Ferromagnetism out of charge fluctuation of strongly correlated electrons in $\kappa$-(BEDT-TTF)$_2$Hg(SCN)$_2$Br

We perform magnetic susceptibility and magnetic torque measurements on the organic $\kappa$-(BEDT-TTF)$_2$Hg(SCN)$_2$Br, which is recently suggested to host an exotic quantum dipole-liquid in its low-temperature insulating phase. Below the metal-insulator transition temperature, the magnetic susceptibility follows a Curie-Weiss law with a positive Curie-Weiss temperature, and a particular $M\propto \sqrt{H}$ curve is observed. The emergent ferromagnetically interacting spins amount to about 1/6 of the full spin moment of localized charges. Taking account of the possible inhomogeneous quasi-charge-order that forms a dipole-liquid, we construct a model of antiferromagnetically interacting spin chains in two adjacent charge-ordered domains, which are coupled via fluctuating charges on a Mott-dimer at the boundary. We find that the charge fluctuations can draw a weak ferromagnetic moment out of the spin singlet domains.


INTRODUCTION
Typical phase transitions in condensed matter accompany either universal critical singularities or the competitions between two different orderings. The former is easily converted to the latter when additional degrees of freedom become relevant. In reality, there often appear intermediate situations where the interplay of several degrees of freedom affects the nature of the growth of correlations and low-lying excitations. In such cases, the phase transitions at low temperatures can be easily masked, and the order parameters suffer intrinsic inhomogeneities. Historical examples are the dynamically disordered charge stripes in high-T c superconducting cuprates [1,2], and the orbital-disorders that trigger the colossal magnetoresistance in manganites [3]. The complexity of dealing with multiple correlated degrees of freedom such as charge, spin, orbital, and lattice often makes it difficult to pin down their dominant mechanism.
Organic κ-(BEDT-TTF) 2 X materials ( Fig. 1(a) and (b)) become an ideal playground to study such an issue in a simpler setup. These materials form quarter-filled two-dimensional strongly correlated electronic systems, where molecular dimer (BEDT-TTF) 2 connected by a large transfer integral t d (see Fig. 1(b)) serves as a lattice site of a Mott insulator by hosting one charge per dimer. In these Mott insulators, the spin-1/2 interact antiferromagnetically as shown Fig. 1(c) and a quantum spin liquid phase is observed in κ-(BEDT-TTF) 2 Cu 2 (CN) 3 (abbreviated as κ-CN) as well as a typical antiferromagnetism in κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl (κ-Cu-Cl) [4,5]. If the degree of dimerization, namely the ratio of t d to other inter-dimer transfer integrals (t B , t p , and t q ), is weakened, the charges would no longer stay on the dimer-orbital, but rather localize to one side of the dimerized molecules to gain the inter-molecular Coulomb interaction V i j , result-ing in a charge-ordered phase. A charge degree of freedom enclosed in the dimer is interpreted as quantum electric dipole [6,7], which is detected by anomalous frequencydependence of dielectricity in many materials including κ-CN [7], κ-Cu-Cl [8], and EtMe 3 Sb[Pd(dmit) 2 ] 2 [9,10]. In that context, the dimer Mott and charge-ordered phases are interpreted as para and ferroelectricity [6,11], separated by a typical Ising type second-order phase transition (see the phase diagram in Fig. 1(d)). However, when the universal criticality of dipoles couples to magnetism or lattice degrees of freedom, this transition can be masked and some inhomogeneous phases may emerge [12]. Indeed, the subtleties of the transition are recently disclosed by the fresh members of this family, κ-(BEDT-TTF) 2 Hg(SCN) 2 Br (κ-Hg-Br) and κ-(BEDT-TTF) 2 Hg(SCN) 2 Cl (κ-Hg-Cl), which have a relatively weak dimerization [13] and fill the empty region of materials parameter space. In contrast to a simple Mott insulator, which shows a crossover from the high-temperature metallic regime [4], these compounds show an abrupt increase of resistivity at the metal-insulator (MI) transition [14]. Raman spectroscopy reveals a distinct charge order in κ-Hg-Cl in the temperature range 15-30 K [15] whereas κ-Hg-Br does not show any sign of regular charge ordering down to lowest temperature [16]. The absence of magnetic order in κ-Hg-Br is also shown by the specific heat measurements down to 100 mK [16]. A picture of "quantum dipole liquid" is provided as an interpretation to the latter intriguing phase [16], possibly consisting of dynamical charge-ordered domains enclosing electric dipole moments maximally amounting to 0.1e per dimer.
