Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.19403

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

Abrar Ahmed Naqash, Salman Sajad Wani, Saif Al-Kuwari

quant-phSubmitted Aug 19, 2026• Announced Aug 21, 2026• 0 citations
arXivHTMLPDF

Abstract

For pure state unitary dynamics, the Mandelstam Tamm (MT) lower-bound functional compares the endpoint Fubini Study return angle with path averaged energy dispersion. Their distinct size and temporal dependences obscure the origin of period two MT structure in discrete time crystal (DTC) like dynamics and its relation to the spin response. For binary Floquet drives, we derive an exact segment resolved MT expression without assuming commuting segment Hamiltonians and apply it to a disordered Floquet Ising chain. Absolute and uniform summability of connected covariances of local energy terms implies an $O(\sqrt{L})$ upper bound on the path-averaged energy dispersion. Endpoint data for four system sizes are consistent with this leading behavior and support the corresponding rescaling of the MT functional. At a representative point in the finite-size region with a locked spin response, odd and even rescaled MT branches remain separated throughout the $10^2$ period observation window. The return angle alternates strongly, whereas the size-normalized path-averaged energy dispersion shows little discernible parity dependence, indicating that endpoint geometry is the main source of the branch splitting. Across the interacting parameter grid, the period-two MT component has a strong partial Spearman rank correlation with the locked spin response after controlling for pulse error and interaction strength. The rescaled MT functional characterizes the global return geometry of finite-size period-two dynamics and quantifies its association with the locked spin response.

Comments
21 pages, 9 figs

Related papers

Discrete time crystal order in spin chains enabled by Floquet flat bands

Mahbub Rahaman, Anindita Roy

We propose a novel protocol to realize discrete time-crystal (DTC) order in clean, periodically driven spin-$1/2$ chains. In each drive cycle, a global spin flip is followed by a two-tone flat-band segment. This flat-band segment engineers a fully degenerate Floquet quasienergy spectrum, suppresses thermalization, and stabilizes a robust period-doubled subharmonic response. Using exact time evolution, we identify a pronounced subharmonic peak at half the drive frequency in the Fourier spectrum of the order parameter, thereby providing clear evidence for the emergence of stable DTC. The resulting phase is insensitive to system size, interaction strength, and interaction range; however, it remains sensitive to spin-rotation errors ($\varepsilon_r$), which can destabilize the subharmonic response. Compared with disorder-induced many-body localized (MBL) and disorder-free dynamically many-body localized (DMBL) DTCs, we find that the exact flat-band protocol offers a broader tunability of drive parameters, whereas MBL and DMBL based DTCs are more resistant to $\varepsilon_r$. In particular, the $\varepsilon_r$ sensitivity can be suppressed by incorporating additional spin-spin interactions that have modest deviations from the ideal flat-band protocol. This manifests itself in a robust DTC response over a finite window of spin-coupling strengths and drive frequencies. Our results establish flat-band driving as a versatile and experimentally relevant route to DTC order in disorder-free spin systems and motivate further exploration of nonequilibrium phases.

arXiv:2603.14307•Physical review. B./Physical review. B•Jul 1, 2026•0 citations

Discrete time crystals enabled by Floquet strong Hilbert space fragmentation

Ling-Zhi Tang, Xiao Li, Z. D. Wang +1 more

Discrete time crystals (DTCs) are non-equilibrium phases of matter that break the discrete time-translation symmetry and is characterized by a robust subharmonic response in periodically driven quantum systems. Here, we explore the DTC in a disorder-free, periodically kicked XXZ spin chain, which is stabilized by the Floquet strong Hilbert space fragmentation. We numerically show the period-doubling response of the conventional DTC order, and uncover a multiple-period response with beating dynamics due to the coherent interplay of multiple $π$-pairs in the Floquet spectrum of small-size systems. The lifetime of the DTC order exhibits independence of the driving frequency and a power-law dependence on the ZZ interaction strength. It also grows exponentially with the system size, as a hallmark of the strong fragmentation inherent to the Floquet model. We analytically reveal the approximate conservation of the magnetization and domain-wall number in the Floquet operator for the emergent strong fragmentation, which is consistent with numerical results of the dimensionality ratio of symmetry subspaces. The rigidity and phase regime of the DTC order are identified through finite-size scaling of the Floquet-spectrum-averaged mutual information, as well as via dynamical probes. Our work establishes the Floquet Hilbert space fragmentation as a disorder-free mechanism for sustaining nontrivial temporal orders in out-of-equilibrium quantum many-body systems.

quant-pharXiv:2512.14182•Physical review. B./Physical review. B•Dec 17, 2025•3 citations

