*4.1. Anion Ordering in [(TMTSF)*1−*xTMTTFx]ReO*<sup>4</sup>

As a first application of the above model, we analyze the role of CDW correlations in the mechanism of anion ordering (AO) in the concrete case of the ReO4 sulfur-selenide alloys [(TMTSF)1−*x*TMTTF*x*]ReO4.

#### 4.1.1. Experimental Features

In both (TMTTF)2*X* and (TMTSF)2*X* series of hole 1/4-filled and dimerized compounds, with non centrosymmetric anions like *X* = ReO4, NO3, BF4 . . . , the possibility arises for the anion lattice to form superstructures below the ordering temperature *T*<sup>0</sup> **<sup>q</sup>**<sup>A</sup> [22,23]. These superstructures superimpose displacement and orientation of the anions [10]. In the high-1D-temperature domain, the most frequent staggered anion ordering wave vectors are **<sup>q</sup>***A*<sup>1</sup> = ( <sup>1</sup> 2 , 1 <sup>2</sup> ) and **<sup>q</sup>***A*<sup>2</sup> = ( <sup>1</sup> <sup>2</sup> , 0), expressed in units of the reciprocal lattice vector (*a*∗, *b*∗) in the *ab* plane of the materials (here the longitudinal part <sup>1</sup> <sup>2</sup> *a*<sup>∗</sup> = 2*kF*).

In the case of the Bechgaard salt (TMTSF)2ReO4, for instance, ( <sup>1</sup> 2 , 1 <sup>2</sup> ) AO takes place at *T*0 1 2 , 1 2 177 K [22], which also coincides with a metal-insulator transition [48]. A similar AO superstructure is found for the Fabre salt (TMTTF)2ReO4 at *T*<sup>0</sup> 1 2 , 1 2 154 K, a slightly lower temperature that falls within the Mott insulating state found below *T<sup>ρ</sup>* 230 K [30,49].

An interesting situation is found for the alloys [(TMTSF)1−*x*(TMTTF)*x*]2ReO4 in solid solution [26]. At *x* = 0.55, the salt is ordered with TMTSF and TMTTF molecules alternating along the stacking *a* axis. Remarkably, the AO transition temperature drops and reaches the minimum, *T*<sup>0</sup> 1 2 , 1 2 82 K, a value significantly lower than the pure limits at *x* = 0 and *x* = 1, as shown in Figure 7 [26]. By contrast, the Mott scale reaches instead a maximum at *T<sup>ρ</sup>* > 325 K in the *x* = 0.55 alloy [26], a value significantly higher than the two pure limits. It is worthwhile to note that the variation of AO temperature in the alloys differs from the one found in the hybrid salt (TMDTDSF)2ReO4, in which each organic molecule is composed of two sulfur and two selenium atoms [50–52]. For the latter, *T*<sup>0</sup> 1 2 , 1 2 165 K, an ordering temperature which for this salt essentially falls on the midpoint between the two limits. As to the Mott scale *T<sup>ρ</sup>* 210 K, it lies below the one found in the pure sulfur limit (*x* = 1).

**Figure 7.** Anion ordering critical temperature for the ReO4 salt in different families of organic conductors and their alloys. After Ilakovac et al. [26] and references there cited.

#### 4.1.2. Electron–Anion Interaction

To examine the interplay between electronic and translational anionic degrees of freedom, we add an electron–anion interaction *HA* to the purely electron part (18). Following Riera and Poilblanc [25], this interaction can be written in the form

$$H\_{a} = \frac{1}{2} \sum\_{r,j} K\_{a} \delta\_{r,j}^{2} + \sum\_{r,j,\sigma} \left[ (\delta\_{r,j} - \lambda \delta\_{r,j-1}) m\_{r,\sigma} \right. $$

$$+ (\lambda \delta\_{r,j} - \delta\_{r,j-1}) m\_{r,\sigma} \big], \tag{43}$$

where the electron charge on even and odd sites in the dimer at *r* and chain *j* is coupled to the anionic displacement *δr*,*j*, as shown in Figure 8. The constant *λ* takes into account the two inequivalent distances between the anions and the molecular sites in the strength of the coupling strength (0 ≤ *λ* ≤ 1). Here the anion displacements are treated in the classical harmonic approximation for which the spring constant *Ka* has been rescaled *Ka*/*g*<sup>2</sup> → *Ka* in order to incorporate the anion-electron coupling strength *g*. By using the transformations (6) and (7), the expression of *Ha* for the lower band electrons, when coupled to the relevant anionic distortions *δ* <sup>1</sup> <sup>2</sup> ,*q*<sup>⊥</sup> in Fourier space, is given by

$$\begin{split} H\_{\mathfrak{a}} &= \frac{1}{2} \sum\_{q\_{\perp}} \mathbb{K}\_{\mathfrak{a}} \, \delta^{2}\_{\frac{1}{2}, q\_{\perp}} \\ &+ \frac{1}{\sqrt{N}} \sum\_{p,k,q\_{\perp}, \sigma} \xi^{p}\_{\gamma q\_{\perp}} \, \delta\_{\frac{1}{2}, q\_{\perp}} d^{\dagger}\_{p,k,\sigma} d\_{-p,k-2p\mathbf{k}\_{F}\sigma} . \end{split} \tag{44}$$

