Effective field theory for one-dimensional valence-bond-solid phases and their symmetry protection

We investigate valence-bond-solid (VBS) phases in one-dimensional spin systems by an effective field theory developed by Schulz [Phys. Rev. B 34, 6372 (1986)]. While the distinction among the VBS phases is often understood in terms of different entanglement structures protected by certain symmetries, we adopt a different but more fundamental point of view, that is, different VBS phases are separated by a gap closing under certain symmetries. In this way, the effective field theory reproduces the known three symmetries: time reversal, bond-centered inversion, and dihedral group of $\pi$ spin rotations. It also predicts that there exists another symmetry: site-centered inversion combined with a spin rotation by $\pi$. We demonstrate that the last symmetry gives distinct trivial phases, which cannot be characterized by their entanglement structure, in terms of a simple perturbative analysis in a spin chain. We also discuss several applications of the effective field theory to the phase transitions among VBS phases in microscopic models and an extension of the Lieb-Schultz-Mattis theorem to non-translational-invariant systems.

Figure 1

Various spin ladders considered in this paper. (a) Four-leg spin tube and (b) conventional (open) ladder with perpendicular interchain couplings. (c) Regarding a spin-2 as four spin-$\frac{1}{2}$'s, a spin-2 chain can be mapped onto a four-leg spin-$\frac{1}{2}$ ladder model with diagonal interchain couplings.

Figure 2

Schematic pictures of the VBS phases appearing in the XXZ chain for (a) $S=1$ and (b) $S=2$. The black dot represents a spin-$\frac{1}{2}$ and the black circle means the symmetrization of enclosed spin-$\frac{1}{2}$'s. Two spin-$\frac{1}{2}$'s linked by the red solid line form a singlet, while spin-$\frac{1}{2}$'s enclosed by the black rectangle are frozen to the $S^z=0$ state.

Figure 3

Schematic pictures of the VBS phases appearing in the open spin ladder with the columnar dimerization for (a) $N=2$ and (b) $N=3$. The black dot and red solid line represent a spin-$\frac{1}{2}$ and singlet bond, respectively.

Figure 4

An expected phase diagram for the spin-2 chain (47). The horizontal and vertical axes correspond to the staggered magnetic field $h$ and some function of $D_{2n}^z$, respectively. The VBS pictures for three Haldane phases realized at $h=0$ are shown in the left, while those for three trivial phases realized in the presence of $h$ are shown in the right. The arrows in the VBS pictures represent polarized spin-$\frac{1}{2}$'s along the staggered magnetic field.