Section 13.2 Time Crystals: Spontaneous Breaking of Time Translation Symmetry

“Spatial crystals break spatial translation symmetry, so can time crystals break time translation symmetry? The answer is both ‘no’ and ‘yes’—it depends on where we ask the question.” —— Unified Framework of Time Crystals in GLS Theory

Introduction: From Wilczek’s Dream to the Equilibrium Denial

The Bold Conception of Time Crystals

In 2012, Nobel laureate Frank Wilczek published a groundbreaking paper in Physical Review Letters, proposing the concept of quantum time crystals.

Core Idea:

Ordinary crystals (such as salt, diamond) exhibit periodic structure in space, spontaneously breaking spatial translation symmetry:

$$\rho (\mathbf{r}+\mathbf{a})=\rho (\mathbf{r})$$

where $\mathbf{a}$ is a lattice vector. Can there exist a state of matter that exhibits periodicity in the time dimension, spontaneously breaking time translation symmetry?

$$\rho (t+T)=\rho (t)$$

Even when the system’s Hamiltonian is time-independent $H\ne H(t)$?

Wilczek’s Proposal:

Consider a rotating superfluid ring. In the ground state, particles in the ring rotate at constant angular velocity ${\omega }_0$, forming a persistent current. From the laboratory reference frame, the system exhibits periodic motion in time with period $T=2\pi /{\omega }_0$.

This seems to satisfy the definition of a time crystal!

The Death Sentence for Equilibrium Time Crystals

However, theoretical physicists quickly discovered problems. In 2013, Patrick Bruno proved a no-go theorem:

Theorem 2.1 (Bruno No-Go Theorem, 2013)

In quantum systems with short-range interactions, the ground state $|{\psi }_0⟩$ of a time-independent Hamiltonian $H$ cannot spontaneously break time translation symmetry. That is, there exist no time-periodic ground state observables.

Proof Sketch:

If the ground state satisfies $⟨{\psi }_0|O(t)|{\psi }_0⟩=⟨{\psi }_0|e^{iHt}O{e}^{-iHt}|{\psi }_0⟩$ exhibiting periodic oscillation, then since $H|{\psi }_0⟩=E_0|{\psi }_0⟩$ (ground state energy), we have:

$$|⟨{\psi }_0|O(t)|{\psi }_0⟩=⟨{\psi }_0|O|{\psi }_0⟩$$

That is, the ground state expectation value is time-independent and cannot oscillate periodically. $\square$

In 2015, Watanabe and Oshikawa further generalized this, proving that similar no-go results hold for finite-temperature canonical ensembles:

Theorem 2.2 (Watanabe-Oshikawa No-Go Theorem, 2015)

In quantum systems with local interactions, any equilibrium state (ground state, thermal state, generalized Gibbs state) of a time-independent Hamiltonian cannot spontaneously break time translation symmetry.

This seemed to pronounce the death sentence for time crystals. Had Wilczek’s beautiful dream been shattered?

The Turnaround in Non-Equilibrium: Floquet Time Crystals

In 2016, a turnaround appeared. Else, Bauer, and Nayak proposed: if we abandon the “equilibrium” requirement and consider periodically driven non-equilibrium systems, the situation is completely different!

Key Insight:

For periodically driven systems, the Hamiltonian itself has periodicity:

$$H(t+T)=H(t)$$

Here, “time translation symmetry” refers to translation symmetry of the drive period $T$. Time crystals correspond to states with evolution period $mT$ ($m>1$, typically $m=2$), i.e.:

$$\rho (t+mT)=\rho (t),\rho (t+T)\ne \rho (t)$$

This is a subharmonic response to the drive frequency, analogous to a forced oscillator’s response at $\omega /2$.

In 2017, two milestone experiments were simultaneously published in Nature:

1. Zhang et al. (University of Maryland): Observed discrete time crystals in trapped ion systems
2. Choi et al. (Harvard University): Observed discrete time crystals in room-temperature diamond spin systems

This marked the transition of Discrete Time Crystals (DTC) from theory to reality!

GLS Theory’s Contribution: Unified Framework and Unified Time Scale

GLS theory provides a unified framework for time crystals, incorporating four seemingly different types of time crystals into the same theoretical system:

graph TD
    A["Unified Framework of Time Crystals"] --> B["Prethermal DTC"]
    A --> C["MBL-DTC"]
    A --> D["Open-System Dissipative Time Crystals"]
    A --> E["Topological Time Crystals"]

    B --> F["High-Frequency Drive omega >> J"]
    C --> F

    B --> G["Exponentially Long Lifetime"]
    C --> H["Eigenstate Order"]
    D --> I["Limit Cycle Attractor"]
    E --> J["Logical Operator Order Parameter"]

    K["Unified Time Scale kappa"] --> G
    K --> H
    K --> I
    K --> J

    G --> L["tau_* ~ exp(c omega/J)"]
    H --> M["pi Spectrum Pairing"]
    I --> N["Liouvillian Spectral Gap"]
    J --> O["Topological Entanglement Entropy gamma"]

Core Ideas:

1. Prethermal DTC: Under high-frequency driving, the system maintains subharmonic locking for exponentially long times, with lifetime controlled by the unified time scale $\kappa (\omega )$

2. MBL-DTC: In strongly disordered localized systems, subharmonic response becomes an eigenstate property, independent of initial state

3. Open-System Dissipative Time Crystals: In Lindblad open systems, time crystals are realized through limit cycles protected by Liouvillian spectral gap

4. Topological Time Crystals: Using non-local logical operators of topological codes (such as surface codes) as order parameters, achieving topologically protected DTC

All these cases are connected through the unified time scale!

