Entanglement entropy as a probe of topological phase transitions

Abstract

Entanglement entropy (EE) provides a powerful probe of quantum phases, yet its role in identifying topological phase transitions in disordered systems remains underexplored. We introduce an exact EE-based framework that captures topological phase transitions even in the presence of disorder. Specifically, for a class of Su-Schrieffer-Heeger (SSH) model variants, we show that the difference in EE between half-filled and near-half-filled ground states, $ΔS^{\mathcal{A}}$, vanishes in the topological phase but remains finite in the trivial phase, a direct consequence of edge-state localization. This behavior persists even in the presence of quasiperiodic or binary disorder. By analyzing domain-wall configurations in the SSH chain, we further show how subsystem tuning allows one to distinguish genuine topological zero-energy eigenstates from trivial localized states. Exact phase boundaries, derived from Lyapunov exponents via transfer matrices, agree closely with numerical results from $ΔS^{\mathcal{A}}$ and the topological invariant $\mathcal{Q}$, with instances where $ΔS^{\mathcal{A}}$ outperforms $\mathcal{Q}$. Our results highlight EE as a robust diagnostic tool and a potential bridge between quantum information and condensed matter approaches to topological matter.

Figures (11)

• Figure 1: SSH chain where the green dashed box shows subsystem $\mathcal{A}$ composed of a few unit cells deep inside the bulk. In this paper, the subsystems are denoted by calligraphic letters $\mathcal{A}$ and $\mathcal{B}$, while the lattice sites within each unit cell are labeled by $A$ and $B$.
• Figure 2: Phase diagrams of the SSH model variants. In (a)-(c) the color scale denotes $\Delta S^{\mathcal{A}} = | S^{\mathcal{A}}_\text{hf} - S^{\mathcal{A}}_{\text{hf}+1} |$ with $\text{hf}=N/2$, and the subsystem size $N_{\mathcal{A}}=50$. In (d)–(f) the color scale denotes topological quantum number $\mathcal{Q}$, which is defined in Appendix \ref{['Quantum number']}. For the disordered cases with $t=1$, the results shown are averaged over 100 disorder realizations. For all the figures, system size $N=400$. Yellow curves mark analytical phase boundaries. On the diagonal line $t=-\lambda$ in (a), the intracell hopping vanishes, causing the system to break into disconnected segments. Therefore the state just above half filling becomes localized on a few sites, reducing its contribution to the entanglement entropy and hence yielding a smaller $\Delta S^{\mathcal{A}}$.
• Figure 3: Spatial distribution of $\Delta \Omega = \left| \Omega_{\text{hf}} - \Omega_{\text{hf}+1} \right|$ for a system of size $N=400$ at half-filling ($\text{hf}=200$). The hopping amplitude is $t=1$ for all cases. Subfigures (a)--(c) correspond to the topological phase, while (d)--(f) correspond to the trivial phase.
• Figure 4: Entanglement-entropy difference $\Delta S^{\mathcal{A}}_i = \left| S^{\mathcal{A}}_i - S^{\mathcal{A}}_{i+1} \right|$ for all eigenstates in (a) the AAH model and (b) the clean SSH model. In (a), the inset shows that only a single eigenstate exhibits vanishing $\Delta S^{\mathcal{A}}_i$, whereas in (b), the SSH model features pairs of states with $\Delta S^{\mathcal{A}}_i = 0$, consistent with its topological edge modes.
• Figure 5: Schematic illustration of situations in which domain walls arise in the SSH chain: (a) a topological SSH chain joined to a trivial SSH chain by a weak bond; (b) two topological SSH chains joined by a weak bond; and (c) two topologically trivial SSH chains joined by a strong bond.