Abelian Floquet symmetry-protected topological phases in one dimension

Time-dependent systems have recently been shown to support novel types of topological order that cannot be realised in static systems. In this paper, we consider a range of time-dependent, interacting systems in one dimension that are protected by an Abelian symmetry group. We classify the distinct topological phases that can exist in this setting and find that they may be described by a bulk invariant associated with the unitary evolution of the closed system. In the open system, nontrivial phases correspond to the appearance of edge modes in the many-body quasienergy spectrum, which relate to the bulk invariant through a form of bulk-edge correspondence. We introduce simple models which realise nontrivial dynamical phases in a number of cases, and outline a loop construction that can be used to generate such phases more generally.Comment: 13 pages, 1 figure; Published versio

Fractional Chern insulators in bands with zero Berry curvature

Even if a noninteracting system has zero Berry curvature everywhere in the Brillouin zone, it is possible to introduce interactions that stabilize a fractional Chern insulator. These interactions necessarily break time-reversal symmetry (either spontaneously or explicitly) and have the effect of altering the underlying band structure. We outline a number of ways in which this may be achieved and show how similar interactions may also be used to create a (time-reversal-symmetric) fractional topological insulator. While our approach is rigorous in the limit of long-range interactions, we show numerically that even for short-range interactions a fractional Chern insulator can be stabilized in a band with zero Berry curvature

Localization renormalization and quantum Hall systems

The obstruction to constructing localized degrees of freedom is a signature of several interesting condensed matter phases. We introduce a localization renormalization procedure that harnesses this property, and apply our method to distinguish between topological and trivial phases in quantum Hall and Chern insulators. By iteratively removing a fraction of maximally-localized orthogonal basis states, we find that the localization length in the residual Hilbert space exhibits a power-law divergence as the fraction of remaining states approaches zero, with an exponent of $\nu=0.5$. In sharp contrast, the localization length converges to a system-size-independent constant in the trivial phase. We verify this scaling using a variety of algorithms to truncate the Hilbert space, and show that it corresponds to a statistically self-similar expansion of the real-space projector. This result accords with a renormalization group picture and motivates the use of localization renormalization as a versatile numerical diagnostic for quantum Hall insulators.Comment: 10+9 pages, 7+4 figure

Ecophysiology of seed dormancy and the control of germination in early spring-flowering Galanthus nivalis and Narcissus pseudonarcissus (Amaryllidaceae)

Seed dormancy induction and alleviation in the winter-flowering moist temperate woodland species Galanthus nivalis and Narcissus pseudonarcissus are complex and poorly understood. Temperature, light and desiccation were investigated to elucidate their role in the germination ecophysiology of these species. Outdoor and laboratory experiments simulating different seasonal temperatures, seasonal durations, and temperature fluctuations; the presence of light during different seasons; and intermittent drying (during the summer period) over several ‘years’ investigated the importance of these factors in germination. Warm summer-like temperatures (20°C) were necessary for germination at subsequent cooler autumn-like temperatures (greatest at 15°C in G. nivalis and 10°C in N. pseudonarcissus). As the warm temperature duration increased so did germination at subsequent cooler temperatures; further germination occurred in subsequent ‘years’ at cooler temperatures following a second, and also third, warm period. Germination was significantly greater in darkness, particularly in G. nivalis. Dormancy increased with seed maturation period in G. nivalis, because seeds extracted from green capsules germinated more readily than those from yellow. Desiccation increased dormancy in an increasing proportion of N. pseudonarcissus seeds the later they were dried in ‘summer’. Seed viability was only slightly reduced by desiccation in N. pseudonarcissus but was poor and variable in G. nivalis. Shoot formation occurred both at the temperature at which germination was greatest and also if 5°C cooler. In summary, continuous hydration of seeds of both species during warm summer-like temperatures results in the gradual release of seed dormancy; thereafter, darkness and cooler temperatures promote germination. Cold temperatures, increased seed maturity (G. nivalis), and desiccation (N. pseudonarcissus) increase dormancy while light inhibits germination

Universal localization-delocalization transition in chirally-symmetric Floquet drives

Periodically driven systems often exhibit behavior distinct from static systems. In single-particle, static systems, any amount of disorder generically localizes all eigenstates in one dimension. In contrast, we show that in topologically non-trivial, single-particle Floquet loop drives with chiral symmetry in one dimension, a localization-delocalization transition occurs as the time $t$ is varied within the driving period ($0 \le t \le T_\text{drive}$). We find that the time-dependent localization length $L_\text{loc}(t)$ diverges with a universal exponent as $t$ approaches the midpoint of the drive: $L_\text{loc}(t) \sim (t - T_\text{drive}/2)^{-\nu}$ with $\nu=2$. We provide analytical and numerical evidence for the universality of this exponent within the AIII symmetry class.Comment: 17 + 5 pages, 7 figure