**1. Introduction**

The discovery of the amazing variety of electronic and structural phases in the Bechgaard and Fabre charge transfer salts series (TMTSF)2*X* and (TMTTF)2*X* over the last four decades or so continues to arouse great interest in the field of organic conductors [1–13]. This research paper is devoted to explore some theoretical aspects about the role played by lattice commensurability in the origin and the development of these phases.

The part played by lattice commensurability was very early suspected to be among the determinant factors behind the occurence of electronic and lattice instabilities found in the phase diagram of these charge transfer salts. This is the case of the importance attributed to the weak but finite dimerization of the organic stacks. It is characterized by a lattice periodicity of wave vector 4*kF* (*kF* being the Fermi wave vector along the stacks) that coincides with the one of the anion lattice *X* [14,15]. This superstructure superimposes to the basic quarter-filled commensurability of the hole band structure, which is fixed by the complete charge transfer and stoichiometry of the salts. This introduces half-filling type of Coulomb Umklapp scattering along the organic stacks, known to have a strong impact on low energy electron correlations in one spatial dimension (1D) [16–18]. Its magnitude was shown to be a key element in controlling the strength of magnetism and Mott insulating behavior across the whole phase diagram of both series of materials [15,19–21].

Another source of lattice periodicity for charge carriers of organic stacks comes from degrees of freedom associated to the anion *X* lattice, whose 4*kF* periodicity is intrinsically linked to that of dimerization [15]. Non-centro-symmetrical anions like *X* = ReO− <sup>4</sup> , ClO<sup>−</sup> <sup>4</sup> , BF<sup>−</sup> <sup>4</sup> , .... and to a certain extend centrosymmetric ones like *X* = PF− <sup>6</sup> , AsF<sup>−</sup> <sup>6</sup> ... can order and form lattice superstructures also called anion orderings (AO) [9,10,22–24]. The coupling between anion degrees of freedom and charge carriers makes AO inevitably entangled to various electronic instabilities of organic stacks [10,25].

The interplay between superimposed lattice periodicities and electronic instabilities has been further revealed by chemistry. A notable example that we will focus on in the present work combines both families of compounds in the form of the alloys [(TMTSF)1−*x*(TMTTF)*x*]2ReO4 with the asymmetrical anion ReO− <sup>4</sup> . Remarkably, in the case where x∼0.5, the alloy is found to be ordered [26]. The alternation of different organic molecules along the stacks then acts as a distinct site potential with the same 4*kF* periodicity found for the bond centered dimerization [26–29]. Transport and structural studies have showed that these superimposed types of commensurability have a strong impact on both the Mott insulating state and the staggered ReO− <sup>4</sup> AO found in the pure limits [26].

An alternating site periodicity can also be generated dynamically in pure systems of the (TMTTF)2*X* salts. This occurs for the charge ordering (CO) phase transition [30–32], which introduces a charge disproportionation in the dimerized unit cell that breaks the inversion symmetry and gives rise to a ferroelectric state [29]. The importance of Coulomb interaction between carriers of different molecular units in the unit cell has pointed to the key role of underlying quarter-filling commensurability component of Umklapp scattering in the electronic origin of this phase [33–36].

The existence of a CO state in the (TMTTF)2*X* series has been found to interfere with the formation of another ordered state involving both spin and lattice degrees of freedom. This the case of the spin-Peierls (SP) instability which corresponds to a 2*kF* bond like tetramerization of the organic stacks that opens a gap in the spin sector [37,38]. On experimental grounds [39], the presence of the CO state is found to particularly weaken the strength of the SP order, an effect as we will see that results from the interplay between different types of commensurability [35,40–43].

In the present paper we shall address theoretically the combined influence of different types of lattice commensurability in the framework of a generalized extended Hubbard model. This will be examined in the 1D regime that characterizes both series of materials at relatively high temperature. The combined influence of site and bond dimerization lattice potentials together with the respective modulation of Coulomb interaction terms are considered. Their impact on excitation gaps and singular correlations of the phase diagram is analyzed using a two-loop renormalization group (RG) method in the electron gas limit. The 2*kF* bond and site density-wave correlations with their phase relation to the underlying alternated lattice are determined. The model is integrated to the one proposed by Riera and Poilblanc for the mechanism of anion ordering in the (TMTSF)2*X* and (TMTTF)2*X* salts. The coupling of anions displacement to charge carriers in systems like the alloys [(TMTSF)1−*x*(TMTTF)*x*]2ReO4 is derived and the reduction (magnification) of the staggered anion ordering (Mott temperature scale) is obtained and shown to compare favorably with previous experiments performed on these materials [26]. Finally, the model is also applied to the influence of CO state on the pressure profile of the spin-Peierls instability occurring in (TMTTF)2*X* with centrosymmetric anions. By modeling CO in terms of a site lattice potential, it is found that lattice coupling responsible for the SP tetramerization is strongly altered by the presence of CO which mainly governs the variation of the temperature scale of the SP instability observed under pressure.

#### **2. Alternating Extended Hubbard Model**

#### *The Model and Its Continuum Limit*

We consider a 1D model of interacting electrons on a bipartite lattice. The Hamiltonian *H* = *H*<sup>0</sup> + *HI* consists of the one-body term

$$H\_0 = -\sum\_{r,\sigma} \left[ (t + \delta t) a\_{r,\sigma}^\dagger b\_{r,\sigma} + (t - \delta t) a\_{r,\sigma}^\dagger b\_{r-1,\sigma} + \text{H.c.} \right]$$

$$+ \epsilon\_0 \sum\_{r,\sigma} (m\_{r,\sigma} - n\_{r,\sigma})\tag{1}$$

and the interaction term,

$$\begin{split} H\_{I} &= \frac{1}{2} \sum\_{r,\sigma} \Big[ (\mathcal{U} + \delta \mathcal{U}) m\_{r,\sigma} m\_{r,-\sigma} + (\mathcal{U} - \delta \mathcal{U}) n\_{r,\sigma} n\_{r,-\sigma} \Big] \\ &+ \sum\_{r,\sigma\_{1,2}} (V + \delta V) m\_{r,\sigma\_{1}} n\_{r,\sigma\_{2}} + (V - \delta V) m\_{r,\sigma\_{1}} n\_{r-1,\sigma\_{2}}. \end{split} \tag{2}$$

