# Time-reversal symmetry breaking hidden order in Sr2(Ir,Rh)O4

Jaehong Jeong1 Yvan Sidis1 Alex Louat2 Véronique Brouet2 & Philippe Bourges1

###### Abstract

Layered 5$d$ transition iridium oxides, Sr2(Ir,Rh)O4, are described as
unconventional Mott insulators with strong spin-orbit coupling. The undoped
compound, Sr2IrO4, is a nearly ideal two-dimensional pseudospin-$1/2$
Heisenberg antiferromagnet, similarly to the insulating parent compound of
high-temperature superconducting copper oxides. Using polarized neutron
diffraction, we here report a hidden magnetic order in pure and doped
Sr2(Ir,Rh)O4, distinct from the usual antiferromagnetic pseudo-spin ordering.
We find that time-reversal symmetry is broken while the lattice translation
invariance is preserved in the hidden order phase. The onset temperature
matches that of the odd-parity hidden order recently highlighted using optical
second harmonic generation experiments. The novel magnetic order and broken
symmetries can be explained by the loop-current model, previously predicted
for the copper oxide superconductors.

Laboratoire Léon Brillouin, CEA-CNRS, Université Paris-Saclay, CEA Saclay,
91191 Gif-sur-Yvette, France

Laboratoire de Physique des Solides, Université Paris-Sud, Université Paris-
Saclay, 91405 Orsay, France

In the $5d$ layered perovskite material, Sr2IrO4, spin-orbit coupling and
strong electron correlations combine to give rise to a spin-orbit coupled Mott
insulator with a pseudo-spin $J=1/2$ antiferromagnetic (AFM) state [1, 2]. It
exhibits close structural [3, 4], electronic [5, 6] and magnetic [3, 4]
similarities with the $3d$ layered perovskite material, La2CuO4, which evolves
from a spin $S=1/2$ antiferromagnetic Mott insulator to a high temperature
superconductor upon doping. Doped Sr2IrO4 has then become a quite promising
material to discover new states of matter, including unconventional
superconductivity.

The crystal structure of the layered perovskite Sr2IrO4 is characterized by
the stacking of IrO2 and SrO layers. The Ir4+ ion is at the center of an
oxygen octahedron, rotated by $\theta=11^{\circ}$ in the basal $ab$ plane.
(Fig. 1c). For a long time, Sr2IrO4 has been described as possessing a
tetragonal centrosymmetric structure with four-fold rotational symmetry about
the $c$-axis, corresponding to a space group $I4_{1}/acd$ [7]. However, a
structural distortion exists even above room temperature as shown by optical
second harmonic generation studies [8] yielding to the space group $I4_{1}/a$
where the $c$\- and $d$-glide planes are lost. That gives rise to additional
weak Bragg spots such as $(1,0,2n+1)$ that have been reported using neutron
diffraction [3, 4]. In this material, crystal field effects, spin-orbit
coupling, Coulomb repulsion and the bending of the Ir-O-Ir bonds play an
important role to understand electronic and magnetic properties.

Within the octahedral crystal field, the Ir $5d$ electronic levels split into
eg and t2g states. Under strong spin-orbit coupling, the t2g states split into
a $J=1/2$ doublet and $J=3/2$ quartet, so that, among the five $5d$ electrons
of the Ir4+ ion, only one remains in the $J=1/2$ doublet state. A large enough
effective Coulomb repulsion finally localizes the $J=1/2$ electron and one is
left with a $J=1/2$ pseudo-spin model. Below $T_{\rm{N}}\sim$230 K [1], an AFM
order develops, characterized by a magnetic propagation wavevector
$\mathbf{q}_{m}=(1,1,1)$ and magnetic moments at the Ir sites aligned in the
basal $ab$-plane [3, 4, 9] (Fig. 1c). The directions of the staggered magnetic
moments are tied to IrO6 octaedra and follow their rotation, $\theta$, giving
rise to a canting of the AFM structure in each IrO2 layer with a ferromagnetic
component along the $b$-axis. As shown in Fig. 1c, the magnetic structure of
Sr2IrO4 has a staggered stacking along the $c$-axis of the canted AFM layers
(defined as $+$ or $-$ depending of the sign $\pm\theta$ of the tilt). In
other words, the weak ferromagnetic component in each layer is stacked to give
a $-++-$ structure along $c$-axis (AF-I phase) [3, 4], so it cancels out
globally. However, the AFM interaction along the $c$-axis is rather weak, thus
it is easily transformed to the $++++$ stacking order (AF-II phase) by
applying an external magnetic field [2]. At variance with the AF-I order, the
AF-II stacking is then characterized by a magnetic propagation vector
$\mathbf{q}_{m}=\mathbf{0}$ [10] and a net (but weak) ferromagnetic moment.
Thorough that paper, we refer to these AFM phases, as the one depicted in Fig.
1c, as conventional AFM phases.

Interestingly, the AFM transition temperature is suppressed under the
substitution of Rh for Ir. While the Rh substitution could be thought to be
iso-electronic, it should be stressed that Rh is likely to play the role of an
acceptor (Rh3+/Ir5+) and effectively give rise to a hole-doping of the
material [10]. Fig. 1a shows the magnetic phase diagram in Sr2Ir1-xRhxO4. From
bulk magnetization [12, 13, 11] and neutron diffraction measurements [3, 14],
the conventional AFM transition decays almost linearly as a function of Rh
substitution up to $x_{c}\approx 0.15$. Actually, x-ray studies [10] indicate
a slightly larger critical doping $x_{c}\approx 0.17$ with the occurence of
another transition at lower temperature when the magnetic correlation lengths
are effectively diverging. Further, upon Rh-substitution, the AFM order
undergoes a transition from the AF-I to the AF-II phase, even at zero magnetic
field [10, 14].

Recently, a hidden broken symmetry phase, developing prior to the AFM state,
has been reported in Sr2(Ir,Rh)O4 using rotational anisotropy optical second
harmonic generation (SHG) measurements [15]. The hidden broken symmetry phase
was observed distinctively at a few K above $T_{\mathrm{N}}$ for the pure
sample, and far above for the doped systems [15]. These data highlight an odd-
parity hidden order as both the inversion and four-fold rotational symmetries
are broken below a temperature $T_{\Omega}$ distinct from the Néel temperature
$T_{\mathrm{N}}$ (Fig. 1a) [15]. From the symmetry analysis, the SHG results
could be in principle explained by a triclinic distortion of the crystal
structure. However there is no experimental evidence using x-ray and neutron
scattering of any structural distortion [2, 3, 4]. It should be nevertheless
stressed that these diffraction studies use too modest spatial resolution to
definitively prove a lack of symmetry lowering.

Alternatively, the SHG signal could be due to a magnetic ordering although
there is no proof of time-reversal symmetry breaking at $T_{\Omega}$.
Actually, the AF-I ground state of pure Sr2IrO4 preserves the parity inversion
symmetry and thus cannot explain the SHG signal. Instead, a few magnetic point
groups that preserve the translation symmetry of the lattice were proposed to
account for the SHG signal, such as $2^{\prime}/m$ or $m1^{\prime}$ [15, 16].
In particular, a so-far non-observed AFM state of $2^{\prime}/m$ symmetry,
corresponding to a stacking $+-+-$ along the $c$ axis of AFM planes, would
produce the SHG signal [16].

Among the magnetic point groups, it is argued [15] that the new broken
symmetries can be caused by a loop current (LC) phase [17, 18], theoretically
proposed to account for the pseudo-gap physics of superconducting cuprates.
The existence of such a magneto-electric state has gained support from the
detection in several cuprate families of its magnetic fingerprint by polarized
neutron diffraction [19, 20, 21, 22, 23]. Using the same technique for
Sr2(Ir,Rh)O4, we here report at the temperature of the odd-parity order,
$T_{\Omega}$, the appearance of a hidden magnetic order, which breaks time
reversal symmetry while preserving lattice translation invariance. Among the
magnetic models inferred from the SHG data [15], only the co-planar LC order
[17, 18] produces a magnetic diffraction pattern consistent with our polarized
neutron data. Our results show that exotic magnetic orders with the same
symmetry properties as the LC phase exist in both iridates and cuprates.

