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Physical review. B./Physical review. B

Charging of Majorana edge modes caused by interaction: Exact results

Andrei A. Zvyagin

Published Mar 9, 2022• 6 citations
Publisher

Abstract

The ground-state behavior of a spinless fermion chain with pairing (the Kitaev chain) and an interaction between fermions at neighboring sites is studied for free open boundaries. Using the exact quantum solution, it has been shown that there can exist boundary bound states for many values of the interaction. The interaction produces charging of the vacua of the model and charging of the boundary bound states. The theory also describes the behavior of an XYZ spin-1/2 chain with free open edges.

DOI
10.1103/physrevb.105.115406

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•Aug 30, 2016•0 citations
The ground-state behavior of a spinless fermion chain with pairing (the Kitaev chain) and an interaction between fermions at neighboring sites is studied for free open boundaries. Using the exact quantum solution, it has been shown that there can exist boundary bound states for many values of the interaction. The interaction produces charging of the vacua of the model and charging of the boundary bound states. The theory also describes the behavior of an XYZ spin-1/2 chain with free open edges.
The ground state behavior of the biaxial spin-1/2 chain with free open edges is studied. Using the exact Bethe ansatz solution we show that there can exist boundary bound states for many finite values of the exchange coupling constants. The non-trivial interaction between spins produces charging of the vacua of the model and boundary bound states. Our theory also describes the behavior of the spinless fermion chain with pairing (the Kitaev chain) and an interaction between fermions at neighboring sites for free open boundaries. Therefore, the simple case of noninteracting fermions simplest boundary states are Majorana edge modes.
The ground state behavior of the spinless fermion chain with an interaction between fermions at neighboring sites is studied for free open boundaries. For the strong enough repulsion boundary gapless states (bound state of Majorana operators from opposite sites of the chain) can exist inside the gap for bulk excitations, i.e., in the topological insulator regime. We propose to use those Majorana zero modes as topological qubits, similar to the ones in one-dimensional topological superconductors. Possible physical realizations of the considered model are discussed.
We study a system of interacting spinless fermions in one dimension that, in the absence of interactions, reduces to the Kitaev chain [Kitaev, Phys. Usp. 44, 131 (2001)]. In the noninteracting case, a signal of topological order appears as zero-energy modes localized near the edges. We show that the exact ground states can be obtained analytically even in the presence of nearest-neighbor repulsive interactions when the on-site (chemical) potential is tuned to a particular function of the other parameters. As with the noninteracting case, the obtained ground states are twofold degenerate and differ in fermionic parity. We prove the uniqueness of the obtained ground states and show that they can be continuously deformed to the ground states of the noninteracting Kitaev chain without gap closing. We also demonstrate explicitly that there exists a set of operators each of which maps one of the ground states to the other with opposite fermionic parity. These operators can be thought of as an interacting generalization of Majorana edge zero modes.
We study Kitaev model in one-dimension with open boundary condition by using exact analytic methods for non-interacting system at zero chemical potential as well as in the symmetric case of Δ = t, and by using density-matrix-renormalization-group method for interacting system with nearest neighbor repulsion interaction. We suggest and examine an edge correlation function of Majorana fermions to characterize the long range order in the topological superconducting states and study the phase diagram of the interating Kitaev chain.
The expectation value of the edge Majorana operator is calculated for the one-dimensional model of spinless fermions with nearest-neighbor interactions and with open boundary conditions. The consideration is performed for the regime of the topological insulator (gapped bulk excitations and edge modes inside the gap), and for the normal metal regime (gapless bulk excitations). We show that the expectation value of the edge Majorana operator can be formed by two contributions, from the bulk states, and from the localized edge modes.
Majorana zero modes, spontaneous Z2fZ_2^{f}Z2f​ symmetry breaking, and long range edge correlation in Kitaev chains: analytic solutions and density-matrix-renormalization-group study
We study Kitaev model in one-dimension with open boundary condition by using exact analytic methods for non-interacting system in the half filled limit as well as in the symmetric case of Δ=t\Delta=tΔ=t, and by using density-matrix-renormalization-group method for interacting system with nearest neighbor repulsion. We show explicitly that the emergence of Majorana zero mode is associated with the spontaneous symmetry broken of fermion number parity operator Z2fZ_2^fZ2f​, which is +1+1+1 for even number of fermions and −1-1−1 for odd number of fermions. Our results are in support of recent proposal by Klassen and Wen\cite{Klassen}, who proposed such spontaneous symmetry broken in systems with Majorana zero modes. We suggest and examine an edge correlation function of Majorana fermions to characterize the long range order in the topological superconducting states and study the phase diagram of the interating Kitaev chain.