Chapter 1: Floquet Quantum Cellular Automaton Theory

Source Theory: euler-gls-info/17-time-crystals-null-modular-z2-holonomy.md, §2-3

Introduction

In the previous overview, we preliminarily understood time crystals—a strange quantum phase with “period doubling” in the time dimension. This chapter will delve into the microscopic mechanism of time crystals: how to rigorously describe them using Floquet Quantum Cellular Automata (Floquet-QCA).

Core Questions of This Chapter:

1. What is a Quantum Cellular Automaton (QCA)?
2. How to introduce periodic driving (Floquet evolution) in QCA?
3. What is the mathematical definition of time crystals?
4. What is the microscopic mechanism of period doubling?

Everyday Analogy:

• QCA: Quantum chess on a board—each cell has quantum state, updated by rules
• Floquet Drive: Timer rings once per second, each ring triggers one step according to rules
• Time Crystal: Game returns to similar configuration every two steps, not every step

1. Computational Universe and QCA Foundations

1.1 Computational Universe Axioms

Review Computational Universe Axiom System (Source Theory §2.1):

$$U_{\mathrm{comp}}=(X,\mathsf{T},\mathsf{C},\mathsf{I})$$

Four-Tuple Meaning:

Reversibility Assumption: $\mathsf{T}$ is reversible, i.e., there exists inverse mapping ${\mathsf{T}}^{-1}$, guaranteeing no information loss.

Mermaid Computational Universe Structure

graph TD
    A["Configuration Space X"] --> B["State x_1"]
    A --> C["State x_2"]
    A --> D["State x_3"]

    B -->|"T, C(x1,x2)"| C
    C -->|"T, C(x2,x3)"| D
    D -->|"T^-1, C(x3,x2)"| C

    E["Information Function I"] -.->|"Quality Assessment"| B
    E -.->|"Quality Assessment"| C
    E -.->|"Quality Assessment"| D

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#e1ffe1

Everyday Analogy:

• $X$: All possible “chess positions”
• $\mathsf{T}$: Legal “move rules”
• $\mathsf{C}$: “Thinking time” for each move
• $\mathsf{I}$: “Advantage score” of current position

1.2 Quantum Cellular Automaton (QCA)

Classical Cellular Automaton:

• Lattice set $\Lambda$ (e.g., one-dimensional chain $\mathbb{Z}$, two-dimensional grid ${\mathbb{Z}}^2$)
• Each lattice site $x\in \Lambda$ has finite state set $S_x$
• Global configuration $s\in {\prod }_{x\in \Lambda }{S}_x$
• Local update rule $s\to s^{\prime }$

Quantum Version (QCA):

• Lattice sites $x\in \Lambda$
• Each site has finite-dimensional Hilbert space ${\mathcal{H}}_x$ (e.g., ${\mathcal{H}}_x\cong {\mathbb{C}}^2$ for single qubit)
• Global Hilbert space $\mathcal{H}={⨂}_{x\in \Lambda }{\mathcal{H}}_x$
• Reversible unitary operator $U:\mathcal{H}\to \mathcal{H}$

Locality Condition: $U$ can be decomposed into finite-depth local unitary gate sequences, guaranteeing causality.

Correspondence Between QCA and Computational Universe:

• Configuration $x\in X$: Normalized basis vector $|x⟩\in \mathcal{H}$
• Update relation $(x,y)\in \mathsf{T}$: $⟨y|U|x⟩\ne 0$
• Cost $\mathsf{C}(x,y)$: Physical time required to execute $U$ (given by unified time scale)

Mermaid QCA Structure

graph LR
    A["Lattice Set Lambda"] --> B["Site x_1<br/>H_x1"]
    A --> C["Site x_2<br/>H_x2"]
    A --> D["Site x_3<br/>H_x3"]

    B --> E["Global Hilbert Space<br/>H = H_x1 ⊗ H_x2 ⊗ H_x3"]
    C --> E
    D --> E

    E --> F["Unitary Operator U<br/>Locally Reversible"]
    F --> G["Evolution n Steps<br/>U^n"]

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#e1ffe1
    style F fill:#ffe1f5
    style G fill:#f5e1ff

Everyday Analogy:

