Non-Abelian tensor Berry connections in multi-band topological systems

Abstract

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 degenerate-band models. Firstly, 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 to the Higgs field. Secondly, 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 an unifying framework for different multi-band 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.

Extracting non-Abelian quantum metric tensor and its related Chern numbers

“…Note that the formulas in Eqs. (5)(6)(7)(8)(9) naturally recover to those for the non-degenerate case when N = 1. Even in the degenerate case, the distance dS 2 also corresponds to the transition probability from the degenerate ground state |Ψ 0 (λ) to excited states after a sudden quench of the parameter from λ to λ+dλ.…”

“…For degenerate quantum states, the complete geometry is characterized by the non-Abelian QGT [2,4,5,[7][8][9]. When the degenerate states are N -fold, the non-Abelian QGTs are naturally defined on the U (N ) vector bundle and contain the non-Abelian generalizations of the quantum metric and Berry curvature.…”