Modeling bond-order wave instabilities in doped frustrated antiferromagnets: Valence bond solids at fractional filling

We explore both analytically and numerically the properties of doped $t\text{−}J$ models on a class of highly frustrated lattices, such as the kagomé and the pyrochlore lattices. Focusing on a particular sign of the hopping integral and antiferromagnetic exchange, we find a generic symmetry-breaking instability toward a twofold-degenerate ground state at a fractional filling below half filling. These states show modulated bond strengths and only break lattice symmetries. They can be seen as a generalization of the well-known valence bond solid states to fractional filling.

Figure 1

(Color online) Four different bisimplex lattices. The kagomé lattice (a) and the pyrochlore lattice (b) together with their lower-dimensional analogs the kagomé strip (c) and the checkerboard lattice (d). The two types of corner-sharing units (up vs down) are distinguished by the linewidth. They correspond to the bond-order wave symmetry-breaking pattern occurring at $n=2∕3$ on the triangle-based lattices and at $n=1∕2$ on the tetrahedron-based lattices.

Figure 2

The kagomé bands and the density of states per unit cell and spin. The energy is measured in units of $t$.

Figure 3

The pyrochlore bands and the density of states per unit cell and spin. The energy is measured in units of $t$.

Figure 4

(Color online) Correlation function of the kinetic energy [Eq. (18)] of a 21- site kagomé sample at $n=2∕3$ and $J∕t=0.4$. The black, empty bonds denote the same reference bond, the red, full bonds negative and the blue, dashed bonds positive correlations. The line strength is proportional to the magnitude of the correlations.

Figure 5

(Color online) Excitation gaps of the $t\text{−}J$ model on the kagomé lattice at $J∕t=1$ as a function of the parameter $\alpha$, which denotes the ratio of the intertriangle to the intratriangle couplings. The gaps are obtained by the CORE method for different sample sizes (and geometries). For the smaller samples (18, v1; 18, v2; 21) ED data are shown for comparison at $\alpha =1$.

Figure 6

(Color online) DMRG results for a $L=24$ kagomé ladder at $J∕t=0.4$ and $n=2∕3$. (a) Local bond strength deviation of the kinetic term. Red, full bonds are stronger (lower in energy) than the average kinetic energy per bond. Blue, dashed bonds are weaker than average bonds. (b) Local bond strength deviation of the exchange term. The color pattern is the same as in the upper panel. The thickness of the bond denotes the deviation from the average value per bond. Note that the patterns of the kinetic and exchange terms are in phase.

Figure 7

(Color online) DMRG results for the alternation of the bond strength of the kinetic term and the spin exchange term as a function of inverse system size $1∕L$, for different values of $J∕t$.

Figure 8

(Color online) DMRG results for the spin gap and single-particle gap for $J∕t=0,0.4,1,2$, as a function of inverse system size $1∕L$.

Figure 9

(Color online) DMRG results for the kinetic energy alternation for Hubbard kagomé strips at $n=2∕3$ of lengths $L=32$ and 48. The modulation is nonmonotonic as a function of $U∕t$ and shows a maximum around $U∕t\approx 10–15$.

Figure 10

The kagomé strip bands and the density of states per unit cell and spin. The energy is measured in units of $t$.