Figure 1: Magnetic phase diagram and loop-current order (a) Magnetic phase
diagram of Sr2(Ir,Rh)O4. The obtained transition temperatures,
$T_{\mathrm{N}}$ and $T_{\mathrm{mag}}$ in this study are represented by black
and red empty circles for the conventional AFM and the hidden magnetism
preserving the lattice translation, respectively. The hidden broken symmetry
phase, $T_{\Omega}$, observed by SHG [15] (orange squares) is also represented
as well as the AFM phase determined by magnetization measurements [12, 13, 11]
(blue triangles) and by neutron diffraction [3, 14] (green diamonds). (b) A
schematic picture for co-planar LC state in a single IrO6 octahedron. The blue
and red arrows denote magnetic moments generated by circulating currents and
the anapole, $\Omega$, respectively. (c) Atomic and AFM structures of Sr2IrO4
with a space group $I4_{1}/a$ (origin choice 2) (Ir atoms are represented in
green, Sr in yellow and oxygen in red). (d) The co-planar LC ordered state in
the basal plane. The red arrows denote the anapole vector, $\Omega$, as shown
in (b), and the plus/minus signs correspond to the orbital magnetic moments
perpendicular to the $ab$-plane. This nearly-ferro-toroidal order preserves
translational symmetry of lattice but breaks parity inversion, four-fold
rotational and time-reversal symmetries.

Results

Let us first describe the co-planar loop current order. It is characterized by
two circulating currents turning clockwise and anti-clockwise within the same
plane inside the IrO6 octahedron (Fig. 1b), and belongs to a $2^{\prime}/m$
point group symmetry. It breaks time-reversal and inversion symmetries but not
their product. The LCs produce two opposite orbital magnetic moments within
each IrO6 octahedron. A toroidal pseudo-vector, or anapole, is defined as an
order parameter by
$\mathbf{\Omega}=\sum_{i}\mathbf{r}_{i}\times\mathbf{m}_{i}$, where
$\mathbf{r}_{i}$ and $\mathbf{m}_{i}$ stand for the position and the orbital
magnetic moment, respectively, in one octahedron (Fig. 1b). It is similar to
the toroidal moment in multiferroic systems [24]. When the anapole vectors are
parallel along the $c$-axis (ferro-toroidal coupling) as shown in Fig. 1d, the
ordered structure does not break translational symmetry but breaks parity
inversion and four-fold rotational symmetries [15]. Since the direction of the
anapole is bound to the orientation of each IrO6 octahedron, the resulting
order is a nearly-ferro-toroidal order (i.e. weakly distorted) (Fig. 1d).

As explained in the Supplementary Note 1, the Fourier transform of the
magnetic correlation function associated with the LC phase is located in
iridates at nuclear Bragg peaks, such as $(1,1,2+4n)$. These Bragg peaks
respect both the original body-centered tetragonal structure condition
$H+K+L=2n$ and the $2H+L=4m$ condition due to the $(1/2,0,1/4)$ translation.
(The wave-vector is denoted by $\mathbf{Q}=(H,K,L)$, see Method). This
produces a very specific magnetic diffraction pattern, which can be probed by
magnetic sensitive diffraction technique, such as neutron scattering
technique.

Figure 2: AFM order in Sr2IrO4 (a) Momentum transfer $\mathbf{Q}$ positions
(projected in the reciprocal $HK$-plane) investigated with neutron scattering:
$(1,0,L=1,2)$ corresponds to the AFM Bragg peaks whereas $(1,1,2)$ is where a
magnetic scattering is expected for the loop current (LC) phase. (b) Thermal
evolution of the peak intensity at the AFM Bragg peak $(1,0,2)$ measured by
unpolarized neutron (black), and the full polarization analysis at $T=100$ and
250 K (red) (see text). (c,d) The $(H,0,L=1,2)$ scans at 100 K for both the
non-spin-flip (NSF) (empty symbols) and spin-flip (SF) (full symbols) channels
(diamond symbols for $L$=1 and square symbols for $L$=2). Whilst the $(1,0,2)$
peak (black squares) shows purely magnetic behaviour, the $(1,0,1)$ (green
diamonds) shows a purely nuclear signal.

We have investigated both the conventional AFM orders and the LC magnetic
order using neutron scattering diffraction experiments. Note that the momentum
transfers $\mathbf{Q}$ are different for both types of phase: $(1,0,L)$ for
AFM peaks and $(1,1,L)$ for the LC peaks (see Supplementary Note 1). Three
major wave-vectors positions, $\mathbf{Q}$, have been examined: their
projection onto the reciprocal $HK$-plane are shown in Fig. 2a. We have then
studied each phase in a different scattering plane: $H00$-$00L$ for
conventional AFM orders and $HH0$-$00L$ to study LC order (see Method). We
have investigated two iridate samples: a pure Sr2IrO4 and a 7% Rh-doped one.
Small single crystals have been co-aligned to get a large enough sample mass
in order to perform the polarized neutron experiment. The Supplementary Note 2
describes the sample preparation and shows magnetization measurements, which
characterize the samples.

We have first studied the conventional AF-II order by repeating unpolarized
neutron diffraction measurements [3, 4]. We report in Fig. 2b the AFM Bragg
peak at $\mathbf{Q}=(1,0,2)$ in the pure system where the magnetic Bragg
intensity shows a sharp AFM transition at $T_{\mathrm{N}}=232$ K. The fitted
critical exponent $2\beta\approx 0.41$ is consistent to 0.36 in the previous
report [3]. Next, we have performed polarized neutron experiment to study the
$\mathbf{Q}=(1,0,L)$ Bragg peaks to disentangle the magnetic and nuclear
contributions of these peaks. The polarized neutron experiment setup is
presented in the Supplementary Note 3. It has been already used in previous
measurements in cuprates and described in Refs. [19, 20, 21, 22, 23]. For a
given neutron polarization $\mathbf{P}$, the neutron intensities,
$I_{\mathrm{SF}}$ and $I_{\mathrm{NSF}}$, are measured in both spin-flip (SF)
and non-spin-flip (NSF) channels, respectively. Full magnetic signal appears
in the SF channel when $\mathbf{P}\parallel\mathbf{Q}$ [21, 23] whereas the
nuclear intensity occurs in the NSF chanel. The Fig. 2c,d shows the wave-
vector scans along $H$ across the magnetic peak $\mathbf{Q}=(1,0,2)$ for the
NSF and SF channels. That proves the magnetic origin of the peak as it is only
seen in the SF channel (Fig. 2d). In Fig. 2b, the magnetic intensity
$I_{\mathrm{mag}}=I_{\mathrm{SF}}-BG$ is determined at two temperatures, 100 K
and 250 K, below and above $T_{\mathrm{N}}$ (where $BG$ stands for the flat
background of Fig. 2d). A clear magnetic intensity is sizeable at 100 K,
whereas no magnetic intensity is seen at 250 K. Further, we report at 100 K in
Fig. 2b the magnetic intensity for three different neutron polarization
states: the polarizations $\mathbf{P}\parallel\mathbf{Q}$ and
$\mathbf{P}\perp\mathbf{Q}$ are in the scattering plane (see Method), whereas
$\mathbf{P}\parallel\mathbf{z}$ is perpendicular to the scattering plane. In
the given geometry, zero magnetic intensity with
$\mathbf{P}\parallel\mathbf{z}$ proves that the AFM moments are confined in
the $ab$-plane. This is in agreement with the previous studies of the neutron
structure factors [3, 4].

In the same scattering plane, the Bragg peaks at $\mathbf{Q}=(1,0,L)$ for
$L=1$ and $3$ were also measured using polarized neutron. They correspond to
the forbidden peaks of the original $I4_{1}/acd$ structure giving rise to the
space group $I4_{1}/a$ where the glide planes are lost [8]. As it has been
already discussed [3, 4], their temperature dependence exhibits no anomaly
neither at the Néel temperature nor at the onset of the odd-parity hidden
order ($T_{\Omega}$) [15]. The origin of that scattering is non-magnetic, as
shown in Fig. 2c,d. For the Bragg peak at $\mathbf{Q}=(1,0,1)$, no signal is
sizeable in the SF channel and only a NSF signal is observed above the
background. These results exclude the possibility of an AFM arrangement of
$+-+-$ type, where the magnetic signal should be only at odd $L$-value. It
casts serious doubts that such an AFM stacking can explain the origin of the
SHG signal [16]. The same conclusion holds for the $++++$ arrangement, not
observed in pure Sr2IrO4, in full agreement with the literature [3, 4, 9]. To
account for the occurrence of an odd-parity hidden order in SHG measurements
in pure Sr2IrO4, we are left with two scenarii. Either, the missing AFM orders
are light-induced metastable states during the SHG measurements, as suggested
in Ref. 16, or another kind of magnetic order exists. Let us consider now this
second scenario.

