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DOE OSTI2023

Tuning nonequilibrium phase transitions with inertia

In striking contrast to equilibrium systems, inertia can profoundly alter the structure of active systems. Here, we demonstrate that driven systems can exhibit effective equilibrium-like states with increasing particle inertia, despite rigorously violating the fluctuation–dissipation theorem. Increasing inertia progressively eliminates motility-induced phase separation and restores equilibrium crystallization for active Brownian spheres. This effect appears to be general for a wide class of active systems, including those driven by deterministic time-dependent external fields, whose nonequilibrium patterns ultimately disappear with increasing inertia. The path to this effective equilibrium limit can be complex, with finite inertia sometimes acting to accentuate nonequilibrium transitions. The restoration of near equilibrium statistics can be understood through the conversion of active momentum sources to passive-like stresses. Unlike truly equilibrium systems, the effective temperature is now density dependent, the only remnant of the nonequilibrium dynamics. Further, this density-dependent temperature can in principle introduce departures from equilibrium expectations, particularly in response to strong gradients. Our results provide additional insight into the effective temperature ansatz while revealing a mechanism to tune nonequilibrium phase transitions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

Development of dispensing hardware for safe fueling of heavy duty vehicles

The development of safe dispensing equipment for the fueling of heavy duty (HD) vehicles is critical to the expansion of this newly and quickly expanding market. This paper discusses the development of a HD dispenser and nozzles assembly (nozzle, hose, breakaway) for these new, larger vehicles where flow rates are more than double compared to light duty (LD) vehicles. This equipment must operate at nominal pressures of 700 bar, -40o C gas temperature, and average flow rate of 5-10 kg/min at a high throughput commercial hydrogen fueling station without leaking hydrogen. The project surveyed HD vehicle manufacturers, station developers, and component suppliers to determine the basic specifications of the dispensing equipment and nozzle assembly. The team also examined existing codes and standards to determine necessary changes to accommodate HD components. From this information, the team developed a set of specifications which will be used to design the dispensing equipment. In order to meet these goals, the team performed computational fluid dynamic, pressure modelling, and temperature analysis in order to determine the necessary parameters to meet existing safety standards modified for HD fueling. The team also considered user, operational, and maintenance requirements, such as freeze lock which has been an issue which prevents the removal of the nozzle from LD vehicles. The team also performed a failure mode and effects analysis (FMEA) to identify the possible failures in the design. The dispenser and nozzle assembly will be tested separately, and then installed on an innovative, HD fueling station which will use a HD vehicle simulator to test the entire system.

08 HYDROGEN↗

DOE OSTI2023

Transition in helium bubble strengthening of copper from quasi-static to dynamic deformation

Damage from low-temperature irradiation and the subsequent degradation of materials performance pose significant challenges for the storage of radioactive materials and for peripheral components in some nuclear reactor designs. Fully understanding the mechanical behavior of such materials requires test data for strain rates in both the quasi-static (< 10/s) and dynamic (>> 10/s) regimes. While dynamic testing has generally been avoided in the past for neutron irradiated (contamination concerns) and ion irradiated (insufficient volume) materials, surface-sensitive Richtmyer-Meshkov instability (RMI) tests were used in the present work to overcome these limitations. Here, nanopillar compression, nanoindentation, and RMI testing data from a helium implanted surface layer (~10 µm thick) were compiled to explore the effects of helium bubbles on the materials strength of high-purity copper at strain rates of 0.001/s – 10 8 /s. While nano-mechanical testing revealed increases in yield strength and hardness with increasing helium dose from 1000 to 4000 appm He, RMI indicated no significant changes in strength as compared to unimplanted copper. Here, this discrepancy in behavior was rationalized through a combination of recent literature and follow-on molecular dynamics (MD) simulations, leading to the conclusion that the nanoscale helium bubbles acting as dispersed barriers to dislocation motion at quasi-static strain rates collapse under shock loading and cease to be effective barriers at high strain rates.

