Entanglement spectrum of a topological phase in one dimension

We show that the Haldane phase of $S=1$ chains is characterized by a double degeneracy of the entanglement spectrum. The degeneracy is protected by a set of symmetries (either the dihedral group of $\pi$ rotations about two orthogonal axes, time-reversal symmetry, or bond centered inversion symmetry), and cannot be lifted unless either a phase boundary to another, “topologically trivial,” phase is crossed, or the symmetry is broken. More generally, these results offer a scheme to classify gapped phases of one-dimensional systems. Physically, the degeneracy of the entanglement spectrum can be observed by adiabatically weakening a bond to zero, which leaves the two disconnected halves of the system in a finitely entangled state.

Figure 1

(Color online) The colormaps show the entanglement entropy $S$ for different spin-1 models: Panel (a) shows the data for Hamiltonian $H_0$ in Eq. (3), panel (b) for $H_0$ plus a term which breaks the $e^{-i\pi S^y}\times \text{TR}$ symmetry [Eq. (17)], and panel (c) for $H_0$ plus a term which breaks inversion symmetry [Eq. (18)]. The blue lines indicate a diverging entanglement entropy as a signature of a continuous phase transition. The phase diagrams contain four different phases: a TRI phase for large $U_{zz}$, two symmetry breaking antiferromagnetic phases $Z_2^z$ and $Z_2^y$, and a Haldane phase (which is absent in the last panel).

Figure 2

(Color online) Entanglement spectrum of Hamiltonian $H_0$ in Eq. (3) for $B_x=0$ (only the lower part of the spectrum is shown). The dots show the multiplicity of the Schmidt values, which is even in the entire Haldane phase.

Figure 3

(Color online) The colormaps show the difference between the two largest Schmidt values $\left|{\lambda }_1-{\lambda }_2\right|$ for different spin-1 models. Panel (a) corresponds to the original Hamiltonian $H_0$ in Eq. (3), panel (b) to $H_0$ plus a term that breaks $e^{i\pi S^y}\times \text{TR}$ [Eq. (17)], and panel (c) to $H_0$ plus a term which breaks inversion symmetry [Eq. (18)]. The quantity $\left|{\lambda }_1-{\lambda }_2\right|$ is zero only in the Haldane phase.

Figure 4

(Color online) Half-chain entanglement entropy of the model Hamiltonian (3) at a bond which is slowly weakened as a function of time $J_{\text{weak}}=J-t\Gamma$ and $\Gamma =J^2/40$. The entanglement entropy of the resulting state is ln2 in the Haldane phase and zero otherwise.