Strange correlations in spin-1 Heisenberg antiferromagnets

We study the behavior of the recently proposed “strange correlator” [Y.-Z. You, Z. Bi, A. Rasmussen, K. Slagle, and C. Xu, Phys. Rev. Lett. 112, 247202 (2014)] in spin-1 Heisenberg antiferromagnetic chains with uniaxial single-ion anisotropy. Using projective quantum Monte Carlo, we are able to directly access the strange correlator in a variety of phases, as well as to examine its critical behavior at a quantum phase transition between trivial and nontrivial symmetry protected topological phases. After finding the expected long-range behavior in these two symmetry conserving phases, we go on to verify the topological nature of two-leg and three-leg spin-1 Heisenberg antiferromagnetic ladders. This demonstrates the power of the strange correlator in distinguishing between trivial and nontrivial symmetry protected topological phases.

Figure 1

Illustration of the three lattice geometries considered in this work: (a) chain, (b) two-leg ladder, and (c) three-leg ladder. The spin exchange couplings $J$ and $K$ are shown as red and blue lines, respectively. Empty black circles represent lattice sites.

Figure 2

Strange correlator in the Haldane $\left(D=0\right)$ and large-$D$ $\left(D=2\right)$ phases, as well as in the vicinity of the critical point $\left(D=1\right)$. The long-range order and exponential decay in the Haldane and large-$D$ phases, respectively, are readily apparent in the log-normal plot of the upper panel, while the lower panel illustrates algebraic decay near $D_c$ using a log-log plot. Solid lines are fits to $C\left(r\right)=C\left(\infty \right),C\left(r\right)\sim r^{-\eta }+{(L-r)}^{-\eta }$, and $C\left(r\right)\sim e^{-r/\xi }+e^{-(L-r)/\xi }$ in the region $L<4r<3L$ (black, red and blue lines, respectively). Taking $J=1$, we find $2m=L^2$ is sufficient to converge to the ground-state limit.

Figure 3

Strange correlator in the two-leg spin-1 ladder with $D=0$ and $J=K=1$. We find $2m=L^2$ is sufficient to converge to the ground-state limit.

Figure 4

Strange correlator in the three-leg spin-1 ladder with $D=0$ and $J=K=1$. Correlations involving the middle leg of the ladder are shown in the left panels, while those involving only the outer legs are shown in the right panels. We find $2m=3L^2$ is sufficient to converge to the ground-state limit.

Figure 5

Finite-size scaling of the order parameter for the strange correlator in chain (left panel) and ladder (right panel) geometries. Lines are obtained as fits to the data using the finite-size scaling forms derived in the Appendix. Parameters are the same as in earlier figures.

Figure 6

Finite-size scaling of the order parameter across the Gaussian critical point of the Heisenberg AFM chain with single-ion anisotropy for the normal Green's function (top panels) and the strange correlator (bottom panels). Lines are shown as guides to the eye. We find $2m=4L^2$ is sufficient to converge to the ground-state limit.