Non-Abelian Tensor Berry Connections in Multiband Topological Systems

Here, we introduce and apply non-Abelian tensor Berry connections to topological phases in multiband systems. These gauge connections behave as non-Abelian antisymmetric tensor gauge fields in momentum space and naturally generalize Abelian tensor Berry connections and ordinary non-Abelian (vector) Berry connections. We build these novel gauge fields from momentum-space Higgs fields, which emerge from the degenerate band structure of multiband models. First, we show that the conventional topological invariants of two-dimensional topological insulators and three-dimensional Dirac semimetals can be derived from the winding number associated with the Higgs field. Second, through the non-Abelian tensor Berry connections we construct higher-dimensional Berry-Zak phases and show their role in the topological characterization of several gapped and gapless systems, ranging from two-dimensional Euler insulators to four-dimensional Dirac semimetals. Importantly, through our new theoretical formalism, we identify and characterize a novel class of models that support space-time inversion and chiral symmetries. Our work provides a unifying framework for different multiband topological systems and sheds new light on the emergence of non-Abelian gauge fields in condensed matter physics, with direct implications on the search for novel topological phases in solid-state and synthetic systems.

Figure 1

(Left) Band structure of the topological phase in the BHZ model at $m=1.3$. (Right) Plot of the integrand of the winding number $w({\mathbb{T}}^2)$ for the BHZ model at $m=1.3$. The maximum of the function is at the $\mathrm{\Gamma }$ point.

Figure 2

(upper left) Band structure of the 3D gapped topological phase with $\mathcal{I}\mathcal{T}$ and $\mathcal{S}$ symmetries at $m=0.1$ and $k_z=0$. (upper right) Plot of ${\mathbf{C}}_{xyz}$ associated with ${\mathrm{\Upsilon }}_{\mathbf{C}}({\mathbb{T}}^3)=-8$ at $m=0.1$. The maximum of the function is at the $\mathrm{\Gamma }$ point. (lower left) Band structure of the same model at $m=1.9$ and $k_z=0$. (lower right) Plot of ${\mathbf{C}}_{xyz}$ associated to ${\mathrm{\Upsilon }}_{\mathbf{C}}({\mathbb{T}}^3)=4$ at $m=1.9$. The maximum of the function is at $X$ points.