Illustrations of vacuum polarization by solitons

Finite temperature induced fermion number for quarks in a chiral field

“…The first term is proportional to the (topological) winding number (9), and as shown in (22,23), the temperature dependent prefactor smoothly reduces to one (plus exponentially decaying corrections) as T → 0, and we recover the well-known zero temperature answer that the fermion number is equal to the winding number of the background field U . The second set of terms in (27) (coming from N (5) T ), are clearly nontopological.…”

“…The total derivative terms go to zero after doing the spatial integrals. The remaining terms in (25) are clearly nontopological and cannot be expressed in terms of the algebraic structure of the winding number density in (9). These terms give a nonzero contribution to the spatial integral and depend sensitively on the profile of the field U , not just on the asymptotic behavior of U .…”

“…In the low temperature limit, this leading-order term (22) reduces smoothly to the zero temperature fermion number N 0 in (9):…”

“…At zero temperature, the fermion number of the vacuum is a topological quantity (up to spectral flow effects), and is related to the spectral asymmetry of the relevant Dirac operator, which counts the difference between the number of positive and negative energy states of the fermion spectrum. Rigorous mathematical results, such as index theorems and Levinson's theorem, show that the fractional part of the vacuum fermion number is a topological invariant; i.e, it depends only on the asymptotic values of the background field, and is invariant under local deformations of the background [6,9,10,11]. This topological character of the fermion number is important for various applications in model field theories, such as soliton models for the nucleon since it allows the fermion number to be kept fixed in a variational calculation that minimizes the energy [12,13,14].…”

An investigation of the Casimir energy for a fermion coupled to the sine-Gordon soliton with parity decomposition

“…More generally, the interaction of fermions with solitonic backgrounds could produce or affect a variety of interesting physical phenomena such as charge and fermion number fractionalization [79][80][81][82][83], hadron physics [68,69,[84][85][86], superfluidity [87,88], superconductivity [89], BoseEinstein condensation [90,91], conducting polymers [83,[92][93][94] and localization of fermions [95][96][97][98]. The spectrum of the fermion can in general be distorted due to the presence of such a background; bound states can appear and continuum states can change as compared with the free fermion.…”