Magnetic phase transitions protected by topological quantum geometry transitions: effects of electron-electron interactions in the Creutz ladder system

TL;DR

This work addresses how electronic correlations modify topological phases in a 1D lattice by studying the Creutz ladder with on-site Hubbard interactions $U$. Using a self-consistent mean‑field approach, the authors derive spin‑resolved bands, Zak phases, and a Fubini–Study metric, revealing a first‑order transition where the ground state switches from AF to FM and the Zak phase jumps from $\pm\pi$ to $0$. This magnetic/topological transition is accompanied by divergences in the quantum geometric tensor, and comprehensive phase diagrams in $(\varepsilon,\lambda)$ map the critical $U_c$ and metastable branches. The results establish the Creutz–Hubbard ladder as a minimal platform for interaction‑driven topological phenomena with potential implications for spintronics and quantum information processing.

Abstract

The interplay between electronic correlations and band topology is a central theme in modern condensed matter physics. In this work, we investigate the effects of on-site Hubbard interactions on the topological, magnetic, and quantum geometric properties of the Creutz ladder, a paradigmatic model of a one-dimensional topological insulator. Using a self-consistent mean-field approach, we uncover a first-order, interaction-driven phase transition that is simultaneously magnetic and topological. We demonstrate that as the Hubbard interaction $U$ is increased, the system's ground state abruptly switches from an anti-ferromagnetic (AF) configuration to a ferromagnetic (F) one. This magnetic transition coincides with a topological transition, marked by a quantized jump in the Zak phase from $\pmπ$ to $0$. We systematically compute the phase diagrams in the parameter space of on-site energy staggering ($ε$) and inter-chain hopping asymmetry ($λ$), revealing the critical interaction strength $U_c$. Furthermore, we analyze the quantum geometry of the Bloch states by calculating the Fubini-Study metric, demonstrating that its components exhibit divergences that precisely signal the topological phase transition. By analyzing the full energy spectrum, we distinguish the true ground state from metastable excited states that emerge past the critical point. Our results establish the Creutz-Hubbard ladder as a minimal model for studying interaction-induced topological phenomena and suggest a potential route for controlling magnetic, topological, and geometric properties via electronic correlations.

Figures (8)

• Figure 1: Schematic of the one-dimensional bipartite Creutz ladder. Red circles and blue squares represent sites on sublattices A and B, respectively.
• Figure 2: Band structure for the values $\overline{\varepsilon}=0,\lambda=1$ and $\overline{u}=4.0$. The blue and red lines correspond to spins $\downarrow$ and $\uparrow$, respectively. (a) For $\overline{n}_{a,\uparrow}=\overline{n}_{b,\uparrow}=0.5$ and $\overline{n}_{a,\downarrow}=\overline{n}_{b,\downarrow}=0.3$, non-degenerate flat bands arise because $\chi_{\sigma,-}=0$ (see Eq. \ref{['eq:flat-bands-spin']}). (b) Introducing an imbalance only in one spin, i.e., $\overline{n}_{a,\uparrow}=0.7,\overline{n}_{b,\uparrow}=0.5$ and $\overline{n}_{a,\downarrow}=\overline{n}_{b,\downarrow}=0.3$, yields dispersive bands for spin $\uparrow$ while spin $\downarrow$ remains flat. (c) As discussed in Eq. \ref{['eq:eigenvalues-at-k-zero']}, for $\overline{n}_{a,\uparrow}=0.0,\overline{n}_{b,\uparrow}=1.0$ and $\overline{n}_{a,\downarrow}=0.6,\overline{n}_{b,\downarrow}=0.3$, the condition $\chi_{\uparrow,-}=-2$ is satisfied, forming a Dirac cone for spin $\uparrow$ while opening a gap for spin $\downarrow$, rendering it an insulator; hence the system exhibits half-metallic behavior.
• Figure 3: Topological phase diagram for each spin block Hamiltonian \ref{['eq:hamiltonian-spin-blocks']}, calculated numerically and in agreement with Eq. \ref{['eq:winding-spin']}. The phase boundaries are determined by $|\Sigma_{\sigma}|=2$, indicating the emergence of linear (Dirac) dispersion in the band structure.
• Figure 4: Contour plot of the Fubini–Study metric components $g_{kk,\sigma\sigma}^{\eta\eta}$ (Eq. \ref{['eq:fubini-study-detailed']}) as a function of $k$ and $\lambda$ for $\overline{\varepsilon}=0$, $l=1$, and (a) $\chi_{\sigma,-}=-1.0$, (b) $\chi_{\sigma,-}=0.0$, and (c) $\chi_{\sigma,-}=1.0$. In case (b), when the spin densities are equal ($\overline{n}_{a,\sigma}=\overline{n}_{b,\sigma}$), the system retains flat bands for $|\lambda|=1$, whereas in the other cases the geometry deforms due to spin-density imbalance. White regions indicate Dirac cones forming near $k=0$ or $k=\pm\pi/l$, as detailed further in the Supplementary Material A .
• Figure 5: Total energy per site as a function of the normalized interaction $\overline{u}/\lambda$ for fixed $\bar{\epsilon}=0.0$ and $\lambda=0.5$, shown for two different temperatures: (a) low temperature ($kT=0.25$) and (b) high temperature ($kT=1.0$). The color of each point corresponds to the Zak phase (Eq. \ref{['eq:Zak_phase']}). At low temperature, a sharp change indicates a first-order transition. At higher temperature, the transition becomes smoother, suggesting thermal fluctuations smear the sharp boundary.