Correlations, spectral gap, and entanglement in harmonic quantum systems on generic lattices

We investigate the relationship between the gap between the energy of the ground state and the first excited state and the decay of correlation functions in harmonic lattice systems. We prove that in gapped systems, the exponential decay of correlations follows for both the ground state and thermal states. Considering the converse direction, we show that an energy gap can follow from algebraic decay and always does for exponential decay. The underlying lattices are described as general graphs of not necessarily integer dimension, including translationally invariant instances of cubic lattices as special cases. Any local quadratic couplings in position and momentum coordinates are allowed for, leading to quasi-free (Gaussian) ground states. We make use of methods of deriving bounds to matrix functions of banded matrices corresponding to local interactions on general graphs. Finally, we give an explicit entanglement-area relationship in terms of the energy gap for arbitrary, not necessarily contiguous regions on lattices characterized by general graphs.Comment: 26 pages, LaTeX, published version (figure added

Changing local recombination patterns in Arabidopsis by CRISPR/Cas mediated chromosome engineering

Chromosomal inversions are recurrent rearrangements that occur between different plant isolates or cultivars. Such inversions may underlie reproductive isolation in evolution and represent a major obstacle for classical breeding as no crossovers can be observed between inverted sequences on homologous chromosomes. The heterochromatic knob (hk4S) on chromosome 4 is the most well-known inversion of Arabidopsis. If a knob carrying accession such as Col-0 is crossed with a knob-less accession such as Ler-1, crossovers cannot be recovered within the inverted region. Our work shows that by egg-cell specific expression of the Cas9 nuclease from Staphylococcus aureus, a targeted reversal of the 1.1 Mb long hk4S-inversion can be achieved. By crossing Col-0 harbouring the rearranged chromosome 4 with Ler-1, meiotic crossovers can be restored into a region with previously no detectable genetic exchange. The strategy of somatic chromosome engineering for breaking genetic linkage has huge potential for application in plant breeding