Z₂ Parity Transition and Topological Index

From Integer Spectral Flow to Two-Valued Topological Invariant

Introduction

In previous two chapters, we proved existence of π-steps: Each time delay parameter crosses quantization step, scattering phase transitions $\pm \pi$. But if system continuously crosses multiple steps, how does phase accumulate? Does there exist topological quantity more “fundamental” than total phase?

This chapter introduces Z₂ topological index $\nu (\tau )\in \{0,1\}$, which only records parity of step transitions, ignoring specific count and direction. We will see this seemingly simplified index actually reveals deepest topological essence of self-referential structure.

From Integer Spectral Flow to Z₂ Reduction

Definition of Spectral Flow Counting

Recall: When delay $\tau$ increases from some reference value ${\tau }_0$ to $\tau$, may cross multiple quantization steps $\{{\tau }_k\}$.

Define spectral flow count:

$$N(\tau )=\sum _{k:{\tau }_0<{\tau }_k<\tau }\Delta n_k$$

where $\Delta n_k=\Delta {\phi }_k/\pi =\pm 1$ is transition direction of $k$-th step.

This is an integer topological invariant, recording “net number of steps”:

• Forward transition ($+\pi$) contributes $+1$
• Backward transition ($-\pi$) contributes $-1$

Motivation for Z₂ Reduction

Although $N(\tau )$ contains complete topological information, in many physical situations, only its parity is observable or essential.

Definition (Z₂ Topological Index)

$$\nu (\tau )=N(\tau )\mathrm{mod}2\in \{0,1\}$$

That is:

• $\nu =0$: Experienced even number of steps (including 0)
• $\nu =1$: Experienced odd number of steps

This is a Z₂ invariant, indicating system is in one of two “topological sectors”.

Why is Parity More Fundamental?

From mathematical perspective:

• Integer group $\mathbb{Z}$ is infinite, needs to “remember” all history
• Z₂ group $\{0,1\}$ has only two elements, only needs to “remember” parity

From physical perspective:

• Phase definitions of many systems only modulo $2\pi$ (e.g., quantum mechanical wave functions)
• But “sign” or “parity” of certain systems is physically observable (e.g., fermion exchange)

Core Insight:

Z₂ parity is finest topological invariant that “can still be well-defined in phase space modulo $2\pi$”.

Z₂ Flipping Rule and Evolution Equation

Transition Rule

Each time $\tau$ crosses a step ${\tau }_k$, topological index flips:

$$\nu ({\tau }_k+0^{+})=\nu ({\tau }_k-0^{-})\oplus 1$$

Here $\oplus$ is modulo 2 addition (XOR operation):

• $0\oplus 1=1$
• $1\oplus 1=0$

Represented diagrammatically:

graph LR
    A["ν=0<br/>Sector A"] -->|"Cross τ₁"| B["ν=1<br/>Sector B"]
    B -->|"Cross τ₂"| C["ν=0<br/>Sector A"]
    C -->|"Cross τ₃"| D["ν=1<br/>Sector B"]
    D -->|"Cross τ₄"| E["ν=0<br/>Sector A"]
    style A fill:#e1f5ff
    style B fill:#ffe1f5
    style C fill:#e1f5ff
    style D fill:#ffe1f5
    style E fill:#e1f5ff

System jumps back and forth between two topological sectors, like a topological bistable system.

Discrete Evolution Equation

If step positions are equally spaced: ${\tau }_k={\tau }_0+k\Delta \tau$, then $\nu (\tau )$ as function of $\tau$ presents periodic square wave:

$$\nu (\tau )=\{\begin{array}{ll}0,& \tau \in [2n\Delta \tau ,(2n+1)\Delta \tau )\\ 1,& \tau \in [(2n+1)\Delta \tau ,(2n+2)\Delta \tau )\end{array}$$

This is a Z₂-valued step function, period $2\Delta \tau$.

Analogy with Modulated Clock

Imagine a special “topological clock” with only two marks: 0 and 1. Each tick, pointer flips from 0 to 1, or from 1 to 0.

This clock doesn’t record “how many ticks total” (that requires infinite precision), only records “current is odd or even number of ticks”.

Self-referential scattering network’s $\nu (\tau )$ is exactly such a pointer of topological clock.

Double Cover Space and Lifted Paths

Base Space and Covering Space

In topology, covering space is construction that “splits” each point of “base space” into multiple points.

