Continuous similarity transformation for critical phenomena: Easy-axis antiferromagnetic XXZ model

We apply continuous similarity transformations (CSTs) to the easy-axis antiferromagnetic XXZ model on the square lattice. The CST flow equations are truncated in momentum space by the scaling dimension $d$ so that all contributions with $d\le 2$ are taken into account. The resulting quartic magnon-conserving effective Hamiltonian is analyzed in the zero-, one-, and two-magnon sector. In this way, a quantitative description of the ground-state energy, the one-magnon dispersion, and its gap as well as of two-magnon bound states is gained for anisotropies ranging from the gapped Ising model to the gapless Heisenberg model. We discuss the critical properties of the gap closing as well as the evolution of the one-magnon roton mininum. The excitation energies of two-magnon bound states are calculated, and their decay into the two-magnon continuum is determined via the inverse participation ratio.

Figure 1

Magnetic Brillouin zone (MBZ) with different kinds of discretizing mesh. In panel (a) important points such as $\mathit{k}=(0,0)$ are sampled directly; this kind of mesh is denoted by $N_0$. For $N_0$, the calculations for $\lambda <1$ or the two-magnon sector do not require additional interpolations. But at $\mathit{k}=(0,0)$ divergences in the flow occur for $\lambda \lesssim 1$. They can be avoided by a mesh of the type shown in panel (b) for odd $L$; this mesh is denoted by $N_{\varnothing }$. We use it especially for the analysis of the critical behavior of the spin gap. Due to the point group symmetries only lattice sites in the marked blue area need to be taken into account.

Figure 2

Generic extrapolations in $1/L$ for $\lambda =0.7, \lambda =0.96$, and $\lambda =0.995$. For small $\lambda$, calculations on the meshes $N_{\varnothing }$ and $N_0$ converge; see panel (a). At higher values of $\lambda$ [see panels (b) and (c)], values from $N_{\varnothing }$ are monotonically increasing, while values from $N_0$ are monotonically decreasing. For the whole range, a monotonic quadratic fit for the $N_{\varnothing }$ data is used to determine the values for $L\to \infty$. The error estimate depicted in yellow results from the difference between the values from $N_{\varnothing }$ and $N_0$ for the highest reached value of $L$.

Figure 3

Ground-state energy per site $e$ as a function of $\lambda$. Red crosses are CST data, which are compared to results of a series expansion(SE) [9] drawn as blue line; the green star marks the QMC results for the isotropic Heisenberg model [14]. As expected from second-order perturbation theory the ground-state energy displays a monotonically decreasing behavior with negative curvature.

Figure 4

CST result of the dispersion $\omega \left(\mathit{k}\right)$ for $N_0$ and $L=16$ for various $\lambda$. For $\lambda =0$, the XXZ model is the Ising model with a flat dispersion, while it is the antiferromagnetic Heisenberg model for $\lambda =1$ with a gapless spectrum and a distinct roton minimum at $\mathit{k}=(\pi ,0)$ [4, 7]. For $0<\lambda <1$ the CST results interpolate smoothly between these two limits.

Figure 5

CST results for the one-magnon gap $\mathrm{\Delta }=\omega (\mathit{k}=0)$ of the XXZ model for $0\le \lambda \le 1$ compared to third-order spin wave theory (SWT) [12, 21], data from coupled cluster method (CCM) [43], results from series expansion (SE) about the Ising limit [10, 11], and the critical power law extracted from SE [10]. A quantitative comparison of the critical behavior is shown in the inset underlining very good agreement.

Figure 6

CST results for the ground-state correlation length $\xi$ for $0.7\le \lambda \le 1$. A comparison of the correlation length with the shortest wrap around $\sqrt{2}L$ in the largest cluster with $L=22$ (dashed line) is shown.

Figure 7

CST results for the roton mininum $\mathit{k}=(\pi ,0)$ and the dispersion maximum $\mathit{k}=(\pi /2,\pi /2)$ for $0\le \lambda \le 1$ compared to data from series expansion (SE) [10], DMRG [7], and QMC [15]. All methods agree well for dispersion maximum. For the roton mininum, all methods predict an inflection point for $\lambda \gtrsim 0.8$. The values for the depth of the roton mode differ slightly.

Figure 8

The upper panel shows the energies of the four magnon-magnon bound states ${\tau }_i$ calculated by CST as a function of the anisotropy $\lambda$ and the corresponding lower edge $2\mathrm{\Delta }$ of the two-magnon continuum. The inset compares data from truncated CST to data from pCUT up to terms $\propto {\lambda }^4$ [11]. One discerns a certain small deviation in quadratic order. In the lower panel, the inverse participation ratio IPR of the four bound states is plotted; only three curves are shown because the eigenwave functions of the two degenerate bound states are complex conjugates with the same IPR $I$. The vertical dashed lines mark the decay points for the corresponding bound state determined by a crossing of $I$ with the threshold $I={10}^{-4}$ (horizontal dashed line).