36 MATERIALS SCIENCE↗

DOE OSTI2022

The temperatures of ejecta transporting in vacuum and gases

In this work, we measure continuous thermal radiance from evolving clouds of liquid metal fragments ejected into vacuum, nonreactive, and reactive gas. We implement a model for the thermalization of the ejecta and gas and use this to constrain the absolute temperature of the ejecta cloud. This model enables further analyses of ejecta thermal behavior under a variety of conditions.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

Carrier Dynamics of Polar, Semipolar, and Nonpolar InGaN/GaN LEDs Measured by Small-Signal Electroluminescence

The carrier dynamics in InGaN/GaN light-emitting diodes (LEDs) are directly tied to their efficiency and maximum modulation speed, which are important metrics for solid-state lighting, displays, and optical communication. In this work, we measure the carrier dynamics of a variety of InGaN/GaN LEDs using small-signal electroluminescence methods [1]. A rate equation approach and associated small-signal circuit are used to model carrier injection, recombination in the active region, recombination in the cladding regions, and carrier escape. The model is fit to the measured optical frequency response (S21) and input impedance (S11) of the LEDs to extract the various carrier lifetimes, the carrier density, and the radiative and non-radiative recombination rates. We specifically study planar nonpolar and semipolar LEDs, which show record-high modulation speeds for III-nitride LEDs and present the modulation characteristics of core-shell nanowire-based LEDs. The planar nonpolar m-plane ($101\bar{0}$) micro-LEDs achieve a record-high -3dB modulation bandwidth for a III-nitride LED of 1.5 GHz [2]. The -3dB response of an electrically injected nanowire-based micro-LED with nonpolar facets is also reported, showing a -3dB bandwidth of 1.2 GHz [3]. The high speed is attributed to the shorter carrier lifetime associated with the nonpolar orientation. We also study the carrier dynamics in semipolar ($20\bar{2}\bar{1}$) LEDs for various temperatures [4]. Finally, we present carrier dynamics measurements on commercial-grade c-plane epitaxy for various active region designs, including a wavelength series and a growth quality series. The wavelength series offers insight into the contributions of the quantum confined Stark effect (QCSE) and InGaN material quality on the green gap [5]. The growth quality series investigates the role of non-radiative centers on the LED performance. Extraction of the carrier dynamics using small-signal electroluminescence offers insight into the factors limiting the efficiency and high-speed performance of III-nitride emitters and can be leveraged to ultimately improve the devices.

LEDs, InGaN, electroluminescence, bandwidth, modul↗

DOE OSTI2023

Poly(diallylmethylammonium) proton conducting membranes with high ionic conductivity at intermediate temperatures

High temperature proton exchange membrane fuel cells are being lately investigated because of their high energy efficiency, their superior heat/water management, CO tolerance, and electrode reaction kinetics. To further advance this technology, the polymer membrane portfolio and performance should be improved for intermediate or high temperature operation (>100 °C). In this work we present new poly(diallylmethylammonium) proton conducting membranes with high ionic conductivity at 120 °C. First, new protic ionic liquids, hereafter called DAMAH + X - , were synthesized leading to diallylmethylammonium monomers with different counter-anions. By radical cyclopolymerization through thermal and photoinitiation mechanisms, self-standing protic polymeric membranes of poly(diallylmethylammonium X - ) were obtained. Membranes showed good thermal stability (>250 °C) and mechanical properties without the need of additives such as (protic) ionic liquids, solvents or inorganic charges. Great attention was paid to understand the effect of the different counter-anions on the membrane properties. As a general trend, fluorinated anions coming from strong acids confer high ionic conductivity and allow to reduce the hygroscopic properties on the protic polymeric membranes. Proton structural and dynamical stability at different temperatures and humidification conditions were investigated by Neutron Scattering (QENS and NR). The optimized poly(diallylmethylammonium X - ) shows similar ionic conductivity values than Nafion 212 under varying relative humidity conditions at 80 °C. Furthermore, it shows a high ionic conductivity value of 1.9 × 10 -3 Scm -1 at 120 °C under dry conditions.

