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Two-dimensional quarter-filled organic solids are a promising class of materials to realize the strongly correlated insulating states called dimer Mott insulator and charge order. In their conducting layer, the molecules form anisotropic triangular lattices, harboring geometrical frustration effect, which could give rise to many interesting states of matter in the two insulators and in the metals adjacent to them. This review is concerned with the theoretical studies on such issue over the past ten years, and provides the systematic understanding on exotic metals, dielectrics, and spin liquids, which are the consequences of the competing correlation and fluctuation under frustration.

The study on

The electronic systems have the charge and the orbital degrees of freedom besides the spins. If each electron is localized on a single atomic orbital, the system is reduced to the above mentioned magnets with short range interactions. However, as the charges fluctuate between different orbitals, the spins couple to the orbital degrees of freedom. Further, when electrons start to delocalize, the charge degrees of freedom play a major role. Thus, either the spin exchange interactions, coupled orbital-spin exchanges, and the Coulomb interactions, or their combinations come into play, depending on their energy scales. Frustration could work on all of them, which is the key distinction from the simple magnets. In the normal metallic state well described in reciprocal-space, the frustration often based on the real space geometry of interactions is not of direct importance. For this reason, the insulating states driven by the strong correlation and the metallic states in their very vicinity are our main focus, wherein the frustration effect is amenable to laboratory studies.

Two-dimensional organic solids could offer great possibility on this matter because of the following potentials they hold; they reveal a class of insulators called the charge order and the dimer Mott, which are the consequences of strong electronic interactions and of their particular carrier number called “quarter-filling". Besides, the molecules in the conducting layers provide a variety of lattice geometry, which are mostly based on a triangular motif, reminiscent of the geometrical frustration. Thus, the two insulating states, when exposed under the frustration effect, yield many interesting and often exotic phenomena. Many theories and experiments during this decade could indeed be understood in this context.

In the present article, the term “strong coupling" refers to the electronic interactions being far stronger than the kinetic energy scale, and the “weak coupling" to the opposite case. The plan of this review is as follows. In

In lattice models with long range interactions, the electrons may localize to maximally avoid each other, when the commensurability of the carrier number to the lattice is satisfied. A classical sketch in _{e}/N_{e}

Schematic illustration of the insulating states on a lattice at 1/(2_{e}/N

The “quarter-filling" of electrons is typically realized in a family of organic materials called charge transfer salts consisting of two kinds of molecules, _{2}

Let us present an overall picture of what we could expect in these material systems. The band structures in the non-interacting limit provides the starting point to understand the features of each actual organic material. They are complicated due to a variety of transfer integrals (

(_{d}. The axis of “anisotropy" of the crystals in (

Now, if we consider a strong inter-molecular Coulomb interactions, _{d}, rules what degrees of freedom participate in the frustration effect. When the dimerization is absent, the charge degrees of freedom under competing

As the dimerization (_{d}) develops, the electron becomes confined within the dimer (dimer Mott insulator), but is still delocalized within the dimer, giving rise to _{d} and

Further stronger dimerization will effectively kill the dielectric moments, and the spin degrees of freedom start to play a major role. This time, by weakening the interaction strength, and in approaching the metal-insulator boundary (not shown), the virtual spacial fluctuation (by

Thus, which kind of interactions (e.g.,

The standard model to describe such three-quarter-filled strongly correlated electronic states is the extended Hubbard model. In two-dimension particularly under frustration, the extreme difficulty of dealing with even this relatively simple model requires a strategy, e.g., to put the problem to some controlled limit, which often means taking some extreme limit of the model wherein the problem simplifies and becomes solvable. One needs to identify which of the specific limit privileges which physical aspect, and choose that limit best adapted to the physical phenomenon under consideration. Several different strong coupling models we discuss here are thus obtained as the limiting cases of the three-quarter-filled extended Hubbard model, as can be viewed in a sketch of

Relationships between the extended Hubbard model and other strong coupling effective models derived from it. These models are categorized into spinless models (_{d} and _{d} being those between the dimerized two sites. _{eff}, _{eff}, _{eff} are the ones on/between dimers.

The Hamiltonian of the extended Hubbard model is given explicitly as,

Here, _{i}_{i↑} + _{i↓}. _{ij} are the on-site and inter-site Coulomb interactions respectively, and _{ij} is the transfer integral between sites _{ij} as those between nearest neighbor molecules (sites) unless otherwise noted.

