# One-Dimensional Alternating Extended Hubbard Model at Quarter-Filling and Its Applications to Structural Instabilities of Organic Conductors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Alternating Extended Hubbard Model

#### The Model and Its Continuum Limit

## 3. Renormalization Group Results

#### 3.1. Formulation and Coupling Constants

#### 3.2. Response Functions and Phase Diagram

## 4. Applications

#### 4.1. Anion Ordering in [(TMTSF)${}_{1-x}$TMTTF${}_{x}$]ReO${}_{4}$

#### 4.1.1. Experimental Features

#### 4.1.2. Electron–Anion Interaction

#### 4.1.3. Anion Ordering

#### 4.1.4. Theory and Experiment

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

#### 4.2.1. Experiments

#### 4.2.2. Electron–Lattice Coupling

#### 4.2.3. Spin-Peierls Instability

#### 4.2.4. Theory and Experiment

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**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.

**Figure 3.**Contour plot of the normalized one-loop charge gap ${\mathsf{\Delta}}_{\rho}/2{E}_{F}(\equiv {e}^{-{\ell}_{\rho}})$ as a function of the alternating site (${\u03f5}_{0}$) and bond ($\delta t$) potentials. The calculations are done for repulsive interactions $U/t=V/t=0.5$ and $\delta U/t=\delta V/t=0.1$.

**Figure 4.**Site (+) and bond $(-)$ $2{k}_{F}$ charge-density-wave and their respective phase (${\theta}_{\pm}$) relative to the alternated lattice.

**Figure 5.**Site (+) and bond $(-)$ $2{k}_{F}$ spin-density-wave and their respective phase (${\theta}_{\pm}$) relative to the alternated lattice.

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

**Figure 7.**Anion ordering critical temperature for the ReO${}_{4}$ salt in different families of organic conductors and their alloys. After Ilakovac et al. [26] and references there cited.

**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 $(\frac{1}{2},\frac{1}{2})$ $\left[(\frac{1}{2},0)\right]$ ordering.

**Figure 9.**Iso$-\lambda $ phase boundaries between $(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0)$ anion orderings, as a function of the normalized site potential and dimerization.

**Figure 10.**Calculated critical temperature for the ($\frac{1}{2},\frac{1}{2}$) anion ordering (

**green**) and Mott scale (

**blue**), as a function of TMTTF concentration x in [(TMTSF)${}_{1-x}$(TMTTF)${}_{x}$]${}_{2}$ReO${}_{4}$ alloys.

**Figure 11.**Critical temperatures of spin-Peierls and charge ordered states as a function of applied pressure in (TMTTF)${}_{2}$AsF${}_{6}$, as determined from NMR experiments. From Zamborszky et al. [39].

**Figure 12.**(

**top**): Calculated variation of the mean-field SP ordering temperature ${T}_{\mathrm{SP}}^{0}$ and of the Mott scale ${T}_{\rho}$ with the amplitude of site potential ${\u03f5}_{0}$ due to charge ordering; (

**bottom**): Calculated variations of the mean-field 1D (3D) spin-Peierls temperature ${T}_{\mathrm{SP}}^{0}$ (${T}_{\mathrm{SP}}\sim {T}_{\mathrm{SP}}^{0}/3)$, as a function of the tuning (pressure) parameter x. The dashed red line is a linear parametrization of the charge ordering temperature ${T}_{\mathrm{CO}}={\u03f5}_{0}\left(x\right)/2$. All temperature scales are normalized by the average hopping t along the stacks.

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**MDPI and ACS Style**

Ménard, M.; Bourbonnais, C.
One-Dimensional Alternating Extended Hubbard Model at Quarter-Filling and Its Applications to Structural Instabilities of Organic Conductors. *Crystals* **2020**, *10*, 942.
https://doi.org/10.3390/cryst10100942

**AMA Style**

Ménard M, Bourbonnais C.
One-Dimensional Alternating Extended Hubbard Model at Quarter-Filling and Its Applications to Structural Instabilities of Organic Conductors. *Crystals*. 2020; 10(10):942.
https://doi.org/10.3390/cryst10100942

**Chicago/Turabian Style**

Ménard, M., and C. Bourbonnais.
2020. "One-Dimensional Alternating Extended Hubbard Model at Quarter-Filling and Its Applications to Structural Instabilities of Organic Conductors" *Crystals* 10, no. 10: 942.
https://doi.org/10.3390/cryst10100942