We report the experimental evidence of intrinsic ferromagnetic exchange interactions emerging in the clean bulk crystal The long molecular axis of BEDT-TTF is tilted by ∼17 • from the a-axis. (b) BEDT-TTF molecules in the b-c plane and the transfer integrals estimated as (t d , t B , t p , t q ) = (126, 83, 60, 40) meV in Ref. [13,17]. (c) Schematic illustration of the dimer Mott insulator and charge order. Charges on Mott-dimers (red circles) carry spin-1/2, and the exchange interactions between them J AF ∝ t 2 B (vertical) and J AF ∝ (t p − t q ) 2 (diagonal directions) form an antiferromagnetic triangular lattice (red lines). In the charge ordered state, the charges localized on one side of the dimer (green circles) form an antiferromagnetic quasi-one-dimensional spin-1/2 chain of J AF ∝ t 2 q (green line). Inset: the Mott-dimer state is the linear combination of charge located on the left/right molecule, supported by large t d . The charge order keeps the charge on one side of the dimerized pair to avoid the inter-site (inter-dimer) Coulomb interaction V i j = V B , V p , V q (indices follow those of t i j ). (d) Phase diagram (schematic) [18] of the present system for low energy effective model of charges proposed in Ref. [19], where (V p − V q ) and V B account for the Coulomb interactions between charges on different dimers in diagonal and vertical directions, respectively. When V i j /2t d 0.5-1, the charge order is realized. According to the first principles-based evaluation [20], charges contribute. We find that the M-H curve at low temperature follows M ∝ √ H, showing a very rapid onset with small field. Although the square-root onset of the M-H curve is well-known for a gapped quasi-one-dimensional quantum magnet near the critical field [21,22], it is qualitatively different from the present gapless M ∝ √ H that continues up to a large field. It does not resemble any of the M-H profile of the magnetism known so far such as the H-linear antiferromagnetic magnetization or the paramagnetic Brillouin curve. Such robust ferromagnetic Curie-Weiss law just above the antiferromagnetic singlet ground state can be scarcely found in nature, except for those originating from magnetic impurities or a spin glass, both of which are excluded in the present case by the lack of remnant filed or hysteresis in magnetization. Since no existing theory on bulk magnetism both for the localized spins or itinerant electrons can be applied, we construct a synergetic quantum-spin model that includes the effect of charge fluctuation. The starting point is the low temperature inhomogeneous state of charges that appear by masking the phase transition in Fig. 1(d). We take account of already existing idea of a short charge correlation length and the robustly remaining charge fluctuation at the simplest level [16,18,19]. The model represents spins on two charge-ordered domains which couple to dimer-spins carried by fluctuating charges at the domain boundary, and successfully shows how ferromagnetic behavior can originate from the charge fluctuation. The theory thus explains the properties disclosed by the magnetic susceptibility and torque measurements.  Fig. 2(a)). Above this temperature, χ M (T ) shows a Pauli-paramagnetic behavior, while below T MI it starts to increase abruptly on lowering the temperature. A Curie-Weiss fit for 20-70 K (the solid line in Fig. 2(b)) gives a positive Θ CW ∼ 16 K with the Curie constant C = 0.060 emu K mol −1 . This positive Θ CW provides strong evidence of a ferromagnetic interaction between spins. The C value shows that ∼1/6 of the total spins contribute to the Curie-Weiss paramagnetism with the ferromagnetic interaction. The 1/6-concentration is intrinsic to the ferromagnetic behavior, as χ M (T ) does not depend on the measured field strength below 5 T in this temperature range (see Fig. S2 in Supplemental Material (SM) [23]). Similar χ M (T ) was observed previously [24]. However, our data shows the ferromagnetic Θ CW more clearly in a wider temperature range (see Section A in SM [23] for a comparison). To the best of our knowledge, this compound is the first to show such ferromagnetic behavior in a family of organic Mott insulators κ-(BEDT-TTF) 2 X and X[Pd(dmit) 2 ] 2 . If one interprets this χ as the one from the ferromagnetic Heisenberg chain [25], the effective ferromagnetic coupling constant is evaluated as J F ∼ Θ CW /Θ = 53 K, with Θ = 0.3036.

Magnetization measurements
The particular ferromagnetic behavior is also found in the field dependence of the magnetization M. The linear M-H ! "  Brillouin curve (the dashed line in Fig. 2(c)). Remarkably, we find that M exhibits a particular field dependence of M ∝ √ H as shown in the solid lines in Fig. 2(c) and the inset.

Magnetic torque measurements
This √ H dependence of M is further confirmed to persist up to 17.5 T by our magnetic torque measurements done for one single crystal. Figure 3(a) shows the field dependence of the magnetic torque obtained from a fixed-angle high-field torque measurements at 0.12 and 1.7 K. As shown in Fig. 3(a), the magnetic torque shows the field dependence of H 3/2 (the dashed line in Fig. 3(a)). Given the form of the magnetic torque M × H, the field dependence of H 3/2 shows M ∝ √ H. Note that the free impurity spins are not respon-sible for this magnetization, since otherwise the saturation should take place at ∼ 1 T for 0.12 K, which is not observed in our data. Throughout the whole sweep of H, the M-H curve shows neither a remnant field nor a hysteresis (Fig. 2(c)). The absence of hysteresis is further confirmed down to the lowest fields by our magnetic torque measurements ( Fig. 3(b)). Our data excludes the spin-glass based weak ferromagnetism picture presented in the previous study [24,26], because both a ferromagnetic state [27] and a spin glass state [28] is known to exhibit clear hystereses in the torque measurements from a remnant field and a frozen moment, respectively.