The form factors for the electron–anion interaction read

$$\xi\_0^p = (\lambda - 1) \frac{(t + \delta t)\epsilon\_0 / E\_F + ip(t - \delta t)}{\sqrt{2t^2 + 2\delta t^2}},\tag{45}$$

and

$$
\xi\_{\frac{1}{2}}^p = (\lambda + 1) \frac{(t + \delta t) + ip(t - \delta t)\epsilon\_0 / E\_F}{\sqrt{2t^2 + 2\delta t^2}},\tag{46}
$$

for the ( <sup>1</sup> <sup>2</sup> , 0) and ( <sup>1</sup> 2 , 1 <sup>2</sup> ) anion orderings. One observes that due to the parity of the electron–anion coupling and the site potential, both factors have a real and an imaginary parts. This will then introduce a phase dependent coupling to CDW.

**Figure 8.** Electron–anion (×) interaction of the Riera-Poilblanc model [25] in systems like (TMTSF)2*X*, (TMTTF)2*X* and their alloys. The arrows depict anion displacements for the ( <sup>1</sup> 2 , 1 <sup>2</sup> ) [( <sup>1</sup> <sup>2</sup> , 0)] ordering.

When the anionic part of the Hamiltonian is incorporated into the action *S*, it becomes

$$S = S\_0 + S\_I - \frac{\beta}{2} \sum\_{q\_\perp} K\_a \delta\_{\frac{1}{2}, q\_\perp}^2 + S\_{ea}.\tag{47}$$

The electron–anion coupling can be recast in the form

$$S\_{ea} = -\sqrt{\beta} \sum\_{q\_\perp} \delta\_{\frac{1}{2}, q\_\perp} e^{-i \frac{\theta\_+}{2}} \left\{ z\_{\theta\_+}^{\varepsilon} O\_{\theta\_+} \text{Re} \left[ \xi\_{q\_\perp}^{x+} e^{-i \frac{\theta\_+}{2}} \right] \right. $$

$$+ \left. i z\_{\theta\_-}^{\varepsilon} O\_{\theta\_-} \text{Re} \left[ \xi\_{q\_\perp}^{x+} e^{-i \frac{\theta\_-}{2}} \right] \right\} \tag{48}$$

where the renormalization factors *<sup>z</sup>θ*<sup>±</sup> = 1 at = 0.

Therefore in the presence of a site potential 0, which breaks the inversion symmetry within the dimer, the anion order parameter *δ* <sup>1</sup> <sup>2</sup> ,*q*<sup>⊥</sup> is coupled to the two independent CDW*θ*<sup>±</sup> of the electron gas, as shown in Figure 4. However, there is a definite form factor for each CDW which in the end plays an important role in the type of AO stabilized. This is examined next.

#### 4.1.3. Anion Ordering

The linear coupling (48) between the anion order parameter and the CDW*θ*<sup>±</sup> composite field is similar to the one encountered in Section 3.2 for the coupling of electrons to external source fields in calculation of susceptibilities *χμ*. This can be exploited to generate a Landau free energy expansion of the anion order parameter. Thus considering *Sea* as a weak perturbation besides *SI*, the successive partial integrations of electron degrees of freedom by the RG yield the effective action at the scale Λ():

$$\begin{split} \mathcal{S}[\boldsymbol{\upmu}^{\*}, \boldsymbol{\upmu}]\_{\ell} = \mathcal{S}\_{0}[\boldsymbol{\upmu}^{\*}, \boldsymbol{\upmu}]\_{\ell} + \mathcal{S}\_{I}[\boldsymbol{\upmu}^{\*}, \boldsymbol{\upmu}]\_{\ell} + \mathcal{S}\_{ea}[\boldsymbol{\upupmu}^{\*}, \boldsymbol{\upupmu}]\_{\ell} \\ &- \mathcal{S} \boldsymbol{\upupmu}[\boldsymbol{\uplomu}]\_{\ell}. \end{split} \tag{49}$$