Here, *a* (†) *<sup>r</sup><sup>σ</sup>* and *b* (†) *<sup>r</sup><sup>σ</sup>* are the annihilation (creation) operators of electrons of spin *<sup>σ</sup>* at even and odd sites of a bipartite lattice; *mr<sup>σ</sup>* = *a*† *<sup>r</sup>σar<sup>σ</sup>* and *nr<sup>σ</sup>* = *b*† *<sup>r</sup>σbr<sup>σ</sup>* stand for the number operators on the corresponding sites. At quarter-filling, we have *mr<sup>σ</sup>* <sup>=</sup> *nr<sup>σ</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> . The standard extended Hubbard part of *H* is described by the hopping integral *t*, and the intra- and inter-site interaction terms *U* and *V*. The model includes the influence of two different alternating potentials. We have first a small lattice dimerization which modulates the hopping integral by *δt* [14,15] and the nearest-neighbor interaction *V* by a corresponding positive *δV*, whether the electrons interact on shorter or longer bonds as expected from quantum chemistry calculations on dimerized chains materials like the Fabre and the Bechgaard salts [44]. The second potential to be examined is an alternating site potential of amplitude <sup>0</sup> [27]. The latter can be found in practice in quasi-1D organic ordered alloys in which the molecular species alternates from site to site [26], or as a result of charge ordering which in systems like the Fabre salts introduces a charge disproportionation modulated along the stacks [31,32]. Besides the modulation of site energy, a molecular alternation is expected to modify Coulomb integral *U* from site to site, which is taken into account by adding (substracting) a negative *δU* when the site energy is increased (reduced) by <sup>0</sup> (see Figure 1).

$$
\begin{array}{ccccc}
\cdot & \cdot & \cdot \\
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\end{array}
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\cdot & \cdot \\
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\end{array}
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\end{array}
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\end{array}
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\end{array}
$$

**Figure 1.** One dimensional alternating extended Hubbard model. The small and big full circles depict higher and lower potential energies, and crosses refer to the positions of anions in systems like (TMTSF)2*X*, (TMTTF)2*X* and their alloys.

Using the Fourier transforms of each sublattice operator,

*ar*,*<sup>σ</sup>* = 2 *<sup>N</sup>* ∑ *k akσe ikr*, (3)

$$b\_{r\mathcal{F}} = \sqrt{\frac{2}{N}} \sum\_{k} b\_{kr} e^{ikr} \,, \tag{4}$$

the one-body part of the Hamiltonian is diagonalized in the form

$$H0 = -\sum\_{k,\sigma} E\_k d\_{k,\sigma}^\dagger d\_{k,\sigma} + \sum\_{k,\sigma} E\_k f\_{k,\sigma}^\dagger f\_{k,\sigma\prime} \tag{5}$$

which is written in terms of the new operators *d* (†) *<sup>k</sup>*,*<sup>σ</sup>* and *f* (†) *<sup>k</sup>*,*<sup>σ</sup>* for the lower and upper bands following the transformations

$$a\_{k, \sigma}^{\dagger} = e^{-i\left(\frac{k}{4} - \frac{\gamma\_k}{2}\right)} \left(\sin\frac{\gamma\_k}{2} d\_{k, \sigma}^{\dagger} + \cos\frac{\gamma\_k}{2} f\_{k, \sigma}^{\dagger}\right),\tag{6}$$

$$b\_{k,r}^{\dagger} = e^{i\left(\frac{k}{4} - \frac{\gamma\_k}{2}\right)} \left(\cos\frac{\gamma\_k}{2} d\_{k,r}^{\dagger} - \sin\frac{\gamma\_k}{2} f\_{k,r}^{\dagger}\right) ,\tag{7}$$

where the phase factors obey the relations

$$
\epsilon\_0 = E\_k \cos \gamma\_{k\prime} \tag{8}
$$

$$2t\cos\frac{k}{2} = E\_k \sin\gamma\_k \cos\nu\_{k\prime} \tag{9}$$

$$
\hbar 2\delta t \sin \frac{k}{2} = E\_k \sin \gamma\_k \sin \nu\_k. \tag{10}
$$

The amplitude of the spectrum for each band is given by

$$E\_k = 2\sqrt{t^2 \cos^2 \frac{k}{2} + \delta t^2 \sin^2 \frac{k}{2} + \left(\frac{\epsilon\_0}{2}\right)^2}. \tag{11}$$

At quarter filling the lower band is occupied up to the Fermi points ±*kF* = ±*π*/2, corresponding to the Fermi level −*EkF* ≡ −*EF* where

$$E\_F = \sqrt{2t^2 + 2\delta t^2 + \epsilon\_{0'}^2} \tag{12}$$

is the Fermi energy. The lower and upper bands in Figure 2 are separated by a gap Δ = 2*E<sup>π</sup>* = 2 ' 4*δt*<sup>2</sup> + <sup>2</sup> 0.

