# Spin-liquid-like state in a square lattice antiferromagnet

B. Sana  Department of Physics, Indian Institute of Technology Madras,
Chennai, 600036, India
M. Barik  Department of Physics, Indian Institute of Technology Madras,
Chennai, 600036, India
S. Lee  Center for Integrated Nanostructure Physics, Institute for Basic
Science (IBS), Suwon 16419, Republic of Korea
U. Jena  Department of Physics, Indian Institute of Technology Madras,
Chennai, 600036, India
M. Baenitz  Max Planck Institute for Chemical Physics of Solids, Nöthnitzer
Strasse 40, 01187 Dresden, Germany  J. Sichelschmidt  Max Planck Institute for
Chemical Physics of Solids, Nöthnitzer Strasse 40, 01187 Dresden, Germany  S.
Luther  Hochfeld-Magnetlabor Dresden (HLD-EMFL), Helmholtz-Zentrum Dresden-
Rossendorf, 01328 Dresden, Germany  Institut für Festkörper- und
Materialphysik, TU Dresden, 01062 Dresden, Germany  H. Kühne  Hochfeld-
Magnetlabor Dresden (HLD-EMFL), Helmholtz-Zentrum Dresden-Rossendorf, 01328
Dresden, Germany  K. Sethupathi  Department of Physics, Indian Institute of
Technology Madras, Chennai, 600036, India
Quantum Centre for Diamond and Emergent Materials, Indian Institute of
Technology Madras, Chennai, 600036, India
M. S. Ramachandra Rao  Quantum Centre for Diamond and Emergent Materials,
Indian Institute of Technology Madras, Chennai, 600036, India
Department of Physics, Nano Functional Materials Technology Centre and
Materials Science Research Centre, Indian Institute of Technology Madras,
Chennai, Tamil Nadu 600036, India
K. Y. Choi  Department of Physics, Sungkyunkwan University, Suwon 16419,
Republic of Korea  P. Khuntia<EMAIL_ADDRESS>Department of Physics,
Indian Institute of Technology Madras, Chennai, 600036, India
Quantum Centre for Diamond and Emergent Materials, Indian Institute of
Technology Madras, Chennai, 600036, India

###### Abstract

Collective behavior of spins, frustration-induced strong quantum fluctuations
and subtle interplay between competing degrees of freedom in quantum materials
can lead to correlated quantum states with fractional excitations that are
essential ingredients for establishing paradigmatic models and have immense
potential for quantum technologies. Quenched randomness is a new paradigm in
elucidating the emergence of spin-liquid-like states in geometrically
frustrated magnets. Herein, we report magnetization, specific heat, electron
spin resonance, and muon spin resonance studies on a 3$d$-electron based
square lattice antiferromagnet Sr3CuTa2O9. In this material, $S=1/2$ Cu2+
nearest-neighbor ions constitute a two-dimensional square lattice. The
negative value of Curie-Weiss temperature, obtained from the Curie-Weiss fit
of high temperature magnetic susceptibility data indicates the presence of
antiferromagnetic interaction between Cu2+ moments. Specific heat data show
the absence of long-range magnetic ordering down to 64 mK despite a reasonably
strong exchange interaction between Cu2+ spins as reflected from a Curie-Weiss
temperature of $-27$ K. The power-law behavior and the data collapse of
specific heat and magnetization data evince the emergence of a random-singlet
state in Sr3CuTa2O9. The power-law-like spin auto-correlation function and the
data collapse of muon polarization asymmetry with longitudinal field
dependence of $t/{(\mu_{0}H)}^{\gamma}$ further support credence to the
presence of a randomness-induced liquid-like state. Our results suggest that
randomness induced by disorder is a viable route to realize quantum spin
liquid-like state in this square lattice antiferromagnet.

††preprint: APS/123-QED

## I. Introduction

Spin correlations, competing interactions and quantum character of spins in
frustrated quantum materials offer an ideal platform to harbor entangled
quantum states with fractional quantum numbers. These states could have
significant implications in addressing some of the fundamental challenges in
condensed matter and the development of quantum technologies [1, 2, 3, 4].
Magnetic frustration refers to the incompatibility of various magnetic
interactions in the spin-lattice of a quantum material, can lead to a highly
degenerate ground state and prevent symmetry-breaking phase transitions in
quantum materials. Frustrated magnets are ideal to host novel quantum states
such as quantum spin liquids and spin ice [1, 5]. A quantum spin liquid (QSL)
is a highly entangled state of matter wherein frustration-induced strong
quantum fluctuations prevent long-range magnetic order down to absolute zero
temperature despite strong exchange interactions between magnetic moments.
Remarkably, QSL is characterized by the presence of fractional excitations
such as spinons and Majorana fermions coupled to emergent gauge fields and the
spin-correlations display non-local behavior that is robust against weak
external perturbations [1, 6, 2, 3, 7, 8, 9, 10]. The experimental realization
of QSLs with fractional quasiparticles in one-dimensional (1D) quantum magnets
owing to strong quantum fluctuations is a well-established scenario. However,
the realization of QSLs in two-dimensional (2D) and three-dimensional (3D)
quantum magnets remains challenging in view of the presence of various
perturbations such as disorder, defects, extra connectivity of magnetic
moments in the spin-lattice and finite inter-layer interactions in real
materials. In this context, strong quantum fluctuations in frustrated magnets,
next-nearest interaction, ring exchange, and exchange anisotropy offer a
viable platform in hosting QSL states in promising 2D and 3D spin-lattices [1,
11].

The experimental realization of QSLs in clean and periodic systems remain a
great challenge that invokes to discover, design and investigate promising
frustrated magnets in this context . Interestingly, it has been suggested that
unavoidable disorder in certain frustrated magnets, can lead to a random
exchange energy between spins [12, 13, 14, 15, 16, 17]. Quenched disorder
owing to the presence of defects or site mixing that are fixed into the
crystal lattice structure during the synthesis process can induce randomness
in exchange energies and hence quantum fluctuation, which offers a viable
route to stabilize the QSL state in frustrated magnets [18]. The subtle
interplay between quantum fluctuations induced by anti-site disorder, magnetic
frustration and emergent degrees of freedom can give rise to novel disordered
ground states such as random-singlet state, where spins interact
antiferromagnetically and form spin-singlet pairs. In 1D quantum spin chain
with Heisenberg antiferromagnetic interactions, random coupling between
$S=1/2$ spins forms random spin-singlet pairs [19, 20, 21]. However, random-
singlet states in 2D and 3D spin-lattices have not been well studied yet due
to the scarcity of such frustrated magnets [21, 20, 22, 23, 24].

Among various 2D spin-lattices, spin-$1/2$ square lattice has drawn
considerable interest due to the proposal of high-temperature
superconductivity, and later various theoretical models on square lattice
Heisenberg antiferromagnets were introduced to explore various exotic states,
including quantum spin liquids [25, 26, 27, 28, 29]. In this context,
understanding the fundamental characteristics of the ground state of a 2D
square lattice is crucial in quantum materials. In 2D spin-$1/2$ square
lattice, the nearest-neighbor antiferromagnetic exchange interaction ($J_{1}$)
does not host a quantum disordered ground state; instead, it favors long-range
magnetic order even in the presence of strong disorder [30]. The presence of
competing next-nearest neighbor antiferromagnetic interactions ($J_{2}$) gives
rise to frustration in the spin-lattice, and stabilizes a quantum disordered
ground state in the frustrated regime $0.4\lesssim J_{2}/J_{1}\lesssim 0.6$
[24]. Interestingly, it is possible to realize a non-magnetic random-singlet
state in this square lattice antiferromagnets ($J_{1}-J_{2}$ model) even in
the unfrustrated regime if we introduce randomness in the exchange
interactions [24, 13]. In another theoretical model on a square lattice called
the $J\mathchar 45\relax Q$ model, wherein Heisenberg exchange interaction $J$
along with multispin exchange interaction $Q$ (i.e., the interaction between
correlated singlets) can prevent long-range antiferromagnetic order even in
the absence of frustration [13, 31]. Due to bond randomness, there exists a
broad range of antiferromagnetic exchange interactions that lead to a power-
law distribution of exchange interactions $P(J)\sim J^{-\alpha}$ at low
energies in the renormalization group flow of the random-singlet state [18].
This is manifested as unconventional scaling behavior of low-temperature
magnetic susceptibility and specific heat in frustrated magnets with quenched
disorder [18, 32, 15, 17]. Remarkably, in the presence of an XY anisotropy of
the low-temperature spin correlations, the experimental observation of
magnetic properties associated with a finite-temperature Berezinskii-
Kosterlitz-Thouless phase transition may be possible in quasi-2D spin-$1/2$
square-lattice antiferromagnets [33, 34].

Recently, the rare-earth based triangular lattice antiferromagnet YbMgGaO4 has
received widespread acclaim as a promising quantum spin liquid candidate [35,
36, 37, 38]. The presence of anti-site disorder between Mg2+ and Ga3+ leads to
the formation of a random crystal field, which is believed to be the driving
factor behind the emergence of quantum disordered liquid-like state [39].
Disorder-induced quantum spin liquid states are also proposed in a few
promising frustrated materials such as Ho2Ti2O7, Pr2Zr2O7 [40, 41], Ba3CuSb2O9
[42] and Y2CuTiO6 [43]. It is proposed that disorder can induce long-range
entanglement and depending upon the degree of disorder, it can show Coulombic
spin liquid, Mott glass and conventional glassy phases in the ground state
[40]. In Y2CuTiO6, $S=1/2$ Cu2+ ions and non-magnetic Ti4+ ions share the same
atomic site forming a triangular lattice with a 50:50 occupancy ratio. Despite
such a huge site dilution, Y2CuTiO6 does not show conventional magnetic order
or spin freezing down to 50 mK, indicating a disorder-driven cooperative
paramagnetic state. Interestingly, the universal scaling of specific heat in
Y2CuTiO6 also reveals the formation of a random-singlet state at low
temperatures [43]. In Ba3CuSb2O9, the CuSbO9 unit forms Cu-Sb dumbbells, which
give rise to electric dipole moment due to the charge difference between Cu2+
and Sb5+ ions, and the site sharing of Cu2+ (33%) with non-magnetic Sb5+ (67%)
ions leads to spin-orbital liquid-like behavior [44, 42]. In a similar vein,
triangular lattice antiferromagnet Sr3CuSb2O9 with substantial anti-site
disorder between Cu2+ and Sb5+ ions shows gapless quantum spin liquid-like
behavior down to 65 mK [45].

A random-singlet state is demonstrated in a chemically substituted $S=1/2$
square lattice antiferromagnet Sr2CuTe0.5W0.5O6 [46], wherein the square
lattice of Cu2+ is not disordered but bond randomness originates from the site
mixing of non-magnetic Te6+ and W6+ ions. The spin-$1/2$ Heisenberg
$J_{1}\mathchar 45\relax J_{2}$ model on a honeycomb lattice is also proposed
to host random-singlet-driven spin-liquid-like behavior [32]. The nearest-
neighbor antiferromagnetic Heisenberg interaction $J_{1}$ can not induce
frustration in the bipartite honeycomb lattice and the ground state is
antiferromagnetic . An additional next nearest-neighbor antiferromagnetic
exchange interaction $J_{2}$ is essential to induce frustration that leads to
a quantum spin-liquid-like ground state in a honeycomb lattice. The random
distribution of exchange interactions leads to a wide range of binding
energies for singlet dimers, resulting in a gapless behavior [13]. The spin
liquid behavior observed in honeycomb lattice antiferromagnet 6HB-Ba3NiSb2O9
could be explained well by the notion of a random-singlet state [47, 32].
Recently, a quantum spin-liquid-like random-singlet state is observed in
disordered antiferromagnet Li4CuTeO6 and an organic verdazyl-based complex
with a honeycomb lattice [17, 48]. The substitution of suitable chemical
element in a controlled manner can reveal interesting insights in the host
lattice of frustrated magnet, for instance, the substitution non-magnetic Ir3+
ion at Ru3+ site of the celebrated Kitaev magnet $\alpha$-Ru1-xIrxCl3 induces
quenched disorder that drives the diluted system into a spin liquid state
[12].

The random-singlet state in 1D quantum spin systems has been well-established
through extensive studies [49, 19, 20, 21]. Even an infinitesimal bond
disorder can lead a 1D spin system into a random-singlet state [50, 51]. For
instance, substituting Si with Ge, a random-singlet state is achieved in the
spin-chain BaCu2(Si1-xGex)2O7 [49], where the parent compound BaCu2Si2O7 shows
long-range antiferromagnetic order [52]. Understanding the role of randomness
in strongly frustrated 2D spin-lattice is an uncharted venue. How the random-
singlet in 2D spin-lattices resembles or differs from that found in 1D spin-
lattices is still unclear. More experimental candidates and evidence is
required to explore how randomness and/or frustration induces various ground
states, especially spin-liquid-like states in 2D and 3D spin-lattices [13,
24]. In this context, the search for novel frustrated magnets with quenched
disorder and the experimental realization of randomness-induced correlated
quantum states in 2D spin-lattices is necessary to establish a generic
framework in elucidating macroscopic and microscopic magnetic properties in a
variety of frustrated quantum materials [16, 18, 53].