*Crystals* **2020**, *10*, 942

This procedure then generates the Landau free energy F[*δ*] of the anionic order parameter. Up to the quadratic level it takes the form

$$\mathcal{G}[\delta]\_{\ell} = \sum\_{q\_{\perp}} \left[ \mathcal{K}\_{a}/2 - \chi\_{\frac{1}{2}, q\_{\perp}}(\ell) \right] \delta\_{\frac{1}{2}, q\_{\perp}}^{2} + \mathcal{O}(\delta^{4}),\tag{50}$$

where

$$\chi\_{\frac{1}{2},q\_{\perp}}(\ell) = \sum\_{\mu=\pm} \text{Re} \left[ \zeta\_{q\_{\perp}}^{+} e^{-i \frac{\theta\_{\mu}}{2}} \right] \chi\_{\theta\_{\mu}}^{\varepsilon}(\ell) \tag{51}$$

is the effective susceptibility involved in the AO at ( <sup>1</sup> <sup>2</sup> , *q*⊥). Relating the loop variable = ln *EF*/*T* to temperature, the zero of the quadratic coefficient leads to the following condition for the AO critical temperature *T*<sup>0</sup> 1 <sup>2</sup> ,*q*<sup>⊥</sup> ,

$$\frac{\mathbb{K}\_{\mathfrak{a}}}{2} = \chi\_{\frac{1}{2}\mathfrak{q}\_{\perp}\perp}(T^{0}\_{\frac{1}{2}\mathfrak{q}\_{\perp}}).\tag{52}$$

The strongest anion susceptibility will lead to the highest *T*<sup>0</sup> 1 <sup>2</sup> ,*q*<sup>⊥</sup> . For repulsive interactions *U* = 2*V* > 0 and non zero *Ka*, the regions of stability of both ( <sup>1</sup> 2 , 1 <sup>2</sup> ) and ( <sup>1</sup> <sup>2</sup> , 0) AO in the (*δt*, 0) plane are shown in Figure 9 for different values of *λ*.

**Figure 9.** Iso−*<sup>λ</sup>* phase boundaries between ( <sup>1</sup> 2 , 1 <sup>2</sup> ) and ( <sup>1</sup> <sup>2</sup> , 0) anion orderings, as a function of the normalized site potential and dimerization.

It is useful to first consider limiting cases for the asymmetry parameter *λ* related to the position of the anions in the unit cell. Thus for *λ* → 1, each anion is aligned with the center of a dimer and *ξ*+ <sup>0</sup> <sup>→</sup> 0, so that only the ( <sup>1</sup> 2 , 1 <sup>2</sup> ) ordering is stabilized through a coupling to both CDW*θ*<sup>±</sup> . Interestingly, the most important coupling is to the CDW*θ*<sup>+</sup> correlations which according to Figure 6 are not singular, at variance with CDW*θ*<sup>−</sup> . This indicates that the driving force of the ( <sup>1</sup> 2 , 1 <sup>2</sup> ) AO is not the result of a 2*kF* instability of the electron gas, as it is for the ordinary Peierls mechanism.

In the opposite decentered limit, where *λ* → 0, both anion couplings are finite. At *δt* = 0, they becomes equal in amplitude and couple identically to both CDW*θ*<sup>±</sup> . The phase boundary then merges with the <sup>0</sup> axis at *δt* = 0, where *θ*<sup>+</sup> = *π*/2 and *θ*<sup>−</sup> = −*π*/2 (or 3*π*/2) and anions are coupled to completely site centered CDW (see Figure 4). It follows that for such a *λ*, the ( <sup>1</sup> <sup>2</sup> , 0) phase is stable over the whole (*δ<sup>t</sup>* > 0, 0) plane. In this case, the singular CDW*θ*<sup>−</sup> correlations are the main driving force of AO.

When *λ* grows from zero in Figure 9, the region of stability of ( <sup>1</sup> 2 , 1 <sup>2</sup> ) phase starts to increase against that of ( <sup>1</sup> <sup>2</sup> , 0). According to (45) and (46), this is the consequence of a reinforcement (weakening) of the effective anion coupling Re [*ξ*<sup>+</sup> 1 2 *e*−*iθ*±/2] (Re [*ξ*<sup>+</sup> <sup>0</sup> *<sup>e</sup>*−*iθ*±/2]) to the CDW*θ*<sup>±</sup> . For the whole range of *<sup>λ</sup>*, the ( <sup>1</sup> 2 , 1 <sup>2</sup> ) AO is essentially due to the coupling to the non singular CDW*θ*<sup>+</sup> , whereas the ( <sup>1</sup> <sup>2</sup> , 0) AO is mainly coupled to the singular CDW*θ*<sup>−</sup> .

