_{xy}=\sqrt{3}(J_{x}J_{y}+J_{y}J_{x})/2$ | $C_{66}$
| $\epsilon_{v}=\epsilon_{xx}-\epsilon_{yy}$ | $O_{v}=\sqrt{3}(J_{x}^{2}-J_{y}^{2})/2$ | $C_{66}$
$\Gamma_{2}$(A2g) | $\omega_{\rm xy}$ | $H_{z}^{\alpha}=\sqrt{35}(J_{+}^{4}+J_{-}^{4})/4i$ | $C_{66}$

Table SII: Fit parameters for the elastic constant $C_{66}$ in the present
analysis

CEF models | $C^{0}_{66{\rm(PM)}}({\rm J/m^{3}})$ | $s$ |  | $t$ | $\left|g_{\Gamma 6{\rm(PM)}}\right|({\rm K})$ | $g^{\prime}_{\Gamma 6{\rm(PM)}}({\rm K})$ | $C^{0}_{66{\rm(AFM)}}({\rm J/m^{3}})$ | $\left|g_{\Gamma 6{\rm(AFM)}}\right|$ | $g^{\prime}_{\Gamma 6{\rm(AFM)}}$ |
---|---|---|---|---|---|---|---|---|---|---
Scheme 0 | 9.1062 | 0.28 |  | 180 | 30 | - | - | - | - | Oyamada et al. [27]
Scheme 1 and 2 | 9.1062 | 0.28 |  | 180 | 15.5 | +0.42 | 9.101 | 21.0 | -0.045 | present work

