Chemically tunable quantum magnetism on the anisotropic triangular lattice in rhenium oxyhalides

Abstract

The spin-1/2 Heisenberg antiferromagnet on an anisotropic triangular lattice (ATL) is an archetypal spin system hosting exotic quantum magnetism and dimensional crossover. However, the progress in experimental research on this field has been limited due to the scarcity of ideal model materials. Here, we show that rhenium oxyhalides A₃ReO₅X₂, where spin-1/2 Re⁶⁺ ions form a layered structure of ATLs, allow for flexible chemical substitution in both cation A²⁺ (A = Ca, Sr, Ba, Pb) and anion X⁻ (X = Cl, Br) sites, leading to seven synthesizable compounds. By combining magnetic susceptibility and high-field magnetization measurements with theoretical calculations using the orthogonalized finite-temperature Lanczos method, we find that the anisotropy ${J}^{{\prime} }/J$ ranges from 0.25 to 0.45 depending on the chemical composition. Our findings demonstrate that A₃ReO₅X₂ is an excellent platform for realizing diverse effective spin Hamiltonians that differ in the strength of the anisotropy ${J}^{{\prime} }/J$ as well as the relevance of perturbation terms such as the Dzyaloshinskii-Moriya interaction and interlayer exchange coupling.

Introduction

Frustrated quantum magnets have long drawn considerable attention due to the possibility of realizing quantum spin-liquid (QSL)¹ and fractional quasi-particle excitations². Although recent advances in analytical and numerical approaches to describe many-body quantum physics have promoted our understanding of various magnetic phenomena, many issues remain unresolved³, highlighting the need for further experimental insights. To this end, it is desirable to develop model compounds with a simple effective spin Hamiltonian, an accessible energy scale (i.e., temperature and magnetic field), and a broad tunability of perturbative terms as well as exchange couplings through chemical substitution.

The spin-1/2 Heisenberg antiferromagnet on a triangular lattice is a prototypical frustrated spin system to explore exotic ground states^4,5,6. A related derivative is the anisotropic triangular lattice (ATL) magnet, where two distinct nearest-neighbor antiferromagnetic (AFM) exchange couplings, ${J}^{{\prime} {\prime}}$ and ${J}^{{\prime} }$, act along the intrachain and interchain directions, respectively^{7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23}. By varying the ratio ${J}^{{\prime} }/J$, this model provides an interpolation between a regular triangular lattice (${J}^{{\prime} }/J=1$) and a set of decoupled one-dimensional (1D) chains (${J}^{{\prime} }/J=0$), with known ground states of the 120^∘ long-range order (LRO)^24,25 and the Tomonaga–Luttinger liquid (TLL)^26,27, respectively. Theoretically, the TLL–like gapless QSL phase is predicted to be stable over a wide parameter range of $0 < {J}^{{\prime} }/J\lesssim 0.6$^12,16,21, as a manifestation of dimensional reduction driven by geometrical frustration¹⁵. On the other hand, an alternative scenario based on an analytical renormalization group study proposes a collinear AFM ground state in this ${J}^{{\prime} }/J$ regime¹⁴. The nature of the ground state in the intermediate region $0.6\lesssim {J}^{{\prime} }/J\lesssim 0.8$ also remains controversial. Some theoretical studies suggest a direct transition from the TLL-like QSL to a spiral LRO^{7,13,18,19,21}, while others propose the emergence of an intermediate phase such as a gapped QSL^12,16.

Experimentally, two types of compounds have been intensively studied as candidates for the ATL quantum antiferromagnet: Cs₂CuX₄ (X = Cl, Br)^{28,29,30,31,32,33,34,35,36,37,38,39,40,41,42} and organic Mott insulators represented by κ-(BEDT-TTF)₂Cu₂(CN)₃^{43,44,45,46,47,48,49,50}. The magnetic properties of Cs₂CuX₄ have been unveiled in detail thanks to relatively weak AFM exchange couplings: J/k_B = 4.3 K and ${J}^{{\prime} }/J=0.30$ (J/k_B = 14.9 K and ${J}^{{\prime} }/J=0.41$) for X = Cl (Br)³⁶, where k_B is the Boltzmann constant. However, Cs₂CuX₄ undergoes a magnetic transition into an incommensurate spiral LRO at low temperatures^28,39 due to the presence of the Dzyaloshinskii-Moriya (DM) interaction¹⁷, which interfere with the observation of essential properties expected for the ideal ATL quantum antiferromagnet. For the organic Mott insulators, the exchange couplings are much stronger, on the order of several hundred kelvin^{10,43,44,45,46}, making the accurate estimation of ${J}^{{\prime} }/J$ elusive. The contribution of multi-spin exchange and spin-lattice coupling can be important in the organic salts¹⁰, as suggested from a pressure-induced insulator-metal transition^47,48 and a magnetic-field-induced AFM transition reminiscent of a spin-Peierls phase⁴⁹, respectively.

Compared to the aforementioned compounds, recently-discovered rhenium oxychlorides A₃ReO₅Cl₂ (A = Ca, Sr, Ba)^{51,52,53,54,55,56} potentially hold an advantage in the search for low-temperature magnetism of the ATL quantum antiferromagnet. The Re⁶⁺ ion has a 5d¹ electronic configuration, and the unique crystal field resulting from the surrounding mixed anions causes the d_xy orbital to become the lowest-energy level without degeneracy. As a consequence, A₃ReO₅Cl₂ can be regarded as a spin-1/2 Heisenberg system with quenched orbital angular momentum. ${J}^{{\prime} }/J$ is estimated to be 0.32, 0.43, and 0.47 for A = Ca, Sr, and Ba, respectively, based on theoretical fits to the temperature dependence of magnetic susceptibility using high-temperature series expansion^52,53. A₃ReO₅Cl₂ exhibits TLL-like characteristics indicative of one-dimensionality at low temperatures, as evidenced by several experimental observations such as the Bonner–Fisher-like magnetic susceptibility⁵⁷, large T-linear magnetic heat capacity^52,53, and spinon and triplon excitations in inelastic neutron scattering (INS)⁵⁴ and Raman spectra⁵⁶. The frustration factor, defined as f ≡ ∣Θ_W∣/T_N, where Θ_W and T_N represent a Weiss temperature and an ordering temperature, respectively, is larger in A₃ReO₅Cl₂ than in Cs₂CuX₄^52,53. This suggests that additional interactions, aside from J and ${J}^{{\prime} }$, play a less significant role in A₃ReO₅Cl₂ than in Cs₂CuX₄. Further development of chemical substitutes for A₃ReO₅Cl₂, along with the evaluation of their effective spin models, should open up the route for a systematic understanding of the fundamental properties of the ATL quantum antiferromagnet.