Discrete time crystals in disordered anisotropic Heisenberg chains

Francesco Formicola, Grazia Di Bello, Antonio De Candia +2 more

A discrete time crystal is an out-of-equilibrium phase of matter characterized by the spontaneous breaking of discrete time-translation symmetry. Using extensive numerical simulations based on matrix-product-state methods, we provide evidence for discrete time-crystalline behavior in strongly disordered spin chains with Heisenberg interactions, including the isotropic point, subject to periodic driving. Starting from a many-body localized regime, we observe that rotations induced by delta kicks produce a pronounced subharmonic response at half the drive frequency in spin observables. We investigate the stability of this response against rotation-angle errors through entanglement entropy, quantum Fisher information, short-range spin correlations, and restricted-control ergotropy. Increasing the rotation error reveals an intermediate dynamical regime separating the time-crystalline and Floquet-localized responses. In this regime, most observables exhibit signatures of weakly correlated dynamics reminiscent of Anderson localization.

quant-pharXiv:2609.04037•Sep 4, 2026•0 citations

Channel-resolved stability selection of subharmonic response in a Floquet Ising chain — audit code, data, and figure pipeline (v1.0-pre-submission)

andrei ursachi

Finite-time return measurements can assign the same stability label to different dynamical responses. We document this ambiguity in a periodically kicked Ising chain initialized in a Neel product state, and replace an earlier numerical claim with a reproducible finite-model benchmark. For ten spins, even-period return highlights identity-like pulse settings as well as a period-doubled cycle. Signed odd-period and Fourier diagnostics distinguish these cases. At the perfect-pulse point, diagonal phase perturbations preserve the basis-state cycle exactly. Under fresh Gaussian transverse rotations, the alternating ensemble-mean local correlation has envelope exp(-2σ²n), independently of the diagonal Ising coupling. Independent local channels also give an elementary product expression for the ensemble cycle fidelity; a shared pulse-angle fluctuation has the same local mean but can have a different fidelity. These ensemble identities do not determine nonlinear per-realization scores, whose averaging order must be retained. All simulation results derive from one newly seeded state-vector dataset with raw traces and source hashes; analytic controls are evaluated separately. The resulting shelves and sampled noise crossings describe a specified preparation, channel, diagnostic and observation window. They supply no evidence for a universal discreteness mechanism, interaction protection from these exact-center ensemble observables, or a thermodynamic time-crystal phase. Corrective release with the corrected manuscript, code, data and figure pipeline. Related Research Square record: https://www.researchsquare.com/article/rs-8790240 . This version replaces the earlier audit bundle with the reviewed reproducibility package. Archived claims are retained for provenance and are not endorsed. This preprint is not peer reviewed.

•May 1, 2026•0 citations

Subharmonic spin correlations and spectral pairing in Floquet time crystals

Alexander-Georg Penner, Harald Schmid, L. I. Glazman +1 more

Floquet time crystals are characterized by the subharmonic behavior of temporal correlation functions. Studying the paradigmatic time crystal based on the disordered Floquet quantum Ising model, we show that its temporal spin correlations are directly related to spectral characteristics and that this relation provides analytical expressions for the correlation function of finite chains, which compare favorably with numerical simulations. Specifically, we show that the disorder-averaged temporal spin correlations are proportional to the Fourier transform of the splitting distribution of the pairs of eigenvalues of the Floquet operator, which differ by π to exponential accuracy in the chain length. We find that the splittings are well described by a log-normal distribution, implying that the temporal spin correlations are characterized by two parameters. We discuss possible implications for the phase diagram of Floquet time crystals.

arXiv:2501.18760•Physical review. B./Physical review. B•May 8, 2025•7 citations

Analytical quantification of strongly disordered discrete time crystals

Anonymous, Biao Huang

Discrete time crystals (DTCs) have emerged as a novel non-equilibrium phenomenon. Nevertheless, analytical quantification for disordered DTCs remains challenging, leaving uncertainties about their behaviors in large scale systems amid current debates on the stability of (Floquet-)many-body localization. In this work, we introduce an analytical framework to calculate the values of key observables in a strongly disordered DTC without fitting parameter. The perturbatively obtained closed-form formulae show quantitative agreement with numerical simulations for inverse participation ratios for eigenstate localization in Fock space, Edwards-Anderson parameters for spin-glass orders, mutual information for long-range entanglement, and the steady-state amplitudes of autocorrelators for period-doubled oscillations. Meanwhile, we demonstrate that eigenstate resonances render the scaling for the deviation of physical observables from their unperturbed values as $O(λ)$, in contrast to non-resonant situations with suppressed deviation $O(λ^2)$. Our newly developed scheme adopts resolvent perturbation formalism, which can directly prescribe arbitrarily higher-order corrections without iterations. With such advantages, we analytically prove that quasienergy corrections for pairwise cat eigenstates are identical up to order $O(λ^{(L/n_{\text{op}})-1})$, where perturbations of strength $λ$ involve at most $n_{\text{op}}$-spin terms. Such spectral pairing deviations quantify the DTC lifetime as $τ_* \sim (1/λ)^{L/n_{\text{op}}}$. We further show that the analytical framework applies to generic DTC models with dominant Ising interaction and a given number of qubits, offering a unified scheme to quantify physical observables for systems beyond numerically accessible sizes, with or without disorders.

