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arXiv:2201.05468 · Physical review. B./Physical review. B

Theory of collective topologically protected Majorana fermion excitations of networks of localized Majorana modes

Kedar Damle

Published Jun 15, 2022• 4 citations
arXivHTMLPDF

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\'asz and Anderson on the dimension of the topologically protected null space of real skew-symmetric (or pure-imaginary Hermitian) adjacency matrices of general graphs. Our approach to Lov\'asz 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 nonbipartite lattices) of Kitaev's honeycomb model.

DOI
10.1103/physrevb.105.235118

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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'asz and Anderson on the dimension of the topologically protected null space of real skew-symmetric (or pure-imaginary Hermitian) adjacency matrices of general graphs. Our approach to Lov'asz 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 nonbipartite lattices) of Kitaev's honeycomb model.
We show that networks of superconducting topological nanowires can realize the physics of exactly solvable Kitaev spin models on trivalent lattices. This connection arises from the low-energy theory of both systems being described by a tight-binding model of Majorana modes. In Kitaev spin models the Majorana description provides a convenient representation to solve the model, whereas in an array of Josephson junctions of topological nanowires it arises from localized physical Majorana modes tunneling between the wire ends. We explicitly show that an array of junctions of three wires---a setup relevant to topological quantum computing with nanowires---can realize the Yao-Kivelson model, a variant of Kitaev spin models on a decorated honeycomb lattice. Employing properties of the latter, we show that the network can be constructed to give rise to two-dimensional collective topological states characterized by Chern numbers $\ensuremath{\nu}=0,\phantom{\rule{0.16em}{0ex}}\ifmmode\pm\else\textpm\fi{}1$, and $\ifmmode\pm\else\textpm\fi{}2$, and that defects in the array can be associated with vortex-like quasiparticle excitations. In addition we show that the collective states are stable in the presence of disorder and superconducting phase fluctuations. When the network is operated as a quantum information processor, the connection to Kitaev spin models implies that decoherence mechanisms can in general be understood in terms of proliferation of the vortex-like quasiparticles.
The classic combinatorial construct of {\em maximum matchings} probes the random geometry of regions with local sublattice imbalance in a site-diluted bipartite lattice. We demonstrate that these regions, which host the monomers of any maximum matching of the lattice, control the localization properties of a zero-energy quantum particle hopping on this lattice. The structure theory of Dulmage and Mendelsohn provides us a way of identifying a complete and non-overlapping set of such regions. This motivates our large-scale computational study of the Dulmage-Mendelsohn decomposition of site-diluted bipartite lattices in two and three dimensions. Our computations uncover an interesting universality class of percolation associated with the end-to-end connectivity of such monomer-carrying regions with local sublattice imbalance, which we dub {\em Dulmage-Mendelsohn percolation}. Our results imply the existence of a monomer percolation transition in the classical statistical mechanics of the associated maximally-packed dimer model and the existence of a phase with area-law entanglement entropy of arbitrary many-body eigenstates of the corresponding quantum dimer model. They also have striking implications for the nature of collective zero-energy Majorana fermion excitations of bipartite networks of Majorana modes localized on sites of diluted lattices, for the character of topologically-protected zero-energy wavefunctions of the bipartite random hopping problem on such lattices, and thence for the corresponding quantum percolation problem, and for the nature of low-energy magnetic excitations in bipartite quantum antiferromagnets diluted by a small density of nonmagnetic impurities.
A quantum computer based on Majorana qubits would contain a large number of zero-energy Majorana states. This system can be modelled as a connected network of the Ising-Kitaev chains alternating the “trivial” and “topological” regions, with the zero-energy Majorana fermions localized at their interfaces. The low-energy sector of the theory describing such a network can be formulated in terms of leading-order couplings between the Majorana zero modes. I consider a minimal model exhibiting effective couplings between four Majorana zero modes – the nonuniform Ising-Kitaev chain, containing two “topological” regions separated by a “trivial” region. Solving the model exactly, I show that for generic values of the model parameters the four zero modes are localized at the four interface points of the chain. In the special case where additional inversion symmetry is present, the Majorana zero modes are “delocalized” between two interface points. In both cases, the low-energy sector of the theory can be formulated in terms of the localized Majorana fermions, but the couplings between some of them are independent of their respective separations: the exact solution does not support the “nearest-neighbor” form of the effective low-energy Hamiltonian.