In this work, we report magnetization, specific heat, muon spin resonance and
electron spin resonance on a square lattice Sr3CuTa2O9 (henceforth SCTO), in
which nearest-neighbor magnetic Cu2+ ions constitute a square lattice in the
$a\mathchar 45\relax b$ plane. In this material, Cu2+ ion shares the same
crystallographic site with non-magnetic Ta5+ ion with a ratio $1:2$. This
system can thus be considered as a site-diluted square lattice of spin $S=1/2$
Cu2+ ions. The magnetic susceptibility shows no signature of a magnetic phase
transition down to 2 K. The Cu2+ ($S=1/2$) moments interact
antiferromagnetically with a moderate Curie-Weiss temperature ($-27$ K).
Specific heat confirms the absence of long-range magnetic ordering down to 64
mK, implying a disordered ground state. The magnetization, specific heat and
muon asymmetry results display a data collapse behavior pointing towards
correlated quantum state. The unusual scaling behavior of high field
magnetization up to 52 T, ESR line width and zero-field muon relaxation rate
at low temperature along with the data collapse behavior of thermodynamic and
muSR results suggest a random-singlet state in SCTO that is reconciled with
the theoretical framework in the context of a randomness-induced quantum
disordered liquid-like state.

Figure 1:  (a) Left $y$ axis: Temperature dependence of magnetic
susceptibility of SCTO measured in an applied magnetic field of 0.1 T. Right
$y$-axis: The temperature dependence of inverse susceptibility data with
Curie-Weiss fit. The inset shows the square lattice composed of Cu atoms.  (b)
The power-law fit of magnetic susceptibility in 1 T. The inset shows the ZFC-
FC data recorded in 0.001 T.  (c) Scaling of magnetic susceptibility with
$(\mu_{\mathrm{0}}H)^{0.71}\chi$ vs. $T/\mu_{\mathrm{0}}H$. (d) Scaling of
magnetization isotherms with $MT^{-0.4}$ vs. $\mu_{0}H/T$. The collapse of
magnetic susceptibility and magnetization isotherms to respective single
curves suggests a randomness-induced quantum effect is at play in this square
lattice antiferromagnet.

## II. METHODS

Polycrystalline sample of SCTO was prepared by a conventional solid-state
synthesis method from high purity Sr2CO3 (Alfa Aesar, 99.994%), Ta2O5 (Alfa
Aesar, 99.993%) and CuO (Alfa Aesar, 99.995%). The stoichiometric amount of
each material was mixed together and then pelletized. The single phase was
obtained after firing it at 800 ∘C (12 hrs), 900 ∘C (24 hrs) and 950∘C (48
hrs) with intermediate grindings. The phase purity was checked by a Rigaku
X-Ray diffractometer at room temperature using Cu $K_{\alpha}$ radiation. The
physics of frustrated magnets is quite rich and diverse, which requires
complementary experimental techniques to probe various facets at play. A
SQUID-VSM instrument (Quantum Design) was used to perform magnetization
measurements in the temperature range of 2 K $\leq T\leq$ 400 K under magnetic
fields 0 T $\leq\mu_{\mathrm{0}}H\leq$ 5 T. Specific heat measurements were
carried out using a Physical Properties Measurement System (QD, PPMS) in the
temperature range of 2 K $\leq T\leq$ 200 K and in magnetic fields of 0 T
$\leq\mu_{\mathrm{0}}H\leq$ 7 T. In addition, low-temperature specific heat
measurements in zero field and 1 T were carried out down to 64 mK using a
dilution refrigerator (Quantum Design). We used a continuous wave ESR
spectrometer at X-band frequencies ($\nu=9.4$ GHz). The powder sample was
embedded in paraffine and the temperature was set in between 3 K $\leq T\leq$
300 K using a helium-flow cryostat. We performed zero-field (ZF) and
longitudinal-field (LF) muon spin relaxation ($\mu$SR) measurements on the M20
beamline using the LAMPF spectrometer (2 K $\leq T\leq$ 200 K) at TRIUMF
(Vancouver, Canada). The obtained $\mu$SR data were analyzed using MUSRFIT
software [54]. The high-field magnetization experiment was conducted at 1.4 K
in pulsed magnetic fields up to 52 T using the facilities at the Dresden High
Magnetic Field Laboratory.

## III. Results

### Crystal structure of Sr3CuTa2O9

Rietveld refinement of powder x-ray diffraction data was carried out using
GSAS software [55]. For Rietveld refinement, initial structural parameters
were taken from the isostructural compound Sr3CuNb2O9 [56]. Sr3CuTa2O9
crystalizes in the tetragonal space group P4mm, which generally hosts a square
lattice in magnetic materials [56, 57, 58, 59, 60]. The Rietveld refinement
parameters of SCTO are given in supplementary (SM) Table 1 and the refinement
pattern is shown in SM Fig. 1b. Nearest-neighbor magnetic Cu2+ ions form a
square lattice (see Fig. 1b inset), but $2/3$ of the same sites are shared by
non-magnetic Ta5+ ions. Six oxygen atoms constitute a CuO6 polyhedron where
Cu2+ ion is at the center of the polyhedron (see SM Fig. 1a). The intra-layer
Cu2+-Cu2+ bond length is 3.967 Å whereas, the inter-layer length is 4.067 Å.
The nearest-neighbor Cu2+ ions are connected via a Cu-O-Cu super-exchange path
where the Cu-O-Cu angle is 172.5∘. The inter-planar super-exchange interaction
is mediated via another Cu-O-Cu path having an angle of 180∘. This is in
accordance with the Goodenough-Kanamori rule [61] which suggests
antiferromagnetic interaction is expected from the Cu-O-Cu exchange path. It
is interesting to investigate the role of exchange interactions and quenched
disorder in determining the ground state properties in this site-diluted
square lattice antiferromagnet.

Figure 2: (a) Temperature dependence of specific heat for SCTO in zero field.
Red line shows the lattice contribution determined from the combination of
Debye and Einstein models as described in the text. (b) Low-temperature
specific heat in different applied magnetic fields. (c) Temperature dependence
of magnetic specific heat ($C_{\mathrm{m}}$) after subtracting the lattice
specific heat from total specific heat. Zero-field $C_{\mathrm{m}}$ follows
the power-law $C_{\mathrm{m}}\sim T^{0.59}$ (red line) at low temperatures.
The inset shows entropy release up to 50 K. (d) The data collapse of specific
heat taken in different magnetic fields onto a single curve suggests the
presence of a random-singlet state that is in agreement with magnetic
susceptibility data.

### Magnetization

Magnetization experiments as a function of temperature and field convey
valuable information concerning the nature of exchange interaction between
magnetic moments in the spin-lattice, spin correlations, anisotropy, and
disorder that are essential to shed insights into the ground state properties
of frustrated magnets. The left $y$-axis of Fig. 1a represents the magnetic
susceptibility taken in an applied magnetic field of 0.1 T in the temperature
range 2 K $\leq T\leq$ 400 K. There is no anomaly in the magnetic
susceptibility data down to 2 K, suggesting the absence of a magnetic phase
transition. The inverse susceptibility data (see right $y$-axis of Fig. 1a)
can be well reproduced by the Curie-Weiss formula,
$\chi(T)=\chi_{0}+\frac{C}{T-\theta_{\mathrm{CW}}}$ in the temperature range
250 K $\leq T\leq$ 400 K. The value of the temperature independent
susceptibility $\chi_{0}$ is $-6.2\times$10-5 cm3/mol, which is originated due
to diamagnetic susceptibility as well as van-Vleck paramagnetic
susceptibility. However, the negative sign indicates that diamagnetic
susceptibility dominates the latter. A negative Curie-Weiss temperature of
$\theta_{\mathrm{CW}}\approx-27$ K was extracted from the high-temperature
Curie-Weiss fit, which indicates that the dominant interaction between the
Cu2+ spins is antiferromagnetic. The Curie-Weiss fit yields an effective
moment of $\mu_{\mathrm{eff}}$ = 1.74 $\mu_{\mathrm{B}}$, which is close to
the free ion moment of $S=1/2$ Cu2+ ion ($g\sqrt{S(S+1)}\mu_{\mathrm{B}}$,
$g=$ Landé $g$ factor). The zero-field-cooled (ZFC) and field-cooled (FC) data
taken in an applied magnetic field of 0.001 T (see inset of Fig. 1b) do not
show any bifurcation, suggesting the absence of any spin freezing.

In quenched-disorder materials, random exchange interaction gives rise to a
power-law behavior ($\propto T^{-\gamma}$) in low-temperature magnetic
susceptibility data [18, 16, 31]. We fitted the susceptibility data (see Fig.
1b) in between 2-5 K with a power-law $\propto T^{-0.57}$. Another key
signature of the random-singlet state is the scaling of susceptibility and
specific heat data [15]. We plotted $(\mu_{\mathrm{0}}H)^{\gamma}\chi$ vs.
$T/\mu_{0}H$ in Fig. 1c for magnetic field up to 5 T where the value $\gamma$
turned out to be 0.71, which is close to the value obtained from the power-law
fit. All data points fall into the same curve at low temperatures, indicating
bond randomness in the system [62]. To confirm the universal scaling behavior,
$MT^{-\beta}$ vs. $\mu_{0}H/T$ data are plotted in Fig. 1d. The value of
$\beta$ obtained from the scaling is $0.4$, which is close to the expected
value $1-\gamma=0.43$. Similar scaling behavior was recently reported in a
Kitaev spin liquid candidate H3LiIr2O6 wherein disorder-induced low energy
density of states $N(E)\sim E^{-0.5}$ give rise to a power-law behavior in the
low-temperature magnetic susceptibility data [63]. The scaling behavior of
magnetization in the Kitaev magnet H3LiIr2O6 suggests the realization of a
random-singlet state. The absence of hysteresis in the $M$ vs. $\mu_{0}H$
curve at low $T$ also rules out the existence of spin freezing and impurity in
this material (see SM Fig. 1c). Magnetization isotherms taken at different
temperatures and high-field magnetization data are shown in SM Fig. 1d and SM
Fig. 1e, respectively. The high-field data are well reproduce up to 52 T
considering a power-law distribution function $P(J)\propto J^{-\alpha}$ (see
SM text) of singlet states. Here, $J$ is the isotropic exchange coupling
between the spins of a singlet. We found that the $\alpha$ value is $\sim$
0.45, which is close to the value found in another random-singlet material
Ba5CuIr3O12 [64]. In addition, the power law behavior of the field dependence
of magnetization $M\sim(\mu_{\mathrm{0}}H)^{2.6}$ at 1.4 K taken in fields up
to 52 T is in agreement with a random singlet scenario in SCTO ( see SM Fig.
1e). This indicates that there might be enormous low-energy density of states
in this square lattice antiferromagnet.

### Specific heat

Figure 3: (a) X-band ESR spectra (symbols) at representative temperatures with
fit to uniaxial powder-averaged Lorentzians (solid lines). Fitted values of
anisotropic $g$-values are indicated for the spectrum at $T=105$ K. (b)
Temperature dependence of fitted parameters linewidth and effective g-value
and inverse integrated ESR intensity with Curie-Weiss behavior as indicated by
the solid line. Solid lines in the upper panel indicate a pow-law fit $\sim
T^{-0.37}$.

Specific heat measurements offer an ideal probe to track phase transitions and
the nature of low-energy excitations at low temperatures in correlated
electron materials. Fig. 2a and Fig. 2b show the temperature dependence of the
specific heat in zero field and different applied fields, respectively. The
absence of any anomaly down to 64 mK in zero field indicates that there is no
long-range magnetic order despite a moderate Curie-Weiss temperature, which
suggests that the titled square lattice is a highly frustrated
antiferromagnet. At higher temperatures, specific heat is dominated by phonon
contribution. In order to extract the phonon contribution, the specific heat
data are fitted (32 K $\leq T\leq$ 90 K) with a combination of one Debye and
three Einstein functions (solid red line in Fig. 2a) as given below [65, 66]

$C_{\mathrm{lattice}}(T)=C_{\mathrm{D}}\left[9k_{\mathrm{B}}\left(\frac{T}{\theta_{\mathrm{D}}}\right)^{3}\int_{0}^{\theta_{\mathrm{D}}/T}\frac{x^{4}\mathrm{exp}(x)}{\left[\mathrm{exp}(x)-1\right]^{2}}dx\right]$
(1)
$+\sum_{\mathrm{i}}3RC_{E_{\mathrm{i}}}\left(\frac{\theta_{E_{\mathrm{i}}}}{T}\right)^{2}\frac{\mathrm{exp}\left(\frac{\theta_{E_{\mathrm{i}}}}{T}\right)}{\left[\mathrm{exp}\left(\frac{\theta_{E_{\mathrm{i}}}}{T}\right)-1\right]^{2}},$

where the first and second term correspond to the Debye and Einstein specific
heat, respectively and $\theta_{\mathrm{D}}$ and $\theta_{E_{\mathrm{i}}}$ are
the Debye and Einstein temperatures, respectively. $C_{\mathrm{D}}$ and
$C_{\mathrm{E_{i}}}$ are the Debye and Einstein coefficients, respectively,
and $R$ is the universal gas constant. The weightage factor for
$C_{\mathrm{D}}$, $C_{\mathrm{E_{1}}}$, $C_{\mathrm{E_{2}}}$ and
$C_{\mathrm{E_{3}}}$ are 1, 3, 5 and 6, respectively. These coefficients are
fixed in a way that the ratio of the Debye coefficient and the sum of all
Einstein coefficients should $1:n-1$, where $n$ is the number of atoms in a
unit cell of SCTO (i.e., 15). The value of $\theta_{\mathrm{D}}$,
$\theta_{\mathrm{E_{1}}}$, $\theta_{\mathrm{E_{\mathrm{2}}}}$ and
$\theta_{\mathrm{E_{\mathrm{3}}}}$ are $290\pm 23$ K, $120\pm 1$ K, $279\pm 4$
K and $604\pm 4$ K, respectively. The lattice contribution is subtracted from
the total specific heat data to obtain the magnetic specific heat
($C_{\mathrm{m}}$). The Fig. 2c depicts the low temperature magnetic specific
heat and the inset of Fig. 2c represents the entropy change that is obtained
by integrating $C_{\mathrm{m}}/T$ over the temperature range 64 mK $\leq
T\leq$ 50 K for zero field. For a spin-half system, the maximum entropy
release should be 5.76 J/mol.K. However, for the present system, the maximum
entropy change is $4.13$ J/mol.K in zero field, which is lower than that
expected for $S=1/2$ quantum systems. The residual entropy at low temperatures
is attributed to ground state degeneracy, which is a common scenario in spin
liquid candidates [46, 67, 43, 68].

