Gapless edge states and their stability in two-dimensional quantum magnets

We study the nature of edge states in extrinsically and spontaneously dimerized states of two-dimensional spin-$\frac{1}{2}$ antiferromagnets, by performing quantum Monte Carlo simulation. We show that a gapless edge mode emerges in the wide region of the dimerized phases, and the critical exponent of spin correlators along the edge deviates from the value of Tomonaga-Luttinger liquid (TLL) universality in large but finite systems at low temperatures. We also demonstrate that the gapless nature at edges is stable against several perturbations such as external magnetic field, easy-plane XXZ anisotropy, Dzyaloshinskii-Moriya interaction, and further-neighbor exchange interactions. The edge states exhibit non-TLL behavior, depending strongly on model parameters and kinds of perturbations. Possible ways of detecting these edge states are discussed. Properties of edge states we show in this paper could also be used as reference points to study other edge states of more exotic gapped magnetic phases such as spin liquids.

Figure 1

(a) 2D dimerized model with bond alternation along the $y$ direction. (b) 2D $J{Q}_3$ model with six-spin interaction term. The modified coupling constant $Q_3^{\prime }$ on edges is explained in Sec. 2.

Figure 2

Spin-spin correlation $|S_{\mu \mu }\left(r_x\right)|=|\langle S_{(r_x,0)}^{\mu }{S}_{(0,0)}^{\mu }\rangle |$ on the edge in the dimerized phase of (a) dimerized model and (b) $J{Q}_3$ model. Insets: Results along the bulk direction: $|B_{\mu \mu }\left(r_y\right)|=|\langle S_{(L/2,r_y)}^{\mu }{S}_{(L/2,0)}^{\mu }\rangle |$. We normalize the values of $|S_{xx}\left(1\right)|$ and $|B_{xx}\left(1\right)|$ to be unity. In the inset, we depict only the results for $B_{xx}$ because of the $SU\left(2\right)$ symmetry of the models. Errors are drawn on all symbols, but they are less than the symbol size.

Figure 3

Perturbations in the dimerized model: (a) DM interaction, (b) next-nearest-neighbor interaction, and (c) bond alternation. An open boundary condition is imposed for both $x$ and $y$ directions, and the edge state appears parallel to the $x$ direction. In panel (a), cross circles on $J$ bonds indicate the direction of the ${\mathbf{D}}_{i,j}$ vector.

Figure 4

Edge spin correlations $S_{\mu \mu }\left(r_x\right)$ of (a) dimerized model and (b) $J{Q}_3$ model under a magnetic field $H$. Circles and triangles are the transverse ($\mu =x$) and longitudinal ($\mu =z$) correlators, respectively. We have eliminated a constant part of ${\langle S_j^z\rangle }^2$ from the data of $\mu =z$. Like Fig. 2, we have used the normalization $S_{xx}\left(1\right)=1$.

Figure 5

Magnetization profile $m\left(r_y\right)=\langle S_{L/2,r_y}^z\rangle$ of the dimerized model with $J^{\prime }/J=2$ under magnetic fields $H$ at $r_x=L/2$.

Figure 6

Effects of (a) a DM interaction (equal to XXZ anisotropy), (b) a next-nearest-neighbor interaction, and (c) an extra bond alternation along the edge direction on the edge spin correlations of the dimerized model. Circles and triangles are the results of $\mu =x$ and $\mu =z$, respectively. In panel (c), we have used the semilog scale and have set a larger $J^{\prime }/J=5$ and a lower temperature $T=1/\left(2L\right)$ to depict a clear exponential-decay form of the edge correlation. In all of the panels, we have used the normalization $S_{xx}\left(1\right)=1$.

Figure 7

Schematic inelastic-neutron-scattering spectrum of a 2D spin-Peierls phase with a gapless edge mode in the space of frequency $\omega$ and the wave number $k_x$. The wave number $k_y$ is fixed to $\pi$. The weight in the low-energy region $\omega \lesssim J$ is the contribution from the edge state, the lower bound of which would be similar to the des Cloizeaux-Peason mode $\sim J\left|\sin \right(k_x\left)\right|$. The weight in the higher-energy region $\omega \gtrsim J$ comes from the gapped triplet excitations in the bulk.