Table SIII: CEF parameters and wave functions for CEF Scheme 0 [27]: Hexagonal ($J=9/2$) CEF parameters | Scheme 0 |  |  |  |  $B_{2}^{0}$(K) |  $B_{2}^{2}$(K) |  $B_{4}^{0}$(K) |  $B_{4}^{2}$(K) |  $B_{6}^{0}$(K) |  $B_{6}^{2}$(K) |  $B_{6}^{6}$(K) | Ref.
---|---|---|---|---|---|---|---|---|---|---|---|---
Hexagonal | $J=9/2$ |  |  |  | +1.0 | - | -0.135 | - | +0.02 | - | +0.3 |
Eigenvalues | Wave functions |  |  |  |  |  |  |  |  |  |  |
E (K) | Symmetry |  | $\left|+9/2\right>$ | $\left|+7/2\right>$ | $\left|+5/2\right>$ | $\left|+3/2\right>$ | $\left|+1/2\right>$ | $\left|-1/2\right>$ | $\left|-3/2\right>$ | $\left|-5/2\right>$ | $\left|-7/2\right>$ | $\left|-9/2\right>$
0 | $\Gamma_{8}^{(1)}$ |  |  | $-\kappa$ |  |  |  |  |  | $\lambda$ |  |
0 | $\Gamma_{8}^{(1)}$ |  |  |  | $\lambda$ |  |  |  |  |  | $-\kappa$ |
616.2 | $\Gamma_{7}$ |  |  |  |  |  | +1 |  |  |  |  |
616.2 | $\Gamma_{7}$ |  |  |  |  |  |  | +1 |  |  |  |
982.6 | $\Gamma_{9}^{(2)}$ |  | $\nu$ |  |  |  |  |  | $\mu$ |  |  |
982.6 | $\Gamma_{9}^{(2)}$ |  |  |  |  | $\mu$ |  |  |  |  |  | $\nu$
3006 | $\Gamma_{9}^{(1)}$ |  | $\mu$ |  |  |  |  |  | $-\nu$ |  |  |
3006 | $\Gamma_{9}^{(1)}$ |  |  |  |  |  | $\nu$ |  |  |  |  | $\mu$
3649 | $\Gamma_{8}^{(2)}$ |  |  |  | $\kappa$ |  |  |  |  |  | $\lambda$ |
3649 | $\Gamma_{8}^{(2)}$ |  |  | $\lambda$ |  |  |  |  |  | $\kappa$ |  |
$\kappa=0.8831$ |  |  |  |  |  |  |  |  |  |  |  |
$\lambda=0.4692$ |  |  |  |  |  |  |  |  |  |  |  |
$\nu=0.7768$ |  |  |  |  |  |  |  |  |  |  |  |
$\mu=-0.6298$ |  |  |  |  |  |  |  |  |  |  |  |
Table SIV: CEF parameters and wave functions for CEF Scheme 1 (Present work): Hexagonal ($J=4$) CEF parameters | Scheme 1 |  |  $B_{2}^{0}$(K) |  $B_{2}^{2}$(K) |  $B_{4}^{0}$(K) |  $B_{4}^{2}$(K) |  $B_{4}^{4}$(K) |  $B_{6}^{0}$(K) |  $B_{6}^{2}$(K) |  $B_{6}^{4}$(K) |  $B_{6}^{6}$(K)
---|---|---|---|---|---|---|---|---|---|---|---
Hexagonal | $J=4$ |  | +1.0 | - | -0.02 | - | - | -0.005 | - | - | +0.32
Eigenvalues | Wave functions |  |  |  |  |  |  |  |  |  |
E (K) | Symmetry |  | $\left|+4\right>$ | $\left|+3\right>$ | $\left|+2\right>$ | $\left|+1\right>$ | $\left|0\right>$ | $\left|-1\right>$ | $\left|-2\right>$ | $\left|-3\right>$ | $\left|-4\right>$
0 | $\Gamma_{5}^{(2)}$ |  | $-\alpha$ |  |  |  |  |  | $+\beta$ |  |
0 | $\Gamma_{5}^{(2)}$ |  |  |  | $-\beta$ |  |  |  |  |  | $-\alpha$
19.1 | $\Gamma_{4}$ |  |  | +1/$\sqrt{2}$ |  |  |  |  |  | -1/$\sqrt{2}$ |
652.1 | $\Gamma_{6}$ |  |  |  |  | +1 |  |  |  |  |
652.1 | $\Gamma_{6}$ |  |  |  |  |  |  | -1 |  |  |
770.6 | $\Gamma_{1}$ |  |  |  |  |  | +1 |  |  |  |
1225 | $\Gamma_{5}^{(1)}$ |  | $-\beta$ |  |  |  |  | $-\alpha$ |  |  |
1225 | $\Gamma_{5}^{(1)}$ |  |  |  | $+\alpha$ |  |  |  |  |  | $+\beta$
1632 | $\Gamma_{3}$ |  |  | +1/$\sqrt{2}$ |  |  |  |  |  | +1/$\sqrt{2}$ |
$\alpha=0.6718$ |  |  |  |  |  |  |  |  |  |  |
$\beta=0.7408$ |  |  |  |  |  |  |  |  |  |  |
Table SV: CEF parameters and wave functions for CEF Scheme 2 (Present work): Orthorhombic ($J=4$) CEF parameters | Scheme 1 |  |  $B_{2}^{0}$(K) |  $B_{2}^{2}$(K) |  $B_{4}^{0}$(K) |  $B_{4}^{2}$(K) |  $B_{4}^{4}$(K) |  $B_{6}^{0}$(K) |  $B_{6}^{2}$(K) |  $B_{6}^{4}$(K) |  $B_{6}^{6}$(K)
---|---|---|---|---|---|---|---|---|---|---|---
Ortho. | $J=4$ |  | +1.0 | +0.030 | -0.02 | +0.001 | 0 | -0.005 | 0 | 0 | +0.32
Eigenvalues | Wave functions |  |  |  |  |  |  |  |  |  |
E (K) | Symmetry |  | $\left|+4\right>$ | $\left|+3\right>$ | $\left|+2\right>$ | $\left|+1\right>$ | $\left|0\right>$ | $\left|-1\right>$ | $\left|-2\right>$ | $\left|-3\right>$ | $\left|-4\right>$
0 | $\Gamma_{1}^{(52)}$ |  | $-\alpha^{\prime}$ |  | $+\beta^{\prime}$ |  | -2$\delta$ |  | $+\beta^{\prime}$ |  | $-\alpha^{\prime}$
0.79 | $\Gamma_{2}^{(52)}$ |  | $-\alpha^{\prime}$ |  | $-\beta^{\prime}$ |  |  |  | $+\beta^{\prime}$ |  | $+\alpha^{\prime}$
19.49 | $\Gamma_{4}$ |  |  | -1/$\sqrt{2}$ |  | +4$\delta$ |  | -4$\delta$ |  | +1/$\sqrt{2}$ |
652.4 | $\Gamma_{2}^{6}$ |  |  | -4$\delta$ |  | -1/$\sqrt{2}$ |  | +1/$\sqrt{2}$ |  | +4$\delta$ |
652.6 | $\Gamma_{1}^{6}$ |  |  | -2$\delta$ |  | +1/$\sqrt{2}$ |  | +1/$\sqrt{2}$ |  | -2$\delta$ |
771.0 | $\Gamma_{1}^{(1)}$ |  | +3$\delta$ |  | +$\delta$ |  | -1 |  | +$\delta$ |  | +3$\delta$
1225 | $\Gamma_{1}^{(51)}$ |  | $+\beta^{\prime}$ |  | $-\alpha^{\prime}$ |  |  |  | $+\alpha^{\prime}$ |  | $-\beta^{\prime}$
1226 | $\Gamma_{2}^{(51)}$ |  | $+\beta^{\prime}$ |  | $+\alpha^{\prime}$ |  | +4$\delta$ |  | $+\alpha^{\prime}$ |  | $+\beta^{\prime}$
1632 | $\Gamma_{3}$ |  |  | -1/$\sqrt{2}$ |  | -2$\delta$ |  | -2$\delta$ |  | +1/$\sqrt{2}$ |
$\alpha^{\prime}=0.4750$ |  |  |  |  |  |  |  |  |  |  |
$\beta^{\prime}=0.5238$ |  |  |  |  |  |  |  |  |  |  |
$\delta=0.0001$ |  |  |  |  |  |  |  |  |  |  |

Then, the free energy of the local $5f$-electronic states in the CEF can be
written as

$\displaystyle F=U-Nk_{\rm
B}T\ln\sum_{i}\exp\\{-E_{i}(\epsilon_{\Gamma})/k_{\rm B}T\\}.$ (13)

Here, $N$ is the number of U ions in a unit volume,
$E_{n}$($\epsilon_{\Gamma}$) is a perturbed CEF level as a function of strain
and $n$ is a collective index for $J$ multiplets and their degenerate states.
$U$ gives the internal energy of the strained system in zero magnetic field,
which is written in terms of the symmetry strains and elastic constants listed
in TABLE SII as,

$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigl{[}C_{B}\epsilon_{B}^{2}+C_{Bu}\epsilon_{B}\epsilon_{u}+C_{u}\epsilon_{u}^{2}+C_{66}(\epsilon_{xy}^{2}+\epsilon_{v}^{2})$
(14) $\displaystyle+C_{44}(\epsilon_{yz}^{2}+\epsilon_{zx}^{2})\bigr{]}.$

Here,
$C_{Bu}=-\frac{(C_{11}^{0}+C_{12}^{0}-C_{13}^{0}-C_{14}^{0})}{\sqrt{3}}$. The
contribution of the electric hexadecapole $H_{z}^{\alpha}$ with $\Gamma_{2}$
symmetry is not considered for zero magnetic field, since it will couple to
the transverse $C_{66}$ mode only in a finite magnetic field due to the
rotational invariance [24, 25]. In the framework of second-order perturbation
theory, the temperature dependence of the elastic constant is given by

