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A review is given for recent theoretical studies on phase transitions in quasi-one-dimensional molecular conductors with a quarter-ﬁlled band. By lowering temperature, charge transfer salts exhibit a variety of transitions accompanying symmetry breaking, such as charge ordering, lattice dimerization, antiferromagnetic transition, spin-Peierls distortion, and so on. Analyses on microscopic quasi-one-dimensional models provide their systematic understandings, by the complementary use of different analytical and numerical techniques; they can reproduce ﬁnite-temperature phase transitions, whose results can be directly compared with experiments and give feedbacks to material design.

Low-dimensional molecular conductors exhibit interplay between electron correlation highlighted by enhanced fluctuation and coupling to the lattice degree of freedom. In the last three decades or so, charge transfer-type salts which are represented as _{2}_{2}_{2}^{−}, then a three quarter-filled HOMO band of TM molecule is realized, ^{+} such as Ag and Li, a quarter-filled LUMO band of DCNQI is realized.

Typical symmetry broken states seen in quarter-filled quasi-one-dimensional molecular conductors. The size of circles and the thickness of bonds represent charge density and the absolute value of transfer integrals, while arrows and broken line ellipses represent ordered spin moments and spin singlet formation, respectively. _{F} is the Fermi momentum, which is a measure for the periodicity for each state. CDW, SDW, CO, DM, and SP stand for charge-density-wave, spin-density-wave, charge ordered, dimer-Mott insulating, and spin-Peierls states, respectively. CO+SP and DM+SP are coexisting states of two orders.

In purely one-dimensional (1D) electronic systems, it is known that any phase transition does not occur at finite temperature (_{F} is measured for a regular 1D chain with lattice constant

2_{F} charge density wave (CDW) state: This is a coexistence of modulation of charge density and lattice distortion (bond order) with the period of 4_{F} in the case of a 1D band. Sometimes the term CDW is used just to represent a charge density modulation, regardless of its physical origin, such as the 4

2_{F} spin density wave (SDW) state: This indicates a Peierls instability-induced state as well, mostly, but with magnetic moments which are modulated with the period of 4

4_{F} charge order (CO): The intersite Coulomb interaction leads to CO with charge localized on every other site. This is essentially a strongly correlated insulator, where localized spins show up on the “charge rich” sites. We simply call this as the CO state in this review.

4_{F} bond order: 2

SP state: In the CO and DM insulators, the localized spin degree of freedom is described by the Heisenberg model. 1D Heisenberg chains are susceptible to SP states where spin singlets are aligned periodically. In the quarter-filling case their period is 4_{F} bond order emerges, while 2_{F} charge modulation is induced as well.

In this review, we will introduce theoretical results for such broken symmetry states in quarter-filled Q1D materials. First, in

In most of the charge transfer salts, low energy properties can be well described by effective models based on one MO of the molecule consisting the valence band: for the _{2}

Let us start with a very basic model for the quarter-filled Q1D molecular conductors, the 1D EHM. It consists of an array of lattice sites with nearest-neighbor transfer integral

where the operator

On the other hand, in many _{2}

For example, in the TM_{2}

We note that the mutual interaction between the next nearest-neighbor site, _{2}-term shows several coexistent states [

There are also many studies on 1D interacting electronic systems coupled to the lattice degree of freedom. Such situations implicitly include the dimensionality effect, since crystal, then the lattice structure, is a three-dimensional object. Therefore, as seen in the following sections, finite-

At quarter-filling, the electron-lattice couplings give rise to various kinds of symmetry breaking. For instance, instabilities in the charge degree of freedom include the 2_{F} CDW state by the Peierls mechanism, and the DM insulator which can be generated spontaneously out of a uniform system. Besides this, the spin degree of freedom in the insulating states is also susceptible to a lattice-coupled instability: the SP transition. Such electron-lattice coupled states are stabilized, e.g., in 1D EHM coupled to translational and/or rotational displacement of the molecules described by the modulation in the transfer integrals: the so-called Su–Schrieffer–Heeger or Peierls-type electron-phonon interaction. The model is sometimes called as the (extended) Peierls–Hubbard model. Another important coupling is that with molecular deformations,

Here we just show one typical example in

where the modulation in the transfer integral _{F} CDW [

Ground state phase diagram of the 1D EHM coupled to Peierls-type electron-lattice coupling [

The bosonization is one of the most powerful method to investigate 1D electronic systems, by which the quantum fluctuation can be fully taken into account [

First, as a basis, we briefly introduce the bosonization approach to the quarter filled 1D EHM [Equation (1)]. The low energy Hamiltonian is separated into the charge part

where _{F}-Umklapp scattering and leads to the insulating ground state with CO. On the other hand, the spin sector is essentially the same as the effective Hamiltonian of a Heisenberg chain. Therefore, the parameters

The low-energy properties of the bosonized Hamiltonians Equations (4,5) can be systematically investigated by the RG approach, where the long length scale properties are analyzed by solving the derived RG equations. Those for the charge part are written as

and for the spin part,

where

Whether the ground state is the metallic TLL or the CO insulating state is judged by the solution of the RG equations for the charge part, Equations (7,8). The metallic TLL state is indicated by

