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arXiv:2504.02157 · Physical review. B./Physical review. B

Topological phase transition in the two-leg Hubbard model: Emergence of the Haldane phase via diagonal hopping and strong interactions

João Pedro Gama D'Elia, Thereza Paiva

Published Jul 16, 2025• 1 citation
arXivHTMLPDF

Abstract

We investigate the two-leg Hubbard model with diagonal hopping to explore the interplay between geometrical frustration and strong electron-electron interactions. Using the Density Matrix Renormalization Group (DMRG) method, we demonstrate the emergence of a topological Haldane phase, which results explicitly from the complementary effects of diagonal hopping-induced frustration and strong on-site Coulomb repulsion. The topological phase transition from a trivial insulator to the nontrivial Haldane phase is characterized by significant changes in magnetic properties, edge correlations, and the appearance of a nonzero string order parameter. Furthermore, we confirm the topological nature of this phase through a detailed analysis of the spin gap and entanglement spectrum, demonstrating clear signatures of symmetry-protected topological order.

DOI
10.1103/rspn-4cyr

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Topological insulators are materials that behave as insulators in the bulk, but present conducting edge states.Despite the growing interest in such materials due to their possible applications on spintronic devices and quantum computing, several open questions still remain regarding the roles of both electronic interactions and dimensionality in these systems.The goal of this work is to tackle some of these questions.To do so, we study a quasi-1D system presenting a topologically non-trivial phase and electronic interactions: the so-called Kane-Mele-Hubbard ladders.Here, we apply the density matrix renormalization group (DMRG) algorithm in the tensor network formalism in order to calculate some physical properties of this system.The DMRG calculations allow us to reach the system's ground state and enable the efficient computation of observables and of the entanglement entropy.The Kane-Mele-Hubbard model is an interacting version of the Kane-Mele model, which displays topologically non-trivial phases.The interest on this system is due to the combination of effects caused by the spin-orbit coupling and the electronic interactions.For the non-interacting case we have a topological insulator, whereas in the highly interacting picture we have a Mott insulator with localized electrons.We explore the phase transition between these regimes, looking at the conditions needed for the appearance of spin-polarized edge states.We also propose the entanglement entropy between two halves of the system as an indicator of the topological phase transition, as it presents a particular pattern when the phase transition takes place.

