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arXiv:2606.26096

Higher Berry curvature, second Chern numbers and magnetoelectric coupling in crystalline insulators

Niclas Heinsdorf, Ken Shiozaki

cond-mat.str-elquant-phSubmitted Jun 24, 2026• Announced Jun 25, 2026• 0 citations
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Abstract

We rewrite a lattice model of the four-dimensional Chern insulator as a family of translationally-invariant infinite chains over the three-dimensional Brillouin zone and compute its higher three-form Berry curvature using infinite matrix product states (iMPS). We calculate the topological phase diagram of the associated Dixmier--Douady--Kapustin--Spodyneiko (DDKS) number as a function of the model's mass term, and show that it is exactly congruent to the phase diagram in terms of the second Chern number, the analytic expression of which is known for this particular model. This agreement demonstrates that higher Berry curvature can be used to compute second Chern numbers in a manifestly quantized manner. Motivated by the connection between the second Chern form and the Chern--Simons axion coupling, we study magnetoelectric coupling in three dimensions and its relation to higher Berry phases.

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We rewrite a lattice model of the four-dimensional Chern insulator as a family of translationally-invariant infinite chains over the three-dimensional Brillouin zone and compute its higher three-form Berry curvature using infinite matrix product states (iMPS). We calculate the topological phase diagram of the associated Dixmier--Douady--Kapustin--Spodyneiko (DDKS) number as a function of the model's mass term, and show that it is exactly congruent to the phase diagram in terms of the second Chern number, the analytic expression of which is known for this particular model. This agreement demonstrates that higher Berry curvature can be used to compute second Chern numbers in a manifestly quantized manner. Motivated by the connection between the second Chern form and the Chern--Simons axion coupling, we study magnetoelectric coupling in three dimensions and its relation to higher Berry phases.
We study the ground-state topology and quasiparticle properties in bosonic Mott insulators with two-dimensional spin-orbit couplings in cold atomic optical lattices. We show that the many-body Chern and spin-Chern number can be expressed as an integral of the quasihole Berry curvatures over the Brillouin zone. Using a strong-coupling perturbation theory, for an experimentally feasible spin-orbit coupling, we compute the Berry curvature and the spin Chern number and find that these quantities can be generated purely by interactions. We also compute the quasiparticle dispersions, spectral weights, and the quasimomentum space distribution of particle and spin density, which can be accessed in cold-atom experiments and used to deduce the Berry curvature and Chern numbers.
Band geometry plays a substantial role in topological lattice models. The Berry curvature, which resembles the effect of magnetic field in reciprocal space, usually fluctuates throughout the Brillouin zone. Motivated by the analogy with Landau levels, constant Berry curvature has been suggested as an ideal condition for realizing fractional Chern insulators. Here we show that while the Berry curvature cannot be made constant in a topological two-band model, lattice models with three or more degrees of freedom per unit cell can support exactly constant Berry curvature. However, contrary to the intuitive expectation, we find that making the Berry curvature constant does not always improve the properties of fractional Chern insulator states. In fact, we show that an ``ideal flatband'' cannot have constant Berry curvature, equivalently, we show that the density algebra of Landau levels cannot be realised in any tight-binding lattice system.
The higher Berry curvature was introduced by Kapustin and Spodyneiko as an extension of the Berry curvature in quantum mechanical systems with finite degrees of freedom to quantum many-body systems in finite spatial dimensions. In this paper, we propose an alternative formulation of the higher Berry curvature using translationally invariant matrix product states. They are the ground states of a set of gapped Hamiltonians which are evolved adiabatically through a discretized parameter space. Because matrix product states transform under a projective representation, evaluating the Berry curvature on a closed loop through parameter space is not sufficient to fix all the gauge degrees of freedom. To obtain a gauge-invariant real quantity, the higher-dimensional Berry curvature is evaluated on small tetrahedra in parameter space. Our numerical calculations confirm that the higher Berry curvature varies continuously throughout an adiabatic evolution and becomes quantized over a closed 3-dimensional parameter space.
The geometry and topology of quantum systems have deep connections to quantum dynamics. In this Letter, I show how to measure the non-Abelian Berry curvature and its related topological invariant, the second Chern number, using dynamical techniques. The second Chern number is the defining topological characteristic of the four-dimensional generalization of the quantum Hall effect and has relevance in systems from three-dimensional topological insulators to Yang-Mills field theory. I illustrate its measurement using the simple example of a spin-3/2 particle in an electric quadrupole field. I show how one can dynamically measure diagonal components of the Berry curvature in an overcomplete basis of the degenerate ground state space and use this to extract the full non-Abelian Berry curvature. I also show that one can accomplish the same ideas by stochastically averaging over random initial states in the degenerate ground state manifold. Finally, I show how this system can be manufactured and the topological invariant measured in a variety of realistic systems, from superconducting qubits to trapped ions and cold atoms.
A time-reversal invariant topological insulator is defined by its topological magnetoelectric response that is robust against disorder. The response formula, defined on a Brillouin torus, defines a mathbbZ2mathbb{Z}_2mathbbZ2​ invariant and classifies the topological phase. However, in the presence of disorder or the magnetic field, the notion of Brillouin torus is destroyed and the response formula is no longer well-defined. This has been a challenging open problem, and it is essental in defining a topological insulator. This thesis proposes a topological response theory that is free from this fundamental deficiency. We derived the magnetoelectric response formula in position space for a generic three dimensional model under disorder and finite magnetic field. For time-reversal invariant systems, we connected the result to the 2nd Chern number in Noncommutative Geometry. We developed the noncommutative theory of Chern numbers and showed that the quantization of the magnetoelectric response is robust against disorder. Numerical studies on serveral disodered topological models in 1D and 3D are presented.
Abstract Topological insulators are exotic material that possess conducting surface states protected by the topology of the system. They can be classified in terms of their properties under discrete symmetries and are characterized by topological invariants. The latter has been measured experimentally for several models in one, two and three dimensions in both condensed matter and quantum simulation platforms. The recent progress in quantum simulation opens the road to the simulation of higher dimensional Hamiltonians and in particular of the 4D quantum Hall effect. These systems are characterized by the second Chern number, a topological invariant that appears in the quantization of the transverse conductivity for the non-linear response to both external magnetic and electric fields. This quantity cannot always be computed analytically and there is therefore a need of an algorithm to compute it numerically. In this work, we propose an efficient algorithm to compute the second Chern number in 4D systems. We construct the algorithm with the help of lattice gauge theory and discuss the convergence to the continuous gauge theory. We benchmark the algorithm on several relevant models, including the 4D Dirac Hamiltonian and the 4D quantum Hall effect and verify numerically its rapid convergence.