Here, we report physical properties of four novel Re-based quantum ATL antiferromagnets, A₃ReO₅Br₂ (A = Ca, Sr, Ba) and Pb₃ReO₅Cl₂. These compounds are (nearly) isostructural with A₃ReO₅Cl₂ (Fig. 1a–c) and exhibit one-dimensional character at low temperatures as well^51,52,53. We also investigate magnetization processes of all the seven A₃ReO₅X₂ families in pulsed high magnetic fields of up to 130 T⁵⁸. On the basis of theoretical calculations using the orthogonalized finite-temperature Lanczos method (OFTLM)^22,59, ${J}^{{\prime} }/J$ is found to vary between 0.25 and 0.45 from compound to compound. We propose that, considering the difference in their crystal structures and spatial distributions of electronic density, A₃ReO₅X₂ can be categorized into three groups in terms of an effective spin Hamiltonian (Fig. 1d): (1) A = Ca with the DM interaction, (2) A = Pb with the relatively strong interlayer exchange coupling, and (3) A = Sr and Ba with no dominant perturbations.

Fig. 1: Structural and magnetic properties of A₃ReO₅X₂.

a Two kinds of crystal structures. Ca₃ReO₅X₂ crystallizes in Type I with the space group Pnma (left), whereas the system with A = Sr, Ba, and Pb crystallizes in Type II with the space group Cmcm (right). b Schematic of an anisotropic triangular lattice (ATL), where upward and downward ReO₅ square pyramids running along the b (a) direction are alternately arranged along the c direction in the Type I (II) structure. Exchange couplings between nearest-neighbor and next-nearest-neighbor Re ions with the bond length d and ${d}^{{\prime} }$ are defined as J and ${J}^{{\prime} }$, respectively. The illustrations in panels (a) and (b) are drawn with VESTA software⁸⁰. c Plots of d and ${d}^{{\prime} }$ for the seven compounds. Elements of A and X are written next to each marker. The data of A₃ReO₅Cl₂ (A = Ca, Sr, Ba) are taken from refs. ^51,53. d Summary of effective spin Hamiltonians with the ATL quantum Heisenberg model. The anisotropy ${J}^{{\prime} }/J$ for each compound is revealed in the present work and displayed in the left axis. The DM interaction and interlayer exchange coupling dominate the low-temperature magnetism for A = Ca and Pb, respectively, whereas there are no significant effects of such perturbations for A = Sr and Ba.

Results

Crystal structures

We first present the crystal structures of newly synthesized compounds: Ca₃ReO₅Br₂ (CROB), Sr₃ReO₅Br₂ (SROB), Ba₃ReO₅Br₂ (BROB), and Pb₃ReO₅Cl₂ (PROC). Regarding the previously reported oxychlorides, Ca₃ReO₅Cl₂ (CROC) crystallizes in an orthorhombic structure with the space group Pnma (No. 62)⁵¹, whereas Sr₃ReO₅Cl₂ (SROC) and Ba₃ReO₅Cl₂ (BROC) with the space group Cmcm (No. 63)⁵³, as depicted in Fig. 1a. Hereafter, we refer to these two structures as Type I and Type II, respectively. Both structures consist of alternating layers of A₃ReO₅ and Cl slabs. The coordination numbers of the A²⁺ ions are 5 or 6 in the Type I structure, whereas they are 8 or 9 in the Type II structure. This difference arises due to the smaller ionic radius of Ca²⁺ (1.00 Å in the 6-fold coordination) compared to Sr²⁺ and Ba²⁺ (1.26 and 1.42 Å, respectively, in the 8-fold coordination). We find that the substitution of Cl⁻ (ionic radius 1.81 Å) with Br⁻ (ionic radius 1.96 Å) does not impact the original crystal structure; CROB crystallizes in Type I, while SROB and BROB crystallize in Type II. PROC also crystallizes in Type II (the ionic radius of Pb²⁺ is 1.29 Å in the 8-fold coordination). Further details on the crystal structures are provided in Supplementary Note 2.

An ATL composed of magnetic Re⁶⁺ ions is depicted in Fig. 1b. The ReO₅ square pyramids, which point upward and downward alternately along the interchain direction, do not share any oxygen ions. Consequently, each Re⁶⁺ ion is well separated from one another by more than 5 Å. Figure 1c compares the lengths of the nearest-neighbor and next-nearest-neighbor Re–Re bonds, d and ${d}^{{\prime} }$, respectively, among all the seven A₃ReO₅X₂ compounds. In common with A = Ca, Sr, and Ba, the substitution of Cl with Br leads to the elongation of both d and ${d}^{{\prime} }$; the increase rates are 1.4% and 0.7% for A = Ca, 1.2% and 1.7% for A = Sr, and 1.4% and 1.5% for A = Ba, respectively.

One dimensionality of the low-temperature magnetism

Next, we show the basic physical properties of A₃ReO₅Br₂ (A = Ca, Sr, Ba) and PROC. Figure 2a–d shows the temperature dependence of magnetization divided by a magnetic field M/H measured at 7 T. All the M/H data above 150 K are well fitted by the Curie–Weiss law M/H = C₀/(T − Θ_W) + χ₀, where C₀ is the Curie constant, and χ₀ is the temperature-independent term, as shown by black lines in Fig. 2a–d. The fitting parameters and the effective magnetic moment μ_eff calculated from C₀ are listed in Table 1. The obtained χ₀ is comparable to the diamagnetic contribution of core electrons χ_dia for each compound: χ_dia = −1.76 × 10⁻⁴,  −2.02 × 10⁻⁴,  −2.24 × 10⁻⁴ and  −2.18 × 10⁻⁴ cm³ mol⁻¹ for CROB, SROB, BROB, and PROC, respectively. The reduction of μ_eff from 1.73 μ_B expected for the pure S = 1/2 case would be attributed to the effect of the spin-orbit interaction, which cancels the spin and orbital angular momentum with each other in the 5d¹ electronic configuration.

Fig. 2: Basic physical properties of four novel compounds.

Temperature dependence of the magnetic susceptibility M/H measured at 7 T (left axis) and the inverse susceptibility ${(M/H-{\chi }_{0})}^{-1}$ (right axis), where a T-independent component χ₀ is subtracted, for CROB (a), SROB (b), BROB (c), and PROC (d). The black lines represent the Curie–Weiss law, i.e., M/H = C/(T − Θ_W) + χ₀, fitted to the experimental data between 150 and 400 K. Heat capacity divided by temperature C/T as a function of T below 100 K for CROB (e), SROB (f), BROB (g), and PROC (h). The inset of each panel displays a C/T versus T² plot, where the black line represents a fitting C/T = α + βT² to the experimental data between 2 and 10 K for CROB and SROB, between 2 and 4 K for BROB, and between 7 and 10 K for PROC.