To investigate the magnetic state below T MI in detail, we measured the magnetic torque curves by rotating the magnetic field in the a-c plane ( Fig. 4(a)), where θ denotes the angle between the field and the a-axis (see Fig. 4(d)). The magnetic torque signal, τ mag = τ 2θ sin 2(θ − φ 2θ ), is obtained after subtracting the sin θ component that comes from the gravity of the sample mass (see Section C in SM [23] for details). These magnetic torque curve measurements allow us to detect the magnitude of the magnetic anisotropy, which is proportional to the amplitude of the 2θ component divided by H 2 ( Fig. 4(b)), and the direction of the magnetic principle axis by the phase φ 2θ (Fig. 4(c)). In the metallic T > T MI phase, φ 2θ stays at around −20 degree, which is close to the angle between the long axis of BEDT-TTF molecules and the a-axis (see Fig. 1(a)), showing that the magnetic anisotropy comes from the spins on the BEDT-TTF dimers [29][30][31].
At T MI , φ 2θ shows a sharp jump which is followed by a rapid shift of φ 2θ toward zero, while at the same time τ 2θ stays nearly temperature independent in contrast to the increase of χ M (T ). These contrasting temperature dependences indicate that the magnetic principle axis varies concomitantly with the decrease of the magnetic anisotropy below T MI . Since Raman [16] and IR [14] vibration measurements observed no change of the phonon spectrum below T MI , the change of φ 2θ cannot be attributed to the rotation of the BEDT-TTF molecules. Therefore, this φ 2θ shift is given by an emergence of a magnetic easy axis parallel to the a axis caused by the ferromagnetic interaction appearing below T MI . A similar but much smaller phase shift has been observed in κ-CN [31], which may be ascribed to an additional moment from valence bond defects [32]. We further find a characteristic temperature T * ∼ 24 K. Below T * , φ 2θ drops, τ 2θ /H 2 increases, and both τ 2θ /H 2 and φ 2θ starts to depend on the field strength. The increase of the magnetic anisotropy particularly developing below T * is consistent with the anisotropy of χ M (T ) observed in the previous measurement [24], supporting the magnetic origin of these temperature changes. This temperature-dependent change is larger for lower fields; as we saw in Fig. 3(b) the torque data at |H| 0.5 T changes its sign below 20 K. Another bump-like feature in φ 2θ is observed around 7 K, implying a further change of the magnetic state. These features might be related to the changes of the relaxation times observed in NMR measurements done at higher fields [33]. We thus observed a distinct change in the magnetic property already starting below T MI via two torque parame-ters. Further magnetic torque measurements performed in a dilution refrigerator reveal no change in the magnetic torque below 2 K ( Fig. 3(a)), showing a saturation of the temperature dependence.

DISCUSSION
Our magnetic measurements on κ-Hg-Br disclose an unconventional magnetic state, which to our best knowledge has never been observed, in the other family members of κ- H at low temperatures, and the large change in the direction of the magnetic principle axis.
Let us first start by elucidating the way the charges are localized at T < T MI . Most of the previously known κ-(BEDT-TTF) 2 X become a dimer Mott insulator depicted schematically in Fig. 1(c). In a Mott phase, the dominant magnetic interactions between the spins carried by the localized charge are always antiferromagnetic as they originate from the kinetic exchange as, where t i j and V i j are the transfer integrals and inter-molecular Coulomb interaction along the exchange bond, and U is the on-molecular Coulomb interaction. Then, the antiferromagnetic order of κ-Cu-Cl and quantum spin liquid nature of κ-CN are roughly understood by the square-like and triangular lattice geometry of J AF which amounts to 500 K [34] and 250 K [35], respectively [36]. Therefore, the positive Θ CW observed in κ-Hg-Br cannot be explained by the magnetism of a dimer Mott insulator.
In fact, the abrupt increase of resistivity just below T MI in both κ-Hg-Br and κ-Hg-Cl is different from the crossover behavior usually observed in dimer Mott materials [4], signaling some sort of translational symmetry breaking of charge distribution. However, the Raman spectroscopy measurements indicate the absence of static charge ordering in κ-Hg-Br [16]. A scenario compatible with all these findings is the dynamical and inhomogeneous charge distribution in between the dimer Mott and charge-ordered state. The intra-dimer transfer integral from the first-principles calculation on κ-Hg-Br is t d ∼ 120 meV [13,17], much weaker than the typical value ∼ 200 meV of the κ-salts [37], and thus a quasi-charge-order by the inter-dimer Coulomb interactions is a reasonable expectation.