$C_{\Gamma}(T,H)=C_{\Gamma}^{0}-Ng_{\Gamma}^{2}\chi_{\Gamma}(T,H).$ (15)

Here, $C_{\Gamma}^{0}$ is the background of the elastic constant. The single-
ion quadrupolar susceptibility $\chi_{\Gamma}$ is defined as the second
derivative of the free energy in the limit $\epsilon_{\Gamma}\rightarrow 0$,

$\displaystyle-g_{\Gamma}^{2}\chi_{\Gamma}$ $\displaystyle\equiv$
$\displaystyle\frac{\partial^{2}F}{\partial\epsilon_{\Gamma}^{2}}\Bigr{|}_{\epsilon_{\Gamma}\rightarrow
0}$ $\displaystyle=$
$\displaystyle-\left<\frac{\partial^{2}E_{i}}{\partial\epsilon_{\Gamma}^{2}}\right>+\frac{1}{k_{\rm
B}T}\Bigl{[}\left<\Bigl{(}\frac{\partial
E_{i}}{\partial\epsilon_{\Gamma}}\Bigr{)}^{2}\right>-\left<\frac{\partial
E_{i}}{\partial\epsilon_{\Gamma}}\right>^{2}\Bigr{]}.$

Here, the brackets $\left<\;\;\right>$ indicate the thermal average. In
addition to the strain-quadrupole interaction (Eq. 12), the intersite
quadrupole-quadrupole interaction can also be added by using the molecular-
field approximation of the quadrupolar moment $O_{\Gamma}$ by considering
sublattices $\alpha$, as

$\displaystyle\mathcal{H}_{\rm MM}$ $\displaystyle=$
$\displaystyle-\sum_{\alpha}\sum_{\Gamma}g^{\prime}_{\Gamma}\left<O_{\Gamma}\right>O_{\Gamma}^{\alpha}$
(17) $\displaystyle=$ $\displaystyle-g^{\prime}_{\Gamma
1}\left<O_{2}^{0}\right>O_{2}^{0(\alpha)}-g^{\prime}_{\Gamma
5}\Bigl{\\{}\left<O_{yz}\right>O_{yz}^{(\alpha)}+\left<O_{zx}\right>O_{zx}^{(\alpha)}\Bigr{\\}}$
$\displaystyle-g^{\prime}_{\Gamma
6}\Bigl{\\{}\left<O_{xy}\right>O_{xy}^{(\alpha)}+\left<O_{2}^{2}\right>O_{2}^{2(\alpha)}\Bigr{\\}},$
$\displaystyle\mathcal{H}_{\rm MS}+\mathcal{H}_{\rm MM}$ $\displaystyle=$
$\displaystyle-\sum_{\alpha}\Bigl{\\{}-g_{\Gamma}\sum_{\Gamma\gamma}O_{\Gamma\gamma}^{(\alpha)}\epsilon_{\Gamma\gamma}+g^{\prime}_{\Gamma}\sum_{\Gamma\gamma}\left<O_{\Gamma\gamma}\right>O_{\Gamma\gamma}^{\alpha}\Bigr{\\}}$
(18) $\displaystyle=$ $\displaystyle-
g_{\Gamma}\sum_{\alpha}\Bigl{\\{}\sum_{\Gamma\gamma}O_{\Gamma\gamma}^{(\alpha)}\Bigl{[}\epsilon_{\Gamma\gamma}+\frac{g^{\prime}_{\Gamma}}{g_{\Gamma}}\left<O_{\Gamma}\right>\Bigr{]}\Bigr{\\}}$
$\displaystyle=$ $\displaystyle-
g_{\Gamma}\sum_{\alpha}\Bigl{\\{}\sum_{\Gamma\gamma}O_{\Gamma\gamma}^{(\alpha)}\epsilon_{\Gamma\gamma}^{\rm
eff}\Bigr{\\}}.$ $\epsilon_{\Gamma\gamma}^{\rm
eff}=\epsilon_{\Gamma\gamma}+\frac{g^{\prime}_{\Gamma}}{g_{\Gamma}}\left<O_{\Gamma}\right>.$
(19)

The temperature dependence of the elastic constant (Eq. 15) can be rewritten
as,

$C_{\Gamma}(T,H)=C_{\Gamma}^{0}-\frac{Ng_{\Gamma}^{2}\chi_{\Gamma}(T,H)}{1-g^{\prime}_{\Gamma}\chi_{\Gamma}(T,H)}.$
(20)

In the present analysis, the second term is divided into two terms with regard
to the different U sites: 2/3 U ions at the UAFM sites [Fig. 1(b) in the main
text] and 1/3 of U ions at the UPM1 and UPM2 sites. We defined two independent
sets of coupling constants for each U-ion contribution as

$\displaystyle C_{\Gamma}(T,H)$ $\displaystyle=$ $\displaystyle C_{66(\rm
PM)}-\frac{\frac{2}{3}Ng_{\Gamma 6(\rm PM)}^{2}\chi_{\Gamma
6}(T)}{1-g^{\prime}_{\Gamma 6(\rm PM)}\chi_{\Gamma 6}(T)}$ (21)
$\displaystyle-\frac{\frac{1}{3}Ng_{\Gamma 6(\rm AFM)}^{2}\chi_{\Gamma
6}(T)}{1-g^{\prime}_{\Gamma 6(\rm AFM)}\chi_{\Gamma 6}(T)}.$

