Instability of the Haldane Phase: Roles of Charge Fluctuations and Hund's Coupling

Abstract

We systematically investigate the stability of the symmetry-protected topological (SPT) Haldane phase in spin-1/2 Heisenberg and half-filled Hubbard ladders coupled by ferromagnetic Hund's interactions. Using density-matrix renormalization group (DMRG) method, we analyze key signatures of the Haldane phase: long-range string order, finite spin gap, and characteristic entanglement spectrum degeneracies. In spin-only Heisenberg ladders, we find immediate onset and continuous strengthening of the Haldane phase with increasing Hund's coupling. In contrast, the inclusion of charge fluctuations in Hubbard ladders leads to a nontrivial stability regime, revealing a robust yet bounded region where SPT order persists despite significant charge fluctuations. We identify distinct boundaries separating a trivial insulating phase from the Haldane SPT phase, governed by both Coulomb repulsion and Hund's coupling. Our results highlight the subtle interplay of spin and charge degrees of freedom in correlated itinerant systems and establish essential criteria for observing Haldane physics experimentally in fermionic ladder materials.

Figures (5)

• Figure 1: Lattice structures of (a) the two-leg Heisenberg ladder and (b) the two-leg Hubbard ladder, with FM Hund’s coupling $J_H$ connecting the chains. Indices $\gamma$ and $i$ label the leg and site positions, respectively. Removing the dashed bonds corresponds to replacing the edge spins with spin-1/2 degrees of freedom in the effective spin-1 Heisenberg chain representation.
• Figure 2: String order in the Heisenberg ladder. (a) String correlation function $\mathcal{O}_\mathrm{str}(|i-j|)$ plotted as a function of distance $|i-j|$ for several values of $J_{\rm H}/J$. The dotted line indicates the string order parameter of the $S=1$ Heisenberg chain. (b) String order parameter $|\mathcal{O}_{\rm str}(-\infty,\infty)|$ as a function of $J_{\rm H}/J$. Inset: $|\mathcal{O}_{\rm str}(-\infty,\infty)|$ plotted against $\sqrt{J_{\rm H}/J}$ in the small $J_{\rm H}/J$ regime; the dashed line shows a fit near $J_{\rm H}/J=0$.
• Figure 3: Charge fluctuations in the Hubbard ladder. Local charge variance $\delta n^2$ plotted as a function of $t/U$ and $t/J_H$. The solid line indicates the phase boundary separating the trivial Mott insulating phase from the topological Haldane phase.
• Figure 4: Topological string order in the Hubbard ladder. String correlation function $O_{\text{str}}(i,j)$ versus $|i-j|$ for (a) various $J_H/t$ at fixed $U/t=4$ and (b) various $U/t$ at fixed $J_H/t = 4$. Log-log plots of $O_{\text{str}}(i,j)$ versus $|i-j|$ for (c) various interaction strengths with $U/t = J_H/t$, and (d) various $J_H/t$ at fixed $U/t=0$. String order parameter as a function of (e) $U/t=J_H/t$ and (f) $J_H/t$ at fixed $U/t=0$. Dotted lines mark the spin-1 Heisenberg chain value.
• Figure 5: (a) Spin gap $\Delta_{\rm s}/J$ for the Heisenberg ladder as a function of $J_{\rm H}/J$. The dashed line marks the spin gap of the spin-1 Heisenberg chain [Eq. \ref{['eq:s1hcain']}]. Inset: Enlarged view of the small-$J_{\mathrm{H}}/J$ region. (b) $\Delta_{\rm s}/t$ for the Hubbard ladder as a function of $J_{\rm H}/t$ for several values of $U/t$. (c) $\Delta_{\rm s}/t$ as a function of $J_{\rm H}/t$ at $U=0$. Dashed lines in (b) and (c) indicate strong-coupling estimates. (d) Entanglement spectrum across the topological transition at $U/t=J_{\rm H}/t \approx 1.2$. Numbers beside the lines denote entanglement level degeneracies. (e,f) Spin–spin correlation function $\langle \mathbf{S}_{\gamma,i} \cdot \mathbf{S}_{\gamma',j} \rangle$ as a function of distance $r$, shown on (e) log-log and (f) semi-log scales for $U/t=J_{\rm H}/t=0.8$ and 1.6, respectively.