Temperature-driven transition from the Wigner crystal to the bond-charge-density wave in the quasi-one-dimensional quarter-filled band

It is known that within the interacting electron model Hamiltonian for the one-dimensional $\frac{1}{4}$-filled band, the singlet ground state is a Wigner crystal only if the nearest-neighbor electron-electron repulsion is larger than a critical value. We show that this critical nearest-neighbor Coulomb interaction is different for each spin subspace, with the critical value decreasing with increasing spin. As a consequence, with the lowering of temperature, there can occur a transition from a Wigner crystal charge-ordered state to a spin-Peierls state that is a bond-charge-density wave with charge occupancies different from the Wigner crystal. This transition is possible because spin excitations from the spin-Peierls state in the $\frac{1}{4}$-filled band are necessarily accompanied by changes in site charge densities. We apply our theory to the $\frac{1}{4}$-filled band quasi-one-dimensional organic charge-transfer solids, in general, and to 2:1 tetramethyltetrathiafulvalene (TMTTF) and tetramethyltetraselenafulvalene cationic salts, in particular. We believe that many recent experiments strongly indicate the Wigner crystal to bond-charge-density Wave transition in several members of the TMTTF family. We explain the occurrence of two different antiferromagnetic phases but a single spin-Peierls state in the generic phase diagram for the 2:1 cationic solids. The antiferromagnetic phases can have either the Wigner crystal or the bond-charge-spin-density wave charge occupancies. The spin-Peierls state is always a bond-charge-density wave.

Figure 1

Schematic of the proposed $T$ versus $P$ phase diagram for ${\left(\mathrm{TMT}C\mathrm{F}\right)}_{2}X$, where $C=\mathrm{T}$ or S, along with the charge occupancies of the sites in the low $T$ phases as determined in this work. The $P$ axis reflects the extent of interchain coupling. Open (filled) arrows indicate the ambient pressure locations of TMTTF (TMTSF) salts.

Figure 2

$V_c$ for Eq. (1a) in the limit of zero e-p interactions $(\alpha =g=0)$ as a function of $S_z$. Results are for a $N=32$ site periodic ring with $U=8$. For $N=32$ $S_z=8$ corresponds to the fully polarized (spinless fermion) limit. (a) Luttinger Liquid exponent $K_{\rho }$ as a function of $V$. $K_{\rho }=\frac{1}{4}$ determines the boundary for the ⋯1010⋯ CO phase. (b) $V_c$ plotted vs. $S_z$.

Figure 3

(Color online) QMC results for the temperature-dependent charge susceptibilities for a $N=64$ site periodic ring with $U=8$, $V=2.75$, $\alpha =0.15$, ${\omega }_S=0.1$, $g=0.5$, and ${\omega }_H=0.5$. (a) and (b) Wave-vector-dependent charge and bond-order susceptibilities. (c) $2k_F$ and $4k_F$ charge susceptibilities as a function of temperature. (d) $2k_F$ and $4k_F$ bond-order susceptibilities as a function of temperature. If error bars are not shown, statistical error bars are smaller than the symbol sizes. Lines are guides to the eye.

Figure 4

(Color online) Same as Fig. 3, but with parameters $U=8$, $V=2.25$, $\alpha =0.27$, ${\omega }_S=0.1$, and ${\omega }_H=0.5$. In panels (a) and (b), data are for $g=0.50$ only. In panels (c) and (d), data for both $g=0.50$ and $g=0.75$ are shown. Arrows indicate temperature where ${\chi }_{\rho }\left(2k_F\right)={\chi }_{\rho }\left(4k_F\right)$ (solid and broken arrows correspond to $g=0.50$ and 0.75, respectively.)

Figure 5

(Color online) Same as Fig. 3, but with parameters $U=8$, $V=2.75$, $\alpha =0.27$, ${\omega }_S=0.1$, $g=0.5$, and ${\omega }_H=0.5$. Arrow indicates temperature where ${\chi }_{\rho }\left(2k_F\right)={\chi }_{\rho }\left(4k_F\right)$.

Figure 6

(Color online) Same as Fig. 3, but with parameters $U=8$, $V=3.5$, $\alpha =0.24$, ${\omega }_S=0.1$, $g=0.5$, and ${\omega }_H=0.5$.

Figure 7

[(a) and (b)] $S=1$ VB diagram in the Heisenberg SP chain. The SP bond order in between the $S=\frac{1}{2}$ solitons in (b) is opposite to that elsewhere. (c) $\frac{1}{4}$-filled band equivalent of (a); the singlet bonds in the background BCDW are between units containing a pair of sites but a single electron (see text). [(d) and (e)] $\frac{1}{4}$-filled band equivalents of (b). Defect units with two different charge distributions are possible. The charge distribution will depend on $V$.

Figure 8

(a) Self-consistent site charges (vertical bars) and spin-spin correlations (points, lines are to guide the eye) in the $N=20$ periodic ring with $U=8$, $V=2.75$, ${\alpha }^2∕K_S=1.5$, $g^2∕K_H=0.32$ for $S_z=2$. (b) Same parameters, except $S_z=3$. Black and white circles at the bottoms of the panels denote charge-rich and charge-poor sites, respectively, while gray circles denote sites with population nearly exactly 0.5. The arrows indicate locations of the defect units with nonzero spin densities.