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Exact Solution of Quadratic Fermionic Hamiltonians for Arbitrary Boundary Conditions.

Abhijeet Alase1, Emilio Cobanera1, Gerardo Ortiz2

  • 1Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, New Hampshire 03755, USA.

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Summary

Researchers developed a method to analyze fermionic Hamiltonians, revealing a bulk-boundary correspondence. This work accurately describes Majorana modes and predicts conditions for observing fractional Josephson effects.

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Area of Science:

  • Condensed Matter Physics
  • Quantum Mechanics
  • Materials Science

Background:

  • Quadratic fermionic Hamiltonians are crucial for understanding many-body quantum systems.
  • Analyzing systems with mixed boundary conditions (open in one dimension, periodic in others) is complex.
  • Understanding bulk-boundary correspondence is key to topological phases of matter.

Purpose of the Study:

  • To develop an exact diagonalization procedure for finite-range quadratic fermionic Hamiltonians.
  • To establish a method for separating bulk and boundary properties of these Hamiltonians.
  • To create a computable indicator for bulk-boundary correspondence.

Main Methods:

  • A Hamiltonian-dependent separation of bulk and boundary contributions.
  • Development of a matrix function to characterize solutions.
  • Application to a specific model: time-reversal-invariant s-wave two-band superconductor in a Josephson ring.

Main Results:

  • The procedure allows exact diagonalization of the specified Hamiltonians.
  • A matrix function is identified that fully characterizes system solutions.
  • The method correctly describes zero-energy Majorana modes.
  • A fractional 4π-periodic Josephson effect is predicted to occur only with an odd number of Majorana pairs per boundary.

Conclusions:

  • The developed procedure provides a powerful tool for analyzing fermionic systems.
  • The bulk-boundary separation is effective for understanding mixed boundary conditions.
  • The findings have implications for understanding topological superconductivity and fractional Josephson effects.