Imaging Bond Order near Nonmagnetic Impurities in Square-Lattice Antiferromagnets

We study the textures of generalized “charge densities” (scalar objects invariant under time reversal), in the vicinity of nonmagnetic impurities in square-lattice quantum antiferromagnets, by order parameter field theories. Our central finding is the structure of the vortex in the generalized density wave order parameter centered at the nonmagnetic impurity. Using exact numerical data from quantum Monte Carlo simulations on an antiferromagnetic spin model, we are able to verify the results of our field theoretic study. We extend our phenomenological approach to the period-4 bond-centered density wave found in the underdoped cuprates.

Figure 1

Cartoons of predicted modulations of VBS “vortices” that form around nonmagnetic impurities. In both columnar and plaquette VBS phases, four domains form and are separated by domain walls of the complementary VBS pattern. Unlike the “pinwheels” depicted in earlier work [15, 16], these structures do not break any symmetry of the impurity Hamiltonian.

Figure 2

Comparison of exact quantum Monte Carlo data with saddle-point mean-field theory in the Néel phase. (a) Dimerization $D(x_{ij})$ of $H_{JQ}$ with a nonmagnetic impurity (a missing spin) on $17\times 17$ system with open boundary conditions. $Q/J=4$, $T/J=0.02$. Despite the seemingly large value of $Q$, the system is deep in the Néel phase with a sublattice magnetization of about 20% of the classical value. (b) Mean-field results for the generalized charge-density $\delta {\rho }_V(r)$ evaluated on the bonds of a $17\times 17$ open square lattice, evaluated from the saddle-point value of $V(\mathbf{r})$ (which contains an impurity-centered vortex) from ${\mathcal{S}}_V+{\mathcal{S}}_{\mathrm{imp},V}$. We have set $a=0.1$, ${\tilde{K}}_1=1$, ${\tilde{K}}_2=0$, $\tilde{s}=-0.01$, $\tilde{u}=0.3$, $\tilde{w}=0.1$, and the impurity coupling ${\lambda }_1=0.5$. The width and darkness of the bonds are related linearly to the plotted quantities. From Fig. 3 it is clear that there is a large contribution from the boundary [25]. Both the effect of the boundary and the single impurity are captured well by the mean-field theory. The mean-field theory results are “generic” and the couplings were not picked to tune the agreement with Monte Carlo simulations.

Figure 3

VBS order parameter on a $32\times 32$ system with periodic boundary conditions from QMC simulations on the JQ model in the Néel phase. On the bonds, the dimerization, $D(x_{ij})$ is plotted in exactly the same way as Fig. 2. From the $D(x_{ij})$ data, we can construct a site-centered complex VBS order parameter, according to Eq.(5) [the VBS order parameter is undefined on the nearest neighbors of the impurity]. The arrow on the sites shows the phase of this order parameter, clearly demonstrating a VBS vortex around the impurity. The magnitude of the VBS order parameter (not shown) decays exponentially on a scale set by the finite VBS correlation length; making the extracted phase noisy far from the impurity. $Q/J$ and $T/J$ are identical to Fig. 2.

Figure 4

CDW and VBS order obtained by minimizing the free energy for a $20\times 20$ lattice with $a=0.15$; the left panel has a vortex, while the right has a domain wall in only one direction. The parameters are ${\tilde{K}}_1=1$, ${\tilde{K}}_2=0$, $\tilde{s}=0.5(1)$, $\tilde{u}=1$, $\tilde{w}=0.1(0.05)$, $K_1=1$, $K_2=1$, $s=-0.7(1)$, $u=4$, $v=0.1$, $w=0$, $\kappa =4(8)$ and the impurity couplings ${\lambda }_1=3(8)$ and ${\lambda }_2=0$ for the left (right) panels.