Figure 3: Hidden magnetic order in Sr2Ir1-xRhxO4 (x=0, 0.07) (a,c) Temperature
dependence of $I_{\mathrm{SF}}$ and a bare SF intensity
$I_{\mathrm{SF}}^{0}=I_{\mathrm{NSF}}/\mathrm{FR}_{0}(T)$ from the
polarization leakage. The temperature-dependent bare flipping ratio
$\mathrm{FR}_{0}(T)$ is taken from the reference Bragg peak $(2,2,0)$ and
rescaled to correct the NSF intensity at $(1,1,2)$ (see Supplementary Note 4).
(b,d) Temperature dependence of the magnetic intensity
$I_{\mathrm{mag}}=I_{\mathrm{SF}}-I_{\mathrm{NSF}}/\mathrm{FR}_{0}(T)$ at the
nuclear Bragg peak $(1,1,2)$. Red curves are guide to the eye and grey areas
represent temperature uncertainty in the determination of $T_{\mathrm{mag}}$.

The overall experimental procedure to extract the magnetic signal associated
with the LC order is detailed in the Supplementary Note 4 and it follows
methods established in previous works on superconducting cuprates [19, 21].
The measured spin-flip intensity, $I_{\mathrm{SF}}$, is reported as a function
of temperature in Fig. 3a for the pure sample. It is compared with the
background baseline $I_{\mathrm{SF}}^{0}$ determined from the measured NSF
intensity (see the Fig. 3a caption and Supplementary Note 4). While
$I_{\mathrm{SF}}^{0}$ decreases monotonically as temperature decreases, the SF
intensity instead departs from $I_{\mathrm{SF}}^{0}$ below
$T_{\mathrm{mag}}\approx 240\pm 30$ K. The difference,
$I_{\mathrm{SF}}-I_{\mathrm{SF}}^{0}$, evidences a spontaneous magnetic order
whose magnetic intensity, $I_{\mathrm{mag}}$, is reported in Fig. 3b. It has a
different symmetry compared to the conventional AFM phases discussed above and
the position corresponds to where magnetic intensity is expected for the LC
phase. For the 7% Rh-substituted sample, the same analysis is given in Fig. 3c
and d. While the AFM transition is suppressed down to $T_{\mathrm{N}}\approx
100$ K, the novel magnetic order is observed at much higher temperature
$T_{\mathrm{mag}}\approx 240\pm 30$ K. Within error bars, the transition
temperature, $T_{\mathrm{mag}}$ and its normalized magnetic intensity do not
show a significant difference between the pure system and 7% Rh-substitution.
This is consistent with the estimate of onset of the hidden order,
$T_{\mathrm{\Omega}}$, from SHG (orange squares in Fig. 1a) [15], which does
not change appreciably with Rh substitution. Using the calibration of nuclear
Bragg peaks intensities, one can deduce the magnetic cross section of the
hidden magnetic order as $\sim 2$ mbarns/f.u., which is less than $\sim
10^{-3}$ of the strongest nuclear Bragg peak. The normalized magnitude of
$I_{\mathrm{mag}}$ for the hidden magnetic order is similar in both samples
and is $\sim 5$ times smaller than the one for the AFM order, as shown in Fig.
3.

Discussion

To understand the spontaneous magnetic order at $T_{\mathrm{mag}}$, a model is
needed to explain the broken symmetries of the hidden order. Concomitant with
the SHG data, time-reversal, parity inversion and 4-fold rotational symmetries
are broken. It preserves the translational symmetry of the underlying lattice,
as the magnetic scattering appears on top of the nuclear Bragg peak. The loop-
current model [17, 18] is a good candidate, since it can explain all these
broken symmetries. Nevertheless, more detailed and quantitative study is
required to establish the exact order parameter in the hidden ordered phase.
The weak magnetic cross section shows how difficult it is to detect and why it
was not reported with typical unpolarized neutron diffraction. Due to the
experimental limitations, a precise determination of the direction of induced
magnetic moments and magnetic structure requires further works.

Actually, other magnetic orders, which could potentially account for the SHG
data, are not consistent with our finding as they would give rise to distinct
scattering patterns. For instance, the proposed structure with $m1^{\prime}$
point group [15] yields a different relation $H+K+L=2n+1$, as the
configuration is opposite when one considers the $(1/2,1/2,1/2)$ translation.
This breaks the original body-centered nuclear structure [7], and would give
rise to magnetic superstructure peaks at $(1,0,L=2n)$ and no magnetic
intensity at any $(1,1,L)$ position. With Rh doping, one also does not observe
magnetic superstructures at $(1,0,L)$ with even $L$ [10, 14]. Both points
dismiss the proposal of the $m1^{\prime}$ point group [15].

Alternatively, the SHG measurements have been re-interpreted considering
conventional AFM orders [16] with a different stacking along the $c$-axis. We
detail in the Supplementary Note 5 why the different AFM structures discussed
in the literature cannot actually explain our observation. First, since the
hidden order occurs at different $\mathbf{Q}$ positions compared to the AFM
order, it is clearly distinct with the AF-I order. Second, the AFM state of
$2^{\prime}/m$ symmetry with a $+-+-$ stacking along the $c$-axis of AFM
planes is argued to explain SHG data [16]. This phase will give rise to
magnetic contributions at positions such as $(1,0,L=2n+1)$ and nothing at
$(1,1,L)$, where we observe the hidden magnetic intensity. Further, the non-
magnetoelectric $++++$ phase (AF-II) (described above) could also account for
the SHG data [16]. This AF-II phase is reported in the Rh-doped system [10,
14] but is absent in the pure system [3, 4]. It would also exhibit the largest
magnetic contribution at $(1,0,L=2n+1)$ as well as tiny magnetic contributions
at Bragg peaks, such as $(1,1,L=2+4n)$, due to its weak ferromagnetism. This
interpretation can be excluded in both samples we have studied. First, in the
pure sample, the ferromagnetic order is absent (see Fig. 1) [3, 4] and can be
only induced by an applied magnetic field of about 0.2 T (2000 Oe) [2].
Second, under Rh substitution, such a ferromagnetic order indeed develops, but
only below $T_{\mathrm{N}}$ [10], as it results from the canting of the AFM
order, clearly lower than $T_{\mathrm{mag}}$. Moreover, the neutron intensity
due to ferromagnetism would be $\sim\mathrm{tan}^{2}\theta\leq 10^{-2}$
smaller than the AFM one, i.e at least one order of magnitude smaller than the
observed magnetic scattering we report here. All these arguments allow us to
rule out the weak ferromagnetism derived from the canted AFM order as a
candidate to account for the observed magnetic scattering at
$\mathbf{Q}=(1,1,2)$. Therefore, our observation of a magnetic signal at
$\mathbf{Q}=(1,1,2)$ is not consistent with any kind of stackings along the
$c$-axis of the pseudo-spin AFM orders considered to explain the SHG signal in
Ref. [16].

Using polarized neutron diffraction, we have experimentally addressed all
these alternative phases and found out evidence for a hidden magnetic order in
Sr2Ir1-xRhxO4. It is a translation-invariant but time-reversal symmetry broken
phase that is consistent with the LC order of $2^{\prime}/m$ point group
symmetry, concomitantly compatible with the SHG signal. In that model, all
IrO6 octahedra, which are the building blocks of the material, are identically
decorated by the same set of staggered magnetic moments, whose magnetism
cancels out on each octahedron (as depicted in Fig. 1b). This magnetic order
is then clearly distinct from the AFM one, where each octahedron has a single
pseudo-spin on the Ir site and where the nearest octahedra have staggered
moments.

In conclusion, we report the first evidence of an unconventional magnetic
order in Sr2(Ir,Rh)O4, which breaks time-reversal symmetry but preserves
translational symmetry of the underlying lattice. By analogy with
superconducting cuprates, where a similar kind of order is observed, one can
refer to it as an intra-unit-cell order. The new magnetic phase overlaps with
parity inversion and rotational symmetry broken phase recently reported using
SHG [15]. Both observations can be described by the loop-current order [17,
18] proposed for the pseudogap state in cuprates, where it is well supported
by polarized neutron measurements [19, 21]. Further, the neutron observation
in cuprates is confirmed as well by recent SHG measurements that show a global
broken inversion symmetry in YBa2Cu3O6+x [25]. This may provide more analogy
between the iridates and the high-$T_{\mathrm{c}}$ cuprates, in spite of a
different nature of $5d$ and $3d$ orbitals. A noticeable difference is here
that the loop order occurs in the insulating compounds at half-filling,
whereas in cuprates it is observed in the doped metallic (superconducting)
state. Our report generalizes the existence of loop-current electronic states
in oxides.

