Number-conserving superconducting wavefunction:\\Intact U(1) symmetry and absence of Majorana zero modes

Traditional theories of superconductivity are usually based on number-fluctuating order parameters that spontaneous breaks the U(1) symmetry. On the contrary, in real solid materials the number fluctuation is, in fact, only a negligibly portion of the system. Here, based on the rigorous many-body structure of `eigen-particles’ that constitutes general eigenstates of the system, we show that quasi-particle excitations adds to the system exactly one real particle (not a superposition of particle and hole), and correspondingly the typical superconducting ground state is \textit{non-degenerate}. That it, strictly speaking, superconductivity (and similarly superfluidity) does not spontaneously break U(1) symmetry, contrary to the standard description. Similarly, zero modes in vortices cannot possibly be Majorana modes, hence inapplicable for braiding quantum information.

We further propose approximate wavefunctions of such number-conserving eigen-particles, which, upon contracting with the standard (number-fluctuating) mean-field, reproduces the typical Bogoliubov quasi-particles. The corresponding number-conserving BCS wavefunction is expected to greatly improve the description of conventional superconductors, resonant valance bond superconductors, and nuclei of even numbers of nucleons.