Identifying symmetry-protected topological order by entanglement entropy

According to the classification using projective representations of the SO(3) group, there exist two topologically distinct gapped phases in spin-1 chains. The symmetry-protected topological (SPT) phase possesses half-integer projective representations of the SO(3) group, while the trivial phase possesses integer linear representations. In the present work, we implement non-Abelian symmetries in the density matrix renormalization group (DMRG) method, allowing us to keep track of (and also control) the virtual bond representations and to readily distinguish the SPT phase from the trivial one by evaluating the multiplet entanglement spectrum. In particular, using the entropies $S^I$ ($S^H$) of integer (half-integer) representations, we can define an entanglement gap $G=S^I-S^H$, which equals 1 in the SPT phase, and $-1$ in the trivial phase. As an application of our proposal, we study the spin-1 models on various one-dimensional (1D) and quasi-1D lattices, including the bilinear-biquadratic model on a single chain, and the Heisenberg model on a two-leg ladder and a three-leg tube. Among others, we confirm the existence of a SPT phase in the spin-1 tube model and reveal that the phase transition between the SPT and the trivial phase is a continuous one. The transition point is found to be critical, with conformal central charge $c=3$ determined by fits to the block entanglement entropy.

Figure 1

(a) Spin chain with coupling constant $J$. (b) Two-leg spin ladder model with $J_L$ and $J_R$ for couplings along chain and rung directions, respectively. (c) Three-leg spin tube model; $J_L$ is the coupling along the leg. Each isosceles triangle contains two kinds of couplings, $J_R$ for the two equal sides and $\alpha J_R$ for the third. (d) SU(2)-invariant matrix product state describing ground state of (a) spin-1 chain, (b) ladder, or (c) tube. $|S=n\rangle$ represents a multiplet with quantum number $S=n$. For the spin-1 model, each local space is a $|S=1\rangle$ triplet. The input bond multiplets on both open ends can be controlled, and three common choices are shown in panel (d). Panel (e) shows the phase diagram of the spin-1 BLBQ chain,[13, 14] H, C, FM, and D stand for Haldane, critical, ferromagnetic, and dimerized phases, respectively. There is a narrow region near $\theta \approx -3/4\pi$ with possible spin nematic order, whose existence still remains debatable.

Figure 2

(a) The multiplet entanglement spectrum of a spin-1 BLBQ chain, calculated using the protocol algorithm. Multiplets $|S=0\rangle \oplus |S=1/2\rangle$ are put on the end bonds, and the converged spectra are obtained after several DMRG sweeps. $\theta =0$ (asterisks) is in the Haldane phase, with half-integer-spin bond multiplets; $\theta =-\pi /2$ (circles) is in the trivial phase, with integer-spin bond multiplets. Every datum represents a multiplet (not as usual a single state within a multiplet). Therefore, a multiplet with symmetry label $S$ corresponds to $2S+1$ degenerate states. (b) Multiplet spectrum of integer and half-integer representations for the Haldane phase calculated for $\theta =0$ and using $|S=1/2\rangle$ (asterisks) or $|S=0\rangle$ (circles), respectively, on the end bonds. The half-integer spectrum has been shifted by ${log}_{10}\left(2\right)$, in order to reveal the one-to-one correspondence between each multiplet in the half-integer spectrum and a pair of degenerate multiplets in the integer spectrum.

Figure 3

Integer and half-integer entanglement entropies, $S^I$ and $S^H$, of the spin-1 BLBQ model, for $L=200$ (dash-dotted lines) and $L=500$ (solid lines). Results for different system sizes coincide for $\theta$ values far from the critical point at ${\theta }_c=-\pi /4$ (vertical dash-dotted line) but differ in the intermediate region between the two vertical dashed lines. $S^I$ and $S^H$ cross at a “pseudotransition” point ${\theta }_c^L$, which moves towards the critical point as the system size is increased [shown in panel (d), the extrapolated point is very close to the true critical one]. In the above calculations, 400 multiplets (about 1600 individual states) have been retained, the truncation errors are of the order ${10}^{-10}$ around critical point, and are negligible (${10}^{-14}$ or less) for the rest parameters. The entanglement entropies are evaluated at the center of the chain. Panel (a) also shows the entanglement entropy obtained by the iTEBD algorithm[31] (asterisks), which favors the minimally entangled states, and always follows the lower entanglement entropies. (b) The entanglement gap $G=S^I-S^H$, which equals $\pm 1$ in the SPT phase and the trivial phase respectively. The dashed vertical lines mark the intermediate region, where $G$ is not a constant owing to finite size effects. Panel (c) shows the string order parameter (SOP $O_z$) of Eq. (2), obtained by iTEBD calculations, which retain up to 200 states.