13.2.1 Prethermal Discrete Time Crystals: Origin of Exponentially Long Lifetime

Floquet Theory Foundation

Consider a periodically driven system with Hamiltonian satisfying $H(t+T)=H(t)$. Define the Floquet operator (one-period evolution operator):

$$F=\mathcal{T}\exp \left(-i{\int }_0^{T}H(t)dt\right)$$

where $\mathcal{T}$ denotes time ordering.

The spectral decomposition of the Floquet operator defines the quasienergy spectrum:

$$F|{\varphi }_{\alpha }⟩=e^{-i{\epsilon }_{\alpha }T}|{\varphi }_{\alpha }⟩$$

where ${\epsilon }_{\alpha }\in (-\pi /T,\pi /T]$ is the quasienergy, and $|{\varphi }_{\alpha }⟩$ is the Floquet eigenstate.

Key Property: Even though $H(t)$ depends on time, $F$ is a time-independent unitary operator, whose eigenstates and eigenvalues describe the system’s “steady-state” properties.

High-Frequency Expansion and Effective Hamiltonian

When the drive frequency $\omega =2\pi /T$ is much larger than the system’s intrinsic energy scale $J$ ($\omega \gg J$), we can perform a Floquet-Magnus expansion:

$$F=\exp \left(-iT\sum _{n=0}^{\infty }{\Omega }_n\right)$$

where:

$${\Omega }_0=\frac{1}{T}{\int }_0^{T}H(t)dt$$

$${\Omega }_1=\frac{1}{2iT}{\int }_0^{T}d{t}_2{\int }_0^{t_2}d{t}_1[H(t_2),H(t_1)]$$

$$⋮$$

Truncating at optimal order $n_{\ast }\sim \alpha \omega /J$, define the effective Hamiltonian:

$$H_{\ast }=\sum _{n\le n_{\ast }}{\Omega }_n$$

Then we have the approximation:

$$F\approx e^{-i{H}_{\ast }T}$$

The error $|F-e^{-i{H}_{\ast }T}|\lesssim C{e}^{-c\omega /J}$ is exponentially small in frequency!

Near-$\pi$ Pulse and ${\mathbb{Z}}_2$ Symmetry

Consider a piecewise drive protocol:

$$H(t)=\{\begin{array}{ll}{H}_X& 0<t<{\tau }_x\\ H_Z& {\tau }_x<t<T\end{array}$$

where $H_X={\sum }_j{h}^x{\sigma }_j^x$ (global transverse field), $H_Z={\sum }_j{J}_{jk}{\sigma }_j^z{\sigma }_k^z$ (longitudinal interactions).

If we choose $h^x{\tau }_x=\pi (1+ϵ)$ ($ϵ$ small), then one-period evolution can be approximately written as:

$$F\approx U_X{e}^{-i{H}_{Z}T}=X{e}^{iϵH_X{\tau }_x}{e}^{-i{H}_{Z}T}$$

where $X={\prod }_j{\sigma }_j^x$ is the global ${\mathbb{Z}}_2$ symmetry generator ($X^2=1$).

Using high-frequency expansion and quasi-local unitary transformation $U_{\ast }$, we can further simplify to:

$$F=U_{\ast }X{e}^{-i{H}_{\ast }T}{U}_{\ast }^{†}+\Delta$$

where $H_{\ast }$ commutes (or approximately commutes) with $X$, and $|\Delta |\lesssim C{e}^{-c\omega /J}$.

Mechanism of Subharmonic Locking

Key Observation: For any local operator $O$ satisfying $XOX=-O$ (odd under ${\mathbb{Z}}_2$ transformation), we have:

$$|⟨O(nT)⟩=⟨{\psi }_0|F^{†n}O{F}^n|{\psi }_0⟩$$

Substituting the structure of $F$:

$$F^n\approx U_{\ast }{X}^n{e}^{-in{H}_{\ast }T}{U}_{\ast }^{†}$$

Since $X^2=1$, $X^n$ oscillates between even and odd $n$:

$$X^n=\{\begin{array}{ll}1& n\text{ even}\\ X& n\text{ odd}\end{array}$$

Therefore:

$$|⟨O(nT)⟩\approx \{\begin{array}{ll}⟨{\psi }_0|U_{\ast }{e}^{-in{H}_{\ast }T}O{e}^{in{H}_{\ast }T}{U}_{\ast }^{†}|{\psi }_0⟩& n\text{ even}\\ -⟨{\psi }_0|U_{\ast }{e}^{-in{H}_{\ast }T}O{e}^{in{H}_{\ast }T}{U}_{\ast }^{†}|{\psi }_0⟩& n\text{ odd}\end{array}$$

That is, $⟨O(nT)⟩$ oscillates with period $2T$, exhibiting subharmonic response!

Prethermal Lifetime and Unified Time Scale

Theorem 2.3 (Exponential Lifetime of Prethermal DTC)

Let the drive protocol satisfy the high-frequency condition $\omega \gg J$, and let there exist a near-$\pi$ global ${\mathbb{Z}}_2$ pulse. Then for any local operator $O$ odd under ${\mathbb{Z}}_2$ transformation, the lifetime of subharmonic locking is:

$${\tau }_{\ast }\sim \exp \left(\frac{c\omega }{J}\right)$$

For any polynomial time $t\lesssim {\tau }_{\ast }$, the subharmonic signal does not decohere.