In molecular solids, the roles played by atomic orbitals in inorganics are replaced with those by molecular orbitals, whereby the transfer integrals are the overlaps of molecular orbitals often provided by the extended Hückel method [

The estimate of the Coulomb interactions is more challenging. Empirical quantum chemistry calculations are performed [^{bare} ~ 4 − 4.5 eV [^{bare}/_{eff}/_{eff }~ 10–15, _{eff}/U_{eff}_{eff }^{bare}, but are still definitely in a strong coupling regime. Note that the downfolding estimation of _{eff}/_{eff} is somewhat larger than expected (see the discussions in [

Going back to the models, one way to simplify the problem is to neglect the spin degrees of freedom, by reducing the extended Hubbard model to a

The physical situation of three electrons (one hole) per two sites implies a half-filled band of spinless fermions,

The strong coupling limit of the _{ij}/_{ij} = ∞, is the Ising model in terms of classical particles,

where the particle number _{i}_{ij} = 0 -limit of the _{i}_{i}

(_{0} = 0), one (_{d}. The approximation to the half-filled Hubbard model (dimer approximation) corresponds to confining the basis of a dimer to those belonging to

To capture another important aspect of the extended Hubbard model, we introduce an intrinsic (non-spontaneous) dimerization, represented by the large intra-dimer transfer integral, _{d}. It is natural to look at the system in such a way that the dimers serve as a unit of a lattice instead of the sites.

On _{d} = 0 (and _{ij} = 0), a half-filled single band Hubbard model

is derived by taking the four lowest eigen-energy-states of zero- (_{eff}_{d}, at _{d} > 0, the above discussion may easily break down. There are two problems. Firstly, the on-dimer Coulomb interactions are modified to

and for realistic values of _{d}, this naturally exceeds the empirical estimation, 2_{d}, adopted to materials. The second problem is more serious. The energy level, _{d} to near

Straightforward way to avoid these problems is to further make _{d }→ ∞, _{eff}^{z}_{d} = ∞, by _{d} > 0 yields [

which is nothing but a transverse Ising model. Here, _{ij}’s connecting the sites _{d}) of electron,

In Equation (6), all the spin configurations were degenerate, which is lifted by a higher order perturbation from the extended Hubbard model. The second order perturbation in terms of _{ij} > 0 (≠ _{d}) gives,

Besides the first Heisenberg term, the direct coupling of dipolar and spin degrees of freedom appears. Particularly, the second term reminds of a Kugel–Khomskii Hamiltonian studied in manganites [

All the above models described in

where _{i}_{i}_{i}^{4},u_{i}^{4}

We present here some introductory key issues in frustrated systems. The system is “frustrated" whenever it cannot minimize its total energy without sacrificing some of the local interaction energies. To be phenomenological, the “frustrated system" shall be characterized by the dense low energy structure of its many body states. The simplest way to capture it is to put the classical particles on the vertices of a unit of triangle or a square and count their energies according to Equation (3). If the two particles locate in both edges of a solid/broken bond, the repulsive/attractive interaction _{ij} =

Note here that restricting the discussion to the nearest neighbor (local) interactions is not intrinsic. The classical degeneracy occurs on a geometrically unfrustrated square lattice when the longer range interactions are added. In fact, it is known that the localized spin systems with long range exchange interactions are intrinsically frustrated, irrespective of the lattice dimensionality or geometry of the lattice [

One of the highlights in frustrated systems is an “order by disorder mechanism" [_{ij}, respectively. These effects are the origins of the exotic metallic states in

In the frustrated systems, a severe competition among such dense low energy states often make it extremely difficult to determine the stable lowest energy states. Then, the above classical Ising particle system provides “the controlled limit" (see the beginning of

(

Besides the basic extended Hubbard model at 3/4-filling, we apply the strong coupling effective models to focus on the role of frustration. In the first part, the topics on frustrated metals are discussed by the spinless Ising and

The charges are frustrated when their inter-site Coulomb interactions compete with each other. A good starting point to understand such effect is the Ising model of classical and spinless particles in Equation (3) on an anisotropic triangular lattice, with two-different types of nearest-neighbor interactions, _{ij} = ^{N}^{/3} fold degeneracy. Now, take another hexagon edged by gray and white sites, and turn all the three grays black. Then, one could again flip the black inside to white, which costs no energy. This game could generate numerous contingent states of the same energy one by one, which is the origin of the macroscopic degeneracy.

