Topological order-by-disorder in orbitally degenerate dipolar bosons on a zigzag lattice

Spinor bosons offer a conceptually simple picture of macroscopic quantum behavior of topological order-by-disorder: The paramagnetic state of two-component dipolar bosons in an orbitally degenerate zigzag lattice is unstable against infinitesimal quantum fluctuations of orbitals, $\lambda$, towards developing nonlocal hidden order. Adjacent to the topological state a locally correlated exact ground state with spontaneously a quadrupoled lattice constant is realized for the broad parameter regime. The topological order is extremely robust surviving the $\lambda \to \infty$ limit where the ground state evolves into the Majumdar-Ghosh state of a frustrated spin-$\frac{1}{2}$ chain.

Figure 1

Geometry of the orbitally degenerate zigzag lattice with the intersite hopping $t$ (between similar orbitals) and on-site hopping between the orthogonal orbitals $\lambda$.

Figure 2

Exact analytical ground state phase diagram of spin-orbital model Eq. (3) obtained in thermodynamic limit. We employ (spin,orbital) notation of different phases. For the ground-state configurations of (Q,$\downarrow \uparrow \uparrow \downarrow$) and (iH,AF) phases see Fig. 3 and for denotations of phases consult text. Inset shows ground-state orbital configuration of (P,F) phase and the effect in this phase of infinitesimal quantum fluctuations in orbitals $\lambda$. Dotted contours encircle 2 sites forming effective spins-1 ${\mathbf{T}}_i={\mathbf{S}}_{2i}+{\mathbf{S}}_{2i+1}$.

Figure 3

Ground-state orbital configurations in (a) (Q,$\downarrow \uparrow \uparrow \downarrow$) and (b) (iH,AF) phases. Only occupied orbitals are displayed per site. Dotted contour in (a) encircles cluster of 4 spins decoupled from the rest of the system. Continuous line indicates ferromagnetic Heisenberg exchange between spins at neighboring sites $-8{\mathbf{S}}_i{\mathbf{S}}_{i+1}$ and dashed lines indicate AF exchange $4\Delta {\mathbf{S}}_i{\mathbf{S}}_{i+1}$.

Figure 4

Bulk short-range spin correlation function's dependence on $\lambda$ in (H,F) phase. Due to extensive degeneracy of ground states in (P,F) phase, numerically we cannot approach arbitrarily close to $\lambda =0$, but the tendency is evident. Symmetry with respect to translations on 2 sites of (H,F) state imposes $\langle {\mathbf{S}}_{2i}{\mathbf{S}}_{2i+1}\rangle =\langle {\mathbf{S}}_{2i+2}{\mathbf{S}}_{2i+3}\rangle$ and $\langle {\mathbf{S}}_{2i}{\mathbf{S}}_{2i+2}\rangle =\langle {\mathbf{S}}_{2i+1}{\mathbf{S}}_{2i+3}\rangle$.

Figure 5

(a) Blue symbols: Bulk Néel order and string order of spin-orbital model (SOM) in (H,F) phase for $\lambda \to 0$ (here $\lambda =0.1$, $\alpha =1$, and $\Delta =0.1$). Red symbols: Corresponding order parameters of spin-1 Haldane chain. (b) Magnetization profile in $S^z=1$ Kennedy-Tasaki ground state of spin-1 Haldane chain on $L=48$ sites (red symbols) is nearly identical to magnetization profile of $\langle T_i^z\rangle =\langle S_{2i}^z+S_{2i+1}^z\rangle$ of SOM on $L=96$ sites (blue symbols) in (H,F) phase for $\lambda \to 0$ (here $\lambda =0.1$, $\alpha =1$, and $\Delta =0.1$). Inset (green circles) shows site-resolved magnetization profile of SOM $\langle S_{2i}^z\rangle \simeq \langle S_{2i+1}^z\rangle \simeq \langle T_i^z\rangle /2$.

Figure 6

Effect of $\lambda$ on ground-state phases presented in Fig. 2 for $\alpha =1$. Dashed line between (H,F) and (D,F) phases represents a boundary phase transition, where edge spins at the end of the open chain disappear in (D,F) phase. All second-order phase transition lines are obtained with extrapolation procedure from finite system size data to infinite size.

Figure 7

FS with respect to $\lambda$ per site for $\Delta =1.5$, $\alpha =1$. Linear scaling of fidelity susceptibility peak per site with system size confirms Ising nature of quantum phase transition from the (Q,$\downarrow \uparrow \uparrow \downarrow$) to (D,F) state.