Proof Sketch:

The error accumulation rate of the high-frequency expansion is:

$$\frac{d⟨H_{\ast }⟩}{dt}\lesssim C^{\prime }{e}^{-c\omega /J}$$

Therefore, energy absorption within time ${\tau }_{\ast }$ is at most:

$$\Delta E\sim {\tau }_{\ast }\cdot C^{\prime }{e}^{-c\omega /J}$$

When $\Delta E$ reaches the system’s intrinsic energy scale $J$, prethermalization ends:

$${\tau }_{\ast }\cdot C^{\prime }{e}^{-c\omega /J}\sim J\Rightarrow {\tau }_{\ast }\sim \frac{J}{C^{\prime }}{e}^{c\omega /J}$$

That is, exponentially long lifetime! $\square$

Role of Unified Time Scale:

In Floquet systems, the average $\bar{\kappa }(\epsilon )$ of the unified time scale $\kappa (\omega )$ over energy window $I$ controls the density of states of the effective Hamiltonian $H_{\ast }$. The prethermal lifetime can be expressed as:

$${\tau }_{\ast }\sim \frac{1}{\bar{\kappa }(\epsilon )}{e}^{c\omega /J}$$

That is, the smaller the unified time scale, the longer the prethermal lifetime!

Experimental Realization: Trapped Ion Systems

In 2021, Kyprianidis et al. reported in Science a prethermal DTC realized on 51 trapped ions, observing stable subharmonic response for over 100 drive periods.

Experimental Protocol:

1. Initial State Preparation: Prepare all ions in $|\downarrow ⟩$ state
2. Drive Sequence:
   • $\pi /2$ pulse ($R_y(\pi /2)$): Rotate spins to the equator
   • Ising interaction: $H_Z={\sum }_{j<k}{J}_{jk}{\sigma }_j^z{\sigma }_k^z$ (realized by laser-induced long-range interactions)
   • Near-$\pi$ pulse ($R_x(\pi +ϵ)$): Realize global ${\mathbb{Z}}_2$ flip
3. Readout: Measure longitudinal magnetization $M_z(n)=N^{-1}{\sum }_j⟨{\sigma }_j^z(nT)⟩$

Observations:

$$M_z(n)\approx (-1{)}^n{M}_0{e}^{-n/{\tau }_{\ast }}$$

where $M_0\approx 0.9$ (close to maximum value 1), ${\tau }_{\ast }\approx 150T$ (corresponding to $\omega /J\approx 5$).

The lifetime is consistent with the theoretically predicted exponential scaling!

Metaphorical Understanding:

Prethermal DTC is like a pendulum periodically pushed:

• Equilibrium: Pendulum rests at lowest point (no periodic motion)
• Periodic Drive: Push the pendulum every time $T$
• Subharmonic Response: Pendulum oscillates with period $2T$ (each push completes half an oscillation cycle)
• Prethermal Lifetime: Even after stopping the pushes, the pendulum continues oscillating for a long time due to “memory effect” (${\tau }_{\ast }\sim e^{c\omega /J}$)

High-frequency driving ($\omega$ large) corresponds to “gentle but frequent” pushes, which actually extends the pendulum’s “memory” lifetime!

13.2.2 MBL Time Crystals: Eigenstate Order and $\pi$ Spectrum Pairing

Introduction to Many-Body Localization (MBL)

In strongly disordered systems, quantum states may remain localized, even at finite temperature without thermalization. This phenomenon is called Many-Body Localization (MBL).

Features of MBL:

1. $l$-bit Diagonalization: There exists a quasi-local unitary $U$ that diagonalizes the Hamiltonian as:

$$U{H}_0{U}^{†}=\sum _i{h}_i{\tau }_i^z$$

where $\{{\tau }_i^z\}$ are quasi-local “$l$-bits” (localized bits), and $h_i$ are local energy levels

2. Logarithmic Entanglement Growth: Entanglement entropy grows logarithmically with time $S(t)\sim \log t$, rather than linear growth in thermalizing systems

3. ETH Violation: Eigenstates do not satisfy the Eigenstate Thermalization Hypothesis, retaining memory of initial state

4. Area-Law Entanglement: Eigenstate entanglement entropy satisfies area law $S\sim L^{d-1}$, rather than volume law $S\sim L^d$

Construction of MBL-DTC

Adding periodic driving and near-$\pi$ global flips to MBL systems can realize eigenstate time crystal order.

Key Construction (Khemani et al., 2016):

Consider a one-dimensional spin chain with random longitudinal field $h_i\in [-W,W]$ ($W$ large), periodic driving:

$$H(t)=\{\begin{array}{ll}{\sum }_i{h}_i{\sigma }_i^z+{\sum }_{i}J{\sigma }_i^z{\sigma }_{i+1}^z& 0<t<T/2\\ {\sum }_i\frac{\pi +{ϵ}_i}{T/2}{\sigma }_i^x& T/2<t<T\end{array}$$

where ${ϵ}_i$ are small random deviations.

Structure of Floquet Operator:

Under strong disorder ($W\gg J$), there exists a quasi-local unitary $U$ such that:

$$F\simeq \tilde{X}{e}^{-i{H}_{\mathrm{MBL}}T}$$

where:

• $X=UX{U}^{†}\approx {\prod }_i{\sigma }_i^x$ is a quasi-local ${\mathbb{Z}}_2$ symmetry generator
• $H_{\mathrm{MBL}}={\sum }_i{h}_i{\tau }_i^z$ is the effective MBL Hamiltonian, diagonalized by $l$-bits

Key Property: $[\tilde{X},H_{\mathrm{MBL}}]=0$ (exact commutation).

