ℤ₂ Holonomy: Observable Sign of Topological Time Anomaly

In previous section, we defined relative cohomology class $[K]$ and gave its three-term decomposition. Now we turn focus to most direct physical criterion for $[K]=0$—ℤ₂ holonomy.

ℤ₂ holonomy is a simple ±1-valued quantity, but it encodes profound topological information: mod 2 jump of quantum phase along loops.

Precise Definition of ℤ₂ Holonomy

Square Root Problem of Scattering Determinant

Given scattering matrix $S(\omega )$ (unitary), its determinant: $$\det S(\omega )\in U(1)\subset \mathbb{C}$$

Can be written in phase form: $$\det S(\omega )=e^{2i\varphi (\omega )}$$

Problem: How to define $\sqrt{\det S(\omega )}$?

Naive choice: $$\sqrt{\det S(\omega )}=e^{i\varphi (\omega )}$$

But this has a problem: $\varphi (\omega )$ itself is not single-valued! It has integer $n$ freedom under $\varphi \to \varphi +2\pi n$.

Branch Selection of Square Root

Mathematically, $\sqrt{\det S}$ is a two-valued function:

• If one branch $e^{i\varphi (\omega )}$ is chosen
• Other branch is $e^{i(\varphi (\omega )+\pi )}=-e^{i\varphi (\omega )}$

Along closed path $\gamma$ in parameter space, when continuously choosing square root, branch switching may occur: $$\sqrt{\det S}{|}_{\gamma \text{end}}=\pm \sqrt{\det S}{|}_{\gamma \text{start}}$$

Define ℤ₂ holonomy: $${\nu }_{\sqrt{\det S}}(\gamma ):=\frac{\sqrt{\det S}{|}_{\gamma \text{end}}}{\sqrt{\det S}{|}_{\gamma \text{start}}}=\pm 1$$

graph LR
    A["Start<br/>√det S = e^(iφ₀)"] -->|Continuous Evolution Along γ| B["End<br/>√det S = e^(iφ₁)"]

    B -.->|ν=+1| C["Same Branch<br/>φ₁=φ₀"]
    B -.->|ν=-1| D["Flipped Branch<br/>φ₁=φ₀+π"]

    style C fill:#6bcf7f
    style D fill:#ff6b6b

Relationship with Winding Number

ℤ₂ holonomy ${\nu }_{\sqrt{\det S}}(\gamma )$ directly relates to winding number of determinant:

Theorem: Let $\mathrm{deg}(\det S{|}_{\gamma })=\frac{1}{2\pi }{\oint }_{\gamma }d(\arg \det S)$ be winding number (integer), then: $${\nu }_{\sqrt{\det S}}(\gamma )=(-1{)}^{\mathrm{deg}(\det S{|}_{\gamma })}$$

Proof: Along one cycle of $\gamma$, phase change: $$\Delta \varphi =\varphi (\text{end})-\varphi (\text{start})=\frac{1}{2}\Delta (2\varphi )=\frac{1}{2}\cdot 2\pi \mathrm{deg}(\det S{|}_{\gamma })=\pi \mathrm{deg}(\det S{|}_{\gamma })$$

Therefore: $$e^{i\Delta \varphi }=e^{i\pi \mathrm{deg}(\det S{|}_{\gamma })}=(-1{)}^{\mathrm{deg}(\det S{|}_{\gamma })}$$

This is exactly definition of ${\nu }_{\sqrt{\det S}}(\gamma )$.

Physical Meaning:

• Even winding: $\mathrm{deg}=2k$ → $\nu =+1$ → square root single-valued
• Odd winding: $\mathrm{deg}=2k+1$ → $\nu =-1$ → square root flips branch

Small Semicircle/Return Rule

Handling Discriminant Set: Non-Crossable Singularities

In parameter space $X^{\circ }$, discriminant set $D$ (e.g., degeneracy points ${\Sigma }_{3|2}$) is part we removed. Therefore, physically allowed paths cannot cross $D$.

But in defining holonomy, we need loops. If loops must bypass discriminant points, how to canonically choose paths?