The bare parameters in the bosonized Hamiltonian, Equations (4,5), can be analytically obtained from the standard bosonization procedure, where only the matrix elements of the mutual interaction between the one-particle states close to the Fermi level are taken into account. It should be remarked that careful treatment is necessary for the 8_{F}-Umklapp scattering, namely the _{F}-Umklapp scattering comes from the third order processes, which are shown in

Lowest order diagrams expressing the 8_{F}-Umklapp scattering [

The quantitative discrepancy between the phase diagrams derived by the analytical bosonization procedure and the numerical method is ascribable, not to the procedure of using the bosonized Hamiltonian together with the RG equations, but to the choice of the initial conditions of the RG equations. As is already noted, the one-particle states around _{F}-Umklapp scattering leading to the CO state noted above. However, it is difficult to obtain the accurate values of the initial conditions by such analytical procedures. Instead, the exact diagonalization method for finite size clusters can be used, as demonstrated for the 1D EHM [

Strictly 1D models do not show any phase transition at finite

where

which enables us to investigate the finite transition temperature

We assume the CO pattern to be anti-phase between the adjacent chains,

where

Here the second term expresses the

where

where

Three distinct regions classified by the appearance of CO insulating states at the finite

The bosonization and RG technique as discussed tells us how to incorporate the correlation effects into the non-interacting system in a controllable way. However, it is not straightforward to extend it to calculate quantitatively physical observables, particularly for the strong-coupling region. Here, we introduce a newly developed scheme combined with numerical methods, by which we can calculate quantities not only above but also below

In the presence of CO, the charge part of the effective Hamiltonian is written by Equation (14 ). The RG equations are given as

The initial condition for

Temperature dependences of the spin susceptibility

In order to discuss the finite

where _{F}-Umklapp scattering due to the gap in the energy dispersion at

Overall _{2}Ag [_{2}AsF_{6} seems to be lower than

Recent experimental progresses showing intricate phase competitions and coexistences have prompted us to explore issues beyond the quarter-filled 1D EHM, where additional terms including the dimensionality effects and electron-lattice couplings give rise to rich phase diagrams. Numerical methods are powerful in investigating and have been applied to reveal the finite-

Although there are a few approximation-free results in higher dimensions, numerical approaches such as exact diagonalization [

We here discuss finite-

where

We apply two approximations to the model. The lattice displacements

Phase diagrams in the plane of

Temperature dependences of charge densities, lattice distortions, and charge and magnetic susceptibilities for

The results are summarized in the

In some molecular conductors showing SP transition, such as in MEM-(TCNQ)_{2}, it is insisted that the lattice displacements are not adiabatic [

Since we have developed well-organized ways of theoretically describing finite-_{2}Ag [

TMTTF_{2}_{6} compound drawn based on NMR measurements [

Now, several Q1D compounds without dimerization have been synthesized where a CO transition is suggested, such as (_{2}Br [_{2})_{2}_{6} and Br] [_{2} and (BCPTTF)_{2}_{6}, AsF_{6}) [_{2}PF_{6} [

From the theoretical point of view, several future directions can be listed. For example, the bosonization method introduced in

Another direction is the coupling between charge and spin degrees of freedom. In _{2}

The spin-charge coupling also appears more directly in some Q1D molecular systems, where the constituent molecule itself contains both the localized spin and itinerant carriers. A typical example is found in phthalocyanine (Pc) compounds such as TPP[Fe(Pc)(CN)_{2}]_{2} [

where the first three terms represent the 1D EHM for _{2}_{2} (_{2} and itinerant carriers on quarter-filled perylene chains interact with each other; in fact, the magnetic response suggests strong coupling of the SP and CDW states [

In summary, we have reviewed recent progresses in theoretical works on the quasi-one-dimensional molecular conductors. The studies on CO as an origin of insulating behavior in the 2:1 charge transfer salts pointed to the importance of the strong-correlation effects, not only

We are deeply grateful to Yukitoshi Motome and Masahisa Tsuchiizu for continued collaborations.

_{2}

_{2}Ag, where DI-DCNQI is 2,5-diiodo-

_{2}AsF

_{6}

_{F}and 4

_{F}Instabilities in the One-Dimensional Electron Gas

_{F}and 4

_{F}instabilities in a one-quarter-filled-band Hubbard model

_{F}and 4

_{F}instabilities in the one-dimensional Hubbard model

_{2}Ag

_{2}AgDetected by Synchrotron Radiation X-Ray Diffraction

_{2}Ag

_{2}SbF

_{6}

_{2}

_{2}]

_{2}

_{2}and Br

_{2})

_{2}

_{6}

_{2}]

_{2}PF

_{6}

_{2}SbF

_{6}

_{2}]

_{2}{TPP = tetraphenylphosphonium and [Fe(Pc)(CN)

_{2}] = dicyano(phthalocyaninato)iron(iii)}

_{2}]

_{2}

_{2}M(mnt)

_{2}, M = Au, Pt, Pd, Ni, Cu, Co and Fe

_{2}[Pt(mnt)

_{2}]