Table 1 Magnetic and thermal parameters for A₃ReO₅X₂ (A = Ca, Sr, Ba, Pb; X = Cl, Br)

The large negative Weiss temperatures indicate the dominance of AFM exchange interactions in these compounds. The smaller value of ∣Θ_W∣ for BROB (Θ_W = − 17.2(2) K) compared to CROB (−36.5(2) K) and SROB (−37.4(4) K) can be attributed to the relatively long Re–Re bond lengths, d and ${d}^{{\prime} }$, in BROB (Fig. 1c). This trend is common with the A₃ReO₅Cl₂ case: Θ_W = −37.8(1), −49.5(1) and  −21.6(1) K for CROC, SROC, and BROC, respectively^52,53. Furthermore, the substitution of Cl with Br leads to a decrease in ∣Θ_W∣; the decrease rates are 3.4%, 24.4%, and 20.4% for A = Ca, Sr, and Ba, respectively. The much smaller decrease rate for A = Ca compared to A = Sr and Ba would be due to the difference in their crystal structures, as mentioned above (Fig. 1a).

On further cooling below 100 K, all the M/H–T curves gradually deviate from the Curie–Weiss behavior and exhibit a broad peak around 10–20 K (Fig. 2a–d), signaling the development of AFM short-range correlations. In the ATL Heisenberg antiferromagnet with moderate anisotropy ${J}^{{\prime} }/J$, a dimensional crossover from 2D to 1D, i.e., the so-called one-dimensionalization, is expected at low temperatures, where the interchain zigzag couplings ${J}^{{\prime} }$ are effectively canceled out due to geometrical frustration, and consequently a TLL-like disordered state would be realized^{52,53,54,55,56}.

Figure 2e–h shows the temperature dependence of heat capacity divided by temperature C/T for each compound. We also display C/T versus T² plots in each inset. For A₃ReO₅Br₂, C/T exhibits a linear T² dependence at low temperatures with a finite intercept, consistent with gapless spin excitations expected in a TLL-like disordered state. In addition, a hump structure appears around 5 K in the C/T–T curve of BROB (Fig. 2g). This anomaly is more likely to reflect the entropy release associated with the development of magnetic short-range correlations rather than a transition into magnetic LRO, since the magnetic susceptibility exhibits a peak at a distinct temperature of T_p ≈ 8 K (Fig. 2c). By fitting the C/T–T² curve to the equation C/T = α + βT², where the first and second terms originate from magnetic and phonon contributions, respectively, in a temperature range between 2 and 10 K for CROB and SROB and between 2 and 4 K for BROB, we obtain the values α and β for each compound as summarized in Table 1 (Note that the estimation of β for BROB has an uncertainty of ~±30% depending on the fitting range). In the 1D chain limit (${J}^{{\prime} }=0$), the T-linear contribution to the heat capacity is given by 2RT/3J⁵⁷, where R is the gas constant, so that α is inversely related to J. Indeed, α in BROB is much larger than those in CROB and SROB. This is compatible with the much weaker exchange couplings J in BROB, as suggested from a smaller value of ∣Θ_W∣. Furthermore, we find that α in A₃ReO₅Br₂ is a bit larger than that in the oxychloride counterpart A₃ReO₅Cl₂^52,53 (Table 1). This tendency also agrees with the decrease of ∣Θ_W∣ by the substitution of Cl with Br. In contrast, C/T approaches zero toward the low-temperature limit for PROC. This is owing to the occurrence of a magnetic LRO at T_N = 6.6 K, as discussed below. Accordingly, we perform a linear fit to the C/T–T² curve of PROC between 7 and 10 K (Table 1). The magnitudes of α and ∣Θ_W∣ in PROC are both intermediate between those in CROC and SROC.

High-field magnetization processes

Here, we move on to the in-field properties of all the seven A₃ReO₅X₂ families, which are useful for estimating the strengths of J and ${J}^{{\prime} }$. Figure 3 summarizes the magnetization data of BROC and BROB. The magnetic susceptibility data for BROC (Fig. 3a) are taken from ref. ⁵³, and those for BROB (Fig. 3b) are identical to Fig. 2c. The M–H curves in Fig. 3c–f represent quasi-isothermal processes obtained using a non-destructive pulsed magnet (for additional data, see Supplementary Note 3). The saturation field at 1.3 K is μ₀H_sat = 55(1) T for BROC and 42(1) T for BROB. The M–H curves exhibit a concave shape followed by an inflection point immediately below H_sat, which is a typical behavior of low-dimensional quantum magnets. The experimental saturation moment is approximately 0.84 and 0.85 μ_B/Re for BROC and BROB, respectively. These values are in good agreement with the effective magnetic moments μ_eff estimated from the Curie–Weiss fits (Table 1).

Fig. 3: Comparison of magnetization data of Ba₃ReO₅X₂ with theoretical calculations based on the OFTLM.

Panels (a) and (b) show the temperature dependence of the magnetic susceptibility measured at 7 T⁵³, where a T-independent component χ₀ is subtracted. Panels c–f show quasi-isothermal magnetization curves in the field-increasing process measured at 4.2 K (c, d) and 1.3 K (e, f) using a non-destructive pulsed magnet. Here, the field derivative of the magnetization dM/d(μ₀H) is displayed in the right axis. Experimental curves are shown by cyan lines, and theoretical curves fitted to each experimental data are shown by black lines. Theoretical M–H curves represent isothermal curves at temperatures corresponding to the experimental ones. The fitting parameters for each compound are summarized in Table 2.

Figure 4 summarizes the magnetization data of CROC, CROB, SROC, SROB, and PROC. The magnetic susceptibility data for CROC (Fig. 4a) and SROC (Fig. 4c) are taken from refs. ⁵² and ⁵³, respectively. Those for CROB (Fig. 4b), SROB (Fig. 4d), and PROC (Fig. 4e) are identical to Fig. 2a, b, and d, respectively. The M–H curves in Fig. 4f–j represent quasi-adiabatic processes with the initial temperature of T_ini = 5 K obtained using a single-turn coil system. The absolute values of M are calibrated using the M–H curves at 4.2 K obtained in a non-destructive pulsed magnet (Supplementary Note 3). For all the M–H curves, an inflection point indicative of the saturation is observed at μ₀H_sat = 83(3) T for CROC, 78(3) T for CROB, 95(5) T for SROC, 80(5) T for SROB, and 115(5) T for PROC.

Fig. 4: Comparison of magnetization data of A₃ReO₅X₂ (A = Ca, Sr, Pb) with theoretical calculations based on the OFTLM.

