Nonrelativistic fermions whose motion is constrained by an infinite potential step (hard wall) are considered. Relative to the bulk, a net fermionic charge is induced at the wall due to the vanishing of the many-particle wavefunctions. The charge, linear charge density and the fluctuations from their mean values are calculated.
Charge fluctuations of solitons with arbitrary fractional fermion number in 1 + 1 dimensions are calculated, generalizing the result of Kivelson and Schrieffer for solitons with fermion number \textonehalf{}. The soliton charge is measured by a sampling function f(x) such that $f(x)\ensuremath{\simeq}1$ over a region of width L around the soliton and then falls to zero in a distance l. It is shown that vacuum fluctuations vanish as ${l}^{\ensuremath{-}1}$ for large l while the additional fluctuations due to the presence of a soliton vanish as either $\mathrm{exp}(\ensuremath{-}\frac{L}{\ensuremath{\xi}})$ or $\mathrm{exp}(\ensuremath{-}2{\ensuremath{\Delta}}_{0}L)$; $\ensuremath{\xi}$ is the soliton width and ${\ensuremath{\Delta}}_{0}$ is the mass gap of a ground state. This result establishes that the fractional fermion charge is a well-defined observable.
Fermion number fractionization in quantum field theory on a finite interval is studied for a (1 + 1) dimensional fermion-soliton system with explicit charge conjugation symmetry breaking. The effects of boundary conditions on the fractional fermion number and the connection with the corresponding open space problem are investigated. It is argued that the open space fractional charges can be correctly reproduced from the finite interval results only through a careful definition of what is meant by the soliton charge. This definition of the charge distinguishes between the fermionic and boundary induced charges in the system, and isolates the soliton from possibly other charged topological objects in the system. It therefore gives a true measure of the localized fractional fermion number induced on the soliton of interest. It is then rigorously proven that the corresponding charge fluctuations vanish, and hence that the induced fractional charge on the soliton is a quantum observable.
Abstract We consider the spatial quantum and thermal fluctuations of non-interacting Fermi gases of N particles confined in d -dimensional non-smooth potentials. We first present a thorough study of the spherically symmetric pure hard-box potential, with vanishing potential inside the box, both at T = 0 and T > 0. We find that the correlations near the wall are described by a ‘hard edge’ kernel, which depend both on d and T , and which is different from the ‘soft edge’ Airy kernel, and its higher d generalizations, found for smooth potentials. We extend these results to the case where the potential is non-uniform inside the box, and find that there exists a family of kernels which interpolate between the above ‘hard edge’ kernel and the ‘soft edge’ kernels. Finally, we consider one-dimensional singular potentials of the form with . We show that the correlations close to the singularity at x = 0 are described by this ‘hard edge’ kernel for while they are described by a broader family of ‘hard edge’ kernels known as the Bessel kernel for and, finally by the Airy kernel for . These one-dimensional kernels also appear in random matrix theory, and we provide here the mapping between the 1 d fermion models and the corresponding random matrix ensembles. Part of these results were announced in a recent letter, Lacroix-A-Chez-Toine et al (2017 Europhys. Lett . 120 10006).
Using the Jordan-Wigner transformation and duality relations, a method for bosonization for nonrelativistic spinless fermions is introduced. The same method is used for one- and two-dimensional cases. In one dimension a cubic potential is obtained. After linearization the usual bosonization results for relativistic fermions are reproduced. In two dimensions the nonlocal potential induces vortices which determines the effective potential between the charged particles. We find that the low-energy excitations are acoustic charge-density waves.