Gutzwiller projected wave functions in the fermionic theory of $S=1$ spin chains

We study in this paper a series of Gutzwiller projected wave functions for $S=1$ spin chains obtained from a fermionic mean-field theory for general $S>1/2$ spin systems [Liu, Zhou, and Ng, Phys. Rev. B 81, 224417 (2010)] applied to the bilinear-biquadratic ($J$-$K$) model. The free-fermion mean-field states before the projection are 1D paring states. By comparing the energies and correlation functions of the projected pairing states with those obtained from known results, we show that the optimized Gutzwiller projected wave functions are very good trial ground-state wave functions for the antiferromagnetic bilinear-biquadratic model in the regime $K<J$ ($-3\pi /4<\theta <\pi /4$). We find that different topological phases of the free-fermion paring states correspond to different spin phases: the weak pairing (topologically nontrivial) state gives rise to the Haldane phase, whereas the strong pairing (topologically trivial) state gives rise to the dimer phase. In particular, the mapping between the Haldane phase and Gutwziller wave function is exact at the Affleck-Kennedy-Lieb-Tasaki (AKLT) point $K/J=1/3$ [$\theta ={\tan }^{-1}(1/3)$]. The transition point between the two phases determined by the optimized Gutzwiller projected wave function is in good agreement with the known result. The effect of $Z_2$ gauge fluctuations above the mean-field theory is analyzed.

Figure 1

The phase diagram of the $S=1$ bilinear-biquadratic model in one dimension. The region $-3\pi /4<\theta ⩽\pi /4$ is of our interest. Red dotted points are studied and the results are summarized in Table .

Figure 2

The spin-spin correlation function $C^{\alpha }\left(r\right)=\langle S_i^{\alpha }{S}_{i+r}^{\alpha }\rangle$ in the optimized projected wave function $|1,0.9767,1.7810\rangle$ at $K=0$. The staggered sign ${(-1)}^r$ shows the short-range antiferromagnetic order of the ground state. The error bar of $C^x\left(r\right)$ is smaller then that of $C^z\left(r\right)$, because in calculating the former, more spin configurations are involved. The dashed lines are exponential fitting of the data. The fitting shows that the correlation length is about 5.92 units of lattice constant.

Figure 3

Spin-Peierls correlation functions at $K=-2$. The blue circled line shows the Spin-Peierls correlation function $\langle P_i^z{P}_{i+r}^z\rangle$ of the projected self-consistent mean-field ground state $P_G{|\psi (\chi =0,\Delta =0.9203,\lambda =0.9110)\rangle }_{\mathrm{mf}}$ with $K=-2$. Here, the spin-Peierls order is defined as $P_i^z=S_i^z{S}_{i+1}^z-S_{i+1}^z{S}_{i+2}^z$. The red squared line shows the Spin-Peierls correlation of the projected optimal trial mean-field state $P_G{|\psi (\chi =1,\Delta =1.1532,\lambda =2.0661)\rangle }_{\mathrm{trial}}$ with $K=-2$. For comparison, the black crossed line is the spin-Peiels correlation of the projected optimal trial mean-field state of the Heisenberg model ($K=0$).

Figure 4

Projected optimal trial mean-field states near the phase transition point. Here, we set $J=1$ and $\chi =1$. The spin-Peierls correlation qualitatively shows the expected result that $\lambda -2\chi$ have differen sign in different phases. When $\lambda -2\chi >0$, the projected wave function have finite spin-Peierls correlation, indicating they are dimerized. Otherwise, when $\lambda -2\chi <0$, the spin-Peierls correlation approaches zero, indicating the states are not dimerized. The point where $\lambda -2\chi =0$ almost overlaps with the point where $\langle P_i^z{P}_{i+\frac{L}{2}}^z\rangle \to 0$. This result verifies our conclusion that the topology of the states distinguishes different phases.

Figure 5

The mean-field excitation spectrum $E_i$ of each flavor for an open chain. Due to particle-hole symmetry, the 200 modes correspond to $L=100$ fermion modes. The two Majorana zero modes manifest themselves. The energy of these two Majorana modes exponentially approaches to zero with increasing $L$.

Figure 6

Projected singlet ground state. The spin-spin correlation function of $\langle S_{1\to N}^z{S}_{L-N+1\to L}^z\rangle \approx -1/4$ when $N⩾15$ shows that the edge states carry spin-1/2, here $S_{1\to N}^z={\sum }_{i=1}^N{S}_i^z$ and $S_{L-N+1\to L}^z={\sum }_{j=1}^N{S}_{L+1-j}^z$.

Figure 7

Projected spin-triplet ground state with fermion parity distribution ($N_x,N_y,N_z$) $=$ (odd, odd, even). The spin-spin correlation function of $\langle S_{1\to N}^z{S}_{L-N+1\to L}^z\rangle \approx -1/4$ when $N⩾15$ shows that the edge states carry spin 1/2.