Small Semicircle Rule

Rule: When closed path $\gamma$ approaches discriminant point $p\in D$, locally adopt small semicircle detour:

• Take small disk $B_{\epsilon }(p)$ in normal direction near $p$
• Path goes around half circle of $\partial B_{\epsilon }(p)$
• Direction determined by right-hand rule

graph LR
    A["Path Approaches<br/>Discriminant Point p"] --> B["Small Disk<br/>B_ε(p)"]

    B --> C["Small Semicircle Detour<br/>∂B_ε⁺"]

    C --> D["Leave Discriminant Point"]

    style B fill:#ff6b6b
    style C fill:#ffd93d

Mathematical Formulation: Let $\gamma :[0,1]\to X^{\circ }$ be closed path, $\gamma (t)$ approaches $D$ at $t=t_0$. Then in $(t_0-\delta ,t_0+\delta )$: $$\gamma (t)=p+\epsilon e^{i\theta (t)}n(p)$$

where $n(p)$ is normal vector of $D$, $\theta (t)$ changes from 0 to π (semicircle).

Return Rule

For open paths (start and end different), if need to reverse at discriminant point:

Rule: Return without crossing at discriminant point:

• Path reaches boundary of discriminant set
• Slide along boundary for small segment
• Return along original path

graph LR
    A["Path Reaches<br/>∂D"] --> B["Slide Along Boundary"]
    B --> C["Return"]
    C --> D["Return Along Original Path"]

    style A fill:#ffd93d
    style C fill:#ff6b6b

Physical Meaning of Rules

Small semicircle/return rule ensures:

1. Continuity: Path is continuous in $X^{\circ }$
2. Differentiability: Avoids non-differentiable behavior at discriminant points
3. Topological Stability: Small perturbations don’t change holonomy value

Theorem (Robustness of Holonomy): Under small semicircle/return rule, ℤ₂ holonomy ${\nu }_{\sqrt{\det S}}(\gamma )$ remains unchanged under small perturbations of path (not crossing $D$).

Topological Time Anomaly: Physical Consequences of π Phase Jump

What Is Topological Time Anomaly?

In unified time scale framework, time is defined by scattering phase derivative $\kappa (\omega )={\phi }^{\prime }(\omega )/\pi$. If scattering phase acquires π jump along some loop, it causes:

Sign Flip of Time: $$\Delta \tau ={\int }_{\gamma }\kappa (\omega )d\omega ={\int }_{\gamma }\frac{{\phi }^{\prime }(\omega )}{\pi }d\omega =\frac{1}{\pi }[\phi {]}_{\gamma }$$

If $[\phi {]}_{\gamma }=\pi$ (mod $2\pi$), then: $$\Delta \tau =1\text{(time unit)}$$

But if we take equivalent path ${\gamma }^{\prime }$ (homotopic to $\gamma$ but around other side of discriminant point), we might get: $$[\phi {]}_{{\gamma }^{\prime }}=\pi +\pi =2\pi \Rightarrow \Delta {\tau }^{\prime }=2$$

Contradiction! Time reading depends on path choice.

Topological Anomaly of Time Arrow

Deeper problem: If ${\nu }_{\sqrt{\det S}}(\gamma )=-1$, it means along $\gamma$, quantum weights of “future” and “past” undergo sign flip.

In quantum mechanics, time evolution operator: $$U(t)=e^{-iHt}$$

If square root of scattering determinant flips: $$\sqrt{\det S}\to -\sqrt{\det S}$$

then effective Hamiltonian acquires shift of $\pi /T$ ($T$ is period): $$H_{\text{eff}}\to H_{\text{eff}}+\pi /T$$

This is topological origin of time crystal phenomenon!

graph TD
    A["ν=-1 Holonomy Anomaly"] --> B["Scattering Phase π Jump"]
    B --> C["Time Reading Ambiguity"]
    B --> D["Effective Hamiltonian Shift"]

    C --> E["Topological Time Anomaly"]
    D --> F["Time Crystal Order"]

    style A fill:#ff6b6b
    style E fill:#d4a5a5
    style F fill:#ffd93d

[K]=0 Eliminates Anomaly

Requiring $[K]=0$ is equivalent to: $$\forall \gamma \in {\mathcal{C}}_{\text{adm}}:{\nu }_{\sqrt{\det S}}(\gamma )=+1$$

