Abstract
Observation of a quantum spin liquid (QSL) state is one of the most important goals in condensedmatter physics, as well as the development of new spintronic devices that support nextgeneration industries. The QSL in two dimensional quantum spin systems is expected to be due to geometrical magnetic frustration, and thus a kagomebased lattice is the most probable playground for QSL. Here, we report the first experimental results of the QSL state on a squarekagome quantum antiferromagnet, KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl. Comprehensive experimental studies via magnetic susceptibility, magnetisation, heat capacity, muon spin relaxation (μSR), and inelastic neutron scattering (INS) measurements reveal the formation of a gapless QSL at very low temperatures close to the ground state. The QSL behavior cannot be explained fully by a frustrated Heisenberg model with nearestneighbor exchange interactions, providing a theoretical challenge to unveil the nature of the QSL state.
Introduction
Magnetic phases of lowdimensional magnets have been studied both theoretically and experimentally in the last half century. Intensive studies of onedimensional (1D) spin systems have successfully captured the exotic quantum states, such as the Tomonaga–Luttinger spinliquid state^{1} and the Haldane state^{2}. Recent progress in synthesising ideal 1D magnets has evolved this research field^{3}. On the other hand, in 2D spin systems, the spin1/2 kagome antiferromagnet is an excellent choice for searching for the QSL state induced by geometrical frustration^{4}. A possible compound for QSL in the kagome antiferromagnets was herbertsmithite, which has the Cu^{2+} layers with ideal kagome geometry sandwiched by nonmagnetic Zn^{2+} layers^{5}. To date, no longrange order has been found at any temperature, and a continuum of spin excitations was observed by INS experiments. However, the lowenergy magnetic excitation is still unclear as seen in a controversy on gapless^{6} or gapped^{7} excitation. This is related to the fact that herbertsmithite is obtained by selectively replacing magnetic Cu^{2+} ions with nonmagnetic Zn^{2+} ions on the triangularlattice planes of its parent compound clinoatacamite^{8}, Cu_{2}(OH)_{3}Cl. This replacement inevitably causes site mixing^{9}. Other materials with the kagome lattice exhibit longrange magnetic or valencebond crystal (VBC) orders caused by lattice distortions, the DM interaction and further neighbour interactions^{10,11,12,13,14}. The lack of a suitable model material exhibiting the QSL hinders observations of the QSL state in the 2D spin1/2 systems.
Another highly frustrated 2D quantum spin system expected to be a QSL state is a compound with the squarekagome lattice (SKL). The SKL was introduced by Siddharthan et al.^{15}. It has the same coordination number as the kagome lattice (z = 4), but with a composition of two inequivalent sublattices in contrast to the kagome lattice. Richter et al. reported that the ground state of the spin1/2 SKL with three equivalent exchange interactions (the case of J_{1} = J_{2} = J_{3} and J_{X} = 0 in Fig. 1c) is similar to that of the kagome lattice^{16}. The ground state of the spin1/2 J_{1}–J_{2} SKL antiferromagnet (the case of J_{2} = J_{3} and J_{X} = 0 in Fig. 1c) was calculated by Rousochatzakis et al.^{17}. It has been predicted to be a crosseddimer VBC state and a square pinwheels VBC state, depending on J_{2}/J_{1}. Moreover, there is a possibility that the QSL ground states are realised in the SKL with three nonequivalent exchange interactions (the case of J_{X} = 0 in Fig. 1c), which lead to the melting of these VBC states^{18}. Very recently, it has also predicted to be a topological nematic spinliquid state^{19}. In the magnetic field, the existence of the magnetisation plateaus of M/M_{sat} = 1/3 and 2/3 has theoretically clarified^{16,17,18,20}, where M_{sat} is the saturation magnetisation. These plateau phases exhibit VBC, up–up–down structure, and alternate trimerized states. In the high magnetic field and lowtemperature regime, a magneticfielddriven Berezinskii–Kosterlitz–Thouless phase transition exists^{21}. However, the lack of a model compound for the SKL system has obstructed a deeper understanding of its spin state. Motivated by the present status on the study of the SKL system, we searched for compounds with the SKL containing Cu^{2+} spins, and synthesised the first compound of a SKL antiferromagnet, KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl, successfully. Here, we use thermodynamic, muon spin relaxation and neutronscattering experiments on powder samples of KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl, to demonstrate the absence of magnetic ordering and the presence of gapless continuum of spin excitations.