• Classical CA: Conway’s Game of Life—cells live/die updated by rules
• QCA: Quantum Game of Life—each cell is a qubit, updated by quantum gates

1.3 Manifestation of Unified Time Scale in QCA

On the physical side, one step of QCA evolution corresponds to some physical time increment $\Delta t$. In the unified time scale framework:

$$\Delta t=\int \kappa (\omega )\mathrm{d}\omega$$

where $\kappa (\omega )$ is unified time scale density (review Chapter 20):

$$\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\mathrm{rel}}(\omega )=\frac{1}{2\pi }\mathrm{tr}Q(\omega )$$

For QCA, can define group delay matrix (Source Theory §2.2):

$$Q(\omega )=-\mathrm{i}U(\omega {)}^{†}{\partial }_{\omega }U(\omega )$$

where $U(\omega )$ is frequency-dependent evolution operator (manifested through driving spectrum and system response).

Complexity Cost $\mathsf{C}$: Complexity cost of single-step QCA evolution can be defined as:

$$\mathsf{C}(x,y)={\int }_{\Omega }w(\omega )\kappa (\omega )\mathrm{d}\omega$$

where $w(\omega )$ is weight function, $\Omega$ is relevant frequency band.

2. Floquet Systems: Periodically Driven Quantum Dynamics

2.1 Time-Dependent Hamiltonian

Floquet System: Hamiltonian $H(t)$ satisfies periodicity

$$H(t+T)=H(t),\forall t$$

where $T$ is driving period.

Classical Examples:

• Atoms driven by periodic laser pulses
• Superconducting circuits driven by alternating current
• Optical lattices with periodic modulation

Evolution Operator: Within one period $[0,T]$, system evolves as:

$$U_F=\mathcal{T}\exp \left(-\mathrm{i}{\int }_0^{T}H(t)\mathrm{d}t\right)$$

where $\mathcal{T}$ is time-ordering operator.

Floquet Theorem: Eigenstates of $U_F$ are called Floquet states $|{\psi }_{\alpha }⟩$, with eigenvalues:

$$U_F|{\psi }_{\alpha }⟩={\mathrm{e}}^{-\mathrm{i}{\epsilon }_{\alpha }T}|{\psi }_{\alpha }⟩$$

where ${\epsilon }_{\alpha }\in (-\pi /T,\pi /T]$ is quasienergy.

Mermaid Floquet Evolution

graph LR
    A["Initial State<br/>psi(0)"] --> B["Drive H(t)<br/>t in [0,T]"]
    B --> C["After One Period<br/>psi(T) = U_F psi(0)"]
    C --> D["After Two Periods<br/>psi(2T) = U_F^2 psi(0)"]
    D --> E["After n Periods<br/>psi(nT) = U_F^n psi(0)"]

    F["Quasienergy<br/>epsilon_alpha"] -.->|"Eigenvalue"| B

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#e1ffe1
    style F fill:#ffe1f5

Everyday Analogy:

• Driving Period $T$: Swing period of pendulum
• Floquet Operator $U_F$: “Equivalent rotation” per period
• Quasienergy ${\epsilon }_{\alpha }$: Equivalent rotation angle divided by period

2.2 Quasienergy Spectrum and Band Structure

Quasienergy ${\epsilon }_{\alpha }$ modulo $2\pi /T$ is equivalent (similar to Brillouin zone):

$${\epsilon }_{\alpha }\sim {\epsilon }_{\alpha }+\frac{2\pi }{T}$$

Therefore usually choose first Brillouin zone ${\epsilon }_{\alpha }\in (-\pi /T,\pi /T]$.

Band Structure: In lattice systems, quasienergy varies with quasimomentum $k$ forming bands ${\epsilon }_{\alpha }(k)$.

Band Gap: Energy difference between adjacent bands is called Floquet band gap:

$${\Delta }_{\mathrm{F}}=\underset{k}{\min }|{\epsilon }_{\alpha +1}(k)-{\epsilon }_{\alpha }(k)|$$

Connection Between Time Crystals and Band Structure: Period-doubling time crystals usually appear when band structure has “symmetric splitting”: two bands differ by $\pi /T$.

