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[Masashi Hase](https://orcid.org/0000-0003-2717-461X), Koji Kaneko, Chihiro Tabata, Hiroki Yamauchi, [Naohito Tsujii](https://orcid.org/0000-0002-6181-5911), [Andreas Dönni](https://orcid.org/0000-0002-7300-9175)

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[Spin-singlet ground state with spin gap in <math>  <mrow>    <msub>      <mi>S</mi>      <mi>eff</mi>    </msub>    <mo>=</mo>    <mfrac>      <mn>1</mn>      <mn>2</mn>    </mfrac>  </mrow></math> antiferromagnetic tetramer compound <math>  <mrow>    <msub>      <mi>Yb</mi>      <mn>2</mn>    </msub>    <msub>      <mi>SiO</mi>      <mn>5</mn>    </msub>  </mrow></math>](https://mdr.nims.go.jp/datasets/25bde590-f070-4b44-ba91-a50c79ce5c8e)

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Submission to Phys. Rev. BSpin-singlet ground state with spin gap in Seff = 1/2antiferromagnetic tetramer compound Yb2SiO5Masashi Hase1,∗ Koji Kaneko2,3, Chihiro Tabata2,3,Hiroki Yamauchi2,3, Naohito Tsujii1, and Andreas Dönni11Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science (NIMS),1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan2Materials Science Research Center,Japan Atomic Energy Agency (JAEA),2-4 Shirakata, Tokai, Naka, Ibaraki 319-1195, Japan3Advanced Science Research Center,Japan Atomic Energy Agency (JAEA), 2-4 Shirakata,Tokai, Naka, Ibaraki 319-1195, Japan(Dated: January 17, 2025)1AbstractWe studied the magnetic properties of Yb2SiO5 with pseudospins of Seff = 1/2 using magneticsusceptibility, specific heat, and inelastic neutron scattering (INS) measurements. A broad max-imum appears at Tmax ∼ 6 K in the plot of the magnetic susceptibility [χ(T )] as a function oftemperature (T ), indicating a low-dimensional antiferromagnetic (AFM) spin system. The sus-ceptibility decreases rapidly with decreasing T below Tmax, suggesting a spin-singlet ground statewith a spin gap. We observed cluster-like excitations for energy transfers of ω ∼ 0.7, 1.2, and 1.9meV in the INS measurements. The T dependence of the intensity [I(T )] indicates that the threeexcitations are magnetic ones. We determined that the spin system in Yb2SiO5 is the Seff = 1/2AFM tetramer with two exchange interactions of J1 = 0.74 meV and J2 = 0.95 meV. The isolatedspin tetramer model can explain the excitation energies, χ(T ), magnetic specific heat at T < 15 K,and the magnitude of the scattering vector (Q) dependence of the intensity [I(Q)] at 0.7 and 1.2meV. This model, however, cannot well explain the I(Q) at 1.9 meV and the I(T ) of the threeexcitations, likely because of the effects of weak inter-tetramer interactions.PACS numbers:∗Electronic address: HASE.Masashi@nims.go.jp2I. INTRODUCTIONA spin system determined using inelastic neutron scattering (INS) experiments is some-times different from that expected from a crystal structure. For example, (VO)2P2O7 wasconsidered to have a two-leg antiferromagnetic (AFM) Heisenberg spin ladder with chainsrunning in the crystallographic a direction [1]. Dispersion relations of magnetic excita-tions were obtained from INS experiments on a single crystal [2]. The results indicate that(VO)2P2O7 is best described as an AFM alternating spin chain directed along the b direction.In CuWO4, short Cu-Cu pairs form an alternating spin chain running in the b direction [3],whereas the INS results on a single crystal show that an AFM alternating spin chain runsin the [2,−1, 0] direction [4].We can determine s spin system from the Q dependence of the INS intensity [I(Q)] ofa powder sample. For example, CaCuGe2O6 has a spin-singlet ground state (GS) with aspin gap [5]. The first- and second-nearest neighbor (1NN and 2NN) Cu–Cu pairs with theCu–Cu lengths of 3.072 and 5.213 Å, respectively [6], can form a frustrated spin chain ifthe 2NN exchange interaction is AFM. A spin-singlet GS with a spin gap is