Motivated by the prospect of attaining Majorana modes at the ends of nanowires, we analyze interacting Majorana systems on general networks and lattices in an arbitrary number of dimensions, and derive universal spin duals. We prove that these interacting Majorana systems, quantum Ising gauge theories, and transverse-field Ising models with annealed bimodal disorder are all dual to one another on general planar graphs. This leads to an interesting connection between heavily disordered annealed Ising systems and uniform Ising theories with nearest-neighbor interactions. As any Dirac fermion (including electronic) operator can be expressed as a linear combination of two Majorana fermion operators, our results further lead to dualities between interacting Dirac fermionic systems on rather general lattices and graphs and corresponding spin systems. Such general complex Majorana architectures (other than those of simple square or other crystalline arrangements) might be of empirical relevance. As these systems display low-dimensional symmetries, they are candidates for realizing topological quantum order. The spin duals allow us to predict the feasibility of various standard transitions as well as spin-glass-type behavior in interacting Majorana fermion or electronic systems. Several systems that can be simulated by arrays of Majorana wires are further introduced and investigated: (1) the XXZ honeycomb compass model (intermediate between the classical Ising model on the honeycomb lattice and Kitaev's honeycomb model), (2) a checkerboard lattice realization of the model of Xu and Moore for superconducting (p+ip)(p+ip)(p+ip) arrays, and a (3) compass-type two-flavor Hubbard model with both pairing and hopping terms. By the use of our dualities (tantamount to high-dimensional fermionization), we show that all of these systems lie in the three-dimensional Ising universality class. We further discuss how the existence of topological orders and bounds on autocorrelation times can be inferred by the use of symmetries and also propose to engineer quantum simulators via such Majorana wire networks.
A quantum computer based on Majorana qubits would contain a large number of zero-energy Majorana states. This system can be modelled as a connected network of the Ising-Kitaev chains alternating the "trivial" and "topological" regions, with the zero-energy Majorana fermions localized at their interfaces. The low-energy sector of the theory describing such a network can be formulated in terms of leading-order couplings between the Majorana zero modes. I consider a minimal model exhibiting effective couplings between four Majorana zero modes - the nonuniform Ising-Kitaev chain, containing two "topological" regions separated by a "trivial" region. Solving the model exactly, I show that for generic values of the model parameters the four zero modes are localized at the four interface points of the chain. In the special case where additional inversion symmetry is present, the Majorana zero modes are "delocalized" between two interface points. In both cases, the low-energy sector of the theory can be formulated in terms of the localized Majorana fermions, but the couplings between some of them are independent of their respective separations: the exact solution does not support the "nearest-neighbor" form of the effective low-energy Hamiltonian.
We introduce a Hamiltonian coupling Majorana fermion degrees of freedom to a quantum dimer model. We argue that, in three dimensions, this model has deconfined quasiparticles supporting Majorana zero modes obeying nontrivial statistics. We introduce two effective field theory descriptions of this deconfined phase in which the excitations have Coulomb interactions. A key feature of this system is the existence of topologically nontrivial fermionic excitations, called ``Hopfions'' because, in a suitable continuum limit of the dimer model, such excitations correspond to the Hopf map and are related to excitations identified by Ran et al. [Phys. Rev. B 84, 184501 (2011)]. We identify corresponding topological invariants of the quantum dimer model (with or without fermions) that are present even on lattices with trivial topology. The Hopfion energy gap depends upon the phase of the model. We briefly comment on the possibility of a phase with a gapped, deconfined Z2{\mathbb{Z}}_{2}Z2​ gauge field, as may arise on the stacked triangular lattice.