Mermaid Quasienergy Spectrum

graph TD
    A["Quasienergy<br/>epsilon_alpha"] --> B["First Brillouin Zone<br/>(-pi/T, pi/T]"]

    B --> C1["Band 1<br/>epsilon_1(k)"]
    B --> C2["Band 2<br/>epsilon_2(k)"]
    B --> C3["Band 3<br/>epsilon_3(k)"]

    C1 -.->|"Band Gap Delta_F"| C2
    C2 -.->|"Band Gap"| C3

    D["Time Crystal Condition<br/>epsilon_2 ≈ epsilon_1 + pi/T"] -.-> C1
    D -.-> C2

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C1 fill:#f5e1ff
    style C2 fill:#fff4e1
    style C3 fill:#e1ffe1
    style D fill:#ffcccc

2.3 Floquet Version of Unified Time Scale

For Floquet evolution operator $U_F(\omega )$ (frequency dependence manifested through driving spectrum), define group delay matrix (Source Theory §2.2):

$$Q_F(\omega )=-\mathrm{i}{U}_F(\omega {)}^{†}{\partial }_{\omega }{U}_F(\omega )$$

Local Unified Time Scale Density Increment:

$${\kappa }_F(\omega )=\frac{1}{2\pi }\mathrm{tr}{Q}_F(\omega )$$

Single-Period Time Increment:

$$\Delta {\tau }_F={\int }_{{\Omega }_F}{w}_F(\omega ){\kappa }_F(\omega )\mathrm{d}\omega$$

This embeds Floquet evolution into unified time scale framework, making it consistent with scattering theory (Chapter 20) and causal diamonds (Chapter 21).

3. Floquet-QCA Object: Time Crystals in Computational Universe

3.1 Defining Floquet-QCA Computational Universe

Definition 3.1 (Source Theory §3.1):

A Floquet-QCA Computational Universe Object is a four-tuple

$$\boxed{U_{\mathrm{FQCA}}=(X,U_F,{\mathsf{C}}_T,\mathsf{I})}$$

Four-Tuple Meaning:

Event Layer Representation: One Floquet evolution step is represented on event layer $E=X\times \mathbb{Z}$ as:

$$(x,n)\mapsto (y,n+1),⟨y|U_F|x⟩\ne 0$$

where $n$ is discrete time label (which period).

Mermaid Floquet-QCA Object

graph TD
    A["Floquet-QCA Object<br/>U_FQCA"] --> B["Configuration Space X"]
    A --> C["Floquet Operator U_F"]
    A --> D["Complexity C_T"]
    A --> E["Information Function I"]

    B --> B1["Hilbert Space Basis<br/>ket x"]
    C --> C1["Period T<br/>Unitary Evolution"]
    D --> D1["Time Scale Cost<br/>kappa_F Integral"]
    E --> E1["Task Quality<br/>Observable Expectation"]

    F["Event Layer<br/>E = X × Z"] --> G["(x,n) → (y,n+1)"]
    B --> F
    C --> G

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#e1ffe1
    style F fill:#ffe1f5

Everyday Analogy:

• $X$: All possible “quantum chess position” configurations
• $U_F$: Move rules for each “round” (including driving)
• ${\mathsf{C}}_T$: “Computation time” per round
• $\mathsf{I}$: “Strategic value” of current position

3.2 Discrete Time Translation Symmetry

In Floquet-QCA, time translation is discrete, generator is $U_F$:

$$\text{Time Translation:}n\mapsto n+1$$

Corresponding evolution: $$|{\psi }_n⟩=U_F^n|{\psi }_0⟩$$

Time Translation Group: $\mathbb{Z}$ (integer addition group), action is $n\mapsto n+1$.

Symmetry: If system Hamiltonian (or evolution operator) is invariant under all time translations, it is said to have time translation symmetry.

Spontaneous Breaking: System evolution operator $U_F$ has symmetry, but evolution trajectories of certain initial states do not have full symmetry.