possible in thefrustrated spin chain [7–10]. The magnetic susceptibility, however, can be explained well byan isolated AFM spin dimer [5]. Therefore, it was inferred that the exchange interactionin the third-nearest neighbor Cu–Cu pair (5.549 Å) is dominant and form the AFM spindimer. The inference was proved to be correct by I(Q) of the powder sample [11]. InVODPO4 − 12D2O, the I(Q) result showed that the origin of the spin-singlet GS with thespin gap is the AFM spin dimer with the V–V length of 4.43 Å instead of the dimer withthe V–V length of 3.09 Å [12]. In β-AgCuPO4, the 1NN and 2NN Cu–Cu pairs (3.10 and3.37 Å, respectively) form a distorted honeycomb lattice [13]. According to the predictionbased on spin-dimer analysis [14], however, a long Cu–Cu pair (5.20 Å) has the strongestexchange interaction and two strong interactions form an AFM alternating spin chain. TheINS results on the powder sample proved that this prediction is correct [15].To increase the evidence that the method to determine a spin system from I(Q) of apowder sample is useful, we are paying attention to Yb3+-based substances because variousspin systems described below have been found. A spin system that cannot be uniquelydetermined from a crystal structure must exist in Yb3+-based substances. In a Yb3+ (J =7/2) ion with low point symmetry, the ground-state multiplet is split into four Kramers3doublets by crystalline electric fields (CEFs). Here, J denotes the magnitude of the totalangular momentum. It is considered that a Yb3+ ion at low temperatures has pseudospinof 1/2 (Seff = 1/2) at a Kramers doublet with the lowest energy if the energy separationbetween the lowest and second-lowest Kramers doublets are larger than the temperatures.Examples of various spin systems are a spin chain in NaYbTe2O7 [16] and YbAlO3 [17], atriangular lattice in YbMgGaO4 [18–20], YbZnGaO4 [21], YbBO3 [22], and NaYbO2 [23],a honeycomb lattice in YbOCl [24] and YbCl3 [25, 26], a distorted honeycomb lattice inYb2Si2O7 [27] and BiYbGeO5 [28], a breathing pyrochlore lattice in Ba3Yb2Zn5O11 [29], aShastry–Sutherland lattice in Yb2Be2GeO7 [30], and a hyperkagome lattice in Li3Yb3Te2O12[31] and Yb3Sc2Ga3O12 [32].To find an intriguing spin system in Yb3+-based substances, we have been focusing onYb2SiO5 in this study. Figures 1(a) and (b) show the crystal structure. The space groupis monoclinic C2/c (No. 15) [33–35]. The lattice constants at room temperature are a =14.28, b = 6.653, c = 10.28 Å, and β = 122.2◦ [34]. Distorted tetrahedra of SiO4 and distortedoctahedra of YbO6 are formed. Figures 1(c) and (d) depict the positions of Yb3+ ions [33–35]. There are two crystallographic Yb3+-ion sites (Yb1 and Yb2); they are indicated by redand blue circles, respectively. Both of the Yb3+ sites are 8f sites, and the site symmetry is 1,meaning that no point symmetry exists. Thus, pseudospins with Seff = 1/2 can be realized.The first- to fifth-nearest neighbor (1NN to 5NN) Yb–Yb pairs have two Yb–O–Yb paths.The five types of Yb–Yb pairs are represented by bars in Fig. 1, and the Yb–Yb lengths andYb–O–Yb angles are reported in Table I. We expected that the four exchange interactionsexcept for the 4NN interaction formed a two-leg spin ladder with chains running in the cdirection as surrounded by the dashed rectangle. Two-leg spin ladders are connected by the4NN interaction (3.571 Å). The magnetic properties of Yb2SiO5 have not been reported andit is not determined which Yb–Yb pairs have dominant exchange interactions. Accordingly,we investigated the magnetic properties of Yb2SiO5 using magnetic susceptibility, specificheat, and INS measurements.II. EXPERIMENTAL METHODSWe synthesized crystalline powders of Yb2SiO5 and nonmagnetic Lu2SiO5 via a solid-state reaction. The starting materials were Yb2O3, Lu2O3, and SiO2 powders with 99.95 %,4(d)��������1NN 3NN2NN4NN5NN(c)Yb1Yb24NN����������������(e)2NN ��3NNO(b)(a)Yb1Yb2SiOYb1Yb2FIG. 1: (a)(b) The crystal structure of Yb2SiO5, as drawn using VESTA [36]. The solid rectanglerepresents the unit cell. Red, blue, light-blue, and black circles denote Yb1, Yb2, Si, and O sites.Distorted tetrahedra of SiO4 and distorted octahedra of YbO6 are also shown. (c)(d) Schematicshowing the positions of Yb3+ ions. The solid rectangle in (c) and parallelogram in (d) representthe unit cell. The five types of Yb–Yb pairs that have two Yb–O–Yb paths are represented bybars. 