Here, the phonon background in the PM phase is redefined as the
phenomenological fit
$C_{66{\rm(PM)}}(T)=C_{66{\rm(PM)}}^{0}-\frac{s}{\exp(t/T)-1}$ with parameters
$C_{66{\rm(PM)}}^{0}$, $s$, and $t$ (listed in TABLE SII). $\left|g_{\Gamma
6{\rm(PM)}}\right|$ and $\left|g_{\Gamma 6{\rm(AFM)}}\right|$ are the coupling
constants of the quadrupolar-strain interaction for the PM and AFM phases,
respectively. The coupling constant of the quadrupolar intersite interaction
$g^{\prime}_{\Gamma 6(\rm(PM)}$ and $g^{\prime}_{\Gamma 6(\rm AFM)}$ are
defined as the same manner. For the AFM phase, we assume that the second term
of Eq. (21), the quadrupolar contribution from U(AFM) is a temperature-
independent constant value, and redefine the temperature-independent
background as $C^{0}_{66{\rm(AFM)}}$.

$C_{\Gamma}(T,H)=C_{66(\rm AFM)}^{0}-\frac{\frac{1}{3}Ng_{\Gamma 6(\rm
AFM)}^{2}\chi_{\Gamma 6}(T)}{1-g^{\prime}_{\Gamma 6(\rm AFM)}\chi_{\Gamma
6}(T)}.$ (22)

## E. Definition of Multipolar Moments and Equivalent Operator Expressions

The electric multipolar operators are defined by multipolar expansion of the
electrostatic potential [3] as

$Q_{lm}\equiv e\sum_{j=1}^{n_{f}}r_{j}^{i}Z_{lm}^{*}(r_{j}).$ (23)

Here, $e<0$ is the electron charge, $n_{f}$ is the number of $f$ electrons.
$Z_{lm}(r_{j})$ is written by using spherical harmonics $Y_{lm}(r_{j})$ as

$Z_{lm}(r_{j})\equiv\sqrt{4\pi/(2l+1)}Y_{lm}(r_{j}).$ (24)

Eq. (24) can be rewritten by replacing ($x,y,z$) in $Z_{lm}$ with spherical
tensor operators $J_{lm}$ with following transformations,

$x^{n_{x}}y^{n_{y}}z^{n_{z}}\rightarrow\frac{n_{x}!n_{y}!n_{z}!}{(n_{x}+n_{y}+n_{z})!}\sum_{\mathcal{P}}\mathcal{P}(J_{x}^{n_{x}}J_{y}^{n_{y}}J_{z}^{n_{z}}).$
(25)

Here, $\mathcal{P}$ is a sum of all possible permutations. The operator
$J_{lm}$ has the following commutation relations, with the ladder operator
$J_{\pm}=J_{x}\pm iJ_{y}$:

$J_{ll}=(-1)^{l}\sqrt{\frac{(2l-1)!!}{(2l)!!}}(J_{+})^{l},$ (26)
$[J_{-},J_{lm}]=\sqrt{(l+m)(l-m+1)}J_{lm-1}.$ (27)

Following are quadrupolar and hexadecapolar operators, which are used in the
present analysis:
i) Rank 2 (Quadrupole)

$\displaystyle\Gamma_{5}(\rm E_{1g}):$ $\displaystyle O_{yz}=$
$\displaystyle\frac{i}{\sqrt{2}}[J_{21}+J_{2-1}]=\frac{\sqrt{3}}{2}(J_{y}J_{z}+J_{z}J_{y})$
(28) $\displaystyle\Gamma_{5}(\rm E_{1g}):$ $\displaystyle O_{zx}=$
$\displaystyle\frac{1}{\sqrt{2}}[-J_{21}+J_{2-1}]=\frac{\sqrt{3}}{2}(J_{z}J_{x}+J_{x}J_{z})$
(29)
$\displaystyle\Gamma_{6}(\rm E_{\rm 2g}):$ $\displaystyle O_{xy}=$
$\displaystyle\frac{i}{\sqrt{2}}[-J_{22}+J_{2-2}]=\frac{\sqrt{3}}{2}(J_{x}J_{y}-J_{y}J_{x})$
(30) $\displaystyle\Gamma_{6}(\rm E_{\rm 2g}):$ $\displaystyle O_{v}=$
$\displaystyle\frac{\sqrt{3}}{2}O_{2}^{2}=\frac{i}{\sqrt{2}}[J_{22}+J_{2-2}]=\frac{\sqrt{3}}{2}(J_{x}^{2}+J_{y}^{2})$
(31)

ii) Rank 4 (Hexadecapole)

$\displaystyle\Gamma_{2}(\rm A_{\rm 2g}):$ $\displaystyle H_{z}^{\alpha}$
$\displaystyle=$ $\displaystyle\frac{\sqrt{35}}{4i}[-J_{44}+J_{4-4}]$ (32)
$\displaystyle=$
$\displaystyle\frac{\sqrt{35}}{8}\\{(J_{x}^{3}J_{y}+J_{x}^{2}J_{y}J_{x}+J_{x}J_{y}J_{x}^{2}+J_{y}J_{x}^{3})$
$\displaystyle-$
$\displaystyle(J_{x}J_{y}^{3}+J_{y}^{2}J_{x}J_{y}+J_{y}J_{x}J_{y}^{2}+J_{x}J_{y}^{3})\\}$

## F. Matrix Elements of Magnetic dipole and electric quadrupoles

Here, we show the matrix elements of magnetic dipole $J_{x}$ and $J_{z}$, and
electric quadrupoles $O_{xy}$ and $O_{yz}$, which are calculated by using
present CEF models. Especially, the finite diagonal- and off-diagonal-matrix
elements of $J_{x}$ and $O_{xy}$ are correspond to the transitions indicated
by arrows in Fig. 3(a) of the main text.