$\pi$ Spectrum Pairing and Eigenstate Order

Since $\tilde{X}$ commutes with $H_{\mathrm{MBL}}$, Floquet eigenstates can be simultaneously diagonalized:

$$F|{\psi }_{\alpha }⟩=e^{-i{\epsilon }_{\alpha }T}|{\psi }_{\alpha }⟩,\tilde{X}|{\psi }_{\alpha }⟩=\pm |{\psi }_{\alpha }⟩$$

For the two eigenspaces of $\tilde{X}$ ($+1$ and $-1$), the Floquet operator establishes a one-to-one mapping between them:

$$F|{\psi }_{+}⟩=e^{-i{\epsilon }_{+}T}X|{\psi }_{+}⟩,F(X|{\psi }_{+}⟩)=e^{-i({\epsilon }_{+}+\pi )T}|{\psi }_{+}⟩$$

That is, each eigenstate $|{\psi }_{+}⟩$ pairs with $\tilde{X}|{\psi }_{+}⟩$ to form a $\pi$ pair, with quasienergies differing by $\pi$ (mod $2\pi$):

$${\epsilon }_{-}={\epsilon }_{+}+\pi (\mathrm{mod}2\pi /T)$$

Definition of Eigenstate Order:

For any initial state $|{\psi }_0⟩$ (regardless of energy window), expand it in the Floquet eigenbasis:

$$|{\psi }_0⟩=\sum _{\alpha }{c}_{\alpha }|{\psi }_{\alpha }⟩$$

Due to $\pi$ pairing of the spectrum, time evolution:

$$|\psi (nT)⟩=F^n|{\psi }_0⟩=\sum _{\alpha }{c}_{\alpha }{e}^{-in{\epsilon }_{\alpha }T}|{\psi }_{\alpha }⟩$$

For local operator $O$ odd under $\tilde{X}$ transformation, expectation value:

$$|⟨O(nT)⟩=\sum _{\alpha ,\beta }{c}_{\alpha }^{\ast }{c}_{\beta }{e}^{in({\epsilon }_{\alpha }-{\epsilon }_{\beta })T}⟨{\psi }_{\alpha }|O|{\psi }_{\beta }⟩$$

Due to $\pi$ pairing and commutation, diagonal terms ($\alpha =\beta$) contribute zero, off-diagonal terms have opposite signs for even and odd $n$:

$$|⟨O(nT)⟩\approx (-1{)}^{n}A$$

where $A$ is a constant independent of $n$!

Key Insight: This is a state-independent subharmonic response—regardless of initial state, it exhibits the same $2T$ periodic oscillation! This is exactly the manifestation of eigenstate order.

Experimental Verification: Superconducting Quantum Processors

In 2022, Mi et al. reported in Nature an MBL-DTC realized on Google Sycamore processor (57 superconducting qubits).

Experimental Highlights:

1. Spectrum Measurement: By preparing multiple eigenstates and measuring their quasienergies, directly verified $\pi$ spectrum pairing

2. Spectrum-Dynamics Consistency: Simultaneously measured spectrum structure ($\pi$ pairing) and dynamics (subharmonic response), confirming consistency

3. Robustness Tests: Verified existence of DTC phase under different disorder strengths, interaction strengths, and system sizes

Metaphorical Understanding:

MBL-DTC is like a “frozen” pendulum system:

• MBL Ground State: Pendulums are strongly “pinned” at different positions (corresponding to different $l$-bit configurations)
• $\pi$ Pairing: Each pendulum configuration has a “mirror configuration” (obtained by $\tilde{X}$ flip)
• Eigenstate Order: Regardless of initial pendulum state, under driving they oscillate with period $2T$ between configuration and mirror
• State Independence: This oscillation is not a “memory effect” (like prethermal DTC), but a necessary consequence of the system’s eigenstructure

13.2.3 Open-System Dissipative Time Crystals: Limit Cycles and Liouvillian Spectral Gap

Lindblad Master Equation and Open-System Dynamics

When a quantum system interacts with an environment, evolution is described by the Lindblad master equation:

$$\frac{d\rho }{dt}={\mathcal{L}}_t(\rho )=-i[H(t),\rho ]+\sum _{\mu }\left(L_{\mu }\rho L_{\mu }^{†}-\frac{1}{2}\{L_{\mu }^{†}{L}_{\mu },\rho \}\right)$$

where:

• $H(t)$: System Hamiltonian (may be time-dependent)
• $L_{\mu }$: Lindblad operators (jump operators), describing dissipative processes (such as spontaneous emission, decoherence)
• ${\mathcal{L}}_t$: Liouvillian superoperator

If $H(t)$ and $\{L_{\mu }\}$ both vary with period $T$, then ${\mathcal{L}}_{t+T}={\mathcal{L}}_t$.

One-Period Quantum Channel and Perron-Frobenius Theory

Define the one-period quantum channel (stroboscopic map):

$$\mathcal{E}=\mathcal{T}\exp \left({\int }_0^T{\mathcal{L}}_{t}dt\right)$$

This is a completely positive trace-preserving (CPTP) map, mapping density matrices to density matrices.

The spectral structure of $\mathcal{E}$ determines the system’s long-time evolution. Similar to classical Perron-Frobenius theory, the spectrum of quantum channels satisfies:

1. Maximum eigenvalue is 1: ${\lambda }_{\max }=1$ (corresponding to steady state)
2. Other eigenvalues have modulus $\le 1$: $|{\lambda }_j|\le 1$
3. Eigenvalues at spectral radius determine long-time behavior

Sufficient Conditions for Dissipative Time Crystals

Theorem 2.4 (Open-System Dissipative Time Crystals)

For the one-period channel $\mathcal{E}$ of a periodic Lindblad semigroup, if:

1. Unique Modulus Group: The spectral radius (eigenvalues with modulus 1) consists only of $m$ pure phases $\{e^{2\pi ik/m}{\}}_{k=0}^{m-1}$

2. Spectral Gap: Eigenvalues with modulus $<1$ satisfy $|{\lambda }_j|\le 1-{\Delta }_{\mathrm{Liouv}}$, where ${\Delta }_{\mathrm{Liouv}}>0$ is the Liouvillian spectral gap

3. Irreducibility: Jordan blocks corresponding to the modulus group are non-splitting

Then almost all initial states converge in the long-time limit to a limit cycle attractor family with period $mT$:

$$\rho (t+mT)=\rho (t),\rho (t+T)\ne \rho (t)$$

Manifesting as an $m$-subharmonic dissipative time crystal.

