Haldane insulator protected by reflection symmetry in the doped Hubbard model on the three-legged ladder

We demonstrate the existence of an insulating phase in the three-legged Hubbard ladder at two-thirds filling. In this phase chargons are bound because the physics within a unit cell favors the formation of triplets. The resultant moments lead to a ground state in the Haldane phase, a symmetry protected topological state of matter. In this purely fermionic model, reflection is protecting but time-reversal and dihedral symmetries are not, in contrast to spin models.

Figure 1

Three-legged ladder in ${\mathrm{Mo}}_3{\mathrm{S}}_7{\left(\mathrm{dmit}\right)}_3$. The electronic structure of a single molecule is described by three Wannier orbitals that have mostly Mo-dmit hybrid character [11]. The hopping integral between the Wannier orbitals on a single molecule are denoted $t_r$, while the hopping between equivalent Wanniers on different molecules displaced along the $c$ axis is denoted $t_l$. Interference effects suppress interchain interactions in both the noninteracting [11] and strongly interacting [16] limits. This results in a three-legged ladder with hopping parameters $t_r$ $\left(t_{\ell }\right)$ along the rungs (legs) and periodic boundary conditions in the rungs. The shaded region indicates the three-site unit cell. Based on structural data from Ref. [10]. Crystal structure visualized with vesta [17].

Figure 2

Charge fluctuations $\mathrm{[}\mathrm{Var}\left({\widehat{n}}_i\right)$; (b)] cause the effective spin per unit cell $[\mathcal{S}$; (a)] to be suppressed from the $\left(\mathcal{S}=1\right)$ triplet that is the ground state of an isolated three-site cluster. $\mathcal{S}$ is given by the solution of $\mathcal{S}(\mathcal{S}+1)=〈{\widehat{\mathit{S}}}_i·{\widehat{\mathit{S}}}_i〉$, where ${\widehat{\mathit{S}}}_i^{}={\sum }_{\alpha }{\widehat{\mathit{S}}}_{i\alpha }^{}$ is the net spin of the $i\mathrm{th}$ rung, ${\widehat{\mathit{S}}}_{i\alpha }^{}={\sum }_{\sigma {\sigma }^{\prime }}{\widehat{c}}_{i\alpha \sigma }^{†}{\mathbf{\tau }}_{\sigma {\sigma }^{\prime }}^{}{\widehat{c}}_{i\alpha {\sigma }^{\prime }}^{}$, and $\mathbf{\tau }$ is the vector of Pauli matrices. $\mathrm{Var}\left({\widehat{n}}_i\right)=〈{\widehat{n}}_i^2〉-{〈{\widehat{n}}_i〉}^2$, where ${\widehat{n}}_i={\sum }_{\alpha \sigma }{\widehat{n}}_{i\alpha \sigma }$.

Figure 3

Correlation lengths for the charge $\left(1/{\xi }_c\right)$ and spin $\left(1/{\xi }_s\right)$ excitations calculated from the transfer matrix of the MPS. $1/{\xi }_c$ is proportional to the charge gap $\left({\mathrm{\Delta }}_c\right)$ while $1/{\xi }_s$ is proportional to the spin gap $\left({\mathrm{\Delta }}_s\right)$. Inset: Charge string order parameter. $O_P={lim}_{|i-j|\to \infty }〈{\mathbb{1}}_{i}exp\left(i\pi {\sum }_{l=i}^j{\widehat{n}}_l\right){\mathbb{1}}_j〉$, where ${\mathbb{1}}_i^{}$ is the identity operator on the $i\mathrm{th}$ set of rungs and ${\widehat{n}}_l={\sum }_{\sigma }{\widehat{n}}_{l\sigma }$. $O_P=0$ in the metallic phase, but $O_P\ne 0$ for a charge ordered state, indicating that molecular (on-rung) chargons are bound in the insulating phase.

Figure 4

String order parameters $O_i={lim}_{|i-j|\to \infty }\langle {\mathbb{1}}_i^{}exp\left(i\pi {\sum }_{l=i+1}^{j-1}{\widehat{S}}_l^z\right){\mathbb{1}}_j^{}\rangle$ and $O_s=-{lim}_{|i-j|\to \infty }\langle {\widehat{S}}_i^{z}exp\left(i\pi {\sum }_{l=i+1}^{j-1}{\widehat{S}}_l^z\right){\widehat{S}}_j^z\rangle$. As expected, $O_i$ and $O_s$ disappear as the spin gap ${\mathrm{\Delta }}_s$ closes. As $U/t_{\ell }\to \infty$ and $t_{\ell }/t_r\to \infty$, the string order parameters approach values found in the ground state of the spin-one Heisenberg chain, where $O_i=0$ [27] and $O_s=0.374325096\left(2\right)$ [28].

Figure 5

The low-lying eigenvalues ${\lambda }_{\alpha }$ of the entanglement spectrum, i.e., the eigenvalues of the reduced density matrix on tracing out half of the system, for $t_{\ell }/t_r=0.6667$. Here, (N,S,k) label particle, spin, and momentum around a triangular molecule quantum numbers, respectively. The entanglement spectrum remains even-fold degenerate for all nonzero $U/t_r$; degeneracies are indicated by the numbers on the right axis. At $U=0$, odd-fold degeneracies are also found consistent with the expected trivial metal. The even-fold degeneracy is a robust signature of an SPT state.

Figure 6

Order parameters $O_{Z_2\times Z_2}=〈U_y^{}{U}_z^{}{U}_y^{†}{U}_z^{†}〉,O_{\mathrm{TR}}=〈U_{\mathrm{TR}}^{}{U}_{\mathrm{TR}}^{*}〉$, and $O_{\mathcal{I}}=〈U_{\mathcal{I}}^{}{U}_{\mathcal{I}}^{*}〉$, where $U$ is a unitary matrix resulting from transforming the iMPS state by dihedral $\left(D_2\cong Z_2\times Z_2\right)$, time-reversal (TR), or reflection $\left(\mathcal{I}\right)$ symmetries, respectively. Note that $O_{\mathrm{TR}}=O_{Z_2\times Z_2}$ because the time-reversal transformation is $\mathcal{K}exp\left(-i\pi {\widehat{S}}_y\right)$, where $\mathcal{K}$ is complex conjugation. $O_g=1$ in the trivial phase and $O_g=-1$ in the nontrivial SPT phase, whereas $|O_g|<1$ if the symmetry is not protecting. Only reflection symmetry stabilizes the nontrivial SPT phase (Haldane phase) for finite $U/t_r$, even under strong charge fluctuations.