⚛️ quantum physics

Coexistence of inequivalent time-crystalline orders in a Floquet collective spin system

This paper demonstrates that spatially non-uniform periodic driving in collective spin systems, such as the Lipkin-Meshkov-Glick model, enables the coexistence of inequivalent discrete time-crystalline orders and chimera phases in both finite-size and thermodynamic limits.

Original authors: Shashank Mishra, Sayan Choudhury

Published 2026-06-23

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Shashank Mishra, Sayan Choudhury

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

Imagine a giant dance floor filled with thousands of dancers (the "spins"). In this specific dance, everyone is holding hands with everyone else, meaning every dancer can feel what every other dancer is doing. This is a "collective spin system."

Usually, if you play music to get them dancing, you play the same beat for everyone. If the beat is right, the whole crowd might start doing a synchronized move that repeats every two beats, even though the music only changes every one beat. This is called a Time Crystal: a pattern in time that refuses to sync up perfectly with the music, creating its own rhythm.

The Big Twist in This Paper
The researchers asked a simple question: What happens if we play two different beats for two different groups of dancers on the same dance floor?

They split the crowd in half.

Group A hears a beat with a specific rhythm (Field $h_1$ ).
Group B hears a slightly different beat (Field $h_2$ ).

Even though everyone is still holding hands and influencing each other, the researchers discovered something amazing: The two groups can decide to dance to completely different rhythms at the same time.

The "Time Crystal" Party
Here is what they found when they tuned the two beats:

The "Double-Beat" vs. "Triple-Beat" Party:
Sometimes, Group A settles into a rhythm where they repeat a move every 2 music cycles, while Group B settles into a rhythm where they repeat a move every 6 music cycles. They are dancing in the same room, holding hands, but they have established two different, stable time-crystal orders. They are "coexisting" without forcing each other to change.
The "Chimera" Dance:
In some cases, one group is doing a wild, complex time-crystal dance (repeating every few beats), while the other group is just bobbing along perfectly in sync with the music (a "synchronized" state). The paper calls this a "Chimera DTC." It's like a party where half the room is doing a complex, chaotic dance, and the other half is just standing still and nodding to the beat, yet they are all part of the same connected group.
The "Tunable" Rhythm:
When the two beats are very similar, the whole crowd can switch between different global rhythms together. It's like a volume knob that lets you smoothly transition the entire dance floor from one type of time-crystal dance to another.

Does This Work in the Real World?
The researchers first did the math for an infinite number of dancers (the "thermodynamic limit"). But they also ran simulations with a smaller, realistic number of dancers (100 spins). They found that even with this smaller group and the "noise" of quantum mechanics (which usually messes up delicate patterns), these strange coexisting rhythms still survived.

Why It Matters
The paper proves that you don't need to build two separate machines to get two different time-crystal behaviors. You can create a single system where different parts naturally settle into different, stable, non-equilibrium rhythms just by applying a slightly different "push" to different sections.

In a Nutshell:
Think of it like a choir where the conductor gives a different tempo to the sopranos and the basses. Even though the singers are all listening to each other, the sopranos might settle into a slow, repeating melody, while the basses settle into a fast, repeating melody. The paper shows that this "split personality" in time is not only possible but robust, even when the singers are tightly connected.

Technical Summary: Coexistence of Inequivalent Time-Crystalline Orders in a Floquet Collective Spin System

Problem Statement
While the classification of equilibrium phases has been extended to periodically driven (Floquet) systems, and Discrete Time Crystals (DTCs) have been established as robust non-equilibrium phases exhibiting discrete time-translation symmetry breaking (TTSB), existing research has predominantly focused on global DTC orders emerging under spatially uniform driving. A critical open question remains: can spatially non-uniform driving induce qualitatively new forms of DTC order, specifically the coexistence of distinct dynamical phases within a single, globally coupled system?

Methodology
The authors investigate this question using the paradigmatic infinite-range interacting Lipkin-Meshkov-Glick (LMG) model. The system consists of $N$ spins partitioned into two equal sub-regions. The model is subjected to a spatially non-uniform periodic drive defined by a Hamiltonian $H(t)$ with two distinct phases per period $T$ :

Interaction Phase ( $0 < t \leq T/2$ ): An all-to-all Ising-like interaction $J$ acts globally on the collective spin operators.
Drive Phase ( $T/2 < t \leq T$ ): The two sub-regions are driven by distinct transverse magnetic fields, $h_1$ and $h_2$ , respectively.

The study employs two complementary approaches:

Thermodynamic Limit ( $N \to \infty$ ): The dynamics are analyzed using semiclassical equations of motion derived via mean-field theory. The stability of phases is characterized using a decorrelator $D(nT)$, which measures the sensitivity to initial conditions (distinguishing non-ergodic phases where $D \sim 0$ from chaotic phases where $D \sim 1/\sqrt{2}$ ).
Finite-Size Quantum Systems ( $N=100$ ): To verify the robustness of these phases against quantum fluctuations, the authors perform exact diagonalization within the symmetric collective-spin subspace. They utilize the Fidelity Out-of-Time-Ordered Correlator (FOTOC) as the quantum analog of the decorrelator and analyze the Fourier spectra of regional and total magnetizations.

Key Contributions and Results
The study establishes that minimal spatial inhomogeneity in the driving field ( $h_1 \neq h_2$ ) induces a rich landscape of dynamical phase coexistence, even in the presence of infinite-range interactions:

Coexistence of Inequivalent DTCs: The system can self-organize into states where distinct sub-regions exhibit different temporal orders. For example, one region may exhibit a period-doubling ( $2T$ ) DTC while the other exhibits a higher-order (HO) DTC (e.g., $3T$ , $6T$ , or $14T$ ). These coexisting phases are characterized by two sharp, robust peaks in the Fourier spectrum of the total magnetization, distinct from the single peak of a global DTC.
Chimera DTCs: The authors demonstrate the emergence of "chimera" states where a period-doubling DTC coexists with a Floquet-synchronized (period- $T$ ) or oscillatory phase.
Global DTC Tunability: When $h_1$ and $h_2$ are proximate, the system realizes a tunable family of global DTC phases, allowing transitions between different global periods by adjusting the field strengths.
Finite-Size Robustness: While quantum fluctuations in finite systems ( $N=100$ ) suppress certain higher-order DTC regions compared to the semiclassical limit, the primary phenomenology of coexisting DTCs, chimera DTCs, and coexistence with oscillatory phases persists. The authors note that the suppression of specific HO-DTCs at $N=100$ is a finite-size effect, as these phases re-emerge at larger system sizes ( $N \sim 1000$ ).

Significance and Claims
The paper claims to establish spatially structured driving as a powerful and versatile route to engineer non-equilibrium phase coexistence in collective spin systems. The primary significance lies in demonstrating that global interactions do not preclude the formation of spatially distinct dynamical orders when the drive is inhomogeneous.

Specifically, the authors assert:

The coexistence of inequivalent time-crystalline orders is a genuine feature of the thermodynamic limit and is not an artifact of the semiclassical approximation.
These phases are observable in experimentally relevant finite-size systems (specifically citing trapped-ion platforms with $N \sim 100$ ), making the proposal amenable to near-term experimental realization.
The boundary between sub-regions hosting distinct temporal orders can be viewed as a temporal analog of a spatial domain wall, suggesting new avenues for exploring temporal topological defects.

The work generalizes previous studies on chimera states in finite spin networks and provides a concrete diagnostic (Fourier response of global magnetization) for the experimental certification of these coexisting phases.

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