Low energy structure of the classical Ising model, _{ising} = _{ij}V_{ij}n_{i}n_{j}_{d}_{d}

Energetics of the classical Ising model (Equation (3)) on the anisotropic triangular lattice. (_{d}/N

(

Another series of degenerate states are found on the other side of the axis, ^{L}

Still, the classical story does not end until we refer to the other series of the degenerate excited states which form dense energy levels, in between the three-sublattice group and the chain-stripe group. We start from one of the chain stripe ground states, and introduce a “defect" into a charge order, _{d }/ L_{d}/N

Hotta _{d}

The second example of “order by disorder" is the one driven by the quantum fluctuation, ^{2}_{d}

Nishimoto and Hotta analyzed the

Does the inclusion of the spin degrees of freedom blur the above exotic solid-liquid picture? This question, not simple enough, is later on studied by Cano-Cortes ^{−3}^{−4}^{−4}

Numerical results of the

Numerical results of the 3/4-filled extended Hubbard model on the anisotropic triangular lattice. (

_{A}_{B}_{C}

The latest exact diagonalization study by Cano-Cortes ^{2} and ^{2}/3 with _{e}/N_{A}_{B}_{C}_{A}_{B}_{C}_{A}_{B}_{C}^{2}/3 = 0.08. The boundary between the two,

Numerical results of the 3/4-filled extended Hubbard model on the anisotropic triangular lattice, by Cano-Cortes ^{2} = (0.75/2)^{2} = 0.5625, which increases to 0.66 in the three-fold state. ^{2}/3 and ^{2} for the pinball and the three-fold charge order, respectively; (

There still remains some intriguing issues on the homogeneous metallic state in the weaker coupling region of the phase diagram, _{h}_{↑}_{↓}

The excited states of the frustrated systems include as much rich physics as their ground states. Back to the Ising model in

Now, if we include a higher order perturbation, the separation of defects increases the energy linear to the distance, _{d}_{4}, by the fourth order ring exchange process on a plaquette, shown schematically in _{d}ε_{4}

(_{e}_{d}_{k}

When the doped particle propagates along the stripes with no classical energy cost, none of the above particular features appear. It just behaves as a one-dimensional free particle. Thus, at

Besides the above charged excitation, the neutral excitations have the collective nature. Again, recall the 2^{L}_{4}. The same operation generates the equally spaced (

Excitation spectra of (_{ij} = _{i}_{j}

While this review is focused on the anisotropic triangular lattice, there are other cases providing the physics of frustration. One example can be found on a square lattice (_{ij}’s in the extended Hubbard model to longer range Coulomb interaction, _{ij} = _{0}/ | _{i}_{j}_{MI}, which is generally expected to happen only in the strong coupling insulators. The origin of this unusual collective excitation is depicted in

Experimental studies started to reveal that many organic crystals are potential dielectrics, insulators which respond to the external electric field. Here, we particularly have in mind the spontaneous polarization of electric moment, ferroelectricity and antiferroelectricity. The realization of “electronic ferroelectricity", which is the one not driven by the ionic displacement of the lattices but originates from the modification of the electronic wave function, is actually discussed in the context of charge ordering in one of the charge transfer salts [

Yoshioka _{d} are enough large, the ground states in one-dimension (

(_{d} > _{1D}_{1D}_{d} in bold red bonds) _{2}

We now extend the discussions to the two-dimensional dimerized lattices, e.g., so-called _{d }> 0 while keeping other _{ij} zero (or small enough), one could specify the one-dimensional dimerized chains running along the _{d} costs a classical energy of Δ_{d} and Δ

(

Hotta proposed a transverse Ising model (Equation (6)) on an effective lattice in unit of dimers to study the above issue directly [_{d}, to the classical Ising model (see _{m}_{i}_{d}

The correlated nature of dipoles influence the magnetic property, which is studied by combining the above transverse Ising terms with the Kugel–Khomskii-type of term (Equation (7)), which we call the dipole-spin model [_{m} ∙ S_{n}

To consider its origin, focus on two sites belonging to different dimers connected by a single bond, as shown in _{z}^{2}_{z}

A quantum disordered “spin-liquid" state breaking no global spin or lattice symmetries has been pursued ever since Anderson proposed an exotic resonating valence bond liquid of spin singlets [_{4} is the ring-exchange coupling constant [_{4} in the localized spin system is generally considered to be weaker by the orders of magnitude than this critical value, a more realistic model was in need.

A breakthrough was brought by the inclusion of the charge degrees of freedom. Morita

The current consensus here could at least arrive at the presence of three phases, where NMI exists over a relatively wide region, _{x}^{2}_{−y2}

(^{2} The phase boundary between the Neél and the spiral orders by the spin wave theory [

_{c}_{c}_{c}_{1} ~ 7.4_{c}_{c}_{1}, at which point a true first order transition takes place.