Proof Sketch:

Decompose the spectrum of $\mathcal{E}$ as:

$$\mathcal{E}=\sum _{k=0}^{m-1}{e}^{2\pi ik/m}{P}_k+\sum _{|{\lambda }_j|<1}{\lambda }_j{P}_j$$

where $P_k$ are projection operators corresponding to $e^{2\pi ik/m}$. Iterating $n$ times:

$${\mathcal{E}}^n=\sum _{k=0}^{m-1}{e}^{2\pi ikn/m}{P}_k+\sum _{|{\lambda }_j|<1}{\lambda }_j^n{P}_j$$

When $n\to \infty$, the second term decays exponentially as $(1-{\Delta }_{\mathrm{Liouv}}{)}^n$, leaving:

$$\underset{n\to \infty }{\lim }{\mathcal{E}}^n{\rho }_0=\sum _{k=0}^{m-1}{e}^{2\pi ikn/m}{P}_k{\rho }_0$$

This is a limit cycle with period $m$. $\square$

Experimental Realization: Rydberg Atom Gas

In 2024, Wu et al. reported in Nature Physics observation of dissipative time crystals in strongly interacting Rydberg gas.

Experimental System:

• Atoms: Room-temperature cesium atom vapor
• Driving: Continuous optical pumping and Rydberg excitation
• Dissipation: Spontaneous emission, collision decoherence

Effective Lindblad Equation:

$$\frac{d\rho }{dt}=-i[\Omega (t)(S^{+}+S^{-}),\rho ]+\gamma (2S^{-}\rho S^{+}-\{S^{+}{S}^{-},\rho \})$$

where:

• $\Omega (t)={\Omega }_0\cos (\omega t)$: Periodic drive field
• $S^{\pm }$: Collective spin operators
• $\gamma$: Decoherence rate

Observations:

1. Parameter Phase Diagram: By tuning drive strength ${\Omega }_0$ and decoherence rate $\gamma$, observed three phase regions:

   • Disordered Phase: No stable limit cycle
   • Time Crystal Phase: Stable limit cycle with period $2T$
   • Multi-Stable Phase: Multiple coexisting limit cycles

2. Liouvillian Spectral Gap: In the time crystal phase, measured ${\Delta }_{\mathrm{Liouv}}\approx 0.1\gamma$, consistent with theoretical prediction

3. Robustness: Time crystal phase is robust to perturbations, with clear phase boundaries

Metaphorical Understanding:

Open-system dissipative time crystals are like a damped but continuously driven pendulum:

• Open Environment: Pendulum experiences air resistance (corresponding to dissipation $\{L_{\mu }\}$)
• Periodic Drive: Push the pendulum every time $T$
• Limit Cycle: Pendulum stabilizes on an oscillation orbit with period $2T$ under the balance of “push—damping”
• Liouvillian Spectral Gap: Orbit has “attractiveness” to perturbations, quickly recovering after deviation (convergence rate $\sim e^{-{\Delta }_{\mathrm{Liouv}}t}$)
• Robustness: Even if pendulum parameters (length, mass) slightly change, as long as within the time crystal phase region, the limit cycle still exists

13.2.4 Topological Time Crystals: Non-Local Order Parameters and Topological Protection

Introduction to Topological Quantum Error-Correcting Codes

Surface Code is a two-dimensional topological quantum error-correcting code, defined on a square lattice, with one qubit on each lattice point and edge.

Stabilizers:

$$A_s=\prod _{j\in \text{star}(s)}{\sigma }_j^x,B_p=\prod _{j\in \text{boundary}(p)}{\sigma }_j^z$$

where:

• $A_s$: Star stabilizer (acts on four qubits around vertex $s$)
• $B_p$: Plaquette stabilizer (acts on four qubits on boundary of plaquette $p$)

Code Subspace: Subspace satisfying all stabilizers equal $+1$:

$${\mathcal{H}}_{\mathrm{code}}=\{|\psi ⟩:A_s|\psi ⟩=B_p|\psi ⟩=|\psi ⟩,\forall s,p\}$$

Code subspace dimension is 4 (under periodic boundary conditions), encoding 2 logical qubits.

Logical Operators:

$${\overline{X}}_L=\prod _{j\in \text{horizontal chain}}{\sigma }_j^x,{\overline{Z}}_L=\prod _{j\in \text{vertical chain}}{\sigma }_j^z$$

These are non-local operators (acting on chains traversing the entire system), but realize logical operations within the code subspace.

Construction of Topological Time Crystals

Key Idea (Wahl 2024 review):

Implement periodic driving on surface code such that logical operators (rather than physical qubit operators) exhibit subharmonic response.

Protocol:

1. Hamiltonian Engineering: Construct effective Hamiltonian $H_{\mathrm{code}}$ acting on code subspace

2. Logical $\pi$ Flip: Periodically apply logical $X$ gate:

$$F_{\mathrm{logical}}\approx {\overline{X}}_L{e}^{-i{H}_{\mathrm{code}}^{\mathrm{top}}T}$$

where $H_{\mathrm{code}}^{\mathrm{top}}$ is the effective topological Hamiltonian

Topologically Protected Order Parameter:

• Local Operators: For any local operator $O_{\mathrm{loc}}$ on physical qubits, expectation value does not exhibit subharmonic response:

$$|⟨O_{\mathrm{loc}}(nT)⟩\approx \text{constant}$$

• Logical Operators: For non-local logical operator ${\overline{Z}}_L$, expectation value exhibits subharmonic response:

$$|⟨{\overline{Z}}_L(nT)⟩\approx (-1{)}^{n}A$$

Topological Entanglement Entropy:

Define entanglement entropy of subregion $A$: $S(A)=-\mathrm{Tr}({\rho }_A\log {\rho }_A)$. In topologically ordered states, there exists a topological contribution:

$$S(A)=\alpha |\partial A|-\gamma +\cdots$$

where:

• $\alpha |\partial A|$: Area law term
• $\gamma$: Topological entanglement entropy ($\gamma >0$ indicates topological order)

In topological time crystals, $\gamma$ co-occurs with subharmonic response: when $⟨{\overline{Z}}_L(nT)⟩$ exhibits subharmonic, $\gamma >0$; when subharmonic disappears, $\gamma \to 0$.

