Temperature-Fermion Number Correlations in Finite Paired Systems

We investigate finite systems of N paired fermions, common in atomic nuclei, for example. These systems exhibit quantum mechanical features akin to those of superconductors. We discover, however, some specific N dependent effects that can not be attained in the thermodynamics limit of ordinary superconductivity. In particular, an important fact is uncovered: there is a strong correlation between the temperature T and the number of fermions N. A certain temperature increase ΔT produces, in thermal quantifiers (such as the entropy), quite different effects if N=4 or N=25. In fact, whether a given temperature value should be regarded as high or low can not be ascertained independent of the N value.

Fermion number 1/2 of sphalerons and spectral mirror symmetry

We present a rederivation of the baryon and lepton numbers $$\frac{1}{2}$$ 1 2 of the $$SU(2)_L$$ S U ( 2 ) L S sphaleron of the standard electroweak theory based on spectral mirror symmetry. We explore the properties of a fermionic Hamiltonian under discrete transformations along a noncontractible loop of field configurations that passes through the sphaleron and whose endpoints are the vacuum. As is well known, CP transformation is not a symmetry of the system anywhere on the loop, except at the endpoints. By augmenting CP with a chirality transformation, we observe that the Dirac Hamiltonian is odd under the new transformation precisely at the sphaleron, and this ensures the mirror symmetry of the spectrum, including the continua. As a consistency check, we show that the fermionic zero mode presented by Ringwald in the sphaleron background is invariant under the new transformation. The spectral mirror symmetry which we establish here, together with the presence of the zero mode, are the two necessary conditions whence the fermion number $$\frac{1}{2}$$ 1 2 of the sphaleron can be inferred using the reasoning presented by Jackiw and Rebbi or, equivalently, using the spectral deficiency $$\frac{1}{2}$$ 1 2 of the Dirac sea. The relevance of this analysis to other solutions is also discussed.

Massive Nambu–Goldstone fermions and bosons for non-relativistic superconformal symmetry: Jackiw–Pi vortices in a trap

We discuss a supersymmetric extension of a non-relativistic Chern–Simons matter theory, known as the supersymmetric Jackiw–Pi model, in a harmonic trap. We show that the non-relativistic version of the superconformal symmetry, called the super-Schrödinger symmetry, is not spoiled by an external field including the harmonic potential. It survives as a modified symmetry whose generators have explicit time dependences determined by the strength of the trap, the rotation velocity of the system, and the fermion number chemical potential. We construct 1/3 Bogomol'nyi–Prasad–Sommerfield (BPS) states of trapped Jackiw–Pi vortices preserving part of the modified superconformal symmetry and discuss fluctuations around static BPS configurations. In addition to the bosonic massive Nambu–Goldstone modes, we find that there exist massive Nambu–Goldstone fermions associated with broken generators of the modified super-Schrödinger symmetry. Furthermore, we find that eigenmodes form supermultiplets of a modified supersymmetry preserved by the static BPS backgrounds. As a consequence of the modified supersymmetry, infinite towers of explicit spectra can be found for eigenmodes corresponding to bosonic and fermionic lowest Landau levels.

Zero-Energy Modes, Fractional Fermion Numbers and The Index Theorem in a Vortex-Dirac Fermion System

Physics of topological materials has attracted much attention from both physicists and mathematicians recently. The index and the fermion number of Dirac fermions play an important role in topological insulators and topological superconductors. A zero-energy mode exists when Dirac fermions couple to objects with soliton-like structure such as kinks, vortices, monopoles, strings, and branes. We discuss a system of Dirac fermions interacting with a vortex and a kink. This kind of systems will be realized on the surface of topological insulators where Dirac fermions exist. The fermion number is fractionalized and this is related to the presence of fermion zero-energy excitation modes. A zero-energy mode can be regarded as a Majorana fermion mode when the chemical potential vanishes. Our discussion includes the case where there is a half-flux quantum vortex associated with a kink in a magnetic field in a bilayer superconductor. A normalizable wave function of fermion zero-energy mode does not exist in the core of the half-flux quantum vortex. The index of Dirac operator and the fermion number have additional contributions when a soliton scalar field has a singularity.