In Fig. 1(d), we locate κ-Hg-Br and κ-Hg-Cl according to the first-principles based evaluations [20]. In the uniform charge-ordered case possibly realized in κ-Hg-Cl, J AF forms long quasi-one-dimensional (1D) chains (see the green lines in Fig. 1(c)). Here the vertical stripe charge configuration is possibly favored for the Coulomb-interaction-strength of V q < V p , V B of the material [20].
When the static and bulk charge order is no longer stabilized in κ-Hg-Br, these chains shall break up into short fragments separated by Mott-dimers, as shown in Fig. 5(a). The way to construct the domain is not really random; we assume that the chain length N roughly corresponds to the correlation length of charges, and a Mott-dimer is inserted between the chains running in the t q direction, while in reality sometimes there will be a connections with the dimers through t p in the other directions. Inside the 1D fragment the spins interact along the t q -bonds via J AF ∼ 170-300 K, which will give t d /J AF ∼ 4-8 (see Section D in SM [23]). The charge on a Mott-dimer fluctuates, with fluctuation parameters defined by t d values. During these fluctuations, the charge (and relevant spin) occupies either left or right molecule on the dimer, and interacts with S = 1/2(green circle) at the adjacent left/right end of the chain via J AF . For the charge configuration shown in Fig. 5(a), J AF shown in green and red bonds have the same amplitude.
To elucidate how these quantum fluctuations modify the dominant antiferromagnetism, we construct a synergetic model [38] consisting of two open chains with N L and N R spin-1/2's and a single electron with S = 1/2 (which we call dimer-spin) as shown in Fig. 5(d). The Hamiltonian is given as whereŜ iγ is the spin on site-i on left and right chain (γ = L/R), c † L/R and c L/R are the creation and annihilation operator of charges on the left/right molecule of the dimer with its number operator n L/R = c † L/R c L/R , andŜ d is the dimer spin. This model cuts out the locally interacting manifold of spins shown in Fig. 5(a). Such charge configurations behind the model are expected for κ-Hg-Br at temperatures less than T MI , where we find no indication of long range order of both charges and spins. The details of electronic state below T * are not really known, but the present model does not contradict with the experimental reports given so far.
The model is solved numerically by combining the exact diagonalization calculation [23]. Since total-S z of Eq. (S2) is a conserved quantity, we analyze the model by dividing the Hilbert space into different S z -sectors, and evaluating the lowest energy levels for each sector. Figure 5(c) shows S L · S R between spins on left and right chains, S γ = j∈γ S j for several different series of N γ and system length N. One finds a dominant ferromagnetic correlation ( S L · S R > 0) for large portions of the two lowest excited states. Representative spatial distribution of spin moments for slightly polarized state is shown in Fig. 5(d); the contribution from the constituents of the wave function with dimer-spin on the right and left are separately drawn. The left-upper panel is a typical spin distribution with two-fold periodic Friedel oscillation generated by the two open edges of the chain [39]. The dimer-spin hops back and forth, and mixes quantum mechanically with spins on closer edges of the chains and suppresses their moments. The moments are redistributed throughout the chains and are accumulated densely on the further edges from the center. They point in the same orientation mediated by fluctuating spins closer to the dimer-spin (top panel of Fig. 5(d)). This interplay of t d and J AF generates a robust quantum ferro- The theory explains the square-root behavior of M-H curve at low temperature in κ-Hg-Br. Our calculations show that the ground state of Eq. (S2) with even N is always nonmagnetic and has dominant antiferromagnetic correlation. Let us consider exciting a magnetic moment by applying a magnetic field. Suppose that for an isolated chain with fixed N L and N R , the lowest eigenenergy of Eq. (S2) for each S z sector is given as E(S z ). In an applied field H, the system acquires a finite magnetization S z that gives the minimum of energy E H = min S z E(S z ) − S z H . The "magnetization curve" at finite N L , N R is given as such that H(S z ) = ∆E/∆S z , where ∆E is the energy difference E(S z ) − E(S z − 1) for ∆S z = 1. As mentioned earlier, the magnetism of the short range charge ordered phase shall be described by the assemblage of small magnetic subsystems, interacting with each other, connected with more than two neighboring subsystems. Since the information on the distribution of the chain length is missing, and since the calculation is dealing with only two interacting segments, the direct comparison of the theory and experiments may seem difficult. However, we find that an unbiased comparison is possible as shown in Fig. 5(e), where we plotted the magnetization density S z /N as a function of ∆E/N of the subsystems with various different N L and N R . Here, since all the data form a universal square-root curve regardless of the chain length N, it can be interpreted as a stochastic magnetization curve against magnetic field H/J AF . As found in the logarithmic plot, the functional form, ∆E/N ∼ C S z /N, always holds regardless of the length of the chains, while the constant C may depend on the ratio of N L and N R . The universal square-root behavior insensitive to N means that the energy is determined locally. Accordingly, if we consider a bulk assemblage of segments of chains connected by Mottdimers, their energy shall be an extensitve quantity, i.e. the summation of local energy gains. Therefore, we consider this functional form to be intrinsic.