i) CEF Scheme 0 ($J=9/2$)

$\begin{array}[]{cll}&{\rm Energy}&{\rm\hskip 45.0ptWavefunction}\\\
\hline\cr&0\,{\rm K}&\left|\Gamma_{8}^{(1)}\pm\right>=-\kappa\left|\pm
7/2\right>+\lambda\left|\mp 5/2\right>\\\ &616.2\,{\rm
K}&\left|\Gamma_{7}\pm\right>=\left|\pm 1\right>\\\ &&\\\ &&\kappa=0.8831\\\
&&\lambda=0.4692\\\ &&\nu=0.7768\\\ &&\mu=-0.6298\\\ \end{array}$ (33)
$\left<J_{x}\right>=\left[\begin{array}[]{rcc|cc}&\left|\Gamma_{8}^{(1)}+\right>&\left|\Gamma_{8}^{(1)}-\right>&\left|\Gamma_{7}+\right>&\left|\Gamma_{7}-\right>\\\
\left<\Gamma_{8}^{(1)}+\right|&0&-1.675&0&0\\\
\left<\Gamma_{8}^{(1)}-\right|&-1.675&0&0&0\\\
\hline\cr\left<\Gamma_{7}+\right|&0&0&-0.5&0\\\
\left<\Gamma_{7}-\right|&0&0&0&+0.5\\\ \end{array}\right]$ (34)

$\left<J_{z}\right>=\left[\begin{array}[]{rcc|cc}&\left|\Gamma_{8}^{(1)}+\right>&\left|\Gamma_{8}^{(1)}-\right>&\left|\Gamma_{7}+\right>&\left|\Gamma_{7}-\right>\\\
\left<\Gamma_{8}^{(1)}+\right|&-2.179&0&0&0\\\
\left<\Gamma_{8}^{(1)}-\right|&0&+2.179&0&0\\\
\hline\cr\left<\Gamma_{7}+\right|&0&0&0&+2.5\\\
\left<\Gamma_{7}-\right|&0&0&+2.5&0\\\ \end{array}\right]$ (35)

$\left<O_{xy}\right>=\left[\begin{array}[]{rcc|cc}&\left|\Gamma_{8}^{(1)}+\right>&\left|\Gamma_{8}^{(1)}-\right>&\left|\Gamma_{7}+\right>&\left|\Gamma_{7}-\right>\\\
\left<\Gamma_{8}^{(1)}+\right|&0&0&0&-5.267\\\
\left<\Gamma_{8}^{(1)}-\right|&0&0&+5.267&0\\\
\hline\cr\left<\Gamma_{7}+\right|&0&-5.267&0&0\\\
\left<\Gamma_{7}-\right|&+5.267&0&0&0\\\ \end{array}\right]\times i$ (36)

$\left<O_{yz}\right>=\left[\begin{array}[]{rcc|cc}&\left|\Gamma_{8}^{(1)}+\right>&\left|\Gamma_{8}^{(1)}-\right>&\left|\Gamma_{7}+\right>&\left|\Gamma_{7}-\right>\\\
\left<\Gamma_{8}^{(1)}+\right|&0&0&0&0\\\
\left<\Gamma_{8}^{(1)}-\right|&0&0&0&0\\\
\hline\cr\left<\Gamma_{7}+\right|&0&0&0&0\\\
\left<\Gamma_{7}-\right|&0&0&0&0\\\ \end{array}\right]\times i$ (37)

ii) CEF Scheme 1 ($J=4$)

For CEF Scheme 1 and 2, the matrix element of the $\Gamma_{6}$ doublet at
around 652 K is omitted because its effect on the low-temperature properties
is negligible.

$\begin{array}[]{cll}&{\rm Energy}&{\rm\hskip 45.0ptWavefunction}\\\
\hline\cr&0\,{\rm K}&\left|\Gamma_{5}^{(2)}\pm\right>=-\alpha\left|\pm
4\right>\pm\beta\left|\mp 2\right>\\\ &19.1\,{\rm
K}&\left|\Gamma_{4}\right>=\frac{1}{\sqrt{2}}(\left|+3\right>-\left|-3\right>)\\\
&&\\\ &&\alpha=0.6718\\\ &&\beta=0.7408\end{array}$ (38)
$\left<J_{x}\right>=\left[\begin{array}[]{rcc|c}&\left|\Gamma_{5}^{(2)}+\right>&\left|\Gamma_{5}^{(2)}-\right>&\left|\Gamma_{4}\right>\\\
\left<\Gamma_{5}^{(2)}+\right|&0&0&-1.652\\\
\left<\Gamma_{5}^{(2)}-\right|&0&0&-1.652\\\
\hline\cr\left<\Gamma_{4}\right|&-1.652&-1.652&0\\\ \end{array}\right]$ (39)

$\left<J_{z}\right>=\left[\begin{array}[]{rcc|c}&\left|\Gamma_{5}^{(2)}+\right>&\left|\Gamma_{5}^{(2)}-\right>&\left|\Gamma_{4}\right>\\\
\left<\Gamma_{5}^{(2)}+\right|&-0.708&0&0\\\
\left<\Gamma_{5}^{(2)}-\right|&0&+0.708&0\\\
\hline\cr\left<\Gamma_{4}\right|&0&0&0\\\ \end{array}\right]$ (40)

$\left<O_{xy}\right>=\left[\begin{array}[]{rcc|c}&\left|\Gamma_{5}^{(2)}+\right>&\left|\Gamma_{5}^{(2)}-\right>&\left|\Gamma_{4}\right>\\\
\left<\Gamma_{5}^{(2)}+\right|&0&+5.266&0\\\
\left<\Gamma_{5}^{(2)}-\right|&-5.266&0&0\\\
\hline\cr\left<\Gamma_{4}\right|&0&0&0\\\ \end{array}\right]\times i$ (41)

