Chiral Symmetry Breaking in QED induced by an External Magnetic Field

Abstract

We simulate Lattice QED in a constant and homogeneous external magnetic field using the Rational Hybrid Monte-Carlo (RHMC) algorithm developed for Lattice QCD. Our current simulations are directed towards observing chiral symmetry breaking in the limit of zero electron bare mass as predicted by approximate (Schwinger-Dyson) methods. Our earlier simulations were performed on a $36^4$ lattice at the fine structure constant $\alpha=1/137$, close to its physical value, with `safe' electron masses $m=0.1$ and $m=0.2$. At this $\alpha$, the dynamical electron mass produced by the external magnetic field, which is an order parameter for this chiral symmetry breaking, is predicted to be far too small to be measurable. Hence we are now simulating at the larger $\alpha=1/5$, where the predicted dynamical electron mass at strong external magnetic fields accessable on the lattice is large enough to be measurable. However this requires electron masses down to $m=0.001$. Such a small $m$ requires lattices larger than $36^4$, but at magnetic fields large enough to produce measurable dynamical electron masses, $36$ is an adequate spatial extent for the lattice in the plane orthogonal to the magnetic field because the electrons preferentially occupy the lowest Landau level. We are therefore performing finite size analyses using $36^2 \times N_\parallel^2$ lattices with $N_\parallel \geq 36$. We measure the chiral condensate $\langle\bar{\psi}\psi\rangle$ as our order parameter for chiral symmetry breaking, since it should remain finite as $m \rightarrow 0$ if chiral symmetry is broken by the magnetic field, but vanish otherwise. Our preliminary results strongly suggest that chiral symmetry {\it is} broken by the external magnetic field. In all our simulations, as well as measuring other observables during these simulations, we are storing configurations at regular intervals for further analysis. One such measurement planned for these stored configurations is the determination of the effects that an external magnetic field has on the coulomb field of a charged particle placed in this magnetic field.