Panels a–e show the temperature dependence of the magnetic susceptibility measured at 7 T^52,53, where a T-independent component χ₀ is subtracted. Panels f–j show quasi-isentropic magnetization curves in the field-increasing (solid lines) and field-decreasing (dashed lines) processes measured at the initial temperature of T_ini = 5 K using a single-turn coil megagauss generator. The field derivative of the magnetization dM/d(μ₀H) in the field-decreasing process is displayed in the right axis. Experimental curves are shown by colored lines, and theoretical curves fitted to each experimental data are shown by black lines. Theoretical M–H curves represent isentropic curves for T_ini = 5 K. The fitting parameters for each compound are summarized in Table 2.

Comparison with theoretical calculations based on the OFTLM

The accurate estimation of J and ${J}^{{\prime} }$ in the quantum ATL antiferromagnet is generally challenging. One might consider achieving this by relating the Weiss temperature and the saturation field to the exchange interactions using the equations ${\Theta }_{{{{\rm{W}}}}}=-(J/2+{J}^{{\prime} })$ and $g{\mu }_{{{{\rm{B}}}}}{H}_{{{{\rm{sat}}}}}=2J{(1+{J}^{{\prime} }/2J)}^{2}$, respectively^10,36. However, the experimentally-determined Θ_W tends to underestimate the actual value, especially for small ${J}^{{\prime} }/J$, even when performing the Curie–Weiss fit to the magnetic susceptibility data in a high-temperature range¹⁰. Besides, H_sat is also highly temperature-dependent.

Recently, the OFTLM was developed⁵⁹ and applied to the calculation of the magnetic susceptibility and magnetization process in the quantum ATL antiferromagnet²². This technique has an advantage over the standard FTLM⁶⁰ in accurately evaluating the magnetic entropy at low temperatures, which is crucial for the calculation of the adiabatic magnetization process²². Here, we adopt the OFTLM to the Hamiltonian of the quantum ATL antiferromagnet in a magnetic field,

$${{{\mathcal{H}}}}=J{\sum}_{\langle i,j\rangle }{{{{\bf{S}}}}}_{i}\cdot {{{{\bf{S}}}}}_{j}+{J}^{{\prime} }{\sum}_{\langle i,k\rangle }{{{{\bf{S}}}}}_{i}\cdot {{{{\bf{S}}}}}_{k}-h{\sum}_{i}{S}_{i}^{z},$$

(1)

where S_i denotes a spin-1/2 operator localized on the ATL, 〈i, j〉 and 〈i, k〉 indicate pairs of neighboring sites along the intrachain and interchain directions, respectively, and h is a magnetic field applied along the z axis. Further details of the calculation are found in the Method section. Importantly, the peak temperature of the magnetic susceptibility T_p normalized to J shifts to a higher temperature side as ${J}^{{\prime} }/J$ decreases^10,22. Thanks to this feature, the strengths of J and ${J}^{{\prime} }$ can be uniquely determined once we reveal experimental values of the Landé g-factor, T_p, and H_sat, which are listed in Table 2. Here, we set the g-value using a relation ${\mu }_{{{{\rm{eff}}}}}=g\sqrt{S(S+1)}$ with S = 1/2, where μ_eff is estimated from the Curie–Weiss fit (Table 1).

Table 2 Experimental values of the peak temperature of the magnetic susceptibility T_p, the saturation magnetic field H_sat, and the Landé g factor as well as the exchange couplings J and ${J}^{{\prime} }$ estimated based on theoretical calculations using the orthogonalized finite-temperature Lanczos method in the seven A₃ReO₅X₂ compounds

For BROC and BROB, all the magnetization data can be simultaneously reproduced by theoretical curves with a single parameter set, as shown in Fig. 3: J/k_B = 20 K and ${J}^{{\prime} }/J=0.40$ for BROC, and J/k_B = 14 K and ${J}^{{\prime} }/J=0.45$ for BROB. ${J}^{{\prime} }/J$ for BROB is almost the same with that for BROC. This is reasonable given that the increase rates of d and ${d}^{{\prime} }$ by the substitution of Cl with Br are almost the same. No signatures of a 1/3-magnetization plateau are observed for both compounds in contrast to the theoretical M–H curves at 1.3 K. Note that in the OFTLM calculations, the finite-size effects are pronounced at low temperatures of k_BT/J < 0.2²², leading to fluctuations in the theoretical M–H curves for T = 1.3 K on the low-field side below the 1/3-magnetization plateau, as shown in Fig. 4e, f. We infer that the appearance of the 1/3-magnetization plateau is blurred by averaging over the M–H curves on randomly-oriented polycrystals in the present experiments using polycrystalline powder samples. Indeed, a previous ESR experiment on a CROC single crystal revealed a weak anisotropy: g = 1.88, 1.85 and 1.92 for H∥a, b, and c, respectively⁵⁴.

Similarly, we calculate magnetic susceptibility and adiabatic magnetization curves for the rest five compounds, as shown in Fig. 4. The exchange parameter set which can reproduce the M/H–T curve as well as the saturation field H_sat is obtained for each compound as follows: J/k_B = 42 K and ${J}^{{\prime} }/J=0.25$ for CROC, J/k_B = 39.5 K and ${J}^{{\prime} }/J=0.25$ for CROB, J/k_B = 47 K and ${J}^{{\prime} }/J=0.35$ for SROC, J/k_B = 40 K and ${J}^{{\prime} }/J=0.25$ for SROB, and J/k_B = 46.5 K and ${J}^{{\prime} }/J=0.35$ for PROC. The deviation between the experimental M–H curves and theoretical ones in the high-field region would be mainly due to the imperfect background subtraction in the experimental data (Note that the detection sensitivity of the magnetization pickup coil is significantly reduced near the maximum field). Given that the error in the experimental H_sat is ~±5 T, the error in the estimated value of ${J}^{{\prime} }/J$ is at most  ±0.05.

