[Paper Review] Z_2 indices of odd symmetric Fredholm operators

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

Topological Materials and Phenomena3 citations

TL;DR

This paper introduces a Z₂-index theory for odd symmetric Fredholm operators—bounded linear operators on a separable complex Hilbert space satisfying I∗TI = T∗ under a real unitary I with I² = −1. While the classical Noether index vanishes, the parity of the kernel dimension is shown to be a stable homotopy invariant, leading to a Z₂-index that generalizes the Gohberg-Krein theorem and provides a topological invariant for two-dimensional topological insulators with odd time-reversal symmetry, where nontrivial Z₂-index implies non-zero spin Chern numbers.

ABSTRACT

A bounded operator T on a separable, complex Hilbert space is said to be odd sym-metric if I∗T tI = T where I is a real unitary satisfying I2 = −1 and T t denotes the transpose of T. The Noether index of an odd symmetric Fredholm operator vanishes, but the parity of the dimension of its kernel is shown to be a homotopy invariant that is stable under compact perturbations. The class of real skew-adjoint Fredholm operators for which Atiyah and Singer defined Z2-indices is a subset of infinite codimension within the set of odd symmetric Frehholm operators. As first example for an odd Z2-index theo-rem, a Z2-version of the Gohberg-Krein theorem is presented. An even Z2-index theorem leads to a phase label for two-dimensional topological insulators with odd time-reversal symmetry, for which non-trivial Z2-index enforces non-zero spin Chern numbers. 1 Resumé Let H be a separable, complex Hilbert space and I a real skew-adjoint unitary operator on H. Skew-adjointness of I is equivalent to I2 = −1 and implies that the spectrum of I is {−ı, ı}. Such an I exists if and only if H is even or infinite dimensional. One may assume to be I in the normal form I = ( 0 −11 0), see Proposition 6 below. This paper is about bounded linear operators T ∈ B(H) on H which are odd symmetric w.r.t. I in the sense that I ∗ T I = T ∗ ⇐ ⇒ I ∗ T t I = T, (1) where the complex conjugate T is defined in terms of complex conjugation C on H by T = CTC, and the transpose of T is T t = (T)∗. The set of bounded odd symmetric operators is denoted by B(H, I). Condition (1) looks like a quaternionic condition, but actually a quaternionic operator rather satisfies I∗TI = T and the set of quaternionic operators forms a multiplicative group, while B(H, I) does not. However, the following can easily be checked. Proposition 1 B(H, I) is a linear space and

Motivation & Objective

• To define and study a Z₂-index for odd symmetric Fredholm operators, where the classical Noether index vanishes.
• To establish the parity of the kernel dimension as a stable homotopy invariant under compact perturbations.
• To generalize the Gohberg-Krein theorem to a Z₂ setting for odd symmetric operators.
• To connect the Z₂-index to topological invariants in condensed matter physics, particularly spin Chern numbers in two-dimensional topological insulators with odd time-reversal symmetry.
• To clarify the relationship between odd symmetric Fredholm operators and the class of real skew-adjoint Fredholm operators used in Atiyah-Singer index theory.

Proposed method

• Define odd symmetric operators T ∈ B(H) via the condition I∗TI = T∗, where I is a real unitary with I² = −1.
• Use the transpose operation Tᵗ = (T)∗ and complex conjugation to express the symmetry condition in terms of Tᵗ.
• Show that the kernel dimension parity is invariant under compact perturbations, establishing a Z₂-index.
• Construct a Z₂-version of the Gohberg-Krein theorem as a first example of an odd Z₂-index theorem.
• Demonstrate that the Z₂-index for even operators leads to a phase label in 2D topological insulators with odd time-reversal symmetry.
• Relate the Z₂-index to spin Chern numbers, showing that nontrivial index implies non-zero spin Chern numbers.

Experimental results

Research questions

• RQ1 Can the parity of the kernel dimension of an odd symmetric Fredholm operator serve as a stable homotopy invariant despite the vanishing Noether index?
• RQ2 How does the Z₂-index for odd symmetric Fredholm operators generalize the classical Gohberg-Krein theorem?
• RQ3 What is the relationship between the class of odd symmetric Fredholm operators and the real skew-adjoint Fredholm operators used in Atiyah-Singer theory?
• RQ4 How does the Z₂-index for even operators lead to a topological phase label in two-dimensional topological insulators with odd time-reversal symmetry?
• RQ5 Does a nontrivial Z₂-index in such systems necessarily imply non-zero spin Chern numbers?

Key findings

• The parity of the kernel dimension of an odd symmetric Fredholm operator is a stable homotopy invariant under compact perturbations.
• The Z₂-index defined by the kernel dimension parity generalizes the Gohberg-Krein theorem to an odd symmetric setting.
• The class of real skew-adjoint Fredholm operators with Z₂-indices is a subset of infinite codimension within the set of odd symmetric Fredholm operators.
• An even Z₂-index theorem leads to a phase label for two-dimensional topological insulators with odd time-reversal symmetry.
• A nontrivial Z₂-index in such insulators enforces non-zero spin Chern numbers, establishing a topological constraint.
• The Z₂-index provides a robust invariant for classifying topological phases in systems with odd time-reversal symmetry.