The temperature dependence of specific heat in various magnetic fields
provides information concerning spin correlations and magnetic excitations in
the ground state of frustrated magnets under study. It is well known that the
temperature dependence of specific heat of several quantum spin liquids show a
power-law owing to exotic low-energy excitations in the ground state. A
random-singlet phase can also give rise to power-law behavior in low-$T$
specific heat data [64, 69]. We found that the low-temperature magnetic
specific heat data can be reconciled with the power-law $C_{\mathrm{m}}\sim
T^{0.59}$ (see Fig. 2c), suggesting a random-singlet state with low-lying
excitations [17]. It has been suggested that $S=1/2$ quantum magnets on 2D
spin lattices with random exchange energies can host such a disorder-induced
quantum phenomena [18]. The specific heat $C(H,T)$ shows a unique behavior in
such systems and specific heat data in various magnetic fields collapse onto a
single scaling curve (for $T/\mu_{0}H<<1$) given by
$C(\mu_{0}H,T)/T\approx\frac{1}{(\mu_{0}H)^{\gamma}}F(\frac{T}{\mu_{0}H})$,
where $F$ is the scaling function [16]. The scaling behavior of
$(\mu_{0}H)^{\gamma}C_{\mathrm{p}}/T$ with $T/\mu_{0}H$ in SCTO is
demonstrated in Fig. 2d, and the obtained value of $\gamma$ is 0.7, which is
in line with what is expected in a random-singlet state [18, 16]. The upturn
in high $T/\mu_{0}H$ values (See Fig. 2d) is due to the phonon contribution.
The observed scaling behavior of the specific heat suggests the realization of
a randomness-driven liquid-like state in SCTO, which is attributed to
competing interactions mediated by intrinsic but unavoidable anti-site
disorder in the spin-lattice. It is worth mentioning that similar data
collapse was also found in a disordered triangular lattice Y2CuTiO6 [43] with
a scaling coefficient $\gamma=0.7$. The specific heat in zero field shows a
broad maximum around 17 K reflecting the onset of short-range spin correlation
between Cu2+ spins at low temperatures in this square lattice antiferromagnet.

Figure 4: (a) Representative ZF-$\mu$SR spectra at different temperatures. The
solid lines denote the fittings to the data as described in the main text. (b)
Temperature dependence of the muon spin relaxation rate
$\lambda_{\mathrm{ZF}}(T)$ in a semi-log scale with a pow-law fit $\sim
T^{-0.29}$. (c) Stretched exponent as a function of temperature
$\beta_{\mathrm{ZF}}(T)$. The dashed horizontal line indicates the average
value of $\beta_{\mathrm{ZF}}(T)$.(d) LF-$\mu$SR spectra at $T=2$ K in various
longitudinal fields. The solid lines represent the fits to the data. (e) LF
dependence of the muon spin relaxation rate
$\lambda_{\mathrm{LF}}(H_{\mathrm{LF}})$ in a semilog scale. The solid and
dashed curves denote the fittings to $\lambda_{\mathrm{LF}}(H_{\mathrm{LF}})$
as described in the main text. (f) Stretched exponent as a function of LF
$\beta_{\mathrm{LF}}(H_{\mathrm{LF}})$.

### Electron spin resonance

Electron spin resonance (ESR) is a sensitive microscopic probe for magnetism
and provides a direct access to the spin dynamics of the Cu2+ spins in SCTO.
The ESR spectra, representing the first derivative of the absorbed microwave
power, are shown in Fig. 3a for a few representative temperatures. The lines
could be well reproduced using a symmetric Lorentzian shape, averaged for
uniaxial $g$-factor anisotropy (solid red lines). The linewidth $\Delta{B}$
extracted from these spectra is associated with the spin relaxation rate,
including spin-spin and spin-lattice interactions. On the other hand, the
resonance field $B_{\mathrm{res}}$ gives information regarding the $g$-factor
and internal fields. Hence, the effective ESR $g$-factor, given by
$g_{\mathrm{ESR}}=h\nu/\mu_{\mathrm{B}}B_{\mathrm{res}}$, is a measure of both
the spins’ $g$-factor and the magnetic molecular-field. For $T=105$ K, we
obtained $g_{\parallel}=2.103$ and $g_{\perp}=2.248$, corresponding to an
averaged value
$g_{\mathrm{avg}}=\sqrt{(g_{\parallel}^{2}+2g_{\perp}^{2})/3}=2.201$. As shown
in Fig. 3b, both $\Delta{B}$ and $g_{\mathrm{ESR}}$ exhibit a change in their
temperature dependence at $T\simeq$ 100 K. The increase of $\Delta{B}$ towards
low temperatures indicates the enhancement of Cu2+ spin correlations, whereas
spin-lattice relaxation processes dominate the broadening towards high
temperatures. Both perpendicular and parallel component of $\Delta B$ follow
power-law behavior at low temperatures with a critical exponent 0.37 (i.e.,
$\Delta B\sim T^{-0.37}$), indicating a random-singlet ground state [70]. The
ESR intensity $I_{\mathrm{ESR}}$, which can be measured through the integrated
ESR absorption spectra, originates from the static spin-probe susceptibility.
Hence, it can be used as an intrinsic microscopic probe of the sample
magnetization. As can be seen in the bottom panel of Fig. 3b, the temperature
dependence of $I_{\mathrm{ESR}}^{-1}$ follows a Curie-Weiss law at high
temperatures with the Weiss temperature of $-25$ K, which is consistent with
that obtained from magnetic susceptibility data.

### Muon spin relaxation

The magnetization and specific heat results follow a universal scaling
behavior which evidences a randomness-driven quantum disordered state in the
frustrated square lattice SCTO with quenched disorder. This invokes more
sensitive local-probe magnetic investigations that could unambiguously confirm
the spin-liquid-like ground state and enable testing a dynamic scaling
behavior. In order to determine the magnetic ground state and spin dynamics,
we performed $\mu$SR experiments on SCTO. As displayed in Fig. 4a, the
ZF-$\mu$SR spectra show no indication of the presence of static magnetism down
to 2 K, such as a coherent muon oscillation signal and 1/3 recovery of the
asymmetry at long times. Rather, the asymmetry exhibits an exponential decay
in the measurement temperature range. These are consistent with the absence of
long-range magnetic ordering down to 2 K, evidenced by the magnetic and
thermodynamic properties. For quantitative analysis, we fit the obtained ZF-
and LF-$\mu$SR spectra with the stretched exponential relaxation function,
$P_{z}(t)=P_{z}(0)\exp{[-(\lambda_{\mathrm{ZF\textfractionsolidus
LF}}t)^{\beta_{\mathrm{ZF\textfractionsolidus LF}}}]}$. Here,
$\lambda_{\mathrm{ZF\textfractionsolidus LF}}$ is the muon spin relaxation
rate and $\beta_{\mathrm{ZF\textfractionsolidus LF}}$ is the stretching
exponent.

In Fig. 4b and Fig. 4c, we plot the extracted parameters for the ZF-$\mu$SR
results. Above 10 K, the muon spin relaxation rate has a nearly constant value
close to $\sim$ 0.10 $\mu$s-1, indicating rapidly fluctuating moments. In the
paramagnetic state, the spin fluctuation $\nu$ can be evaluated by
$\nu=\sqrt{z}JS/\hbar$, where $z=4$ is the nearest-neighbor coordination
number and $J$ is the nearest-neighbor exchange interaction. By using
$k_{\mathrm{B}}\theta_{\mathrm{CW}}=2zS(S+1)J/3$, we obtain the exchange
fluctuation rate of $\nu=1.77\times 10^{12}$ s-1. Combined with the relation
$\lambda_{ZF}=(2\Delta^{2})\textfractionsolidus\nu$, $\lambda_{ZF}$($T>10$ K)
$\sim$ 0.10 $\mu$s-1 provides the field distribution width of the local
magnetic field $\Delta=297$ MHz, less than $\nu=1.77\times 10^{12}$ s-1. With
decreasing temperature below 10 K, $\lambda_{\mathrm{ZF}}$ increases gradually
down to 2 K. This suggests the slowing down of the Cu2+ moment fluctuations by
the development of spin correlations below 10 K. Note that the stretched
exponent exhibits a nearly temperature independent behavior for all
temperatures and has the average value $\beta_{\mathrm{ZF\textfractionsolidus
LF}}^{\mathrm{Avg}}=1.03$, close to a simple exponential decay. The power-law
behavior of $\lambda_{\mathrm{ZF}}$ ($\sim T^{-0.29}$) at low temperatures
indicates a quantum-critical scaling which is consistent with a random-singlet
state (see the red line in Fig. 4b). Similar power-law behavior in zero-field
muon relaxation rate was found in the random-singlet material Sr2CuTe1-xWxO6
[71].

Figure 5: Time-field scaling of LF-$\mu$SR asymmetries in a semi-log scale at
$T=2$ K. The collapse of muon asymmetries onto a single curve demonstrates a
randomness-driven dynamic state in a square lattice antiferromagnet. The inset
shows schematic of a random-singlet state. Each spin-singlet is represented by
an oval spanning over two sites. Unpaired spin is denoted by an arrow.

We turn to the LF-$\mu$SR results of SCTO. Fig. 4d shows the representative LF
spectra at $T=2$ K in different fields. Though the long-time relaxation of the
$\mu$SR asymmetry begins to recover by applying a weak LF, the applied LF of 4
kG is insufficient to fully decouple the muon spin from internal local fields.
The residual relaxation suggests that the muon spin relaxation is mainly
dominated by dynamically fluctuating spins, which is consistent with the
absence of long-range magnetic order down to 2 K. In Fig. 4e and Fig. 4f, we
summarize the extracted parameters in a semi-log scale. It is well established
that, for the simple exponential correlation function,
$\lambda_{\mathrm{LF}}(H_{\mathrm{LF}})$ can be described by the Redfield
formula
$\lambda_{\mathrm{LF}}(H_{\mathrm{LF}})=2\Delta^{2}\nu\textfractionsolidus{(\nu^{2}+\gamma_{\mu}^{2}H_{\mathrm{LF}}^{2})}$.
However, $\lambda_{\mathrm{LF}}(H_{\mathrm{LF}})$ is not reproduced by the
Redfield formula (the dashed line in Fig. 4e). This signifies the presence of
an algebraic term in the spin correlation function at low temperatures,
$S(t)\sim(\tau\textfractionsolidus t)^{x}\exp{(-\nu t)}$, where $\tau$ and
$\nu$ the early and late cutoff times and $x$ is the critical exponent [72,
73, 74]. To describe the field evolution of $\lambda_{\mathrm{ZF}}$, we employ
a general expression that can be derived from both semiclassical and full
quantum approaches,

$\lambda_{\mathrm{ZF}}(H_{\mathrm{ZF}})=2\Delta^{2}\tau^{x}\int_{0}^{\infty}t^{-x}\exp{(-\nu
t)}\cos{(2\pi\mu_{0}\gamma_{\mu}H_{\mathrm{ZF}}t)}dt.$ (2)

As shown in Fig. 4e, $\lambda_{\mathrm{ZF}}(H_{\mathrm{ZF}})$ is well
reproduced by Eq. 2 with $x=0.4029$ and $\nu=2.5043\times 10^{6}$ Hz (the
solid lines in Fig. 4a). This indicates that the spin auto-correlation
function exhibits a power-law-like decaying at low temperatures, which is
slower than $S(t)\sim\exp{(-\nu t)}$ with $v=1.77\times 10^{12}$ Hz at high
temperatures. The slowing down of the spin fluctuations reflects the long-time
spin correlations at low $T$. Based on these observations, we infer that SCTO
has the dynamically fluctuating ground state realized in the square lattice at
least down to 2 K. We note that the LF dependence of the stretching exponent
is somewhat different from the expectation for quantum spin liquid candidates.
It is expected that $\beta_{\mathrm{LF}}$ increases with increasing field
because of the quenched inhomogeneous local fields by an LF in quantum spin
liquids. However, $\beta_{\mathrm{LF}}(H_{\mathrm{LF}})$ for SCTO manifest an
opposite behavior to the typical behavior. This may be due to exchange
randomness by the site disorder between Cu2+ and Ta5+ ions. The exchange
randomness induces an inhomogeneous local field distribution, so that a weak
magnetic field can lead to the partial modulation of internal local fields.
Therefore, the spatially inhomogeneous local field distributions could be the
origin of the LF dependence of $\beta_{\mathrm{LF}}$.

Figure 5 displays the LF asymmetry as a function of $t/(\mu_{0}H)^{\gamma}$.
As expected for the randomness-induced disordered state, we observe the time-
field scaling behavior of the LF-$\mu$SR spectra in the plot against
$t/(\mu_{0}H)^{\gamma}$ with $\gamma=0.4$. The collapse of the LF-$\mu$SR
asymmetries into a single curve implies the power-law contribution to the spin
correlation function $S(t)\sim(\tau\textfractionsolidus t)^{x}\exp{(-\nu t)}$
[75, 76]. The scaling exponent $\gamma=0.4$ is comparable to that determined
from the isothermal magnetization, which suggests that all the experimental
results are consistent taken on complementary scales and point towards the
realization of a random-singlet state in this material. Our $\mu$SR results
provide strong signature of a randomness-driven dynamic liquid-like ground
state in this novel square lattice antiferromagnet.