The experimental data is plotted together in Fig. 5(e) where we add the magnetic torque data (data in Fig. 3(a) divided by H) into the field dependence of the magnetization data by SQUID (the data in Fig. 2(c) as it is) so that M estimated from τ 2θ /H coincides to M of the SQUID data at 5 T. The horizontal axis of the experimental data is determined by the value of J AF , and is illustrated for the two parameter choices of J AF /k B = 170 K and 300 K discussed in Section D of the SM [23]. The shaded region represents the vertical range over which the absolute value of the magnetic moment may vary if the distribution of chain lengths has a large variance, and hence one may state that the theory shows good qualitative agreement with the experiment for any comparable choice of J AF .
An extrapolation of the experimental data in Fig. 5(e) shows that the magnetization reaches µ B /6 at about 20-30 T. Therefore, approximately, the field strength of 20-30 T which is comparable to T * = 24 K, gives the energy scale to excite the 1/6 magnetic moment from the nonmagnetic ground state. At the temperature range T * < T < T MI , such 1/6 moment is thermally excited and contributes to the ferromagnetism; the ferromagnetically coherent orientation of the moment would contribute to the phase shift of φ 2θ . There, M-H curve no longer has a square-root, because the low energy magnetic excitations are smeared out. The energy scale of µ B H ∼ 0.1J AF to have the µ B /6 moment is consistent with 20-30 T.
Also, the preserved SU(2) symmetry in Eq. (1) matches with the restored isotropy in the magnetic torque at T < T MI . Notice that this ferromagnetic phase is not a long-range order but a correlation because of the one-dimensionality, as can also be suggested from the lack of the hysteresis. Below T * the nonmagnetic ground state component becomes dominant. From Raman spectroscopy measurements, the static charge ordering is excluded, whereas the broad peak in ν 2 mode is still compatible with quasi-charge-ordered domains with a variant charge disproportionation maximally amounting to ±0.1e, which are coherently fluctuating together inside the domain with a frequency estimated as 1.3 THz [14,16].
While evaluating the precise character of the charge distribution is beyond the scope of any theory currently available, in Section F of SM [23] we provide a phenomenological treatment performed by assuming a functional form for ξ(T ) that is valid throughout the critical regime. Within this approach, we show that χ manifests a Curie-Weiss-like behavior that reflects the ferromagnetic correlations between thermally excited spins at temperatures T > T * . Further experimental information concerning the functional form of ξ(T ) is required to verify this type of treatment.
Once the temperature falls below T * , the dipole (charge) degrees of freedom become correlated over a length scale ξ whose T -dependence saturates, and fluctuates slowly together at a corresponding timescale. These fluctuations can safely be integrated out (see Ref. [6]), leading to the effective model of Eq. (1) for spins with antiferromagnetic interactions on chain segments of average length ξ.
One may suspect that the spin models with extrinsic impurities can also explain the phenomena. Although the possibility of spin glass is experimentally excluded, its quantum analogue, the random singlet phase may share similar feature with the present magnetism [40]; most of the spin moments form a singlet and the remaining spins may contribute to the magnetism. However, for such state to happen one needs a large amount of static randomness in the distribution of J AF = J(1 ± ∆) that amounts to ∆ 0.6 [40], which cannot happen in the present system.
Naively, our ferromagnetism can be viewed as a local double-exchange; a single charge hops back and forth inside the dimer, and since it interacts with the spin chains on both sides, it is favorable to have both point in the same direction to maximally gain the antiferromagnetic exchange interaction. The difference from the double-exchange is that the kinetics of charge is local, keeping the insulating character, and the weakly coupled one-dimensional fragments of spin chains do not allow the development of coherent ferromagnetic long range order.
By designing a three-dimensional critical phase in between the charge order and dimer Mott insulator a two dimensional ferromagnet might be available, in which case the ferromagnetic long-range order is allowed at a finite temperature.

Materials and Methods:
Single crystals of κ-(BEDT-TTF) 2 Hg(SCN) 2 Br were grown by electrochemical oxidation of the BEDT-TTF solution [16]. In this synthesis, many single crystals with a variety of the size were obtained. We used a batch of the smaller crystals (powder samples, 3.73 mg) for the SQUID measurements and picked up larger ones for the magnetic torque measurements.