Methods: Polarized neutron diffraction

The polarized neutron experiments were performed on the triple axis
spectrometer 4F1 (Orphée, Saclay, France) (see Supplementary Note 3). All
measurements were done in two different scattering planes, either $H00$-$00L$
or $HH0$-$00L$, where the scattering wave-vector is quoted as
$\mathbf{Q}=H\mathbf{a}^{*}+K\mathbf{b}^{*}+L\mathbf{c}^{*}\equiv(H,K,L)$ with
$a^{*}=b^{*}=1.15~{}$Å-1 and $c^{*}=0.24$ Å-1. As emphasized in Supplementary
Note 1, in Sr2(Ir,Rh)O4, the conventional AFM order is expected in the
$H00$-$00L$ plane and the LC phase in the $HH0$-$00L$ plane. For each wave-
vector $\bf{Q}$, the scattered intensity is measured in both spin-flip (SF)
and non-spin-flip (NSF) channels. The neutron measurements have been performed
with a neutron polarization $\mathbf{P}\parallel\mathbf{Q}$ where the magnetic
signal appears entirely in the SF channel [19, 21]. The AF-I order was further
studied with a polarization $\mathbf{P}\perp\mathbf{Q}$ (but still within the
$H00$-$00L$ plane) as well as with a polarization
$\mathbf{P}\parallel\mathbf{z}$ which is perpendicular to the scattering plane
(along $0K0$ in the given case). The Sr2(Ir,Rh)O4 single crystals preparation
and caracterization are described in the Supplementary Note 2.

We wish to thank S. Di Matteo, D. Hsieh, B.J. Kim, M.R. Norman and C.M. Varma
for fruitful discussions. We also acknowledge financial supports from the
projects NirvAna (contract ANR-14-OHRI-0010) and SOCRATE (ANR-15-CE30-0009-01)
of the ANR French agency.

The authors declare that they have no competing financial interests.

Correspondence and requests for materials should be addressed to Jaehong Jeong
(email<EMAIL_ADDRESS>or Philippe Bourges (email:
philippe.bourges@cea.fr).

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##  Supplementary Note 1

Reciprocal space comparison with cuprates

Figure 4: MO2 plane in cuprates (M=Cu) and in Sr2IrO4 (M=Ir). (Left figure) In
cuprates, the CuO2 plane is typically a weakly distorted square lattice that
one generally represents as a regular square lattice, so it could be described
by the minimal unit cell (dashed blue). (Right figure) In contrast, in
iridates, due to a large in-plane rotation of the IrO6 octahedra, the
corresponding unit cell is doubled and rotated by 45∘ (black). (Center figure)
Planar reciprocal space where a few interesting $\mathbf{Q}$-positions are
compared in both systems (written in italic blue for cuprates and in black for
iridates).

The Sr2IrO4 exhibits a very similar in-plane structure compared to
superconducting cuprates. In cuprates, corner-shared CuO6 octahedra or CuO5
pyramids make a (nearly) square lattice of CuO2 plaquettes on the basal
$ab$-plane. On the other hand in iridates, the IrO6 octahedra are also corner-
shared but rotated by $\theta\sim 11^{\circ}$ in the basal plane. Due to this
in-plane rotation, the unit cell for iridates is doubled and rotated by 45∘ as
shown in Fig. 4. Despite a difference between the spin $S=1/2$ of Cu2+ and
pseudo-spin $J=1/2$ of Ir4+ states, the antiferromagnetic (AFM) structure is
also almost the same in both systems, so the magnetic Bragg conditions are
also similar. However, under the 45∘ unit-cell transformation, actual
$(H,K,L)$ Miller indices are different. In the center of Fig. 4, we compare a
few interesting Bragg $\mathbf{Q}$-positions in the planar reciprocal
$HK$-plane for iridates (black) and cuprates (blue). For instance, the
$(1/2,1/2,L)$ for the AFM order in cuprates is transformed to $(1,0,L)$ or
$(0,1,L)$ for iridates, and the $(1,0,L)$ in cuprates, where the loop-current
(LC) order has been reported [1, 2], corresponds to $(1,1,L)$ for iridates.

##  Supplementary Note 2

Coaligned single crystals and magnetization measurements

We have investigated the pure and 7% Rh-doped single crystals grown by flux
method at Laboratoire de Physique des Solides (Orsay). Owing to a rectangular
cuboid shape of the crystals, several tiny crystals could be coaligned in
order to increase the total mass, as shown in Fig. 5a. To address the
conventional antiferromagnetic (AFM) order, the magnetization was measured
under a magnetic field $H=1$ T using Magnetic Property Measurement System
(MPMS). The pure system originally exhibits an AFM order, but it can easily
show ferromagnetism, by applying a small external magnetic field $H\approx
0.2$ T ($\simeq$2000 Oe) [3]. For instance, the reported magnetic moment
deduced from magnetization measurements shown in Fig. 5b corresponds to a net
ferromagnetic moment under 1 T. This ferromagnetism originates from a canting
of the AFM structure. The transition temperature is taken at the maximum slope
in $M(T)$ curve, i.e., the minimum of the 1st derivative of magnetization,
$dM(T)/dT$ shown in Fig. 5b. The pure system shows a sharp transition at
$T_{\mathrm{N}}\approx 232$ K and the saturated ferromagnetic moment per Ir
ion is $\sim 0.08\mu_{\mathrm{B}}$ at 1T. On the other hand, the doped one
shows a broad transition near $T_{\mathrm{N}}\approx 100$ K and the moment is
also reduced to $\sim 0.04\mu_{\mathrm{B}}$. To map out the phase diagram
(Fig.1a in the main text), the transition temperatures for different doping
levels were determined from the previous literatures [4, 5], following the
same procedure.

Figure 5: Coaligned single crystals and measured magnetic moments (a) Total 18
single crystals of the 7% Rh-doped sample were coaligned on a thin Al plate.
Two major crystallographic directions are denoted: $(1,1,0)$ and $(0,0,1)$.
(b) Ferromagnetic moments deduced from magnetization measurements at $H=1$ T.
By substituting Rh for Ir, the magnetic transition temperature and the
saturated moment are rapidly suppressed.

##  Supplementary Note 3

Experimental neutron scattering setup

Polarized neutron diffraction experiments were performed on the triple-axis
spectrometer 4F1 located at the Orphée reactor in Saclay (France). The
polarized neutron setup was similar to the one used previously for studying
cuprates [1, 6, 7]. A polarized incident neutron beam with $E_{i}=13.7$ meV
($k_{i}=2.57$ Å-1) is obtained by a polarizing supermirror (bender) and
scattered neutrons are measured with a Heusler analyzer which determines as
well the final neutron polarization. A pyrolytic graphite filter is put before
the bender to remove high harmonics. A small magnetic field of typically 10 G
is applied using a Helmholtz-like coil. It is used to change adiabatically the
direction of the neutron polarization at the sample position. A Mezei flipper
is located before the sample position to flip the neutron spin. For each Bragg
position, the scattered neutron intensity is measured in both spin-flip (SF)
and non-spin-flip (NSF) channels, that corresponds to two different states of
the Mezei flipper, flipper-off and flipper-on, respectively. One defines the
flipping ratio $\mathrm{FR}=I_{\mathrm{NSF}}/I_{\mathrm{SF}}$ of the Bragg
peaks intensities in both NSF and SF channels. It determines the polarization
efficiency of the apparatus. Due to unavoidable neutron polarization leakage
from the NSF to the SF channel (imperfect polarization), a value for FR was
obtained between 30–50 for the pure sample and 50–65 for the doped sample. In
order to measure a small magnetic signal on top of the large nuclear peak, it
is essential to keep very stable and homogeneous neutron polarization through
the whole measurement. Thus all the data have been measured at a fixed
configuration of the spectrometer while changing the temperature.