#### 4.1.4. Theory and Experiment

One can proceed to the application of the above model to the variation of anion ordering *T*<sup>1</sup> 2 , 1 2 and Mott insulating scale *T<sup>ρ</sup>* in the alloys [(TMTTF)1−*x*TMTSF*x*]ReO4. First, we fix the various parameters of the model. The average longitudinal hopping *t* of the model can be set from the results of band calculations [53,54], *t* = [364(1 − *x*) + 200*x*] meV, interpolating between the pure Fabre (*x* = 1) and the Bechgaard (*x* = 0) limits. From these results, one can also determine the modulation of the hopping, *δt* = [26(1 − *x*) + 14.5*x*] meV. From photoemission experiments [55], the difference in ionization energy between the TMTSF and TMTTF organic molecules leads to a site modulation energy of <sup>0</sup> 200 meV, in order to use 0(*x*) = 4*x*(1 − *x*)200 meV, as the effective alternating site potential as a function of *x* (this amplitude for the site potential <sup>0</sup> does not include the contribution coming from charge ordering, which is observed in (TMTTF)2ReO4 at *x* = 1 [30,49] and should be present at finite *x*. Although this contribution is not known in the alloy, its input in the calculations would further suppress anion ordering transition in Figure 10). Regarding interactions, we shall take *U* = [200(1 − *x*) + 268*x*] meV giving a stronger (weaker) site repulsive in the Fabre (Bechgaard) case. According to quantum chemistry calculations [44], the value of the nearest-neighbor interaction can be fixed at *<sup>V</sup>* <sup>=</sup> *<sup>U</sup>*/2 for all *<sup>x</sup>*. Finally, we shall link the small modulations of interactions, *<sup>δ</sup><sup>U</sup>* <sup>=</sup> <sup>−</sup> <sup>0</sup> *EF U* 3 and *δV* = *<sup>t</sup>δ<sup>t</sup> E*2 *F U* <sup>3</sup> , to those of 0, *U* and *δt*. In this way for instance, the values *U* ± *δU* and *V* ± *δV* at *x* = 0.5 corresponds to the interaction values in the Bechgaard (+) and Fabre (−) cases.

**Figure 10.** Calculated critical temperature for the ( <sup>1</sup> 2 , 1 <sup>2</sup> ) anion ordering (**green**) and Mott scale (**blue**), as a function of TMTTF concentration *x* in [(TMTSF)1−*x*(TMTTF)*x*]2ReO4 alloys.

With the above figures, the ( <sup>1</sup> 2 , 1 <sup>2</sup> ) AO will dominate for not too small *λ* and essentially arbitrary values of anion lattice stiffness *Ka*. We shall fix *Ka*/[2(1 + *λ*)2] = 1.75 (eV)−<sup>1</sup> to give from (52), the experimental value *T*<sup>0</sup> 1 2 , 1 2 180 K in the Bechgaard *x* = 0 limit [22]. Thus when the site potential 0(*x*) grows with *x*, alongside the effect of dimerization *δt*/*t*, *T*<sup>0</sup> 1 2 , 1 2 decreases due to the weakening of the coupling to non singular CDW*θ*<sup>+</sup> correlations in (51) and (52); it reaches a minimum at *x* = 0.5, namely where <sup>0</sup> is maximum, in qualitative agreement with the results of Figure 7. As *x* grows further, <sup>0</sup> goes down and *T*<sup>0</sup> 1 2 , 1 2 starts to increase and evolve toward the limiting value of 190 K in the *x* = 1 Fabre case. This value is nearly the same than for *x* = 0, but higher than the experimental value [30,49,56]. The minimum value *T*<sup>0</sup> 1 2 , 1 2 105 K at *x* = 0.5 corresponds to a decrease of 40% or so from the pure *x* = 0 value, compared with nearly 55% for experiments in Figure 7, showing a qualitative agreement (see the above note in Section 4.1.4).

In contrast to *T*<sup>0</sup> 1 2 , 1 2 , the value of the Mott scale *T<sup>ρ</sup>* in Figure 10 reaches a maximum at more than twice its value found in the *x* = 1 Fabre limit. This is a consequence of positive <sup>0</sup> and *δt*/*t* alternating potentials, whose influences add in quadrature in the *g*<sup>3</sup> expression in (19) and the value of *Tρ*. This conclusion for *Tρ* agrees with those previous works [26–29].

#### *4.2. Interplay between the Spin-Peierls and Charge Ordered States*

As a second application of our model, we examine the influence of charge ordering on the spin-Peierls instability of weakly localized 1D Mott insulators of the Fabre salts series.