The magnetic susceptibility of powder samples was measured by SQUID for 1.6-300 K. The magnetic torque measurements were carried out for one single crystal with dimensions 0.75×0.57×0.13 mm 3 attached to a piezo-resistive cantilever by a tiny amount of grease (see Fig. S3 Figure S1 shows the data of the magnetic susceptibility χ M (T ) of Fig. 2 in the main text (red circles), that of sample 2 (blue circles), and those from the previous report [24] consisting of two different series of data (grey line and squares). To compare to our polycrystal results of χ M (T ), the single crystal data of previous report [24] is averaged for all axes, showing a large sample variance in their results. A Curie-Weiss fit of sample 2 (the blue dashed line in Fig. S1(b)) also gives a positive Curie-Weiss temperature Θ CW ∼ 13 K with the Curie constant C = 0.069 emu K mol −1 , showing a good reproducibility of our result of sample 1. As shown in Fig. S1(b) (the dashed lines), our results for both samples show a positive Curie-Weiss temperature more clearly in a Curie-Weiss fitting for a wider temperature range. A positive Curie-Weiss temperature can also be deduced from the polycrystal data of Ref. [24] by fitting in a narrower temperature range at lower temperatures. Instead, Ref. [24] focused on the field dependence of their χ M (T ) data only in the limited range below ∼ 20 K off the Curie-Weiss region, and together with the results from the ESR measurements, argued that there exists a spin glass state with a weak ferromagnetic moment. In our case, we confirmed a reproducibility of our torque measurements done in other samples for different magnetic field orientations, confirming that there is no intrinsic sample dependence in our results.

Temperature dependence of the M-H curve at high temperatures
The temperature dependence of the M-H curve of sample 2 is shown in Fig. S2(a). As shown in Fig. S2(a), the M-H curve is linear above 20 K, which becomes non-linear at lower temperatures. At 1.7 K, the field dependence of M is well fitted to M ∝ √ H (see the solid lines in Fig. S2(a) and (b)), showing a good reproducibility of the sample 1 data shown in the main text.

Calibration of the piezo-resistive cantilever by the gravity signal
In this section, we explain how we calibrated the temperature dependence of the sensitivity of the torque cantilever by using the gravity signal. We measured the angle dependence of the torque by rotating the sample attached to a piezo- FIG. S1. The temperature dependence of the magnetic susceptibility (a) and the invserse of the magnetic susceptibility (b) of our data (filled circles) shown with the data in the previous report taken from Fig. 4 (single crystal, grey diamonds) and Fig. 5 (polycrystals, grey squares) in Ref. [24]. For the single crystal data from Fig. 4 in Ref. [24], we averaged the data taken for H a, b, and c to compare the data of polycrystals. Only the data obtained by field cool (FC) are shown.
resisitive cantilever (Fig. S3(a)) in a magnetic field. The torque signal is given by The first sin θ term represents the gravity torque coming from the sample mass, and the second sin 2θ term represents the magnetic torque (τ mag = M × H). Figure S3(b) shows a typical torque curve which consists of the gravity torque (the blue line) and the magnetic torque (the pink line). As shown in Fig. S3(b), the different oscillation frequency allows one to clearly separate these two signals. The accuracy in the estimation of the magnetic torque signal is limited by the noise of the torque signal itself (∼ 3%), which is mainly caused by the irregular motions of the rotator.
The temperature dependence of the gravity signal (Fig. S3(c)) reflects the temperature dependence of the sensitivity of the piezo-resistive cantilever, which is used to calibrate the magnetic torque signal obtained at different temperatures. We note that, although the gravity signal shows a small field dependence at low temperatures (up to ∼ 3% of the data), the ambiguity owing to this field dependence is so small (comparable to the symbol size of the plot) that the field dependence can be safely ignored. Analysis on the effective model Eq. (1) We analyze the effective Hamiltonian Eq. (1) in the main text which we rewrite here: We consider two quantum spin chains consisting of N L and N R sites, where the spins on one edge of both chains can interact also with the adjacent dimer-spin S = 1/2 when it is on the left and right side of the dimerized two molecules. The antiferromagnetic interaction, J AF is evaluated as J AF ∼ 4t 2 q /(U − V q ) where t q is the transfer integral connecting the green bond with index-q in Fig. 1(b), and U and V q are the on-molecule and inter-dimer Coulomb interaction, respectively.
The model parameters of the materials are evaluated based on the first principles calculation. First, we consider as a reference a dimer Mott insulator, κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl and κ-CN, where spin-1/2 is localized on each dimer, forming a quantum spin-1/2 Heisenberg system on a triangular lattice. The fit of the experimentally measured susceptibility by the high-temperature expansion gives J AF ∼ 500 K [34] and ∼ 250 K [35], respectively. Independently, from the interdimer transfer integral t, one can evaluate J AF = 4t 2 /U dimer with t ∼ 70 meV for κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl and 50 meV for κ-CN [37], which gives the above experimentally derived values if we take U dimer ∼ 460 meV. Simultaneously, from the first principles and ab initio and cRPA study [49] giving U ∼ 0.83 eV V ∼ 0.4 eV, and t d ∼ 200 meV, and using the formula (see Ref. [12] in the main text), we find U dimer ∼ 0.65 eV. Therefore, in reality the Coulomb interaction effect has ambiguity and may be properly reduced by about 30-40% from the ab initio and cRPA values. Here, notice that previously the Coulomb interaction on an isolated dimer was evaluated as U dimer ∼ 2t d , and since t d differs much between materials, so is U dimer in that context. However, this evaluation is valid in the limit of very large U and V = 0 in Eq. (S3), which is an unrealistic situation. Recent theoretical studies [37,49] revealed that U dimer does not depend much on materials, because the face-to-face distances between dimerized molecules, and U dimer is insensitive to the relative angles between molecules (unlike t).