Mermaid Time Translation Symmetry

graph LR
    A["Time Translation Group Z"] --> B["Generator<br/>n → n+1"]
    B --> C["Evolution Operator<br/>U_F"]

    C --> D1["Symmetric State<br/>U_F psi = psi"]
    C --> D2["Symmetry-Broken State<br/>U_F^m psi = psi<br/>(m > 1)"]

    D1 --> E1["Period T<br/>Ordinary Floquet"]
    D2 --> E2["Period mT<br/>Time Crystal"]

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D1 fill:#e1ffe1
    style D2 fill:#ffcccc
    style E1 fill:#f0f0f0
    style E2 fill:#ffe1f5

3.3 Rigorous Definition of Time Crystals

Definition 3.2 (Source Theory §3.2):

In Floquet-QCA computational universe $U_{\mathrm{FQCA}}$, if there exist:

1. Local Observable $O$ (e.g., local operator $O_x$ acts only on finite region)
2. Integer $m\ge 2$
3. Initial State Family ${\mathcal{R}}_0$ (satisfying finite density and finite correlation length conditions)

such that:

(Condition 1) Long-Term Periodicity: For almost all ${\rho }_0\in {\mathcal{R}}_0$, there exists sufficiently large $n_0$ such that for all $n\ge n_0$ we have

$$\boxed{⟨O{⟩}_{n+m}=⟨O{⟩}_n}$$

where $$⟨O{⟩}_n=\mathrm{tr}({\rho }_0{U}_F^{†n}O{U}_F^n)$$

(Condition 2) Minimal Periodicity: No $1\le m^{\prime }<m$ satisfies the same condition.

Then $U_{\mathrm{FQCA}}$ is said to be in period $mT$ time crystal phase.

Special Case: $m=2$ is called period-doubling time crystal.

Mermaid Time Crystal Definition

graph TD
    A["Floquet-QCA<br/>U_FQCA"] --> B["Local Observable O"]
    A --> C["Initial State Family R_0"]

    B --> D["Evolution Sequence<br/>O_n = tr(rho_0 U_F^n O U_F^-n)"]
    C --> D

    D --> E{" Period m Exists? "}

    E -->|"O_n+m = O_n<br/>m=1"| F["Ordinary Floquet<br/>No Symmetry Breaking"]
    E -->|"O_n+m = O_n<br/>m≥2"| G["Time Crystal<br/>Symmetry Breaking"]

    G --> G1["m=2<br/>Period Doubling"]
    G --> G2["m=3,4,...<br/>Higher Periods"]

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#ffcccc
    style F fill:#f0f0f0
    style G fill:#ffe1f5
    style G1 fill:#ffaaaa

Everyday Analogy:

• $O$: “Measuring instrument” for some local region (e.g., measuring a spin)
• $⟨O{⟩}_n$: Reading of measuring instrument at round $n$
• Period $m=2$: Reading repeats every two rounds (not every round)

4. Period Doubling Mechanism: Symmetric Splitting of Quasienergy Bands

4.1 Two-Subspace Model

Consider simplest period-doubling time crystal model: Hilbert space decomposes into two subspaces

$$\mathcal{H}={\mathcal{H}}_A\oplus {\mathcal{H}}_B$$

Action of Floquet Operator:

$$U_F:\{\begin{array}{l}{\mathcal{H}}_A\to {\mathcal{H}}_B\\ {\mathcal{H}}_B\to {\mathcal{H}}_A\end{array}$$

Therefore:

• One period: $U_F$ maps ${\mathcal{H}}_A$ to ${\mathcal{H}}_B$
• Two periods: $U_F^2$ maps ${\mathcal{H}}_A$ back to ${\mathcal{H}}_A$

Local Observable: Choose $O$ such that $⟨O{⟩}_{{\mathcal{H}}_A}\ne ⟨O{⟩}_{{\mathcal{H}}_B}$.

Result: $$⟨O{⟩}_n=\{\begin{array}{ll}⟨O{⟩}_A,& n\text{even}\\ ⟨O{⟩}_B,& n\text{odd}\end{array}$$

Period is $2$!

Mermaid Two-Subspace Alternation

graph LR
    A["Subspace H_A"] -->|"U_F"| B["Subspace H_B"]
    B -->|"U_F"| A

    A1["Observable<br/>O_A"] -.->|"Expectation"| A
    B1["Observable<br/>O_B"] -.->|"Expectation"| B

    C["Period n=0,2,4,...<br/>O_n = O_A"] -.-> A
    D["Period n=1,3,5,...<br/>O_n = O_B"] -.-> B

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#e1ffe1
    style D fill:#ffe1f5

4.2 $\pi /T$ Splitting of Quasienergy Bands

In more general models, two subspaces correspond to splitting of quasienergy bands.