Experimental Realization: Superconducting Quantum Processors

In 2024, Xiang et al. reported in Nature Communications realization of long-lived topological time crystals on superconducting quantum processors.

Experimental Parameters:

• Number of Qubits: $N=36$ ($6\times 6$ square lattice)
• Encoding: One logical qubit (surface code distance $d=3$)
• Drive Period: $T=1\mu \mathrm{s}$
• Observation Time: Over 1000 periods

Key Observations:

1. Subharmonic Response of Logical Channel:

$$|⟨{\overline{Z}}_L(nT)⟩\approx 0.7\times (-1{)}^n{e}^{-n/{\tau }_{\ast }}$$

Lifetime ${\tau }_{\ast }\approx 500T$

2. No Subharmonic in Physical Channel:

$$|⟨{\sigma }_j^z(nT)⟩\approx \text{constant}\pm \text{noise}$$

3. Non-Zero Topological Entanglement Entropy:

$$\gamma \approx 0.3\pm 0.1$$

Remains non-zero throughout the subharmonic locking period

GLS Theory Explanation:

The stability of topological time crystals is provided by two layers of protection mechanisms:

1. Prethermal Protection: Exponentially long lifetime ${\tau }_{\ast }\sim e^{c\omega /J}$ (similar to prethermal DTC)

2. Topological Protection: Subharmonic response of logical operators is insensitive to local perturbations (local errors can be detected and corrected by error-correcting codes)

In the GLS framework, topological entanglement entropy $\gamma$ can be expressed through boundary K-theory index ${\mu }_K$. Topological time crystals correspond to:

$${\mu }_K[{\overline{Z}}_L(nT)]=(-1{)}^n{\mu }_K[{\overline{Z}}_L(0)]$$

That is, the K-theory index oscillates in time!

Metaphorical Understanding:

Topological time crystals are like an “invisible” pendulum system:

• Physical Qubits: Ordinary “visible” pendulums, appearing stationary
• Logical Qubits: “Invisible” pendulums formed by “collective modes” of multiple physical pendulums
• Topological Protection: Even if individual physical pendulums fail (corresponding to qubit errors), the invisible pendulum can still oscillate normally
• Non-Locality: The “position” of the invisible pendulum cannot be obtained by measuring a single physical pendulum, must measure the entire chain
• Topological Entanglement Entropy $\gamma$: “Existence proof” of the invisible pendulum—even if we can’t see it, we can infer its existence through special structure of quantum entanglement

13.2.5 Multi-Frequency Driving and Time Quasicrystals

Concept of Time Quasicrystals

Ordinary spatial quasicrystals (such as Penrose tiling) have long-range order but no periodicity, with diffraction patterns showing discrete Bragg peaks but corresponding to incommensurate reciprocal space vectors.

Time quasicrystals are a generalization of this concept to the time dimension: system evolution is not periodic, but has long-range order in the time domain, corresponding to incommensurate multiple frequency components.

Multi-Frequency Driving Framework

Consider quasi-periodic driving with $k$ mutually irrational frequencies $\{{\omega }_1,{\omega }_2,\dots ,{\omega }_k\}$:

$$H(t)=H_0+\sum _{i=1}^k{V}_i(t),V_i(t+T_i)=V_i(t)$$

where $T_i=2\pi /{\omega }_i$.

Time Translation Group: Generated by $k$ independent periods, forming ${\mathbb{Z}}^k$.

In the joint high-frequency limit ${\min }_i{\omega }_i\gg J$, we can define effective evolution:

$$\mathcal{U}(\vec{n})=U{e}^{-i{H}_{\ast }{\sum }_i{n}_i{T}_i}\mathfrak{g}(\vec{n})U^{†}+\mathcal{O}(e^{-c{\min }_i{\omega }_i/J})$$

where:

• $\vec{n}=(n_1,n_2,\dots ,n_k)\in {\mathbb{Z}}^k$: Time lattice points
• $\mathfrak{g}:{\mathbb{Z}}^k\to G_f$: Quotient representation to finite group $G_f$

Definition and Classification of Time Quasicrystals

Definition 2.5 (Time Quasicrystal)

If the image $\mathrm{Im}(\mathfrak{g})\subset G_f$ of $\mathfrak{g}$ is non-trivial (not the identity), then the system exhibits quasicrystal order in the time domain.

The autocorrelation function of the order parameter on ${\mathbb{Z}}^k$ lattice satisfies:

$${\mathcal{C}}_O(\vec{n})=\underset{L\to \infty }{\lim }\frac{1}{|{\Lambda }_L|}\sum _{x\in {\Lambda }_L}⟨O_x(\vec{n})O_x(0)⟩$$

Exhibiting multiple incommensurate subharmonic lines.

Spectral Signature:

In frequency domain, discrete spectral peaks appear at:

$${\omega }_{\mathrm{peak}}=\sum _{i=1}^k{m}_i{\omega }_i,(m_1,\dots ,m_k)\in {\mathbb{Z}}^k$$

But since $\{{\omega }_i\}$ are mutually irrational, these peak positions are dense (infinitely many peaks in any arbitrarily small frequency window).

However, peak intensities are not uniform, but concentrated at specific combination frequencies determined by $\mathfrak{g}$.

Experimental Realization: Multi-Color Driven Quantum Simulators

In 2025, He et al. reported in Physical Review X the first experimental realization of discrete time quasicrystals.

