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arXiv:1004.4229 · Physical Review Letters

Topological Order Parameters for Interacting Topological Insulators

Zhong Wang, Xiao-Liang Qi, Shoucheng Zhang

Published Dec 13, 2010• 206 citations
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Abstract

We propose a topological order parameter for interacting topological insulators, expressed in terms of the full Green's functions of the interacting system. We show that it is exactly quantized for a time-reversal invariant topological insulator, and it can be experimentally measured through the topological magneto-electric effect. This topological order parameter can be applied to both interacting and disordered systems, and used for determining their phase diagrams.

DOI
10.1103/physrevlett.105.256803

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We propose a topological order parameter for interacting topological insulators, expressed in terms of the full Green's functions of the interacting system. We show that it is exactly quantized for a time-reversal invariant topological insulator, and it can be experimentally measured through the topological magneto-electric effect. This topological order parameter can be applied to both interacting and disordered systems, and used for determining their phase diagrams.
For interacting Z2{Z}_{2}Z2​ topological insulators with inversion symmetry, we propose a simple topological invariant expressed in terms of the parity eigenvalues of the interacting Green's function at time-reversal invariant momenta and zero frequency. We derive this result from our previous formula involving the integral over the frequency-momenta space. This formula greatly simplifies the explicit calculation of Z2{Z}_{2}Z2​ topological invariants in inversion-symmetric insulators with strong interactions.
We study the effect of interactions on the time-reversal invariant topological insulators in four and three spatial dimensions. Their topological indices are expressed by the interacting Green's functions. Under the local self-energy approximation, we find that interaction could induce nontrivial frequency-domain winding numbers and change the topological classes of the system. Our results suggest that the topological phases could be destroyed without developing long-range orders. Practical issues on the accurate frequency-momentum integration combined with dynamical-mean-field theory and diagrammatic calculations of the interacting Green's functions are also addressed.
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A time-reversal invariant topological insulator is defined by its topological magnetoelectric response that is robust against disorder. The response formula, defined on a Brillouin torus, defines a mathbbZ2mathbb{Z}_2mathbbZ2​ invariant and classifies the topological phase. However, in the presence of disorder or the magnetic field, the notion of Brillouin torus is destroyed and the response formula is no longer well-defined. This has been a challenging open problem, and it is essental in defining a topological insulator. This thesis proposes a topological response theory that is free from this fundamental deficiency. We derived the magnetoelectric response formula in position space for a generic three dimensional model under disorder and finite magnetic field. For time-reversal invariant systems, we connected the result to the 2nd Chern number in Noncommutative Geometry. We developed the noncommutative theory of Chern numbers and showed that the quantization of the magnetoelectric response is robust against disorder. Numerical studies on serveral disodered topological models in 1D and 3D are presented.