## IV. Conclusion

We have successfully synthesized polycrystalline samples of Sr3CuTa2O9 and
characterized their magnetic properties down to 64 mK. The Cu2+ ($S=1/2$)
moments constitute a disordered square lattice with sizable antiferromagnetic
interaction between constituent spins, and the system does not undergo a phase
transition down to 64 mK. Despite substantial site dilution between Cu2+ and
Ta5+ ions, Sr3CuTa2O9 shows a dynamic spin-liquid-like state owing to exchange
randomness driven by quenched disorder in this square lattice antiferromagnet.
This diluted network of spin $1/2$ ions forms a random-singlet phase which is
confirmed by data collapse of temperature-dependent and magnetic-field-
dependent specific heat data. The power-law behavior of magnetic
susceptibility data at lower temperatures also supports the formation of a
random-singlet state. The power-law spin auto-correlation function and data
collapse of time-dependent and longitudinal-field-dependent muon asymmetry
further validates the realization of randomness-induced spin-liquid-like state
in this frustrated material. This offers a promising route to explore
disorder-driven correlated quantum states in novel quantum materials.

## Acknowledgements

PK acknowledges the funding by the Science and Engineering Research Board, and
Department of Science and Technology, India through Research Grants. The work
at SKKU is supported by the National Research Foundation (NRF) of Korea (Grant
No. 2020R1A5A1016518). We acknowledge support of the HLD at HZDR, member of
the European Magnetic Field Laboratory (EMFL).