1NN, 2NN, 3NN, 4NN, and 5NN denote the first-, second-, third-, fourth-, and fifth-nearestneighbor Yb–Yb pairs, respectively. As indicated by the dashed rectangle, a two-leg spin ladderwith chains running in the c direction are formed by the four exchange interactions except for the4NN interaction. Two-leg spin ladders are connected by the 4NN interaction. (e) The actual spinsystem (spin tetramer) formed by 2NN and 3NN Yb–Yb pairs indicated by blue and pink lines,respectively. The Hamiltonian is expressed as H = J2(S1 ·S2 +S3 ·S4) + J1S2 ·S3. The positionsof S1 and S2 are denoted by r⃗1 and r⃗2, respectively, where the origin is the center of two Yb2 sites.99.9 %, and 99.9 % purities, respectively. A stoichiometric mixture of the powders was sin-tered in air at 1523 K for Yb2SiO5 or 1573 K for Lu2SiO5 with intermediate grinding. X-raypowder diffraction patterns were recorded at room temperature using an X-ray diffractome-ter (RINT-TTR III; Rigaku). Within the experimental accuracy, the obtained samples weresingle-phase. The lattice constants evaluated by us are almost the same as those in [34].We measured the magnetic susceptibility and specific heat using a Quantum Designmagnetic property measurement system and a Quantum Design Dynacool, respectively. Weperformed INS measurements using the low-energy triple-axis spectrometer (LTAS) installedat the JRR-3 reactor at the Japan Atomic Energy Agency (JAEA). The energy of the final5TABLE I: Yb–Yb length and Yb–O–Yb angle in the first- to fifth-nearest neighbor (1NN to 5NN)Yb–Yb pairs [34]. Yb1 and Yb2 sites are surrounded by six O2− ions with Yb–O lengths of2.183–2.339 Å and 2.167–2.260 Å, respectively.Yb-Yb (Å) Yb-O-Yb (degrees)1NN Yb1-Yb2 3.358 95.66 98.172NN Yb2-Yb2 3.431 101.56 ×23NN Yb1-Yb2 3.477 99.21 100.924NN Yb2-Yb2 3.571 105.55 ×25NN Yb1-Yb1 3.599 105.42 ×2neutrons was fixed at 2.6 meV. Higher-order beam contamination was effectively eliminatedusing a cooled Be filter positioned in front of the sample. The energy resolution was 0.1meV (full-width at half-maximum, FWHM) for an energy transfer of ω = 0 meV. Theresolution was determined from incoherent scattering from the sample. A powder sample ofapproximately 8.7 g was mounted in a top-load cryostat [37].III. RESULTSThe red circles in Fig. 2 show the temperature (T ) dependence of the magnetic suscep-tibility [χ(T )] of Yb2SiO5 powder measured in a magnetic field of µ0H = 0.01 T. A broadmaximum is observed at Tmax ∼ 6 K [Fig. 2(a)], indicating a low-dimensional AFM spin sys-tem. The susceptibility decreases rapidly with decreasing T at temperatures less than Tmax,suggesting a spin-singlet GS with a spin gap. A spin-singlet GS with a spin gap appearsin Yb2Si2O7 (AFM dimer) [27], BiYbGeO5 (AFM dimer) [28], and Ba3Yb2Zn5O11 (AFMtetrahedron) [29]. The calculated χ(T ) values indicated by lines will be explained later.Red and blue circles in Fig. 3(a) show the T dependence of the magnetic specific heatCmag(T ) of Yb2SiO5 powder under 0 and 9 T magnetic fields, respectively. The magneticspecific heat was obtained by subtracting the specific heat of nonmagnetic isostructuralLu2SiO5 at 0 T from that of Yb2SiO5. A maximum of Cmag(T ) is observed at approximately4 and 8 K in the data recorded under 0 and 9 T fields, respectively. No transition appearsat T ≥ 2 K and µ0H ≤ 14 T. The calculated Cmag(T ) indicated by lines will be