$\left<O_{yz}\right>=\left[\begin{array}[]{rcc|c}&\left|\Gamma_{5}^{(2)}+\right>&\left|\Gamma_{5}^{(2)}-\right>&\left|\Gamma_{4}\right>\\\
\left<\Gamma_{5}^{(2)}+\right|&0&0&-0.198\\\
\left<\Gamma_{5}^{(2)}-\right|&0&0&-0.198\\\
\hline\cr\left<\Gamma_{4}\right|&+0.198&+0.198&0\\\ \end{array}\right]\times
i$ (42)

iii) CEF Scheme 2 ($J=4$)

$\begin{array}[]{clll}&{\rm Energy}&{\rm\hskip 45.0ptWavefunction}\\\
\hline\cr&0\,{\rm
K}&\left|\Gamma_{1}^{(52)}\right>=-\alpha^{\prime}(\left|+4\right>+\left|-4\right>)\\\
&&\hskip
45.0pt+\beta^{\prime}(\left|+2\right>+\left|-2\right>)-2\delta\left|0\right>\\\
&0.79\,{\rm
K}&\left|\Gamma_{2}^{(52)}\right>=-\alpha^{\prime}(\left|+4\right>-\left|-4\right>)\\\
&&\hskip 45.0pt-\beta^{\prime}(\left|+2\right>-\left|-2\right>)\\\
&19.49\,{\rm
K}&\left|\Gamma_{4}\right>=-\frac{1}{\sqrt{2}}(\left|+3\right>-\left|-3\right>+4\delta(\left|+1\right>-\left|-1\right>)\\\
&&\\\ &&\alpha^{\prime}=0.4750\\\ &&\beta^{\prime}=0.5238\\\
&&\delta=0.0001\end{array}$ (43)

$\left<J_{x}\right>=\left[\begin{array}[]{rcc|c}&\left|\Gamma_{1}^{(52)}\right>&\left|\Gamma_{2}^{(52)}\right>&\left|\Gamma_{4}\right>\\\
\left<\Gamma_{1}^{(52)}\right|&0&0&0\\\
\left<\Gamma_{1}^{(52)}\right|&0&0&+2.351\\\
\hline\cr\left<\Gamma_{4}\right|&0&+2.351&0\\\ \end{array}\right]$ (44)

$\left<J_{z}\right>=\left[\begin{array}[]{rcc|c}&\left|\Gamma_{1}^{(52)}\right>&\left|\Gamma_{2}^{(52)}\right>&\left|\Gamma_{4}\right>\\\
\left<\Gamma_{1}^{(52)}\right|&0&-0.708&0\\\
\left<\Gamma_{1}^{(52)}\right|&-0.708&0&0\\\
\hline\cr\left<\Gamma_{4}\right|&0&0&0\\\ \end{array}\right]$ (45)

$\left<O_{xy}\right>=\left[\begin{array}[]{rcc|c}&\left|\Gamma_{1}^{(52)}\right>&\left|\Gamma_{2}^{(52)}\right>&\left|\Gamma_{4}\right>\\\
\left<\Gamma_{1}^{(52)}\right|&0&+5.264&0\\\
\left<\Gamma_{1}^{(52)}\right|&-5.264&0&0\\\
\hline\cr\left<\Gamma_{4}\right|&0&0&0\\\ \end{array}\right]\times i$ (46)

$\left<O_{yz}\right>=\left[\begin{array}[]{rcc|c}&\left|\Gamma_{1}^{(52)}\right>&\left|\Gamma_{2}^{(52)}\right>&\left|\Gamma_{4}\right>\\\
\left<\Gamma_{1}^{(52)}\right|&0&0&0\\\
\left<\Gamma_{1}^{(52)}\right|&0&0&+0.282\\\
\hline\cr\left<\Gamma_{4}\right|&0&-0.282&0\\\ \end{array}\right]\times i$
(47)

Figure S3:  (a) Measured magnetic susceptibility of UNi4B for
$H\parallel[01\bar{1}0]$ (open circles) [17] and calculated magnetic
susceptibility using the CEF level Schemes 0, 1, and 2. (b) Inverse magnetic
susceptibility of UNi4B for $H\parallel[0001]$ and $H\parallel[01\bar{1}0]$
with experimental (open circles) and calculated data.

## G. Consistency Test for CEF Calculation of the Magnetic Susceptibility

In the main text, we propose new CEF level schemes (Scheme 1 and Scheme 2) to
reproduce the elastic response of the $C_{66}$ mode in UNi4B. Here, we show
the consistency test using these CEF models by comparing the calculated with
the experimentally obtained magnetic susceptibility. Figs. S3(a) and S3(b)
show the calculated magnetic susceptibility of $H\parallel[01\bar{1}0]$, and
the inverse magnetic susceptibilities for $H\parallel[01\bar{1}0]$ and [0001]
compared with the experimental data. We notice that Scheme 0 reproduces the
data for the high-temperature region $T>50$ K well, while Schemes 1 and 2
deviate slightly from the experimental data. On the other hand, Schemes 1 and
2 reproduce the low-temperature region below $\sim 50$ K well, including the
AFM-ordered phase, and mimic the leveling off and decrease of the data below
10 K for $H\parallel[01{\bar{1}}0]$, while Scheme 0 suggests a continuous
increase toward absolute zero. The leveling off and decreasing susceptibility
imply the presence of PM U ions in the AFM phase, so the calculated results of
Schemes 1 and 2 are consistent with the experimentally obtained magnetic
response. This consistency suggests that the change in the Fermi surface due
to the AFM order enhances the localized character of the U ions in the AFM
phase, and this localized electronic state is reproduced well by the $J=4$
state. Otherwise, we need to consider a novel coupling mechanism between the
strain field and the toroidal magnetic moment (augmented multipole) that
originates from parity mixing.

