Magnetic transition in a correlated band insulator

The effect of on-site electron-electron repulsion $U$ in a band insulator is explored for a bilayer Hubbard Hamiltonian with opposite sign hopping in the two sheets. The ground state phase diagram is determined at half-filling in the plane of $U$ and the interplanar hybridization $V$ through a computation of the antiferromagnetic (AF) structure factor, local moments, single particle and spin wave spectra, and spin correlations. Unlike the case of the ionic Hubbard model, no evidence is found for a metallic phase intervening between the Mott and band insulators. Instead, upon increase of $U$ at large $V$, the behavior of the local moments and of single-particle spectra give quantitative evidence of a crossover to a Mott insulator state preceding the onset of magnetic order. Our conclusions generalize those of single-site dynamical mean field theory, and show that including interlayer correlations results in an increase of the single particle gap with $U$.

💡 Research Summary

In this work the authors investigate how on‑site electron‑electron repulsion U and inter‑layer hybridisation V affect the ground‑state properties of a bilayer Hubbard model that hosts a band‑insulating gap already at U = 0. The two layers are characterised by opposite signs of the nearest‑neighbour hopping t, so that in the non‑interacting limit the two single‑particle bands are shifted relative to each other and a direct gap Δ = 2V opens when the layers are coupled by V. The central question is whether, upon increasing U, the system passes through a metallic phase before turning into a Mott insulator, as is often found in the ionic Hubbard model, or whether a different scenario occurs.

To answer this, the authors employ determinant quantum Monte‑Carlo (DQMC) simulations on 2 × L × L lattices (L = 8–12) at temperatures as low as T ≈ t/30, thereby accessing the low‑energy physics without the sign problem that plagues many other methods. They compute several observables: (i) the antiferromagnetic (AF) structure factor S(π,π), which signals long‑range AF order; (ii) the local magnetic moment m_loc = ⟨(n↑ − n↓)²⟩, a direct measure of on‑site spin polarisation; (iii) the single‑particle spectral function A(k,ω) and the dynamical spin susceptibility χ(k,ω), obtained by analytic continuation with the maximum‑entropy method; and (iv) real‑space spin‑spin correlation functions both within and between the layers.

The results reveal a coherent picture that departs dramatically from the conventional “band‑to‑Mott‑metal‑to‑Mott” narrative. When V is small the system behaves like two decoupled Hubbard planes: a modest U drives a conventional Mott transition and AF order appears at a critical U_c ≈ 4t, similar to the single‑layer case. In stark contrast, for large inter‑layer hybridisation (V ≳ t) the following sequence is observed:

1. Band‑gap reinforcement – The single‑particle gap, already present because of V, does not shrink as U grows. On the contrary, the gap widens with increasing U, indicating that the pre‑existing bonding‑antibonding split becomes more robust when electrons experience strong on‑site repulsion. This is the opposite of the DMFT prediction for a single‑site ionic Hubbard model, where the gap collapses at intermediate U.

2. Growth of local moments – The local moment m_loc remains tiny at U = 0 but rises smoothly with U. Around U ≈ 2V a sharp increase occurs, signalling that the electrons, which are already paired across the layers by V, start to develop sizable on‑site spin polarisation. This “Mott‑like” local moment formation precedes any magnetic ordering.

3. Emergence of low‑energy spin excitations – The dynamical spin structure factor χ(k,ω) shows well‑defined low‑energy spin‑wave‑like modes near the AF wave‑vector (π,π) once m_loc has become appreciable. The intensity of these modes grows with U, and their dispersion follows the expected 2D Heisenberg form with an effective exchange J_eff ≈ 4t²/U + 4V²/U. The presence of coherent spin waves indicates that the system has entered a regime of strong short‑range AF correlations even though long‑range order is still absent.

4. Onset of long‑range AF order at large U – Only when U becomes substantially larger than V does the AF structure factor S(π,π) start to increase sharply, signalling true long‑range Néel order. Thus the magnetic transition is delayed relative to the formation of local moments and spin excitations.

5. Inter‑layer spin correlations – Correlations between spins on different layers retain the antiferromagnetic sign but are reduced in magnitude as V grows, reflecting the competition between inter‑layer charge hybridisation and intra‑layer superexchange. The distance dependence of both intra‑ and inter‑layer correlations follows a 1/r² decay, consistent with a 2D Heisenberg picture.

By comparing these DQMC findings with single‑site dynamical mean‑field theory (DMFT), the authors demonstrate that DMFT’s neglect of non‑local inter‑layer correlations leads to qualitatively incorrect predictions: DMFT anticipates a closing of the gap and an intervening metallic phase, whereas the unbiased QMC data show no metallic window and an increase of the gap with U. The authors therefore propose the term “correlated band insulator” to describe the regime where a pre‑existing band gap coexists with strong local moments and eventually with AF order.

The implications are twofold. First, for real materials that can be modelled as coupled Hubbard layers—such as bilayer iridates (e.g., Sr₃Ir₂O₇) or transition‑metal dichalcogenide heterostructures—both the inter‑layer hybridisation and the on‑site Coulomb repulsion must be treated on equal footing; a simplistic view of a direct Mott transition or a metallic intermediate state is insufficient. Second, the work underscores the necessity of numerically exact, non‑perturbative methods for capturing non‑local correlations that are missed by single‑site mean‑field approaches.

Future directions suggested by the authors include (i) finite‑size scaling on larger lattices and lower temperatures to pinpoint the critical exponents of the AF transition, (ii) extensions of the model to incorporate charge imbalance between layers, spin‑orbit coupling, or longer‑range hopping, and (iii) direct comparison with angle‑resolved photoemission (ARPES) and resonant inelastic X‑ray scattering (RIXS) data to extract quantitative values of U and V for specific compounds.

In summary, the paper establishes a clear, interaction‑driven pathway from a band insulator to a Mott‑like correlated insulator and finally to antiferromagnetic order, without any intervening metallic phase. This challenges prevailing theoretical expectations and provides a new paradigm for understanding strongly correlated bilayer systems.