**Figure 2.** Electron spectrum of the one-dimensional extended alternated Hubbard model.

For the interacting part of the Hamiltonian, we shall focus on the weak coupling case by considering only low energy scattering processes taking place within the lower '*d*' band (perturbative corrections coming from the upper *f* band at higher energy have been considered by Penc and Mila [27]). This yields

$$H\_I = \frac{1}{2N} \sum\_{\{k, \sigma\}} g(k\_1, k\_2, k\_3, k\_4) \delta\_{k\_4 + k\_3, k\_1 + k\_2 + G} $$

$$\times \, d\_{k\_4, \sigma\_4}^\dagger d\_{k\_3, \sigma\_3}^\dagger d\_{k\_2, \sigma\_2} d\_{k\_1, \sigma\_1} + \dots \, \tag{13}$$

where *G* = 0 (*G* = ±2*π*) stands for normal (Umklapp) scattering processes. The interaction amplitude reads

$$\begin{split} \log(k\_1, k\_2, k\_3, k\_4) &= \quad (\mathcal{U} + \delta \mathcal{U}) e^{l(2\gamma |\mathbf{x}| - \frac{\mathbf{q}}{\Delta})} \prod\_{l=1}^4 \sin\frac{\gamma\_{\mathbf{k}\_l}}{2} \\ &\quad + (\mathcal{U} - \delta \mathcal{U}) e^{-l(2\gamma |\mathbf{x}| - \frac{\mathbf{q}}{\Delta})} \prod\_{l=1}^4 \cos\frac{\gamma\_{\mathbf{k}\_l}}{2} \\ &\quad + (2V \cos\frac{k\_4 - k\_1}{2} - 2i\delta V \sin\frac{k\_4 - k\_1}{2}) e^{l(\frac{\mathbf{q}}{\Delta} + 2\gamma\_{\mathbf{(l})})} \sin\frac{\gamma\_{\mathbf{k}\_l}}{2} \cos\frac{\gamma\_{\mathbf{k}\_l}}{2} \sin\frac{\gamma\_{\mathbf{k}\_l}}{2} \\ &\quad + (2V \cos\frac{k\_4 - k\_1}{2} + 2i\delta V \sin\frac{k\_4 - k\_1}{2}) e^{-l(\frac{\mathbf{q}}{\Delta} + 2\gamma\_{\mathbf{(l})})} \cos\frac{\gamma\_{\mathbf{k}\_l}}{2} \sin\frac{\gamma\_{\mathbf{k}\_l}}{2} \sin\frac{\gamma\_{\mathbf{k}\_l}}{2} \cos\frac{\gamma\_{\mathbf{k}\_l}}{2}, \end{split} \tag{14}$$

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

where

$$\nu\_{[1,3]} \quad = \quad \frac{\nu\_{k\_4} \mp \nu\_{k\_3} \pm \nu\_{k\_2} - \nu\_{k\_1}}{4}. \tag{15}$$

In weak coupling, the lattice model can be transposed into the continuum limit of the electron gas model. This standard procedure first consists in the linearization of the spectrum around the Fermi points *pkF* = ±*kF*,

$$E\_F - E\_k \approx E\_p(k) = \upsilon\_F (pk - k\_F),\tag{16}$$

where

$$v\_F = \frac{t^2 - \delta t^2}{\sqrt{2t^2 + 2\delta t^2 + \epsilon\_0^2}} \text{'} \tag{17}$$

is the Fermi velocity (*h*¯ = 1 throughout). The operators for left and right moving electrons become *<sup>d</sup>*±,*k*,*<sup>σ</sup>* (*d*† <sup>±</sup>,*k*,*σ*). The linearized spectrum is limited by an energy cut-off *<sup>E</sup>*0/2 <sup>=</sup> *EF* on either side of the Fermi level, where *E*<sup>0</sup> is the total width of the lower band. The interaction is defined with respect to the Fermi points giving rise to four possible coupling constants, commonly noted as *gi*=1−<sup>4</sup> [16,17,45]. Together with the one-electron part, this leads to the usual form for the Hamiltonian of the electron gas model. For the lower band, one gets

*H* = ∑ *k*,*p*,*σ Ep*(*k*)*d*† *<sup>p</sup>*,*k<sup>σ</sup>dp*,*k<sup>σ</sup>* + 1 *<sup>N</sup>* ∑ {*k*,*q*,*σ*} *g*<sup>1</sup> *d*† +,*k*1+*q*,*σ*<sup>1</sup> *d*† −,*k*2−*q*,*σ*<sup>2</sup> *d*+,*k*2,*σ*<sup>2</sup> *d*−,*k*1,*σ*<sup>1</sup> + *g*<sup>2</sup> *d*† +,*k*1+*q*,*σ*<sup>1</sup> *d*† −,*k*2−*q*,*σ*<sup>2</sup> *d*−,*k*2,*σ*<sup>2</sup> *d*+,*k*1,*σ*<sup>1</sup> + 1 <sup>2</sup> (*g*<sup>+</sup> <sup>3</sup> *<sup>d</sup>*† +,*k*1+*q*,*σ*<sup>1</sup> *d*† +,*k*2−*q*+*G*,*σ*<sup>2</sup> *<sup>d</sup>*−,*k*2,*σ*<sup>2</sup> *<sup>d</sup>*−,*k*1,*σ*<sup>1</sup> + H.c.) + 1 <sup>2</sup> *<sup>g</sup>*<sup>4</sup> *<sup>d</sup>*† +,*k*1+*q*,*σ*<sup>1</sup> *d*† +,*k*2−*q*,*σ*<sup>2</sup> *d*+,*k*2,*σ*<sup>2</sup> *d*+,*k*1,*σ*<sup>1</sup> + 1 <sup>2</sup> *<sup>g</sup>*<sup>4</sup> *<sup>d</sup>*† −,*k*1+*q*,*σ*<sup>1</sup> *d*† −,*k*2−*q*,*σ*<sup>2</sup> *d*−,*k*2,*σ*<sup>2</sup> *d*−,*k*1,*σ*<sup>1</sup> , (18)