explained600.050.10 100 200 300Susceptibility (emu/mol Yb)Temperature (K)00.050.10 10 20 30Susceptibility (emu/mol Yb)Temperature (K)(a)0.01 T(b)0.01 TFIG. 2: The T dependence of the magnetic susceptibility χ(T ) of Yb2SiO5 powder under a 0.01 Tmagnetic field (circles). The solid line indicates χ(T ) values calculated for the spin-1/2 tetramerwith J1 = 0.74 meV, J2 = 0.95 meV, g = 4.2, and a constant term χ0 = 0.01 (emu/mol Yb). Thedashed line indicates the sum of χ(T ) values calculated for two isolated spin-1/2 dimers with theexchange interactions of 0.7 and 1.2 meV. We used g = 4.0 and a constant term χ0 = 0.01 (emu/molYb) for the χ(T ) values.later. Figure 3(b) shows the magnetic entropy Smag(T ) of Yb2SiO5. The value of entropyreaches R ln 2 and R ln 4 at T ∼ 23 and ∼ 150 K, respectively.Figure 4 shows the INS intensity I(Q,ω) map of Yb2SiO5 powder at 1.8 K, where Qand ω are the magnitude of the scattering vector and the energy transfer, respectively. Weobserved excitations at ω ∼ 0.7, 1.2, and 1.9 meV. The excitations are dispersionless andexist over a wide Q range, indicating cluster excitations.Figure 5(a) depicts the ω dependence of the INS intensity I(ω) of Yb2SiO5 powder atthe four investigated temperatures. The line shows the sum of three Gaussians each with aFWHM of 0.1 meV, which was evaluated from the incoherent scattering. The three peaks701230 5 10 15 20Cmag (T) (J/K mol Yb)T (K)0 T9 T(a)05101510 100Smag (T) (J/K mol Yb)T (K)(b) 0 T� ln 2� ln 4FIG. 3: (a) The T dependence of the magnetic specific heat Cmag(T ) of Yb2SiO5 under 0 and9 T magnetic fields (circles). The solid lines indicate Cmag(T ) values calculated for the spin-1/2tetramer with J1 = 0.74 meV, J2 = 0.95 meV, and g = 4.2 for the 9 T data. The dashed lineindicates the sum of Cmag(T ) values at 0 T calculated for two isolated spin-1/2 dimers with theexchange interactions of 0.7 and 1.2 meV. (b) The T dependence of the magnetic entropy Smag(T )of Yb2SiO5 under a 0 T magnetic field.are slightly wider than the resolution, consistent with the picture of the cluster excitations.Figure 5(b) represents the T dependence of the integrated intensity I(T ) of the 0.7, 1.2,and 1.9 meV excitations. As T is raised, the intensities of the 0.7 and 1.2 meV excitationsdecrease at T ≤ 30 K and are almost constant at T ≥ 30 K. The 1.9 meV excitation isobserved only at low temperatures. The T dependence indicates that the three excitationsare magnetic in nature. The peak positions are almost independent of T . The calculatedI(T ) values indicated by the lines will be explained later.Figure 6 shows the Q dependence of the INS intensity I(Q) at ω = 0.7, 1.2, and 1.98Q (Å-1)�(meV)1.8 K160012008004000FIG. 4: The INS intensity I(Q,ω) map of Yb2SiO5 powder at 1.8 K. The energy of the finalneutrons is 2.6 meV. The vertical key on the right shows the INS intensity in arbitrary units.meV. A broad maximum is observed at Q ∼ 1.25 Å−1 in the three experimental curves,although the Q dependence of the INS intensity is weak at the 1.9 meV excitation. Asshown in Fig. 6(b), I(Q) at ω = 1.9 meV cannot be explained by the square of the magneticform factor [f(Q)2] of Yb3+ ions [38]. The experimental Q dependence indicates that thethree excitations are not CEF excitations of Yb3+ ions because the intensity of the CEFexcitations decreases monotonically with increasing Q as f(Q)2. The calculated I(Q) valuesindicated by the solid lines will be explained later.IV. ANALYSES AND DISCUSSIONWe consider a spin system that can explain the experimental results for Yb2SiO5. Thespin system is a spin cluster because the magnetic excitations are dispersionless. The sim-plest cluster is an AFM dimer. As is shown in Fig. 1 and Table I, two types of Seff = 12AFMdimers are possible: Yb1–Yb1 and Yb2–Yb2 dimers can be formed by the 5NN Yb–Yb pairand the 2NN or 4NN pair, respectively. In this case for two