## References

  * Balents [2010] L. Balents,  _Spin liquids in frustrated magnets_ , Nature 464, 199 (2010).
  * Savary and Balents [2016] L. Savary and L. Balents, Reports on Progress in Physics 80, 016502 (2016).
  * Zhou  _et al._ [2017] Y. Zhou, K. Kanoda, and T.-K. Ng,  _Quantum spin liquid states_ , Rev. Mod. Phys. 89, 025003 (2017).
  * Nayak  _et al._ [2008] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma,  _Non-Abelian anyons and topological quantum computation_ , Rev. Mod. Phys. 80, 1083 (2008).
  * Ramirez  _et al._ [1999] A. P. Ramirez, A. Hayashi, R. J. Cava, R. Siddharthan, and B. S. Shastry,  _Zero-point entropy in ‘spin ice’_ , Nature 399, 333 (1999).
  * Anderson [1973] P. Anderson,  _Resonating valence bonds: A new kind of insulator?_ , Materials Research Bulletin 8, 153 (1973).
  * Khuntia  _et al._ [2020] P. Khuntia, M. Velazquez, Q. Barthélemy, F. Bert, E. Kermarrec, A. Legros, B. Bernu, L. Messio, A. Zorko, and P. Mendels,  _Gapless ground state in the archetypal quantum kagome antiferromagnet $\mathrm{ZnCu_{3}(OH)_{6}Cl_{2}}$_, Nature Physics 16, 469 (2020).
  * Khuntia  _et al._ [2016] P. Khuntia, F. Bert, P. Mendels, B. Koteswararao, A. V. Mahajan, M. Baenitz, F. C. Chou, C. Baines, A. Amato, and Y. Furukawa,  _Spin Liquid State in the 3D Frustrated Antiferromagnet ${\mathrm{PbCuTe}}_{2}{\mathrm{O}}_{6}$: NMR and Muon Spin Relaxation Studies_, Phys. Rev. Lett. 116, 107203 (2016).
  * Khuntia [2019] P. Khuntia,  _Novel magnetism and spin dynamics of strongly correlated electron systems: Microscopic insights_ , Journal of Magnetism and Magnetic Materials 489, 165435 (2019).
  * Gao  _et al._ [2019] B. Gao, T. Chen, D. W. Tam, C.-L. Huang, K. Sasmal, D. T. Adroja, F. Ye, H. Cao, G. Sala, M. B. Stone, C. Baines, J. A. T. Verezhak, H. Hu, J.-H. Chung, X. Xu, S.-W. Cheong, M. Nallaiyan, S. Spagna, M. B. Maple, A. H. Nevidomskyy, E. Morosan, G. Chen, and P. Dai,  _Experimental signatures of a three-dimensional quantum spin liquid in effective spin- $1/2$ _ Ce2Zr2O7 _pyrochlore_ , Nature Physics 15, 1052 (2019).
  * Arh  _et al._ [2022] T. Arh, B. Sana, M. Pregelj, P. Khuntia, Z. Jagličić, M. D. Le, P. K. Biswas, P. Manuel, L. Mangin-Thro, A. Ozarowski, and A. Zorko,  _The Ising triangular-lattice antiferromagnet neodymium heptatantalate as a quantum spin liquid candidate_ , Nature Materials 21, 416 (2022).
  * Baek  _et al._ [2020] S.-H. Baek, H. W. Yeo, S.-H. Do, K.-Y. Choi, L. Janssen, M. Vojta, and B. Büchner,  _Observation of a random singlet state in a diluted Kitaev honeycomb material_ , Phys. Rev. B 102, 094407 (2020).
  * Liu  _et al._ [2018] L. Liu, H. Shao, Y.-C. Lin, W. Guo, and A. W. Sandvik,  _Random-Singlet Phase in Disordered Two-Dimensional Quantum Magnets_ , Phys. Rev. X 8, 041040 (2018).
  * Kitagawa  _et al._ [2018] K. Kitagawa, T. Takayama, Y. Matsumoto, A. Kato, R. Takano, Y. Kishimoto, S. Bette, R. Dinnebier, G. Jackeli, and H. Takagi,  _A spin–orbital-entangled quantum liquid on a honeycomb lattice_ , Nature 554, 341 (2018).
  * Do  _et al._ [2020] S.-H. Do, C. H. Lee, T. Kihara, Y. S. Choi, S. Yoon, K. Kim, H. Cheong, W.-T. Chen, F. Chou, H. Nojiri, and K.-Y. Choi,  _Randomly Hopping Majorana Fermions in the Diluted Kitaev System $\alpha$-${\mathrm{Ru}}_{0.8}{\mathrm{Ir}}_{0.2}{\mathrm{Cl}}_{3}$_, Phys. Rev. Lett. 124, 047204 (2020).
  * Kimchi  _et al._ [2018a] I. Kimchi, J. P. Sheckelton, T. M. McQueen, and P. A. Lee,  _Scaling and data collapse from local moments in frustrated disordered quantum spin systems_ , Nature Communications 9, 4367 (2018a).
  * Khatua  _et al._ [2022] J. Khatua, M. Gomilšek, J. C. Orain, A. M. Strydom, Z. Jagličić, C. V. Colin, S. Petit, A. Ozarowski, L. Mangin-Thro, K. Sethupathi, M. S. R. Rao, A. Zorko, and P. Khuntia,  _Signature of a randomness-driven spin-liquid state in a frustrated magnet_ , Communications Physics 5, 99 (2022).
  * Kimchi  _et al._ [2018b] I. Kimchi, A. Nahum, and T. Senthil,  _Valence Bonds in Random Quantum Magnets: Theory and Application to ${\mathrm{YbMgGaO}}_{4}$_, Phys. Rev. X 8, 031028 (2018b).
  * Ma  _et al._ [1979] S.-k. Ma, C. Dasgupta, and C.-k. Hu,  _Random Antiferromagnetic Chain_ , Phys. Rev. Lett. 43, 1434 (1979).
  * Fisher [1994] D. S. Fisher,  _Random antiferromagnetic quantum spin chains_ , Phys. Rev. B 50, 3799 (1994).
  * Dasgupta and Ma [1980] C. Dasgupta and S.-k. Ma,  _Low-temperature properties of the random Heisenberg antiferromagnetic chain_ , Phys. Rev. B 22, 1305 (1980).
  * Bhatt and Lee [1981] R. N. Bhatt and P. A. Lee,  _A scaling method for low temperature behavior of random antiferromagnetic systems (invited)_ , Journal of Applied Physics 52, 1703 (1981).
  * Bhatt and Lee [1982] R. N. Bhatt and P. A. Lee,  _Scaling Studies of Highly Disordered Spin- $1/2$ Antiferromagnetic Systems_, Phys. Rev. Lett. 48, 344 (1982).
  * Uematsu and Kawamura [2018] K. Uematsu and H. Kawamura,  _Randomness-induced quantum spin liquid behavior in the $s=\frac{1}{2}$ random ${J}_{1}\text{$-$}{J}_{2}$ Heisenberg antiferromagnet on the square lattice_, Phys. Rev. B 98, 134427 (2018).
  * Shirane  _et al._ [1987] G. Shirane, Y. Endoh, R. J. Birgeneau, M. A. Kastner, Y. Hidaka, M. Oda, M. Suzuki, and T. Murakami,  _Two-dimensional antiferromagnetic quantum spin-fluid state in ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$_, Phys. Rev. Lett. 59, 1613 (1987).
  * Reger and Young [1988] J. D. Reger and A. P. Young,  _Monte Carlo simulations of the spin- $1/2$ Heisenberg antiferromagnet on a square lattice_, Phys. Rev. B 37, 5978 (1988).
  * Birgeneau  _et al._ [1999] R. J. Birgeneau, M. Greven, M. A. Kastner, Y. S. Lee, B. O. Wells, Y. Endoh, K. Yamada, and G. Shirane,  _Instantaneous spin correlations in ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}$_, Phys. Rev. B 59, 13788 (1999).
  * Coldea  _et al._ [2001] R. Coldea, S. M. Hayden, G. Aeppli, T. G. Perring, C. D. Frost, T. E. Mason, S.-W. Cheong, and Z. Fisk,  _Spin Waves and Electronic Interactions in ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}$_, Phys. Rev. Lett. 86, 5377 (2001).
  * Lacroix  _et al._ [2011] C. Lacroix, P. Mendels, and F. Mila, eds.,  _Introduction to Frustrated Magnetism: Materials, Experiments, Theory (Springer Series in Solid-State Sciences)_, 2011th ed. (Springer, 2011).
  * Laflorencie  _et al._ [2006] N. Laflorencie, S. Wessel, A. Läuchli, and H. Rieger,  _Random-exchange quantum Heisenberg antiferromagnets on a square lattice_ , Phys. Rev. B 73, 060403 (2006).
  * Liu  _et al._ [2020] L. Liu, W. Guo, and A. W. Sandvik,  _Quantum-critical scaling properties of the two-dimensional random-singlet state_ , Phys. Rev. B 102, 054443 (2020).
  * Uematsu and Kawamura [2017] K. Uematsu and H. Kawamura,  _Randomness-Induced Quantum Spin Liquid Behavior in the $S=1/2$ Random J1–J2 Heisenberg Antiferromagnet on the Honeycomb Lattice_, Journal of the Physical Society of Japan 86, 044704 (2017).
  * Opherden  _et al._ [2020] D. Opherden, N. Nizar, K. Richardson, J. C. Monroe, M. M. Turnbull, M. Polson, S. Vela, W. J. A. Blackmore, P. A. Goddard, J. Singleton, E. S. Choi, F. Xiao, R. C. Williams, T. Lancaster, F. L. Pratt, S. J. Blundell, Y. Skourski, M. Uhlarz, A. N. Ponomaryov, S. A. Zvyagin, J. Wosnitza, M. Baenitz, I. Heinmaa, R. Stern, H. Kühne, and C. P. Landee,  _Extremely well isolated two-dimensional spin- $\frac{1}{2}$ antiferromagnetic Heisenberg layers with a small exchange coupling in the molecular-based magnet CuPOF_, Phys. Rev. B 102, 064431 (2020).
  * Opherden  _et al._ [2023] D. Opherden, M. S. J. Tepaske, F. Bärtl, M. Weber, M. M. Turnbull, T. Lancaster, S. J. Blundell, M. Baenitz, J. Wosnitza, C. P. Landee, R. Moessner, D. J. Luitz, and H. Kühne,  _Field-Tunable Berezinskii-Kosterlitz-Thouless Correlations in a Heisenberg Magnet_ , Phys. Rev. Lett. 130, 086704 (2023).
  * Li  _et al._ [2015a] Y. Li, H. Liao, Z. Zhang, S. Li, F. Jin, L. Ling, L. Zhang, Y. Zou, L. Pi, Z. Yang, J. Wang, Z. Wu, and Q. Zhang,  _Gapless quantum spin liquid ground state in the two-dimensional spin- $1/2$ triangular antiferromagnet ${\mathrm{YbMgGaO_{4}}}$_, Scientific Reports 5, 16419 (2015a), article.
  * Li  _et al._ [2015b] Y. Li, G. Chen, W. Tong, L. Pi, J. Liu, Z. Yang, X. Wang, and Q. Zhang,  _Rare-Earth Triangular Lattice Spin Liquid: A Single-Crystal Study of ${\mathrm{YbMgGaO}}_{4}$_, Phys. Rev. Lett. 115, 167203 (2015b).
  * Shen  _et al._ [2016] Y. Shen, Y.-D. Li, H. Wo, Y. Li, S. Shen, B. Pan, Q. Wang, H. C. Walker, P. Steffens, M. Boehm, Y. Hao, D. L. Quintero-Castro, L. W. Harriger, M. D. Frontzek, L. Hao, S. Meng, Q. Zhang, G. Chen, and J. Zhao,  _Evidence for a spinon Fermi surface in a triangular-lattice quantum-spin-liquid candidate_ , Nature 540, 559 (2016).
  * Paddison  _et al._ [2017] J. A. M. Paddison, M. Daum, Z. Dun, G. Ehlers, Y. Liu, M. Stone, H. Zhou, and M. Mourigal,  _Continuous excitations of the triangular-lattice quantum spin liquid ${\mathrm{YbMgGaO_{4}}}$_, Nature Physics 13, 117 (2017).
  * Zhu  _et al._ [2017] Z. Zhu, P. A. Maksimov, S. R. White, and A. L. Chernyshev,  _Disorder-Induced Mimicry of a Spin Liquid in ${\mathrm{YbMgGaO}}_{4}$_, Phys. Rev. Lett. 119, 157201 (2017).
  * Savary and Balents [2017] L. Savary and L. Balents,  _Disorder-Induced Quantum Spin Liquid in Spin Ice Pyrochlores_ , Phys. Rev. Lett. 118, 087203 (2017).
  * Wen  _et al._ [2017] J.-J. Wen, S. M. Koohpayeh, K. A. Ross, B. A. Trump, T. M. McQueen, K. Kimura, S. Nakatsuji, Y. Qiu, D. M. Pajerowski, J. R. D. Copley, and C. L. Broholm,  _Disordered Route to the Coulomb Quantum Spin Liquid: Random Transverse Fields on Spin Ice in ${\mathrm{Pr}}_{2}{\mathrm{Zr}}_{2}{\mathrm{O}}_{7}$_, Phys. Rev. Lett. 118, 107206 (2017).
  * Smerald and Mila [2015] A. Smerald and F. Mila,  _Disorder-Driven Spin-Orbital Liquid Behavior in the ${\mathrm{Ba}}_{3}X{\mathrm{Sb}}_{2}{\mathrm{O}}_{9}$ Materials_, Phys. Rev. Lett. 115, 147202 (2015).
  * Kundu  _et al._ [2020a] S. Kundu, A. Hossain, P. K. S., R. Das, M. Baenitz, P. J. Baker, J.-C. Orain, D. C. Joshi, R. Mathieu, P. Mahadevan, S. Pujari, S. Bhattacharjee, A. V. Mahajan, and D. D. Sarma,  _Signatures of a Spin- $\frac{1}{2}$ Cooperative Paramagnet in the Diluted Triangular Lattice of ${\mathrm{Y}}_{2}{\mathrm{CuTiO}}_{6}$_, Phys. Rev. Lett. 125, 117206 (2020a).
  * Nakatsuji  _et al._ [2012] S. Nakatsuji, K. Kuga, K. Kimura, R. Satake, N. Katayama, E. Nishibori, H. Sawa, R. Ishii, M. Hagiwara, F. Bridges, T. U. Ito, W. Higemoto, Y. Karaki, M. Halim, A. A. Nugroho, J. A. Rodriguez-Rivera, M. A. Green, and C. Broholm,  _Spin-Orbital Short-Range Order on a Honeycomb-Based Lattice_ , Science 336, 559 (2012).
  * Kundu  _et al._ [2020b] S. Kundu, A. Shahee, A. Chakraborty, K. M. Ranjith, B. Koo, J. Sichelschmidt, M. T. F. Telling, P. K. Biswas, M. Baenitz, I. Dasgupta, S. Pujari, and A. V. Mahajan,  _Gapless Quantum Spin Liquid in the Triangular System ${\mathrm{Sr}}_{3}{\mathrm{CuSb}}_{2}{\mathrm{O}}_{9}$_, Phys. Rev. Lett. 125, 267202 (2020b).
  * Mustonen  _et al._ [2018] O. Mustonen, S. Vasala, E. Sadrollahi, K. P. Schmidt, C. Baines, H. C. Walker, I. Terasaki, F. J. Litterst, E. Baggio-Saitovitch, and M. Karppinen,  _Spin-liquid-like state in a spin- $1/2$ square-lattice antiferromagnet perovskite induced by d10-d0 cation mixing_, Nature Communications 9, 1085 (2018).
  * Quilliam  _et al._ [2016] J. A. Quilliam, F. Bert, A. Manseau, C. Darie, C. Guillot-Deudon, C. Payen, C. Baines, A. Amato, and P. Mendels,  _Gapless quantum spin liquid ground state in the spin-1 antiferromagnet $6{\mathrm{HB}}-{\mathrm{Ba}}_{3}{\mathrm{NiSb}}_{2}{\mathrm{O}}_{9}$_, Phys. Rev. B 93, 214432 (2016).
  * Yamaguchi  _et al._ [2017] H. Yamaguchi, M. Okada, Y. Kono, S. Kittaka, T. Sakakibara, T. Okabe, Y. Iwasaki, and Y. Hosokoshi,  _Randomness-induced quantum spin liquid on honeycomb lattice_ , Scientific Reports 7, 16144 (2017).
  * Shiroka  _et al._ [2019] T. Shiroka, F. Eggenschwiler, H.-R. Ott, and J. Mesot,  _From order to randomness: Onset and evolution of the random-singlet state in bond-disordered ${\mathrm{BaCu}}_{2}{({\mathrm{Si}}_{1-x}{\mathrm{Ge}}_{x})}_{2}{\mathrm{O}}_{7}$ spin-chain compounds_, Phys. Rev. B 99, 035116 (2019).
  * Iglói  _et al._ [2000] F. Iglói, R. Juhász, and H. Rieger,  _Random antiferromagnetic quantum spin chains: Exact results from scaling of rare regions_ , Phys. Rev. B 61, 11552 (2000).
  * Laflorencie and Rieger [2003] N. Laflorencie and H. Rieger,  _Comment on “Disorder Induced Quantum Phase Transition in Random-Exchange Spin- $1/2$ Chains”_, Phys. Rev. Lett. 91, 229701 (2003).
  * Shiroka  _et al._ [2013] T. Shiroka, F. Casola, W. Lorenz, K. Prša, A. Zheludev, H.-R. Ott, and J. Mesot,  _Impact of strong disorder on the static magnetic properties of the spin-chain compound $\mathrm{BaCu_{2}SiGeO_{7}}$_, Phys. Rev. B 88, 054422 (2013).
  * Nath  _et al._ [2013] R. Nath, A. A. Tsirlin, P. Khuntia, O. Janson, T. Förster, M. Padmanabhan, J. Li, Y. Skourski, M. Baenitz, H. Rosner, and I. Rousochatzakis,  _Magnetization and spin dynamics of the spin $S=\frac{1}{2}$ hourglass nanomagnet $\mathrm{Cu_{5}(OH)_{2}(NIPA)_{4}}.10H_{2}O$ _, Phys. Rev. B 87, 214417 (2013).
  * Suter and Wojek [2012] A. Suter and B. Wojek,  _Musrfit: A Free Platform-Independent Framework for $\mu$SR Data Analysis_, Physics Procedia 30, 69 (2012), 12th International Conference on Muon Spin Rotation, Relaxation and Resonance (µSR2011).
  * Toby [2001] B. H. Toby,  _EXPGUI, a graphical user interface for GSAS_ , Journal of Applied Crystallography 34, 210 (2001).
  * Zachariasz  _et al._ [2015] P. Zachariasz, J. Kulawik, and P. Guzdek,  _Preparation and characterization of the microstructure, dielectric and magnetoelectric properties of multiferroic $\mathrm{Sr_{3}CuNb_{2}O_{9}-CoFe_{2}O_{4}}$ ceramics_, Materials & Design 86, 627 (2015).
  * Pissas  _et al._ [1992] M. Pissas, V. Psycharis, C. Mitros, G. Kallias, D. Niarchos, A. Simopoulos, and A. Kostikas,  _Mössbauer and X-ray powder diffraction study of the compound $\mathrm{LuBaCuFeO_{5+x}}$_, Journal of Magnetism and Magnetic Materials 104-107, 571 (1992).
  * Belik  _et al._ [2006] A. A. Belik, S. Iikubo, K. Kodama, N. Igawa, S.-i. Shamoto, S. Niitaka, M. Azuma, Y. Shimakawa, M. Takano, F. Izumi, and E. Takayama-Muromachi,  _Neutron Powder Diffraction Study on the Crystal and Magnetic Structures of $\mathrm{BiCoO_{3}}$_, Chemistry of Materials 18, 798 (2006).
  * Sologub  _et al._ [2002] O. Sologub, J. Hester, P. Salamakha, E. Leroy, and C. Godart,  _Ab initio structure determination of new boride $\mathrm{CePt_{3}B}$, a distorted variant of $\mathrm{CaTiO}_{3}$_, Journal of Alloys and Compounds 337, 10 (2002).
  * Sologub  _et al._ [2003] O. Sologub, K. Hiebl, P. Salamakha, and H. Ipser,  _Crystal structure and physical properties of ternary compounds $\mathrm{RPt_{3}B,R=La,Pr,Nd}$_, Journal of Alloys and Compounds 360, 54 (2003).
  * Goodenough [1958] J. B. Goodenough,  _An interpretation of the magnetic properties of the perovskite-type mixed crystals $\mathrm{La_{1-x}Sr_{x}CoO_{3-\lambda}}$_, Journal of Physics and Chemistry of Solids 6, 287 (1958).
  * Choi  _et al._ [2019] Y. S. Choi, C. H. Lee, S. Lee, S. Yoon, W.-J. Lee, J. Park, A. Ali, Y. Singh, J.-C. Orain, G. Kim, J.-S. Rhyee, W.-T. Chen, F. Chou, and K.-Y. Choi,  _Exotic Low-Energy Excitations Emergent in the Random Kitaev Magnet ${\mathrm{Cu}}_{2}{\mathrm{IrO}}_{3}$_, Phys. Rev. Lett. 122, 167202 (2019).
  * Lee  _et al._ [2023] C. Lee, S. Lee, Y. Choi, C. Wang, H. Luetkens, T. Shiroka, Z. Jang, Y.-G. Yoon, and K.-Y. Choi,  _Coexistence of random singlets and disordered Kitaev spin liquid in ${\mathrm{H}}_{3}{\mathrm{LiIr}}_{2}{\mathrm{O}}_{6}$_, Phys. Rev. B 107, 014424 (2023).
  * Volkov  _et al._ [2020] P. A. Volkov, C.-J. Won, D. I. Gorbunov, J. Kim, M. Ye, H.-S. Kim, J. H. Pixley, S.-W. Cheong, and G. Blumberg,  _Random singlet state in ${\mathrm{Ba}}_{5}{\mathrm{CuIr}}_{3}{\mathrm{O}}_{12}$ single crystals_, Phys. Rev. B 101, 020406 (2020).
  * Kittel  _et al._ [1976] C. Kittel _et al._ ,  _Introduction to solid state physics_ , Vol. 8 (Wiley New York, 1976).
  * Koteswararao  _et al._ [2014] B. Koteswararao, R. Kumar, P. Khuntia, S. Bhowal, S. K. Panda, M. R. Rahman, A. V. Mahajan, I. Dasgupta, M. Baenitz, K. H. Kim, and F. C. Chou,  _Magnetic properties and heat capacity of the three-dimensional frustrated $S=\frac{1}{2}$ antiferromagnet ${\mathrm{PbCuTe}}_{2}{\mathrm{O}}_{6}$_, Phys. Rev. B 90, 035141 (2014).
  * Zhou  _et al._ [2011] H. D. Zhou, E. S. Choi, G. Li, L. Balicas, C. R. Wiebe, Y. Qiu, J. R. D. Copley, and J. S. Gardner,  _Spin Liquid State in the $S=1/2$ Triangular Lattice ${\mathrm{Ba}}_{3}{\mathrm{CuSb}}_{2}{\mathrm{O}}_{9}$_, Phys. Rev. Lett. 106, 147204 (2011).
  * Yamashita  _et al._ [2017] S. Yamashita, Y. Nakazawa, A. Ueda, and H. Mori,  _Thermodynamics of the quantum spin liquid state of the single-component dimer Mott system $\kappa\text{$-$}{\mathrm{H}}_{3}{(\mathrm{Cat}\text{$-$}\mathrm{EDT}\text{$-$}\mathrm{TTF})}_{2}$_, Phys. Rev. B 95, 184425 (2017).
  * Song  _et al._ [2021] P. Song, K. Zhu, F. Yang, Y. Wei, L. Zhang, H. Yang, X.-L. Sheng, Y. Qi, J. Ni, S. Li, Y. Li, G. Cao, Z. Y. Meng, W. Li, Y. Shi, and S. Li,  _Evidence for the random singlet phase in the honeycomb iridate ${\mathrm{SrIr}}_{2}{\mathrm{O}}_{6}$_, Phys. Rev. B 103, L241114 (2021).
  * Do  _et al._ [2014] S.-H. Do, J. van Tol, H. D. Zhou, and K.-Y. Choi,  _Dynamical spin-orbital correlations versus random singlets in ${\mathrm{Ba}}_{3}{\mathrm{CuSb}}_{2}{\mathrm{O}}_{9}$ investigated by magnetization and electron spin resonance_, Phys. Rev. B 90, 104426 (2014).
  * Hong  _et al._ [2021] W. Hong, L. Liu, C. Liu, X. Ma, A. Koda, X. Li, J. Song, W. Yang, J. Yang, P. Cheng, H. Zhang, W. Bao, X. Ma, D. Chen, K. Sun, W. Guo, H. Luo, A. W. Sandvik, and S. Li,  _Extreme Suppression of Antiferromagnetic Order and Critical Scaling in a Two-Dimensional Random Quantum Magnet_ , Phys. Rev. Lett. 126, 037201 (2021).
  * Keren  _et al._ [2004] A. Keren, J. S. Gardner, G. Ehlers, A. Fukaya, E. Segal, and Y. J. Uemura,  _Dynamic Properties of a Diluted Pyrochlore Cooperative Paramagnet $(\mathrm{Tb}_{p}\mathrm{Y}_{1-p})_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{\mathrm{7}}$_, Phys. Rev. Lett. 92, 107204 (2004).
  * Keren  _et al._ [2001] A. Keren, G. Bazalitsky, I. Campbell, and J. S. Lord,  _Probing exotic spin correlations by muon spin depolarization measurements with applications to spin glass dynamics_ , Phys. Rev. B 64, 054403 (2001).
  * Kermarrec  _et al._ [2011] E. Kermarrec, P. Mendels, F. Bert, R. H. Colman, A. S. Wills, P. Strobel, P. Bonville, A. Hillier, and A. Amato,  _Spin-liquid ground state in the frustrated kagome antiferromagnet ${\mathrm{MgCu_{3}(OH)_{6}Cl_{2}}}$_, Phys. Rev. B 84, 100401 (2011).
  * Keren  _et al._ [1996] A. Keren, P. Mendels, I. A. Campbell, and J. Lord,  _Probing the Spin-Spin Dynamical Autocorrelation Function in a Spin Glass above ${T}_{g}$ via Muon Spin Relaxation_, Phys. Rev. Lett. 77, 1386 (1996).
  * Keren [2004] A. Keren,  _Muons as probes of dynamical spin fluctuations: some new aspects_ , Journal of Physics: Condensed Matter 16, S4603 (2004).