Magnetic phase transition in Ca₃ReO₅Br₂ and Pb₃ReO₅Cl₂

The ground state of the ATL quantum antiferromagnet is highly sensitive to weak perturbations such as the DM interaction and interlayer exchange couplings¹⁷. Cs₂CuCl₄ undergoes a magnetic transition into a spiral LRO phase with a magnetic propagation vector of q = (0, 0.475, 0)²⁸, which is attributed to a combined effect of magnetic frustration and the DM interaction³⁴. As for A₃ReO₅X₂, the DM interaction is prohibited for A = Sr, Ba, and Pb with the Type II structure (Supplementary Note 4), whereas this is not the case for A = Ca with the Type I structure. Indeed, a neutron scattering experiment on CROC confirmed the appearance of a spiral LRO with q = (0, 0.465, 0) below T_N = 1.13 K^52,54. Therefore, a similar magnetic transition is expected for CROB, which is isostructural with CROC. Figure 5a shows the temperature dependence of C/T measured down to 0.24 K for CROB. A clear lambda-shaped peak is observed at T_N = 1.15 K, indicating a second-order transition. Figure 5b illustrates the DM vectors in one ATL layer allowed for the Type I structure (space group Pnma)¹⁷. As discussed in ref. ¹⁷, the c component of the DM vector on the intrachain bond D_c and the a component on the diagonal bond ${D}_{a}^{{\prime} }$ are most relevant in the ground state selection. For Cs₂CuCl₄, it is proposed that the contribution of ${D}_{a}^{{\prime} }$ is significant³⁰, resulting in the spiral moment lying almost within the bc plane²⁸. On the other hand, a recent electron-spin-resonance (ESR) study suggested that the intrachain DM interaction D_c is the primary source for stabilizing the spiral LRO in CROC, as indicated by the observed shift of the soft mode in the disordered phase⁵⁵. Despite the slightly weaker exchange interactions in CROB than in CROC, they exhibit nearly the same transition temperature, suggesting that the intrachain DM interaction is slightly enhanced in CROB due to the spin-orbit coupling.

Fig. 5: Magnetic transition and its microscopic origin in CROB and PROC.

a Temperature dependence of the heat capacity divided by temperature C/T measured at 0 T using ³He for CROB. b Schematic of symmetrically-allowed DM vectors in one ATL layer in CROB¹⁷. D_i (${D}_{i}^{{\prime} }$) (i = a, b, c) indicates the i component of the DM vector on the J (${J}^{{\prime} }$) bond. In the neighboring ATL layer, all the DM components except for D_c orient in the opposite direction. Temperature dependence of the magnetic susceptibility M/H measured at various magnetic fields (c) and the heat capacity divided by temperature C/T at 0 and 7 T (d) for PROC. The inset of panel c shows a magnetization curve at 2 K measured in a static magnetic field. The gray dashed line is a guide for the eye to clarify the slope change of the magnetization curve. e Schematic of the Wannier orbitals around the Pb/Sr and Re atoms for PROC (top) and BROC (bottom) revealed by the first-principles calculations. The paths of intralayer exchange ${J}^{{\prime} }$ and interlayer exchange ${J}^{{\prime\prime}}$ and ${J}^{{\prime\prime} {\prime} }$ couplings are depicted by red arrows. The illustrations in panels (b) and (e) are drawn with VESTA software⁸⁰.

In addition, we observe a magnetic transition at T_N = 6.6 K for PROC. As shown in Fig. 5c, the M/H–T curve at 0.1 T exhibits a sudden drop below T_N, indicating an AFM phase transition. M/H below T_N gradually rises with increasing an applied magnetic field up to 7 T, suggesting a spin-flop transition (see also the M–H curve at 2 K shown in the inset of Fig. 5c). The C/T–T curve exhibits a broad lambda-shaped anomaly indicative of a second-order transition, as shown in Fig. 5d. This anomaly persists at almost the same temperature until 7 T, indicating that the AFM or field-induced spin-canted phases are robust against a higher magnetic field. The magnetic contribution of C is found to be proportional to T³ below T_N (Supplementary Note 5), suggesting the presence of 3D magnetic correlations, i.e., interlayer exchange couplings. This is confirmed by the first-principles calculations: the strengths of interlayer exchange couplings, ${J}^{{\prime\prime}}$ and ${J}^{{\prime\prime} {\prime} }$, defined as shown in Fig. 5e, are estimated to ${J}^{{\prime\prime}}$/J = 0.006 and ${J}^{{\prime\prime} {\prime} }/J=0.190$ for PROC, whereas both ${J}^{{\prime\prime}}$/J and ${J}^{{\prime\prime} {\prime} }/J$ are much smaller than 0.01 for other compounds (Supplementary Note 6). The relatively strong ${J}^{{\prime\prime} {\prime} }$ coupling can be attributed to the electron hopping via Pb 6p orbital.

Absence of magnetic LRO down to 60 mK in Ba₃ReO₅Cl₂

As the DM interaction is absent and the interlayer exchange couplings are negligibly weak, A₃ReO₅X₂ with A = Ba or Sr may serve as an ideal ATL quantum antiferromagnet, in which magnetic LRO is theoretically not predicted^12,16,21. This scenario is supported by muon spin relaxation (μSR) measurements on BROC down to 60 mK, as shown in Fig. 6a (see Supplementary Note 7 for additional data at higher temperatures). When cooled below 1 K in zero magnetic field, a gradual increase of the spin relaxation rate was observed. However, the spin relaxing behavior is not accompanied by spin precession, which typically indicates magnetic ordering, and the spin depolarization is recovered when a relatively weak longitudinal magnetic field of 10 mT is applied. More notably, the time spectra measured under longitudinal fields cannot be adequately described by the conventional dynamical Kubo–Toyabe (KT) function. We therefore employ the so-called undecouplable Gaussian KT function⁶¹ (solid lines in Fig. 6a), a model that accounts for dynamical local spin fluctuations arising from sporadic breakdowns of the spin-singlet state (Supplementary Note 7). Our analysis reveal that the μSR time spectra observed below 1 K can be interpreted as a slowing down of local spin fluctuations with decreasing temperature. These results strongly suggest the presence of a dynamically fluctuating local magnetic field persisting down to the lowest measured temperature of 60 mK, rather than a static internal magnetic field.

Fig. 6: Spin dynamics and excitation spectra in BROC.

a μSR time spectra measured at 60 mK under longitudinal fields of 0, 2, 5, and 10 mT. The solid lines represent the undecouplable Gaussian Kubo–Toyabe (uGKT) relaxation functions fitted to the corresponding experimental data. Each dataset and its fitted curve are shown in the same color. b Color contour map of INS intensities as a function of wavevector Q, measured at 4 K with an incident neutron energy of E_i = 9.70 meV. c Constant-Q cuts of panel (b), sliced at ∣Q∣ = [0.35, 0.45], [0.45, 0.55], [0.55, 0.65], [0.65, 0.75], [0.75, 0.85], and [0.85, 0.95] Å⁻¹ (the average ∣Q∣ value is indicated in the graph). Each Plot is vertically shifted by 0.2 for clarity.