•Dec 20, 2019•0 citations
We investigate the two-leg Hubbard model with diagonal hopping to explore the interplay between geometrical frustration and strong electron-electron interactions. Using the Density Matrix Renormalization Group (DMRG) method, we demonstrate the emergence of a topological Haldane phase, which results explicitly from the complementary effects of diagonal hopping-induced frustration and strong on-site Coulomb repulsion. The topological phase transition from a trivial insulator to the nontrivial Haldane phase is characterized by significant changes in magnetic properties, edge correlations, and the appearance of a nonzero string order parameter. Furthermore, we confirm the topological nature of this phase through a detailed analysis of the spin gap and entanglement spectrum, demonstrating clear signatures of symmetry-protected topological order.
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 the density-matrix renormalization group 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.
Abstract One of the most famous quantum systems with topological properties, the spin S=1{{{{{{{\mathcal{S}}}}}}}}=1S=1 S = 1 antiferromagnetic Heisenberg chain, is well-known to display exotic S=1/2{{{{{{{\mathcal{S}}}}}}}}=1/2S=1/2 S = 1 / 2 edge states. However, this spin model has not been analyzed from the more general perspective of strongly correlated systems varying the electron-electron interaction strength. Here, we report the investigation of the emergence of the Haldane edge in a system of interacting electrons – the two-orbital Hubbard model—with increasing repulsion strength U and Hund interaction JH. We show that interactions not only form the magnetic moments but also form a topologically nontrivial fermionic many-body ground-state with zero-energy edge states. Specifically, upon increasing the strength of the Hubbard repulsion and Hund exchange, we identify a sharp transition point separating topologically trivial and nontrivial ground-states. Surprisingly, such a behaviour appears already at rather small values of the interaction, in a regime where the magnetic moments are barely developed.
We investigate the effects of electronic correlations on the Bernevig-Hughes-Zhang model using the real-space density matrix renormalization group (DMRG) algorithm. We introduce a method to probe topological phase transitions in systems with strong correlations using DMRG, substantiated by an unsupervised machine learning methodology that analyzes the orbital structure of the real-space edges. Including the full multi-orbital Hubbard interaction term, we construct a phase diagram as a function of a gap parameter (mmm) and the Hubbard interaction strength (UUU) via exact DMRG simulations on $N\ifmmode\times\else\texttimes\fi{}4$ cylinders. Our analysis confirms that the topological phase persists in the presence of interactions, consistent with previous studies, but it also reveals an intriguing phase transition from a paramagnetic to a stripey antiferromagnetic topological insulator. The combination of the magnetic structure factor, strength of magnetic moments, and the orbitally resolved density, provides real-space information on both topology and magnetism in a strongly correlated system.
We investigate the ground-state phase diagram of a modified spinless Haldane-Hubbard model, with broken threefold rotational symmetry, employing exact diagonalization calculations. The interplay of asymmetry, interactions, and topology gives rise to a rich phase diagram. The noninteracting limit of the Hamiltonian exhibits a higher-order topological insulator characterized by the existence of corner modes, in contrast to known chiral edge metallic states of the standard Haldane model. Our investigation demonstrates that these symmetry-protected states are robust to the presence of finite interactions. Furthermore, in certain regimes of parameters, we show that a topological Mott insulator exists in this model, where a nontrivial topological bulk coexists with an interaction-driven charge-density wave, whose emergence is characterized by a Z2{Z}_{2}Z2​-symmetry breaking within the 3d-Ising3d\text{-Ising}3d-Ising universality class.
We investigate, by means of a field-theory analysis combined with the density-matrix renormalization group (DMRG) method, a theoretical model for a strongly correlated quantum system in one dimension realizing a topologically-ordered Haldane phase ground state. The model consists of a spin-1/2 Heisenberg chain coupled to a tight-binding chain via two competing Kondo exchange couplings of different type: a ``sss-wave'' Kondo coupling (JKsJ^s_{K}JKs​), and a less common ``ppp-wave'' (JKpJ^p_{K}JKp​) Kondo coupling. While the first coupling is the standard Kondo interaction studied in many condensed-matter systems, the latter has been recently introduced by Alexandrov and Coleman [Phys. Rev. B 90, 115147 (2014)] as a possible mechanism leading to a topological Kondo-insulating ground state in one dimension. As a result of this competition, a topological quantum phase transition (TQPT) occurs in the system for a critical value of the ratio JKs/JKpJ^s_{K}/J^p_{K}JKs​/JKp​, separating a (Haldane-type) topological phase from a topologically trivial ground state where the system can be essentially described as a product of local singlets. We study and characterize the TQPT by means of the magnetization profile, the entanglement entropy and the full entanglement spectrum of the ground state. Our results might be relevant to understand how topologically-ordered phases of fermions emerge in strongly interacting quantum systems. Received: 1 July 2016, Accepted: 30 August 2016; Edited by: K. Hallberg; Reviewed by: D. C. Cabra, Instituto de Física La Plata, La Plata, Argentina; DOI: http://dx.doi.org/10.4279/PIP.080005Cite as: F T Lisandrini, A M Lobos, A O Dobry, C J Gazza, Papers in Physics 8, 080005 (2016)This paper, by F T Lisandrini, A M Lobos, A O Dobry, C J Gazza, is licensed under the Creative Commons Attribution License 3.0.
Topological insulators are materials that behave as insulators in the bulk, but present conducting edge states.Despite the growing interest in such materials due to their possible applications on spintronic devices and quantum computing, several open questions still remain regarding the roles of both electronic interactions and dimensionality in these systems.The goal of this work is to tackle some of these questions.To do so, we study a quasi-1D system presenting a topologically non-trivial phase and electronic interactions: the so-called Kane-Mele-Hubbard ladders.Here, we apply the density matrix renormalization group (DMRG) algorithm in the tensor network formalism in order to calculate some physical properties of this system.The DMRG calculations allow us to reach the system's ground state and enable the efficient computation of observables and of the entanglement entropy.The Kane-Mele-Hubbard model is an interacting version of the Kane-Mele model, which displays topologically non-trivial phases.The interest on this system is due to the combination of effects caused by the spin-orbit coupling and the electronic interactions.For the non-interacting case we have a topological insulator, whereas in the highly interacting picture we have a Mott insulator with localized electrons.We explore the phase transition between these regimes, looking at the conditions needed for the appearance of spin-polarized edge states.We also propose the entanglement entropy between two halves of the system as an indicator of the topological phase transition, as it presents a particular pattern when the phase transition takes place.