08 HYDROGEN↗

DOE OSTI2024

Investigation of experimental signatures of spin glass transition temperature

We present a series of temperature and field-dependent magnetization studies of large single-crystal spin glass samples, focusing on both field-cooled (FC) and zero-field-cooled (ZFC) magnetization studies, as well as ac susceptibility measurements. Using the above experimental techniques we aim to understand the nature of spin glass transition in presence of a field, a key factor in understanding the properties of these systems. Building on previous studies that have explored magnetic signatures indicative of spin glass transitions, our research employs a systematic approach to refine the identification of this transition temperature. Through static and dynamic measurements, we aim to shed light on the open issues regarding the key markers of spin glass transitions, enhancing our understanding of these complex systems.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Mechanical and Thermal Properties for Uranium and U–6Nb Alloy from First-Principles Theory

Elasticity, lattice dynamics, and thermal expansion for uranium and U–6Nb alloy (elastic moduli) are calculated from density functional theory that is extended to include orbital polarization (DFT+OP). Introducing 12.5 at.% of niobium, substitutionally, in uranium softens all the cii elastic moduli, resulting in a significantly softer shear modulus (G). Combined with a nearly invariant bulk modulus (B), the quotient B/G increases dramatically for U–6Nb, suggesting a more ductile material. Lattice dynamics from a harmonic model coupled with a DFT+OP electronic structure is applied for α uranium, and the obtained phonon density of states compares well with inelastic neutron-scattering measurements. The Debye temperature associated with the lattice dynamics falls within the range of experimentally observed Debye temperatures and it also validates our quasi-harmonic (QH) phonon model. The QH Debye–Grüneisen phonon method is combined with a DFT+OP electronic structure and used to explore the anisotropic thermal expansion in α uranium. The anomalous negative thermal expansion (contraction) of the b lattice parameter of the α-phase orthorhombic cell is relatively well reproduced from a free-energy model consisting of QH-phonon and DFT+OP electronic structure contributions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2024

Shifting groundwater fluxes in bedrock fractures: Evidence from stream water radon and water isotopes

Geologic features (e.g., fractures and alluvial fans) can play an important role in the locations and volumes of groundwater discharge and degree of groundwater-surface water (GW-SW) interactions. However, the role of these features in controlling GW-SW dynamics and streamflow generation processes are not well constrained. GW-SW interactions and streamflow generation processes are further complicated by variability in precipitation inputs from summer and fall monsoon rains, as well as declines in snowpack and changing melt dynamics driven by warming temperatures. Using high spatial and temporal resolution radon and water stable isotope sampling and a 1D groundwater flux model, we evaluated how groundwater contributions and GW-SW interactions varied along a stream reach impacted by fractures (fractured-zone) and below the fractured hillslope (non-fractured zone) in Coal Creek, a Colorado River headwater stream affected by summer monsoons. During early summer, groundwater contributions from the fractured zone dominated, but declined throughout the summer. Groundwater contributions from the non-fractured zone were constant throughout the summer and became proportionally more important later in the summer. We hypothesize that groundwater in the non-fractured zone is dominantly sourced from a high-storage alluvial fan at the base of a tributary that is connected to Coal Creek throughout the summer and provides consistent groundwater influx. Water isotope data revealed that Coal Creek responds quickly to incoming precipitation early in the summer, and summer precipitation becomes more important for streamflow generation later in the summer. We quantified the change in catchment dynamic storage and found it negatively related to stream water isotope values, and positively related to modeled groundwater discharge and the ratio of fractured zone to non-fractured zone groundwater. Importantly, we interpret these relationships as declining hydrologic connectivity throughout the summer leading to late summer streamflow supported predominantly by shallow flow paths, with variable response to drying from geologic features based on their storage. As groundwater becomes more important for sustaining summer flows, quantifying local geologic controls on groundwater inputs and their response to variable moisture conditions may become critical for accurate predictions of streamflow.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2023

Profile observations of the Arctic atmospheric boundary layer with the BELUGA tethered balloon during MOSAiC