Results
Crystal structure
The synthesis of KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl was conceived following the identification of the naturally occurring mineral atlasovite, KCu_{6}FeBiO_{4}(SO_{4})_{5}Cl^{22}. The space group and structural parameters of KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl are determined as P 4/ncc, (the same space group as atlasovite) and a = 9.8248(9) Å, c = 20.5715(24) Å, respectively (see Supplementary Note 1). As shown Fig. 1a and b, the SKL in the crystal structure of KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl comprises the sixcoordinated Cu^{2+} ions. In each SK unit, the square is enclosed by four scalene triangles. From this crystal structure, it is recognised that KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl has three types of first neighbour interactions, J_{1}, J_{2} and J_{3}, as shown in Fig. 1c. The orbital arrangements can be reasonably deduced from the oxygen and chloride positions around the Cu^{2+} ions. Judging from the \({d}_{{x}^{2}{y}^{2}}\) orbitals arranged on the SKL, the nearestneighbour (NN) magnetic couplings J_{i} (i = 1–3) are superexchange interactions occurring through Cu–O–Cu bonds: J_{1} through the Cu1–O–Cu1 bond with a bond of angle 112.62°, and J_{2} and J_{3} through Cu1–O–Cu2 with bond angles of 120.12° and 108.61°, respectively. Since the Cu–O–Cu angle significantly influences on the value of the exchange interactions, the variation of the angles can give strong bonddependent exchange interactions^{23}. Therefore, J_{2} with the largest angle is expected to be the largest antiferromagnetic interaction, while J_{3} with the smallest angle is considered to be the smallest antiferromagnetic interaction among the three interactions. One prominent and important feature of the present structure is the occupancy of nonmagnetic atoms in the interlayer space of the unit cell (Fig. 1b), which elongate the interlayer spacing. Furthermore, the Cu^{2+} ions and nonmagnetic ions have different valence numbers in KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl, avoiding site mixing, unlike the Cu^{2+} and Zn^{2+} site mixing observed in herbertsmithite (for more details, see Supplementary Notes 1 and 2). Therefore, the crystal perfectness and high twodimensionality of KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl are ideal for studying the intrinsic magnetism on frustrated 2D magnets. However, the obtained INS experimental results are inconsistent with the calculated results for the J_{1}J_{2}J_{3} SKL model (discussed below).
Magnetic and thermodynamic properties
Figure 2a presents the temperature dependence of the magnetic susceptibility χ(T) and the inverse magnetic susceptibility 1/χ(T) of KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl in the temperature range 1.8–300 K. With decreasing temperature, the magnetic susceptibility gradually increases. This feature suggests the absence of any longrange order down to 1.8 K. From 1/χ(T) with the Curie–Weiss law C/(T–θ_{CW}), between 200 K and 300 K, we estimated the Curie constant and Weiss temperature to be C = 2.86(1) and θ_{CW} = −237(2) K, respectively. The C corresponds to an effective moment of 1.96 μ_{B}, consistent with the spin S = 1/2 of Cu^{2+}. The large negative Weiss temperature and the absence of longrange orders suggest an antiferromagnetic frustrated system.
The magnetisation curve measured at 1.8 K, as shown in the inset of Fig. 2b, has two components: an intrinsic component M_{bulk} and free spin component M_{free}. Following the analysis for herbertsmithite^{24}, a saturated magnetisation of M_{free} can be estimated by subtraction of the linear M_{bulk} from the measured total magnetisation M_{obs.}. The M_{free} can be fitted a Brillouin function for a spin1/2, suggesting the component is attributed to the paramagnetic impurity or the unpaired spins on surface of powder particles. The saturated value of M_{free} indicates the presence of free spins with about 2.4% in total Cu^{2+} ions in our sample. The M_{bulk} at high magnetic field is only ~0.15 μ_{B}/Cu^{2+} at 60 T, indicating that strong antiferromagnetic exchange interaction dominates in this system (see Fig. 2b).
A Schottkylike peak in the heat capacity is observed at around T^{*} ≈ 2 K, as shown in Fig. 2c. As the released magnetic entropy at 15 K is only 16% of the expected total entropy, which is similar to that of herbertsmithite. In herbertsmithite, this behaviour was attributed to weakly coupled spins residing on the interlayer sites^{9}. However, in KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl, it is difficult to assign the 16% entropy to the site mixing because of the valence of nonmagnetic ions different from Cu^{2+}. Rather the observed peak can be attributed easily to the development of shortrange spin correlations. Similar characteristics are observed in the calculated specific heat and entropy of the the spin1/2 kagome antiferromagnet^{25}. Small broad peak appear at around T ≈ J/100, and the released entropy at around this temperature is about 20%. As discussed below, the magnitude of the exchange interaction J_{av} ≡ (J_{1} + J_{2} + J_{3})/3 = 137 K for KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl, namely, J_{av}/100 ≈ T^{*}. However, careful consideration is necessary about what origin of this peak is. We therefore conclude that the longrange magnetic and VBCordering behaviours are not observed in magnetic susceptibility, magnetisation and specific heat.
