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The $\mathbb{Z}_2$ -Index of a Pair of Pure States and the Topology of Interacting 1D Superconductors

This paper defines a relative $\mathbb{Z}_2$ -index for parity-invariant pure states on $C^*$ -algebras and applies it to one-dimensional self-dual CAR algebras to construct a many-body Majorana number that completely classifies symmetric automorphic-path components of interacting superconductors.

Original authors: Anna Mazhar, Jacob Shapiro

Published 2026-09-01

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CC BY 4.0

Original authors: Anna Mazhar, Jacob Shapiro

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of quantum materials, scientists have long been fascinated by a peculiar kind of order that does not rely on the arrangement of atoms, but rather on the invisible, tangled relationships between particles. Imagine a long chain of atoms where the electrons behave not as individuals, but as a single, collective wave. In certain conditions, specifically in one-dimensional superconductors, this collective state can become "topologically" distinct from its neighbors. This means the material possesses a hidden property that cannot be changed by simply stretching or twisting the chain; it can only be altered by a dramatic, global event that tears the chain apart. This distinction is crucial because the edges of such a chain often host exotic particles called Majorana modes, which are their own antiparticles and hold promise for building stable quantum computers. However, most theories describing these states assume the particles do not interact with one another, a simplification that breaks down in real, messy materials where electrons constantly bump into each other.

A team of researchers at Princeton University has now developed a new mathematical tool to describe these complex, interacting systems without relying on that simplifying assumption. They focused on the fundamental question of how to tell two different quantum states apart when the particles are interacting. In the simpler, non-interacting world, scientists have a reliable way to count a specific feature of the system, often called a Majorana number, which acts like a switch that is either on or off. The new work proves that this switch concept survives even when particles interact, provided the system has a specific kind of symmetry related to the balance between matter and antimatter. The researchers defined a precise way to compare two quantum states, showing that if you try to smoothly transform one state into another, you will hit an impassable barrier if their "switches" are set differently. This barrier is not a physical wall, but a topological obstruction that forces the system to jump between distinct categories.

The core of their discovery is a new index, a kind of topological fingerprint, that they can calculate for any pure quantum state in a one-dimensional superconductor. They demonstrated that this index is robust: it does not change if you wiggle the system slightly or if you apply a local transformation that respects the system's symmetry. More importantly, they proved that this index completely classifies the possible states of the system. If two states have the same index, the researchers showed that there exists a continuous, symmetry-preserving path connecting them, meaning they are fundamentally the same type of matter. If the indices differ, no such path exists. This result settles a long-standing question about whether the simple, non-interacting classification of these materials holds up in the complex, interacting reality. The authors proved that the familiar "Majorana number" used for simple chains is indeed the correct descriptor for the complex, interacting chains, as long as the states are sufficiently well-behaved locally.

However, the paper also reveals a subtle limitation in how we view these states. While the new index successfully groups states into two broad categories, it does not catch every possible difference between them. The researchers found that two states can have the same index and be connected by a valid path, yet still differ in a very specific, localized way at the very edge of the system. This difference is a "parity obstruction," a tiny mismatch in the number of particles at the boundary that the main index overlooks. It is a reminder that while the bulk of the material is well-understood, the edges can still hold secrets that require a more detailed look. The work also clarifies which mathematical tools are appropriate for studying these systems. The authors showed that using a very loose definition of "closeness" between states, which works well for some problems, fails here because it allows the system to slip through the topological barriers that should keep them apart. Instead, they proved that a stricter definition of closeness is necessary to preserve the integrity of the topological classification.

By establishing these rules, the paper provides a solid foundation for understanding the phases of matter in interacting one-dimensional superconductors. It confirms that the topological protection of Majorana modes is a real feature of interacting systems, not just a mathematical artifact of simplified models. The researchers achieved this by constructing a rigorous framework that treats the quantum states as objects that can be compared and transformed, much like comparing two shapes to see if one can be morphed into the other without tearing. Their findings suggest that the "on" and "off" switch of the Majorana number is a reliable guide for navigating the landscape of interacting quantum matter, offering a clear path forward for identifying and manipulating these exotic states in future experiments. The work stands as a proof that the deep, abstract mathematics of topology can be successfully applied to the messy, interacting reality of quantum materials, bridging the gap between idealized theory and physical possibility.

Technical Summary: The Z2-Index of a Pair of Pure States and the Topology of Interacting 1D Superconductors

Problem Statement
The paper addresses the classification of topological phases in one-dimensional interacting Fermionic systems, specifically class D superconductors. While the topological invariant for non-interacting systems (the Majorana number) is well-understood via single-body Fredholm indices or projection pairs, extending this to interacting many-body systems requires a state-theoretic formulation. Existing operator-algebraic approaches often rely on the "split property" (strong locality) or are restricted to quasi-free states. The authors aim to define a relative topological index for pairs of pure states that is robust under interactions, requires minimal locality assumptions, and provides a complete classification of the resulting phase space under symmetric local automorphisms.

