1D Many-body localized Floquet systems II: Symmetry-Broken phases

Recent work suggests that a sharp definition of `phase of matter' can be given for periodically driven `Floquet' quantum systems exhibiting many-body localization. In this work we propose a classification of the phases of interacting Floquet localized systems with (completely) spontaneously broken symmetries -- we focus on the one dimensional case, but our results appear to generalize to higher dimensions. We find that the different Floquet phases correspond to elements of $Z(G)$, the centre of the symmetry group in question. In a previous paper we offered a companion classification of unbroken, i.e., paramagnetic phases.Comment: Published versio

Defining Time Crystals via Representation Theory

Time crystals are proposed states of matter which spontaneously break time translation symmetry. There is no settled definition of such states. We offer a new definition which follows the traditional recipe for Wigner symmetries and order parameters. Supplementing our definition with a few plausible assumptions we find that a) systems with time independent Hamiltonians should not exhibit TTSB while b) the recently studied $\pi$ spin glass/Floquet time crystal can be viewed as breaking a global internal symmetry and as breaking time translation symmetry as befits its two names

Absolute Stability and Spatiotemporal Long-Range Order in Floquet systems

Recent work has shown that a variety of novel phases of matter arise in periodically driven Floquet systems. Among these are many-body localized phases which spontaneously break global symmetries and exhibit novel multiplets of Floquet eigenstates separated by quantized quasienergies. Here we show that these properties are stable to all weak local deformations of the underlying Floquet drives -- including those that explicitly break the defining symmetries -- and that the models considered until now occupy sub-manifolds within these larger "absolutely stable" phases. While these absolutely stable phases have no explicit global symmetries, they spontaneously break Hamiltonian dependent emergent symmetries, and thus continue to exhibit the novel multiplet structure. The multiplet structure in turn encodes characteristic oscillations of the emergent order parameter at multiples of the fundamental period. Altogether these phases exhibit a form of simultaneous long-range order in space and time which is new to quantum systems. We describe how this spatiotemporal order can be detected in experiments involving quenches from a broad class of initial states.Comment: Published version. Minor typos corrected, some discussions expande

Preference and usage of pasture versus free-stall housing by lactating dairy cattle

The aim of the current study was to assess if cows preferred pasture or indoor housing, and how diurnal and environmental factors affected this preference. Lactating dairy cows (n = 5 groups, each containing 5 cows) were sequentially housed either in a free-stall barn on pasture, or given the choice between the 2 environments. Each group was tested 3 times under each condition, for a total of 21 d, to assess the effects of varying climatic conditions (outdoor temperature ranged from 9.9 to 28.2°C and daily rainfall from 0 to 65 mm/d over the course of the experiment). When provided the choice, cows spent on average (± SD) 13.0 ± 0.6 h/d on pasture, mainly at night. The time cows spent on pasture during the day decreased with the temperature-humidity index (R2 = 0.55); time on pasture at night decreased with rainfall (R2 = 0.12). When provided a choice, cows spent more of their lying time on pasture (69.4 ± 0.02% of the total lying time/d) than indoors in the free-stalls. Cows also spent more time in total lying down when provided a choice than when confined to pasture [0.6 h/d more lying time; standard error of the difference (SED) = 0.21 h/d] and spent even more time lying down when confined indoors (1.1 h/d more time; SED = 0.21 h/d). Cows used the indoor housing especially for feeding; feeder use peaked when cows returned from morning and afternoon milkings. However, cows with free access to pasture spent 1.0 h/d (SED = 0.09 h/d) less time eating the TMR available indoors, resulting in a decline in intake of 2.9 kg of dry matter/d (SED = 0.36 kg of dry matter/d). How cows used the indoor housing differed when cows were provided a choice; for example, cows spent a greater percentage of their time indoors at the feed alley both during the day (47% of the total time spent indoors, versus 41% for cows confined indoors, SED = 0.02%) and at night (22 vs. 5%, SED = 0.04%). In conclusion, under the housing and environmental conditions tested, cows showed a strong preference for access to pasture at night and for access to indoor housing during the day when temperature and humidity increased