##  Supplementary Note 4

Analysis for the flipping ratio

According to the (co-planar) loop current (LC) model, the magnetic scattering
intensity should be observed at Bragg reflections $(1,1,2+4n)$, on top of the
nuclear scattering. As the LC phase respects the translation symmetry of the
lattice, it corresponds to an intra-unit-cell magnetic order. Thus, the
detection of such a magnetic order critically depends on the ability to
disentangle nuclear and magnetic scatterings. This difficulty can be overcomed
by using polarized neutron scattering technique [1, 2]. Once the neutron spin
polarization $\mathbf{P}$ is set parallel to the transferred momentum,
$\mathbf{P}\parallel\mathbf{Q}$, the magnetic scattering purely appears in
spin flip (SF) channel and the nuclear one in the non-spin-flip (NSF) channel.
In principle, probing separately the SF and NSF scattering channels allows one
to determine the magnetic and nuclear scatterings. However, polarized neutron
experiments are always limited by the quality of the neutron spin
polarization, which is given by $\mathrm{FR}$, that should go to infinity for
a perfectly polarized neutron beam. In practice, $\mathrm{FR}$ is finite and a
fraction $\mathrm{1/FR}$ of the nuclear (i.e. NSF) scattering goes into the SF
channel: it is referred to as the polarization leakage that determines a
corresponding bare flipping ratio $\mathrm{FR}_{0}$. Another important point
is to determine possible temperature dependence of the bare flipping ratio
$\mathrm{FR}_{0}(T)$. Indeed, when changing the temperature, very tiny changes
of the experimental set-up may occur[2], producing a continuous drift of
$\mathrm{FR}_{0}(T)$ that needs to be calibrated.

Figure 6: Comparison between the $(1,1,2)$ and the reference $(2,2,0)$ Bragg
peaks. (a) (H,H,L) scattering plane where the $(1,1,2)$ (red circle) and
$(2,2,0)$ (blue square) Bragg peaks positions are underlined. (b) 1/FR
measured at the reference postion (2,2,0) for both pure (full squares) and
doped (empty squares) samples, showing a small drift of 1/FR0 in temperature.
(c,d) The relative deviation of 1/FR at $(1,1,2)$ from the scaled $(2,2,0)$
for both systems. Figure 7: $H$-scan of the $(1,1,2)$ Bragg peak in the NSF
channel. The integrated intensity for the NSF channel at $(1,1,2)$ in the
doped sample is about 1.6 times smaller than one in the pure sample.

In order to reveal a small magnetic signal on top of the nuclear polarization
leakage in the SF channel, one has to pick up the weakest nuclear Bragg peaks
(with the appropriate symmetry for the LC phase) to get the better signal-to-
background ratio. In addition, the magnetic form factor is generally
suppressed at a high momentum transfer. Considering these constraints, the
$(1,1,2)$ reflection has been studied, which has the lowest $|\mathbf{Q}|$ of
the allowed nuclear Bragg peaks $(1,1,2+4n)$ for the LC phase. Further, in
order to determine the temperature dependence of the bare flipping ratio,
$\mathrm{FR}_{0}(T)$, a non-magnetic reference Bragg peak is measured where
the magnetic scattering is zero or small enough and where the spectrometer
geometry is kept similar. Here, we chose the $(2,2,0)$ Bragg peak (see Fig.
6a) as the reference because a magnetic signal is expected to be considerably
reduced by the magnetic form factor and the spectrometer geometry is not
changed much with respect to $\mathbf{Q}=(1,1,2)$. Using $\mathrm{FR}_{0}(T)$,
obtained from the non-magnetic reference, one can then determine the intrinsic
polarization leakage $I_{\mathrm{SF}}^{0}=I_{\mathrm{NSF}}/\mathrm{FR}_{0}(T)$
and next extract from the SF intensity $I_{\mathrm{SF}}$ the true magnetic one
at $\mathbf{Q}=(1,1,2)$,
$I_{\mathrm{mag}}=I_{\mathrm{SF}}-I_{\mathrm{SF}}^{0}$.

For the pure sample, the $\mathrm{FR}_{0}^{220}(T)$ at the $(2,2,0)$ is
$\sim$39 at 300 K and it increases up to $\sim$45 at 100 K, meaning that
$1/\mathrm{FR}_{0}(T)$ (reported in Fig. 6b) decreases upon cooling. That
smooth decrease indicates no magnetic signal because one should observe a
sudden increase of 1/FR in case of a magnetic order. The
$\mathrm{FR}_{0}^{220}(T)$ for the doped sample is higher ($\sim$50) (1/FR
$\sim 0.019$ reported in Fig. 6b) and much stable in temperature. That proves
as well no magnetic signal at the $(2,2,0)$ peak in the doped sample. The
observed temperature-dependent $1/\mathrm{FR}^{220}(T)$ can therefore be taken
as the bare $1/\mathrm{FR}_{0}(T)$ for the polarization calibration.

In contrast, the $\mathrm{FR}^{112}(T)$ at the $(1,1,2)$ shows a clear change
of slope around 200–250 K as shown in Fig. 6c,d. The $\mathrm{FR}^{112}(T)$ is
$\sim$33–38 and $\sim$50–63 for the pure and doped sample, respectively. This
main difference of the magnitude of FR originates from different experimental
conditions such as sample mosaicity, the number of blades on the analyzer and
optimization of the guide field. In order to estimate the bare
$\mathrm{FR}_{0}^{112}(T)$ for the $(1,1,2)$, the $\mathrm{FR}_{0}^{220}(T)$
is scaled by a factor that gives the same value with the
$\mathrm{FR}^{112}(T)$ at high temperature above 250 K. Then, as shown in Fig.
6c-d, we clearly observe a departure of $\mathrm{FR}^{112}(T)$ from
$\mathrm{FR}_{0}^{112}(T)$ at low temperature. Using this
$\mathrm{FR}^{112}_{0}(T)$, the bare polarization leakage,
$I_{SF}^{0}(T)=I_{NSF}(T)/FR^{112}_{0}(T)$ in the SF channel can be determined
at $\mathbf{Q}=(1,1,2)$. By subtracting it from the measured $I_{SF}(T)$, the
magnetic intensity can then be reported as,
$I_{mag}(T)=I_{SF}(T)-I_{SF}^{0}(T)=I_{SF}(T)-I_{NSF}(T)/FR_{0}(T)$. We
applied this method in the data analysis presented in Fig. 3 of the main
manuscript.

Note that, for a quantitative comparison of the magnetic intensities of both
samples, all raw intensities have been background-subtracted, normalized to
the same monitor counts and weighted by an estimated sample weight. The Fig. 7
depicts scans of the nuclear Bragg peak intensity at $\mathbf{Q}=(1,1,2)$
along ${\it H}$ in both samples. Due to a broader mosaicity of coaligned
crystals, the peak width is broader in the doped sample. From Fig. 7, one
deduces the integrated intensity of the Bragg peak, which is in the doped
sample $\sim$1.6 times smaller than in the pure sample. Taking that into
account, one could compare the magnetic intensity of the observed hidden order
around 240 K for both the pure and doped samples. We found a similar amount of
the magnetic intensity in both samples as shown in Fig. 3b-c of the
manuscript.

##  Supplementary Note 5

Can an antiferromagnetic structure explain the hidden magnetic order at
$\mathbf{Q}=(1,1,2)$ ?

The fact that we have ascribed the magnetic intensity at $\mathbf{Q}=(1,1,2)$
to a hidden magnetic order deserves additional comments. Briefly, the hidden
order at $\mathbf{Q}=(1,1,2)$ cannot be described by any stacking of the
planar pseudo-spin, $J_{eff}=1/2$, antiferromagnetic (AFM) pattern. Below, we
discuss different AFM models considered in the literature:

First, in the AFM $-++-$ model (AF-I), the magnetic intensity is expected at
some $\mathbf{Q}=(1,1,L)$. However, it should be zero for $L$=2 because it
does not preserve the body-centered symmetry of the unit cell, so $H+K+L$
should be odd or $L=2n+1$. As a matter of fact, a tiny magnetic intensity
(about a hundred times weaker than at $\mathbf{Q}=(1,0,2)$) has been reported
at $\mathbf{Q}=(1,1,1)$ in pure ${\rm Sr_{2}IrO_{4}}$ [8]. That contribution
is much weaker than at $\mathbf{Q}=(1,0,L)$ because the magnetic intensity at
$\mathbf{Q}=(1,1,L)$ is proportional to the non-collinear pattern of the Ir
moment. It is typically proportional to the square of the planar tilt angle of
the IrO6 octahedra, $\sim\theta^{2}$ with $\theta\simeq 11^{\circ}$.

