Dynamical structure factor of the $J_1-J_2$ Heisenberg model in one dimension: The variational Monte Carlo approach

The dynamical spin structure factor is computed within a variational framework to study the one-dimensional $J_1-J_2$ Heisenberg model. Starting from Gutzwiller-projected fermionic wave functions, the low-energy spectrum is constructed from two-spinon excitations. The direct comparison with Lanczos calculations on small clusters demonstrates the excellent description of both gapless and gapped (dimerized) phases, including incommensurate structures for $J_2/J_1>0.5$. Calculations on large clusters show how the intensity evolves when increasing the frustrating ratio and give an unprecedented accurate characterization of the dynamical properties of (nonintegrable) frustrated spin models.

Figure 1

Top: accuracy of the DFS, WFA, and WFB wave functions for a chain of $L=30$ sites. $\mathrm{\Delta }E$ is the difference between the variational energy $E_0^{\mathrm{var}}$ and the exact ground-state energy $E_0$, obtained with Lanczos diagonalizations. For $J_2/J_1>0.5$, the accuracy of the DFS state is not shown since it rapidly deteriorates. Bottom: overlap $|\langle {\mathrm{\Psi }}_0|{\mathrm{{\rm Y}}}_0\rangle |$ between the variational wave functions, either WFA or WFB, and the exact one.

Figure 2

Dynamical structure factor for $L=30$ and $J_2=0$. The Lanczos results are reported in the left panel. The variational calculations in the middle and right panels were obtained using the Ansätze WFA and WFB, respectively.

Figure 3

Comparison between Lanczos and variational calculations for $S^z(q,\omega )$ at different momenta $q=2\pi /L\times n$, with $n$ being an integer specified in the figure. Here, we consider $L=30$ and $J_2=0$. The $\delta$ functions in Eqs. (1) and (9) have been replaced by normalized Gaussians with $\sigma =0.05J_1$. Statistical errors are negligible within the present scale.

Figure 4

The same as in Fig. 3, but for $J_2/J_1=0.45$.

Figure 5

The overlaps between the exact and the variational excitations are reported for a chain of $L=30$ sites. The cases with $J_2=0$ (left panel) and $J_2/J_1=0.45$ (right panel) are shown. The size of the circles indicates the magnitude of the overlap, while their colors represent the value of the variational spectral weights $|\langle {\mathrm{\Psi }}_n^q|S_q^z|{\mathrm{\Psi }}_0{\rangle |}^2$. For a detailed description of the calculations of the overlaps, see the main text.

Figure 6

Dynamical spin structure factor $S^z(q,\omega )$ for a chain of $L=30$ sites: comparison of variational and Lanczos results for different values of the frustrating ratio. The $\delta$ functions in Eqs. (1) and (9) have been replaced by normalized Gaussians with $\sigma =0.1J_1$.

Figure 7

Dynamical spin structure factor $S^z(q,\omega )$ for different values of the frustrating ratio and $L=198$ sites. The $\delta$ functions in Eq. (9) have been replaced by normalized Gaussians with $\sigma =0.1J_1$. The white dashed lines for $J_2=0$ indicate the lower and upper limits of the two-spinon contributions, Eqs. (18) and (19).