#### 4.2.1. Experiments

The spin-Peierls instability is found in some members of the Fabre (TMTTF)2*X* series with *X* = PF6 and AsF6. These quarter-filled band but weakly dimerized systems show a 1D Mott insulating behavior below some temperature scale *T<sup>ρ</sup>* 220 K [57,58]. In (TMTTF)2AsF6, for instance, the Mott insulating behavior is followed at lower temperature by a continuous transition toward the formation of charge ordered (CO) state at *T*CO 103 K [29,31]. Below *T*CO, there is a charge disproportionation in the dimer unit cell leading to a finite static alternated site potential 0. Within the CO state, there is an additional instability that involves both spins and lattice degrees of freedom. According to x-ray diffuse scattering [37,39], the lattice becomes unstable with the onset of 1D lattice fluctuations at the wave vector 2*kF* below the characteristic temperature scale *T*<sup>0</sup> SP. For the AsF6 compound, *T*0 SP 40 K [10,37]. Occurring well below the Mott scale *<sup>T</sup>ρ*, *<sup>T</sup>*<sup>0</sup> SP takes place in the presence of strong antiferromagnetic correlations and then refers to a spin-Peierls (SP) instability. A true 3D SP ordering, however, occurs only at a much lower temperature, namely, *T*SP 11 K [37,39]. The latter obeys the empirical rule *<sup>T</sup>*SP ∼ *<sup>T</sup>*<sup>0</sup> SP/3 for the reduction of the ordering temperature by 1D fluctuations in weakly coupled Peierls and spin-Peierls chains.

From NMR experiments under pressure [39], *T*SP is found to increase while *T*CO is steadily decreasing. This indicates that a reduction of the charge disproportionation, that is the site alternated potential 0, enhances the SP ordering. The increase of *T*SP with pressure carries on until its amplitude reaches *T*CO, defining a critical pressure above which *T*SP undergoes a steady decrease (see Figure 11).

**Figure 11.** Critical temperatures of spin-Peierls and charge ordered states as a function of applied pressure in (TMTTF)2AsF6, as determined from NMR experiments. From Zamborszky et al. [39].

#### 4.2.2. Electron–Lattice Coupling

The formation of a bond superstructure in the dimerized chains results from a coupling of the lattice degrees of freedom to bond charge density-wave correlations of electrons. By analogy with a Peierls electron–lattice coupling in a tight-binding band [59,60], the coupling develops from the modulation of the interdimer hopping integral *t*<sup>2</sup> = *t* − *δt* by the displacement *φ<sup>r</sup>* of the dimer from its equilibrium position. The expansion of *t*2(*φ*) with respect to *φ* leads to an additional electron–lattice part to the Hamiltonian (5) which is of the form,

$$\begin{split} H\_{\text{ep}} + H\_{\text{p}}^{0} &= \sum\_{r,r} t\_{2}' [a\_{r,r}^{\dagger} b\_{r-1,r} (\phi\_{r-1} - \phi\_{r}) + \text{H.c.}] \\ &\quad + \frac{K\_{D}}{2} \sum\_{r} (\phi\_{r-1} - \phi\_{r})^{2}. \end{split} \tag{53}$$

Here *t* <sup>2</sup> = *dt*2/*d<sup>φ</sup>* and *KD* is the spring constant of the harmonic potential energy, *<sup>H</sup>*<sup>0</sup> p, of interdimer lattice modes in the static limit. From the canonical transformation (6) and (7), the Hamiltonian for the coupling of the tetramerization order parameter, *φ*2*kF* , to electrons close to Fermi level is given by

$$H\_{\rm cp} + H\_{\rm p}^{0} = \frac{1}{\sqrt{N}} \sum\_{k, \sigma} [g\_{\rm cp}(\epsilon\_0) \, d\_{+, k, \sigma}^{\dagger} d\_{-, k - 2k\_F, \sigma} \phi\_{2k\_F} + \text{H.c.}]$$

$$+ 2K\_D |\phi\_{2k\_F}|^2\tag{54}$$

where

$$\mathcal{g}\_{\text{CP}}(\varepsilon\_0) = -2i \, t\_2' \sin \gamma\_{k\_F} = -2i \, t\_2' \sqrt{1 - \frac{\epsilon\_0^2}{E\_F^2}} \tag{55}$$

is the electron–lattice coupling constant showing an explicit dependence on the potential amplitude <sup>0</sup> of CO, which reduces the strength of the coupling.

Transposing this term into the action allows to write at = 0,

$$S = S\_0 + S\_I - 2\beta K\_D |\phi\_{2k\_F}|^2 + S\_{\text{ep}} \tag{56}$$

where

$$\begin{split} S\_{\mathsf{CP}} &= -\sqrt{\beta} \, g\_{\mathsf{CP}}(\epsilon \mathbf{e}) \Big[ \Big( z\_{\theta\_{+}}^{\mathsf{c}} \, O\_{\theta\_{+}} \sin \frac{1}{2} \theta\_{+} \\ & \qquad - i z\_{\theta\_{-}}^{\mathsf{c}} \, O\_{\theta\_{-}} \sin \frac{1}{2} \theta\_{-} \Big) \phi\_{2k\_{F}} + \text{c.c.} \Big], \tag{57}$$

in which we have decomposed the BOW composite field in terms of the two independent CDW at *θ*<sup>±</sup> = *θg*<sup>3</sup> (*θg*<sup>3</sup> ± *π*), and for *q*<sup>0</sup> = (2*kF*, 0). Here the initial conditions for the pair vertex renormalization factors at = 0 are *z<sup>c</sup> <sup>θ</sup>*<sup>±</sup> <sup>=</sup> 1.