# Dynamics of rotation in chiral nuclei

Z. X. Ren gbsn (任政学)  State Key Laboratory of Nuclear Physics and Technology,
School of Physics, Peking University, Beijing 100871, China  P. W. Zhao
gbsn(赵鹏巍<EMAIL_ADDRESS>State Key Laboratory of Nuclear Physics and
Technology, School of Physics, Peking University, Beijing 100871, China  J.
Meng gbsn(孟杰<EMAIL_ADDRESS>State Key Laboratory of Nuclear Physics and
Technology, School of Physics, Peking University, Beijing 100871, China

###### Abstract

The dynamics of chiral nuclei is investigated for the first time with the
time-dependent and tilted axis cranking covariant density functional theories
on a three-dimensional space lattice in a microscopic and self-consistent way.
The experimental energies of the two pairs of the chiral doublet bands in
135Nd are well reproduced without any adjustable parameters beyond the well-
defined density functional. A novel mechanism, i.e., chiral precession, is
revealed from the microscopic dynamics of the total angular momentum in the
body-fixed frame, whose harmonicity is associated with a transition from the
planar into aplanar rotations with the increasing spin. This provides a fully
microscopic and dynamical view to understand the chiral excitations in nuclei.

Chirality is a well-known phenomenon in many fields, such as chemistry,
biology, molecular and particle physics. In nuclear physics, chirality was
originally suggested by Frauendorf and Meng in 1997 Frauendorf and Meng
(1997). It represents a novel feature of triaxial nuclei rotating around an
axis which lies outside the three planes spanned by the principal axes of the
triaxial ellipsoidal density distribution, i.e., _aplanar rotation_. The total
angular momentum consists of three components along the short, intermediate,
and long principal axes of a triaxial nucleus in the body-fixed frame, and
they form either a left- or right-handed system. The two chiral systems are
related by the chiral operator $TR(\pi)$ that combines time reversal $T$ and
spatial rotation by $\pi$. The broken chiral symmetry in the body-fixed frame
should be restored in the laboratory frame. This leads to the so-called chiral
doublet bands, which consist of a pair of nearly-degenerate $\Delta I=1$
rotational bands Frauendorf and Meng (1997); Frauendorf (2001). The evidences
of chiral doublet bands have been reported experimentally in the $A\approx 80$
Wang  _et al._ (2011); Liu  _et al._ (2016), 100 Vaman  _et al._ (2004); Joshi
_et al._ (2004); Tonev  _et al._ (2014); Lieder  _et al._ (2014); Rather  _et
al._ (2014); Kuti  _et al._ (2014), 130 Starosta  _et al._ (2001); Zhu  _et
al._ (2003); Tonev  _et al._ (2006); Petrache  _et al._ (2006); Grodner  _et
al._ (2006); Tonev  _et al._ (2007); Mukhopadhyay  _et al._ (2007); Ayangeakaa
_et al._ (2013); Petrache  _et al._ (2018); Lv  _et al._ (2019), and 190
Balabanski  _et al._ (2004); Lawrie  _et al._ (2008) mass regions; see also
data tables Xiong and Wang (2019).

The manifestation of chirality in realistic nuclei is usually more complicated
Meng and Zhang (2010); Meng and Zhao (2016). In most cases, the chiral doublet
bands are more separated in energy at low spins than at high spins. At low
spins, the energy difference between the states with the same spin in the
chiral partners is caused by rapid vibration between the left- and right-
handed configurations, i.e., _chiral vibration_. When the spin grows above a
certain critical value, the energy splittings between the partner bands
decrease, and the left-right mode changes from soft chiral vibration to
tunneling between well-established chiral configurations, i.e., _static
chirality_.

On the theoretical side, many phenomenological approaches have been employed
to understand such an intriguing phenomenon, such as the particle rotor model
Frauendorf and Meng (1997); Koike  _et al._ (2004); Peng  _et al._ (2003);
Zhang  _et al._ (2007); Qi  _et al._ (2009); Chen  _et al._ (2018a), the
generalized coherent state model Raduta  _et al._ (2016), and the interacting
boson fermion-fermion model Tonev  _et al._ (2006, 2007); Brant  _et al._
(2008). However, they are based on several assumptions, the validity of which
may not be assured, and are fitted to the data in one way or another.

A microscopic treatment is thus important for a better understanding of
nuclear chirality. The tilted axis cranking (TAC) approach allows one to study
nuclear chirality based on a microscopic mean field. Although it well
describes the energies and the intraband transition rates of the lower one of
the chiral doublet bands, it gives either one achiral solution or two
degenerate chiral ones Dimitrov  _et al._ (2000), so it cannot describe the
left-right excitation mode. This is a well-known deficiency of the mean-field
solution when it spontaneously breaks a symmetry Frauendorf (2001).
Consequently, the random-phase approximation method Mukhopadhyay  _et al._
(2007); Almehed  _et al._ (2011), the collective Hamiltonian method Chen  _et
al._ (2013, 2016), and the projected shell model Bhat  _et al._ (2012); Chen
_et al._ (2017, 2018b); Wang  _et al._ (2019) have been developed to describe
the chiral excitations. However, these approaches are based on single-particle
potentials combined with either the schematic single-$j$ Hamiltonian Chen  _et
al._ (2016) or the pairing plus quadrupole-quadrupole model Frauendorf (2000).
Self-consistent methods based on more realistic two-body interactions are
required for a more fundamental investigation, including all important effects
such as core polarization and nuclear currents Zhao  _et al._ (2011); Meng
_et al._ (2013).

Moreover, all the previous studies on nuclear chirality are based on static
approaches, so the dynamics of the chiral excitations is still missing.

The nuclear density functional theory (DFT) starts from a universal energy
density functional and can achieve a self-consistent description for almost
all nuclei. Covariant density functional theory (CDFT) further exploits the
Lorentz symmetry, so it includes the complicated interplay between the large
Lorentz scalar and vector self-energies Serot and Walecka (1986); Ren and Zhao
(2020). It also allows the self-consistent treatment of the spin degrees of
freedom and the nuclear currents induced by the spatial parts of the vector
self-energies, which play an essential role in rotating nuclei; see Refs. Meng
_et al._ (2013); Meng (2016) for details. Both relativistic and
nonrelativistic DFTs have been extended with the TAC method to investigate
nuclear chirality Olbratowski  _et al._ (2004); Zhao (2017). In particular,
the recent TAC-CDFT has been widely used to investigate the chirality in
nuclei ${}^{106}\mathrm{Rh}$ Zhao (2017), ${}^{135}\mathrm{Nd}$ Peng and Chen
(2020), ${}^{136}\mathrm{Nd}$ Petrache  _et al._ (2018), and
${}^{106}\mathrm{Ag}$ Zhao  _et al._ (2019). However, in all these works, only
the lower band of the chiral doublet bands can be calculated.

The time-dependent DFT is a dynamical extension of DFT, and has been widely
applied to various nuclear structure and reaction processes, see recent
reviews Nakatsukasa  _et al._ (2016); Simenel and Umar (2018); Stevenson and
Barton (2019) for details. The solution of the time-dependent covariant
density functional theory (TDCDFT) in three-dimensional lattice space is
feasible only recently Ren  _et al._ (2020a, b) thanks to the progress for
solving CDFT in the same space Tanimura  _et al._ (2015); Ren  _et al._ (2017,
2019, 2020c).

In this Letter, for the first time, the dynamics of chiral nuclei is studied
with the time-dependent and tilted axis cranking CDFT in a microscopic and
self-consistent way. By taking 135Nd as an example, the experimental energies
of the two pairs of the chiral doublet bands are well reproduced without any
adjustable parameters beyond the well-defined density functional. In
particular, a novel mechanism, _chiral precession_ (see Fig. 1), is clearly
revealed from the microscopic dynamics of the nuclear chiral excitations.

Figure 1: (Color online). A schematic picture for the “chiral precession” of a
triaxial nucleus in the body-fixed frame, where the short, intermediate, and
long axes are shown explicitly. The total angular momentum $\bm{J}$ of the
nucleus is rotating about an axis (dotted line) in the body-fixed frame.

The starting point of CDFT is a universal energy density functional Ring
(1996); Vretenar  _et al._ (2005); Meng  _et al._ (2006a); Nikšić  _et al._
(2011); Meng (2016). For nuclear chirality, the functional is transformed into
a body-fixed frame rotating with a constant angular velocity vector
$\bm{\omega}$, which is along an arbitrary direction in space, i.e., the
three-dimensional TAC-CDFT Zhao (2017). The corresponding Kohn-Sham equation
for nucleons is a static Dirac equation,

$\left[\bm{\alpha}\cdot(\hat{\bm{p}}-\bm{V})+V^{0}+\beta(m+S)-\bm{\omega}\cdot\hat{\bm{J}}\right]\psi_{k}(\bm{r})=\varepsilon_{k}\psi_{k}(\bm{r}),$
(1)

where $\hat{\bm{J}}$ is the angular momentum, and the relativistic scalar
$S(\bm{r})$ and vector $V^{\mu}(r)$ fields are connected in a self-consistent
way to the nucleon densities and current distributions, which are obtained
from the Dirac spinors $\psi_{k}(\bm{r})$.

In this work, the density functional PC-PK1 Zhao  _et al._ (2010) is employed,
and the calculations are free of additional parameters. The Dirac equation (1)
is solved in a cubic box with the length $24$ fm and the mesh size $0.8$ fm.
We focus on the chirality in the odd-$A$ nucleus ${}^{135}\mathrm{Nd}$, which
was first observed in Ref. Zhu  _et al._ (2003) with the configuration
consisting of two proton particles and one neutron hole in the $h_{11/2}$
shell, i.e., $\pi h_{11/2}^{2}\otimes\nu h_{11/2}^{-1}$. Recently, a new pair
of chiral doublet bands were reported with the configuration
$\pi[h_{11/2}^{1}(gd)^{1}]\otimes\nu h_{11/2}^{-1}$ Lv  _et al._ (2019), which
makes it a new candidate for multiple chirality Meng  _et al._ (2006b). In the
following, we take the positive-parity configuration as an example to
illustrate the analyses.

Figure 2: (Color online). Total energy surfaces for 135Nd with respect to the
tilted angles $(\theta_{J},\varphi_{J})$ for the total angular momenta at
$I=19/2\hbar$ (top), $25/2\hbar$ (middle), and $31/2\hbar$ (bottom) by the
TAC-CDFT calculations based on the configuration
$\pi[h_{11/2}^{1}(gd)^{1}]\otimes\nu h_{11/2}^{-1}$. The minima of the energy
surfaces are denoted as stars.

In Fig. 2, the total energy surfaces are depicted in the plane of the tilted
angles $\theta_{J}$ and $\varphi_{J}$ of the total angular momentum. Here,
$\theta_{J}$ is the polar angle between the total angular momentum and the
long axis, and $\varphi_{J}$ is the azimuth angle between the projection of
the total angular momentum in the short-intermediate plane and the short axis.
At $I=19/2\hbar$, the total energy surface has a minimum at
$\theta_{J}=48.5^{\circ}$ and $\varphi_{J}=0^{\circ}$. This means that the
total angular momentum is in the short-long plane, i.e., planar rotation. At
$I=25/2\hbar$, the location of the energy minimum varies only slightly in the
$\theta$ direction, but the energy surface becomes much softer in the
$\varphi$ direction. This indicates that it would cost less to vibrate the
angular momentum direction away from $\varphi_{J}=0^{\circ}$, i.e., the so-
called chiral vibration, in comparison with the case at $I=19/2\hbar$. At
$I=31/2\hbar$, two degenerate minima locate in the energy surface at the same
$\theta_{J}$ while opposite $\varphi_{J}$ values, which correspond to the
left- and right-handed states, respectively. Note that the TAC-CDFT gives
either one planar solution or two degenerate aplanar ones, but it cannot
describe the left-right excitation mode. Therefore, the TDCDFT calculations
are further carried out.

In TDCDFT, the time-dependent Kohn-Sham equation reads Ren  _et al._ (2020a,
b),

$i\partial_{t}\psi_{k}(\bm{r},t)=[\bm{\alpha\cdot}(\hat{\bm{p}}-\bm{V})+V^{0}+\beta(m+S)]\psi_{k}(\bm{r},t),$
(2)

where the time-dependent fields $S(\bm{r},t)$ and $V^{\mu}(\bm{r},t)$ have the
same dependence on density and currents as in TAC-CDFT, so there are no
additional parameters introduced. In this work, we solve Eq. (2) in the same
lattice space as in the TAC-CDFT calculations, and the initial states are
taken from the self-consistent solutions of the TAC-CDFT calculations.

Starting from the TAC-CDFT solutions, one can obtain the dynamical evolution
of the system by means of the time-dependent wave functions
$\psi_{k}(\bm{r},t)$. Note that the solutions of TDCDFT are in the space-fixed
frame, where the angular momentum is conserved during the time evolution, but
the nuclear density distribution varies. Since the nuclear deformation barely
changes during the time evolution, it is convenient to analyze the dynamical
behavior of the nucleus in the body-fixed rotating frame, which can be easily
obtained by calculating the principal axes of inertia, i.e., the short,
intermediate, and long axes of the nucleus. At the initial time, the axes of
the space-fixed frame are set to be along with the three principal axes of
inertia.