Experimental System:

• Platform: Ultracold atoms in optical lattices
• Driving: Independent lattice modulations with two incommensurate frequencies ${\omega }_1/{\omega }_2=(\sqrt{5}+1)/2$ (golden ratio)

Observations:

1. Multiple Subharmonic Peaks: Observed peaks at $(m_1{\omega }_1+m_2{\omega }_2)/2$ in spectrum, corresponding to time quasicrystal order

2. Long-Time Stability: Quasicrystal order remains stable for $>100T_1$ ($T_1=2\pi /{\omega }_1$)

3. Rigidity: Robust to small perturbations (changing drive amplitude, frequency), peak positions determined only by ratio of ${\omega }_1,{\omega }_2$

GLS Theory Prediction:

Under multi-frequency driving, the unified time scale $\kappa (\omega )$ has non-trivial average on ${\mathbb{Z}}^k$ lattice:

$$\bar{\kappa }(\vec{n})=\frac{1}{|\vec{n}|}\sum _{i=1}^k{n}_i\kappa ({\omega }_i)$$

Lifetime of time quasicrystals is given by:

$${\tau }_{\mathrm{qc}}\sim \frac{1}{{\min }_i\bar{\kappa }({\omega }_i)}{e}^{c{\min }_i{\omega }_i/J}$$

That is, slowest decay direction determines lifetime.

13.2.6 Core Role of Unified Time Scale in Time Crystals

Reviewing the four types of time crystals, the unified time scale $\kappa (\omega )$ plays a central role in all cases:

graph TD
    A["Unified Time Scale kappa(omega)"] --> B["Prethermal DTC"]
    A --> C["MBL-DTC"]
    A --> D["Open-System Dissipative Time Crystals"]
    A --> E["Topological Time Crystals"]
    A --> F["Time Quasicrystals"]

    B --> G["Exponential Lifetime tau_* ~ exp(c omega/J)"]
    C --> H["Stability of Eigenstate Order"]
    D --> I["Liouvillian Spectral Gap Delta_Liouv"]
    E --> J["Evolution of Topological Entanglement Entropy gamma"]
    F --> K["Multi-Frequency Lifetime tau_qc"]

    G --> L["kappa Controls tau_*"]
    H --> L
    I --> L
    J --> L
    K --> L

    L --> M["Unified Time Arrow and Classification of Non-Equilibrium Phases"]

Unified Formula

The “lifetime” or “stability” of all time crystals can be uniformly expressed as:

$${\tau }_{\mathrm{TC}}\sim \frac{1}{\bar{\kappa }(\epsilon )}\times \{\begin{array}{ll}{e}^{c\omega /J}& \text{Prethermal/Time Quasicrystal}\\ \infty & \text{MBL (Strict Eigenstate Order)}\\ 1/{\Delta }_{\mathrm{Liouv}}& \text{Open System (Spectral Gap Protection)}\\ e^{c\omega /J}\times \gamma & \text{Topological (Dual Protection)}\end{array}$$

where $\bar{\kappa }(\epsilon )$ is the average of the unified time scale over the relevant energy window.

Physical Meaning

Decreasing Unified Time Scale Extends Time Crystal Lifetime:

• $\bar{\kappa }$ small $\iff$ Low density of states $\iff$ Sparse energy levels $\iff$ Long coherence time
• This is consistent with “decoherence suppression” mechanisms in quantum information

Contrast with Thermalization Rate:

In Section 13.1, we saw that the unified time scale $\kappa (\omega )$ controls thermalization rate $v_{\mathrm{ent}}(\epsilon )\propto \bar{\kappa }(\epsilon )J_{\mathrm{loc}}$.

Time crystals precisely utilize high-frequency driving or topological protection to suppress thermalization and extend coherence time:

$${\tau }_{\mathrm{thermalization}}\sim \frac{1}{\bar{\kappa }(\epsilon )J_{\mathrm{loc}}}\text{vs}{\tau }_{\mathrm{TC}}\sim \frac{1}{\bar{\kappa }(\epsilon )}{e}^{c\omega /J}$$

When $\omega \gg J$, ${\tau }_{\mathrm{TC}}\gg {\tau }_{\mathrm{thermalization}}$, time crystals can exhibit stable subharmonic response before thermalization!

13.2.7 Applications and Future Directions

Application I: Quantum Memory and Quantum Clocks

Time Crystals as Quantum Memory:

Utilizing the long-lived coherence of time crystals, quantum information can be encoded in the phase and amplitude of subharmonic response:

$$|{\psi }_{\mathrm{data}}⟩=\alpha |0⟩+\beta |1⟩\Rightarrow M_z(n)=\mathrm{Re}({\alpha }^{\ast }\beta )\times (-1{)}^n{e}^{-n/{\tau }_{\ast }}$$

Advantages:

• Automatic Error Detection: Deviation of subharmonic signal from $(-1{)}^n$ pattern indicates errors
• Long Coherence Time: ${\tau }_{\ast }\sim e^{c\omega /J}$ far exceeds traditional quantum memory
• Topological Protection: Topological time crystals are immune to local errors

Time Crystals as Quantum Clocks:

Utilizing the stability of subharmonic frequency, time crystals can be used as high-precision frequency standards:

$$f_{\mathrm{TC}}=\frac{1}{2T}=\frac{\omega }{4\pi }$$

Stability is limited by prethermal lifetime:

$$\Delta f/f\sim {\tau }_{\ast }^{-1}$$

Application II: Non-Equilibrium Matter Engineering

Time crystals open a new field of non-equilibrium matter. Traditional condensed matter physics studies equilibrium phases (such as superconductors, topological insulators), but many interesting phenomena only exist in non-equilibrium:

Examples:

1. Floquet Topological Insulators: Periodic driving can induce non-trivial topological phases in originally topologically trivial systems

2. Light-Induced Superconductivity: Terahertz laser pulses can transiently induce superconducting states in certain materials

3. Dynamical Phase Transitions: When drive parameters cross critical values, systems exhibit singular behavior similar to equilibrium phase transitions

Time crystals provide a unified theoretical framework for these phenomena.