Second, in the AFM $++++$ model (AF-II), a magnetic intensity at
$\mathbf{Q}=(1,1,2)$ could exist in principle due to the ferromagnetic
component. However, this interpretation can be excluded in both samples we
have studied. First, in the pure sample, the ferromagnetic order is absent
(see Fig.1 of the manuscript) [8, 9] and can be only induced by an applied
magnetic field of about 2000 Oe [3]. Second, under Rh substitution, such a
ferromagnetic order indeed develops but only below $T_{\mathrm{N}}$ [10] as it
results from the canting of the AFM order, clearly lower than
$T_{\mathrm{mag}}$. We therefore rule out the weak ferromagnetism derived from
the canted AFM order as a candidate to account for the observed magnetic
scattering at $\mathbf{Q}=(1,1,2)$.

Third, in the hypothethic $-+-+$ model considered by Di Matteo and Norman
[11], a magnetic intensity will be also present at some $(1,1,L)$ positions.
The structure factor for the $-+-+$ stacking is not zero only for even $L$ as
it keeps the body-centered symmetry of the unit-cell. Further, due to the
glide plane, the structure factor is proportional to $\sin(\pi/2(H+L/2))$.
Actually, it gives zero intensity at $(1,1,2)$ and then is not consistent with
our finding. Further, such an AFM phase, $-+-+$, would also lead to magnetic
contributions at $(1,0,L)$ for odd $L$, which are not observed in any neutron
diffraction reports [8, 9] in pure ${\rm Sr_{2}IrO_{4}}$ as well as shown in
the Fig. 2d of the manuscript.

Finally, one should stress that magnetic contribution from any of these AFM
phases at $(1,1,L)$ would be negligible as it is systematially proportional to
$\sim\mathrm{tan}^{2}\theta$ with $\theta\simeq 11^{\circ}$. Then, any effect
around that $\bf{Q}$-position would be quantitatively too weak to be measured
in our experiment. In absence of a planar tilt, it would even be strictly
zero, although the signal for the LC phase will be still not zero.

## References

  * [1] Fauqué, B. _et al._ Magnetic order in the pseudogap phase of high-$T_{\mathrm{c}}$ superconductors.  _Physical Review Letters_ 96, 197001 (2006).
  * [2] Bourges, P. & Sidis, Y.  Novel magnetic order in the pseudogap state of high-$T_{\mathrm{c}}$ copper oxides superconductors.  _Comptes Rendus Physique_ 12, 461–479 (2011).
  * [3] Kim, B. _et al._ Phase-sensitive observation of a spin-orbital mott state in Sr2IrO4.  _Science_ 323, 1329–1332 (2009).
  * [4] Qi, T. _et al._ Spin-orbit tuned metal-insulator transitions in single-crystal Sr2Ir1-xRhxO4 $(0\leq x\leq 1)$.  _Physical Review B_ 86, 125105 (2012).
  * [5] Ye, F. _et al._ Structure symmetry determination and magnetic evolution in Sr2Ir1-xRhxO4.  _Physical Review B_ 92, 201112 (2015).
  * [6] Mook, H., Sidis, Y., Fauqué, B., Balédent, V. & Bourges, P.  Observation of magnetic order in a superconducting YBa2Cu3O6.6 single crystal using polarized neutron scattering.  _Physical Review B_ 78, 020506 (2008).
  * [7] Li, Y. _et al._ Unusual magnetic order in the pseudogap region of the superconductor HgBa2CuO4+δ.  _Nature_ 455, 372–375 (2008).
  * [8] Ye, F. _et al._ Magnetic and crystal structures of Sr2IrO4: A neutron diffraction study.  _Physical Review B_ 87, 140406 (2013).
  * [9] Dhital, C. _et al._ Neutron scattering study of correlated phase behavior in Sr2IrO4.  _Physical Review B_ 87, 144405 (2013).
  * [10] Clancy, J. _et al._ Dilute magnetism and spin-orbital percolation effects in Sr2Ir1-xRhxO4.  _Physical Review B_ 89, 054409 (2014).
  * [11] Di Matteo, S. & Norman, M. R.  Magnetic ground state of Sr2IrO4 and implications for second-harmonic generation.  _Phys. Rev. B_ 94, 075148 (2016).


# Cm15+ and Bk16+ ion clocks with enhanced sensitivity to new physics

V. A. Dzuba and V. V. Flambaum  School of Physics, University of New South
Wales, Sydney 2052, Australia

###### Abstract

We perform calculations of electronic structure of Cm15+ and Bk16+ ions and
demonstrate that they have transitions which combine the features of atomic
optical clocks with the enhanced sensitivity to the variations of the fine
structure constant $\alpha$. The high sensitivity is due to large nuclear
charge $Z$, high ionisation degree $Z_{i}$ and the effect of level crossing,
which enables optical transitions between states of different configurations.
These are the $6p_{1/2}-5f_{5/2}$ and $6p_{1/2}-5f_{7/2}$ transitions in the
single valence electron approximation. Variation of $\alpha$ may be due to the
interaction with scalar and pseudoscalar (axion) dark matter fields.
Therefore, Cm15+ and Bk16+ clocks are promising candidates to search for these
fields.

Presently the fractional precision of the frequency measurements in optical
clock transitions of Sr, Yb, Al+, Hg, Hg+, and Yb+ atomic systems reached an
unprecedented level of 10-18 (see e.g. w181 ; w182 ; w183 ; w184 ; w185 ; w186
; Ye22 ). It is natural to use the advantage of so high accuracy of the
measurements in search for the physics beyond the standard model, for example,
search for the space-time variation of the fundamental constants such as the
fine structure constant $\alpha$ ($\alpha=e^{2}/\hbar c$). Variation of
$\alpha$ may be due to the interaction of electromagnetic field with scalar
and pseudoscalar (axion) dark matter fields Arvanitaki ; Stadnik ; Stadnik2 .
Therefore, the measurements of $\alpha$ variation is one of the instruments to
search for dark matter.

The search for the slow drift of $\alpha$ can be performed by comparing the
frequencies of atomic transitions with different sensitivities to the
variation of $\alpha$ over long period of time. The laboratory limits on the
present-day time variation of $\alpha$ have already passed $10^{-18}$ per year
(see e.g. the measurements in Refs. Rosenband ; Leefer ; Godun ; Filzinger ;
Sherrill  and review Safronova1 ).

The sensitivity of the atomic transitions to the variation of $\alpha$ used in
these studies is not very high webbPRL ; webb ; Canadien . This is also true
for all working atomic clocks. It was suggested in Ref. HCI  to use optical
transitions in highly-charged ions (HCI) to achieve significantly stronger
sensitivity to the $\alpha$ variation. Indeed, effects of $\alpha$ variation
in electron energies are due to the relativistic corrections which increase
proportional to $Z^{2}(Z_{i}+1)^{2}$ webbPRL ; webb ; HCI . Therefore, to have
large effects we should consider heavy highly charged ions. To avoid
cancellation between the relativistic shifts of upper and lower levels in an
electronic transition we should consider transitions between different
electronic configurations. However, frequencies of such transitions in HCI
increase proportional to $(Z_{i}+1)^{2}$ and are usually far outside the laser
range. The solution is based on the fact that ordering of the electron states
depends on $Z_{i}$ and by removing electrons we may achieve so called level
crossing brining the frequencies of the transitions between states of
different configurations into optical region crossing . The biggest effects
happen near the crossing between $s$ and $f$, or $p_{1/2}$ and $f$ orbital
energies webbPRL ; webb ; HCI ; crossing . The search for appropriate
transitions in HCI is now a popular area of theoretical hci1 ; Cf11 ; Cf1 ;
hci4 ; Ho14 ; Cf2 ; Cf3 ; Ir17t ; Os  and experimental Bekker ; Ho14e ; Ir17
research (see also reviews Safronova2 ; Sahoo ).

The highest sensitivity to $\alpha$ variation has been found in Cf17+ and
Cf15+ ions Cf1 ; Cf11 ; Cf2 ; Cf3 . These ions have all important factors of
enhancement: high $Z$ ($Z=98$), high ionisation degree, and optical
transitions between states of different configurations which correspond to the
$6p_{1/2}-5f$ transitions. The disadvantage of the use of Cf is its
instability. The most long-lived isotopes of Cf are 249Cf (the half-life is
351 years) and 251Cf (the half-life is 898 years) isotopes . Isotope 250Cf
with zero nuclear spin lives only 13 years.