## H. Elastic Constant $C_{66}$ and Calculated Quadrupolar Susceptibility at
Low Temperatures ($<1$ K)

Here, we show additional low-temperatures results ($<1$ K) obtained using a
dilution refrigerator. Figs. S4 and S5 show the relative change of the elastic
constant $C_{66}$ as a function of magnetic field and temperature,
respectively. The data in Fig. S4 were taken for both increasing and
decreasing magnetic field, as indicated by arrows, and the data are vertically
shifted for better visualization. The data in Fig. S5 are shifted vertically
based on the magnetic-field dependence of $\Delta C_{66}$ to compare the
response between $H\parallel[01\bar{1}0]$ and $H\parallel[2\bar{1}\bar{1}0]$.
The right- and left-headed horizontal arrows on the side of the magnetic field
values indicate heating and cooling processes, respectively. The arrowhead at
around 0.3 K indicates the inflection point, defined by the crossing of the
linear extrapolations from the low- and high-temperature side. The shallow
minimum that appears around 0.3 K in the temperature dependence and around 4 T
in the magnetic-field dependence for $H\parallel[2\bar{1}\bar{1}0]$ are twice
as large as similar anomalies observed for $H\parallel[01\bar{1}0]$. The
anisotropy is roughly consistent with the calculation based on the CEF Scheme
2. Figure S6 shows the calculated $\Gamma_{6}$(E2g)-quadrupolar susceptibility
as a function of temperature at various magnetic fields and as a function of
magnetic field at 0.1 K for (a) $H\parallel[01\bar{1}0]$ and (b)
$H\parallel[2\bar{1}\bar{1}0]$. Here, Figs. S5 and S6 are displayed in a
similar style with corresponding colors. It appears that the local minimum is
caused by a level crossing in the magnetic-field dependence of the quadrupolar
susceptibility for $H\parallel[2\bar{1}\bar{1}0]$ but not for
$H\parallel[01\bar{1}0]$. This anisotropic response of the CEF effect is
roughly consistent with the tendency of the local minimum in $C_{66}$, as
shown in Fig. S4, where the magnitude of the local minimum for
$H\parallel[2\bar{1}\bar{1}0]$ is twice as large as that for
$H\parallel[01\bar{1}0]$. In order to reproduce the magnetic-field dependence
and anisotropy of $T^{*}$ to higher fields, a more detailed calculation
considering molecular-field models is needed, which is out of the scope of
this work.

Figure S4:  Relative change of the elastic constant $C_{66}$ of UNi4B as a
function of magnetic field for (a) $H\parallel[01\bar{1}0]$ and (b)
$H\parallel[2\bar{1}\bar{1}0]$ at various fixed temperatures.  Figure S5:
Relative change of the elastic constant $C_{66}$ of UNi4B as a function of
temperature for (a) $H\parallel[01\bar{1}0]$ and (b)
$H\parallel[2\bar{1}\bar{1}0]$ at various magnetic fields. Figure S6:
Calculated $\Gamma_{6}$ (E2g)-quadrupolar susceptibility of $O_{xy}$ using the
CEF Scheme 2 as a function of temperature (bottom axis) and magnetic field
(top axis) at 0.1 K for (a) $H\parallel[01\bar{1}0]$ and (b)
$H\parallel[2\bar{1}\bar{1}0]$ at various fixed magnetic fields. The inset in
each figure shows the magnetic-field dependence of the energy niveaus of the
ground-state doublet and the first excited singlet.

## I. Anisotropy of the $H$-$T$ Phase Diagram and the 330 mK Anomaly

In Figs. S7 and S8, we display the $H$–$T$ phase diagrams of the low-
temperature region on a log and linear $T$ scale for in-plane magnetic fields.
The inflection points (Fig. S4), local minima, and elastic anomalies are
compiled into the phase diagrams. Open blue symbols represent the phase
boundaries for $H\parallel[2\bar{1}\bar{1}0]$, closed red symbols the phase
boundaries for $H\parallel[01\bar{1}0]$. The marker (+) with dashed lines
indicates the broad minimum in the magnetic-field dependence of $C_{66}$.
Figure S8 shows the $H$–$T$ phase diagram for the low-temperature and low-
magnetic field region on an enlarged linear-$T$ scale to emphasize the
magnetic-field dependence of the $T^{*}\sim 0.33$ K anomaly. The closed red
circles ($\bullet$) and open blue squares ($\Box$) represent the temperature
where the leveling off or broad minimum appears in $C_{66}$ vs. $T$ in Fig.
S5. Open black symbols ($\circ$ and $\Box$) filled with ($+$) or ($\times$)
are positions of the specific-heat anomalies previously reported by Movshovich
et al. [22], which are displayed for comparison.

Figure S7:  Magnetic field–temperature phase diagrams of UNi4B for fields
aligned along $[2\bar{1}\bar{1}0]$ and $[01\bar{1}0]$ on a (a) log-$T$ and (b)
linear-$T$ scale.  Figure S8:  Magnetic field–temperature phase diagram of
UNi4B for the low-temperature and low-magnetic-field region on a linear-$T$
scale.

##  I J. Elastic Response of the $C_{66}$ Mode under Pulsed Magnetic Fields
above 30 T

In Fig. S9, we display the relative change of $C_{66}$ obtained in pulsed
magnetic fields. Since the data between 30–60 T shows no further anomalies, we
only present the static-field results in the main text. Here, the gray dashed
curves marked by, 10 K*, 7 K*, 5 K*, and 1.7 K* show the data obtained in
static magnetic fields up to 17 T for comparison.