During the Multidisciplinary drifting Observatory for the Study of Arctic Climate (MOSAiC) expedition, the Balloon-bornE moduLar Utility for profilinG the lower Atmosphere (BELUGA) was deployed from an ice floe drifting in the Fram Strait from 29 June to 27 July 2020. The BELUGA observations aimed to characterize the cloudy Arctic atmospheric boundary layer above the sea ice using a modular setup of five instrument packages. The in situ measurements included atmospheric thermodynamic and dynamic state parameters (air temperature, humidity, pressure, and three-dimensional wind), broadband solar and terrestrial irradiance, aerosol particle microphysical properties, and cloud particle images. In total, 66 profile observations were collected during 33 balloon flights from the surface to maximum altitudes of 0.3 to 1.5 km. The profiles feature a high vertical resolution of 0.01 m to 1 m, including measurements below, inside, and above frequently occurring low-level clouds. This publication describes the balloon operations, instruments, and the obtained data set. We invite the scientific community for joint analysis and model application of the freely available data on PANGAEA.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Photo-Switching of Protein Dynamical Collectivity

We examine changes in the picosecond structural dynamics with irreversible photobleaching of red fluorescent proteins (RFP) mCherry, mOrange2 and TagRFP-T. Measurements of the protein dynamical transition using terahertz time-domain spectroscopy show in all cases an increase in the turn-on temperature in the bleached state. The result is surprising given that there is little change in the protein surface, and thus, the solvent dynamics held responsible for the transition should not change. A spectral analysis of the measurements guided by quasiharmonic calculations of the protein absorbance reveals that indeed the solvent dynamical turn-on temperature is independent of the thermal stability/photostate however the protein dynamical turn-on temperature shifts to higher temperatures. This is the first demonstration of switching the protein dynamical turn-on temperature with protein functional state. The observed shift in protein dynamical turn-on temperature relative to the solvent indicates an increase in the required mobile waters necessary for the protein picosecond motions, that is, these motions are more collective. Melting-point measurements reveal that the photobleached state is more thermally stable, and structural analysis of related RFP’s shows that there is an increase in internal water channels as well as a more uniform atomic root mean squared displacement. These observations are consistent with previous suggestions that water channels form with extended light excitation providing O2 access to the chromophore and subsequent fluorescence loss. We report that these same channels increase internal coupling enhancing thermal stability and collectivity of the picosecond protein motions. The terahertz spectroscopic characterization of the protein and solvent dynamical onsets can be applied generally to measure changes in collectivity of protein motions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

Thermodynamic stability of a spin microemulsion in Rashba spin-orbit-coupled bosons

Recent finite-temperature numerical simulations have unveiled a quantum “spin” microemulsion analog, found by raising the temperature of a stripe supersolid phase in a Rashba spin-orbit coupled Bose gas. This microemulsion state is a highly correlated, isotropic normal fluid where atoms self-arrange based on their internal pseudospin into patterns that resemble bicontinuous microemulsions. This finding leaves several open questions regarding the broader accessibility of this phase in experiments. Here, we use equilibrium finite-temperature numerical simulations based on a coherent-state path integral representation to perform a computational investigation into the thermodynamic stability of the spin microemulsion state. Numerical simulations emphasize the requirement of a nearly, but not perfectly, isotropic spin-orbit coupling in order to achieve the microemulsion phase in cold-atom experiments. Moreover, the microemulsion state exists independent of miscibility of the pseudospin components and for a wide range of pseudospin population imbalance, suggesting a high degree of flexibility in choosing the atom and hyperfine states in an experimental realization. Lastly, we demonstrate this feasibility by mimicking a Rashba spin-orbit-coupled 87 Rb experiment in an isotropic harmonic trap, where we confirm the microemulsion's existence via its density profile and equilibrium quasimomentum distribution.

Complex Langevin dynamics↗

DOE OSTI2021

Metastable piezoelectric group-IV monochalcogenide monolayers with a buckled honeycomb structure