Quantum spin fluctuations in KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl
To confirm the absence of spin ordering caused by quantum fluctuations, we performed μSR measurements. Figure 3a shows the weak longitudinalfield (LF) (=50 G) μSR spectra at various temperatures. The weak LF was applied to quench the depolarisation due to random local fields from nuclear magnetic moments. The spectra are well fitted by the exponential function
where a_{1} is an intrinsic initial asymmetry a_{1} = 0.133, a_{BG} is a constant background a_{BG} = 0.047 (see Supplementary Note 2), λ is the muon spin relaxation rate. Hartree potential calculation predicted a local potential minimum in the lattice (see Fig. 3d, e)^{26,27,28}. A muon site corresponding to a local potential minimum is located at the 16g site. Quantum fluctuations of the Cu^{2+} spins down to 58 mK without spin ordering/freezing are evidenced by the longtime μSR spectra. The weak LF signals at the lowest temperature (58 mK) decrease continuously without oscillations up to 15 μs, as shown in Fig. 3b. If this spectrum is due to static magnetism, the internal field (estimate as λ_{ZF}/γ_{μ}, where γ_{μ} is the muon gyromagnetic ratio) should be less than 20 G. (see Supplementary Note 3). However, relaxation is clearly observed in the LF spectrum even at 0.395 T, which is evidence for the fluctuation of Cu^{2+} electron spins without spin ordering/freezing (see Fig. 3c). As shown in Fig. 3f, the increase of λ at around T^{*} renders evidence for a slowing down of the spin fluctuation resulting from the development of shortrange correlations. In addition, they exhibit a plateau with weak temperature dependence at low temperature, which has been found in other QSL candidates^{29}. The LF spectra measured at 58 mK under several magnetic fields are also fitted by Eq. (1). Using the power law represented by 1/(a + bH^{α}) with α = 0.46, where a and b depend on the fluctuation rate and fluctuating field, we obtain a good fitting to the LF dependence of the muon spin relaxation rate λ, as shown in Fig. 3g. Incidentally, the 1/(a + bH^{2}) is a standard case that the λ obeys the Redfield equation. In ordinary disordered spin systems, the muon spin relaxation rate exhibits a field inverse square dependence. Such a spectralweight function is commonly used to describe classical fluctuations in the paramagnetic regime. The observed values, α = 0.46, are inconsistent with the existence of a single timescale and suggest a more exotic spectral density, such as the one at play in a QSL. All of these μSR results strongly support the formation of a QSL at very low temperature close to the ground state in KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl^{30,31}.
Gapless continuum of spin excitations
The quantum statistics of quasiparticle excitations depend on the type of QSL, in particular, the nature of their excitation. To grasp the whole picture of the spin excitation, first we performed the INS experiment in a wide energy range. As shown in Fig. 4a, streaklike excitation at Q = 0.8 Å^{−1} and flat signals at around E = 7 and 10 meV are observed at 5 K. The Edependence of the INS intensity can be fitted well by two or three Gaussian functions and linear baseline, and the corresponding integrated intensities are obtained (for more details, see Supplementary Note 4). As shown in Fig. 4b, the peak positions of excitations are estimated to be 10.1(1) meV, 9.4(3) meV, and 7.3(1) meV, respectively. The signal due to magnetic excitation is generally enhanced at lowQ values, whereas phonon excitation is dominant at highQ. As shown in Fig. 4c, the baseline increase with increasing with Q. Therefore, the baseline may well comes from a number of phonon excitations in a multielement material KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl. The peak at 9.4 meV also increases with increasing with Q, indicating that it comes from phonon excitation. On the other hand, the flat signals have a characteristic feature of magnetic excitation. In order to investigate whether the spin excitation is gapless or gapped, we performed the INS experiments in the lowenergy region. These signals are also observed at 0.3 K, as shown in Fig. 4d, there are the streaklike excitation and flat signals are also observed. As shown in Fig. 4e and g, the INS spectra exhibit the feature of a gapless continuum of spin excitations. Streaklike excitation at Q = 0.8 Å^{−1} is clearly visible down to the elastic line, and its intensity increases continuously without signature of energy gap at least within the instrumental resolution (FWHM = 0.05 meV for E_{i} = 1.69 meV). The excitation persist up to at least T = 30 K (see Supplementary Fig. 5), which is consistent with the exchange constants estimated later. The Qdependence of the INS intensity after integration over a finite energy interval is shown in Fig. 4h. There are three peaks at Q = 0.8, 1.25, and 1.58 Å^{−1} at 0.3 K, and the peaks are observed even at low temperatures close to the ground state (48 mK). As discussed below, this result is inconsistent with the calculated dynamical spin structure factor S(q, ω) in the J_{1}–J_{2}–J_{3} SKL antiferromagnet with parameters, which reproduce magnetic susceptibility and magnetisation process. These INS data are consistent with a gapless continuum of spinon excitations. From the above, the flat signals at approximately 10 and 7 meV probably indicate a van Hove singularity of spinon continuum edges at this energy.