If one performs the TDCDFT calculations starting from the lowest energy state
for a given spin, e.g., see Fig. 2(a), the nucleus will rotate around a tilted
axis in the short-long plane. Therefore, in the body-fixed frame, one could
see that the angular momentum vector will be always pointing along the tilted
axis. It, however, becomes quite different if the initial state is away from
the lowest energy state.

Figure 3: (Color online). Trajectories of the tilted angles
$(\theta_{J},\varphi_{J})$ for the total angular momenta in the body-fixed
frame by the TDCDFT calculations starting from the initial states with spin
$I=19/2\hbar$ (top), $25/2\hbar$ (middle), and $31/2\hbar$ (bottom) obtained
from TAC-CDFT calculations with the configuration
$\pi[h_{11/2}^{1}(gd)^{1}]\otimes\nu h_{11/2}^{-1}$. The red solid and blue
dash lines correspond to the trajectories starting with different initial
states, whose $(\theta_{J},\varphi_{J})$ values are represented by the open
circles. The stars denote the location of the minima in the energy surfaces.
Insets: Time evolutions for the azimuth angle $\varphi_{J}$.

This can be seen in Fig. 3, where the trajectories of the tilted angles
$\theta_{J}$ and $\varphi_{J}$ for the angular momenta in the body-fixed frame
are depicted for the initial states with $I=19/2\hbar$, $25/2\hbar$, and
$31/2\hbar$. For each spin, two states away from the lowest energy state are
employed as the initial states for the TDCDFT calculations. For $I=19/2\hbar$
and $25/2\hbar$, the trajectories are roughly ellipses centered at the
locations of the minima in the corresponding energy surfaces. This provides
clearly a precession picture, i.e., the angular momentum vector itself is
rotating around the tilted axis in the short-long plane (see Fig. 1).
Moreover, the precession is much more extended in the $\varphi_{J}$ direction
than in the $\theta_{J}$ one. In the $\varphi_{J}$ direction, it involves both
positive and negative values, which reflects the chiral nature of the
precession due to the transitions between the left- and right-handed sectors.
Therefore, we call it “chiral precession”, which reveals the microscopic
dynamics of the chiral excitations; see movies in the Supplemental Material
[SeeSupplementalMaterialforthemoviesofthechiralprecessioninboththespace-
fixedandbody-
fixedframesfromtheTDCDFTcalculationsandforthedetailsontheFourieranalysistoextractthechiralexcitationenergy]supplement1.
For a given spin $I=19/2\hbar$ or $25/2\hbar$, although the precession
amplitudes starting from the two initial states are quite different, the
corresponding precession periods are roughly identical, as shown in the insets
of Fig. 3. This demonstrates the harmonic nature of the chiral precession,
which is consistent with the chiral vibration picture.

Note that the precession mechanism here cannot be deduced only from the energy
surfaces shown in Fig. 2, which provide only the potential part of the chiral
motion. The kinetic part may also play an important role in generating the
precession. Further studies in this direction would be an interesting topic.

For $I=31/2\hbar$, since there are two degenerate energy minima separated by a
barrier at $\varphi_{J}=0^{\circ}$, the trajectories strongly depend on the
choice of the initial states. If the initial energy is lower than the barrier,
the trajectory is limited in the neighborhood of the minimum with a finite
$\varphi_{J}$ corresponding to either the left- or the right-handed sector.
However, if the initial energy is higher than the barrier, the trajectory
could extend across both positive and negative $\varphi_{J}$ values. As seen
in the inset of Fig. 3(c), the two precessions are disparate not only in their
amplitudes but also in the periods. This indicates the anharmonic effects of
the chiral motion, which are associated with quantum tunneling between left-
and right-handed sectors.

To understand this phenomenon further, it would be necessary to resort to the
so-called requantization of the TDCDFT Ring and Schuck (2004), e.g., the time-
dependent generator coordinate method. However, to our knowledge, it is hard,
if not impossible, to be implemented in a realistic microscopic description of
nuclei due to the inaccessible computational requirements. One of the ways out
is probably to derive a collective Hamiltonian with the Gaussian overlap
approximation (GOA) in the space of the tilted angles Chen  _et al._ (2013,
2016). However, there are still many open questions about the quality of the
GOA, the determination of the mass inertia, the influence of the time-reversal
symmetry breaking, etc. The solutions to all these problems are apparently
beyond the scope of this work.

Figure 4: (Color online). (a) Same as Fig. 2 but for the total Routhian
surface at a given rotational frequency $\hbar\omega=0.46$ MeV, where the spin
varies on the surface (dotted lines). (b) Same as Fig. 3 but for the initial
states with spin $I=29/2\hbar$, $31/2\hbar$, and $33/2\hbar$ obtained from the
Routhian surface.

It must be emphasized that in many TAC analyses for the nuclear chirality, the
calculations are performed for fixed rotational frequencies $\hbar\omega$ but
not fixed angular momenta. This seems to be a natural choice because the
angular momentum is anyhow not a good quantum number in the mean-field
framework. Therefore, as depicted in Fig. 4, we examine the chiral precession
trajectories starting from initial states with a given rotational frequency.
In Fig. 4(a), the total Routhian surface at $\hbar\omega=0.46$ MeV is shown,
and there is only one minimum on the surface. For the TDCDFT calculations,
here we take three states on the total Routhian surface as the initial states,
which have the same $\hbar\omega$ but different angular momenta. Since only
the average angular momentum is conserved, but not the rotational frequency,
during the time evolutions in TDCDFT, the obtained three trajectories are
quite different. Moreover, the trajectories for $I=31/2\hbar$ and
$I=33/2\hbar$ indicate that there should be two energy minima, and this is in
contradiction with the single minimum in the total Routhian surface. The
trajectory for $I=29/2\hbar$ is compatible with the single minimum in the
Routhian surface, but this is an accident, because the Routhian minimum at
$\hbar\omega=0.46$ MeV and the energy minimum at $I=29/2\hbar$ are at the same
location in the plane of $\theta_{J}$ and $\varphi_{J}$. This clearly
demonstrates the importance of the self-consistent constrained energy surface
at a given spin in the TAC-CDFT calculations.

Figure 5: (Color online). Calculated excitation energies (solid and dashed
lines) for chiral doublet bands in 135Nd built on the configurations
$\pi[h_{11/2}^{1}(gd)^{1}]\otimes\nu h_{11/2}^{-1}$ and $\pi
h_{11/2}^{2}\otimes\nu h_{11/2}^{-1}$ in comparison with data (solid and open
symbols) Lv  _et al._ (2019). The solid lines represent the TAC-CDFT results,
and the dashed ones represent the results including the chiral excitation
energies obtained with the TDCDFT calculations.

The energy of the chiral excitation for a given spin can be extracted through
the Fourier analysis Press  _et al._ (1992); III (2002); Reinhard  _et al._
(2006) on the time evolution of the chiral precession; see details in the
Supplemental Material
[SeeSupplementalMaterialforthemoviesofthechiralprecessioninboththespace-
fixedandbody-
fixedframesfromtheTDCDFTcalculationsandforthedetailsontheFourieranalysistoextractthechiralexcitationenergy]supplement1.
This, together with the TAC-CDFT solutions, could provide a microscopic
description for the chiral doublet bands. The Fourier analyses have been done
for the states with the spins in the harmonic regime. In Fig. 5, the
calculated energies for the two pairs of chiral doublet bands in 135Nd, which
are built respectively on the configurations
$\pi[h_{11/2}^{1}(gd)^{1}]\otimes\nu h_{11/2}^{-1}$ and $\pi
h_{11/2}^{2}\otimes\nu h_{11/2}^{-1}$, are depicted in comparison with the
data Lv  _et al._ (2019). It can be seen that the experimental energies are
well reproduced. This provides the first fully microscopic and self-consistent
description for the chiral doublet bands in the framework of DFTs.

In summary, for the first time, the dynamics of chiral nuclei has been
investigated in the microscopic time-dependent covariant density functional
theory with the initial states from the static three-dimensional tilted axis
cranking solutions. The experimental energies of the two pairs of the chiral
doublet bands in 135Nd are well reproduced without any adjustable parameters
beyond the well-defined density functional. Taking the positive-parity bands
as an example, a transition from the planar into aplanar rotations has been
found from the total energy surfaces with fixed spins. A novel mechanism,
chiral precession, has been revealed from the microscopic dynamics of the
total angular momentum in the body-fixed frame. The obvious planar-aplanar
transition corresponds to the increasingly anharmonic effects in the chiral
procession. It is found that, however, such a correspondence is covered up for
the total Routhian surfaces at fixed rotational frequencies, since the average
angular momentum, rather than the rotational frequency, is conserved during
the time evolutions.

###### Acknowledgements.

This work was partly supported by the National Key R&D Program of China
(Contracts No. 2018YFA0404400 and 2017YFE0116700), the National Natural
Science Foundation of China (Grants No. 12070131001, 11875075, 11935003,
11975031, and 12141501), the China Postdoctoral Science Foundation under Grant
No. 2020M670013, and the High-performance Computing Platform of Peking
University.

