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Topological Phase Diagram of Generalized SSH Models with Interactions

Yuxiao Hang, Stephan Haas

TL;DR

The paper investigates how interactions modify topological phases in generalized SSH chains with two- and three-site unit cells using DMRG at half- and two-thirds fillings. It maps phase diagrams by sweeping the interaction $J_z$ and dimerization $δ$, and characterizes transitions through entanglement entropy scaling and local magnetization, extracting central charges to identify Ising, Luttinger, and KT universality classes. A key finding is an interaction-induced intermediate antiferromagnetic phase in the 2-site model and its absence in the 3-site model, with distinct edge-state behavior: zero-energy edge modes in the 2-site case remain robust up to finite $J_z$, while edge modes in the 3-site case reside at finite energy and are not protected, though edge-localized polarization persists. The work clarifies the interplay between topology and many-body correlations in 1D systems and provides generalizable insights for how unit-cell size dictates edge-state robustness and phase structure in interacting SSH-like models.

Abstract

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.

Topological Phase Diagram of Generalized SSH Models with Interactions

TL;DR

The paper investigates how interactions modify topological phases in generalized SSH chains with two- and three-site unit cells using DMRG at half- and two-thirds fillings. It maps phase diagrams by sweeping the interaction and dimerization , and characterizes transitions through entanglement entropy scaling and local magnetization, extracting central charges to identify Ising, Luttinger, and KT universality classes. A key finding is an interaction-induced intermediate antiferromagnetic phase in the 2-site model and its absence in the 3-site model, with distinct edge-state behavior: zero-energy edge modes in the 2-site case remain robust up to finite , while edge modes in the 3-site case reside at finite energy and are not protected, though edge-localized polarization persists. The work clarifies the interplay between topology and many-body correlations in 1D systems and provides generalizable insights for how unit-cell size dictates edge-state robustness and phase structure in interacting SSH-like models.

Abstract

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 and dimerization , 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 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 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.

Paper Structure

This paper contains 17 sections, 8 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Generalized SSH Models with Interactions and Variable Unit Cell Size
  3. 2-Site Unit Cell interacting SSH Model
  4. 3-Site Unit Cell interacting SSH Model
  5. DMRG Analysis of Generalized SSH Chains with Interactions
  6. The 2-site unit cell interacting SSH model
  7. Entanglement Entropy
  8. Magnetization Profiles
  9. Finite-Size Scaling and Central Charge
  10. The 3-site unit cell interacting SSH model
  11. Entanglement Entropy
  12. Magnetization Profiles
  13. Finite-Size Scaling and Central Charge
  14. Conclusions
  15. Numerical Details
  16. ...and 2 more sections

Figures (11)

  • Figure 1: Illustration of the generalized SSH chains with (a) a 2-site unit cell and (b) a 3-site unit cell. The bonds represent cyclically alternating couplings: $J_1, J_2$ in (a), and $J_1, J_2, J_3$ in (b). The boxes indicate the bipartition of the system used to compute the von Neumann entanglement entropy of the ground state. In (a), the system is cut at the midpoint ($L = N/2$), while in (b), the cut is placed on a $J_3$ bond at subsystem length $L = N/3$.
  • Figure 2: Bipartite von Neumann entanglement entropy in the interacting 2-site unit cell SSH model evaluated along various cuts in parameter space spanned by $( J_z, \delta)$. Here we consider $\delta$ sweeps at fixed interaction strengths, where each color represents one fixed strength(Red: L=100, Blue: L=200, Orange: L=300, Green: L=400)(a) $J_z$=4.0 (strong repulsive interaction) (b) $J_z$=0.0 (no interaction) (c) $J_z$=-0.8 (moderate attractive interaction. In (d) we fix $\delta$=0.30 and sweep through $J_z$. The observed singularities allow us to map out the phase diagram shown in Fig. \ref{['figure:Phase Diagram']}.
  • Figure 3: Magnetization profiles in the two-site unit cell SSH model with repulsive interactions. Here we show the local expectation values of the magnetization $\sigma^z_i$ at each site for (a) $J_z$=4.0, $\delta$=-0.9 (topologically non-trivial regime with surface states) (b)$J_z$=4.0, $\delta$=0.0 (classical antiferromagnetic regime) (c)$J_z$=4.0, $\delta$=0.9 (topologically trivial regime).
  • Figure 4: The scaling of S with L at critical point for (a) $J_z$=4.0, $\delta$=0.3415 (b) $J_z$=-0.5, $\delta$=0.0 (c) $J_z$=4.0, $\delta$=-0.3415, and (d) $J_z$=-0.74, $\delta$=0.10.
  • Figure 5: Bipartite von Neumann entanglement entropy in the interacting 3-site unit cell SSH model at 2/3 filling, evaluated along various cuts in parameter space spanned by $( J_z, \delta)$. Here we consider $\delta$ sweeps at fixed interaction strengths (a) $J_z$=4.0 (b) $J_z$=0.0 (c) $J_z$=-0.8. In (d) we fix $\delta$=-0.30 and sweep through $J_z$. The observed singularities allow us to map out the phase diagram shown in Fig. 8. Each color represents one fixed strength(Red: L=30, Blue: L=60, Orange: L=90, Green: L=120).
  • ...and 6 more figures