Multiple two-dimensional materials are being naïvely termed stable on the grounds of displaying phonon dispersions with no negative frequencies and of not collapsing on molecular dynamics calculations at fixed volume. But, if these phases do not possess the smallest possible structural energy, how does one understand and establish their actual meta stability? To answer this question, twelve two-dimensional group-IV monochalcogenide monolayers (SiS, SiSe, SiTe, GeS, GeSe, GeTe, SnS, SnSe, SnTe, PbS, PbSe, and PbTe) with a buckled honeycomb atomistic structure—belonging to symmetry group P3m1—displaying an out-of-plane intrinsic electric polarization are shown to be metastable by three independent methods. First, we uncover a coordination-preserving structural transformation from the low-buckled honeycomb structure onto the lower-energy Pnm2 1 (or Pmmn for PbS, PbSe, and PbTe) phase to estimate energy barriers E B that must be overcome during such structural transformation. Using the curvature of the local minima and E B as inputs to Kramers escape formula, large escape times are found, implying the structural metastability of the buckled honeycomb phase (with the exception of PbS and PbSe, these phases display escape times ranging from 700 years to multiple times the age of the universe and can be considered “stable” for practical purposes in that relative sense). The second demonstration is provided by phonon dispersion relations that include the effect of long-range Coulomb forces and display no negative vibrational modes. The third and final demonstration of structural metastability is furnished by room-temperature ab initio molecular dynamics for selected compounds. Here, the magnitude of the electronic band gap evolves with chemical composition. Different from other binary two-dimensional compounds such as transition metal dichalcogenide monolayers and hexagonal boron nitride monolayers which only develop an in-plane piezoelectric response, the twelve group-IV monochalcogenide monolayers with a buckled honeycomb structure also display out-of-plane piezoelectric properties.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Topography and canopy cover influence soil organic carbon composition and distribution across a forested hillslope in the discontinuous permafrost zone

Abstract Topography and canopy cover influence ground temperature in warming permafrost landscapes, yet soil temperature heterogeneity introduced by mesotopographic slope positions, microtopographic differences in vegetation cover, and the subsequent impact of contrasting temperature conditions on soil organic carbon (SOC) dynamics are understudied. Buffering of permafrost‐affected soils against warming air temperatures in boreal forests can reflect surface soil characteristics (e.g., thickness of organic material) as well as the degree and type of canopy cover (e.g., open cover vs. closed cover). Both landscape and soil properties interact to determine meso‐ and microscale heterogeneity of ground warming. We sampled a hillslope catena transect in a discontinuous permafrost zone near Fairbanks, Alaska, to test the small‐scale (1 to 3 m) impacts of slope position and cover type on soil organic matter composition. Mineral active layer samples were collected from backslope, low backslope, and footslope positions at depths spanning 19 to 60 cm. We examined soil mineralogical composition, soil moisture, total carbon and nitrogen content, and organic mat thickness in conjunction with an assessment of SOC composition using Fourier‐transform ion cyclotron resonance mass spectrometry (FT‐ICR‐MS). Soils in the footslope position had a higher relative contribution of lignin‐like compounds, whereas backslope soils had more aliphatic and condensed aromatic compounds as determined using FT‐ICR‐MS. The effect of open versus closed tree canopy cover varied with the slope position. On the backslope, we found higher oxidation of molecules under open cover than closed cover, indicating an effect of warmer soil temperature on decomposition. Little to no effect of the canopy was observed in soils at the footslope position, which we attributed, in part, to the strong impact of soil moisture content in SOC dynamics in the water‐gathering footslope position. The thin organic mat under open cover on the backslope position may have contributed to differences in soil temperature and thus SOC oxidation under open and closed canopies. Here, the thinner organic mat did not appear to buffer the underlying soil against warm season air temperatures and thus increased SOC decomposition as indicated by the higher oxidation of SOC molecules and a lower contribution of simple molecules under open cover than the closed canopy sites. Our findings suggest that the role of canopy cover in SOC dynamics varies as a function of landscape position and soil properties, namely, organic mat thickness and soil moisture. Condition‐specific heterogeneity of SOC composition under open and closed canopy cover highlights the protective effect of canopy cover for soils on backslope positions.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Dynamics in Flexible Pillar[ n ]arenes Probed by Solid-State NMR

