Ground States of a Frustrated Quantum Spin Chain with Long-Range Interactions

The ground state of a spin-$1/2$ Heisenberg chain with both frustration and long-range interactions is studied using Lanczos exact diagonalization. The evolution of the well-known dimerization transition of the system with short-range frustrated interactions (the $J_1\mathrm{\text{−}}{J}_2$ chain) is investigated in the presence of additional unfrustrated interactions decaying with distance as $1/r^{\alpha }$. It is shown that the continuous (infinite-order) dimerization transition develops into a first-order transition between a long-range ordered antiferromagnetic state and a state with coexisting dimerization and critical spin correlations at wave number $k=\pi /2$. The relevance of the model to real systems is discussed.

Figure 1

Approximate phase diagram as a function of the frustration strength $g$ and the inverse of the long-range exponent $\alpha$. The dashed curves indicate continuous phase transitions, whereas the thick solid curve represents a first-order transition. The curve for $g<0$ corresponds to the interaction studied in 10. At ${\alpha }^{-1}=0$, the dominant spin correlations in the VBS state change from $k=\pi$ to $\pi /2$ at $g\approx 0.52$ [9]. This transition (or cross over) evolves to the point where all the phase boundaries come together.

Figure 2

Low-energy levels of a 16-spin system at $\alpha =4$ (upper panel) and $\alpha =1$ (lower panel). The spin $S$ and the momentum $k$ of the states are indicated in the upper panel. The inset in the lower panel shows the avoided level crossing of the two $k=0$ singlets in greater detail.

Figure 3

Dependence on the inverse chain length of the singlet-triplet crossing point $g_{\mathrm{cross}}$ and the location $g_{\mathrm{peak}}$ of the ground state energy maximum for different long-range interaction exponents $\alpha$. The two lines show extrapolations of the $g=2.0$ numerical data to $N=\infty$.

Figure 4

Size dependence of the second derivative of the ground state energy with respect to the frustration parameter $g$ at the point $g_{\mathrm{peak}}$ where the ground state energy takes its maximum value.

Figure 5

Spin (upper panel) and dimer (lower panel) correlations in a 32-spin chain at $\alpha =1$. At $g=0.25$ and 0.45, the system is in the AFM and $\mathrm{VBS}\mathrm{\text{−}}\mathrm{QLRO}(\pi /2)$ phases, respectively. A first-order transition between these states occurs at $g\approx 0.39$.