Statistical physics  |  Percolation theory  |  Disordered systems  |  Graph theory  |  Network science

Summary of my research  

The theory of low-temperature phases and their response in idealized condensed matter systems whose Hamiltonian is translationally invariant is well-developed. There is a rich history of ideas and results on the corresponding behaviour in systems with static impurities, which lead to the presence of ‘quenched disorder’ in the Hamiltonian. However, the wide variety of possibilities left open by the physics of quenched disorder means that problems in this area have remained at the forefront of modern developments. 

In the first project with Prof. Kedar Damle and fellows Sounak Biswas and Md. Mursalin Islam, I studied the effects of vacancy disorder on the random-hopping problem on bipartite lattices. We made use of an elegant connection between the spatial extent of zero energy wavefunctions and certain connected components of the lattice obtained from the structure theory of bipartite graphs developed by Dulmage and Mendelsohn. Dulmage-Mendelsohn decomposition theorem employs maximum-matching of a graph, and I implemented maximum matching algorithms and compared their performance to execute the large-scale numerics on diluted bipartite lattices. I used a burning algorithm to find the connected components of the Dulmage-Mendolsohn decomposition. We found that the low-dilution limit turns out to be critical in two dimensions, that is, the random

geometry of these connected components exhibits universal scaling behavior with a diverging lengthscale. Next, we embarked on a study of the random-hopping problem in three dimensions as well as initiated a more detailed scaling analysis of the two-dimensional case to try and pin down the universality class of this critical behaviour [Phys. Rev. X 12, 021058 ]. In three dimensions, we found a percolation-like phase transition in the universality class of three-dimensional percolation, while our two-dimensional study provided us with a detailed understanding of this corresponding universality class of incipient percolation [Phys. Rev. X 12, 021058 ].

The bipartite nature of the lattice has been a key ingredient in all of the above. A natural question to ask is -“Is there a natural generalization of these questions for non-bipartite lattices?”. Recent work  [Phys. Rev. B 105, 235118] suggests the answer is yes. Motivated by this, we studied analogous questions in the non-bipartite case in two and three dimensions. Employing the structure theory of maximum matchings by Gallai and Edmonds, we studied triangular lattice and Shastry-Sutherland lattice (Snub square tiling) in two dimensions with a nonzero density n_v of static vacancies at random locations [arXiv:2311.05634]. We identify a novel percolation phenomenon associated with monomer (unmatched sites in a maximum matching) carrying regions, namely R-type regions, and their perfectly matched counterparts, namely P-type regions, that controls the large length scale properties of this random geometry at low dilution. This unique percolation phase has two populations of samples, one where R-type regions are dominant and the other where P-type regions are dominant. This percolation phenomenon, entitled as Gallai-Edmonds percolation, lies in a new universality class. $\calR$-type regions carry topologically robust collective Majorana excitations, and we use the results of  [Phys. Rev. B 105, 235118] to draw conclusions about the spatial wavefunctions of topologically protected collective Majorana fermion excitations of networks of coupled Majorana modes. In the case of non-bipartite lattices, the matching is found via the celebrated Edmonds blossom algorithm. Finding a maximum matching in a general graph is harder compared to bipartite graphs due to the presence of odd loops. Then, I use a generic burning algorithm to find connected components of Gallai-Edmonds decomposition and study their properties.

Motivated by the results in two-dimensional non-bipartite lattices, we studied the random geometry of maximum matchings of randomly diluted 3D non-bipartite lattices such as the stacked triangular lattice and corner-sharing octahedron lattice with a nonzero density n_v of static random dilution. Here, we found four unique phases of percolation characterized by the nature of the percolating cluster and how often they percolate in the random samples. We also found that the phenomenon of Gallai-Edmonds percolation at the neighbourhood of the threshold of these transitions can be described by universal critical behaviour.

Snapshots of two different samples of the diluted triangular lattice, both with L = 5000 and nv = 0.48. In each sample, the vertices belonging to the largest R-type region are colored brown, those belonging to the second-largest Rtype region are colored bright pink, those belonging to the largest P-type region are colored cyan, those belonging to the second-largest P-type region are colored green, and those not belonging to any of these four clusters are left uncolored. (Left panel) The sample on the left has one very large R-type region that appears to almost span the whole sample. In comparison, the largest two P-type regions are both small. The secondlargest R-type region is even smaller. (Right panel) The sample on the right has completely different morphology, with the largest P-type region almost spanning the whole lattice, the largest and second-largest R-type regions being rather small, and the second-largest P-type region being the smallest of the four regions displayed.