Operator Spreading in Quantum Maps

Operators in ergodic spin-chains are found to grow according to hydrodynamical equations of motion. The study of such operator spreading has aided our understanding of many-body quantum chaos in spin-chains. Here we initiate the study of "operator spreading" in quantum maps on a torus, systems which do not have a tensor-product Hilbert space or a notion of spatial locality. Using the perturbed Arnold cat map as an example, we analytically compare and contrast the evolutions of functions on classical phase space and quantum operator evolutions, and identify distinct timescales that characterize the dynamics of operators in quantum chaotic maps. Until an Ehrenfest time, the quantum system exhibits classical chaos, i.e. it mimics the behavior of the corresponding classical system. After an operator scrambling time, the operator looks "random" in the initial basis, a characteristic feature of quantum chaos. These timescales can be related to the quasi-energy spectrum of the unitary via the spectral form factor. Furthermore, we show examples of "emergent classicality" in quantum problems far away from the classical limit. Finally, we study operator evolution in non-chaotic and mixed quantum maps using the Chirikov standard map as an example.Comment: 21 pages, 7 figures v2: References adde

Observation of discrete time-crystalline order in a disordered dipolar many-body system

Understanding quantum dynamics away from equilibrium is an outstanding challenge in the modern physical sciences. It is well known that out-of-equilibrium systems can display a rich array of phenomena, ranging from self-organized synchronization to dynamical phase transitions. More recently, advances in the controlled manipulation of isolated many-body systems have enabled detailed studies of non-equilibrium phases in strongly interacting quantum matter. As a particularly striking example, the interplay of periodic driving, disorder, and strong interactions has recently been predicted to result in exotic "time-crystalline" phases, which spontaneously break the discrete time-translation symmetry of the underlying drive. Here, we report the experimental observation of such discrete time-crystalline order in a driven, disordered ensemble of $\sim 10^6$ dipolar spin impurities in diamond at room-temperature. We observe long-lived temporal correlations at integer multiples of the fundamental driving period, experimentally identify the phase boundary and find that the temporal order is protected by strong interactions; this order is remarkably stable against perturbations, even in the presence of slow thermalization. Our work opens the door to exploring dynamical phases of matter and controlling interacting, disordered many-body systems.Comment: 6 + 3 pages, 4 figure

Dichromatic state sum models for four-manifolds from pivotal functors

A family of invariants of smooth, oriented four-dimensional manifolds is defined via handle decompositions and the Kirby calculus of framed link diagrams. The invariants are parametrised by a pivotal functor from a spherical fusion category into a ribbon fusion category. A state sum formula for the invariant is constructed via the chain-mail procedure, so a large class of topological state sum models can be expressed as link invariants. Most prominently, the Crane-Yetter state sum over an arbitrary ribbon fusion category is recovered, including the nonmodular case. It is shown that the Crane-Yetter invariant for nonmodular categories is stronger than signature and Euler invariant. A special case is the four-dimensional untwisted Dijkgraaf-Witten model. Derivations of state space dimensions of TQFTs arising from the state sum model agree with recent calculations of ground state degeneracies in Walker-Wang models. Relations to different approaches to quantum gravity such as Cartan geometry and teleparallel gravity are also discussed

Effects of degree and timing of social housing on reversal learning and response to novel objects in dairy calves

Rodents and primates deprived of early social contact exhibit deficits in learning and behavioural flexibility. They often also exhibit apparent signs of elevated anxiety, although the relationship between these effects has not been studied. To investigate whether dairy calves are similarly affected, we first compared calves housed in standard individual pens (n = 7) to those housed in a dynamic group with access to their mothers (n = 8). All calves learned to approach the correct stimulus in a visual discrimination task. Only one individually housed calf was able to re-learn the task when the stimuli were reversed, compared to all but one calf from the group. A second experiment investigated whether this effect might be explained by anxiety in individually housed animals interfering with their learning, and tested varying degrees of social contact in addition to the complex group: pair housing beginning early (approximately 6 days old) and late (6 weeks old). Again, fewer individually reared calves learned the reversal task (2 of 10 or 20%) compared to early paired and grouped calves (16 of 21 or 76% of calves). Late paired calves had intermediate success. Individually housed calves were slower to touch novel objects, but the magnitude of the fear response did not correlate with reversal performance. We conclude that individually housed calves have learning deficits, but these deficits were not likely associated with increased anxiety