isolated dimers, two spin-gapexcitations appear. Thus, the three magnetic excitations cannot be explained.Even if the weak 1.9 meV excitation is ignored, any spin model formed by two types ofAFM dimers cannot explain the difference in the intensity between the 0.7 and 1.2 meV90501001 10 100Intensity (arb. units)Temperature (K)050010000 0.5 1 1.5 2 2.5Intensity (arb. unis)Q = 1.0 (Å−1)�1.8, �10, �30,�60 K(a)� (meV)(b)�0.7 meV�1.2�1.9204060801001.6 1.8 2 2.2FIG. 5: (a) The ω dependence of the INS intensity I(ω) of Yb2SiO5 powder at Q = 1.0 Å−1 andvarious temperatures. The energy of the final neutrons is 2.6 meV. The line shows the sum ofthree Gaussians each with a FWHM of 0.1 meV. The inset is an enlarged figure clearly showingthe 1.9 meV excitation. (b) The T dependence of the integrated intensity of the 0.7, 1.2, and 1.9meV excitations. The red, blue, and black lines indicate the integrated intensity of the excitationsfrom GS (|02⟩) to 1ES (|13⟩), 2ES (|11⟩), and 4ES (|12⟩), respectively, calculated for the spin-1/2tetramer with J1 = 0.74 meV and J2 = 0.95 meV. The three lines are multiplied by the same value.excitations. The powder-averaged INS intensity at the dimer-gap energy is given by [11]I(Q) = r20k′kNpf(Q)2(1− sin(2Qd)2Qd). (1)Here, k and k′ are the incident and scattered neutron wave numbers, respectively, andr0 = −0.54× 10−12 cm. The parameters N , p, f(Q), and 2d denote the number of dimers,the thermal population factor for the ground state [39], the magnetic form factor of Yb3+100501001502000 0.5 1 1.5 2Intensity (arb. units)0500100015000 0.5 1 1.5 2Intensity (arb. units)1.8 K�0.7 meV, 02 to 13�1.2 meV, 02 to 11�1.9 meV, 02 to 12(a)Q (Å-1)1.8 K1.9 meV, 02 to 12� � �(b)FIG. 6: (a) The Q dependence of the INS intensity I(Q) at 1.8 K for Yb2SiO5 powder at 0.7, 1.2,and 1.9 meV. The red, blue, and black lines indicate I(Q) for the excitations from GS (|02⟩) to 1ES(|13⟩), 2ES (|11⟩), and 4ES (|12⟩), respectively, calculated for the spin-1/2 tetramer formed by 3NNand 2NN Yb–Yb pairs with J1 = 0.74 meV and J2 = 0.95 meV. The three lines are multiplied bythe same value. (b) Enlarged figure of (a) for the 1.9 meV excitation. The square of the magneticform factor of a Yb3+ ion is shown by the dashed line.ions, and the Yb–Yb length in a dimer, respectively. We measured the INS intensity usingthe fixed final energy (2.6 meV) and a fixed beam monitor. The value of k′/k was thenalready accounted for. The difference in intensity between the 0.7 and 1.2 meV excitationsis generated by k′k, p, and 2d. The values of k′kand p at 1.8 K were calculated to be 0.888and 0.968, respectively, for the 0.7 meV excitation and 0.827 and 0.999, respectively, forthe 1.2 meV excitation. Thus, k′kp is 0.860 and 0.827 for the 0.7 and 1.2 meV excitations,11TABLE II: Yb–Yb length in six tetramers formed by two types of Yb–Yb pairs. As shown inFig. 1(c), |r⃗1 − r⃗2|, |2r⃗2|, |r⃗1 + r⃗2|, and |2r⃗1| correspond to the lengths between S1 and S2 (alsoS3 and S4), between S2 and S3, between S1 and S3 (also S2 and S4), and between S1 and S4,respectively.tetramer |r⃗1 − r⃗2| (Å) |2r⃗2| (Å) |r⃗1 + r⃗2| (Å) |2r⃗1| (Å)S1 and S2 S2 and S3 S1 and S3 S1 and S4S3 and S4 S2 and S41NN-2NN Yb1-Yb2-Yb2-Yb1 3.357 3.431 4.096 6.4581NN-4NN Yb1-Yb2-Yb2-Yb1 3.357 3.571 6.183 9.2871NN-5NN Yb2-Yb1-Yb1-Yb2 3.357 3.599 3.696 6.0773NN-2NN Yb1-Yb2-Yb2-Yb1 3.477 3.431 3.696 3.9863NN-4NN Yb1-Yb2-Yb2-Yb1 3.477 3.571 5.693 8.7323NN-5NN Yb2-Yb1-Yb1-Yb2 3.477 3.599 4.096 6.692respectively. As shown in Table I, the values of 2d for Yb1–Yb1 and Yb2–Yb2 dimers areclose to each other. Consequently, the intensities of the 0.7 and 1.2 meV excitations mustbe similar to each other, which is inconsistent with the I(Q) results in Fig. 6.The next simplest cluster that has a spin-singlet ground state is a tetramer (four-spinsystem). When we select two types of Yb–Yb pairs in Table I, six types of spin tetramersare possible, as listed in Table II. In any tetramer, the Hamiltonian is expressed asH = J2(S1 · S2 + S3 · S4) + J1S2 · S3. (2)There are two ST = 0 states (|01⟩ and |02⟩), three ST = 1 states (|11⟩, |12⟩, and |13⟩),and one ST = 2 state (|21⟩). ST is the magnitude of the sum of the spin operators in thetetramer. The eigenstates |ij⟩ are explicitly given in Ref.