Studies of braided non-Abelian anyons using anyonic tensor networks

The content of this thesis can be broadly summarised into two categories: first, I constructed modified numerical algorithms based on tensor networks to simulate systems of anyons in low dimensions, and second, I used those methods to study the topological phases the anyons form when they braid around one another. In the first phase of my thesis, I extended the anyonic tensor network algorithms, by incorporating U(1) symmetry to give a modified ansatz, Anyon-U(1) tensor networks, which are capable of simulating anyonic systems at any rational filling fraction. In the second phase, I used the numerical methods to study some models of non-Abelian anyons that naturally allows for exchange of anyons. I proposed a lattice model of anyons, which I dubbed anyonic Hubbard model, which is a pair of coupled chains of anyons (or simply called anyonic ladder). Each site of the ladder can either host a single anyonic charge, or it can be empty. The anyons are able to move around, interact with one another, and exchange positions with other anyons, when vacancies exist. Exchange of anyons is a non-trivial process which may influence the formation of different kinds of new phases of matter. I studied this model using the two prominent species of anyons: Fibonacci and Ising anyons, and made a number of interesting discoveries about their phase diagrams. I identified new phases of matter arising from both the interaction between these anyons and their exchange braid statistics.Comment: 150 pages, PhD thesis, Macquarie University, Sydney. Chapter 6 of this thesis titled "Phase transitions in braided non-Abelian anyonic system" contains results which are yet to be finalised and publishe

Local Nonparametric Estimation of Scalar Diffusions

This paper studies the functional estimation of the drift and diffusion functions for recurrent scalar diffusion processes from equally spaced observations using the local polynomial kernel approach. Almost sure convergence and a CLT for the estimators are established as the sampling frequency and the time span go to infinity. The asymptotic distributions follow a mixture of normal laws. This theory covers both positive and null recurrent diffusions. Almost sure convergence rates are sometimes path dependent but expected rates can always be characterized in terms of regularly varying functions. The general theory is specialized for positive recurrent diffusion processes, and it is shown in this case that the asymptotic distributions are normal. We also obtain the limit theory for kernel density estimators when the process is positive recurrent, namely, requiring only that the invariant probability measure exists. Nonetheless, it is also shown that such an estimator paradoxically vanishes almost surely when the invariant measure is fat tailed and nonintegrable, that is, in the null recurrent case

B1_TreeRef6_MrBayesFile_Morph1-25

File_B1 (plain text). The MrBayes [10] executable file to infer branch lengths for the 25 candidate morphological partitions, using the total-evidence Maximum Likelihood topology from [6]. The matrix consists of only the 46 extant taxa from [6], since molecular branch lengths (to which these morphological branch lengths have to be compared) cannot be ascertained for extinct taxa

C10_8clocks_topolConvergence

File_C10 (pdf). Convergence diagnostics for tree topologies sampled in C8. (A) AWTY [20] plots demonstrating relatively good correlation (albeit with substantial variance) for all clades across 4 runs, and (A) high variation at different stages in a single run, but no obvious directional trends. The topology convergence statistics from MrBayes and AWTY (standard deviation of split frequencies across runs) were also relatively good ie being <0.093 across all comparisons (see top right cells of panel A). These results are consistent with convergence or near-convergence, i.e. runs sampling similar distributions but cycling very slowly through parameter space

C1_PartitionFinder

File_C1 (zipped, plain text). Matrix with molecular data for 46 extant taxa (extracted from [6]), in Phylip format (mammals.phy); PartitionFinder [12] command file (cfg) with 71 candidate partitions (by genes and by codons, with noncoding genes treated as single candidate partitions); best scheme with 7 partitions requiring separate substitution models, found by PartitionFinder using the Bayesian Information Criterion with unlinked branch lengths

C11_8clocks_param_summary

File_C11 (plain text). Convergence diagnostics for numerical parameters from MrBayes [10] for the analysis in Files C7-9. PSRFs (ratio of within-run to between-run variance) is very 1 for most numerical parameters, but approaches ~1.7 for a single parameter (due to variance in 1 run). This single outlier is slightly higher than desirable. These diagnostics do not indicate convergence, though are consistent with convergence being approached

B5_TreeRef9_MrBayesFile_Codons1-3

File_B5 (plain text). The MrBayes executable file to infer branch lengths for the 3 candidate molecular partitions (and overall morphological branch lengths), using the topology from [9]. The matrix consists of the 46 extant taxa from [6]

C2_1clock - MrBayes file

File_C2 (plain text). MrBayes executable file for a total-evidence dating analysis of all 86 taxa in the matrix in [6], using a single relaxed (independent gamma rates) clock for all traits (morphological and molecular). The sampled ancestor birth-death tree prior [17], and the Markov model of morphological evolution [18,19], were used. Optimal substitution models and substitution-model-partitions were found with PartitionFinder for molecular data (see C1) and with stepping-stone analysis in MrBayes for morphological data as implemented in [10]. Numerous (>20) MCMC runs were initially performed for 5 million generations to investigate tuning, mixing and convergence. The final analysis was then performed with 4 runs of 20 million generations, with the first 30% of samples discarded as burnin

A1_CharacterPartitions

File_A1 (Excel). Table describing the 28 partitions (25 morphological, 3 molecular) and listing the included characters. Character numbering is based on the matrix from [6], available as MorphoBank Project 773 (http://dx.doi.org/10.7934/P773)

C7_8clocks MrBayes file

File_C7 (plain text). MrBayes executable file for a total-evidence dating analysis of all 86 taxa in the matrix in [6], using 8 separate relaxed clocks (independent gamma rates; 7 for morphology and 1 for molecular data), as found in analysis B-3. The sampled ancestor birth-death tree prior [17], and the Markov model of morphological evolution [18,19], were used. Optimal substitution models and substitution-model-partitions were found with PartitionFinder for molecular data (see C1) and with stepping-stone analysis in MrBayes for morphological data [10]. Numerous (>20) MCMC runs were initially performed for 20 million generations to investigate tuning, mixing and convergence. The final analysis was then performed with 4 runs and the trees from 10 million post-burnin generations retained Note: The burnin for each run varies considerably due to variation in time to reach (apparent) stationarity, from 20 million to 50 million. For computational efficiency, the 4 runs were performed separately and in all cases, the last 10 million steps (after stationarity) were retained. However, the step numbers have been readjusted in file C8 below so they are all identical across runs (ie to a common burnin of 50 million), to facilitate downstream analyses e.g. generating summary statistics in MrBayes