Figure 4

(a) Integer and half-integer entanglement entropies $S^I$ and $S^H$ for the spin-1 tube model. The critical point estimated from their crossing point is ${\alpha }_c=0.571\left(1\right)$. (b) Entanglement gap $G=S^I-S^H$. $G=1$ when $\alpha <{\alpha }_c$, identifying the existence of a SPT phase, and $G=-1$ when $\alpha <{\alpha }_c$, corresponding to a trivial phase. The system size is $3\times 100$, 400 multiplets are reserved, which lead to maximum truncation error about ${10}^{-8}$ (at the critical point).

Figure 5

Analysis of the block entanglement entropy $S\left(x\right)$ of the ground state of the spin-1 three-leg tube Heisenberg model in the vicinity of ${\alpha }_c$. Panels (a) and (b) show the entanglement entropy between boundary block of length $x$ and the rest of the system, for several different system sizes and $\alpha$ values, on (a) a linear scale and (b) a log(sin) scale on the horizontal axis. Curves are vertically offset by 1 unit for clarity. Here we show the data on one of the three sublattices in the tube model, which contains entanglement entropies cut at the $i$th bond [$\mathrm{mod}(\mathrm{i},3)=1$]; the other two curves give the same fitting results and are not present here. The conformal central charge is determined as $c\simeq 3$, panel (c) shows how the fitted $c$'s vary with $\alpha$ for three fixed system sizes. Panels (d) and (e) show, respectively, the maximal $c$ values and corresponding $\alpha$ values obtained for 5 different $N$ values. The system size ranges from $N=50\times 3$ to $100\times 3$ ($N$ is the total site number), and up to 450 bond multiplets are reserved in the calculations. A half-integer bond representation was adopted in the calculations; the fittings of integer-representation entanglement entropies lead to the same conclusion.

Figure 6

(a) The ground-state energy per site, $e_0$, of a spin-1 tube versus the coupling ratio $\alpha$. The system size varies from $60\times 3$ to $120\times 3$. For the largest size $120\times 3$, 500 SU(2) multiplets [$\approx 2000$ equivalent U(1) states] are retained in the calculations, truncation errors are less than ${10}^{-9}$. The inset shows the first-order derivatives of energies $d{e}_o/d\alpha$, which are substantially converged with different system sizes and are shown clearly to be continuous through the critical point. (b) The second-order derivative $d^2{e}_0/d{\alpha }^2$, which shows a diverging peak at ${\alpha }_c=0.5715\left(5\right)$. The inset in panel (b) shows $d^2{e}_0/d{\alpha }^2$ in the vicinity of critical point on a log-log scale. The data points fall into two linear lines (except for the points very close to the critical point ${\alpha }_c$, owing to the finite-size effects near the critical point), which implies algebraic divergence. The dashed lines in the inset are fits to the form $d^2{e}_0/d{\alpha }^2\propto {(\alpha -{\alpha }_c)}^{-\nu }$, with $\nu \simeq 0.87$ and $0.6$, approaching critical point from the left and right sides, respectively.

Figure 7

The entanglement entropies of spin-1 two-leg ladder system with system size $80\times 2$. Two hundred multiplets are retained in the calculations, and the maximum truncation errors $\approx {10}^{-8}$. (a) $S^I$ and $S^H$ represent integer and half-integer entropies, respectively. $S^I\left(0\right)$ or $S^I\left(1\right)$ means that a $|S=0\rangle$ or $|S=1\rangle$ is the lowest multiplet in the multiplet spectrum, respectively. The dashed line is a guide for the eye.