Key Condition (Source Theory §3.3): There exist two quasienergy bands ${\epsilon }_{\alpha }(k)$ and ${\epsilon }_{\beta }(k)$ satisfying

$${\epsilon }_{\beta }(k)\approx {\epsilon }_{\alpha }(k)+\frac{\pi }{T}$$

Physical Meaning:

• Quasienergy difference $\pi /T$ corresponds to phase difference $\pi$ (within period $T$)
• After two periods, phase difference accumulates to $2\pi$ (return to original position)

Quasienergy Spectrum and Time Crystals:

$$U_F=\sum _{\alpha }{\mathrm{e}}^{-\mathrm{i}{\epsilon }_{\alpha }T}|{\psi }_{\alpha }⟩⟨{\psi }_{\alpha }|$$

If ${\epsilon }_{\beta }={\epsilon }_{\alpha }+\pi /T$, then:

$${\mathrm{e}}^{-\mathrm{i}{\epsilon }_{\beta }T}={\mathrm{e}}^{-\mathrm{i}({\epsilon }_{\alpha }+\pi /T)T}=-{\mathrm{e}}^{-\mathrm{i}{\epsilon }_{\alpha }T}$$

Phase flips sign after one period!

Mermaid Quasienergy Splitting

graph TD
    A["Quasienergy Spectrum"] --> B["Band alpha<br/>epsilon_alpha"]
    A --> C["Band beta<br/>epsilon_beta = epsilon_alpha + pi/T"]

    B --> D["Phase<br/>exp(-i epsilon_alpha T)"]
    C --> E["Phase<br/>exp(-i epsilon_beta T) = -exp(-i epsilon_alpha T)"]

    D --> F["One Period<br/>Phase Unchanged"]
    E --> G["One Period<br/>Phase Flipped"]

    F -.->|"Ordinary Floquet"| H["Period T"]
    G -.->|"Time Crystal"| I["Period 2T"]

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#e1ffe1
    style H fill:#f0f0f0
    style I fill:#ffe1f5

Everyday Analogy:

• Quasienergy: “Equivalent frequency” of pendulum clock
• $\pi /T$ Splitting: Two pendulum clocks differ by half beat
• Phase Flip: Each drive flips phase relationship between two clocks
• Period Doubling: Need two drives to synchronize two clocks back to original position

4.3 Microscopic Picture of Spontaneous Symmetry Breaking

Symmetry Operator: Discrete time translation ${\mathcal{T}}_1:n\mapsto n+1$.

Symmetric State: $$U_F|{\psi }_{\mathrm{sym}}⟩=|{\psi }_{\mathrm{sym}}⟩$$ (quasienergy $\epsilon =0$)

Symmetry-Broken State: $$U_F|{\psi }_{\mathrm{SSB}}⟩=-|{\psi }_{\mathrm{SSB}}⟩$$ (quasienergy $\epsilon =\pi /T$)

After two periods: $$U_F^2|{\psi }_{\mathrm{SSB}}⟩=|{\psi }_{\mathrm{SSB}}⟩$$

Ground State Degeneracy: In ideal case, two symmetry-broken states $|{\psi }_{+}⟩$ and $|{\psi }_{-}⟩$ have same energy (or quasienergy differs by $\pi /T$), forming degeneracy.

Initial State Preparation: If initial state is some superposition of $|{\psi }_{+}⟩$ or $|{\psi }_{-}⟩$, evolution will exhibit period $2T$ oscillation.

5. Spin Chain Floquet-QCA Model Example

5.1 Model Definition

Consider one-dimensional spin chain (Source Theory Appendix A.1):

Lattice: $\Lambda =\mathbb{Z}$ (one-dimensional chain)

Local Hilbert Space: ${\mathcal{H}}_x\cong {\mathbb{C}}^2$ (one spin-1/2 per site)

Global Hilbert Space: $\mathcal{H}={⨂}_{x\in \mathbb{Z}}{\mathbb{C}}^2$

Two-Step Floquet Evolution: $$U_F=U_2{U}_1$$

where:

First Step $U_1$: Pairwise spin-flip gates acting between even-odd sites $$U_1=\prod _{x\text{even}}\exp \left(-\mathrm{i}J{\sigma }_x^z{\sigma }_{x+1}^z\right)$$

Second Step $U_2$: Similar gates acting between odd-even sites $$U_2=\prod _{x\text{odd}}\exp \left(-\mathrm{i}{J}^{\prime }{\sigma }_x^z{\sigma }_{x+1}^z\right)$$

Parameters: $J,J^{\prime }$ are coupling strengths.

