[Paper Review] A Lorentz Invariant Pairing Mechanism: Relativistic Cooper Pairs

A. Bermúdez, M. A. Martín-Delgado|ArXiv.org|Jan 10, 2008

Atomic and Subatomic Physics Research51 references3 citations

TL;DR

This paper proposes a Lorentz-invariant pairing mechanism for two relativistic spin-1/2 fermions via a Dirac string coupling, which induces bound states analogous to Cooper pairs in BCS superconductivity. In the weak coupling regime, the system exhibits s-wave-like, spin-singlet bound states with energy gaps scaling as ℏω, resembling conventional Cooper pairs; in the strong coupling regime, phonons become dynamically active, leading to multiple bound states with mixed singlet-triplet character and non-trivial orbital entanglement.

ABSTRACT

We study a Lorentz invariant pairing mechanism that arises when two relativistic spin-1/2 fermions are subjected to a Dirac string coupling. In the weak coupling regime, we find remarkable analogies between this relativistic bound system and the well known superconducting Cooper pair. As the coupling strength is raised, quenched phonons become unfrozen and dynamically contribute to the gluing mechanism, which translates into novel features of this relativistic superconducting pair.

Motivation & Objective

• To establish a Lorentz-invariant mechanism for fermion pairing in a relativistic two-body system using Dirac string coupling.
• To investigate whether such a mechanism produces bound states analogous to Cooper pairs in conventional BCS superconductivity.
• To explore the role of dynamical phonons in the pairing mechanism, especially in the strong coupling regime.
• To compare the weak and strong coupling limits in terms of binding energy, spin structure, and orbital excitation.
• To determine whether the relativistic bound states exhibit properties consistent with unconventional superconducting order parameters.

Proposed method

• Formalism based on the two-body Dirac oscillator Hamiltonian in 2D, derived from a non-minimal coupling to a Dirac string potential.
• Use of chiral creation/annihilation operators (a_r, a_l) to diagonalize the Hamiltonian and analyze the energy spectrum.
• Solution of the relativistic two-body problem in the center-of-mass frame using a Fock space representation of vibrational modes.
• Analysis of eigenstates and energy levels via exact diagonalization in the Fock basis, identifying stable bound states at specific quantum numbers (n_r, n_l).
• Comparison of weak and strong coupling regimes by varying the coupling strength ζ and observing changes in phonon occupation and spin-orbit entanglement.
• Use of the Dirac string frequency ω as a proxy for the Debye frequency ω_D in BCS theory, enabling direct comparison of energy gap scaling.

Experimental results

Research questions

• RQ1 Can a Lorentz-invariant two-fermion pairing mechanism be constructed using a Dirac string coupling, and does it yield bound states resembling Cooper pairs?
• RQ2 How does the energy gap of the bound state scale with the Dirac string frequency ω in the weak coupling regime, and does it match BCS scaling?
• RQ3 What happens to the phonon degrees of freedom in the strong coupling regime—do they remain frozen or become dynamically active?
• RQ4 How does the spin and orbital structure of the bound states evolve from weak to strong coupling, and does it break spin-singlet dominance?
• RQ5 Can the relativistic pairing mechanism produce multiple stable bound states with mixed singlet-triplet character, and what does this imply for superconducting order parameters?

Key findings

• In the weak coupling regime, the system forms a single stable bound state with energy gap scaling as ΔE⁻ ∼ ℏω, closely mirroring the BCS energy gap scaling.
• The weakly coupled bound state is a spin singlet with spherically symmetric, onion-like probability density distribution in relative coordinate space.
• In the strong coupling regime, multiple stable bound states emerge (e.g., |E_{211}⟩ and |E_{311}⟩), each with non-trivial entanglement between spin and orbital Fock states.
• Phonons become dynamically active in the strong coupling limit, with vibrational modes transitioning between Fock states |n_r, n_l⟩, indicating unfrozen phonons that actively contribute to pairing.
• The strong coupling bound states are not pure singlets but linear superpositions of singlet and triplet spin states, suggesting potential for spin-polarized or unconventional superconducting order.
• The inter-particle distance in the bound states remains finite, with expectation values ⟨Γ⟩_2 and ⟨Γ⟩_3 depending on coupling strength ζ and quantum numbers (n_r, n_l), confirming stable, finite-size pairs.