#### 4.2.3. Spin-Peierls Instability

The presence of the SP order parameter *φ*2*kF* then linearly couples as an external field to the two independent CDW for a complex *g*3. In weak coupling *S*ep can be taken as a perturbative term alongside *SI*. Thus the successive partial integrations of the RG transformation following (26), down to the energy distance Λ0() from the Fermi level will lead to the effective action at step :

$$\begin{split} \mathcal{S}[\psi^\*, \psi]\_{\ell} &= \mathcal{S}\_0[\psi^\*, \psi]\_{\ell} + \mathcal{S}\_I[\psi^\*, \psi]\_{\ell} + \mathcal{S}\_{\text{cp}}[\psi^\*, \psi, \phi]\_{\ell} \\ &\quad - \beta \mathfrak{F}[\phi] .\end{split} \tag{58}$$

The RG then generates a dependent Landau free energy F[*φ*] of the SP order parameter. At the quadratic level, it takes the form

$$\mathfrak{F}[\mathfrak{\mathfrak{g}}]\_{\ell} = \left[2K\_{D} - |g\_{\text{ep}}(\mathfrak{e}\_{0})|^{2} \sum\_{\mu=\pm} (\sin^{2}\frac{1}{2}\theta\_{\mu}) \chi^{c}\_{\theta\_{\mu}}(q\_{0\prime}, \ell) \right] |\mathfrak{g}\_{2k\_{F}}|^{2}$$

$$+ \quad \mathcal{O}(\mathfrak{\mathfrak{g}}^{4}). \tag{59}$$

The mean field criteria for the SP 'transition' temperature at *T*<sup>0</sup> SP is obtained from the zero of the quadratic term at <sup>0</sup> SP = ln *EF*/*T*<sup>0</sup> SP. This leads to

$$\frac{2K\_D}{|\mathcal{g}\_{\rm ep}(\varepsilon\_0)|^2} = \sum\_{\mu=\pm} (\sin^2 \frac{1}{2} \theta\_{\mu}) \chi\_{\theta\_{\mu}}^{\varepsilon} (q\_{0\prime} T\_{\rm SP}^0). \tag{60}$$

The right-hand-side of this expression is essentially dominated by the CDW*θ*<sup>−</sup> susceptibility *<sup>χ</sup><sup>c</sup> θ*− which is more centered on bonds. It presents a power law divergence with decreasing temperature, while *χ<sup>c</sup> <sup>θ</sup>*<sup>+</sup> , which is more site-centered saturates at a small value. According to the results of Section 3.2, the bond susceptibility takes the form

$$
\chi\_{\theta\_{-}}^{\varepsilon}(q\_{0\prime}T) \sim (\pi \upsilon\_F)^{-1} \mathbb{C}\_{\theta\_{-}}(T/T\_{\rho})^{-\gamma^\*},
$$

where *<sup>γ</sup>*<sup>∗</sup> = 3/2 for the fixed point behavior below the Mott scale *<sup>T</sup><sup>ρ</sup>* [47]. Here *<sup>C</sup>θ*<sup>−</sup> is a positive constant that gives the contribution to the susceptibility from all energy scales above *Tρ*.

By singling out the dominant *θ*<sup>−</sup> part, we obtain the approximate result for the mean field SP temperature

$$T\_{\rm SP}^{0} \approx \left[ \mathbb{C}\_{\theta\_{-}} (\sin^{2}{\frac{1}{2}} \theta\_{-}) |\mathbb{g}\_{\rm rep}(\epsilon\_{0})|^{2} / 2K\_{\rm D} \right]^{\frac{2}{3}} T\_{\theta\_{\prime}} \tag{61}$$

where |*g*¯ep(0)| <sup>2</sup> = |*g*ep(0)| 2/*πvF*. Although from (21) an increase of the site potential <sup>0</sup> raises the value of |*g*3| and then *Tρ*, <sup>0</sup> is a major source of reduction of the electron-phonon matrix element *<sup>g</sup>*¯ep(0), which together with the shift of CDW*θ*<sup>−</sup> off the bonds, leads to an overall decrease of *<sup>T</sup>*<sup>0</sup> SP with 0. This is shown in Figure 12 (top). The charge imbalance between sites of neighbouring dimers is therefore acting as the main source of reduction of the lattice coupling to 2*kF* bond correlations.