where the scattering amplitudes in the standard terminology [16,17], are *g*<sup>1</sup> ≡ *g*(+*kF*, −*kF*, +*kF*, −*kF*), *g*<sup>2</sup> ≡ *g*(+*kF*, −*kF*, −*kF*, +*kF*), *g*<sup>±</sup> <sup>3</sup> ≡ *g*(±*kF*, ±*kF*, ∓*kF*, ∓*kF*) and *g*<sup>4</sup> ≡ *g*(±*kF*, ±*kF*, ±*kF*, ±*kF*), which in order stand for backward, forward, half-filling Umklapp, and inner branch forward scattering processes. According to the expression (14) for the lattice model, the bare *gi* amplitudes are given by

$$\mathbf{g}\_1 = \mathcal{U}(1 + \epsilon\_0^2 / E\_F^2) - 2\delta \mathcal{U} \epsilon\_0 / E\_F + 4t \delta V \delta t / E\_F^2 \tag{19}$$

$$\begin{split} \mathbf{g}\_{2} &= \mathcal{U}(1 + \epsilon\_{0}^{2}/E\_{F}^{2}) - 2\delta\mathcal{U}\epsilon\_{0}/E\_{F} + 2V(1 - \epsilon\_{0}^{2}/E\_{F}^{2}) \\ \mathbf{g}\_{3}^{p} &= \left[\mathcal{U}(1 + \epsilon\_{0}^{2}/E\_{F}^{2}) - 2\delta\mathcal{U}\epsilon\_{0}/E\_{F}\right] \frac{2t\delta t}{t^{2} + \delta t^{2}} \\ &+ 2\delta V(1 - \epsilon\_{0}^{2}/E\_{F}^{2}) \\ &+ ip\left[2\mathcal{U}t\epsilon\_{0}/E\_{F} - \delta\mathcal{U}(1 + \epsilon\_{0}^{2}/E\_{F}^{2})\right] \frac{t^{2} - \delta t^{2}}{t^{2} + \delta t^{2}} \end{split} \tag{20}$$

$$\begin{aligned} & \quad \mathbf{r} \cdot \mathbf{r}\_F | \mathbf{a} \cdot \mathbf{r}\_0 \rangle \cdot \mathbf{r}\_F \quad \mathbf{r} \cdot (\mathbf{r}\_0 \cdot \mathbf{r}\_{F/1})^2 t^2 + \delta t^2 \\ & \equiv |g\_3| e^{i p \theta} \end{aligned} \tag{21}$$

$$\mathbf{g}\_4 = \mathbf{g}\_2.\tag{22}$$

From these expressions, we first observe that alongside the bond alternation of hopping *δt* and the modulation of on-site energy 0, both local (*δU* < 0) and intersite (*δV* > 0) modulations of interactions contribute to an increase of backscattering and half-filling Umklapp. Furthermore, the local components of the modulation contribute to an imaginary part of *g<sup>p</sup>* <sup>3</sup> <sup>=</sup> <sup>|</sup>*g*3|*eipθ*, Ref. [28] whose argument *θ* plays an important role for the relative phase of charge/spin density-wave correlations with respect to the lattice. This will be analyzed in Section 3.2.

#### **3. Renormalization Group Results**

#### *3.1. Formulation and Coupling Constants*

We apply the renormalization group approach to our effective electron gas model described above. In the following, we sketch out only the main steps of the procedure that will be useful later on for applications [17,46,47]. We follow Ref. [46] and write the partition function in the functional integral form,

*Z* = <sup>D</sup>*ψ*∗D*<sup>ψ</sup> <sup>e</sup> S*[*ψ*∗,*ψ*] , (23)

over a set anticommuting fermion variables {*ψ*∗, *ψ*}, where the measure is

$$\mathfrak{D}\psi^\*\mathfrak{D}\psi = \prod\_{k,p,\sigma} d\psi^\*\_{p,\sigma}(\vec{k})d\psi\_{p,\sigma}(\vec{k}),$$

and ¯ *k* = *k*, *ω<sup>n</sup>* = (2*n* + 1)*πT* (*kB* = 1 throughout). The action *S* = *S*<sup>0</sup> + *SI* splits into a free-quadratic-(*S*0) and an interacting-quartic-(*SI*) parts,

$$\begin{split} \mathbb{S}\left[\Psi^{\ast},\Psi\right] &= \sum\_{\vec{k},\vec{p},\sigma} z^{-1} \left[G^{0}\_{\vec{p}}(\vec{k})\right]^{-1} \Psi^{\ast}\_{p,\sigma}(\vec{k}) \,\psi\_{p,\sigma}(\vec{k}) \\ &- \frac{T}{N} \sum\_{\vec{k}\_{1},\vec{k}'\_{1},\vec{k}\_{2},\vec{k}'\_{2},\sigma^{\prime}} \sum\_{\sigma,\sigma'} \delta\_{\vec{k}\_{1}+\vec{k}\_{2}\vec{k}'\_{1}+\vec{k}'\_{2}(\pm \mathcal{L})} \\ &\times \left\{ z\_{1}g\_{1}\psi^{\ast}\_{+,\sigma}(\mathcal{K}\_{1})\psi^{\ast}\_{-,\sigma'}(\mathcal{K}\_{2})\psi\_{+,\sigma'}(\vec{k}\_{2})\,\psi\_{-,\sigma'}(\vec{k}\_{1}) \\ &+ z\_{2}g\_{2}\psi^{\ast}\_{+,\sigma'}(\bar{k}\_{1})\psi^{\ast}\_{-,\sigma'}(\vec{k}\_{2})\,\psi\_{-,\sigma'}(\vec{k}\_{2})\,\psi\_{+,\sigma'}(\vec{k}\_{1}) \\ &+ \frac{1}{2}z\_{3}(g\_{3}^{+}\psi^{\ast}\_{+,\sigma}(\mathcal{K}\_{1})\,\psi^{\ast}\_{+,\sigma'}(\vec{k}\_{2})\,\psi\_{-,\sigma'}(\vec{k}\_{2})\,\psi\_{-,\sigma'}(\vec{k}\_{1}) \\ &+ \frac{1}{2}g\_{4}\psi^{\ast}\_{+,\sigma'}(\mathcal{K}\_{1})\,\psi\_{+,\sigma'}^{\ast}(\mathcal{K}\_{2})\,\psi\_{+,\sigma'}(\vec{k}\_{2})\,\psi\_{-,\sigma'}(\vec{k}\_{1})\right\}, \tag{24}$$

where *z* and *z*1,2,3 are in order the renormalization factors for the one-particle propagator

$$G\_p^0(\vec{k}) = [i\omega\_n - E\_p(k)]^{-1} \tag{25}$$

and the four-points electron-electron vertices Γ1,2,3. The *zi* are combined to *z* to give the renormalization factors *z*<sup>2</sup>*zi* for each coupling *gi*=1,2,3. At the bare level, the couplings *gi* are defined at the band edge energy cutoff Λ0(≡ *EF*) above and below the Fermi level where both *z* and the *zi*'s equal unity.