If we also adopt U ∼ 0.83 eV and V q ∼ 0.4 eV in our material κ-Hg-Br, and use the first principles results t q = 40 meV [50], we find J AF = 4t 2 q /(U − V q ) ∼ 15 meV ∼ 170 K. Here, U − V q is the energy difference between the Mott state and the excited state that has doubly electron-occupied molecule, where we set (U − V q ) ∼ 430 meV. If we reduce the Coulomb interaction energy in the numerator by 40%, the value will become J AF ∼ 290 K. Notice that our evaluation does not agree with ∼70 K in Ref. [24] estimated using magnetization data between 90 and 50 K, and fitting them with negative Curie T , which is not enough precise because it depends on the fitting range. The inter-dimer transfer integral is t d = 126 meV from the same first-principles evaluation. Based on this consideration, we take J AF = 170-300 K, t d /J AF ∼ 4-8 which we adopt in the following. Our numerical results remain almost quantitatively unchanged by the variation of t d /J AF within this range.
The model (S2) is solved in a two-fold manner. We first diagonalize the Hamiltonian of a simple spin chain of length N γ with open boundary given as in unit of J AF = 1 and obtain few lowest eigeneneriges n (N γ , S z γ ) (n = 1, 2, 3 · · · ) and |N γ , S z γ n , for each fixed value of the quantized z-component of total spin, S z γ (which is integer/half-integer for even/odd N γ ). Along with this, we elso prepare a set of eigenstates, |N γ , S z γ ; ↑ n and |N γ , S z γ ; ↓ n , of where the dimer-spin S z d =↑, ↓ is attached to one edge of the spin chain. By using these low energy eigenstates as building blocks one can construct the low energy basis of Eq. (S2).
In Eq. (S2), the total S z = S z L + S z d + S z R of the whole system of size N = N L + N R + 1 is a conserved quantity, so that its low energy basis is a combination of different choices of (S z L , S z d , S z R ) in each total S z -sector. Also, depending on whether the dimer-spin S d is interacting with the left or right chain, the basis includes a variety of states. The off diagonal terms of Eq. (S2) between these basis are given for example as, which are partly displayed in Fig. S4(a). By diagonalizing the representation of Eq. (S2) spanned by the low energy basis we obtain the eigenstates as superpositions of these basis states. We denote the energy and eigenstate as E n (N L , N R , S z ) and |N L , N R , S z n , n = 0, 1, 2 · · · for each given S z sector.
Magnetic properties of the effective model Eq. (1) When the two spin chains are disconnected from the dimerspin, the lowest energy state of even-N L/R chain is a singlet, and the first excited state carries spin-1. For odd-N L/R chain, the lowest energy state already hosts spin-1/2. When they are coupled by the dimer-spins in Eq. (S2), the lowest energy state based on these singlets still remains almost nonmagnetic, whereas all the states based on states with finite magnetic moments on both chains have robust ferromagnetic correlations, which we explain here in more detail. Figure S4(b) shows E(N L , N R , S z ) for several chices of N L = N R = 8-24 chains at t d /J AF = 8. The lowest S z starts from 0 and 1/2 for even and odd N = N L + N R +1, respectively. As mentioned above, the smallest S z = 0 or 1/2 has the lowest energy, E(S z = 0 or 1/2) or for each chain length. While the shorter chain has lower E(S z = 0 or 1/2), it does not mean that the shorter N's are realized, because the length of the chain is determined in advance by the energetics of the charge degrees of freedom. Here, we are interested in the magnetic excitation energy ∆E = E(N L , N R , S z ) − E(N L , N R , S z = 0 or 1/2). We mark as shaded region above the lowest energy level up to the 1/6 concentration of the full moment. As shown for the selected three sets of (N L , N R ) in Fig. S4(c), ∆E/N follows a universal functional form. Since the derivatives of ∆E against the excited magnetic moment ∆S z gives the M-H curve, we also find a universal curve in Fig. 4(e), which follows a squareroot behavior.
