[Paper Review] Z_2 indices and factorization properties of odd symmetric Fredholm operators

Hermann Schulz‐Baldes|arXiv (Cornell University)|Nov 2, 2013

Advanced Topics in Algebra15 references21 citations

TL;DR

This paper establishes a factorization theorem for odd symmetric Fredholm operators on separable Hilbert spaces, proving they can be written as $ T = I^*A^tIA $ for some bounded operator $ A $. The key contribution is the introduction of a $ \mathbb{Z}_2 $-index defined by the parity of the kernel dimension, which classifies the connected components of the space of such operators and provides a topological invariant applicable to systems with time-reversal symmetry, such as topological insulators.

ABSTRACT

A bounded operator $T$ on a separable, complex Hilbert space is said to be odd symmetric if $I^*T^tI=T$ where $I$ is a real unitary satisfying $I^2=-1$ and $T^t$ denotes the transpose of $T$. It is proved that such an operator can always be factorized as $T=I^*A^tIA$ with some operator $A$. This generalizes a result of Hua and Siegel for matrices. As application it is proved that the set of odd symmetric Fredholm operators has two connected components labelled by a $Z_2$-index given by the parity of the dimension of the kernel of $T$. This recovers a result of Atiyah and Singer. Two examples of $Z_2$-valued index theorems are provided, one being a version of the Noether-Gohberg-Krein theorem with symmetries and the other an application to topological insulators.

Motivation & Objective

• To establish a general factorization property for odd symmetric Fredholm operators on separable Hilbert spaces.
• To define and analyze a $ \mathbb{Z}_2 $-index based on the parity of the kernel dimension of such operators.
• To recover and generalize the Atiyah-Singer index theorem in the context of time-reversal symmetric systems.
• To provide a mathematical framework for $ \mathbb{Z}_2 $-valued index theorems in condensed matter physics, particularly for topological insulators.
• To demonstrate the stability of topological invariants under time-reversal symmetry breaking perturbations using spectral and Fredholm theory.

Proposed method

• Uses the existence of a real skew-adjoint unitary operator $ I $ with $ I^2 = -\mathbf{1} $ to define the class of odd symmetric operators satisfying $ I^*T^tI = T $.
• Proves that any odd symmetric operator $ T $ admits a factorization $ T = I^*A^tIA $ for some $ A \in \mathbb{B}(\mathcal{H}) $, generalizing a result by Hua and Siegel.
• Applies the factorization to show that the set of odd symmetric Fredholm operators has two connected components, labeled by the $ \mathbb{Z}_2 $-index $ \mathrm{Ind}_2(T) = \dim(\ker T) \mod 2 $.
• Analyzes the spectral properties of such operators, showing that generalized eigenspaces have even dimension, generalizing Kramers degeneracy.
• Constructs Riesz projections $ P_\pm $ onto the positive and negative spectra of $ PAP $, where $ P $ is the Fermi projection, and uses them to define Fredholm operators on subspaces.
• Demonstrates that the Noether indices of $ P_\pm F P_\pm $ are equal modulo 2 and stable under perturbations that break time-reversal symmetry, leading to a robust $ \mathbb{Z}_2 $ invariant.

Experimental results

Research questions

• RQ1 Can every odd symmetric Fredholm operator be factorized as $ T = I^*A^tIA $ for some bounded operator $ A $?
• RQ2 What topological invariant classifies the connected components of the space of odd symmetric Fredholm operators?
• RQ3 How does the $ \mathbb{Z}_2 $-index relate to the spectral degeneracy of such operators, particularly in finite-dimensional and compact cases?
• RQ4 Can the $ \mathbb{Z}_2 $-index be interpreted as a stable invariant in systems where time-reversal symmetry is broken?
• RQ5 What is the physical significance of the $ \mathbb{Z}_2 $-index in topological insulators, particularly in relation to spin Chern numbers?

Key findings

• Every odd symmetric Fredholm operator $ T \in \mathbb{F}(\mathcal{H}, I) $ admits a factorization $ T = I^*A^tIA $ for some $ A \in \mathbb{B}(\mathcal{H}) $, with $ \ker A = \ker T $ when $ \dim(\ker T) $ is even or infinite.
• The space of odd symmetric Fredholm operators has exactly two connected components, classified by the $ \mathbb{Z}_2 $-index $ \mathrm{Ind}_2(T) = \dim(\ker T) \mod 2 $.
• For compact odd symmetric operators $ K $, the dimension of the generalized eigenspace $ d_k(K, \lambda) $ is even for all $ \lambda \neq 0 $ and $ k \geq 1 $.
• In finite dimensions, the generalized eigenspace dimensions $ d_k(T, \lambda) $ and $ d_1(T^*T, \lambda) $ are even for all $ \lambda \in \mathbb{C} $, generalizing Kramers degeneracy.
• The Noether indices of the restricted operators $ P_\pm F P_\pm $ satisfy $ \mathrm{Ind}(P_+ F P_+) = -\mathrm{Ind}(P_- F P_-) $, and their values modulo 2 define the $ \mathbb{Z}_2 $-invariant.
• The $ \mathbb{Z}_2 $-invariant $ \mathrm{Ind}_2(T_P) $ is stable under perturbations that break time-reversal symmetry, such as magnetic fields, as shown via a homotopy argument in $ \mathbb{F}(\mathcal{H}, I) $.