Methodology
The work is grounded in the theory of unital $C^*$ -algebras and the self-dual Canonical Anticommutation Relations (CAR) algebra. The methodology proceeds in three stages:

Abstract Index Definition: The authors define a relative $\mathbb{Z}_2$ -index, $N(\omega_1, \omega_2)$ , for a pair of locally-comparable, parity-invariant pure states on a $C^*$ -algebra equipped with an involutive automorphism $\theta$ (representing Fermionic parity). Two states are "locally-comparable" if they are related by an inner automorphism (conjugation by a unitary $u$ ). The index is defined as $N(\omega_1, \omega_2) := \omega_1(u^*\theta(u)) \in \{\pm 1\}$ .
Application to 1D Systems: The framework is specialized to the one-dimensional self-dual CAR algebra, $A = \text{CAR}_{\text{sd}}(H, \Xi)$ , where $\Xi$ is particle-hole conjugation. A "locality automorphism" $\sigma$ is introduced, induced by the sign of the position operator (a Bogoliubov automorphism flipping the sign of operators on the left half-chain). A state $\omega$ is defined as " $\sigma$ -local" if $\omega$ and $\omega \circ \sigma$ are locally-comparable. The many-body Majorana number is then defined as $N_\sigma(\omega) := N(\omega, \omega \circ \sigma)$ .
Classification via Automorphic Paths: Recognizing that the norm topology on the state space is too fine (preserving superselection sectors) and the weak- $*$ topology is too coarse (washing out the index), the authors introduce "symmetric local automorphism paths." These are point-norm continuous paths of automorphisms $\alpha_t$ that commute with parity and satisfy a specific commutator condition with $\sigma$ (preserving the locality structure). The classification problem is reformulated as determining when two states are connected by such a path.

Key Contributions and Results

Well-Definedness and Properties of the Relative Index: Theorem 2.3 establishes that $N(\omega_1, \omega_2)$ is well-defined, multiplicative, invariant under parity-preserving automorphisms, and locally constant in the norm topology. Crucially, if $\|\omega_1 - \omega_2\| < 2$ , the index is $+1$ .
Correspondence with Single-Body Indices: Theorem 2.6 and Theorem 3.10 prove that for quasi-free states associated with basis projections $P, Q$ satisfying the Hilbert–Schmidt condition ( $P-Q \in J_2(H)$ ), the many-body index $N(\omega_P, \omega_Q)$ coincides with the standard mod-2 index of the pair of projections, $(-1)^{\dim \text{im } P \cap \ker Q}$ . Furthermore, for a $\sigma$ -local state $\omega_P$ derived from a single-body projection $P$ , $N_\sigma(\omega_P)$ equals the single-body Majorana number.
Topology of the State Space: Section 3.6 demonstrates that the index is not continuous in the weak- $*$ topology. Example 3.11 constructs a weak- $*$ continuous path of $\sigma$ -local states where the index jumps from $-1$ to $+1$ at the limit, illustrating that the weak- $*$ topology fails to detect the topological obstruction carried by transport to spatial infinity.
Complete Classification: The central result (Theorem 4.8 and Corollary 4.9) states that two pure, parity-invariant, $\sigma$ -local states $\omega_1$ and $\omega_2$ are "automorphic-path-connected" (via a symmetric local automorphism path) if and only if $N_\sigma(\omega_1) = N_\sigma(\omega_2)$ . This provides a complete classification of the automorphic-path-components of the state space.
Refinement of Locality: The paper introduces $\sigma$ -locality as a condition weaker than the Matsui split property. Appendix B constructs a pure state that is $\sigma$ -local but not split, demonstrating that the authors' framework captures a broader class of states than those satisfying the strict split property.
SRE and LGAs: Appendix C shows that Short-Range Entangled (SRE) states generated by Locally Generated Automorphisms (LGAs) with symmetric generators have a trivial index ( $N_\sigma = +1$ ). It further proves that Kitaev chains with indices differing by 2 are connected by such automorphisms, consistent with the $\mathbb{Z}_2$ nature of the invariant.

Significance and Claims
The paper claims to provide a rigorous operator-algebraic foundation for the many-body Majorana number in interacting systems. Its primary significance lies in:

Minimality: The definition of $\sigma$ -locality is the weakest known condition sufficient to define the index, extending beyond the split property.
Completeness: The index $N_\sigma$ completely classifies the phases of 1D class D superconductors under symmetric local automorphisms, distinguishing the two topological sectors.
Distinction of Obstructions: The work clarifies the relationship between the bulk obstruction (captured by $N_\sigma$ ) and relative zero-dimensional parity obstructions. It notes that while $N_\sigma$ classifies the bulk phase, two states with the same $N_\sigma$ may still have a non-trivial relative index $N(\omega_1, \omega_2) = -1$ , indicating a local parity obstruction that is "forgotten" by the bulk classification but detectable by the relative pair index.
Topological Robustness: By utilizing the norm topology and automorphic paths, the authors ensure the index is stable against perturbations that preserve the superselection sector and locality structure, avoiding the pathologies of the weak- $*$ topology.

The authors explicitly state that their approach is complementary to existing literature, focusing on the minimal requirements necessary to make the indices well-defined and to exhibit the topological structure of the problem, rather than relying on specific physical models or quasi-free assumptions.

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