Yang _{1,0}) is the four-spin interactions on a diamond plaquette (

Numerical results by Yang

Still, the detailed nature of the NMI remains unclear. The QP-PIRG study indicates an unconventional magnetically gapless state, consisting of degenerate states with different wave numbers and different total spins [

The amount of theoretical works dedicated to the organic material systems over the past ten years are the approaches to the extended Hubbard model sometimes including the electron-lattice terms from a weak-coupling point of view. Their two main issues should be (1) How the shape of the Fermi surface, or the energy dispersions near the Fermi level, is deformed by the inclusion of Coulomb interactions, and (2) How the electron-lattice coupling modify the electronic states. The deformation of the Fermi surface has been discussed in Hubbard models in the context of high-_{c}_{F}_{F}

The weak coupling approach starts from the non-interacting band and treats the interaction effect perturbatively. The static charge susceptibility in the random phase approximation (RPA) is given in the form,

where _{ij}, and the Green’s function of the phonons

In contrast, in the fluctuation exchange (FLEX) approximation, _{c}_{0} (

Yoshimi _{0} (_{c}_{c}_{1}, as can bee seen from a growth of peak in _{1} characterizes the three-fold charge ordering, which is basically the one discussed in the context of charge frustration in

(_{c}_{0} and _{c}_{1} and Q_{2} wave numbers.

The electron-lattice coupling aids the instability at the other wave vector, Q_{2}, to compete with Q_{1}. Udagawa and Motome studied the electron-lattice coupling effect to the extended Hubbard model by the RPA [_{1}, but could deal with the competition between the two energy scales in a simpler manner (see _{c}_{1})-peak with no doubt originates from the _{1}) -peak, which reflects the spacial geometry of _{c}_{2})-peak is due to a bare nesting instability in _{0} (_{c}_{2}) is significantly enhanced, and this could be ascribed to the electron-lattice coupling, _{H} (Equation (8)), through the _{2}) -term in Equation (10).

Recall that the ground state of the extended Hubbard model is a diagonal stripe (Q_{2}) or a checkerboard stripe charge order (see _{2} _{P}) of lattice modulations, and found one Peierls coupling which stabilizes the horizontal stripe charge order at _{ij} along charge rich/poor horizontal stripe. Accordingly, the exchange coupling of spin chains (_{P} >0, this modulation is stable even in the three-fold charge ordered state, in which the texture of charges show coexistence of two periods [_{1} and Q_{2}, when the electronic correlation is taken into account.

An interplay of spins with charges and bonds is pursued in the exact diagonalization study by Dayal _{i}_{i}_{i}

Exact diagonalization results by Dayal _{P} = 1.1 (square type along _{H} = 0.1, and _{P}_{H}_{i}_{i}

Most of the theories we went through so far have been motivated by the experimental studies on two families of organic crystals, _{2} _{2}

What triggered increasing interests on frustration effect in organic crystals was a story of an “organic thyristor" in _{2}CsZn(SCN)_{4}—a nonlinear conductance characterized by the downturn of resistivity near the onset of the I-V curve [_{1} = (2/3,_{2} = (0,^{13}-NMR measurement [_{2}-X-ray intensity both show the linearly decreasing behavior against the current density, identifying the origin of the I-V characteristics [_{1}-peak is related to the three-fold charge ordered metallic state and the q_{2}-peak to the insulating horizontal stripe charge order (for corresponding theories, see _{1},q_{2},q_{1}^{′} be explained in a consistent/unified manner? ; (4) What is the most important energy scale, and is it due to disorders or related to phonons ?

The answer to (1) is given by the experiment [_{2}RbZn(SCN)_{4}. This salt displays intrinsically the same competition but at a higher temperature and with somewhat different q_{1}^{′} = (1/3,_{1}^{′}, q_{2}-volumes are exclusive, namely separating in space. The growth of the size of the q_{2}-domain (see the peak intensity) was detected by the rods of the X-ray; it develops along the interlayer direction—

Theories must rather be responsible for the remaining issues. Regarding (2), the static nonequilibrium mean-field theories confirm that the horizontal stripe order is taken over by the three-fold charge order under the applied electric field as well as at higher temperature; Yukawa and Ogata dealt with the extended Hubbard model without lattice distortions, and showed that the solid component of the three-fold charge order partially melts by the electric field, but the phase itself is stabilized thermodynamically against the horizontal charge ordering [