Pillar[n]arenes are supramolecular assemblies that can perform a range of technologically important molecular separations which are enabled by their molecular flexibility. Here, we probe dynamical behavior by performing a range of variable-temperature solid-state NMR experiments on microcrystalline perethylated pillar[n]arene (n = 5, 6) and the corresponding three pillar[6]arene xylene adducts in the 100–350 K range. This was achieved either by measuring site-selective motional averaged 13 C 1 H heteronuclear dipolar couplings and subsequently accessing order parameters or by determining 1 H and 13 C spin–lattice relaxation times and extracting correlation times based on dipolar and/or chemical shift anisotropy relaxation mechanisms. We demonstrate fast motional regimes at room temperature and highlight a significant difference in dynamics between the core of the pillar[n]arenes, the protruding flexible ethoxy groups, and the adsorbed xylene guest. Additionally, unexpected and sizable 13 C 1 H heteronuclear dipolar couplings for a quaternary carbon were observed for p-xylene adsorbed in pillar[6]arene only, indicating a strong host–guest interaction and establishing the p-xylene location inside the host, confirming structural refinements.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

Comparing first-principles density functionals plus corrections for the lattice dynamics of YBa 2 Cu 3 O 6

The enigmatic mechanism underlying unconventional high-temperature superconductivity, especially the role of lattice dynamics, has remained a subject of debate. Theoretical insights have long been hindered due to the lack of an accurate first-principles description of the lattice dynamics of cuprates. Recently, using the r2SCAN meta-generalized gradient approximation (meta-GGA) functional, we have been able to achieve accurate phonon spectra of an insulating cuprate YBa 2 Cu 3 O 6 and discover significant magnetoelastic coupling in experimentally interesting Cu–O bond stretching optical modes [Ning et al., Phys. Rev. B 107, 045126 (2023)]. We extend this work by comparing Perdew–Burke–Ernzerhof and r2SCAN performances with corrections from the on-site Hubbard U and the D4 van der Waals (vdW) methods, aiming at further understanding on both the materials science side and the density functional side. We demonstrate the importance of vdW and self-interaction corrections for accurate first-principles YBa 2 Cu 3 O 6 lattice dynamics. Since r2SCAN by itself partially accounts for these effects, the good performance of r2SCAN is now more fully explained. Finally, in addition, the performances of the Tao–Mo series of meta-GGAs, which are constructed in a different way from the strongly constrained and appropriately normed (SCAN) meta-GGA and its revised version r2SCAN, are also compared and discussed.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

Extended anharmonic collapse of phonon dispersions in SnS and SnSe

The lattice dynamics and high-temperature structural transition in SnS and SnSe are investigated via inelastic neutron scattering, high-resolution Raman spectroscopy and anharmonic first-principles simulations. We uncover a spectacular, extreme softening and reconstruction of an entire manifold of low-energy acoustic and optic branches across a structural transition, reflecting strong directionality in bonding strength and anharmonicity. Further, our results solve a prior controversy by revealing the soft-mode mechanism of the phase transition that impacts thermal transport and thermoelectric efficiency. Our simulations of anharmonic phonon renormalization go beyond low-order perturbation theory and capture these striking effects, showing that the large phonon shifts directly affect the thermal conductivity by altering both the phonon scattering phase space and the group velocities. These results provide a detailed microscopic understanding of phase stability and thermal transport in technologically important materials, providing further insights on ways to control phonon propagation in thermoelectrics, photovoltaics, and other materials requiring thermal management.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Absolutely Stable Time Crystals at Finite Temperature

Here, we show that locally interacting, periodically driven (Floquet) Hamiltonian dynamics coupled to a Langevin bath support finite-temperature discrete time crystals (DTCs) with an infinite autocorrelation time. By contrast to both prethermal and many-body localized DTCs, the time crystalline order we uncover is stable to arbitrary perturbations, including those that break the time translation symmetry of the underlying drive. Our approach utilizes a general mapping from probabilistic cellular automata to open classical Floquet systems undergoing continuous-time Langevin dynamics. Applying this mapping to a variant of the Toom cellular automaton, which we dub the "π-Toom time crystal," leads to a 2D Floquet Hamiltonian with a finite-temperature DTC phase transition. We provide numerical evidence for the existence of this transition, and analyze the statistics of the finite temperature fluctuations. Finally, we discuss how general results from the field of probabilistic cellular automata imply the existence of discrete time crystals (with an infinite autocorrelation time) in all dimensions, d ≥ 1.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