arXiv:2512.20707•Physical review. B./Physical review. B•Sep 11, 2026•0 citations
For pure state unitary dynamics, the Mandelstam Tamm (MT) lower-bound functional compares the endpoint Fubini Study return angle with path averaged energy dispersion. Their distinct size and temporal dependences obscure the origin of period two MT structure in discrete time crystal (DTC) like dynamics and its relation to the spin response. For binary Floquet drives, we derive an exact segment resolved MT expression without assuming commuting segment Hamiltonians and apply it to a disordered Floquet Ising chain. Absolute and uniform summability of connected covariances of local energy terms implies an O(L)O(\sqrt{L})O(L​) upper bound on the path-averaged energy dispersion. Endpoint data for four system sizes are consistent with this leading behavior and support the corresponding rescaling of the MT functional. At a representative point in the finite-size region with a locked spin response, odd and even rescaled MT branches remain separated throughout the 10210^2102 period observation window. The return angle alternates strongly, whereas the size-normalized path-averaged energy dispersion shows little discernible parity dependence, indicating that endpoint geometry is the main source of the branch splitting. Across the interacting parameter grid, the period-two MT component has a strong partial Spearman rank correlation with the locked spin response after controlling for pulse error and interaction strength. The rescaled MT functional characterizes the global return geometry of finite-size period-two dynamics and quantifies its association with the locked spin response.
We propose a novel protocol to realize discrete time-crystal (DTC) order in clean, periodically driven spin-1/21/21/2 chains. In each drive cycle, a global spin flip is followed by a two-tone flat-band segment. This flat-band segment engineers a fully degenerate Floquet quasienergy spectrum, suppresses thermalization, and stabilizes a robust period-doubled subharmonic response. Using exact time evolution, we identify a pronounced subharmonic peak at half the drive frequency in the Fourier spectrum of the order parameter, thereby providing clear evidence for the emergence of stable DTC. The resulting phase is insensitive to system size, interaction strength, and interaction range; however, it remains sensitive to spin-rotation errors (εr\varepsilon_rεr​), which can destabilize the subharmonic response. Compared with disorder-induced many-body localized (MBL) and disorder-free dynamically many-body localized (DMBL) DTCs, we find that the exact flat-band protocol offers a broader tunability of drive parameters, whereas MBL and DMBL based DTCs are more resistant to εr\varepsilon_rεr​. In particular, the εr\varepsilon_rεr​ sensitivity can be suppressed by incorporating additional spin-spin interactions that have modest deviations from the ideal flat-band protocol. This manifests itself in a robust DTC response over a finite window of spin-coupling strengths and drive frequencies. Our results establish flat-band driving as a versatile and experimentally relevant route to DTC order in disorder-free spin systems and motivate further exploration of nonequilibrium phases.
Discrete time crystals (DTCs) are non-equilibrium phases of matter that break the discrete time-translation symmetry and is characterized by a robust subharmonic response in periodically driven quantum systems. Here, we explore the DTC in a disorder-free, periodically kicked XXZ spin chain, which is stabilized by the Floquet strong Hilbert space fragmentation. We numerically show the period-doubling response of the conventional DTC order, and uncover a multiple-period response with beating dynamics due to the coherent interplay of multiple πππ-pairs in the Floquet spectrum of small-size systems. The lifetime of the DTC order exhibits independence of the driving frequency and a power-law dependence on the ZZ interaction strength. It also grows exponentially with the system size, as a hallmark of the strong fragmentation inherent to the Floquet model. We analytically reveal the approximate conservation of the magnetization and domain-wall number in the Floquet operator for the emergent strong fragmentation, which is consistent with numerical results of the dimensionality ratio of symmetry subspaces. The rigidity and phase regime of the DTC order are identified through finite-size scaling of the Floquet-spectrum-averaged mutual information, as well as via dynamical probes. Our work establishes the Floquet Hilbert space fragmentation as a disorder-free mechanism for sustaining nontrivial temporal orders in out-of-equilibrium quantum many-body systems.