Mermaid Spin Chain Floquet Evolution

graph LR
    A["Spin Chain<br/>...x-1, x, x+1, x+2..."] --> B["Step 1 U_1<br/>Even-Odd Pairs"]
    B --> C["Step 2 U_2<br/>Odd-Even Pairs"]
    C --> D["One Period U_F=U_2 U_1"]

    B --> B1["Site Pairs<br/>(0,1), (2,3), ..."]
    C --> C1["Site Pairs<br/>(1,2), (3,4), ..."]

    E["Spin Operator<br/>sigma_x^z"] -.->|"Coupling"| B
    E -.->|"Coupling"| C

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#e1ffe1

5.2 Existence of Time Crystal Phase

Theorem A.1 (Source Theory Appendix A.2):

In the above spin chain Floquet-QCA model, there exist parameter regions $(J,J^{\prime })$ and initial state families ${\mathcal{R}}_0$ (e.g., spontaneously symmetry-broken antiferromagnetic state mixtures), such that there exists local observable $O={\sigma }_0^z$ satisfying time crystal condition, with period $2T$.

Intuitive Understanding:

1. Initial State: Antiferromagnetic order $|\uparrow \downarrow \uparrow \downarrow \cdots ⟩$
2. First Step $U_1$: Flips some spin pairs, forms $|\downarrow \uparrow \downarrow \uparrow \cdots ⟩$
3. Second Step $U_2$: Flips again, returns to $|\uparrow \downarrow \uparrow \downarrow \cdots ⟩$
4. Observable: $⟨{\sigma }_0^z⟩$ alternates values at odd/even periods

Stability: This time crystal phase is robust under parameter tuning and local noise, as long as:

• Floquet band gap ${\Delta }_{\mathrm{F}}>0$
• Noise correlation length finite

5.3 Numerical Verification Scheme

Step 1: Prepare Initial State $${\rho }_0=|\uparrow \downarrow \uparrow \downarrow \cdots ⟩⟨\uparrow \downarrow \uparrow \downarrow \cdots |$$

Step 2: Floquet Evolution $${\rho }_n=U_F^n{\rho }_0{U}_F^{†n}$$

Step 3: Measure Observable $$⟨{\sigma }_0^z{⟩}_n=\mathrm{tr}({\rho }_n{\sigma }_0^z)$$

Step 4: Check Periodicity Determine whether $⟨{\sigma }_0^z{⟩}_n$ satisfies: $$⟨{\sigma }_0^z{⟩}_{n+2}=⟨{\sigma }_0^z{⟩}_n$$

Expected Results:

• For $n=0,2,4,\dots$: $⟨{\sigma }_0^z{⟩}_n=+1$
• For $n=1,3,5,\dots$: $⟨{\sigma }_0^z{⟩}_n=-1$

Mermaid Numerical Verification Flow

graph TD
    A["Prepare Initial State<br/>rho_0"] --> B["Floquet Evolution<br/>U_F^n"]
    B --> C["Measure Observable<br/>sigma_z"]
    C --> D["Record Sequence<br/>O_n, n=0,1,2,..."]

    D --> E{" Check Period? "}

    E -->|"O_n+2 = O_n"| F["Time Crystal Phase<br/>Period 2T"]
    E -->|"O_n+1 = O_n"| G["Ordinary Floquet<br/>Period T"]

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#ffcccc
    style F fill:#aaffaa
    style G fill:#f0f0f0

6. Unique Insights from Computational Universe Perspective

6.1 Discrete Time Structure of Event Layer

In computational universe framework, time is not a continuous parameter, but discrete label $n$ on event layer $E=X\times \mathbb{Z}$.