In present work we study the Cm15+ and Bk16+ ions. They are very similar to
the Cf17+ in terms of electronic spectra and the sensitivity to the variation
of $\alpha$ but have much longer-living isotopes. For example, the half-life
of 247Cm is 15.6 million years, 248Cm with zero nuclear spin has half-life
0.348 million years, the half-life of 247Bk is 1380 years isotopes . In
addition, the long-living isotopes of Cm and Bk have lower value of atomic
number $A$ than the long living isotopes of Cf, which probably means that it
is easier to produce them. The Cm15+, Bk16+ and Cf17+ ions share another
important advantage - relatively simple electronic structure. They have only
one electron above closed subshells and only two states above the ground state
in the optical region. This means that the high accuracy of the calculations
is possible and interpretation of the experimental spectra may be relatively
simple.

Figure 1: Energy level diagram for the Cm15+ and Bk16+ ions (approximately in
scale). Possible clock transitions are shown as red double arrows.

The energy diagram for the lowest states of Cm15+ and Bk16+ is presented on
Fig. 1. It is based on the results of our calculations. We perform the
calculations with the use of the combination of the configuration interaction
(CI) with the single-double (SD) coupled-cluster methods SD+CIa ; SD+CIb . The
effect of various external fields (e.g., electromagnetic multipole fields for
calculating transition amplitudes, hyperfine interaction, etc.) are included
within the well-known random-phase approximation (RPA). The details of the
calculations are presented in the Appendix. The results for the energy levels
and other parameters for the lowest states of Cm15+ and Bk16+ are presented in
Table 1. The other states of the odd parity are at very high excitation
energies, $E\sim$ 500000 cm-1.

Table 2 presents the lowest energy levels of even states. Electric dipole
transitions from these states to $6s^{2}6p$ and $6s^{2}5f$ may, in principle,
help to measure frequency of very weak clock transitions, which can be found
as the difference of frequencies of E1 transitions from one of high even state
to the $6s^{2}6p_{1/2}$ and $6s^{2}5f_{5/2}$ states. Moreover, the measurement
of any energy interval between the states of $6s^{2}6p$ and $6s6p5f$
configurations may help to significantly improve accuracy of the theoretical
prediction of small energy interval between the clock states. Indeed, this
interval is much smaller than the removal energy of electron. For example, the
calculated ionisation potential of Cm15+ is 2,168,971 cm-1, which is 57 times
bigger than the energy interval between the $5f_{5/2}$ and $6p_{1/2}$ states.
This energy interval may be treated as the difference of removal energies of
$5f_{5/2}$ and $6p_{1/2}$. Therefore, one may expect that the relative
theoretical error for this energy interval is 50 times bigger than the
relative error in the ionisation potential. However, energy intervals within
the same configuration (e.g. the fine splitting) have been calculated very
accurately and reliably since there are no any cancellations here. For
example, if the energy of optical transition between $6s^{2}6p_{3/2}$ and any
$6s6p5f$ states will be measured, we may significantly reduce theoretical
error in the energy interval between the $5f_{5/2}$ and $6p_{1/2}$ states.

To calculate sensitivity of the atomic frequencies to the variation of
$\alpha$ we present the them in the form

$\omega(x)=\omega_{0}+q\left[\left(\frac{\alpha}{\alpha_{0}}\right)^{2}-1\right]\equiv\omega_{0}+qx,$
(1)

where $\omega_{0}$ and $\alpha_{0}$ are present values for the frequency and
the fine structure constant, $q$ is the sensitivity coefficient. The values of
$q$ are found as a numerical derivative of the calculated values of the
frequencies. It is also convenient to have the so called enhancement factor
$K$ ($K=2q/\omega$). It relates the rates of the changing frequencies and
$\alpha$. For the relative variation of the ratio of the frequencies
$\omega_{1}/\omega_{2}$ we have

$\frac{\dot{\omega_{1}}}{\omega_{1}}-\frac{\dot{\omega_{2}}}{\omega_{2}}=(K_{1}-K_{2})\frac{\dot{\alpha}}{\alpha}.$
(2)

The calculated values of $q$ and $K$ for the lowest states of Cm15+ and Bk16+
ions are presented in Table 1. The values are given with respect to the ground
state. These values indicate very high sensitivity of the ionic frequencies to
the variation of $\alpha$. The values of $q$ and $K$ are comparable to those
for Cf17+ Cf11  and much bigger than values of $q$ and $K$ in existing atomic
clocks Canadien .

Note however, that larger values of $q$ are more important than larger values
of $K$. This is because the enhancement factor $K$ can get large value due to
small value of the transition frequency (like the $6p_{1/2}-5f_{5/2}$
transition in Bk16+). Having small value of the transition frequency not
always brings significant advantage; see Ref. Schiller  for a detailed
discussion.

The lifetimes of the $5f_{5/2}$ states of the Cm15+ and Bk16+ ions presented
in Table 1 are determined by the electric quadrupole (E2) transition to the
ground state. The lifetimes of the $5f_{7/2}$ states are strongly dominated by
the magnetic dipole (M1) transition to the $5f_{5/2}$ states. See Appendix for
the calculation of the amplitudes. The $5f_{5/2}$ state of Bk16+ has very long
lifetime of $\sim 800$ s due to the small frequency of the transition to the
ground state (transition rate $\propto\omega^{5}$). The lifetimes of other
states in Table 1 are also relatively large. This means that the high accuracy
of the measurements is possible.

Table 1:  Parameters of the lowest states of Cm15+ and Bk16+ ions: excitation energies ($E$), sensitivity coefficients ($q$), enhancement factors ($K$), lifetimes ($\tau$), static dipole polarisabilities ($\alpha_{0}$), electric quadrupole moments ($Q$), and magnetic dipole hyperfine structure constants ($A$); $g_{I}=\mu/I$, where $I$ is nuclear spin. State | $E$ | $q$ | $K$ | $\tau$ | $\alpha_{0}$ | $Q$ | $A/g_{I}$
---|---|---|---|---|---|---|---
| cm-1 | cm-1 |  |  | $a_{0}^{3}$ | $|e|a_{0}^{2}$ | GHz
| Cm15+
$6p_{1/2}$ | 0 | 0 | 0 |  | 1.1998 | 0 | 154
$5f_{5/2}$ | 38375 | 367000 | 15 | 576 ms | 1.2460 | -0.459 | 1.55
$5f_{7/2}$ | 54730 | 380000 | 13 | 20 ms | 1.2433 | -0.550 | 0.689
$6p_{3/2}$ | 182359 |  |  |  |  |  |
| Bk16+
$6p_{1/2}$ | 0 | 0 | 0 |  | 1.0685 | 0 | 174
$5f_{5/2}$ | 8880 | 403000 | 91 | 787 s | 1.1091 | -0.507 | 1.74
$5f_{7/2}$ | 27520 | 419000 | 30 | 13 ms | 1.1069 | -0.604 | 0.772
$6p_{3/2}$ | 202403 |  |  |  |  |  |
Table 2: Lowest even energy levels (cm-1) of Cm15+ and Bk16+. Conf. | $J$ | $E$ | Conf. | $J$ | $E$
---|---|---|---|---|---
Cm15+ | Bk16+
$6s6p^{2}$ | 1/2 | 220159 | $6s6p5f$ | 5/2 | 215399
$6s6p5f$ | 5/2 | 234517 | $6s6p5f$ | 7/2 | 228430
$6s6p5f$ | 7/2 | 246936 | $6s6p5f$ | 3/2 | 229257
$6s6p5f$ | 3/2 | 250537 | $6s6p^{2}$ | 1/2 | 229273
$6s6p5f$ | 5/2 | 253281 | $6s5f^{2}$ | 7/2 | 232137
$6s6p5f$ | 7/2 | 255862 | $6s6p5f$ | 5/2 | 233227
$6s6p5f$ | 5/2 | 274931 | $6s6p5f$ | 7/2 | 238672
$6s6p5f$ | 7/2 | 275226 | $6s5f^{2}$ | 3/2 | 245731
$6s5f^{2}$ | 7/2 | 281399 | $6s5f^{2}$ | 5/2 | 249308
$6s5f^{2}$ | 3/2 | 290189 | $6s6p5f$ | 7/2 | 256126
$6s5f^{2}$ | 5/2 | 294383 | $6s6p5f$ | 5/2 | 257240
$6s5f^{2}$ | 1/2 | 314906 | $6s5f^{2}$ | 1/2 | 270551
$6s5f^{2}$ | 3/2 | 319212 | $6s5f^{2}$ | 3/2 | 274819
$6s5f^{2}$ | 3/2 | 325976 | $6s5f^{2}$ | 3/2 | 282097
$6s5f^{2}$ | 1/2 | 339563 | $6s5f^{2}$ | 1/2 | 295155

We consider now some systematic effects in the frequency measurements.