For Z₂ case, most important is double cover:

graph TB
    subgraph Covering Space M-
        A1["τ₀, ν=0"] --> B1["τ₁, ν=1"]
        B1 --> C1["τ₂, ν=0"]
        C1 --> D1["τ₃, ν=1"]
    end
    subgraph Base Space M
        A["τ₀"] --> B["τ₁"]
        B --> C["τ₂"]
        C --> D["τ₃"]
    end
    A1 -.Projection.-> A
    B1 -.Projection.-> B
    C1 -.Projection.-> C
    D1 -.Projection.-> D

Base Space $M$: One-dimensional axis of delay parameter $\tau$ Covering Space $\tilde{M}$: Each $\tau$ splits into two points $(\tau ,0)$ and $(\tau ,1)$, labeling different topological sectors

Lifted Paths and Holonomy

In base space $M$, path from ${\tau }_a$ to ${\tau }_b$ can be lifted to covering space $\tilde{M}$:

• If path crosses even number of steps, lifted path’s start and end in same sector (closed)
• If path crosses odd number of steps, lifted path’s start and end in different sectors (flipped)

This property of “whether lifted path closes” is called holonomy.

For Z₂ double cover, holonomy only has two values:

• $\mathrm{hol}=+1$: Lifted path closes
• $\mathrm{hol}=-1$: Lifted path flips

And:

$$\mathrm{hol}({\tau }_a\to {\tau }_b)=(-1{)}^{\nu ({\tau }_b)-\nu ({\tau }_a)}=(-1{)}^{\Delta \nu }$$

Principal Bundle Perspective

More abstractly, covering space can be understood as a Z₂ principal bundle:

$$\pi :\tilde{M}\to M$$

Fiber is Z₂ group $\{+1,-1\}$.

Along closed path in base space, “rotation” (actually flip) of fiber characterized by holonomy.

Self-referential scattering network’s $\nu (\tau )$ is exactly index of parallel transport of this Z₂ principal bundle.

Self-Reference Degree: Intrinsic Topological Label of Loops

Classification of Closed Loops

In parameter space, consider closed path $\gamma :[0,T]\to M$, satisfying $\gamma (0)=\gamma (T)={\tau }_0$.

Such loops can be classified in two ways:

Integer Classification: Net number of steps loop crosses $N(\gamma )\in \mathbb{Z}$

Z₂ Classification: Parity of loop $\sigma (\gamma )=N(\gamma )\mathrm{mod}2\in \{0,1\}$

Definition of Self-Reference Degree

For self-referential scattering network, define self-reference degree:

$$\sigma (\gamma )=\nu ({\tau }_{\mathrm{final}})-\nu ({\tau }_{\mathrm{initial}})\mathrm{mod}2$$

For closed loops (${\tau }_{\mathrm{final}}={\tau }_{\mathrm{initial}}$), this is equivalent to:

$$\sigma (\gamma )=\text{(number of steps inside loop)}\mathrm{mod}2$$

Physical Meaning:

• $\sigma =0$: System “completely returns to itself” (even number of flips, net effect identity)
• $\sigma =1$: System “returns to itself in flipped way” (odd number of flips, net effect sign flip)

This has deep connection with fermion double-valuedness (detailed in next chapter).

Homotopy Classes of Loops

In topology, two loops that can be continuously deformed into each other are called homotopy equivalent.

For parameter space $M$, homotopy classes of all closed loops based at point ${\tau }_0$ constitute fundamental group ${\pi }_1(M,{\tau }_0)$.

In simple cases (like $M=\mathbb{R}$ or $M=S^1$), fundamental group is trivial or cyclic. But self-reference degree $\sigma$ assigns these loops an additional Z₂ label:

$$[\gamma ]\in {\pi }_1(M)\Rightarrow ([\gamma ],\sigma (\gamma ))\in {\pi }_1(M)\times {\mathbb{Z}}_2$$

This is an enhanced topological invariant, completely characterizing topological type of self-referential loops.

Connection with Levinson Theorem

Classical Levinson Theorem

In quantum scattering theory, Levinson theorem establishes relationship between scattering phase shift and number of bound states:

$$\delta (E=\infty )-\delta (E=0)=\pi \cdot N_{\text{bound}}$$

where $\delta (E)$ is phase shift at energy $E$, $N_{\text{bound}}$ is number of bound states supported by potential.

Delay-Driven “Topological Levinson Theorem”

In self-referential scattering network, delay $\tau$ plays role of “parameterized energy”. We can establish similar relationship:

$$\phi (\tau =\infty )-\phi (\tau =0)=\pi \cdot N_{\text{crossings}}$$

where $N_{\text{crossings}}$ is total number of times poles cross real axis (signed).

More precisely, for periodic steps $\{{\tau }_k\}$:

$$N(\tau )=\frac{1}{\pi }[\phi (\tau )-\phi ({\tau }_0)]$$

This is topological pairing of spectral flow and phase.