This ensures:

• Time reading single-valued: Independent of path choice
• Causal consistency: No topological flip of future-past direction
• Thermodynamic arrow: Entropy increase direction globally consistent

Calculation Examples

Example 1: One-Dimensional δ Potential Scattering

Consider one-dimensional δ potential: $$V(x)=g\delta (x)$$

Scattering matrix (reflection amplitude $r$): $$r(\omega )=\frac{-ig}{2\omega -ig}$$

When $\omega$ winds once around complex plane pole ${\omega }_0=ig/2$:

Calculate Winding Number: Pole is in upper half plane, take counterclockwise small circle $\gamma$ around ${\omega }_0$.

$$\mathrm{deg}(r{|}_{\gamma })=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{dr}{r}=\frac{1}{2\pi i}{\oint }_{\gamma }d(\log r)$$

By residue theorem: $$\log r(\omega )=\log (-ig)+\log \frac{1}{2\omega -ig}$$

Winding once around ${\omega }_0$, logarithm acquires $2\pi i$: $$\mathrm{deg}(r{|}_{\gamma })=1\text{(odd)}$$

ℤ₂ Holonomy: $${\nu }_{\sqrt{r}}(\gamma )=(-1{)}^1=-1$$

Physical Meaning: Winding around bound state pole in complex plane, scattering square root flips branch! This $-1$ is exactly sign of topological time anomaly.

Example 2: ℤ₂ Version of Aharonov-Bohm Effect

Consider magnetic flux in two-dimensional plane (origin removed): $$A_{\theta }=\frac{\alpha }{r},\alpha \in [0,1)$$

Circular path $\gamma$ around origin, scattering phase: $$\Delta \varphi ={\oint }_{\gamma }{A}_{\theta }d\theta =2\pi \alpha$$

Half-Flux Point $\alpha =1/2$: $$\Delta \varphi =\pi \Rightarrow \mathrm{deg}(\det S{|}_{\gamma })=1$$

Therefore: $${\nu }_{\sqrt{\det S}}(\gamma )=-1\text{when}\alpha =1/2$$

Physical Phenomenon: When magnetic flux is exactly half quantum, quantum interference pattern reverses! This can be observed in Aharonov-Bohm experiments.

graph LR
    A["Magnetic Flux α"] -->|α<1/2| B["ν=+1<br/>Normal Interference"]
    A -->|α=1/2| C["Discriminant Point"]
    A -->|α>1/2| D["ν=-1?<br/>Interference Reversed"]

    C --> E["Topological Phase Transition"]

    style C fill:#ff6b6b
    style E fill:#ffd93d

Example 3: ℤ₂ Index of Topological Superconductor Endpoint

At one-dimensional endpoint of topological superconductor (Class D), reflection matrix $r(\omega )$: $$r(\omega )=\frac{\omega -i\Delta }{\omega +i\Delta }$$

where $\Delta$ is pairing gap.

Zero Energy Point $\omega =0$: $$r(0)=\frac{-i\Delta }{i\Delta }=-1$$

Phase jump π!

Holonomy Around Zero Point: Take small circle $\gamma$ around $\omega =0$: $$\mathrm{deg}(r{|}_{\gamma })=1$$

$${\nu }_{\sqrt{r}}(\gamma )=-1$$

Topological Interpretation: This $-1$ is exactly topological invariant ℤ₂ index of Majorana zero mode!

Dictionary Between ℤ₂ Holonomy and Relative Cohomology

From Holonomy to Cohomology Class

Given family of closed loops $\{{\gamma }_{\alpha }\}$ generating $H_1(X^{\circ },\partial X^{\circ };{\mathbb{Z}}_2)$ (relative homology), define map: $$\Psi :H_1(X^{\circ },\partial X^{\circ };{\mathbb{Z}}_2)\to {\mathbb{Z}}_2$$ $$\Psi ([\gamma ])=\{\begin{array}{ll}0& \text{if}{\nu }_{\sqrt{\det S}}(\gamma )=+1\\ 1& \text{if}{\nu }_{\sqrt{\det S}}(\gamma )=-1\end{array}$$

Theorem (Poincaré-Lefschetz Duality): $\Psi$ is linear functional, corresponds to element of $H^1(X^{\circ },\partial X^{\circ };{\mathbb{Z}}_2)$.

