Abstract
Using cluster perturbation theory we calculate Green's functions, quasi-particle energies and topological invariants for interacting electrons on a 2D honeycomb lattice, with intrinsic spin–orbit coupling and on-site e–e interaction. This allows us to define the parameter range (Hubbard U versus spin–orbit coupling) where the 2D system behaves as a trivial insulator or quantum spin Hall insulator. This behavior is confirmed by the existence of gapless quasi-particle states in honeycomb ribbons. We have discussed the importance of the cluster symmetry and the effects of the lack of full translation symmetry typical of CPT and of most quantum cluster approaches. Comments on the limits of applicability of the method are also provided.
Highlights
Topological invariants are widely recognized as a powerful tool to characterize different phases of matter; in particular they turn out to be useful in the classification of topological insulators [1, 2]
In the same way as the Thouless–Kohmoto–Nightingale–den Nijs (TKNN) [9] topological invariant was defined for the integer quantum Hall effect, the above 2 invariant was defined for the topological insulator in terms of band eigenvectors and, as such, only applies to non-interacting systems
In recent years a new class of many-body approaches has been developed to calculate the one-particle Green’s function of extended systems solving the many body problem in a subsystem of finite size and embedding it within an infinite medium. These methods gather under the name of quantum cluster theories [21] and include cluster perturbation theory (CPT) [22, 23], dynamical cluster approach (DCA) [24], variational cluser approximation (VCA) [25], cellular dynamical mean field theory (CDMFT) [26]
Summary
Topological invariants are widely recognized as a powerful tool to characterize different phases of matter; in particular they turn out to be useful in the classification of topological insulators [1, 2]. In recent years a new class of many-body approaches has been developed to calculate the one-particle Green’s function of extended systems solving the many body problem in a subsystem of finite size and embedding it within an infinite medium These methods gather under the name of quantum cluster theories [21] and include cluster perturbation theory (CPT) [22, 23], dynamical cluster approach (DCA) [24], variational cluser approximation (VCA) [25], cellular dynamical mean field theory (CDMFT) [26]. In this paper we consider the Kane–Mele–Hubbard model [30,31,32,33] describing a 2D honeycomb lattice with both local e–e interaction and spin–orbit coupling and we adopt an approach based on CPT to determine the one-particle propagator, the topological HamiltonianG−1(k, ω = 0) and its eigenvectors This allows us to identify a general procedure that can be extended to any quantum cluster approach and to investigate how Greens function-based topological invariants can be effectively calculated.
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