Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle (1808.05375v5)

Abstract: We show that the Wannier obstruction and the fragile topology of the nearly flat bands in twisted bilayer graphene at magic angle are manifestations of the nontrivial topology of two-dimensional real wave functions characterized by the Euler class. To prove this, we examine the generic band topology of two dimensional real fermions in systems with space-time inversion $I_{ST}$ symmetry. The Euler class is an integer topological invariant classifying real two band systems. We show that a two-band system with a nonzero Euler class cannot have an $I_{ST}$-symmetric Wannier representation. Moreover, a two-band system with the Euler class $e_{2}$ has band crossing points whose total winding number is equal to $-2e_2$. Thus the conventional Nielsen-Ninomiya theorem fails in systems with a nonzero Euler class. We propose that the topological phase transition between two insulators carrying distinct Euler classes can be described in terms of the pair creation and annihilation of vortices accompanied by winding number changes across Dirac strings. When the number of bands is bigger than two, there is a $Z_{2}$ topological invariant classifying the band topology, that is, the second Stiefel Whitney class ($w_2$). Two bands with an even (odd) Euler class turn into a system with $w_2=0$ ($w_2=1$) when additional trivial bands are added. Although the nontrivial second Stiefel-Whitney class remains robust against adding trivial bands, it does not impose a Wannier obstruction when the number of bands is bigger than two. However, when the resulting multi-band system with the nontrivial second Stiefel-Whitney class is supplemented by additional chiral symmetry, a nontrivial second-order topology and the associated corner charges are guaranteed.

• The paper introduces the Euler class as a key topological invariant to classify two-band systems and demonstrate Wannier obstruction.
• It reveals that a nonzero Euler class causes the failure of the Nielsen-Ninomiya theorem by generating band crossings with net winding numbers.
• The study shows that adding trivial bands transforms the topology, rendering the obstruction fragile and enabling higher-order topological phases in TBG.

Failure of Nielsen-Ninomiya Theorem and Fragile Topology in Two-Dimensional Systems

This paper explores the topological properties of twisted bilayer graphene (TBG) at the magic angle, particularly focusing on its nearly flat bands and the associated Wannier obstruction and fragile topology. The authors investigate these phenomena using the Euler class, a topological invariant class for two-dimensional real wave functions, in systems with space-time inversion symmetry ($I_{ST}$). The research demonstrates the failure of the conventional Nielsen-Ninomiya theorem in certain two-dimensional systems and proposes a generalized framework to understand the band topology under such conditions.

Key Findings

1. Euler Class as a Topological Invariant: The paper emphasizes the importance of the Euler class, an integer topological invariant, in classifying real two-band systems. The authors show that a two-band system with a nonzero Euler class cannot possess an $I_{ST}$-symmetric Wannier representation, evidencing a Wannier obstruction.
2. Violation of the Nielsen-Ninomiya Theorem: It is shown that in systems with non-trivial Euler classes, the traditional Nielsen-Ninomiya theorem, which predicts the appearance of Dirac points with opposite winding numbers to ensure a zero net winding number, cannot be applied. Instead, band crossing points arise with total winding numbers directly related to the Euler class.
3. Fragile Topology in Multi-band Systems: The research highlights that adding trivial bands to a two-band system can change its topological classification to a $Z_{2}$ invariant, described by the second Stiefel-Whitney class $w_2$. This alteration renders the Wannier obstruction fragile, implying the obstruction can disappear when trivial bands are added.
4. Higher-order Topological Phases: When additional chiral symmetries are present, multi-band systems exhibiting non-trivial second Stiefel-Whitney class can host higher-order topological phases, characterized by corner charges while maintaining a multi-band Wanier representation.

Implications and Future Directions

The findings on the Euler class and its interaction with space-time inversion symmetries open new pathways in studying topological materials, especially those without inversion symmetry but with robust time-reversal symmetry. The revelation that the Nielsen-Ninomiya theorem fails under these conditions leads to new theoretical frameworks for non-trivial topologies in two-dimensional materials beyond the standard topological classifications.

In practice, these insights are particularly relevant for understanding the superconducting and insulating states observed in TBG. The concept of fragile topology may also influence the design of novel electronic devices, where topological properties can be engineered by altering band structures via trivial band addition.

Future research directions include the exploration of different symmetry constraints on the band topology and the corresponding Euler classes in higher-dimensional systems, particularly in materials with potent electron-electron interactions. Furthermore, verifying these theoretical predictions through experimental observations in TBG and similar materials will be pivotal in confirming these new topological insights. This foundational work is crucial for advancing the theoretical framework of topological phases of matter, especially in complex materials where traditional approaches fail.