To confirm the TLL-like behavior, INS experiments were performed on a polycrystalline sample of BROC. Figure 6b shows the INS intensity map as a function of wavevector ∣Q∣, measured at E_i = 9.70 meV at 4 K. The INS intensity is concentrated around ∣Q∣ = 0.6–0.7 Å⁻¹, indicating the development of short-range AFM correlations. This feature is consistent with expectations for a 1D antiferromagnet, where powder-averaged spin excitation spectrum exhibits a prominent gapless character above ∣Q₀∣ = π/d, where d is the intrachain atomic distance. For BROC, ∣Q₀∣ of 0.54 Å⁻¹ obtained from d = 5.79 Å⁵³ well agrees with the position of the observed gapless excitation. In addition, a signature of weak dispersion along the interchain direction is observed. As shown in the constant-Q cuts of the INS spectra in Fig. 6c, the intensity peak slightly shifts toward higher energy transfer at ∣Q∣ < 0.6 Å⁻¹. This trend is also observed in measurements at higher E_i, confirming its intrinsic origin (Supplementary Note 8). In a purely 1D antiferromagnet, constant-Q cuts are expected to show a broad peak at a fixed energy transfer of πJ/2, where the spinon density of states is maximal. The peak position and its shift are reproduced by simulating the spin excitation spectrum based on the Random phase approximation^2,54 (Supplementary Note 8). The exchange parameters J and ${J}^{{\prime} }/J$ are estimated to be 21 K and 0.25, respectively. The value of J is in good agreement with that obtained from the magnetization measurements (Table 2). On the other hand, the simulation tends to underestimate ${J}^{{\prime} }/J$; the values of 0.15 for CROC⁵⁴ and 0.25 for BROC are roughly 60% of those listed in Table 2. This discrepancy may be attributed to the renormalization effect on ${J}^{{\prime} }$, which is neglected in the simulation.

Discussion

We demonstrate that all three previously reported compounds A₃ReO₅Cl₂ (A = Ca, Sr, Ba)^51,53 can accommodate the substitution of Cl with Br. The halide substitution results in the elongation of both d and ${d}^{{\prime} }$ in the ATL composed of Re⁶⁺ ions (Fig. 1c). These elongations lead to a weakening of the exchange couplings J and ${J}^{{\prime} }$, as indicated by the decrease in ∣Θ_W∣ and the increase in α ( ∝ J⁻¹) (Table 1). Notably, this tendency is in contrast with another model compound of an ATL quantum antiferromagnet Cs₂CuCl₄, where the substitution of Cl with Br leads to the enhancement of J and ${J}^{{\prime} }$³⁶. We also reveal that the anisotropy ${J}^{{\prime} }/J$ in A₃ReO₅X₂ ranges from 0.25 to 0.45, although the low-temperature magnetism is commonly characterized by one-dimensionalization irrespective of the types of the A cation and X anion. The halide substitution does not significantly modify ${J}^{{\prime} }/J$ for A = Ca and Ba, while a substantial decrease in ${J}^{{\prime} }/J$ (from 0.35 to 0.25) is found for A = Sr (Table 2). This can be attributed to a relatively larger increase in ${d}^{{\prime} }$ (~1.7%) compared to d (~1.2%) for A = Sr (Fig. 1c).

The most fascinating aspect of A₃ReO₅X₂ is the controllability of perturbative interactions, which is responsible for selecting the ground state (Fig. 1d). CROC and CROB (Group 1) undergo a magnetic transition at T_N = 1.13⁵² and 1.15 K (Fig. 5a), respectively, driven by the uniform DM interaction allowed by the crystallographic symmetry. Further ESR studies are needed to quantitatively evaluate the strength of the DM interaction relative to the exchange couplings in CROB^34,55,62. The DM-induced incommesurate spiral order is expected as the ground state of CROC⁵⁴ and CROB. Theoretically, the application of a magnetic field on the ATL quantum antiferromagnet with the DM interaction can induce successive phase transitions associated with the emergence of versatile LRO phases, such as commensurate AFM and incommensurate cone phases, aside from the low-field spiral and forced ferromagnetic phases^14,17. Indeed, Cs₂CuCl₄ exhibits multiple phase transitions for H∥b and c³³, and notably, the phase diagram becomes more complicated under pressure³⁷. While the interchain DM interaction ${D}_{a}^{{\prime} }$ plays a crucial role in determining the magnetic structure of Cs₂CuCl₄, it is the intrachain DM interaction D_c that is expected to be more relevant for the magnetic structures of CROC and CROB. Comprehensive magnetization measurements on CROC and CROB single crystals below T_N will provide direct insight into the role of the interchain DM interaction in shaping the field-induced phases of ATL quantum antiferromagnets. PROC (Group 2) undergoes a magnetic transition at T_N = 6.6 K due to the interlayer exchange couplings via the Re–O–Pb–O–Re superexchange path, which arises from the hybridization of the O 2p and Pb 6p orbitals. The exchange paths of the interlayer ${J}^{{\prime\prime}}$ and ${J}^{{\prime\prime} {\prime} }$ couplings in PROC (Fig. 5e) are more complex than those considered in the previous theoretical work¹⁷. It is necessary to construct a new theoretical model in line with the spin Hamiltonian of PROC. The rest four A₃ReO₅X₂ compounds with A = Sr or Ba (Group 3) are close to the ideal ATL quantum antiferromagnet free from the interlayer exchange and DM interactions. Since ${J}^{{\prime} }/J$ for these compounds is smaller than 0.6, their ground states are expected to be a TLL-like gapless QSL^12,16,21. The diversity in ${J}^{{\prime} }/J$ values (0.25–0.45) offers an excellent opportunity to meticulously investigate the impact of magnetic frustration on the TLL state. Our μSR experiments confirm the absence of a magnetic transition down to 60 mK in BROC. TLL-like behavior is also observed in the INS spectra obtained from polycrystalline BROC samples. A detailed characterization of magnetic excitations, such as the search for triplon, in the disordered ground state, using single-crystalline BROC with a relatively higher ${J}^{{\prime} }/J$ ratio (${J}^{{\prime} }/J\approx 0.40$), would be essential for elucidating the nature of the spin liquid state in the 2D quantum antiferromagnet.

Methods

Sample preparation

Polycrystalline samples of A₃ReO₅X₂ (A = Ca, Sr, Ba, Pb; X = Cl, Br) and an isostructural nonmagnetic compound Pb₃WO₅Cl₂ were synthesized by conventional solid-state reaction. AO, AX₂, and ReO₃ (WO₃) were mixed in a molar ratio of 2:1:1 in an argon-filled glovebox. The mixture was pressed into a pellet and wrapped in gold foil, and then sealed in an evacuated quartz tube. All the samples were sintered twice with an intermediate grinding step. The sintering conditions for each compound are listed in Supplementary Table S1. Single crystals of A₃ReO₅Br₂ (A = Ca, Sr, Ba) with a size of ~50 μm were grown by partially melting a stoichiometric mixture in an evacuated quartz tube at 1050, 1030, 900 ^∘C for 20 h for A = Ca, Sr, and Ba, respectively.