Let us examine these low energy states by classifying them to three different groups; (N L , N R ) consisting of the combination of (even, even), (odd, odd) and (even, odd) numbers, and for each of them we examine the lowest and second-lowest energy states. Figure S5(a)-(c) show the spatial distribution of magnetization density, where the upper and lower panels are the ones separately calculated for the groups of basis that have dimer-spin on the right and left part of the dimer, respectively. (Fig. S4(a) left panel is the same as Fig. 5 in the main text). The wave function consists of the anti-bonding superposition of these two groups of the basis of equal weight. By further examining the composition of the basis one can simply depict the major configurations that have dominant contributions, which we show schematically on the upper part of these panels. For example, in (even,even) chain the lowest energy state S z = 1/2 (left part of (a)) consists of linear combination of three manifolds of states, |0, ↓, ⇑ , |0, ↑, 0 , | ⇑, ↓, 0 , where the three arrows/0 indicate the spins that are carried by the three parts of the system; left, dimer, and right chain, and thin ↑ / ↓ are spin-1/2, ⇑ / ⇓ are spin-1 and bold arrows are the spin-3/2. In each manifold, the dimer-spins fluctuate back and forth and exchange with spins on both sides when it is present.
The lowest energy state of (even, even) chain on the left part of Fig. S5(a) has a moment that simply fluctuates back and forth while not contributing much to the magnetization. The first excited state S z = 3/2 of (even, even) chain on the right part of Fig. S5(a) has a strong ferromagnetic correlation between S L and S R . For the (odd, odd) chain the lowest energy state (left part of Fig. S5(b)) carries spin-1/2 on both chains so that the ferromagnetic correlation develops mediated by the dimer-spin, which is further enhanced in the S z = 3/2 excited state(right part of Fig. S5(b)). The (even, odd) case has a singlet lowest energy state where all the spins die out, but the excited state has a strong ferromagnetic correlation between all spins. Figure S5(d) shows the magnetization S L of the two lowest energy states, for different N and different series of N L = N R =even, odd and N L + 1 = N R , which correspond to the above mentioned three cases. The correlation between these moments are mostly ferromagnetic, S L ·S R > 0, which is displayed in Fig. 5(c) in the main text for the same parameters by the same symbols.

Phenomenological treatment
The lack of experimental information on the correlation length ξ of charge ordering makes it difficult to precisely evaluate the thermodynamic quantities in theory. Here, we will make reasonable assumption that the correlation length grows rapidly on lowering the temperature as ξ ∝ (k B T ) −ν , where ν = 1 is the critical exponent of the two-dimensional Ising universality class which the charge ordering transition belongs to. As we discussed in the introduction part of the main text, this second order phase transition is masked at low temperature in the real material. Therefore, although the form ξ ∝ (k B T ) −ν diverges with T → 0, the increase of true ξ will gradually slow down and should stop at some temperature. In the following, we make use of ξ ∝ (k B T ) −ν , while supposing that this assumption may apply only at T T * ∼ 0.15J AF . Although the thermodynamic susceptibility we derive here in a phenomenological manner is fragile, we discuss it here because we find that it may help the understanding of the possible behavior of χ that originates from the ferromagnetic correlation of the excited state of the model (1).
Let us start with the square-root form of the experimental M-H curve, which agrees well with the energetics of our microscopic model; in Fig.5(e) in the main text, we showed the comparison between the experimental and theoretical data. From the latter data, we evaluate the functional form as M/N = α(∆E/N) 1/2 , with α = 0.56. Here, we denote the magnetization M as a continuous variable, which can be identified as discrete variable S z used in the main text.  Fig. S6(a). One finds that the envelope of the energy landscape is nearly flat at M 3, particularly when N is large, indicating that the system is indeed close to the partially polarized ferromagnetic long range ordered phase.
Let us regard the system as an ensemble of these chains. Although each spin chain is correlated with more than two other chains in reality, we use a rough assumption that the Ninsensitive functional form of the two-chain calculation shown in Fig. S6(a) will hold. Within this assumption χ is determined by a typical length scale N of the chains. N can be read off as ξ since the length of the antiferromagnetically coupled chain corresponds to the length scale of the charge ordering. We assume the gaussian distribution of N as  Figure S6(b) shows the susceptibility χ obtained by varying A = 5-10, where k B T ∼ 0.5 corresponds to about 90 K when we interpret the energy unit as J AF = 170 K. One finds that χ −1 ∝ (T − T c ) in the intermediate temperature region that resembles the experimental observation (Fig. 2(c) in the main text), while the actual value of T c depends on the parameter A and also slightly on σ. The inverse of temperature dependence ofN = N max N=N min NP(N) ∼ ξ we adopted for a given A and σ is shown in the inset of Fig. S6(b). At the highest temperature k B T ∼ 0.5, the chain length is as short as ξ ∼ 10-20 for A = 5-10, and at k B T ∼ 0.1 it grows up to ξ ∼ 50-100, where we took (N min , N max ) = (4, 500). We mask the low temperature part in the figure since the as-sumption on the functional form of ξ may at most hold only at k B T 0.15J AF , which is approximately the region T T * , whereN 70 remains short.