Figure S9:  Magnetic-field dependence of the relative change of $C_{66}$ in
pulsed magnetic fields for (a) $H\parallel[01\bar{1}0]$ and (b)
$H\parallel[2\bar{1}\bar{1}0]$ at various temperatures. The gray dashed curves
marked by 10 K*, 7 K*, 5 K* and 1.7 K* show the data obtained in static
magnetic fields up to 17 T for comparison. The lower panels show the sound-
attenuation change $\Delta\alpha_{66}$ vs. $H$. All data were taken with both
up- and down-field sweeps.


# Formatting Instructions for ICLR 2021
Conference Submissions

Antiquus S. Hippocampus, Natalia Cerebro & Amelie P. Amygdale
Department of Computer Science
Cranberry-Lemon University
Pittsburgh, PA 15213, USA
<EMAIL_ADDRESS>
&Ji Q. Ren & Yevgeny LeNet
Department of Computational Neuroscience
University of the Witwatersrand
Joburg, South Africa
<EMAIL_ADDRESS>
Coauthor
Affiliation
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$\displaystyle p({\textnormal{a}})$ |  A probability distribution over a continuous variable, or over a variable whose type has not been specified
$\displaystyle{\textnormal{a}}\sim P$ |  Random variable a has distribution $P$
$\displaystyle\mathbb{E}_{{\textnormal{x}}\sim P}[f(x)]\text{ or }\mathbb{E}f(x)$ |  Expectation of $f(x)$ with respect to $P({\textnormal{x}})$
$\displaystyle\mathrm{Var}(f(x))$ |  Variance of $f(x)$ under $P({\textnormal{x}})$
$\displaystyle\mathrm{Cov}(f(x),g(x))$ |  Covariance of $f(x)$ and $g(x)$ under $P({\textnormal{x}})$
$\displaystyle H({\textnormal{x}})$ |  Shannon entropy of the random variable x
$\displaystyle D_{\mathrm{KL}}(P\|Q)$ |  Kullback-Leibler divergence of P and Q
$\displaystyle\mathcal{N}({\bm{x}};{\bm{\mu}},{\bm{\Sigma}})$ |  Gaussian distribution over ${\bm{x}}$ with mean ${\bm{\mu}}$ and covariance ${\bm{\Sigma}}$

Functions

$\displaystyle f:{\mathbb{A}}\rightarrow{\mathbb{B}}$ |  The function $f$ with domain ${\mathbb{A}}$ and range ${\mathbb{B}}$
---|---
$\displaystyle f\circ g$ |  Composition of the functions $f$ and $g$
$\displaystyle f({\bm{x}};{\bm{\theta}})$ |  A function of ${\bm{x}}$ parametrized by ${\bm{\theta}}$. (Sometimes we write $f({\bm{x}})$ and omit the argument ${\bm{\theta}}$ to lighten notation)
$\displaystyle\log x$ |  Natural logarithm of $x$
$\displaystyle\sigma(x)$ |  Logistic sigmoid, $\displaystyle\frac{1}{1+\exp(-x)}$
$\displaystyle\zeta(x)$ |  Softplus, $\log(1+\exp(x))$
$\displaystyle||{\bm{x}}||_{p}$ |  $L^{p}$ norm of ${\bm{x}}$
$\displaystyle||{\bm{x}}||$ |  $L^{2}$ norm of ${\bm{x}}$
$\displaystyle x^{+}$ |  Positive part of $x$, i.e., $\max(0,x)$
$\displaystyle\bm{1}_{\mathrm{condition}}$ |  is 1 if the condition is true, 0 otherwise

##  6 Final instructions

Do not change any aspects of the formatting parameters in the style files. In
particular, do not modify the width or length of the rectangle the text should
fit into, and do not change font sizes (except perhaps in the References
section; see below). Please note that pages should be numbered.

##  7 Preparing PostScript or PDF files

Please prepare PostScript or PDF files with paper size “US Letter”, and not,
for example, “A4”. The -t letter option on dvips will produce US Letter files.

Consider directly generating PDF files using `pdflatex` (especially if you are
a MiKTeX user). PDF figures must be substituted for EPS figures, however.

Otherwise, please generate your PostScript and PDF files with the following
commands:

    dvips mypaper.dvi -t letter -Ppdf -G0 -o mypaper.ps
    ps2pdf mypaper.ps mypaper.pdf

###  7.1 Margins in LaTeX

Most of the margin problems come from figures positioned by hand using
`\special` or other commands. We suggest using the command `\includegraphics`
from the graphicx package. Always specify the figure width as a multiple of
the line width as in the example below using .eps graphics

       \usepackage[dvips]{graphicx} ...
       \includegraphics[width=0.8\linewidth]{myfile.eps}

or

       \usepackage[pdftex]{graphicx} ...
       \includegraphics[width=0.8\linewidth]{myfile.pdf}

for .pdf graphics. See section 4.4 in the graphics bundle documentation
(http://www.ctan.org/tex-archive/macros/latex/required/graphics/grfguide.ps)

A number of width problems arise when LaTeX cannot properly hyphenate a line.
Please give LaTeX hyphenation hints using the `\-` command.

#### Author Contributions

If you’d like to, you may include a section for author contributions as is
done in many journals. This is optional and at the discretion of the authors.

#### Acknowledgments

Use unnumbered third level headings for the acknowledgments. All
acknowledgments, including those to funding agencies, go at the end of the
paper.

##  Appendix A Appendix

You may include other additional sections here.