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arXiv:2504.00853 · Physical review. B./Physical review. B

Universality of the topological phase transition in the interacting Haldane model

Simone Fabbri, Alessandro Giuliani, Robin Reuvers

Published Sep 18, 2025• 0 citations
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Abstract

The Haldane model is a standard tight-binding model describing electrons hopping on a hexagonal lattice subject to a transverse, dipolar magnetic field. We consider its interacting version for values of the interaction strength that are small compared to the bandwidth. We study the critical case at the transition between the trivial and ``topological'' insulating phases, and we rigorously establish that the transverse conductivity on the dressed critical line is quantized at a half-integer multiple of ${e}^{2}/h$: this is the average of the integer values of the Hall conductivity in the insulating phases on either side of the dressed critical line. Together with previous results, this fully characterizes the nature of the phase transition between different Hall plateaus and proves its universality with respect to many-body interactions. The proof is based on a combination of constructive renormalization group methods and exact lattice Ward identities.

DOI
10.1103/1lkl-775k

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The Haldane model is a standard tight-binding model describing electrons hopping on a hexagonal lattice subject to a transverse, dipolar magnetic field. We consider its interacting version for values of the interaction strength that are small compared to the bandwidth. We study the critical case at the transition between the trivial and ``topological'' insulating phases, and we rigorously establish that the transverse conductivity on the dressed critical line is quantized at a half-integer multiple of e2/h{e}^{2}/he2/h: this is the average of the integer values of the Hall conductivity in the insulating phases on either side of the dressed critical line. Together with previous results, this fully characterizes the nature of the phase transition between different Hall plateaus and proves its universality with respect to many-body interactions. The proof is based on a combination of constructive renormalization group methods and exact lattice Ward identities.
lattice model exhibiting the integer quantum Hall effect. We consider an interacting version of the model, and prove that for short-range interactions, smaller than the bandwidth, the Hall conductivity is quantized, for all the values of the parameters outside two critical curves, across which the model undergoes a 'topological' phase transition: the Hall coefficient remains integer and constant as long as we continuously deform the parameters without crossing the curves; when this happens, the Hall coefficient jumps abruptly to a different integer. Previous works were limited to the perturbative regime, in which the interaction is much smaller than the bare gap, so they were restricted to regions far from the critical lines. The non-renormalization of the Hall conductivity arises as a consequence of lattice conservation laws and of the regularity properties of the current-current correlations. Our method provides a full construction of the critical curves, which are modified ('dressed') by the electron-electron interaction. The shift of the transition curves manifests itself via apparent infrared divergences in the naive perturbative series, which we resolve via renormalization group methods.
We study the Haldane-Hubbard model by exact renormalization group techniques. We analytically construct the topological phase diagram, for weak interactions. We predict that many-body interactions induce a shift of the transition line: in particular, repulsive interactions enlarge the topologically nontrivial region. The presence of new intermediate phases, absent in the noninteracting case, is rigorously excluded at weak coupling. Despite the nontrivial renormalization of the wave function and of the Fermi velocity, the conductivity is universal: at the renormalized critical line, both the discontinuity of the transverse conductivity and the longitudinal conductivity are independent of the interaction, thanks to remarkable cancellations due to lattice Ward identities. In contrast to the quantization of the transverse conductivity, the universality of the longitudinal conductivity cannot be explained via topological arguments.
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 study strong correlation effects in topological insulators via the Lanczos algorithm, which we utilize to calculate the exact many-particle ground-state wave function and its topological properties. We analyze the simple, noninteracting Haldane model on a honeycomb lattice with known topological properties and demonstrate that these properties are already evident in small clusters. Next, we consider interacting fermions by introducing repulsive nearest-neighbor interactions. A first-order quantum phase transition was discovered at finite interaction strength between the topological band insulator and a topologically trivial Mott insulating phase by use of the fidelity metric and the charge-density-wave structure factor. We construct the phase diagram at T=0T=0T=0 as a function of the interaction strength and the complex phase for the next-nearest-neighbor hoppings. Finally, we consider the Haldane model with interacting hard-core bosons, where no evidence for a topological phase is observed. An important general conclusion of our work is that despite the intrinsic nonlocality of topological phases their key topological properties manifest themselves already in small systems and therefore can be studied numerically via exact diagonalization and observed experimentally, e.g., with trapped ions and cold atoms in optical lattices.
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.
In this Rapid Communication, we construct a large-NNN exactly solvable model to study the interplay between interaction and topology, by connecting the Sachdev-Ye-Kitaev (SYK) model with constant hopping. The hopping forms a band structure that can exhibit both topologically trivial and nontrivial phases. Starting from a topologically trivial insulator with zero Hall conductance, we show that the interaction can drive a phase transition to a topologically nontrivial insulator with quantized nonzero Hall conductance, and a single gapless Dirac fermion emerges when the interaction is fine tuned to the critical point. The finite temperature effect is also considered, and we show that the topological phase with a stronger interaction is less stable against temperature. Our model provides a concrete example to illustrate the interacting topological phases and phase transitions, and can shed light on similar problems in physical systems.