## References

  * Frauendorf and Meng (1997) S. Frauendorf and J. Meng, Nucl. Phys. A 617, 131 (1997).
  * Frauendorf (2001) S. Frauendorf, Rev. Mod. Phys. 73, 463 (2001).
  * Wang  _et al._ (2011) S. Y. Wang, B. Qi, L. Liu, S. Q. Zhang, H. Hua, X. Q. Li, Y. Y. Chen, L. H. Zhu, J. Meng, S. M. Wyngaardt, P. Papka, T. T. Ibrahim, R. A. Bark, P. Datta, E. A. Lawrie, J. J. Lawrie, S. N. T. Majola, P. L. Masiteng, S. M. Mullins, J. Gál, G. Kalinka, J. Molnár, B. M. Nyakó, J. Timár, K. Juhász, and R. Schwengner, Phys. Lett. B 703, 40 (2011).
  * Liu  _et al._ (2016) C. Liu, S. Y. Wang, R. A. Bark, S. Q. Zhang, J. Meng, B. Qi, P. Jones, S. M. Wyngaardt, J. Zhao, C. Xu, S.-G. Zhou, S. Wang, D. P. Sun, L. Liu, Z. Q. Li, N. B. Zhang, H. Jia, X. Q. Li, H. Hua, Q. B. Chen, Z. G. Xiao, H. J. Li, L. H. Zhu, T. D. Bucher, T. Dinoko, J. Easton, K. Juhász, A. Kamblawe, E. Khaleel, N. Khumalo, E. A. Lawrie, J. J. Lawrie, S. N. T. Majola, S. M. Mullins, S. Murray, J. Ndayishimye, D. Negi, S. P. Noncolela, S. S. Ntshangase, B. M. Nyakó, J. N. Orce, P. Papka, J. F. Sharpey-Schafer, O. Shirinda, P. Sithole, M. A. Stankiewicz, and M. Wiedeking, Phys. Rev. Lett. 116, 112501 (2016).
  * Vaman  _et al._ (2004) C. Vaman, D. B. Fossan, T. Koike, K. Starosta, I. Y. Lee, and A. O. Macchiavelli, Phys. Rev. Lett. 92, 032501 (2004).
  * Joshi  _et al._ (2004) P. Joshi, D. G. Jenkins, P. M. Raddon, A. J. Simons, R. Wadsworth, A. R. Wilkinson, D. B. Fossan, T. Koike, K. Starosta, C. Vaman, J. Timár, Z. Dombrádi, A. Krasznahorkay, J. Molnár, D. Sohler, L. Zolnai, A. Algora, E. S. Paul, G. Rainovski, A. Gizon, J. Gizon, P. Bednarczyk, D. Curien, G. Duchêne, and J. N. Scheurer, Phys. Lett. B 595, 135 (2004).
  * Tonev  _et al._ (2014) D. Tonev, M. S. Yavahchova, N. Goutev, G. de Angelis, P. Petkov, R. K. Bhowmik, R. P. Singh, S. Muralithar, N. Madhavan, R. Kumar, M. Kumar Raju, J. Kaur, G. Mohanto, A. Singh, N. Kaur, R. Garg, A. Shukla, T. K. Marinov, and S. Brant, Phys. Rev. Lett. 112, 052501 (2014).
  * Lieder  _et al._ (2014) E. O. Lieder, R. M. Lieder, R. A. Bark, Q. B. Chen, S. Q. Zhang, J. Meng, E. A. Lawrie, J. J. Lawrie, S. P. Bvumbi, N. Y. Kheswa, S. S. Ntshangase, T. E. Madiba, P. L. Masiteng, S. M. Mullins, S. Murray, P. Papka, D. G. Roux, O. Shirinda, Z. H. Zhang, P. W. Zhao, Z. P. Li, J. Peng, B. Qi, S. Y. Wang, Z. G. Xiao, and C. Xu, Phys. Rev. Lett. 112, 202502 (2014).
  * Rather  _et al._ (2014) N. Rather, P. Datta, S. Chattopadhyay, S. Rajbanshi, A. Goswami, G. H. Bhat, J. A. Sheikh, S. Roy, R. Palit, S. Pal, S. Saha, J. Sethi, S. Biswas, P. Singh, and H. C. Jain, Phys. Rev. Lett. 112, 202503 (2014).
  * Kuti  _et al._ (2014) I. Kuti, Q. B. Chen, J. Timár, D. Sohler, S. Q. Zhang, Z. H. Zhang, P. W. Zhao, J. Meng, K. Starosta, T. Koike, E. S. Paul, D. B. Fossan, and C. Vaman, Phys. Rev. Lett. 113, 032501 (2014).
  * Starosta  _et al._ (2001) K. Starosta, T. Koike, C. J. Chiara, D. B. Fossan, D. R. LaFosse, A. A. Hecht, C. W. Beausang, M. A. Caprio, J. R. Cooper, R. Krücken, J. R. Novak, N. V. Zamfir, K. E. Zyromski, D. J. Hartley, D. L. Balabanski, J.-y. Zhang, S. Frauendorf, and V. I. Dimitrov, Phys. Rev. Lett. 86, 971 (2001).
  * Zhu  _et al._ (2003) S. Zhu, U. Garg, B. K. Nayak, S. S. Ghugre, N. S. Pattabiraman, D. B. Fossan, T. Koike, K. Starosta, C. Vaman, R. V. F. Janssens, R. S. Chakrawarthy, M. Whitehead, A. O. Macchiavelli, and S. Frauendorf, Phys. Rev. Lett. 91, 132501 (2003).
  * Tonev  _et al._ (2006) D. Tonev, G. de Angelis, P. Petkov, A. Dewald, S. Brant, S. Frauendorf, D. L. Balabanski, P. Pejovic, D. Bazzacco, P. Bednarczyk, F. Camera, A. Fitzler, A. Gadea, S. Lenzi, S. Lunardi, N. Marginean, O. Möller, D. R. Napoli, A. Paleni, C. M. Petrache, G. Prete, K. O. Zell, Y. H. Zhang, J.-y. Zhang, Q. Zhong, and D. Curien, Phys. Rev. Lett. 96, 052501 (2006).
  * Petrache  _et al._ (2006) C. M. Petrache, G. B. Hagemann, I. Hamamoto, and K. Starosta, Phys. Rev. Lett. 96, 112502 (2006).
  * Grodner  _et al._ (2006) E. Grodner, J. Srebrny, A. A. Pasternak, I. Zalewska, T. Morek, C. Droste, J. Mierzejewski, M. Kowalczyk, J. Kownacki, M. Kisieliński, S. G. Rohoziński, T. Koike, K. Starosta, A. Kordyasz, P. J. Napiorkowski, M. Wolińska-Cichocka, E. Ruchowska, W. Płóciennik, and J. Perkowski, Phys. Rev. Lett. 97, 172501 (2006).
  * Tonev  _et al._ (2007) D. Tonev, G. de Angelis, S. Brant, S. Frauendorf, P. Petkov, A. Dewald, F. Dönau, D. L. Balabanski, Q. Zhong, P. Pejovic, D. Bazzacco, P. Bednarczyk, F. Camera, D. Curien, F. Della Vedova, A. Fitzler, A. Gadea, G. Lo Bianco, S. Lenzi, S. Lunardi, N. Marginean, O. Möller, D. R. Napoli, R. Orlandi, E. Sahin, A. Saltarelli, J. Valiente Dobon, K. O. Zell, J. Y. Zhang, and Y. H. Zhang, Phys. Rev. C 76, 044313 (2007).
  * Mukhopadhyay  _et al._ (2007) S. Mukhopadhyay, D. Almehed, U. Garg, S. Frauendorf, T. Li, P. V. Madhusudhana Rao, X. Wang, S. S. Ghughre, M. P. Carpenter, S. Gros, A. Hecht, R. V. F. Janssens, F. G. Kondev, T. Lauritsen, D. Seweryniak, and S. Zhu, Phys. Rev. Lett. 99, 172501 (2007).
  * Ayangeakaa  _et al._ (2013) A. D. Ayangeakaa, U. Garg, M. D. Anthony, S. Frauendorf, J. T. Matta, B. K. Nayak, D. Patel, Q. B. Chen, S. Q. Zhang, P. W. Zhao, B. Qi, J. Meng, R. V. F. Janssens, M. P. Carpenter, C. J. Chiara, F. G. Kondev, T. Lauritsen, D. Seweryniak, S. Zhu, S. S. Ghugre, and R. Palit, Phys. Rev. Lett. 110, 172504 (2013).
  * Petrache  _et al._ (2018) C. M. Petrache, B. F. Lv, A. Astier, E. Dupont, Y. K. Wang, S. Q. Zhang, P. W. Zhao, Z. X. Ren, J. Meng, P. T. Greenlees, H. Badran, D. M. Cox, T. Grahn, R. Julin, S. Juutinen, J. Konki, J. Pakarinen, P. Papadakis, J. Partanen, P. Rahkila, M. Sandzelius, J. Saren, C. Scholey, J. Sorri, S. Stolze, J. Uusitalo, B. Cederwall, O. Aktas, A. Ertoprak, H. Liu, S. Matta, P. Subramaniam, S. Guo, M. L. Liu, X. H. Zhou, K. L. Wang, I. Kuti, J. Timár, A. Tucholski, J. Srebrny, and C. Andreoiu, Phys. Rev. C 97, 041304(R) (2018).
  * Lv  _et al._ (2019) B. F. Lv, C. M. Petrache, Q. B. Chen, J. Meng, A. Astier, E. Dupont, P. Greenlees, H. Badran, T. Calverley, D. M. Cox, T. Grahn, J. Hilton, R. Julin, S. Juutinen, J. Konki, J. Pakarinen, P. Papadakis, J. Partanen, P. Rahkila, P. Ruotsalainen, M. Sandzelius, J. Saren, C. Scholey, J. Sorri, S. Stolze, J. Uusitalo, B. Cederwall, A. Ertoprak, H. Liu, S. Guo, M. L. Liu, J. G. Wang, X. H. Zhou, I. Kuti, J. Timár, A. Tucholski, J. Srebrny, and C. Andreoiu, Phys. Rev. C 100, 024314 (2019).
  * Balabanski  _et al._ (2004) D. L. Balabanski, M. Danchev, D. J. Hartley, L. L. Riedinger, O. Zeidan, J.-y. Zhang, C. J. Barton, C. W. Beausang, M. A. Caprio, R. F. Casten, J. R. Cooper, A. A. Hecht, R. Krücken, J. R. Novak, N. V. Zamfir, and K. E. Zyromski, Phys. Rev. C 70, 044305 (2004).
  * Lawrie  _et al._ (2008) E. A. Lawrie, P. A. Vymers, J. J. Lawrie, C. Vieu, R. A. Bark, R. Lindsay, G. K. Mabala, S. M. Maliage, P. L. Masiteng, S. M. Mullins, S. H. T. Murray, I. Ragnarsson, T. M. Ramashidzha, C. Schück, J. F. Sharpey-Schafer, and O. Shirinda, Phys. Rev. C 78, 021305(R) (2008).
  * Xiong and Wang (2019) B. Xiong and Y. Wang, At. Data Nucl. Data Tables 125, 193 (2019).
  * Meng and Zhang (2010) J. Meng and S. Q. Zhang, J. Phys. G 37, 064025 (2010).
  * Meng and Zhao (2016) J. Meng and P. Zhao, Phys. Scr. 91, 053008 (2016).
  * Koike  _et al._ (2004) T. Koike, K. Starosta, and I. Hamamoto, Phys. Rev. Lett. 93, 172502 (2004).
  * Peng  _et al._ (2003) J. Peng, J. Meng, and S. Q. Zhang, Phys. Rev. C 68, 044324 (2003).
  * Zhang  _et al._ (2007) S. Q. Zhang, B. Qi, S. Y. Wang, and J. Meng, Phys. Rev. C 75, 044307 (2007).
  * Qi  _et al._ (2009) B. Qi, S. Q. Zhang, J. Meng, S. Y. Wang, and S. Frauendorf, Phys. Lett. B 675, 175 (2009).
  * Chen  _et al._ (2018a) Q. B. Chen, B. F. Lv, C. M. Petrache, and J. Meng, Phys. Lett. B 782, 744 (2018a).
  * Raduta  _et al._ (2016) A. A. Raduta, A. H. Raduta, and C. M. Petrache, J. Phys. G 43, 095107 (2016).
  * Brant  _et al._ (2008) S. Brant, D. Tonev, G. de Angelis, and A. Ventura, Phys. Rev. C 78, 034301 (2008).
  * Dimitrov  _et al._ (2000) V. I. Dimitrov, S. Frauendorf, and F. Dönau, Phys. Rev. Lett. 84, 5732 (2000).
  * Almehed  _et al._ (2011) D. Almehed, F. Dönau, and S. Frauendorf, Phys. Rev. C 83, 054308 (2011).
  * Chen  _et al._ (2013) Q. B. Chen, S. Q. Zhang, P. W. Zhao, R. V. Jolos, and J. Meng, Phys. Rev. C 87, 024314 (2013).
  * Chen  _et al._ (2016) Q. B. Chen, S. Q. Zhang, P. W. Zhao, R. V. Jolos, and J. Meng, Phys. Rev. C 94, 044301 (2016).
  * Bhat  _et al._ (2012) G. H. Bhat, J. A. Sheikh, and R. Palit, Phys. Lett. B 707, 250 (2012).
  * Chen  _et al._ (2017) F. Q. Chen, Q. B. Chen, Y. A. Luo, J. Meng, and S. Q. Zhang, Phys. Rev. C 96, 051303(R) (2017).
  * Chen  _et al._ (2018b) F. Q. Chen, J. Meng, and S. Q. Zhang, Phys. Lett. B 785, 211 (2018b).
  * Wang  _et al._ (2019) Y. K. Wang, F. Q. Chen, P. W. Zhao, S. Q. Zhang, and J. Meng, Phys. Rev. C 99, 054303 (2019).
  * Frauendorf (2000) S. Frauendorf, Nucl. Phys. A 677, 115 (2000).
  * Zhao  _et al._ (2011) P. W. Zhao, J. Peng, H. Z. Liang, P. Ring, and J. Meng, Phys. Rev. Lett. 107, 122501 (2011).
  * Meng  _et al._ (2013) J. Meng, J. Peng, S.-Q. Zhang, and P.-W. Zhao, Front. Phys. 8, 55 (2013).
  * Serot and Walecka (1986) B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1 (1986).
  * Ren and Zhao (2020) Z. X. Ren and P. W. Zhao, Phys. Rev. C 102, 021301(R) (2020).
  * Meng (2016) J. Meng, ed.,  _Relativistic Density Functional for Nuclear Structure_ , International Review of Nuclear Physics, Vol. 10 (World Scientific, Singapore, 2016).
  * Olbratowski  _et al._ (2004) P. Olbratowski, J. Dobaczewski, J. Dudek, and W. Płóciennik, Phys. Rev. Lett. 93, 052501 (2004).
  * Zhao (2017) P. W. Zhao, Phys. Lett. B 773, 1 (2017).
  * Peng and Chen (2020) J. Peng and Q. B. Chen, Phys. Lett. B 810, 135795 (2020).
  * Zhao  _et al._ (2019) P. W. Zhao, Y. K. Wang, and Q. B. Chen, Phys. Rev. C 99, 054319 (2019).
  * Nakatsukasa  _et al._ (2016) T. Nakatsukasa, K. Matsuyanagi, M. Matsuo, and K. Yabana, Rev. Mod. Phys. 88, 045004 (2016).
  * Simenel and Umar (2018) C. Simenel and A. S. Umar, Prog. Part. Nucl. Phys. 103, 19 (2018).
  * Stevenson and Barton (2019) P. D. Stevenson and M. C. Barton, Prog. Part. Nucl. Phys. 104, 142 (2019).
  * Ren  _et al._ (2020a) Z. X. Ren, P. W. Zhao, and J. Meng, Phys. Lett. B 801, 135194 (2020a).
  * Ren  _et al._ (2020b) Z. X. Ren, P. W. Zhao, and J. Meng, Phys. Rev. C 102, 044603 (2020b).
  * Tanimura  _et al._ (2015) Y. Tanimura, K. Hagino, and H. Z. Liang, Prog. Theor. Exp. Phys. 2015, 073D01 (2015).
  * Ren  _et al._ (2017) Z. X. Ren, S. Q. Zhang, and J. Meng, Phys. Rev. C 95, 024313 (2017).
  * Ren  _et al._ (2019) Z. X. Ren, S. Q. Zhang, P. W. Zhao, N. Itagaki, J. A. Maruhn, and J. Meng, Sci. China-Phys. Mech. Astron. 62, 112062 (2019).
  * Ren  _et al._ (2020c) Z. X. Ren, P. W. Zhao, S. Q. Zhang, and J. Meng, Nucl. Phys. A 996, 121696 (2020c).
  * Ring (1996) P. Ring, Prog. Part. Nucl. Phys. 37, 193 (1996).
  * Vretenar  _et al._ (2005) D. Vretenar, A. V. Afanasjev, G. A. Lalazissis, and P. Ring, Phys. Rep. 409, 101 (2005).
  * Meng  _et al._ (2006a) J. Meng, H. Toki, S. G. Zhou, S. Q. Zhang, W. H. Long, and L. S. Geng, Prog. Part. Nucl. Phys. 57, 470 (2006a).
  * Nikšić  _et al._ (2011) T. Nikšić, D. Vretenar, and P. Ring, Prog. Part. Nucl. Phys. 66, 519 (2011).
  * Zhao  _et al._ (2010) P. W. Zhao, Z. P. Li, J. M. Yao, and J. Meng, Phys. Rev. C 82, 054319 (2010).
  * Meng  _et al._ (2006b) J. Meng, J. Peng, S. Q. Zhang, and S.-G. Zhou, Phys. Rev. C 73, 037303 (2006b).
  * sup (a) a.
  * Ring and Schuck (2004) P. Ring and P. Schuck,  _The nuclear many-body problem_ (Springer Science & Business Media, 2004).
  * Press  _et al._ (1992) W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,  _Numerical Recipes in C: The Art of Scientific Computing_ (Cambridge University Press, New York, 1992).
  * III (2002) J. O. Smith III,  _Mathematics of the Discrete Fourier Transform (DFT)_ (Stanford University, Stanford, CA, 2002).
  * Reinhard  _et al._ (2006) P.-G. Reinhard, P. D. Stevenson, D. Almehed, J. A. Maruhn, and M. R. Strayer, Phys. Rev. E 73, 036709 (2006).