Symmetry-protected topological order in magnetization plateau states of quantum spin chains

A symmetry-protected topologically ordered phase is a short-range entangled state, for which some imposed symmetry prohibits the adiabatic deformation into a trivial state which lacks entanglement. In this paper we argue that magnetization plateau states of one-dimensional antiferromagnets which satisfy the conditions $S-m\in$ odd integer, where $S$ is the spin quantum number and $m$ the magnetization per site, can be identified as symmetry-protected topological states if an inversion symmetry about the link center is present. This assertion is reached by mapping the antiferromagnet into a nonlinear sigma model type effective field theory containing a novel Berry phase term (a total derivative term) with a coefficient proportional to the quantity $S-m$, and then analyzing the topological structure of the ground state wave functional which is inherited from the latter term. A boson-vortex duality transformation is employed to examine the topological stability of the ground state in the absence/presence of a perturbation violating link-center inversion symmetry. Our prediction based on field theories is verified by means of a numerical study of the entanglement spectra of actual spin chains, which we find to exhibit twofold degeneracies when the aforementioned condition is met. We complete this study with a rigorous analysis using matrix product states.

Figure 1

Schematic pictures of $S=1$ Haldane (VBS) and large-$D$ phases. The links connecting spin-1/2's (balls) represent singlet bonds.

Figure 2

Schematic pictures of the spin Berry phase for magnetization plateau states in spherical coordinates.

Figure 3

Following Ref. [29], we take a summation over spin Berry phases with an alternating sign (vertical arrows), which amounts to counting the vorticity in every other row. The horizontal arrows cancel due to the contributions from adjacent plaquettes.

Figure 4

Application of a staggered magnetic field along the $z$ axis alters the local magnetization from $S-m$ to $S-m+{(-)}^j\delta m$, which clearly results in the breakdown of the structure (13). This perturbation can be prohibited by imposing a link-center inversion symmetry on the system.

Figure 5

An illustration of an FFAF chain. Each circle represents a spin-$S$ object. Two successive ferromagnetic couplings and one antiferromagnetic coupling residing on the nearest neighbor links are repeated as depicted.

Figure 6

Magnetization curves of (a) $S=1/2$ and (b) $S=1$ FFAF chains. (c) Entanglement spectra obtained by the bipartition at an antiferromagnetic bond under $H/J=0.4$ for $S=1/2,H/J=1.0$ and 2.5 for $S=1$ [shown by arrows in panels (a), (b)]. Calculations are performed using iTEBD.

Figure 7

(a) The VBS picture of a model plateau state (15) at $m=1/2$ in a $S=3/2$ spin chain. (b) Large-$D$ plateau.

Figure 8

Schematic pictures of $S=2$ Haldane, intermediate-$D$, and large-$D$ phases. The links connecting spin-1/2's (balls) represent singlet bonds.