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

Technical Summary: Higher Berry Curvature, Second Chern Numbers, and Magnetoelectric Coupling in Crystalline Insulators

Problem Statement
Topological invariants are essential for classifying quantum many-body ground states. While the Berry curvature (a two-form) and the associated Chern number successfully characterize families of zero-dimensional ground states (e.g., Bloch states in the Brillouin zone), recent theoretical developments suggest a generalization to spatially extended systems. In this framework, families of short-range entangled states in $d$ spatial dimensions are characterized by higher-degree differential forms (e.g., a three-form for 1D systems). Specifically, for one-dimensional systems parametrized by a three-dimensional space, the relevant invariant is the Dixmier–Douady–Kapustin–Spodyneiko (DDKS) number, $\nu$, derived from a higher Berry curvature.

A significant gap exists in the explicit, manifestly quantized computation of these higher invariants for generic interacting or free-fermion systems. While analytical expressions exist for specific free-fermion models (like the four-dimensional Dirac model), numerical methods for computing the second Chern number ($ch_2$) in four dimensions typically rely on lattice gauge theory or real-space magnetic field responses. These standard methods often lack "manifest quantization," meaning they converge to integer values only in the continuum limit or require extensive mesh refinement, rather than returning integers by construction. Furthermore, the physical interpretation of the DDKS number in three spatial dimensions, particularly its relation to orbital magnetoelectric coupling and the Chern-Simons axion angle ($\theta_{CS}$), remains an open question.

Methodology
The authors address these issues by rewriting the lattice model of a four-dimensional (4D) Chern insulator as a family of translationally invariant infinite one-dimensional (1D) chains parametrized by the three-dimensional Brillouin zone ($T^3$).

1. Model Reformulation: Starting from the 4D Dirac Hamiltonian, the authors perform a partial Fourier transform on one momentum component ($k_4$), resulting in a family of 1D Hamiltonians $\{H_\lambda\}$ dependent on parameters $\lambda = (\lambda_1, \lambda_2, \lambda_3)$. These Hamiltonians describe 1D chains with two orbitals per unit cell.
2. Numerical Framework (iMPS): The ground states of these 1D chains are approximated using infinite Matrix Product States (iMPS). The authors utilize the algorithm proposed in Ref. [13], which generalizes the Fukui-Hatsugai-Suzuki method to compute higher Berry curvature.
3. Discrete Higher Berry Curvature:
   • The parameter space $T^3$ is discretized into a grid and triangulated into tetrahedra.
   • For each vertex, an iMPS ground state is obtained via iDMRG.
   • Transition functions between neighboring iMPS tensors are computed using the dominant eigenvectors of mixed transfer matrices.
   • A gauge-invariant phase $\phi(\Delta_2)$ is defined for triangular faces, weighted by the Schmidt spectrum to handle varying bond dimensions.
   • The higher Berry curvature $F(\Delta_3)$ for a tetrahedron is defined as the oriented sum of these phases over its boundary.
   • The DDKS number $\nu$ is obtained by summing $F(\Delta_3)$ over the entire triangulation of $T^3$, yielding a manifestly quantized integer.
4. Comparative Analysis: The authors compute the second Chern number $ch_2$ for the same model using a standard lattice gauge theory approach (Ref. [34]) and compare the resulting phase diagrams.
5. Magnetoelectric Coupling: The study investigates the relationship between the DDKS number and the Chern-Simons axion angle $\theta_{CS}$, which governs orbital magnetoelectric coupling in 3D. They compare the variation of $\theta_{CS}$ (derived analytically from the second Chern form) with the variation of a "higher Berry phase" $\theta_{HBP}$ (derived from the iMPS construction) across a pumping cycle.

Key Contributions and Results

• Manifestly Quantized Computation of $ch_2$: The authors demonstrate that the DDKS number $\nu$, computed via the iMPS-based higher Berry curvature algorithm, generates a topological phase diagram for the 4D Chern insulator that is exactly congruent to the phase diagram of the second Chern number $ch_2$. This agreement holds across different mass regimes ($m$), including regions where standard lattice gauge theory calculations of $ch_2$ suffer from significant finite-grid errors. The iMPS method yields exact integers even on coarse grids (e.g., $4^3$), whereas the lattice gauge method requires much finer grids ($10^4$ points) to converge.
• Theoretical Connection via Künneth Formula: The paper establishes a rigorous link between the 4D second Chern number and the 3D DDKS number using the Künneth formula. By decomposing the 4D torus $T^4$ as $S^1 \times T^3$, the authors show that the second Chern class on $T^4$ corresponds to the product of the generator of $H^1(S^1, \mathbb{Z})$ and the class in $H^3(T^3, \mathbb{Z})$ measured by $\nu$. Thus, for this specific decomposition, $ch_2 = \nu$.
• Magnetoelectric Coupling in the 4D Dirac Model: For the 4D Dirac model, the authors find that the change in the Chern-Simons axion angle $\Delta\theta_{CS}$ along a pumping cycle is reproduced precisely by the change in the higher Berry phase $\Delta\theta_{HBP}$. This suggests a direct correspondence between the higher Berry phase and the magnetoelectric response in systems where the first Chern numbers of BZ slices vanish.
• Counterexample in the Hopf Insulator: The authors test this correspondence on the Hopf insulator, a two-band model with a non-trivial Hopf invariant. In this case, while the Chern-Simons angle is quantized by the Hopf invariant, the computed higher Berry phase $\theta_{HBP}$ is neither quantized nor equal to $\theta_{CS}$. The authors attribute this discrepancy to the "delicate" nature of the Hopf invariant, which is strictly well-defined only for two-band models in momentum space, a condition that may not be preserved in the real-space iMPS description of the thermodynamic limit.

Significance and Claims
The paper claims to provide the first explicit, manifestly quantized numerical computation of the second Chern number for a 4D lattice model using higher Berry curvature and iMPS. This validates the theoretical framework that higher Berry curvatures can serve as probes for higher-degree topological invariants in families of spatially extended states.

The authors position their results within the context of quantum geometry and topological properties, noting that while the correspondence between higher Berry phases and magnetoelectric coupling holds for the 4D Dirac model, it is not universal, as evidenced by the Hopf insulator case. They emphasize that their method is particularly suited for interacting systems where single-particle Bloch descriptions fail, though they currently focus on free-fermion benchmarks. The work highlights the potential of synthetic dimensions and quasi-periodic driving in experimental platforms to realize and probe these higher-dimensional topological phenomena.

The paper concludes modestly, stating that the relationship between higher Berry curvature and physical response in three dimensions remains an open question, particularly regarding the universality of the connection to $\theta_{CS}$. It suggests that higher Berry phases may capture geometric information distinct from standard Chern-Simons responses, warranting further investigation into their role in nonlinear transport and ground-state physics.