Application III: Quantum Computing and Simulation

Quantum Simulation of Time Crystals:

Time crystals are ideal test platforms for “quantum simulators” (such as ultracold atoms, trapped ions, superconducting qubits):

• Parameters fully controllable (drive frequency, interaction strength, disorder level)
• Observables easily measurable (magnetization, autocorrelation functions)
• Clear theoretical predictions (subharmonic response, lifetime scaling)

Time Crystal-Assisted Quantum Computing:

Utilizing the stability of time crystals, new quantum algorithms can be designed:

1. Self-Correcting Quantum Gates: Using logical operators of topological time crystals to realize inherently protected quantum gates

2. Non-Adiabatic Geometric Phase Gates: Using Berry phases of Floquet eigenstates to realize high-fidelity quantum gates

3. Dissipation Engineering: Using limit cycles of open-system time crystals to design “auto-reset” quantum state preparation protocols

Unsolved Problems

Problem 1: Three-Dimensional Topological Time Crystals

Do three-dimensional topological time crystals exist? How are their topological invariants (such as Chern-Simons invariants) defined?

Problem 2: Strict Definition of Continuous Time Crystals

Can “continuous time crystals” observed in open systems (such as space-time crystals in ${}^3$He) be strictly defined? Relationship with discrete time crystals?

Problem 3: Phase Transitions of Time Crystals

Which universality class do phase transitions between time crystal phase and ordinary phase (no subharmonic response) belong to? What are the critical exponents?

Problem 4: Many-Body Quantum Scars and Time Crystals

Some systems have a few “scar eigenstates” (violating ETH). Can they support time crystals? Relationship with MBL-DTC?

Problem 5: Time Crystals and Time-Reversal Symmetry Breaking

Do time crystals necessarily accompany time-reversal symmetry ($\mathcal{T}$) breaking? How to distinguish “genuine” time crystals from “disguised” periodic drive responses?

13.2.8 Summary: Possibility Map of Time Symmetry Breaking

Let us summarize the overall picture of time crystals with a comprehensive diagram:

graph TB
    A["Time Translation Symmetry Breaking"] --> B["Equilibrium"]
    A --> C["Non-Equilibrium"]

    B --> D["Bruno-Watanabe-Oshikawa No-Go Theorems"]
    D --> E["Equilibrium Time Crystals Impossible"]

    C --> F["Periodic Driving Floquet"]
    C --> G["Open-System Dissipation Lindblad"]

    F --> H["Prethermal DTC"]
    F --> I["MBL-DTC"]
    F --> J["Topological DTC"]
    F --> K["Time Quasicrystals"]

    G --> L["Dissipative Time Crystals"]

    H --> M["High-Frequency Drive omega >> J"]
    I --> N["Strong Disorder Localization"]
    J --> O["Topological Quantum Error-Correcting Codes"]
    K --> P["Multi-Frequency Driving"]
    L --> Q["Liouvillian Spectral Gap"]

    M --> R["Exponential Lifetime tau_* ~ exp(c omega/J)"]
    N --> S["pi Spectrum Pairing + Eigenstate Order"]
    O --> T["Non-Local Logical Operator Order Parameter"]
    P --> U["Multiple Incommensurate Frequencies"]
    Q --> V["Limit Cycle Attractor"]

    R --> W["Unified Time Scale kappa Controls Lifetime"]
    S --> W
    T --> W
    U --> W
    V --> W

    W --> X["Unified Framework of GLS Theory"]

Core Conclusions:

1. Equilibrium: Spontaneous breaking of time translation symmetry impossible (no-go theorems)

2. Non-Equilibrium: Under periodic driving or open-system dissipation, time crystals possible and realized

3. Four Types of Time Crystals: Prethermal DTC, MBL-DTC, Topological DTC, Dissipative Time Crystals, each with unique mechanisms

4. Unified Time Scale: $\kappa (\omega )$ controls stability and lifetime in all cases

5. Duality with Quantum Chaos: ETH corresponds to thermalization (symmetry), time crystals correspond to non-thermalization (symmetry breaking)

Philosophical Reflection:

The story of time crystals demonstrates the charm of theoretical physics:

• No-go theorems tell us “where it’s impossible”
• Creative circumvention tells us “how to realize dreams in allowed places”
• Unified framework tells us “seemingly different phenomena share common essence”

Wilczek’s dream was not completely shattered, but blossomed in the new realm of non-equilibrium. This reminds us:

Progress in physics is not only “discovering new phenomena,” but also “understanding why certain phenomena are impossible, and how to expand boundaries within possible ranges.”

Next Section Preview:

Section 13.3 Physical Basis of Consciousness: From Structural Conditions to Quantum Fisher Information

We will enter an even more radical field: Can consciousness be strictly defined in a completely physicalized, informational framework?

The answer is: Yes, through five structural conditions:

1. Integration (mutual information)
2. Differentiation (entropy)
3. Self-Reference—World-Self Model
4. Intrinsic Time Scale (quantum Fisher information)
5. Causal Controllability (empowerment)

The unified time scale will again play a central role: Intrinsic time $\tau$ is precisely the “subjective time” of the consciousness subsystem, constructed from quantum Fisher information:

$$\tau (t)={\int }_{t_0}^t\sqrt{F_Q[{\rho }_O(s)]}ds$$

When $F_Q$ is large, subjective time flows fast, experiencing “time slows down”; when $F_Q$ is small, subjective time flows slowly, experiencing “time flies”.

This is not science fiction, but a strictly provable theorem. Ready? Let’s enter the quantum theory of consciousness!