Exact Solution of Quadratic Fermionic Hamiltonians for Arbitrary Boundary Conditions

TL;DR

This paper introduces an exact diagonalization method for finite-range quadratic fermionic Hamiltonians with arbitrary boundary conditions, enabling precise analysis of bulk-boundary phenomena and Majorana modes in superconducting systems.

Contribution

The paper develops a novel, exact solution technique for quadratic fermionic Hamiltonians with arbitrary boundary conditions, advancing the analysis of topological phases and boundary states.

Findings

Accurately describes zero-energy Majorana modes in superconductors.

Predicts fractional 4 pi Josephson effect only in phases with odd Majorana pairs.

Provides a bulk-boundary indicator for topological characterization.

Abstract

We present a procedure for exactly diagonalizing finite-range quadratic fermionic Hamiltonians with arbitrary boundary conditions in one of D dimensions, and periodic in the remaining D-1. The key is a Hamiltonian-dependent separation of the bulk from the boundary. By combining information from the two, we identify a matrix function that completely characterizes the solutions, and allow to identify an indicator of bulk-boundary correspondence. As an illustration, we show how our method correctly describes the zero-energy Majorana modes of a time-reversal-invariant s-wave superconductor in a Josephson ring configuration, and predicts that a fractional 4 pi Josephson effect can only be observed in phases hosting an odd number of Majorana pairs.