## Blackbody radiation (BBR) shift.

The BBR shift of the transition frequency (in Hz) is given by (see, e.g.
Derevianko )

$\delta\nu_{\rm BBR}=-8.611\times
10^{-3}\left(\frac{T}{300K}\right)^{4}\Delta\alpha_{0}.$ (3)

Here $\Delta\alpha_{0}$ is the difference in the static scalar
polarizabilities of two states (in a.u.). We calculate the polarizabilities as
described in the Appendix. The results are presented in Table 1. Using these
numbers we get at T=300K the values for the relative frequency shift which are
presented in Table 3.

Table 3:  Relative frequency shift due to BBR in Cm15+ and Bk16+ Ion | Transition | $\delta\nu_{\rm BBR}/\nu$
---|---|---
Cm15+ | $6s_{1/2}-5f_{5/2}$ | $-3.4\times 10^{-19}$
| $5f_{5/2}-5f_{7/2}$ | $2.2\times 10^{-20}$
Bk16+ | $6s_{1/2}-5f_{5/2}$ | $-1.5\times 10^{-18}$
| $5f_{5/2}-5f_{7/2}$ | $3.4\times 10^{-20}$

## Quadrupole shift.

The $5f$ states have sufficiently large values of the total angular momentum
$J$ to make them sensitive to the gradient of electric field $\varepsilon$ via
quadrupole interaction. Corresponding energy shift is given by

$\Delta
E_{Q}=\frac{J_{z}^{2}-J(J+1)}{2J(2J-1)}Q\frac{\partial\varepsilon_{z}}{\partial
z},$ (4)

where $Q$ is atomic quadrupole moment defined as doubled expectation value of
the E2 operator in the stretched state

$Q=2\langle J,J_{z}=J|E2|J,J_{z}=J\rangle.$ (5)

The calculated values of the quadrupole moment $Q$ for low states of Cm15+ and
Bk16+ are presented in Table 1. These values are close to those calculated for
Cf15+ and Cf17+ in Ref. Cf3 . Therefore, the same estimate is valid:
$\delta\nu/\nu\sim 10^{-16}$. These shifts can be further suppressed by up to
four orders of magnitude by averaging over projections of the total angular
momentum $J$ Sr+ .

## Hyperfine structure.

Hyperfine structure may complicate the work with the ions. In particular, it
leads to enhancement of the second-order Zeeman shift, since small hyperfine
intervals go into denominators of the expression for the shift. This
complications can be easily avoided for Cm15+ ion. The isotope 248Cm lives 348
thousand years and has zero nuclear spin. Bk has no long lifetime isotopes
with zero nuclear spin since it has odd number of protons. In any case, it is
useful to know the values of the hyperfine constants for future analysis. We
calculate magnetic dipole constant $A$ as described in the Appendix. The
results are presented in Table 1 in a form of $A/g_{I}$ ($g_{I}=\mu/I$, $I$ is
nuclear spin), which is approximately the same for all isotopes.

## Sympathetic cooling.

Ion-based optical clocks are susceptible to thermal motion due to the finite
ion temperature. This motion can be reduced by applying sympathetic cooling of
the clock ion via the cotrapped logic ion. The most efficient sympathetic
cooling occurs when the charge-to-mass ratio of the clock ion is equal to that
of the logic ion cooling . The $Z_{i}/A$ ratio is 0.0605 for 248Cm15+ and
0.0648 for 247Bk16+. A possible logic ion in both cases is 24Mg+ where the
$Z_{i}/A$ ratio is 0.042.

In summary, we state that the Cm15+ and Bk16+ ions may serve as optical clocks
which are not sensitive to external perturbations (BBR shift, quadrupole
shift, etc.) but very sensitive to the hypothetical time variation of the fine
structure constant and dark matter fields.

###### Acknowledgements.

This work was supported by the Australian Research Council Grants No.
DP230101058 and DP200100150.

##  Appendix A Method of calculations.

We treat the Cm15+ and Bk16+ ions as systems with three valence electrons
above the closed-shell $[1s^{2}\dots 5d^{10}]$ core. Calculations start from
the relativistic Hartree-Fock (RHF) procedure for the closed-shell core. The
RHF Hamiltonian has the form

$\displaystyle\hat{H}^{\rm
RHF}=c\mathbf{\alpha}\cdot\mathbf{p}+(\beta-1)mc^{2}+$ (6) $\displaystyle
V_{\rm nuc}+V_{\rm Breit}+V_{\rm QED}+V_{\rm core},$

where $c$ is the speed of light, $\mathbf{\alpha}$ and $\beta$ are Dirac
matrixes, $\mathbf{p}$ is the electron momentum, $V_{\rm nuc}$ is the nuclear
potential obtained by integrating the Fermi distribution of nuclear charge,
$V_{\rm Breit}$ is the potential due to the Breit interaction Breit , $V_{\rm
QED}$ is the potential describing quantum electrodynamic corrections QED ,
$V_{\rm core}$ is the self-consistent RHF potential created by all core
electrons. The single-electron basis states for valence electrons are
calculated in the field of the frozen core using the B-spline technique
B-spline . These basis states are used in all consequent calculations. This
corresponds to the so-called $V^{N-M}$ approximation VN-M , where $M$ is the
number of valence electrons ($M=3$ in our case).

The single-double coupled-cluster method (SD) is used (we use the version
described in Ref. SD+CIb ) to include the correlations between valence and
core electrons. Solving the SD equations involves iterations for the core and
for the valence states until the full convergence is achieved. As a result,
the all-order correlation operators $\hat{\Sigma}_{1}$ and $\hat{\Sigma}_{2}$
are produced. The $\hat{\Sigma}_{1}$ operator is the single-electron operator
which describes the correlation interaction of a particular valence electron
with the core. The $\hat{\Sigma}_{2}$ operator is the two-electron operator
which describes the screening of the Coulomb interaction between valence
electrons by the core electrons. The resulting effective CI Hamiltonian has
the form

$\hat{H}^{\rm CI}=\sum_{i=1}^{M}\left(\hat{H}^{\rm
RHF}+\hat{\Sigma}_{1}\right)_{i}+\sum_{i<j}^{M}\left(\frac{e^{2}}{r_{ij}}+\hat{\Sigma}_{2ij}\right).$
(7)

The energy and wave function of the many-electron state $a$ are found by
solving the CI equation

$\left(\hat{H}^{\rm CI}-E_{a}\right)X_{a}=0,$ (8)

where $X_{a}$ contains coefficients of the expansion of the valence wave
function over single-determinant basis states.

To calculate transition amplitudes or hyperfine structure we need to include
an additional operator of external field, such as magnetic dipole or electric
quadrupole laser field, magnetic dipole nuclear field, etc. This is done in
within the random-phase approximation (RPA).

The RPA equations have the form (see e.g. TDHF )

$(\hat{H}^{\rm RHF}-\epsilon_{c})\delta\psi_{c}=-(\hat{F}+\delta V_{\rm
core})\psi_{c},$ (9)

where $\hat{H}^{\rm RHF}$ is given by (6), index $c$ numerates single-electron
states in the core, $\psi_{c}$ and $\delta\psi_{c}$ are corresponding single-
electron functions and corrections due to the operator of external field
$\hat{F}$, and $\delta V_{\rm core}$ is the change of the self-consistent
Hartree-Fock potential due to the change in all core functions. Solving Eqs.
(9) self-consistently allows to determine $\delta V_{\rm core}$. Then the
transition amplitude is given by is given by

$A_{ab}=\langle X_{b}|\sum_{i=1}^{M}(\hat{F}+\delta V_{\rm
core})_{i}|X_{a}\rangle,$ (10)

where the wave functions for states $a$ and $b$ come from solving the CI
equation (8). For energy shift (e.g. in calculating the hyperfine constant)
$a=b$ in (10).

Polarizabilities are calculated using the wave functions found as described
above and the techniques of summations over complete set of intermediate
states as described in Refs pol1 ; pol2 .

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