The RG transformation is standard and consists in the succesive integrations of electronic degrees of freedom, denoted by *ψ*¯(∗), in outer energy shell of thickness Λ()*d* on both sides of the Femi level of the lower band, where Λ() = Λ0*e*− is the cutoff at the step of the RG procedure. The integration of degrees of freedom from step to + *d* is achieved perturbatively. This recursive transformation can be written in the form

$$\begin{split} Z \sim & \iint [\mathfrak{D}\psi^\* \mathfrak{D}\psi]\_{\ell+d\ell} e^{S[\psi^\*,\mathfrak{p}]\_{\ell}} \iint [\mathfrak{D}\psi^\* \mathfrak{D}\psi]\_{d\ell} e^{S\_0[\psi^\*,\mathfrak{p}]\_{d\ell}} \\ & \qquad \times e^{S\_1[\psi^\*,\mathfrak{p},\mathfrak{p}^\*,\mathfrak{p}]\_{d\ell}} \\ \sim & \iint [\mathfrak{D}\psi^\* \mathfrak{D}\psi]\_{\ell+d\ell} e^{S[\psi^\*,\mathfrak{p}]\_{\ell}} e^{\sum\_n \frac{1}{\pi} \langle S\bar{\chi}[\psi^\*,\mathfrak{p},\bar{\varphi}^\*,\bar{\mathfrak{p}}]\_{d\ell} \rangle\_{d\ell}} \\ \sim & \iint [\mathfrak{D}\psi^\* \mathfrak{D}\psi]\_{\ell+d\ell} e^{S[\psi^\*,\mathfrak{p}]\_{\ell+d\ell}} \,. \end{split} \tag{26}$$

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

where *<sup>S</sup><sup>n</sup> <sup>I</sup>* 0,*d* are outer shell free (loop) averages with external fermion legs in the inner energy shells at Λ ≤ Λ( + *d*). The effective-renormalized-action *S*[*ψ*∗, *ψ*]+*d* at + *d* leads to the recursion transformation for the *z s*.

Thus for the one-particle propagator, *z*( + *d*) = *z*()*z*(*d*), which leads to the familiar result at the two-loop level [17,46,47],

$$\frac{d\ln z}{d\ell} = -\frac{1}{16} (\pi v\_F)^{-2} [(2\text{g}\_2 - \text{g}\_1)^2 + 3\text{g}\_1^2 + 2\text{g}\_3^+ \text{g}\_3^-]\_\prime \tag{27}$$

which is independent of the phase of the Umklapp term. The recursion relations *gi*( + *d*) = *zi*(*d*)*z*2(*d*)*gi*() for the coupling constants lead to the two-loop flow equations

$$\frac{d}{d\ell}\mathfrak{g}\_1 = -\mathfrak{g}\_1^2 - \frac{1}{2}\mathfrak{g}\_1^3. \tag{28}$$

$$\frac{d}{d\ell}(2\mathfrak{g}\_2 - \mathfrak{g}\_1) = \mathfrak{g}\_3^+ \mathfrak{g}\_3^- - \frac{1}{2}\mathfrak{g}\_3^+ \mathfrak{g}\_3^- (2\mathfrak{g}\_2 - \mathfrak{g}\_1),\tag{29}$$

$$\frac{d}{d\ell}\mathfrak{g}\_3^p = \mathfrak{g}\_3^p(2\mathfrak{g}\_2 - \mathfrak{g}\_1)$$

$$\frac{1}{2}\mathfrak{z}^p[(2\tilde{\mathfrak{z}}\_2 - \tilde{\mathfrak{z}}\_1)^2 + \tilde{\mathfrak{z}}^+\tilde{\mathfrak{z}}^-] \tag{20}$$

$$-\frac{1}{4}\bar{\mathbf{g}}\_3^p[(2\bar{\mathbf{g}}\_2-\bar{\mathbf{g}}\_1)^2+\bar{\mathbf{g}}\_3^+\bar{\mathbf{g}}\_3^-]\_\text{J}\tag{30}$$

$$\frac{d\theta}{d\ell} = 0.\tag{31}$$

The first equation for *g*¯1 is connected to spin degrees of freedom and is decoupled from (2*g*¯2 − *g*¯1, *g*¯ *p* <sup>3</sup> ), which is connected to the charge. These extend the known flow equations of the electron gas model [17,46,47] to the case of a complex *g<sup>p</sup>* <sup>3</sup> . Note that only the amplitude of Umklapp <sup>|</sup>*g*3<sup>|</sup> renormalizes, whereas its phase *<sup>θ</sup>* remains scale invariant [Im(*d* ln *<sup>g</sup><sup>p</sup>* <sup>3</sup> ) = 0] and is then fixed at the bare level by the expression (21). The renormalization of *g*<sup>4</sup> is here neglected. However the influence of this coupling has been incorporated through the normalization *g*¯1 = *g*1/*πv<sup>σ</sup>* and (2*g*¯2 − *g*¯1, *g*¯ *p* 3 ) = (2*g*<sup>2</sup> <sup>−</sup> *<sup>g</sup>*1, *<sup>g</sup><sup>p</sup>* <sup>3</sup> )/*πv<sup>ρ</sup>* for the decoupled spin (*σ*) and charge (*ρ*) interactions, respectively [17], where *vσ*,*<sup>ρ</sup>* = *vF* ∓ *g*4/2*π* are the spin and charge velocities.