**Figure 12.** (**top**): Calculated variation of the mean-field SP ordering temperature *T*<sup>0</sup> SP and of the Mott scale *T<sup>ρ</sup>* with the amplitude of site potential <sup>0</sup> due to charge ordering; (**bottom**): Calculated variations of the mean-field 1D (3D) spin-Peierls temperature *T*<sup>0</sup> SP ( *<sup>T</sup>*SP <sup>∼</sup> *<sup>T</sup>*<sup>0</sup> SP/3), as a function of the tuning (pressure) parameter *x*. The dashed red line is a linear parametrization of the charge ordering temperature *T*CO = 0(*x*)/2. All temperature scales are normalized by the average hopping *t* along the stacks.

It is worth noticing that the mean-field prediction *T*<sup>0</sup> SP does not coincide with a real transition temperature, but refers to the temperature scale for the onset of strong 2*kF* 1D lattice SP fluctuations [61,62], such as those seen for instance in diffuse x-ray scattering [37]. For a true 3D transition to take place, interchain coupling *V*<sup>⊥</sup> (e.g., electron–lattice, Coulomb, ...) is necessary to stabilize long-range ordering for a one-component SP order parameter. The generalization of the above approach to the case of weakly coupled Landau-Ginzburg chains is well known [63]; it leads to the relation

$$T\_{\rm SP} \simeq \frac{T\_{\rm SP}^0}{\ln \left(T\_{\rm SP}^0 / V\_\perp\right)} \sim T\_{\rm SP}^0 / n\_\prime \tag{62}$$

between the *T*<sup>0</sup> SP and *<sup>T</sup>*SP scales. For *<sup>V</sup>*<sup>⊥</sup> *<sup>T</sup>*<sup>0</sup> SP, fluctuations reduce *<sup>T</sup>*<sup>0</sup> SP by a factor *α*, which is typically around 3.

#### 4.2.4. Theory and Experiment

We are now in a position to apply our model to the evolution of the SP instability in the presence of CO for a compound like (TMTTF)2AsF6 under pressure (top of Figure 12). We consider the hopping modulation *δt*(*x*) = 0.2 − 0.5*x* (normalized by *t* = 1), tuned downward by the pressure parameter *x*. Here, 0.2 is a typical band calculation value of *δt* for a compound like (TMTTF)2AsF6 at ambient pressure (*x* = 0) [64]. We fix the interaction parameters *U* = 1 and *V* = 0.5 in order to obtain a Mott temperature scale, *T<sup>ρ</sup>* ∼ 100 K (*T<sup>ρ</sup>* ∼ 0.09 in units of *t*), consistent with the range found in experiments for a value of *t* = 1300 K at ambient pressure [64]. Charge ordering is responsible for the onset of a static site potential, which we parametrize by the linear profile 0(*x*) = 0(0)(1 − *x*/*xc*). Here we shall use the relation 0(0) = ΔCO ∼ 2*T*CO between the CO gap and the observed critical temperature *T*CO ≈ 100 K at ambient pressure. Finally, we let the normalized small variations *δU* = −0(*x*)/*EF* and *δV* = *tδt*(*x*)/*E*<sup>2</sup> *<sup>F</sup>* evolving under pressure following 0(*x*) and *δt*(*x*), respectively.

The solution of (60) for *T*<sup>0</sup> SP from the use of (42) and (55) as a function of the tuning parameter *x*, is displayed in the bottom panel of Figure 12. We see that to the fall of 0(*x*) under pressure corresponds an increase of *T*<sup>0</sup> SP, the latter being a consequence of the boost in the electron lattice-coupling *g*ph(0) that overcomes the impact of the reduction of *T<sup>ρ</sup>* on *T*<sup>0</sup> SP under pressure. A maximum of *<sup>T</sup>*<sup>0</sup> SP is reached at *xc* where 0(*xc*) vanishes, leading to near 50% of increase of *T*<sup>0</sup> SP from its ambient pressure value. Beyond *xc*, 0(*x* > *xc*) = 0 and *T*<sup>0</sup> *SP* undergoes a monotonic decreases with *x*, which according to (60) and (61) is governed by that of *T<sup>ρ</sup>* under pressure—when only bond order wave correlations with *θ*<sup>−</sup> = *π* couple to the lattice (at sufficiently high but intermediate pressure, interchain antiferromagnetic exchange coupling, which is present but not considered in the present high temperature calculations, enters into play and introduces a competition between the SP and the magnetic Néel states, from which the latter state ends up to be favored [20,21,61]. This competition is also documented on experimental grounds in the similar compound (TMTTF)2PF6 − see for instance Refs. [65,66]. In the very high pressure range, it is the turn of the Mott scale *T<sup>ρ</sup>* to become irrelevant in (TMTTF)2*X*, with the Néel state merging into an itinerant SDW state when interchain exchange evolves towards coherent interchain hopping, as found in (TMTSF)2*X* at low pressure [1,4]. In these conditions the absence of a tetramerization-Peierls-instability, despite the absence of charge ordering, may take its origin from the relatively weak amplitude of Umklapp scattering. This suppresses *T<sup>ρ</sup>* and strongly reduces the growth of bond CDW correlations for the Peierls scale *T*<sup>0</sup> *<sup>P</sup>* in (60). The long-range Peierls ordering is likely to be preempted when conflicting sources of interchain coupling are added to the model such as single electron hopping and Coulomb interactions, along with the emergence of SDW long-range order [10]).