X-ray diffraction measurement and structural analysis

The synthesized polycrystalline samples were characterized by X-ray diffraction (XRD) measurements using a Rigaku RINT-2500 diffractometer with Cu Kα radiation. All samples used for physical property measurements were confirmed to be single phase without any noticeable impurities. The crystal structures of Ca₃ReO₅Br₂ and Sr₃ReO₅Br₂ were determined by single-crystal XRD measurements using a Rigaku R-AXIS RAPID IP diffractometer with monochromated Mo Kα radiation. The structures were solved by direct methods and refined by full-matrix least-squares methods on ∣F²∣ using the SHELXL2013 software. The crystal structure of Ba₃ReO₅Br₂ was determined by powder XRD measurements using a Rigaku SmartLab diffractometer with Cu Kα1 radiation monochromated by a Ge(111)-Johansson-type monochromator. The structure of Pb₃ReO₅Br₂ was determined by powder XRD using synchrotron radiation (wavelength λ = 0.4013 Å) on BL02B2 at a synchrotron facility SPring-8 in Japan. Rietveld refinements for Ba₃ReO₅Br₂ and Pb₃ReO₅Cl₂ were performed using the RIETAN-FP program⁶³, based on the structural model of Ba₃ReO₅Cl₂⁵³. All measurements were performed at room temperature.

Physical property measurements

Magnetization measurements up to 7 T were performed using a commercial magnetometer (MPMS-3, Quantum Design). Heat capacity measurements were performed in zero field by a thermal relaxation method using a commercial cryostat equipped with a superconducting magnet (PPMS, Quantum Design) down to 2 K and by a quasi-adiabatic heat-pulse method in a ³He cryostat down to 0.24 K. Magnetization measurements up to  ~60 T and  ~130 T were performed by the induction method using a non-destructive pulsed magnet (~4 ms duration) and a horizontal single-turn coil system (~8 μs duration)⁵⁸, respectively. Polycrystalline samples were used in all the magnetization measurements. A₃ReO₅Br₂ (A = Ca, Sr, Ba) single crystals and Pb₃ReO₅Cl₂ polycrystalline samples were used in the heat capacity measurements.

Muon spin relaxation experiment

μSR experiments were performed at the muon D1 area of the Materials and Life Science Experimental Facility (MLF) at J-PARC, Japan. Polycrystalline samples of Ba₃ReO₅Cl₂ were glued onto a silver plate sample holder using GE varnish and mounted on a dilution refrigerator for cooling. The sample was irradiated with a surface muon beam that was nearly 100% polarized along the beam axis. A time histogram of decay positron events, corresponding to the muon lifetime of ~2.2 μs, was recorded using detectors in the μSR spectrometer. The asymmetry of decay events between the forward and backward counters, which is proportional to the muon spin polarization, was recorded and converted into a time-dependent spin polarization. A longitudinal magnetic field of up to 10 T was applied along the beam axis using a Helmholtz coil installed in the μSR spectrometer. For zero-field measurements, the magnetic field near the sample was actively maintained below 10 μT by using a triaxial correction coil and four triaxial fluxgate magnetometers, which monitored the local magnetic environment to compensate for stray fields originating from beamline magnets and other sources.

Inelastic neutron scattering experiment

Inelastic neutron scattering (INS) experiments were performed using the cold-neutron disk chopper spectrometer AMATERAS installed in the MLF⁶⁴. We used polycrystalline samples of Ba₃ReO₅Cl₂ with the weight of 11.63 g. To reduce absorption at low-scattering angles, the polycrystalline samples were shaped in 9 platelike blocks with the dimension of 8.8 × 8.8 × 3.5 mm³. Incident neutrons were set 11^∘ away from the perpendicular direction of the plane. Incident neutron energies were set to E_i = 20.9, 9.70, 5.57, and 3.61 meV (ΔE = 0.85, 0.29, 0.13, and 0.070 meV, respectively) and E_i = 42.1, 15.2, 7.75, and 4.69 meV (ΔE = 1.0, 0.28, 0.12, and 0.068 meV, respectively) using repetition multiplication⁶⁵. The sample was cooled down to 4K using a bottom-loading closed-cycle referigerator. All the data collected were analyzed using the software suite UTSUSEMI⁶⁶. The effect of absorption is corrected from a density estimated from the weight and the dimension⁶⁷. The observed spectra were fitted by the weakly coupled 1D model based on the random phase approximation^2,54. See Supplementary Note 8 for details^68,69,70.

Orthogonalized finite-temperature Lanczos method

The finite-temperature Lanczos method (FTLM) is useful for the analysis of frustrated quantum spin systems⁶⁰, though the orthogonalized finite-temperature Lanczos method (OFTLM) is a more accurate method than the standard FTLM, particularly at low temperatures^22,59. In the OFTLM, we first calculate several low-lying exact eigenvectors $\left\vert {\Psi }_{i,m}\right\rangle$ with N_V levels, where we define the order of the corresponding eigenvalues {E_i,m} as ${E}_{0,m}\le {E}_{1,m}\le \cdots \le {E}_{{N}_{V}-1,m}$. By using a normalized random initial vector with ${S}_{{{{\rm{tot}}}}}^{z}=m$, $\left\vert {V}_{r,m}\right\rangle$, we calculate the modulated random vector

$$\left\vert {V}_{r,m}^{{\prime} }\right\rangle=\left[I-{\sum }_{i=0}^{{N}_{V}-1}\left\vert {\Psi }_{i,m}\right\rangle \left\langle {\Psi }_{i,m}\right\vert \right]\left\vert {V}_{r,m}\right\rangle,$$

(2)

with normalization

$$\left\vert {V}_{r,m}^{{\prime} }\right\rangle \Rightarrow \frac{\left\vert {V}_{r,m}^{{\prime} }\right\rangle }{\sqrt{\langle {V}_{r,m}^{{\prime} }| {V}_{r,m}^{{\prime} }\rangle }}.$$

(3)

The partition function of the OFTLM is obtained as follows:

$$Z{(T,h)}_{{{{\rm{OFTL}}}}}={\sum }_{m=-{M}_{{{{\rm{sat}}}}}}^{{M}_{{{{\rm{sat}}}}}}\left[\frac{{N}_{{{{\rm{st}}}}}^{(m)}-{N}_{V}}{R}{\sum }_{r=1}^{R}{\sum }_{j=0}^{{M}_{L}-1}\right.\\ \left.{e}^{-\beta {\epsilon }_{j,m}^{(r)}(h)}| \langle {V}_{r,m}^{{\prime} }| {\psi }_{j,m}^{r}\rangle {| }^{2}+{\sum }_{i=0}^{{N}_{V}-1}{e}^{-\beta {E}_{i,m}(h)}\right],$$

