Topological lumps and Dirac zero modes inSU(3)lattice gauge theory on the torus

Abstract

We compute eigenmodes of the lattice Dirac operator for quenched SU(3) gauge configurations on the 4-torus with topological charge ±1. We find a strong dependence of the zero modes on the boundary conditions which we impose for the Dirac operator. The lumps seen by the eigenmodes often change their position when changing the boundary conditions, while the local chirality of the lumps remains the same. Our results show that the zero mode of a charge ±1 configuration can couple to more than one object. We address the question whether these objects could be fractionally charged lumps.

Laplacian modes probing gauge fields

“…In a similar way the zero modes on equilibrium configurations have been observed to localize to different locations on the lattice when scanning through the boundary conditions [8]. This effect has been reported even for symmetric lattices representing 'zero temperature' [14]. Given the intuition from calorons, one expects these modes to detect carriers of topological charge, including such of fractional charge.…”

“…As Fig. 17 shows, the ad- with the maximum at (x 1 , x 3 ) = (6, 6) occurring in the first interval as seen in the second row; middle: plane through (x 2 , x 4 ) = (4, 4) with the maximum at (x 1 , x 3 ) = (14,14) occurring in the second interval as seen in the third row; right: plane through (x 2 , x 4 ) = (6, 3) with the maximum at (x 1 , x 3 ) = (6, 11) occurring in the third interval as seen in the fourth row; such that the plots contain the respective global maximum. The vertical scale is 0.1 for the adjoint and 0.05 for the fundamental plots, respectively.…”

“…There seems to be no correlation of this distance to the ζ-value at which the jump occurs. Instead, the fact that the jump distribution is rather flat around half the linear extension of the lattice (here 8), may point to a random distribution of pinning centres, as suggested for the hopping of fermions in [14]. In order to make such a statement more precise, one would need a model also for the small jumps.…”

Calorons and dyons at the thermal phase transition analyzed by overlap fermions

“…We gave up, on the other hand, the exclusive focus on the zero mode(s) [52,56] of the configurations being under investigation. We have concentrated instead on the effect of changing the fermionic boundary conditions on the whole overlap-based topological charge density mapped out by a given (not too large) number of eigenmodes.…”

“…What was common to both techniques was inspired by the theoretically known behavior of zero modes of caloron-like configurations [54]. Thus, particular emphasis was first given to the zero modes (or the real modes in case of the Wilson-Dirac operator), which must be present in configurations with topological charge Q = 0, and to the effect on them of changing the boundary conditions for the Dirac operator [52,53,55,56]. Confronting the zeromode pattern with the picture revealed by smearing, it became clear [55] that the zero modes are part of the topological structure of a typical Monte Carlo configuration but cannot exhaustively explain it.…”