Comparison with theory
To determine the magnetic parameters and to clarify the magnetic properties of KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl, we calculated the magnetic susceptibility, the magnetisation curve at zero temperature, and the magnetic excitation at zero temperature by mean of the exact diagonalization (ED), finite temperature Lanczos (FTL)^{32} and densitymatrix renormalization group (DMRG) method. We succeeded in reproducing the magnetic susceptibility and magnetisation curve of KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl with the J_{1}–J_{2}–J_{3} SKL model with J_{1} = 135 K, J_{2} = 162 K, J_{3} = 115 K and g = 2.11, as shown in Fig. 5a and b, where g is the gyromagnetic ratio. In the magnetisation process, the magnetisation plateaus of M/M_{sat} = 1/3 and 2/3 were confirmed at around 150 T and 270 T, respectively. This indicates the possibility to observe magnetisation plateaus experimentally if the measurement of the magnetisation process in a further strong magnetic field is performed. However, the result of inelastic neutron scattering that is the most important evidence of QSL cannot be reproduced in the J_{1}–J_{2}–J_{3} SKL with these parameters. In the inelastic neutronscattering experiment, in the lowenergy region, the strongest intensity become around Q = 0.8 Å^{−1} as shown Fig. 4f, g, while in the dynamical DMRG method, it is around Q = 1.3 Å^{−1} as shown in Fig. 5c. To eliminate this discrepancy, we also calculated the SKL model with nextnearestneighbour (NNN) interaction J_{X} in the diagonal direction of the Cu^{2+} square. We calculated this SKL model with various values of the parameters, but we could not reproduce the experimental results. Therefore, in order to understand the experiment correctly, we need to calculate the model with further interactions.
Discussion
We have synthesised a SKL spin1/2 antiferromagnet KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl without site disorder, thus providing a first candidate to investigate the SKL magnetism. The μSR measurement shows no longrange ordering down to 58 mK, roughly three orders of magnitude lower than the NN interactions. The INS spectrum exhibits a streaklike gapless excitation and flat dispersionless excitation, consistent with powderaveraged spinon excitations. Our experimental results strongly suggest the formation of a gapless QSL in KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl at very low temperature close to the ground state; however, they are inconsistent with the theoretical studies based on the J_{1}–J_{2}–J_{3} SKL Heisenberg model. In the J_{1}–J_{2}–J_{3} SKL Heisenberg model, the VBC and Néel order stats are expected with high probability. In fact, the VBC state is the ground state of the J_{1}–J_{2} SKL antiferromagnet regardless of the magnitude relation of J_{1}/J_{2}^{17}. Thus, to realise the QSL state in the SKL, we must impose an additional condition such as longerrange exchange interactions. Further theoretical study would reveal the conditions inducing the QSL state in SKL antiferromagnets.
Methods
Sample synthesis
Single phase polycrystalline KCu_{6}AlBiO_{4}(SO_{4})_{5}Cl was synthesised by the solidstate reaction in which highpurity KAl(SO_{4})_{2}, CuCl_{2}, CuSO_{4}, CuO and Bi_{2}O_{3} powders were mixed in a molar ratio of 2:1:6:5:1, followed by heating at 600 °C for 3 days and slow cooling in air.
Xray diffraction
Synchrotron powder XRD data were collected using an imaging plate diffractometer installed at the BL8B of the Photon Factory. The incident synchrotron Xray energy of 18.0 keV (0.68892 Å) was selected.
Magnetic susceptibility and lowfield magnetisation
Magnetic susceptibility and lowfield magnetisation measurements were performed using a commercial superconducting quantum interference device magnetometer (MPMSXL7AC: Quantum Design).
Highfield magnetisation
Highfield magnetisation measurements up to 60 T were conducted using an induction method in a pulsed magnetic field at the International MegaGauss Science Laboratory, The University of Tokyo.
Heat capacity
The specific heat was measured between 0.2 and 20 K using a PPMS (physical property measurement system; Quantum Design).