(4)

where M_sat is the saturation magnetization, ${N}_{{{{\rm{st}}}}}^{(m)}$ is the dimension of the Hilbert subspace with ${S}_{{{{\rm{tot}}}}}^{z}=m$, R denotes the number of random samplings of the OFTLM, M_L denotes the dimension of the Krylov subspace, β is the inverse temperature 1/k_BT, and $\left\vert {\psi }_{j,m}^{r}\right\rangle$ [${\epsilon }_{j,m}^{(r)}(h)$] are the eigenvectors (eigenvalues) in the M_L-th Krylov subspace with ${S}_{{{{\rm{tot}}}}}^{z}=m$. The energy E(T, h)_OFTL, magnetization M(T, h)_OFTL, magnetic susceptibility χ(T)_OFTL, and magnetic entropy S_mag(T, h)_OFTL are obtained as follows:

$$E{(T,h)}_{{{{\rm{OFTL}}}}}= \frac{1}{Z{(T,h)}_{{{{\rm{OFTL}}}}}}{\sum }_{m=-{M}_{{{{\rm{sat}}}}}}^{{M}_{{{{\rm{sat}}}}}}\left[\frac{{N}_{{{{\rm{st}}}}}^{(m)}-{N}_{V}}{R}\right.\\ \times {\sum }_{r=1}^{R}{\sum }_{j=0}^{{M}_{L}-1}{\epsilon }_{j,m}^{(r)}(h){e}^{-\beta {\epsilon }_{j,m}^{(r)}(h)}| \langle {V}_{r,m}^{{\prime} }| {\psi }_{j,m}^{r}\rangle {| }^{2}\\ +\left.{\sum }_{i=0}^{{N}_{V}-1}{E}_{i,m}(h){e}^{-\beta {E}_{i,m}(h)}\right],$$

(5)

$$M{(T,h)}_{{{{\rm{OFTL}}}}}= \frac{1}{Z{(T,h)}_{{{{\rm{OFTL}}}}}}{\sum }_{m=-{M}_{{{{\rm{sat}}}}}}^{{M}_{{{{\rm{sat}}}}}}\left[\frac{{N}_{{{{\rm{st}}}}}^{(m)}-{N}_{V}}{R}\right.\\ \times {\sum }_{r=1}^{R}{\sum }_{j=0}^{{M}_{L}-1}m{e}^{-\beta {\epsilon }_{j,m}^{(r)}(h)}| \langle {V}_{r,m}^{{\prime} }| {\psi }_{j,m}^{r}\rangle {| }^{2}\\ +\left.{\sum }_{i=0}^{{N}_{V}-1}m{e}^{-\beta {E}_{i,m}(h)}\right],$$

(6)

$$\chi {(T)}_{{{{\rm{OFTL}}}}}= \frac{1}{{k}_{{{{\rm{B}}}}}TZ{(T,0)}_{{{{\rm{OFTL}}}}}}{\sum }_{m=-{M}_{{{{\rm{sat}}}}}}^{{M}_{{{{\rm{sat}}}}}}\left[\frac{{N}_{{{{\rm{st}}}}}^{(m)}-{N}_{V}}{R}\right.\\ \times {\sum }_{r=1}^{R}{\sum }_{j=0}^{{M}_{L}-1}{m}^{2}{e}^{-\beta {\epsilon }_{j,m}^{(r)}}| \langle {V}_{r,m}^{{\prime} }| {\psi }_{j,m}^{r}\rangle {| }^{2}\\ +\left.{\sum }_{i=0}^{{N}_{V}-1}{m}^{2}{e}^{-\beta {E}_{i,m}}\right],$$

(7)

$${S}_{{{{\rm{mag}}}}}{(T,h)}_{{{{\rm{OFTL}}}}}=\frac{E{(T,h)}_{{{{\rm{OFTL}}}}}}{T}-{k}_{{{{\rm{B}}}}}\ln Z{(T,h)}_{{{{\rm{OFTL}}}}}.$$

(8)

Since the last terms in Eqs. (4)–(7) are exact values, they are more accurate than those obtained using the standard FTLM, particularly at low temperatures.

In the adiabatic processes, the entropy remains constant while the temperature changes. The temperature under the adiabatic processes can be determined based on the following relationship with h:

$${S}_{{{{\rm{mag}}}}}{(T,h)}_{{{{\rm{OFTL}}}}}-{S}_{{{{\rm{mag}}}}}{({T}_{{{{\rm{ini}}}}},0)}_{{{{\rm{OFTL}}}}}=0,$$

(9)

where ${S}_{{{{\rm{mag}}}}}{({T}_{{{{\rm{ini}}}}},0)}_{{{{\rm{OFTL}}}}}$ corresponds to the initial entropy, and T_ini represents the initial temperature. We solved Eq. (9) for T using the Newton–Raphson method, and then substituted the obtained T into Eq. (6) to calculate the magnetization under the adiabatic processes.

We performed OFTLM calculations for a cluster of 36 sites under periodic boundary conditions with R = 10, N_V = 4, and M_L = 100. Our calculations have revealed that there are almost no finite-size effects in the magnetization for k_BT/J > 0.1 and in the magnetic susceptibility for k_BT/J > 0.2²². Therefore, the analysis of the magnetic susceptibility and magnetization curves in this study is sufficiently accurate.

First-principles calculations

First-principles calculations are performed based on density functional theory (DFT) using the program package Quantum ESPRESSO⁷¹, which employs plane waves and pseudopotentials to describe the Kohn–Sham orbitals and the crystalline potential, respectively. We set the plane-wave cutoff for a wave function to 120 Ry, suitable for the optimized norm-conserving pseudopotentials⁷² provided in the SG15 pseudopotential library⁷³. The exchange and correlation effect is described with the Perdew–Burke–Ernzerhof functional⁷⁴ based on the generalized gradient correction. We perform the Brillouin-zone integral on 5 × 10 × 5 (10 × 10 × 5) k-point grids for the charge-density calculation of the Type I (Type II) structure by using the optimized tetrahedron method⁷⁵. The fluctuated position of halogen atoms is fixed to the middle of the fluctuation to make the structure suitable for the DFT calculation. We computed the maximally localized Wannier function (MLWF)⁷⁶ associated with the 5d_xy orbitals of the Re atoms. The onsite screened Coulomb integral (U) is computed with the constrained random phase approximation (cRPA)⁷⁷. For the calculations of MLWF and cRPA, we use the RESPACK⁷⁸ program package. The susceptibility for the screened Coulomb interaction in Type I (II) materials is computed with 878 (439) empty bands whose upper limit is 35 eV from the Fermi level. The plane-wave cutoff for the susceptibility is set to 12 eV.