Quantized edge magnetizations and their symmetry protection in one-dimensional quantum spin systems

The bulk electric polarization works as a nonlocal order parameter that characterizes topological quantum matters. Motivated by a recent paper [H. Watanabe et al., Phys. Rev. B 103, 134430 (2021)], we discuss magnetic analogs of the bulk polarization in one-dimensional quantum spin systems, that is, quantized magnetizations on the edges of one-dimensional quantum spin systems. The edge magnetization shares the topological origin with the fractional edge state of the topological odd-spin Haldane phases. Despite this topological origin, the edge magnetization can also appear in topologically trivial quantum phases. We develop straightforward field theoretical arguments that explain the characteristic properties of the edge magnetization. The field theory shows that a U(1) spin-rotation symmetry and a site-centered or bond-centered inversion symmetry protect the quantization of the edge magnetization. We proceed to discussions that quantum phases on nonzero magnetization plateaus can also have the quantized edge magnetization that deviates from the magnetization density in bulk. We demonstrate that the quantized edge magnetization distinguishes two quantum phases on a magnetization plateau separated by a quantum critical point. The edge magnetization exhibits an abrupt stepwise change from zero to $1/2$ at the quantum critical point because the quantum phase transition occurs in the presence of the symmetries protecting the quantization of the edge magnetization. We also show that the quantized edge magnetization can result from the spontaneous ferrimagnetic order.

Figure 1

Schematic $x$ dependence of $\varphi \left(x\right)$ of sine-Gordon model (3). The shaded areas depict the left and right edges, $\mathcal{L}$ and $\mathcal{R}$.

Figure 2

Meron configuration of staggered moment, $\mathit{n}\left(x_j\right)={(-1)}^j{\mathit{S}}_j$, with skyrmion number $+1/2$.

Figure 3

Site $r$ dependence of the “charge” $C_r$ of Eq. (44) with $N=2$ (filled balls) and $\delta m^z\left(r\right)$. The edge magnetizations on the left and right edges are quantized as $\pm 0.25000$. We used parameters $J_{\perp }=1, J_{\parallel }=0.5, J_s=0.05, h_u=1.4$, and $2L=160$. (Inset) Magnetization curve. The shaded area highlights the $1/2$ magnetization plateau.

Figure 4

(a) Spin-1 chain (49) with bond-inversion center. (b) Magnetization curve for $J_4=0$. (c) Site $r$ dependence of $\delta m^z\left(r\right)$ (d) $J_4$ dependence of lowest-energy excitation gap ($+$ markers) and edge magnetizations (47) (circles and triangles). The edge magnetizations are quantized as $M_{\mathrm{left}}^z=M_{\mathrm{right}}^z=0.499999999$ for $J_4>J_{4c}\approx 0.295$. We used parameters $J=1, \alpha =0.2, h_u=1.5$. The system size is $L=162$ for (b) an (c) and 242 for (d).

Figure 5

(a) $J_4$ dependence of lowest-energy excitation gap for $h_u/J=1.5, \alpha =0.2$, and $L=122$ (circles) and $L=242$ ($+$ markers). (b) Numerically calculated entanglement entropy for $h_u/J=1.5, \alpha =0.2$, and $L=242$ (circles) and the fitting function (56) with $(a_s,c)=(0.52,0.94)$. The central charge $c\approx 1$ is consistent with the sine-Gordon theory (55).

Figure 6

(a) Union-jack strip. (b) Site $r$ dependence of $\langle C_j\rangle$ and $\delta m^z\left(r\right)$ with $N=3$ and $m=1/6$. The edge magnetizations are quantized as $M_{\mathrm{right}}^z=-M_{\mathrm{left}}^z=0.5000000$. We used $J_1=1, J_4=4, h_u=0.1$, and $3L=240$.

Figure 7

A vortex with $\nu =1$ resting a bond connecting two points $x_j$ and $x_{j+1}$ is shown. The space-time is discretized to the rectangular lattice. The vorticity is defined on a rectangular plaquette indicated by a gray circle with an arrow. The $\theta$ field changes by $\pi$ along the vertical line with arrows. The vorticity on the plaquette is $\nu =\left[\theta \right(x,\tau +\delta \tau )-\theta (x,\tau )-\theta (x+\delta x,\tau +\delta \tau )+\theta (x+\delta x,\tau +\delta \tau \left)\right]/2\pi =1$.

Figure 8

Comparison of edge magnetizations $M_{\mathrm{right}}^z$ and $M_{\mathrm{left}}^z$ for $L=242$ (circles and triangles) and for $L=240$ (squares and pentagons). The other parameters are fixed to $h_u/J=1.5$ and $\alpha =0.2$. The former case has the $I_b$ symmetry and the latter has not. The inversion symmetry quantizes the edge magnetizations except in the vicinity of the quantum critical point $J_4/J=0.295$. Without the inversion symmetry, the edge magnetization shows continuous changes even away from the quantum critical point.

Figure 9

$J_4$ dependence of polarization amplitude $\langle U\rangle$ (circles) and total charge ${\sum }_{j=1}^L\langle C_j\rangle$ (triangles) for $h_u/J=1.5, \alpha =0.2$ with system size (a) $L=242$ and (b) $L=240$. (a) When the ${\mathcal{I}}_b$ symmetry is present, the polarization amplitude and the total charge show abrupt changes at $J_4/J\approx 0.295$ thanks to the ${\mathcal{I}}_b$ symmetry. (b) On the other hand, when the ${\mathcal{I}}_b$ symmetry is absent, the total charge changes continuously though the accuracy of the numerical data is extremely lowered in the vicinity of the quantum critical point $J_4/J\approx 0.295$. Note that apart from $\langle U\rangle$ and the total charge, the other thermodynamic quantities behave similarly regardless of the presence of the absence of the ${\mathcal{I}}_b$ symmetry.