Floquet Evolution: $$(x,n)\to (y,n+1),⟨y|U_F|x⟩\ne 0$$

Time Crystal: “Super-periodic” structure on event layer—events separated by $m$ steps are equivalent.

Mermaid Event Layer Structure

graph TD
    A["Event Layer<br/>E = X × Z"] --> B["Time Slice<br/>n=0"]
    A --> C["Time Slice<br/>n=1"]
    A --> D["Time Slice<br/>n=2"]
    A --> E["Time Slice<br/>n=3"]

    B --> B1["Configuration x_0"]
    C --> C1["Configuration x_1"]
    D --> D1["Configuration x_2"]
    E --> E1["Configuration x_3"]

    B1 -->|"U_F"| C1
    C1 -->|"U_F"| D1
    D1 -->|"U_F"| E1

    F["Time Crystal<br/>x_0 ≈ x_2 ≈ x_4..."] -.-> B1
    F -.-> D1

    style A fill:#e1f5ff
    style B fill:#ffe1e1
    style C fill:#f5e1ff
    style D fill:#fff4e1
    style E fill:#e1ffe1
    style F fill:#ffe1f5

6.2 Time Crystals in Complexity Geometry

Complexity Cost ${\mathsf{C}}_T$ given by unified time scale integral:

$${\mathsf{C}}_T={\int }_{{\Omega }_F}{w}_F(\omega ){\kappa }_F(\omega )\mathrm{d}\omega$$

Complexity Characteristics of Time Crystals:

• Single-period cost ${\mathsf{C}}_T$ fixed
• But “effective period” is $m{\mathsf{C}}_T$ ($2{\mathsf{C}}_T$ when $m=2$)

Complexity Geometry Interpretation: Time crystals correspond to special closed loops on control manifold $(\mathcal{M},G)$, where complexity increment and time increment decouple.

6.3 Connection with Causal Diamond Chains

Each Floquet Period $\leftrightarrow$ One Causal Diamond ${◊}_{F,k}$

Diamond Chain: $$\{{◊}_{F,k}{\}}_{k\in \mathbb{Z}}$$

Time Crystal: “Markov non-totally-ordered” structure on diamond chain—every $m$ diamonds form one complete period.

This will be detailed in next chapter (02-Z₂ Holonomy).

7. Chapter Summary

7.1 Core Concepts Review

Floquet-QCA Object: $$U_{\mathrm{FQCA}}=(X,U_F,{\mathsf{C}}_T,\mathsf{I})$$

Time Crystal Definition: $$⟨O{⟩}_{n+m}=⟨O{⟩}_n,m\ge 2$$

Quasienergy Spectrum Condition: $${\epsilon }_{\beta }\approx {\epsilon }_{\alpha }+\frac{\pi }{T}$$

Spin Chain Model: $$U_F=U_2{U}_1,U_1=\prod _{x\text{even}}{\mathrm{e}}^{-\mathrm{i}J{\sigma }_x^z{\sigma }_{x+1}^z}$$

7.2 Key Insights

1. Discrete Time Translation Symmetry Breaking: Time crystals are spontaneous symmetry breaking $\mathbb{Z}\to m\mathbb{Z}$

2. Quasienergy Band Splitting is Microscopic Mechanism: ${\epsilon }_{\beta }-{\epsilon }_{\alpha }=\pi /T$ causes phase flip

3. Computational Universe Provides Discretization Framework: Event layer $E=X\times \mathbb{Z}$, complexity cost ${\mathsf{C}}_T$

4. Spin Chain Model is Realizable: Two-step Floquet evolution $U_F=U_2{U}_1$ exhibits period doubling at appropriate parameters

7.3 Preview of Next Chapter

Next chapter (02-time-crystal-z2.md) will discuss:

• Realization of Null-Modular double cover on Floquet chain
• Mod-2 phase label ${ϵ}_F$
• Precise correspondence between Z₂ holonomy and time crystal parity
• Proof of Theorem 4.1

Core Formula Preview: $$\boxed{\text{Period-Doubling Time Crystal}\iff {\mathrm{hol}}_{{\mathbb{Z}}_2}({\Gamma }_F)=1}$$

End of Chapter

Source Theory: euler-gls-info/17-time-crystals-null-modular-z2-holonomy.md, §2-3