The properties of the above flow equations are standard [17,47] and can be summarized as follows. In the spin sector for instance, the negative (positive) sign of *g*<sup>1</sup> determines the conditions for the flow to strong (weak) attractive coupling, *g*¯∗ <sup>1</sup> → −2 (*g*¯<sup>∗</sup> <sup>1</sup> → 0), as → ∞. In the attractive case, this indicates the emergence of a spin gap Δ*σ*, whose scale is of the order of the cutoff energy 2Λ(*σ*) at which the flow of *<sup>g</sup>*¯1 in (28) becomes singular at the one-loop O(*g*¯<sup>2</sup> <sup>1</sup>) level, namely <sup>Δ</sup>*<sup>σ</sup>* <sup>∼</sup> <sup>2</sup>*EFe*−1/|*g*¯1<sup>|</sup> .

If we now consider the charge sector, the magnitude of 2*g*¯2 − *g*¯1 with respect to |*g*¯3| at the bare level determines the conditions for strong coupling or a charge gap Δ*ρ*. Thus for *g*¯1 − 2*g*¯2 ≥ |*g*¯3|, charge degrees of freedom remain gapless since |*g*¯<sup>∗</sup> <sup>3</sup> | → 0 and *g*¯<sup>∗</sup> <sup>1</sup> − 2*g*¯<sup>∗</sup> <sup>2</sup> is non universal as → ∞. In the whole region where *g*¯1 − 2*g*¯2 < |*g*¯3|, both 2*g*¯<sup>∗</sup> <sup>2</sup> − *g*¯<sup>∗</sup> <sup>1</sup> → 2 and |*g*¯<sup>∗</sup> <sup>3</sup> | → 2 are marginally relevant and scale to strong coupling when → ∞. An order of magnitude for the charge gap Δ*<sup>ρ</sup>* can be readily given by the singularities encountered at a finite *<sup>ρ</sup>* in (29) and (30) at the one-loop, O(*g*¯2), level [16]. For −|*g*¯3| < *<sup>g</sup>*¯1 − <sup>2</sup>*g*¯2 < |*g*¯3|, one has Δ*<sup>ρ</sup>* ∼ 2Λ(*ρ*) = 2*EF* exp[−*c*/ &|*g*¯3|<sup>2</sup> − (2*g*¯2 − *<sup>g</sup>*¯1)2], where *<sup>c</sup>* = arccos[(2*g*¯2 − *<sup>g</sup>*¯1)/|*g*¯3|]; for *g*¯1 − 2*g*¯2 = −|*g*¯3|, Δ*<sup>ρ</sup>* = 2*EF* exp[−(1/|*g*¯3|)]; and finally for *g*¯1 − 2*g*¯2 < −|*g*¯3|, one has Δ*<sup>ρ</sup>* = 2*EF* exp[−*c* / &(2*g*¯2 <sup>−</sup> *<sup>g</sup>*¯1)<sup>2</sup> − |*g*¯3|2], where *<sup>c</sup>* <sup>=</sup> cosh−1[(2*g*¯2 <sup>−</sup> *<sup>g</sup>*¯1)/|*g*¯3|].

We display in Figure 3 the contour plot of the scale for the charge gap *e*−*<sup>ρ</sup>* = Δ*ρ*/2*EF* at the one-loop level in the (0, *δt*) plane of alternating potentials and for repulsive (*U*, *V*) interactions and smaller modulations (*δU*, *δV*). In the first quadrant where both *δt* and <sup>0</sup> are positive, the variation of Δ*<sup>ρ</sup>* is not monotonous; it first increases with *δt* and <sup>0</sup> and then undergoes a smooth decreases. According to (21), both the real and imaginary parts of Umklapp increase at relatively small *δt* and 0; this is responsible for the increase of Δ*ρ*. At sufficiently large *δt*, however, a reduction of the imaginary

part of *g<sup>p</sup>* <sup>3</sup> becomes apparent and leads to the decrease of Δ*ρ*. A similar variation of charge gap has been obtained in the bosonization approach to the alternating Hubbard model with positive *U* [28].

**Figure 3.** Contour plot of the normalized one-loop charge gap <sup>Δ</sup>*ρ*/2*EF*(<sup>≡</sup> *<sup>e</sup>*−*<sup>ρ</sup>* ) as a function of the alternating site (0) and bond (*δt*) potentials. The calculations are done for repulsive interactions *U*/*t* = *V*/*t* = 0.5 and *δU*/*t* = *δV*/*t* = 0.1.

Interestingly, if we broaden the analysis situation where positive *δU* > 0 and *δV* > 0 are considered, the competition between a positive <sup>0</sup> and negative *δt* can bring both the real and imaginary parts of *g<sup>p</sup>* <sup>3</sup> and in turn Δ*<sup>ρ</sup>* to zero, as shown in the second quadrant of Figure 3. Around this point, the behavior of Equations (29) and (30), as → *<sup>ρ</sup>* at the one-loop level shows that the gap vanishes following the power law <sup>Δ</sup>*ρ*/2*EF* <sup>∼</sup> [ <sup>1</sup> <sup>2</sup> <sup>|</sup>*g*¯3|/(2*g*¯2 <sup>−</sup> *<sup>g</sup>*¯1)]1/(2*g*¯2−*g*¯1).

