Manifestation of topological behaviors in interacting Weyl systems: one-body verse two-body correlations

TL;DR

This paper investigates the limitations of using single-particle Green's functions to identify topological phases in interacting Weyl semimetals, highlighting discrepancies with many-body invariants in strongly correlated regimes.

Contribution

It clarifies the relationship between single-particle and many-body topological invariants and demonstrates the breakdown of Green's function methods in strongly correlated phases.

Findings

Green's function topological index can be nonzero in non-topological phases.

Many-body Chern number correctly identifies topological phases.

Surface states are linked to particle-hole excitations, not quasiparticles.

Abstract

Understanding correlation effects in topological phases of matter is at the forefront of current research in condensed matter physics. Here we try to clarify some subtleties in studying topological behaviors of interacting Weyl semimetals. It is well-known that there exist two topological invariants defined to identify their topological character. One is the many-body Chern number, which can be directly linked to the Hall conductivity and thus to the two-particle correlations. The other is the topological index constructed from the single-particle Green's functions. Because the information of Green's functions is easier to be achieved than the many-body wavefunctions, usually only the latter is employed in the literature. However, the approach based on the single-particle Green's function can break down in the strongly correlated phase. For illustration, an exactly solvable two-orbital model with momentum-local two-body interactions is discussed, in which both topological invariants can be calculated analytically. We find that the topological index calculated from the Green's function formalism can be nonzero even for a non-topological strongly correlated phase with vanishing many-body Chern number. In addition, we stress that the physical surface states implied by nonzero many-body Chern numbers should be the edge modes of particle-hole collective excitations, rather than those of quasiparticle nature derived from the Green's function formalism. Our observations thus demonstrate the limitation of the validity of Green's function formalism in the investigations of interacting topological materials.