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arXiv:2501.18036 · Nature Communications

Realization of two-dimensional discrete time crystals with anisotropic Heisenberg coupling

Eric D. Switzer, Niall Robertson, Nathan Keenan, Ángel Rodríguez-Alcaraz, Andrea D'Urbano, Bibek Pokharel, Talat Shahnaz Rahman, Oles Shtanko, Sergiy M. Zhuk, Nicolás Lorente

Published Jan 28, 2026• 1 citation
arXivHTMLPDF

Abstract

A discrete time crystal (DTC) is an out-of-equilibrium phase of matter that spontaneously breaks discrete time-translation symmetry. Previous studies have been limited to a set of models with Ising-like couplings - and mostly only in one dimension - thus precluding our understanding of the existence (or not) of DTCs in models with more realistic interactions. In this work, by combining the latest generation of IBM quantum processors with state-of-the-art tensor network methods, we demonstrate the existence of a DTC in a two-dimensional system governed by anisotropic Heisenberg interactions. We uncover a rich phase diagram encompassing spin-glass, ergodic, and time-crystalline phases, and identify the interplay of initialization, interaction anisotropy, and driving protocols in stabilizing the DTC phase. By extending the study of Floquet matter beyond simplified models, we lay the groundwork for exploring how driven systems bridge the gap between quantum coherence and emergent non-equilibrium thermodynamics.

DOI
10.1038/s41467-025-67787-1

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A discrete time crystal (DTC) is an out-of-equilibrium phase of matter that spontaneously breaks discrete time-translation symmetry. Previous studies have been limited to a set of models with Ising-like couplings - and mostly only in one dimension - thus precluding our understanding of the existence (or not) of DTCs in models with more realistic interactions. In this work, by combining the latest generation of IBM quantum processors with state-of-the-art tensor network methods, we demonstrate the existence of a DTC in a two-dimensional system governed by anisotropic Heisenberg interactions. We uncover a rich phase diagram encompassing spin-glass, ergodic, and time-crystalline phases, and identify the interplay of initialization, interaction anisotropy, and driving protocols in stabilizing the DTC phase. By extending the study of Floquet matter beyond simplified models, we lay the groundwork for exploring how driven systems bridge the gap between quantum coherence and emergent non-equilibrium thermodynamics.
The discrete time crystal (DTC) is a nonequilibrium phase of matter that spontaneously breaks time-translation symmetry. Disorder-induced many-body localization can stabilize the DTC phase by breaking ergodicity and preventing thermalization. Here, we observe the hallmark signatures of the many-body–localized DTC using a quantum simulation platform based on individually controllable carbon-13 nuclear spins in diamond. We demonstrate long-lived period-doubled oscillations and confirm that they are robust for generic initial states, thus showing the characteristic time-crystalline order across the many-body spectrum. Our results are consistent with the realization of an out-of-equilibrium Floquet phase of matter and introduce a programmable quantum simulator based on solid-state spins for exploring many-body physics.
We report on the first experimental observation of a non-equilibrium phase of matter, the discrete time crystal (DTC). A DTC breaks time-translational symmetry and displays spatio-temporal quantum order in all of its eigenstates, a feature dubbed “eigenstate order”. We implement Floquet dynamics on a 1D chain of 20 superconducting qubits [2]. Engineered disorders in the two-qubit couplings allow many-body localization (MBL) to occur despite strong external drive, thereby stabilizing the non-equilibrium phase [3]. We carefully validate the phase structure of the DTC by probing the average response of all eigenstates belonging to the Floquet unitary. Using a suitable choice of order parameter, we further identify the location of the MBL-ergodicity crossover via experimentally observed finite-size effects. These results open a direct path to studying quantum phase transitions and critical phenomena on NISQ quantum processors.
Discrete time crystals (DTC) have emerged as a significant phase of matter for out-of-equilibrium many-body systems. In this work, we study how random long-range interactions contribute to the stability of the DTC phase. Generally, a stable DTC phase is believed to be realized in disordered systems with short-range interactions. Here, we explore a periodically driven quantum Sherrington-Kirkpatrick (SK) model of Ising spin-glass in which all spins are coupled to each other randomly. We investigate the possibilities of the DTC phase in the SK model within three different driving protocols and found that the quantum SK model exhibits a robust DTC phase, although systems with uniform long-range interactions can exhibit only prethermal DTC phase. Further, we demonstrate that disorder in the local XYXYXY term or transverse field is crucial for stabilizing a broad DTC phase, despite random couplings in the SK model. Our analysis shows that the stability of the DTC phase is determined by the non-ergodic nature of the Floquet eigenstates.
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.
Discrete-Time Crystals (DTC) are a non-equilibrium phase of matter characterized by the breaking of time-translation symmetry in periodically driven quantum systems. In this work, we present a detailed thermodynamic analysis of a DTC in a one-dimensional spin-1/2 chain coupled to a thermal bath. We derive a master equation from the microscopic model, and we explore key thermodynamic quantities, such as work, heat, and entropy production. Our results reveal that the DTC signature inevitably decays in the presence of environmental noise, but we show that a periodic measurement scheme can mitigate the effects of decoherence, stabilizing the subharmonic oscillations of the DTC for extended periods. These findings provide insights into the robustness of time-crystalline phases and potential strategies for protecting them in experimental settings.
Periodically driven quantum systems host a range of non-equilibrium phenomena which are unrealizable at equilibrium. Discrete time-translational symmetry in a periodically driven many-body system can be spontaneously broken to form a discrete time crystal, a putative quantum phase of matter. We present the observation of discrete time crystalline order in a driven system of paramagnetic PPP -donor impurities in isotopically enriched 28Si^{28}Si28Si cooled below 101010 K. The observations exhibit a stable subharmonic peak at half the drive frequency which remains pinned even in the presence of pulse error, a signature of DTC order. We propose a theoretical model based on the paradigmatic central spin model which is in good agreement with experimental observations, and investigate the role of dissipation in the stabilisation of the DTC. Both experiment and theory indicate that the order in this system is primarily a dissipative effect, and which persists in the presence of spin-spin interactions. We present a theoretical phase diagram as a function of interactions and dissipation for the central spin model which is consistent with the experiments. This opens up questions about the interplay of coherent interaction and dissipation for time-translation symmetry breaking in many-body Floquet systems.