Magnetization plateaus of an easy-axis kagome antiferromagnet with extended interactions

We investigate the properties in finite magnetic field of an extended anisotropic $XXZ$ spin-1/2 model on the kagome lattice, originally introduced by Balents, Fisher, and Girvin [Phys. Rev. B 65, 224412 (2002)]. The magnetization curve displays plateaus at magnetization $m=1/6$ and $1/3$ when the anisotropy is large. Using low-energy effective constrained models (quantum loop and quantum dimer models), we discuss the nature of the plateau phases, found to be crystals that break discrete rotation and/or translation symmetries. Large-scale quantum Monte Carlo simulations were carried out in particular for the $m=1/6$ plateau. We first map out the phase diagram of the effective quantum loop model with an additional loop-loop interaction to find stripe order around the point relevant for the original model as well as a topological ${\mathbb{Z}}_2$ spin liquid. The existence of a stripe crystalline phase is further evidenced by measuring both standard structure factor and entanglement entropy of the original microscopic model.

Figure 1

Kagome lattice with the interactions between the first (black), second (red), and third (blue) neighbors within the same hexagonal plaquettes, as included in the model (1). Typical examples of the third-nearest-neighbor interaction between different hexagons [that are not included in the Hamiltonian (1)] are shown by green dashed lines. Others are obtained by space-group operations.

Figure 2

Ground-state magnetization curves for a system of size $L=6$ and different values of $J_{xy}$, in the ground state ($J_z=1$ is taken as the unit of energy). Inset: Magnetization curve for $J_z=0$ (Ising limit). The three plateaus at magnetizations $m=0,1/6,1/3$ are clearly visible for the lowest value of the off-diagonal coupling constant $J_{xy}=-0.06$, and gradually disappear as $|J_{xy}|$ gets larger.

Figure 3

Representation of one of the ground states of model (1) for $J_{xy}=0$ and $m=1/6$ in terms of dimers (red) on the underlying triangular lattice (blue). Loop configurations are formed due to the constraint of two dimers per triangular lattice site.

Figure 4

Off-diagonal second-order process of $H_{xy}$ on a flippable bow-tie plaquette (green) for $n_{\downarrow }=3$. The antiparallel spins of two flippable pairs are exchanged, corresponding to a flip of two parallel dimers on the underlying triangular lattice (blue).

Figure 5

Phase diagram of the QLM Eq. (5) as a function of $V/t$, containing the three different phases discussed in Sec. 3a. The boundary between stripe phase and ${\mathbb{Z}}_2$ spin liquid is determined using topological degeneracy and the stripe structure factor.

Figure 6

Top left: Triangular clusters used in the QLM study. Bottom left: notations for the loop segment occupation number. Right: First Brillouin zone of the triangular lattice, with the reciprocal space vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$. The high symmetry points $K=(4\pi /3,0),M=(\pi ,\pi /\sqrt{3})$, and $A=(\pi ,0)$ required for the nonflippable states are represented.

Figure 7

Drawing two lines on the triangular lattice allows us to define two conserved ${\mathbb{Z}}_2$ invariants $(p_x,p_y)$ associated with the parity of the number of loop segments crossed by the line. In this example, the number of loop segments pointing in the vertical $y$ (horizontal $x$) direction which cross the line is odd (even), corresponding to $p_y=-1$ ($p_x=1$). This number does not depend on the precise location of the line (parallel lines shifted by one or several lattice units result in the same parity). The flipping terms of the Hamiltonian, acting for instance on the shaded plaquette (other flips are possible in this configuration), do not change these two parities. Note that a loop segment located on a diagonal bond of a plaquette contributes to both $p_x$ and $p_y$.

Figure 8

Topological gap ${\mathrm{\Delta }}_{\text{T}}$ as a function of $V/t$, for different sample sizes $L$. The finite-size effects are different for even (${\mathrm{\Delta }}_{\text{T}}^{\text{even}}$; left panel) and odd (${\mathrm{\Delta }}_{\text{T}}^{\text{odd}}$; right panel) values of $L$. In the thermodynamic limit, we expect that in both cases, ${\mathrm{\Delta }}_{\text{T}}$ if finite for the stripe phase $V/t<{(V/t)}_{\text{c}}$, and vanishes for the ${\mathbb{Z}}_2$ spin liquid phase ${(V/t)}_{\text{c}}\le V\le 1$. Left inset: Finite-size scaling as a function of $1/L$ for the even-$L$ samples provides an estimate ${(V/t)}_{\text{c}}\simeq 0.70\left(5\right)$ of the value where the gap first vanishes. Right inset: Zoom close to the RK point for odd-$L$ samples.

Figure 9

The stripe structure factor $S_{\mathrm{stripe}}$ as a function of inverse system size, for different values of $V/t$.

Figure 10

Expectation value $W_3$ of the quantifier of $Z_3$ anisotropy of histograms of loop occupations, as a function of $V/t$ for three even lattices sizes $L=4,6,8$. Insets: Color-coded normalized histograms of loop occupations numbers (using a color interpolation scheme), deep in the stripe phase (left inset, $L=6,V/t=-0.3$) and closer to the critical point (right inset, $L=6,V/t=0.3$). Vertical axis of histograms is $H_y$ (ranging from $-\sqrt{3}/2$ to $\sqrt{3}/2$), horizontal axis is $H_x$ (from $-1/2$ to 1). The histograms are enclosed in an equilateral triangle pictured close to the right inset.

Figure 11

Finite-size scaling of the $m=1/6$ plateau width. The plateau is found to disappear around $J_{xy}\simeq -0.081$.

Figure 12

Left: Connected correlation function ${\langle S_i^z{S}_j^z\rangle }_{\text{c}}=\langle S_i^z{S}_j^z\rangle -\langle S_i^z\rangle \langle S_j^z\rangle$ on the $m=1/6$ plateau (for $J_{xy}=-0.078,T=J_{\mathrm{ring}}/8$ on a $L=8$ cluster with periodic boundary conditions). Reference site is indicated as a black circle. Blue (red) denote positive (negative) values, and the radius is proportional to the absolute value of the correlation. Data close to the reference site have not been plotted for readability. Right: Cartoon representation of the spin and dimer configurations in one state of the stripe phase, with the kagome lattice in black and the underlying triangular lattice in blue.

Figure 13

Finite-size scaling of the (scaled) structure factor $S\left(\mathbf{Q}\right)/N$ inside the magnetization plateau. Despite the relatively small available sizes, we can reach the regime of saturation, confirming the striped solid anticipated from the QLM.

Figure 14

Rényi block entanglement entropies $S_q\left(A\right)$ ($q=2,3$) on the $m=1/6$ plateau, as a function of $L$, on various systems with $N=3L^2$ sites. Simulation parameters are $J_{xy}=-0.078,T=J_{\mathrm{ring}}/4$, and $h=1.03$. In each case, the $A$ block corresponds to a half-system cylinder containing $(L/2)\times L$ unit cells (see inset). The positive correction $log\left(3\right)$ is shown by a star.