Muon spin relaxation (μSR)
The μSR experiments were performed using the spinpolarised pulsed surfacemuon (μ^{+}) beam at the D1 beamline of the Materials and Life Science Experimental Facility (MLF) of the Japan Proton Accelerator Research Complex (JPARC). The spectra were collected in the temperature range from 58 mK to 300 K using a dilution refrigerator and ^{4}He cryostat.
Inelastic neutron scattering (INS)
The highenergy INS experiment was performed on the HRC^{33}, installed at BL12 beamline at MLF of JPARC. At the HRC, white neutrons are monochromatised by a Fermi chopper synchronised with the production timing of the pulsed neutrons. The energy transfer was determined from the timeofflight of the scattered neutrons detected at position sensitive detectors. A 200Hz Fermi chopper was used to obtain a high neutron flux. A GMtype closed cycle cryostat was used to achieve 5 K. The energy of incident neutrons were E_{i} = 45.95 meV. The data collected by HRC were analysed using the software suite HANA^{34}. The lowenergy INS experiments were performed using the coldneutron timeofflight spectrometer PELICAN at the OPAL reactor at ANSTO^{35}. The instrument was aligned for an incident energy E_{i} = 2.1 meV. The sample was held in an oxygenfree copper can and cooled using a dilution insert installed in a toploading cryostat and data collected at 25 K, 15 K and 48 mK. The sample was corrected for background scattering from an empty can and normalised to the scattering from a vanadium standard. The PELICAN data corrections were performed using the freely available LAMP software. The INS spectra in a wide momentumenergy range were measured using the coldneutron disk chopper spectrometer AMATERAS installed in the MLF at JPARC^{36}. The sample was cooled to 0.3 K using a ^{3}He refrigerator. The scattering data were collected with a set of incident neutron energies, E_{i} = 1.69, 3.14 and 15.16 meV. The data collected by AMATERAS were analysed using the software suite UTSUSEMI^{37}.
Calculations
Magnetic susceptibility of the SKL is calculated by the full ED method for 18site and FTL method for 36site under the periodic boundary condition (PBC). The result of the FTL method is deduced by the statistical average of 40 sampling. The magnetisation curve at T = 0 K is calculated by the Lanczostype ED calculations for a 36site PBC cluster and the DMRG method for a 60site PBC cluster. The truncation number in the DMRG calculation is 6000 and resulting truncation errors are less than 2 × 10^{−5}. The dynamical spin structure factor S(q, ω) is calculated using the dynamical DMRG^{38} method for a 48site PBC cluster. The truncation number m = 6000 and the truncation error are less than 5 × 10^{−3}. S(q, ω) is defined as follows:
where q is the momentum, \(\left0\right\rangle\) is the ground state with energy E_{0}, η is a broadening factor and \({S}_{{\bf{q}}}^{z}={N}^{1/2}{\sum }_{i}{e}^{i{{\bf{q}}r}_{i}}{S}_{i}^{z}\) with r_{i} being the position of spin i and \({S}_{i}^{z}\) being the z component of S_{i}. The value of η is taken to be 1.16 meV.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The μSR and INS experiments were performed at the MLF of JPARC under a user programme (Proposal Nos. 2017B0019, 2017B0039, 2017B0049 and 2018B0068). Synchrotron powder XRD measurements were performed with the approval of the Photon Factory Program Advisory Committee (Proposal No. 2016G030). Theoretical study is in part supported by Creation of new functional devices and highperformance materials to support nextgeneration industries (GCDMSI) to be tackled by using postK computer and by MEXT HPCI Strategic Programs for Innovative Research (SPIRE) (hp160222, hp170274). This study is partly supported by the GrantinAid for Scientific Research (No. 17K14344) from MEXT, Japan.
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M.F. and K.M. conceived the study. H.S. and M.F. performed the synchrotron XRD experiment. T.K. performed the specific heat measurement. A.M. and K.K. performed the highfield magnetisation measurement. A.K., H.O., H.L. and M.F. performed the μSR experiments. K.M. and T.T. performed numerical calculations by ED and DMRG developed by S.S. R.M., D.Y., S.Y., S.I., T.H., T.M., S.O.K., K.N., M.F. and S.M. performed the neutronscattering experiments. All the authors contributed to the writing of the paper.
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Fujihala, M., Morita, K., Mole, R. et al. Gapless spin liquid in a squarekagome lattice antiferromagnet. Nat Commun 11, 3429 (2020). https://doi.org/10.1038/s4146702017235z
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Further reading

Probing resonating valence bond states in artificial quantum magnets
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Physical Review B (2021)
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