⚛️ quantum physics

Rescaled Mandelstam Tamm characterization of discrete time crystal response in a disordered Floquet Ising chain

This paper introduces a rescaled Mandelstam-Tamm functional to characterize discrete time crystal dynamics in a disordered Floquet Ising chain, demonstrating that the observed period-two structure arises primarily from endpoint geometry rather than energy dispersion and strongly correlates with locked spin responses.

Original authors: Abrar Ahmed Naqash, Salman Sajad Wani, Saif Al-Kuwari

Published 2026-08-21

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CC0 1.0

Original authors: Abrar Ahmed Naqash, Salman Sajad Wani, Saif Al-Kuwari

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

In the quantum world, time is usually treated as a smooth, unbroken river. However, when scientists subject a collection of atoms to a rhythmic, repeating push and pull, something strange can happen. Instead of settling into a chaotic, overheated mess, the system can lock into a rigid pattern that repeats at a slower pace than the driving force. This phenomenon, known as a discrete time crystal, represents a new state of matter that defies the usual tendency of systems to lose their structure over time. To understand why this happens, physicists often look at how far a quantum system travels through its possible states and how much energy it fluctuates along the way. A classic rule, known as the Mandelstam–Tamm relation, sets a limit on how fast a quantum system can change based on its energy spread. For decades, this rule has been a reliable tool for understanding time, but its application to these exotic, repeating quantum systems has remained murky, particularly regarding how the system's size affects the measurement.

A team of researchers has now brought clarity to this picture by developing a precise method to track the journey of a quantum system through a repeating cycle. They focused on a specific model: a chain of magnetic atoms, or spins, arranged in a line with random imperfections and subjected to a two-step pulse sequence. In this setup, the system is first allowed to interact with its neighbors, and then it is given a global flip. The researchers wanted to see if the system could maintain a stable, repeating rhythm despite the disorder and the imperfect nature of the pulses. To do this, they derived a new, exact formula that breaks down the system's evolution into its individual steps, allowing them to calculate the total "distance" the system travels in its quantum state space without making simplifying assumptions about whether the steps commute or cancel each other out.

The study revealed that the behavior of these time crystals is deeply connected to the geometry of the system's path. When the researchers analyzed chains of different lengths, ranging from eight to sixteen atoms, they found that the energy fluctuations along the path grow in a predictable way as the system gets larger. Specifically, the total energy variation scales with the square root of the number of atoms. This discovery allowed them to rescale their measurements, effectively normalizing the data so that systems of different sizes could be compared fairly. Once this scaling was applied, a clear pattern emerged: in the regions where the system exhibited a locked, time-crystal response, the path the system took split into two distinct branches. One branch corresponded to the odd-numbered steps in the cycle, and the other to the even-numbered steps. These two branches remained separated throughout the entire observation window of one hundred cycles, never merging into a single path.

What drives this separation is not the energy fluctuations themselves, but rather the final position of the system at the end of each step. The researchers found that the system's return angle—how far it has moved away from its starting point—alternates strongly between odd and even steps. In contrast, the normalized energy spread along the path showed very little difference between the odd and even steps. This indicates that the distinct two-step rhythm of the time crystal is primarily a geometric feature of where the system ends up, rather than a result of how much energy it is jiggling with along the way. The study confirms that this geometric signature is tightly linked to the system's ability to maintain a locked spin response, which is the hallmark of a time crystal.

The researchers also tested how this geometric signature relates to other ways of identifying time crystals, such as looking at the spacing between energy levels. They found that while the geometric path and the spin response move in perfect lockstep, showing a very strong correlation, the connection to the energy level spacing is much weaker and depends heavily on the specific conditions of the experiment. This suggests that the geometric path provides a more direct and reliable way to characterize the time-crystal behavior in finite systems than looking at the energy spectrum alone. By focusing on the actual path the system takes and how it returns to its starting point, the researchers have provided a clearer, more intuitive picture of how these quantum systems resist disorder and maintain their rhythmic structure. The work does not claim to have solved the mystery of time crystals entirely, but it offers a robust, size-independent tool for measuring their stability and understanding the specific role of geometry in their formation.

Technical Summary: Rescaled Mandelstam–Tamm Characterization of Discrete Time-Crystal Response

Problem Statement
The paper addresses the characterization of discrete time-crystal (DTC) dynamics in disordered Floquet systems, specifically focusing on the geometric and energetic origins of period-two structures in the Mandelstam–Tamm (MT) lower-bound functional. While the MT relation provides a geometric lower bound on evolution time based on the Fubini–Study return angle and path-averaged energy dispersion, the distinct size and temporal dependencies of these two quantities have obscured the origin of the period-two MT structure observed in DTC-like dynamics and its specific relation to the locked spin response. Existing literature has utilized spectral pairing, gap ratios, and nonanalyticities in return probability to diagnose DTC order, but a geometric characterization linking the MT functional directly to the subharmonic spin response in finite-size systems remains to be fully elucidated.

