[Paper Review] Inversion symmetry protected topological insulators and superconductors

Yuan-Ming Lu, Dung‐Hai Lee|arXiv (Cornell University)|Mar 21, 2014

Topological Materials and Phenomena8 citations

TL;DR

This paper classifies three-dimensional topological insulators and superconductors protected by spatial inversion symmetry using K-theory, identifying new topological phases that are trivial without inversion symmetry. It constructs lattice models and identifies quantized response functions—such as the 'half' spin quantum Hall effect on surfaces—as experimental signatures of these phases.

ABSTRACT

Three dimensional topological insulator represents a class of novel quantum phases hosting robust gapless boundary excitations, which is protected by global symmetries such as time reversal, charge conservation and spin rotational symmetry. In this work we systematically study another class of topological phases of weakly interacting electrons protected by spatial inversion symmetry, which generally don't support stable gapless boundary states. We classify these inversion-symmetric topological insulators and superconductors in the framework of K-theory, and construct their lattice models. We also discuss quantized response functions of these inversion-protected topological phases, which serve as their experimental signatures.

Motivation & Objective

• To classify topological insulators and superconductors protected by spatial inversion symmetry in three dimensions.
• To identify new topological phases that are only stable when inversion symmetry is preserved.
• To construct explicit lattice models for these inversion-protected topological phases.
• To derive quantized response functions that serve as experimental signatures of these phases.
• To understand how inversion symmetry modifies the standard 10-fold way classification of topological phases.

Proposed method

• Uses K-theory to classify topological phases protected by both global symmetries and spatial inversion symmetry.
• Applies the extension problem of complex Clifford algebras to classify mass matrices in Dirac Hamiltonians.
• Derives the classifying space for mass matrices as $ C_d $, corresponding to $ \pi_0(C_d) $, for inversion-protected systems.
• Constructs effective Dirac Hamiltonians with mass domain walls to model surface states in inversion-protected topological superconductors.
• Analyzes surface states via zero-energy solutions of the Dirac equation with a spatially varying mass term.
• Computes the spin quantum Hall conductance on the surface to identify the 'half' spin quantum Hall effect as a signature.

Experimental results

Research questions

• RQ1 How does spatial inversion symmetry modify the classification of topological insulators and superconductors beyond the standard 10-fold way?
• RQ2 What new topological phases emerge that are only stable when inversion symmetry is preserved?
• RQ3 Can inversion-protected topological phases be realized in explicit lattice models?
• RQ4 What are the quantized response functions that distinguish these phases experimentally?
• RQ5 Why do certain topological phases (e.g., TRI triplet superconductors) become trivial when inversion symmetry is imposed?

Key findings

• Inversion symmetry protects a new class of topological insulators and superconductors that are trivial without this symmetry, such as 3D singlet superconductors (class C) and magnetic insulators.
• The classification of inversion-protected topological phases is given by $ \pi_0(C_d) $, derived from the extension of complex Clifford algebras.
• For the $ \nu = 1 $ inversion-protected 3D singlet superconductor, the surface exhibits a gapped state with a Chern number $ C_s = \pm 1 $, leading to a quantized spin Hall conductance.
• The surface response shows a 'half' spin quantum Hall effect, with $ \sigma_{xy}^{\text{spin}} = \pm \frac{\hbar}{8\pi} $, which is not possible in conventional 2D singlet superconductors.
• Inversion symmetry can forbid certain topological phases—such as TRI triplet superconductors (class DIII)—by trivializing their classification to $ \mathbb{Z}_2 = 0 $ when inversion is imposed.
• The presence of a special inversion symmetry with $ \boldsymbol{I}^2 = -1 $, realized via $ C_2 $ rotation in 1D/2D systems, leads to distinct topological classifications compared to standard inversion.