[39].We evaluated the J1 and J2 exchange interactions using eigenenergies given in Ref. [39].Only the set in which J1 = 0.74 meV (8.6 K) and J2 = 0.95 meV (11.0 K) can explainthe three excitation energies (0.7, 1.2, and 1.9 meV). The results are shown in Table III.When J1 = 0.74 meV and J2 = 0.95 meV, the ground, first-excited, second-excited, third-excited, fourth-excited, and fifth-excited states (GS, 1ES, 2ES, 3ES, 4ES, and 5ES) are |02⟩,|13⟩, |11⟩, |01⟩, |12⟩, and |21⟩ states, respectively. The 0.7, 1.2, and 1.9 meV excitations12TABLE III: The excitation energy from GS determined experimentally and calculated for the spintetramer with J1 = 0.74 meV and J2 = 0.95 meV. The excitation from GS to 3ES or 5ES isforbidden (F).eigenstate ST Excitation energy (meV)exp. calc.GS |02⟩ 0 0 01ES |13⟩ 1 0.7 0.712ES |11⟩ 1 1.2 1.223ES |01⟩ 0 F 1.684ES |12⟩ 1 1.9 1.935ES |21⟩ 2 F 2.18correspond to the excitations from GS to 1ES, 2ES, and 4ES, respectively. The excitationsfrom GS to 3ES and 5ES are forbidden.We investigated whether the spin tetramer can explain the other results. The solid line inFig. 2 indicates χ(T ) calculated for the spin tetramer with J1 = 0.74 meV, J2 = 0.95 meV,and g = 4.2. We added a constant susceptibility χ0 = 0.01 (emu/mol Yb). The calculatedχ(T ) is close to the experimental χ(T ). The large value of χ0 is likely attributable to VanVleck paramagnetism of Yb3+ ions. The red solid line in Fig. 3 represents Cmag(T ) calculatedfor the spin tetramer and is consistent with the experimental Cmag(T ) at T < 15 K. Theblue solid line in Fig. 3 indicates Cmag(T ) calculated for the spin tetramer when µ0H = 9 Tand g = 4.2, which was used in the calculation of χ(T ). The experimental and calculatedCmag(T ) curves corresponding to 9 T are close to each other. We also calculated χ(T ) andCmag(T ) for two isolated dimers with the exchange interactions of 0.7 and 1.2 meV. Thedashed line in Fig. 2 indicates the sum of χ(T ) curves of the two dimers and can explain theexperimental χ(T ). The dashed line in Fig. 3(a) indicates the sum of Cmag(T ) curves at 0T of the two dimers and is slightly larger than the experimental Cmag(T ) at around 4 K. Wecannot exclude the two-isolated-dimer model by the results of χ(T ) and Cmag(T ). However,the two-isolated-dimer model cannot be applied to Yb2SiO5 mainly because of the differencein the intensities at 0.7 and 1.2 meV described in the 2nd paragraph in the section IV.The experimental and calculated Cmag(T ) curves corresponding to 0 T differ with each13other in the temperature range T > 15 K. This difference indicates the existence of the CEFexcitations. As evident in Fig. 3(b), Smag(T ) reaches R ln 2 and R ln 4 at T ∼ 23 and ∼ 150K, respectively. R ln 2 and R ln 4 correspond to the entropy of the lowest Kramers doubletand that of the lowest and second-lowest Kramers doublets, respectively. We infer that thedifference in energy between the lowest and the second-lowest doublets is 100 K (8.62 meV)or larger. Thus, the three excitations at 0.7, 1.2, and 1.9 meV are not CEF excitations.Unfortunately, we could not investigate the CEF excitations using the LTAS because of thesmall energy transfers.Next, we consider which spin tetramer can explain the experimental I(Q). The I(Q)curve is given by the formulaI(Q) = f(Q)2[A12(1− sin(Q|r⃗1 − r⃗2|)Q|r⃗1 − r⃗2|)+A23(1− sinQ|2r⃗2|Q|2r⃗2|)+A13(1− sin(Q|r⃗1 + r⃗2|)Q|r⃗1 + r⃗2|)+A14(1− sinQ|2r⃗1|Q|2r⃗1|)]. (3)Each parameter A12, A23, A13, or A14 is determined by J1, J2, the initial