A discrete time crystal is an out-of-equilibrium phase of matter characterized by the spontaneous breaking of discrete time-translation symmetry. Using extensive numerical simulations based on matrix-product-state methods, we provide evidence for discrete time-crystalline behavior in strongly disordered spin chains with Heisenberg interactions, including the isotropic point, subject to periodic driving. Starting from a many-body localized regime, we observe that rotations induced by delta kicks produce a pronounced subharmonic response at half the drive frequency in spin observables. We investigate the stability of this response against rotation-angle errors through entanglement entropy, quantum Fisher information, short-range spin correlations, and restricted-control ergotropy. Increasing the rotation error reveals an intermediate dynamical regime separating the time-crystalline and Floquet-localized responses. In this regime, most observables exhibit signatures of weakly correlated dynamics reminiscent of Anderson localization.
Finite-time return measurements can assign the same stability label to different dynamical responses. We document this ambiguity in a periodically kicked Ising chain initialized in a Neel product state, and replace an earlier numerical claim with a reproducible finite-model benchmark. For ten spins, even-period return highlights identity-like pulse settings as well as a period-doubled cycle. Signed odd-period and Fourier diagnostics distinguish these cases. At the perfect-pulse point, diagonal phase perturbations preserve the basis-state cycle exactly. Under fresh Gaussian transverse rotations, the alternating ensemble-mean local correlation has envelope exp(-2σ²n), independently of the diagonal Ising coupling. Independent local channels also give an elementary product expression for the ensemble cycle fidelity; a shared pulse-angle fluctuation has the same local mean but can have a different fidelity. These ensemble identities do not determine nonlinear per-realization scores, whose averaging order must be retained. All simulation results derive from one newly seeded state-vector dataset with raw traces and source hashes; analytic controls are evaluated separately. The resulting shelves and sampled noise crossings describe a specified preparation, channel, diagnostic and observation window. They supply no evidence for a universal discreteness mechanism, interaction protection from these exact-center ensemble observables, or a thermodynamic time-crystal phase. Corrective release with the corrected manuscript, code, data and figure pipeline. Related Research Square record: https://www.researchsquare.com/article/rs-8790240 . This version replaces the earlier audit bundle with the reviewed reproducibility package. Archived claims are retained for provenance and are not endorsed. This preprint is not peer reviewed.
Floquet time crystals are characterized by the subharmonic behavior of temporal correlation functions. Studying the paradigmatic time crystal based on the disordered Floquet quantum Ising model, we show that its temporal spin correlations are directly related to spectral characteristics and that this relation provides analytical expressions for the correlation function of finite chains, which compare favorably with numerical simulations. Specifically, we show that the disorder-averaged temporal spin correlations are proportional to the Fourier transform of the splitting distribution of the pairs of eigenvalues of the Floquet operator, which differ by π to exponential accuracy in the chain length. We find that the splittings are well described by a log-normal distribution, implying that the temporal spin correlations are characterized by two parameters. We discuss possible implications for the phase diagram of Floquet time crystals.
Discrete time crystals (DTCs) have emerged as a novel non-equilibrium phenomenon. Nevertheless, analytical quantification for disordered DTCs remains challenging, leaving uncertainties about their behaviors in large scale systems amid current debates on the stability of (Floquet-)many-body localization. In this work, we introduce an analytical framework to calculate the values of key observables in a strongly disordered DTC without fitting parameter. The perturbatively obtained closed-form formulae show quantitative agreement with numerical simulations for inverse participation ratios for eigenstate localization in Fock space, Edwards-Anderson parameters for spin-glass orders, mutual information for long-range entanglement, and the steady-state amplitudes of autocorrelators for period-doubled oscillations. Meanwhile, we demonstrate that eigenstate resonances render the scaling for the deviation of physical observables from their unperturbed values as O(λ)O(λ)O(λ), in contrast to non-resonant situations with suppressed deviation O(λ2)O(λ^2)O(λ2). Our newly developed scheme adopts resolvent perturbation formalism, which can directly prescribe arbitrarily higher-order corrections without iterations. With such advantages, we analytically prove that quasienergy corrections for pairwise cat eigenstates are identical up to order O(λ(L/nop)−1)O(λ^{(L/n_{\text{op}})-1})O(λ(L/nop​)−1), where perturbations of strength λλλ involve at most nopn_{\text{op}}nop​-spin terms. Such spectral pairing deviations quantify the DTC lifetime as τ∗∼(1/λ)L/nopτ_* \sim (1/λ)^{L/n_{\text{op}}}τ∗​∼(1/λ)L/nop​. We further show that the analytical framework applies to generic DTC models with dominant Ising interaction and a given number of qubits, offering a unified scheme to quantify physical observables for systems beyond numerically accessible sizes, with or without disorders.