Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bogoliubov quasi-particles in superconductors are integer-charged particles inapplicable for braiding quantum information

Zhiyu Fan, Wei Ku

Abstract

We present a rigorous proof that under a number-conserving Hamiltonian, one-body quasi-particles generally possess quantized charge and inertial mass identical to the bare particles. It follows that, Bogoliubov zero modes in the vortex (or on the edge) of superconductors $\textit{cannot}$ be their own anti-particles capable of braiding quantum information. As such, the heavily pursued Majorana zero mode-based route for quantum computation requires a serious re-consideration. This study further reveals the conceptual challenge in preparing and manipulating braid-able quantum states via physical thermalization or slow external fields. These profound results should reignite the long-standing quest for a number-conserving theory of superconductivity and superfluidity without fictitiously breaking global U(1) symmetry.

Bogoliubov quasi-particles in superconductors are integer-charged particles inapplicable for braiding quantum information

Abstract

We present a rigorous proof that under a number-conserving Hamiltonian, one-body quasi-particles generally possess quantized charge and inertial mass identical to the bare particles. It follows that, Bogoliubov zero modes in the vortex (or on the edge) of superconductors be their own anti-particles capable of braiding quantum information. As such, the heavily pursued Majorana zero mode-based route for quantum computation requires a serious re-consideration. This study further reveals the conceptual challenge in preparing and manipulating braid-able quantum states via physical thermalization or slow external fields. These profound results should reignite the long-standing quest for a number-conserving theory of superconductivity and superfluidity without fictitiously breaking global U(1) symmetry.

Paper Structure

This paper contains 11 sections, 47 equations, 2 figures.

Table of Contents

  1. 1. One-body Green's function represented by eigen-particles
  2. 2. Particle conservation in systems with spontaneously broken symmetry
  3. a. Apparent "exception" with spontaneously broken symmetry?
  4. b. proper consideration on quantum state thermalization
  5. c. Eigen-particle picture and number conservation of one-body quasi-particles
  6. d. An example of number non-conserving systems
  7. e. Lack of external preference to coherently break number conservation
  8. 3. Proper and improper Bogoliubov quasi-particles
  9. a. Proper Bogoliubov quasi-particles
  10. b. Improper Bogoliubov quasi-particles
  11. c. An error in previous attempts toward proper Bogoliubov quasi-particles

Figures (2)

  • Figure 1: Matrix representation of a number-conserving many-body Hamiltonian $H$, with elements $\bra{I}H\ket{I^\prime}$ in a general basis $\ket{I}$ of Fock space, showing independent diagonalization of each sector of fixed particle numbers in $\tilde{H}$ that results in conservation of $N$ in eigenstates. In contrast, $H_\mathrm{eff}$ for standard (effective field) theories of superconductivity contains approximate (problematic) couplings (purple blocks) between sectors of different particle numbers, resulting in artificial coherence in the eigenstates beyond the physical dynamics described by $H$.
  • Figure S1: Sketches of (a) $\tilde{u}_k$ and $\tilde{v}_k$, and (b) $u_k$ and $v_k$, as a function of bare one-body energy $\epsilon_k$. Notice that in our picture, $|\tilde{v}_k|^2 < \frac{1}{2}$ always reflects the strength of number impairing fluctuations, while in the standard picture, $u_k$ and $v_k$ switch their roles in reflecting the fluctuation for $\epsilon_k \le E_F$ and $\epsilon_k > E_F$, respectively. Not accounting for this role switching obscures the previous attempts for repairing number conservation.