(_{2}CsZn(SCN)_{4}, adapted with permission from [_{1}, q_{1}^{′} and q_{2} peaks of CsZn/Co and RbZn salts; (

Kuroki [_{1} and q_{1}^{′} (issue (3)) on the RPA basis, by introducing several extra _{ij}’s between those separated by two molecular spacing. He showed that the peak positions of the charge susceptibility _{c}

The answer to the final issue may depend on what approach one takes. The bulk coherence of the phases are lost in these materials, a situation difficult to treat in theories. We ourselves speculate from the strong coupling point of view that the disorders embedded in the frustration of

A distinguished example of possible collective charge excitations is proposed in _{2}I_{3} by Ivek _{c}_{F}_{ij} may remain small enough to allow for such frustration induced excitations.

While this review is focused on the organic materials, there are some relevant inorganic materials sharing common physics. Kimber _{3}NaRu_{2}O_{9} [_{2}^{5.5+}O_{9} forms an ideally frustrated isotropic triangular lattice. A sharp metal-insulator transition accompanied by a structural distortion at _{2}^{5+}O_{9}) and (Ru_{2}^{6+}O_{9}), which is concluded from the experimental evaluation of bond-length and by the LDA + U calculation. This is an almost classical charge ordering. They also find a melting of charge ordering by the irradiation of X-rays, which lasted for periods of several hours at low temperature. The X-rays impinging on the sample generate a low concentration of holes by the photoelectric effect, which could cause exactly the same situation found theoretically in

(_{2}I_{3} along the zigzag direction. The unit cell includes four independent sites, A, A′, B, C, where A and A′ have inversion crystallographically; (_{2}I_{3} at

Most of the strongly correlated insulators become a magnet, a spin glass, or form valence bonds (spin gapped), and at present, the only remaining possibility of realizing a triangular lattice spin liquid points toward the organic candidates, _{2}Cu_{2}(CN)_{2} [_{3}Sb[Pd(dmit)_{2}]_{2} [^{1}H-NMR study on _{2}Cu_{2}(CN)_{3} pointing out the absence of magnetic orders down to 32 mK [_{3}Sb[Pd(dmit)_{2}]_{2} is also intriguing; specific heat [

Very recently, it turned out that these materials have something more than a simple quantum antiferromagnet could offer. Abdel-Jawad _{2}Cu_{2}(CN)_{3} as shown in _{c}_{2}Cu[N(CN)_{2}]Cl, at around 27 K from a dimer Mott insulator to charge ordering. At almost the same temperature, the long range antiferromagnetic ordering sets in, pointing toward the possibility of realizing multiferroics in the organics for the first time. Many experiments are thus ongoing, and at present, there are so many issues to be solved. (1) Is the “6 K anomaly" a phase transition or not, and is it related only to charges or also to spins?; (2) What is the ground state of the system?; (3) Do the low energy thermodynamic excitations include the charge degrees of freedom, and are they magnetic or nonmagnetic?

(_{2}Cu_{2}(CN)_{2} along the _{2}Cu_{2}(CN)_{2} by Mana

We proposed a new object, “quantum electric dipole" tightly bound to dimers, as a microscopic origin of the dielectric anomaly, which emerges in the dimer Mott insulating state [

The trials to reconcile the real materials with lattice models by the ab-initio downfolding calculations are performed on these material systems. Nakamura _{2}Cu(NCS)_{2} and _{2}Cu_{2}(CN)_{3} as _{eff}/t_{eff}_{eff}/t_{eff}_{eff}/t_{eff}_{eff}/t_{eff}_{2}Cu[N(CN)_{2}]Cl, EtMe_{3}Sb[Pd(dmit)_{2}]_{2}, and also the ones on the molecular-based models (not in unit of dimers) are awaited. At present, the semi-empirical evaluation of the transfer integrals at hand [_{d}/t_{B}_{2}Cu_{2}(CN)_{3}, _{2}Cu[N(CN)_{2}]Cl and EtMe_{3}Sb[Pd(dmit)_{2}]_{2}, respectively. For example, if one assumes _{B}_{d}/t_{B}_{eff}/t_{eff}_{3}Sb[Pd(dmit)_{2}]_{2} is a “clean" Mott insulator well approximated by the single band Hubbard model, whereas _{2}Cu_{2}(CN)_{3} is a “dirty" Mott insulator since the intra-dimer degrees of freedom must be accounted for in such cases where _{d}/V