Through boundary map $\partial :H^1(\partial X^{\circ })\to H^2(X^{\circ },\partial X^{\circ })$, this 1-cohomology class lifts to 2-cohomology class, exactly part of $[K]$!

Holonomy Criterion for [K]=0

Theorem (Holonomy Criterion): $$[K]=0⟺\forall \gamma \in H_1(X^{\circ },\partial X^{\circ };{\mathbb{Z}}_2):\Psi ([\gamma ])=0$$

That is: $$[K]=0⟺\text{On all allowed loops}{\nu }_{\sqrt{\det S}}(\gamma )=+1$$

graph TD
    A["All Loops<br/>ν=+1"] <==> B["Holonomy Functional Ψ=0"]
    B <==> C["Relative Cohomology<br/>[K]=0"]

    C --> D["No Topological Time Anomaly"]
    C --> E["Scattering Winding Even"]
    C --> F["Quantum Phase Single-Valued"]

    style A fill:#6bcf7f
    style C fill:#ffd93d

Experimental Detection Schemes for ℤ₂ Holonomy

Scheme 1: Purification Interference Loop

Setup: Quantum state $\rho (\lambda )$ evolves with parameter $\lambda$, forming closed loop $\gamma$.

Steps:

1. Prepare initial purification $|\psi ({\lambda }_0)⟩$
2. Adiabatic evolution along $\gamma$
3. Measure Berry phase: ${\varphi }_B=i{\oint }_{\gamma }⟨\psi |d\psi ⟩$
4. If $e^{i{\varphi }_B}=-1$, then $\nu =-1$

Precision Requirements:

• Phase noise $\delta \varphi \lesssim 0.25$ rad
• Number of shots $N_{\text{shots}}\gtrsim 30$

Scheme 2: Time Crystal Order Parameter

Setup: Periodically driven system, Floquet operator $F=e^{-i{H}_{\text{eff}}T}$.

Steps:

1. Modulate driving parameters along closed loop $\gamma$
2. Measure subharmonic response $2T$ peak intensity $I_{2T}$
3. Compare different paths: If $I_{2T}(\gamma )\ne I_{2T}({\gamma }^{\prime })$, suggests $\nu$ flip

Criteria:

• ℤ₂ time crystal: $I_{2T}/I_T>0.1$
• Holonomy flip: $\Delta I_{2T}/I_{2T}\gtrsim 0.5$

Scheme 3: Phase Readout of Topological Qubit

Setup: Majorana zero mode or topological superconductor qubit.

Steps:

1. Encode quantum information into topological subspace
2. Adiabatic transport along $\gamma$
3. Readout quantum state: If flipped, then $\nu =-1$

Advantages:

• Topologically protected, low decoherence
• Direct readout of ℤ₂ index

Summary: Triple Role of ℤ₂ Holonomy

ℤ₂ holonomy ${\nu }_{\sqrt{\det S}}(\gamma )$ plays triple role in topological constraints:

Core Insight:

ℤ₂ holonomy translates abstract topological constraint $[K]$ into observable physical quantity—±1 sign of path phase.

Requiring $\nu =+1$ on all physical loops is exactly experimental criterion for $[K]=0$.

Next Step: Standard Model Group Structure S(U(3)×U(2))

Now we understand mathematical language of topological constraints (relative cohomology $[K]$) and physical criterion (ℤ₂ holonomy $\nu$).

Next section will show most stunning application of topological constraints: Direct derivation of gauge group of Standard Model from punctured structure of 5D density matrix manifold!

$$S(U(3)\times U(2))\cong \frac{SU(3)\times SU(2)\times U(1)}{{\mathbb{Z}}_6}$$

This is not coincidence, but result of topological necessity.