[Paper Review] Theory of collective topologically-protected Majorana fermion excitations of networks of localized Majorana modes

Kedar Damle|arXiv (Cornell University)|Jan 14, 2022

Topological Materials and Phenomena37 references3 citations

TL;DR

This paper develops a theoretical framework for characterizing collective, topologically protected zero-energy Majorana fermion excitations in networks of localized Majorana modes. By applying the Gallai-Edmonds decomposition to the network's graph, the authors construct a maximally-localized basis for the topologically protected null space of skew-symmetric hopping matrices, linking it to the number of monomers in maximum matchings—offering a local, graph-theoretic proof of the Lovász-Anderson theorem and enabling detection of zero modes in disordered systems.

ABSTRACT

Predictions of localized Majorana modes, and ideas for manipulating these degrees of freedom, are the two key ingredients in proposals for physical platforms for Majorana quantum computation. Several proposals envisage a scalable network of such Majorana modes coupled bilinearly to each other by quantum-mechanical mixing amplitudes. Here, we develop a theoretical framework for characterizing collective topologically protected zero-energy Majorana fermion excitations of such networks of localized Majorana modes. A key ingredient in our work is the Gallai-Edmonds decomposition of a general graph, which we use to obtain an alternate ``local'' proof of a ``global'' result of Lov{\'a}sz and Anderson on the dimension of the topologically protected null space of {\em real skew-symmetric} (or pure-imaginary hermitean) adjacency matrices of general graphs. Our approach to Lov{\'a}sz and Anderson's result constructs a maximally-localized basis for the said null-space from the Gallai-Edmonds decomposition of the graph. Applied to the graph of the Majorana network in question, this gives a method for characterizing basis-independent properties of these collective topologically protected Majorana fermion excitations, and relating these properties to the correlation function of monomers in the ensemble of maximum matchings (maximally-packed dimer covers) of the corresponding network graph. Our approach can also be used to identify signatures of zero-energy excitations in systems modeled by a free-fermion Hamiltonian with a hopping matrix of this type; an interesting example is provided by vacancy-induced Curie tails in generalizations (on non-bipartite lattices) of Kitaev's honeycomb model.

Motivation & Objective

• To develop a theoretical framework for collective topologically protected Majorana fermion excitations in networks of localized Majorana modes.
• To provide a local, constructive proof of the Lovász-Anderson theorem on the dimension of the null space of real skew-symmetric matrices.
• To relate the topological protection of zero modes to the structure of maximum matchings in the network graph.
• To identify signatures of zero-energy excitations in free-fermion Hamiltonians with disordered hopping amplitudes.
• To extend prior results on topological zero modes from bipartite to general graphs, including non-bipartite lattices.

Proposed method

• Utilizes the Gallai-Edmonds decomposition of a general graph to analyze the structure of the topologically protected null space of skew-symmetric adjacency matrices.
• Constructs a maximally-localized basis for the null space directly from the decomposition, ensuring topological protection depends only on connectivity pattern.
• Applies the framework to Majorana networks with both bond disorder (random hopping amplitudes) and site disorder (deleted nodes) to model physical imperfections.
• Establishes a correspondence between the number of monomers in any maximum matching and the dimension of the null space, proving the Lovász-Anderson result in a local, constructive way.
• Generalizes previous results on topological zero modes from bipartite to non-bipartite graphs, extending the applicability to models like Kitaev’s honeycomb model on non-bipartite lattices.
• Uses the formalism to analyze vacancy-induced Curie tails in generalized Kitaev models, linking them to monomer correlations in maximum matchings.

Experimental results

Research questions

• RQ1 How can the dimension of the topologically protected null space of a skew-symmetric hopping matrix be characterized in terms of graph structure?
• RQ2 What is the role of the Gallai-Edmonds decomposition in constructing a maximally-localized basis for topological zero modes?
• RQ3 How does the presence of bond and site disorder affect the existence and localization of zero-energy Majorana excitations?
• RQ4 Can the Lovász-Anderson theorem be proven constructively via graph decomposition, independent of spectral analysis?
• RQ5 What physical signatures in correlation functions or thermodynamic responses indicate the presence of topologically protected zero modes?

Key findings

• The dimension of the topologically protected null space of a real skew-symmetric matrix equals the number of monomers in any maximum matching of the associated graph, confirming the Lovász-Anderson theorem via a local construction.
• The Gallai-Edmonds decomposition provides a systematic method to construct a maximally-localized basis for the null space, independent of the specific values of nonzero hopping amplitudes.
• Topological protection of zero modes depends solely on the pattern of nonzero connections, not on the magnitude of hopping amplitudes.
• The framework identifies signatures of zero-energy excitations in disordered systems, such as vacancy-induced Curie tails in generalized Kitaev models.
• The method generalizes prior results from bipartite to general graphs, enabling analysis of topological zero modes in non-bipartite lattices like the kagome or triangular lattice.
• The correlation function of monomers in maximum matchings serves as a direct physical observable linked to the topological protection of zero modes.