Recently, the studies on dielectric properties of the organic materials by the optical second-harmonic generations (SHG) are developing. Yamamoto _{2}I_{3} discussed in the previous subsection, and found a clear SHG signal below the charge order transition at 135 K [_{2}IBr_{2} [

Transfer integrals of (_{2}Cu_{2}(CN)_{3} [_{2}Cu[N(CN)_{2}]Cl [_{3}Sb[Pd(dmit)_{2}]_{2} [_{eff}_{B}_{r}_{eff} = (_{p} −t_{q}_{p} −2t_{q}

In frustrated quantum magnets, a building block of nonmagnetic states is a pair of spins forming singlets called valence bond. The spins therein are highly correlated with other spin singlets but are still strongly fluctuating, and the degree of localization of singlet pairs could be a measure of the spin gap — spanning from a gapped valence bond solid to a possibly gapless spin liquid often represented by the terminology “the resonating valence bonds".

In the quarter-filled electronic systems, the “valence bonds" of charges on a dimer could be a key ingredient. When each charge is localized on a single “valence bond orbital" of a dimer, the charge gap opens, forming a dimer Mott insulator. If the charge gap is large enough, the electronic spins conform to a localized Heisenberg spin description. But this does not apply to the dimer Mott insulators in organic _{2} _{c}_{2}Cu_{2}(CN)_{3}, even a clear Mott gap itself is vanishing spectroscopically [_{2}Cu_{2}(CN)_{3} mainly in the latter context. These are something not attained in a laboratory study of simple frustrated magnets.

The finite structural dimerization allows for a particular classes of dielectric properties in the families of organic materials, the strongly dimerized _{2}

Finally, the simplest form of charge frustration occurs in non-dimerized quarter-filled systems, as a competition of charge orders of numerous types of stripe textures, and as an emergence of three-fold exotic metals with coexisting solid and liquid components of charges. They are the products of frustrated Coulomb interactions, and could be a source of inhomogeneity found in the _{2}

In this way, classified by the degree of dimerization, the effects of frustrations on the quarter-filled electronic systems with strong interactions could be viewed systematically, as we proposed in the preliminaries (

I acknowledge the collaboration with Nobuo Furukawa, Satoshi Nishimoto, Frank Pollmann, Kenn Kubo, Akihiko Nakagawa, Shin Miyayara, and Mitake Miyazaki, which this review is based on. I attained a great deal from the fruitful discussions and communications with Yoshio Nogami, Ichiro Terasaki, Takahiko Sasaki, Tetsuaki Itou, and Shinichiro Iwai. I thank Jaime Merino, Kaoru Yamamoto, Takashi Koretsune, and Martin Drressel for many helpful discussions in writing this article. During the studies related to this review, the author was supported by Grant-in-Aid for Scientific Research (No. 21110522, 19740218, 22014014) from the Ministry of Education, Science, Sports and Culture of Japan.

_{2}X

_{2}I

_{3}

_{2}Cu

_{2}(CN)

_{3}

In one dimension, taking

_{2}Cu

_{2}(CN)

_{3}

In the variational Monte Carlo study in [29], the plane wave function is used as a trial wave function, which is disadvantageous of realizing such long range order, to estimate the upper bound of the phase boundary, _{c}_{c}^{′}_{c}_{c}^{′}

_{2}

_{2}

The VMC could afford a largest system size of ^{−2}

_{2}

_{2}

_{2}Cu

_{2}(CN)

_{3}

The VMC calculation by Watanabe

_{2}Cu

_{2}(CN)

_{3}

_{2}Cu

_{2}(CN)

_{3}

_{2}

_{2}RbZn(SCN)

_{4}at 90K

_{2}

_{2}

_{2}

_{2}CsZn(SCN)

_{4}(

_{0.7}Zn

_{0.3})

_{2}CsZn(SCN)X

_{4}

_{2}

_{2}I

_{3}

_{2}I

_{3}

_{2}I

_{3}by synchrotron X-ray diffraction

_{3}NaRu

_{2}O

_{9}

_{2}Cu

_{2}(CN)

_{3}

_{2}Cu

_{2}(CN)

_{3}

_{2}Cu

_{2}(CN)

_{3}

_{2}]

_{2}(dmit = 1,3-dithiole-2-thione-4,5-dithiolate)

_{2}Cu[N(CN)

_{2}]

_{2}I

_{3}

_{2}S

_{4}a multiferroic relaxor?

_{2}Cu

_{2}(CN)

_{3}