The results of the above model for *T*<sup>0</sup> SP and by extension *T*SP fairly bear the comparison with the experiments shown in Figure 11. The present model could also offer an explanation as to the absence of a spin-Peierls transition in a compound like (TMTTF)2SbF6, which rather exhibits a Néel transition around 7K [67]. The SbF6 Fabre salt is known to develop charge ordering at a temperature *T*CO ≈ 160 K [29,30,68], which is sizably higher than for other SP salts like PF6 and AsF6. Following (55), a significantly larger *T*CO or 0, together with a relatively weaker value of *T<sup>ρ</sup>* for SbF6 [58], would further suppress the electron-phonon matrix element *g*ep(0) and from (61) the scales *T*0 SP and *T*SP. Including the influence of interchain exchange interaction between spins, which is present in practice but has been disregarded in this work (see for instance Refs. [20,21,61]), will favor the competition between the SP and the Néel states and may be responsible for the absence of long-range SP ordering and the relatively small Néel temperature for this salt (see also the note in the preceding paragraph). The observation of a reentrant SP state in (TMTTF)2SbF6 at higher pressure is consistent with that view [67]. These results obtained above are also compatible with those using the bosonization technique [40], and numerics [35,40].

#### **5. Concluding Remarks**

In the work developed above, we inquired into the properties of the one-dimensional extended Hubbard model at quarter-filling with superimposed dimerization, site and anion alternating lattice potentials, as they can be found in practice in low-dimensional charge transfer salts with a 2:1 stochiometry. The renormalization group method was applied to the continuum or electron gas limit of the model in order to determine the influence of dimerization and site commensurability potentials on low energy interactions, in particular effective half-filling type of Umklapp scattering that emerges from the lattice. The impacts of Umklapp on singular correlations and different excitation gaps of the model have been determined, along with spin and charge density-wave correlations and their specific phase relations to the underlying alternated lattice.

By coupling electrons to anionic displacements in systems like the (TMTTF)2*X* and (TMTSF)2*X* charge transfer salts, the mechanism of anion ordering in these materials has been investigated. It was found that a site alternated potential can significantly reduce the coupling of charge density wave correlations to anion displacements and in turn the critical temperature of staggered anion ordering. This occurs while the Mott insulating character of electrons is on the contrary strongly enhanced by site and bond commensurabilities. These opposed effects were found to be consistent with observations about anionic order and Mott insulating state in [(TMTSF)1−*x*(TMTTF)*x*]2ReO4 alloys.

The influence of the charge ordered state on the spin-Peierls ordering has also been investigated for members of the Fabre salts series at low applied pressure. Acting as alternate site commensurability, charge order was found to mainly reduce the inter-dimer tight-binding electron-phonon matrix element. The coupling of the lattice to bond density-wave correlations is then lessened by charge disproportionation and a competition between the spin-Peierls and the charge ordered states takes place. The interplay gives rise to a characteristic dome of the spin-Peierls ordering temperature as a function of the suppression of charge order under pressure, a result congruent with observations made in the Fabre salts series.

Allowing for anionic displacements besides lattice degrees of freedom would be an interesting possible extension of the latter calculations. This could provide the opportunity to check if the spin-Peierls lattice distortion is accompanied by staggered anion ordering, as suspected on experimental grounds for spin-Peierls systems of the Fabre salts series [11].

It is worth mentioning another straightforward application of the model of anion ordering which was not considered above. This concerns the anion displacement that goes with the charge ordering transition in the (TMTTF)2*X* series. Such anion displacement is known to be uniform in character [10]. The corresponding order parameter *δ*0,0 will thus be linearly coupled to uniform charge density of the stacks, in a way similar to the expression given in (48). Since uniform charge susceptibility is a quantity proportional to the dielectric constant, as the latter is being singular at the charge ordering transition [32,68], it will drive a collective shift of the anionic position as apparently found experimentally [9,68,69].

**Author Contributions:** M.M. performed the calculations and the comparisons with experiments in the framework of a PhD thesis at Université de Sherbrooke (2017). C.B. wrote the paper. Both authors contributed to the discussion of the results. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Science and Engineering Research Council of Canada (NSERC) and the Réseau Québcois des Matériaux de Pointe (RQMP).

**Acknowledgments:** The authors thank G. Chitov for useful comments on many aspects of this work.

**Conflicts of Interest:** The authors declare no conflict of interest.