Z₂ Version: Parity Levinson Theorem

Taking modulo 2:

$$\nu (\tau )=N(\tau )\mathrm{mod}2=\frac{1}{\pi }[\phi (\tau )-\phi ({\tau }_0)]\mathrm{mod}2$$

This can be rewritten as:

$$\nu (\tau )=\{\begin{array}{ll}0,& \phi (\tau )-\phi ({\tau }_0)\in 2\pi \mathbb{Z}\\ 1,& \phi (\tau )-\phi ({\tau }_0)\in \pi +2\pi \mathbb{Z}\end{array}$$

That is: Topological index only depends on whether phase difference is odd multiple of $\pi$.

This is Z₂ reduction version of Levinson theorem, applicable to double cover structure of self-referential network.

Experimental Measurement and Index Reconstruction

Measurement Protocol

To experimentally determine Z₂ index $\nu (\tau )$, can adopt following protocol:

Step 1: Phase Scan

• Fix frequency $\omega ={\omega }_{\ast }$
• Scan delay parameter $\tau \in [{\tau }_{\min },{\tau }_{\max }]$
• Measure scattering phase $\phi ({\omega }_{\ast };\tau )$

Step 2: Step Identification

• Unwrap phase data, remove $2\pi$ periodicity
• Identify jumps of magnitude $\pm \pi$ on $\phi (\tau )$ curve
• Record step positions $\{{\tau }_k\}$ and jump directions $\{\Delta n_k\}$

Step 3: Index Calculation

• Start from initial value $\nu ({\tau }_{\min })=0$
• Accumulate step by step: $\nu ({\tau }_{k+1})=\nu ({\tau }_k)\oplus 1$
• Obtain complete $\nu (\tau )$ curve

Step 4: Robustness Check

• Since $\nu \in \{0,1\}$ only has two values, naturally robust to noise
• Even if individual step positions have errors, as long as correctly judge “parity”, index is correct
• Can use “majority voting” from multiple measurements

Frequency Window Integral Method

Another more robust method uses scale identity:

$$I(\tau )={\int }_{{\omega }_1}^{{\omega }_2}\kappa (\omega ;\tau )d\omega =\frac{1}{\pi }[\phi ({\omega }_2;\tau )-\phi ({\omega }_1;\tau )]$$

When $\tau$ crosses step, $I(\tau )$ jumps $\pm 1$.

Define cumulative index:

$$\nu (\tau )=\left[\sum _k\Delta I_k\right]\mathrm{mod}2$$

This avoids precisely locating step positions, only need to judge parity transitions of frequency window integral.

Group Delay Fingerprint

Can also use group delay double-peak merger as “soft index”:

• In Sector A ($\nu =0$): Group delay single peak, smaller peak value
• Approaching step: Double peaks appear and converge
• Crossing step into Sector B ($\nu =1$): Peak flips or disappears
• Next step: Repeat above process, return to Sector A

By identifying periodicity of double-peak merger, can reconstruct $\nu (\tau )$ without directly measuring phase.

Relationship with Fundamental Group

Topologization of Configuration Space

From perspective of computational universe, self-referential scattering network corresponds to closed loops in configuration graph.

Topologize configuration graph as two-dimensional complex $\mathcal{X}$, homotopy classes of closed loops constitute fundamental group ${\pi }_1(\mathcal{X})$.

Each loop $\gamma \in {\pi }_1(\mathcal{X})$ corresponds to a delay evolution path, its self-reference degree $\sigma (\gamma )$ is a topological invariant.

Homotopy Equivalence and Index Invariance

Lemma: If two loops ${\gamma }_1$ and ${\gamma }_2$ are homotopy equivalent (can be continuously deformed into each other), then their self-reference degrees are same:

$$[{\gamma }_1]=[{\gamma }_2]\Rightarrow \sigma ({\gamma }_1)=\sigma ({\gamma }_2)$$

Proof: Homotopy deformation corresponds to continuous change of paths in parameter space. As long as deformation process doesn’t cross new steps, self-reference degree remains unchanged. And in connected component of parameter space, step set is discrete, continuous deformation won’t “accidentally cross” steps.

Therefore, $\sigma :{\pi }_1(\mathcal{X})\to {\mathbb{Z}}_2$ is a group homomorphism.

Z₂ Cohomology and Chern Class

From more advanced topological perspective, $\sigma$ defines a Z₂ cohomology class on parameter space $M$:

$$[\sigma ]\in H^1(M;{\mathbb{Z}}_2)$$

This is a one-dimensional Z₂ Chern class, characterizing non-triviality of double cover bundle.

For simple parameter spaces (like $M=S^1$), have $H^1(S^1;{\mathbb{Z}}_2)={\mathbb{Z}}_2$, exactly corresponding to two possible double covers:

• Trivial cover: $S^1\times \{0,1\}$ (two disconnected circles)
• Non-trivial cover: Möbius strip (topologically non-orientable)

Z₂ index of self-referential scattering network is exactly judging “which cover does system live on”.

