[Paper Review] Effect of Particle Number Conservation on the Berry Phase Resulting from Transport of a Bound Quasiparticle around a Superfluid Vortex

Yiruo Lin, Anthony J. Leggett|arXiv (Cornell University)|Aug 8, 2017

Topological Materials and Phenomena1 references6 citations

TL;DR

This paper investigates how particle number conservation affects the Berry phase of a bound quasiparticle transported around a superfluid vortex in an s-wave superfluid. Using a one-dimensional annular geometry with a Zeeman trap and magnetic flux to tune superfluid velocity, the authors show that particle number-conserving many-body theory—beyond the standard Bogoliubov-de Gennes (BdG) framework—yields non-universal Berry phases that depend on system parameters and trap geometry, with the superfluid condensate contributing significantly to the phase beyond BdG predictions, especially at non-zero superfluid velocity.

ABSTRACT

Motivated by understanding Majorana zero modes in topological superfluids in particle-number conserving framework beyond the present framework, we study the effect of particle number conservation on the Berry phase resulting from transport of a bound quasiparticle around a superfluid vortex. We find that particle-number non-conserving calculations based on Bogoliubov-de Gennes (BdG) equations are unable to capture the correct physics when the quasiparticle is within the penetration depth of the vortex core where the superfluid velocity is non-zero. Particle number conservation is crucial for deriving the correct Berry phase in this context, and the Berry phase takes non-universal values depending on the system parameters and the external trap imposed to bind the quasiparticle. Of particular relevance to Majorana physics are the findings that superfluid condensate affects the part of the Berry phase not accounted for in the standard BdG framework, and that the superfluid many-body ground state of odd number of fermions involves superfluid condensate deformation due to the presence of the bound quasiparticle - an effect which is beyond the description of the BdG equations.

Motivation & Objective

• To examine the validity of the particle-number non-conserving Bogoliubov-de Gennes (BdG) framework in calculating the Berry phase of a bound quasiparticle around a superfluid vortex.
• To investigate how superfluid condensate deformation and particle number conservation influence the Berry phase, particularly in the presence of non-zero superfluid velocity.
• To construct a many-body ground state ansatz beyond BdG that satisfies the f-sum rule and continuity condition, ensuring consistency with exact many-body quantum mechanics.
• To quantify the contribution of the superfluid condensate to the Berry phase, especially in odd-fermion-number systems hosting Majorana zero modes.
• To derive analytical expressions for the Berry phase in the presence of a weak Zeeman trap and tunable magnetic flux, revealing non-universal behavior not captured by standard BdG theory.

Proposed method

• Modeling a superfluid in an annular geometry with a localized Zeeman potential to bind a quasiparticle, using a one-dimensional effective Hamiltonian with s-wave pairing.
• Applying the BdG equations in a particle-number non-conserving approximation to compute the Berry phase, serving as a baseline for comparison.
• Relating the Berry phase to the system's angular momentum via linear response theory, identifying contributions from both quasiparticle motion and condensate deformation.
• Introducing a many-body ground state ansatz that explicitly conserves fermion number and satisfies the f-sum rule, enabling consistent calculation of the Berry phase beyond BdG.
• Using boundary matching conditions at the trap edge to solve the BdG equations analytically in the wide-trap limit, deriving energy eigenstates and their flux dependence.
• Deriving the energy spectrum as a function of magnetic flux λ, and computing dE/dλ to extract the Berry phase via the geometric phase formula.

Experimental results

Research questions

• RQ1 How does particle number conservation alter the Berry phase of a bound quasiparticle transported around a superfluid vortex compared to the standard BdG framework?
• RQ2 What is the contribution of the superfluid condensate to the Berry phase, particularly when the superfluid velocity is non-zero?
• RQ3 Can a many-body ground state ansatz beyond BdG be constructed that satisfies the f-sum rule and correctly captures the continuity condition for the Berry phase?
• RQ4 How does the presence of an odd number of fermions lead to superfluid condensate deformation that affects the Berry phase?
• RQ5 What is the analytical form of the Berry phase for a bound quasiparticle in a weak Zeeman trap with tunable magnetic flux, and how does it deviate from universal values predicted by BdG?

Key findings

• Particle number conservation leads to a non-universal Berry phase that depends on system parameters such as trap width, superfluid velocity, and pairing gap, in contrast to the universal π/2 phase predicted by BdG theory.
• The superfluid condensate contributes significantly to the Berry phase beyond the BdG description, especially at non-zero superfluid velocity, with this contribution being non-perturbative and system-dependent.
• In the odd-fermion-number ground state, the presence of a bound quasiparticle induces a deformation of the superfluid condensate, a many-body effect absent in the BdG framework.
• The Berry phase is derived analytically as dE/dλ = 2p_F near λ = 0, where p_F is the Fermi momentum (in units with ħ = 1), showing a direct link between energy level splitting and geometric phase.
• In the wide-trap limit, the bound state energy is E = √((nπv_F/θ_L)² + Δ²) − V, which correctly accounts for Andreev reflection and quantized momentum due to the trap size.
• The standard BdG framework fails to capture the correct physics when the quasiparticle is within the vortex core penetration depth, where superfluid velocity is non-zero and condensate effects dominate.