Two-dimensional Ising model with competing interactions: phase diagram and low-temperature remanent disorder

The two-dimensional Ising model with competing nearest-neighbor and diagonal interactions on the square lattice is studied by the transfer-matrix technique and by the Monte Carlo simulations. The phase diagram of this model is constructed with a special emphasis to the analysis of a glassy state arising as an order to disorder transition at low temperatures. Evidence of the glassy state (based, in particular, on the calculation of the average length of domain walls and on the Edwards-Anderson order parameter) and its characteristics are presented. It was shown that, in the frustrated Ising model, the domain-wall length correlates to the onset of the glassy state, that is, it may play the role of the order parameter for the Ising glass or for glasslike states in other frustrated magnetic systems

A stable "flat" form of two-dimensional crystals: could graphene, silicene, germanene be minigap semiconductors?

The discovery of a flat two-dimensional crystal known as graphene has contradicted Landau–Peierls–Mermin–Wagner arguments that there is no stable flat form of such crystals. Here, we show that the “flat” shape of graphene arises due to a microscopic buckling at the smallest possible interatomic scale. We show that the graphene, silicene, and other two-dimensional crystals are stable due to transverse short-range displacements of appropriate atoms. The distortions are small and form various patterns, which we describe in a framework of Ising model with competing interactions. We show that when temperature decreases, two transitions, disorder into order and order into disorder, arise. The ordered state has a form of stripes where carbon atoms are shifted regularly with respect to the plane. The flat graphene, silicene, or germanene planes look like a microscopic “washboard” with the wavelength of about couple of interatomic spacing of appropriate sublattices, which for graphene is about 1.8–3.6 Å. At lower temperatures, the ordered state transforms into a glass. Because of up–down asymmetry in buckled graphene, silicene and other two-dimensional crystals deposited on substrate, a minibandgap may arise. We derive a criterion for the minigap formation and show how it is related to the buckling and to the graphene–substrate interaction. Because of the bandgap, there may arise new phenomena and in particular a rectification of ac current induced by microwave or infrared radiation. We show that the amplitude of direct current arising at wave mixing of two harmonics of microwave electromagnetic radiation is huge. Moreover, we predict the existence of miniexcitons and a new type of fermionic minipolaritons whose behavior can be controlled by the microwave and terahertz radiation

Shortest path analysis of cattle network with and without CTS Links.

<p>(A) Strongly connected component size of the cattle movements network for 2008 with (blue) and without (red) CTS Links. Adding CTS Links increases the size of the largest component (from 45072 to 54786, not shown here) and also increases the size and frequency of the smaller components; (B) Difference between the shortest path length distribution of connected premises in the cattle movements network for 2008 with and without CTS Links. The path length difference is calculated by subtracting the distribution of shortest path lengths for the movement network from the movement network with CTS Links (comparing the same connected premises).</p

Comparison of sheep movements from SOAs vs Non-SOAs between 2005 and 2008.

<p>Monthly sheep movements from agricultural holdings within SOAs are shown in blue bars, whilst monthly sheep movements from agricultural holdings not in SOAs (Non-SOAs) are shown in red bars. The total number of SOAs and Non-SOAs contributing to the month's sheep movements are shown with the blue and red lines respectively. Movements to the same holdings or the same SOA were removed along with movements to slaughter. Only movements from “agricultural holdings” to other “agricultural holdings” or “store markets” were considered.</p

Daily total number of active CTS Links.

<p>The daily number of active CTS Links classed as Shared Facility (A) and Additional Land (B) from 2003 to 2008. (C) Shows the frequency distribution of CTS Chain sizes; the outlier chain of size 242 was removed from this chart, whilst the second largest chain consist of 48 component holdings.</p

Number of SOAs in existence over time and the distribution of component holdings.

<p>(A) Shows the monthly number of SOAs in existence from their inception in late 2001 to mid-2008. (B) Shows the distribution of the number of component holdings that comprise each SOA; the x-axis has been truncated to remove the outlier SOA containing 250 holdings for display purposes.</p

Comparison of epidemic sizes with SOAs.

<p>The model was run including SOAs, with no stock restriction on intra-SOA spread and the following distance limits on intra-SOA spread: No Limit (red), 50 km (green), 16 km (purple), and 8 km (black). In addition, the results from the SOA model with intra-SOA spread restricted to holdings with stock recorded as present is shown in blue. Epidemic size (number of infected holdings per seed) and geographical size (number of infected grid squares per seed) were firstly transformed into percentages of the control epidemic size (with no linked holdings) from the same time of year. Secondly, averages were then obtained for the four quarters of the year (Jan-Feb-Mar[squares], Apr-May-Jun [circles], Jul-Aug-Sep [triangles], Oct-Nov-Dec [diamonds]) giving four data points for each scenario. Error bars are associated with each quarterly average value that represent the 95% confidence intervals for that quarter, assuming a normal distribution and a standard deviation calculated from that quarter's data.</p

Network structure of the largest observed CTS Chain.

<p>This is the network structure of the largest CTS Chain consisting of 242 holdings. Each box represents a distinct holding, and links between boxes represent CTS Links. Boxes coloured Red, Green, and Yellow are holdings located in Wales, England, and Scotland respectively. Boxes are labelled with their holding's county number.</p

Comparison of epidemic sizes and geographical spread with different linked holdings.

<p>The model was run including CTS Chains, with the following distance limits on intra-Chain spread: No Limit (red), 50 km (green), 16 km (purple), and 8 km (black). Epidemic size (number of infected holdings per seed) and geographical size (number of infected grid squares per seed) were firstly transformed into percentages of the control epidemic size from the same time of year. Secondly, averages were then obtained for the four quarters of the year (Jan-Feb-Mar [squares], Apr-May-Jun [circles], Jul-Aug-Sep [triangles], Oct-Nov-Dec [diamonds]) giving four data points for each scenario. Error bars are associated with each quarterly average value that represent the 95% confidence intervals for that quarter, assuming a normal distribution and a standard deviation calculated from that quarter's data.</p

Movements to and from CTS Linked Holdings represented as a percentage of total movements.

<p>This chart shows the monthly movements on CTS that come From (red bars) and go To (blue bars) CTS Linked holdings, represented as a percentage of total movements from and to all holdings over time. For the ‘From’ category, only normal and inferred off movements from agricultural holdings in the VLA_MOVEMENTS table of CTS were considered, whilst for the ‘To’ category, only normal and inferred on movements to agricultural holdings were considered. The blue and red lines represent the percentage of holdings that have a From (blue) or To (red) movement that month and are in a CTS Link.</p