state, and the finalstate [39]. As shown in Fig. 1(c), the vectors r⃗1 and r⃗2 denote the positions of S1 and S2,respectively. The values of Yb–Yb pairs are reported in Table II.Figure 7 shows I(Q) calculated for the six types of spin tetramers. The I(Q) curvesfrom |02⟩ to |13⟩ (from GS to 1ES) and from |02⟩ to |11⟩ (from GS to 2ES) depend on thetetramer, whereas the I(Q) curves from |02⟩ to |12⟩ (from GS to 4ES) are similar to oneanother. Only the spin tetramer formed by 3NN and 2NN Yb–Yb pairs can explain theexperimental I(Q) curves.The lines in Fig. 6(a) indicate I(Q) calculated for the 3NN–2NN tetramer. The threelines are multiplied by the same value. The experimental I(Q) values at 0.7 and 1.2 meVare consistent with the I(Q) values calculated for the |02⟩ to |13⟩ and |02⟩ to |11⟩ excita-tions, respectively, when we assume that the background intensity is 74. The isolated spintetramer model can explain qualitatively the weak intensity of the 1.9 meV excitation. Theexperimental I(Q) value at 1.9 meV, however, is smaller than the I(Q) value calculated for|02⟩ to |12⟩ excitation even if the background intensity is 0. The lines in Fig. 5(b) indicateI(T ) calculated for the 3NN–2NN tetramer. The three lines are multiplied by the samevalue. The overall T dependence of the experimental curves for the |02⟩ to |13⟩ and |02⟩to |11⟩ excitations (0.7 and 1.2 meV excitations, respectively) are captured by the present141NN-2NN1NN-4NN1NN-5NN3NN-2NN3NN-4NN3NN-5NN(a) 02 to 130123450 1 2 3 4Intensity (arb. units)(b) 02 to 1101230 1 2 3 4Intensity (arb. units)(c) 02 to 1200.20.40.60 1 2 3 4Intensity (arb. units)Q (Å-1)FIG. 7: The Q dependence of the INS intensity I(Q) calculated for the six types of spin tetramerswith J1 = 0.74 meV and J2 = 0.95 meV: (a) the excitation from |02⟩ to |13⟩ (from GS to 1ES),corresponding to the 0.7 meV excitation, (b) the excitation from |02⟩ to |11⟩ (from GS to 2ES),corresponding to the 1.2 meV excitation, and (c) the excitation from |02⟩ to |12⟩ (from GS to 1ES),corresponding to the 1.9 meV excitation.model, while the experimental I(T ) value is smaller than the calculated I(T ) value for the|02⟩ to |12⟩ excitation.The spin-1/2 tetramer with J1 = 0.74 meV and J2 = 0.95 meV can explain the excitationenergies, χ(T ), and Cmag(T ) at T < 15 K for Yb2SiO5. These values are determined byeigenenergies. The I(Q) curves calculated for the |02⟩ to |13⟩ and |02⟩ to |11⟩ excitationsare consistent with the experimental I(Q) curves, indicating that the eigenstates are almostcorrect. However, the spin tetramer cannot well explain the I(Q) at 1.9 meV correspondingto the |02⟩ to |12⟩ excitation and the I(T ) value. Probably, weak inter-tetramer interactionsaffect the eigenstates. The maximum of the half-width at half-maximum (HWHM) at the0.7 meV excitation is 0.07 meV that is 0.02 meV larger than HWHM of the energy resolution,15suggesting that the maximum of inter-tetramer interactions is 0.02 meV. The experimentaland calculated Cmag(T ) under a 0 T field differ from each other at T > 15 K. We willinvestigate the CEF excitations using INS measurements with higher ω values and willconfirm whether CEF-excitation energies can explain Cmag(T ) at T > 15 K.The spin tetramer is an intriguing spin system. For example, Bose-Einstein conden-sation of triplons in magnetic fields was found in the spin-1 tetramer antiferromagnetK2Ni2(MoO4)3 [40]. If a magnetic order occurs in spin-tetramer materials, we can expecttwo types of magnetic excitations such as gapless transverse-mode (Nambu-Goldstone mode)[41] and gapped longitudinal-mode (amplitude Higgs mode) [42] excitations corresponding tofluctuations in directions perpendicular and parallel to ordered moments, respectively [43].Several spin-tetramer materials have been reported, including CaV4O9 (S = 12) [44, 45],Cu2PO4 (S = 12) [39], Cu2Te2O5Br2 (S = 12) [46], Cu2CdB2O6 (S = 12) [47, 