Methodology
The authors develop a theoretical framework and apply it to a disordered Floquet Ising chain with open boundary conditions.

Derivation of Segment-Resolved MT Functional: For binary Floquet drives (where Hamiltonians $H_1$ and $H_2$ act sequentially), the authors derive an exact expression for the pure-state MT lower-bound functional without assuming that the segment Hamiltonians commute. They express the time-integrated energy dispersion as a duration-weighted sum of dispersions evaluated in the states entering each segment. This involves "dressing" the second segment Hamiltonian ( $H_2$ ) by the propagator of the first ( $U_1$ ), yielding an operator $H_2^{\langle B \rangle} = U_1^\dagger H_2 U_1$ .
Scaling Analysis: Theoretical arguments based on the absolute and uniform summability of connected covariances of local energy terms suggest an $O(\sqrt{L})$ upper bound on the path-averaged energy dispersion for local Hamiltonians. Consequently, the authors propose rescaling the MT functional by $\sqrt{L}$ to create a size-normalized diagnostic.
Numerical Simulation: The model is simulated for system sizes $L \in \{8, 10, 12, 16\}$ with $N_{dis} = 50$ disorder realizations. The observation window spans $N=100$ periods. The study classifies parameter points as "DTC-like" if they exhibit a locked period-two spin response and localized-like adjacent-gap statistics.
Statistical Correlation: The authors employ Spearman rank correlation and partial rank correlation (controlling for pulse error $\epsilon$ and interaction strength $J_z$ ) to quantify the relationship between the rescaled MT component, the locked spin response, and the adjacent-gap ratio.

Key Contributions

Exact Segment-Resolved Expression: The paper provides an exact, non-perturbative expression for the MT functional in piecewise-constant binary drives, applicable even when $[H_1, H_2] \neq 0$ . This allows for the evaluation of the path-averaged energy dispersion directly on the stroboscopic orbit.
Size Normalization Validation: Numerical data across four system sizes confirm that the path-averaged energy dispersion scales as $\sqrt{L}$ , supporting the use of $\sqrt{L}$ as the leading normalization factor for the MT diagnostic in finite-size regimes.
Decomposition of Period-Two Structure: The study decomposes the period-two structure of the rescaled MT sequence into its geometric (endpoint angle) and energetic (path-averaged dispersion) components.
Correlation with Spin Response: The work establishes a strong statistical link between the period-two component of the rescaled MT sequence and the locked spin response, while demonstrating a weaker, control-dependent link to spectral indicators like the adjacent-gap ratio.

Results

Scaling Behavior: The disorder-averaged path-averaged energy dispersion is consistent with $\sqrt{L}$ scaling across the tested sizes. The relative range of the normalized dispersion is small (median ~1.0%), validating the rescaling.
Origin of Branch Splitting: In DTC-like regions, the rescaled MT sequence separates into distinct odd and even branches. The analysis reveals that this separation is driven primarily by the alternation of the endpoint Fubini–Study return angle (which is determined by return fidelity). The size-normalized path-averaged energy dispersion shows little discernible parity dependence, indicating that the geometric endpoint properties, rather than the accumulated path length, are the main source of the branch splitting.
Correlation with DTC Signatures:
- Spin Response: There is a very strong partial Spearman rank correlation (approx. 0.97) between the locked MT component and the locked spin response after controlling for pulse error and interaction strength. This indicates that the MT sequence tracks the subharmonic structure of the spin dynamics.
- Spectral Statistics: The correlation between the locked MT component and the disorder-averaged adjacent-gap ratio is significantly weaker and highly dependent on the control variables. The sign of the correlation can even change depending on the residualization method used.
Finite-Size Limitations: The authors note that the unrescaled MT bound remains far from saturation (values are orders of magnitude below the physical time), meaning the rescaled quantity serves as a state-dependent diagnostic of the trajectory rather than a tight bound on elapsed time. Furthermore, the data do not support a controlled pointwise extrapolation to the thermodynamic limit due to sensitivity to finite-size correction forms.

Significance and Claims
The paper claims that the rescaled Mandelstam–Tamm functional provides a state-dependent geometric characterization of period-two dynamics in finite-size systems. It quantifies the association between the global return geometry of the quantum trajectory and the locked spin response characteristic of DTC order.

Crucially, the authors are modest in their claims:

They explicitly state that the MT diagnostic is not an additional DTC order parameter, nor does it provide independent evidence for DTC order, as it is calculated from the same trajectories and uses the same Fourier projection as the spin response.
The work does not claim to define a thermodynamic phase diagram; rather, it identifies finite-size DTC-like, localized non-DTC, and thermal-like sectors.
The primary contribution is the decomposition of the MT functional, showing that in the DTC regime, the period-two structure is dominated by endpoint geometry (return fidelity) rather than the path-averaged energy dispersion, and that this geometric structure is tightly coupled to the subharmonic spin response.

Technical Summary: Rescaled Mandelstam–Tamm Characterization of Discrete Time-Crystal Response

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