Symmetry-protected topological order and negative-sign problem for $\mathrm{SO(}N)$ bilinear-biquadratic chains

Using a generalized Jordan-Wigner transformation combined with the defining representation of the SO($N$) spin, we map the SO($N$) bilinear-biquadratic (BLBQ) spin chain into the $N$-color bosonic particle model. We find that, when the Jordan-Wigner transformation disentangles the symmetry-protected topological entanglement, this bosonic model becomes negative-sign-free in the context of quantum Monte Carlo simulation. For the SO(3) case, moreover, the Kennedy-Tasaki transformation for the $S=1$ BLBQ chain, which is also a topological disentangler, derives the same bosonic model through the dimer-R bases. We present the temperature dependence of the energy, entropy, and string order parameter for the SO($N=3,4,5$) BLBQ chains by a world-line Monte Carlo simulation for the $N$-color bosonic particle model.

Figure 1

Diagrammatic representation of the interaction terms in the Hamiltonian (10). The indices $n$ and $n^{\prime }$ represent colors of particles. The superscripts of $\Gamma$ indicate abbreviations of cross, repulsive, and horizontal, respectively.

Figure 2

The relations among the various representations and transformations. The transformations in the vertical direction are nonlocal, while those in the horizontal direction are local.

Figure 3

($N=3$)-color world-line configuration of Eq. (60) with $L=4$ with a periodic boundary. The gray box denotes an imaginary-time slice with a local Boltzmann weight $\langle n(t+1)\left|[1-({\tilde{h}}_{i,i+1}-C)\delta \tau ]\right|n\left(t\right)\rangle$. The solid up and down triangles are a worm's head (black) and tail (white).

Figure 4

Energy per site for SO(3), SO(4), and SO(5) BLBQ models at finite temperatures. The system size $L$ is 256. Horizontal dotted lines are the exact values of the ground-state energy.

Figure 5

Entropy per site of the SO(3) BLBQ model at finite temperatures. The system size $L$ is 256.

Figure 6

Entropy per site of the SO(3), SO(4), and SO(5) BLBQ models at the VBS points [$\alpha =(N-2)/N$]. The system size $L$ is 256.

Figure 7

Two-point correlation function of the off-diagonal order in the mapped system, which corresponds to the string correlation function of the AKLT model. The system size $L$ is 256. The inset is a semilogarithmic plot.

Figure 8

Correlation length of the off-diagonal order, which corresponds to the string correlation length at the AKLT point of the $S=1$ BLBQ chain. We use the second moment of the correlation function to estimate the correlation length.