48], SeCuO3(S = 12) [49], CuInVO5 (S = 12) [50], Ba3Yb2Zn5O11 (Seff = 12) [29], and Rb2Ni2(MoO4)3(S = 1) [43]. For some of the materials, the values of J1, J2, spin gap, and AFM transitiontemperature are summarized in Table I in [43]. The J1 and J2 values in Yb2SiO5 are smallerthan those in the materials, reflecting that an exchange interaction in a Yb–Yb pair is usu-ally weak. To our knowledge, Yb2SiO5 is the first material based on Yb3+ ions with spintetramers formed by the two types of exchange interactions (i.e., J1 and J2 interactions).Materials based on Yb3+ ions have the following advantages when used to investigatespin systems. An exchange interaction between two pseudospins is usually weak, leading toa low energy scale. Phase transitions in magnetic fields can appear even under low magneticfields. In the Yb3+-based antiferromagnets shown in the Introduction and Yb2SiO5, a Yb-Yblength in a Yb-Yb pair where a dominant exchange interaction exists is short as shown inTable I. The length is comparable to that in a pair of two 3d ions where a dominant exchangeinteraction exists. In INS experiments, it is easier to investigate magnetic excitations in asmall Q range for Yb3+ compounds than for 3d compounds. Spin systems formed by Yb3+ions will be studied extensively in the future.V. CONCLUSIONWe investigated the magnetic properties of Yb2SiO5 with pseudospins of Seff = 1/2using magnetic susceptibility, specific heat, and INS measurements. A broad maximum16appears at Tmax ∼ 6 K in the χ(T ) plot, indicating a low-dimensional AFM spin system.The susceptibility decreases rapidly with decreasing T below Tmax, suggesting a spin-singletground state with a spin gap. A maximum was observed at approximately 4 and 8 K inCmag(T ) under 0 and 9 T fields, respectively. We observed excitations at ω ∼ 0.7, 1.2,and 1.9 meV in INS spectra. The excitations are dispersionless and exist over a wide Qrange, indicating cluster excitations. As T is raised, the intensities of the 0.7 and 1.2 meVexcitations decrease at T ≤ 30 K and are almost constant at T ≥ 30 K. The 1.9 meVexcitation is observed only at low temperatures. The T dependence indicates that the threeexcitations are magnetic in nature. A broad maximum was observed at Q ∼ 1.25 Å−1 inI(Q) for the three excitations, although the Q dependence of the INS intensity is weak atthe 1.9 meV excitation. Although a quasi-one-dimensional spin system is expected, the spinsystem in Yb2SiO5 is the Seff = 1/2 tetramer formed by the 2NN and 3NN Yb–Yb pairswith J1 = 0.74 meV and J2 = 0.95 meV. The 0.7, 1.2, and 1.9 meV excitations correspondto the excitations from GS to 1ES, 2ES, and 4ES, respectively. The excitations from GS to3ES and 5ES are forbidden. The isolated spin tetramer model can explain the excitationenergies, χ(T ), Cmag(T ) at T < 15 K, and I(Q) at 0.7 and 1.2 meV. This model, however,cannot well explain the I(Q) at 1.9 meV and I(T ) for the three excitations, likely becauseof the effects of weak inter-tetramer interactions.AcknowledgmentsThis work was supported by the Japan Society for the Promotion of Science (JSPS)KAKENHI Grant Number 24K06947 and by the World Premier International ResearchCenter Initiative (WPI), Ministry of Education, Culture, Sports, Science and Technology(MEXT), Japan. One of the authors (Naohito Tsujii) has been supported by JSPS KAK-ENHI (24K00590). This work is partially based on experiments performed on the LTAS(Proposal 2024A-A06) at the JRR-3 reactor, Japan Atomic Energy Agency (JAEA). Wethank Manami Ikeda at the National Institute for Materials Science (NIMS) for the samplesynthesis and X-ray diffraction measurements. We are grateful to Masamichi Nishino, Kazu-nari Yamaura, Alexei Belik, and Yoshihiro Tsujimoto at NIMS for their fruitful discussions.17[1] D. C. Johnston, J. W. Johnson, D. P. Goshorn, and A. J. 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