#### *3.2. Response Functions and Phase Diagram*

In order to analyze the nature of correlations and the possible phases of the above model, we proceed to the calculation of susceptibilities. To do so, we follow Ref. [46] and add to the action a linear coupling to a set of infinitesimal source fields {*h*} to fermion pair fields. These are associated to susceptibilities that can become singular in the 2*kF* density-wave and superconducting channels. As infinitesimal terms, they can be combined at the bare level to the interaction term *SI* and treated as a perturbation. The action becomes

$$\mathcal{S}[\psi^\*, \psi, \{h\}] = \mathcal{S}\_0[\psi^\*, \psi] + \mathcal{S}\_I[\psi^\*, \psi] + \mathcal{S}\_h[\psi^\*, \psi, \{h\}],\tag{32}$$

where

$$\begin{split} \mathcal{S}\_{h}[\boldsymbol{\upmu}^{\*},\boldsymbol{\upmu},\{h\}] &= \sum\_{\boldsymbol{\upmu}} \left[ \sum\_{\mu} z\_{\mu}^{\text{c}} h\_{\mu}^{\text{s}\*} (\boldsymbol{\upeta}) \Delta\_{\mu} (\boldsymbol{\upeta}) \\ &+ \sum\_{\mu=\pm} z\_{\theta\_{\mu}}^{\text{c}} h\_{\mu}^{\text{c\*}\*} (\boldsymbol{\upeta}) \mathcal{O}\_{\theta\_{\mu}} (\boldsymbol{\upeta}) \\ &+ \sum\_{\mu=\pm} z\_{\theta\_{\mu}}^{\text{c\*}} \vec{h}\_{\mu}^{\text{c\*}\*} (\boldsymbol{\upeta}) \cdot \vec{\mathcal{S}}\_{\theta\_{\mu}} (\boldsymbol{\upeta}) + \text{"c.c."} \right]. \end{split} \tag{33}$$

In the superconducting channel, the pair fields are

$$\Delta\_{\mu}(\vec{q}) = \sqrt{\frac{T}{N}} \sum\_{\vec{k}, \alpha, \beta} a \psi\_{-, -\alpha}(-k) \sigma\_{\mu}^{a\beta} \psi\_{+, \beta}(\vec{k} + \vec{q}) \tag{34}$$

for *μ* = 0 singlet (SS) and *μ* = 1, 2, 3 triplet (TS) superconductivity. Here *σ*<sup>0</sup> = **1**, *σ*1,2,3 are the Pauli matrices, and *q*¯ = (*q*, *ω<sup>m</sup>* = 2*πmT*). The initial pair renormalization factors at = 0 are *z*<sup>s</sup> *<sup>μ</sup>* = 1.

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

In the 2*kF* density-wave channel, the presence of a complex Umklapp interaction term *<sup>g</sup><sup>p</sup>* 3 in (21), which can be written as *g*± <sup>3</sup> <sup>=</sup> ±|*g*3|*eiθ*<sup>±</sup> , where *<sup>θ</sup>*<sup>+</sup> <sup>=</sup> *<sup>θ</sup>* and *<sup>θ</sup>*<sup>−</sup> <sup>=</sup> *<sup>θ</sup>* <sup>−</sup> *<sup>π</sup>*, introduces spin and charge density-wave correlations with a particular phase relation with respect to the lattice. For 2*kF* charge-density-wave (CDW), the pair field can be written in terms of two independent stationary waves,

$$O\_{\theta\_{\pm}}(\vec{q}) = e^{i\theta\_{\pm}}O^\*(\vec{q}) + O(\vec{q}),\tag{35}$$

where

$$O(\bar{q}) = \sqrt{\frac{T}{N}} \sum\_{\bar{k}, \alpha} \psi\_{-,\alpha}^\*(\bar{k} - \bar{q}) \psi\_{+,\alpha}(\bar{k})$$

for *<sup>q</sup>*¯ = (*q*, *<sup>ω</sup>m*). The phase relation of CDW*θ*<sup>±</sup> maxima and minima with respect to the lattice is shown in Figure <sup>4</sup> at *<sup>q</sup>* <sup>=</sup> <sup>2</sup>*kF*. In the absence of <sup>0</sup> and *<sup>δ</sup>U*, for instance, the imaginary part of *<sup>g</sup><sup>p</sup>* <sup>3</sup> vanishes and *<sup>θ</sup>*<sup>−</sup> = *<sup>π</sup>* and CDW*θ*<sup>−</sup> correlations are centered on bonds between dimers, whereas *<sup>θ</sup>*<sup>+</sup> = 0 refers to CDW*θ*<sup>+</sup> whose maxima are centered on dimers. In the presence of a finite site potential <sup>0</sup> and/or *δU*, the inversion symmetry within the dimers is broken and the position of maxima for CDW*θ*<sup>±</sup> move accordingly (see Figure 4).

**Figure 4.** Site (+) and bond (−) 2*kF* charge-density-wave and their respective phase (*θ*±) relative to the alternated lattice.

A similar decomposition can be made for 2*kF* spin-density-wave (SDW*θ*<sup>±</sup> ) by introducing

$$
\vec{S}\_{\theta\_{\pm}}(\vec{q}) = \epsilon^{i\theta\_{\pm}} \vec{S}^\*(\vec{q}) + \vec{S}(\vec{q}) \tag{36}
$$

for SDW*θ*<sup>±</sup> , where

$$\mathcal{S}(\vec{q}) = \sqrt{\frac{T}{N}} \sum\_{\vec{k}, \vec{a}, \vec{\theta}} \Psi^\*\_{-,a}(\vec{k} - \vec{q}) \vec{\sigma}^{a\beta} \Psi\_{+,a}(\vec{k}) \tag{37}$$

is the spin field at *<sup>q</sup>*¯ = (*q*, *<sup>ω</sup>m*). When <sup>0</sup> and *<sup>δ</sup><sup>U</sup>* are absent, *<sup>g</sup><sup>p</sup>* <sup>3</sup> is real and *<sup>θ</sup>*<sup>±</sup> <sup>=</sup> <sup>0</sup>(*π*), so that *Sθ*<sup>±</sup> describe 2*kF* SDW with spin maxima centered on (between) the dimers, as shown in Figure 5. In the same Figure, for finite and positive <sup>0</sup> and/or *δU*, *θ*<sup>±</sup> moves away from 0(*π*) alongside the maxima of spin density that move in (between) the unit cell.