Physical Example: Topological Sectors of Optical Microring

Experimental Setup

Consider an integrated photonic microring resonator, changing effective round-trip delay $\tau$ via thermo-optic modulation.

graph LR
    A["Input Waveguide"] --> B["Coupler κ"]
    B --> C["Through Port"]
    B --> D["Microring"]
    D --> E["Thermo-Optic Phase Modulator"]
    E --> F["Tunable Delay τ"]
    F --> D
    style D fill:#e1f5ff
    style F fill:#ffe1f5

Parameters:

• Coupling coefficient $\kappa =0.3$
• Loop length $L=100\mu \mathrm{m}$
• Measurement wavelength $\lambda =1550\mathrm{nm}$

Step Spacing Prediction

Free Spectral Range:

$$\mathrm{FSR}=\frac{c}{n_{\text{eff}}L}\approx 600\mathrm{GHz}$$

Corresponding delay quantization spacing:

$$\Delta \tau =\frac{2\pi }{\omega }=\frac{\lambda }{c}\approx 5.2\mathrm{fs}$$

By changing voltage of phase modulator, can precisely scan $\tau$ and observe π-steps.

Observation of Topological Sectors

Sector A ($\nu =0$):

• Transmission spectrum shows single resonance peak
• Phase slowly changes
• Moderate group delay

Approaching Step:

• Transmission spectrum double peaks appear
• Phase steeply rises
• Group delay double peaks converge

Sector B ($\nu =1$):

• Transmission spectrum peak shifts
• Phase transition $\pi$
• Group delay single peak recovers (but different peak value)

By real-time monitoring transmission spectrum and phase, can clearly identify which topological sector system currently in.

Chapter Summary

Core Concepts

Z₂ Topological Index: $$\nu (\tau )=N(\tau )\mathrm{mod}2\in \{0,1\}$$ Only records parity of step transitions, is more “fundamental” topological invariant than integer spectral flow.

Flipping Rule: $$\nu ({\tau }_k+0)=\nu ({\tau }_k-0)\oplus 1$$ Each crossing of step, index flips between 0 and 1.

Double Cover Structure: Parameter space $M$ lifted to double cover $\tilde{M}$, each point splits into two topological sectors.

Self-Reference Degree: Z₂ label $\sigma (\gamma )$ of closed loops, characterizing “in what way loop returns to itself”.

Parity Levinson Theorem: $$\nu (\tau )=\frac{1}{\pi }[\phi (\tau )-{\phi }_0]\mathrm{mod}2$$ Topological index determined by parity of phase difference.

Physical Picture

Z₂ topological index is “intrinsic parity” of self-referential system: It doesn’t record “how many steps taken”, only records “odd or even number of steps”. This extremely simple binary classification precisely reveals deepest topological structure.

Why is Z₂ So Important?

1. Mathematically: Z₂ is smallest non-trivial group, is “atomic unit” of topological classification
2. Physically: Many fundamental phenomena (fermion exchange, time reversal, parity) are Z₂ symmetries
3. Experimentally: Binary index robust to noise, easy to measure and verify
4. Philosophically: Z₂ characterizes “two ways of identity”: Completely return to self vs return to self in flipped way

This is exactly theme of next chapter: Fermions as “self-referential fingerprint of universe”!

Thought Questions

1. Group Structure: How many subgroups does Z₂ group have? How many quotient groups? What is relationship with “irreducibility” of topological index?

2. Covering Degree: Do there exist self-referential networks with “Z₃ cover” or “Z_n cover”? What physical phenomena do they correspond to?

3. Chern Number Generalization: On two-dimensional parameter space $({\tau }_1,{\tau }_2)$, does Z₂ index generalize to some kind of “two-dimensional Chern number”?

4. Experimental Design: If your phase measurement only has $\pi /10$ precision, can you still reliably reconstruct $\nu (\tau )$? (Hint: Use statistics of multiple measurements)

5. Quantum Entanglement Analogy: Are “two sectors” in double cover space similar to qubit’s $|0⟩$ and $|1⟩$? Is flipping of topological index some kind of “topological quantum gate”?

Preview of Next Chapter

Z₂ topological index is not only mathematically elegant structure, but also has deep connection with fundamental particles of physical world:

Self-Referential Explanation of Fermion Origin

We will:

• Establish precise correspondence between Z₂ of self-referential scattering network ↔ fermion exchange statistics
• Explain why “sign flip after $2\pi$ rotation” and “π-step transition” are essentially isomorphic
• Explore relationship between spin double cover $\mathrm{Spin}(n)\to \mathrm{SO}(n)$ and Null-Modular double cover
• Propose bold hypothesis: Fermions may be topologically inevitable product of universe as self-referential system

Let us enter this profound exploration connecting topology, statistics, and fundamental structure of universe!