Lattice topologies

Thus, in so far as energetics of low-lying excited states of odd and even carbon systems and doped systems are concerned, once electron correlations are introduced, the notion of solitons as elementary electronic excitations becomes somewhat blurred [128]. Moreover, it is important to evaluate whether the associated topological features of the low-lying excitations survive when we introduce realistic electron-electron correlations. The purpose of this chapter is to study the equilibrium geometries of excitations in the Peierls-PPP model and discover if there is a relation between the relaxed geometries and the topological lattice distortions associated with solitons in the presence of realistic electron correlations. 

Cataldo, F., Ori, O., Iglesias-Groth, S. (2010). Topological lattice descriptors of graphene sheets with fullerene-like nanostructures. Mol. Simul. 36, 341-353. 

There are cases where non-regular lattices may be of advantage [36,37]. The computational effort, however, is substantially larger, which makes the models less flexible concerning changes of boundary conditions or topological constraints. Another direction, which may be promising in the future, is the use of hybrid models, where for example local attachment kinetics are treated on a microscopic atomistic scale, while the transport properties are treated by macroscopic partial differential equations [5,6]. 

Structurally Dynamics CA. Most of the CA that we will encounter throughout this book (indeed, most that are currently being studied ) assume that the underlying lattice remains a passive and static object. The lattice is thus typically an arena for activity, not an active participant in the dynamics. What if the lattice were somehow made an integral part of the dynamics That is to say, what if the topology - the sites and connections among sites -- evolved alongside the value states Structurally dynamic CA are discussed in Chapter 8. 

Chapter 5 provides some examples of purely analyti( al tools useful for describing CA. It discusses methods of inferring cycle-state structure from global eigenvalue spectra, the enumeration of limit cycles, the use of shift transformations, local structure theory, and Lyapunov functions. Some preliminary research on linking CA behavior with the topological characteristics of the underlying lattice is also described. 

Figure 2.9 shows a typical Gc topology, computed for a one-dimensional lattice consisting of four viu tices and evolving according to totalistic rule T2. 

The explicit form of the Gc topology is analytically accessible for only a very few specific systems, most notably those defined by additive rules and relatively simple lattices. The method of calculation of these topologies will be presented in some detail iii chapter 5. 

Given any lattice C and arbitrary transition rule we define a natural topology... 

What is the dynamical response of a system to a sequence of systematically applied minimal topological deformations of a l-dirn lattice ... 

Fig. 3.44 Example of the range deperident behavior of rule T20 the system evolves as class-2 for r=l, class-4 for r = 2 and class-3 for r 3. The various possible intermediate (or transitional) behaviors can be studied by successively applying we.ll-defined minimal topological deformations to the initial lattice.

Fig. 3.45 Time evolution of rule T12 on (a) r — 2 lattice, (b,c) intermediate lattices, defined by populating an r=2 lattice with a fraction p of vertices that have 6 nearest-neighbors, with p6 0.15, pc 0.30, and (d) r = 3. We see that the class-3 behavior on the pure range-r graphs in (a) and (b) can become effectively class-2 on certain intermediate (or hybrid) topologies.

Fig. 3.46 Dynamical pivfiles for graph sequences Gs (defined in section 3.3.2), representing averages over Ng sequence samples. The x-axis labels each g G Gs, dashed lines denote pure range-r topologies r with gi = range-1, 1-dira lattice and vertical bars give the mean absolute deviations of a particular rneasiire. Each system has size. N = 12, with Ng and rules TZ as follows (a) Ng = 50, K = OTIO, (b) Ng = 25, Ti= OT26, (c) Ng = 50, 7 = T16, (d) dg = 50, 7 = T4.

There is a strong dependence on the exact lattice structure intermediate topologies typically have large measure fluctuations. 

If we define systems as being complex or chaotic by their having only small numbers of cycles with long periods, then, for most of the small size systems studied here, it is upon the intermediate topologies that the most complex dynamics takes place when compared with the behaviors on surrounding graphs, dynamical complexity appears to be inhibited by r lattices. 

A partial analogy between the dynamics of CA and the behaviors of continuous dynamical systems may be obtained by exploiting a fundamental property of CA systems namely, continuity in the Cantor-set Topology. We recall from section 2.2.1 that the collection of all one-dimensional configurations, or the CA phase space, r = where E = 0,1,..., fc 9 cr and Z is the set of integers by which each site of the lattice is indexed, is a compact metric space homeomorphic to the Cantor set under the metric... 

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