**Figure 5.** Site (+) and bond (−) 2*kF* spin-density-wave and their respective phase (*θ*±) relative to the alternated lattice.

Making the substitution *SI* → *SI* + *Sh* in the RG transformation (26), the renormalized action at Λ() reads

$$\begin{split} S[\psi^\*, \psi, \{h\}]\_\ell &= S\_0[\psi^\*, \psi]\_\ell + S\_l[\psi^\*, \psi]\_\ell + S\_h[\psi^\*, \psi, \{h\}]\_\ell \\ &+ \sum\_{\mu, \tau} \chi^r\_\mu(\bar{q}\_\mu, \ell) h^{r\*}\_\mu(\bar{q}\_\mu) h^r\_\mu(\bar{q}\_\mu) + \dots \ . \tag{38}$$

Here, the flows of renormalization factors in *S*<sup>0</sup> and *SI* coincide with those obtained previously in (27)–(30), whereas the *z<sup>r</sup> <sup>μ</sup>*'s associated to the pair vertices in (33) are governed at the two-loop level by an equation of the form [17,46,47]

$$\frac{d\ln z\_{\mu}^{r}}{d\ell} = \frac{1}{2\pi v v\_{F}} g\_{\mu}^{r} - \frac{1}{4\pi^{2}v\_{F}^{2}} [g\_{1}^{2} + g\_{2}^{2} - g\_{1}g\_{2} + g\_{3}^{+}g\_{3}^{-}/2],\tag{39}$$

where for superconducting correlations (*r* = s) the combinations of couplings *g<sup>r</sup> <sup>μ</sup>* are *g*<sup>s</sup> SS = −*g*<sup>1</sup> − *g*<sup>2</sup> and *g*<sup>s</sup> TS = *g*<sup>1</sup> − *g*<sup>2</sup> for singlet and triplet superconductivity; for density-wave correlations, one has *g*c *<sup>θ</sup>*<sup>±</sup> <sup>=</sup> *<sup>g</sup>*<sup>2</sup> <sup>−</sup> <sup>2</sup>*g*<sup>1</sup> ∓ |*g*3<sup>|</sup> for CDW*θ*<sup>±</sup> in the charge sector (*<sup>r</sup>* <sup>=</sup> c); and *<sup>g</sup><sup>σ</sup> <sup>θ</sup>*<sup>±</sup> <sup>=</sup> *<sup>g</sup>*<sup>2</sup> ± |*g*3<sup>|</sup> for SDW*θ*<sup>±</sup> in the spin sector (*r* = *σ*). The second term of (39), which is common to all pair vertices, refers to the self-energy corrections of Equation (27). According to (39), the behavior of *z<sup>r</sup> <sup>μ</sup>* is well known and follows the power law

$$z\_{\mu}^{r} \sim [\Lambda(\ell)]^{-\frac{1}{2}\gamma\_{\mu}^{r\*}},\tag{40}$$

at large . It signals a singularity when the exponent *γr*<sup>∗</sup> *<sup>μ</sup>* > 0. The expression for <sup>1</sup> 2*γr*<sup>∗</sup> *<sup>μ</sup>* coincides with the right side expression of (39) evaluated at the fixed points values of scaling Equations (28)–(30). A singular behavior will also be found in the corresponding expressions for susceptibilities, which are given by the quadratic field terms of (38). These are generated by the RG transformation (26) and take the form,

$$\chi\_{\mu}^{r}(\vec{q}\_{\mu}) = (\pi \upsilon\_{F})^{-1} \int\_{0}^{\ell} [z\_{\mu}^{r}(\ell)]^{2} d\ell,\tag{41}$$

which are defined positive and evaluated in the static limit *q*¯*<sup>μ</sup>* = (*qμ*, 0), for *q*SS,TS = 0 and *q*CDW,SDW = 2*kF*. At large , the susceptibilities will be governed by a power law

$$
\pi \upsilon \upsilon\_{\mathsf{F}} \chi^r\_{\mu}(\vec{q}\_{\mu}) \approx A^r\_{\mu}(\ell) [\Lambda(\ell)]^{-\gamma^{r\*}\_{\mu}} + c^r\_{\mu}.\tag{42}
$$

where *c<sup>r</sup> <sup>μ</sup>* is a positive constant.

The phase diagram determined by the dominant and subdominant singularities in the susceptibilities *χ<sup>r</sup> <sup>μ</sup>* is shown in Figure 6, as a function of initial *gi*. Its structure necessarily presents many similarities with the known two-loop RG results of the electron gas model [17,47], but also some differences due to the presence of a complex *g<sup>p</sup>* <sup>3</sup> . In Figure 6 the massive (Δ*<sup>ρ</sup>* = 0) charge sector, delimited by the separatrix *g*<sup>1</sup> − 2*g*<sup>2</sup> = |*g*3|, is enlarged with *δt*, 0, *δV* and negative *δU*, which is detrimental to the region of singular superconducting correlations on the left of this line. At *g*<sup>1</sup> > 0, this is also concomitant with the strengthening of dominant dimer or site like SDW*θ*<sup>+</sup> , and subdominant interdimer or bond like CDW*θ*<sup>−</sup> singular correlations. In the attractive region where *<sup>g</sup>*<sup>1</sup> < 0, only the reinforcement of CDW*θ*<sup>−</sup> singular correlations is found on the right-hand side of the separatrix, where a gap in both spin and charge degrees of freedom occurs.

**Figure 6.** Phase diagram of the 1D alternated extended Hubbard model in the continuum electron gas limit.

#### **4. Applications**
