Quantum entanglement and topological order in hole-doped valence-bond solid states

We present a detailed analysis of topological properties of the valence-bond solid (VBS) states doped with fermionic holes. As concrete examples, we consider the supersymmetric extension of the SU(2) and the SO(5) VBS states, dubbed UOSp(1$|$2) and UOSp(1$|$4) supersymmetric VBS states, respectively. Specifically, we investigate the string-order parameters and the entanglement spectra of these states to find that, even when the parent states (bosonic VBS states) do not support the string order, they recover it when holes are doped and the fermionic sector appears in the entanglement spectrum. These peculiar properties are discussed in light of the symmetry-protected topological order. To this end, we characterize a few typical classes of symmetry-protected topological orders in terms of supermatrix product states (SMPS). From this, we see that the topological order in the bulk manifests itself in the transformation properties of the SMPS in question and thereby affects the structure of the entanglement spectrum. Then, we explicitly relate the existence of the string order and the structure of the entanglement spectrum to explain the recovery and the stabilization of the string order in the supersymmetric systems.

Figure 1

The $r\to \infty$ limit of $\mathcal{S}=1$ SVBS state. Filled circles denote the bosonic qubits [$S=\frac{1}{2}$ spins for UOSp(1$|$2) and four-dimensional SO(5) spinors for UOSp(1$|$4)]. On a chain with even number of sites, the MPS is block diagonal with the (1,1) block ${\mathcal{B}}_{1,1}$ and the (2,2) block ${\mathcal{B}}_{2,2}$ corresponding to states A and B, respectively.

Figure 2

The string-order parameter ${\mathcal{O}}_{\text{string}}^{\infty }$ for several values of superspin $\mathcal{S}$ plotted as a function of $r$ (Ref. 27). Note that ${\mathcal{O}}_{\text{string}}^{\infty }(r=0)=0$ for even $\mathcal{S}$ corresponding to the vanishing of string-order parameter for even $S$.

Figure 3

The infinite-distance limit of the string correlation function for UOSp(1$|$4) states. At $r=0$, the string order reproduces the known result (Ref. 28) $9/25=0.36$. Also plotted is the string order of the $\mathcal{S}=2$ ($M=2$) state. As in the UOSp(1$|$2) case, the string order vanishes at $r=0$ and revives after doping.

Figure 4

The behavior of entanglement spectrum of the $\mathcal{S}=1$ UOSp(1$|$2) SVBS state (the inset is for the bosonic-pair VBS state). “B” and “F” denote bosonic and fermionic parts of the spectrum, respectively.

Figure 5

The behavior of the entanglement entropy of the $\mathcal{S}=1$ UOSp(1$|$2) SVBS state. (The inset is for the bosonic-pair VBS state.)

Figure 6

The entanglement spectrum and the entanglement entropy (inset) of the $\mathcal{S}=2$ UOSp(1$|$2) SVBS chain. “B” and “F” denote bosonic and fermionic parts of the spectrum, respectively.

Figure 7

The entanglement spectrum of the UOSp(1$|$4) SVBS state (37). The entanglement entropy of the same state is shown in the inset.

Figure 8

Diagrammatic representation of the main part of string correlation function $\left\{\left(T_{\text{string}}^z{\mathbf{V}}_{\text{R},1}^{\left(u\right)}\right)\left({\mathbf{V}}_{\text{L},1}^{\left(u\right)}{T}^z\right)\right\}$.

Figure 9

Rewriting the boundary factor (for $a=z$) using ${\widehat{u}}_x$. When $U_x$ and $U_z$ are anticommuting, the minus sign coming from ${\widehat{u}}_x{S}^z{\widehat{u}}_x^{†}=-S^z$ is canceled and an overall plus sign is recovered.

Figure 10

Rewriting the boundary factor (for $a=z$) using ${\widehat{u}}_x$. When $U_x$ and $U_z$ are anticommuting, the minus sign coming from ${\widehat{u}}_x{S}^z{\widehat{u}}_x^{†}=-S^z$ is canceled and an overall plus sign is recovered. Note that an extra $P$ matrix appears in the SUSY case.

Figure 11

The $r\to \infty$ limit of the bulk superspin $\mathcal{S}=2$ UOSp(1$|$4) SVBS state is given by the superposition of the two partially dimerized states related to each other by one-site translation. When we make an entanglement cut at an arbitrary bond (shown by wavy lines), we always have two different kinds of sections: one with 4 “fermionic” edge states (upper) and the one with 10 “bosonic” edge states (lower). These two different sections respectively yield fourfold- and tenfold-degenerate entanglement levels.