Topological Phase Diagram of Generalized SSH Models with Interactions

We investigate interacting Su-Schrieffer-Heeger (SSH) chains with two- and three-site unit cells using density matrix renormalization group (DMRG) simulations. By selecting appropriate filling fractions and sweeping across interaction strength ( J_z ) and dimerization ( δ), we map out their phase diagrams and identify transition lines via entanglement entropy and magnetization measurements. In the two-site model, we observe the emergence of an interaction-induced antiferromagnetic intermediate phase between the topologically trivial and non-trivial regimes, as well as a critical region at negative ( J_z ) with suppressed magnetization and finite-size scaling of entanglement entropy. In contrast, the three-site model lacks an intermediate phase and exhibits asymmetric edge localization and antiferromagnetic ordering in both positive and negative ( J_z ) regimes. We further examine the response of edge states to Ising perturbations. In the two-site model, zero-energy edge modes are topologically protected and remain robust up to a finite interaction strength. However, in the three-site model, where the edge states reside at finite energy, this protection breaks down. Despite this, the edge-localized nature of these states survives in the form of polarized modes whose spatial profiles reflect the non-interacting limit.

💡 Research Summary

The authors investigate interacting Su‑Schrieffer‑Heeger (SSH) chains with two‑site and three‑site unit cells using large‑scale density matrix renormalization group (DMRG) simulations. By fixing the total spin to half‑filling, they ensure that edge modes in the topologically non‑trivial regime are part of the ground state and thus accessible to the numerical method. The Hamiltonians are written in the XXZ spin representation, with alternating XY couplings (J_{xy}^{(i)}) that encode the dimerization parameter (\delta) and longitudinal Ising‑type couplings (J_z^{(i)} = J_z (1\pm\delta)) (or a constant for the third bond in the three‑site case). The parameter space explored consists of the dimerization (\delta) and the interaction strength (J_z), which can be repulsive ((J_z>0)) or attractive ((J_z<0)).

For the two‑site model the authors sweep (\delta) at several fixed values of (J_z) and compute the bipartite von Neumann entanglement entropy (EE) as well as local magnetization (\langle\sigma^z_i\rangle). When (J_z>1) two distinct transition lines appear: a transition from the trivial dimerized phase ((\delta>0)) to an intermediate antiferromagnetic (AFM) phase, and a second transition from this AFM phase to the topologically non‑trivial phase ((\delta<0)). Finite‑size scaling of the EE yields a central charge (c=1/2) for both transitions, consistent with Ising criticality. At the SU(2)‑symmetric point (J_z=1) the critical theory is a level‑1 Wess–Zumino–Witten model with (c=1). For (-1/\sqrt{2}<J_z<1) only a single transition at (\delta=0) remains, again with (c=1) (Luttinger‑liquid behavior). When (J_z) becomes negative, a narrow region (-1<J_z<-1/\sqrt{2}) shows non‑universal finite‑size corrections and a Kosterlitz‑Thouless (KT) transition, characterized by an exponentially diverging correlation length. For (J_z<-1) the EE collapses to zero, indicating a classical ferromagnetic phase. Magnetization profiles corroborate these findings: the trivial and non‑trivial phases exhibit near‑zero bulk magnetization with edge‑localized zero‑energy modes (for (\delta<0)), while the intermediate AFM phase displays staggered (\uparrow\downarrow) order throughout the chain.

The three‑site model, which in the non‑interacting limit hosts a single edge state at finite energy (near the (2N/3)‑th band), behaves differently under interactions. The phase diagram contains only a single transition line at (\delta=0); no intermediate AFM phase appears for any (J_z). Entanglement entropy again shows a logarithmic divergence at the transition, with central charge (c=1) (Luttinger liquid) or (c=1/2) (Ising) depending on the side of the line. Magnetization remains staggered across the whole parameter range, reflecting persistent AFM correlations. Edge localization is asymmetric: for (\delta>0) the edge mode sits on the left end, for (\delta<0) on the right. Because the edge state resides at finite energy, it is not protected by chiral symmetry; an Ising perturbation quickly shifts its energy, yet the spatial profile retains a polarized character reminiscent of the non‑interacting limit.

Overall, the work maps out detailed topological phase diagrams for interacting SSH chains with different unit‑cell sizes, identifies interaction‑induced antiferromagnetic phases, characterizes critical behavior via entanglement scaling and central charges, and clarifies how edge‑state protection depends on both the underlying band topology and the strength of many‑body interactions. These results deepen our understanding of how strong correlations reshape topological phases in one‑dimensional quantum systems.